Post on 07-May-2022
Research ArticleDesign of Digital IIR Filter with Conflicting Objectives UsingHybrid Gravitational Search Algorithm
D S Sidhu1 J S Dhillon2 and Dalvir Kaur3
1Department of Electronics and Communication Engineering Giani Zail Singh Punjab Technical University CampusBathinda 151001 India2Department of Electrical and Instrumentation Engineering Sant Longowal Institute of Engineering and TechnologyLongowal 148106 India3Department of Electronics and Communication Engineering Punjab Institute of Technology Kapurthala 144601 India
Correspondence should be addressed to D S Sidhu dssidhuyahoocom
Received 17 May 2015 Revised 14 September 2015 Accepted 15 September 2015
Academic Editor Erik Cuevas
Copyright copy 2015 D S Sidhu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In the recent years the digital IIR filter design as a single objective optimization problem using evolutionary algorithms has gainedmuch attention In this paper the digital IIR filter design is treated as a multiobjective problem by minimizing the magnituderesponse error linear phase response error and optimal order simultaneously along with meeting the stability criterion Hybridgravitational search algorithm (HGSA) has been applied to design the digital IIR filter GSA technique is hybridized with binarysuccessive approximation (BSA) based evolutionary search method for exploring the search space locallyThe relative performanceof GSA and hybrid GSA has been evaluated by applying these techniques to standard mathematical test functions The aboveproposed hybrid search techniques have been applied effectively to solve the multiparameter and multiobjective optimizationproblem of low-pass (LP) high-pass (HP) band-pass (BP) and band-stop (BS) digital IIR filter designThe obtained results revealthat the proposed technique performs better than other algorithms applied by other researchers for the design of digital IIR filterwith conflicting objectives
1 Introduction
The electrical signal is the best method to transfer infor-mation to longer distances As the signal becomes complexwe require to process the signal to extricate the informa-tion available in the signal Generally signals are classifiedinto two categories that is continuous-time and discrete-time signals Digital signal processing (DSP) is a numericalmanipulation of discrete-time signal data With the advent ofdigital circuit technology cheaper simple and faster digitalcomputers with huge memory storage capability are able toperform complex real time digital signal processing In therecent period digital signal processing (DSP) is the area ofinterest for engineers and scientists due to their accuracy reli-ability flexibility low cost and small physical size Filteringis an essential part of digital signal processing for differentapplications such as telecommunications speech processingimage processing consumer electronics biomedical systems
industrial applications andmilitary electronics As comparedto the analog filters digital filters are flexible reliable andversatile The sampled values of input signal and the transferfunction of digital filter can be easily stored in the memory[1 2]
Digital filters are classified as finite impulse response(FIR) filters and infinite impulse response (IIR) filters IIRfilter provides better response than FIR filters with the sameorder or same number of coefficients The stability of thedigital IIR filter can be obtained by limiting the parameterspace in a suitable range Multimodal error surface of IIRfilter may force gradient based algorithm to get stuck inlocal minima [3] In order to avoid the above problem andto find the global minima global optimization techniqueslike genetic algorithm (GA) particle swarm optimization(PSO) differential evolution (DE) predator prey optimiza-tion (PPO) artificial bee colony (ABC) and so forth havebeen used by the researchers [3ndash7] to design digital IIR filters
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 282809 16 pageshttpdxdoiorg1011552015282809
2 Mathematical Problems in Engineering
The synergy between exploration and exploitation is athrust area in the optimization process The generalizationof heuristics allows the wide applicability but it limitsthe efficiency of heuristics and simply delivers mediocreperformance The hybridization of general heuristics withproblem-specific heuristics provides better results for morecomplicated problems The technique which uses generalheuristics as global search and a specific heuristic as localsearch is generally called hybrid algorithm With a propertuning hybrid algorithms with a good explorative ability asa population-based global search algorithm also provide agood exploitive performance as a local search algorithm [8]Many researchers have applied various hybrid optimizationtechniques for design of digital IIR and FIR filters Singhet al [9] had applied DE hybridized with pattern searchtechnique for the design of digital IIR filter DE has beenused for global search technique and pattern search methodas local search technique Kaur and Dhillon [10] had pro-posed integrated cat swarm optimization (CSO) and DEfor digital IIR filter design So Kaur and Dhillon discussedhybridization of two global search techniques Hua et al[11] have hybridized DE and PSO for designing 2D FIRfilter Vasundhara et al [12] had proposed hybridization ofadaptive DE and PSO for FIR filter design Both researchershave also considered the hybridization of two differentglobal search techniques Bindiya and Elias [13] had applieddifferent modified metaheuristic algorithms for the designof multiplier-less nonuniform channel filters and comparedtheir results They concluded that gravitational search algo-rithm (GSA) is the fastest and also gives acceptable valuesof the frequency-response characteristics and complexitySaha et al [14] had applied GSA for design and simulationof FIR band-pass and band-stop filters and compared theresults with results obtained by DE PSO RGA and BBOIt has been concluded that GSA converges faster than othertechniques Saha et al [15] had also applied GSAwith waveletmutation for design of IIR filter and concluded that GSAWMprovides better results as compared to other techniques Theabove researchers considered the design problem as a singleobjective optimization problem and applied the techniquesfor minimizing the magnitude response error of digital filteronly with supplementary conditions
Yu and Xinjie [16] had applied cooperative coevolu-tionary genetic algorithm to design the digital IIR filterMagnitude response and phase response are simultaneouslyoptimized along with the minimum filter order Wang et al[17] applied local search operator enhanced multiobjectiveevolutionary algorithm (LS-MOEA) for design of digital IIRfilter withmultiple objectivesWang et al [17] has also appliednondominated sorted based genetic algorithm II (NSGA-II)for the above problem Kaur et al [18] applied real codedgenetic algorithm (RCGA) for design of digital IIR filterwith multiple objectives The literature survey reveals thatmetaheuristicmethods generally suffer from either the explo-ration or exploitation capabilities leading to lack of diversityand stagnation to local minima respectively Involvementof many parameters leads to parameter tuning problemsand becomes time consuming Mostly the aforementioned
methods lack to maintain balance between exploration andexploitation
Any proposed optimizationmethodmay be a good choiceto design digital filters thatmay have exclusive advantages likerobust global search capability little information require-ment ease of implementation parallelism no requirement ofcontinuous and differentiable objective function The intentof this paper is to propose the global search techniquegravitational search algorithm (GSA) to design the digital IIRfilter The global search technique is then hybridized withevolutionary search method for exploring the search spacelocally With the increase in number of filter coefficientsthe number of comparisons increases exponentially in evolu-tionary search technique during exploration of search spaceTo reduce such comparisons a unique binary successiveapproximation (BSA) strategy has been implemented whileexploring the search space locally The opposition-basedlearning strategy is applied to have a start with good initialpopulation In this paper the design of digital IIR filter istreated as a multiobjective problem by simultaneously min-imizing the magnitude response error linear phase responseerror and optimal order along with meeting the stabilitycriterion The proposed hybrid technique is applied to solvethemultiparameter andmultiobjective optimization problemof low-pass high-pass band-pass and band-stop digital IIRfilter design Owing to the imprecise nature of decisionmakerjudgement it is presumed that decision maker may havefuzzy or imprecise goals for each objective [19]HybridGSA isused to simulate trade-off between conflicting objectives thatis magnitude response error and phase response Once thetrade-off has been obtained fuzzy set theory helps decisionmaker to decide the optimal design of IIR digital filter overthe noninferior trade-off curve
This paper is divided into five sections In Section 2digital IIR filter problem is described The gravitationalsearch algorithm (GSA) and binary successive approxima-tion (BSA) based evolutionary search method for optimalIIR filter design are explained in Section 3 The GSA andhybrid GSA have been applied to standard mathematical testfunctions in Section 4 to evaluate the relative performanceIn Section 5 digital IIR filter has been designed by applyingthe proposed hybrid heuristic search technique and obtainedresults are evaluated and compared with results of otherauthors The final conclusion and discussions are outlined inSection 6
2 IIR Filter Design Problem
Digital infinite impulse response (IIR) filter design requiresthe optimization of set of filter coefficients which meetvarious specification of the filter such as pass-band width andgain stop-band width and attenuation edge frequencies ofpass band and stop band peak ripples in pass band and stopband and linear phase response in pass band and transitionband
The cascaded digital IIR filter has several first order andsecond order sections together and is easy to implement
Mathematical Problems in Engineering 3
Putting 119911 = 119890119895120596 the structure of cascading type IIR filter can
be expressed as [20]
119867(120596 119909)
= 119860(
119872
prod
119894=1
1 + 119886119894119890minus119895120596
1 + 119887119894119890minus119895120596
)(
119873
prod
119896=1
1 + 119901119896119890minus119895120596
+ 119902119896119890minus2119895120596
1 + 119903119896119890minus119895120596
+ 119904119896119890minus2119895120596
)
(1)
where
119909 = [119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873 119904119873]119879 (2)
Vector 119909 denotes the set of filter coefficients of first orderand second order sections of dimensions119863times1with119863 = 2119872+
4119873 + 1The coefficients of the transfer function 119867(120596 119909) are
approximated during the design of filter The transfer func-tion119867(120596 119909) is compared this with the ideal transfer function119867119868(120596) The magnitude error function 1198641(119909) is obtained fromthese two values as given below
Minimize 1198641 (119909) =
119896
sum
119894
1003816100381610038161003816119867119889 (120596119894) minus
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816
1003816100381610038161003816 (3a)
where
119867119889 (120596119894) =
1 for 120596119894 isin pass band
0 for 120596119894 isin stop band(3b)
The ripple magnitude of pass-band 1205751(119909) and of stop-band 1205752(119909) are defined as below
1205751 (119909) = max120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816 minus min120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816
for 120596119894 isin pass band(4a)
1205752 (119909) = max120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816 for 120596119894 isin stop band (4b)
During the design of filter the linear phase response isalso optimized for both pass band and transition band Theexpression for phase is defined as below
0 = tanminus1 imaginary part of numerator
real part of numerator
minus tanminus1 imaginary part of denominator
real part of denominator
(5)
The first order difference in phase is determined as
Δphase = Δ01 Δ02 Δ0119899minus1 (6)
where
Δ0119894 = Δ0119894+1 minus Δ0119894 (119894 = 1 2 119899 minus 1) (7)
119873 is the total number of sampling points in pass bandand transition band The phase response will be linear if allthe elements of Δphase have the equal values [18] So the
H(120596119901)
1 minus 120575p1 1 1 + 120575p2
120583n(H(120596119901))
Hmin(120596119901)
1
0
Hmax(120596119901)
Figure 1Membership function ofmagnitude response during pass-band
second objective function is to minimize variance of phasedifferences and is expressed as below
Minimize 1198642 = var Δ0119899
0119899 isin pass band cup transition band(8)
Consider all these objectives subject to the followingstability constraints
1 + 119887119894 ge 0 (119894 = 1 2 119872) (9a)
1 minus 119887119894 ge 0 (119894 = 1 2 119872) (9b)
1 minus 119904119896 ge 0 (119896 = 1 2 119873) (9c)
1 + 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9d)
1 minus 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9e)
For the design of digital IIR filter with the above mul-tiobjective optimization criterion the fuzzy membershipfunctions are assigned to magnitude response error func-tion and phase response error function along with stabilityconstraints The fuzzy sets are stated by their membershipfunctions Suchmembership functions describe the degree ofmembership in certain fuzzy sets using values from 0 to 1Themembership function value 0 indicates incompatibility withthe sets whereas 1 means full compatibility [19] The memberfunctions are used to convert minimization to maximizationoptimization problem and constraints are converted intoobjective function to be maximized
21 Magnitude Response Error The main objective to designdigital IIR filter is to minimize the magnitude responseerror in a predefined pass-band and stop-band and withinprescribed permissible ripples In the prescribed pass-bandthe signal is allowed to pass and in the prescribed stop-bandthe signal is restricted
Themembership function ofmagnitude response in pass-band is described in Figure 1
4 Mathematical Problems in Engineering
120575s
0
1
H(120596119904)
120583n(H(120596119904))
Hmax(120596119904)
Figure 2 Membership function of magnitude response during stopband
Mathematical representation of membership function ofmagnitude response during pass-band is given as below
119906119899 (119867(120596119901))
=
0 119867(120596119901)le 119867
min(120596119901)
or 119867(120596119901) ge 119867max(120596119901)
119867(120596119901)minus 119867
min(120596119901)
(1 minus 1205751199011) minus 119867min(120596119901)
119867min(120596119901)
lt 119867(120596119901)lt 1 minus 1205751199011
119867max(120596119901)
minus 119867(120596119901)
119867max(120596119901)
minus (1 + 1205751199012)
1 + 1205751199012 lt 119867(120596119901)lt 119867
max(120596119901)
1 1 minus 1205751199011 le 119867(120596119901)le 1 + 1205751199012
(10)
where 120596119901 is the frequency in pass-band 119867(120596119901) is the magni-tude response in pass-band119867min
(120596119901)is theminimummagnitude
in pass band 119867max(120596119901)
is maximum magnitude in the pass-band 1205751199011 and 1205751199012 are the maximum tolerable deviations inmagnitude in pass-band 1205751199012 is kept almost nearer to zero119906119899(119867(120596119901)
) is the membership function of magnitude responsein pass-band 119899 isin pass band
Themembership function ofmagnitude response in stop-band is described in Figure 2
Mathematical expression of membership function ofmagnitude response during stop-band is given as below
120583119899 (119867(120596119904)) =
0 119867(120596119904)ge 119867
max(120596119904)
119867max(120596119904)
minus 119867(120596119904)
119867max(120596119904)
minus 120575119904
120575119904 lt 119867(120596119904)lt 119867
max(120596119904)
1 119867(120596119904)le 120575119904
(11)
where120596119904 is the frequency in stop band119867(120596119904) is themagnituderesponse in stop-band 119867max
119904 is maximum magnitude in thestop-band 120575119904 is the maximum tolerable deviations in magni-tude in stop-band 119906119899(119867(120596119904)) is the membership function ofmagnitude response in stop-band 119899 isin stop band
120583bi1
1
01 + bi1
Figure 3 Membership function of stability constraint
The main objective is to maximize the membershipfunction of magnitude response in both pass-band and stop-band and is represented as below
1205831 = min min 120583119899 (119867(120596119901)) 119899 isin pass band
min 120583119899 (119867(120596119904)) 119899 isin stop band
(12)
The maximum value of 1205831 provides the design of digitalfilter having minimum value of magnitude response error
22 Phase Response Error As described in (8) the phaseresponse error 1198642 is to be minimized This minimiza-tion problem is converted into maximization problem asdescribed as below
1205832 =1
1 + 1198642
(13)
where 1205832 is the objective function of phase response Maxi-mum value of objective function 1205832 provides the minimumvalue of linear phase response error 1198642
23 Membership Function of Stability Constraints The stabil-ity constraints for the design of digital IIR filter are obtainedby using the Jury method [21] on the filter coefficients givenin (9a) Using fuzzy set theory the membership function ofstability constraints given in (9a) is shown in Figure 3
Mathematically membership function of stability con-straint given in (9a) is given below
1205831119894 =
0 if (1 + 119887119894) le 0
1 if (1 + 119887119894) gt 0
(119894 = 1 2 119872) (14)
Similarly membership function for stability constraintsgiven in (9b) to (9e) can be describedThe overall objective tomeet these above five stability constraints is mathematicallygiven below and is to be maximized
1205833
=
1
5
2
sum
119896=1
[
1
119872
(
119872
sum
119894=1
120583119896119894)] +
5
sum
119896=3
[
[
1
119873
(
119873
sum
119895=1
120583119896119895)]
]
(15)
If the value 1205833 is 1 that means all constraints are satisfiedOtherwise it gives the percentage level of satisfaction ofconstraints
Mathematical Problems in Engineering 5
24 Multiobjective Problem Formulation The task of digitalIIR filter design is to find an optimum structure withminimum magnitude response error and minimum linearphase response error while satisfying the stability constraintsBy applying fuzzy set theory multiobjective constrainedproblem is converted intomultiobjective unconstrained opti-mization problem and is stated as below
Maximize [1205831 (119883) 1205832 (119883) 1205833 (119883)]119879 (16)
where 1205831(119883) is the membership function of magnituderesponse error given in (12) 1205832(119883) is the membershipfunction of linear phase response error given in (13) 1205833(119883)
is the membership function of stability constraints given in(15)119883 is a vector decision variable of dimensions119863 times 1 with119863 = 2119872 + 4119873 + 1
The objective is to find the value of filter coefficients beingdecision variables 119883 which maximizes the entire objectivefunctions simultaneously The value of membership functionindicates how much the solution satisfies the 120583119894 objectiveon the scale from 0 to 1 The maximum satisfaction ofmembership function for any filter coefficient combinationis obtained by taking the intersection of the membershipfunctions of participating objectives and is expressed asbelow
fit119895 (119905) = min 1205831119895 1205832119895 1205833119895 (119895 = 1 2 119873119875) (17)
25 Optimal Order of Digital Filter The expression foroptimal order of digital IIR filter is described below
119874 =
119872
sum
119895=0
120572119895 + 2
119873
sum
119896=0
120573119896 (18)
where 120572119895 and 120573119896 are 119895th and 119896th control genes of corre-sponding first order and second order blocks respectivelyand the value of genes will be either 1 or 0 119872 and 119873 arethe maximum number of first order and second order blocksrespectively Maximum order of the digital IIR filter will be119872 + 2119873 The order of the filter has been determined bycontrol genes as shown in Figure 4 The coding method hasbeen taken from Yu and Xinjie [16]The value of control genedetermines whether the particular block will be consideredfor filter design or notThe block will be considered activatedwhen the corresponding control gene is 1 The number ofbinary bits used to generate control genes depends upon thevalue of 119872 and 119873 The decision vector 119909 shown in (2) ismodified as below119909 = [119880119881 119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873
119904119873]119879
(19)
The variable 119880 is a positive integer with maximum valueof (2119872
minus 1) and the variable 119881 is a positive integer withmaximum value of (2119873 minus 1) The variable 119880 and 119881 will alsobe optimized along with the coefficients of the filter
3 Optimization Technique
31 Gravitational Search Algorithm This optimization algo-rithm is based on the law of gravity In this algorithm each
1205721 1205722 1205723 1205731 1205732 1205733 1205734
1 + a2zminus1
1 + b2zminus1
1 + a1zminus1
1 + b1zminus1
1 + a3zminus1
1 + b3zminus1
1 + p1zminus1 + q1z
minus2
1 + r1zminus1 + s1z
minus2
1 + p3zminus1 + q3z
minus2
1 + r3zminus1 + s3z
minus2
1 + p2zminus1 + q2z
minus2
1 + r2zminus1 + s2z
minus2
1 + p4zminus1 + q4z
minus2
1 + r4zminus1 + s4z
minus2
Figure 4 Activation and deactivation of filter coefficients withcontrol genes
candidate of the population is considered as an object (mass)Each object has four specifications that is active gravitationalmass passive gravitational mass inertia mass and positionGravitational and inertia masses are determined by thefitness function and position of the object corresponds to thesolution of the problemAll the objects attract each otherwitha gravitational force Candidates having good solutions haveheavy masses Heavy masses move slowly and attract lightermasses towards good solutionsAt the endof iterationmassesare updated as per their new positionsWith the lapse of timewe consider that all the masses be attracted by the heaviestmass and this mass will represent the optimum solution [22]
In GSA population is considered as an isolated universeof objects (masses) Value of gravitational mass and inertiamass of an object has been assumed equal Each object obeysthe Newtonian laws of gravitation and motion as describedbelow
Law of Gravitation Each object attracts every other objectwith gravitational force The gravitational force between twoobjects is directly proportional to the product of their masses(119872) and inversely proportional to the square of the distance(119877) between them But in GSA only distance 119877 is takeninstead of R2 because it gives better results This is thedeviation of GSA from the Newtonian laws of gravitation
Law of Motion The current velocity of any object is equal tothe fraction of previous velocity of the object and accelerationof the object Acceleration of the object is equal to the forceapplied on the object divided by the inertiamass of the object
Algorithm has been initialized with random populationof 119863 dimensional 119873119875 objects (masses) and each object isdescribes as
119883119894 = (1199091119894 119909
119889119894 119909
119863119894 )
for (119894 = 1 2 119873119875 119889 = 1 2 119863)
(20)
where 119909119889119894 represents the position of 119894th object in the 119889th
dimension
6 Mathematical Problems in Engineering
At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below
119865119889119894119895 (119905) = 119866 (119905)
119872119894 (119905) times 119872119895 (119905)
119877119894119895 (119905) + 120576
(119909119889119895 (119905) minus 119909
119889119894 (119905)) (21)
where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below
119877119894119895 (119905) =
10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)
100381710038171003817100381710038172
(22)
To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by
119865119889119894 (119905) =
119873119875
sum
119895=1119895 =119894
rand (119895) 119865119889119894119895 (119905) (23)
where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object
As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below
119886119889119894 (119905) =
119865119889119894 (119905)
119872119894 (119905) (24)
The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow
V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)
(25)
119909119889119894 (119905 + 1) = 119909
119889119894 (119905) + V119889119894 (119905 + 1) (26)
where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below
119866 (119905) = 119866119900119890minus120572119905119879
(27)
where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations
Masses of the objects are updated as follows
119898119894 (119905) =
fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)
119872119894 (119905) =
119898119894 (119905)
sum119873119895=1119898119895 (119905)
for (119894 = 1 2 119873)
(28)
where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows
best (119905) = min119895isin(1119873119875)
fit119895 (119905)
worst (119905) = max119895isin(1119873119875)
fit119895 (119905) (29)
To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows
119865119889119894 (119905) = sum
119895isin119896119887119890119904119905119895 =119894
rand (119895) 119865119889119894119895 (119905) (30)
where kbest is the set of objects with best fitness value andmaximummass
If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below
V119905119894119889 =
Vmin119889 V119905119894119889 lt Vmin
119889
Vmax119889 V119905119894119889 gt Vmax
119889
V119905119894119889 no violation of limits
(31)
Similarly if elements of object position 119909119905119894119889 violate their
limits their values are updated as below
119909119905119894119889 =
119909min119889 119909
119905119894119889 lt 119909
min119889
119909max119889 119909
119905119894119889 gt 119909
max119889
119909119905119894119889 no violation of limits
(32)
32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove
The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2
119863 feasible solutions are generated A 119863
dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current
point of hypercube The better solution is obtained from
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
The synergy between exploration and exploitation is athrust area in the optimization process The generalizationof heuristics allows the wide applicability but it limitsthe efficiency of heuristics and simply delivers mediocreperformance The hybridization of general heuristics withproblem-specific heuristics provides better results for morecomplicated problems The technique which uses generalheuristics as global search and a specific heuristic as localsearch is generally called hybrid algorithm With a propertuning hybrid algorithms with a good explorative ability asa population-based global search algorithm also provide agood exploitive performance as a local search algorithm [8]Many researchers have applied various hybrid optimizationtechniques for design of digital IIR and FIR filters Singhet al [9] had applied DE hybridized with pattern searchtechnique for the design of digital IIR filter DE has beenused for global search technique and pattern search methodas local search technique Kaur and Dhillon [10] had pro-posed integrated cat swarm optimization (CSO) and DEfor digital IIR filter design So Kaur and Dhillon discussedhybridization of two global search techniques Hua et al[11] have hybridized DE and PSO for designing 2D FIRfilter Vasundhara et al [12] had proposed hybridization ofadaptive DE and PSO for FIR filter design Both researchershave also considered the hybridization of two differentglobal search techniques Bindiya and Elias [13] had applieddifferent modified metaheuristic algorithms for the designof multiplier-less nonuniform channel filters and comparedtheir results They concluded that gravitational search algo-rithm (GSA) is the fastest and also gives acceptable valuesof the frequency-response characteristics and complexitySaha et al [14] had applied GSA for design and simulationof FIR band-pass and band-stop filters and compared theresults with results obtained by DE PSO RGA and BBOIt has been concluded that GSA converges faster than othertechniques Saha et al [15] had also applied GSAwith waveletmutation for design of IIR filter and concluded that GSAWMprovides better results as compared to other techniques Theabove researchers considered the design problem as a singleobjective optimization problem and applied the techniquesfor minimizing the magnitude response error of digital filteronly with supplementary conditions
Yu and Xinjie [16] had applied cooperative coevolu-tionary genetic algorithm to design the digital IIR filterMagnitude response and phase response are simultaneouslyoptimized along with the minimum filter order Wang et al[17] applied local search operator enhanced multiobjectiveevolutionary algorithm (LS-MOEA) for design of digital IIRfilter withmultiple objectivesWang et al [17] has also appliednondominated sorted based genetic algorithm II (NSGA-II)for the above problem Kaur et al [18] applied real codedgenetic algorithm (RCGA) for design of digital IIR filterwith multiple objectives The literature survey reveals thatmetaheuristicmethods generally suffer from either the explo-ration or exploitation capabilities leading to lack of diversityand stagnation to local minima respectively Involvementof many parameters leads to parameter tuning problemsand becomes time consuming Mostly the aforementioned
methods lack to maintain balance between exploration andexploitation
Any proposed optimizationmethodmay be a good choiceto design digital filters thatmay have exclusive advantages likerobust global search capability little information require-ment ease of implementation parallelism no requirement ofcontinuous and differentiable objective function The intentof this paper is to propose the global search techniquegravitational search algorithm (GSA) to design the digital IIRfilter The global search technique is then hybridized withevolutionary search method for exploring the search spacelocally With the increase in number of filter coefficientsthe number of comparisons increases exponentially in evolu-tionary search technique during exploration of search spaceTo reduce such comparisons a unique binary successiveapproximation (BSA) strategy has been implemented whileexploring the search space locally The opposition-basedlearning strategy is applied to have a start with good initialpopulation In this paper the design of digital IIR filter istreated as a multiobjective problem by simultaneously min-imizing the magnitude response error linear phase responseerror and optimal order along with meeting the stabilitycriterion The proposed hybrid technique is applied to solvethemultiparameter andmultiobjective optimization problemof low-pass high-pass band-pass and band-stop digital IIRfilter design Owing to the imprecise nature of decisionmakerjudgement it is presumed that decision maker may havefuzzy or imprecise goals for each objective [19]HybridGSA isused to simulate trade-off between conflicting objectives thatis magnitude response error and phase response Once thetrade-off has been obtained fuzzy set theory helps decisionmaker to decide the optimal design of IIR digital filter overthe noninferior trade-off curve
This paper is divided into five sections In Section 2digital IIR filter problem is described The gravitationalsearch algorithm (GSA) and binary successive approxima-tion (BSA) based evolutionary search method for optimalIIR filter design are explained in Section 3 The GSA andhybrid GSA have been applied to standard mathematical testfunctions in Section 4 to evaluate the relative performanceIn Section 5 digital IIR filter has been designed by applyingthe proposed hybrid heuristic search technique and obtainedresults are evaluated and compared with results of otherauthors The final conclusion and discussions are outlined inSection 6
2 IIR Filter Design Problem
Digital infinite impulse response (IIR) filter design requiresthe optimization of set of filter coefficients which meetvarious specification of the filter such as pass-band width andgain stop-band width and attenuation edge frequencies ofpass band and stop band peak ripples in pass band and stopband and linear phase response in pass band and transitionband
The cascaded digital IIR filter has several first order andsecond order sections together and is easy to implement
Mathematical Problems in Engineering 3
Putting 119911 = 119890119895120596 the structure of cascading type IIR filter can
be expressed as [20]
119867(120596 119909)
= 119860(
119872
prod
119894=1
1 + 119886119894119890minus119895120596
1 + 119887119894119890minus119895120596
)(
119873
prod
119896=1
1 + 119901119896119890minus119895120596
+ 119902119896119890minus2119895120596
1 + 119903119896119890minus119895120596
+ 119904119896119890minus2119895120596
)
(1)
where
119909 = [119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873 119904119873]119879 (2)
Vector 119909 denotes the set of filter coefficients of first orderand second order sections of dimensions119863times1with119863 = 2119872+
4119873 + 1The coefficients of the transfer function 119867(120596 119909) are
approximated during the design of filter The transfer func-tion119867(120596 119909) is compared this with the ideal transfer function119867119868(120596) The magnitude error function 1198641(119909) is obtained fromthese two values as given below
Minimize 1198641 (119909) =
119896
sum
119894
1003816100381610038161003816119867119889 (120596119894) minus
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816
1003816100381610038161003816 (3a)
where
119867119889 (120596119894) =
1 for 120596119894 isin pass band
0 for 120596119894 isin stop band(3b)
The ripple magnitude of pass-band 1205751(119909) and of stop-band 1205752(119909) are defined as below
1205751 (119909) = max120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816 minus min120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816
for 120596119894 isin pass band(4a)
1205752 (119909) = max120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816 for 120596119894 isin stop band (4b)
During the design of filter the linear phase response isalso optimized for both pass band and transition band Theexpression for phase is defined as below
0 = tanminus1 imaginary part of numerator
real part of numerator
minus tanminus1 imaginary part of denominator
real part of denominator
(5)
The first order difference in phase is determined as
Δphase = Δ01 Δ02 Δ0119899minus1 (6)
where
Δ0119894 = Δ0119894+1 minus Δ0119894 (119894 = 1 2 119899 minus 1) (7)
119873 is the total number of sampling points in pass bandand transition band The phase response will be linear if allthe elements of Δphase have the equal values [18] So the
H(120596119901)
1 minus 120575p1 1 1 + 120575p2
120583n(H(120596119901))
Hmin(120596119901)
1
0
Hmax(120596119901)
Figure 1Membership function ofmagnitude response during pass-band
second objective function is to minimize variance of phasedifferences and is expressed as below
Minimize 1198642 = var Δ0119899
0119899 isin pass band cup transition band(8)
Consider all these objectives subject to the followingstability constraints
1 + 119887119894 ge 0 (119894 = 1 2 119872) (9a)
1 minus 119887119894 ge 0 (119894 = 1 2 119872) (9b)
1 minus 119904119896 ge 0 (119896 = 1 2 119873) (9c)
1 + 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9d)
1 minus 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9e)
For the design of digital IIR filter with the above mul-tiobjective optimization criterion the fuzzy membershipfunctions are assigned to magnitude response error func-tion and phase response error function along with stabilityconstraints The fuzzy sets are stated by their membershipfunctions Suchmembership functions describe the degree ofmembership in certain fuzzy sets using values from 0 to 1Themembership function value 0 indicates incompatibility withthe sets whereas 1 means full compatibility [19] The memberfunctions are used to convert minimization to maximizationoptimization problem and constraints are converted intoobjective function to be maximized
21 Magnitude Response Error The main objective to designdigital IIR filter is to minimize the magnitude responseerror in a predefined pass-band and stop-band and withinprescribed permissible ripples In the prescribed pass-bandthe signal is allowed to pass and in the prescribed stop-bandthe signal is restricted
Themembership function ofmagnitude response in pass-band is described in Figure 1
4 Mathematical Problems in Engineering
120575s
0
1
H(120596119904)
120583n(H(120596119904))
Hmax(120596119904)
Figure 2 Membership function of magnitude response during stopband
Mathematical representation of membership function ofmagnitude response during pass-band is given as below
119906119899 (119867(120596119901))
=
0 119867(120596119901)le 119867
min(120596119901)
or 119867(120596119901) ge 119867max(120596119901)
119867(120596119901)minus 119867
min(120596119901)
(1 minus 1205751199011) minus 119867min(120596119901)
119867min(120596119901)
lt 119867(120596119901)lt 1 minus 1205751199011
119867max(120596119901)
minus 119867(120596119901)
119867max(120596119901)
minus (1 + 1205751199012)
1 + 1205751199012 lt 119867(120596119901)lt 119867
max(120596119901)
1 1 minus 1205751199011 le 119867(120596119901)le 1 + 1205751199012
(10)
where 120596119901 is the frequency in pass-band 119867(120596119901) is the magni-tude response in pass-band119867min
(120596119901)is theminimummagnitude
in pass band 119867max(120596119901)
is maximum magnitude in the pass-band 1205751199011 and 1205751199012 are the maximum tolerable deviations inmagnitude in pass-band 1205751199012 is kept almost nearer to zero119906119899(119867(120596119901)
) is the membership function of magnitude responsein pass-band 119899 isin pass band
Themembership function ofmagnitude response in stop-band is described in Figure 2
Mathematical expression of membership function ofmagnitude response during stop-band is given as below
120583119899 (119867(120596119904)) =
0 119867(120596119904)ge 119867
max(120596119904)
119867max(120596119904)
minus 119867(120596119904)
119867max(120596119904)
minus 120575119904
120575119904 lt 119867(120596119904)lt 119867
max(120596119904)
1 119867(120596119904)le 120575119904
(11)
where120596119904 is the frequency in stop band119867(120596119904) is themagnituderesponse in stop-band 119867max
119904 is maximum magnitude in thestop-band 120575119904 is the maximum tolerable deviations in magni-tude in stop-band 119906119899(119867(120596119904)) is the membership function ofmagnitude response in stop-band 119899 isin stop band
120583bi1
1
01 + bi1
Figure 3 Membership function of stability constraint
The main objective is to maximize the membershipfunction of magnitude response in both pass-band and stop-band and is represented as below
1205831 = min min 120583119899 (119867(120596119901)) 119899 isin pass band
min 120583119899 (119867(120596119904)) 119899 isin stop band
(12)
The maximum value of 1205831 provides the design of digitalfilter having minimum value of magnitude response error
22 Phase Response Error As described in (8) the phaseresponse error 1198642 is to be minimized This minimiza-tion problem is converted into maximization problem asdescribed as below
1205832 =1
1 + 1198642
(13)
where 1205832 is the objective function of phase response Maxi-mum value of objective function 1205832 provides the minimumvalue of linear phase response error 1198642
23 Membership Function of Stability Constraints The stabil-ity constraints for the design of digital IIR filter are obtainedby using the Jury method [21] on the filter coefficients givenin (9a) Using fuzzy set theory the membership function ofstability constraints given in (9a) is shown in Figure 3
Mathematically membership function of stability con-straint given in (9a) is given below
1205831119894 =
0 if (1 + 119887119894) le 0
1 if (1 + 119887119894) gt 0
(119894 = 1 2 119872) (14)
Similarly membership function for stability constraintsgiven in (9b) to (9e) can be describedThe overall objective tomeet these above five stability constraints is mathematicallygiven below and is to be maximized
1205833
=
1
5
2
sum
119896=1
[
1
119872
(
119872
sum
119894=1
120583119896119894)] +
5
sum
119896=3
[
[
1
119873
(
119873
sum
119895=1
120583119896119895)]
]
(15)
If the value 1205833 is 1 that means all constraints are satisfiedOtherwise it gives the percentage level of satisfaction ofconstraints
Mathematical Problems in Engineering 5
24 Multiobjective Problem Formulation The task of digitalIIR filter design is to find an optimum structure withminimum magnitude response error and minimum linearphase response error while satisfying the stability constraintsBy applying fuzzy set theory multiobjective constrainedproblem is converted intomultiobjective unconstrained opti-mization problem and is stated as below
Maximize [1205831 (119883) 1205832 (119883) 1205833 (119883)]119879 (16)
where 1205831(119883) is the membership function of magnituderesponse error given in (12) 1205832(119883) is the membershipfunction of linear phase response error given in (13) 1205833(119883)
is the membership function of stability constraints given in(15)119883 is a vector decision variable of dimensions119863 times 1 with119863 = 2119872 + 4119873 + 1
The objective is to find the value of filter coefficients beingdecision variables 119883 which maximizes the entire objectivefunctions simultaneously The value of membership functionindicates how much the solution satisfies the 120583119894 objectiveon the scale from 0 to 1 The maximum satisfaction ofmembership function for any filter coefficient combinationis obtained by taking the intersection of the membershipfunctions of participating objectives and is expressed asbelow
fit119895 (119905) = min 1205831119895 1205832119895 1205833119895 (119895 = 1 2 119873119875) (17)
25 Optimal Order of Digital Filter The expression foroptimal order of digital IIR filter is described below
119874 =
119872
sum
119895=0
120572119895 + 2
119873
sum
119896=0
120573119896 (18)
where 120572119895 and 120573119896 are 119895th and 119896th control genes of corre-sponding first order and second order blocks respectivelyand the value of genes will be either 1 or 0 119872 and 119873 arethe maximum number of first order and second order blocksrespectively Maximum order of the digital IIR filter will be119872 + 2119873 The order of the filter has been determined bycontrol genes as shown in Figure 4 The coding method hasbeen taken from Yu and Xinjie [16]The value of control genedetermines whether the particular block will be consideredfor filter design or notThe block will be considered activatedwhen the corresponding control gene is 1 The number ofbinary bits used to generate control genes depends upon thevalue of 119872 and 119873 The decision vector 119909 shown in (2) ismodified as below119909 = [119880119881 119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873
119904119873]119879
(19)
The variable 119880 is a positive integer with maximum valueof (2119872
minus 1) and the variable 119881 is a positive integer withmaximum value of (2119873 minus 1) The variable 119880 and 119881 will alsobe optimized along with the coefficients of the filter
3 Optimization Technique
31 Gravitational Search Algorithm This optimization algo-rithm is based on the law of gravity In this algorithm each
1205721 1205722 1205723 1205731 1205732 1205733 1205734
1 + a2zminus1
1 + b2zminus1
1 + a1zminus1
1 + b1zminus1
1 + a3zminus1
1 + b3zminus1
1 + p1zminus1 + q1z
minus2
1 + r1zminus1 + s1z
minus2
1 + p3zminus1 + q3z
minus2
1 + r3zminus1 + s3z
minus2
1 + p2zminus1 + q2z
minus2
1 + r2zminus1 + s2z
minus2
1 + p4zminus1 + q4z
minus2
1 + r4zminus1 + s4z
minus2
Figure 4 Activation and deactivation of filter coefficients withcontrol genes
candidate of the population is considered as an object (mass)Each object has four specifications that is active gravitationalmass passive gravitational mass inertia mass and positionGravitational and inertia masses are determined by thefitness function and position of the object corresponds to thesolution of the problemAll the objects attract each otherwitha gravitational force Candidates having good solutions haveheavy masses Heavy masses move slowly and attract lightermasses towards good solutionsAt the endof iterationmassesare updated as per their new positionsWith the lapse of timewe consider that all the masses be attracted by the heaviestmass and this mass will represent the optimum solution [22]
In GSA population is considered as an isolated universeof objects (masses) Value of gravitational mass and inertiamass of an object has been assumed equal Each object obeysthe Newtonian laws of gravitation and motion as describedbelow
Law of Gravitation Each object attracts every other objectwith gravitational force The gravitational force between twoobjects is directly proportional to the product of their masses(119872) and inversely proportional to the square of the distance(119877) between them But in GSA only distance 119877 is takeninstead of R2 because it gives better results This is thedeviation of GSA from the Newtonian laws of gravitation
Law of Motion The current velocity of any object is equal tothe fraction of previous velocity of the object and accelerationof the object Acceleration of the object is equal to the forceapplied on the object divided by the inertiamass of the object
Algorithm has been initialized with random populationof 119863 dimensional 119873119875 objects (masses) and each object isdescribes as
119883119894 = (1199091119894 119909
119889119894 119909
119863119894 )
for (119894 = 1 2 119873119875 119889 = 1 2 119863)
(20)
where 119909119889119894 represents the position of 119894th object in the 119889th
dimension
6 Mathematical Problems in Engineering
At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below
119865119889119894119895 (119905) = 119866 (119905)
119872119894 (119905) times 119872119895 (119905)
119877119894119895 (119905) + 120576
(119909119889119895 (119905) minus 119909
119889119894 (119905)) (21)
where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below
119877119894119895 (119905) =
10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)
100381710038171003817100381710038172
(22)
To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by
119865119889119894 (119905) =
119873119875
sum
119895=1119895 =119894
rand (119895) 119865119889119894119895 (119905) (23)
where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object
As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below
119886119889119894 (119905) =
119865119889119894 (119905)
119872119894 (119905) (24)
The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow
V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)
(25)
119909119889119894 (119905 + 1) = 119909
119889119894 (119905) + V119889119894 (119905 + 1) (26)
where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below
119866 (119905) = 119866119900119890minus120572119905119879
(27)
where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations
Masses of the objects are updated as follows
119898119894 (119905) =
fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)
119872119894 (119905) =
119898119894 (119905)
sum119873119895=1119898119895 (119905)
for (119894 = 1 2 119873)
(28)
where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows
best (119905) = min119895isin(1119873119875)
fit119895 (119905)
worst (119905) = max119895isin(1119873119875)
fit119895 (119905) (29)
To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows
119865119889119894 (119905) = sum
119895isin119896119887119890119904119905119895 =119894
rand (119895) 119865119889119894119895 (119905) (30)
where kbest is the set of objects with best fitness value andmaximummass
If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below
V119905119894119889 =
Vmin119889 V119905119894119889 lt Vmin
119889
Vmax119889 V119905119894119889 gt Vmax
119889
V119905119894119889 no violation of limits
(31)
Similarly if elements of object position 119909119905119894119889 violate their
limits their values are updated as below
119909119905119894119889 =
119909min119889 119909
119905119894119889 lt 119909
min119889
119909max119889 119909
119905119894119889 gt 119909
max119889
119909119905119894119889 no violation of limits
(32)
32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove
The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2
119863 feasible solutions are generated A 119863
dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current
point of hypercube The better solution is obtained from
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Putting 119911 = 119890119895120596 the structure of cascading type IIR filter can
be expressed as [20]
119867(120596 119909)
= 119860(
119872
prod
119894=1
1 + 119886119894119890minus119895120596
1 + 119887119894119890minus119895120596
)(
119873
prod
119896=1
1 + 119901119896119890minus119895120596
+ 119902119896119890minus2119895120596
1 + 119903119896119890minus119895120596
+ 119904119896119890minus2119895120596
)
(1)
where
119909 = [119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873 119904119873]119879 (2)
Vector 119909 denotes the set of filter coefficients of first orderand second order sections of dimensions119863times1with119863 = 2119872+
4119873 + 1The coefficients of the transfer function 119867(120596 119909) are
approximated during the design of filter The transfer func-tion119867(120596 119909) is compared this with the ideal transfer function119867119868(120596) The magnitude error function 1198641(119909) is obtained fromthese two values as given below
Minimize 1198641 (119909) =
119896
sum
119894
1003816100381610038161003816119867119889 (120596119894) minus
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816
1003816100381610038161003816 (3a)
where
119867119889 (120596119894) =
1 for 120596119894 isin pass band
0 for 120596119894 isin stop band(3b)
The ripple magnitude of pass-band 1205751(119909) and of stop-band 1205752(119909) are defined as below
1205751 (119909) = max120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816 minus min120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816
for 120596119894 isin pass band(4a)
1205752 (119909) = max120596119894
1003816100381610038161003816119867 (120596119894 119909)
1003816100381610038161003816 for 120596119894 isin stop band (4b)
During the design of filter the linear phase response isalso optimized for both pass band and transition band Theexpression for phase is defined as below
0 = tanminus1 imaginary part of numerator
real part of numerator
minus tanminus1 imaginary part of denominator
real part of denominator
(5)
The first order difference in phase is determined as
Δphase = Δ01 Δ02 Δ0119899minus1 (6)
where
Δ0119894 = Δ0119894+1 minus Δ0119894 (119894 = 1 2 119899 minus 1) (7)
119873 is the total number of sampling points in pass bandand transition band The phase response will be linear if allthe elements of Δphase have the equal values [18] So the
H(120596119901)
1 minus 120575p1 1 1 + 120575p2
120583n(H(120596119901))
Hmin(120596119901)
1
0
Hmax(120596119901)
Figure 1Membership function ofmagnitude response during pass-band
second objective function is to minimize variance of phasedifferences and is expressed as below
Minimize 1198642 = var Δ0119899
0119899 isin pass band cup transition band(8)
Consider all these objectives subject to the followingstability constraints
1 + 119887119894 ge 0 (119894 = 1 2 119872) (9a)
1 minus 119887119894 ge 0 (119894 = 1 2 119872) (9b)
1 minus 119904119896 ge 0 (119896 = 1 2 119873) (9c)
1 + 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9d)
1 minus 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9e)
For the design of digital IIR filter with the above mul-tiobjective optimization criterion the fuzzy membershipfunctions are assigned to magnitude response error func-tion and phase response error function along with stabilityconstraints The fuzzy sets are stated by their membershipfunctions Suchmembership functions describe the degree ofmembership in certain fuzzy sets using values from 0 to 1Themembership function value 0 indicates incompatibility withthe sets whereas 1 means full compatibility [19] The memberfunctions are used to convert minimization to maximizationoptimization problem and constraints are converted intoobjective function to be maximized
21 Magnitude Response Error The main objective to designdigital IIR filter is to minimize the magnitude responseerror in a predefined pass-band and stop-band and withinprescribed permissible ripples In the prescribed pass-bandthe signal is allowed to pass and in the prescribed stop-bandthe signal is restricted
Themembership function ofmagnitude response in pass-band is described in Figure 1
4 Mathematical Problems in Engineering
120575s
0
1
H(120596119904)
120583n(H(120596119904))
Hmax(120596119904)
Figure 2 Membership function of magnitude response during stopband
Mathematical representation of membership function ofmagnitude response during pass-band is given as below
119906119899 (119867(120596119901))
=
0 119867(120596119901)le 119867
min(120596119901)
or 119867(120596119901) ge 119867max(120596119901)
119867(120596119901)minus 119867
min(120596119901)
(1 minus 1205751199011) minus 119867min(120596119901)
119867min(120596119901)
lt 119867(120596119901)lt 1 minus 1205751199011
119867max(120596119901)
minus 119867(120596119901)
119867max(120596119901)
minus (1 + 1205751199012)
1 + 1205751199012 lt 119867(120596119901)lt 119867
max(120596119901)
1 1 minus 1205751199011 le 119867(120596119901)le 1 + 1205751199012
(10)
where 120596119901 is the frequency in pass-band 119867(120596119901) is the magni-tude response in pass-band119867min
(120596119901)is theminimummagnitude
in pass band 119867max(120596119901)
is maximum magnitude in the pass-band 1205751199011 and 1205751199012 are the maximum tolerable deviations inmagnitude in pass-band 1205751199012 is kept almost nearer to zero119906119899(119867(120596119901)
) is the membership function of magnitude responsein pass-band 119899 isin pass band
Themembership function ofmagnitude response in stop-band is described in Figure 2
Mathematical expression of membership function ofmagnitude response during stop-band is given as below
120583119899 (119867(120596119904)) =
0 119867(120596119904)ge 119867
max(120596119904)
119867max(120596119904)
minus 119867(120596119904)
119867max(120596119904)
minus 120575119904
120575119904 lt 119867(120596119904)lt 119867
max(120596119904)
1 119867(120596119904)le 120575119904
(11)
where120596119904 is the frequency in stop band119867(120596119904) is themagnituderesponse in stop-band 119867max
119904 is maximum magnitude in thestop-band 120575119904 is the maximum tolerable deviations in magni-tude in stop-band 119906119899(119867(120596119904)) is the membership function ofmagnitude response in stop-band 119899 isin stop band
120583bi1
1
01 + bi1
Figure 3 Membership function of stability constraint
The main objective is to maximize the membershipfunction of magnitude response in both pass-band and stop-band and is represented as below
1205831 = min min 120583119899 (119867(120596119901)) 119899 isin pass band
min 120583119899 (119867(120596119904)) 119899 isin stop band
(12)
The maximum value of 1205831 provides the design of digitalfilter having minimum value of magnitude response error
22 Phase Response Error As described in (8) the phaseresponse error 1198642 is to be minimized This minimiza-tion problem is converted into maximization problem asdescribed as below
1205832 =1
1 + 1198642
(13)
where 1205832 is the objective function of phase response Maxi-mum value of objective function 1205832 provides the minimumvalue of linear phase response error 1198642
23 Membership Function of Stability Constraints The stabil-ity constraints for the design of digital IIR filter are obtainedby using the Jury method [21] on the filter coefficients givenin (9a) Using fuzzy set theory the membership function ofstability constraints given in (9a) is shown in Figure 3
Mathematically membership function of stability con-straint given in (9a) is given below
1205831119894 =
0 if (1 + 119887119894) le 0
1 if (1 + 119887119894) gt 0
(119894 = 1 2 119872) (14)
Similarly membership function for stability constraintsgiven in (9b) to (9e) can be describedThe overall objective tomeet these above five stability constraints is mathematicallygiven below and is to be maximized
1205833
=
1
5
2
sum
119896=1
[
1
119872
(
119872
sum
119894=1
120583119896119894)] +
5
sum
119896=3
[
[
1
119873
(
119873
sum
119895=1
120583119896119895)]
]
(15)
If the value 1205833 is 1 that means all constraints are satisfiedOtherwise it gives the percentage level of satisfaction ofconstraints
Mathematical Problems in Engineering 5
24 Multiobjective Problem Formulation The task of digitalIIR filter design is to find an optimum structure withminimum magnitude response error and minimum linearphase response error while satisfying the stability constraintsBy applying fuzzy set theory multiobjective constrainedproblem is converted intomultiobjective unconstrained opti-mization problem and is stated as below
Maximize [1205831 (119883) 1205832 (119883) 1205833 (119883)]119879 (16)
where 1205831(119883) is the membership function of magnituderesponse error given in (12) 1205832(119883) is the membershipfunction of linear phase response error given in (13) 1205833(119883)
is the membership function of stability constraints given in(15)119883 is a vector decision variable of dimensions119863 times 1 with119863 = 2119872 + 4119873 + 1
The objective is to find the value of filter coefficients beingdecision variables 119883 which maximizes the entire objectivefunctions simultaneously The value of membership functionindicates how much the solution satisfies the 120583119894 objectiveon the scale from 0 to 1 The maximum satisfaction ofmembership function for any filter coefficient combinationis obtained by taking the intersection of the membershipfunctions of participating objectives and is expressed asbelow
fit119895 (119905) = min 1205831119895 1205832119895 1205833119895 (119895 = 1 2 119873119875) (17)
25 Optimal Order of Digital Filter The expression foroptimal order of digital IIR filter is described below
119874 =
119872
sum
119895=0
120572119895 + 2
119873
sum
119896=0
120573119896 (18)
where 120572119895 and 120573119896 are 119895th and 119896th control genes of corre-sponding first order and second order blocks respectivelyand the value of genes will be either 1 or 0 119872 and 119873 arethe maximum number of first order and second order blocksrespectively Maximum order of the digital IIR filter will be119872 + 2119873 The order of the filter has been determined bycontrol genes as shown in Figure 4 The coding method hasbeen taken from Yu and Xinjie [16]The value of control genedetermines whether the particular block will be consideredfor filter design or notThe block will be considered activatedwhen the corresponding control gene is 1 The number ofbinary bits used to generate control genes depends upon thevalue of 119872 and 119873 The decision vector 119909 shown in (2) ismodified as below119909 = [119880119881 119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873
119904119873]119879
(19)
The variable 119880 is a positive integer with maximum valueof (2119872
minus 1) and the variable 119881 is a positive integer withmaximum value of (2119873 minus 1) The variable 119880 and 119881 will alsobe optimized along with the coefficients of the filter
3 Optimization Technique
31 Gravitational Search Algorithm This optimization algo-rithm is based on the law of gravity In this algorithm each
1205721 1205722 1205723 1205731 1205732 1205733 1205734
1 + a2zminus1
1 + b2zminus1
1 + a1zminus1
1 + b1zminus1
1 + a3zminus1
1 + b3zminus1
1 + p1zminus1 + q1z
minus2
1 + r1zminus1 + s1z
minus2
1 + p3zminus1 + q3z
minus2
1 + r3zminus1 + s3z
minus2
1 + p2zminus1 + q2z
minus2
1 + r2zminus1 + s2z
minus2
1 + p4zminus1 + q4z
minus2
1 + r4zminus1 + s4z
minus2
Figure 4 Activation and deactivation of filter coefficients withcontrol genes
candidate of the population is considered as an object (mass)Each object has four specifications that is active gravitationalmass passive gravitational mass inertia mass and positionGravitational and inertia masses are determined by thefitness function and position of the object corresponds to thesolution of the problemAll the objects attract each otherwitha gravitational force Candidates having good solutions haveheavy masses Heavy masses move slowly and attract lightermasses towards good solutionsAt the endof iterationmassesare updated as per their new positionsWith the lapse of timewe consider that all the masses be attracted by the heaviestmass and this mass will represent the optimum solution [22]
In GSA population is considered as an isolated universeof objects (masses) Value of gravitational mass and inertiamass of an object has been assumed equal Each object obeysthe Newtonian laws of gravitation and motion as describedbelow
Law of Gravitation Each object attracts every other objectwith gravitational force The gravitational force between twoobjects is directly proportional to the product of their masses(119872) and inversely proportional to the square of the distance(119877) between them But in GSA only distance 119877 is takeninstead of R2 because it gives better results This is thedeviation of GSA from the Newtonian laws of gravitation
Law of Motion The current velocity of any object is equal tothe fraction of previous velocity of the object and accelerationof the object Acceleration of the object is equal to the forceapplied on the object divided by the inertiamass of the object
Algorithm has been initialized with random populationof 119863 dimensional 119873119875 objects (masses) and each object isdescribes as
119883119894 = (1199091119894 119909
119889119894 119909
119863119894 )
for (119894 = 1 2 119873119875 119889 = 1 2 119863)
(20)
where 119909119889119894 represents the position of 119894th object in the 119889th
dimension
6 Mathematical Problems in Engineering
At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below
119865119889119894119895 (119905) = 119866 (119905)
119872119894 (119905) times 119872119895 (119905)
119877119894119895 (119905) + 120576
(119909119889119895 (119905) minus 119909
119889119894 (119905)) (21)
where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below
119877119894119895 (119905) =
10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)
100381710038171003817100381710038172
(22)
To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by
119865119889119894 (119905) =
119873119875
sum
119895=1119895 =119894
rand (119895) 119865119889119894119895 (119905) (23)
where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object
As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below
119886119889119894 (119905) =
119865119889119894 (119905)
119872119894 (119905) (24)
The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow
V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)
(25)
119909119889119894 (119905 + 1) = 119909
119889119894 (119905) + V119889119894 (119905 + 1) (26)
where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below
119866 (119905) = 119866119900119890minus120572119905119879
(27)
where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations
Masses of the objects are updated as follows
119898119894 (119905) =
fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)
119872119894 (119905) =
119898119894 (119905)
sum119873119895=1119898119895 (119905)
for (119894 = 1 2 119873)
(28)
where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows
best (119905) = min119895isin(1119873119875)
fit119895 (119905)
worst (119905) = max119895isin(1119873119875)
fit119895 (119905) (29)
To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows
119865119889119894 (119905) = sum
119895isin119896119887119890119904119905119895 =119894
rand (119895) 119865119889119894119895 (119905) (30)
where kbest is the set of objects with best fitness value andmaximummass
If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below
V119905119894119889 =
Vmin119889 V119905119894119889 lt Vmin
119889
Vmax119889 V119905119894119889 gt Vmax
119889
V119905119894119889 no violation of limits
(31)
Similarly if elements of object position 119909119905119894119889 violate their
limits their values are updated as below
119909119905119894119889 =
119909min119889 119909
119905119894119889 lt 119909
min119889
119909max119889 119909
119905119894119889 gt 119909
max119889
119909119905119894119889 no violation of limits
(32)
32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove
The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2
119863 feasible solutions are generated A 119863
dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current
point of hypercube The better solution is obtained from
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
120575s
0
1
H(120596119904)
120583n(H(120596119904))
Hmax(120596119904)
Figure 2 Membership function of magnitude response during stopband
Mathematical representation of membership function ofmagnitude response during pass-band is given as below
119906119899 (119867(120596119901))
=
0 119867(120596119901)le 119867
min(120596119901)
or 119867(120596119901) ge 119867max(120596119901)
119867(120596119901)minus 119867
min(120596119901)
(1 minus 1205751199011) minus 119867min(120596119901)
119867min(120596119901)
lt 119867(120596119901)lt 1 minus 1205751199011
119867max(120596119901)
minus 119867(120596119901)
119867max(120596119901)
minus (1 + 1205751199012)
1 + 1205751199012 lt 119867(120596119901)lt 119867
max(120596119901)
1 1 minus 1205751199011 le 119867(120596119901)le 1 + 1205751199012
(10)
where 120596119901 is the frequency in pass-band 119867(120596119901) is the magni-tude response in pass-band119867min
(120596119901)is theminimummagnitude
in pass band 119867max(120596119901)
is maximum magnitude in the pass-band 1205751199011 and 1205751199012 are the maximum tolerable deviations inmagnitude in pass-band 1205751199012 is kept almost nearer to zero119906119899(119867(120596119901)
) is the membership function of magnitude responsein pass-band 119899 isin pass band
Themembership function ofmagnitude response in stop-band is described in Figure 2
Mathematical expression of membership function ofmagnitude response during stop-band is given as below
120583119899 (119867(120596119904)) =
0 119867(120596119904)ge 119867
max(120596119904)
119867max(120596119904)
minus 119867(120596119904)
119867max(120596119904)
minus 120575119904
120575119904 lt 119867(120596119904)lt 119867
max(120596119904)
1 119867(120596119904)le 120575119904
(11)
where120596119904 is the frequency in stop band119867(120596119904) is themagnituderesponse in stop-band 119867max
119904 is maximum magnitude in thestop-band 120575119904 is the maximum tolerable deviations in magni-tude in stop-band 119906119899(119867(120596119904)) is the membership function ofmagnitude response in stop-band 119899 isin stop band
120583bi1
1
01 + bi1
Figure 3 Membership function of stability constraint
The main objective is to maximize the membershipfunction of magnitude response in both pass-band and stop-band and is represented as below
1205831 = min min 120583119899 (119867(120596119901)) 119899 isin pass band
min 120583119899 (119867(120596119904)) 119899 isin stop band
(12)
The maximum value of 1205831 provides the design of digitalfilter having minimum value of magnitude response error
22 Phase Response Error As described in (8) the phaseresponse error 1198642 is to be minimized This minimiza-tion problem is converted into maximization problem asdescribed as below
1205832 =1
1 + 1198642
(13)
where 1205832 is the objective function of phase response Maxi-mum value of objective function 1205832 provides the minimumvalue of linear phase response error 1198642
23 Membership Function of Stability Constraints The stabil-ity constraints for the design of digital IIR filter are obtainedby using the Jury method [21] on the filter coefficients givenin (9a) Using fuzzy set theory the membership function ofstability constraints given in (9a) is shown in Figure 3
Mathematically membership function of stability con-straint given in (9a) is given below
1205831119894 =
0 if (1 + 119887119894) le 0
1 if (1 + 119887119894) gt 0
(119894 = 1 2 119872) (14)
Similarly membership function for stability constraintsgiven in (9b) to (9e) can be describedThe overall objective tomeet these above five stability constraints is mathematicallygiven below and is to be maximized
1205833
=
1
5
2
sum
119896=1
[
1
119872
(
119872
sum
119894=1
120583119896119894)] +
5
sum
119896=3
[
[
1
119873
(
119873
sum
119895=1
120583119896119895)]
]
(15)
If the value 1205833 is 1 that means all constraints are satisfiedOtherwise it gives the percentage level of satisfaction ofconstraints
Mathematical Problems in Engineering 5
24 Multiobjective Problem Formulation The task of digitalIIR filter design is to find an optimum structure withminimum magnitude response error and minimum linearphase response error while satisfying the stability constraintsBy applying fuzzy set theory multiobjective constrainedproblem is converted intomultiobjective unconstrained opti-mization problem and is stated as below
Maximize [1205831 (119883) 1205832 (119883) 1205833 (119883)]119879 (16)
where 1205831(119883) is the membership function of magnituderesponse error given in (12) 1205832(119883) is the membershipfunction of linear phase response error given in (13) 1205833(119883)
is the membership function of stability constraints given in(15)119883 is a vector decision variable of dimensions119863 times 1 with119863 = 2119872 + 4119873 + 1
The objective is to find the value of filter coefficients beingdecision variables 119883 which maximizes the entire objectivefunctions simultaneously The value of membership functionindicates how much the solution satisfies the 120583119894 objectiveon the scale from 0 to 1 The maximum satisfaction ofmembership function for any filter coefficient combinationis obtained by taking the intersection of the membershipfunctions of participating objectives and is expressed asbelow
fit119895 (119905) = min 1205831119895 1205832119895 1205833119895 (119895 = 1 2 119873119875) (17)
25 Optimal Order of Digital Filter The expression foroptimal order of digital IIR filter is described below
119874 =
119872
sum
119895=0
120572119895 + 2
119873
sum
119896=0
120573119896 (18)
where 120572119895 and 120573119896 are 119895th and 119896th control genes of corre-sponding first order and second order blocks respectivelyand the value of genes will be either 1 or 0 119872 and 119873 arethe maximum number of first order and second order blocksrespectively Maximum order of the digital IIR filter will be119872 + 2119873 The order of the filter has been determined bycontrol genes as shown in Figure 4 The coding method hasbeen taken from Yu and Xinjie [16]The value of control genedetermines whether the particular block will be consideredfor filter design or notThe block will be considered activatedwhen the corresponding control gene is 1 The number ofbinary bits used to generate control genes depends upon thevalue of 119872 and 119873 The decision vector 119909 shown in (2) ismodified as below119909 = [119880119881 119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873
119904119873]119879
(19)
The variable 119880 is a positive integer with maximum valueof (2119872
minus 1) and the variable 119881 is a positive integer withmaximum value of (2119873 minus 1) The variable 119880 and 119881 will alsobe optimized along with the coefficients of the filter
3 Optimization Technique
31 Gravitational Search Algorithm This optimization algo-rithm is based on the law of gravity In this algorithm each
1205721 1205722 1205723 1205731 1205732 1205733 1205734
1 + a2zminus1
1 + b2zminus1
1 + a1zminus1
1 + b1zminus1
1 + a3zminus1
1 + b3zminus1
1 + p1zminus1 + q1z
minus2
1 + r1zminus1 + s1z
minus2
1 + p3zminus1 + q3z
minus2
1 + r3zminus1 + s3z
minus2
1 + p2zminus1 + q2z
minus2
1 + r2zminus1 + s2z
minus2
1 + p4zminus1 + q4z
minus2
1 + r4zminus1 + s4z
minus2
Figure 4 Activation and deactivation of filter coefficients withcontrol genes
candidate of the population is considered as an object (mass)Each object has four specifications that is active gravitationalmass passive gravitational mass inertia mass and positionGravitational and inertia masses are determined by thefitness function and position of the object corresponds to thesolution of the problemAll the objects attract each otherwitha gravitational force Candidates having good solutions haveheavy masses Heavy masses move slowly and attract lightermasses towards good solutionsAt the endof iterationmassesare updated as per their new positionsWith the lapse of timewe consider that all the masses be attracted by the heaviestmass and this mass will represent the optimum solution [22]
In GSA population is considered as an isolated universeof objects (masses) Value of gravitational mass and inertiamass of an object has been assumed equal Each object obeysthe Newtonian laws of gravitation and motion as describedbelow
Law of Gravitation Each object attracts every other objectwith gravitational force The gravitational force between twoobjects is directly proportional to the product of their masses(119872) and inversely proportional to the square of the distance(119877) between them But in GSA only distance 119877 is takeninstead of R2 because it gives better results This is thedeviation of GSA from the Newtonian laws of gravitation
Law of Motion The current velocity of any object is equal tothe fraction of previous velocity of the object and accelerationof the object Acceleration of the object is equal to the forceapplied on the object divided by the inertiamass of the object
Algorithm has been initialized with random populationof 119863 dimensional 119873119875 objects (masses) and each object isdescribes as
119883119894 = (1199091119894 119909
119889119894 119909
119863119894 )
for (119894 = 1 2 119873119875 119889 = 1 2 119863)
(20)
where 119909119889119894 represents the position of 119894th object in the 119889th
dimension
6 Mathematical Problems in Engineering
At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below
119865119889119894119895 (119905) = 119866 (119905)
119872119894 (119905) times 119872119895 (119905)
119877119894119895 (119905) + 120576
(119909119889119895 (119905) minus 119909
119889119894 (119905)) (21)
where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below
119877119894119895 (119905) =
10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)
100381710038171003817100381710038172
(22)
To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by
119865119889119894 (119905) =
119873119875
sum
119895=1119895 =119894
rand (119895) 119865119889119894119895 (119905) (23)
where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object
As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below
119886119889119894 (119905) =
119865119889119894 (119905)
119872119894 (119905) (24)
The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow
V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)
(25)
119909119889119894 (119905 + 1) = 119909
119889119894 (119905) + V119889119894 (119905 + 1) (26)
where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below
119866 (119905) = 119866119900119890minus120572119905119879
(27)
where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations
Masses of the objects are updated as follows
119898119894 (119905) =
fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)
119872119894 (119905) =
119898119894 (119905)
sum119873119895=1119898119895 (119905)
for (119894 = 1 2 119873)
(28)
where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows
best (119905) = min119895isin(1119873119875)
fit119895 (119905)
worst (119905) = max119895isin(1119873119875)
fit119895 (119905) (29)
To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows
119865119889119894 (119905) = sum
119895isin119896119887119890119904119905119895 =119894
rand (119895) 119865119889119894119895 (119905) (30)
where kbest is the set of objects with best fitness value andmaximummass
If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below
V119905119894119889 =
Vmin119889 V119905119894119889 lt Vmin
119889
Vmax119889 V119905119894119889 gt Vmax
119889
V119905119894119889 no violation of limits
(31)
Similarly if elements of object position 119909119905119894119889 violate their
limits their values are updated as below
119909119905119894119889 =
119909min119889 119909
119905119894119889 lt 119909
min119889
119909max119889 119909
119905119894119889 gt 119909
max119889
119909119905119894119889 no violation of limits
(32)
32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove
The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2
119863 feasible solutions are generated A 119863
dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current
point of hypercube The better solution is obtained from
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
24 Multiobjective Problem Formulation The task of digitalIIR filter design is to find an optimum structure withminimum magnitude response error and minimum linearphase response error while satisfying the stability constraintsBy applying fuzzy set theory multiobjective constrainedproblem is converted intomultiobjective unconstrained opti-mization problem and is stated as below
Maximize [1205831 (119883) 1205832 (119883) 1205833 (119883)]119879 (16)
where 1205831(119883) is the membership function of magnituderesponse error given in (12) 1205832(119883) is the membershipfunction of linear phase response error given in (13) 1205833(119883)
is the membership function of stability constraints given in(15)119883 is a vector decision variable of dimensions119863 times 1 with119863 = 2119872 + 4119873 + 1
The objective is to find the value of filter coefficients beingdecision variables 119883 which maximizes the entire objectivefunctions simultaneously The value of membership functionindicates how much the solution satisfies the 120583119894 objectiveon the scale from 0 to 1 The maximum satisfaction ofmembership function for any filter coefficient combinationis obtained by taking the intersection of the membershipfunctions of participating objectives and is expressed asbelow
fit119895 (119905) = min 1205831119895 1205832119895 1205833119895 (119895 = 1 2 119873119875) (17)
25 Optimal Order of Digital Filter The expression foroptimal order of digital IIR filter is described below
119874 =
119872
sum
119895=0
120572119895 + 2
119873
sum
119896=0
120573119896 (18)
where 120572119895 and 120573119896 are 119895th and 119896th control genes of corre-sponding first order and second order blocks respectivelyand the value of genes will be either 1 or 0 119872 and 119873 arethe maximum number of first order and second order blocksrespectively Maximum order of the digital IIR filter will be119872 + 2119873 The order of the filter has been determined bycontrol genes as shown in Figure 4 The coding method hasbeen taken from Yu and Xinjie [16]The value of control genedetermines whether the particular block will be consideredfor filter design or notThe block will be considered activatedwhen the corresponding control gene is 1 The number ofbinary bits used to generate control genes depends upon thevalue of 119872 and 119873 The decision vector 119909 shown in (2) ismodified as below119909 = [119880119881 119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873
119904119873]119879
(19)
The variable 119880 is a positive integer with maximum valueof (2119872
minus 1) and the variable 119881 is a positive integer withmaximum value of (2119873 minus 1) The variable 119880 and 119881 will alsobe optimized along with the coefficients of the filter
3 Optimization Technique
31 Gravitational Search Algorithm This optimization algo-rithm is based on the law of gravity In this algorithm each
1205721 1205722 1205723 1205731 1205732 1205733 1205734
1 + a2zminus1
1 + b2zminus1
1 + a1zminus1
1 + b1zminus1
1 + a3zminus1
1 + b3zminus1
1 + p1zminus1 + q1z
minus2
1 + r1zminus1 + s1z
minus2
1 + p3zminus1 + q3z
minus2
1 + r3zminus1 + s3z
minus2
1 + p2zminus1 + q2z
minus2
1 + r2zminus1 + s2z
minus2
1 + p4zminus1 + q4z
minus2
1 + r4zminus1 + s4z
minus2
Figure 4 Activation and deactivation of filter coefficients withcontrol genes
candidate of the population is considered as an object (mass)Each object has four specifications that is active gravitationalmass passive gravitational mass inertia mass and positionGravitational and inertia masses are determined by thefitness function and position of the object corresponds to thesolution of the problemAll the objects attract each otherwitha gravitational force Candidates having good solutions haveheavy masses Heavy masses move slowly and attract lightermasses towards good solutionsAt the endof iterationmassesare updated as per their new positionsWith the lapse of timewe consider that all the masses be attracted by the heaviestmass and this mass will represent the optimum solution [22]
In GSA population is considered as an isolated universeof objects (masses) Value of gravitational mass and inertiamass of an object has been assumed equal Each object obeysthe Newtonian laws of gravitation and motion as describedbelow
Law of Gravitation Each object attracts every other objectwith gravitational force The gravitational force between twoobjects is directly proportional to the product of their masses(119872) and inversely proportional to the square of the distance(119877) between them But in GSA only distance 119877 is takeninstead of R2 because it gives better results This is thedeviation of GSA from the Newtonian laws of gravitation
Law of Motion The current velocity of any object is equal tothe fraction of previous velocity of the object and accelerationof the object Acceleration of the object is equal to the forceapplied on the object divided by the inertiamass of the object
Algorithm has been initialized with random populationof 119863 dimensional 119873119875 objects (masses) and each object isdescribes as
119883119894 = (1199091119894 119909
119889119894 119909
119863119894 )
for (119894 = 1 2 119873119875 119889 = 1 2 119863)
(20)
where 119909119889119894 represents the position of 119894th object in the 119889th
dimension
6 Mathematical Problems in Engineering
At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below
119865119889119894119895 (119905) = 119866 (119905)
119872119894 (119905) times 119872119895 (119905)
119877119894119895 (119905) + 120576
(119909119889119895 (119905) minus 119909
119889119894 (119905)) (21)
where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below
119877119894119895 (119905) =
10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)
100381710038171003817100381710038172
(22)
To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by
119865119889119894 (119905) =
119873119875
sum
119895=1119895 =119894
rand (119895) 119865119889119894119895 (119905) (23)
where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object
As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below
119886119889119894 (119905) =
119865119889119894 (119905)
119872119894 (119905) (24)
The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow
V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)
(25)
119909119889119894 (119905 + 1) = 119909
119889119894 (119905) + V119889119894 (119905 + 1) (26)
where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below
119866 (119905) = 119866119900119890minus120572119905119879
(27)
where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations
Masses of the objects are updated as follows
119898119894 (119905) =
fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)
119872119894 (119905) =
119898119894 (119905)
sum119873119895=1119898119895 (119905)
for (119894 = 1 2 119873)
(28)
where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows
best (119905) = min119895isin(1119873119875)
fit119895 (119905)
worst (119905) = max119895isin(1119873119875)
fit119895 (119905) (29)
To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows
119865119889119894 (119905) = sum
119895isin119896119887119890119904119905119895 =119894
rand (119895) 119865119889119894119895 (119905) (30)
where kbest is the set of objects with best fitness value andmaximummass
If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below
V119905119894119889 =
Vmin119889 V119905119894119889 lt Vmin
119889
Vmax119889 V119905119894119889 gt Vmax
119889
V119905119894119889 no violation of limits
(31)
Similarly if elements of object position 119909119905119894119889 violate their
limits their values are updated as below
119909119905119894119889 =
119909min119889 119909
119905119894119889 lt 119909
min119889
119909max119889 119909
119905119894119889 gt 119909
max119889
119909119905119894119889 no violation of limits
(32)
32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove
The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2
119863 feasible solutions are generated A 119863
dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current
point of hypercube The better solution is obtained from
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below
119865119889119894119895 (119905) = 119866 (119905)
119872119894 (119905) times 119872119895 (119905)
119877119894119895 (119905) + 120576
(119909119889119895 (119905) minus 119909
119889119894 (119905)) (21)
where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below
119877119894119895 (119905) =
10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)
100381710038171003817100381710038172
(22)
To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by
119865119889119894 (119905) =
119873119875
sum
119895=1119895 =119894
rand (119895) 119865119889119894119895 (119905) (23)
where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object
As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below
119886119889119894 (119905) =
119865119889119894 (119905)
119872119894 (119905) (24)
The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow
V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)
(25)
119909119889119894 (119905 + 1) = 119909
119889119894 (119905) + V119889119894 (119905 + 1) (26)
where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below
119866 (119905) = 119866119900119890minus120572119905119879
(27)
where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations
Masses of the objects are updated as follows
119898119894 (119905) =
fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)
119872119894 (119905) =
119898119894 (119905)
sum119873119895=1119898119895 (119905)
for (119894 = 1 2 119873)
(28)
where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows
best (119905) = min119895isin(1119873119875)
fit119895 (119905)
worst (119905) = max119895isin(1119873119875)
fit119895 (119905) (29)
To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows
119865119889119894 (119905) = sum
119895isin119896119887119890119904119905119895 =119894
rand (119895) 119865119889119894119895 (119905) (30)
where kbest is the set of objects with best fitness value andmaximummass
If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below
V119905119894119889 =
Vmin119889 V119905119894119889 lt Vmin
119889
Vmax119889 V119905119894119889 gt Vmax
119889
V119905119894119889 no violation of limits
(31)
Similarly if elements of object position 119909119905119894119889 violate their
limits their values are updated as below
119909119905119894119889 =
119909min119889 119909
119905119894119889 lt 119909
min119889
119909max119889 119909
119905119894119889 gt 119909
max119889
119909119905119894119889 no violation of limits
(32)
32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove
The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2
119863 feasible solutions are generated A 119863
dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current
point of hypercube The better solution is obtained from
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Table 1 Coefficient vector at the corners of hypercube
Hyper cube corners
Possiblecombinations of
3-bit1198622 1198621 1198620
Distance of hypercubefrom centre point
1199091198881198943 1199091198881198942 1199091198881198941
Possible coefficient pattern of the IIR filter at the corner of hypercube
0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 minus Δ 1198941
5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 minus Δ 1198942 119909
1198881198941 + Δ 1198941
6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 minus Δ 1198941
7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909
1198881198942 + Δ 1198942 119909
1198881198941 + Δ 1198941
3(011) 7(111)
2(010) 6(110)
0(000) 4(100)
5(101)1(001)
xci3 xci2 x
ci1
o
Figure 5 Three-dimensional hypercube representing corners in decimals
4(100)
2(010) 6(110)
1(001) 3(011) 5(101) 7(111)
minus21
minus20
+21
+20minus20 +20
Figure 6 BSA for 3-bit code
the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]
33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]
The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below
119862119905119894119895 =
1 for (119895 = 1)
0 for (119895 = 2 3 4 119863)
(33)
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 2 Comparison of number of function evaluations
Value of 119872 and 119873Number of committed coefficients
119863
Number of corners of hypercube(2119863)
Number of comparisons byBSA method
(2 times 119863)1 1 7 128 142 2 13 8192 263 3 19 524288 384 4 25 33554432 505 5 31 2147483648 62
(1) Search space identification(2) Randomized initialization of population(3) Fitness evaluation of objects using (17)WHILE (stopping criterion is not met)(4) Update 119866(119905) best(119905) worst(119905) and 119872119894(119905) (119894 = 1 2 119873119875)(5) Calculation of the total force in different directions using (23)(6) Calculation of acceleration and velocity using (24) and (25) respectively(7) Updating objectsrsquo position using (26)(8) Fitness evaluation of objects using (17)(9) Apply exploratory move to improve the fitness value using Algorithm 2ENDDOSTOP
Algorithm 1 Hybrid GSA
Two corners with reference to above selected corner arecreated for comparison as below
1198621199051198941119895 =
1 for 119894 + 1
119862119905119894119895 for 119895 = 1 2 119894 (119894 + 2) 119863
(34)
1198621199051198942119895
=
0 for 119894
1198621199051198941119895 for 119895 = 1 2 (119894 minus 1) (119894 + 1) 119863
(35)
In reference to these two corners coefficient vectors are gen-erated as shown in Table 1 Mathematically it is representedin the generalized form
119909119905119894119898119895 = 119909
119862119905119894119895 + Δ
119905119894119898119895
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(36)
where
Δ119905119894119898119895 =
+Δ 119894119895 if 119862119905119894119898119895 = 1
minusΔ 119894119895 if 119862119905119894119898119895 = 0
(119898 = 1 2 119895 = 1 2 119863 119894 = 1 2 119873119875)
(37)
The initial increment to coefficients is decided by
Δ 119895 =
119909max119895 minus 119909
min119895
120575
(38)
Objective functions at 1199091198961119895 and 1199091198962119895 are evaluated using (17)
as follows
119865119905119894119898 = 119891 (119909
119905119894119898119895) (119898 = 1 2) (39)
The minimum value of these two is selected to be com-puted with the rest of the corners generated subsequentlyand the selected corner for the generation of the next twocorner is
119862119905119894119895 =
1198621199051198941119895 if 1198651199051198941 lt 119865
1199051198942
1198621199051198942119895 if 1198651199051198941 gt 119865
1199051198942
(119895 = 1 2 119863) (40)
This process is repeated till all the corners of the hyper-cube are explored and the overall minimum is selected tofind the new centre point for the next iteration When thelast element of 119862
119905119894119895 vector contains one of the last branches
of BSA tree that is reached which ensures that all cornersof hypercube are explored the procedure is terminated InBSAmethod the number of computations is reduced by largeamount as elaborated in Table 2
34 Algorithms The different steps of the proposed algo-rithms are shown in Algorithms 1 and 2
4 Validation of Proposed Technique
The proposed GSA technique alone and then hybridizedwith BSA based evolutionary technique has been applied tostandard unimodal test functions multimodal test functions
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
(1) Enter with filter coefficients being decision variable as 1199090119894 1198650= 119891(119909
0119894 ) and set 119896 = 0
DO(21) 119909
119888119894 = 119909119896119894 (119894 = 1 2 119863)
(22) Set 119862119896119894 (119894 = 1 2 119863) using (33) 119896 = 119896 + 1(23) Compute 119862
1198961119894 (119894 = 1 2 119863) using (34)
(24) Compute 1198621198962119894 (119894 = 1 2 119863) using (35)
(25) Obtain 119909119896119898119894
and Δ119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (36) and (37) respectively(26) Evaluate fitness 119865119896 from (39)DO(271) 119895 = 119895 + 1
(272) Obtain 119862119896119898119894 119909119896119898119894 and Δ
119896119898119894
(119898 = 119894 2) (119894 = 1 2 119863) from (34) (35) (36) and (37) respectively(273) Evaluate fitness 119865119896 from (39)(274) Obtain 119862
119896119894 (119894 = 1 2 119863) from (40)
WHILE (119895 lt 119863)
IF (119865119896lt 119865119896minus1
) THEN(281) 119909119896119894 = 119909
119896minus1119894 (119894 = 1 2 119863) and 119865
119896= 119865119896minus1
ELSE(282) Δ 119894 = Δ 119894120590 (119894 = 1 2 119863)
ENDWHILE (Δ le 120598)
RETURN
Algorithm 2 BSA
Table 3 Unimodal test functions
Name Functions Dimension Search range
Sphere 11989101 =
119863
sum
119894=1
1199091198942 30 [minus100 100]
119863
SumSquare 11989102 =
119863
sum
119894=1
1198941199091198942 30 [minus10 10]
119863
SumPower 11989103 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
(119894+1) 30 [minus1 1]119863
Step 11989104 =
119863
sum
119894=1
(lfloor119909119894 + 05rfloor)2 30 [minus100 100]
119863
Elliptic 11989105 =
119863
sum
119894=1
1199091198942(106)
[(119894minus1)(119863minus1)]30 [minus5 10]
119863
Exponential 11989106 = exp(05
119863
sum
119894=1
119909119894) minus 1 30 [minus128 128]119863
Quartic 11989107 =
119863
sum
119894=1
1198941199091198944+ random [0 1) 30 [minus128 128]
119863
Schwelfel 222 11989108 =
119863
sum
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816+
119863
prod
119894=1
1003816100381610038161003816119909119894
1003816100381610038161003816
30 [minus10 10]119863
Schwelfel 12 11989109 =
119863
sum
119894=1
(
119894
sum
119895=1
119909119895)
2
30 [minus100 100]119863
Schwelfel 221 11989110 = max 10038161003816100381610038161199091
1003816100381610038161003816 1 le 119894 le 119863 30 [minus100 100]
119863
with large and small dimensional variables shown in Tables3 4 and 5 respectively The population has been taken as50 Maximum number of iterations has been set to 200 In(27) 119866119900 the initial value of gravitational constant has beentaken as 200 A predefined constant 120572 has been taken as20 The GSA is hybridized with BSA based on evolutionary
search technique for searching the variable space locally Theexploratory move has been repeated 50 times The algorithmis made to run 500 times independently to justify the globalsolution
The desired and obtained values of test functions areshown in Table 6 It is observed that for test functions with
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table4Multim
odaltestfunctio
nswith
larged
imensio
nalvariables
Nam
eFu
nctio
nsDim
ensio
nSearch
rang
e
Ackly
11989111=
20exp(
minus02radic
119863minus1
119863
sum 119894=1
1199091198942)
minusexp(119863minus1119863
sum 119894=1
cos2
120587119909119894)
+20+
11989030
[minus3232]119863
Greiwangk
11989112=
sum119863 119894=11199091198942
4000
minus
119863
prod 119894=1
cos(
119909119894
radic119894
)+
130
[minus600600]119863
Penalized-1
11989113=
120587 119863
119863minus1
sum 119894=1
(119910119894minus
1)2[1+
10sin2(120587119910119894+1)]+
10sin2(1205871199101)+
(119910119863
minus1)2
+
119863
sum 119894=1
119906(119909119894101004)
30[minus5050]119863
Penalized-2
11989114=
01
119863minus1
sum 119894=1
(119909119894minus
1)2[1+sin2(3120587119909119894+1)]+sin2(31205871199091)+
(119909119863
minus1)2[1+sin2(2120587119909119863)]
+
119863
sum 119894=1
119906(11990911989451004)
30[minus5050]119863
Alpine
11989115=
119863
sum 119894=1
1003816 1003816 1003816 1003816119909119894sin
119909119894+
011199091198941003816 1003816 1003816 1003816
30[minus1010]119863
Bohachevsky2
11989116=
119863minus1
sum 119894=1
[1199091198942
+2119909119894+12minus
03cos(
3120587119909119894)cos(
3120587119909119894+1)+
03]
30[minus100100]119863
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 5 Multimodal test functions with small dimensional variables
Name Functions Dimension Search range
Foxholes 11989117 = (
1
500
+
25
sum
119895=1
(119895 +
2
sum
119894=1
(119909119894 minus 119886119894119895)6)
minus1
)
minus1
2 [minus65536 65536]119863
Kowalik 11989118 =
11
sum
119894=1
[119886119894 minus
1199091 (1198871198942+ 1198871198941199092)
1198871198942+ 1198871198941199093 + 1199094
] 4 [minus5 5]119863
Six-humpCamel-back 11989119 = 41199091
2minus 211199091
4+
1
3
11990916+ 11990911199092 minus 41199092
2+ 41199092
4 2 [minus5 5]119863
Branin 11989120 = (1199092 minus51
4120587211990912+
5
120587
1199091)
2
+ 10 (1 minus
1
8120587
) cos (1199091) + 10 2 [minus5 10] [0 15]
Goldstien-price
11989121 = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909
21 minus 141199092 + 611990911199092 + 3119909
22)]
times [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909
21 + 481199092 minus 3211990911199092 + 27119909
22)]
2 [minus2 2]119863
Hartman-1 11989122 = minus
4
sum
119894=1
119888119894 exp(minus
4
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 4 [0 1]
119863
Hartman-2 11989123 = minus
4
sum
119894=1
119888119894 exp(minus
6
sum
119895=1
119886119894119895 (119909119895 minus 119901119894119895)2) 6 [0 1]
119863
Shekelrsquos-1 11989124 = minus
5
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-2 11989125 = minus
7
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
Shekelrsquos-3 11989126 = minus
10
sum
119894=1
[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]
minus1
4 [0 10]119863
large dimensional variables hybridization of GSA with BSAtechnique gives better results as compared to GSA in terms ofachieved function values
Wilcoxon Signed Rank test has been applied to validatethe performance of hybrid algorithm at a significance level005
Result 1 (119885-value) 119885-values are expressed in terms of meanand standard deviations of test statics It is observed that 119885-value is minus25424 and its corresponding 119901 value is 001108 Sothe result is significant at 119901 le 005
Result 2 (119882-value) The119882-value is 12The critical value of119882for119873 = 14 at 119901 le 005 is 21Therefore the result is significantat 119901 le 005
From the above results it is concluded that proposedGSA and hybrid GSA can be applied to unimodal as well asmultimodal test functions Since error surface of IIR filter isa multimodal function so the above proposed technique canbe applied to design the digital IIR filters
5 Design of DigitalIIR Filter and Comparisons
The design of digital IIR filter in cascaded form has beenimplemented using proposed GSA and hybrid GSA (HGSA)techniques by searching the filter coefficients in such amanner so that the fitness value (17) approaches 1 Theperformance of the designed digital IIR filter is measuredbased on pass-band and stop-band ripples phase responseerror and order of the filter Low-pass (LP) high-pass (HP)
band-pass (BP) and band-stop (BS) IIR filters have beenconsidered for the design In this paper the order of digitalIIR filter is a variable in the optimization process and isoptimized simultaneously along with 1205831 1205832 and 1205833 objectivefunctions The maximum order for LP HP BP and BS filtershas remained 12 as shown in Table 7 Hence the maximumvalue of119872 and119873 is kept as 4 for LP and HP filters and 0 and6 respectively for BP and BS filtersThe design conditions forpass-band and stop-band normalized frequencies of LP HPBP and BS filters are also shown in Table 7 where 120596 is thenormalized frequency of the signal and varies from0 to120587Theresults of the digital IIR filter design given by Yu and Xinjie[16] Wang et al [17] and Kaur et al [18] are referred to tocompare with design obtained by proposed HGSA approach
The phase response error (8) pass-band as well as stop-band ripples (4a) and (4b) obtained from the proposed GSAand HGSA techniques for LP HP BP and BS filters arecompared with CCGA NSGA-II LS-MOEA and RCGA inTable 8 From Table 8 it is concluded that the proposed GSAand HGSA techniques offer better performance in termsof phase response error pass-band ripples as well as stop-band ripples for LP HP BP and BS filters It is also revealedHGSA designs have better IIR filter than GSA The filterdesign is performed for 500 independent trial runs to achieveminimum maximum average and standard deviation offitness function described in (17) and is depicted in Table 9Very small value of standard deviation proves the robustnessof the proposed hybrid search technique to achieve globalsolution
The designed filters obtained by the proposed HGSAtechnique for LP HP BP and BS are given by (41) (42) (43)and (44) respectively
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 6 Results for test functions
Function Desired optimal valueof test function
GSA GSA-BSAObtainedvalue of testfunction
Standarddeviation
Obtainedvalue of testfunction
Standarddeviation
11989101 0 615 times 10minus15
338 times 10minus01 994 times 10minus16 498 times 10
minus01
11989102 0 126 times 10minus13
291 times 10minus05 127 times 10minus14 135 times 10
minus05
11989103 0 157 times 10minus13
132 times 10minus10 110 times 10minus13 133 times 10
minus10
11989104 0 0 156 times 10+00
0 859 times 10minus01
11989105 0 124 times 10+02
118 times 10+02 424 times 10+01 108 times 10
+02
11989106 0 0 786 times 10minus08
0 202 times 10minus08
11989107 0 125 times 10minus06
141 times 10minus04 815 times 10minus17 457 times 10
minus06
11989108 0 344 times 10minus07
515 times 10minus04 134 times 10minus07 227 times 10
minus08
11989109 0 113 times 10minus12
427 times 10+00 230 times 10minus14 103 times 10
+01
11989110 0 228 times 10minus02 958 times 10minus01
173 times 10minus01
111 times 10+00
11989111 0 736 times 10minus01
704 times 10minus02 287 times 10minus01 173 times 10
minus01
11989112 0 124 times 10+00
381 times 10minus01 110 times 10+00 215 times 10
minus01
11989113 0 165 times 10minus02
160 times 10minus01 118 times 10minus02 110 times 10
minus01
11989114 0 236 times 10minus01
100 times 10+00 220 times 10minus01 932 times 10
minus01
11989115 0 767 times 10minus04
121 times 10minus02 171 times 10minus08 121 times 10
minus02
11989116 0 0 440 times 10+00
0 249 times 10+00
11989117 10 09980 628 times 10minus01 09980 617 times 10
minus01
11989118 0000308 0000370 735 times 10minus05 0000364 755 times 10
minus05
11989119 minus10316 minus10316 660 times 10minus05
minus10316 128 times 10minus04
11989120 03979 03979 558 times 10minus17 03979 558 times 10
minus17
11989121 30 30 134 times 10minus15 30 134 times 10
minus15
11989122 minus38628 minus38628 893 times 10minus15
minus38628 893 times 10minus15
11989123 minus33224 minus33852 125 times 10minus02
minus33852 125 times 10minus02
11989124 minus101532 minus101532 209 times 10+00
minus101532 248 times 10+00
11989125 minus104029 minus104029 281 times 10minus01
minus104029 179 times 10minus14
11989126 minus105364 minus105364 161 times 10minus14
minus105364 161 times 10minus14
Table 7 Prescribed design conditions for LP HP BP and BS filters
Filter typeMaximumorder offilter
Pass band Stop band
LP 12 0 le 120596 le 02120587 03120587 le 120596 le 120587
HP 12 08120587 le 120596 le 120587 0 le 120596 le 07120587
BP 12 04120587 le 120596 le 061205870 le 120596 le 025120587
075120587 le 120596 le 120587
BS 12 0 le 120596 le 025120587
075120587 le 120596 le 12058704120587 le 120596 le 06120587
The magnitude response and phase response of LP HPBP and BS are shown in Figure 7 The pole-zero plots for LPHP BP and BS are shown in Figure 8 It is observed that thedesigned filters follow the stability constraints applied duringdesigning process as all the poles lie inside the unit circle
From the above results it is concluded that the proposedhybrid heuristic search technique is effective and robust
technique for design of digital IIR filter with better phaseresponse error pass-band ripples and stop-band ripples
119867LP (119911) = 0201016 times
(119911 + 0216875)
(119911 minus 0406875)
times
(1199112minus 0949082119911 + 0959715)
(1199112minus 1230432119911 + 0649521)
(41)
119867HP (119911) = 0149117 times
(119911 minus 0597477)
(119911 + 0402310)
times
(1199112+ 0877094119911 + 0918969)
(1199112+ 1220268119911 + 0634219)
(42)
119867BP (119911) = 0238985 times
(1199112minus 1608518119911 + 0968449)
(1199112+ 0622686119911 + 0530151)
times
(1199112+ 1616206119911 + 0984543)
(1199112minus 0638647119911 + 0527260)
(43)
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
Table 8 Comparison of design results for LP HP BP and BS Filters
Technique Lowest order of filter Pass-band magnitude ripple Stop-band magnitude ripple Phase response errorLP filter
CCGA 3 09034 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01699 14749 times 10
minus4
NSGA-II 3 09117 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01719 12662 times 10
minus4
LS-MOEA 3 09083 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01586 10959 times 10
minus4
RCGA 3 09141 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01556 11788 times 10
minus4
GSA 3 09201 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01471 10391 times 10
minus4
HGSA 3 09208 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01475 10375 times 10minus4
HP filterCCGA 3 09044 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01749 97746 times 10
minus5
NSGA-II 3 08960 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01769 91419 times 10
minus5
LS-MOEA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01746 96251 times 10
minus5
RCGA 3 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01742 95757 times 10
minus5
GSA 3 09052 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01674 75431 times 10
minus5
HGSA 3 09051 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01698 73734 times 10minus5
BP filterCCGA 4 08920 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01654 81751 times 10
minus5
NSGA-II 4 09100 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01771 36503 times 10
minus4
LS-MOEA 4 09285 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01734 60371 times 10
minus5
RCGA 4 09333 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01641 59070 times 10
minus5
GSA 4 09354 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01587 60121 times 10
minus5
HGSA 4 09398 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01634 56440 times 10minus5
BS filterCCGA 4 08966 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01733 16198 times 10
minus4
NSGA-II 4 08917 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01770 15190 times 10
minus4
LS-MOEA 4 08967 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01725 15084 times 10
minus4
RCGA 4 08975 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01708 15144 times 10
minus4
GSA 4 09004 le
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 10
10038161003816100381610038161003816119867(119890119895120596
)
10038161003816100381610038161003816le 01481 14103 times 10
minus4
HGSA 4 09009 le
10038161003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 10 1003816
1003816100381610038161003816H(ej120596)1003816100381610038161003816
1003816le 01459 13827 times 10minus4
Table 9 Variation of fitness function for LP HP BP and BS filters
Filter Minimum Maximum Average Standard DeviationLP 09948596 09998963 09998357 36140 times 10
minus4
HP 07615849 09999232 08609298 84818 times 10minus2
BP 09900417 09999436 09998430 73827 times 10minus4
BS 09993995 09998618 09998390 22359 times 10minus5
119867BS (119911) = 0455623 times
(1199112+ 0358866119911 + 0913907)
(1199112minus 0738038119911 + 0471956)
times
(1199112minus 0363914119911 + 0943718)
(1199112+ 0741348119911 + 0475340)
(44)
6 Conclusion
This paper proposes the design of digital IIR filter withconflicting objectives by using hybrid GSA where GSA and
BSA based evolutionary search algorithms are applied asglobal and local search optimization techniques respec-tively HGSA maintains the balance between explorationand exploitation Lesser parameters are required to tuneThe results reveal that the proposed HGSA techniques offerbetter performance in terms of phase response error andpass-band as well as stop-band ripples for LP HP BPand BS filters as compared to CCGA and NSGA-II LS-MOEA and RCGA methods It is also revealed that HGSAoutperforms GSA to optimize unimodal multimodal higherand lower dimensions standard test problems as well as forthe design of digital IIR filters The proposed method inde-pendently designs digital filter whether it is LP HP BP or BSfilter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
LPF (HGSA) LPF (HGSA)
HPF (HGSA) HPF (HGSA)
BPF (HGSA) BPF (HGSA)
BSF (HGSA) BSF (HGSA)
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
02
04
06
08
10
Mag
nitu
de
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
10
20
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
05
10
15
20
25
30
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
0
Phas
e
0 02 03 04 05 06 07 08 09 1001
Normalized frequency
minus30
minus25
minus20
minus15
minus10
minus05
minus30
minus20
minus10
minus30
minus20
minus10
Figure 7 Magnitude and phase responses of LP HP BP and BS filters
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
0
05
10
Real part
Imag
inar
y pa
rt
minus05
minus10
10050minus05minus10
LPF (HGSA) HPF (HGSA)
BPF (HGSA) BSF (HGSA)
Figure 8 Pole and zero plots of LP HP BP and BS filters
References
[1] J G Proakis and D G Manolakis Digital Signal ProcessingPrinciples Algorithms and Applications Pearson EducationSingapore 2013
[2] B A Shenoi Introduction to Digital Signal Processing and FilterDesign Wiley-Interscience Hoboken NJ USA 2006
[3] N Karaboga and M B Centinkaya ldquoA novel and efficientalgorithm for adaptive filtering artificial bee colony algorithmrdquoTurkish Journal of Electrical Engineering amp Computer Sciencesvol 19 no 1 pp 175ndash190 2011
[4] C-W Tsai C-H Huang and C-L Lin ldquoStructure-specifiedIIR filter and control design using real structured geneticalgorithmrdquo Applied Soft Computing vol 9 no 4 pp 1285ndash12952009
[5] S Chen and B L Luk ldquoDigital IIR filter design using particleswarm optimisationrdquo International Journal of Modelling Identi-fication and Control vol 9 no 4 pp 327ndash335 2010
[6] N Karaboga ldquoDigital IIR filter design using differential evo-lution algorithmrdquo Eurasip Journal on Applied Signal Processingvol 2005 Article ID 856824 2005
[7] B Singh J S Dhillon and Y S Brar ldquoPredator prey optimiza-tionmethod for the design of IIR filterrdquoWSEAS Transactions onSignal Processing vol 9 no 2 pp 51ndash62 2013
[8] J-Y Lin and Y-P Chen ldquoAnalysis on the collaboration betweenglobal search and local search in memetic computationrdquo IEEETransactions on Evolutionary Computation vol 15 no 5 pp608ndash623 2011
[9] B Singh J S Dhillon and Y S Brar ldquoA hybrid differentialevolution method for the design of IIR digital filterrdquo ACEEEInternational Journal of Signal amp Image Processing vol 4 no 1pp 1ndash10 2013
[10] K Kaur and J S Dhillon ldquoDesign of digital IIR filters usingintegrated cat swarm optimization and differential evolutionrdquoInternational Journal of Computer Applications vol 99 no 4pp 28ndash43 2014
[11] J HuaW Kuang Z Gao LMeng and Z Xu ldquoImage denoisingusing 2-D FIR filters designed with DEPSOrdquo Multimedia Toolsand Applications vol 69 no 1 pp 157ndash169 2014
[12] Vasundhara D Mandal R Kar and S P Ghoshal ldquoDigital FIRfilter design using fitness based hybrid adaptive differential evo-lution with particle swarm optimizationrdquo Natural Computingvol 13 no 1 pp 55ndash64 2014
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[13] T S Bindiya and E Elias ldquoModified metaheuristic algorithmsfor the optimal design of multiplier-less non-uniform channelfiltersrdquoCircuits Systems and Signal Processing vol 33 no 3 pp815ndash837 2014
[14] S K Saha R Kar D Mandal and S P Ghoshal ldquoDesignand simulation of FIR band pass and band stop filters usinggravitational search algorithmrdquoMemetic Computing vol 5 no4 pp 311ndash321 2013
[15] S Saha R Kar D Mandal and S Ghoshal ldquoOptimal IIRfilter design using Gravitational Search AlgorithmwithWaveletMutationrdquo Journal of King Saud UniversitymdashComputer andInformation Sciences vol 27 no 1 pp 25ndash39 2015
[16] Y Yu and Y Xinjie ldquoCooperative coevolutionary geneticalgorithm for digital IIR filter designrdquo IEEE Transactions onIndustrial Electronics vol 54 no 3 pp 1311ndash1318 2007
[17] Y Wang B Li and Y Chen ldquoDigital IIR filter design usingmulti-objective optimization evolutionary algorithmrdquo AppliedSoft Computing vol 11 no 2 pp 1851ndash1857 2011
[18] R Kaur M S Patterh and J S Dhillon ldquoReal coded geneticalgorithm for design of IIR digital filter with conflicting objec-tivesrdquoAppliedMathematics amp Information Sciences vol 8 no 5pp 2635ndash2644 2014
[19] Y S Brar J S Dhillon and D P Kothari ldquoInteractive fuzzysatisfyingmulti-objective generation schedulingrdquoAsian Journalof Information Technology vol 3 no 11 pp 973ndash982 2004
[20] J-T Tsai J-H Chou and T-K Liu ldquoOptimal design of digitalIIR filters by using hybrid Taguchi genetic algorithmrdquo IEEETransactions on Industrial Electronics vol 53 no 3 pp 867ndash8792006
[21] I Jury Theory and Application of the Z-Transform MethodWiley New York NY USA 1964
[22] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009
[23] J S Dhillon J S Dhillon and D P Kothari ldquoEconomic-emission load dispatch using binary successive approximation-based evolutionary searchrdquo IET Generation Transmission andDistribution vol 3 no 1 pp 1ndash16 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of