Jointly multi-user detection and channel estimation with ... · user channel estimation (MuChE) and...

WIRELESS COMMUNICATIONS AND MOBILE COMPUTINGWirel. Commun. Mob. Comput. 2011; 11:767–782

Published online 18 January 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.920

RESEARCH ARTICLE

Jointly multi-user detection and channel estimationwith genetic algorithm†

Fernando Ciriaco1, Taufik Abrão2*, Antonio Fischer de Toledo1 and Paul Jean E. Jeszensky1

1 Department of Telecommunications and Control Engineering, Escola Politécnica of University of São Paulo (PTC-EPUSP), São Paulo,SP 05508-900, Brazil2 Electrical Engineering Department, State University of Londrina (DEEL-UEL), Londrina, PR 86051-990, Brazil

ABSTRACT

This work aims at proposing the use of the evolutionary computation methodology in order to jointly solve the multi-user channel estimation (MuChE) and detection problems at its maximum-likelihood, both related to the direct sequencecode division multiple access (DS/CDMA). The effectiveness of the proposed heuristic approach is proven by comparingperformance and complexity merit figures with that obtained by traditional methods found in literature. Simulation resultsconsidering genetic algorithm (GA) applied to multipath, DS/CDMA and MuChE and multi-user detection (MuD) showthat the proposed genetic algorithm multi-user channel estimation (GAMuChE) yields a normalized mean square errorestimation (nMSE) inferior to 11%, under slowly varying multipath fading channels, large range of Doppler frequenciesand medium system load, it exhibits lower complexity when compared to both maximum likelihood multi-user channelestimation (MLMuChE) and gradient descent method (GrdDsc). A near-optimum multi-user detector (MuD) based on thegenetic algorithm (GAMuD), also proposed in this work, provides a significant reduction in the computational complexitywhen compared to the optimum multi-user detector (OMuD). In addition, the complexity of the GAMuChE and GAMuDalgorithms were (jointly) analyzed in terms of number of operations necessary to reach the convergence, and comparedto other jointly MuChE and MuD strategies. The joint GAMuChE–GAMuD scheme can be regarded as a promisingalternative for implementing third-generation (3G) and fourth-generation (4G) wireless systems in the near future. Copyright© 2010 John Wiley & Sons, Ltd.

KEYWORDS

near-optimum multi-user detectors; fading channel estimation; heuristic approach; genetic algorithm; DS-CDMA systems

*Correspondence

Taufik Abrão, Electrical Engineering Department, State University of Londrina (DEEL-UEL), Londrina, PR 86051-990, Brazil.E-mail: [email protected]

1. INTRODUCTION

Even though there are several works dealing with channelestimation, very few have analyzed the channel from thepoint of view of heuristic approaches [1--3]. It is well knownthat the heuristic method does not always find the optimaloperation point, however, it is very efficient at achievingnear-optimal solutions, and faster than any conventionalpoint-by-point exhaustive search technique. Based on themaximum likelihood (ML) rule, in this paper an alternativeapproach, which uses the genetic algorithm multi-userchannel estimation (GAMuChE) is revealed. The proposedGAMuChE is capable of estimating the user’s complexchannel coefficients from the statistics provided by theoutput of matched filters bank (MFB) at the receiver.

†The material in this paper was presented in part at the ISSSTA’06 - IEEE International Symposium on Spread Spectrum Techniques and Applications,Manaus, Brazil, 2006

Other heuristic techniques, for instance, the simulatingannealing, tabu search, and ant colony optimization [4,5],particle swarm, and local search [6,7], could be successfullyemployed with similar trade-off between performance andcomplexity. However, in this work the genetic algorithm(GA) was selected due to its powerful optimization charac-teristic, including mechanisms to escape from local optima.

Most of the analyses found in the literature about channelestimation consider an ML approach, which consistsof determining the multi-user signal correlation matrixinversion. However, the computational effort necessary toachieve such optimization, grows prohibitively when thenumber of users (K), paths (L), or both increase. With thepurpose of overcoming such difficulties, approximations areimplemented. The heuristic approach to estimate channel

Copyright © 2010 John Wiley & Sons, Ltd. 767

Jointly multi-user detection and channel estimation with GA F. Ciriaco et al.

parameters, proposed in this paper, endeavors to reducethe exponential computational complexity inherent to theML method, i.e., proportional to 2KL, while maintainingthe nMSE of the coefficients at acceptable levels.

In Reference [1], a genetic approach was proposedin order to jointly estimate the information bits andchannel coefficients for a flat channel in a synchronousdirect sequence code division multiple access (DS/CDMA)system. In Reference [2], a micro GA-based inter-symbol interference (ISI) channel estimation technique wasintroduced, where the Viterbi algorithm was employed fordata detection in a single-user receiver over an additivewhite Gaussian noise (AWGN) channel.

More recently, other no heuristic methods have beenunveiled. In Reference [8], jointly symbol timing, complexamplitude of a flat fading channel and transmittedsymbols estimation have been explored from the Bayesianperspective, using particle-filtering techniques. A similarproblem considering single-input multiple-output (SIMO)wireless channels scenarios and sphere decoding to a jointlyML channel estimation and signal detection was reportedin Reference [9]. It was shown that the complexity wasequivalent for both the sphere decoding and heuristictechniques, over a wide range of systems parameters,at single and multi-user detectors. In contrast, for abroad range of signal-to-noise ratio (SNR), the spheredecoding approach provides a better performance. Multi-carrier CDMA systems, multi-user detection (MuD) andchannel estimation under time-varying frequency-selectivefading channels and narrowband interference (NBI) wasanalyzed in Reference [10]. A weighted recursive at the leastM-estimate channel estimator and a robust decorrelatingdetector were exploited in order to jointly suppress NBIand multiple access interference (MAI), respectively. InReference [11], channel estimates for long-code CDMAsystems over multipath channels using training sequences,and based on the maximum likelihood multi-user channelestimation (MLMuChE) approach, was reported. Adoptinga static channel (i.e., no Doppler spread), it was reported thatreliable estimates for the channel coefficients were obtainedafter processing an excessive frame length (I ≥ 50 bits), andan nMSE equal to 6%.

It has been shown that heuristics are a viable approachto separately deal with channel coefficients in Reference[3] and MuD in References [7,12,13] for DS/CDMAsystems, yielding good performance and low computationalcomplexity.

For any learning method, the codifying of solutioncandidates is very important, and one of the mostcrucial factors for the heuristic algorithm success. ManyGA applications have used fixed length vectors andunalterable bit order to codify solution candidates.However, more recently, other coding techniques have beenexamined, including coding for multivariable problems(i.e., simultaneous optimization of several characteristics)and coding in the complex domain [14].

The favored choice of coding implementation, inmost optimization cases, is the binary representation,

for historical and contextual reasons [15]. Regardlessthe lengthy vector candidate produced by the binaryrepresentation when compared to the decimal coding,the former offers advantages in terms of convergencespeed, overall quality, and ease of execution (e.g.,implementation). In addition, the binary coding facilitatesthe GA implementation in computational language,exploring efficiently the digital processors capacity.

The organization of this paper is as follows. Section 2presents the system model. In Section 3, a brief introductionof the channel estimation metric is provided. Section 4outlines the optimization problem, and discusses the choiceof the GA parameters. The simulation results are exploredextensively in Section 5, and computational complexityaspects are exposed in Section 6. Finally, conclusions aredrawn in Section 7.

2. DS/CDMA SYSTEM MODEL

In a DS/CDMA system with binary phase-shift keying(BPSK) modulation shared with K asynchronous users, thekth user transmitted signal corresponding to an informationsequence of length I, is given by

xk(t) =√

2Pk

I∑i=1

b(i)k sk(t − iTb) cos(ωct) (1)

where Pk = A2k/2 represents the kth user transmitted power;

b(i)k ∈ {−1, +1} is the ith BPSK symbol with period Tb and

bit energy Ebk= PkTb; ωc is the carrier frequency; sk is the

signature sequence assigned to the kth user and is given by

sk(t) =N−1∑n=0

ps(t − nTc)ck,n (2)

where sk corresponds to the spreading sequence defined inthe interval [0, Tb), and zero outside; ck,n ∈ {−1; 1} is thenth chip of the sequence with length N assigned to the kthuser; Tc is the chip period and the processing gain, and (i.e.,Tb/Tc), is equal to N. The pulse shaping ps(t) is assumedrectangular with unitary amplitude in the interval [0; Tc),and zero outside.

Assuming that the signal xk(t) of each user propagatesover L independent slow Rayleigh fading paths, thebaseband received signal at the base station is given by

r(t) =K∑

k=1

L∑�=1

wk,�xk(t − τk,�) + η(t) (3)

where wk,� is the complex attenuation factor, and τk,� is thedelay with respect to the timing reference at the receiver onthe �th path of the kth user. The channel attenuations anddelays are assumed to be constant during the estimationprocess. The random delay, τk,�, takes into account theasynchronous nature of the transmission, dk, as well as the

768 Wirel. Commun. Mob. Comput. 2011; 11:767–782 © 2010 John Wiley & Sons, Ltd.DOI: 10.1002/wcm

F. Ciriaco et al. Jointly multi-user detection and channel estimation with GA

propagation delay, that is �k,�, for kth user, in the �th path,resulting in

τk,� = �k,� + dk (4)

where η(t) in Equation 3 represents the AWGN. It isassumed that the �-path of the complex attenuation factorfor the kth user over the ith bit interval is

w(i)k,� = β

(i)k,�e

jφ(i)k,�

where the phase φk,� ∈ [0, 2π[ and the channel coefficientsamplitude βk,� represents the small scale fading envelopefollowing a Rayleigh distribution.‡ In addition, it isassumed that the channel gain is normalized for allusers:

∑L

�=1 E[β2k,�] = 1 ∀ k, where E[·] is the expectation

operator.

3. METRICS FOR CHANNELESTIMATION AND MULTI-USERDETECTION

3.1. ML and heuristic channel estimation

The received signal was discretized (at the receiver) bysampling r(t) at the chip rate. The observation vectors wereformed by collecting N successive outputs of r(nt) with thetime interval between samples, equal to one chip period (Tc),starting at an arbitrary timing reference at the receiver. Thepath delays from all users were assumed enclosed in onesymbol period; hence in an observation window, equal toTb, there were, at most, two symbols for any user-receivedsignal. Using vectorial notation, Equation (3) can be statedas [11]

ri = UZbi + ni (5)

where ri is the ith N × 1 observation vector, U is anN × 2KN spreading matrix, Z is a 2KN × 2K channelresponse matrix, bi is 2K × 1 symbol vector, and ni is aN × 1 complex Gaussian zero-mean random vector withindependent elements and bilateral power density, givenby σ2 = N0/2. The spreading matrix is constructed fromshifted versions of the spreading codes, corresponding tothe ith and (i + 1)th symbols of each user. For the shortcode case [3], the observation vector can be written as

U = [UR

1 UL1 UR

2 UL2 . . . UR

k ULk . . . UR

K ULK

](6)

‡ With probability density function (PDF) given by: px(x|σ) =xσ2 exp

(− x2

2σ2

), where σ is the Rayleigh parameter. Given n

independent and identically distributed Rayleigh samples, a maximum

likelihood estimate of σ is: σ =√

12n

∑n

i=1x2

i .

where

URk =

ck,1 ck,2 · · · ck,N

ck,2 ck,3 · · · 0

......

. . ....

ck,N−1 ck,N · · · 0

ck,N 0 · · · 0

and

ULk =

0 0 · · · 0

0 0 · · · ck,1

......

. . ....

0 0 · · · ck,N−2

0 ck,1 · · · ck,N−1

are constructed from the right and left parts ofthe kth user spreading code, respectively. The chan-nel response is a matrix in the form of Z =diag(z1, z1, z2, z2, . . . zK, zK), where zk is the N × 1channel response vector for the kth user. In order to simplifythe analysis, it is assumed that the entire energy of the�th path is captured at the instant τk,� = qk,�Tc, i.e., at qk,�

positioned in zk. For example, when an user includes twopaths with the delays τk,1 = 3Tc and τk,2 = 5Tc, then

zk = [0 0 0 wk,1 0 wk,2 0 . . . 0

]�(7)

The non-zero coefficients locations determine the pathdelays. Finally, the symbol vector has the form:

bi = [b1,i b1,i+1 b2,i b2,i+1 . . . bK,i bK,i+1

]�(8)

With the purpose of estimating the channel parameters,Equation (5) can be rewritten as

ri = UBiz + ni (9)

where the NK × 1 channel response vector is given by

z = [z�

1 z�2 . . . z�

K

]�, and the 2KN × NK multi-user

information matrix is defined as

Bi =

b(i)1 0 · · · 0

b(i+1)1 0 · · · 0

0 b(i)2 · · · 0

0 b(i+1)2 · · · 0

......

. . ....

0 0 · · · b(i)K

0 0 · · · b(i+1)K

⊗ IN (10)

Wirel. Commun. Mob. Comput. 2011; 11:767–782 © 2010 John Wiley & Sons, Ltd. 769DOI: 10.1002/wcm


where ⊗ denotes the Kronecker product and IN is the Nidentity matrix. Thus, it is possible to estimate N channelparameters for each user. In this paper, it is assumed that thepath delays for each user are within one symbol duration.Then, the number of non-zero coefficients in the effectivechannel response vector is determined by the number ofpaths and delays as in Equation (7).

The objective is to find the vector z in Equation (9)according to the ML rule related to the channel responsefunction, using the knowledge of their spreading codes andthe transmitted bits from all users. These bits could bemade available from either the preamble before the dataor the pilot channel. Furthermore, in the estimation phase,training or pilot sequences are assumed, and in the trackingphase, data decisions from the detector are fed back to theestimator.

The joint conditional distribution of I receivedobservation vectors, given the knowledge of the spreadingsequences, the channel and the information bits can beexpressed as

p(r1, r2, . . . , rI |U, z, B1,B2, . . . ,BI ) = −1

(πσ2)NI×

× exp

{−1

σ2

I∑i=1

(ri − UBiz)H(ri − UBiz)

}(11)

with (·)H = [(·)∗]� representing the transposed hermitianoperator.

The estimate that maximizes the ML function satisfiesthe Equation [11], and can be written as{

I∑i=1

(UBi)H(UBi)

}zML(I) =

I∑i=1

(UBi)Hri (12)

Defining the NK × NK multi-user signal correlation matrixas

RI = 1

I

I∑i=1

(UBi)H(UBi) (13)

and the NK × 1 vector as

yI = 1

I

I∑i=1

(UBi)Hri (14)

the length of information sequence should satisfy thecondition I ≥ K + �K/N� in order to RI be consideredfull rank.§ Hence the ML channel estimation can be written

§ Assuming that random spreading codes are used, and these spreadingcodes are linearly independent over the length of the informationsequence I.

as

zML(I) = R−1I yI (15)

where zML(I) is jointly Gaussian with mean z, and thecovariance matrix that satisfies the condition σ2

IR−1

I [11].Heuristic algorithms (e.g., GA) can be employed to

compute recursively the ML estimate in an optimizationsubspace search. The direct computation of the exact MLchannel estimate, involving lengthy calculations to solve thecorrelation matrix, R−1

I yI are computationally intensive.On the other hand, the potential gain in speeding up theexecution time, by approximating the ML solution to aheuristic-interactive procedure, is substantial, for the reasonthat the updating of the channel estimate is carried outconcomitantly with the receiving of the preamble data,therefore it is no longer necessary to wait for the completionof the preamble in order to perform the calculations. Inthis paper, the GA optimization techniques are appliedin order to estimate multi-user channel parameters (i.e.,GAMuChE).

The GAMuChE allows the estimation of the vector z (i.e.,module and phase) carrying out the minimization of meansquared error of the candidate-vectors, z, which represent asmall subspace of the total search space; and the procedureenables a substantial complexity reduction when comparedto the usual exhaustive search procedure. Hence, the jointlychannel coefficients estimates for the mth bit of all users isobtained by computing the squared error minimization asfollows:

z(m)GA = min

z∈RNK×1α(z) (16)

= minz∈RNK×1

m∑i=m−I

(ri − UBiz)H(ri − UBiz)

where I is the processing window and the estimates z(m)GA are

obtained for each symbol period. This cost function is thenegative of the log-likelihood function (LLF) (11), ignoringconstants that are independent of the channel z. Therefore,admitting that the set of initial information bits (10) andthe spread sequences matrix (6) are known, the channelcoefficients for all users can be estimated jointly. For thesake of simplicity, the delays associated to the significantmultipaths of all users were assumed to be known.

3.2. Metric for heuristic multi-userdetection

Using a vectorial notation, the Equation (3) can be stated as

r(t) =I−1∑i=0

s�(t − iTb)aw(i)b(i) + η(t) (17)



where s(t) = [s1(t − τ1,1), s1(t − τ1,2), . . . , s1(t − τ1,L),. . . , sk(t − τk,�), . . . , sK(t − τK,L)]� is the users’ signaturesequence vector. The diagonal matrix for the averagereceived users’ amplitude, including the path lossesand shadowing effects, is a = diag

[√P

′1I,

√P

′2I, . . . ,√

P′KI], where IL×L is the identity matrix with dimension

L; w(i) = diag[ω

(i)1,1, . . . , ω

(i)1,L, ω

(i)2,1, . . . , ω

(i)2,L, . . . , ω

(i)K,L

]is the diagonal channel gain matrix, and the data

vector is given by b(i) = [b(i)

1 ,b(i)2 , . . . ,b(i)

K

]�with

b(i)k representing the 1 × L kth user bit vector. For

simplicity, and without loss of generality, the randomdelays were arranged in a crescent order, such that0 ≤ τ1,1 ≤ τ1,2 ≤ · · · τ1,L ≤ τ2,1 ≤ · · · ≤ τK,L < Tb.

A Rake receiver, suitable for multipath fading channels,composed of a bank of KL filters matched to the usersignature sequence, yielded the coherent reception of the kthuser, corresponding to the �th multipath component (e.g.,‘finger’), that was sampled at the end of the ith bit intervalaccording to the following expression:

y(i)k,� =

∫ +∞

−∞r(t)sk(t − iTb − τk,�) dt

=√

P′kTbβ

(i)k,�b

(i)k + SI(i)

k,� + I(i)k,� + n

(i)k,� (18)

where the first term corresponds to the desired signal, thesecond term to the self-interference (SI), the third term tothe MAI over the �th multipath component of kth user, andthe last term to the filtered AWGN.

The output of the matched filter bank, at the ith symbolinterval, can be written using a vector notation as

y(i) = [y

(i)1,1, y

(i)1,2, . . . , y

(i)1,L, . . . , y

(i)K,1, . . . , y

(i)K,L

]�(19)

= R�[1]ac(i+1)b(i+1) + R[0]ac(i)b(i) + · · ·

+R[1]ac(i−1)b(i−1) + n(i)

where the matrices R[0] and R[1] with LK × LK

dimension are defined by the elements:

Rjk[0] =

1, if j = k

Rjk(τjk, 0), if j < k

Rkj(τjk, 0), if j > k

(20)

and

Rjk[1] ={

0, if j ≥ k

Rkj(τjk, 0), if j < k(21)

with the partial cross-correlation elements Rjk given by

Rj,k(τ, i) =∫ Tb

0

sj(t)sk(t + iTb + τ) dt (22)

where i = 0 and the filtered noise vector n(i) has anautocorrelation matrix

E[n(i)n(j)�] =

0.5N0R

�[1], if j = i + 1;

0.5N0R[0], if j = i;

0.5N0R[1], if j = i − 1;

0, otherwise

(23)

The conventional detector for frequency selective chan-nels consists in combining the available MFB outputs ofeach user (i.e., fingers) in a coherent way and weighting it byeach channel gain [16]. The MRC combines the D correla-tors’ output signals, followed by an abrupt decision circuit:

d(i)k =

D∑�=1

Re{y

(i)k,�(s)β

(i)k,�

}(24)

b(i)k = sign

(d

(i)k

)(25)

where D ≤ L represents the number of correlators (e.g.,one for each user) of the receiver. This technique, alsoknown as Rake diversity, requires an estimation of thefollowing user-specific parameters: channel coefficients,β, power, P

′ , delay, τ, (and therefore correlations, R), andphase, φ. Furthermore, it is necessary to consider that theRake receiver performance may be degraded when thenumber of users is increased (e.g., increasing the MAI),and, or the power of the interference is raised (e.g., near-fareffect). One possible solution is to adopt joint decisionrelated to multi-user strategies, such as the maximumlikelihood sequence detector [17,18].

Joint optimum decisions are obtained by the optimummulti-user detector (OMuD) that selects the most likelysequence of transmitted bits, once given the observations atreceiver. Any joint decision strategy made on the ith bits ofthe K users has to take into account at least the decisions oneither the (i − 1)th bit or the (i + 1)th of the each user. Forthe joint decision of all bits from all users, it was adoptedthe one-shot approach in asynchronous channels [18]. Inthis context the K-user, L-paths, I-frame, and asynchronouschannel scenario can be viewed as a KLI-user synchronouschannel scenario, and then the KLI-user vector B can bewritten as

B = [b(0)� ,b(1)� ,b(2)� , . . . ,b(I−1)�]�

(26)

The objective is to compute the KLI-vector B thatmaximizes [18]

g{y(t), t ∈ [0, (I − 1)Tb]|B}

= exp

(−∫ (I−1)Tb

0

[y(t) − S(B)]2 dt

)(27)



where

S(B) =I−1∑i=0

K∑k=1

L∑�=1

√P

′kb

(i)k sk(t − τk,�) (28)

Based on the matched filter observations, vector y(i)

in Equation (19), the maximization of Equation (27) isequivalent to selecting of the vector B, that maximizes theso-called LLF [18]:

�(B) = 2Re{B�CHAY} − B�CARACHB (29)

where the coefficients and amplitudes diagonal matrices,with dimension KLI, are defined, respectively, byC = diag[w(0),w(1),w(2), . . . ,w(I−1)] and A = diag[a,a,

a, . . . ,a]; where the MFB outputs matrix (applied toall K users, L fingers, and I bits) is Y = [

y(0)� ,y(1)� ,

y(2)� , . . . ,y(I−1)�]�

, and the block-tridiagonal, block-Toeplitz cross correlation matrix R, with the samedimension, can be defined as [18]:

R =

R[0] R�[1] 0 · · · 0 0

R[1] R[0] R�[1] · · · 0 0

0 R[1] R[0] · · · 0 0

· · · · · · · · · . . . · · · · · ·0 0 0 · · · R[1] R[0]

(30)

Therefore, the complete frame, with the estimatedtransmitted bits associated with all K users, can be obtainedwith the optimization of Equation (29), resulting in

b = arg

{max

B∈{+1,−1}IK[�(B)]

}(31)

The OMuD attempts to find the best vector of data bits in aset with all possibilities. This is a well known combinatorialproblem of selecting the most likely transmitted datastream, given the signal energies, cross-correlations, andthe signals of the MFB outputs, that results in a non-deterministic polynomial-time hard (NP-hard) in numberof users [19], and where the traditional algorithms areinefficient. Restricting the search space, the heuristicalgorithms implement search mechanisms aims to find asolution based on an objective function (or fitness value,in terms of Biology parlance) [4--6,20], which is ableto quantify the improvement tendency in the sense ofminimum Euclidean distance between any pair of distinctmulti-user signals. For the MuD problem over frequencyselective channels [1,7,12,21], the fitness value can beexpressed as Equation (29). Therefore, each heuristicalgorithm, through distinct search mechanisms, seeks outthe maximization of LLF by testing distinct frames ofcandidate bits in each new iteration. These attempts aimto maximize the DS/CDMA system average performanceover K active users. The increasing of number of attempts

and/or the enlarging in a search of sub-space, approachesthe performance obtainable from OMuD.

4. GA CODING AND CHOICE OFPARAMETERS

Considering that the data vector to be optimized is binaryin its natural form, the coding for the MuD problemis intrinsically binary. Therefore, it is not necessary toaccomplish an additional coding of the candidates solutionsfor the binary form. Those vectors candidates will berepresented directly by the bits of transmitted information,resulting in an individual with Qindiv = KI bits long.

Whereas the coding for the MuD problem is intrinsicallybinary (e.g., the data vector to be optimized is binary), theMuChE is substantially no binary: the channel responsevectors to be optimized can be mapped onto the complexnumber domain. On the other hand, the receiver architecturebecomes simpler when the estimator and the detector use thesame GA structure to accomplish their optimizations. Thatis only possible if both problems present the same coding,and in this case it should be binary.

Considering the advantages associated to the binaryrepresentation in terms of allowing a simplified architecturefor the receiver, three different possible binary codingfor the coefficients and delay estimate problem, namedhereafter Type-I, -II, and -III, were investigated. Note thatthe number of possible coding strategies for the MuChEproblem cannot be uniquely defined, which leaves enoughmargin to allow for implementation and comparison withother coding possibilities. The Type-I, -II, and III coding aredescribed in Table I, and comparisons of figures of merit forthese coding are made in Section 5 (e.g., nMSE, or accuracy,versus complexity, or number of operations).

4.1. Binary coding types for the MuChEproblem

Independent of the coding method, the GAMuChEobjective is to minimize the cost function α(z) in Equation(16). All possible combinations of z ∈ C, which givesan infinite number of solutions/possibilities, characterizethe search universe of this MuChE. For convenience, thebinary domain is employed, thus allowing the selectionof an adequate number of quantization levels, in orderto reduce the number of possible solutions, to increasethe processing speed and to expedite the implementation.Table I summarizes the important characteristics for thethree binary coding methods.

Table I shows, for the Type-I binary coding, that thechannel response zk,� may be separated into real andimaginary parts, resulting in two real values, �{.} and {.},representing respectively the real and imaginary parts of acomplex number. In addition, z�

k,� and z k,� are separated

further into integer and fractional parts, and where theoperator �n� denotes the greatest integer not exceeding n.



Table I. Characteristics of binary coding types.

Characteristic Type-(I) Type-(II) Type-(III)

ChannelRepresentation

{z�k,�

=∣∣�{zk,�}∣∣ and z

k,�=∣∣ {zk,�}∣∣{

��k,�

=⌊z�k,�

⌋and �

k,�= z�

k,�− ��

k,�,

� k,�

=⌊z k,�

⌋and

k,�= z

k,�− �

k,�

{z�k,�

=∣∣�{zk,�}∣∣

z k,�

=∣∣ {zk,�}∣∣

|zk,�| =√(

�{zk,�})2 +

( {zk,�})2

�zk,� = arctg(

{zk,�}�{zk,�}

)Digitalization(ADC)

{��k,�

= ADC[��k,�

]Qint,

�k,�

= ADC[

round(

1 �k,�

)]Qfrac

z�k,�

= ADC[Amax

z�k,�

]Qint

z k,�

= ADC[Amax

z k,�

]Qint

{|zk,�| = ADC

[Amax|zk,� |

]Qabs

�zk,� = ADC[

2��zk,�

]Qphs

IndividualRepresentation

�(I)k,�

=[��k,� �k,�� k,� k,�

]�(II)k,�

=[z�k,�

z k,�

]�(III)k,�

=[|zk,�| �zk,�

]Size (bits)

Qindiv = 2KL(Qint +Qfrac

)Qindiv = 4KLQint Qindiv = 2KL

(Qabs +Qphs

)Sign size (bits)

Qsign = 2KL Qsign = 2KL −

The integer and fractional parts, χ�k,� and ψ�

k,�, are thendigitized by the ADC[·]q operator yielding the binaryversion χ�

k,�and ψ�

k,�, i.e., the value of the argument into

a binary vector with q bits. Qint and Qfrac represent theamount of bits required for the input parameters of thealgorithm, and determine the precision and complexity ofthe GAMuChE. A similar procedure is also applied to theimaginary part.

In the GA class of evolutionary algorithms, a set ofattributes (or genes) is termed as a chromosome. Therefore,for the Type-I coding, the individual representation, orchromosome, is the vector constructed from the integer andfractional parts computed as above. Hence, the ith individualis composed of KL chromosomes, resulting in a binarycolumn vector given by

zi = [�

(·)1,1 . . . �

(·)1,L �

(·)k,� . . . �

(·)K,1 . . . �

(·)K,L

]�(32)

where the superscript (·) could expresses Type-I, -II, or-III individuals. Therefore, each individual size will beproportional to the total number of users, paths and bitresolution (including the integer and fractional parts); forinstance, adopting Type-I coding, the individual size resultsQindiv = 2KL(Qint + Qfrac) bits. In addition, the sign of thecomplex number, expressed by just one bit, is considered.Hence, the second characteristic to be optimized by theGAMuChE algorithm is the signal of each individual,wherein the computing of all users and all paths yieldsQsign = 2KL bits.

Type-II binary coding differs in one key respect fromType-I coding: in the former, the fractional and integerparts of the channel response (real and imaginary parts) aredigitized jointly. The objective of this procedure is to assurethat the estimator does not get saturated. In this codingprocedure, the received signal strength, or amplitude, at the

ith bit interval is obtained by defining:

Amax = ‖ri‖

where Amax represents both the largest possible signalstrength value and channel estimates, characterizingindirectly the amount of energy received in the ith bitinterval, and assuring the non-saturation of the obtainedestimate.

Similarly to the Type-I coding, in the Type-II, theamount of bits, Qint, represents an input parameter andcontributes to the levels of precision and complexity of GAimplementation. Hence, the processing chromosome unit isdefined as the vector formed by the real and imaginary partsof the channel response, zk,�; the ith individual is formedfrom KL chromosomes.

In Type-III coding, the channel coefficients to beestimated are codified in polar form, which simplifies theimplementation (e.g., there is no need to consider thecomplex number signs). Furthermore, as shown in Table I,the digitized module and phase are not separated entities.As in the previous case, there is still a need of finding anunsaturated estimator. The phase is found just consideringthe principal value of the angle. Next, both module andphase of zk,� are digitized through the ADC[·]q operator. Asin the previous coding strategies, each individual’s size inthe Type-III coding will be proportional to the total numberof users, processed paths, and number of resolution bits forthe module and phase representation.

4.2. GA parameters

In this paper, in order to estimate population size for theMuD and MuChE cases, the following adapted Equations



[21,22] was used:

p = 10 · ⌊0.3454(√

π(Qindiv − 1) + 2)⌋

(33)

The suitable population sizes for the GAMuD andGAMuChE optimization problem, p1 and p2, respectively,were computed in the initialization stage and maintainedconstant in all subsequent generations.

The estimates for the GA’s first population were obtainedthrough the MFB outputs, for the MuD problem, and by arandomly fashion for the channel estimates, as describedbelow.

4.2.1. First population in GAMuD.

The outputs of the conventional detector were used asinitial estimates:

B1 =[b(0)� , b(1)� , . . . , b(I−1)�

]�(34)

where from Equation (25) results b(i = sign(d(i)

). The

other terms were obtained through the initial individual (B1)with mutation operator, Equation (37).

4.2.2. First population in GAMuChE.

The estimate of the coefficients for the first symbol andfirst population, with dimension Qindiv × P ,

Z = [z1 z2 z3 ... zP

](35)

was obtained randomly.However, for the coefficient estimates of the second

symbol period, the first individual of the Z(m)

populationconsists in the best individual found in the previous symbolperiod estimation:

z(m)1 = z(m−1)

GA (36)

The other p2 − 1 individuals of the first population wererandomly generated.

In the MuChE and MuD problem context, the individual’saptitude is measured through the squared error function (16)and the LLF (29); wherein the life or death decision-makingprocess of individuals takes place. The selection processchooses the best M (i.e., mating pool size) individualsfrom the population pi, i = 1; 2, as the parents for thenext generation. Consequently, thepi − Mi individuals withlower fitness scores are removed from the reproductionstage. The mating pool size is selected in such a way thatensures the convergence of the velocity and quality of thefinal solution [14]. Based on non-exhaustive algorithmsperformance testing and literature reporting, the matingpool sizes for both MuD and MuChE problems, werechosen to be respectively M1 = 0.1p1 and M2 = 0.2Qindiv.Furthermore, for both implementations (MuD and MuChE),mutation based on noise and uniform crossing over [14] was

used in order to generate a second-generation population ofsolutions. The mutation operator is described as

newindiv = sign[indiv + N

(0, σ2

mut

)](37)

where N(

0, σ2mut

)represents a Gaussian distribution with

standard deviation σmut and expectation zero. The standarddeviation is strongly related with the mean mutationrate[13,21].

The replacement strategy used in this work was of aglobal elitism, wherein only the best pi individuals from thejoint population of parents and offsprings are maintained forthe next generation.

Finally, the optimization process is finished after a fixednumber of generations (g), with g1 and g2 representing thegeneration numbers for the MuD and MuChE problem,respectively. Thus, the vector of bits B that maximizesEquation (29) and the vector of coefficients in a binaryform that minimizes Equation (16), z(m)

best, are selected.The coefficient estimates vector is then obtained bydigital to analog conversion z(m)

GA = DAC[z(m)

best

]Qint,Qfrac

. TheAlgorithms 1 and 2 show the pseudo-code for the GAMuDand GAMuChE, respectively.

5. NUMERICAL RESULTS

This section presents numerical results and discusses theapplicability and feasibility of the GA technique in thesearch of more reliable users’ channel impulse responsecoefficients/estimates and the use of these estimates forMuD. It is also included some Monte Carlo simulationresults that show the improvements on performanceobtained by jointly applying the GAMuD and GAMuChEalgorithms.

5.1. Merit figures and parameters setup

In order to implement the Monte Carlo simulations, thefollowing parameters were adopted:

� either pseudo-noise (PN) spreading sequences withprocessing gain N = 16 or Gold sequence with N =31;

� number of active asynchronous users:K = 4 orK = 8,equivalent to a system loading U = K

N;

� adopted transmission rate Rb = 1Tb

= 9.6 kbps;� two or three-paths slow Rayleigh channels with rays

delayed randomly and uniformly distributed in the[0; N − 1]Tc interval;

� uniform power profile, resulting E[β2

k,1

] = . . . =E[β2

k,L

] = 1L, ∀ k;

� received average SNR: γk = Eb

N0

L∑�=1

E[β2

k,�

] = 10 dB,

∀ k, except for Section 5.3, where γk ∈ [0; 20] dB;



Table II. Main system parameters.

System K Seq N U I L Eb/N0

S1 4 PN 16 0.25 10 2 10 dBS2 4 PN 16 0.25 10 3 10 dBS3 8 PN 16 0.50 10 2 10 dBS4 8 Gold 31 ≈ 0.25 10 2 10 dB

Table III. GAMuChE parameters.

System p2 pm pc M g2 Qint Qfrac

S1 80 0.625% 50% 8 200 3 bits 7 bitsS2 80 0.625% 50% 8 200 3 bits 7 bitsS3 110 0.830% 50% 11 200 3 bits 7 bitsS4 110 0.284% 50% 11 200 3 bits 8 bits

� all K users were considered with uniformly distributedvelocity in the interval [1; vmax] (Km/h);

� carrier frequency of fc = 1λc

= 2 GHz;� and as a result, the maximum Doppler frequency

attainable was fD ∈ [1.9; 277.8] Hz, where fD =vmaxλc

.

Four systems, S1–S4, were selected according to theabove parameters for the simulations; and Table IIsynthesizes these parameters corresponding to the fourdifferent systems. Table III and IV show the equivalentparameters for the GAMuChE and GAMuD algorithmsrespectively, and the pseudocodes of these two GAimplementations are described in Algorithms 1 and 2.

Algorithm 1 GAMuD

Input: p1, B1, M, g1 Output: B1

begin1. initialize first population B; g = 0;2. evaluate the fitness(B);3. while g < g1 then;4. Bselected = Selection(B, T );5. Bcross = Crossover(Bselected);6. Bnew = Mutation(Bcross);7. evaluate the fitness(Bnew);8. B = Replacement(B ∪ Bnew);9. end10. B1 = B(:, 1)end

The channel response estimates were evaluated consid-ering two different figures of merit: the nMSE and the meanpercentage error (MPE). For the ith bit interval, the nMSEmeasures the joint effect of module and phase coefficients

Table IV. GAMuD parameters.

System p1 pm pc M g1

S1 40 2.5% 50% 4 20S4 60 1.6% 50% 6 20

Algorithm 2 GAMuChE

Input: p2, z1, M, g2 Output: zbestbegin1. initialize first population Z; g = 0;2. Z = DAC

(Z)

Qint,Qfrac;

3. evaluate the fitness(Z);4. while g < g2 then;5. Zselected = Selection(Z, M);6. Zcross = Crossover(Zselected );7. Znew = Mutation(Zcross);

8. Znew = DAC(Znew

)Qint,Qfrac

;

9. evaluate the fitness(Znew);10. Z = Replacement(Z ∪ Znew);11. end10. zbest = Z(:, 1)end

errors, and is defined as

ζ(m)GA = 1

T

T∑t=1

‖z(m,t)GA − z‖2

‖z‖2(38)

where T is the number of realizations and ‖ · ‖ indicatesvectorial norm operator. Alternatively, the MPE is apercentile measure of the difference between the true andthe estimated values for both phase and module separately,and is defined as

|ξ(m)GA | = 1

T

T∑t=1

|z(m,t)GA | − |z|

|z| × 100 [%] (39a)

�ξ(m)GA = 1

T

T∑t=1

�z(m,t)GA − �z

�z× 100 [%] (39b)

While useful as a first approach, the nMSE, which isextensively used in the specialized literature, contains aserious deficiency. Because it only expresses the joint effectof module and phase coefficient errors, it cannot be usedto model the sub-optimum multi-user and conventionalreceiver in which the performance can be more sensitive tothe phase than to the module errors [23]. Herein, both thenMSE and MPE figures of merit were used as a baselinefor performance evaluation of several channel estimationtechniques.

5.2. Coefficients channel estimation

5.2.1. MSE performance.

The nMSE was used to evaluate the predictive perfor-mance of the following three channel estimate methods:the proposed GaMuChE, the traditional MLMuChE, andthe GrdDsc (Gradient-Descent) [11].

Figure 1 shows these results as a function of theframe length, I, and maximal Doppler frequency, fD. Itwas considered T = 30 realizations, and in the GrdDsc



algorithm case, µ was equal to 0.002 [11]. Table Vhighlights the nMSE achieved by each estimator in the range1.85 ≤ fD ≤ 277.78 Hz. For a specified maximum Dopplerfrequency, fD, the bold face value indicates the minimumnMSE among the three estimators. It can be noted fromthe table that the MLMuChE and the proposed GAMuChEestimators reach approximately the same performance in thesubrange 15 ≤ fD ≤ 95 Hz, resulting in E[ζGA] ≈ E[ζML];however, the frame length of the GAMuChE estimator, IGA,was always smaller. It is also interesting to note that forhigh values of Doppler, fD > 100 Hz, the GAMuChE offersbetter performance with smaller frame sizes.

Only in quasi-static fading channels (fD � 10 Hz), theMLMuChE yields better nMSE regarding to the proposedmethod. On the other hand, the GrdDsc channel estimatorhas approximately twice the minimum GA averagesquared error (E[ζGD] � 2 × E[ζGA]), in the entire Dopplerfrequency range considered, but with larger frame lengths.Only for quasi-static channels, GrdDsc estimator results ina compatible nMSE, but at cost of a significant increase inthe frame processing. In summary, for realistic preamblelengths, which are necessary to obtain a normalized meansquare of at least E[ζ] ≤ 0.1 (10%), all the three estimatemethods performance could be compared as follows:

0100

200300

0

50

100

0

0.05

0.1

0.15

0.2

0.25

0.3

Frame LengthMaximum Doppler Frequency

Nor

mal

ized

MSE

GrdDscMLMuChE

a)

0100

200300 0

1020

30

0

0.05

0.1

0.15

0.2

0.25

0.3

Frame LengthMaximum Doppler Frequency

Nor

mal

ized

MSE

GrdDscMLMuChEGAMuChE

b)

Figure 1. nMSE versus frame length (I) and maximum Dopplerfrequency, for K = 8 users, 2 paths: (a) Gradient Descent andML algorithms, considering frame length in the range 1 ≤ I ≤100; (b) zoom in 1 ≤ I ≤ 30, including GAMuChE algorithm

performance.

Table V. Normalized MSE and associated frame length for theestimators.

Max. Doppler GrdDsc MLMuChE GAMuChE

Freq., fD [Hz] nMSE IGD nMSE IML nMSE IGA

1.85 0.078 100 0.015 40 0.055 2018.51 0.121 100 0.052 30 0.057 1546.29 0.136 80 0.060 30 0.060 1592.59 0.144 50 0.074 20 0.076 12

138.88 0.169 40 0.114 20 0.079 15185.18 0.182 30 0.176 20 0.084 9277.78 0.202 20 0.268 20 0.108 12

� under quasi-static channels (fD < 5 Hz), the threeestimate methods reach the performance requirement;

� under medium Doppler frequency scenarios (10 �fD � 100 Hz), both ML and GA methods could beemployed;

� for high Doppler frequencies (100 � fD � 270 Hz)only GAMuChE achieves the nMSE requirement.

Figure 2 shows a typical nMSE evolution for GAMuChEoptimization along G = 200 generations, considering 16paths, K = 8 users, L = 2 slow paths, and fD = 17 Hz.Because there are no previous estimates for the firstcoefficient estimate, the process demands a larger numberof generations in order to obtain an acceptable ζ

(m)GA. After the

first 10 estimates, the nMSE approaches to E[ζ(10)GA ] = 0.08.

However, slightly smaller values for nMSE can be obtainedby simply increasing m in ζ

(m)GA (e.g., ζ

(197)GA ).

Figure 3 shows the Type-I GAMuChE algorithmaccuracy for two channel response vectors, under Eb/N0 =10 dB and fD = 17 Hz. The coefficients estimation wascarried out during the duration of 670 bits; the framelength was I = 10 bits and the information bits alongthe 670 Tb was assumed to be known. The Type-I codingGAMuChE was able to track the channel variations

0 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Generations

Nor

mal

ized

MSE

ζGA(1)

ζGA(2)

ζGA(5)

ζGA(197)

Figure 2. nMSE evolution for K = 8 users as a function of thenumber of generations G; representatives nMSE are shown: forthe initial coefficients

(z (1), z (2), and z (5)

)and for an intermediate

coefficient(

z (197))

, from a group of 670 estimates, Figure 3.



accurately. Similar performances could be observed forother channel coefficients of the same user and for the otherusers’ channel estimates as well.

5.2.2. MPE performance.

Figure 4 compares the three GAMuChE coding typesdiscussed in Section 4; that figure shows the PDFsof the MPE, measured under a specific condition ofsystem operation (S1, as presented in Table II). In thesesimulated conditions, the figure shows that the Type-Icoding performed better, obtaining smaller values of MPEfor the module and phase coefficients estimates and with asmaller standard deviation for the module when comparedto the Type-II and -III coding. More specifically, the meanand standard deviation values of the MPE (module andphase) can be visualized in the Table VI. Examination ofboth the graph and table shows a clear difference amongthe coding types, wherein the Type-I was consistentlymore reliable than the others. Therefore, Type-I codingwas adopted for the subsequent analysis of some otherperformance figures.

0 100 200 300 400 500 600 7000

0.5

1

1.5

Abs

olut

e V

alue

0 100 200 300 400 500 600 7000

90

180

270

360

Number of bits

Phas

e V

alue

GAMuChEActual

(a)

(b)

0 100 200 300 400 500 600 7000

0.5

1

1.5

Abs

olut

e V

alue

0 100 200 300 400 500 600 7000

90

180

270

360

Number of bits

Phas

e V

alue

GAMuChEActual

Figure 3. Tracking performance (module and phase) over 670 bitsfor two users; system with K = 8 users, L = 2, Doppler spreadfD = 17 Hz, NFR = 0 dB, preamble length I = 10 bits and Type-I

coding GAMuChE. (a) user 3, path 1; (b) user 7, path 2.

−20 0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

MPE (module & phase)

PDF

Module, cod. IPhase, cod. IModule, cod. IIPhase, cod. IIModule, cod. IIIPhase, cod. III

Module

Phase

Figure 4. PDF of the mean percentage error (MPE, module, andphase) for the Type-I, II, and III coding, system S1.

Table VII allows comparison between GAMuChE andMLMuChE estimators’ accuracy through the PDFs ofthe MPE under S2 system operation conditions (referto Table II). The proposed GAMuChE strategy reachessmaller MPE values (mean and standard deviation) for bothmodule and phase coefficient estimates, and yields betterperformance than the ML estimator for a small preamblelengths (I ≤ 15 bits). The results shown in Figure 1also corroborates this finding: the ML estimator reachesacceptable nMSE levels only for large values of preamblelength (e.g., frame length IML ≥ 30 bits). Moreover, theGAMuChE yields estimates with smaller standard deviationwhen compared to the MLMuChE implementation.

In addition, examining more carefully the data at TablesII, VI, and VII, and Figure 4, it is evident an increase in theaccuracy when there was an increment in multipath diversity(e.g., from L = 2 to L = 3). Nevertheless, it should beemphasized that the GAMuChE complexity grows with theincreasing of the multipath diversity.

When there is a possibility of having non-resolvable paths(e.g., S3 operation in Table II), the GA channel estimatesyield insufficient accuracy, as shown in Table VII. The

Table VI. MPE values for each coding type; system S1.

Coding |�GA| ± �|�GA | ��GA ± ��GA

Type I 25.55 ± 5.78 % 29.21 ± 16.78 %Type II 46.26 ± 8.94 % 68.18 ± 21.81 %Type III 44.69 ± 6.96 % 112.80 ± 7.48 %

Table VII. MPE values for four systems with Type-I coding.

System |�GA| ± ��GA ��GA ± ��GA

S1 25.55 ± 5.78 % 29.21 ± 16.78 %S2 23.49 ± 4.24 % 22.71 ± 10.15 %S3∗ 419.49 ± 88.66 % 177.22 ± 43.94 %S4 29.00 ± 2.28 % 28.00 ± 8.88 %ML (S2) 25.49 ± 7.96 % 41.70 ± 18.21 %∗With no resolvable paths.



GAMuChE strategy considers that it is possible to estimateonly one path for each component of the vector zk inEquation (7). Consequently, when two or more overlappingmultipath develops, the estimator performance is degradedconsiderably. That loss is clearly observed through thevalues of the MPE, for the module and phase, when theloading system rises from U = 25%, system S1 (resolvablepaths only) to U = 50%, system S3 (non-resolvable paths).It is worth noting that the Monte Carlo simulations, for theS3 system parameters (Table II), generated non-resolvablepaths among user’s paths as well as temporary pathsoverlaps for the same user’s paths.

Existence of resolvable paths at the receiver and highimmunity to MAI‖ are the necessary conditions to increaseconsiderably the accuracy of the GAMuChE estimator, andto achieve the values presented in Figures 1 and 2, andTable VII. Besides, Table VII portrays the value of the MPEfor the S4 system, using deterministic spreading sequencesand all other parameters identical to S1 system, as shown inTable II, which assures a larger immunity to MAI and to SI,i.e., they provide low cross-correlation and low out-of-phaseauto-correlation characteristics. Comparison of values inTable VII, reveals smaller phase MPE and similar moduleMPE values for the S4 system in relation to those obtainedwith the use of PN sequences (S1 system).

Acquisition techniques capable of estimating path delayswith errors of less than one chip period, Tc, are notuncommon [11,24]. The several simulations, that have beencarried out in this work, have shown that the GAMuChE wascapable of accomplishing the tracking within similar marginof error during the acquisition stage, and consequently,able to find the correct delay values for all multipaths.For the sake of simplicity, only integer multiples of Tc

were considered in modeling the users and multipathdelays. Therefore, in order to increase the performanceand accuracy, the tracking stages of GAMuChE must beperformed within a fraction of Tc (i.e., τk,� < Tc), at theexpense of increasing the vector size¶ zk given by Equation(7). In addition, considering that the users’ received signalstrength were jointly estimated with the correspondingcomplex channel coefficients, the estimates were alsoaffected by the resolution of the tracking stage.

5.3. Jointly Multi-user detection andchannel coefficients estimates

This section presents the bit error rate (BER) performanceresults for the joint operation of the proposed GAMuChEand GAMuD algorithms. S1 and S4 systems (Table II)outline the base set for the Monte Carlo simulationsthat were conducted. The MPE values obtained from the

‖ Through the use of the deterministic spreading sequences, or theprocessing gain increase, or both.¶ By consequence, computational complexity and memory require-ments are also increased.

simulations previously described (Table VII) were usedfor the performance study of the GA-based multi-userdetector. In order to characterize the robustness of theGAMuD, errors in the module and phase of the channelcoefficients estimates were modeled according to twoGaussian distributions N(m, σ2), using the MPE values forS1 and S4 systems (Table VII), i.e., errors with a distributionN(|ξGA|, σ2

ξGA) for the module, and N(�ξGA, σ2

�ξGA) for

the phase of channel coefficients were introduced, and theperformance’s degradation was evaluated; the results areshown in Figure 5.

The figure illustrates the relationship between receivedaverage SNR, γ , and BER for several MuDs: Rake, ananalytical analysis for a system with only one active user,or SuB (i.e., single user bound), the MLMuChE plusDecorrelator Detector [18], the GAMuChE in the absenceof channel errors estimates, and GAMuChE plus GAMuDalgorithms. Perfect or ideal channel estimates, i.e., |ξ(m)

GA| =�ξ

(m)GA = 0 in Equation (39a), were adopted for the ‘Rake

without Errors’ and ‘GAMuChE without Errors’ detectors.The results indicate a clear difference, in terms of

BER performance, among the four different methods. Withno exceptions, the GAMuD with the channel predictionsprovided by GAMuChE (i.e., GAMuChE + GAMuD),consistently prevailed over both, the Rake with perfectchannel estimates and the ML + Decorrelator detector,for the S1 and S4 cases. It is also interesting to notethat intensive computation is required in order to obtainthe inverse of the correlation matrix for the joint ML +Decorrelator method.

Furthermore, it may be stated that if more reliableestimates for the channel coefficients could be obtained(e.g., by the increasing of the number of bits for the codingprocess, or by improving the immunity to MAI and SI), thenthe GAMuChE plus GAMuD would also yield better BERperformance.

Hence, GAMuChE plus GAMuD can be an attractiveoption, capable of achieving a near-optimum performance,particularly when more reliable channel estimates can beobtained.

6. COMPLEXITY ANALYSIS OF THEPROPOSED ALGORITHMS

The computational complexity of the proposed heuristicstructure was determined in terms of the average (e.g.,necessary) number of operations, or flops (i.e., float pointoperations) [25], considering the joint detection and channelestimation algorithms. The analysis consisted in evaluatingthe complexity of the channel estimate stage accomplishedby GAMuChE, with Type-I coding associated with thedetection stage, performed by the GAMuD, in order tocompare with other solutions.

The comparative computational costs of the proposedType-I GAMuChE plus GAMuD, the GrdDsc [11] andthe maximum likelihood (MLMuChE plus OMuD) [18]algorithms are shown in detail on the right side in



0 5 10 15 2010

−4

10−3

10−2

10

(a) (b)−1

γ [dB]

BE

R

Rake without Errors

MLMuChE + Decorrelator

GAMuChE + GAMuD

GAMuD without Errors

SuB (BPSK)

0 5 10 15 2010

−4

10−3

10−2

10−1

γ [dB]

Rake without Errors

MLMuChE + Decorrelator

GAMuChE + GAMuD

GAMuD without Errors

SuB (BPSK)

S2 systemS1 system

Figure 5. GAMuD BER performance with channel predictions obtained by GAMuChE. (a) S1, (b) S4 system.

Table VIII. The parameters pi and gi (with i = 1, 2) referto the population size and number of generations of thedetection stage (GAMuD, i = 1) and channel estimationstage (GAMuChE, i = 2).

Initially, considering the parameters given in Tables IIand III, the computational complexity of the GAMuChEand GrdDsc algorithms were examined. Figure 6 showsthat, for a system load equal to 50% approximately, bothalgorithms exhibit similar degree of complexity, for theprocessing gain N ≤ 170. For higher values of processinggain (i.e., N > 170), the GAMuChE yields a progressivelylower computational complexity.

The number of operations required for the jointdetection and channel estimation in the GA, GrdDscand ML strategies respectively was determined using theexpressions given in Table VIII, and the numerical resultsare shown in Table IX. In order to gain further insight intothe computational complexity, an efficient implementationfor the decorrelator detector, based on preconditioned

Table VIII. Computational complexity for the heuristic jointlydetection and channel estimates.

Jointly detector Number of operations

MLMuChE + OMuD 2Qindiv (N2K + 2N +Qindiv)++2KI

((KID)2 + 3KID

)GrdDsc + OMuD 2KI (KID)2

(1 + 3

KID

)+

+K 2N2(

5N + K+1K

)GrdDsc + itDec [26] ≈ 1.5I(KD)3+

+K 2N2(

5N + K+1K

)GAMuChE + GAMuD p1g1(KID)(KID + 11)+

+p2g2K(N2 + 2D(Qint +Qfrac + 6)

)

conjugate gradient algorithm [26], labeled hereafter ‘itDec’,was also analyzed and the expression and resulting numberof operations can be seen in Tables VIII and IX, respectively.The itDec was combined to the GrdDsc channel estimatorto produce the GrdDsc plus itDec structure. As shownin the tables, the proposed structure yields a significantreduction in computational complexity when comparedto the MLMuChE plus OMuD and GrdDsc plus OMuDstructures. On the other hand, the number of operationsfor the proposed heuristic structure is approximately oneto two orders of magnitude larger than the obtained for theGrdDsc plus itDec structure. Furthermore, the complexitiesof the proposed and GrdDsc plus itDec structures becomesimilar when the processing gain N increases (e.g., N2

compares N3, as shown in Table VIII). Nevertheless, ascan be seen in Figure 5, the BER performance advantage ofthe proposed GAMuChE plus GAMuD structure over the

0 50 100 150 200 2500

1

2

3

4

5x 1011

Processing Gain (N)

Num

ber o

f Ope

ratio

ns

Transition point (N ≈ 170)

GAMuChEGrdDsc

Figure 6. The computational complexity increases as a functionof the processing gain (≈ 50% system load).



Table IX. Number of operations for the jointly detection and channel estimation strategies.

System S1 S2 S3 S4

ML + OMuD 15.19 × 1080 2.28 × 1075 5.12 × 1099 1.72 × 10100

GrdDsc + OMuD 3.15 × 1028 1.62 × 1016 3.15 × 1028 3.15 × 1028

GrdDsc + itDec 3.48 × 105 3.84 × 105 1.45 × 106 9.72 × 106

GAMuChE + GAMuD 3.6 × 107 4.76 × 107 1.20 × 108 2.46 × 108

alternative GrdDsc plus itDec, whose BER performance isbounded by the MLMuChE plus Decorrelator, is evident,and in addition, the former strategy presents a slightlygreater complexity than the latter, for small to mediumprocessing gain scenarios.

Finally, in order to demonstrate the feasibility of theGAMuChE plus GAMuD, an illustrative computationalcomplexity analysis, of the joint detection and channelestimation strategies, was carried out taking into accountthe TIA/EIA/IS-2000-2 wireless communication systemstandard [27], considering as an example the CDMA2000radio Configuration 3 under basic transmission rate (i.e.,9.6 kbps, which results in a processing gain, N, equal to128). According to the TIA/EIA/IS-2000-2 specification,in the Configuration 3 mode each data frame consists of192 bits, with the first 20 bits indicating the quality andset and unset operations of the registers. Therefore, for theheuristic channel estimation procedure, up to a maximumof 20 bits could be used as preamble for the GAMuChE toperform the channel estimates at each frame duration.

In order to compare the four algorithms fairly (i.e., theproposed GAMuChE plus GAMuD, the MLMuChE plusOMuD, GrdDsc plus OMuD, and GrdDsc plus itDec allwith overall analytical complexities given in Table VIII), thefollowing specified CDMA2000 parameters were adopted:basic data rate of 9.6 kbps, processing gain equal to 128,a processing preamble size for the GA channel estimatorequal to 10 bits (while holding the preamble size equal to 20

0 10 20 30 40 50 60100

10100

10200

Num

ber o

f ope

ratio

ns

0 10 20 30 40 50 60108

1010

1012

Number of users (K)

Num

ber o

f ope

ratio

ns

MLMuChE + OMuDGrdDsc + OMuDGrdDsc + itDecGAMuChE + GAMuD

GrdDsc + OMuDGrdDsc + itDecGAMuChE + GAMuD

b)

a)

Figure 7. Computational complexity represented by the numberof operations as function of number of users; (a) Number ofoperations given in logarithmic scale; (b) Zoon in the proposed

GAMuChE plus GAMuD versus GrdDsc plus itDec.

bits for the both GrdDsc plus OMuD and MLMuChE plusOMuD algorithms, c.f., Figure 1), and the number of codingbits equal to 16 (i.e., Qint plus Qfrac from Table I). As canbe seen in Figure 7, the computational complexity variedmarkedly among the three first structures, but, only slightlyamong the proposed and GrdDsc plus itDec structures (c.f.Table VIII).

Although the number of operations usually increasedwith the number of users, Figure 7a also shows thatthere was a significant and progressive increase in theseparation between the line of the graph representing thejoint structure GAMuChE plus GAMuD and each one ofthe two other lines representing the joint ML detection andML or GrDsc channel estimation strategies, respectively.From the results obtained in the simulations and the analysisdescribed above, it is clear that, for a vast number of activeusers, the proposed joint GAMuChE plus GAMuD schemewas consistently superior to any other scheme. Thus, itmay be concluded that the proposed detection strategyyields an excellent performance versus complexity trade-off when appropriately used, being actually possible forimplementation through high performance computers, andrepresents an attractive solution for the 3G and 4G wirelesscommunication systems.

7. CONCLUSIONS

A heuristic approach, based on the genetic evolution theory,applied to multipath MuChE and MuD, both related tothe DS/CDMA systems, was proposed in this paper. Theeffectiveness of the evolutionary computation methodologyin solving the jointly MuD and channel estimation problemswas demonstrated by the simulation results that wereobtained, such as BER, nMSE, MPE, and number ofoperations. These results are very promising in terms ofperformance versus complexity trade-off in all analyzedscenarios.

The proposed heuristic channel estimation procedurealso revealed remarkable low computational complexityand exceptional detecting capability with high resolution.Moreover, the GAMuChE estimates accuracy versuscomplexity can be improved even further with the evolutionof digital signal processing technology.

The three channel estimate methods compared in thispaper (i.e., the proposed GAMuChE, the traditionalMaximum Likelihood, and the GrdDsc) yielded very similarmean-squared error of channel estimation; however, both



the computational cost and latency (e.g., smaller framelength) were significantly smaller for the GAMuChE.In particular, when compared with the GrdDsc channelestimator algorithm, the Type-I GAMuChE exhibitedsmaller normalized mean squared error and similarcomplexity for processing gain N ≤ 170, and expressivelysmaller complexity results for N >> 170.

This study has also unveiled that the MPE figure of meritcriterion is more effective in quantifying separately theerrors contained in both the module and phase estimateswhen compared to the more extensively used nMSE.Simulation results indicated that for practical preamble size(10 bits), the proposed Type-I GAMuChE estimator reachedsmaller MPE than that obtained by the MLMuChE andGrdDsc multi-user channel estimators.

The joint characteristics of the GAMuD plus GAMuChEalgorithm yielded an excellent performance versuscomplexity trade-off when adequately used, indicatingthe feasibility of this joint structure for the 3G and4G wireless communications systems. The relationshipbetween received average SNR and BER was alsoinvestigated for a number of different MuDs. The proposedType-I GAMuChE plus GAMuD algorithm consistentlyprevailed over the Rake with perfect channel estimates.Additionally, if more reliable estimates for the channelcoefficients could be obtained, then the GAMuChE plusGAMuD BER performance would considerably improve.

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AUTHORS’ BIOGRAPHIES

Fernando Ciriaco received the B.S.and M.Sc. degrees in Electrical Engi-neering from UEL - Londrina StateUniversity, Brazil in 2004 and 2006,respectively. He is currently a Ph.D.student at EPUSP - Escola Politécnicaof University of São Paulo, Brazil,Coordinator and Professor at theElectrical Engineering Department of

Faculdade Pitágoras, Brazil and also acts as a coordinatorof the Wireless Biosignal Development Systems Group inGELT Tecnologia, Brazil sponsored by FINEP. His researchinterests are multi-user detection, heuristic algorithms,MIMO, MC-CDMA systems, wireless communications,wireless biosignals and signal processing.

Taufik Abrão received the B.S., M.Sc.,and Ph.D., all in Electrical Engineeringfrom EPUSP - Escola Politécnica ofUniversity São Paulo, Brazil, in 1992,1996, and 2001, respectively. Dr Abrãois currently an Associate Professor atthe Electrical Engineering Departmentof UEL - State University of Londrina,Brazil. From 2007 to 2008, he was

a Postdoctoral Research Fellow for 16-month periodat the TSC/UPC - Department of Signal Theory andCommunications, Universitat Politécnica de Catalunya,Barcelona, Spain, sponsored by a fellowship from theCAPES-Brazil. His research interests include multi-userdetection, MC-CDMA and MIMO systems, heuristicand optimization aspects of DS-CDMA systems and 4G

systems. He is author or co-author of more than 60 refereedresearch papers published in specialized periodicals andkey conferences in the area of wireless communication andnetworking.

Antonio Fischer de Toledo, aftera period in industry, joined theDepartment of Telecommunicationsand Control of the Polytechnic Schoolof the University of São Paulo (i.e.,EPUSP) as an assistant professor,in 2003. He obtained his B.Sc.degree in electrical engineering fromEscola de Engenharia Mauá. After his

graduation in 1973, he was employed by TELESP (i.e.,Telecommunications of São Paulo), where he was, initially,asked to elaborate the proceedings, i.e., the specification,planning and implementation of tests on cables that werenewly installed by contractors. During the 1970s and1980s, he became involved in several other aspects ofthe outside plant construction, including research effortsdealing with more advanced materials. Between 1987and 1992, he was with the Department of ElectricalEngineering and Electronics at the University of Liverpool,United Kingdom, where he received his M.Sc.(Eng.) andPh.D. degrees. During this time, he was involved withsoftware investigation of time diversity techniques, andradio propagation research for cellular communicationsystems. In August 1992, he returned to TELESP, and joinedthe cellular department that was created at that time. He isthe author of several technical papers in the area of indoorand outdoor radio propagation and diversity techniques.

Paul Jean Etienne Jeszensky receivedthe B.S., M.S., and Ph.D., all inElectrical Engineering from EPUSP -Escola Politécnica of University of SãoPaulo, Brazil, in 1972, 1981, and 1989,respectively. Since 1990 he has beenwith EPUSP where he is Full Professorand Researcher in CommunicationSystems. He was visiting professor at

UPC - Universitat Politécnica de Catalunya, Barcelona,Spain in 1995 and at TUB - Technical University ofBudapest, Hungary in 2001. He is author of the bookSistemas Telefônicos (in Portuguese), Editora Manole,2003 and his current research interests include CDMAsystems, multi-user detection, code sequences analysis andrelated topics.


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