Digital Filter Structures - The Catholic University of...

1Copyright © 2001, S. K. Mitra

Digital Filter StructuresDigital Filter Structures

• The convolution sum description of an LTIdiscrete-time system can, in principle, beused to implement the system

• For an IIR finite-dimensional system thisapproach is not practical as here the impulseresponse is of infinite length

• However, a direct implementation of the IIRfinite-dimensional system is practical



• Here the input-output relation involves afinite sum of products:

• On the other hand, an FIR system can beimplemented using the convolution sumwhich is a finite sum of products:

∑∑ == −+−−= Mk k

Nk k knxpknydny 01 ][][][

∑ = −= Nk knxkhny 0 ][][][



• The actual implementation of an LTI digitalfilter can be either in software or hardwareform, depending on applications

• In either case, the signal variables and thefilter coefficients cannot be representedwith finite precision


Digital Filter StructuresDigital Filter Structures• However, a direct implementation of a digital

filter based on either the difference equationor the finite convolution sum may notprovide satisfactory performance due to thefinite precision arithmetic

• It is thus of practical interest to developalternate realizations and choose the structurethat provides satisfactory performance underfinite precision arithmetic


Digital Filter StructuresDigital Filter Structures• A structural representation using

interconnected basic building blocks is thefirst step in the hardware or softwareimplementation of an LTI digital filter

• The structural representation provides thekey relations between some pertinentinternal variables with the input and outputthat in turn provides the key to theimplementation


Block Diagram RepresentationBlock Diagram Representation

• In the time domain, the input-outputrelations of an LTI digital filter is given bythe convolution sum

or, by the linear constant coefficientdifference equation

∑∑ == −+−−= Mk k

Nk k knxpknydny 01 ][][][

∑∞−∞= −= k knxkhny ][][][


Block Diagram RepresentationBlock Diagram Representation• For the implementation of an LTI digital

filter, the input-output relationship must bedescribed by a valid computational algorithm

• To illustrate what we mean by acomputational algorithm, consider the causalfirst-order LTI digital filter shown below



• The filter is described by the differenceequation

• Using the above equation we can computey[n] for knowing the initial condition

and the input x[n] for :

]1[][]1[][ 101 −++−−= nxpnxpnydny

0≥n1−≥n]1[−y



• We can continue this calculation for anyvalue of the time index n we desire

]1[]0[]1[]0[ 101 −++−−= xpxpydy]0[]1[]0[]1[ 101 xpxpydy ++−=]1[]2[]1[]2[ 101 xpxpydy ++−= ..

.

..

.


Block Diagram RepresentationBlock Diagram Representation• Each step of the calculation requires a

knowledge of the previously calculatedvalue of the output sample (delayed value ofthe output), the present value of the inputsample, and the previous value of the inputsample (delayed value of the input)

• As a result, the first-order differenceequation can be interpreted as a validcomputational algorithm


Basic Building BlocksBasic Building Blocks• The computational algorithm of an LTI

digital filter can be convenientlyrepresented in block diagram form using thebasic building blocks shown below

x[n] y[n]

w[n]

+ Ax[n] y[n]

y[n]1−zx[n]x[n] x[n]

x[n]

Adder

Unit delay

Multiplier

Pick-off node


Basic Building BlocksBasic Building Blocks

Advantages of block diagram representation• (1) Easy to write down the computational

algorithm by inspection• (2) Easy to analyze the block diagram to

determine the explicit relation between theoutput and input


Basic Building BlocksBasic Building Blocks

• (3) Easy to manipulate a block diagram toderive other “equivalent” block diagramsyielding different computational algorithms

• (4) Easy to determine the hardwarerequirements

• (5) Easier to develop block diagramrepresentations from the transfer functiondirectly


Analysis of Block DiagramsAnalysis of Block Diagrams• Carried out by writing down the expressions

for the output signals of each adder as a sumof its input signals, and developing a set ofequations relating the filter input and outputsignals in terms of all internal signals

• Eliminating the unwanted internal variablesthen results in the expression for the outputsignal as a function of the input signal andthe filter parameters that are the multipliercoefficients


Analysis of Block DiagramsAnalysis of Block Diagrams• Example - Consider the single-loop feedback

structure shown below

• The output E(z) of the adder is

• But from the figure)()()()( 2 zYzGzXzE +=

)()()( 1 zEzGzY =


Analysis of Block DiagramsAnalysis of Block Diagrams

• Eliminating E(z) from the previous twoequations we arrive at

which leads to)()()()]()(1[ 121 zXzGzYzGzG =−

)()(1)(

)()()(

21

1zGzG

zGzXzYzH

−==



• Example - Analyze the cascaded latticestructure shown below where the z-dependence of signal variables are notshown for brevity



• The output signals of the four adders aregiven by

• From the figure we observe

21 SXW α−=112 SWW δ−=

213 WSW ε+=

21 SWY γ+β=

31

2 WzS −=2

11 WzS −=


Analysis of Block DiagramsAnalysis of Block Diagrams• Substituting the last two relations in the first

four equations we get

• From the second equation we get and from the third

equation we get

31

1 WzXW −α−=2

112 WzWW −δ−=

221

3 WWzW ε+= −

31

1 WzWY −γ+β=

)1/( 112

−δ+= zWW2

13 )( WzW −+ε=



• Combining the last two equations we get

• Substituting the above equation in

we finally arrive at

113 11WW

zz−

−

δ++ε=

,31

1 WzXW −α−= 31

1 WzWY −γ+β=

21

21

)(1)()( −−

−−

α+αε+δ+γ+γε+βδ+β==zzzz

XYzH


The Delay-Free Loop ProblemThe Delay-Free Loop Problem• For physical realizability of the digital filter

structure, it is necessary that the blockdiagram representation contains no delay-free loops

• To illustrate the delay-free loop problemconsider the structure below


The Delay-Free Loop ProblemThe Delay-Free Loop Problem• Analysis of this structure yields

which when combined results in

• The determination of the currentvalue of y[n] requires the knowledge of thesame value

][][][ nynwnu +=])[][(][ nAunvBny +=

( )])[][(][][ nynwAnvBny ++=


The Delay-Free Loop ProblemThe Delay-Free Loop Problem• However, this is physically impossible to

achieve due to the finite time required tocarry out all arithmetic operations on adigital machine

• Method exists to detect the presence ofdelay-free loops in an arbitrary structure,along with methods to locate and removethese loops without the overall input-outputrelation


The Delay-Free Loop ProblemThe Delay-Free Loop Problem• Removal achieved by replacing the portion

of the overall structure containing the delay-free loops by an equivalent realization withno delay-free loops

• Figure below shows such a realization ofthe example structure described earlier


Canonic and Canonic and NoncanonicNoncanonicStructuresStructures

• A digital filter structure is said to becanonic if the number of delays in the blockdiagram representation is equal to the orderof the transfer function

• Otherwise, it is a noncanonic structure


Canonic andCanonic and Noncanonic NoncanonicStructuresStructures

• The structure shown below is noncanonic asit employs two delays to realize a first-orderdifference equation

]1[][]1[][ 101 −++−−= nxpnxpnydny


Equivalent StructuresEquivalent Structures

• Two digital filter structures are defined tobe equivalent if they have the same transferfunction

• We describe next a number of methods forthe generation of equivalent structures

• However, a fairly simple way to generate anequivalent structure from a given realizationis via the transpose operation



Transpose Operation• (1) Reverse all paths• (2) Replace pick-off nodes by adders, and

vice versa• (3) Interchange the input and output nodes• All other methods for developing equivalent

structures are based on a specific algorithmfor each structure



• There are literally an infinite number ofequivalent structures realizing the sametransfer function

• It is thus impossible to develop allequivalent realizations

• In this course we restrict our attention to adiscussion of some commonly usedstructures



• Under infinite precision arithmetic anygiven realization of a digital filter behavesidentically to any other equivalent structure

• However, in practice, due to the finitewordlength limitations, a specificrealization behaves totally differently fromits other equivalent realizations


Equivalent StructuresEquivalent Structures• Hence, it is important to choose a structure

that has the least quantization effects whenimplemented using finite precisionarithmetic

• One way to arrive at such a structure is todetermine a large number of equivalentstructures, analyze the finite wordlengtheffects in each case, and select the oneshowing the least effects



• In certain cases, it is possible to develop astructure that by construction has the leastquantization effects

• We defer the review of these structures aftera discussion of the analysis of quantizationeffects

• Here, we review some simple realizationsthat in many applications are quite adequate


Basic FIR Digital FilterBasic FIR Digital FilterStructuresStructures

• A causal FIR filter of order N is characterizedby a transfer function H(z) given by

which is a polynomial in• In the time-domain the input-output relation

of the above FIR filter is given by

∑ =−= N

nnznhzH 0 ][)(

1−z

∑ = −= Nk knxkhny 0 ][][][


Direct Form FIR Digital FilterDirect Form FIR Digital FilterStructuresStructures

• An FIR filter of order N is characterized byN+1 coefficients and, in general, requireN+1 multipliers and N two-input adders

• Structures in which the multipliercoefficients are precisely the coefficients ofthe transfer function are called direct formstructures



• A direct form realization of an FIR filter canbe readily developed from the convolutionsum description as indicated below for N =4



• An analysis of this structure yields

which is precisely of the form of theconvolution sum description

• The direct form structure shown on theprevious slide is also known as a tappeddelay line or a transversal filter

][][][][ 4433 −+−+ nxhnxh][][][][][][][ 22110 −+−+= nxhnxhnxhny

][][][][ 4433 −+−+ nxhnxh



• The transpose of the direct form structureshown earlier is indicated below

• Both direct form structures are canonic withrespect to delays


Cascade Form FIR DigitalCascade Form FIR DigitalFilter StructuresFilter Structures

• A higher-order FIR transfer function canalso be realized as a cascade of second-order FIR sections and possibly a first-ordersection

• To this end we express H(z) as

where if N is even, and if Nis odd, with

∏ =−− ++= K

k kk zzhzH 12

21

110 )(][)( ββ

2NK =

21+= NK

02 =Kβ


Cascade Form FIR DigitalCascade Form FIR DigitalFilter StructuresFilter Structures

• A cascade realization for N = 6 is shownbelow

• Each second-order section in the abovestructure can also be realized in thetransposed direct form


PolyphasePolyphase FIR Structures FIR Structures

• The polyphase decomposition of H(z) leadsto a parallel form structure

• To illustrate this approach, consider a causalFIR transfer function H(z) with N = 8:

4321 43210 −−−− ++++= zhzhzhzhhzH ][][][][][)(8765 8765 −−−− ++++ zhzhzhzh ][][][][


PolyphasePolyphase FIR Structures FIR Structures• H(z) can be expressed as a sum of two

terms, with one term containing the even-indexed coefficients and the othercontaining the odd-indexed coefficients:

)][][][][][()( 8642 86420 −−−− ++++= zhzhzhzhhzH)][][][][( 7531 7531 −−−− ++++ zhzhzhzh

)][][][][][( 8642 86420 −−−− ++++= zhzhzhzhh)][][][][( 6421 7531 −−−− ++++ zhzhzhhz



• By using the notation

we can express H(z) as

3211 7531 −−− +++= zhzhzhhzE ][][][][)(

43210 86420 −−−− ++++= zhzhzhzhhzE ][][][][][)(

)()()( 21

120 zEzzEzH −+=



• In a similar manner, by grouping the termsin the original expression for H(z), we canreexpress it in the form

where now)()()()( 3

223

113

0 zEzzEzzEzH −− ++=

210 630 −− ++= zhzhhzE ][][][)(

211 741 −− ++= zhzhhzE ][][][)(

212 852 −− ++= zhzhhzE ][][][)(



• The decomposition of H(z) in the form

or

is more commonly known as the polyphasedecomposition

)()()()( 32

231

130 zEzzEzzEzH −− ++=

)()()( 21

120 zEzzEzH −+=



• In the general case, an L-branch polyphasedecomposition of an FIR transfer functionof order N is of the form

where

with h[n]=0 for n > N

)()( Lm

Lm

m zEzzH ∑ −=

−= 10

∑+

=

−+=LN

n

mm zmLnhzE

/)(

][)(1

0


PolyphasePolyphase FIR Structures FIR Structures• Figures below show the 4-branch, 3-branch,

and 2-branch polyphase realization of atransfer function H(z)

• Note: The expression for the polyphasecomponents are different in each case)(zEm



• The subfilters in the polyphaserealization of an FIR transfer function arealso FIR filters and can be realized usingany methods described so far

• However, to obtain a canonic realization ofthe overall structure, the delays in allsubfilters must be shared

)( Lm zE


PolyphasePolyphase FIR Structures FIR Structures• Figure below shows a canonic realization of

a length-9 FIR transfer function obtainedusing delay sharing


Linear-Phase FIR StructuresLinear-Phase FIR Structures• The symmetry (or antisymmetry) property of a

linear-phase FIR filter can be exploited toreduce the number of multipliers into almosthalf of that in the direct form implementations

• Consider a length-7 Type 1 FIR transferfunction with a symmetric impulse response:

321 3210 −−− +++= zhzhzhhzH ][][][][)(654 012 −−− +++ zhzhzh ][][][


Linear-Phase FIR StructuresLinear-Phase FIR Structures• Rewriting H(z) in the form

we obtain the realization shown below

)]([)]([)( 516 110 −−− +++= zzhzhzH342 32 −−− +++ zhzzh ][)]([


Linear-Phase FIR StructuresLinear-Phase FIR Structures

• A similar decomposition can be applied to aType 2 FIR transfer function

• For example, a length-8 Type 2 FIR transferfunction can be expressed as

• The corresponding realization is shown onthe next slide

)]([)]([)( 617 110 −−− +++= zzhzhzH

)]([)]([ 4352 32 −−−− ++++ zzhzzh



• Note: The Type 1 linear-phase structure fora length-7 FIR filter requires 4 multipliers,whereas a direct form realization requires 7multipliers



• Note: The Type 2 linear-phase structure fora length-8 FIR filter requires 4 multipliers,whereas a direct form realization requires 8multipliers

• Similar savings occurs in the realization ofType 3 and Type 4 linear-phase FIR filterswith antisymmetric impulse responses

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