Filter Banks (Mertins - Signal Analysis (Revised Edition))

Signal Analysis: Wavelets,Filter Banks, Time-Frequency Transforms andApplications. Alfred MertinsCopyright 0 1999 John Wiley& Sons Ltdprint ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4Chapter 6Filter BanksFilter banks are arrangementsof low pass, bandpass, and highpass filters usedfor the spectral decomposition and composition of signals. They play an important role in many modern signal processing applications such as audio andimage coding. The reason for their popularity is the fact that they easily allowthe extractionof spectral components of a signal while providing very efficientimplementations. Since most filter banks involve various sampling rates, theyare also referred to as multirate systems. T o give an example,Figure6.1shows an M-channel filter bank.

The input signal is decomposed into M socalled subb and signalsby applying M analysis filters with different passbands.Thus, each of the subband signals carries information on the input signal ina particular frequency band. The blocks with arrows pointing downwards inFigure 6.1 indicate downsampling (subsampling) by factor N, and the blockswith arrows pointing upwards indicate upsampling by N. Subsampling by Nmeans that only every N t h sample is taken. This operation serves t o reduceor eliminate redundancies in the M subband signals. Upsampling by N meansthe insertion of N - 1 consecutive zeros between the samples. This allows usto recover the original sampling rate. The upsamplers are followed by filterswhich replace the inserted zeros with meaningful values.

In the case M = Nwe speak of critical subsampling, because this is the maximum downsamplingfactor for which perfect reconstruction can be achieved. Perfect reconstructionmeans that the output signal is a copy of the input signal with no furtherdistortion than a time shift and amplitude scaling.143144BanksAnalysis filter bankISubbandsignalsChapter 6. FilterSynthesis filter bankiFigure 6.1. M-channel filter bank.From the mathematical point of view, a filter bank carries out a seriesexpansion, wherethe subbandsignals are thecoefficients, and thetime-shiftedvariants gk:(n- i N ) , i E Z,of the synthesis filter impulse responsesgk (n)formthe basis.

The maindifference to theblock transforms is that thelengths of thefilter impulse responses are usually larger than N so that the basis sequencesoverlap.6.16.1.1Basic Multirate OperationsDecimation andInterpolationInthis section, we derive spectralinterpretations for the decimationandinterpolation operations that occur in every multirate system. For this, weconsider the configuration in Figure 6.2. The sequence ).(Wresultsfrominserting zeros into ~ ( r n )Because.of the different sampling rates we obtainthe following relationship between Y ( z )and V ( z ) :Y ( P )= V ( z ) .(6.1)After downsampling and upsampling by N the values w(nN) and u ( n N )are still equal, while all other samples of ).(Ware zero.

Using the correspondencee j 2 m h / N = 1 for n / N E Z,0 otherwise,N 2=0.{the relationship between).(WandU(.)can be written as. N-l1456.1. Basic MultirateOperationsFigure 6.2. Typical components of a filter bank.The z-transform is given bycccV(z) =w(n)zPn=-ccN-l~=cc.l N--1-NCU(W&z).i=OThe relationship between Y ( z )and V ( z )is concluded from (6.1) and (6.5):With (6.6) and V ( z ) = H ( z ) X ( z ) we have the following relationshipbetween Y ( 2 ) and X ( z ) :N-lFrom (6.1) and (6.7) we finally concludeX(z)= G ( z )Y ( z N ). N-l=l-XG(z)H(W&z)X(W&z).N a=O.Chapter 6. Filter Banks146-2.G... -2nn-R2.GRIhi(ej0)I-RnI2nR...*WFigure 6.3.

Signal spectra for decimation and interpolationaccordingstructure in Figure 6.2 (non-aliased case).... ...-2.Rtothe' u(ej0)1I-RR2.G* WFigure 6.4. Signal spectra in the aliased case.The spectraof the signals occurring in Figure 6.2 are illustrated in Figure 6.3for the case of a narrowband lowpass input signal z(n), which does not leadto aliasing effects. This means that the products G(z)(H(W&z)X(W&z))in(6.8) are zero for i # 0.

The general case with aliasing occurs when thespectra become overlapping. This is shown in Figure 6.4, where the shadedareas indicate the aliasing components that occur due to subsampling. It isclear that z(n) can only be recovered fromy(m) if no aliasing occurs. However,the aliased case is the normal operation mode in multirate filter banks. Thereason why such filter banks allow perfect reconstruction lies in the fact thatthey can be designed in such a way that the aliasing components from allparallel branches compensate at the output.1476.1. Basic MultirateOperationsFigure 6.5.

Type-l polyphase decomposition for M = 3.6.1.2Polyphase DecompositionConsider the decomposition of a sequence ).(Xinto sub-sequences xi(rn), asshownin Figure 6.5. Interleaving all xi(rn) again yields the original X(.)This decomposition iscalled a polyphase decomposition, and the xi(rn) arethe polyphase components of X(.).Several types of polyphase decompositionsare known, which are briefly discussed below.Type-l. A type-l polyphase decomposition of a sequence ).(Xcomponents is given bycinto it4M-lX(2)=2-eX&M),e=owhere&(z)ze(n) = z(nM + l ) .t)(6.10)Figure 6.5 shows an example of a type-l decomposition.Type-2.

The decomposition into type-2 polyphase components is given bycM-lX ( 2 )=z-(M-l-l)X;(.M)7(6.11)e=owherex;(2)t)X;(.)= z(nit4+ it4 - 1- l ) .(6.12)148BanksChapter 6. FilterThus, the only difference between a type-l and a type-2 decomposition lies inthe indexing:X&) = XL-,-&).(6.13)Type-3. A type-3 decomposition readscM-lX(z) =ze X&"),(6.14)l=OwhereX&)z:e(n)= z(nM -e).t)(6.15)The relation to the type-lpolyphase components isPolyphase decompositions are frequently used for both signals and filters.In the latter case we use the notation H i k ( z ) for the lcth type-l polyphasecomponent of filter H i ( z ) .

The definitions for type-2 and type-3 componentsare analogous.6.26.2.1Two-Channel Filter BanksPR ConditionLet us consider the two-channel filter bank in Figure 6.6. The signals arerelated asY0(Z2) =[ H o b ) X(z) + Ho(-z) X(-z)l,:Y1(z2) =X(z);[ H l ( Z ) X ( z ) + H1(-z)= [Yo(z2)Go(.)X(-z)l,(6.17)+ Y1(z2)Gl(z)] .Combining these equations yields the input-output relationX(Z)=; [Ho(z)Go(.) + HI(z)Gl(z)]X(z)++[Ho(-z) Go(z) + H1(-z) Gl(z)] X(-z).(6.18)The first term describes the transmission of the signal X ( z ) through thesystem, while the second term describes the aliasing component at the output6.2.

Two-Channel Filter Banks149Figure 6.6. Two-channel filter bank.of the filter bank. Perfect reconstruction is givenif the outputsignal is nothingbut a delayed version of the input signal. That is, the transfer function forthe signal component, denoted as S ( z ) ,must satisfyand the transfer function F ( z ) for the aliasing component must be zero:+F ( z ) = Ho(-z)Go(z) H~(-z)G ~ ( z=) 0.(6.20)If (6.20) is satisfied, the output signal contains no aliasing, but amplitude distortions may be present. If both (6.19) and (6.20) are satisfied, the amplitudedistortions also vanish. Critically subsampled filter banks that allow perfectreconstruction are also known as biorthogonal filter banks.

Several methodsfor satisfying these conditions either exactly or approximately can be foundin the literature. The following sections give a brief overview.6.2.2QuadratureMirrorFiltersQuadrature mirror filter banks (QMF banks) provide complete aliasing cancellation at the output, but condition (6.19) is only approximately satisfied.The principle was introduced by Esteban and Galand in [52]. In QMF banks,H o ( z ) is chosen as a linear phase lowpass filter, and the remaining filters areconstructed as= Hob)Go(.)Hl(Z)= Ho(-z)G ~ ( z ) = -H~(z).(6.21)150BanksChapter 6.

FilterFigure 6.7. QMF bank in polyphase structure.As can easily be verified, independent of the filter H o ( z ) ,the condition F ( z ) =0 is structurally satisfied, so that one only has to ensure that S ( z ) = H i ( z )H:(-z) M 22-4. The name QMF is due to the mirror image property+IHl(,.G- q = IHo(&+qwith symmetry around ~ / 2 .QMF bank prototypes with good coding properties havefor instance beendesigned by Johnston [78].One important property of the QMF banks is their efficient implementation dueto themodulated structure,where the highpass and lowpass filters arerelated as H l ( z ) = Ho(-z). For the polyphase components this means thatHlo(z) = Hoo(z) and H l l ( z ) = -Hol(z).

The resulting efficient polyphaserealization is depicted in Figure 6.7.6.2.3GeneralPerfect Reconstruction Two-ChannelFilter BanksA method for the construction of PR filter banks is to choose(6.22)Is is easily verified that (6.20) is satisfied. Inserting the above relationshipsinto (6.19) yieldsUsing the abbreviationT ( z )= Go(2) H o ( z ) ,(6.24)6.2.BanksTwo-Channel Filter1.51.5~~~~~~~~~e1-t151m0.5 -~~~~~~~~~.1.-vt0.5 -v*u0-0.5- 3 ' 000'''''02468'''10 1412'16 18-0.5*J'-70c2''0 42'6'0.0'8 141210'''1816n(4(b)Figure 6.8. Examples of Nyquist filters T ( z ) ; (a) linear-phase; (b) shortoveralldelay.this becomes+ (-1)"l22-4 = T ( z )T(-z).(6.25)+Note that i [ T ( z ) T ( - z ) ] is the z-transform of a sequence that only hasnon-zero even taps, while i [ T ( z )- T ( - z ) ] is the z-transform of a sequencethat only has non-zero oddtaps.

Altogether we can saythat in order to satisfy(6.25), the system T ( z ) has to satisfyn=qn=q+21,l#Oarbitrary n = q 21 1.+ +e a(6.26)In communications, condition (6.26) is known as the first Nyquist condition.Examples of impulse responses t(n)satisfying the first Nyquist condition aredepicted in Figure 6.8. The arbitrary taps are thefreedesign parameters,which may be chosen in order to achieve good filter properties. Thus, filterscan easily be designed by choosing a filter T ( z )and factoring it intoHo(z) andGo(z). This can be done by computing the roots of T ( z ) and dividing theminto two groups, which form the zeros of Ho(z) and Go(z).