Filter Banks (779443), страница 2
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The remainingfilters are then chosen according to (6.24) in order to yield a PR filter bank.This design method is known as spectral factorization.6.2.4MatrixRepresentationsMatrix representations are a convenient and compact way of describing andcharacterizing filter banks. In the following we will give a brief overview ofthe most important matrices and their relation to the analysis and synthesisfilters.152BanksChapter 6. FilterModulation Matrix.
The input-output relations of the two-channel filterbank may also be written in matrix form.For this, we introduce the vectors(6.27)1(6.28)and the so-called modulation matrix or alias component (AC) matrax(6.29)which contains the filters Ho(z) and H I ( z ) andtheirmodulatedHo(-z) and H l ( - z ) . We getversionsPolyphase Representation of the Analysis Filter Bank. Letusconsider the analysis filter bankin Figure 6.9(a). The signals yo(m) and y1 (m)may be written asandy1(m)= C h 1 ( n ) x ( 2 m - n)~n= C h l O ( k ) zo(m - Ic)k+ C h l l ( k ) z1(m - L),k(6.33)6.2. Two-Channel Filter Banks1531Figure 6.9.
Analysis filter bank. (a) direct implementation; (b) polyphase realiza-tion.where we used the following polyphase components:boo@)=how),hOl(k) = ho(2k+ 11,hlO(k) = h1(2k),hll(k)=+ 11,SO(k) = 2(2k),51(k)= 2(2k - 1).Thelast rows of (6.32),and (6.33) respectively, show thatthe completeanalysis filter bank can be realized by operating solely with the polyphasecomponents, as depicted in Figure 6.9(b). The advantage of the polyphaserealization compared to the direct implementation in Figure 6.9(a) is thatonly the required output values are computed.When looking at the firstrows of (6.32) and (6.33) this soundstrivial, because theseequations areeasily implemented anddonotproduceunneeded values. Thus, unlike inthe QMF bank case, the polyphase realization does not necessarily lead tocomputational savings compared t o a proper direct implementation of theanalysis equations. However, it allows simple filter design, gives more insightinto the properties of a filter bank, and leads to efficient implementationsbased on lattice structures; see Sections 6.2.6 and 6.2.7.It is convenient to describe (6.32) and (6.33) in the z-domain using matrixnotation:2/P(Z) = E ( z )% ( z ) ,(6.34)(6.35)(6.36)154BanksChapter 6.
FilterMatrix E ( z ) is called the polyphase matrix of the analysis filter bank. As caneasily be seen by inspection, it is related to the modulation matrix as follows:(6.37)with’(6.38)z-l](6.39)W = [ ’1 -11 1and=[Here, W is understood as the 2x2-DFT matrix. In view of the general M channel case, we use the notation W-’ = ;WH for the inverse.Polyphase Representation of the Synthesis Filter Bank. We considerthe synthesis filter bank in Figure 6.10(a).
The filters Go(z) and Gl(z) canbe written in terms of their type-2 polyphase components asandGl(z) = z-’G:O(Z’)+ G:,(Z’).(6.41)This gives rise to the following z-domain matrix representation:The corresponding polyphase realization is depicted in Figure 6.10. Perfectreconstruction up to an overall delay of Q = 2mo 1 samples is achieved if+R ( z ) E ( z )= 2-0The PR condition for an even overall delay ofI.Q(6.43)= 2mo samples is(6.44)6.2.BanksTwo-Channel Filter155(4(b)Figure 6.10. Synthesis filterbank.realization.(a) direct implementation; (b) polyphaseParaunitaryTwo-Channel Filter Banks6.2.5The inverse of a unitary matrixis given by the Hermitian transpose.A similarproperty can be stated for polyphase matrices as follows:(6.45)E-yz) = E ( z ) ,wherek ( z )= ( E ( z ) y ,= 1,(6.46)E ( z ) k ( z )= k ( z )E ( z )= I .(6.47)121such that~for transposing the matrixAnalogous to ordinary matrices, ( E ( z ) )standsand simultaneously conjugating the elements:In the case of real-valued filter coefficients we havethat B ( z ) = ET(zP1)andfiik(z)= Hik(z-l), suchE ( z ) ET(z-1) = ET(z-1) E ( z ) = I .(6.48)Since E ( z ) is dependent on z , and since the operation (6.46) has to be carriedout on the unit circle, and not at some arbitrary point in the z plane, a matrixE ( z ) satisfying (6.47) is said to be paraunitary.Modulation Matrices.
As can be seen from (6.37) and (6.47), we haveH m ( z ) R m ( z )= R m ( z ) H m ( z )= 2 I(6.49)for the modulation matrices of paraunitary two-channel filter banks.Matched Filter Condition. From (6.49) we may conclude that the analysisand synthesis filters in a paraunitary two-channel filter bank are related asG ~ ( z=) f i k ( ~ )t)gk(n) = h i ( - n ) ,L = 0,l.(6.50)156BanksChapter 6.
FilterThis means thatan analysis filter andits corresponding synthesis filtertogether yield a Nyquist filter (cf. (6.24)) whose impulse responseis equivalentto the autocorrelation sequence of the filters in question:(6.51)Here we find parallels to data transmission, where the receiver input filter ismatched to the output filter of the transmitter such that the overall resultis the autocorrelation sequence of the filter. This is known as the matchedfiltercondition. The reason for choosing this special input filter is that ityields a maximum signal-to-noise ratio if additive white noise interferes onthe transmission channel.Power-Complementary Filters.
From (6.49) we conclude+2 = Ho(z)fio(z) Ho(-z)fio(-z),(6.52)which for z = eJ" implies the requirement2 = IHO(ej")l2+IHo(ej ( W + 4 ) 12.(6.53)powerWe observe thatthe filters Ho(ejW) and Ho(ej(w+")) mustbecomplementary to one another. For constructing paraunitary filter banks wetherefore have to find a Nyquist filter T ( z ) which can be factored intoT ( 2 )= Ho(z) f i O ( 2 ) .(6.54)Note that a factorization is possible only if T(ej") is real and positive.
Afilter that satisfies this condition is said to be valid. Since T ( e J Whas) symmetryaround W = 7r/2 such a filter is also called a valid halfband filter.This approachwas introduced by Smith and Barnwell in [135].Given Prototype. Given an FIR prototype H ( z ) that satisfies condition(6.53), the required analysis and synthesis filters can be derived as(6.55)Here, L is the number of coefficients of the prototype.6.2.BanksTwo-Channel Filter157Number of Coefficients. Prototypes for paraunitary two-channel filterbanks have even length. This is seen by formulating (6.52)in the time domainand assuming an FIR filter with coefficients ho(O),. .
. ,ho(25):c2kse0=h0(n)h;;(n- 24.(6.56)n=OFor C = k , n = 25, 5 # 0, this yields the requirement 0 = h0(25)h:(O),whichfor ho(0) # 0 can only be satisfied by ho(2k) = 0. This means that the filterhas to have even length.Filter Energies. It is easily verified that all filters in a paraunitary filterbank have energy one:2222(6.57)llhollez = llhllle, = llgoIle, = llglllez = 1.Non-Linear Phase Property. We will show that paraunitary two-channelfilter banks are non-linear phase with one exception. The following proof isbased on Vaidyanathan [145].We assume that two filters H ( z ) and G ( z ) arepower-complementary and linear-phase:c2B(z)+= H(z)fi(z) G(z)G(z)= eja z L ~ ( z ) ,G ( z ) = ejp z L G ( z ) ,pER1(6.58)(linear-phase property).We conclude(H(z)ejal'+ jG(z)ejp/') (H(z)ej"/'- jG(z)ejp/') = c2z-~.(6.59)Both factors on the left are FIR filters, so thatAdding and subtracting both equationsshows that H ( z ) and G ( z )must havethe form(6.61)in order to be both power-complementary and linear-phase.
In other words,power-complementary linear-phase filters cannot have more than two coefficients.1586.2.6BanksChapter 6. FilterParaunitary Filter BanksinLatticeStructureParaunitary filter banks can be efficiently implemented in a lattice structure[53], [147]. For this, we decompose the polyphase matrix E ( z ) as follows:(6.62)Here, the matrices B k , k = 0,. . .,N-1 are rotation matrices:(6.63)and D ( z ) is the delay matrixD=[;(6.64)zlll]It can be shown that such a decomposition is always possible [146].Provided cos,&# 0, k = (),l,... ,N-1, we can also write(6.65)withN-l1(6.66)k=OThis basically allows us to reduce the total number of multiplications.
Therealization of the filter bank by means of the decomposed polyphase matrixis pictured in Figure 6.11(a). Given a k , k = 0,. . . ,N - 1, we obtain filters oflength L = 2 N .Since this lattice structure leads to a paraunitary filter bank for arbitrarywe can thus achieve perfect reconstruction even if the coefficients must bequantized due to finite precision. In addition, this structure may be used foroptimizing the filters. For this, we excite the filter bank with zeuen(n)= dn0and ~ , d d ( n ) = dnl and observe the polyphase components of Ho(z) and H l ( z )at the output.ak,The polyphase matrix of the synthesis filter bankhasthefactorization:R(2)= BTD’(2)BT .
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