Oversampled Wilson-type Cosine Modulated Filter Banks With Linear ...
in Proc. 30th Asilomar Conf. Signals, Systems, Computers, Pacific Grove (CA), Nov. 1996, pp. 998–1002 Copyright IEEE 1996
Oversampled Wilson-Type Cosine Modulated Filter Banks with Linear Phase* Helmut Bolcskei and Franz Hlawatsch
INT€ET,Vienna University of Technology Gusshausstrasse29389,A-1040 Vienna, Austria Phone: +43 1 58801 3518;Fax: 4 3 1 587 05 83 email: [email protected] Abstract We introduce Wilson filter banks (WFBs) as a new type of cosine modulated filter banks (CMFBs) corresponding to the discrete-time Wilson expansion WFBs albw linear phase filters in all chunnels. Weformulate perfect reconstruction (PR) conditionsfor oversampled and critically sampled WFBs and show that PR WFBs correspond to PR DFTflter b a d s with twice the oversamplingfactor: Generalizing WFBs, we then pmpose the new family of “even-stacked” CMFBs allowing both PR and linear phase filters in all channels. This CMFB family contains WFBs as well as CMFBs recentty introduced by Lin and Vaidyanathan. Finally, after extending conventional (“oddstacked ”) CMFBs to the oversampled case, we fonnulute un@ied PR conditionsfor both even- and odd-stacked oversampled and critically sampled CMFBs. We show that PR CMFBs are always related to PR DFTfilfer banks of the same stacking type and with twice the oversampling factoz
1. Introduction Recent interest in oversampled filter banks (FBs) [ 11-[SI is mainly due to their increased design freedom and noise immunity [ 1. 3.41. Oversampled DF” F B s [61-[81, [31 and oversampled cosine modulated FBs ( C M F B s ) [9]-[15], [8] are especially attractive as they allow an efficientimplementation. An advantage of C M F B s over DFT F B s is the fact that their subband signals are real for real input signals and real analysis prototype. This paper introduces and studies a new (possibly oversince sampled) CMFB which we call Wilson FB -) it corresponds to the discrete-time Wlson expansion [16]. Wilson expansions are based on cosine and sine modulation of a prototype function and can be constructed to have good time and frequency localization besides being orthonormal [171-[ 191, [161. An important advantage (especially in image coding applications) of W F B s over conven-
2. Wilson Filter Banks We consider a WFB with 2N channels and decimation factor 2M. The WFB is critically sampled if N = M and oversampled if N > M. The analysis FB consists of two partial F B s with impulse responses { h k [ n ] } k = O , . . , ~ and {&[n]}k=1,..,~-1, respectively, that are derived from an analysis prototype h[n]as k=O Nnl, h [ n ]= JZh[n] COS(%(^-$)) , k = 1,...,N-1 h[n- sM] k =N and
{
&[n]=
998
fih[n- MI sin(
$( n - M - E2 ) )
fork = 1,...,N-1. Heres = 0 for N even and s = 1for N odd. Similarly, the synthesis FB consists of the two partial F B S {fk[n]}k=O ....N and { f . . [ n ] } k = i , . . , ~ - ~ derived from a synthesis prototype f[n]as
f [nl,
k=O & f [ n ]cos($(n+$)) , k = 1,...,N-1 f[n+ sM] (-l)n+rM, k = N and fk[[.]
‘Funding by FWF grant P10531-6PH.
1058-6393197$10.00 0 1997 IEEE
tional C M F B s [9]-[15] is that they have linear phase filters in all channels if the prototypes have linear phase. Organization of paper. After the introduction of (possibly oversampled) WFBs in Section 2, Section 3 shows a close relation between W F B s and DFT FBs with twice the oversampling factor. Section 4 provides perfect reconsmction (PR) conditions for (possibly oversampled) W F B s and shows that PR W F B s can always be derived from PR DFT FBs. Section 5 presents a new generalized framework for linear phase CMFBs that contains WFBs and a CMFB type recently introduced by Lin and Vaidyanathan [20].
= - J Z f [ n + M ] sin
(2 -
(n
+ M + ;)
for k = 1,...,N-1 . Note that there are 2N channels but only N+l differentchanneIfrequencies8k = & (k = 0, ...,N) as depicted schematically in Fig. l(a). In particular, the k = 0 channel is centered at frequency 00 = 0, which is a difference from conventional CMFBs (see Section 5). If the analysis prototype has linear phase, i.e., h[2ZN - n] = h[n] with some 1 E 2Z, we have
hk[21N-n] = (-l)k hk[n], k = 0,..,N-l,
where
k = 0,..,N fk,zi[n] = fk[n= fk[n - %MI, k = 1,.., N - 1 ,
fk,2i-i[n]
and hk,2i[n] = h;[2iM .- n ] , k = 0,..,N hk,2i-i[n] = K;[2iM 4, k = 1,. . , N - 1 .
-
Note that (1) describes an expansion of the reconstructed signal 2[n]into the synthesis functions fk,m[n];this expansion can alternatively be written as
hjV[2(ZN+sM)-n] = hN[n], &[2(ZN+M)-n] = ( - l ) k f l K k [ n ] , k = l , . . , N - 1 . Thus, all analysis filters have linear phase as well. Similarly, for a linear phase synthesis prototype f[n] all synthesis filters f k [n]and fk [n]have linear phase. This is an important advantage of WFBs over conventional CMFBs [9]-[15].
k = l m=--oo
r=--m
Similarly, the input-output relation in an even-stacked DFT FB [6] with 2N channels and decimation factor M , using analysis prototype h[n]and synthesis prototype f [ n ] is , 2N-1
00
1 Ho I Ho HI
U 4N
4N
4N
4N 2
Figure 1. TrMsferfunctions of the channelfilters in (a)a WFB ol; more generally, an even-stacked CMFB with 2N channels, (b)an N-channel odd-stacked CMFB.
3 Relation to DFT Filter Banks We shall next show that there is a close relationship between W F B s and DFI' F B s with twice the oversampling factor. The basic idea-combining positive and negative frequencies in a DIT FE3 to obtain a CMFB-has been introduced by Daubechies et al. [ 17 in a signal expansion context, and has also been used in filter bank theory for many years [141 (in filter bank theory, however, emphasis has been placed on the near-PR case). The input-output relation in a WFB with 2N channels and decimation factor 2M is
= f i f [n - 2iM] cos(,(n k?r
N a , k=O
N-1
k=l
+ N -2iM))
= f k [ n - 2iM] = fk,2&].
it-00
00
Form = 2i - 1,Eq.(4) can be verified in a similar way, and verification of ( 5 ) and (6) is straightforward.
i=-CO
999
Similarly, the WFB's analysis functions can be derived from the DFT FB's analysis functions as
hk,m[n]= -pt,,[n] DFI' (-l)m&
Jz
+ l g ! k , m [ ne+$ ]
k = 1 , ...,N - 1,
ho,ai[nI = h 3 n I
1, (7)
(8)
7
DFT (9) h ~ , z i [ n=] hN,ai-$[n]. Inserting the relations (4)-(6) and (7)-(9) in (2) and arranging terms leads to the following important result.
Theorem [21]. For a WFB, the reconstructed signal can be decomposed as 1 (10) +I = 5 [(Sz)[nI+ ( T z ) [ n I l
With (12),the first PR condition in (1l), S = 21, is satisfied if and only if
5
f[n - m M ] h ( - n + m M
m=-w
where 411 is the unit sample. This is equivalent to the PR condition for a DFT FB with 2N channels and decimation factor M , i.e., with twice the C M F B ' s oversampling factor. In general, (13) does not lead to a similarly simple condition for the second PR condition T = 0. In the case of integer oversampling, however, i.e., N = K M with K E N, it can be shown [21] that T = 0 is satisfied if and only if 00
(-1)iK
9
where the operators S and T are defined as IN-l
k=O m = - w 00
k=O m = - w
Note that the DFT FB input-output relation in (3) can be written as 5 ~ 4 2 1= ( S z ) [ n ]and , thus the operator S corresponds to an even-stacked DFT FB with 2N channels and decimation factor M , i.e., with twice the oversampling factor of the CMFl3.
5 A Unified Framework for C
4 Perfect Reconstruction Conditions With (lo), it can be shown [21] that a necessary and suf, ficient condition for PR with zero delay, 5[n]= z [ n ] is where I and 0 denote the identity and zero operator, respectively. For T = 0 the WFB's input-output relation (10)reduces to 2[n]= ( S z ) [ n ]i.e., , the input-output relation of an even-stacked DFT FB with 2N channels and decimation factor M . Thus, PR WFBs correspond to even-stacked PR DFT FBs with twice the oversampling factol: In particular, it can be shown [21] that paraunitary [ 141 W F B s correspond to paraunitary DFT F B s (here, f [ n ]= h'[-n]). The operators S and T can be expressed in the time do-
i
main as
h[-72
+ m M + 2",
i
h[n- TM], k = 0
, k = 1 , ..,N - 1 hi[.] = JZh[n] COS(^^ + 6 ) h[n- SM]( - l ) n - $ M , k =N and for k = 1,..,N - 1. Similarly, the synthesis FB consists of partial FBs { f ~ [ n ] } k = ~and , . . {, =~f ; [ n ] } k = ~ , . . , ~ - - ldefined in terms of a synthesis prototype f [ n ]as
m=-w
I=-00
In a WFB all channel filters have linear phase if the prototypes have linear phase. We now introduce a generalized framework for "linear-phase" CMF'Bs. In analogy to even-stacked DFT FBs, we call this new CMFB class evenstacked. We show that the class of even-stacked CMFBs contains WFBs and the linear phase CMF'Bs recently introduced by Lin and Vaidyanathan [20].Subsequently, we extend the conventional "odd-stacked" CMFBs to the oversampled case, and finally we present unified PR conditions that are valid for both even- and odd-stacked CMFBs. Even-stacked CMFBs. The analysis FB in an evenstacked CMFB with 2N channels and decimation factor 2M consists of two partial F B s {h![n]}k=o,..,~ and {hi[7Z]}k=1,..,N,r derived from an analysis prototype h[n],
G[n]= f i h [ n - M ] s i n ( $ ( n - M ) +&)
m
m
+i K ) M ]
for I = 0, ...,K-1 and -CO < m < 00. Note that critical sampling, N = M , is a special case with K = 1. The operators S and T can also be expressed in the frequency and in the polyphase domains; this leads to comesponding formulations of the PR conditions in ( 1 1 ) [21].
(&,2)fkq3nI, 2N-1
f[n- ( I
.h[n-IM+(2m+i+l)KM] = 0
00
(Sz)[nl =
+ 21N] = z16 [ I ] ,
(12)
f[n+rM], k = O
- (21+1)N]f[n - m M ] h[n- m M
+ (21 + l ) W .
(13)
1000
J 2 f [ n ] COS(%^ f [ n+ s M ]
- +!),
k =l,..,N-l k =N
and
where the operators S and T are defined as 2N-1
for k = 1 , ..,N - 1. Here, we define the phases as
a A 4; = -cr---k+r2N 2
k=O m=-w
2N-1
with a E Z;
furthermore r , s E {0,1} withs = r foraevenands = 1r for a odd. Thus, even-stacked CMFBs are parameterized in terms of the two parameters CY E P and r E {0,1}. For any choice of a E P and r E { 0,l}, all arualysisfilters hove linearphase if h[a+ (21 - l ) N - n] = h[n] with some
1 E z. Similarly, all synthesis filters have linear phase if f[-a- (21 - l ) N - n] = f[n]. Note that even-stacked CMFBs have 2N channels but only N + 1different channel frequencies& = (k = 0 , ...,N ) ,with the k = 0 channel centered at frequency = 0 (see Fig. l(a)). A special even-stacked CMFB 'is the WFB introduced in Section 2; it is obtained by choosing the parameters as a = N and r=O. Another important special case is the CMFB recently introduced (for critical sampling) by Lin and Vaidyanathan [20];its generalization to arbitrary oversampling is obtained for a = 0 and r = 0. Odd-stacked CMFBs. Odd-stackedCMFBs are the traditional CMFB type previously defined for critical sampling [9]-[15].In the general case of an odd-stacked CMFB with N channels and decimation factor M (note that the CMFB is oversampled for N > M), we define the analysis and synthesis filters respectively as
&
ha[.]
eo
= JZh[n]cos(
(k + 1/2)a
m
n + 4a),
-
fork = 0, ...,N 1. Here, h[n]and f[n] denote the analysis and synthesis prototype, respectively, and the phases are defined as
m
(The operators S,T in Sections 3 and 4 are a special case.) In the even-stacked case we have fEz[n] = f [ n mM1 -k(n-mM), h f ) [ n ]= h' [mM- 4 wgp4-n) 4 k = 4: and b, = (-l)*. In the odd-stackedcase, and h:?[n] are obtained by formally replacing k with k +1/2 in the above expressions; furthermore (6k = 4; and b, = 1. For an even-stacked (odd-stacked) CMFB the operator S correspondsto an even-stacked (odd-stacked) DFT FB [a] with 2N channels and decimation factor M; this DFT FB has thus the same stacking type and twice the oversampling factor of the CMFB. PR conditions. We next provide unified PR conditions for oversampled and critically sampled, even- and oddstacked CMFBs. These PR conditions generalize the PR conditions for WFBs given in Section 4. For PR with zero delay, 5[n]= z[n],it is necessary and sufficient that [21]
w,, ftz[n]
S=21
and
T=O.
(14)
(Note that (11) is a special case.) For T = 0 the CMFB's input-output relation reduces to 2[n] = (Sz)[n],which is the input-output relation of the corresponding DFT FB. Hence, PR CMFBs correspond to PR DFT FBs ofthe same stacking type and with twice the oversamplingfacto. In particular, paraunitary CMFBs can be shown to correspond to paraunitary DFI' FBs. The operators S and T can be expressed in the time domain as
3
00
00
( S Z ) [ ~= ] 2N
d i ~ [ n -2 l q
. h[-n + m M
this extends the phase definition given in [ 1I] for the special
f[n-mM] m=-m
1=-00
+2 z q ,
(15)
0 0 0 0
case of critical sampling. Note that the channel frequencies are O k = %j$; in particular, the k = 0 channel is centered at frequency 80 = 2 (see Fig. l(b)). A disadvan4N tage of odd-stacked CMFBs is that the channel filters do not have linear phase even if the prototypes have linear phase [91-[151. Relation to DFT FBs. For both even- and odd-stacked CIvlFBs, the following decomposition of the reconstructed signal (generalizing (10)) can be shown [2I],
(Tz)[n] = (-1)+2N
d1amz[-n+2mM-
- 21N - a]f[n- m M ]h[n - m M + a + 21N] ,
(16)
with dl = 1 and a, = (-1)'" in the even-stacked case and di = (-1)l and a, = 1 in the odd-stacked case. With (15). the first PR condition in (14),S = 21, is satisfied if and only if 00
1 5[n] = 2 [(Sz)[n]+ (Tz)[nl],
f[n ,=-W
1001
- n M ]h[-n + m M +"2
=
1
ap].
Note that this condition is independent of the stacking type; it is the PR condition for a DFT FB (even- or odd-stacked) with 2N channels and decimation factor M , i.e., twice the CMFB’s oversampling factor. Furthermore, for integer oversampling,N = K M , it follows from (16) that T = 0 is satisfied if and only if 00
pi f[n- (ZtiK)M]h[” - (Z-2mK-iK)M+ a]= 0 k-00
for 1 = 0, ..., K-1 and -cu < m < 30, wherepi = ( - l ) i K in the even-stacked case and pi = (-l)iin the odd-stacked case. Note that critical sampling, A- = M , is a special case with K = 1.
6 Conclusion We introduced and studied Wilsonfilterbanks (WFBs), a new type of cosine modulated filter banks (CMFEs) that corresponds to the discrete-time Wilson expansion and allows both perfect reconstruction (PR) and linear phase filters in all channels. We formulated PR conditions for oversampled and critically sampled WFBs, and we showed that PRWFBs are associated to PR DFT FBs. Generalizing WFBs, we then defined the new class of even-stacked C M F B s that all have the desirable property of allowing both PR and linear phase filters in all channels. The C M F B s recently introduced for critical sampling by Lin and Vaidyanathan were extended to the oversampled case and shown to be a further special case (besides WFBs) of evenstacked CMFBs. The conventional (“odd-stacked”) C M F B s were also extended to the oversampled case. Finally, we presented a unified set of PR conditions that applies to both even- and oddstacked, oversampled and critically sampled CMFBs. These PR conditions showed that PR CMFBs are always related to PR DFT filter banks of the same stacking type and with twice the oversampling factor. We note that frame-theoretic properties of oversampled C M F B s are discussed in [Zl].
References
[5] A. J. E. M. Janssen, “Density theorems for filter banks,” Nat. Lab. Report 6858, Eindhoven (The Netherlands), Apr. 1995. [6] R. E. Crochiere and L. R. Rabiner, Multirate D i g i d Signal Processing. Prentice-Hall, 1983. [7] K. Swaminathan and P. P.Vaidyanathan, ‘Theoryand design of uniform DFT. parallel, quadrature mirror filter banks,” IEEE Trans. Circuits and Systems, Vol. CAS-33, No. 12. Dec.1986, pp. 1170-1191. [8] H. BiSlcskei and F. Hlawatsch. “Oversampled modulated filter banks,” to appear in GaborAnalysis: Theory. Algorithm, Md Applications, H. G. Feichtinger and T. Strohmer, eds., Birkt&user, June 1997. [9] H. S. Malvar, Signal Processing with Lapped Tranrfom. Artech House, 1992. [lo] R. D. Koilpillai and P. P. Vaidyanathan, “Cosine-modulated FIR filter banks salisfying perfect reconstruction,” IEEE Trans. Signal Processing, vol. 40,no. 4, April 1992, pp. 770783. [ 111 R. A. Gopinath and C. S.Burms, “Some results in the theory
modulated filter banks and modulated wavelet tight frames,” Applied and Complctarional Harmonic Analysis, vol. 2,1995, pp. 303-326. [121 T. Q. Nguyen and R. D. Koilpillai, ‘The theory and design of
[ 131
[14] [15] [16]
[17]
[18]
[l] H. Belcskei, F. Hlawatsch. and H. G. Feichtinger, “Frametheoretic analysis of filter banks,“ submitted to IEEE Tram. Signal Processing, Feb. 1996. [Z] Z. CvetkoviCand M.Vetterli, “Oversampled filter banks,” to appear in IEEE Trans. Signal Processing. [3] H. Bolcskei, E Hlawatsch, and H. G. Feichtinger, “Ovcrsampled FIR and W DFT filter banks and Weyl-Heisenberg frames,” P m . IEEE ICASSP-96, Atlanta (GA), May 1996, vol. 3, pp. 1391-1394. [4] H. Biilcskei, E Hlawatsch, and H. G. Feichtinger, ‘‘Frametheoretic analysis and design of oversampled filter banks.” P m . IEEE ISCAS-96. Atlanta (GA), May 1996, vol. 2, pp. 409-412.
1002
[191
arbitrary-length cosine-modulatedfilter banks and wavelets, satisfymg perfect reconstruction,” IEEE Trans. Signal Processing, vol. 44, no. 3, pp. 473-483, March 1996. T. A. Ramstad and J. P. Tanem. “Cosine-modulated analysissynthesis filter bank with critical sampling and perfect reconstruction,” P m . IEEE ICASSP-9I. Toronto, Canada, May 1991, pp. 1789-1792. P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice-Hall, 1993. M. Vetterli and J. KovaEeviC. Wavelets and Subband Coding. Prentice-Hall, 1995. H. Bolcskei, H. G. Feichtinger, K. Grijchenig, and F. Hlawatsch, “Discrete-time Wilson expansions,” Pmc. IEEESP Int. Sympos. Ti-Frequency T i - S c a l e Analysis, Paris (France), June 1996, pp. 525-528. I. Daukhies, S.Jaffard. and J. L. Joum6, “A simple Wilson orthonormal basis with exponential decay,” SLAM J. Math A d . vol. 22,1991, pp. 554-572. H. G. Feichtinger, K. Grtkhenig, and D. Walnut,‘Wilson bases and modulation spaces,” Math Nachrichten, vol. 155, 1992, pp. 7-17. P. Auscher, “Remarks on the local Fourier bases,” in Wavelets: Mathemafics and Applicarionr, eds. J. J. Bendetto and M. W. Frazier. Boca Raton a ) CRC :Press, 1993, pp. 203218.
1203 Y.-P. Lin and P.P. Vaidyanathan, ‘Zinear phase cosine modulated maximally decimated filter banks with perfect reconstruction,” IEEE T m . Signal Processing, vol. 42. no. 11, NOV.1995, pp. 2525-2539.
[213 H. BiSlcskei and F. Hlawatsch, “Oversampled cosine modulated filter banks.“ to be submitted to IEEE Trans. CircuitS and Systems.
Oversampled Wilson-Type Cosine Modulated Filter Banks with Linear Phase* Helmut Bolcskei and Franz Hlawatsch
INT€ET,Vienna University of Technology Gusshausstrasse29389,A-1040 Vienna, Austria Phone: +43 1 58801 3518;Fax: 4 3 1 587 05 83 email: [email protected] Abstract We introduce Wilson filter banks (WFBs) as a new type of cosine modulated filter banks (CMFBs) corresponding to the discrete-time Wilson expansion WFBs albw linear phase filters in all chunnels. Weformulate perfect reconstruction (PR) conditionsfor oversampled and critically sampled WFBs and show that PR WFBs correspond to PR DFTflter b a d s with twice the oversamplingfactor: Generalizing WFBs, we then pmpose the new family of “even-stacked” CMFBs allowing both PR and linear phase filters in all channels. This CMFB family contains WFBs as well as CMFBs recentty introduced by Lin and Vaidyanathan. Finally, after extending conventional (“oddstacked ”) CMFBs to the oversampled case, we fonnulute un@ied PR conditionsfor both even- and odd-stacked oversampled and critically sampled CMFBs. We show that PR CMFBs are always related to PR DFTfilfer banks of the same stacking type and with twice the oversampling factoz
1. Introduction Recent interest in oversampled filter banks (FBs) [ 11-[SI is mainly due to their increased design freedom and noise immunity [ 1. 3.41. Oversampled DF” F B s [61-[81, [31 and oversampled cosine modulated FBs ( C M F B s ) [9]-[15], [8] are especially attractive as they allow an efficientimplementation. An advantage of C M F B s over DFT F B s is the fact that their subband signals are real for real input signals and real analysis prototype. This paper introduces and studies a new (possibly oversince sampled) CMFB which we call Wilson FB -) it corresponds to the discrete-time Wlson expansion [16]. Wilson expansions are based on cosine and sine modulation of a prototype function and can be constructed to have good time and frequency localization besides being orthonormal [171-[ 191, [161. An important advantage (especially in image coding applications) of W F B s over conven-
2. Wilson Filter Banks We consider a WFB with 2N channels and decimation factor 2M. The WFB is critically sampled if N = M and oversampled if N > M. The analysis FB consists of two partial F B s with impulse responses { h k [ n ] } k = O , . . , ~ and {&[n]}k=1,..,~-1, respectively, that are derived from an analysis prototype h[n]as k=O Nnl, h [ n ]= JZh[n] COS(%(^-$)) , k = 1,...,N-1 h[n- sM] k =N and
{
&[n]=
998
fih[n- MI sin(
$( n - M - E2 ) )
fork = 1,...,N-1. Heres = 0 for N even and s = 1for N odd. Similarly, the synthesis FB consists of the two partial F B S {fk[n]}k=O ....N and { f . . [ n ] } k = i , . . , ~ - ~ derived from a synthesis prototype f[n]as
f [nl,
k=O & f [ n ]cos($(n+$)) , k = 1,...,N-1 f[n+ sM] (-l)n+rM, k = N and fk[[.]
‘Funding by FWF grant P10531-6PH.
1058-6393197$10.00 0 1997 IEEE
tional C M F B s [9]-[15] is that they have linear phase filters in all channels if the prototypes have linear phase. Organization of paper. After the introduction of (possibly oversampled) WFBs in Section 2, Section 3 shows a close relation between W F B s and DFT FBs with twice the oversampling factor. Section 4 provides perfect reconsmction (PR) conditions for (possibly oversampled) W F B s and shows that PR W F B s can always be derived from PR DFT FBs. Section 5 presents a new generalized framework for linear phase CMFBs that contains WFBs and a CMFB type recently introduced by Lin and Vaidyanathan [20].
= - J Z f [ n + M ] sin
(2 -
(n
+ M + ;)
for k = 1,...,N-1 . Note that there are 2N channels but only N+l differentchanneIfrequencies8k = & (k = 0, ...,N) as depicted schematically in Fig. l(a). In particular, the k = 0 channel is centered at frequency 00 = 0, which is a difference from conventional CMFBs (see Section 5). If the analysis prototype has linear phase, i.e., h[2ZN - n] = h[n] with some 1 E 2Z, we have
hk[21N-n] = (-l)k hk[n], k = 0,..,N-l,
where
k = 0,..,N fk,zi[n] = fk[n= fk[n - %MI, k = 1,.., N - 1 ,
fk,2i-i[n]
and hk,2i[n] = h;[2iM .- n ] , k = 0,..,N hk,2i-i[n] = K;[2iM 4, k = 1,. . , N - 1 .
-
Note that (1) describes an expansion of the reconstructed signal 2[n]into the synthesis functions fk,m[n];this expansion can alternatively be written as
hjV[2(ZN+sM)-n] = hN[n], &[2(ZN+M)-n] = ( - l ) k f l K k [ n ] , k = l , . . , N - 1 . Thus, all analysis filters have linear phase as well. Similarly, for a linear phase synthesis prototype f[n] all synthesis filters f k [n]and fk [n]have linear phase. This is an important advantage of WFBs over conventional CMFBs [9]-[15].
k = l m=--oo
r=--m
Similarly, the input-output relation in an even-stacked DFT FB [6] with 2N channels and decimation factor M , using analysis prototype h[n]and synthesis prototype f [ n ] is , 2N-1
00
1 Ho I Ho HI
U 4N
4N
4N
4N 2
Figure 1. TrMsferfunctions of the channelfilters in (a)a WFB ol; more generally, an even-stacked CMFB with 2N channels, (b)an N-channel odd-stacked CMFB.
3 Relation to DFT Filter Banks We shall next show that there is a close relationship between W F B s and DFI' F B s with twice the oversampling factor. The basic idea-combining positive and negative frequencies in a DIT FE3 to obtain a CMFB-has been introduced by Daubechies et al. [ 17 in a signal expansion context, and has also been used in filter bank theory for many years [141 (in filter bank theory, however, emphasis has been placed on the near-PR case). The input-output relation in a WFB with 2N channels and decimation factor 2M is
= f i f [n - 2iM] cos(,(n k?r
N a , k=O
N-1
k=l
+ N -2iM))
= f k [ n - 2iM] = fk,2&].
it-00
00
Form = 2i - 1,Eq.(4) can be verified in a similar way, and verification of ( 5 ) and (6) is straightforward.
i=-CO
999
Similarly, the WFB's analysis functions can be derived from the DFT FB's analysis functions as
hk,m[n]= -pt,,[n] DFI' (-l)m&
Jz
+ l g ! k , m [ ne+$ ]
k = 1 , ...,N - 1,
ho,ai[nI = h 3 n I
1, (7)
(8)
7
DFT (9) h ~ , z i [ n=] hN,ai-$[n]. Inserting the relations (4)-(6) and (7)-(9) in (2) and arranging terms leads to the following important result.
Theorem [21]. For a WFB, the reconstructed signal can be decomposed as 1 (10) +I = 5 [(Sz)[nI+ ( T z ) [ n I l
With (12),the first PR condition in (1l), S = 21, is satisfied if and only if
5
f[n - m M ] h ( - n + m M
m=-w
where 411 is the unit sample. This is equivalent to the PR condition for a DFT FB with 2N channels and decimation factor M , i.e., with twice the C M F B ' s oversampling factor. In general, (13) does not lead to a similarly simple condition for the second PR condition T = 0. In the case of integer oversampling, however, i.e., N = K M with K E N, it can be shown [21] that T = 0 is satisfied if and only if 00
(-1)iK
9
where the operators S and T are defined as IN-l
k=O m = - w 00
k=O m = - w
Note that the DFT FB input-output relation in (3) can be written as 5 ~ 4 2 1= ( S z ) [ n ]and , thus the operator S corresponds to an even-stacked DFT FB with 2N channels and decimation factor M , i.e., with twice the oversampling factor of the CMFl3.
5 A Unified Framework for C
4 Perfect Reconstruction Conditions With (lo), it can be shown [21] that a necessary and suf, ficient condition for PR with zero delay, 5[n]= z [ n ] is where I and 0 denote the identity and zero operator, respectively. For T = 0 the WFB's input-output relation (10)reduces to 2[n]= ( S z ) [ n ]i.e., , the input-output relation of an even-stacked DFT FB with 2N channels and decimation factor M . Thus, PR WFBs correspond to even-stacked PR DFT FBs with twice the oversampling factol: In particular, it can be shown [21] that paraunitary [ 141 W F B s correspond to paraunitary DFT F B s (here, f [ n ]= h'[-n]). The operators S and T can be expressed in the time do-
i
main as
h[-72
+ m M + 2",
i
h[n- TM], k = 0
, k = 1 , ..,N - 1 hi[.] = JZh[n] COS(^^ + 6 ) h[n- SM]( - l ) n - $ M , k =N and for k = 1,..,N - 1. Similarly, the synthesis FB consists of partial FBs { f ~ [ n ] } k = ~and , . . {, =~f ; [ n ] } k = ~ , . . , ~ - - ldefined in terms of a synthesis prototype f [ n ]as
m=-w
I=-00
In a WFB all channel filters have linear phase if the prototypes have linear phase. We now introduce a generalized framework for "linear-phase" CMF'Bs. In analogy to even-stacked DFT FBs, we call this new CMFB class evenstacked. We show that the class of even-stacked CMFBs contains WFBs and the linear phase CMF'Bs recently introduced by Lin and Vaidyanathan [20].Subsequently, we extend the conventional "odd-stacked" CMFBs to the oversampled case, and finally we present unified PR conditions that are valid for both even- and odd-stacked CMFBs. Even-stacked CMFBs. The analysis FB in an evenstacked CMFB with 2N channels and decimation factor 2M consists of two partial F B s {h![n]}k=o,..,~ and {hi[7Z]}k=1,..,N,r derived from an analysis prototype h[n],
G[n]= f i h [ n - M ] s i n ( $ ( n - M ) +&)
m
m
+i K ) M ]
for I = 0, ...,K-1 and -CO < m < 00. Note that critical sampling, N = M , is a special case with K = 1. The operators S and T can also be expressed in the frequency and in the polyphase domains; this leads to comesponding formulations of the PR conditions in ( 1 1 ) [21].
(&,2)fkq3nI, 2N-1
f[n- ( I
.h[n-IM+(2m+i+l)KM] = 0
00
(Sz)[nl =
+ 21N] = z16 [ I ] ,
(12)
f[n+rM], k = O
- (21+1)N]f[n - m M ] h[n- m M
+ (21 + l ) W .
(13)
1000
J 2 f [ n ] COS(%^ f [ n+ s M ]
- +!),
k =l,..,N-l k =N
and
where the operators S and T are defined as 2N-1
for k = 1 , ..,N - 1. Here, we define the phases as
a A 4; = -cr---k+r2N 2
k=O m=-w
2N-1
with a E Z;
furthermore r , s E {0,1} withs = r foraevenands = 1r for a odd. Thus, even-stacked CMFBs are parameterized in terms of the two parameters CY E P and r E {0,1}. For any choice of a E P and r E { 0,l}, all arualysisfilters hove linearphase if h[a+ (21 - l ) N - n] = h[n] with some
1 E z. Similarly, all synthesis filters have linear phase if f[-a- (21 - l ) N - n] = f[n]. Note that even-stacked CMFBs have 2N channels but only N + 1different channel frequencies& = (k = 0 , ...,N ) ,with the k = 0 channel centered at frequency = 0 (see Fig. l(a)). A special even-stacked CMFB 'is the WFB introduced in Section 2; it is obtained by choosing the parameters as a = N and r=O. Another important special case is the CMFB recently introduced (for critical sampling) by Lin and Vaidyanathan [20];its generalization to arbitrary oversampling is obtained for a = 0 and r = 0. Odd-stacked CMFBs. Odd-stackedCMFBs are the traditional CMFB type previously defined for critical sampling [9]-[15].In the general case of an odd-stacked CMFB with N channels and decimation factor M (note that the CMFB is oversampled for N > M), we define the analysis and synthesis filters respectively as
&
ha[.]
eo
= JZh[n]cos(
(k + 1/2)a
m
n + 4a),
-
fork = 0, ...,N 1. Here, h[n]and f[n] denote the analysis and synthesis prototype, respectively, and the phases are defined as
m
(The operators S,T in Sections 3 and 4 are a special case.) In the even-stacked case we have fEz[n] = f [ n mM1 -k(n-mM), h f ) [ n ]= h' [mM- 4 wgp4-n) 4 k = 4: and b, = (-l)*. In the odd-stackedcase, and h:?[n] are obtained by formally replacing k with k +1/2 in the above expressions; furthermore (6k = 4; and b, = 1. For an even-stacked (odd-stacked) CMFB the operator S correspondsto an even-stacked (odd-stacked) DFT FB [a] with 2N channels and decimation factor M; this DFT FB has thus the same stacking type and twice the oversampling factor of the CMFB. PR conditions. We next provide unified PR conditions for oversampled and critically sampled, even- and oddstacked CMFBs. These PR conditions generalize the PR conditions for WFBs given in Section 4. For PR with zero delay, 5[n]= z[n],it is necessary and sufficient that [21]
w,, ftz[n]
S=21
and
T=O.
(14)
(Note that (11) is a special case.) For T = 0 the CMFB's input-output relation reduces to 2[n] = (Sz)[n],which is the input-output relation of the corresponding DFT FB. Hence, PR CMFBs correspond to PR DFT FBs ofthe same stacking type and with twice the oversamplingfacto. In particular, paraunitary CMFBs can be shown to correspond to paraunitary DFI' FBs. The operators S and T can be expressed in the time domain as
3
00
00
( S Z ) [ ~= ] 2N
d i ~ [ n -2 l q
. h[-n + m M
this extends the phase definition given in [ 1I] for the special
f[n-mM] m=-m
1=-00
+2 z q ,
(15)
0 0 0 0
case of critical sampling. Note that the channel frequencies are O k = %j$; in particular, the k = 0 channel is centered at frequency 80 = 2 (see Fig. l(b)). A disadvan4N tage of odd-stacked CMFBs is that the channel filters do not have linear phase even if the prototypes have linear phase [91-[151. Relation to DFT FBs. For both even- and odd-stacked CIvlFBs, the following decomposition of the reconstructed signal (generalizing (10)) can be shown [2I],
(Tz)[n] = (-1)+2N
d1amz[-n+2mM-
- 21N - a]f[n- m M ]h[n - m M + a + 21N] ,
(16)
with dl = 1 and a, = (-1)'" in the even-stacked case and di = (-1)l and a, = 1 in the odd-stacked case. With (15). the first PR condition in (14),S = 21, is satisfied if and only if 00
1 5[n] = 2 [(Sz)[n]+ (Tz)[nl],
f[n ,=-W
1001
- n M ]h[-n + m M +"2
=
1
ap].
Note that this condition is independent of the stacking type; it is the PR condition for a DFT FB (even- or odd-stacked) with 2N channels and decimation factor M , i.e., twice the CMFB’s oversampling factor. Furthermore, for integer oversampling,N = K M , it follows from (16) that T = 0 is satisfied if and only if 00
pi f[n- (ZtiK)M]h[” - (Z-2mK-iK)M+ a]= 0 k-00
for 1 = 0, ..., K-1 and -cu < m < 30, wherepi = ( - l ) i K in the even-stacked case and pi = (-l)iin the odd-stacked case. Note that critical sampling, A- = M , is a special case with K = 1.
6 Conclusion We introduced and studied Wilsonfilterbanks (WFBs), a new type of cosine modulated filter banks (CMFEs) that corresponds to the discrete-time Wilson expansion and allows both perfect reconstruction (PR) and linear phase filters in all channels. We formulated PR conditions for oversampled and critically sampled WFBs, and we showed that PRWFBs are associated to PR DFT FBs. Generalizing WFBs, we then defined the new class of even-stacked C M F B s that all have the desirable property of allowing both PR and linear phase filters in all channels. The C M F B s recently introduced for critical sampling by Lin and Vaidyanathan were extended to the oversampled case and shown to be a further special case (besides WFBs) of evenstacked CMFBs. The conventional (“odd-stacked”) C M F B s were also extended to the oversampled case. Finally, we presented a unified set of PR conditions that applies to both even- and oddstacked, oversampled and critically sampled CMFBs. These PR conditions showed that PR CMFBs are always related to PR DFT filter banks of the same stacking type and with twice the oversampling factor. We note that frame-theoretic properties of oversampled C M F B s are discussed in [Zl].
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