Author's personal copy ARTICLE IN PRESS. Signal Processing 88 (2008) Contents lists available at ScienceDirect.

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1 Signal Processing 88 (008) Contents lists available at ScienceDirect Signal Processing journal homeage: wwwelseviercom/locate/sigro Critically samled and oversamled comlex filter banks via interleaved DF modulation iao-long Wang, Peng-Lang Shui National Laboratory of Radar Signal Processing, idian University, aibai Load No:, i an, Shaanxi , China article info Article history: Received 19 August 007 Received in revised form May 008 Acceted 11 June 008 Available online 1 June 008 Keywords: Interleaved DF modulated filter bank Oversamled Nearly erfect reconstruction (NPR) Prototye filters abstract his aer resents a new class of comlex modulated filter banks, namely, interleaved discrete Fourier transform (DF) modulated filter banks, where analysis filter bank comrises interleaved exonentially modulated versions of two lowass analysis rototye filters and synthesis filter bank is generated from two lowass synthesis rototye filters in the same manner As comared with DF modulated filter banks, interleaved DF modulated filter banks are more flexible in structure and rovide more degrees of freedom for design It is shown that M-channel critically samled interleaved DF modulated filter banks can be erfect reconstruction (PR) and FIR However, critical samling limits stoband attenuation of the filter banks Further, we investigate the structure of oversamled interleaved DF modulated filter banks and a bi-iterative quadratic rogramming algorithm is develoed for designing oversamled nearly erfect reconstruction (NPR) interleaved DF modulated filter banks Owing to more degrees of freedom for design, oversamled interleaved DF modulated filter banks achieve better total erformance than oversamled DF modulated filter banks do & 008 Elsevier BV All rights reserved 1 Introduction Multirate filter banks lay an imortant role in many signal rocessing areas such as data comression, feature extraction, adative filtering, etc [1,] Generally, signals to be rocessed are first decomosed into lower-rate subband signals by analysis filter bank, and then the subband signals are rocessed, and finally the desired signals are recovered from the rocessed subband signals by the synthesis filter bank Downsamling incurs aliasing and aliasing cancellation deends on the structure of filter banks and aroriate synthesis filter banks If the signals recovered from the non-rocessed subband signals always equal to the inut signals excet a scale factor and a delay, then the filter bank is referred to as erfect reconstruction Corresonding author Fax: , addresses: xlwg@yahoocn (-L Wang), englangshui@yahoocomcn (P-L Shui) (PR) one [1] Sometimes, the PR roerty is relaxed to nearly erfect reconstruction (NPR) in order to comromise other merits of filter banks Modulated filter banks were widely investigated owing to their simle structure and easy design Modulated filter banks contain two tyical classes: discrete Fourier transform (DF) modulated filter banks and cosine modulated filter banks (CMFBs) CMFBs can be designed to satisfy many merits, for instance, the PR roerty, finite imulse resonse (FIR), linear-hase roerty and critically samling [ ] In certain sense, these merits are owing to the double-sided sectrum structure of subband filters helful for aliasing cancellation Unfortunately, the doublesided sectrum structure imlies that CMFBs cannot artition ositive and negative frequency comonents into different subbands Consequently, CMFBs are unsuited for some alications where ositive and negative frequency comonents imly different hysical meaning and require to be rocessed searately In DF modulated filter banks, all the subband filters excet a lowass filter 01-18/$ - see front matter & 008 Elsevier BV All rights reserved doi:10101/jsigro

2 -L Wang, P-L Shui / Signal Processing 88 (008) and a ossible highass filter are of single-sided sectrum Hence, DF modulated filter banks naturally slit ositive and negative frequency comonents into different subbands for searate rocess Moreover, a otential merit of DF modulated filter banks is to generate -D searable DF modulated filter banks that can efficiently extract the directional features in -D images At resent, it is an active area to develo -D filter banks with directional selectivity [,7] One imortant aroach is the tensor roducts of the -band and M-band dual-tree comlex wavelets that can artition ositive and negative frequency comonents into different subbands [8 11] As comared with the dual-tree comlex wavelets, 1-D DF modulated filter banks is a more direct aroach to construct -D searable filter banks with directional selectivity It is well-known that critically samled M-channel DF filter bank with FIR analysis and synthesis filters and the PR roerty must be generated from a rototye filter of length M [1,11], which corresonds to a simle block transform Even though IIR rototye filters are allowed, critically samled DF filter banks of PR suffer from oor stoband attenuation yet [1], which is due to the fact that the single-sided sectrum of the subband filters incur a conflict between stoband attenuation and aliasing cancellation One aroach to avoid the conflict is to design oversamled DF modulated filter banks Oversamled NPR generalized DF filter banks of a single rototye filter were designed by the semi-definite rogramming (SDP) [1], and a single rototye filter indicates that the analysis and synthesis filters are generated from the same lowass filter In [1], the window-based method was develoed for designing rototye filters, which results in a small-scale otimization at the cost of little erformance degradation In [1],a bi-iterative algorithm was roosed to design doublerototye oversamled DF filter banks by iteratively otimize the analysis and synthesis rototye filters Each ste of the bi-iterative algorithm is to solve a quadratic rogramming with linear inequality constraints Another aroach to avoid the conflict between stoband attenuation and aliasing cancellation in critically samled DF modulated filter banks is to emloy the M-channel modified DF modulated filter banks (MDF- FBs) [1] and the modified exonentially modulated filter banks (EMFBs) [17], which can reserve the critically samled roerty and achieve satisfactory stoband attenuation In this aer, we roose a new structure of DF modulated filter banks, namely, the interleaved DF modulated filter banks In the new structure, the analysis filters are obtained by interlaced frequency shifting two lowass analysis rototye filters while the synthesis filters are generated from two lowass synthesis rototye filters in the same manner We rove that the critically samled interleaved DF filter banks can be FIR and PR when the channel number is an even integer Like critically samled DF modulated filter banks, interleaved DF modulated filter banks also suffer from limited stoband attenuation when FIR and PR condition are satisfied Further, oversamled NPR interleaved DF modulated filter banks are considered he bi-iterative algorithm similar to that in [1] is develoed to iteratively otimize two analysis rototye filters and two synthesis rototye filters he new structure rovides more degrees of freedom for design as comared with the double-rototye oversamled DF modulated filter banks in [1] hus, the oversamled interleaved DF modulated filter banks can achieve better total erformance his aer is organized as follows Section resents the structure of interleaved DF modulated filter banks Section gives the sufficient and necessary condition of the erfect reconstruction of M-channel critically samled interleaved DF modulated filter banks Section gives the bi-iterative quadratic rogramming algorithm to design oversamled NPR filter banks Section reorts the numerical examles and the erformance contrast Finally, we conclude our aer Structure of interleaved DF modulated filter banks In an M-channel DF modulated filter bank, the analysis filters and synthesis filters are generated from one or two lowass rototye filters by comlex exonential modulation [1,1 1,18] he analysis and synthesis filters are formulized as follows: h m ðnþ ¼ðnÞe jmn=m, g m ðnþ ¼qðnÞe jmn=m ; m ¼ 0; 1; ; M 1, (1) where (n) and q(n) are analysis and synthesis rototye filters When q(n) ¼ ( n), all the filters are generated from a single rototye and the filter bank is referred to as single-rototye DF modulated filter bank [1,1,1,18] Otherwise, it is referred to as the doublerototye DF modulated filter bank [1] Generally, double-rototye DF modulated filter banks can achieve better overall erformance than single-rototye ones, owing to more degrees of freedom for design he generalized DF (GDF) modulated filter banks introduce frequency-shift and hase-shift constants to increase flexibility in structure [1] he analysis and synthesis filters are h m ðnþ ¼ðnÞe jðmþm 0Þðnþn 0 Þ=M, g m ðnþ ¼ð nþe jðmþm 0Þðn n 0 Þ=M ; m ¼ 0; 1; ; M 1, () where m 0 and n 0 are frequency-shift and hase-shift constants, resectively It is easily observed that all the analysis filters and synthesis filters in DF modulated and GDF modulated filter banks have a same magnitude resonse excet frequency shifts his limits aliasing cancellation In this aer, we roose a new structure of DF modulation As shown in Fig 1, the analysis filter banks are generated from two lowass rototye filters 0 (n) and 1 (n) by the interleaved DF modulation, h k ðnþ ¼ 0 ðnþe jkn=m ; k ¼ 0; 1; ; ½ðM 1Þ=Š, h kþ1 ðnþ ¼ 1 ðnþe jðkþ1þn=m ; k ¼ 0; 1; ; ½M=Š 1, ()

3 880 -L Wang, P-L Shui / Signal Processing 88 (008) where [x] denotes the largest integer no more than x In the similar manner, the synthesis filters are generated from two lowass rototye filters q 0 (n) and q 1 (n), g k ðnþ ¼q 0 ðnþe jkn=m ; k ¼ 0; 1; ; ½ðM 1Þ=Š, g kþ1 ðnþ ¼q 1 ðnþe jðkþ1þn=m ; k ¼ 0; 1; ; ½M=Š 1 () An interleaved DF modulated filter bank is called critically samled when K ¼ M and oversamled when KoM he new structure allows the subband filters to have different shaed magnitude resonses As shown in Fig, the magnitude resonses in even-indexed channels have a same shae excet frequency shifts and so do the magnitude resonses in odd-indexed channels he new modulation structure rovides more degrees of freedom for design More degrees of freedom bring two merits: critically samled filter banks can be PR and FIR when M is an even number and oversamled filter banks achieve better overall erformance M-channel PR critically samled interleaved DF modulated filter banks When channel number M is even integer, critically samled interleaved DF filter banks have a secial olyhase structure, which allows us to design critically samled, FIR, and PR interleaved DF filter banks he section will derive the olyhase structure of the critically samled filter banks and their PR condition Let P 0 (z) and P 1 (z) denote two analysis rototye filters and Q 0 (z) and Q 1 (z) denote two synthesis rototye filters he tye-i olyhase reresentations of the analysis rototye filters and the tye-iii reresentations of the Fig 1 Structure of M-channel interleaved DF modulated filter banks synthesis rototye filters are given as follows: P 0 ðzþ ¼ z l P 0;l ðz M Þ; P 1 ðzþ ¼ z l P 1;l ðz M Þ; Q 0 ðzþ ¼ z l Q 0;l ðz M Þ; Q 1 ðzþ ¼ z l Q 1;l ðz M Þ () In terms of the modulated structure, the tye-i analysis olyhase reresentation and the tye-iii synthesis olyhase reresentation are M W ðm 1Þl M H 0 ðzþ H 0 ðzþ H ðzþ H 1 ðzþ H M ðzþ H M 1 ðzþ ¼ P H 1 ðzþ H M ðzþ H ðzþ H Mþ1 ðzþ 7 7 H ðzþ H ðzþ P z l P 0;l ðz M Þ P z l W l M P 0;lðz M Þ P z l W ðm 1Þl M P 0;l ðz M Þ ¼ P z l W l M P 1;lðz M Þ P z l W l M W l M P 1;lðz M Þ P 7 z l W l P 1;l ðz M Þ ½ W n M ; Wn M ŠP 0 ðz M Þ ¼ ½W n M Dn M ; Wn M Dn M ŠP 1ðz M Þ 1 z 1 z ðþ 7, () Fig Magnitude resonses of the analysis filters H m (z), m ¼ 0,1,y,M 1

4 -L Wang, P-L Shui / Signal Processing 88 (008) G 0 ðzþ G ðzþ G M ðzþ G 1 ðzþ G ðzþ G ðzþ ¼ P 7 G 0 ðzþ G 1 ðzþ G M 1 ðzþ ¼ G M ðzþ G Mþ1 ðzþ 7 G ðzþ P P P P P z l Q 0;l ðz M Þ z l Q 0;l ðz M ÞW l M P z l Q 0;l ðz M ÞW ðm 1Þl M z l Q 1;l ðz M ÞW l M z l Q 1;l ðz M ÞW l M Wl M z l Q 1;l ðz M ÞW l M WðM 1Þl M ¼½1; z; ; z ðþ Š " " # " ## W M D M W M Q 0 ðz M Þ ; Q 1 ðz M Þ, (7) W M D M W M where the suerscrit denotes the conjugate and transose, the suerscrit denotes the transose, and P i ðzþ ¼diag½P i;0 ðzþ; P i;1 ðzþ; ; P i; ðzþš; i ¼ 0; 1, Q i ðzþ ¼diag½Q i;0 ðzþ; Q i;1 ðzþ; ; Q i; ðzþš; i ¼ 0; 1, D M ¼ diag½1; W 1 M ; W M ; ; WðM 1Þ M Š, W M ¼½W kl M Š k;;1;;m 1, 8 j ¼ i; i ¼ 0; 1; ; M 1 >< P M ¼½PŠ i;j ¼ 1; and j ¼ ði MÞþ1; i ¼ M; M þ 1; ; M 1 >: 0; else: P i (z) and Q i (z), i ¼ 0,1 are two airs of diagonal matrices consisting of the olyhase comonents of the analysis and synthesis rototye filters, D M is an M M diagonal matrix, W M is the M-oint DF matrix, and P is a M M ermutation matrix Consequently, the analysis and synthesis olyhase matrices can be written in block matrix form as 7 ½W n HðzÞ ¼P M ; Wn M ŠP 0ðzÞ, ½W n M Dn M ; Wn M Dn M ŠP 1ðzÞ " " # " ## W M D M W M GðzÞ ¼ Q 0 ðzþ ; Q 1 ðzþ P (8) W M D M W M he above exressions show that signal decomosition and reconstruction can be fast imlemented by the flowdiagram in Fig In Fig, the analysis art requires two M-oint inverse DF transforms and the synthesis art requires two M-oint DF transforms Interleaved DF modulated filter banks are of the same comutational efficiency as the DF-FBs, the MDF-FBs [1], and the EMFBs [17] In terms of (8), the transfer matrix of the multirate system in Fig is " " # W M " D M W M ## ðzþ ¼GðzÞHðzÞ ¼ Q 0 ðzþ ; Q 1 ðzþ W M D M W M ½W n PP M ; Wn M ŠP 0ðzÞ ½W n M Dn M ; Wn M Dn M ŠP 1ðzÞ " # " # I M I M I M I M ¼ MQ 0 ðzþ I M I M P 0 ðzþþmq 1 ðzþ I M I M P 1 ðzþ (9) It is known that the filter bank is PR only when the transfer matrix satisfies:ðzþ ¼z n d IM (n d is an integer) [1] From (9), we easily rove the following theorem: heorem M-channel critically samled interleaved DF modulated filter bank is PR if and only if there exist an integer n d such that the olyhase comonents of the analysis and synthesis rototye filters satisfy " P 0;l ðzþ P 1;l ðzþ #" Q 0;l ðzþ Q 0;lþM ðzþ # P 0;lþM ðzþ P 1;lþM ðzþ Q 1;l ðzþ Q 1;lþM ðzþ ¼ z n d M I ; l ¼ 0; 1; ; M 1 (10) Moreover, when the analysis and synthesis rototye filters are causal and suort in {0,1,yML 1} and {1,,y,ML}, resectively, n d must be in the set {1,,y,L 1} and the delay of the filter bank is Mn d In terms of the transformed matrix in (9), the roof of the condition (10) is straightforward In what follows, we consider the delay of the filter banks when the four rototye filters are causal and of a same order aking the determinants of the matrices in (10), we get ½P 0;l ðzþp 1;lþM ðzþþp 0;lþM ðzþp 1;l ðzþš½q 0;l ðzþq 1;lþM ðzþ þ Q 0;lþM ðzþq 1;l ðzþš ¼ z n d M (11) Since the left-hand side of (11) is the roduct of two olynomials of z and the right-hand side is a monomial, the solutions of (11) have the form as P 0;l ðzþp 1;lþM ðzþþp 0;lþM ðzþp 1;l ðzþ ¼c l z ðn dþr l Þ =M, Q 0;l ðzþq 1;lþM ðzþþq 0;lþM ðzþq 1;l ðzþ ¼z ðn d r l Þ =ðc l MÞ, (1) where c l is nonzero constants and r l is integers Noted that c l and r l may be different for each l For simlicity, all the nonzero constants c l are taken as one As a result, the

5 88 -L Wang, P-L Shui / Signal Processing 88 (008) Fig he olyhase imlementation of M-channel interleaved DF modulated filter banks analysis olyhase comonents determine the synthesis ones in terms of " # " # Q 0;l ðzþ Q 0;lþM ðzþ ¼ z n d P 0;l ðzþ P 1;l ðzþ 1 Q 1;l ðzþ Q 1;lþM ðzþ M P 0;lþM ðzþ P 1;lþM ðzþ " # P 1;lþM ðzþ P 1;l ðzþ ¼ z r l (1) P 0;lþM ðzþ P 0;l ðzþ herefore, a M-channel FIR PR interleaved DF modulated filter bank can be reresented by Q 0 ðzþ ¼ M 1 z Mr l ½z l P 1;lþM ðz M Þþz Mþl P 1;l ðz M ÞŠ, Q 1 ðzþ ¼ M 1 z Mr l ½z l P 0;lþM ðz M Þþz ðmþlþ P 0;l ðz M ÞŠ, P 0;l ðzþp 1;lþM ðzþþp 0;lþM ðzþp 1;l ðzþ ¼z ðn dþr l Þ =M, l ¼ 0; 1; ; M 1 (1) Set the analysis rototye filters have the suort {0,1,y,ML 1} hen, each analysis olyhase comonent is a causal filter suorted in {0,1,y,L 1} In order to assure the synthesis rototye filters are causal, all the integers r l must be negative integers Moreover, when all the integers are a same negative integer, the two synthesis rototye filters have the minimal order ML 1 Without loss of generality, we assume that r l ¼ 1 In this case, the two synthesis rototye filters have the suort {1,,y,ML} Further, according to the third equation in (1), the left-hand side is a olynomial suorted in {0,1,y,L } and r l ¼ 1, and then n d must be an integer between 1 and L 1 When n d is determined, the delay of the filter bank is Mn d Let the analysis rototye filters 0 (n), 1 (n), n ¼ 0,1y,ML 1 be causal and of length ML, which are exressed by two column vectors: 0 ¼½ 0 ð0þ; 0 ð1þ; ; 0 ðml Þ; 0 ðml 1ÞŠ, 1 ¼½ 1 ð0þ; 1 ð1þ; ; 1 ðml Þ; 1 ðml 1ÞŠ aking r l ¼ 1 in (1), the two synthesis filters can be rewritten as " Q 0 ðzþ ¼z 1 M 1 z l P 1;M l 1 ðz M Þþ # z l P 1;M l 1 ðz M Þ, l¼m " Q 1 ðzþ ¼z 1 M 1 z l P 1;M l 1 ðz M Þþ # z l P 1;M l 1 ðz M Þ l¼m (1)

6 -L Wang, P-L Shui / Signal Processing 88 (008) In time domain, the two synthesis rototye filters are exressed as two column vectors: q 0 ¼½q 0 ð1þ; q 0 ðþ; ; q 0 ðmlþš, q 1 ¼½q 1 ð1þ; q 1 ðþ; ; q 1 ðmlþš It is easily roved that q 0 ¼ C 1 ; q 1 ¼ C 0, (1) where C is a ML ML ermutation matrix with the form as follows: J M J M C ; J 7 M ¼ c J M where J M is an M M reversal (or anti-diagonal) matrix As a result, the two synthesis rototye filters are locally reversed versions of the two analysis rototye filters Numerical results show that critically samled PR interleaved DF modulated filter banks suffer from oor stoband attenuation Oversamled NPR interleaved DF modulated filter banks Limited stoband attenuation of critically samled PR interleaved DF modulated filter banks origins from the conflict between the stoband attenuation and aliasing cancellation, like in critically samled PR DF modulated filter banks [1,18] Oversamling subband signals can avoid this conflict In this section, we deal with the design of oversamled NPR interleaved DF modulated filter banks here have existed several methods to design oversamled NPR DF modulated filter banks, including the SDP [1], the window-based algorithm [1], and the bi-iterative algorithm [1] Following the bi-iterative algorithm in [1], we develo the bi-iterative quadratic rogramming algorithm to design oversamled NPR interleaved DF modulated filter banks 1 Structure of oversamled interleaved DF modulated filter banks For the interleaved DF modulated filter bank in Fig 1, the inut outut relationshi is formulized as YðzÞ ¼ 1 K M 1 þ 1 K H m ðzþg m ðzþðzþ K 1 M 1 ðzþðzþþ K 1 H m ðzw k K ÞG mðzþðzw k K Þ A k ðzþðzw k KÞ, (17) where (z) is the transfer function of the filter bank and A k (o) is the aliasing transfer function for aliasing comonent (zw K k ) In order to achieve NPR, the transfer function of the filter bank is desired to aroximate a ure delay and all the aliasing transfer functions are desired to be as small as ossible in [,] Without loss of generality, the four rototye filters are assumed to have a same length ML and are denoted by the column vectors, 0 ¼½ 0 ð0þ; 0 ð1þ; ; 0 ðml 1ÞŠ, 1 ¼½ 1 ð0þ; 1 ð1þ; 1 ðml 1ÞŠ, q 0 ¼½q 0 ð1þ; q 0 ðþ; ; q 0 ðmlþš, q 1 ¼½q 1 ð1þ; q 1 ðþ; ; q 1 ðmlþš (18) For convenience in exression, we introduce a vectorvalued function ðoþ ½1; e jo ; ; e jðml 1Þo Š ; o ½ ; Š hen, the frequency resonses of the rototye filters are exressed in vector form by P 0 ðoþ ¼ 0 ðoþ; P 1ðoÞ ¼ 1 ðoþ, Q 0 ðoþ ¼e jo q 0 ðoþ; Q 1ðoÞ ¼e jo q 1ðoÞ (19) he stoband of the rototye filters is secified as the set ½ ; =KŠ[½=K; Š he stoband energy of the rototye filters is formulized as E sb ð i Þ¼ 1 ¼ i E sb ðq i Þ¼ 1 Z =K =K Z =K 1 =K i ðk; LÞ i Z =K =K jp i ðoþj do ðoþ n ðoþdo jq i ðoþj do ¼ q i ðk; LÞq i; i ¼ 0; 1, (0) where (K,L) is a symmetric oelitz matrix ML ML whose first row is a real vector ½tð0Þ; tð1þ; ; tðml 1ÞŠ tðkþ ¼ 1 Z =K =K 8 < 1 1=K; k ¼ 0; cos ko do ¼ : sinðk=kþ ; ka0: k! i (1) In some alications, for examle, subband adative filtering, the analysis filters are also desired to have as small inband aliasing as ossible [1,1] he inband aliasing of a filter H(o) with decimation factor K is measured by [1] E inb_alia ðhþ ¼ 1 Z K 1 jhððo kþ=kþj do: () K For the oversamled interleaved DF modulated filter banks with decimation factor K, the inband aliasing of the

7 88 -L Wang, P-L Shui / Signal Processing 88 (008) analysis filters is measured by E inb_alia ð i Þ¼ 1 Z K 1 K where W ¼ 1 Z K ¼ 1 i K Z K 1 jp i ððo kþ=kþj do o k K o k K do i W i; i ¼ 0; 1, () K 1 o k K i o k do K As in Ref [1], the outut aliasing distortion at each frequency can be controlled by the residual aliasing function AðoÞ ¼ 1 K K 1 ¼ 1 K K 1 M 1 ½ðM 1Þ=Š jh m ðo k=kþg m ðoþj jp 0 ðo k=k m=mþq 0 ðo m=mþj þ 1 K K 1 ½M=Š 1 jp 1 ðo k=k ðm þ 1Þ=MÞQ 1 ðo ðm þ 1Þ=MÞj, () where [x] denotes the largest integer no more than a real number x he overall residual aliasing is measured by the integral of (), E alia ð 0 ; 1 ; q 0 ; q 1 Þ¼ 1 Z AðoÞdo ¼ M þ 1 Z! 1 K 1 K jp 0 ðo k=kþq 0 ðoþj do þ M 1 K Z K 1! jp 1 ðo k=kþq 1 ðoþj do () Deriving Eq (), we utilize the fact that the functions jp 0 ðo k=k m=mþq 0 ðo m=mþj; jp 1 ðo k=k m=mþq 1 ðo m=mþj have the eriod Further, utilizing Eq (19), Eq () is rewritten in two forms as E alia ð 0 ; 1 ; q 0 ; q 1 Þ¼ 0 U 0ðq 0 Þ 0 þ 1 U 1ðq 1 Þ 1 ¼ q 0 K 0ð 0 Þq 0 þ q 1 K 1ð 1 Þq 1, () where U 0 ðq 0 Þ 1 Z M þ 1 K 1 jq K 0 ðoþj ðo k=kþ ðo k=kþdo, U 1 ðq 1 Þ 1 Z M K K 1 jq 1 ðoþj ðo k=kþ ðo k=kþdo, K 0 ð 0 Þ 1 Z M þ 1 K 1 jp K 0 ðo k=kþj ðoþ ðoþdo, K 1 ð 1 Þ 1 Z M K 1 jp K 1 ðo k=kþj ðoþ ðoþdo, (7) where the symbol [x] denotes the largest integer no more than a real number x From (), the overall residual aliasing E alia ( 0, 1, q 0, q 1 ) is a bi-quadratic with resect to the vector air { 0, 1 } and vector air {q 0, q 1 }, strictly seaking, it is a quadratic function on the other vector air when one of the two vector airs is fixed Next, we deal with the transfer function of the filter bank In the frequency domain, it is reresented by ðoþ ¼ 1 K where ¼ 1 K M 1 ½ðM 1Þ=Š þ 1 K H m ðoþg m ðoþ ½M=Š 1 P 0 ðo m=mþq 0 ðo m=mþ P 1 ðo ðm þ 1Þ=MÞQ 1 ðo ðmþ1þ=mþ ¼ 0 Y 0ðoÞq 0 þ 1 Y 1ðoÞq 1, (8) ½ðM 1Þ=Š Y 0 ðoþ ¼ 1 e jðo m=mþ ðo m=mþ K ðo m=mþ, ½M=Š 1 Y 1 ðoþ ¼ 1 e jðo ðmþ1þ=mþ ðo ðm þ 1Þ=MÞ K ðo ðm þ 1Þ=MÞ (9) In terms of (8), the transfer function is a bi-linear function with resect to the vector air { 0, 1 } and vector air {q 0, q 1 }, that is, it is a linear function on the other vector air when one of the two vector airs is fixed Both the bi-quadratic structure () and the bi-linear structure (8) suort an efficient bi-iterative quadratic rogramming algorithm to design oversamled NPR interleaved DF modulated filter banks Bi-iterative quadratic rogramming algorithm Based on the quadratic measures in (0) and () and the bi-quadratic measure in () on the stoband energy, inband aliasing, and overall residual aliasing, as well as the bi-linear transfer function in (8), we emloy the biiterative scheme to iteratively otimize the rototye filter airs { 0, 1 } and {q 0, q 1 } When the synthesis rototye filters {q 0, q 1 } are fixed, we otimize the

8 -L Wang, P-L Shui / Signal Processing 88 (008) analysis rototye filters 0 and 1 by minimizing the weight sum of their stoband energy, inband aliasing, and the overall residual aliasing subject to that the real and imagery arts of the transfer function distortion are in a redefined tolerance range he corresonding otimization is formulized by 8 0 ðk; LÞ 0 þ 1 ðk; LÞ 9 1 >< >= min það U 0 0ðq 0 Þ 0 þ U 1 1ðq 1 Þ 1 Þ 0 ; 1 >: þbð 0 W 0 þ 1 W >; 1Þ s:t: d= 0 ReðY 0 ðoþþq 0 þ 1 ReðY 1ðoÞÞq 1 cosðmn d oþd=, d= 0 ImðY 0 ðoþþq 0 þ 1 ImðY 1ðoÞÞq 1 þ sinðmn d oþd=, o ½ ; Š (0) where d is a redefined small ositive number to control the transfer function distortion and the ositive weights a and b comromise the stoband energy, inband aliasing, and overall residual aliasing his is a tyical semi-infinite rogramming with a quadratic objective and linear inequality constraints [19] he linear semi-infinite constraints assure the transfer function of the filter bank to satisfy jðoþ e jmndo jd; o ½ ; Š (1) he objective function in (0) is a ositive-defined quadratic form on 0 and 1 Moreover, when the frequency o is discretized, the otimization (0) reduces to a quadratic rogramming whose global otimal solution can be achieved When the analysis rototye filters 0 and 1 are fixed, the synthesis rototye filters q 0 and q 1 are otimized by 8 q 0 ðk; LÞq 0 þ q 1 ðk; LÞq 9 1 >< >= min þaðq K 0 0ð 0 Þq 0 þ q L 1 1ð 1 Þq 1 Þ q 0 ;q 1 >: þbð 0 W 0 þ 1 W >; 1Þ s:t: d= 0 ReðY 0 ðoþþq 0 þ 1 ReðY 1ðoÞÞq 1 cosðmn d oþd=, d= 0 ImðY 0 ðoþþq 0 þ 1 ImðY 1ðoÞÞq 1 þ sinðmn d oþd=, o ½ ; Š () Similarly, the otimization () is a semi-infinite quadratic rogramming and can be reduced to a quadratic rogramming by discretizing frequency Starting from an initial synthesis rototye filter air, we iteratively otimize the synthesis and analysis rototye filters and the corresonding algorithm is referred to as the bi-iterative quadratic rogramming algorithm hough each of the otimizations (0) and () can achieve their global otimal solutions, the otimal solutions deend on the fixed filter airs that determine the matrices U 0 and U 1 in (0) or matrices K 0 and K 1 in () and linear inequalities as constraints Generally, the iteration rocess always converges to the four rototye filters with good roerties However, the final solution of the iteration is generally not the otimal solution of the joint otimization of (0) and () Note that the joint otimization of (0) and () is too comlicated to efficiently solve it, due to the fourth-order objective function and quadratic inequality constraints Particularly, the final solution deends on the initial synthesis filter air to start the iterative rocess It is ivotal to select aroriate initial synthesis filters Here, when the delay of the filter bank equals to the length ML of the rototye filters, the initial synthesis rototye filters consist of a linear-hase equirile lowass filter designed by the Parks-McClellan algorithm [0] (see the file firmm in Matlab70) and the rototye filter by the algorithm in [1] (from the web site: htt://wwwecemamasterca/ davidson) he bi-iterative quadratic rogramming algorithm is summarized as follows: (1) Initialization: For given the channel number M and the decimation factor K, the initial synthesis rototye filter q (0) 0 is calculated by the Parks-McClellan algorithm, where the stoband cutoff frequency /K (0) and the assband cutoff frequency /M he other q 1 is obtained from the algorithm in [1] () Otimize analysis rototye filters: Assume the synthesis rototye filters q 0 and q 1 have been (k 1) (k 1) obtained Relace q 0 and q 1 in (0) by q (k 1) 0 and (k 1) q 1 and solve the quadratic rogramming he obtained otimal filters are (k) 0 and (k) 1 () Otimize analysis rototye filters: Relace 0 and 1 (k) (k) in () by 0 and 1 and solve the quadratic (k) rogramming he obtained otimal filters are q 0 and q (k) 1 () Iteration terminate condition: For obtained rototye filters (k) 0, (k) 1,q (k) 0, and q (k) 1, calculate the total erformance measure of the filter bank () c k ¼ P i ¼ 0,1[E sb ( (k) i )+E sb (q (k) i )]+ae alia ( (k) 0, (k) 1,q (k) (k) 0 q 1 +b P i ¼ 0,1E inb_alia ( (k) i ) () If c k c k 1 Z, where Z is a redefined small ositive number, the iteration rocess terminates and (k) 0, (k) 1,q (k) 0,and q (k) 1 are taken as the final rototye filters Otherwise, set k+1-k and go to the ste () for the next iteration When the delay of the filter bank is Mn d not equal to the length ML of the rototye filters, we first design a filter bank with a delay Mn d and rototye filters of length Mn d by the above bi-iterative algorithm When Ln d, the synthesis rototye filters are lengthened to the filters of length ML as the initial synthesis rototye filters to start the iteration rocess by aending zeros in the right-hand side When Lon d, the synthesis rototye filters are truncated for the filters of length ML as the initial synthesis rototye filters to start the iteration rocess by removing the M(n d L) coefficients in the right-hand side Numerical examles In this section, we give a set of numerical examles whose delays equal to the lengths of the rototye filters and a numerical examle of low delay to testify the

9 88 -L Wang, P-L Shui / Signal Processing 88 (008) Fig Frequency resonses of the four rototye filters with M ¼ 1, K ¼ 8, N ¼,the delay and the transfer function distortion level d ¼ 001 effectiveness of the interleaved DF modulation and the roosed algorithm he filter banks are evaluated by the four erformance measures as follows: (1) Inband aliasing of analysis rototye filters, 10 log 10 {E inb_alia ( i )}(db); () the overall residual aliasing, 10 log 10 {E alia ( 0, 1,q 0,q 1 )}(db); () stoband level of the rototye filters, 0 log 10 fmax =Ko ðjhðoþj=jhð0þjþgðdbþ; () the transfer function distortion level d In the first examle, the channel number M ¼ 1, the decimation factor K ¼ 8, the length of the rototye filters N ¼, and the delay of the filter bank is Mn d ¼ N ¼ he weights a ¼ K,b ¼ M in the objective function, the distortion level of the transfer function is d ¼ 001, and the frequency samling interval is Do ¼ (100 M) in the frequency discretization he frequency resonses of the four rototye filters are shown in Fig For comarison in erformance, we demonstrate the frequency resonses of the single-rototye DF modulated filter bank in [1] and the double-rototye DF modulated filter bank in [1] with the same channel number, decimation factor, the length of the rototye filters, and the same transfer function distortion level in Figs and, resectively he erformance measures of the three filter banks are listed in able 1 with the bold fonts As comared with the single-rototye and double-rototye DF modulated filter banks [1,1], the interleaved DF modulated filter Fig Frequency resonse of the single rototye filter in [1] with M ¼ 1, K ¼ 8, N ¼, the delay, and the transfer function distortion level 001 banks achieve better erformance measures, including stoband level, inband aliasing, and overall residual aliasing hese imrovements are owing to the fact that interleaved DF modulation rovides more degrees of freedom for design when the length of the rototye filters is fixed

10 -L Wang, P-L Shui / Signal Processing 88 (008) Fig Frequency resonses of the analysis and synthesis rototye filters in [1] with M ¼ 1, K ¼ 8, N ¼ the delay, and the transfer function distortion level 001 able 1 Comarison of inband aliasing, residual aliasing, and stoband level of oversamled interleaved DF modulated filter banks, single-rototye DF modulated filter banks [1] and double-rototye DF modulated filter banks [1], where the transfer function distortion level d ¼ 001 and the weights a ¼ K, b ¼ M, and the delays of the filter banks are ML equal to the lengths of the rototye filters in the roosed algorithm yes Subband number M Decimation factor K Filter length N Inband aliasing (db) Residual aliasing (db) Stoband level (db) Wilbur [1] 1 8 M Dam [1] 1 8 M 1 his aer 1 8 M Wilbur [1] 1 8 M Dam [1] 1 8 M his aer 1 8 M Wilbur [1] 1 M Dam [1] 1 M 1 17 his aer 1 M Wilbur [1] 1 M Dam [1] 1 M his aer 1 M Wilbur [1] M 99 8 Dam [1] M 7 0 his aer M In able 1, we give more numerical examles for comarison It can be seen that in most cases the oversamled interleaved DF modulated filter banks achieve better overall erformance measure Additionally, we also give an examle with less redundancy and short length where M ¼ 1, K ¼ 8, the length of the filters N ¼ M, and the delay of the filter bank is N ¼ he frequency resonses of the four rototye filters are shown in Fig 7 he inband aliasing of the two analysis rototyes are 011 and 0 db, the overall residual aliasing is 09 db, and the stoband attenuations of the four rototye filters are 7, 7, 71, and

11 888 -L Wang, P-L Shui / Signal Processing 88 (008) Fig 7 Frequency resonses of the rototye filters of oversamled interleaved DF modulated filter bank with M ¼ 1, K ¼ 8, N ¼ M, the transfer function distortion level 001, and the delay of the filter bank is 71 db, resectively In the same case, the inband aliasing of the single-rototye filter bank [1] is 01 db, the residual aliasing is 07 db, and the stoband level is 0 db, as shown in Fig 8 his examle shows that oversamled interleaved DF filter bank rovide more imrovements in erformance measures when the filter banks with low redundancy and rototye filters of short length are designed Finally, we give an examle of low delay, where M ¼ 1, K ¼ 8, N ¼, the delay of the filter bank is 8, and the transfer function distortion level 001 he frequency resonses of the four rototye filters are shown in Fig 9 he erformance measures are as follows: Inband aliasing 789 and 70 db, the overall residual aliasing db, and the stoband levels 0, 898, 10, and db As comared with the filter bank in Fig, we find that the low delay incurs obvious degradation in overall residual aliasing and the stoband levels Conclusions Fig 8 Frequency resonse of the rototye filter of oversamled singlerototye DF modulated filter bank with M ¼ 1, K ¼ 8, N ¼ M, and the distortion level of the transfer function 001 [1] his aer resents the structure and fast decomosition and reconstruction of interleaved DF modulated filter banks and show that the interleaved DF modulated filter banks can be critically samled FIR and erfect reconstruction, when the channel number is an even integer Moreover, the bi-iterative quadratic rogramming algorithm is roosed to design oversamled interleaved DF modulated filter banks he numerical examles show that oversamled interleaved DF modulated filters can achieve better overall erformance than the singlerototye and double-rototye DF modulated filter banks with rototye filters of the same length do

12 -L Wang, P-L Shui / Signal Processing 88 (008) Fig 9 Frequency resonses of the four rototye filters with M ¼ 1, K ¼ 8, N ¼, the delay of the filter bank is 8, the transfer function distortion level 001 Acknowledgment he authors would like to thank the anonymous reviewers and the associate editor for the insightful comments and suggestions References [1] PP Vaidyanathan, Multirate Systems and Filter Banks, Prentice- Hall, Englewood Cliffs, NJ, 199 [] JJ Shynk, Frequency-domain and multirate adative filtering, IEEE Signal Process Mag 9 (1) (199) 1 7 [] Q Nguyen, RD Koilillai, he theory and design of arbitrarylength cosine-modulated filter banks and wavelets, satisfying erfect reconstruction, IEEE rans Signal Process () (199) 7 8 [] JS Mao, SC Chan, KL Ho, heory and design of a class of M- channel IIR cosine-modulated filter banks, IEEE Signal Process Lett 7 () (000) 8 0 [] F Argenti, E Del Re, Design of biorthogonal M-channel cosinemodulated FIR/IIR filter banks, IEEE rans Signal Process 8 () (000) [] FCA Fernandes, RLC van Saendonck, CS Burrus, Multidimensional, maing-based comlex wavelet transforms, IEEE rans Image Process 1 (1) (00) [7] IW Selesnick, RG Baraniuk, NG Kingsbury, he dual-tree comlex wavelet transform, IEEE Signal Process Mag () (00) 1 11 [8] IW Selenick, Hilbert transform airs of wavelet bases, IEEE Signal Process Lett 8 () (001) [9] IW Selenick, he design of aroximate Hilbert transform airs of wavelet bases, IEEE rans Signal Process 0 () (00) [10] IW Selenick, he double-density dual-tree discrete wavelet transform, IEEE rans Signal Process () (00) [11] C Chaux, L Duval, J-C Pesquet, Image analysis using a dual-tree M- band wavelet transform, IEEE rans Image Process 1 (8) (00) 97 1 [1] AK Djedid, Design of stable, causal, erfect reconstruction, IIR uniform DF filter banks, IEEE rans Signal Process 8 () (000) [1] Matthew R Wilbur, imothy N Davidson, James P Reilly, Efficient design of oversamled NPR GDF filterbanks, IEEE rans Signal Process (7) (00) [1] K-F-C Yiu, N Grbic, S Nordholm, K-L eo, Multicriteria design of oversamled uniform DF filter banks, IEEE Signal Process Lett 11 () (00) 1 [1] HH Dam, S Nordholm, A Cantoni, JM de Haan, Iterative method for the design of DF filter bank, IEEE rans Circuits Systems II 1 (11) (00) 81 8 [1] Kar, NJ Fliege, Modified DF filter banks with erfect reconstruction, IEEE rans Circuits Systems II (11) (1999) [17] A Viholainen, M Renfors, Alternative subband signal structures for comlex modulated filter banks with erfect reconstruction, in: ISCAS 00, vol III, 00, 8 [18] PN Heller, Kar, Q Nguyen, A general formulation of modulated filterbanks, IEEE rans Signal Process 7 () (1999) [19] R Hettich, KO Kortanek, Semi-infinite rogramming: theory, methods, and alications, SIAM Rev () (199) 80 9 [0] AV Oenheim, RW Schafer, Discrete-ime Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989,