On the Theory and Design of a Class of Recombination Nonuniform Filter Banks with Low-Delay FIR and Ilk Filters

This paper studies the theory and design of a attenuation than their FIR counterparts. In [3], a class of LP class of recombination nonuniform FBs (RNFB) with low-delay RNFBs with cosine-rolloff transition band was designed using the (LD) FIR and IIR filters. The conditions for suppressing the REMEZ algorithm. It has been known that FBs designed with spurious response and achieving a good frequency cosine-rolloff transition band have reasonably good reconstruction characteristic for these LD FIR/IIR RNFBs are developed. errors and the approximate power complementary [3,8]. In this The proposed LD FIR RNFBs have a lower system delay than paper, the approach in [3] is first extended to RNFBs with FIR their linear-phase counterparts, at the expense of slight filters having lower system delay than their LP RNFB counterparts. increase in phase distortion of the analysis filters and They are then model reduced to an IIR RNFB by using modified arithmetic complexity. By model reducing the LD FIR model reduction technique proposed in [9]. The proposed LD FIR uniform FBs by the modified model reduction method, an IIR and IIR filters have a similar frequency characteristic as the RNFB with a similar characteristic can be readily obtained. A original LP analysis filters in the passband and the transition band design example is given illustrate the effectiveness of the with the group delays in these regions are slightly relaxed. As long proposed method. as the stopband attenuation of the LD and LP analysis filters are comparable, reasonably good reconstruction error will be achieved. INTRODUCTION One advantage of the proposed method using the modified model Filter banks (FBs) have important applications in speech, audio reduction technique is that the stability of the model-reduced filters image and array processing. The theory and design of uniform is guaranteed and the IIR filters obtained closely approximate the perfectreconstruction (PR) andanearlygperfectreconstruction (NPR) properties of the LD FIR filters. Another advantage of the FBs have been widely studied [1]. In applications such as signal proposed LD and IIR RNFBs is that they have approximately analysisa ncoding swith nonuniform frequency spacing have passband LP analysis filters, and hence a lower phase distortion the potential to offer more flexibility in time-frequency partitioning than their IIR nonuniform CMFBs counterparts [9]. and better performance. This has attracted considerable interests in The paper is organized as follows: The theory of RNFBs is designing nonuniform FBs [2-7]. One useful approach is the recalled in Section I. Section II is devoted to the matching indirect method proposed in [7], where certain channels of a condition and the design ofithe low delay FIR and JIR RNFBs. A uniform FB are merged using the synthesis FB of a recombination design example will be given in Section III to show the efficiency FB or a transmultiplexer (TMUX). The resulting FBs are called of the method. Finally conclusions are drawn in Section IV. recombination nonuniform FBs (RNFBs). One advantage of the 1 RECOMBINATION NONUNIFORm FILTER BANKS RNFB is that the PR property is structurally imposed as long as the original uniform and recombination filter banks are PR. Moreover, Figure 1 shows the structure of the RNFB considered in this if the number of channels of the uniform and recombination paper. Consecutive channels of a uniform M-channel analysis FB TMUX are coprime, the analysis and synthesis filters of the are combined using the synthesis filters of another FB having a resulting RNFB admit an equivalent linear time invariant (LTI) smaller number of channels, say mk as shown in the figure. The representation. In other words, the frequency responses of the analysis (synthesis) filters can be optimized directly, which sampling rate after recombination is reduced by a factor ofm considerably simplifies the design procedure. When the number of For simplicity, only the merging of the mk channels of an Mchannels of the uniform and recombination TMUX are nonchannel analysis bank are shown. Further merging of consecutive coprime, the analysis and synthesis filters are linear periodic time channels can be performed. Let the merged outputs be indexed by varying (LPTV). It was shown that Line Phase (LP) RNFB with good frequency characteristics can still be obtained even in the aegerk-the nub of hane non-coprime case by imposing relatively simple conditions on the merged at the k-th output. For critical sampling, we have FBs. A class ofRNFB based on the cosine modulation filter banks YK1 (Mk IM) = 1 . In the synthesis banks, each merged output will (CMFB) was proposed in [2]. Recently, the theory and design of pass through the analysis filters of the recombination FBs and they RNFBs with LP filters were studied in order to achieve RNFBs will be fed to the synthesis bank of the uniform FB for with good frequency characteristic [3]. The form of the analysis LP reconstruction. Each synthesis-analysis structure, involving anm filters is also analyzed in [3]. khne B scle MX t a bevdi 2 hti h In this paper, the design of a class of RNFB with JIR filters is canlF,i aldaTU.I a bevdi 2 hti h studied. Compared to FIR filters, JIR filters have the potential to TMUXs are PR, then they only introduce a signal delay in the offer lower system delay, sharper cutoff and higher stopband subband signal of the uniform FB. If these delays are properly 0-7803-9390-2/06/$20.00 ©2006 IEEE 895 ISCAS 2006 compensated in other subbands, then the system is PR if the response, passband and stopband of the various components, let's original uniform FB is PR. Further, if mk and M are coprime to examine the difference between the starting locations of the each, then the equivalent analysis filter of the k-th output is LTI. In passband edges of the supports of H/k +(c) and GC (c) . It can be the more general case, the analysis filters are LPTV as mk and M seen that all components in D16 ) constitute the passband of the are non-coprime. The discrete-time Fourier transform (DTFT) of LPTV filter A,(6o) if i= qmf y . However, D2(6) will the k-th merged output can be written as: 1he k-hmeredoutut bewrittenas) probably give rise to undesirable spurious response from the Y,(e) M = overlapping of the transition bands of H/k+i (c) and G, (co) . The imkco-2tT imkco-2tT source ofthe spurious response, which is the cross-term in D2() X) 01H +(e M )X(e M (1) causes bumping in the stopband of A,(6 ). Fortunately, it will be where X(ejc) and Yk(eje) are the input and the k-th subband shown below, similar to the result in [3] for the RNFBs with LP output of the RNFB. To save notation, we shall write X(ao) as filters, that the spurious response can be suppressed if the X(eJ.'~) , Y'j() as Y}eJ (e') for the input, output and other related appropriate conditions on the constant cl and a matching condition On the LD FIR or IIR analysis/synthesis filters are imposed. quantities. Here we assume that the merging of the mk channel TMUX starts at the Ik -th channel of the M-channel uniform FB. (1) II. RECOMBINATION NONUNIFORM FILTER BANKS can be written as: WITH LD-FIR AND IIR FILTERS Yk(CO) =I XkolA,A(co)X((mco-2nr)/M) (2) In the proposed FB, the analysis filters are either LD FIR or IIR M =0 filters. For the case of LD FIR filters with filter order N, its where frequency response can be written as H(ej"') = e-i]D/2HR() A,(o ink -i 2ir)/M)o)(3) Aj)=2 ^0 Gkj(o)H/k+((mkco2 ,)1M) (3) where HR(co) is a complex function. If HR(co) approximates a r=0.M ,1. A,(a ) should be designed to extract the frequency real function, then it represents approximately the magnitude components of X(a ) exactly. It is known that the TMUX can be response and the passband group delay is approximately constant and is equal to (D/2), which can be made smaller than (N/2) for its derived from an 1-skewed FB [10]. Let {G(z) Gh(z)} LP counterpart. In the latter case, HR(co) is real-valued. Further, i,..m,be the analysis filters and the synthesis filters of a 1=0.mI an LD FIR filter can be made approximately symmetry/antiuniform FB. The 1-skewed FB with the analysis filters and symmetry if the first (D+1) samples of its impulse response are synthesis filters {Gkj(z),Gkj(z)} satisfy Gkj(z)=Gj(z) and approximately symmetry/anti-symmetry. If filters H/Ak+0() and Gki(z)=z'G (z) For simplicity, we define G (co) in the LD RNFB are approximately symmetric, then their H/k +(w) = H/ + (mkG!o/M) . So (3) can be written as: frequency response can be written as: ~~ffO~~~~~ ink-i ~~~~~~~Hi + (CO) -e]aC0H/k+(c) , G,jco) -e]l8CR,t 6 A, (o) = e jX ok GC (c)H Hk+(co27-rlmk) , (4) k (0)) (6) Similar, if H/k +(co) and GC (co) are approximately anti-symmetric: = 0.,M1. Note that H/k +(w0) and Ci (w) can be written as:R_R k _~~~~~~~~~R --R H/(o /+()+H/+(),Af/k 6 zA/m H1+(w) je-]aOH/ki() I Hjk+i(co) j ) Hl+i () l +i (a)) =Hl+H (n) MmtI'Z/tk +i l (c) e-0Hk+(C)H X0~V i k Hl+'(C GC(co)=G+(co)+G (co), I/M a nI/M . G+(y)--je_ioGR(c(), GC7(co)--je GjR(co), (7) A where, H>R (co) and GR(co) are the corresponding amplitude Assume that the stopband edges of Hik+i(co) and Gj(o) lie ¢ k+l within [R(lk i)lMR 12M,(lk i1)/MR 12M] and responses, a and ,b are the passband group delay. Due to space [Ri/ mk R / 2m R (i 1) / m R / 2m respectively, and the limitation, we only consider LD FBs with filters that are symmetric k-k-I> m/k~ ~"~k] and when I is even and anti-symmetry when I is odd. The cross-term stopband attenuation is sufficiently high that it can be treated as in D () , which is the source of the spurious response and causes zero outside the region. Let's consider the right hand side of (4) 2 without the trivial delay: the bumps in the stopband of AT(a), becomes: ik1 -R D(cS) = n0 G (co)H/k +i(oi-2 nkr1 m>) e '("') u (C H (-2)rr R (C (C/+/k+ (w 2f2'r/k) (w)k =-D1(o) D2( ) D3( ) D4(o), -R j [ki+ t2Tm6,l() 8 where + C/k +/+1 H/k +1+i(co )G k +I (w)) (8) The cross-term can be cancelled if we impose the condition on the D1(o) = ,^ GC+(co)Hk+i(c 2rz/imk) , constants c1: Ck +j = -C/k+i+l, when the uniform LD FB and the D2(o)=6(k C7()H+ (o-2fT/in) (5) recombination LD TMUX are designed to satisfy (the matching Wo))=0Y'nG--(o))H1,i(-2)1mcondition): D3(a) and D4(a) conjugate of D1(a) and D2(a) Defineq LM/mkl and y= [lfr 2], where Lx] denotes the nearest H/k+i(0)=C(co), (9) integer less than or equal to x . In order to determine the spurious whcaloanbwrteas