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Procedía Engineering 29 (20120) 435 - 440

Procedía Engineering

www.elsevier.com/Iocate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE 2012)

Design of Linear Phase Nonuniform Filter Banks with Interpolated Prototype Filters

Ming Wanga*, Jian Wana, Yan Zhaoa

aScience and Technology On Blind Signal Processing Laboratory, No.118, Yihuanluxiyiduan Road, Wuhou District, Chengdu,

610041, China

Abstract

Most of nonuniform filter banks designed by existing methods are not linear phase. In order to solving the problem, a novel method for designing linear-phase nonuniform filter banks is proposed in this paper. By analyzing the filter banks structure with block decimation transformed to rational decimation, we get an interpolated filter banks structure which removes the mirror frequency brought from interpolation. The filter banks are obtained by cosine modulation in this paper, and the prototype filters are interpolated and filtered before modulation. To make sure the significant aliasing distortions are cancelled completely, all prototype filters are designed with consistent transition band performance, using the optimization algorithm. Simulation results demonstrate that the linear-phase nonuniform filter banks designed by the proposed method have small amplitude distortions and aliasing distortions.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology

Keywords: Linear-Phase; Nonuniform Filter Banks; aliasing distortion; cosine modulation; interpolated

1. Introduction

Nonuniform filter banks (NUFBs) are widely used in many signal processing applications because of their flexibility in partitioning subbands. In some specific applications, such as image coding, it is crucial for all filters to have linear-phase (LP) property. This is LP filters can avoid artifacts in the reconstructed images. However most of NUFBs using existing design method are not LP[1-3]. A method of designing LP NUFBs is proposed in [4], where certain subbands of an LP uniform filter banks are recombined by synthesis filters of transmultiplexers, But this method has large implementation complexity and system

* Corresponding author. Tel.: 186-28-8659-5394; fax: 186-28-8659-5365. E-mail address: minnwann2111@nmail.com.

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.12.737

delay. Uniform analysis filters and synthesis filters of transmultiplexers must have matching amplitude response for cancelling aliasing distortion. In [5]-[6], direct subband merging of cosine-modulated filter banks is studied. Because of the conflict between passband flatness and aliasing cancellation, the direct subbands merging have not LP property. Thus, in the context, it is valuable to exploit the efficient design of NUFBs with LP property.

In this paper, a novel method with cosine modulating interpolated prototype filters is presented for the design of LP NUFBs. Thanks to the interpolated filter banks structure, prototype filters can be designed using optimization algorithm with the same optimized parameters except different passband widths. Thus, all prototype filters have the same shape of transition band to cancel aliasing distortion efficiently. As illustrated by example, a LP NUFBs with simple design and excellent performance are present.

2. The structure of interpolated nonuniform filter banks

Fig.1(a) shows the M-band analysis-synthesis nonuniform filter banks system with block decimation factors pk / qk, 0 < k < N -1, N < M . We assume that factors pk and qk are coprime and the NUFBs are critically sampled, that is

± P = i.

k=0 Pk

(a) (b)

Fig.1 (a)An M-band analysis-synthesis nonuniform filter bank system with block decimation; (b)The interpolated NUFBs with rational decimation

Fig. 1(b) is the equivalent form of Fig. 1(a). Define least common multiple of q0,q1,^qN-1 as M , then

mk = pt— , 0 < k < N -1 . In Fig. 1(b), NUFBs with block decimation factors are improved to the

interpolated NUFBs with rational decimation, where we need remove the mirror frequency of Pk ( zmk ) by the lowpass filter Bk ( z ) before modulation. That is

Pk ( z ) = Pk ( zmk ) Bk ( z ). (2)

NUFBs Ht (z), Gt (z) are obtained by cosine modulating prototype Pt (z).

Hk(z) = Ht(zmk )B k(z) = /ekUk(z) + /-BkVl(z) Gt(z) = Gt(zmk )B \(z) = e^U,(z) + /BkVl(z) (3)

where B \ (z) are bandpass filters by shifting frequency of Bk (z). 0k are chosen to satisfy the aliasing

cancellation constraint. In order to ensure the LP property of Hk (z) and Gk (z), dk has to be [0, ±—,n].

Uk (z) = WM+a5)( Nk-1)/2 p (zWM+°-5))

vt (z) = WM+°-5)( Nk-1)/2 p (zWM+°-5))

where the index rk selects where the passband is located. The input-output relationship in the z-domain is given by

11 mk-1 _ — _ — N-1 1 1

M-1 mt-1 _L

X(z) = Z Hk(zmiWmk)G(zmiK)X(z) + Z Z Hk(zmk WWG(zmkWP)X(zwm). (5;

k =0 M mk p=0 k =0 M mk l = 1 p=0

The distortion transfer function is

N -1 1 1 m*-1 _ _L _ _L

T (z)=Z17— Z Hk (zm* wp )G* (zm* wm*).

The aliasing function is

t=0 M mk p=o

N-1 1 1 M-1 mk-1 _ _L _ _L

A( z) = Z"^— Z ZH (zmkWW )Gk (zmk ). k=0 M mk l=1 p=0

Due to (3) , The distortion transfer function and the aliasing function can be written as

N-1 1 1 mk-1 _L

T (z) = Z-^— Z Hk (z)Gk (z) B\ (zmkwpt )2 (s;

k =0 M mk p=0

N-1 1 1 M-1 mk-1 _L _L

A(z) = Z^" Z Z ■Hk (zWMmk )Gk (z)B 'k((zmk Wmk Wi )B k (zmk W^).

k =0 M mk l=1 p=0

From (8)(9), we can see that the amplitude distortion and aliasing error are decided by the performance of Hk (z) and Gk (z), namely the performance of prototype filters Pk (z). If (10) is satisfied and Pk (z) have the same shape of transition band, the amplitude distortion introduced by the filter banks will be eliminated and the significant aliasing can be cancelled completely.

|Hk (z)Gk (z)2 +| Ht+1(z )Gk+1( z)|2 = 1. (10)

(10) is got from the condition of zero amplitude distortion, it can be written to (11) about Pt (eim).

P 7 [P^ ^f

J[m-(——+—

Pt+1(e 2 2 )

]d® = 1 (11)

where msk is the cutoff frequency of Pt (eJm). If Ft (eJm) and Pt+1(eJm) have the same shape of transition band, (11) can be written as

r [ Pt (eJ^)|2 +| Pk (eJ'^-k ))2]d® = 1 • (12)

(12) is the optimization item of designing the prototype filter Pk (eJm).

3. Conditions of near perfect reconstruction

The amplitude response of filters are written as

\Ht (eJm)|2 = Ht (eJm)H't(eJm) = J + e-J2Bk )Ut (eJm)Vt J + UU + V2(eJm). (13)

Ut (eJm) and Vk (eJm) do not overlap significantly except when k = 0 and k = N -1 . 0k have to be [0,±nn], so e2Bk + e12Bk * 0. Thus, the passband of the fist filter around frequencies m = 0 will not be

flat and the same as the last filter around frequencies m = n. T(e'm) will create significant distortions around the m = 0 and m = n in -n < m < n too[7]. Therefore, in order to cancel amplitude distortions and make the passband flat while maintaining the LP property of individual filters, the first and last filters have to be designed separately.

From (3) we known gk (n) = hk (Lk -1 - n) , where Lk is the order of analysis filter hk (n) , so Gk (z) = z-(Lk-1)Ht (z. The distortion transfer function (6) can be written as

n-1 i i m-1 (Lk ^ t(z) = Y—— Yz mk W-p(Lk-1) ( ) ¿0 M —t f-0 m

To eliminate the phase distortion, eq (15) should be fulfilled.

H (z— W-p)

-= C,0 < k < N-1,C eZ . (15)

Assuming the orders of Pk (z) are constant as L, the orders of Bk (z) are LB-t +1. Due to the transition band of Bk (z) is very wide, LB can be a small constant. Bk (z) are all designed by the Parks-McClellan algorithm. From (2) we can get the order of Pk (z) are (L -1) —k + LB-k +1, then C = L -1 + LB.

Analyzing the aliasing function(7), we can decompose A(z) with Akw (z) and Ahkigh (z)[8], that is

AT (z) = - W J23kUk (z)Vk (zWM) + /2BkVk (z)Uk (zWu'k)]

1 _ _ _ _ . (16)

Ahigh (z) = —[/-2BkUk (z)Vk (zWM+1) + 2°kVt (z)Uk (zW-J'k+1))]

In order to cancel significant aliasing distortions, the conditions followed have to be satisfied[7-9]. a)AT(z) i mk +Ak+l(z) i mk+l = 0 b)AkT(z) i mk +A*»(z) i mk+l = 0 c)Ahkigh(z) i mk + AT(z) i mk+1 = 0 d)Af (z) i mk + A«?(z) i mk+1 = 0 where 0 < k < N - 2. In each coupling, if the magnitude responses of two aliasing terms have the same amount at the same frequency point and their phases differ by n , the significant aliasing can be cancelled

completely. That requires 0k are chosen as [0 or n] for symmetry and [— or — ] for antisymmetry

alternately and the prototype filters Pk (z) have the same shape of transition band. The findings up to now can be summarized to two conditions followed.

1) The analysis filters hk (n) , 0 < k < N -1 , satisfy an alternate symmetry property. That is hk (n) are symmetry and antisymmetry alternately.

2) The prototype Pk(z) ,1 < k < N - 2 and H0(z), HN-1(z) have the same characteristic of transition band, namely the same transition bandwidth and transition band attenuation.

An efficient iterative design method for the prototype filter of uniform filter banks is proposed in [10]. Optimization includes two items as (18).

E1 =J [|P(/M)f +|Pk(/(-(n/M)))|2 -1]2da and E2 = J|P(/7'")|2da (18)

where a =-+ Bg , Bg is the required transition bandwidth. Comparing E1 with eq(12), they have the

similar form. So Pk(z) can be designed using the optimization algorithm in reference [10]. Eq(18) are rewritten as

nmk /M n

Ek,1 = J [|P(/a)\2 +|P(/J(a-('mk/M)))f -1]2da and EM =J |Pk(/Ja)\2 da (19)

where ak = —k-+Bg ,1 < k < N - 2, Bg is a constant for all k . The bandwidth of Pk (z) is —k-, Because ' 2M M

Pk (z) have the same order and transition bandwidth, it will keep the same transition band shape after optimization. h0(n) and hN-1(n) are designed using the optimization algorithm too. Actually, h0(n) is a

prototype filter whose bandwidth is-hN-1(n) is obtained by frequency shifting n of a prototype

filter h';\(n) , that is hN 1(n) = (-1)"h""'1(n) , where the bandwidth of h"w1(n) is 2nm"-1 . So h0(n) is

symmetry and hN-1(n) is antisymmetry. When designing h0(n) or h"W1(n), we change the mt in (19) into 2m0 or 2mN-1. Bg is not changed. The orders of H0(z) and H""(z) are L as the same as Pk (z). So H0(z), HNW1(z) and Pk (z) have the same shape of transition bands.

4. Simulation results

The reconstruction performance of NUFBs is decided by the flatness of passband, attenuation of stopband and consistency of transition band of individual filter. In this paper, we design the prototype filters of NUFBs with optimization algorithm to get more excellent reconstruction performance.

The systematic design procedures of the proposed LP NUFBs are summarized below. Pk (z) , 1 < k < N - 2, H0(z), H"W1 (z) are all designed with the optimization algorithm in [10]. Bk (z), 0 < k < N -1 are all design by Parks-McClellan algorithm.

1) Choose the filter order L, LB and constant transition width Bg . Then the order of Pk (zmk) are (L -1)mt +1; the order of Bk (z) are LBmt +1; the order of Pt (z) are (L -1)mt + LBmt +1 and C = L -1 + LB.

2) Design H0(z) and HN-1(z). Design h0(n) and h"11(n) first. Bandwidths of h0(n) and h"11(n) are

respectively 2nm° and 2nm"-1 ; the orders are L ; transition bandwidths are all Bg . Then

hN-1 (n) = (-1)"h^W1(n) , and H0(z) and HN-1(z) are interpolated by m0 and mk respectively. Thus, H0(z) = H0(zm0)B0(z) and HN-1(z) = HN-1(zm"A)BN-1(z) , where B0(z) is lowpass and BN-1(z) is highpass.

3) Design the prototype filters Pk (z) ,1 < k < " - 2 . Bandwidths are —k-; transition bandwidths are

Bg ; the orders are L . Pk(z) are interpolated by the corresponding mk respectively. So Pk (z) = Pk (zmk)Bk (z) ,where Bk (z) are lowpass.

4) Cosine modulate Pt (z) to obtain Hk (z) ,1 < k < N - 2 . Qk are chosen as [0 or n] for symmetry and

n n 1 k-1 [— or — ] for antisymmetry alternately. Frequency shifting factors rk =—V .

2 2 mk i=0

2 2 2 1

Example: In this example, a " = 4 channels LP NUFBs is designed. Sample factors are [y ,—].

The parameters L = 128; LB = 64; Bg = —— ; frequency shifting factors r1 = 1, r2 = 2; phase shifting factors

01 = n / 2, 02 = 0. Fig.2(a) shows amplitude response of 4 channels LP NUFBs. The stopband attenuation is close to -60dB. Fig.2(b) shows the amplitude distortions, that is under 2 x 10-3. Fig.2(c) shows the aliasing distortions, that is under 1.5 x 10-3. From what the Fig.2 show, we can see the LP NUFBs have excellent reconstruction performance using the proposed method. The aliasing error and amplitude distortion are much smaller than that of [4]. The transition bandwidth is much smaller than that of [7]. Because the NUFBs are obtained by modulating several prototype filters, the proposed method has lower implementation complexity than the direct design method.

5. Conclusion

In this paper, a design of cosine modulation LP NUFBs with interpolated prototype filters is proposed. The prototype filters have the same transition bandwidth and order, then they have consistent transition bands performance after optimization, and the aliasing can be cancelled completely. Besides, optimization complexity is lower thanks to filters interpolated. When the NUFBs have great channels and some of them have the same sampling factors, the proposed method is more suitable.

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