Design of Two-Channel Linear Phase QMF Banks Based on Real IIR All-Pass Filters

Design of two-channel linear-phase QMF banks based

on real IIR all-pass filters

J.-H. Lee and Y.-H. Yang

Abstract: The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar’s algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison.

1 Introduction

For many communication and signal processing systems, quadrature mirror filter (QMF) banks have been widely used to achieve the goals of subband coding and short-time spectral analysis[1 – 4]. In these applications, a QMF bank is employed to decompose a signal into subbands and the subband signals in the analysis system are decimated by an integer equal to the number of subbands. Moreover, two-channel QMF banks can easily be used for constructing M-channel QMF banks based on a tree structure. Hence, it is worth exploiting the design problem of two-channel QMF banks with linear phase characteristics.

Several techniques have been presented for designing two-channel QMF banks with IIR analysis filters and approximately linear phase based on the least-squaresðL2Þ error criteria[5 – 12]. These IIR QMF banks are designed with the linear phase property imposed on the analysis filters. In contrast, a technique has been proposed in[13]for designing an IIR QMF bank with arbitrary group delay optimal in the minimax ðL1Þ sense. Recently, the design results for IIR linear-phase QMF banks based on real all-pass sections and complex all-all-pass sections have been reported in [9 – 12] and [14,15], respectively. The main advantage of using all-pass sections is that the designed IIR QMF banks can possess approximately linear phase response without magnitude distortion.

This paper presents a technique based on a variant of Karmarkar’s algorithm[16]for the optimal design of two-channel linear-phase QMF banks using real IIR digital all-pass filters (DAFs). The design problem is formulated by using the minimax error criteria on the phase approximation

to obtain an appropriate objective function that leads to a nonlinear optimisation problem. To effectively tackle the resulting nonlinear minimax optimisation problem, we utilise a nonlinear minimax algorithm [17] to result in a sequence of linear Chebyshev approximation problems. Each of the linear Chebyshev approximations provides the required increment for updating the filter coefficients during each iteration. As a result, the key operation of the proposed technique is to find the linear Chebyshev approximation of a desired phase response from a linear subspace related to the objective function. This can easily be solved by using a variant of Karmarkar’s algorithm [16]. Several design examples showing the effectiveness of the proposed technique are also provided.

2 Two-channel QMF banks with linear phase response

Consider the two-channel filter bank with a system architecture shown in Fig. 1. H0ðzÞ and H1ðzÞ designate the low-pass and high-pass analysis filters, respectively, and F0ðzÞ and F1ðzÞ designate the low-pass and high-pass synthesis filters, respectively. We can easily show that the input – output relationship in the Z-transform is given by

^ X XðzÞ ¼1 2½H0ðzÞF0ðzÞ þ H1ðzÞF1ðzÞXðzÞ þ1 2½H0ðzÞF0ðzÞ þ H1ðzÞF1ðzÞXðzÞ ð1Þ The first term of (1) represents a linear shift-invariant system response, which is the desired signal translation from x(n) to ^xxðnÞ; and the second term represents the aliasing error due to the change of sampling rate in the filter bank. Setting the synthesis filters F0ðzÞ ¼ H1ðzÞ and F1ðzÞ ¼ H0ðzÞ eliminates the aliasing term. As the mirror-image symmetry about the frequency o¼ p=2 exists between H0ðzÞ and H1ðzÞ; we have H0ðzÞ ¼ H1ðzÞ: Therefore (1) becomes ^ X XðzÞ ¼1 2½H 2 0ðzÞ  H 2 0ðzÞXðzÞ ð2Þ qIEE, 2003

IEE Proceedings online no. 20030699 doi: 10.1049/ip-vis:20030699

The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, 106, Taiwan, Republic of China

Paper first received 23rd September 2002 and in revised form 11th April 2003

Setting z¼ expðjoÞ in (2), we get ^ X XðejoÞ ¼1 2½H 2 0ðe jo

Þ  H02ðejðoþpÞÞXðe jo

Þ ð3Þ

Let Tðejo_{Þ denote the frequency response of the QMF bank.} Equation (3) reveals that producing a reconstructed signal ^xxðnÞ that is a delayed replica of x(n) requires

Tðejo_{Þ ¼ H}2 0ðe

jo_{Þ  H}2 0ðe

jðoþpÞ_{Þ ¼ e}jgdo _{for all o ð4Þ} where gdis the system delay of the QMF bank. This imposes constraints not only that H0ðzÞ should be an ideal low-pass filter, but also that its behaviour for all o should satisfy the condition given in (4). The designs of QMF banks using conventional FIR or IIR structures for H0ðzÞ usually induce both magnitude and phase distortions.

3 Problem formulation of two-channel QMF bank design

Here, we consider the two-channel QMF bank with analysis and synthesis structures shown by Figs. 2 and 3, respectively, where A1ðz2Þ and A2ðz2Þ are two real IIR DAFs. The main advantage of using this kind of QMF bank is that the design problem can be focused on the phase approximation without magnitude distortion[11, 12]. In the following, we briefly describe how to construct this two-channel QMF bank from the conventional structure shown inFig. 1. By setting the analysis filters H0ðzÞ and H1ðzÞ of Fig. 1to H0ðzÞ ¼ B1ðzÞ þ B2ðzÞ 2 and H1ðzÞ ¼ B1ðzÞ  B2ðzÞ 2 ð5Þ

respectively, where BiðzÞ are two real IIR DAPs with orders equal to Ni and frequencies given by

Biðe jo Þ ¼ ejNio PNi n¼0 aiðnÞejno PNi n¼0 aiðnÞejno ; for i¼ 1; 2 ð6Þ From (5), we have B1ðzÞ ¼ H0ðzÞ þ H1ðzÞ and B2ðzÞ ¼ H0ðzÞ  H1ðzÞ ð7Þ Hence, H0ðzÞ and H1ðzÞ satisfy the all-pass complementary and power complementary properties. They are termed the doubly-complementary (DC) filter pair [18].

Moreover, the QMF property requires the condition H0ðzÞ ¼ H1ðzÞ: The resulting filter banks are termed DC-based QMF filter banks [9 – 12, 14, 15]. Hence, we have from (7) that

B1ðzÞ ¼ H0ðzÞ þ H0ðzÞ ¼ 2h0þ 2h2z2þ . . . ¼ A1ðz2Þ and

B2ðzÞ ¼ H0ðzÞ  H0ðzÞ ¼ z1½2h1þ 2h3z2þ . . .

¼ z1A2ðz2Þ ð8Þ

where hi; i¼ 0; 1; . . . ; represent the impulse response coefficients of H0ðzÞ: Consequently, (5) can be rewritten as follows: H0ðzÞ ¼ A1ðz2Þ þ z1A2ðz2Þ 2 and H₁ðzÞ ¼ A1ðz 2_{Þ  z}1_A 2ðz2Þ 2 ð9Þ

Substituting (9) into (4) yields the frequency response of the DC-based QMF bank as follows:

TðejoÞ ¼1 2e jo_A 1ðe j2o ÞA2ðe j2o Þ ð10Þ

Equation (10) reveals that the resulting QMF bank possesses perfect magnitude response, i.e. there is no magnitude distortion. Therefore, the design problem is to find real coefficients for the IIR DAFs A1ðz2Þ and A2ðz2Þ such that the resulting phase response ArgfTðejo_{Þg of} the QMF bank can approximate the desired phase characteristic in the minimax sense. From (6), we have the real IIR DAFs A1ðz2Þ and A2ðz2Þ with frequency responses given by A1ðej2oÞ ¼ ej2N1o PN1 n¼0 a1ðnÞej2no PN1 n¼0 a1ðnÞej2no ¼ ej1ðoÞ _and A2ðe j2o Þ ¼ ej2N2o PN2 n¼0 a2ðnÞej2no PN2 n¼0 a2ðnÞej2no ¼ ej2ðoÞ ð11Þ

respectively. Moreover, without loss of generality, both coefficients a1ð0Þ and a2ð0Þ can be set to one. Then, the phase responses _iðoÞ; i ¼ 1; 2; are given by

iðoÞ ¼ 2Nio 2iðoÞ ¼ 2Nio 2tan1 P N_i n¼1 aiðnÞsinð2noÞ 1þP N_i n¼1 aiðnÞcosð2noÞ 8 > > > < > > > : 9 > > > = > > > ; ð12Þ H0(z) H1(z) 2 2 2 2 ˆ x0(n) x (n) x1(n) x(n) F₀(z) F₁(z)

Fig. 1 Two-channel QMF bank

–

Fig. 2 Analysis system for DC-based QMF bank

2 1/2 1/2 2

_–

Substituting (11) into (9) yields H0ðe

jo Þ ¼1

2½e

j1ðoÞ_{þ e}jo_ej2ðoÞ_

¼ exp j1ðoÞ þ 2ðoÞ  o 2

 cos 1ðoÞ  2ðoÞ þ o 2 ð13Þ and H1ðe jo Þ ¼1 2½e

j1ðoÞ_{ e}jo_ej2ðoÞ_ ¼ j exp j1ðoÞ þ 2ðoÞ  o

2

 sin 1ðoÞ  2ðoÞ þ o 2

ð14Þ In order to guarantee that H0ðzÞ and H1ðzÞ are linear-phase low-pass and high-pass filters, respectively, we can impose the following conditions on _iðoÞ; i ¼ 1; 2; based on (13) and (14):

Case (i): For N1¼ N2 ¼ N: 1ðoÞ ¼ 2No  o=2 ₂ðoÞ ¼ 2No þ o=2

; for 0 % o % op

1ðoÞ ¼ 2No  o=2 þ p=2 2ðoÞ ¼ 2No þ o=2  p=2

; for os%o % p ð15Þ Case (ii): For N1¼ N2þ 1:

1ðoÞ ¼ 2N1oþ o=2 2ðoÞ ¼ 2N2o o=2   ; for 0 % o % op 1ðoÞ ¼ 2N1oþ o=2  p=2 2ðoÞ ¼ 2N2o o=2 þ p=2   ; for os%o % p ð16Þ where op and os are the passband and stopband edge frequencies of H0ðzÞ; respectively. Equations (15) and (16) reveal that the above conditions also satisfy the following stability constraints for the real IIR DAFs Aiðz2Þ[15, 18]: _iðoÞ is monotonically decreasing and iðpÞ ¼ ið0Þ  2Nip; for i¼ 1; 2:

Next, from (10), (15) and (16), we note that the frequency response of the DC-based QMF bank becomes

Tðejo_{Þ ¼}1

2exp½jðo þ 1ðoÞ þ 2ðoÞÞ ¼1

2exp½jð2N1þ 2N2þ 1Þo ð17Þ Equation (17) reveals that the resulting DC-based QMF bank possesses a linear phase response with group delay gd ¼ 2N1þ 2N2þ 1 and without magnitude distortion. As a result, the design problem can be formulated as follows. Find filter coefficients aiðnÞ of the real IIR DAPs Aiðz2Þ; i¼ 1; 2; and n ¼ 0; 1; 2; . . . ; Ni; such that the resulting phase responses iðoÞ given by (12) approximate the desired phase responses as shown by (15) and (16) in the minimax sense. Comparing (12), (15) and (16), one can obtain the corresponding design problems for case (i) and case (ii), respectively, as follows:

Case (i): Minimise  tan1 PN_i n¼0 aiðnÞ sinð2noÞ 1þP Ni n¼0 aiðnÞ cosð2noÞ 8 > > > < > > > : 9 > > > = > > > ;  iðoÞ ð18Þ where iðoÞ are given by

1ðoÞ ¼o4; 0 o  op ₁ðoÞ ¼o 4 p 4;os o  p   and 2ðoÞ ¼ o4; 0 o  op 2ðoÞ ¼ o₄þp₄;os o  p   ð19Þ Case (ii): Minimise  tan1 PNi n¼0 aiðnÞ sinð2noÞ 1þP N_i n¼0 a_iðnÞ cosð2noÞ 8 > > > < > > > : 9 > > > = > > > ;  iðoÞ ð20Þ where iðoÞ are given by

1ðoÞ ¼ o4; 0 o  op 1ðoÞ ¼ o4þp4;os o  p   and 2ðoÞ ¼o4; 0 o  op 2ðoÞ ¼o4p4;os o  p   ð21Þ andkxk denotes the Chebyshev norm. Let

MiðoÞ ¼ PNi n¼0 aiðnÞ sinð2noÞ 1þP N_i n¼0 aiðnÞ cosð2noÞ ð22Þ

Then, (18) and (20) can be rewritten as

Minimisek tan1ðMiðoÞÞ þ tan1tanðiðoÞÞk ð23Þ Using the addition formula of inverse tangents, we obtain

Minimise tan1 MiðoÞ þ tanðiðoÞÞ 1 MiðoÞ tanðiðoÞÞ

  _ð24Þ

From (24), an equivalent minimisation problem can be obtained by dropping the inverse tangent operation as follows:

Minimise MiðoÞ þ tanðiðoÞÞ 1 MiðoÞ tanðiðoÞÞ _ð25Þ

because the inverse tangent function is a monotonic function. Performing some algebraic manipulation on (25) yields the following minimisation problem

Minimise PNi n¼1

a_iðnÞ½sinð2noÞ þ ’iðoÞ cosð2noÞ þ ’iðoÞ PNi

n¼1

aiðnÞ½cosð2noÞ  ’iðoÞ sinð2noÞ þ 1 ð26Þ where ’iðoÞ ¼ tanðiðoÞÞ; for i ¼ 1; 2: Therefore, the overall design problem is to find the optimal coefficients aiðnÞ; n ¼ 1; 2; . . . ; Ni; i¼ 1; 2; for the minimisation problem of (26). However, (26) represents a highly nonlinear minimisation problem. Directly solving (26) is not an easy task.

4 Proposed design technique

In this Section, we present a design technique based on the nonlinear minimax algorithm of[17]in conjunction with a variant of Karmarkar’s algorithm [16] for solving the resulting minimisation problem of (26). This is through a frequency sampling and iterative approximation scheme to find the optimal coefficients aiðnÞ; n ¼ 1; 2; . . . ; Ni; i¼ 1; 2; for the real IIR DAFs shown by (11). At the kth iteration, we construct the following function:

Apxðak;oÞ ¼ PNi n¼1

aikðnÞ½sinð2noÞ þ ’iðoÞ cosð2noÞ þ ’iðoÞ PN_i

n¼1

aikðnÞ½cosð2noÞ  ’iðoÞ sinð2noÞ þ 1 ð27Þ and the N_i 1 gradient vector of Apxðak;oÞ given by

HApxðak;oÞ ¼ ½c1ðoÞ c2ðoÞ . .. cNiðoÞT ð28Þ where cnðoÞ represents the nth gradient component of Apxðak;oÞ and is given as follows:

cnðoÞ ¼ @Apxðak;oÞ @a_iðnÞ ¼ ½sinð2noÞ þ ’iðoÞcosð2noÞ  X Ni n0_¼1

aikðn0Þ½cosð2n0oÞ  ’iðoÞsinð2n0oÞ þ 1

!!,

XNi n0_¼1

aikðn0Þ½cosð2n0oÞ  ’iðoÞ sinð2n0oÞ þ 1

!2

 ½cosð2noÞ  ’iðoÞ sinð2noÞ

 X

Ni

n0_¼1

a_ikðn0Þ½sinð2n0oÞ þ ’iðoÞcosð2n0oÞ þ ’iðoÞ

!!,

XNi n0_¼1

a_ikðn0Þ½cosð2n0oÞ  ’iðoÞ sinð2n0oÞ þ 1

!2

ð29Þ

for n¼ 1; 2;. .. ;Ni: Let O¼ fo1¼ 0; o2. . . :; oL¼ pg be a dense grid of frequencies uniformly distributed in the range from o¼ 0 to o ¼ p: The design process of the proposed technique is then performed on O: Next, a linearisation scheme is utilised to approximate the related phase error of (27) due to a perturbation in the filter coefficient vector in the linear subspace spanned by the gradient matrix associated with Apxðak;oÞ: As a result, the approximation for minimising the above peak phase error as shown in (26) can be formulated as finding the increment da of the filter coefficient vector a¼ ½aið1Þ;aið2Þ;. .. ;aiðNiÞT; i¼ 1;2; at the kth iteration such thatkApxðak;olÞdaTHApxðak; olÞk is minimised. Let da be expressed as da ¼ ½daið1Þdai ð2Þ .. .daiðNiÞT; the expressionkApxðak;olÞ  daTHApx ðak;olÞk can be rewritten as follows:

kApxðak;olÞ  XNi

n¼l

daikðnÞcnðolÞk ð30Þ As a result, finding the solution to minimise (30) is equivalent to calculating da in an Ni-dimensional linear subspace C spanned by the set of basis functions

fc1ðoÞ; c2ðoÞ; . . . ; cNiðoÞg to minimise (30). We note that the minimisation of (30) can be achieved by performing the following minimisation problem

Minimise kUdak sk ð31Þ

where U is a L Ni matrix with entries given by cnðolÞ; 1 l  L; 1  n  Ni and s is a L 1 vector with the lth entry given by sðlÞ ¼ Apxðak;olÞ; 1 % l % L; and dak represents the increment computed for updating the coefficient vector ak: Equation (31) represents an equivalent form of the following linear Chebyshev minimisation problem:

Minimise e

Subject to jUdak sj % e1; ð32Þ where 1 is a L 1 vector with all entries equal to 1. The minimisation problem of (32) can be further rewritten as

Minimise e

Subject to fUdak e1  s; Udak e1 %  s ð33Þ This leads to the standard dual form of a linear program-ming (LP) problem as follows:

Maximise bTw Subject to ATw % c ð34Þ where w¼ ½ dak eT; b¼ ½ 0T 1 T; c¼ ½ sT _sT_T_{; A} ¼ U T _UT 1T _1T " # ð35Þ where 0 is a Ni 1 vector with all entries equal to 0. Using definition 3.1 of [19], we can easily obtain the standard primal form of (34) as follows:

Minimise cTx

Subject to Ax¼ b; x ^ 0 ð36Þ

where w and x respectively denote the dual variable and the corresponding primal variable for the above LP problems. In the following, we describe how to adopt the primal-form affine scaling variant of Karmarkar’s algorithm (the PAS algorithm) of[16]for solving (36).

Assume that an initial solution x which satisfies the constraints is given. Then, x is mapped into a vector y0with all entries equal to one as follows:

y¼ D1x x ð37Þ

where Dxdenotes a diagonal matrix containing the entries of x. Based on the mapping, we create

A¼ ADxand c¼ Dxc ð38Þ

To satisfy the equality constraints, we project c onto the null space of A to obtain

cp ¼ Pc; ð39Þ

where

P¼ I  ATðAATÞ1A ð40Þ

denotes the projection operator. Next, we move from the initial y to y1in the directioncpto reduce the transformed objective function in the maximum rate according to

y1¼ 1  x cp maxiðeTicpÞ

where the required step size x2 ð0; 1Þ is chosen so that y1 _{> 0; 1 represents a vector with appropriate size and all} entries equal to one, and eiis a vector with appropriate size and the ith entry equal to one and the others equal to zero. After obtaining y1; we then find a new feasible solution x1 for (36) by performing the inverse mapping.

Consider the projection operator given by (40). Substituting (38) into (40) yields

P¼ I  DxA T

ðAD2xA T

Þ1ADx ð42Þ

Accordingly, we have from (38), (39), and (42) that cp¼ Dx½I  A

T ðAD2xA

T

Þ1AD2xc ð43Þ Next, we define the following two vectors

w¼ ðAD2xA T

Þ1AD2xc and r¼ c  A T

w ð44Þ

where w represents the dual variable vector associated with the primal variable vector x. Hence, the primal variable xkþ1 at theðk þ 1Þth iteration is found by the following equation

xkþ1¼ xk_x gdx ð45Þ where d_x¼ D2xr and g¼ max i ðe T icpÞ ¼ max i ðe T iDxrÞ ¼ max i ðe T iD 2 xr=xiÞ ð46Þ

with xibeing the ith entry of x.

4.1

Determination of initial guess for the filter

coefficient vector

To initiate the design process, we have to make an initial guess a0 for the filter coefficient vector a according to the nonlinear minimax algorithm of [17]. An appropriate manner is to minimize the squared value of the numerator of Apxða; oÞ given by (27), i.e. minimise

XL l¼1

XN_i n¼1

aiðnÞ½sinð2nolÞ þ ’iðolÞ cosð2nolÞ þ ’iðolÞ !2

ð47Þ To solve this minimisation problem, we use a simple approach as follows. Let X be a L Nimatrix whose entries are given by

Xðl; nÞ ¼ sinð2nolÞ þ ’iðolÞ cosð2nolÞ; 1 l  L; 1  n  Ni

ð48Þ Then, it is easy to show that minimising (47) yields the required initial guess a0 as follows:

a₀¼ ðXTXÞ1XTq ð49Þ

where the L 1 vector q has entries given by

qðlÞ ¼ ’iðolÞ; 1 l  L ð50Þ After finding the appropriate initial guess a0and setting the initial filter coefficient increment to a zero vector, we present an iterative procedure based on the above presen-tation to compute the filter coefficient vector a during the design process.

4.2

Iterative procedure

Step 1: Determine the design parameters: the orders N1and N2; passband edge frequency op and stopband edge

frequency os: Make an initial guess a0 for the filter coefficient vector a¼ ½aið1Þ; aið2Þ; . . . ; aiðNiÞ

T

; i¼ 1; 2; from (49) and set the iteration number k¼ 0:

Step 2: Perform a test to stop the iteration process. We set a stopping criterion as follows: if jðmaxfjUdak sjg  maxfjUdakþ1 sjgÞ= maxfjUdak sjgj % n; then the design process is terminated and the filter coefficient vector akobtained contains the designed filter coefficients, where n is a preset small positive number. Otherwise, go to step 3. Step 3: Calculate the increment da of the filter coefficient vector ak¼ ½aikð1Þ; aikð2Þ; . . . ; aikðNiÞ

T

at the kth iteration in an Ni-dimensional linear space to minimise

kApxðak;oÞ  da T

HApxðak;oÞk ð51Þ

Utilising the PAS algorithm presented for solving the standard LP problem (36), we perform the following iterative procedure:

(3.1): Choose an initial guess x0which satisfies the equality constraints ATx0¼ b and x0 > 0 of (36). We can simply choose x0 _{¼ ½x}T 1 xT2 T ; where x1¼ x2 ¼ 1=ð2JÞ; J is the number of sample points in frequency. Set the iteration number m¼ 0:

(3.2): At the mth iteration, compute w¼ ½daT _eT ¼ ðAD2

xATÞ1AD2xc according to the following process: (3.2.1): Construct two diagonal matrices Di¼ diagðxiÞ; i¼ 1; 2:

(3.2.2): Compute M₁¼ D2

1þ D22; M2¼ D21 D22; g¼ 1T M211:

(3.2.3): Compute dam¼ ðUTðM1 M211TM2=gÞUÞ1 ðUT_ðM

1 M211TM2=gÞsÞ and e ¼ ð1TM2ðUdam sÞÞ=g: Step 4: Compute e=maxfjUdam sjg: Set the obtained dam equal to dak and go to step 7 if e= maxfjUdam sjg ^ ; where is a preset small positive number. Otherwise, go to step 5.

Step 5: Compute dx¼ D2xr¼ D2xðc  ATwÞ according to the following process: (5.1): Compute d1¼ D21ðs þ e1  UdamÞ; d2 ¼ D22ðs þ e1þ UdamÞ: (5.2): Compute dx¼ ½dT1 dT2 T :

Step 6: Compute the step size x=g from (46) and update the primal variable vector according to

xmþ1¼ xm_x gdx Then, set m¼ m þ 1 and go to step 3.2.

Step 7: Use the optimal solution dakobtained to find the best increment such that

kApxðakþ bdak;oÞk; 8b ^ 0 ð52Þ is minimised. We adopt the Nelder and Mead simplex algorithm[20]to perform a line search to find the best value of b. Let the best value of b be bk. We update the filter coefficient vector according to a_kþ1¼ akþ bkdak: Then, set k¼ k þ 1 and go to step 2.

5 Simulation results

In this Section, we present simulation results for the design of two-channel linear-phase QMF banks based on real IIR DAFs for illustration and comparison. These designs were performed on a personal computer with Pentium-III CPU using the MATLAB programming language. For compari-son, design results using the proposed technique and using techniques presented in [11] and [12] are presented. The performance for each of the QMF banks designed is evaluated in terms of peak stopband ripple (PSR) of

H0ðzÞ; maximal variation of phase response (MVPR) and maximal variation of group delay (MVGD) in Tðejo_{Þ; and} maximal variation of filter-bank response (MVFBR). They are defined as follows:

PSR¼ 20 log10 max ol2½os;p jH0ðe jo_l Þj ðdBÞ MVPR¼ max ol2½0;p jArgfTðejol_Þg þ ð2N1þ 2N2þ 1Þolj ðradiansÞ MVGD max ol2½0;p jGDfTðejol_Þg  ð2N1þ 2N2þ 1Þj ðsamplesÞ; MVFBR¼ max ol2½0;p 20 log10Tðe jol_{Þ }1 2e jð2N1þ2N2þ1Þol         ðdBÞ ð53Þ where GDðxÞ denotes the group delay of x. The performance for each of the QMF banks designed using the techniques of [11]and[12]are evaluated based on the filter coefficients presented in[11]and[12].

Example 1: This example is the same as that given in ([11], Ex. 1). We use the same specifications for this design: real IIR DAFs A1ðz2Þ and A2ðz2Þ with orders N1 and N2 equal to 3 and 2, respectively, low-pass analysis filter H0ðzÞ with passband edge frequency op¼ 0:4p and stopband edge frequency os¼ 0:6p. Significant design results, namely, PSR, MVPR, MVGD, and MVFBR obtained using the technique of [11] and the proposed technique are shown in Table 1 for comparison. Table 2 lists the filter coefficients obtained after three iterations for both designs of A1ðz2Þ and A2ðz2Þ; using the proposed technique with parameters x¼ 0:97; ¼ 0:99999; and L¼ 8ðNiþ 1Þ: The parameter n was set to 2  108 to design A1ðz2Þ and 4  106 to design A2ðz2Þ: The corresponding magnitude responses of the H0ðejoÞ designed are shown in Fig. 4. The resulting phase errors and group delay deviations of the QMF banks designed are depicted in Figs. 5 and 6, respectively.Figure 7 plots the variations of the designed filter-bank responses. Simulation results show that the proposed technique

provides very satisfactory design results when compared to the technique of [11].

Example 2: This example is the same as ([12], Ex. 1). We use the same specifications for this design: real IIR DAFs A1ðz2Þ and A2ðz2Þ with orders N1¼ N2¼ 2; low-pass analysis filter H0ðzÞ with passband edge frequency

Table 1: Significant design results for design example 1

Proposed technique Technique of[11] PSR (dB) 2 19.965154415 2 18.051817634 MVPR (radians) 0.205953001 0.303082474 MVGD (samples) 1.497994713 2.161264942 MVFBR (dB) 2 19.760593942 2 16.422653793

Table 2: Filter coefficients designed for design example 1

n a1ðnÞ a2ðnÞ 0 1.00000000000000 1.00000000000000 1 0.23809492090228 2 0.23721170338893 2 2 0.07300653565757 0.15616284193843 3 0.03862697338297 0 0.1 0.2 0.3 0.4 0.5 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 normalised frequency magnitude response, dB proposed Lawson [12]

Fig. 4 Magnitude responses of low-pass analysis filters designed for example 1 0 0.1 0.2 0.3 0.4 0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 normalised frequency

phase deviation, rad

proposed Lawson [12]

Fig. 5 Phase deviations of QMF banks designed for example 1

0 0.1 0.2 0.3 0.4 0.5 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0 2.5 normalised frequency

group delay deviation in samples

proposed Lawson [12]

Fig. 6 Group delay deviations of QMF banks designed for example 1

op¼ 0:4p and stopband edge frequency os¼ 0:6p: Significant design results, namely, PSR, MVPR, MVGD, and MVFBR obtained by using the technique of[12]and the proposed technique are shown inTable 3 for comparison. Table 4 lists the filter coefficients obtained after four iterations and three iterations for the designs of A1ðz2Þ and A2ðz2Þ; respectively, using the proposed technique with parameters x¼ 0:97; ¼ 0:99999; and L ¼ 8ðNiþ 1Þ:

The parameter n was set to 2 108 to design A1ðz2Þ and 4 106 _{to design A}

2ðz2Þ: The corresponding magnitude responses of the H0ðejoÞ designed are shown inFig. 8. The resulting phase errors and group delay deviations of the QMF banks designed are depicted in Figs. 9 and 10, respectively.Figure 11plots the variations of the designed filter-bank responses. Simulation results show that the proposed technique provides much better performance than the technique of[12].

Table 3: Significant design results for design example 2

Proposed technique Technique of[12] PSR (dB) 2 18.112979613 2 15.617494878 MVPR (radians) 0.132845763 0.198064914 MVGD (samples) 0.895850891 1.212561067 MVFBR (dB) 2 23.560033741 2 20.098651279

Table 4: Filter coefficients designed for design example 2

n a1ðnÞ a2ðnÞ 0 1.00000000000000 1.00000000000000 1 2 0.23721170339174 0.23401447596825 2 0.15616284195027 2 0.09173259794723 0 0.1 0.2 0.3 0.4 0.5 –55 –50 –45 –40 –35 –30 –25 –20 –15 normalised frequency variation response, dB proposed Lawson [12]

Fig. 7 Variations of filter bank responses designed for example 1

0 0.1 0.2 0.3 0.4 0.5 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 normalised frequency magnitude response, dB proposed Lawson [12]

Fig. 8 Magnitude responses of low-pass analysis filters designed for example 2 0 0.1 0.2 0.3 0.4 0.5 –0.20 –0.15 –0.10 –0.05 0 0.05 0.10 0.15 0.20 normalised frequency

phase deviation, rad

proposed Lawson [12]

Fig. 9 Phase deviations of QMF banks designed for example 2

0 0.1 0.2 0.3 0.4 0.5 –1.0 –0.5 0 0.5 1.0 1.5 normalised frequency

group delay deviation in samples

proposed Lawson [12]

Fig. 10 Group delay deviations of QMF banks designed for example 2 0 0.1 0.2 0.3 0.4 0.5 –55 –50 –45 –40 –35 –30 –25 –20 –15 normalised frequency variation response, dB proposed Lawson [12]

Fig. 11 Variations of filter bank responses designed for example 2

6 Conclusions

This paper has presented a technique for the design of two-channel linear-phase quadrature mirror filter (QMF) banks based on real infinite impulse response (IIR) digital all-pass filters. The design problem is first formulated as a nonlinear optimisation problem of an appropriate objective function for the phase response using the minimax ðL1Þ error criteria. A nonlinear minimax algorithm is used to generate a sequence of linear Chebyshev approximation problems. Solving each of the linear Chebyshev approximations provides the required increment for updating the filter coefficients during each iteration. This leads to the linear Chebyshev approximation of the desired response from a linear subspace related to the objective function, which can easily be solved using a variant of Karmarkar’s algorithm. Computer simulations have shown the effectiveness of the proposed technique.

7 Acknowledgments

This work was supported by the National Science Council under Grant NSC90-2213-E002-096.

8 References

1 Crochiere, R.E.: ‘Digital signal processor: sub-band coding’, Bell Syst. Tech. J., 1981, 60, pp. 1633 – 1653

2 Bellanger, M.G., and Daguet, J.L.: ‘TDM-FDM transmultiplexer: digital polyphase and FFT’, IEEE Trans. Commun., 1974, 22, pp. 1199 – 1204

3 Vary, P., and Heute, U.: ‘A short-time spectrum analyzer with polyphase network and DFT’, Signal Process., 1980, 2, pp. 55 – 65 4 Woods, J.W., and O’Neil, S.D.: ‘Subband coding of images’, IEEE

Trans. Acoust. Speech Signal Process., 1986, 34, pp. 1278 – 1288 5 Basu, S., Chiang, C., and Choi, H.: ‘Wavelets and perfect reconstruction

coding with causal stable IIR filters’, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., 1995, 42, pp. 24 – 38

6 Ekanayake, M.M., and Premaratne, K.: ‘Two-channel IIR QMF banks with approximately linear-phase analysis and synthesis filters’, IEEE Trans. Signal Process., 1995, 43, pp. 2313 – 2322

7 Creusere, C.D., and Mitra, S.K.: ‘Image coding using wavelets based on perfect reconstruction IIR filter banks’, IEEE Trans. Circuits Syst. Video Technol., 1996, 6, pp. 447 – 458

8 Argenti, F., Cappellini, V., Sciorpes, A., and Venetsanopoulos, A.N.: ‘Design of IIR linear-phase QMF banks based on complex allpass sections’, IEEE Trans. Signal Process., 1996, 44, pp. 1262 – 1267

9 Lu, C.-K., Anderson, M., and Summerfield, S.: ‘Design of approxi-mately linear-phase allpass based QMF banks’. Proc. Int. Symp. on Digital signal processing, 1996, London, pp. 56 – 61

10 Summerfield, S., and Lu, C.-K.: ‘Design and VLSI implementation of multirate filter banks based on approximately linear phase allpass sections’. Proc. Int. Symp. on Circuits and systems, Monterey, CA, 1998, pp. 413 – 416

11 Lawson, S.S., and Klouche-Djedid, A.: ‘Technique for design of two-channel approximately linear phase QMF bank and its application to image compression’, IEE Proc., Vis. Image Signal Process., 2001, 148, (2), pp. 85 – 92

12 Lawson, S.S.: ‘Design of IIR-based wavelet filter banks and their application to image coding’. Proc. IEE Seminar on Time-scale and time-frequency analysis and applications, 2000, pp. 7/1 – 7/6

13 Lee, J.-H., and Niu, I.-C.: ‘Minimax design of two-channel IIR QMF banks with arbitrary group delay’, IEE Proc., Vis. Image Signal Process., 2001, 148, (6), pp. 384 – 390

14 Vaidyanathan, P.P., Regalia, P.A., and Mitra, S.K.: ‘Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications’, IEEE Trans. Circuits Syst., Apr. 1987, 34, pp. 378 – 388

15 Ikehara, M., Funaishi, M., and Kuroda, H.: ‘Design of complex all-pass networks using Remez algorithm’, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process, Aug. 1992, 39, pp. 549 – 556

16 Adler, I., Karmarkar, N., Resende, M.G.C., and Veiga, G.: ‘An implementation of Karmarkar’s algorithm for linear programming’, Math. Program., 1989, 44, pp. 297 – 335

17 Osborne, M.R., and Watson, G.A.: ‘An algorithm for minimax approximation in the nonlinear case’, Comput. J., 1969, 12, pp. 63 – 68 18 Vaidyanathan, P.P.: ‘Multirate systems and filter banks’ (Prentice-Hall,

Englewood Cliffs, NJ, USA, 1992)

19 Papadimitriou, C.H., and Steiglitz, K.: ‘Combinatorial optimization: algorithms and complexity’ (Prentice-Hall, Englewood Cliffs, NJ, USA, 1982)

20 Nelder, J.A., and Meade, R.: ‘A simplex method for function minimization’, Comput. J., 1965, 7, pp. 308 – 313