design of two-channel iir filter banks with arbitrary group delay

Design of two-channel IIR filter banks with arbitrary group delay

J.-H.Lee and I.-C.Niu

Abstract: The nonlinear optimisation problem that results from considering the design of a two- channel nonuniform division filter bank is solved. This is through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisaf ion procedure ensures that the reflection coefficients lie in (- 1,l). Simulation examples are provided for illustration.

1 Introduction

Quadrature mirror filter (QMF) banks find applications in sub-band coding of speech signals [ 11, communication systems [2], short-time spectral analysis [3], and sub- band coding of image signals [4]. In these applications, a QMF bank is used to decompose a signal into sub-bands with equal bandwidth, and the sub-band signals in the analysis system are decimated by an integer which is equal to the number of the sub-bands. However, uniform sub- band decomposition is not an appropriate scheme to match the requirements for the sub-band coding of speech and audio signals. The most appropriate decomposition must consider the critical bands of the ear. It has been mentioned previously [SI that these critical bands have nonuniform bandwidths and cannot be easily constructed by a conventional tree structure based on two-channel QMF banks. Therefore, it is worth exploiting the design problem of two- channel nonuniform division filter (NDF) banks.

The conditions on the analysis and synthesis filters of NDF banks, as well as methods to design them, are presented elsewhere [SI. Owing to the difficulty in solving the design problem with nonlinear constraints, a structure and design procedure for pseudo-QMF banks has been presented [6]. The main drawback of this method is that FIR filters with complex coefficients are required by the resulting NDF bank to reduce the aliasing distortion. A structure for two-channel NDF banks has been considered [7 ] , and design methods have been proposed [SI for optimally designing NDF banks based on L1 and L, error criteria.

Designing an NDF bank has been widely considered [S-81. However, linear-phase FIR analysis filters were employed for developing these design techniques.

0 IEE, 2000 IEE Proceedings online no. 20000489 DOZ; 10.1049/ip-vis:20000489 Paper first received 19 May 1999 and in final revised form 28th March 2000 The authors are with the The Department of Electrical Engineering, National Taiwan University, Taipei, 106 Taiwan

534

Although using linear-phase FIR analysis filters leads to a favorable problem formulation for design, the system delay kd of the designed NDF bank is determined by the lengths of FIR filters used; hence, the long overall system delay of an NDF bank with linear-phase FIR analysis filters may prohibit practical applications. It is well known that an IIR filter requires lower order than an FIR filter under the same stopband energy. For a given passband group delay, IIR NDFs have better passband ripple/stopband attenuation and/or lower computational complexity than FIR NDFs under the same stopband energy for the analysis and synthesis filters. Several approaches dealing with the design problem of two-channel IIR uniform division filter banks with conventional filter structures have been reported [9-111. There is, however, practically no published work concerning the design of two-channel IIR NDF banks with arbitrary group delay.

In this paper, we consider the design of two-channel IIR NDF banks with arbitrary group delay in the sense of L, error criteria. Utilising a lattice structure for the denominators of the IIR analysis filters, a design technique based on an approximation scheme is presented for efficiently solving the resulting design problem that is basically a nonlinear optimisation problem. During the design process, the proposed technique finds the tap coefficients for the numerators and the reflection coefficients for the denominators of the IIR analysis filters simultaneously. It ensures the stability of the designed IIR analysis filters by incorporating an efficient stabilisation procedure to make the magnitude of each reflection coefficient within - 1 and +l. Computer simulations show that the IIR NDF banks designed using the proposed technique provide a satisfac- tory performance, although the designed results are only local solutions (instead of the global optimum).

2 Two-channel NDF banks with arbitrary group delay

Consider the two-channel NDF bank, with the architecture given previously [7], which is shown in Fig. 1. The analysis low-pass and high-pass filters are designated by H,(z) and H,(z), respectively, and the synthesis low- pass and high-pass filters are designated by Fo(z) and F , (z) , respectively. B,(z) and B, (z) are low-pass filters

IEE Proc.-vis. Image Signal ProcesJ.. l6f. 147, No. 6, December 2000

4-

T I I

e jxn

a

jxn e

b

Fig. 1 U Analysis system b Synthesis system

Two-channel NDF bunk systenz striictuve

responsible for achieving aliasing-free operation during the rational decimation and interpolation. It can be shown that using the modulations of multiplying expCjnz) in the high- pass sub-band channel leads to the favourable result that B,(z) can be a low-pass filter with real coefficients. The desired magnitude responses for the analysis filters Ho(z) and H I (z) with passband widths equal to Lorc/L and L , z /L , respectively, are shown in Fig. 2. L =Lo + L, , and up and os denote the related band-edge frequencies satisfying

Assume that the associated magnitude responses are op + = 2nLolL.

set to

Further, assume that Ho(z) and H,(z) have zero stopband response. As shown previously [SI, the input/output

relationship of the NDF bank in the frequency domain is given by

where Go and G, are the resulting group delays of the upper and lower channels, respectively. W, =exp( -j2zlL). Substituting L =Lo + L , , Fo(e'") = Ho(e'") and Fl (e jw) = -H,(ej") into eqn. 2 yields

where

The first term of eqn. 3 represents the response of a linear shift-invariant system T(eJW) with input X(eJw) , and the other two terms represent the resulting aliasing distortion. Therefore, perfect reconstruction for a group delay kd requires the following conditions:

PR I : T(eJW) must be equal to e -Jkdw for all o. PR 2: The magnitude V , ( o ) of Vl(eJw) must be zero, i.e.

V , ( o ) = 0, for all o.

PR 3: The magnitude V2(0) of V2(eJu1) must be zero, i.e.

V2(0) = 0, for all CO.

For simplicity, assuming GO = GI , we can neglect the phase term e-Jwco in eqn. 4 and express T(eJf") as

Furthermore, we note from eqns. 5 and 6 and L =Lo + L , that the aliasing distortion can be eliminated if

IEE Proc.-Vis. Image Signal Process., Vol. 147, No. 6, December 2000 535

+

3 Design of two-channel IIR NDF banks in L, sense

T(eJ") = e-Jkdr'',

Ho(eJw) = 0,

Hl(eJW) = 0,

for 0 F o 5 71,

for 0, 5 o 5 n,

for 0 5 CO I wp,

for cop I w 5 w,

1 Ho(e/">Wo(e/(W-O~-".) ) = L ~ l ( e / ( u - o > p - 4 )HI (eJW),

\

LLO LL,

3, I Problem formulation Here, we consider the design of the two-channel IIR NDF as shown in Fig. 1. Let the low-pass analysis filter be an IIR filter with order Mo/No (i.e. MO zeros and No poles) and transfer function Ho(z) = Ao(z)/BNo(z). The numerator Ao(z) is an Moth-order polynomial with tap coefficient vector a. = [aoo, sol,. . . , u, ,~]~; and BN,(z) is an Noth- order FIR lattice filter with reflection coefficient vector

denotes the transpose operation. Similarly, let the high-pass analysis filter be an IIR filter with order Ml/N, and transfer function HI (2) = A , (z)/BN, (2). The numerator A , (z) is an MI th- order polynomial with tap coefficient vector al = [alo, a,,, . . . , a I M I I T , and BN,(z ) is an N,th-order FIR lattice filter with reflection coefficient vector kl = [k , , , k, , , . . . ,k lN,IT. Fig. 3 shows the system structure for BN9(z) and BN,(z), which can be obtained from the following recursive formula [ 121:

= , ko?, . . . , koNoIT, where

Bo@) = Q o ( 4 = 1

Bnk> = Bn-1(z> + knz-'Qn-,(z)

Q,<4 = knB*-,(z) +z- 'Qn- , (4

(10)

Hence, the overall design task is to find the required tap and reflection coefficients {arnl, kin} for the stable IIR filters H,(z), i = O , 1, such that the conditions shown in eqn. 9 must be satisfied. Accordingly, we consider that the overall error function E to be minimised in the L, sense can be expressed as the following weighted sum of four terms:

Qo(z)=I QN.~ (Z)=*

Fig. 3

536

Lattice structure for Nth-order FIR lattice$lter

where a, p, and y represent the relative weights between the four error terms. We note from eqn. 11 that the overall error fimction E is a function of fourth degree in the tap and reflection coefficients. Therefore, minimising E directly leads to a very highly nonlinear programming problem in addition to the stability problem for Ho(z) and HI (z).

3.2 Proposed design technique Let (01 = O , w2,... ,w1=wP, ~ I + I , . . . , ~ I + J , ~ I + J + I =os, . . . , oI+J+K = n} be a grid of equidistant frequencies, distributed in the range of w=O to w=n, for evaluating the magnitude response of the NDF bank and the related error terms defined as above. Moreover, assume that the set SI contains the grid points (0, = 0, w2, . . . , wI =up}, S, contains the grid points {col+,, . . . ,w ,+ ,> , and S, contains the grid points (wI+,+1 =os, o,,+?, . . . , = n}. We then rewrite eqn. 1 1 as the following approximation:

= le-/kdwI - F(el"r>l2

W,ESI us,us,

3.2. I Determination of initial guesses for H, (2) and H,(z): Our design experience shows that using better initial guesses for Ho(ejw) and H I (e'") usually provides better design results. To initiate the design process, we propose a procedure for determining appropriate initial guesses @(z) and q ( z ) for Ho(z) and H,(z), respectively. According to the principle of a two-channel NDF bank, we define two desired frequency responses Do(eJ") and D , (e'") as

a e - j $ l o , for w E LO, wP1 I I -

I [

0, for o E [U,, z]

0, for w E [0, wp]

Two FIR filters G,(z) and G,(z) with orders 2N0 and 2Nl are designed to optimally approximate Do(ejw> and D , (e'"),

IEE Proc.-Vis. Image Signal Process., Vol. 147, No. 6, December 2000

Table 1 : Significant design results for example 1

IIR NDF bank FIR NDF bank

Filter order No. of coefficients PRE in lT(ej"')l (dB) Max. variation of group delay of T(eJo) NPSR (dB) of Ho(ejw) NPSR (dB) of Hl(ej'o) Max. variation of passband group delay of Ho(ejw) Max. variation of passband group delay of Hi (ejo) SEE of HO(ej'lJ)

SEE of H, (e'(") Max. variation of filter bank response No. of iterations

10 /10 ,11 /11 44 0.01 48 0.0583

32.21 32.07 0.01 58

0.0230

3.33 x 10-2 5.68 x 1 0-2

2.27 10-3

37

32, 33 67 0.0238 0.0746

32.10 31.95 0.0160

0.01 63

3.85 x lo-' 5 . 1 2 ~ lo-' 2.76 10-3

9

respectively, using conventional least-squares error criteria. Let the resulting filter coefficients be given by { hoo, h,, , . . . , h0(2M0)} and {hl,, h l l , . . . , h 1 ( 2 N , ) } > respectively.

Through the use of the model reduction algorithm presented previously [ 131, we find two IIR filters where the numerator and denominator are of order No (corresponding to Go@)) and of order N I (corresponding to Gl(z)), respectively. Assume that the IIR filters have denominators CO@) and C , (z) with coefficients {coo = 1, col, . . . , cONO} and {cl0 = 1, c I 1 , . . . , c I N , } , respectively. The initial lattice systems B$o and B;, with reflection coefficients {GI, G,, . . . ,GNo} and {ql, k?,,. . . ,kyN,} corresponding to Co(z) and C,(z), respectively, can then be found since there exists a one-to-one correspondence between {cIo, cI l , . . . , c IN,} and {kp,, ky2,. . . ,vN,f [14] for i = O , 1.

The best L, solution for the corresponding initial numerator AP(z),i = 0,1, can be obtained by solving the following optimisation problem:

AP(eJ"> ,2 minimise lDi(eJ") - ___ B&(eJW) '

Table 2: Tap and reflection coefficients for example 1

For evaluating the related error functions given by eqn. 15, we again take a set of discrete frequency points linearly distributed over S = [0, up] U (cop + us ) /2 U [ws, n]. Let S, = S1 U (up + 0,)/2 U S3 = { w1 = 0, w2, . . . , uI = cop, wI+, =(up -+ ~ $ 2 , 01+2 =us,. . . , w I + ~ + , = n } be the dense grid of frequency points, and U, be a complex ( I + K + 1) x (M, + 1) matrix with its (m, n)th element given by

1 5 n l M , + I

d, is a complex ( I + K + 1) x 1 vector, with its mth element given by

(17)

The initial coefficient vector up = [upo, . . . , uPM,lT o f A:(z), optimal in the L, sense for eqn. 15, can be found by minimising IUpP - d,I2 = (U,u: - d,)H(U,ap - dJ, where

denotes the complex conjugate transpose. Clearly, this leads to the optimal solution given by up = ~Re(Z7~U,)}- l [Re(Ufid,)] , for i = O , 1, where Re(X) represents the real part of the matrix X . After finding the appropriate initial guesses q ( z ) = AP(z)/BL$, we present an iterative procedure step by step for computing Ai(z ) and BN,(z), for i = 0, 1, during the design process.

d,(m) = D,(eJ"-), 1 5 m 5 I + K + 1

3.2.2 Iterative procedure: Step I : At the lth iteration, let the analysis filters be

Utilise a linearisation scheme to rewrite f(e'wi) and ?(ej"1) as follows [15]:

and

0 1 2 3 4

5 6 7 8 9

10 11

8.6657905 x 1 0-3

- 4.041 6626 x 1 0-2

7.21 62064 x 1 0-2

- 5.655851 0 x 1 O-* 3.0787734 x 1 0-'

- 5.9047004 x 1 O-* 2.9256295 x 1 0-2

8.6821333 x lo-' 1.661 3748 x IO-' 1.1964706 x lo-' 8.91 18083 x 1 0-2

- 5.2321871 x lo-' 7.791 1067 x lo-'

- 7.2660247 x 1 OW' 6.7708888 x lo-'

- 6.3779579 x lo-' 5.8390526 x 1 OW'

- 4.5798891 x 1 OW' 2.476871 1 x lo-'

- 6.9787484 x 1 OW2 6.001 3386 x 1 Ow3

- 6.621 2287 x 1 0-3

- 2.6735861 x 1 0-'

- 4.8663890 x I O Y 2 - 3.8462788 x 1 0-2

2.2971 930 x 1 0-'

8.065731 9 x lo-' 1.321 771 0 x 1 OW'

- 2.2281 802 x 1 OW' - 3.2913158 x lo-'

1.3735270 x lo-@ -1.4431611 x10-o

6.2466860 x lo-'

- 1.0973161 x lo-' 6.51 22570 x lo-' 4.7095441 x lo-' 4.0467721 x lo-' 3.551 8737 x lo-' 3.1592069 x lo-' 2.7136095 x lo-' 2.0592667 x 1 OF' 1.2434145~ lo-' 5.2666408 x 1 O-' 1.21 87899 x 1 0-2

IEE Proc -Es. Image Signul Process.. Vol. 147, No. 6, December 2000

-

537

respectively, where the numerators k,(eJ('g) have coefficient vectors ii, = [a",o, a",, , . . .L dlM,IT , for i = 0,1. We then find the optimal numerators A,(eJ"j). First, several matrices are defined as follows: WO is an (Z+J+K) x (MO + 1) matrix, with the (m, n)th entry given by

- 9 -

8 - - 3

$ 7 - Q

m

w , , E S l U S , U S , , n = 1 , 2 ) . . . , M o + l

(19)

W , is an (Z+J+K) x (M, + 1 ) matrix, with the (m, n)th entry given by

I I I I I

W, E SI U S 2 U 5'3, = 1,2, . . . ,MI + 1

(20)

Yo is an K x (MO + 1) matrix, with the (m, n)th entry given by

w,+I+j €5'3, n = 1 , 2 , . . . , Mo+1

Y, is an Zx ( M I + 1) matrix, with the (m, n)th entry given by

e-;(fl-l)(om

Y, (m , n ) = ~ w, E S , , n = 1 , 2 , . . . , M I + 1 Bfy, (eJ"V1) '

(22)

Zo is an ( J+2) x (MO + 1) matrix, with the (m, n)th entry given by

C O , E C O ~ U S ~ U W , , I Z = ~ , ~ , . . . , M o + l

(23)

2, is an ( J+2) x (MI + 1) matrix, with the (m, n)th entry given by

f f ; (eJ("m+/- l -".-"J) I)%+/-,

LL1 NI

Z,(m, n ) = - B' (ejSu+/-l) '

w, E wp us, uo,, n = 1 , 2 , . . . , M I + 1

(24)

Moreover, w= [WO, Wll, yo, = [Yo, 0KX(Ml+l)I, YI1 = [O, Y , ] , where 0, denotes a zero matrix with size shown by its subscript M x N, Z = [Z,, Z , ] , and zd is an (Z+J+K) x 1 complex vector, with the mth entry given by

zd(m) = e - J k d w m , U,,, E S, U S, U S, (25)

As a result, the required coefficient vectors ii, can be found from eqn. 12 by minimising the following error function:

" I ' I

- 6 0 1 \ 0.1 0.2 0.3 0.4 0.5

-70- ' ' ' ' '

normalised frequency

1

6 t r l

5 r l I I I h I I

0 0.1 0.2 0.3 0.4 0.5 normalised frequency

b

Fig. 4 esample 1

~ Hn(e J"')IJ(LL,) _ - - HI (e Jo)IJ(LL,) a Magnitude response b Group delay response

Magnitude and group delay responses of analysis Jilters for

where the vector 2 = [2;, iir]'. Clearly, the optimal solution for eqn. 26 is given by

since eqn. 26 is a quadratic function of the coefficients 8,0, a",,,. . . , E r M , , for i = O , 1. After obtaining the coefficient vectors iii, we update the numerator coefficients of e(=) as follows:

for i = 0, 1, where z(0 < z < 1) is a smoothing parameter with the best value chosen experimentally. Step 2: Compute the gradient matrices Ht = [V?(e~'"~)] for wi E SI US, US,. H, = [VHk(ejWl)j for a, E s,, H , = [VH', (e'"')] for wi E SI, and Ha = [V V'(e;"l)] for wi E cop US, U os, where V represents the gradient operator [ aiakol, a/ako2, . . . , a/akoN,, a/i3aoo, a/&,, , . . . , a/aaoM0, aiak,,, aiak,,, . . . , aiakINll aiaa,,, aiaa,,, . . . , a~aa,,~]. Step 3: Use a linearisation scheme to approximate the frequency response error E' for eqn. 12 due to a perturba- tion in the coefficient vector in the linear subspace spanned

IEE Proc.-Vis. Image Signal Process., Vol. 147, No. 6. December 2000 538

0.025

0.020 2.5 r 0.015

ai 0.010

0.005

m O

'2 -0.005

E -0.010

C

a,

U

L

U

0)

-0.015 1 V

-0.020 -0.025 ~ 0.2 0.3 0.4 0.5 0.1


a

19.10

19.08

- % 19.06

2 19.04

19.02

Q

0 8 19.00 ?? 2 18.98

Q 18.96 2

- a, U

& 18.94

18.92

0 0.1 0.2 0.3 0.4 0.5


b

Fig. 5 a Magnitude response b Group delay response

Magnitude and group delay responses oj T(eJw) for example I

by the gradient matrices H,, H,, H , , and H,. This means that the approximation error given by

,EL = le-~bwi - f / (e /uz) - Vf'(eI"t)vl' W,ESI us,us,

is computed, where the vector v = [Ak,, Au,, A k l , b u l l =

Ak,,, . . . , dk lNl , Aa,, , A a , , , . . . , Aa,, ,]T contains the incre- ments of the independent coefficients to be found.

We solve the minimisation problem of eqn. 29 to obtain the increment coefficient vector v. Let r1 be an ( I+ J f K ) x 1 compl$x vector, with the ith entry given by r1 (i) = e -jkdwr - T'(ejoi), for wi E S , U S2 U S3; r2 be an I x 1 complex vector, with the ith entry given by r2(i )= -H:(ejOi), for w i ~ S 1 ; r3 be a K x I complex vector, with the ith entry given by r3(i) = -f(,(ejwI+I+J),

for oi+l+J E S,; and r, be a ( 5 + 2 ) x 1 complex vector,

[Akol, Ako2,. . . , A k w O , Aaoo, Aaol , . . . , AaoM , Akil,

IEE Proc.-Vis. Image Signal Process., Vol. 147, No. 6, December 2000

@?- 0 X

7-

v

1.5 0 8 ??

3 1

E

a, U

.% K ol

0.5

0 0.1 0.2 0.3 0.4 0.5


Magnitude response of variation of filter bank response for

o " " ' " ' l "

Fig. 6 example 1

with the ith entry given by r4(i)= T?(ejoi+f-l), for oi+, - E up US, U 0,. The optimal solution v for eqn. 29 can then be expressed as

v = {Re[HfH, + M H ~ H , + P H ~ H , + ~ H ; H , I } - '

x Re[HFrl + aHyr , + P H f r , + yHcr,] (30)

Step 4: Perform a line search by using the Nelder-Meade simplex algorithm [16] to find the best step size t with 0 < t < 1, to update the numerators and denominators of H,(z) and Hl(z) such that the following error function reaches its minimum:

+ a lN{+1(ejwi)12 W , G

+ P IHk1(ejWI)12 W,ES3

+ y C lP'+'(ejWi>12 (31) U, EWpUSZUW,

subject to the constraints of maxlkf? + tAk,.l < k,,,,, for i = O , j = 1 , 2 , . . . , N , and i = l , j = 1 , 2 , . . . , N I ; where Hb+'(e@') and #+'(e'") have tap and reflection coefficient vectors given by U;+' = [ab, + tAa,,, ab, + tA ~ 0 1 , . . . , abMn + tAaoM,,IT; = [k',, + tAko1, G2 + tAk02, . . . , kONn + tAkoN,lT; U{+' = [U:, + tAalo , + tAa, . . , a',M, + t A a I M l l T ; and k : + ' = [ $ , + t A k l 1 , $ 2 + t A k 1 2 , . . . , $ N , + t A k l N I l T , respectively. f"+l (e jW) and f'+'(ejci' l) are the corresponding filter bank response and aliasing distortion. Moreover, k,, is a preset maximal absolute value and must be less than 1 for the reflection coefficients, in order to ensure the stability of the designed IIR NDF bank. Step 5: Compute the overall error function of eqn. 12 corresponding to H'o+'(e'"), T:+'(ejw) and V{+'(ejw), which is in fact giyen by eqn, 31. Step 6: Compute the ratio IE' - E'+lI/E'. If this ratio is larger than a preset positive number c, then set 1 = 1 + 1 and go to Step 1. Otherwise, we terminate the design process. Remarks: There are two situations where the gradient matrices H,, Ho,Hl, and H, may degenerate. Case I : The columns of H,, H,, H I , and Ha are not linearly independent. The optimal solution for eqn. 29 is then not

539

Table 3: Significant design results for example 2

IIR NDF bank FIR NDF bank

Filter order No. of coefficients PRE in jT(e1") j (dB) Max. variation of group delay of T ( e 4 NPSR (dB) of Hb(elw)

NPSR (dB) of HI (elw)

Max variation of passband

group delay of H,,(e'") Max. variation of passband group delay of H, (elo) SEE of /-/,(e'") SEE of Hl(eJw) Max. variation of filter bank response No. of iterations

13 /14 ,17 /17 63 0.01 76 0.0703

40.88 40.66 0.0091

0.0247

4.05 10-3 7.42 10-3 2.10 10-3

41

49,49 100 0.0190 0.091 8

40.53 40.42 0.0304

0.0250

2.99 10-3 6.45 10-3

2.57 10-3

10

unique. To find an appropriate optimal solution, we construct matrices G,, Go, G I , and Ga by choosing the independent columns from H,, HO, H I , and H,, and a vector U by choosing the components of v corresponding to the independent columns. Then use G,, Go, G , , Ga and U to replace H,, H,, H I , Ha, and v in eqn. 29. On the other hand, if only Ht (or H0 or H I or Ha) has columns not linearly independent, then we construct a matrix G, (or Go or GI or G,) by replacing the elements of those columns that are not linearly independent with zero elements from Hb (or H, or N, or Ha). Then use Gt (or Go or GI or G,) to replace Ht (or Ho or HI or Ha) in eqn. 29 to obtain an appropriate optimal solution. Case 2: At the Ith iteration, the ith reflection coefficient k, may have the absolute value equal to kma. To tackle this difficulty, we construct a vector U by eliminating AkL of the

Table 4: Tap and reflection coefficients for example 2

0

m -10 U U) t -20 0 Q

-30

.- a -40 a, U

C 0,

E -50

-60

-70 0 0.1 0.2 0.3 0.4 0.5


a

19

18

E 17

6 16

5 15 a $? 14

$ 13

- I (I)

a,

- E

; 12 0

1 1

10

I I I I I I I

1 1 I

0 0.1

. ---

U 0.2 0.3 0.4 0.5


b

Fig. 7 Magnitude and graup delay responses of analysis filters for example 2 ___ Ho(ej")P (U,,) - - - H,(eJ")P(LL, ) a Magnitude response b Group delay response

0 1 2 3 4 5 6 3

8 9

10 11 12 13 14 15 16 17

- 3.6798234 2.6099284

- 9.0796157 2.1 271 854

- 3.8010473 5.4977591

- 6.6896321 6.9851301

- 6.2758842 4.8330749

- 3.0750743 1.5743740

- 5.6882050 1.320964Q

10-3 10-2 10-2

10-1

10-1

10-t 10-1 lo-' 1Q-' 10-1

10-f i o -+ 10-2 10-2

- 8.2994053 9.3802032

- 8.7566468 8.5826413

- 8.1937563 7.4277865

- 7.0493877 7.0366242

- 6.8684451 6.8390959

- 6.6734683 5.5310626-

- 3.Q505849 7.328221 6

lo-' lo-' 10-1 lo-'

lo-' lo-' 10-1 lo-'

10-1

10-1

i o - ' lo-' lo-' 10-2

1.5980566 3.5848626 5.8518981 7.6135849 7.1787741 2.9788998

- 5.9077057 - 1.8788240 - 3.261 5090 - 4.1421895 - 3.5958718 - 1 .go35730

8.9533637 3.4236521 1.8856250

- 6.1255005 5.6539967

- 1.7337973

10-3

10-3 10-3 10-3 10-3 10-3 10-2 10-2

10-2 10-3 10-2 10-1 10-0 10-0

10-0

10-0

10-3

1 W 2

-8.1584134 lo-' 8.4751 020 lo-' 5.4108119 1.1230581 io - ' 1.1278975 lo-' 1.0793339 lo-' 1.0367129 lo-' 9.8855147 lo-* 9.1518793 8.0536860 t a-2

6.6165397 4.9924064 io-' 3.3990564 lo-* 2.0406182 1.04t2525 4.2050121 top3 1.10958031 0-3

540 IEE Pmc - l'is Image Sfgnat Process, VO! I I Z No 6, December 2000

vector v and four matrices G,, Go, G1 ~ and G, by eliminating the columns of HI, Ho, H , , and Ha, corresponding to Aki, respectively. Then, we use G,, Go, GI, G,, and U to replace H I , Ho, If,, Ha, and v in eqn. 29.

4 Simulation examples

In this Section, we present several simulation examples of designing two-channel IIR NDF banks for illustration. For all design examples, the spacing for two adjacent frequency points in [0, x] is set to d299, i.e. the number of grid points taken in [0, n] is 300. The presented design examples are, in fact, related to the optimal design of two- channel IIR NDF banks with low group delay. The performance for each of the designed filter banks is evaluated in terms of the peak reconstruction error (PRE), the normalised peak stopband ripple (NPSR), and the stopband error energies (SEE) of the designed Ho(z) and H,(z). They are given by

PIIE(dB) = max{l2010g,~ l?(oj)ll) for w j E [ 0 , 4

NPSR,(dB) = -20loglO ( ma------ 'H&&l)

NPSR, (dB) = -20 logl, ( max- l H 4 l )

GD(x) denotes the group delay of x. For comparison, we also use the proposed technique for

each design example with a FIR NDF bank and the same group delay kd. For designing FIR NDF banks, the compu- tations required for finding the denominators in the proposed iterative procedure are not necessary and can be discarded. Example 1: We use the following design specifications: the group delay kd is 19; up = 0 . 3 ~ and CO, = 0 . 5 ~ ; Lo = 2 and Ll = 3; MO =No = 10 and M , = N I = 11. In the FIR case, Ho(z) and H,(z) have orders equal to 32 and 33, respectively. Accordingly, the numbers of independent coefficients for the IIR and FIR NDF banks are 44 and 67, respectively. Moreover, the values of a, b, y, z, and E are set to 0.012, 0.012, 0.05, 0.5, and lop3, respectively, for the IIR NDF bank; they are set to 0.015, 0.015, 0.02, 0.5, and I O p 3 , respectively, for the FIR NDF bank.

Table 1 shows the significant design results for both of the designed IIR and FIR NDF banks. The resulting tap and reflection coefficients of the designed IIR NDF bank are

IEE Proc -VIS Image Signal Process, Vol 147, No 6, December 2000

0.025

0.020

0.01 5

0.010 L A

-0 005

-0 010

-0.015 -

-0.020 -

-0.025 I I I I I I I I I

0 0.1 0.2 0.3 0.4 0.5 normalised frequency

a

F 29.10

29.08

29.06

29.04

29.02

29.00

28.98

28.96

28.94

28.92

28.90 0 0.1 0.2 0.3 0,4 0.5


b

Fig. 8 a Magnitude response b Group delay response

Magnitude and group delay response of p e l " ) for example 2

listed in Table 2. Fig. 4 plots the corresponding magnintde responses and the group delays of Ho(eJ")/l/r;Lo and H, (eJW)/2/(LLl). Fig. 5 depicts the magni$de response and the group delay response in samples of Tie@') for the designed IIR NDF bank. The magnitude of e -J "" - T(eJW)

0 0.1 0.2 0.3 0.4 0.5


Magnihcde response of' variation of jilter bank response for Fig. 9 example 2

54 1

is shown in Fig. 6. From the design results of using the proposed technique, we note that the IIR NDF bank provides the advantage of requiring a lower order for achieving a comparable performance than the FIR NDF bank. Example 2: The design specifications used are as follows: the group delay kd is 29; cop = 0 . 1 2 ~ and o, = 0 . 2 8 ~ ; L o = l and L , =4; hf0=13 and N0=14; and M I = N I = 17. In the FIR case, both H,(z) and Hl(z) have orders equal to 49. Accordingly, the numbers of independent coefficients for the IIR and FIR NDF banks are 63 and 100, respectively. Moreover, the values of a, ,B, y , z and t are set to 0.02, 0.02, 0.04, 0.5 and 5 x respectively, for the IIR NDF bank; they are set to 0.1, 0.1, 0.05, 0.5, and respectively, for the FIR NDF bank.

Table 3 shows the significant design results for both of the designed IIR and FIR NDF banks. The resulting tap and reflection coefficients of the designed IIR NDF bank are listed in Table 4. Fig. 7 plots the corresponding magnitude responses and the group delays of H,(eJ”)l &Lo and H,(eJ”)12/LLI. Fig. 8 depicts the magnitude response and the group delay response in samples of T(eJO) forthe designed IIR NDF bank. The magnitude of e -Jkdw - T(eJW) is shown in Fig. 9. We again observe from the design results that the IIR NDF bank provides the advantage of requiring a lower order for achieving a comparable performance than the FIR NDF bank.

5 Conclusions

We have employed recursive analysis filters with a lattice denominator for designing IIR NDF banks. At each iteration, the proposed design technique adjusts the tap and reflection coefficients for the analysis filters, to reduce the resulting squared error and keep the designed IIR analysis filters stable. For the first task, a linearisation scheme has been proposed to solve the resulting highly nonlinear programming problem. For the second task, the stability of the designed recursive analysis filters is ensured by incorporating an efficient stabilisation procedure. Comp- uter simulations have shown the effectiveness of the proposed technique.

6 Acknowledgments

This work was supported by the National Science Council under Grant NSC88-22 18-E002-03 1.

7 References

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542 IEE Proc.-Vis. Image Signal Process.. Vol. 147, No. 6, December 2000

design of two-channel iir filter banks with arbitrary group delay

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