Filter design for polyphase filter banks with arbitrary number of subband channels

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FILTER DESIGN FOR POLYPHASE FILTER BANKS WITH ARBITRARY NUMBER O F SUBBAND CHANNELS

Ju-Hong Lee and Wen-Juh Kang

Department of Electrical Engineering National Taiwan University Taipei, Taiwan. Republic of China

ABSTRACT

This paper concerns the filter design problem for a recent proposed polyphase filter bank with arbitrary number of subband channels. According to the perfect reconstruction condition for I-D case presented in [ I ] , an analytical formula for the filter design is developed. Compared with direct numerical design algorithms. this analytical formula allows one to design the required low-pass FIR prototype filter with much less computational complexity and obtain better filter bank performance. A design example is given for illustration.

I. INTRODUCTION

Polyphase filter banks (PFB) have been successfully used t o constitute a multirate signal processing system for splitting

a signal into N subbands and allowing the resynthesis of the signal from the subbands. Current applications are mainly in subband coding of speech signals ( 2 , 31, TDM-FDM trans- multiplexing systems [4, 5 1 , and short-time spectral analysis [ 6 ] . Consider the filter design for PFB, many results have been reported in the literature (7-141. Most o f these methods employ numerical algorithms directly to minimize the associated error measure in filter design process. [ 151 derived an analytical formula to represent the error measure. As a result. the filter coefficients can be found by a straight nonlinear optimization procedure and saving computations can be achieved. However. all of these methods have focused o n the design o f quadrature mirror filters (QMF).

In [ I ] , based on nonsymmetrical frequency band alloca- tion in constructing band-pass fdter for subband channels, one of the authors has developed a onedimensional (I-D) PFB with N axbitrary. Each constructed band-pass filter is a nonsymmetrically frequency-shifted version o f a low-pass prototype filter. It has been shown in [ 11 that the new PFB

possesses -me advantages over conventional PFB's. In this paper, we concern the filter design problem for the I-D PFB. An analytical formula for finding the coefficients of the required I-D low-pass FIR prototype filter is derived. Using thls formula. one can design the prototype filter with much less computations than direct numerical algorithms. Moreover, the simulation example shows that the resulted reconstruction error is smaller under the same number o f subband channels and filter length.

11. THE PROPOSED ANALYTICAL DESIGN FORMULA The basic structure of a I-D filter bank with N subbands t o be considered is shown in Figure 1. Its efficient structure based on polyphase network and fast Fourier transform has been developed in [ I ] . The band-pass filters H,(w) and H,(w)

are given as

h

respectively. where G i ( w ) = G ( w - in/N) and G ( w ) denotes the I-D low-pass prototype filter. Assume that C ( w ) is linear phase. The associated condition on the magnitude response of G ( w ) for perfect reconstruction is given by [ 1 ]

2 N - 1

k = O Z l C ( w - k n / N ) I 2 = N for 0

<

w S 2 n ( 2 ) T o design the low-pass prototype filter G ( w ) which approximates the condition of (2) in some optimum sense, we define an appropriate approximation error measure E as

E = E,

+

a E , ( 3 )

where E, denotes the ripple energy over the whole frequency range and is given as

2 N - 1

E, = J:( :o l C ( w - Pn/N) l 2 - N

l 2

d w (4)

1720

(2)

E,

denotes the stopband energy of G(w) and is given as

The Q in (3) designates the weight between E, and E,. To solve

the problem o f minimizing E, unconstrained numerical optimization algorithms can be employed t o find the filter coefficients of G ( w ) . However, at each iteration, we have t o compute the Fourier transform of Gi(w), i = 0, 1,

.

. .

.

, 2N - 1, in older t o compute E, and E,. This requires considerable computations and hence leads to substantial computer time requirement

Next, we derive a closed analytical formula to represent (3) and turn the optimization problem of (3) into a direct search problem of fdter coefficients. Assume that G ( w ) is a linear phase

FIR

filter with length M.

Then G ( w ) can be expressed as

= exp (- j (M - 1 ) w/2) i f M i s o d d , or = exp (- j ( M - 1) w/2) ( M - l ) / Z ( Z d ( n ) cos((n- (M - 1)/2) w ) ) , n= 0 if M is even

.

where d (( M - 1)/2) = g ((M - 1)/2)

,

for M odd, d (n) = 2g(n) , for 0

<

n

<

( M - 3)/2 and M odd,

for 0

<

n

<

M/2 - 1 and M even.

using (61, we obtain (217) 333-4789 1721 cos(w (m - (M - 1)/2)) cos(w (n - (M - 11/21] 1 ( M - 1 ) / 2 ( M - 1 ) / 2 = - Z Z 2 m = O n = 0 d ( m ) d ( n ) ( c o s ( w (m - n))+ cos(w ( m + n - M + 1))). (7)

F o r the case of odd M, substituting (7) and the following relationship Z N - 1 k = O Z: exp (j (w - 2nk/2N)

a )

2N exp (jnw) , if n/2N = integer, ={0 , e k e , or equivalently, Z N - 1 k = O Z c o s ( ( w - k n / N ) P ) 2N cos(fw) , if n/2N = integer,

={

0 , else,

into (2) and performing some manipulations yields 2 N - 1 k=O Z I G ( w - k r / N ) l ' ( M - 1 ) / 2 ( M - 1 ) / 2 m = O n = O = N Z Z d ( m ) d(n) (A (m, n) + B (m, n))

4

F ( w ) (8) where c o s ( ( m - n) w ) , if (m - n)/2N = integer,

,

else, A ( m , n ) = c o s ( ( m + n - M + l ) w )

,

0 , e l = , if (m

+

n

-

M

+

1)/2N = integer, for 0

<

m, n

<

(M - 1)/2.

use the result of (8) and the following relationship

Consider theintegral (4). We expand the square term and

n , i f k = O , 0 , else,

1:

c o s ( k w ) d w =

to obtain

~

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E, = J; (F’ ( U ) - ? N F ( w )

+

N 2 ) d w ( M - I ) / 2 n = O = N2 n ( I - ( Z 2d2 (n)) - 2d2 ((M-l)/2)

+

1 ( M - 1 ) / 2 2 I = O J = O r n = ~ n = ~

-

2 B Z B d ( m ) d ( n ) d ( i ) d ( j ) ) (9) where the four integers m, n, i. and j must satisfy one of the following relationships:

a. b. c.

f o r 0 Q m. n,i. j

<

(M - l ) / 1 and integers k l , k 2 , and k , . For the E, of ( 5 ) , using the following relationship

m - n = 2Nkl and i - J = 2 2Nk, m - n = 2Nk, and i + j - M + 1 = + 2 N k 2 m + n - M + 1 = 2Nk, and i + j - M + 1 = + 2 N k , n ( 1 - I/N) (- sin(kn/N))/k , if k # 0, , i f k = 0 (10)

Jz,N

c o s ( k w ) d w = we obtain 1 ( M - 1 ) i . Z ( M - l ) / 2 2 m = ~ n = ~

- -

z

Z d(m) d(n) (C (m - n) + where C ( 0 ) = n ( I - I/N), C (k) = (- sin (kn/N))/k , if k/N # integer C ( k ) = 0 , if k f 0 and k/N = integer.

From (9) and ( 1

i),

we note that the approximation error measure E can be expressed as a closed analytical formula in terms of the filter coefficients, g(O), g( l ) ,

.

, g(u - I ) . On the other hand, for the case of even M , following the similar procedure. we can obtain the similar result. Therefore, com- puting the filter coefficients by minimizing E can be performed by utilizing a simple direct search method.

111. EXPERIMENTAL RESULTS

In this section, we present an example for illustration. The filter length M and the weight (I were set to 28 and I ,

respectively. The number N of subbands was 3. Based on the proposed analytical formula, we employed a simple gradient method to iteratively search the filter coefficients, g(O), g( I),

.

. . .

. , g(27). The number of iterations was 200. For com- parison, the direct numerical optimization of (3) was also performed. Figure 2 shows the frequency responses of C ( w )

using both methods. The corresponding reconstruction errors are shown in Figure 3. Table 1 compares the computational complexity in terms of the number of operations required at each iteration step. Table 2 lists the computed filter coef-

ficients, g(O), g( l), , ,

.

, , , g( 13), for the simulation of linear

phase FIR filter design. From these results, we observe that the proposed analytical design formula is superior to the direct numerical optimization algorithm.

REFERENCES 1. 2. 3. 4. 5 . 6. 7. 8. 9. IO. 1 1 . 12. 13. 4. 5 .

Ju-Hong Lee and Tsai-Tien Young, “New Polyphase Filter Bank-based Analysis/Synthesis Systems”, Submitted to IEEE Trans. ASSP.

R.E. Crochiere, “Digital Signal Processor: Subband Coding”, the Bell System Technical Journal, Vol. 60, No. 7 , pp. 1633-1653, Sept. 1981.

D. Esteban and C. Galand, “‘Application of Quadrature Mirror Filter to Split-Band Voice Coding Schemes”, in Proc. IEEE Int. Conf. o n ASSP, pp. 191-195, May 1977. M. Vetterli, “Perfect Transmultiplexer”, in Proc. IEEE Int. Conf. on ASSP, pp. 2567-2570, April 1986. M.G. Bellanger and J.L. Daguet, “TDM-FDM Transmultk plexer: Digital Polyphase and FFT”, IEEE Trans. Commun., Vol. COM-21, No. 9 , pp. 1199-1204, Sept.

1974.

R.E. Crochiere and L.R. Rabiner, MULTIRATE DIGITAL SIGNAL PROCESSING, Englewood Cliffs, NJ: Prentice- Hall, 1983.

V.K. Jain and R.E. Crochiere, “Quadrature Mirror Filter Design in Time Domain”, IEEE Trans. ASSP, Vol. ASSP- 32, pp. 353-361. April 1984.

F. Mintzer, “Filters for Distortion-Free Two-Band Multirate Filter Banks”, IEEE Trans. ASSP, Vol. ASSP- 33, pp. 6 2 6 6 3 0 , June 1985.

P. Chu, “Quadrature Mirror Filter Design for an Arbitrary Number of Equal Bandwidth Channels”, IEEE Trans. ASSP, Vol. ASSP-33, pp. 203-2 18, February 1985. K. Swaminathan and P.P. Vaidyanathan, “Theory and Design of Uniform D I T . Parallel, Quadrature Mirror Filter Bank”, IEEE Trans. Circuits Syst., Vol. C A S 3 3 , pp. 117-1 191. Dec. 1986.

P.P. Vaidyanathan, “Theory and Design of M-Channel Maximally Decimated Quadrature Mirror Filters with Arbitrary M, Having Perfect Reconstruction Property”, IEEE Trans. ASSP, Vol. ASSP-35, pp. 4 7 6 4 9 2 , April

1987.

P.P. Vaidyanathan and K. Swaminathan, “AliasFree. Real Coefficients m-Band QMF Banks for Arbitrary m”, IEEE Trans. Circuits Syst., Vol. CAS-34, pp. 1485-1496, Dec. 1987.

Z. Doganata, P.P. Vaidyanathan. and T.Q. Nguyen. “General Synthesis Procedure for FIR Lossless Transfer Matrices, for Perfect-Reconstruction Multirate Filter Bank Applications”, IEEE Trans. ASSP, Vol. ASSP-36.

J.D. Johnston, “A Filter Family Designed for Use in Quadrature Mirror Filter Banks”, in Proc. IEEE Int. Conf. ASSP, pp. 291-294, April 1980.

G. Pirani and V. Zingarelli, “An Analytical Formula for the Design of Quadrature Mirror Filters”, IEEE Trans. ASSP, Vol. ASSP-32, pp. 645-648, June 1954.

pp. 1561-1574, Oct. 1988.

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Analysis Filter

B a n k

Synthesis Filter

Bank

Figure 1

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Figure

2.

The Magnitude Response of

G(u)

-

Using The Analytic Design Formula

---

Using

A

Conventional Numerical Algorithm

8 ° - o - o ~

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' ' '

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j

o . o s a 0 . 0 1 0 0 .ooc) 0 . 0 0 0

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1

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s . 0 P I

Figure

3.

Reconstruction Error Bet::een The Output and Input

of 'Ihe System of Fig.

1

-

Using The Analytic Design Formula

Using

A

Conventional Manerica1 Algorithm

1723

(217) 333-4789

Table

1

The Cunputational Canplexity

Algorithn

Analytical

Foxnula

I

1514

1

3723 !

Table

2

The Filter Coefficients

g ( 0 ) = -0.30155435OE-02 g(1) = -0.44925447602 g ( 2 ) -0.314682306602 g ( 3 ) = 0.260439676602 g ( 4 ) = 0.915403406602 g ( 5 ) = 0.899816595602 g ( 6 ) = -0.255303234602 g ( 8 ) = -0.253663825601 g ( 7 ) = -0.198699317601 g ( 9 ) = 0.255687792602 g(10)= 0.750467467601 g(ll)= 0.181366878E 00 g(12)= 0.2886486573 00 g(13)= 0.3562212753 00