Designing filters for polyphase filter banks

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Designing filters for polyphase filter

banks

J.-H. Lee W.-J. Kang

Indexing term.: Filter design, Polyphosefilter banks

Abstract: The paper concerns the filter design problem for a recently-proposed polyphase filter bank with an arbitrary number of subband channels. We develop an analytical formula for the design of a prototype filter used in the polyphase filter bank. Compared with direct numerical design methods, this formula-based method allows the design of the required FIR lowpass prototype filter with a much lower complexity of operation and better filter bank performance. We further extend the theoretical results to two dimensions. Examples are also presented.

1 Introduction

Polyphase filter banks (PFB) based on quadrature mirror filters (QMF) have been successfully used to constitute a multirate signal processing system for splitting a signal into N subbands and allowing the resynthesis of the signal from the subbands. Main applications of PFBs are in subband codng of speech signals [l, 21, TDM-FDM

transmultiplexing systems [3, 41, and short-time spectral analysis [S, 61. Recently, the application of PFBs has

been extended to two dimensions, for example for the subband coding of images [7, 81, and 2-D short-time spectral analysis [SI.

Considering the filter design for I-D PFBs, many results have been reported in the literature [lo-171. Most of these methods employ numerical algorithms directly to minimise the associated error measure in the filter design process. Reference 18 describes an analytical formula to represent the error measure. As a result, the filter coeficients can be found by a straight nonlinear optimisation procedure, and a saving in computational load can be achieved. However, most of the above authors consider the design of QMFs. Moreover, methods for designing nonseparable 2-D prototype filters for use in 2-D PFBs were not available, although the 1-D methods [17, 181

were used for the design of a separable 2-D prototype filter for 2-D PFBs [9]. Recently, Reference 19 proposed an approach for the design of 2-D PFBs based on a con- straint of cascaded Smith forms.

Based on nonsymmetrical frequency-band allocation in constructing bandpass filters for subband channels, Reference 20 describes a I-D PFB with N arbitrary sub- bands. Each bandpass filter is a nonsymmetrical frequency-shifted version of a lowpass prototype filter. It is shown in Reference 20 that the new PFB possesses some advantages over conventional PFBs. A new 2-D Paper 847W (E3), received 20th May 1991

The authors are with the Department of Electrical Engineering, Na- tional Taiwan University, Taipei, Taiwan, Republic of China IEE PROCEEDINGS-G, Vol. 139, N o . 3, J U N E 1992

PFB with nonseparable filters is presented in Reference 21 because using nonseparable filters results in direc- tional capability in the subband decomposition.

In this paper, we consider the filter design problem for the new PFBs. An analytical design formula for obtain- ing the coefficients of the required 1-D FIR lowpass prototype filter is derived in Section 2. Compared with direct numerical design algorithms, the analytical design formula requires much less computational load and results in a smaller reconstruction error for the same number of subbands and filter length. In Section 3, we develop further an analytical formula for the design of a 2-D nonseparable FIR lowpass prototype filter for use in the new 2-D PFB, while in Section 4 we present simula- tion results for illustration.

2

The basic structure of a I-D filter bank with N subbands to be considered is shown in Fig. 1. Its efficient structure

Analytical design formula for 1 -D PFB

a

I

b Fig. 1 I-Dfilter bank a Analysis lilter bank b Synthesis tiller bank

based on a polyphase network and a fast Fourier transform (FFT) has been described in Reference 20. The bandpass filters

Rdo)

and HAW) are represented by

H A 4 = G i b )

+ G~N-I-Aw)

and

~ ~ ~ O J ) = G ~ C O ) - G , , - , - , ( C L I ) i=O,1, 2 ,..., N - l (1) 363

(2)

respectively, where Gdo)

a

G(o - in / N ) and G(o) denotes the 1-D FIR lowpass prototype filter with length

M . The associated conditions for perfect reconstruction

are [ 2 0 ] IG(o)I = 0

I w I

L n / N (2) I G ( w - k n / N ) I 2 = N O $ o < 2 n ( 3 ) Z N - 1 k = O and M = N ( 2 L

+

1 )

+

1 (4)

where L is an integer. To design the FIR lowpass proto- type filter G(o) which approximates the condition of eqn. 3 in some optimum sense, we define an appropriate approximation error measure E as

( 5 )

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

E = E,

+

aE,

E, denotes the stopband energy related to G(o) and is given by

(7)

a in eqn. 5 designates the relative weight between E, and E,. To solve the problem of minimising E, some well- known unconstrained numerical optimisation algorithms can be employed to find the filter coefficients of

WO).

However, at each iteration, we have to compute the Fourier transform of Gdo) for i = 0, 1 ,

...,

2 N - 1, to obtain E, and E,. This requires considerable computation time.

In the following, we derive a closed analytical formula to represent eqn. 5 and turn the minimisation problem into a direct search problem for the filter coeficients. Since G(o) is in linear phase with length M, it can be expressed as G(o) = exp (-j(M - 1)w/2)

}

(8) M D

C

44

COS ((n - ( M - 1)/2)w)

I.=.

where M D = (M - 1)/2, d((M - 1)/2) = g((M - 1)/2), and d(n) = 2g(n) for 0

<

n

<

( M - 3)/2 if M is odd, or M D = M / 2 - 1, d(n) = 2g(n) for 0 G n

<

M / 2 - 1 if M is

even. From eqn. 8, we have

x cos ((n - (M - 1 ) / 2 ) 0 ) 1 M D M D = -

c c

d ( ~ ) d ( ~ M C o S ((m - 2 m = o s = o

+

cos ((m

+

n - M

+

l k ) ) (9) and 1 M D M D Z N - 1 2 m = o n = o k = O = -

1

d(m)d(n) cos ((m - n)@ - k n / N )

+

cos ((m

+

n - M

+

1x0

-

k n / N ) ) )

e

F ( o ) (10)

Next, based on the fact that 2 N - 1

1

~ cos (I(w - k n / N ) )

k = O

2N cos (lo) 1/2N = integer

= I O else

eqn. 10 can be rewritten as

+

r(m

+

n - M

+

1 ) )

where

cos (lo) 1/2N = integer

‘0

= (0 else

Now consider the integral given in eqn. 6. Expanding the squared integrand, we obtain

E, =

!

( N z

-

2 N F ( o )

+

F 2 ( o ) ) d o (13)

Substituting eqn. 11 into eqn. 13 and using the fact that

H k = O

0 else

ff

cos ( k w ) d w =

we can show that the integration of eqn. 13 gives

M D

d2(n)

+

d2((M - 1)/2)) 1 M D M D M D M D

+

- 2 i = o j = o m = O

1

“ = o d ( i ) d ( M M n ) ) (15) In eqn. 15, the term d Z ( ( M - 1)/2) exists only when M is odd and the last sum term exists only when one of the following combinations of the four integers i, j , m, n holds

(a) m - n = 2 N k , and i - j = + 2 N k , (b) m - n = 2 N k , and i

+

j

-

M

+

1 = + 2 N k , (c) m

+

n - M

+

1 = 2 N k 3 and i + j - M

+

1 = + 2 N k 3

Next, consider E, in eqn. 7. Substituting eqn. 9 into for 0 $ i, j , m, n G M D and integers k l , k , , k ,

.

eqn. 7 and allowing for the fact that

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41

- 1/N) k = 0

6-

sin ( k n / N ) ) / k else

lN

cos ( k o ) d o = integration yields where $ 1 - 1 / N ) l = O

$0

= -(sin (ln/N))/l I # 0 and I/N # integer

[o

else

From eqns. 15 and 17, we note that the approximation error measure E of eqn. 5 can be expressed as a closed analytical formula in terms of the filter coefficients

do),

g ( l ) ,

...,

g(M - 1). Hence, the computation of the filter coefficients by minimising E can be performed by utilising a simple direct search method.

IEE PROCEEDINGS-G, Vol. 139, No. 3, J U N E 1992 364

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3

In this section, we extend the above 1-D theoretical results to the 2-D case. Consider the 2-D filter bank shown in Fig. 2. The number of subband channels is

N,N,, where N , and N, are two arbitrary integers. An

efficient structure consisting of a 2-D polyphase network and a 2-D FFT for the 2-D filter bank has been described

i,n Reference 22. The bandpass filters Hi,,(wl, w,) and

Hi, ,(CO,, w,) are represented by

Analytical design f o r m u l a f o r 2-D PFB

Hi,,(w17 O 2 ) Gi.,(wl, O.2)

+

G i , 2 N 2 - l - , ( W l r O2)

+ G 2 N i - l - i . ~ ( w l ,

O 2 )

+ G Z N , - ~ - ~ , Z N ~ - I - , ~ ~ I ~

4

Ho.0

b Fig. 2 2-Dfilter bank a 2-D analysis filter bank b 2-D synthesis filter bank

and fii.jCw19 WZ) = Gi.j(w19 0 2 ) - G i , 2 N z - I - , ( W 1 , 0 2 ) - G Z N ~ - l - i . , ( w I , w 2 )

+

GZN, - 1 - i , Z N ~ - 1 -,(mi,

4

(18) where G i . @ , , 02) G(o, - i n / N , , w , - j n / N , ) , i = 0, 1,

..., N I

- 1 and j = 0, 1,

..., N,

- 1. G(w,, 0,) is a 2-D FIR lowpass prototype lilter with length M I x M , and impulse response g(m, n). Consider that G(w,, w,) is nonseparable. The associated conditions for perfect reconstruction are

I

G(w,, 0,)

I

= 0

I

w1

I

3 z/Nl and

I

w 2

I

3 n / N , (19)

where L , , L , , I,, and I, are integers. Furthermore, with linear phase, G(w,, w,) can be written as

G(wl,

4

= exp { - j W l -

Wl

+

( M 2 - 1)wz)/2}

x cos ((n - ( M , - 1)/2)m2) (23)

where M x = ( M , - 1)/2 if M I is odd and M x = M , / 2

- 1 if M , is even; M y = ( M , - 1)/2 if M , is odd and

M y = M,/2 - 1 if M , is even; d(m, ( M , - 1)/2) = 2g(m, ( M , - 1)/2) if M , is odd and d ( ( M , - 1)/2, n) = 2 g ( ( M ,

- 1)/2, n) if M I is odd; d((M, - 1)/2, ( M , - 1)/2) =

g ( ( M , - 1)/2, ( M , - 1)/2) if M , and M , are odd; and d(m, n) = 4g(m, n) elsewhere.

Using eqn. 21 and performing some algebraic manipu- lations, we obtain

I

G(% w2)

l2

x (cos ((m1 - m2)w1)

+

cos ((ml

+

m2 - M I

+

l)o,)) x (cos ((n1 - n2)wz)

+

cos ( ( n ,

+

nz - M ,

+

IN,))

Further, from eqn. 24, we obtain

(kiwi) kJ2Ni = integer for i = 1,2

else

#(mi, ki) =

According to the proposed approximation error measure E in eqn. 5 , E, and E, are given by

E, =

l

[(N,N, - ~ ( w , , w , ) ~ do, do, (27) 365

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and

respectively for the 2-D case. Expanding the squared

integrand of eqn. 27 yields

E, =

l

l ( N % - 2 N 1 N 2 F(w,, 0,)

(29)

+

FZ(oi, ut)) do, dm2

Using eqns. 14 and 26 and performing algebraic manipu-

lations, we obtain

4

E, = N:N$ r2 - 2N:N$ k2

1

Pi i = 1

+

( N : N $ r 2 / 4 )

1

4m1, n,)d(m,, n2)

x 4i1,

ill

4iz1

i2)

(30)

where m 2 = 0 n , = O f M x

lox

d2(m2, (M2 - 11/21 M , = odd p2 = m 2 = 0 else d2((Ml - 1)/2, n,) M, = odd else d2((M1 - 11/27 (M, - 1)/2) (31)

and the summation of the last term is taken over 0 Q m,,

m 2 , i,, i, Q Mx and 0 Q n,, n , , j,, j, C M y . Further-

more, this summation exists only when one of the following combinations of integer indexes holds.

m , - mz = 2N,k and i, - i, = +2N,k n , - n2 = 2N21 and jl - j2 = f 2 N 2 1 m2 = 2N,k and i, - i, = f 2 N 1 k P 4 = { M I = o d d and M, = o d d 0 else

(4

{

(b) {;I

,

- n , - = 2 N 2 1 a n d j , + j z - MI

+

1 = f 2 N 2 1

{

(f)

{;

(9)

{

(h)

{

m, - m2 = 2N,k and i,

+

i, - M I

+

1 = f 2 N , k m , - m, = 2N,k and i, - i, = f 2 N , k n1

+

n2 - M,

+

1 = 2 N 2 1 andj, - j, = f 2 N 2 1 m , - m2 = 2N,k and i , - i, = f 2 N i k n ,

+

n2 - M,

+

1 = 2 N 2 1

(4

{

n , - n 2 = 2 N 2 1 a n d j , - j 2 = f 2 N 2 1 (e) a n d j , + j , - M , + l = f 2 N 2 1

,

+

m2 - MI

+

1 = 2N,k and i , - i, = f 2 N l k

,

- n2 = 2 N 2 1 and j, - j, = f 2 N 2 1 m, - m2 = 2N,k and i,

+

i, - M I

+

1 = f 2 N l k n1 - n2 = 2 N 2 1 andj, + j 2 - M,

+

1 = + 2 N 2 1 m , - m 2 = 2 N , k a n d i , + i , - M , + 1 = f 2 N , k n , + n , - M,

+

1 = 2 N 2 1 andj, - j , = f 2 N 2 1 m ,

+

m2 - M ,

+

1 = 2N,k and i, - i, = f 2 N , k n,

+

n2 - M ,

+

1 = 2 N 2 1 and jl -j2 = f 2 N 2 1

r

366 (j)

(“,

+

m2 - M ,

+

1 = 2N,k and i ,

-

i , = f 2 N 1 k n1 - n , = 2 N 2 1 and j, + j 2 - M ,

+

1 = f 2 N 2 1 m, - m , = 2 N , k a n d i , + i 2 - M,

+

1 = f 2 N , k n , + n 2 - M , + 1 = 2 N 2 1 andj, +j2 - M,

+

1 = f 2 N 2 1 m ,

+

m2 - M I

+

1 = 2N,k and i, - i, = f 2 N , k

(4

n , + n , - M 2 + 1 = 2 N , 1 and j,

+

j2 - M,

+

1 = f 2 N , 1 m,

+

m2 - M,

+

1 = 2N,k and i,

+

i, - M,

+

1 = +2N,k n, - n2 = 2 N 2 1 and jl - j, = f 2 N 2 1 m,

+

m2 - M I

+

1 = 2N,k and i,

+

i, - M I

+

1 = f 2 N , k n , - n , = 2 N 2 1 a n d j , + j , - M , + 1 = f 2 N 2 1 m,

+

m2 - M,

+

1 = 2N,k n,

+

n2 - M,

+

1 = 2 N 2 1 andj, -j2 = f 2 N 2 1 m ,

+

m2 - M,

+

1 = 2N,k

i

and i,

+

i, - M,

+

1 = f 2 N , k (32) and i,

+

i, - MI

+

1 = +2N,k a n d j , + j , - M 2 + 1 = +2N,l

Next, substituting eqn. 24 into eqn. 28 and using

n ,

+

n2 - M2

+

1 = 2 N 2 1

where k and 1 are integers.

(33) k = O {:f(kr/N))/k else COS ( k o ) d o = r l = O 0 else l = O

0 and Z/Ni # integer

r(l - 1/NJ 1

= o

Ri(l) = -(sin ( l z / N J ) / l 1 # 0 and l/Ni it integer else

(35)

for i = 1,2. Hence, we note from eqns. 30 and 34 that the

approximation error measure E for the 2-D case can be

expressed as a closed analytical formula in terms of the filter coeilicients. Therefore, the filter coefficients

dm,

n)

of G(o,, 0,) can also be found by utilising a simple direct search algorithm. In the following section, evaluation of computational complexity and simulation results are presented to confirm the above theoretical work.

i o

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4 Experimental examples

4.1 1 - D design example

The filter length M and the relative weight a were set to 16 and 0.1, respectively. The number of subbands N was three. Based on the analytical formulas presented in Section 2, we used a steepest-descent algorithm to iter-

size used was 128. Fig. 3 shows the frequency responses of G(w) using both methods. The corresponding reconstruction errors are shown in Fig. 4. Table 1 compares the computational complexities in terms of the number of operations required at each iteration. Table 2 lists the Table 2: Coefficients of designed 1 - D prototype filter

Proposed design method Numerical design method

-1101

0 0 2 0 4 0 6 0 8 1 0

normalised frequency Fig. 3 Magnitude response of

WO)

OW[ 0 02

t

-0.02 -0041 0 0 2 0 4 0 6 0 8 1 0 normalised frequency Fig. 4 Reconstruction error for the I-D case

Table 1 : Comparison of computational complexity for the

1 - D case

Real additions Real multidications

~~

Proposed design method 360 800 Numerical desian method 1 3 x 1 O4 1 5 x lo4

atively search the filter coefficients g(O), g(l),

. . .

, g(15).

The number of iterations was 200. For comparison, the direct numerical optimisation of eqn. 5 based on a steepest-descent algorithm was also performed. The FFT

0.73321 3933 x 10‘: g o = g , s -0.002699048 g , =g,, -0.009523474 g 2 = g , 3 -0.022380432 -0.24281 51 56 x

lo-’

g 3 = g , , -0.003564694 -0.418084883 x

lo-’

-0.105630515 x 10- g r = g l 0.06701 3556 0.662609824 x

lo-’

0.1 76067248 g 5 = g , , 0.175784464 g s = g s 0.288924903 0.2891 41 969 g, = g a 0.361 273963 0.360878687 -n Fig. 5

Table 3: Comparison of computational complexity for the 2- D case

Magnitude response of Ww,, wJ for nomeprable case

Real additions Real multiplications Proposed design method 2.0 x 1 Os Numerical design method 3.2 x

lo8

3.7 x

lo8

8.4 x 1 O4

computed filter coefficients for this case. From these results, we observe that the proposed analytical design method is superior to the direct numerical optimisation algorithm.

4 2 2-0 nonseparable design example

In this case, the 2-D FIR lowpass prototype filter is nonseparable with M , = M, = 16. Both N, and N, were set to three and the relative weight a to 1OOO. Fig. 5 shows the frequency response of G(w,, w 2 ) using the proposed 2-D design method described in Section 3. Table 3 compares the computational complexities for the 2-D case. Owing to the huge computational burden shown in Table 3 when using the numerical method, only the 2-D Nter design was performed. Table 4 lists the designed filter coefficients. To demonstrate the reconstruction capability of the resulting 2-D PFB, we used a 128 x 128 Table 4: Coefficients of the designed 2-D nonsaparable prototype filter

g(m,n) m=O 1 2 3 4 5 6 7 n =O 1 2 3 4 5 6 7 0.00017 0.00033 0.00028 -0.00019 -0.001 18 -0.00253 -0.00386 -0.00469 0.00033 0.00062 0.00054 -0.00033 -0.00218 -0.00472 -0.00720 -0.00875 0.00028 0.00054 0.00047 -0.00028 -0.001 89 -0.00410 -0.00627 -0.00761 -0.00019 -0.00033 -0.00028 0.00020 0.001 19 0.00253 0.00384 0.00465 -0.001 18 -0.0021 8 -0.001 89 0.001 19 0.00765 0.01 650 0.0251 6 0.03053 -0.00253 -0.00472 -0.00410 0.00253 0.01 650 0.03565 0.05438 0.06601 -0.00386 -0.00720 -0.00627 0.00384 0.0251 6 0.05438 0.08297 0.10072 IEE PROCEEDINGS-G, Vol. 139, N o . 3, J U N E 1992

-0.00469 -0.00875 -0.00761 0.00465 0.03053 0.06601 0.1 0072 0.1 2228 361

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2-D image as the input signal. Fig. 6 shows the simula-

tion results. The difference image shown in Fig. 6c is equal to

I

(x(m, n) ~ i ( m , n )

1 ,

where x(m, n) denotes the

original image and i ( m , n) the reconstructed image. For the displayed images, the darkest image pixel has gray value 0, while the brightest image pixel has gray value

0 b

Fig. 6

U Ongmal image h Reconstructed image

c Difference image

S~mulation resultsfor the 2-D nonseparable rase

4

Fig. 7 Magnitude response uJC(o,, w2)for separable case 255. From Fig. 6, we observe that the reconstructed image is almost the same as the original image.

4.3 2 - D separable design example

Here we consider the same design example treated above, except that the 2-D prototype filter is separable. Fig. 7 shows the frequency response of G(w,, w 2 ) designed by using the proposed 1-D design method described in

Section 2. Table 5 lists the designed filter coefficients. Fig.

8 shows the simulation results. Again, the difference image is almost invisible.

o b

Fig. 8

U Reconstructed image

b Difference image

Simulation results for the 2-D separable case

5 Conclusions

In this paper, we have presented an analytical method for the design of the lowpass prototype filter for a newly- developed polyphase filter bank. A closed analytical formula has been derived to represent an appropriately defined approximation error measure in the design process. Based on this formula, computing the filter coefficients by minimising the approximation error can be performed utilising a simple direct search algorithm. This leads to a reduction in the complexity of operations in the filter design process. The 1-D theoretical results have also been extended to two dimensions. An analytical formula for designing the new 2-D polyphase filter bank with a 2-D nonseparable prototype filter is derived. This formula facilitates the design work in the 2-D case. If the

2-D polyphase filter bank is constructed using a 2-D sep-

arable prototype filter, the developed 1-D analytical formula can be employed to further reduce the design complexity. The theoretical work has been confirmed by computer simulations.

6 References

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Table 5: Coefficients of the designed 2-D separable prototype filter

g ( m , n ) m=O 1 2 3 4 5 6 7 n=O 0.00018 1 0.00033 2 0.00029 3 -0.00018 4 -0.00117 5 -0.00252 6 -0.00385 7 -0.00467 0.00033 0.00062 0 00054 -0.00033 -0.0021 8 - 0.0047 1 -0.0071 9 -0.00873 0.00029 0.00054 0.00047 -0.00029 -0.001 90 -0.0041 1 -0.00627 -0.00761 -0.0001 8 -0.00033 -0.00029 0.0001 8 0.001 16 0.00250 0.00382 0.00464 -0.001 17 -0 00218 -0.001 90 0.001 1 6 0.00762 0.01 647 0.0251 4 0.03053 -0.00252 -0.00471 -0.0041 1 0.00250 0.01 647 0.03562 0.05437 0.06600 -0.00385 -0.0071 9 -0.00627 0.00382 0.0251 4 0.05437 0.08279 0.10074 -0.00467 -0.00873 -0.00761 0.00464 0.03053 0.06600 0.10074 0.1 2230 368 ~~ ~

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