Novel efficient approach for the design of equiripple quadrature mirror filters

R.H. Yang Y.C. Lim

Indexing terms: FIRfilters, Quadrature mirrorfilters, Weighted least squares technique

Abstract: Quadrature mirror filters have been used extensively in subband coding of speech signals. In this paper, we introduce a novel effi- cient approach for the design of equiripple quad- rature mirror filters. The new approach is more efficient than the previously proposed design method in terms of computer time and memory requirement.

1 Introduction

Quadrature mirror filters (QMF) have been used exten- sively in speech signal processing [l-91. Fig. 1 illustrates a two-band coding system. The speech signal x(n) is fil- tered by a lowpass filter H,(z) and a highpass filter H2(z) . The outputs of Hl(z) and H 2 ( z ) are decimated by a factor of two, coded and transmitted. The received signals are interpolated with zeros, filtered by H,(z) and H4(z), respectively, and then summed to produce the recon- structed speech s(?~). To eliminate aliasing noise, the above four filters are selected such as in Reference 1

H1(z) = H ~ ( - z ) = H,(z) = - H ~ ( - z ) (1)

Note that eqn. 1 does not guarantee perfect reconstruc- tion. Nevertheless, perfect reconstruction filter banks exist [10-12]. If the frequency response of Hl(z) is given by

Hl(e‘o) = e - j ( N - lId2D(w) (2) where N is the length of Hl(z), the frequency response of the overall system from input x(n) to output s(n) can be written as in Reference 1

(3) [D’(w) + D’(z - w)] ~ ( ~ i a ) = f e - j ( N - l l w

Ideally, we would like D2(w) + D2(n - w) to be equal to unity for 0 < w < 0.571 and D(w) to be equal to zero for 0 . 5 ~ < w, < w < z, where W, is the stopband edge of Hl(z); in practice, this ideal cannot be achieved for finite N . Let k(w) be a given tolerance function, and 6 be a variable to be minimised. The problem of the design of a weighted minimax quadrature mirror filter can now be formulated as a constrained nonlinear optimisation problem of minimising 6 subject to the constraints of

0 IEE, 1994 Paper 99253 (E5), first received 19th March and in revised form 18th October 1993 R.H. Yang is with the Electrical Engineering Department, Singapore Polytechnic, Singapore 0513 Y.C. Lim is with the Electrical Engineering Department, National University of Singapore, Singapore 051 1

IEE Proc.-Vis. Image Signal Process., Vol. 141, N o . 2, April IY94

Fig. 1

eqn. 4.

Two-handfiltenny schemefor split-hand speech coding

I D2(w) + D 2 ( z - W ) ~ 1 I < k(w)6 0 < w < 0 . 5 ~ (4a)

I D(w) I < W, < w < A (46)

This constrained nonlinear optimisation problem is diffi- cult to solve. Lim et al. [7] converted the above con- strained nonlinear optimisation problem into an iterative unconstrained nonlinear optimisation problem, rendering the design of equiripple quadrature mirror filters pos- sible. However, the design time is long, particularly for long filters.

In this paper, we propose a novel and efficient approach for the design of equiripple quadrature mirror filters. We introduce a method for generating a profile of the desired frequency response of the filter. We adopt the conventional weighted least squares (WLS) method [13] and modify it to suit the purpose of the design of equi- ripple quadrature mirror filters.

2 New design approach

Our new algorithm designs an equiripple quadrature mirror filter in two steps. The flowchart is shown in Fig. 2.

The first step of our algorifhm is to generate a profile whose magnitude response D(w) satisfies the condition for distortion-free reconstruction.

b2(o) + D2(x - W ) = 1 ( 5 4

D(w) = 0 w, < w < Z (56)

0 < w < 0 . 5 ~

The method of generating the profile will be discussed in Section 3.

The second step is the iterative weighted least squares procedure. Since H , ( z ) has the property that its impulse response h(n) is symmetrical and its filter length is even, D(w) can be written as

where

a(n) = 2h( r + ?I) = 2h( r - 1 - n )

The crux of the weighted least squares algorithm for the design of an equiripple filter is to derive a proper least squares weighting function which produces an equiripple frequency response of the filter. Let rdw) be the least

T f i rs

initialisation

I . . I

I

evaluate ED(w) secor

find the extremal points

evaluatep,(w)

r,+l (w) =r, (wM,C)

ep

.tep

frequency range where 0 < o < 0 . 5 ~ . The extremal points of I EG(w)I for 0 < w < 0 . 5 ~ and those of I D(o)l for w, < o < x are searched. If they are equiripple, the algo- rithm is terminated; otherwise, the weighting function is updated by using the relationship of eqn. 9.

Ti + I(w) = ri(w)Pi(o) PAW) 0 (9) A is recomputed in the next iteration. The method for deriving &(CO) will be detailed in Section 5 .

Our proposed algorithm is essentially a WLS approach. Nevertheless, there are differences between this proposed algorithm and that presented in Reference 13. In this proposed algorithm, both I D(w) I for w, < w < x and I E,(o) I for 0 < w < x are minimised simultaneously. In the algorithm presented in Reference 13, only the weighted peak frequency response error is minimised. Further details on this aspect will be presented in Section 4.

3 Profile generation

In the design of quadrature mirror filters, there is no don't-care frequency band. Therefore, b(w) in the full fre- quency range from 0 to x must be defined. As H,( z ) is a lowpass filter and 6 ( w ) satisfies eqn. 5 , b(w) for the fre- quency ranges 0 < w < w p , w = 0.5 and w, < w < x are defined in eqn. 10

1.0 o < w < w p b(o) = 1 /42) w = 0.5x i 0.0 0, < 0 < x

where wp is the passband frequency edge and op= x - w,.

b(w) for the two frequency ranges where wp < w < 0 . 5 ~ and where 0% < w < w, is constructed as follows. First. the freauencv bands not defined in ean. 10 are con-

Fig. 2 Flow chart ofthe new algorithm

squares weighting function used in the ith iteration. As in Reference 13, we set the initial least squares weighting function ro(o) equal to unity for all the frequency grid points. In the ith iteration, the values of a(n), for n = 0, 1, . . . , N/2 - 1, can be obtained by the following matrix

( 7 4

( 7 4

WLS algorithm with r(w) set equal to 1 to approximate the specifications defined in eqn. IO. Let the frequency

operations. Define

A = [do), a(l), . . . , a(N/2 - 1)IT

= CB(,0), B(wl), . . ., @wQ- ,)IT

1 ( 7 4

COS (O.500), COS ( I S W ~ ) , . . . , COS [(N/2 - 1)00] COS (0.501), COS ( ~ S C O ~ ) , . . . , COS [(N/2 - 1)00]

u = 2

Lcos (0.50~- I)r cos (1.50~- 1), . . . , cos [ (N/2 - l)op- ,] J and

( 7 4

where Q is the number of frequency grid points. The solu- tion which minimises

1 rAoj)CD(oj) - b(wj)I2 j

is given by Reference 13

A = (UTRU)-'UTRb ( 7 4

E,(,) = D2(w) + D2(n - 0) - 1 (8) After A is obtained, D(w) is evaluated in the frequency range where w, < w < x, and EG(w) is evaluated in the

where UT is the transpose of U. Define

response of this design be b(w) After b(w) is obtained, b(w) for 0% i w < w, is set equa! to d(w) over the same frequency range. The segment of D(o) where 0.5s < w < w, thus obtained will not piece seamlessly with the values of D(o) at w = 0.5n and w 2 w, defined in eqn. 10. To overcome this difficulty, the values of this segment of b(w) at the two grid points nearest to 0.5n and the two grid points nearest to w, are modified by Neville's poly- nomial interpolation algorithm (see Appendix). Finally, b(w) for wp < w < O15x is obtained by using the relation- ship b(w) = J[l - D2(n - 4 3 .

As an example, Fig. 3 shows plots of b(w) and b(w) of a filter of length 24 and w, = 0.586~ for the frequency range where om < w < w,.

Although D(o) may be constructed by using many other methods, from our experience, the above method produces the most satisfactory results.

96 IEE Proc.-Vis. Image Signal Process., Vol. 141, N o . 2, April 1994

4 Weighting function adjustment

In the design of an equiripple quadrature mirror filter, both EG(w) for 0 < w < 0.5n and D(w) for w, < w < n must be equiripple simultaneously. Owing to symmetry,

5 ,(025. 1/42)

wp/2n normalised frequency wJ2n

Fig. 3 Plots o f6 (0 ) and 6(w) of afil ter of length 24 and U, = OJ86n in thefrequency range where w p < w < w,. Points Q,, Q , , Q, and Q, are moved to Q , , Q , , Q , and Q , , respectively, by the polynomial inter- polation algorithm

EG(w) is equiripple for 0.5n < w < n if it is equiripple for 0 < w < 0.5n. Equiripple design is achieved by choosing a suitable least squares weighting function r(w). Our iter- ative technique to arrive at such a suitable r(w) is described below.

For w, < w < 71, the maximum value of I D(w) I should be minimised. Therefore, in this frequency range, our weighting function adjustment strategy is exactly the same as that presented in Reference 13.

Define

ED(w) = D(w) - b(w) (1 1) From eqns. 11 and 8, we have

= CB(O) + E D ( w ) 1 2

+ Cb(7I - w ) + E,(n - w)]2 - 1

0 < w < 0 . 5 ~ (12a) For 0 < w < 0.5n, the maximum value of I E,(w) I should be minimised. In this frequency range, the weighting function adjustment is more complicated. Ignoring the second order terms and taking note of eqn. 5a, eqn. 12a can be rewritten as

EG(W) = 2[E,(w)b(w) + E,@ - w)b(n - w)]

0 < w < 0.5n (126)

For 0 4 w < up, &w) = 1 and b(n - w ) = 0; eqn. 12b can be rewritten as

EG(w) = 2ED(w) 0 < < w p ( 1 3 )

EG(w) 2 24(2)ED(w) w = 0.5n (14)

At the frequency point where w = 0.5n, b(w) = 1/4(2); eqn. 12b can be written as

Thus, EG(w) is proportional to Edw) for 0 < w d wp and for w = 0.571. As a consequence, the minimisation of the peak value of I EG(w) I can be achieved by minimising the

IEE Proc.-Vis. Image Signal Process., Vol. 141, No. 2, April 1994

peak magnitude of I Edw) I using the conventional WLS approach.

We shall subdivide the frequency range where wp < w < 0 . 5 ~ into two cases for further study.

Case I ; EG(w) > 0 Referring to eqn. 12b, if Edw) > 0, decreasing Edw) will cause IEG(w)I to decrease. E,(w) can be decreased by increasing dw). If ED(w) < 0, increasing Edw) will cause I E,@) I to decrease. E&) can be increased by decreasing <U). The effect of E& - w) on I E,@) I is similar to that of E,(w) on I EG(w) I ; r(x - 0) should be adjusted accord- ingly in the next iteration.

Case 2: E,(o) < 0 If ED(w) > 0, increasing Edw) will cause IEG(o)I to decrease. This can be achieved by decreasing 4w). If ED@) < 0, decreasing Edw) w i l n a u s e I E,@) I to decrease. This can be achieved by increasing 4w). The effect of E,(n - CO) on I E,@) I is similar to that of ED@) on I E,(w) I ; r(n - w) should be adjusted accordingly in the next iteration.

Summarising, for wp < w < OSn, if EG(w)ED(w) > 0, r(w) must be increased in the next iteration; otherwise, 4w) must be decreased. Similarly, if E,(o)E,(n - w ) > 0, r(n - w) must be increased in the next iteration; other- wise, r ( ~ - w) must be decreased.

5 Factor p,(w)

In our new algorithm, the weighting function is updated in each iteration by using eqn. 9. The factor PAW) is obtained as follows.

(a) Evaluate E&) for 0 < w < x by using eqn. 11. (b) Evaluate E,(w) for 0 d w Q 0 . 5 ~ by using eqn. 12b. (c) Find the extremal points of IEG(w)I for

0 < w < 0 . 5 ~ and those of I Edw)I for w, d w d x. Denote the magnitude of the extremal point at frequency w in the ith iteration by v(w).

(d) Form an envelope function BXw) consisting of the segments of straight lines passing through all the extre- mal points as shown in Fig. 4. As a quadrature mirror

a. (w)

0.2 0.3 0.4 0.5 normalised frequency

Fig. 4 Plot showing the relationship between B i o ) and Vim)

filter has even order, E,(n) is always equal to zero. To avoid BAw) from getting too close to zero for the fre- quency grid points near w = x, we let VAT) be equal to the magnitude of its neighbouring extremal point. Let

where wI and w, are the frequencies of the neighbouring extremal points to the left and to the right of w, respect- ively.

(e) Let

97

where

9

D

c 4 E 2 -

factor ai in the ith iteration by

(18)

In this example, 6, = 0.0004137. Therefore, we have a, < 0.05 in the sixth iteration. Note that a, = 0.05 cor- responds to 0.43 dB difference between 6, and 6,. Thus,

a . = - - l 6i ' 6,

:\

In eqn. 17, N , is the number of terms that v{:(w)/k(w) are summed. Thus, Y , is the average value of the weighted peak ripple magnitudes. To avoid the mistake of taking 'fake peak' into consideration, two of the smallest weighted peak ripples are discarded in deriving Y,. Therefore, N , is smaller than the number of extrema1 points by two.

(f) Obtain B,(o) from B;(w). (i) for 0 < (ii) for up <

, w = 0 . 5 ~ and w, < w < Z, let B A 4 = PXw)

0.5q if E,(o)E,(w) z 0, Bi(w) = BXw); otherwise BAw) = l/fi:(w). If E,(w)E,,(n - w ) > 0, /?Ax - w) = B;(w); otherwise Bin - 0) = l/@;(w).

As the algorithm converges, the envelope function curve becomes flatter with an increasing number of iterations. Therefore, B,(w)/k(w) approaches a constant (Yi) for all w. It can be seen from eqn. 16 that, if BAw)/k(w) z Y,, PXw) z 1 ; otherwise, BXw) < 1. It can also be seen from eqn. 16 that PXw) approaches 1.0 when BAw)/k(w) approaches Y i . Therefore, p,(o) approaches 1.0 as the number of iterations increases.

6 Example

We choose one of the 16 quadrature mirror filters given in Reference 4 as an example to illustrate our design approach. The filter length N is 64 and the stopband edge is 0.586~. Fig. 5 shows plots of the maximum and

- Q 3 8 '.; % I \

Itemtion count

Fig. 5 ation

a Maximum value and b Minimum ualue of f l iw) in each iter-

minimum values of p,(o) in each iteration. As can be seen from Fig. 5, at the beginning of the iterations, the maximum and minimum values of PXw) are far from 1.0. Subsequently, they approach unity.

Let bi be the value of 5 after the ith iteration. Fig. 6 shows a plot of ai for all i of this example. As can be seen in Fig. 6, ai diverges in the first iteration and then con- verges rapidly. If sufficient iterations have been run, the final value of 6, denoted by 6,, can be obtained. For this example, 6, = 0.0003974. Define a convergence criterion

98

I 1 I I I 2 4 6 8 10

iteration count ob

Fig. 6

this is negligibly small as the stopband attenuation in this example is about -68 dB. As the number of iterations increases, 6 becomes smaller. In this example, 6,, = 0.0004006, resulting in a < 0.01. This corresponds to 0.087 dB difference between 6,, and 6,. Fig. 7 is a plot of the frequency response of the filter after the tenth iter- ation. It can be seen from Fig. 7 that the frequency response is equiripple.

7 Termination of algorithm

There are several ways that our algorithm may be ter- minated. Experiments show that it normally takes fewer than ten iterations to achieve a < 0.05. Just like the algo- rithm in Reference 8, although on the whole d i decreases as the number of iterations increases, occasionally hi may diverge in a few iterations. Therefore, we may let the algorithm run for 20 iterations and choose the best solu- tion which corresponds to the smallest 6,.

Another criterion for the termination of the algorithm is to check for the flatness of Biw) in each iteration. The following criterion can be used.

Value of6 in each iteration

si < (1 + Y)Yi (19) In eqn. 19, y is a user-defined constant, the algorithm may be terminated when eqn. 19 is satisfied. Comparing eqn. 18 and eqn. 19, it is clear that we are attempting to approximate 6, using Y i . The relationship between y and a is unknown. However, from experience, y is larger than 2a for most cases.

8 Computational complexity and memory

As stated in Section 1, the computational complexity of the previous design approach is very high. Our new approach designs quadrature mirror filters by using the modified conventional WLS method. By using the com- putational procedure described in Reference 14, the com-

requirement

I E E Proc.-Vis. Image Signal Process., Vol. 141, No. 2, April I994

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putational complexity of our new approach is O(N2) . As a consequence, the new approach is significantly faster than the previous approach.

. ;:: -100 -8 0 0

1.0006r

0.1 0.2 0.3 normalised frequency

a

0.4 05

I 1 I I I 0 0.1 0.2 0.3 0.4 0.5

b

0.9934l

normalised frequency

Fig. 7 a Frequency response of the quadrature mirror filter alter 10 iterations b Frequency response of the overall system alter 10 iterations

Table 1 compares the computer times and the values of 6 of 16 examples designed by these two approaches. The examples are selected from Reference 4 in which it is claimed that they fill over 95% of the applications. The filter length is from 8 to 64. The design is conducted on a SUN SPARC Station-1 by using double precision arith- metic. The criterion for the termination of the algorithm for both approaches is a < 0.01. As can be seen from Table 1, the new approach is 8 to 75 times faster than the previous approach.

From Table 1, it can also be seen that the maximum deviation obtained by the new approach may be slightly larger than that obtained by the previous approach for some examples. Experiments show that the maximum dif- ference is less than 10%.

As the new approach uses fewer matrices, it requires only 60% to 70% of the memory space of that of the previous approach for N ranging from 8 to 64.

IEE Proc.-Vis. Image Signal Process., Vol. 141, No. 2, April 1994

T a b l e l : M a x i m u m dev ia t ion a n d computer time of 16 e x a m d e s

w, N Previous New algorithm approach

6 computer 6 computer time (s) time (s)

1 0 . 7 8 ~ 8 0.0357820 2 12 0.0046500 3 16 0.0010721 4 0 . 7 ~ 12 0.0247208 5 16 0.0063865 6 24 0.0007577 7 0.62571 16 0.0390702 8 24 0.0079637

1.6 0.0358476 0.2 6.2 0.0046415 0.5

11.6 0.001 1455 0.3 5.8 0.0247530 0.4

13.1 0.0063892 0.5 49.6 0.0007789 3.8 13.0 0.0391211 0.6 43.7 0.0079532 1.8

9 32 0,0016479 154.1 0.0017241 10.6

15231 1.7 07254 7.2

10 11 0 .586~ 24 0.03121546 12 13 14 64 0.0003734 231 3.0 0.0004006 30.8 15 0 .546~ 48 0.0219468 524.3 0.0218958 11.1 16 64 0.0070547 1756.9 0.0070521 69.1

32 0.0107388 156.2

9 Conclusion

A novel and efficient approach for the design of equi- ripple quadrature mirror filters is presented. The compu- tational complexity of the new approach is O(N2) . Compared to the previous design method, the new approach achieves significant savings in computer time and yields reasonably good results.

10 References

I ESTEBAND, D., and GALAND, C.: ‘Application of quadrature mirror filters to split band voice coding schemes’. Proc. ICASSP, 1977, pp. 191-195

2 TRIBOLET, J.M., and CROCHIERE, R.E.: ‘Frequency domain coding of speech, IEEE Trans., 1979, ASSP-27, pp. 512-530

3 JOHNSTON, J.D., and CROCHIERE, R.E.: ‘An all-digital com- mentary grade subband coder’, Journal of the Audio Eng. Soc., 1979, 27, pp. 855-865

4 JOHNSTON, J.: ‘A filter family designed for use in quadrature filter banks’. Proc. ICASSP, 1980, pp. 291-294

5 ROTHWEILER, J.H.: ‘Polyphase quadrature filters: a new subband coding technique’. Proc. ICASSP, 1983, pp. 1280-1283

6 LIM, Y.C., KOH, S.N., and KO, C.C.: ‘A new filtering scheme for split-band voice coding’, IREE Journal of Electrical and Electronic Engineering, Australia, 1989, 9, (l), pp. 27-34

7 LIM, Y.C., YANG, R.H., and KOH, S.H.: ‘The design of weighted minimax quadrature mirror filters’, IEEE Trans., 1993, SP-41, pp. 1780-1789

8 GRENEZ, F.: ‘Design of quadrature mirror filters by linear pro- gramming’. Proc. IEEE Int. Conf. ASSP, Tokyo, 1986, pp. 2615- 2618

9 GRENEZ, F.: ‘Chehyshev design of filters for subband coders’, IEEE Trans., 1988, ASP-36, pp. 182-185

10 SMITH, M.J.T., and BARNWELL, 111, T.P.: ‘Exact reconstruction techniques for tree-structured subband coders’, IEEE Trans., 1986, ASP-34, pp. 434-441

1 I VAIDYANATHAN, P.P.: ‘Theory and design of M-channel maxi- mally decimated quadrature mirror filters with arbitrary M, having the perfect reconstruction property’, IEEE Trans., 1987, ASP-35, pp. 476-492

12 NQUYEN, T.Q., and VAIDYANATHAN, P.P.: ‘Structures for M- channel perfect reconstruction FIR Q M F banks which yield linear phase analysis filters’, IEEE Trans., 1987, ASP-35, pp. 433-446

13 LIM, Y.C., LEE, J.H., CHEN, C.K., and YANG, R.H.: ‘A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design’, IEEE Trans., 1992, SP-40, pp. 551-558

14 YANG, R.H., and LIM, Y.C.: ’Eficient computational procedure for the design of FIR digital filters using WLS technique’, IEE Proc. G, 1993,140, (5 ) , pp. 355-359

I 5 PRESS, W.H., FLANNERY, B.P., TEUKOLSKY, SA., and VET- TERLING, W.T.: ‘Numerical recipes in c‘ (Cambridge University Press, 1988)

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11 Appendix

1 I . 1 Neville’s polynomial interpolation algorithm [ 151 Suppose that the values of y =f(x) are known for M values of x, i.e. (xlr yl), (x,, y,), ..., and (x,, y,) are known, where f(x) is a polynomial of degree M - 1. Suppose also that x, < x2 < ... < xM and that x,, x,, . . . , x, are not necessarily evenly spaced. It is desired to estimate the value off(x) for x, < x < x,.

Let Pi = yi , for i = I , 2, ..., M. Let PI, be the value at x of the unique polynomial of degree one passing through both (x,, y,) and (x2, y,). Likewise define P,,, P, , , ..., Po-,,,. Similarly, for higher order poly- nomials, up to PI,, ... ,. Hence, p,,, ..., is the value of f(x) of the unique polynomial of degree M - 1 passing through all the .%I points, (x,, y,), (x,, y,), ,.., and (XM > Y d . tcu,’-”

The various Ps form a ‘tableau’ with ‘ancestors’ on the left leading to a single ‘descendant’ at the extreme right. For example, with M = 4,

XI : y, = P,

f x,: y 3 = P:’ b2,,

x,: y, = P ,

Neville’s algorithm is a recursive way of calculating the various Ps in the tableau a column at a time, from left to right. It is based on the relationship between a ‘daughter’ P and its two ‘parents’. For the above example. P,,, P,, and P,, are calculated first, by using

(x - Xi+,)Pi + (Xi - X)Pi+I x. - x . Pi(i+ 11 =

I I f 1

for i = 1, 2. 3 (AI )

Next, PI,, and P,,, are calculated by using

for i = 1, 2 (A2) Finally, the desired answer can be obtained by

IEE Proc.-Vis. image Signal Process., Vol. 141, No. 2, April 1994