Design of nonrecursive quadrature mirror filtersS.W. Foo, B.E., M.Sc, C.Eng., M.I.E.E., and Prof. L.F. Turner, B.S

Indexing terms: Nonrecursive structures, Quadrature mirror filters

Abstract: A new method of designing nonrecursive quadrature mirror filters with small ddesired overall response is described. By imposing linear constraints at the midpoint of thscale, in addition to the normal passband and stopband constraints, a solution that is veryoptimum is obtained. Satisfactory designs can thus be obtained without resorting to technisolution of nonlinear constraints. Some results of practical interests are presented.

JUNI 7viation from thefilter frequency

ose LIBRARY

1 Introduction

The subband coding of speech signals has recently receivedconsiderable attention [1—6]. In coding schemes of this type,the speech signal is filtered into several contiguous subbandsand the signals in the different subbands are processed andcoded differently before transmission. In this way a reductionof transmission rate is effected while the quality of the originalspeech signal is preserved.

As part of the signal processing involved within each sub-band, the signals are sampled at a rate less than the Shannonrate, with the result that aliasing can occur when practicalfilters are employed. Aliasing cancellation can be achieved byusing quadrature mirror filters to perform the band splitting[5]. In general, in any situation in which band splitting isperformed and downsampling is employed, the use ofquadrature mirror filters makes it possible, on reconstruction,to remove frequency aliasing completely. Although quadraturemirror filters can be implemented in a recursive form [6], andthe length of filter for a given amount of out-of-bandattenuation can thereby be reduced, there is the disadvantagethat the phase of the signal is distorted. Nonrecursive structureshave linear phase characteristics, and hence they better preservethe phase of the signal.

In this paper, a new method of designing nonrecursivequadrature mirror filters is described. By the inclusion ofadditional linear constraints, the method takes into consider-ation the deviation of the overall system frequency response aswell as the passband ripple and stopband attenuation.

For completeness, a brief description of the designrequirements for quadrature mirror filters is given in Section 2.In Section 3, the general problem of quadrature mirror filterdesign is formulated and the various parameters involved arediscussed. In Section 4, the specific design of quadraturemirror filters with two and four taps is presented, anddiscussions of the specific design of filters of larger orderhaving parameters of particular interest are given in Section5. Some conclusions are presented in Section 6.

2 Theory of quadrature mirror filters

Quadrature mirror filters were proposed by Estaban andGaland [5], and the block diagram of a splitband systememploying quadrature mirror filters is shown in Fig. 1. Theinput signal x(t) is lowpass filtered to a bandwidth of B(=fs/2) and sampled at the Shannon rate/g. The lowpass filterHi and the highpass filter Hu are then used to split thespectrum of x(ri) into lower and upper parts of equal band-width. The filtered signals xt(n) and xu(n) which each occupya bandwidth of approximately half of B are decimated (down-sampled) by retaining only alternate samples. Compressionencoding is usually carried out on the decimated signals yi(n)a nd yu(

n) before transmission to the receiver.

Paper 1841G, first received 19th October 1981 and in revised form18th January 1982The authors are with the Department of Electrical Engineering,Imperial College of Science and Technology, London SW7 2BT,England. Mr. Foo is on leave from the Ministry of Defence, Singapore

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

At the receiver of the system, signals yt(n) and J>U(H) areinterpolated to obtain the signals Wj(«) and uu{n). The inter-polation takes the form of the addition of zeros betweensuccessive samples of signalsyt(fi) and yu(n), respectively. Theinterpolated signals u^ri) and uu(n) are then filtered by Px andPu, respectively, and combined to obtain s(n) which is intendedto be a replica of the original signal x(n).

The design constraints for filters yielding an undistortedreconstruction of the original signal were shown by Estabanand Galand to be:

(a) Hi- a symmetrical nonrecursive lowpass filter of evenorder

(b) Hu(z) = Ht(— z), where Hu{z) is the z-transform of Hu

(c) Pt(z) = Hi(z), where P,(z) is the z-transform of Pt

(d)Pu(z) = —Hu(z), where Pu(z) is the z-transform of Pu

(e)Hf2(ei") + Ht2(eHn-UJ)) = 2, where Hf(eJui) isrelated to the Fourier transform Ht(e

iu}), of //j(z) byH*(ei") = Hl(e

i") exp (j{N- l)co/2).When these constraints are satisfied, the output signal s(n)

is an exact replica of x(n), apart from a delay of (TV— 1)samples. That is,s(n) =x(n —N+ 1).

Constraints (a) to (d) are necessary for the cancellation offrequency aliasing, with the additional constraint (e) beingnecessary in order to ensure a flat overall response. Constraint(e) is difficult to satisfy in practice and it is this constraint thatis considered in detail in this paper.

Fig. 1 Subband coding using quadrature mirror filters

3 Problem formulation

The aim is to design nonrecursive quadrature mirror filterswhich, for a given number of taps and a given transition width,minimise the passband ripple, maximise the stopbandattenuation and minimise the overall system deviation. Onlythe design of the lowpass filters will be considered since thedesign of other filters can be readily derived from the lowpassdesign.

Let Bp be the set of passband frequencies / for which}p and let B8 be the set of stopband frequencies/

for which {Fs < / < 0.5}, where, in the above, Fp and F, arethe cutoff frequencies of the passband and stopband,respectively, of the lowpass filter.

Further, let B represent the set of frequencies / , such that}

If, in the passband, the desired magnitude is \fl, and 8p isthe maximum deviation allowed, where 5P > 0, then it is

0143-7089/82/030061 +08 $01.50/0 61

desired that

\/2-8p H*(e i2nn

f^Bp (1)

If in the stopband, the desired magnitude is 0, and 5S denotesthe maximum allowable ripple, with 8S > 0, then it is desiredthat

- 5 H*(ei2nf)

(2)

Further it is desired that

2-d0 [H*(ei27Tf)]2

2+d0

(3)

where d0 is a nonnegative number which denotes themaximum deviation in the overall response.

The aim is to find the impulse response h(n) that minimisesthe quantity max {5P, 5S, d0} for a given set of TV, Fp, Fp, Fs.

pOn account of the nonlinear constraints set out in eqn. 3

preceding, linear programming methods cannot be used tosolve the global optimisation problem. A method will now bedescribed which provides a better solution than one in whichthe constraints of eqn. 3 are ignored.

At the midpoint of the filter frequency scale, that is, whenfm = / « /4 = 1/4 for/s = 1, the middle term of eqn. 3 becomes

= 2[Hi(eiiT/2)]2

and the constraints of eqn. 3 become

2-d0 < 2[Ht(einn)]2 < 2+d0

or

1 -do/2 < [Htie"12)]2 < 1 +dol2 (4)

On adding (do/4)2 to the three terms in eqn. 4, the followingresult is obtained:

l+(do/4)+(do/4)2

i.e.

(5)

For cases of practical interests,

(do/4f < [ H t ' "

so

(6)

It is important to note that the constraints of eqn. 6 are linearconstraints.

By imposing the linear constraints of eqn. 6 instead of thenonlinear constraints of eqn. 3, a solution can be obtainedwhich minimises d0 as well as 5S and 5 P , although the solutionmay of course not be a global minimum. As will be seen later,the solution is very satisfactory for most practical purposesand, should further optimisation be required, the solution canbe used as a starting point in an iterative process.

The deviation d0 of the overall system for frequencies lessthan Fp is related to 8S and 5P , as is the overall deviation d0

for frequencies greater than Fs. That is, f o r / C 5 p UBs

2+d0 (7)

(8)2-d0 < ( x / 2 - 5 p ) 2 + 5 s2

On expanding eqn. 7, it can be seen that

d0 > 2 V 2 5 p + 5 p + 5 s2

(9)

In most practical cases, 5P + 52 < 2 \ /2 6 p ; thus, the expectedminimum d0 is approximately 2y/28p.

If 5P is set equal to K8S, where A" is a constant, then theproblem can be formulated in a form suitable for solution bylinear programming [7]. A set of frequencies {fpc} and {/„.}are chosen from Bp and Bs, respectively. The greater thenumber of frequencies chosen, that is, the higher the griddensity, the more accurate are the results of the computation.However, the complexity of the computation also increaseswith increasing grid density.

Using the notations of Appendix 9, the problem can bestated as follows:

Find the set of b(/), i = 1, 2 , . . . (TV/2) subject to

(0 > 0 (10)

Nil

(ii) -K8S+ X b (n) cos. {2vf{n - (112))}

<V2Af/2

(iii) -K8S-

/e{/pc}

JV/2

(iv) - 88 + X b(n) cos {2nf(n - (1/2))} < 0

f^ifsc)

N / 2

(12)

03)

04)

(15)

(16)

such that — 5S is maximised, i.e. 6S is minimised.In eqns. 10 to 16 preceding, TV is an even integer and fs =

1.Assume that 8S is the minimum value obtained for a given

set of Fp,Fs and TV, then, from eqn. 9, the lower bound for d0

ismin {d0 } > 2%/2K8s + 82

S+K282S (17)

In the discussions which follow, the solution with constraintsof eqns. 10 to 14 only will be referred to as solution A, andthe solution with the complete set of constraints of eqns. 10to 16 will be referred to as solution B. It should be noted thatthe constraints of eqns. 15 and 16 are based on constraint (e)of Section 2.

(v) - 5S - X b{n) cos {2nf(n - (1/2))} < 0

N/2

(vi) -(\/y/2)K8s+ X b(n) cos {nfs(n

-(l/2))/2}< 1N/2

(vii) - (1 ly/2)K8S - I b («) cos {TIfs (n

62 IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

4 Quadrature mirror filters of two and four taps

The design of quadrature mirror filters with N = 2 andN= 4will now be considered. These designs provide an insight intothe general design of quadrature mirror filters, in addition tobeing valuable in their own right.

When Ar = 2, there is only one design variable, namelyh(0), since h(l) = h(0). In this case, the magnitude of thefrequency response is given by

= 2/2(0) cos (co/2) (17)

and the magnitude of the frequency response of the mirrorfilter is

H*(eio}) = 2/i(0) cos [(IT - C O ) / 2 ]

= 2/2(0) cos [(TT/2)-(CO/2)]

= 2/2(0) sin (co/2)

The magnitude of the overall response is

(18)

= [2/i (0) cos (co/2)]2 + [2/2(0) sin (co/2)]2

= 4/22 (0) [cos2 (to/2) + sin2 (co/2)]

= 4/22(0) (19)

If

M0) = (My/2) (20)

then R*(eicj) = 2, for all co, and a perfect solution isobtained.

2.Or

0.5

0.00.000 0.125 0.250 0.375 0.500

Fig. 2 Frequency response of4-point FIR lowpass filter

When N = 4, there are two variables, h(0) and h(l). Thediscrete frequency spectrum is as shown in Fig. 2. Asexplained in Appendix 9, point 3 in the Figure is fixed at zeroas a characteristic of symmetrical FIR filters. Thus, the valuesat two points only are to be decided upon. As point 2represents magnitude at the midpoint of the filter frequencyscale, this point is chosen to be equal to unity, and point 1,being in the passband, is chosen to be \fl. With this selection,

H*(ei0) = 2/2(0)+ 2/2(1) = y/2 (21)

H*(einn) = 2/2(0) cos (3TT/4)+2/2(1) cos (TT/4)

= [-2/2(0) + 2/2(1)] fy/2 = 1

i.e.

-2/2(0) + 2/2(1) = V2 (22)

On solving eqns. 21 and 22, it is found that h(0)= 0 andh(l) = 1/V5, which reduces to the case for N = 2.

Thus, if the size of the transition width, the stopbandattenuation and the passband ripple are unimportant, then aperfect quadrature mirror filter that gives the required overallresponse can always be obtained.

5 Quadrature mirror filters of larger number of taps

As explained above, perfect quadrature mirror filters areobtainable with the desired overall response if the passbandripple, stopband attenuation and the size of the transitionwidth of the filters are unimportant. In practice, of course,these parameters are important and it is desirable to havefilters that not only keep the overall response close to thedesired value, but also have a low passband ripple, a large stop-band attenuation and a small transition width.

The design technique as proposed and formulated inSection 3 has been used in the design of quadrature mirrorfilters for N = 8, 16 and 32 for various values of transitionwidth and various values of Ks of interest. Filters of any evenorder can be designed using this method, although for large N,the complexity of the problem increases and the time takenfor computer solution increases rapidly. The design for thecase of TV = 16 will now be considered in some detail.

The grid density used in the solution was chosen to be 10;that is, the frequency scale normalised with respect to thesampling frequency fs wasdivided into 16 x 10 = 160 divisionsfor the case in which N = 16.

The transition width (TW), expressed in units for/ s , wasvaried in steps from 1/8 to 1/4 and the cutoff frequency,/^,of the filters was varied over the range of interest.

Of particular interest is the peak-to-peak deviation of theoverall response from the desired value, as well as the stop-band attenuation. The stopband attenuation, denoted £Ts,wascalculated as

Es = 20(log1 05 s- log1 0V2) (23)

A measure of the peak-to-peak deviation of the overallresponse was obtained as follows:

0.8

0.7

0.6

0.5

0.4-

-0.2

Fig. 3

253 256 260264 268504 508 5128 12"248 253 256 260264sample number

Augmented impulse response

The frequency response of the filter was obtained by firstadding 256 — (16/2) = 248 zeros to both ends of the impulseresponse as shown in Fig. 3. The Fourier transform of thisaugmented impulse response gives 512 samples of the frequencyresponse of the filter over the frequency range from zero to1.0. By adding the mirror image of this frequency response tothe response itself, the response of the overall system isobtained. Mathematically,

Rm = (24)

n = 0 , 1 , . . . , 256

IEE PROC, Vol. 129, Pt. G, No. 3, JUNE 1982

Let Rmax and Rmin represent the maximum and the minimum

63

values, respectively, of the resulting overall response over theentire filter frequency range. The maximum peak-to-peakdeviation of the overall response, denoted Do, is defined as

-30

Do = 20(log10Rmax-log10Rmin) (25)

The results of Do and Es could be presented as a function ofthe cutoff frequency. However, a better way of presenting thedata is obtained by transforming the cutoff frequency in termsof percentage of transition width deviation from the midpoint,i.e. fm = 0.25, of the filter frequency scale. This value,denoted fc, is calculated from the normalised cutoff frequency £

fc = (0.25-Fp)*l00/TW (26)

The results of Do and Es for various values of transition width(TW), with K being chosen to be 1, are shown in Figs. 4 and 5,respectively, as a function of/c.

To give some idea of the results obtained when the mid-point constraints are neglected (solution A), the correspondingDo and EB are shown in Figs. 6 and 7.

For various values of K, with TW fixed, the variation of Do

and Es as a function of fc is shown in Figs. 8 and 9 under theconditions for solution B, and in Figs. 10 and 11 under theconditions for solution A.

Fig. 12 provides a comparison of the maximum one-sideddeviation DOs, with K = 1 and TW = 5/32 for solutions A andB, and Fig. 13 provides a comparison of Es with K — 1 andTW = 5/32 for solutions A and B. The maximum one-sideddeviation D08 is calculated using

DOs = ™xof{20logw(RmaJ2);20loglo(2/Rmin)}(21)

and the bounds are calculated using

where

da = (28)

a

20 30 40 50cutoff frequency(percent TW from midpoint)

Fig. 4 Maximum peak-to-peak deviation of overall response asfunction of cutoff frequency expressed in percentage of transitionwidth from fj4 (solution B)

A' = 16, A: — 1Transition width (TW) values:

CD

I -50o

- 6 0

-7 0

20 30 40 50cutoff frequency (percent TW from midpoint)

#

Fig. 5 Stopband attenuation as function of cutoff frequencyexpressed in percentage of transition width from fsj4 (solution B)

N - 16, K = 1Transition width (TW) values:a L +_§_ o-2- X— A i-

8 > + 3 2 ' U 16 > A 3 2 ' 4

CD 6

o 5

E

E 3

20 30 40 50

cutoff frequency (percent T Wf rom midpoint)

Fig. 6 Maximum peak-to-peak deviation of overall response forvarious values of transition width (solution A)

TV = 16, K - 1Transition width (TW) values:

From these Figures, the following important observations canbe made:

(i) The minimum values of Do for a given K and TW occurat about the same cutoff frequency for both solution A andsolution B, but the minimum values of Do are not necessarilythe same at the points of minimum.

64 IEEPROC, Vol. 129, Pt. G, Vo. 3, JUNE 1982

(ii) As TW is increased, the cutoff frequency fc at which theminimum Do occurs moves closer to 50%. It should be notedthat a value of fc of 50% means that the transition band issymmetrical about the midpoint of the filter frequency scale.

(iii) In general, for a given value of K, increasing TW resultsin a decreasing Do for solution B; this is not so for solution A.

(iv) For solution B with a given value of TW, decreasing Kresults in a smaller attenuation (i.e. larger Es) and also adecreasing value of Do. For solution A, decreasing K results ina smaller stopband attenuation but the value of Do does notnecessarily decrease.

(v) Solution B achieves a lower value of DQ at the expense

-40

-60

-70

-30r

20 30 40 50cutoff frequency (percent TW from midpoint)

Fig. 7 Stopband attenuation for various values of transition width(solution A)

N = 16, K = 1Transition width (TW) values:

5-35co

CJ - 40T3Coao

- 4 5

0.150 0.175 0.200 0.225normalised cutoff frequency

0.250

Fig. 9 Stopband attenuation for various values of K (solution B)

OK=1, + K = 2,OK = 3N = 16, TW = £

8 r

0.150 0.175 0.200 0.225normalised cutoff frequency

0.2 50

Fig. 8 Maximum peak-to-peak deviation of overall response forvarious values of K (solution B)

= 1, + K = 2,s

= 3

•5 5

0.150 0.175 0.200 0.225 0.250normalised cutoff frequency

Fig. 10 Maximum peak-to-peak deviation of overall response forvarious values of K (solution A)

n K = l , + K = 2,/>K = 3N = 16, TW = •£

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982 65

of a large increase of Es when compared with solution A. Thegraphs of Es against fc are concave upwards for solution B,whereas those for solution A are concave downwards (convex),

(vi) From Fig. 12, it can be seen that if the midpointconstraints are included, then the minimum one-sideddeviation DOs is very close to the global minimum, since theglobal optimal value of DOs, for the range of/c of interest, liesbetween the curve representing the results for solution B andthe curve for the bound to solution B. The difference betweenthese curves is small. If, however, the midpoint constraints arenot imposed (solution A), then the difference between theresults and the bound is large.

-40

CQ•o

co

1-45

0.150 0.175 0.200 0.225normalised cutoff frequency

0.250

Fig. 11 Stopband attenuation for various values of K (solution A)

aK = l, + K = 2, o K = 3N = 16, TW = ^

0.175 0.200 0.225normalised cutoff frequency

Fig. 12 Comparison of maximum one-sided deviation of overallresponse for solution A and solution B

a results A, + results BA bound B, X bound AN = 16, K = 1, TW = £

66

Simulations carried out for the cases of N = 8 and N = 32show that the above observations also apply in these cases.

As an example, the lowpass filter response and the overallsystem response for a particular case in which N=32 areshown in Figs. 14 and 15, respectively.

6 Conclusions

The introduction of additional linear constraints at the mid-point of the frequency scale in the linear programmingformulation of the design of the quadrature mirror filtersprovides a much better overall response than that obtainedwhen the midpoint constraints are neglected. The deviationof the overall response is very much reduced, but the reductionof the ripple in the overall response has the effect of reducing

-30

2 - 3 5

g-AO

-U 5

0.175 0 200 0.225normalised cutoff frequency

Fig. 13 Comparison of stopband attenuation for solution A andsolution B

a results B, + results AN = 16, K ~ 1, TW = £

10

0

-10

-20

-30

-40

• ° - -50a>

B -60

E -70

-80

-90

-100

-1100 0.05 0.10 0.15 0.20 0.25 0.3C 0.35 C.AC C.A5 C.5C

normal ised f requency

Fig. 14 Frequency response of 32-point lowpass filter

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982

the stopband attenuation of the filter. Different combinationsof stopband attenuation and deviation of the overall responsecan easily be obtained by using a different multiplicationfactor K.

Whereas a global optimal solution to the design involvesnonlinear constraints, and is correspondingly more compli-cated, the method introduced in this paper involves only linearconstraints, is relatively simple, and gives a solution which issatisfactory in most cases of practical interest, and is very closeto the global optimum.

2.08

2.07

2.06

2.05

2.04

I 2.03

E 2.02o

| 2.01

> 2.00

1.99

1.98

1.97

1.960 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

normalised frequency

Fig. 15 Response of overall system based on filters with lowpassfrequency response shown in Fig. 14

9 Appendix: Nonrecursive filters of even order

For symmetrical FIR filters of even order, the impulseresponse coefficients are related by

h(n) = h(N- l-n) 0 < n < (N/2)-l (29)

Under this condition, the frequency response of the filter isgiven by

JV-I

H(ej") = I h(n)e-ju}n

n=0

I h(n)ei"n+ X h(m)e-Jutm

n=0 m=JV/2

by putting m = (N — 1 — n), we have:

H{eiuj) = £ h(n)e-jojn +

0

I h(N-l-n)e-iu}(-N-l-n)

n=(iV72)-l

[h(n)e~juin +h(N-l-n)x- l -n) ]

71=0

n=0

n =02h(n)x

cos (30)

7 Acknowledgment

One of the authors, S.W. Foo, is in receipt of a DSO scholar-ship, and he wishes to thank the Ministry of Defence, Singaporefor this financial support.

8 References

1 CROCHIER, R.E., WEBBER, S.A., and FLANAGAN, J.L.: 'Digitalcoding of speech in sub-bands', Bell Syst. Tech. J., 1976, 55, pp.1069-1085

2 CROCHIERE, R.E.: 'On the design of sub-band coders for low-bitrate speech communication', ibid., 1977,56, pp. 747-770

3 BARABELL, A.J., and CROCHIERE, R.E.: 'Subband coder designincorporating quadrature filters and pitch prediction'. Proceedingsof the 1979 international conference on acoustics, speech and signalprocessing, Washington, DC 1979, pp. 530-533

4 GRAUEL, C. 'Subband coding with adaptive bit allocation' SignalProcess., 1980, 2, pp. 23-30

5 ESTABAN, D., and GALAND, C: 'Application of quadraturemirror filters to split band voice coding schemes". Proceedings ofthe 1977 international conference on acoustics, speech and signalprocessing, Hartford, CT, 1977, pp. 191-195

6 RAMSTAD, T.R., and FOSS, O.: 'Subband coder design usingrecursive quadrature mirror filters'. Proceedings of 1980 EuropeanAssociation for Signal Processing conference, Lausanne, Switzerland,1980, pp. 747-752

7 GASS, S.I.: 'Linear programming: methods and applications'(McGraw-Hill, 1975)

To reduce the argument of the cosine term into a formindependent of N, let

bin) = 2h«Nl2)-n) n = 1,2,...,TV/2 (31)

Then

where

(32)

TV/2

n = i(33)

It should be noted that H*(e10J) is a real function taking onboth positive and negative values, and the magnitude ofH(e]") is the same as that of H* (eiu}). Furthermore, at u =n,H*(eioj) = 0 and is independent of b(n) or h(n). Thisimplies that the filters with a frequency response that is non-zero at co = n (e.g. a highpass filter) cannot be satisfactorilydesigned with this technique.

IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982 67

Foo Say Wei received the B.E. degree inelectrical engineering from the Universityof Newcastle, Australia, and the M.Sc.degree in industrial engineering from theUniversity of Singapore. Since graduatinghe has been working in the Ministry ofDefence, Singapore, first as head of theElectrical Branch and subsequently ashead of a research department. He ispresently on leave and is working for thePh.D. degree in electrical engineering at

the Imperial College of Science and Technology, London.AT*

Laurence Turner served an apprentice-ship with AEI, during which time heobtained ONC and HNC certificates. Hethen studied at the University of Birm-ingham from which he received the B.Sc.and Ph.D. degrees. After a period as aresearch fellow at the University of Birm-ingham, he joined Standard Telecom-munications Laboratories Ltd., where heworked on aspects of pattern recognitionand data transmission. He subsequently

joined the Electrical Engineering Department at ImperialCollege, where he is at present Professor and head of theDigital Communications Section.

68 IEEPROC, Vol. 129, Pt. G, No. 3, JUNE 1982