Multidimensional spherically symmetric recursivedigital filter design satisfying prescribed magnitudeand constant group delay responses

H.K. Kwan, BSc, MPhil, PhD, CEng, MIEEC.L. Chan, BSc(Eng)

Indexing terms: Digital filters, Filters and filtering, Circuit theory and design, Mathematical techniques

Abstract: A computationally efficient techniquefor the design of multidimensional sphericallysymmetric recursive digital filters satisfying pre-scribed magnitude and constant group delay spe-cifications is presented. The denominator and thenumerator of the transfer function are designedseparately. The former is used to approximate thegroup delay response and the latter is used toapproximate the magnitude response. Moreover,this method also makes use of the symmetric con-ditions of the transfer function. Therefore thenumbers of parameters and sample pointsrequired for the optimisation are greatly reduced.As a result, the amount of computation can beminimised and the convergence can also beimproved. Such advantages are extremely signifi-cant for high-order multidimensional filter design.Two examples of the 2-dimensional and oneexample of the 3-dimensional digital filter designare given to illustrate the proposed method. Com-parisons with the results of other methods are alsogiven.

1 Introduction

Two-dimensional and 3-dimensional digital filters arefinding applications in many fields such as image pro-cessing, seismic signal processing, magnetic data pro-cessing and biomedical tomography. In many of theseapplications, a signal does not have any preferred spatialdirection and so the required digital functions can be cir-cularly or spherically symmetric. In the last decade, manymethods [1, 2] have been proposed for 2-dimensionalrecursive digital filter design and some [3-5] for3-dimensional recursive digital filter design. However,few of them satisfy both magnitude and group delaycharacteristics in which the latter is extremely importantin 2- and 3-dimensional image processing [6].

Maria and Fahmy [7, 8] adopted the idea of using a2-dimensional digital filter to approximate a given mag-nitude specification and then cascaded it with a 2-dimensional allpass digital filter to equalise the resultinggroup delay. However, the overall filter is, in general, not

Paper 5458G (E10), first received 3rd June 1986 and in revised form22nd April 1987

The authors are with the Department of Electrical and ElectronicEngineering, University of Hong Kong, Pokfulam Road, Hong Kong

optimal and the filter order becomes higher. Chotteraand Jullien [9] formulated this kind of recursive digitalfilter design as a linear programming problem. However,to use a linear stability constraint, which is only a suffi-cient condition for stability, the method only yields asubclass of possible solutions. Besides, there are alsosome methods [10, 11] which tackle this problem bychoosing a performance index expressed as a linear com-bination of three error functions, i.e. one for the magni-tude response and two for the group delay responses.However, these methods may have the difficulties of com-putational complexity and bad convergence.

Hinamoto and Maeskawa [12] recently proposed amethod for 2-dimensional recursive digital filter design.The design procedure is divided into two parts. The firstpart is to design an all-pole digital filter to approximate agiven group delay specification. The second part is todesign a mirror-image polynomial as the numerator toapproximate a given magnitude specification. Thismethod has the advantages of reduction of the amount ofcalculations, improvement of convergence, stable struc-ture, and reduction in round-off error in cascade realis-ation. However, the general transfer function of themethod does not fully use the properties of circular sym-metry. Therefore, in the design of a high-order circularlysymmetric filter, the computation time of this method willbe very long.

Application of the properties of symmetry [5, 13, 14]in filter design has been tried in References 15 and 16. Inthis paper, we are going to present a new and computa-tionally efficient method for the design of a multidimen-sional spherically symmetric digital filter that meets agiven set of magnitude and constant group delay specifi-cations. This method adopts a similar procedure to thatof Reference 12. However, the transfer function is basedon the format of a separable denominator and an insep-arable linear phase numerator. Moreover, the propertiesof octagonal symmetry are fully used in the optimisationprocedure. Unlike the method used in Reference 11, thetransfer function used consists of a separable denomina-tor and a general inseparable numerator. Other specialcases of a 2nd-order 2-dimensional transfer function canalso be found in References 15 and 16. Because of theproperties of symmetry adopted in our method, theamount of calculation, the number of coefficients to bedetermined, the number of frequency samples required,and the number of iterations in the optimisation can begreatly reduced. Moreover, the method is furtherextended and generalised for a multidimensional spher-ically symmetric digital filter design. This method isextremely suitable for the design of high-order spherically

IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 187

symmetric digital filters. Two examples of 2-dimensionalfilter design and one example of 3-dimensional filterdesign will be used to illustrate this method. Compari-sons with the results of two other 2-dimensional filterdesign methods are also provided.

2 Spherical symmetry properties

Generally speaking, a JV-dimensional filter //(wl5 w 2 , . . . ,wN) that possesses spherical symmetry in magnitudemeans that

I H(wu w2, . . . , wN) | = | H(w'u W2, . . . , w'N) | (1)

for all values of wl5 w2, . . . , vvN and w\, w'2, ..., w'N in therange — n < w,- ̂ n, — 71 < wj ^ 7r, for 1 = 1, 2, . . . , ATsuch that

(wj + w2 + • • • + w2N)112 = (w'2 w'2)1'2

(2)

Obviously, spherical symmetry must imply the followingkinds of symmetry:

\H(wlt w2, . . . , w,-, . . . , wN)\

= \H(w1, w2, . . . , -w,., . . . , WJV)| (3)

and

H(wlt w 2 , . . . , j, . . . , wN)\

= \H(w1,w2,...,wj,...,wi,...,wN)\ (4)

for all i = 1,2,..., N and; = 1,2, ...,N.For a N-dimensional filter, such kinds of symmetry

should be referred as (2NN !)-hedral symmetry. Forexample, it should be referred as 8-hedral (octogonal)symmetry for a 2-dimensional filter and 48-hedral sym-metry for a 3-dimensional filter [4, 5]. In this paper, atransfer function possessing these kinds of symmetry ischosen to approximate the characteristics of sphericalsymmetry.

3 Design procedure

In this N-dimensional filter design method, the first stepis to design a N-dimensional all-pole filter to approx-imate a given group delay specification. Then a linearphase numerator is designed to approxinate a given mag-nitude specification. This method purposely separates thegroup delay optimisation and the magnitude opti-misation to prevent the problems of complex computa-tion and convergency. However, it may lose a certaindegree of freedom compared with that of the simulta-neous optimisation of group delay and magnitude.Though the denominator is only used to approximate thegroup delay response, it provides a recursive part for thetransfer function so that the magnitude approximationcarried out by the numerator can be done more effi-ciently.

3.1 Group delay approximationThe denominator of the transfer function is chosen to beseparable such that the stability problem can be simpli-fied and also the computation can be greatly reduced.

1

where

(5)

(6)

Q(Zi l) is a polynomial in z,- \ z(- 1 = exp (— ;w,), which

can be expressed in the (2/C)th-order as

{id)QbT l)=U (dV + dfzrx + dfz;2)fc=i

or in the (2K + l)th order as

i=\,2,...,N (76)

To ensure stability, d^ must be expressed in terms ofanother set of real nonzero parameters {q[*]}, p = 0, 1,2;» = 1,2:

d(O) = ^(0)2 _ j

d^ = q(?2 + q2k)2 + l (8)

It can be verified that the magnitude of eqn. 6 is (2NN !)-hedrally symmetric. Eqn. 6 can be written as

. _. . . . 1

xexp(jeD(wuw2,...,wN)) (9)

^ K , w 2 , . . . , wN) = O'iwJ + O'(w2) + ••• + 0'(wN) (10)

where ^(w,) is the argument of l/QCzf1), 1 = 1, 2, . . . , N.The group delay functions are defined as

T , ( W 1 5 W 2 , . . . , WN) =

Substituting eqn. 11 into eqn. 10, we have

-d6'(Wi)T,(W15 W2 , . . . , WN) = — = T(W.)

(11)

(12)

The group delay in the direction of w, is a function of asingle variable w,-. Moreover, it can easily be verifiedfrom eqns. 6, 7 and 12 that

T(Wi) = T(-Wl) (13)

and from eqns. 7 and 12

T M , W2 , . . . , wN) = T/W15 W2 , . . . , wN) (14)

where i,j = 1, 2 , . . . , N.

Based on eqns. 12, 13 and 14, the approximation ofgroup delay is reduced to be a 1-dimensional problem.Hence, the number of sample data for the optimisation ofgroup delay specifications can be greatly reduced.

Samples can be chosen only on the wt axis in therange 0 < wt < n. Supposing that M' samples are chosena t w i 9 5 9 = 1> 2, . . . , M' and the corresponding groupdelays are i(wlfl). The statement of the problem is to findthe parameter vector b

= 1, 2; k = 1, 2, . . . , (15)

so as to minimise the group delay performance index J(b)which is defined as

M'

9=1(16)

where <r is the desired constant group delay, which ischosen to be a positive value [11], and ug is a non-

188 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987

negative weighting function and is equal to zero outsidethe passband. The Fletcher-Powell algorithm [17] isused in the optimisation of eqn. 16.

3.2 Magnitude approximationThe next step is to design a constant group delay orlinear phase numerator polynomial so as to approximatethe desired magnitude characteristics. Let the recursivefilter be

where

. 1

/=

My is chosen to be

» Z 2 > - - - > z N ) = z

(17)

(18)

—mtN

/ mr mf mj

x ( I E ••• E « , . . „\oi=0 «2-0 ajv = O

x cos ( a ^ ) • • • cos (aNwN)

where

fli ••• oj ... ajv

cos (a,- vv,) = -L-

_ f(f)

+ z"

... aj ... a;... ON

Uj= 1,2,.. N

(19)

(20)

(21)

and the order of the numerator is 2Lmf, where mf is apositive integer. Because of the (2NN !)-hedral symmetry,the sample data may be taken only in the region R on the(wl5 w 2 , . . . , wN) domain such that

'0

R

w

(22)

Therefore the number of samples required for opti-misation can be greatly reduced.

Supposing that M" discrete samples (wlm, w 2 m , . . . ,wNm), for m = 1, 2, . . . , M", in i? are chosen, the corre-

sponding magnitude and desired magnitude are

\H(e-Jwim, e~jW2m,..., e~jWNm)\

a n d

I \€ ,6 , . . . , £ : )

respectively. For simplicity, they are written as | H(m) \and Y(m), respectively. The statement of the problem is tofind the parameter vector a

i = l , 2 , . . . , N ; f = 0 , 1 , 2 , . . . , L ] T (23)

to minimise the magnitude performance index

M"(24)

where um is a nonnegative weighting function. TheFletcher-Powell algorithm [17] is used for the opti-misation of eqn. 24.

4 Examples

Two examples of a 2-dimensional lowpass and bandpassdigital filter design and one example of a 3-dimensionallowpass digital filter design are illustrated below. In allexamples, for the optimisation of the denominator poly-nomial, discrete frequency samples are taken at intervalsof O.l7i on the wx axis within the passband radius. Forthe optimisation of the numerator polynomial, discretefrequency samples are taken at intervals of 0.1 n in theregion R as defined in eqn. 22. For the two examples ofthe 2-dimensional filter, the weighting functions (Tables1, 2 and 4) of the frequency samples in both optimisationprocedures are chosen to be equivalent to taking at fre-quency samples at intervals of 0.17t on the (wlt w2) planesuch that

— 71 < W2 (25)

for the purpose of obtaining similar conditions for usefulcomparison with other methods [10, 12].

The initial value of each unknown coefficient is chosento be 1. Also for the purpose of comparison, the per-formance of the filter is shown by the relative root meansquare (RMS) errors [12] of the designed filter which aredefined as

fcti

1/2

1/2 x 100 (26)

where

= G + mffor i = 1, 2

and

M" ) 1/2

Z Y(m)2}m = 1

x 100 (27)

\J

4.1 Examples of 2 -dimensional filter designIn the (wl5 w2) plane, actually, the region R is the [0°,45°] sector. Therefore, samples are taken in this sector forthe optimisation of the magnitude specifications. The twoexamples shown below have been tried in References 10and 12. Comparisons of the performance of this methodwith the design methods mentioned in References 10 and12 are also provided.

4.1.1 Example 1: Lowpass filter: This example is todesign a 2-dimensional circularly symmetric lowpassfilter with the following desired magnitude specifications:

y(exp(-;w1),

exp(-7w2))= <

M.O0.80.44

0.14

0.030.002

.0.001

forfor

for

for

forforfor

0.0 $O.ITC <

0.2TI <

0 .3TT<

0.47C <

0.5TC<

0.6TT <

$ r ^ O . I T C

: r < 0.2TT

c r ^ 0.371

c r ^ 0.471

c r ^ 0.5TT

c r ^ 0.67C

c r ^ n

where r = (wj + w2)1/2. The group delay is required to beconstant in the passband where r < 0.3rc.

Let the desired group delay a = 2. The filter is chosento consist of a 4th-order (K = 2) denominator (see eqn.

1EE PROCEEDINGS, Vol. 134, PL G, No. 4, AUGUST 1987 189

la) and a single section (L = 1) 4th-order numerator(my = 2). The weighting function um is chosen accordingto Table 1 and the weighting function ug is chosenaccording to Table 2.

Table 1: Weighting functions um of the samples used in theexamples of 2-dimensional filter design

484

4884

48884

488884

4888884

48888884

488888884

4888888884

14444444442

*0.9*0.8*0.7*0.6TI0.5*0.4*0.3*0.2*0 .1 *

0

0 0 .1 * 0.2* 0.3* 0.4* 0.5* 0.6* 0.7* 0.8* 0.9* *

Table 2: Weighting function ug for example 1

9 w1a ug

1 0 72 O.iTt 103 0.2TI 10

4 0.3TI 2

After 32 iterations, the resultant denominator is

D(zi\ zj2) = Qizi^Qiz^1)

where

Q(zf *) = (1.670937 - 1.503865zr1 + 0.825198zr2)

x (1.577846 - 1.841585zr1 + 0.580569zr2), i = 1, 2

After 28 iterations, the resultant numerator is

where

*Il) =

' 2.255632

-2.910392

3.282184

-2.910392

2.255632

h \ zl2, ,-3

, ̂ r

-2.910392

5.625357

-4.034687

5.625357

-2.910392

3.282184 -2.910392

-4.034687

4.716775

-4.0346873.282184

5.625357-4.034687

5.625357

-2.910392

2.255632-2.910392

3.282184-2.910392

2.255632J

In this example, it was found that the resultant denomi-nator was identical to that given in Reference 12. In thiscase, the performance indices eqns. 16 and 24 have thevalues J{b) = 1.75 x 10 ~9 and E{a) = 0.158, respectively.

Table 3 shows the comparison of the performance of thepresent method with the two other methods reported inReferences 10 and 12. It can be seen from Table 3 thatthe present method is better in both error performance aswell as efficiency of optimisation. The group delay andmagnitude responses of the filter are shown in Fig. 1.

(TT-n)

(TT.-TT)

(TT.-TT)

C-TT.TT)

(-TT.1T)

(-TT.TI)

Fig. 1 Passband group delay and magnitude response of the filtergiven in example 1

a T,(W,, w2) + 2b T2(W,, w2) + 2c Magnitude response

4.1.2 Example 2: Bandpass filter: This example is todesign a 2-dimensional circularly symmetric bandpassfilter with the desired magnitude specifications

y(exp (-M), exp (~

where

= (E/(E

E = (n/c)112 exp {-r2/c) r = (wj + w2)1'2

S = 100 c = 0.5

Table 3: Results of example 1

This methodReference 12Reference 10

et i

1.94 x 1O"4

1.94 x 10-4

9.32

1.94 x 10"4

1.94 x 10-4

8.18

12.8116.9524.36

Number ofindpendentparameters

102933

Number of samplesrequired(denominator)

66220220

Number of samplesrequired(numerator)

41717

190 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987

The group delay is required to be constant in the pass-band where r > 0.6TT, (see Appendix).

Let the desired group delay a — 1. The filter is chosento consist of a 4th-order (K = 2) denominator (see eqn.la) and a single section (L = 1) 4th-order numerator{mf = 2). The weighting function um is chosen accordingto Table 1 and the weighting function ug is chosenaccording to Table 4.

Table 4: Weighting function u for example 2

1 0 112 0.1 n 223 0.2TI 22

4 0.3;r 225 0.4TT 18

6 0.5TT 14

After 24 iterations, the resultant denominator is

D(zr1,z2-2) = e(z1-1)e(z2-1)where

Q{zrl) = (2.408485 - 0.599085z,r1 + 0.992430zr2)

x (1.672211 - 1.752\35zf1

+ 0.575655zr2), i = 1,2

After 18 iterations, the numerator obtained is

where

A =

[Uf1,

0.188264

0.6821420.2054920.6821420.188264

_ - 2 - 3 - 4 -z l » z l ) z l .

0.682142-0.763777-0.607780

-0.7637770.682142

0.205492

-0.607780-1.387962

-0.6077800.205492

]A[l,z;\z;

0.682142

-0.763777

-0.607780

-0.7637770.682142

2 - 3 - - 4 - i T> Z 2 > Z 2 J

0.188264"

0.682142

0.205492

0.682142

0.188264-

In this case, the performance indices eqns. 16 and 24 havethe values J{b) = 5.32 and E(a) = 7.36, respectively. Table5 shows the comparison of the performance of thismethod with the other two methods. It can be seen fromTable 5 that the error performance indices of this methodand the method in Reference 12 are approximately equalbut the efficiency of the optimisations of this method ismuch better. The group delay and magnitude responsesof the filter are shown in Fig. 2.

4.2 Example of 3 -dimensional filter design

4.2.1 Example 3: Lowpass filter: This example is todesign a 3-dimensional spherically symmetric lowpass

filter with the desired magnitude specifications asexample 1 with r = (w\ + w\ 4- w2)1'2. Also the groupdelay is required to be constant in the passband, r

g0.371.

(TT.-TT)

(TT,-TT)

(TT.-TT)

(-TT.TT)

(-TT.TT)

(-TT.TT)

Fig. 2 Passband group delay and magnitude response of the filtergiven in example 2

a T,(W,, w2) + 2b r2{wv w2) + 2c Magnitude response

Let the desired group delay a = 2. The filter is chosento have a 4th-order (K = 2) denominator (see eqn. 7a)and a single section (L = 1) 4th-order numerator (my =2). In this example, all the weighting functions, ug and um

are chosen to be 1. After 24 iterations, the resultantdenominator is

where

Q(zfl) = (1.577846 - 1.841585zf1 + 0.580569zr2)

x (1.670937- 1.5038652," *

+ 0.825198zr2), i = 1,2, 3

Table 5: Results of example 2

This methodReference 12Reference 10

7.367.20

15.11

7.367.20

14.62

Em

18.0718.4558.86

Number ofindependentparameters

102933

Number of samplesrequired(numerator)

66220220

Number of samplesrequired(denominator)

46060

1EE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 191

After 40 iterations, the coefficients of the numerator are

Co.0,0= -1-733318 x 10"1

Co,o.i = 1.109532 x 10"1

C 0 t 0 j 2 = -2.038416 x 10~2

Co, i,i = -1.280308 x 10"x

Co, i , 2 = 4.743257 x 10"3

Co, 2 , 2 = 1.866163 x 10"2

Ci . i . i = 3.097474 x 10"2

C 1 > l i 2 = 7.832095 x 10"3

C i , 2 , 2 = -5.384163 x 10"2

C2 2 > 2 = 1.144929 x 10~2

In this case, the performance indices eqns. 16 and 24 havethe values J{b) = 2.5 x 10"2 2 and E(a) = 0.0877, respec-tively. Table 6 shows the relative RMS errors. The group

Table 6: Results of Example 3

of previous papers. Moreover, a 3-dimensional examplehas also been tried and gives good results. Based on theresults obtained, one can conclude that this method is areliable and computationally efficient method for thedesign of multidimensional spherically symmetric recur-sive digital filter. Furthermore, the efficiency increases asthe order of the digital filter to be designed increases.

6 Acknowledgment

The authors wish to thank the anonymous reviewers fortheir comments and suggestions.

7 References

1 CAPPELLINI, V., CONSTANTINIDES, A.G., and EMILIANI,P.: 'Digital filters and their applications', (Academic Press, NewYork, 1978)

2 DUDGEON, D.E., and MERSEREAU, R.M.: 'Multidimensionaldigital signal processing' (Prentice-Hall, New Jersey, 1984)

3 AHMADI, M., and RAMACHANDRAN, V.: 'A method for thedesign of stable (N-D) analog and digital filters'. Proceedings of

Number of Number of samples Number of samplesindependent required requiredparameters (numerator) (denominator)

This method 1.98 x10 " 1 0 1.98 ><10-10 13.22 14 286

delay and magnitude responses are shown in Fig. 3 whichare very satisfactory.

4.3 RemarksThe proposed design method was implemented on anIBM PC/XT microcomputer (clock rate = 4.77 MHz)using Basic compiled with the Microsoft Basic Compiler.The computation times of examples 1, 2 and 3 were 74minutes, 44 minutes and 10 hours, respectively. However,these computation times can be reduced considerably ifan IBM PC/AT microcomputer and a more efficient lan-guage are used.

The results of this method shown in Tables 3 and 5have some slight differences compared with those of thevalues obtained by Reference 12. It may be due to thedifference in the wordlengths of the computers used.However, such minor differences are reasonable. Theresults obtained using the method in Reference 10 shownin Tables 3 and 5 are obtained directly from Tables 1 and2 of Reference 12.

For a special case of mf = 1, the transfer functioneqns. 6, 7 and 17-19 can then be written as a cascade of1st- and 2nd-order modules. This format is then verysuitable for modular implementation.

5 Conclusions

In this paper, a computationally efficient method hasbeen presented for the design of a multidimensionalspherically symmetric recursive digital filter to approx-imate a given set of magnitude and constant group delayspecifications. The transfer function is inherently stableand (2NN !)-hedrally symmetric. Therefore the number ofindependent coefficients required to be determined andthe number of samples to be used in the optimisationscan be greatly reduced. To test the performance of thismethod, two 2-dimensional examples used in References10 and 12 are used in this paper. This method generallyshows better performance in error indices and, especially,the efficiency of optimization when compared with results

International Conference on Acoustics Speech and Signal Pro-cessing, 1981, pp. 704-707

4 MUTLUAY, H.E., and FAHMY, M.M.: 'Frequency domain designof N-D digital filters', IEEE Trans., 1985, CAS-32, pp. 1226-1233

5 PITAS, I., and VENETSANOPOULOUS, A.N.: 'The use of sym-metries in the design of multidimensional digital filters', ibid., 1986,CAS-33, pp. 863-873

6 HUANG, T.S., BURNETT, J.W., and DECZKY, A.G.: 'The impor-tance of phase in image processing filters', ibid., 1975, ASSP-23,pp. 529-542

7 MARIA, G.A., and FAHMY, M.M.: 'An lp design technique to thetwo-dimensional filters', ibid., 1974, ASSP-22, pp. 15-21

8 MARIA, G.A., and FAHMY, M.M.: 7p approximation of the groupdelay response of one- and two-dimensional filters', ibid., 1974,CAS-21, pp. 431-436.

9 CHOTTERA, A T , and JULLIEN, G.A.: 'Design of two-dimensional recursive digital filters using linear programming', ibid.,1982, CAS-29, pp. 817-826

10 ALY, A.H, and FAHMY, M.M.: 'Design of two-dimensional recur-sive digital filters with specified magnitude and group delay charac-teristics', ibid., 1978, CAS-25, pp. 908-916

11 AHMADI, M., BORARIE, M.T., RAMACHARNDRAM, V., andGARGOUR, C.S.: 'Design of 2-D recursive digital filters with sepa-rable denominator transfer function and a new stability test', ibid.,1985, ASSP-33, pp. 1316-1318

12 HINAMOTO, T , and MAEKAWA, S.: 'Design of two-dimensionalrecursive digital filter using mirror-image polynomials', ibid., 1986,CAS-33, pp. 922-926

13 KARIVARATHARAJAN, P., and SWAMY, M.N.S.: Quadrantalsymmetry associated with two-dimensional digital transfer func-tions', ibid., 1978, CAS-25, pp. 340-343

14 ALY, S.A.H, and FAHMY, M.M.: 'Symmetry in two-dimensionalrectangularly sampled digital filters', ibid., 1981, ASSP-29, pp. 794-805

15 CHARALAMBOUS, C : 'Design of 2-dimensional circularly-symmetric digital filters', IEE Proc. G, Electron. Circuits & Syst.,1982,129, (2), pp. 47-54

16 KARIVARATHA RAJAN, P , and SWAMY, M.N.S.: 'Design ofseparable denominator 2-dimensional digital filters possessing realcircularly symmetric frequency responses', ibid., 1982, 129, (5),pp. 235-240

17 FLETCHER, R., and POWELL, M.J.D.: 'A rapidly convergentdescent method for minimization', Comput. J., 1963, 6, pp. 163-168

8 Appendix

In example 2, the constant group delay specification ofthe 2-dimensional bandpass filter covers the portion of

192 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987

(n-TT,w3)

(TT.-TT.O)

(TT-TT,-TT)

y-cut off edges of/I bandpass filter

(-•n,Tr,w3)

(-TT.TT.O)

(-TT.TT.TT)

Fig. 3 Passband group delay T,(W,, W2) + 2 in the (w,, w2) plane andmagnitude responses of the filter given in example 3

a Passband group delay, - n < w3 < nb Magnitude response with vv3 = 0c Magnitude response with w2 = W3

the stopband which includes the origin. The reason isexplained below.

According to the properties of symmetry (eqns. 12, 13and 14) of the transfer function, for a particular frequencyw',

T(W\ 0) = a

is equivalent to

T(W', W 2 ) = T( — W', W 2 ) = W') =

for — n < w, < 7i and — n < w2 ^ n. Therefore to specifythat the passband of the bandpass filter has a constantgroup delay is equivalent to specifying that the shadedregion in Fig. 4 be of constant group delay, that is to say,T(W) is constant for 0 ^ wt ^ wc).

Fig. 4 Region of constant group delay in specifying T(W,), 0 ^ vv,wc, to be a constant

Hon Keung Kwan received the BSc degreefrom the University of London, UnitedKingdom and the MPhil degree fromChinese University of Hong Kong, HongKong, in 1976 and 1977, respectively. Hethen joined Interquartz Ltd., Hong Kong,as an assistant design engineer workingon the design of electronic consumer pro-ducts; and later Tek-Devices Ltd., HongKong, as a design Engineer working onthe design of memory systems. In 1978, he

went to Imperial College, University of London, UnitedKingdom to do research work and received his DIC and PhDin 1981. He then joined the Department of Electronic Engineer-ing, Hong Kong Polytechnic, Hong Kong, as a lecturer. SinceDecember 1981, he has been a lecturer in the Department ofElectrical and Electronic Engineering, University of HongKong, Hong Kong. His current research interests include VLSIdigital signal processing, 1-dimensional and multidimensionaldigital filter design. Dr. Kwan is currently a member of theDigital Signal Processing Technical Committee of the IEEECircuits and Systems Society and a senior member of the IEEE.

Chun Leung Chan received the BSc(Eng)degree in Electrical Engineering from theUniversity of Hong Kong in 1981. Sincethen, he has been a graduate trainee, athird engineer, and a second engineer inthe China Light & Power Co., Ltd., HongKong. At present, he is an engineer of theHong Kong Telephone Co. Ltd., HongKong. Meanwhile, he has been alsoworking for his PhD degree at the Uni-versity of Hong Kong, Hong Kong. His

main areas of interest are digital signal processing and multidi-mensional digital filter design.

IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 193