NEW DESIGN METHODS FOR TWO-DIMENSIONAL

CIRCULARLY SYMMETRIC DIGITAL FILTERS

Thesis submitted for the degree of

Doctor of Philosophy

CHAN Chun Leung

B.Sc.(Eng.), MIEEE, MIERE

Department of Electrical and Electronic Engineering

University of Hong Kong

July 1988

ABSTRACT

Two-dimensional digital filters have found increasing

applications in various areas. The present thesis

concentrates on the topic of the design of a common kind of

two-dimensional digital filters :- circularly symmetric

filters. Although there are a number of existing design

methods for this kind of filters, many of them can be

further improved to give better performance. Numerous new

efficient design methods with high performance will be

described in this thesis. The design methods aims for three

different kinds of filters : (i) zero phase non-recursive

digital filters, (ii) recursive digital filters and (iii)

linear phase recursive digital filters.

For the design of zero-phase circular symmetric

non-recursive digital filters, a method of an enhanced

McClellan transformation is introduced. It can give good

circular approximation even at large radii of frequency.

Three design methods are introduced for circularly

symmetric recursive digital filters. In order to reduce the

computational time, all these methods make use of the

ii

one-dimensional filters. The first method employs a new

spectral transformation while the other two methods use

non-linear optimization.

For the linear phase circularly symmetric digital filter

design, three new methods are introduced. In order to

reduce the computational time and achieve the convergence of

iteration easily, the numerator and the denominator of the

filter transfer function are designed separately. In the

linear phase approximation, the first method employs the

hypergeometric series. The second method utilizes the

Bessel polynomial while the third method uses non-linear

optimization.

All the newly introduced methods are easy-to-use. Most

of them can give digital filters suitable for implementation

by VLSI second-order digital filter modules. Some of them

use sum .of powers-of-two values as the filter coefficients

so that the designed filters are suitable for multiplierless

filter implementation. Besides, some of the methods have

been further extended for multidimensional ellipsoidally and

spherically symmetric digital filter designs. Finally, for

each method, several typical examples will be given to

illustrate its performance.

iii

REFERENCES

[1] HU, J.V. and RABINER, L.R.: 'Design techniques for

two-dimensional digital filters', IEEE Trans. Circuits

Syst., vol.CAS-33, pp.1022-1026, Oct. 1986.

[2] KAMP, Y. and THIRAN, J.P.: 'Chebyshev approximation for

two dimensional nonrecursive digital filters', IEEE

Trans. Circuits Syst., vol.CAS-22, pp.208-218, Mar.

1975.

[3] HARRIS, D.B. and MERSEREAU, R.M.: 'A comparison of

algorithms for minimax design of two-dimensional

linear-phase FIR digital filters', IEEE Trans. Acoust.,

Speech, Signal Processing, vol.ASSP-25, pp.492-500,

Dec. 1977.

[4] HUANG, T.S.: 'Two dimensional windows', IEEE Trans.

Audio Electroacoust., vol.AU-20, pp.88-89, Mar. 1972.

[5] MCCLELLAN, J.H.: 'The design of two-dimensional filters

by transformations', Proc. 7th Annual Princeton

Conference on Information Systems, 1973, pp.247-251.

[6] MERSEREAU, R.M., MECKLENBRAUKER, F.G. and QUATIERI,

T.F.: 'McClellan transformation for two-dimensional

digital filtering: Part I - Design, Part II -

Implementation', IEEE Trans. CAS-23, pp.405-422, Jul.

1976.

R.1

References

[7] FETTWEIS, A.: 'Symmetry requirements for

multidimensional digital filters', Int. J. Circuit

Theory Appl., pp.343-353, vol.5, 1979.

[8] RAJAN, P.K. and SWAMY, M.N.S.: 'Design of circular

symmetry two-dimensional FIR digital filters employed

transformations with variable parameters', IEEE Trans.

Acoust., Speech, Signal Processing, vol.ASSP-31,

pp.627-642, Jun. 1983.

[9] HAZRA, S.N. and REDDY, M.S.: 'Design of circularly

symmetric low-pass two-dimensional FIR digital filters

using transformation', IEEE Trans. Circuits Syst.,

vol.CAS-33, pp.1022-1026, Oct. 1987.

[10] COSTA, J.M. and VENETSANOPOULOS, A.N.: 'Design of

circularly symmetric two-dimensional recursive

filters', IEEE Trans. Acoust., Speech, Signal

Processing, vol.ASSP-22, pp.432-443, Dec. 1974.

[11] ALI, A.M.: 'Design of the inherently stable

two-dimensional recursive filters imitating the

behaviour of one-diemnsionil analog filters'. Proc. of

Int. Conf. of Acoust., Speech, Signal Processing, 1978,

pp.765-768.

[12] THYAGARAJAN, K.S.: 'Design of 2-dimensional IIR digital

filters with circular symmetry by transformation of the

variable'. Proc. of IEEE Conf. of Acoust., Speech,

Signal Processing, 1981, pp.685-687.

R.2

References

[13] CHARALAMBOUS, C.: 'Design of 2-dimensional

circularly-symmetric digital filters', IEE Proc. Pt.G,

vol.129, pp.47-54, Apr. 1982, .

[14] RAJAN, P.K. and SWAMY, M.N.S.: 'Design of separable

denominator 2-dimensional digital filters possessing

real circularly symmetric frequency responses', IEE

Proc. Pt.G, vol.129, pp.235-240, Oct. 1982.

[15] GOODMAN, D.M.: 'A design technique for circularly

symmetric low-pass filters', IEEE Trans. Acoust.,

Speech, Signal Processing, vol.ASSP-26, pp.290-304,

Aug. 1978.

[16] BERNSTEIN, R.: 'Design of inherently stable

two-dimensional recursive filters from one-dimensional

filters', IEEE Trans. on Acoustic, Speech, Signal

Processing, vol.ASSP-32, pp.164-168, Feb. 1984.

[17] KING, R.A. and KAYRAN, A.: 'A new transformation

technique for the design of 2-dimensional stable

recursive digital filters', in Proc. of IEEE Int. Conf.

on Acoustics, Speech, Signal Processing, 1981,

pp.196-199.

[18] HARN, L. and SENOI, B.A.: 'Design of stable

two-dimensional IIR filters using digital spectral

transformation', IEEE Trans. Circuits Syst.,

vol.CAS-33, pp.483-490, May 1986.

R.3

References

[19] ALY, A.H. and FAHMY, M.M.: 'Design of two-dimensional

recursive digital filters with specified magnitude

and group delay characteristics', IEEE Trans. Circuits

Syst., vol.CAS-25, pp.908-916, Nov. 1978.

[20] FAHMY, M.F. and SOBHY, M.I.: 'A new method for the

design of 2-D filters with guaranteed stability', IEEE

Trans. Circuits Syst., vol.CAS-29, pp.246-251, 1982.

[21] KNOWLES, J.B. and OLCAYTO, E.M.: 'Coefficient accuracy

and digital filter response', IEEE Trans. Circuit

Theory, vol.CT-15, pp.31-41, Mar. 1968.

[22] LIM, Y.C., PARKER, S.R. and CONSTANTINIDES, A.G.:

'Finite word length FIR filter design using integer

programming over a discrete coefficient space', IEEE

Trans. Acoust., Speech, Signal Processing, vol.ASSP-30,

pp.661-664, Aug. 1982.

[23] LIM, Y.C. and PARKER, S.R.: 'FIR filter design over a

discrete power-of-two coefficient space', IEEE Trans.

Acoust., Speech, Signal Processing, vol.ASSP-31,

pp.581-591, Jun. 1983.

[24] LIM, Y.C. and PARKER, S.R.: 'Discrete coefficent FIR

digital filter design based upon LMS criteria', IEEE

Trans. Circuits Syst., vol.CAS-30, pp.723-739, Oct.

1983.

R.4

References

[25] PEI, S.C. and JAW, S.B.: 'Efficient design of 2-D

multiplierless FIR filters by transformation', IEEE

Trans. Circuits Syst., vol.CAS-34, pp.436-438, Apr.

1987.

[26] MCCLELLAN, J.H. and CHAN, D.S.K.: 'A 2-D FIR filter

structure derived from the Chebyshev recursion', IEEE

Trans. Circuits Syst., vol.CAS-24, pp.372-378, Jul.

1977.

[27] MCCLELLAN, J.H., PARKS, T.W. and RABINER, L.R.: 'A

computer program for designing optimum FIR linear phase

digital filters', IEEE Trans. Audio and

Electroacoustics, vol.AU-21, pp.506-526, Dec. 1973.

[28] NGUYEN, D.T. and SWAMY, M.N.S.: 'A class of 2-D

separable denominator filters designed via the

McClellan transformation', IEEE Trans. Circuits Syst.,

vol.CAS-33, pp. 874-881, Sep. 1986.

[29] NAGAMUTHU, N.: 'Design of 2-D recursive digital filters

with cinstant group delay characteristics', Journal of

Franklin Institute, vol. 320, no. 3/4, pp. 191-200,

Sep./Oct., 1985.

[30] PENDERGRASS, N.A., MITRA, S.K. and JURY, E.I.:

'Spectral transformations for two-dimensional digital

filters', IEEE Trans. Circuits Syst., vol.CAS-23,

pp.26-35, Jan. 1976.

R.5

References

[31] CHAKRABARTI, S. and MITRA, S.K.: 'Design of

two-dimensional digital filters via spectral

transformations', Proc. IEEE, vol.65, pp.905-914, Jun.

1977.

[32] HUANG, T.S.: 'Stability of two dimensional recursive

filters', IEEE Trans. Audio and Electroacoustics,

vol.AU-20, pp.158-163, Jun. 1972.

[33] CONSTANINIDES, A.G.: 'Spectral transformations for

digital filters', Proc. IEE, vol.117, pp.1585-1590,

Aug. 1970.

[34] HARN, L. and SHENOI, B.A.: 'On stable digital frequency

transformation for designing 2-D digital filters',

Proc. 27th Midwest Symp. on Circuits and Systems, 1984,

pp. 319-322.

[35] STANLEY, DAUGHERTY, G.R. and DAUGHERTY, R.,

Digital Signal processing, 2nd Ed., 1984, Ch.7, p.176.

[36] SAAL R.: Handbook of filter design, Berlin:

AEG-telefunken, 1979.

[37] DUBOIS, E. and BLOSTEIN, M.L.: 'A circuit analogy

method for the design of recursive two-dimensional

digital filters', Proc. IEEE Int. Sym. on Circuits

Syst., 1975, pp.451-454.

[38] KWAN, H.K.: 'A multi-output second-order digital filter

structure for VLSI implementation', IEEE Trans.

Circuits Syst., vol.CAS-32, pp.108-109, Jan. 1985.

R. 6

References

[39] KARIVARATHARAJAN, P. and SWAMY, M.N.S.: 'Quadrantal

symmetry associated with two-dimensional digital

transfer functions', IEEE Trans. Circuits Syst.,

vol.CAS- 25, pp.340-343, June 1978.

[40] PELED, A. and LIU, B.: Digital signal processing:

theory, design, and implementation (Wiley), Ch.2.

[41] FLETCHER, R. and POWELL, H.J.D.:'A rapidly convergent

descent methdo for minimization', Computer J., vol.6

1963, pp.163-168.

[42] NGUYEN, D.T. and SWAMY, M.N.S.: 'Approximation design

of 2-D digital filters with elliptical magnitude

response of arbitrary orientation', IEEE Trans.

Circuits Syst., vol.CAS-22, pp.597-603, Jun. 1986.

[43] HUANG, T.S., BURNETT, J.W. and DECZKY, A.G.: 'The

importance of phase in image processing filters',

IEEE Trans. Acoust., Speech, Signal Processing,

vol.ASSP-23, pp.529-542, Dec. 1975.

[44] MARIA, G.A., and FAHMY, M.M.: 'An l design technique

to the two-dimensional filters', IEEE Trans. Acoustics,

Speech, Signal Processing, vol.ASSP-22, pp.51-21, Feb.

1974.

[45] MARIA, G.A., and FAHMY, M.M.: 'lp approximation of the

group delay response of one- and two-dimensional

filters', IEEE Trans. Circuits Syst., vol.CAS-21,

pp.431-436, May 1974.

R.7

References

[46] DELCARO, L.G., and SICURANZA, G.L.: 'Design of

two-dimensional recursive digital filters', IEEE Trans.

Acoust., Speech, Signal Processing, vol.ASSP-25,

pp.577-579, Dec. 1977.

[47] LAMPROPOULOUS, G.A., and FAHMY, M.M.: 'A new technique

for the design of two-dimensional FIR and IIR filters',

IEEE Trans. Acoust., Speech, Signal Processing,

vol.ASSP-33, pp.268-280, Feb. 1985.

[48] HINAMOTO, T. and MAEKAWA, S.: 'Design of two-

dimensional recursive digital filter using mirror-

image polynomials', IEEE Trans. Circuits Syst.,

vol.CAS-33, pp.922-926, Aug. 1986.

[49] CHOTTERA, A.T. and JULLIEN, G.A.: 'Design of two-

dimensional recursive digital filters using linear

programming', IEEE Trans. Circuits Syst., vol.CAS-29,

pp.817-826, Dec. 1982.

[50] ABBRAMOWITZ, M. and STEGUN, A.: 'Handbook of

mathematical functions', (Dover, New York, 1972)

[51] THIRAN, J.P.: 'Recursive digital filters with maximally

flat group delay', IEEE Trans. Circuit Thoery,

vol.CT-18, pp.659-664, Nov. 1971.

[52] CHENEY, E.W., Introduction to Approximation Theory (New

York:McGraw-Hill), 1966, pp.72-100.

R.8

References

[53] REMEZ, E.Y.: 'General computational methods of

Tchebysheff approximation', Kiew, 1957 (Atomic Energy

Translation 4491, pp.1-85).

[54] BLUM, E.K., Numerical Analysis and Computation Theory

and Practice (London:Addision-Weley), 1972, pp.287-289.

[55] VALKENBURY, M.E.V.: 'Analog filter design (John Wiley &

Sons, 1982).

[56] STORCH, L.: 'Synthesis of constant-time delay ladder

networks using Bessel polynOmials', Proc. I.R.E.,

vol.42, 1666-1675, 1954.

[57] ORCHARD, H.J.: 'The roots of maximally flat delay

polynomials', IEEE Trans. on Circuit Theory, vol.CT-12,

pp.452-454, 1965.

[58] MUTLUAY, H.E. and FAHMY, M.M.: 'Frequency domain

design of N-D digital filters', IEEE Trans. Circuits

Syst., vol.CAS-32, pp.1226-1233, Dec. 1985.

[59] PITAS, I. and VENETSANOPOULOUS, A.N.: 'The use of

symmetries in the design of multidimensional digital

filters', IEEE Trans. Circuits Syst., vol.CAS-33,

pp.863-873, Sep. 1986.

[60] AHMADI, M., BORARIE, M.T., RAMACHARNDRAM, V. and

GARGOUR, C.S.: 'Design of 2-D recursive digital

filters with separable denominator transfer function

R.9

References

and a new stability test', IEEE Trans. Acoust., Speech,

Signal Processing, vol.ASSP-33, pp.1316-1318, Oct.

1985.

[61] CHAN, C.L. and KWAN, H.K.: 'Narrow transition width

two-dimensional digital fan filter design', Proc. 7th

Conf. on Circuits and System, 1987, pp.22.29-22.32,

Shenchun.

[62] MARIA, G.A. and FAHMY, M.M.: 'A new design technique

for recursive digital filter', IEEE Trans. Circuits

Syst., vol.CAS-23, pp.323-325, May 1976.

[63] AHMADI, M., BORARIE, M.T., RAMACHARNDRAM, V.and

GARGOUR, C.S.: 'Design of 2-D recursive digital filters

with separable denominator transfer function and a

newly stability test', IEEE Trans. Acoust., Speech,

Signal Processing, vol.ASSP-33, pp.1316-1318, Dec.

1985.

[64] HADJIFOTIOU, A. and APPLEBY, D.G.: 'Design of digital

filters with severaly quantized coefficients', The

Radio and Electronic Engineer, Vol.46, No.1, pp.23-28,

Jan. 1976.

R.10