Simple and computationally efficient design of two-dimensional circularly symmetric IIR digital filters

Transcription

1 INT. J. ELECTRONICS, 1988, VOL. 64, No.2, Simple and computationally efficient design of two-dimensional circularly symmetric IIR digital filters C. L. CHANt and H. K. KWANt A simple and computationally efficient method for the design of a two-dimensional circularly symmetric IIR digital filters is introduced. This method has several advantages. Firstly, the digital filter format is a cascade of first- and second-order filter modules and is suitable for the VLSI modular implementation. Secondly, most of the coefficients can be determined directly from an one-dimensional analogue filter table and the stability of the filter is preserved. Finally, the unknown coefficients are very few in number and can be obtained by an unconstrained optimization method. Three examples are shown to illustrate the design method. 1. Introduction In general, direct realization of a two-dimensional (2-D) infinite impulse response (I1R) digital filter is undesirable because of the large number of coefficients and the problems of stability and roundoff noise. Consequently, the idea of expressing a 2-D digital filter in terms of first- and second-order sections in cascade is widely accepted (Aly and Fahmy 1978, Fahmy and Sobhy 1982, Charalambous 1982) because it can simplify the stability problem as well as improve the roundoff noise behaviour. Charalambous (1982) applied this idea to design circularly symmetric 2-D I1R digital filters. However, the number of coefficients is still large for the design of high order digital filters. On the other hand, many methods (Costa and Venetsanopoulous 1974, Goodman 1978, Ali 1978, Thyagarajan 1981) have been introduced on the design of 2-D I1R digital filters from one-dimensional (I-D) digital filters. Most of them utilize different kinds of transformation of variables. However, these methods generally require a cascade of I1R digital filters with different directions of recursion. Rajan and Swamy (1982) introduced a new I1R digital filter format which utilized I-D digital filter coefficients and was octagonally symmetric so that a cascade of digital filters with different directions ofrecursion was not required. But, the employed I-D filter has to be a digital filter of a special format. This paper presents a simple and computationally efficient method for the design of 2-D circularly symmetric I1R digital filter of moderate precision. The filter format employed is a cascade of first- and second-order digital filter modules and octagonally symmetric in magnitude. For the even order case, the filter format is similar to that of Charalambous' (1982) method. In order to reduce the number of coefficients going to be optimized, the coefficients of a I-D analogue filter are utilized. The I-D analogue filter can be a Butterworth, a Chebyshev or an elliptic filter which can be looked up from an analogue filter table (Saal 1979) directly. The circularity in magnitude is approximated by an additional special term in each second order module. Therefore, the coefficients required to be determined are very few in number Received 15 June 1987; accepted 29 June t Department of Electrical and Electronics Engineering, University of Hong Kong, Pokfulam Road, Hong Kong.

2 230 C. L. Chan and H. K. Kwan and can be obtained by an unconstrained optimization method. A similar method of multiplying a polynomial of special terms to the I-D filter can be found in (Dubois and Blostein 1975). Their method was applied for long polynomials which were not suitable for modular implementation and no symmetric properties were applied. Moreover, because of the octagonally symmetric properties, the denominator is separable. Provided that the I-D analogue filter is stable, the resultant filter will be inherently stable. Sections 2--4 present the theory of this method. Section 5 describes the design procedures. Three illustrative examples will be given in I-D analogue filter The general format (Kwan 1985) of the transfer function of a Butterworth, a Chebyshev or an elliptic I-D analogue filter can be expressed as 1 N m SZ + n T1(s) = A.--. f1 z v v (1) s+qo v=1 s +Pvs+qv for odd order, i.e. (2N + l)th order, and N z T1(s)=A' f1 zmvs +n v v=i S +Pvs+qv for even odd, i.e. (2N)th order. N is a positive integer. mv = 0 or 1. A, nv, Pv and qv are real numbers. s is the complex analogue frequency variable. In the following, for simplicity, only the expressions for odd order case are shown. Those for the even order case are similar. (2) 3. New 2-D filter format Based on the transfer function (1), a new 2-D filter can be formulated as z 1 Tz(sl' sz) = A (Sl + qo)(sz + qo) N (. f1 mvsl z + nv)( mvsz+)+ nv xvslz Szz v=1 (si + PvSI + qv)(s~ + Pvsz + qv) This expression (3) has a special property. By putting Sl = 0, we can obtain the response on the WI axis. (3) Similarly, Therefore, the responses of Tz(sl, sz) on the 0 1 and Oz analogue frequency axes preserve the characteristics of T1(sl) and T1(sz). Expression (3) can be transformed into the digital domain by means of the bilinear transformations: (4) (5) and (6)

3 Design of2-d circularly symmetric IIR digital filters 231 where k is determined by the one-to-one correspondence of the critical analogue frequency no and the normalized critical digital frequency wo, 0 :5: Wo :5: n, k = no cot (wo/2) (7) where av = (k 2 mv + nvf + k 4 xv bv = 2(nv - k 2 mj(k 2 mv + nv) - k 4 xvj Cv = 4(nv - k 2 my + k 4 xvj dv = k 2 + kpv + qv (9) ev = 2(qv - k 2 ) fv = k 2 - kpv + qv For the transfer function (8), several points should be emphasized. (a) The property that the responses of the 2-D transfer function on the Wi and W 2 axes preserve the characteristics of the 1-D transfer function is still true. (b) The digital filter is octagonally symmetric in magnitude (Karivaratharajan and Swamy 1978). (c) The transfer function is a product of first- and second-order sections. Therefore, the filter is suitable for the VLSI modular implementation by cascading first- and second-order filter modules. Besides, such an arrangement can also reduce the roundoff noise. (d) Due to the property of bilinear transformation (Peled and Liu 1976), provided that the original I-D analogue filter is stable, the resultant 2-D digital filter must also be stable. 4. Statement of the problem Refer to expression (3), the variables mv, nv, Pv and qv are obtained from the I-D analogue filter which can be looked up from the filter tables (Saal 1979). The only unknowns are Xv, v = I, 2,..., N. Let X be the vector of the unknown Xv t represents the operation of matrix transposition. To solve the problem, we have to choose M number of samples on the (Wi' w 2 ) plane. Assuming the desired magnitude at the frequencies (w li, w 2 ;) be Hiexp(jwd, exp(jw 2 } For simplicity, it is written as Hii), i = 1,..., M. Then, we can define a (10)

4 232 C. L. Chan and H. K. Kwan performance index as the sum of the squares of the errors, M Q(X) = L {IH(exp (jwd, exp (jw 2»I-HiiW i= 1 (11 ) The problem is to find an optimal X, say X*, such that for all X, Q(X*) ~ Q(X) (12) The Fletcher Powell algorithm (Fletcher and Powell 1963) is used for the optimization. 5. Design procedures The design procedures can be summarized into the following steps. Step 1 Design a suitable I-D analogue filter, which could be a Butterworth, a Chebyshev or an elliptic filter, from the analogue filter table (Saal 1979) to approximate the desired response on the W l axis in the format as given by eqns. (1) or (2). Step 2 Obtain the 2-D filter in the format as given by eqn. (3). Step 3 As the filter is octagonally symmetric, M sample points are chosen on each of the arcs of the circles encircling the origin in the 0, 45 ] sector of the (w l, w 2 ) plane (Charalambous 1982), see Fig. 1 for M = 8. Step 4 Minimize the performance index given in eqn. (11) by a non-linear optimization method. 6. Examples Three examples are given below. They are obtained, respectively, from a Butterworth, a Chebyshev and an elliptic I-D analogue filters. As the filters are octagonally symmetric in magnitude, the sample points are chosen in the 0 0, 45 ] sector of the (w l, w 2 ) plane. In each example, besides (0, 0), eight equally spaced sample points are chosen on each of the arcs of 19 different circles encircling the origin. Totally, 153 sample points are taken. The initial values of the variables are all 1 and the Fletcher-Powell algorithm is employed for the optimization Example l-wide-bandfilter A sixth-order 1-D analogue Butterworth filter Tl(s) = (S2 + 0'5176s + 1) (S s + 1). (S s + 1)

5 Design of2-d circularly symmetric IIR digital filters 233 L.c:.._...L.- Figure 1. Sample points used for optimization. ---" w, Tt is chosen for the design. The passband frequency is 0 4n. The resultant solution is H(z, Z2) = x , '00000] 1 ] lz1 1 z1 2 J -3, zi ' zi ] 1 ] lz1 1 z1 2 J -6, ,77024 zi zi 2 The performance index (11) of this case has the value given by Q(X*) = Figure 2 shows the magnitude response of this filter.

6 234 C. L. Chan and H. K. Kwan (TIl TI) Figure 2. Magnitude response of Example Example 2-medium-bandfilter A fourth-order I-D analogue Chebyshev filter 1 1 T 1 (s) = (S s + 0'5125) (S s + 1,2196) is chosen for the design. The passband frequency is 0 25n. The resultant solution is H(Zl' Z2) = x '00000] 1 ] lz1 1 z1 2 J -2,18238 ooסס 4 0-2,18238 zi , zi 2 ( z z1 2 )(l - 1'17613zi zi2) ] 1 l'ooooo]i ooסס 1 0-4,65183 lz1 1 z1 2 J -4, ,65183 ' zi , zi 2 The performance index (11) of this case has the value given by Q(X*) = Figure 3 shows the magnitude response of this filter Example 3-narrow-bandfilter A fourth-order I-D analogue elliptic filter 1 S T 1 (s) = (S s + 0,6259). (S2 + 0'3623s + 1,1948)

7 Design of2-d circularly symmetric IIR digital filters 235 Figure 3. Magnitude response of Example 2. is chosen for the design. The passband frequency is 0 1 n. The resultant solution is H(Z1' Z2) = x 10-2 lz ooסס 1 0-2,00348 l'oooooj 1 J 1 1 z1 2 ] -2,00348 ooסס 4 0-2,00348 Z2 1 ooסס z2 2 (1-1'63479z '68695z1 2 )(1-1'63479z '68695z2 2 ) ooסס 1 0 J 1 J ooסס 1 0 lz1 1 z1 2 ] -1, ,66362 Z2 1 ooסס 1 0 z ooסס 1 0 (1-1'78419z '89445z 1 2 )(1-1'78419z '89445z2 2 ) The performance index (11) of this case has the value given by Q(X*) = Figure 4 shows the magnitude response of this filter Remarks The proposed design method was implemented on an IBM AT microcomputer (clock rate = 6 MHz) using BASIC language compiled with the Microsoft BASIC compiler. The computational times of Examples 1, 2 and 3 were 8 minutes, 6 5 minutes and 4 minutes respectively. Since the number of coefficients for the circularity approximation is small in number, the precision cannot be very high. However, in general, the results obtained are acceptable. 7. Conclusions A method for the design of two-dimensional circularly symmetric recursive digital filter from 1-0 analogue filter has been described. The I-D analogue filter

8 236 C. L. Chan and H. K. K wan (O,3n,O,3n) Figure 4. Magnitude response of Example 3. has a format which meets the general form of the most popular types of analogue filter, including the Butterworth, Chebyshev and elliptic filters. Making use of this advantage, the number of coefficients to be determined can be greatly reduced. In addition, because of the symmetry of the magnitude of the filter, the number of samples required for the optimization is also reduced. Hence, much of the computational time can be saved. Besides, the stability of the filter is also guaranteed. The final but not least advantage of this method is that the filter format is suitable for the VLSI modular implementation. However, it also possesses some disadvantages. Since the number of variables is small in number, the precision cannot be very high. Also, the obtained solution is only a sub-optimal solution. Conclusively, this method is simple, computationally efficient and very suitable for the moderately precise 2-D circularly symmetric IIR filter design. REFERENCES ALI, A. M., 1978, Design of inherently stable two-dimensional recursive filters imitating the behaviour of one-dimensional analog filters. Proceedings of IEEE Conference on Acoustics, Speech and Signal Processing, pp ALY, S. A. R., and FAHMY, M. M., 1978, Design of two-dimensional recursive digital filters with specified magnitude and group delay characteristics. IEEE Transactions on Circuits and Systems, 25, CHARALAMBOUS, C, 1982, Design of 2-dimensional circularly-symmetric digital filters. lee Proceedings Pt. G, 129, COSTA, J. M., and VENETSANOPOULOUS, A. N., 1974, Design of circularly symmetric twodimensional recursive filters. IEEE Transactions on Acoustics, Speech and Signal Processing, 25, DUBOIS, E., and BLOSTEIN, M. L., 1975, A circuit analogy method for the design of recursive two-dimensional digital filters. Proceedings of IEEE International Symposium on Circuits and Systems, FAHMY, M. F., and SOBHY, M. I., 1982, A new method for the design of 2-D filters with guaranteed stability. IEEE Transactions on Circuits and Systems, 29, FLETCHER, R., and POWELL, M. J. D., 1963, A rapidly convergent descent method for minimization. Computer Journal, 6,

9 Design of2-d circularly symmetric IIR digital filters 237 GOODMAK, D. M., 1978, A design technique for circularly symmetric low-pass filters. IEEE Transactions on Acoustics, Speech and Signal Processing, 26, KARIVARATHARAJAN, P., and SWAMY, M. N. S., 1978, Quadrantal symmetry associated with two-dimensional digital transfer functions. IEEE Transactions on Circuits and Systems, 25, KWAN, H. K., 1985, A multi-output second-order digital filter structure for VLSI implementation, IEEE Transactions on Circuits and Systems, 32, PELED, A., and LID, B., 1976, Digital signal processing: theory, design, and implementation (Wiley) ch. 2. RAJAN, P. K., and SWAMY, M. N. S., 1982, Design of separable denominator 2-dimensional digital filters possessing real circularly symmetric frequency responses. lee Proceedings Pt. G, 5, SAAL, R., 1979, Handbook offilter design (Berlin: AEG-Telefunken). THYAGARAJAN, K. S., 1981, Design of 2-D IIR digital filters with circular symmetry by transformation of the variable. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, pp