2-D FIR and IIR Filters design: New Methodologies and New General Transformations

Transcription

1 ikos E. astorakis, Stavros Kaminaris International Journal of Instrumentation easurement -D FIR IIR Filters design: ew ethodologies ew General Transformations ikos E. astorakis* Stavros Kaminaris** *Hellenic aval Academy, Department of omputer Science, Piraeus, GREEE **Piraeus University of Applied Sciences, Department of Electrical Engineering, Egaleo, Athens, GREEE Abstract ew general transformations for designing -D (Two-Dimensional FIR IIR filters will presented in this paper. The present methodology can be viewed as an extension of the clellan Transformations can be applied in several cases of -D FIR IIR filter design. umerical examples illustrate the validity the efficiency of the method. Keywords -D Filters, FIR Filters, IIR Filters, ultidimensional Systems, ultidimensional Filters, Filter Design, clellan Transformations I. ITRODUTIO -D (Two-Dimensional filter design is not a simple task due to the heavy computational load to the non-existence of stability conditions in an explicit form. Roughly speaking, the design of -D filters FIR includes a Fourier method that uses Fourier analysis, where appropriate window Functions can also eliminate the so called Gibbs oscillations as in -D case, a Transformations method which is based on clellan Transformations from appropriate -D filters [],[] an optimiation method i.e. the minimiation of an appropriate norm, [],[]. On the other h, the design of -D filters IIR includes also transformations, irror Image Polynomials, SVD (Singular Value Decomposition Optimiation, [],[]. Several Authors have published works on optimiation-based -D filter design while a great number of papers are dedicated to transformations mainly to clellan Transforms. clellan Transformations were introduced in [3] have been used for the last forty years. A brief overview the various extension of clellan Transformations can be found in [],[],[4],[8],[9]. In general, a clellan Transformation is described by As Harn Shenoi pointed out in [5] as guyen Swamy reported in [6], till now a transformation for IIR filter design analogous to clellan transformation does not exist due to the requirements of -D stability. guyen Swamy in [7] use the usual clellan transformation in the special case of separable denominator. Fundamental results on clellan transformation can be found in [8] [9] while remarkable studies are given [0] [8]. Various useful results for -D IIR Filters design are presented in [9] [5]. In [6], we proposed some transformations for the first-order IIR -D second-order IIR -D notch filters. In [6], we propose the transformation, real numbers or simply ( 0 For first-order second-order IIR -D notch filters design. This paper examines this transformation as well as its generaliation to the general -D filters design. cos( cos( cos( k l is the frequency of the original -D filter, whereas, the frequencies of the -D filter in design. II. THE TRASFORATIO AD ITS GEERALIZATIOS Instead of the classic clellan Transformation cos( cos( cos( where is the k l ISS: Volume, 07

2 ikos E. astorakis, Stavros Kaminaris International Journal of Instrumentation easurement frequency of the original -D filter, whereas, the frequencies of the -D filter in design, we propose here the transformation of [6], real numbers or simply or simply where in [6], for the -D notch filters, we demed As a simple generaliation of this transformation we propose the following transformation not only for the design of otch Filters, but for every -D filter (either IIR or FIR: where, are real numbers 0 Unlike the original clellan Transform cos cos cos where we dem only, in our transformation we dem not only,but also 0. The disadvantage of clellan Transform is that it can be applied in FIR filters i.e. in a filter transfer function A H B =. In this paper the new proposed transformation we can apply it in every - D prototype filter B in general polynomial of. We are ready now to prove the Theorem. Theorem. onsider a prototype -D BIBO stable filter a filter transfer function A H ( Under the transformation ( 0, the prototype -D BIBO of ( gives A, H, (3, where the new transfer function H, is also stable the origin of the axes ( 0 is depicted to the point (, (0,0 Proof. Start first to prove that the origin of the axes ( 0 is depicted to the point (, (0,0 which is j j j obvious because from ( one has e e e or equivalently cos cos cos. Therefore because the solution of the equation cos cos (i.e. ( 0 must be (, (0,0. Hence the origin of the axes ( 0 is depicted to the point (, (0,0., 0for For Stability we have to prove that every inside the unit bi-disk i.e. for every. Assume first that there are some However, in this case we have a B such that such that h, since, 0. B 0, on the other, we have (since 0 that makes our -D filter transfer function A H non-stable (in BIBO sense, but this contradicts to the assumption. So, this completes the Proof. A very interesting extension of this transformation can be the following k0 l0 ( k l, k0 l0 f 0 for all k l k, k 0,,..., l, l 0,,..., the following theorem can be proved. Theorem. onsider a prototype -D BIBO stable filter a filter transfer function A H ( B Under the transformation k l (, k0 l0 f (4 ISS: Volume, 07

3 ikos E. astorakis, Stavros Kaminaris International Journal of Instrumentation easurement 3 k0 l0 0, for all k l k, k 0,,..., l, l 0,,..., the prototype -D BIBO of ( gives A, H, B, where the new transfer function H, (3 is also stable the origin of the axes ( 0 is depicted to the point (, (0,0 Proof. It is easy to prove that necessary sufficient condition for the depiction of the origin of the axes ( 0 to the point (, (0,0 is k0 l0 For Stability we have also to prove that, 0for every inside the unit bidisk i.e. for every Assuming that there are some such that B would have a., 0, in this case we k l such that k0 l0 B 0, on the other h, since, we would have k l k l k0 l0 k0 l0 k0 l0 k0 l0 (all the have the same sign that makes our -D filter transfer function H non-stable (in BIBO sense, but this contradicts to the assumption. So, this completes the Proof. onsider now the most general transformation k l f(, k0 l0 g (, k l D k0 l0 under what circumstances this transformation would transform the prototype -D BIBO stable filter of ( to a stable -D filter? k l f(, k0 l0 k l (, g (, k l k0 l0 D k0 l0 h k l It is easy to verify that a necessary sufficient condition can be h (, all the h (, to k0 l0 have the same sign. It is also known that the -D system k l k0 l0 k l D k0 l0 f(, g (, if k0 l0 h (, K is BIBO stable if only After these preparations we are ready to prove the following theorem Theorem 3. onsider a prototype -D BIBO stable filter a filter transfer function A H ( Under the transformation k l f(, k0 l0 k l (, g (, k l k0 l0 D k0 l0 h k l where k0 l0 k0 l0 D a stable -D system -D BIBO of ( gives (5 k l k0 l0 k l D k0 l0 f(, g (, k0 l0 h (,, the prototype A, H, B, where the new transfer function H, (3 is also stable the origin of the axes ( 0 is depicted to the point (, (0,0 Proof. It is easy to prove that necessary sufficient condition for the depiction of the origin of the axes ( 0 to the point (, (0,0 is also The -D rational function ISS: Volume, 07

4 ikos E. astorakis, Stavros Kaminaris International Journal of Instrumentation easurement 4 h (, which is equivalent from (5 that k0 l0 k0 l0 k0 l0 D For Stability we have also to prove that, 0for every inside the unit bidisk i.e. for every. This is true, because if one assumes that there are some B, 0, we would have a k l k0 l0 such that other h, since such that B 0. On the, we would have k l k l h h k0 l0 k0 l0 (, (, Fig. agnitude response of the filter of (6 onsider now the transformation all the have the same sign that makes our -D filter A H non-stable which is transfer function this contradicts to the assumption. III. UERIAL EXAPLES Example. onsider the example of 6.4 of [7]. A -D (digital IIR three-pole Butterworth filter is described as follows H K 3 ( ( ( where, are real numbers for example / 0 H 3 ( ( ( ( ( ( 0.95 j j e ( 0.95 e that gives the -D (digital IIR filter A, H, =, ( ( ( ( ( 0,9065( 3 3 ( ( j j ( ( ( 0.95 e ( ( 0.95 e ( (7 K / 9000 magnitude response in Fig. (6 the -D magnitude response in Fig.. ISS: Volume, 07

5 ikos E. astorakis, Stavros Kaminaris International Journal of Instrumentation easurement 5 onsider again the transformation ( / one takes H,,, (( 0.84(.09e (.09e 4 j0.633 j (.40e (.40e j0.66 j0.66 Fig. agnitude Response of the -D filter of (7 Example. hebyshev filters ([8] have the property that the magnitude of the frequency response is either equiripple in the passb monotonic in the stopb or monotonic in the passb equiripple in the stopb. The digital filter for this 4th-order hebyshev I digital lowpass filter is expressed as follows: the -D magnitude response in Fig.4. (8 H ( ( ( ( 4 j0.633 j (.09e (.09e j0.66 j (.40e (.40e (8 magnitude response in Fig.3 Fig. agnitude Response of the -D filter of (8 Fig 3. agnitude response of the filter of (8 ISS: Volume, 07

6 ikos E. astorakis, Stavros Kaminaris International Journal of Instrumentation easurement 6 IV. OLUSIO ew general transformations have been introduced for the design of -D (Two-Dimensional FIR IIR filters. It seems that this methodology can be viewed as an extension of the clellan Transformations can be applied in several cases of -D FIR IIR filter design, while the clellan Transformations are applied only for the design of -D FIR filters. Two umerical examples illustrated the validity the efficiency of the method. The proposed methods ensure stability in all the cases due to Theorems,, 3. REFEREES [] Belle A. Shenoi, agnitude Delay Approximation of -D -D Digital Filters, Springer Verlag, Berlin, 999 [] Wu-Sheng Lu, Andreas Antoniou, Two-dimensional digital filters,. Dekker, ew York, 99 [3] J.H.clellan, The Design of two-dimensional digital filters by transformations in Proc. 7th Annual Princeton onf.inf. 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[].E.astorakis, "A method for computing the -D stability margin", IEEE Transactions on ircuits Systems, Part II, Vol.45, o.3, pp , arch 998. [].E.astorakis, "Recursive Algorithms for Two-Dimensional Filters' Spectral Transformations", IEEE Transactions on Signal Processing. Vol.44, o.0, pp , Oct.996. [3] V. ladenov,. astorakis, Design of -Dimensional Recursive Filters by using eural etworks, IEEE Trans. on eural etworks. [4] astorakis,.e., Gonos, I.F., Swamy,..S., Stability of multidimensional systems using genetic algorithms, IEEE Trans. on ircuits Systems, Part I: Fundamental Theory Applications, Volume 50, Issue 7, July 003, pp [5] astorakis,.e. Swamy,..S., "Spectral transformations for two-dimensional filters via FFT" IEEE Transactions on ircuits Systems I: Fundamental Theory Applications, Vol.49, Issue 6, pp.87-83, Jun 00 [6] astorakis,.e. ew method for designing -D (Two- Dimensional IIR otch filters, Submitted to IEEE Transactions on ircuits Systems I (October 00 [7] harles S. Williams, Designing Digital Filters, Prentice Hall, ew Jersey, 986 [8] ational Instruments ISS: Volume, 07