STATE SPACE MODELING OF 2-D RECURSIVE DIGITAL FILTERS

Transcription

1 STTE SPCE MODELING OF -D RECURSIVE DIGITL FILTERS Stela ngelova Stefanova Technology School Electronic System associated with the Technical University of Sofia, Mladost, 750 Sofia, Bulgaria, , In this paper, a novel technique is introduced for the design of recursive digital filters in the state space form. The Roesser s state space model has been used for the spatial domain representation of the -D causal recursive digital filters. This model divides the local state into a horizontal and a vertical state which are propagated in horizontal and vertical directions respectively. The design approach for the -D recursive filter is realized as approimation problem in time domain. This problem is solved by standard gradient type optimization procedure. The effectiveness of this method has been demonstrated by modeling of the first quadrant Gaussian -D IIR filter.. INTRODUCTION Keywords: -D IIR digital filter, state space identification methods There are numerous techniques developed for digital filter design, both in frequency domain and in time domain. Most of these methods are analytical techniques. Typical approimation methods for -D FIR and IIR filters are based on windowing, spectral transformations, error criterion optimization, direct least square or minima procedures [-7]. -D IIR digital filters are usually designed either by spectral transformation of -D filter, linear programming, or nonlinear optimization approaches. design technique based on nonlinear Newton s optimization for -D FIR and IIR digital filters is proposed in [3]. One direct method for approimately linear phase -D IIR digital filters is presented in [4]. Some of the -D IIR digital filter design approaches are realized by solving the weighting lest square optimization problem, where the objective functions are defined as errors between frequency responses and desired target functions [], [5], [6]. The main idea of the other methods is to formulate the -D digital filters approimation in frequency domain as a minima optimization problem [], [7]. Various design techniques have been proposed for -D recursive digital filters, either in frequency domain or in spatial domain [8] -[]. Most of those techniques are for a special class of -D filters called as separable-in-denominator digital filters [8], []. This is due to the fact that these spatial -D filters share some important properties of -D digital filters such as stability and minimality conditions. Therefore, many -D spatial design techniques have been developed using separable-indenominator digital filters as the etensions of corresponding -D techniques [8], []. There are relatively few techniques developed on identification and design of general -D recursive digital filters. However, the problem of general -D identifications using an analytical solution has not been addressed due to its mathematically comple nature. 55

2 n IIR filter can be represented by either difference equation or state space form. The state space form in general involves more numbers of coefficients than a transfer function unless it is represented as one of the canonical forms. However, there are many benefits from using a state space model in the analysis, design, and implementation of digital filters. First, the state space model is more robust than a transfer function representation. In other words, it ehibits less coefficient sensitivity. Second, various forms of state space models possess distinctive properties that are desirable in different applications. Third, the major part of modern control theory is based on the state space model. Furthermore, the difference function representation could be uniquely determined by the state space form representation. In this paper, the state space model will be utilized for the -D IIR digital filter design. The filter's input and resulting output could be selected arbitrarily by the designer in spatial domain. In other words, the proposed technique can be uniformly applied for identification of a -D filter with an impulse response, a step response or a response to a random -D input signal.. -D STTE SPCE MODEL OF DIGITL FILTERS The -D state space models have been mainly used for the spatial domain representation of the -D causal recursive digital filters. Kung et al. [3] have shown that the Roesser s model [] is the most general form and the other representations can be imbedded in the Roesser s model. Roesser s state space model divides the local state into a horizontal and a vertical state which are propagated in horizontal and vertical directions respectively. Consider the transfer function of a stable -D digital filter of order N in z and N in z : N(z,z ( H(z,z = D(z,z The input-output relation of the filter is represented by the following Roesser s statespace equations []: ( h v (n +,n (n,n + = 3 4 h v (n,n (n,n + (n,n y(n,n = d v(n,n where n is an integer-valued horizontal coordinate; n is an integer-valued vertical coordinate; p,n R q,n R N n,n R u(n is the scalar input of the filter; y(n is the scalar output of the filter; h [ c c ] + u(n,n ( is the horizontal state vector; h N 56 b b u(n,n

3 N v ( n,n R is the N vertical state vector. Matrices,, 3, 4, b, b, c, c and d are real coefficient matrices with appropriate dimension. This means that the state-space model has the dimension ( N, N. The block diagram of the -D state-space digital filter is given in Figure. The transfer function H(z,z of the -D digital filter is given in terms of the coefficient matrices as: zi N b (3 H(z,z = [ c c ]. + d I b 3 z N 4 where and are the and N identity matrices, respectively. I N I N N N N 3. DESIGN TECHNIQUE Fig.. Block diagram of a -D state-space digital filter The design approach for the -D recursive filter is realized as approimation problem in time domain. This problem is solved by standard gradient type optimization procedure. The following norm [8]-[0] error criteria are defined: L / / ( h(n,n h I (n,n (h(n,n ( n,n Z ( n,n Z ε = where h(n,n and h I (n,n are the impulse responses of the original system and identified system, respectively, Z = {(n,n 0 < n < N, 0 < n < N } is the given region where the error norms are calculated. 4. RESULTS Eample: First Quarter Gaussian Filter The prototype model used by ly and Fahmy in [9] for designing a -D causal recursive filter is presented in the work. It is a first quadrant Gaussian -D scalar filter described by the following impulse response: 57

4 h(n,n = ep{ 0,0303 [( n 4 + (n The selected region for identification consists of Z = {(n,n 0 < n < 0, 0 < n < 0}. This impulse response is illustrated with the surface shown in Fig.. The identified -D filter in Roesser s model of order (3, 3 is given as: 4 ]} = [ e - 3, -.09e -, -.668e - ] T [ e -, 6.08e -, e - ] T b ; = 3 b = ; ; c = [ e -.940e.890e] ; c = [ e- 9.50e- ] ; d = 9.400e-3. Fig.. Impulse response of the -D Gaussian Filter 58

5 The impulse response of the identified -D IIR filter is shown in Fig. 3. Fig. 3. Impulse response of the identified -D Filter 5. CONCLUSIONS In this paper, a novel technique is introduced for the design of recursive digital filters in the state space form. The proposed method has no limitation on the type of response to be approimated. ny type of responses with sufficient data points could be used as a sample for filter identification. The effectiveness as well as robustness of this method has been demonstrated by modeling of the first quadrant Gaussian -D IIR filter. 6. REFERENCES [] Charalambous C,.and Khreishi H.: Minima design of D FIR linear phase digital filters, IEE Proceedings, 988, Vol. 35, Pt. G, (5, pp [] Zhu W.-P., Omair hmad M. and Swamy M.N.S: least-square design approach for -D FIR filters with arbitrary frequency response, IEEE Trans., 999, CS-46, (8, pp [3] Lampopoulos G.., and Fahmy M.M.: new technique for design of two-dimensional FIR and IIR filters, IEEE Trans., 985, SSP-33, (, pp [4] nderson M. S. and Lawson S. S.: Direct design of approimately linear phase (LP -D IIR digital filters, Electronics Letters, 993, Vol. 9, (9, pp [5] Lu, W.S., S.-C. Pei and Tseng C.C.: Weighted Least Squares Method for the Design of Stable -D and -D IIR digital filters, IEEE Trans. on Signal Processing, 998, Vol-46, (, pp

6 [6] Stefanova S.., least square approach for -D FIR digital filters design, Proc. оf the0- th International Scientific and pplied Science Conference Electronics ЕТ 00, Sozopol, Bulgaria, 6-8 Sept. 00, Book 4. [7]. Stefanova S.., Minima Optimization Technique for Design of -D IIR Digital Filters, Proc. оf the National Conference with International Participation Electronics 00, Sofia, Bulgaria, May 00. [8]. Ramos, J., Subspace lgorithm for Identifying -D Separable in Denominator Filters, IEEE Trans. on Circuits and Syst., Vol. 4, No., 63 67, January, 994. [9]. ly, S.H. and Fahmy, M.M., Spatial-Domain Design of Two-Dimensional Recursive Digital Filters, IEEE Trans. on Circuits and Syst., Vol. 7, No. 0, 89 90, October, 980. [0]. Hinamoto, T. and Maekawa, S., Design of -D Separable in Denominator Filters Using Canonic Local State-Space Models, IEEE Trans. on Circuits and Syst., Vol. CS-33, No. 9, 9 99, September, 986. []. Bose, T. and Chen, M., Design of Two-Dimensional Digital Filters in the Spatial Domain, IEEE Trans. on Signal Processing, Vol. 4 No. 3, , March, 993. []. Roesser, R.P., Discrete State-Space Model for Linear Image Processing, IEEE Trans. on utomatic Control, Vol. C-0, 0, February 975. [3]. Kung, S., Levy, B.C., Morf, M., and Kailath, T., New Results in -D Systems Theory, Part II: -D State-Space Models - Realization and the Notions of Controllability, Observability, and Minimality, Proc. of the IEEE, Vol. 65, No. 6, , June,