Muhammad T. Salam

By Muhammad Tariqus Salam and Venkat Ramachandran

This paper develops a design of two-dimensional (2D) digital filter with monotonic amplitude-frequency responses using Darlington-type gyrator networks by the application of Generalized Bilinear Transformation (GBT). The proposed design provides the stable monotonic amplitude-frequency responses and the desired cutoff frequency of the 2D digital filters. This 2D recursive digital filter design includes 2D digital low-pass, high-pass, band-pass and band-elimination filters. Design examples are given to illustrate the usefulness of the proposed technique. Index Terms— Stability, monotonic response, GBT, gyrator network. 1. Introduction Because of recent growth in the 2D signal processing activities, a significant amount of research work has been done on the 2D filter design [1] and it is seen that monotonic characteristics in frequency response of a filter is getting more popular. The filters with the monotonic characteristics are one of the best filters for the digital image, video and audio (enhancement and restoration) [2]. The filters are widely accepted in the applications of medical science, geographical science and environment, space and robotic engineering [1]. For example, medical applications are concerned with processing of chest X-Ray, cine angiogram, projection of frame axial tomography and other medical images that occurs in radiology, nuclear magnetic resonance (NMR), ultrasonic scanning and magnetic resonance imaging (MRI) etc. and the restoration and enhancement of these images are done by the 2D digital filters [3]. The design of 2D recursive filters is difficult due to the non-existence of the fundamental theorem of algebra in that the factorization of 2D polynomials into lower order polynomials and the testing for stability of a 2D transfer function (recursive) requires a large number of 54 Digital Filters computations. But, the major drawbacks of the recursive filters are their lower-order realizations and computational intensive design techniques. Several design techniques of 2D recursive filter have been reported in the literature [2], [4] – [9] and most of these designs have problems of computational complexity, stability and unable to provide variable magnitude monotonic characteristic. A design technique of 2D recursive filters have been shown which met simultaneously magnitude and group delay specifications [4], although the technique has the advantage of always ensuring the filter stability, the difficulties to be encountered are computational complexity and convergence [5]. In [6], 2D filter design as a linear programming problem has been proposed, but, this tends to require relatively long computation time. In [7], a filter design has been shown using the two specifications as the problem of minimizing the total length of modified complex errors and minimized it by an iterative procedure. Difficulties of the design obtain for two-dimensional stability testing at each iteration during the minimization procedure. One way to ensure a 2D transfer function is stable is if the denominator of the transfer function is satisfied to be a Very Strict Hurwitz Polynomial (VSHP) [8] and that can ensure a transfer function that there is no singularity in the right half of the biplane, which can make a system unstable. In [9]-[11], stable 2D recursive filters have been designed by generation of Very Strict Hurwitz Polynomial (VSHP), but it is not guaranteed to provide the stable monotonic amplitude-frequency responses. Several filter designs with monotonic amplitude frequency response has been reported [12] – [16], but to the best of our knowledge, filter design with variable monotonic amplitude frequency response is not proposed yet. In this paper, 2-D digital filters with variable monotonic amplitude frequency responses are designed starting from Darlington-type networks containing gyrators and doublyterminated RLC-networks. The extension of Darlington-synthesis to two-variable positive real functions is given in [17], [18]; but they do not contain gyrators. From the 2-D stable transfer functions so obtained, the GBT [19] is applied to obtain 2-D digital functions and their properties are studied. The designed filters are used in the image processing application. 2. THE TWO BASIC STRUCTURES CONSIDERED Two filter structures are considered for 2D digital recursive filters design and both structures are taken from Darlington-synthesis [20]. Figures 1(a) and (b) show the two structures considered in this paper. The impedances of the filters are replaced by doubly-terminated RLC filters and the overall transfer function will be of the form H ( s1 , s 2 , g )   N  ( g )s  s  0 0 1 M n Nn 2  0  0   D ( g )s  M d Nd (1)

Part of the book: Digital Filters