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Int. J. Signal and Imaging Systems Engineering, Vol. 3, No. 4, 2010
Design of Two-Dimensional Recursive Digital Filters with Specified Magnitude and Group Delay Characteristics using Taguchi-based Immune Algorithm Mohammed Abo-Zahhad*, Sabah M. Ahmed, Electrical and Electronics Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt. Email:
[email protected],
[email protected]. Email:
[email protected],
[email protected]. *Corresponding author
Ahmad F. Al-Ajlouni Communication Engineering Department, Hijjawi Faculty for Engineering Technology, Yarmouk University, Irbid, Jordan. Irbid 21163 Jordan Email:
[email protected]
Nabil Sabor Electrical and Electronics Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt. Email:
[email protected] Abstract: Over past few years, several studies have been carried out by researchers to describe a general design methods based on modern heuristic optimization algorithms for designing twodimensional recursive digital filters with only specified magnitude characteristic (Mastorakis et al., 2001;2003). In this paper one modern heuristic optimization algorithms, named the Taguchibased Immune Algorithm (TBIA), is adopted to solve the problem of designing two-dimensional recursive digital filters with specified magnitude and group delay characteristics. The TBIA is based on both features of the biological immune system and the Taguchi method which increases the ability of the IA to find the global optimal solution in a nonlinear space. Based on minimizing the magnitude and group delay errors, a multi-criterion combination is employed as the design criterion to obtain optimal recursive filter that satisfies the required specifications. In addition, the TBIA also guarantees the filter stability by satisfying stability constraints in the design process. In this paper, the algorithm is detailed for the design of three recursive filters categories with: 1) predefined magnitude; 2) predefined phase (or group delay) and 3) predefined magnitude and phase (or group delay). The computational experiments show the ability of the proposed TBIA approach to design stable complex filters with better magnitude and group-delay characteristics and to obtain more robust results compared to the previous design methods (Mastorakis et al., 2001; 2003; 2006; Fahmy and Aly, 1978; Hinamoto and Doi, 1996; Tsai et al., 2008). Keywords: Two Dimensional Digital Filters, Immune Algorithm, Taguchi Method, Group Delay. Reference to this paper should be made as follows: Abo-Zahhad, M., Ahmed, S., Al-Ajlouni, A. and Sabor, N. (2010) ‘Design of Two-Dimensional Recursive Digital Filters with Specified Magnitude and Group Delay Characteristics using Taguchi-based Immune Algorithm’, Int. J. Signal and Imaging Systems Engineering, Vol. x, No. x, pp. xx–yy. Biographical notes: Prof. Mohammed Abo-Zahhad (SIEEEM’00) received his B.S.E.E. and M.S.E.E degrees in electrical engineering in 1979 and 1983 respectively, both from Assiut University, Egypt. In 1988, he received Ph. D. degree from the University of Kent at Canterbury, UK and Assiut University (channel system). His research interests include switched-capacitor, optical and digital filters, biomedical signal processing, speech processing, data compression, wavelet-transforms, genetic algorithms, immune algorithms and electronic systems. He has published more than 90 papers in international journals and conferences in the above fields. He is currently a Professor of Electronics and Communication Engineering, since Jan.1999 and a vicedean for graduated studies, Faculty of Engineering, Assiut University, since August 2006. He is a senior IEEE member, 2000 and a member of the European Society of Circuit Theory and Applications, 1998.
Copyright © 2010 Inderscience Enterprises Ltd.
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Design of 2-D Recursive Digital Filters with Specified Magnitude and Group Delay Characteristics using TBIA
Dr. Sabah Mohamed Ahmed received B.S.E.E. (with honors) and M.S.E.E degrees in electrical engineering from Assiut University, Egypt, in 1979 and 1985 respectively. She received her Ph. D. degree from the Technical University of Budapest, Hungry in 1992. In 2000 she joined Computer Science Department at Zarka Private University, Jordan. Dr. Sabah research interests include Speech analysis and synthesis, digital filters, medical signal processing, data compression, wavelettransforms, genetic and immune algorithms and computer applications. She authored over 40 published articles in the above fields. She is currently a Professor of Electronics and Communication Engineering, Since Feb. 2009. Dr. Ahmad Al-Ajlouni (SIEEEM’08) received his Ph.D. degree in electrical and computer engineering from Clarkson University in 1997. Dr. Al-Ajlouni is now an associate professor of Computer and Communication Engineering - Hijjawi Faculty at Yarmouk University. His research interests are in the areas of computer networks, digital signal processing, active noise cancellation and artificial intelligence. Mr. Nabil Sabor was born in Assiut, Egypt, in 1984. He received B.S.E.E. (with honors) in electrical engineering from Assiut University, Egypt, in 2006. He has worked at the Faculty of Engineering, Assiut University as a Demonstrator (2006). Mr. Nabil research interests include signal processing, data compression, wavelet-transforms, genetic and immune algorithms and digital filters.
I. Introduction The two-dimensional (2-D) recursive digital filters design has received growing attention for digital image processing, biomedical data processing, and satellite image processing applications (Kalinli and Karaboga, 2005; Tsai et al., 2005; Lu and Antoniou, 1992). In the digital filter design, there are principally two approaches: transformation approach (Lu and Antoniou, 1992) and optimization approach (Kumar et al., 2009). In the former method, the analog IIR filter must be designed first, and then it is transformed to the digital domain using bilinear transformation. In the optimization approach, various methods have been proposed to obtain optimal filter performance to some extent, where the p norm error, meansquare-error (MSE), group delay and equi-ripple magnitudes of both passband and stopband are usually used as criteria to measure the performances of the designed digital IIR filters (Tsai and Chou, 2006). However, many of the existing methods (Lu and Antoniou, 1992; Fahmy and Maria, 1974) may result in an unstable filter. Various techniques have been proposed in order to overcome these instability problems, but the outcome was often a system with a very small stability margin and may not be of practical importance (Mastorakis et al., 2001; 2003; 2006). The use of the GA for the digital IIR filters design is practical and attractive because of the following advantages: i) the classical analog-to-digital transformation is avoided; ii) the multi-objective functions can be simultaneously solved [4]. In the multi-parameter and multi-criterion optimization problem of designing the 2-D digital filters, the particular challenge is that the GA-based methods may be trapped in the local optima of the multiobjective functions when the number of the parameters is large and there are numerous local optima. Therefore, it is worthy to further develop an efficient evolutionary algorithm to solve the problem of designing the optimal digital IIR filters (Tsai et al., 2004; 2005; Tsai and Chou, 2006). In recent years, there has been an increasing interest in the area of the immune algorithms (IAs) and its applications in designing one and two-dimensional IIR filters with specified magnitude (Kalinli and Karaboga, 2005; Tsai et al., 2004; Tsai and Chou, 2006). Immune algorithms (IAs) are the algorithms that mimic the antigen-antibody reaction of the immune system in mammals. The antigen and the antibody in
the IA are equivalent to the objective function and the feasible solution for a conventional optimization method. The purpose of this paper is to present a novel robust approach named TBIA by integrating the IA and the Taguchi method to solve the 2-D filter design problem. In the TBIA, the clonal proliferation with hypermutation is used to increase antibodies diversity and to improve their capability of recognizing the selective antigens. Moreover, the Taguchi method is used to select the better antibody genes to enhance the immune algorithm. Therefore, the integration of both the clonal proliferation within hypermutation and the Taguchi method for the recombination can improve the immune algorithm, so that the TBIA can be more robust and quickly convergent (Tsai et al., 2004). The paper is organized as follows. Section II describes the problem formulation. The TBIA algorithm for designing 2-D digital IIR filters is described in Section III. In Section IV, we adopt the same examples as that considered in (Mastorakis et al., 2001; 2003; 2006; Fahmy and Aly, 1978; Hinamoto and Doi, 1996) to evaluate the proposed TBIA approach, and to compare its performance with the performance of these references. Finally, Section V offers some conclusions.
II. Problem Formulation For the design purposes, we consider the following 2-D IIR filter transfer function: K
H z1 , z 2 H 0
K
a i 0 j 0
K
1 b z
k 1
i j ij 1 2
zz
c k z 2 d k z1 z 2
, a 00 1
(1)
k 1
where, aij , bk , ck , and d k are the coefficients of the filter, K is filter order,
z1 e
j1
and z 2 e
H0 j 2
is a constant multiplier, with
1 n1 / N1
and
2 n2 / N 2 , ( 1 , 2 [0, π]). The frequency response of the above filter may be expressed by:
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M. Abo-Zahhad et al.
M 1 , 2 H e j1 , e j1 M 1 , 2 e j 1 ,2 where, M 1 , 2
is
the
filter’s
(2)
magnitude
function
and 1 , 2 is the filter’s phase function. The group delay functions of the filter can be defined as
i 1 , 2
1 , 2 , i
i 1,2
(3)
To formulate the design problem, let us consider
M d 1 , 2 and d 1 , 2 , (where i=1, 2) as the desirable
bk ck 1 d k
(8)
d k 1 bk ck
, k 1, 2, 3, , K
II. TBIA Approach for the 2-D recursive filter design This section describes how to solve the 2-D filter design problem by using TBIA approach. Figure (1) and Table (1) illustrate the pseduocode and flow chart of the design algorithm, respectively. In the following, the algorithm is detailed; where solution refers to chromosome or antibody and each filter coefficient refers to one gene.
i
amplitude and group delay responses, respectively. The design task at hand amounts to finding a transfer function H z1 , z 2 in Equation (1) such that the function M 1 , 2 approximates the desired amplitude response M d 1 , 2 and the group delays i 1 , 2 approximates the desired group delays d 1 , 2 . This approximation can be achieved i by minimizing the following multi-objective function: F X Wm Fm X W1 F1 X W 2 F 2 X
(4)
where, X is the parameters vector , defined as X aij , i j 0,..., K , bk , ck , d k , k 1,..., K , H 0
Wm and W i are the relative weights of the components Fm X and F i X , respectively.
gen=0; % initialization of generations counter Chrom=Initial_pop(); % Construct the initial population pool While (gen
