DESIGN OF OPTIMAL FRACTIONAL DELAY-IIR FILTER USING EVOLUTIONARY ALGORITHMS

: This paper presents the optimal design of fractional delay-Infinite Impulse Response filter (IIR) using a meta-heuristic approach called Modified Cuckoo Search Algorithm (MCSA). The Fractional Delay (FD) filters are used to give fraction of delay to signal and the FD-IIR filters are being used in various applications of signal processing. Coefficients of optimized Fractional Delay-Infinite Impulse Filter (FD-IIR) is calculated using Modified Cuckoo Search Algorithm (MCSA) to match the response of fractional delay IIR filter with ideal response of the filter. Different heuristic optimization algorithms such as Cat, Bat, Genetic Algorithm (GA), Simulated Annealing (SA), Cuckoo Search Algorithm (CSA), Particle Swarm Optimization (PSO) etc. have been used to design optimal fractional delay-IIR filter. FD-IIR filter design is a multimodal design problem. Hence Meta-heuristic optimization algorithm is used in the paper. The proposed algorithm is a modification of cuckoo search algorithm and hence called Modified Cuckoo Search Algorithm. It is a meta-heuristic optimization technique based on population of birds and cuckoos behavior. It is simple and is a global optimization algorithm. The performance of MCSA is compared with genetic algorithm, particle swarm optimization and cuckoo search algorithm. It is found that MCSA provides better results than GA, PSO and CSA. The fitness function used to evaluate the algorithm is Weighted Least Square function. The simulation results show that proposed algorithm, MCSA has less absolute magnitude error. The statistical data analysis also shows that MCSA has higher value of percentage improvement in magnitude error and has faster convergence rate than GA, PSO and CSA in terms of execution time


I. INTRODUCTION
Digital signal processing is very useful in the field of science and technology today. Digital signals are used to handle discrete time signals and filters are one of the important parts of this field. Filters are not only for frequency selective purpose but are also used in many applications that is as simple as from reducing ripples to higher sophisticated circuits like reduction in noise level, enhancement in video signal in astrological and biological systems and equalization in graphics [1]. Delay filters provide a fractional amount of delay to the signals where accuracy of the system is very important. Hence, nowadays, a principle field of fractional delay filters is in signal processing [2]. Two categories of digital filters are recursive and non-recursive. FIR filters are non-recursive filters whose output depends on present and past input values only whereas recursive are IIR filters whose output depend on past input and output values and thus provide better performance than non-recursive filters i.e. FIR filter [1]. Some of the limitations of gradient optimization are: continuous and differentiable fitness function is required and provide local minimum solution as it is centered on the suboptimal solution [27]. Thus, design of IIR filters using optimization algorithms makes their design easier and simple. These algorithms are differential evolution (DE) [2], simulated annealing (SA) [3], genetic algorithm (GA) [4], ANT colony optimization [5], Gravitation search algorithm (GSA) [1], BAT algorithm [6], Firefly algorithm [7], Particle Swarm Optimization (PSO) [8], [9], CAT swarm optimization [10], [11], Cuckoo Search Algorithm (CSA) [12], [13], [14]. The proposed algorithm in the paper is a modification to cuckoo search algorithm which provide global minimum solution and has better performance results than CSA. This paper is divided into four sections. First section describe introduction. Then section 2 describes about the FD-IIR filter and the proposed algorithm for the optimal design of filter. In section 3 discusses about the simulation results and statistical data and the comparison between the results is analyzed. Finally, section 4 concludes the paper.

A. Design of Fractional delay IIR filter problem:
Digital fractional delay filter has ideal frequency response as given below [14]: (1) where w is digital frequency, w є [0, π] and v is fractional delay, v є [0, 1]. The discrete signal can be delayed and can be expressed mathematically as (2) where x(n) and y(n) is input signal and output signal respectively and Z is a positive integer. If delay is not integer then it is written as (3) Where Z is integral part of delay and p is fractional delay and its value ranges from 0 to 1, that is, [15]. The integral part of delay can be implemented as chain of Z unit delays but the fractional part of delay is to be subjected to approximation to control the delay value continuously whenever a fraction delay value is needed [16]. There are several techniques used to computecoefficients of fractional delay filter such as Lagrange interpolation [17], Farrow structures [18], minimax design method [19], and least squares and weighted least squares [15].To approximate the results of FD filter weighted least square (WLS) method is used as objective function in this paper. WLS is used to reduce the complexity of the design and it also uses different weighing function over the whole frequency band thus meeting the results of the requirement of the system whereas in least square method it assumes same importance all over the frequency range. Two categories into which Fractional delay filters can be classified is: (i) fixed fractional delay filter in which delay p is fixed. In this the signal is delayed to some pre specified value and, (ii) variable fractional delay filter in which delay is variable [15], that is, in this the desired delay is introduced in the signal as parameter to the filter. The filter design problem is to approximate the desired filter response to the ideal filter response [23], [24], [25]. FD-IIR filter has transfer function as given below: (4) Where M and N are degrees of numerator and denominator, respectively and is the filter order. a v and b v are the real filter coefficients that need to be optimized [14]. FD-IIR filter has difference equation expressed as below: (5) The frequency response of the filter is given by: The error to be minimized is given by: (7) where H i (w,v) and H(w,v) is the ideal and approximated filter response, respectively. The weighted least square objective function is expressed as: (8) Where F(w) is given by: (9) This fitness function is used to minimize the error given in equation (7) using Genetic algorithm, Particle Swarm optimization, Cuckoo SearchAlgorithm [26] and Modified Cuckoo Search algorithm, thus, computing the optimized filter coefficients and using it for design of FD-IIR filter.

B. Proposed Optimization algorithms:
Different optimization techniques are used for filter design problem. Some of them are classical algorithms which can be used for only simple design problems and some are heuristic and meta-heuristic optimization algorithm. Heuristic algorithms are problem specific that is they perform local search and results obtained are not guaranteed to be global solution. Some of these algorithms are gravitation search algorithm, genetic algorithm, orthogonal GA, simulated annealing, differential evolution and PSO whereas Meta-heuristic algorithms can be defined as one stage higher than heuristic algorithms and they can act as guiding approach for the heuristic optimization algorithms [12], [28]. Some of these are Tabu search [20], ant colony optimization [5], and artificial immune system [21].These are meta-heuristic algorithms which are based on population, trajectory and stochastic features. Metaheuristics has two important features: intensification and diversification. Intensification aims to search locally and more intensively around the current best solution and find the best solution from the search whereas diversification determines the best solution by exploring the search space more efficiently and globally so as to get the best result [22]. Proposed algorithm that is used in this paper is described as below: Modified Cuckoo Search Algorithm: The proposed algorithm is a modification to the well-developed algorithm that is cuckoo search. Two modifications are done to the present cuckoo search algorithm. One involves change in value of probability of discovering rate of alien eggs and second one is change in the value of step size in levy flight using Mantegna's algorithm [29]. This modification has shown improvement in performance than cuckoo search and also has shown better results when compared with genetic, particle swarm and cuckoo search algorithms. In cuckoo search algorithm the fraction of nest to be abandoned or the discovery rate of alien egg was kept 0.25 but it is found that by changing the value to 0.75 it gives better results over the cuckoo search algorithm in terms of optimized filter coefficients. Also, according to Mantegna's algorithm which has same behavior as Levy flight distribution, the distribution parameter (β) can be set in the range β ε [0.3, 1.99]. Thus value of β is set as one in the proposed algorithm as opposed to 3/2 in the cuckoo search algorithm and it is found to give better results in terms of minimum magnitude error and faster execution. According to Mantegna's algorithm [29], the distribution is calculated using equation (10): (10) whereβ is distribution parameter. The step size is calculated using equation (11).
Step value is multiplied with factor 0.01 considering L/100 to be step size of walk and so that the egg doesn't move out of search space. (m-best) factor remains unchanged when the best solution is obtained; otherwise the difference tells that solution obtained is not best. step size=0.01*step.*(m-best) (11) where m is given by m=m + step size.* randn (size (m))(12) The algorithm provides efficient result when parameter β = 1 and it is observed that calculation becomes faster with an integer value. Also the execution time decreases when value of β increases from lower fraction values. Firstly, solution iscalculated using random nest, then by using Mantegna's algorithm and both solutions are compared. Out of both functions whichever has lesser value that will be the new solution. The discovery rate of alien eggs is fixed to 0.75 and the best solution is kept. The flowchart for the modified cuckoo search algorithm is shown in Fig. 1.  . The discovery rate of alien eggs is pa = 0.75 and distribution parameter β = 1. Graphs for magnitude response and error response for different orders are represented. Analysis of graphs depicts that MCSA has lesser error in magnitude, higher percentage improvement, and lesser execution time. The response using different algorithm is recorded in Fig. 2 -5. Numerator and denominator coefficients with optimized value for the design of filter are calculated using GA, PSO, CSA and MCSA are given in Table I.

B. Analogy between different algorithms on the basis of magnitude response and their error:
Magnitude response of different algorithms for different order of filter is analyzed by simulations. The order of filter taken for inspection is 2, 3, 4 and 5. Comparison of Magnitude response of each order for different algorithms and their magnitude error is computed and plotted as shown in Fig. 2-Fig. 5. Analysis of graphs shows that MCSA has low magnitude error as compared to other optimization algorithms.   C. Analysis of Statistical data Analysis of Statistical data is done using different filter order and different optimization techniques to get the best optimal solution for filter design. Table III-X shows comparative study of different algorithms in terms of features like average value, standard deviation, and variance, error of magnitude with maximum and minimum value. From Table III it is seen that for 2 nd order values of mean, standard deviation and variance that is, 0.0012, and , respectively are obtained using MCSA. Table IV shows maximum, minimum and average value of magnitude error for 2 nd order filter which are 0.0051, and 0.0012, respectively computed using MCSA. On similar lines the statistical data and qualitative data for 3 rd , 4 th and 5 th order is analyzed and recorded in Table V-X. It is found that MCSA has lower magnitude error value as compared to other algorithms

IV. CONCLUSION
In this paper modification to the well-developed cuckoo search algorithm is applied to get the optimal filter coefficient. A number ofoptimization algorithms are available to find the optimal solution of fractional delay IIR filter design problem. But as the FD-IIR filter design problem is a multimodal design problem only a few of them can provide the best global solution. The magnitude response of FD-IIR filter approximates the ideal response and it is analyzed that Modified Cuckoo Search Algorithm is superior to cuckoo search algorithm, genetic algorithm and particle swarm optimization algorithm. The filter coefficients are obtained for the filter order 2 nd , 3 rd , 4 th , 5 th using GA, PSO, CSAand MCSA and it is found that MCSA provides optimal solution. As the filter design is a multimodal optimization problem so MCSA is suitable for the problem and better results are achieved using this algorithm.The Modified cuckoo search algorithm gives global optimal solution and is simpler and robust.Also the performance of Modified Cuckoo Search Algorithm becomes more effective when Mantegna's algorithm is used which has similar behavior as Levy flights in cuckoo search algorithm rather than using Gaussian or uniform distribution for each solution. Hence, Modified Cuckoo Search algorithm is proved to have given better results among all other algorithms.