Design of Linear Phase FIR Filter Using Differential Evolution Optimization
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.doc   Design of Linear Phase FIR Filter Using Differential Evolution Optimization2003(2).doc (Size: 242.5 KB / Downloads: 49)

K Rashmi Sekhar
S.Sneha Deepthi

Differential Evolution Algorithm is discussed. The design of a linear phase FIR Filter using DIFFERENTIAL EVOLUTION Algorithm is proposed. The frequency response of the designed LOW PASS FIR filter using Differential Evolution for different population and iterations are realized and compared. The best obtained filter can be used as a pulse-shaping filter in QPSK.

Digital filters are discrete time LTI systems characterized by Linear constant coefficient difference equations. They generally consist of adders, multipliers and flip-flops or delay elements. The amplitude by which each sample of the input to the filter is multiplied is nothing but the coefficients of the impulse response. Depending upon the impulse response of any digital filter, its characteristics will vary. Depending upon the duration of the impulse response coefficients, digital filters are classified as FIR and IIR. It is always possible to design an FIR filter transfer function with an exact linear-phase response, while it is nearly impossible to design a linear-phase IIR filter transfer function. That is why in many practical systems, FIR digital filter is normally used. In any digital communication system, transmitting pulse-shaping filters are generally realized by different types of FIR filter. In this paper, the design of robust stable low pass FIR digital filter using Differential Evolution algorithm is discussed. It has been observed that the performance of Differential Evolution is superior to that of Genetic Algorithm in the filter design problem.. The frequency response of the designed FIR filter for different population size and iterations has been presented. Comparisons of the variation of averaged cost function of the design algorithm with population size and the number of iterations has also been presented. Minimax error norm has been considered as averaged cost function in the discussed algorithm. The proposed FIR digital filter can be used as a pulse-shaping filter in a QPSK modulated system to study its impact on the system performance.

Differential Evolution

In the year 1995, Price and Storn have proposed a new floating-point encoded evolutionary algorithm for global optimization. The name of the algorithm is Differential Evolution (DE) algorithm to signify a special type of differential operator that has been utilized to create new off springs from the parent chromosomes without adopting classical crossover or mutation. It is a new heuristic algorithm for minimizing possibly non-linear and non-differentiable continuous space functions. By means of an extensive test bed, it has been demonstrated that this method converges faster and with more certainly than any other global acclaimed optimization technique. This algorithm can be implemented very easily and it requires negligible parameter tuning, which has made the algorithm reasonably popular very soon. DE is a very simple, powerful, stochastic, population based, robust, easy to use optimization algorithm, which has been developed to optimize real parameter and real valued functions. DE evolutes different generations until the stopping criterion is reached. General problem formulation for DE is for an objective function f: X⊆ RD→R, the goal of the optimization technique is to find x*∈X such that f(x*) ≤ f(x) for all x∈ X [12]. The crucial idea behind DE is a new scheme for generating trial parameter vectors. DE generates new parameter vectors by adding the weighted difference vector between two population members to a third member. If the resulting vector yields a smaller objective function value than a predetermined population member, the newly generated vector replaces the vector with which it was compared. The comparison vector can, but need not be the part of the generation process. In addition, the best parameter vector is evaluated for every generation in order to keep track of the progress that is made during the minimization process. There are different variants of DE scheme, which are now being used in practice. Mutation is that particular step in DE, which is responsible for different types of DE technique. It is a very good global optimizer for function optimization. It has found its applications in design of digital filters, electromagnetic inverse problems, composite materials, antennas and different other mathematical and engineering domains. This algorithm can optimize any function with D real parameters for any positive integer number D. Before the start of the algorithm, the size of the population P should also be selected. P does not change during the minimization process. So, DE will utilize P number of D dimensional vector with each vector having the form: Xi,G, =[X1,i,G; X2,i,G; …;XD,i,G], where i =1,2,…P and G is the iteration number or generation number.

This evolutionary algorithm includes four steps, namely Initialization, Mutation, Recombination and Selection.

A. Initialization
This step indicates the start of the evolutionary algorithm. Initially, the values associated with the parameter vector are chosen randomly and it should cover the entire parameter space. Unless otherwise mentioned, generally a uniform probability distribution for all the random variables is assumed. In case of a preliminary solution is available, the initial population is often generated by adding normally distributed random deviations to the nominal solutions. Let Xi,j,1 denote the ith element of the jth member of population at first iteration. In order to ensure that this value is within a certain limit [XL, XU], Xi,j,1 is chosen according to the following equation: XL ≤ Xi,j,1 ≤ XU,..(1) where XL and XU are the upper and lower bound of the random variable respectively.

B. Mutation
The step mutation actually expands the search space. In each generation or iteration of the algorithm, to change the member of the population Xi,G, a mutant or donor vector Vi,G+1 is created. Depending upon the generation of the donor vectors from the parameter (target) vector, different variants of DE has been developed. For each parameter vector Xi,G, i=1, 2… P; a mutant vector is generated using different equations in different schemes of DE. In the DE/rand/1 scheme, mutant vector of the next generation Vi,G+1 is generated according to the following equation Vi,G+1=Xr1,G+F(Xr2,G-Xr3,G),..(2) where i = 1,2,..,P where i, r1, r2 and r3 are distinct from each other and chosen from the set {1, 2… P}. F is the weighting factor which is in the range [0, 2] and it will control the variation of the amplification of the differential variation ( Xr2,G - Xr3,G). In order to ensure that the running index i is different from the other three distinct indices r1, r2 and r3; the minimum number of population in this particular scheme of Differential Evolution technique must be 4.
DE/rand to best /1 follows the same procedure as that of DE/rand/1. The only difference is that the donor vector Vi,G+1 is created using any two randomly selected parameter vector and the best member of the current generation. This can be represented mathematically, in accordance to the following equation:Vi,G+1 =Xi,G+λ(Xbest,G − Xi,G)+F(Xr2,G - Xr3,G)..(3) where λ is a control parameter from the set [0, 2]. In the scheme DE/best/1 and DE/best/2, mutant vector of any generation is produced without using the corresponding parameter vector. The donor vectors for each of these two schemes are given by (4) and (5) respectively:Vi,G+1 = Xbest,G + F(Xr1,G-Xr2,G),..(4) Vi,G+1 = Xbest,G + F(Xr1,G+Xr2,G−Xr3,G-Xr4,G),..(5).

Another different DE scheme is known as DE/rand/2. Here, in order to construct a donor vector, altogether 5 distinct parameter vectors are randomly chosen from the rest of the population. In this process, two weighted difference vectors are added to the same to generate the donor vector [10], as shown in (6): Vi,G+1 =Xr1,G+F1(Xr2,G – X3,G)+F2(Xr4,G – Xr5,G),..(6) where F1 and F2 are two weighing factors selected in the range from 0 to 1.

C. Recombination
Recombination incorporates successful solutions from the previous generation. In order to increase the potential diversity of the population member, Recombination process plays a very important role. At the end of this step, the trial vector Ui,G+1 is developed from the elements of the target vector Xi,G and the elements of the donor vector Vi,G+1. Elements of the donor vector enter the trial vector with probability RP.


In this paper, the design of an FIR digital filter using DE technique has been discussed. The performance of the proposed filter has been analyzed qualitatively and quantitatively along with standard RC and RRC pulse shaping filter. The proposed filter exhibits better performance as compared to the standard filters with a faster convergence speed. It has also been verified that the FIR filter designed with highest population size and iteration number exhibits the best performance as expected theoretically.


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