IPASJ International Journal of Electronics & Communication (IIJEC) Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm Email:
[email protected] ISSN 2321-5984 A Publisher for Research Motivation........ Volume 3, Issue 4, April 2015 Using Multi-objective Genetic Algorithm Designing High-Pass IIR Filter Sonal Dwivedi1, Asst. Prof. Pankaj Shrivastav2 1 Electronics & communication Dept., MPCT, Gwalior 2 Electronics & communication Dept., MPCT ABSTRACT In this present work, IIR filter is designed by using multi-objective genetic algorithm in Matlab. The reason behind using multi-objective G.A is to find the optimal solution between two objectives that are contradicted of each other. Because of contradiction among two objectives it is difficult to find one optimal solution by simple GA function, so by implementing multiobjective GA it can be easily achieved. As many papers have been published on filters using GA, the following researches points only one parameter. But we have designed a filter, high-pass filter taking two parameters as objective cut-off frequency ( ) and damping ratio ( ) and obtained desired responses such as step and frequency response, poles-zeros and pareto solution. GA proves very helpful in determining the coefficients of high-pass filter also give solution of the pareto front. In this paper, during GA process number of iteration methods evolves to determine the optimal solution of optimal IIR high-pass filter. The modern genetic method is more relevant to provide fast and better response than any other. Keywords: IIR filter design, Genetic algorithm optimization, multi-objects. 1. INTRODUCTION In the field of digital filters, many researchers have done their excellent work as it is basic and needful research topic for the development of new techniques resulting advances in digital filter designing. Digital filters are basically to separate the required signal from the original signal in any pattern [1]. Today’s, DSP processing has become very much flexile, high achievement, fast and better outcome, low yield etc. Digital Filters can be FIR(Finite Impulse Response) filter also called Non-recursive filters or IIR(Infinite Impulse Response) filter also called Recursive filters, depends on the type of pulse response. Inspired by nature, Genetic algorithm is being developed which is widely in use today. It provide algorithm to make models worthiness. To solve any problem by using traditional formula method which has accuracy but have some limitations and these limitations are removed by assumptions, but assumptions make our model less rigorous. So to avoid this problem genetic algorithm is developed which provides better solution than the traditional methods. Consequently, there is requisite to develop techniques which can be used design filters satisfying refine specifications. Yet many papers have published on filter designing using genetic algorithm with better responses than traditional method [2]. In Last few years Paper published on optimization uses single objective fitness function. But we have used the multiple objective fitness function in this paper. The genetic algorithm operations possess population fitness or initialization, selection, cross over and mutation [3]. In this paper we are making IIR high-pass filter using multi-objective Genetic algorithm to get proper and better responses such as step response, frequency response, settling time, desired cut-off frequency and damping ratio by providing sampling frequency. 2. METHODS 2.1 Infinite Impulse Response Filter Design Digital filter is basically a Digital Signal Processing (DSP). Filters can be classified in various forms, depends on criteria for which it is required but the two major types of digital filters are FIR(Finite Impulse Response) and IIR(Infinite Impulse Response). FIR is filter whose impulse response is of finite duration as it does not have any feedback and also get settles to zero in finite duration. It is also known as non-recursive digital filter. Volume 3, Issue 4, April 2015 Page 8 IPASJ International Journal of Electronics & Communication (IIJEC) A Publisher for Research Motivation........ Volume 3, Issue 4, April 2015 Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm Email:
[email protected] ISSN 2321-5984 The infinite impulse response (IIR), they have feedback with infinite impulse responses which can also known as recursive digital filters. The feedback is a recursive part of filter. Where y(n) and x(n) are input and output of filter. And are coefficients of IIR filters. In order to build the structure of filter with desired parameters it is essential to convert the parameter into the filter coefficients. The well- known techniques which is generally used for designing FIR filters namely windowing technique, the frequency sampling and optimal filter design method. And IIR filter design depend on existing analog filters to obtain digital filters. The main purpose behind this designing of filter is to obtain the coefficient of filters. 2.2 GENETIC ALGORITHM Figure 1: Processing of Genetic Algorithm for Digital Designing Genetic algorithms are adaptive algorithms for finding the global optimum solution for an optimization problem. The genetic algorithm developed by Holland in the early seventies as computer programs that mimic the natural evolutionary process which is characterized by binary representation of individual solutions, simple problemindependent crossover and mutation operators, and a proportional selection rule.The G.A is a method for solving both constraint and unconstraint optimization problems that is based on natural selection,the process that derives biological evolution. The genetic algorithm continuously modifies a population of individual solutions genetic. At each step, the genetic algorithm picks individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over many generations, the population generates toward an optimal solution. We can use this genetic algorithm to solve different optimization problem that are not well suitable for standard level of optimization algorithm and also those problems in which the objective function is incoherent, non differentiable, imaginary, or highly impartial. The genetic algorithm can address problems of mixed integer programming, where some components are confined to be integer-valued. Volume 3, Issue 4, April 2015 Page 9 IPASJ International Journal of Electronics & Communication (IIJEC) A Publisher for Research Motivation........ Volume 3, Issue 4, April 2015 Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm Email:
[email protected] ISSN 2321-5984 The three steps of children for next generation created by genetic algorithm are given: Elite child is child with best fitness values in current generation and will proceed automatically to survive for next generation. Crossover children’s are made by combining two different parts of parent cell. Mutation children are basically generated by providing random changes in original parent. Those random changes can be any. 2.3 GAIIR (genetic algorithm infinite impulse response) Genetic algorithm optimization required a fitness function so it can test the population fitness and resulting to next generation evacuation. GA is the key parameter for formulating the fitness function. We represented the whole process of genetic algorithm in the above flow chart. Here we are using multi-objective genetic algorithm to find the optimal solution of our two objectives. First we will consider cut-off frequency and second will be damping ratio for our designing. In GA all genes in chromosomes will get involved in the search of coefficient during genetic operations. Consider the following second order transfer function for analog high pass filter. From this analog high-pass filter we can obtain the digital IIR filter by applying bilinear transform by selecting appropriate sampling frequency ( ) or sampling time. Here we are considering two variables i.e., coefficient of‘s’ and coefficient of ‘s0’. The order of filter which we are using is second order high- pass filter. If we want to build higher order filter it is also possible to build by using separate variables. But less number of variables provide fast response, better stability and low memory space. So one should keep less number of variables in order to get fast and better response. Our fitness value is given below: Where , are desired cut-off frequency and damping and are calculated parameters. MATLAB is high level- language which posses a pre-developed functions or tool boxes from where problems can easily be solved out. Out of many tool boxes matlab consist of pre-defined tool called genetic algorithm which can be used by coding or by direct execution on optimization application and makes the process easier. Proceeding further, we can see GA have two function format first is ‘ga’ and second is ‘gamulti’. Generally ‘ga’ is being used for single objective with best fitness and ‘gamulti’ is used for multi-objective with pareto front. According to number of objects we select which GA function is to be used. In this paper we are assuming two objective cases and the optimal solution is obtained by pareto font. The number of outputs will produce out of which best solution is selected by designer according to desired requirement. 3. RESULTS FOR GENETIC ALGORITM The output of ‘gamulti’ pareto front is shown below in figure (2), In which it can be easily seen that two objects are contradictory to each other, object 1 and object 2. So we have to compromise with either of the two solutions. Either cut-off frequency or step response. This output helps us to design IIR digital filter with multi-objective GA. This fitness test provides multiple optimal solutions after GA operation. Figure 2: Pareto Front for High-pass Filter Volume 3, Issue 4, April 2015 Page 10 IPASJ International Journal of Electronics & Communication (IIJEC) Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm Email:
[email protected] ISSN 2321-5984 A Publisher for Research Motivation........ Volume 3, Issue 4, April 2015 The table (1) shows the total number of outputs which we obtained after applying multi-objective GA to second order high-pass filter. It consist of total eighteen values. But we will select any one value of ‘b’ and ‘a’. We can choose any solution number according to which we obtain the best result. We can compromise with the object , taking object as major requirementor can also choose vice-versa due to the contradictory between two objectives. In figure (3) we have summarized the poles and zeros behavior of high-pass filter. It shows that the location of poleszeros for designed high-pass filter lies within unit circle. It signifies that designed filter is stable. Table 1- GA optimized coefficient and performance index S. No. FVAL a 17 4.05E+06 8.44E+06 1.42E+06 J2 2.54E+01 2.79E+01 2.74E+08 1.51E+01 1.51E+01 3.97E+00 3.97E+00 4.95E+05 3.97E+00 3.97E+00 3.97E+00 3.97E+00 4.95E+05 2.74E+08 3.97E+00 3.97E+00 1.51E+01 18 -42117.5 -25.3989 1 1 2 J1 4.21E+04 1.90E+04 3.26E-01 Value of Coefficients 3 4 5 6 7 1.42E+06 1.42E+06 4.05E+06 1.65E+07 3.26E-01 8 9 10 11 12 1.65E+07 5.32E+06 8.44E+06 5.32E+06 3.26E-01 13 3.26E-01 14 15 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.68391 1.68606 0.92499 1.66805 1.66805 1.61272 1.61272 0.92499 1.61272 1.61272 1.61272 1.61272 0.92499 0.92499 1.61272 1.61272 1.66805 1.68391 0.73133 1 0.73252 9 0.32715 2 0.72312 4 0.72312 4 0.72339 2 0.72340 9 0.32715 1 0.72340 9 0.72340 6 0.72340 8 0.72340 6 0.32715 1 0.32715 2 0.72339 2 0.72340 8 0.72312 4 0.73133 1 b 0.85381 1 0.85464 8 0.56303 6 0.84779 4 0.84779 4 0.83402 8 0.83403 2 0.56303 4 0.83403 2 0.83403 1 0.83403 2 0.83403 1 0.56303 4 0.56303 6 0.83402 8 0.83403 2 0.84779 4 0.85381 1 1.70762 -1.7093 1.12607 1.69559 1.69559 1.66806 1.66806 1.12607 1.66806 1.66806 1.66806 1.66806 1.12607 1.12607 1.66806 1.66806 1.69559 1.70762 0.85381 1 0.85464 8 0.56303 6 0.84779 4 0.84779 4 0.83402 8 0.83403 2 0.56303 4 0.83403 2 0.83403 1 0.83403 2 0.83403 1 0.56303 4 0.56303 6 0.83402 8 0.83403 2 0.84779 4 0.85381 1 We set the sampling frequency of 4Hz and cutoff frequency at 1Hz with absolute gain 0.92 and damping is to 0.707. Designed filter satisfied both the conditions and gives the optimal results as required, as shown in figure (2). Volume 3, Issue 4, April 2015 Page 11 IPASJ International Journal of Electronics & Communication (IIJEC) A Publisher for Research Motivation........ Volume 3, Issue 4, April 2015 Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm Email:
[email protected] ISSN 2321-5984 Figure 3: For all eighteen solutions the Pole zero map and step response given separately using GA. Figure 4: Frequency Response of all Optimal design (i.e., for all eighteen) 4. CONCLUSION The problem arises while taking multi objects separately, performing genetic algorithm and combining both the results then both the outcomes are found contradict of each other. As if we minimize one object other get exceeds or viceversa. So to solve this contradictory we use multi-objective genetic algorithm. In this paper, we have designed highpass infinite impulse response filter with multi-objective genetic algorithm in MATLAB programming language version (R20014a). It is observed that the optimal PARETO solution is achieved also give satisfactory results in response.Traditional approaches follow the direct search method or the formulae based methods. In direct search method there is very difficult to reach to the optimum point of fitness function if there is infinite search space and the traditional formulae based method used assumptions for finding optimum solution which may or may not be optimum. Genetic algorithm we use to design a high-pass filter but it can also be used for other type of filters and also for different topologies. In future by using the constraints (linear or non-linear) we can find the best response for the specific type of filter’s configuration. Volume 3, Issue 4, April 2015 Page 12 IPASJ International Journal of Electronics & Communication (IIJEC) A Publisher for Research Motivation........ Volume 3, Issue 4, April 2015 Web Site: http://www.ipasj.org/IIJEC/IIJEC.htm Email:
[email protected] ISSN 2321-5984 REFERENCES [1]. S. U. Ahmad and A. Antoniou, “Design of Digital Filters Using Genetic Algorithms,” IEEE Transaction on Signal Processing, Vol.1, No. 1, 2006, pp. 1-9. [2]. Ranjeet Singh, Sandeep k. Araya. ” Genetic Algorithm for the Design of Optimal IIR Digital Filters”, journal of Signal and Information Processing, 2012, 3, 286-292 [3]. D. E. Goldberg, “Genetic Algorithm in Search, Optimization and Machine Learning, Pearson Education,” Low Price Edition, Delhi, 2005. [4]. V. K. Ingle and J. G. Proakis, “Digital Signal Processing Using MATLAB,” Thomson Books, New Delhi, 2004. [5]. Karaboga, Nurhan and Bahadir Cetinkaya, (2004), “Performance comparison of Genetic Algorithm based design methods of digital filters with optimal magnitude response and minimum phase”, IEEE. PP. 644-647 [6]. Karaboga, Nurhan and Bahadir Cetinkaya, (2004), “Optimal design of minimum phase digital FIR filters by using Genetic Algorithm”, IEEE. PP. 24-28. [7]. A. Antoniou, “Digital Filters Analysis, Design and Ap-plication,” Tata Mcgraw-Hill Edition, New Delhi, 2005. [8]. F. Argenti and E. Del Re, “Design of IIR Eigen Filters in the Frequency Domain,” IEEE Transactions on Signal processing, Vol. 46, No. 6, 1998, pp. 1694-1700. doi:10.1109/78.678495 [9]. Rosa, Vagner S., Fábio F. Daitx, Eduardo Costa, and Sergio Bampi, (2009), “Design Flow for the Generation of Optimized FIR Filters”, IEEE, p. 1000-1005. [10]. Rao, Nagaraja S., M. N. Giri Prasad and Manoj Kumar Singh, (2009), “The robust design of linear phase FIR filter using mex-mutation evolutionary programming”. ARPN Journal of Engineering and Applied Sciences. VOL. 4, NO. 4, pp. 102-108 Volume 3, Issue 4, April 2015 Page 13