FIR Filter Design using Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach
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FIR Filter Design using Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach

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This paper presents an alternative approach for the ...

This paper presents an alternative approach for the
design of linear phase digital low pass FIR filter using Particle
Swarm Optimization with Constriction Factor and Inertia
Weight Approach (PSO-CFIWA). FIR filter design is a multimodal
optimization problem. The conventional gradient based
optimization techniques are not efficient for digital filter
design. Given the filter specification to be realized, PSO
algorithm generates a set of filter coefficients and tries to
meet the ideal frequency characteristic. In this paper, for the
given problem, the realization of the FIR filters of different
order has been performed. The simulation results have been
compared with the well accepted evolutionary algorithm such
as genetic algorithm (GA). The results justify that the proposed
filter design approach using PSO-CFIWA outperforms to that
of GA, not only in the accuracy of the designed filter but also
in the convergence speed and solution quality.

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    FIR Filter Design using Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach FIR Filter Design using Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach Document Transcript

    • ACEEE Int. J. on Electrical and Power Engineering, Vol. 02, No. 02, August 2011 FIR Filter Design using Particle Swarm Optimization with Constriction Factor and Inertia Weight Approach 1 Rajib Kar, 1Durbadal Mandal, 1Dibbendu Roy, 2Sakti Prasad Ghoshal 1 Department of ECE, National Institute of Technology, Durgapur, India rajibkarece@gmail.com, durbadal.bittu@gmail.com , dibbaroy@gmail.com 2 Department of EE, National Institute of Technology, Durgapur, India spghoshalnitdgp@gmail.comAbstract— This paper presents an alternative approach for the program have to be iterated many times [4]. Different heuristicdesign of linear phase digital low pass FIR filter using Particle optimization algorithms such as genetic algorithm (GA),Swarm Optimization with Constriction Factor and Inertia simulated annealing algorithms etc. have been widely usedWeight Approach (PSO-CFIWA). FIR filter design is a multi- for the optimal design of digital filters. When consideringmodal optimization problem. The conventional gradient based global optimization methods for digital filter design, the GAoptimization techniques are not efficient for digital filter seems to have attracted considerable attention. Filtersdesign. Given the filter specification to be realized, PSOalgorithm generates a set of filter coefficients and tries to designed by GA have the potential of obtaining near globalmeet the ideal frequency characteristic. In this paper, for the optimum solution [5-6]. Although standard Gas (hereingiven problem, the realization of the FIR filters of different referred to as Real Coded GA (RGA)) have a good performanceorder has been performed. The simulation results have been for finding the promising regions of the search space, theycompared with the well accepted evolutionary algorithm such are inefficient in determining the local minimum in terms ofas genetic algorithm (GA). The results justify that the proposed convergence speed and solution quality. Particle Swarmfilter design approach using PSO-CFIWA outperforms to that Optimization (PSO) is an evolutionary algorithm developedof GA, not only in the accuracy of the designed filter but also by Kennedy and Eberhart in 1995 [8-9]. Several attempts havein the convergence speed and solution quality. been made towards the optimization of the FIR Filter [14] andIndex Terms— FIR Filter; PSO; GA; Optimization; Magnitude in other areas [10] also using PSO algorithm. The PSO isResponse; Convergence; Low Pass Filter simple to implement and its convergence may be controlled via few parameters. This paper describes the FIR digital filter I. INTRODUCTION design using the PSO with constriction factor and inertia weight approach (PSO-CFIWA). PSO-CFIWA algorithm tries A digital filter is a system that performs mathematical to find best coefficients that closely match the ideal frequencyoperations on a sampled, discrete-time signal to reduce or response. The rest of the paper is arranged as follows. Inenhance certain aspects of that signal. This is in contrast to section 2, the filter design problem is formulated. Section 3the other major type of electronic filter, the analog filter, which briefly discusses on the real coded GA (RGA). Section 4 showsis an electronic circuit operating on continuous-time analog the algorithms of GA, general PSO and PSO-CFIWA. Sectionsignals. There are two major classes of digital filters namely, 5 describes the simulation result. Finally section 6 concludesfinite impulse response (FIR) filters and infinite impulse the paper.response (IIR) filters depending on the length of the impulseresponse [7]. FIR filter is an attractive choice because of the II. FILTER DESIGNease in design and stability. By designing the filter taps to besymmetrical about the center tap position, a FIR filter can be A digital FIR filter is characterized byguaranteed to have linear phase. Finite impulse response(FIR) digital filters are known to have many desirable featuressuch as guaranteed stability, the possibility of exact linearphase characteristic at all frequencies and digital We assume that hn   0 and h0   0implementation as non-recursive structures. Linear phase FIRfilters are also required when time domain specifications are Where, N is the order of the filter which has N+1 number ofgiven [1]. The most frequently used method for the design of coefficients. h(n) is the filter impulse response. It is calcu-exact linear phase weighted Chebyshev FIR digital filter is lated by applying an impulse signal at the input. The value ofthe one based on the Remez-exchange algorithm proposed h(n) will determine the type of the filter e.g. low pass, highby Parks and McClellan [2]. Further improvements to their pass, band pass etc. The value of h(n) is to be determined inresults have been reported in [3]. The main limitation of this the design process and N represents the order of the polyno-procedure is that the relative values of the amplitude error in mial function. This paper presents the most widely used FIRthe frequency bands are specified by means of the weighting that h(n) is odd symmetric and the order is even. The lengthfunction, and not by the deviations themselves. Therefore, of h(n) is N+1 and the number of coefficients is also N+1. Thein case of designing low-pass filters with a given stop band individual represents h(n) . In each iteration, these individu-deviation, given filter length and cutoff frequencies, the als are updated. Fitness of particles is calculated using the new coefficients. This fitness is used to improve the search 1© 2011 ACEEEDOI: 01.IJEPE.02.02. 162
    • ACEEE Int. J. on Electrical and Power Engineering, Vol. 02, No. 02, August 2011in each iteration, and result obtained after a certain number values from the minimum value.of iterations or after the error is below a certain limit is consid-  Copying of the elite strings over the non-selectedered to be the final result. Because its coefficients are matched, strings.the dimension of the problem reduces by a factor of 2. The  Crossover and mutation to generate off-springs.(N+1)/2 coefficients are then flipped and concatenated to find  Genetic cycle updating.the required N+1 coefficient. The least square (LS) error is  The iteration stops when the maximum number ofused to evaluate the individual. It takes the squared error cycles is reached. The grand minimum Error and itsbetween the frequency response of the ideal and the actual corresponding chromosome string or the desiredfilter. An ideal filter has a magnitude of 1 on the pass band and solution are finally obtained.a magnitude of 0 on the stop band. So the error for this fitnessfunction is the squared difference between the magnitudes of IV. PARTICLE SWARM OPTIMIZATION (PSO)this filter and the filter designed using the evolutionary algo- PSO is a flexible, robust population-based stochasticrithms. The individuals that have higher evaluation values search or optimization technique with implicit parallelism,represent the better filters, the filters with better frequency which can easily handle with non-differential objectiveresponse. The frequency response of the FIR digital filter can functions, unlike traditional optimization methods. PSO isbe calculated as, less susceptible to getting trapped on local optima unlike GA, Simulated Annealing etc. Eberhart and Shi [9] developed PSO concept similar to the behavior of a swarm of birds. PSO is developed through simulation of bird flocking inWhere, w k  2k N   ; H e jw k is the Fourier transform complex multidimensional space. Bird flocking optimizes a certainvector. This is the FIR filter frequency response. objective function. Each agent knows its best value so farThe expression of the LS function is given below: (pbest). This information corresponds to personal experiences of each agent. Moreover, each agent knows the best value so far in the group (gbest) among pbests. Namely, each agent tries to modify its position using the following information:  The distance between the current position andWhere, H i represents the ideal magnitude response of the pbest.filter and is given as  The distance between the current position and gbest. Mathematically, velocities of the particles are modified according to the following equation [11]:  H d e j k represents the filter to be designed, K is the num-ber of samples. Equation (2) represents the fitness functionto be minimized using evolutionary algorithm. III. REAL CODED GENETIC ALGORITHM (RGA) where Vi k is the velocity of agent i at iteration k ; w is the weighting function; Cj is the weighting factor; randi is the GA is mainly a probabilistic search technique, based onthe principles of natural selection and evolution. At each random number between 0 and 1; S ik is the current positiongeneration it maintains a population of individuals where of agent i at iteration k; pbestik is the pbest of agent i; gbest keach individual is a coded form of a possible solution of theproblem at hand called chromosome. Chromosomes are is the gbest of the group. The searching point in the solutionconstructed over some particular alphabet, e.g., the binary space can be modified by the following equation:alphabet {0, 1}, so that chromosomes’ values are uniquelymapped onto the decision variable domain. Each chromosomeis evaluated by a function known as fitness function, which The first term of (4) is the previous velocity of the agent. Theis usually the fitness function or the objective function of second and third terms are used to change the velocity of thethe corresponding optimization problem. Steps of RGA as agent. Without the second and third terms, the agent willimplemented for optimization of spacing between the elements keep on ‘‘flying’’ in the same direction until it hits theand current excitations are [10, 11]: boundary. w, corresponds to a kind of inertia and tries to  Initialization of real chromosome strings of n p explore new areas. For Particle Swarm Optimization with population, each consisting of a set of excitations. Constriction Factor and Inertia Weight Approach Size of the set depends on the number of excitation (PSOCFIWA) [12, 13], the velocity of (4) is manipulated in elements in a particular array design. accordance with (6).  Decoding of strings and evaluation of Error of each string.  Selection of elite strings in order of increasing Error 2© 2011 ACEEEDOI: 01.IJEPE.02.02.162
    • ACEEE Int. J. on Electrical and Power Engineering, Vol. 02, No. 02, August 2011Normally, C1=C2=1.5-2.05 and Constriction Factor CFa  isgiven in (7).Where = C1 + C2 , and  >4.For==2.05, the computed value of = 0.73.The best values of C1, C2, and are found to vary with thedesign sets. In inertia weight approach (IWA), inertia weightat (k+1)th cycle is as given in (8).Where, wmax =1.0; wmin =0.4; k max = Maximum number ofiteration cycles. The solution updating is the same as (5). V. RESULTS AND DISCUSSIONSA. Analysis of Magnitude response of low-pass FIR filters The MATLAB simulation has been performed extensivelyto realize the low pass FIR filter with the order of 20 and 30.Hence the length of the filter coefficients is 21 and 31,respectively. The sampling frequency was chosen as fs = 1Hz.Also, for all the simulations the sampling number was takenas 128. TABLE I GA PARAMETERS The control parameter values GA used in this work aregiven in Table I. Pass band normalized cut off frequency is0.4. Algorithms are run for 25 times. The best possible sets ofcoefficients for the designed FIR filter for order 20 and 30have been shown in Table II and Table III, respectively. TABLE II. OPTIMIZED C OEFFICIENTS OF FIR FILTER OF ORDER 20 Fig. 1 Magnitude response of 20 th order low-pass FIR filters 3© 2011 ACEEEDOI: 01.IJEPE.02.02.162
    • ACEEE Int. J. on Electrical and Power Engineering, Vol. 02, No. 02, August 2011 TABLE III. OPTIMIZED COEFFICIENTS OF FIR FILTER OF ORDER 30As seen from Figure 1, for the pass band region, the new PSO(PSO-CFIWA) produces a better response than that of GA.The filters designed by the PSO algorithm have sharpertransition band responses than that produced by GAalgorithm. For the stop band region, the filters designed bythe PSO-CFIWA method produce better responses than theothers. The best optimized coefficients obtained for thedesigned filter with the order of 20 and 30 have been calculatedby the two methods and given in Table II and III, respectively.B. Comparative effectiveness and convergence profiles ofRGA and PSO-CFIWA In order to compare the algorithms in terms of theconvergence speed, Figure 3 shows the evolution of bestsolutions obtained when GA is employed. Figure 4 showsthe evolution of best solutions obtained when the new PSOis employed. The convergence graph has been shown forthe filter order of 30. A similar plot can be obtained for the FIRfilter of order 20. From the figures drawn for this filter, it isseen that the PSOCFIWA algorithm is significantly fasterthan the GA algorithm for finding the optimum filter. The newPSO converges to a much lower fitness in lesser number ofiterations. The minimum Error values are plotted against thenumber of iteration cycles to get the convergence profilesfor the optimization techniques. Figs. 3-4 show the Fig. 2 Magnitude response of 30 th order low-pass FIR filtersconvergence profiles for RGA and PSOCFIWA for 30th orderlow-pass FIR filters respectively. RGA and PSOCFIWAconverge to their respective minimum ripple magnitude inless than 500 iterations. Further RGA yields suboptimal highervalues of Error but PSOCFIWA yields near optimal (least)Error values consistently in both cases. With a view to theabove fact, it may finally be inferred that the performance ofPSOCFIWA technique is better as compared to RGA. Alloptimization programs are written in MATLAB 7.5 versionon core (TM) 2 duo processor, 3.00 GHz with 2 GB RAM. Fig. 3. Convergence profile for RGA in case of 30 th order low-pass FIR filters. 4© 2011 ACEEEDOI: 01.IJEPE.02.02.162
    • ACEEE Int. J. on Electrical and Power Engineering, Vol. 02, No. 02, August 2011 [5] S. Chen, “IIR Model Identification Using Batch-Recursive Adaptive Simulated Annealing Algorithm,” In Proceedings of 6th Annual Chinese Automation and Computer Science Conference, (2000), pp. 151–155. [6] N.E. Mastorakis, I.F. Gonos, M.N.S. Swamy, “Design of Two Dimensional Recursive Filters Using Genetic Algorithms,” IEEE Transaction on Circuits and Systems I-Fundamental Theory and Applications, 50 (2003), pp. 634–639. [7] L. Litwin, “FIR and IIR digital filters,” IEEE Potential, 0278- 6648, 2000, 28–31. [8] J. Kennedy, R. Eberhart, “Particle Swarm Optimization,” IEEE int. Conf. On Neural Network, 1995. [9] R. Eberhart, Y. Shi, “Comparison between Genetic Algorithms Fig. 4. Convergence profile for PSOCFIWA in case of 30 th order and Particle Swarm Optimization”, Proc. 7 th Ann. Conf. on low-pass FIR filters Evolutionary Computation, San Diego, 2000. [10] D. Mandal, S. P. Ghoshal, and A. K. Bhattacharjee, “A Novel VI. CONCLUSIONS Particle Swarm Optimization Based Optimal Design of Three-Ring Concentric Circular Antenna Array,” IEEE International Conference This paper presents an alternative approach for FIR filter on Advances in Computing, Control, and Telecommunicationdeign using PSOCFIWA. Filters of orders 20 and 30 have Technologies, 2009. (ACT’09), pp. 385-389, 2009.been realized using GA as well as PSOCFIWA. Extensive [11] D. Mandal, S. P. Ghoshal, and A. K. Bhattacharjee,simulation results justify that the proposed algorithm “Application of Evolutionary Optimization Techniques for Findingoutperforms GA in the accuracy of the magnitude response the Optimal set of Concentric Circular Antenna Array,” Expert Systems with Applications, (Elsevier), vol. 38, pp. 2942-2950, 2010.of the filter as well as in the convergence speed. [12] D. Mandal, S. P. Ghoshal, and A. K. Bhattacharjee, “Comparative Optimal Designs of Non-uniformly Excited REFERENCES Concentric Circular Antenna Array Using Evolutionary[1] T.W. Parks, C.S. Burrus, Digital Filter Design, Wiley, New Optimization Techniques,” IEEE Second International ConferenceYork, 1987. on Emerging Trends in Engineering and Technology, ICETET’09[2] T.W. Parks, J.H. McClellan, “Chebyshev approximation for (2009), 619-624.non-recursive digital filters with linear phase,” IEEE Trans. Circuits [13] D. Mandal, S. P. Ghoshal, and A. K. Bhattacharjee, “DesignTheory CT-19 (1972), pp. 189–194. of Concentric Circular Antenna Array With Central Element Feeding[3] J.H. McClellan, T.W. Parks, L.R. Rabiner, “A computer program Using Particle Swarm Optimization With Constriction Factor andfor designing optimum FIR linear phase digital filters,” IEEE Trans. Inertia Weight Approach and Evolutionary ProgrammingAudio Electroacoust., AU-21 (1973) 506–526. Technique,” Journal of Infrared Milli Terahz Waves, (Springer)[4] L.R. Rabiner, “Approximate design relationships for low-pass vol. 31 (6), p. 667–680, 2010.FIR digital filters,” IEEE Trans. Audio Electroacoust, AU-21 (1973), [14] J. I. Ababneh, M.H. Bataineh. “Linear phase FIR filter designpp. 456–460. using particle swarm optimization and genetic algorithm”. Elsevier, Digital Signal Processing, 2006. 5© 2011 ACEEEDOI: 01.IJEPE.02.02.162