Swarm and Evolutionary Computation 7 (2012) 58–64

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Swarm and Evolutionary Computation

2210-65

http://d

n Corr

E-m

durbada

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journal homepage: www.elsevier.com/locate/swevo

Regular Paper

Craziness based Particle Swarm Optimization algorithm forFIR band stop filter design

Rajib Kar a, Durbadal Mandal a,n, Sangeeta Mondal b, Sakti Prasad Ghoshal b

a Department of Electronics & Communication Engineering, National Institute of Technology, Durgapur, Indiab Department of Electrical Engineering, National Institute of Technology, Durgapur, India

a r t i c l e i n f o

Article history:

Received 22 June 2011

Received in revised form

12 March 2012

Accepted 12 May 2012Available online 15 June 2012

Keywords:

FIR band stop filter

RGA

PSO

CLPSO

CRPSO

Parks and McClellan (PM) Algorithm

02/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.swevo.2012.05.002

esponding author. Tel.: þ91 9434788059.

ail addresses: rajibkarece@gmail.com (R. Kar)

l.bittu@gmail.com (D. Mandal), sangeeta.aas

alnitdgp@gmail.com (S.P. Ghoshal).

a b s t r a c t

In this paper, an improved particle swarm optimization technique called Craziness based Particle

Swarm Optimization (CRPSO) is proposed and employed for digital finite impulse response (FIR) band

stop filter design. The design of FIR filter is generally nonlinear and multimodal. Hence gradient based

classical optimization methods are not suitable for digital filter design due to sub-optimality problem.

So, global optimization techniques are required to avoid local minima problem. Several heuristic

approaches are available in the literatures. The Particle Swarm Optimization (PSO) algorithm is a

heuristic approach with two main advantages: it has fast convergence, and it uses only a few control

parameters. But the performance of PSO depends on its parameters and may be influenced by

premature convergence and stagnation problem. To overcome these problems the PSO algorithm has

been modified in this paper and is used for FIR filter design. In birds’ flocking or fish schooling, a bird or

a fish often changes directions suddenly. This is described by using a ‘‘craziness’’ factor and is modelled

in the technique by using a craziness variable. A craziness operator is introduced in the proposed

technique to ensure that the particle would have a predefined craziness probability to maintain the

diversity of the particles. The algorithm’s performance is studied with the comparison of real coded

genetic algorithm (RGA), conventional PSO, comprehensive learning particle swarm optimization

(CLPSO) and Parks and McClellan (PM) Algorithm. The simulation results show that the CRPSO is

superior or comparable to the other algorithms for the employed examples and can be efficiently used

for FIR filter design.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

A digital filter is simply a discrete time, discrete amplitudeconvolver. Filtering is, in essence, the multiplication of the signalspectrum by the frequency domain impulse response filter, e.g. for anideal band stop filter the pass band part of the signal spectrum ismultiplied by one and the stop band part of the signal is by zero.Digital filters are basic building blocks in many digital signal proces-sing systems. They have wide range of applications in communica-tion, image processing, pattern recognition, etc. There are two majorclasses of digital filters, namely, finite impulse response (FIR) filtersand infinite impulse response (IIR) filters depending on the length ofthe impulse response [1]. FIR filter is an attractive choice becauseof the ease in design and stability. By designing the filter taps to besymmetrical about the centre tap position, a FIR filter can beguaranteed to have linear phase. FIR filters are known to have many

ll rights reserved.

,

@gmail.com (S. Mondal),

desirable features such as guaranteed stability, the possibility of exactlinear phase characteristic at all frequencies and digital implementa-tion as non-recursive structures. Linear phase FIR filters are alsorequired when time domain features are specified [2]. The mostfrequently used method for the design of exact linear phase weightedChebyshev FIR digital filter is the one based on the Remez-exchangealgorithm proposed by Parks and McClellan [3] which is popularlyknown as PM algorithm. Further improvements in their results havebeen reported in [4]. The main limitation of this procedure is that therelative values of the amplitude error in the frequency bands arespecified by means of the weighting function, and not by thedeviations themselves. Therefore, in case of designing band stopfilters with a given stop band deviation, filter length and cut-offfrequency, the programme has to be iterated many times [5]. Anumber of models have been developed for the FIR filter techniquesand design optimization methods. This is a thrust research area,aiming at obtaining more general and innovative techniques that areable to solve or optimize new and complex engineering problems [6].In some cases, such initiatives were successful and proven to exhibitbetter performance than the conventional approaches. However,there are few drawbacks associated to these methods, e.g., increased

R. Kar et al. / Swarm and Evolutionary Computation 7 (2012) 58–64 59

computational cost and non-existence of theoretical proof of conver-gence to global optimum in sufficiently general conditions.

The conventional gradient based optimization method [7] is notsufficient to optimize the multimodal and non-uniform objectivefunction, and it cannot converge to the global minimum solution. Theclassical gradient based optimization methods are not suitable forFIR filter optimization because of the following reasons: (i) highlysensitive to starting points when the number of solution variablesand hence the size of the solution space increase, (ii) frequentconvergence to local optimum solution or divergence or revisitingthe same suboptimal solution, (iii) requirement of continuous anddifferentiable objective cost function (gradient search methods),(iv) requirement of the piecewise linear cost approximation (linearprogramming), and (v) problem of convergence and algorithm com-plexity (nonlinear programming).

So, evolutionary methods have been implemented in the design ofoptimal digital filters with better control of parameters and thehighest stop band attenuation. Different heuristic optimization tech-niques are reported in the literatures. When considering globaloptimization methods for digital filter design, the GA [8–10] seemsto have attracted considerable attention. Filters designed by GA havethe potential of obtaining near global optimum solution. Althoughstandard GA (also known as Real Coded GA (RGA)) shows a goodperformance for finding the promising regions of the search space,they are inefficient in determining the global optimum in terms ofconvergence speed and solution quality. In order to overcome theproblems associated with RGA, orthogonal genetic algorithm (OGA)[11], hybrid Taguchi GA (TGA) [12] have been proposed. Tabu search[13], Simulated Annealing (SA) [14], Bee Colony Algorithm (BCA) [15],Differential Evolution (DE) [16,17], Differential cultural algorithm[18], Particle Swarm Optimization (PSO) [19–21], some variants ofPSO like Quantum PSO (QPSO) [22], PSO with Quantum Infusion(PSO-QI) [23,24], Adaptive inertia weight PSO [25], Chaotic mutationPSO (CM PSO) [26,27], some hybrid algorithms like, DE-PSO [28,29]have also been used for the filter design problem.

Most of the above algorithms show the problems of prematureconvergence, stagnation and revisiting of the same solution overand again. In order to get rid of these problems and to maintainthe diversity of the particles and moreover, because in birds’flocking or fish schooling, a bird or a fish often changes directionssuddenly, the authors, in this paper have modified the conven-tional PSO by introducing an entirely new velocity expressionassociated with many random numbers and a ‘‘craziness velocity’’having a predefined probability of craziness.

This paper describes an alternative technique for the FIR bandstop (BP) digital filter design using Craziness based Particle SwarmOptimization Technique (CRPSO). CRPSO technique tries to find thebest coefficients that closely match the ideal frequency response.Based upon the improved PSO approach, this paper presents a good,comprehensive set of results, and states arguments for the superiorityof the algorithm. Simulation results demonstrate the effectivenessand better performance of the proposed designed method.

The rest of the paper is arranged as follows. In Section 2, the FIRfilter design problem is formulated. Section 3 briefly discusses on thealgorithms of real coded genetic algorithm (RGA), classical PSO,CLPSO, and the CRPSO techniques. Section 4 describes the simulationresults obtained for FIR BP filter using PM algorithm, RGA, classicalPSO, CLPSO and the proposed CRPSO. Finally, Section 5 concludesthe paper.

2. Problem formulation

A digital FIR filter is characterized by

HðzÞ ¼XN

n ¼ 0

hðnÞz�n, ð1Þ

where N is the order of the filter which has (Nþ1) number of filterimpulse response coefficients h(n). The values of h(n) will deter-mine the type of the filter e.g. low pass, high pass, band pass, etc.and are to be determined in the design process. This paperpresents optimal design of an even order linear phase FIR bandstop (BS) filter with positive symmetric h(n). The number ofcoefficients h(n) is (Nþ1). Because the coefficients h(n) aresymmetrical, the dimension of the problem is halved. The(Nþ1)/2 coefficients are then flipped and concatenated to findthe required (Nþ1) number of coefficients. The optimizationalgorithm attains the minimum error between the desired fre-quency response and the actual frequency response by determin-ing the optimal h(n) values after a certain maximum number ofiterations. The optimal h(n) values, after concatenation, finallyrepresent the filter with optimal frequency response.

Various filter parameters which are responsible for the optimalfilter design are stop band and pass band normalized edgefrequencies (op, os), pass band and stop band ripples (dp and ds),stop band attenuation and transition width. These parameters aremainly decided by the filter coefficients [29]. In this paper, theproposed CRPSO and other existing techniques (for the sake ofcomparison) as mentioned earlier are individually applied in orderto obtain the desired filter response as close as possible to the idealresponse, where dp, ds, N, op, os are individually specified.

Now for (1), each filter coefficient particle vector is {h0, h1 y hN}.The particle vectors are distributed in a D-dimensional search space,where D¼(Nþ1) for Nth order FIR filter. The frequency response ofthe FIR digital filter can be calculated as

Hðejwk Þ ¼XN

n ¼ 0

hðnÞe�jwkn, ð2Þ

where Hðejwk Þ is the Fourier transform complex vector. This is theFIR filter frequency response. The frequency is sampled in [0, p] withM sampling points; the position of each particle vector inD-dimensional search space represents the same coefficients h(n)of the transfer function (1).

In this paper, the authors have adopted a new error fitnessfunction in order to achieve higher stop band attenuation and tohave an accurate control on the transition width. The error fitnessfunction J used in this paper is given in (3). Using (3), it is foundthat the proposed filter design approach results in considerableimprovement over PM and other optimization techniques:

J¼X

abs absð HðoÞ�� ���1Þ�dp

� �þX

absð HðoÞ�� ���dsÞ

� �ð3Þ

where abs or, 99 indicates the absolute value.

For the first term of (3), oApass band including a portion ofthe transition band and for the second term of (3), oAstop bandincluding a portion of the transition band. The error fitnessfunction given in (3) represents the generalized fitness functionto be minimized using the evolutionary algorithms RGA, conven-tional particle swarm optimization (PSO), CLPSO and CRPSOindividually. Each algorithm tries to minimize this error fitness J

and thus improves the filter performance. Unlike other errorfitness functions as given in [16,19,21,24] which consider onlythe maximum errors, J involves summation of all absolute errorsfor the whole frequency band, and it is experienced that mini-mization of J yields higher stop band attenuation and lesser passband ripples. Transition width is affected a little. Since thecoefficients of the linear phase filter are matched, the dimensionof the problem is thus reduced by a factor of 2. By onlydetermining half of the coefficients, the filter can be designed.This greatly reduces the computational burdens of the algorithms,applied to the design of linear phase FIR filters.

R. Kar et al. / Swarm and Evolutionary Computation 7 (2012) 58–6460

3. Evolutionary techniques employed

3.1. Real coded genetic algorithm (RGA)

Real coded genetic algorithm (RGA) is mainly a probabilisticsearch technique, based on the principles of natural selection andevolution. At each generation, it maintains a population of vectorswhere each vector is a coded form of a possible solution of theproblem at hand called chromosome. Chromosomes are con-structed over some particular alphabet, e.g., the binary alphabet{0, 1}, so that chromosomes’ values are uniquely mapped onto thereal decision variable domain. Each chromosome is evaluated by afunction known as fitness function, which is usually the objectivefunction of the corresponding optimization problem [35].

The basic steps of RGA are shown in Table 1.

3.2. Particle Swarm Optimization (PSO)

PSO is a flexible, robust population-based stochastic search/optimization technique with implicit parallelism, which caneasily handle with non-differential objective functions, unliketraditional optimization methods. PSO is less susceptible togetting trapped on local optima unlike GA, Simulated Annealing,etc. Eberhart et al. [30,31] developed PSO concept similar to thebehaviour of a swarm of birds. PSO is developed through simula-tion of bird flocking in multidimensional space. Bird flockingoptimizes a certain objective function. Each particle (bird) knowsits best value so far (pbest). This information corresponds topersonal experiences of each particle. Moreover, each particleknows the best value so far in the group (gbest) among pbests.Namely, each particle tries to modify its position using thefollowing information:

�

TabSte

In

T

The distance between the current position and the pbest.

� The distance between the current position and the gbest.

Similar to GA, in PSO techniques also, real-coded particlevectors of population np are assumed. Each particle vector con-sists of components or sub-strings as required number of normal-ized filter coefficients, depending on the order of the filter to bedesigned.

Mathematically, velocities of the particle vectors are modifiedas follows:

V ðkþ1Þi ¼wnV ðkÞi þC1nrand1nðpbestðkÞi �SðkÞi ÞþC2nrand2nðgbestðkÞ�SðkÞi Þ

ð4Þ

where V kð Þi is the velocity of ith particle at kth iteration; w is the

weighting function; C1 and C2 are the positive weighting factors;rand1and rand2 are the random numbers between 0 and 1; S kð Þ

i isthe current position of ith particle vector at kth iteration; pbest kð Þ

i

is the personal best of ith particle vector at kth iteration; gbest(k) is

le 1ps for RGA.

RGA, initialize the real chromosome string vectors of np population, each consisting

(þ , �1, respectively) for positive symmetric linear phase even Nth (N¼20 or 28 or

used for the implementation of FIR BS filter are as follows:

Step 1. Initialization.

Step 2.Decoding the strings and evaluation of error fitness value of each string.

Step 3. Selection of elite strings in order of increasing error fitness values from the

Step 4. Copying the elite strings over the non-selected strings.

Step 5. Crossover and mutation generate the off-springs.

Step 6. Genetic cycle updating.

he iteration stops when the maximum number of genetic cycles is reached. The gra

optimal (N/2þ1)h(n) coefficients and finally, (Nþ1) number of optimal filter coeffic

of the filter.

the group best of the group at kth iteration. The searching point inthe solution space may be modified by

Sðkþ1Þi ¼ SðkÞi þV ðkþ1Þ

i ð5Þ

The first term of (4) is the previous velocity of the particlevector. The second and third terms are used to change the velocityof the particle. Without the second and third terms, the particlewill keep on ‘‘flying’’ in the same direction until it hits theboundary. Namely, it corresponds to a kind of inertia representedby the inertia constant, w and tries to explore new areas.

3.3. Comprehensive Learning Particle Swarm Optimization (CLPSO)

Steps of CLPSO as implemented for optimization of h(n)coefficients are adopted from [36]. In this work, initialization ofreal chromosome string vectors of np population, each consistingof a set of h(n) coefficients is made. Size of the set depends on thenumber of coefficients in a particular filter design. The same setsof parameters are used as mentioned in [36].

3.4. Craziness based Particle Swarm Optimization (CRPSO)

In order to get rid of the limitations of classical PSO [32–33]already mentioned and because in birds’ flocking or fish school-ing, a bird or a fish often changes directions suddenly, the authorshave modified the conventional PSO by introducing an entirelynew velocity expression (6) associated with many random num-bers and a ‘‘craziness velocity’’ having a predefined probability ofcraziness. This modified PSO is termed as CRPSO.

The velocity in this case can be expressed as follows [34]:

V ðkþ1Þi ¼ r2nsignðr3ÞnV ðkÞi þð1�r2ÞnC1nr1nfpbestðkÞi �SðkÞi g

þð1�r2ÞnC2nð1�r1ÞnfgbestðkÞ�SðkÞi g ð6Þ

where r1, r2 and r3are the random parameters uniformly takenfrom the interval [0,1] and sign(r3)is a function defined as

signðr3Þ ¼�1 where r3r0:05

¼ 1 where r340:05ð7Þ

The two random parameters rand1 and rand2 of (4) areindependent. If both are large, both the personal and socialexperiences are over used and the particle is driven too far awayfrom the local optimum. If both are small, both the personal andsocial experiences are not used fully and the convergence speed ofthe optimization technique is reduced. So, instead of takingindependent rand1and rand2, one single random number r1 ischosen so that when r1 is large, (1�r1) is small and vice versa.Moreover, to control the balance of global and local searches,another random parameter r2 is introduced. For birds’ flocking forfood, there could be some rare cases that after the position of theparticle is changed according to (5), a bird may not, due to inertia,fly towards a region at which it thinks is most promising for food.

of (N/2þ1) number of h(n) coefficients within maximum and minimum bounds

36 in this work) order filter design. The steps of RGA as adopted from [35] and

minimum value.

nd minimum error fitness value, its corresponding chromosome string having

ients are obtained by concatenation to get the final optimal frequency spectrum

Table 3RGA, PSO, CRPSO parameters.

Parameters RGA PSO CRPSO

Population size 120 120 120

Iteration cycles 100 100 100

Crossover rate 0.8 – –

Crossover Two point crossover – –

Mutation rate 0.001 – –

Selection probability 1/3 – –

C1 – 2.05 2.05

C2 – 2.05 2.05

vmini

– 0.01 0.01

vmaxi – 1.0 1.0

wmax – 1.0 –

wmin – 0.4 –

Pcr – – 0.3

vcraziness – – 0.0001

Table 4Optimized coefficients of FIR BS filter of order 20.

h(N) RGA PSO CLPSO CRPSO

h(1)¼h(21) 0.0045 �0.0079 �0.0079 0.0065

h(2)¼h(20) �0.0366 �0.0277 �0.0306 �0.0324

h(3)¼h(19) 0.0762 0.0662 0.0658 0.0626

h(4)¼h(18) 0.0514 0.0569 0.0559 0.0557

h(5)¼h(17) �0.0732 �0.0674 �0.0595 �0.0585

h(6)¼h(16) �0.0007 �0.0047 �0.0028 0.0050

h(7)¼h(15) �0.0938 �0.0851 �0.0844 �0.0884

h(8)¼h(14) �0.0399 �0.0568 �0.0507 �0.0448

h(9)¼h(13) 0.2985 0.2947 0.2915 0.2926

h(10)¼h(12) 0.0300 0.0309 0.0360 0.0323

h(11) 0.5840 0.5881 0.5898 0.5894

R. Kar et al. / Swarm and Evolutionary Computation 7 (2012) 58–64 61

Instead, it may be leading toward a region which is in oppositedirection of what it should fly in order to reach the expectedpromising regions. So, in the step that follows, the direction of thebird’s velocity should be reversed in order for it to fly back to thepromising region. sign(r3) is introduced for this purpose. In birds’flocking or fish schooling, a bird or a fish often changes directionssuddenly. This is described by using a ‘‘craziness’’ factor and ismodelled in the technique by using a craziness variable. Acraziness operator is introduced in the proposed technique toensure that the particle would have a predefined crazinessprobability to maintain the diversity of the particles. Conse-quently, before updating its position the velocity of the particleis crazed by

V kþ1ð Þ

i ¼ V kþ1ð Þ

i þP r4ð Þnsign r4ð Þnvcrazinessi ð8Þ

where r4 is a random parameter which is chosen uniformly withinthe interval [0,1];

vcrazinessi is a random parameter which is uniformly chosen

from the interval vmini ,vmax

i

� �; and P(r4) and sign(r4) are defined,

respectively, as

Pðr4Þ ¼ 1 when r4rPcr

¼ 0 when r44Pcrð9Þ

signðr4Þ ¼ �1 when r4Z0:5

¼ 1 when r4o0:5ð10Þ

where Pcr is a predefined probability of craziness.Reversal of the direction of bird’s velocity should rarely occur;

to achieve this, r3r0.05 (a very low value) is chosen whensign(r3) will be �1 to reverse the direction. If Pcr is chosen lessor, equal to 0.3, the random number r4will have more probabilityto become more than Pcr, in that case, craziness factor P(r4) will bezero in most cases, which is actually desirable, otherwise heavyunnecessary oscillations will occur in the convergence curve nearthe end of the maximum iteration cycles as referred to (9).vcraziness is chosen very small (¼0.0001) as shown in Table 3.r4Z0.5 or, o0.5 is chosen to introduce equal probability ofdirection reversal of vcrazinessas referred to (8) and (10).

The design objective in this paper is to obtain the optimalcombination of the BS filter coefficients, so as to acquire themaximum stop band attenuation with the least increment intransition width. Here lies the author’s contribution that thisdesign objective has been attained by the proposed CRPSOtechnique. The steps of CRPSO algorithm are given in Table 2.The values of the parameters used for the RGA, PSO and CRPSOtechniques are given in Table 3. The values of the parameters usedfor the CLPSO technique are adopted from [36].

Table 2Steps of CRPSO.

Step 1: Initialization: Population (swarm size) of particle vectors, nP; maximum itera

optimized, (h(n)¼(N/2þ1)) (since the FIR BS filter is positive, symmetric, linear pha

maximum values of filter coefficients, hmin¼�1, hmax¼1; number of samples¼128;

Step 2: Generate initial particle vectors of filter coefficients (N/2þ1) randomly withi

Step 3: Computation of population based minimum error fitness value and computa

vector (hgbest).

Step 4: Updating the velocities as per (6) and (8); updating the particle vectors as per

of the updated error fitness values of the particle vectors and population based mini

Step 5: Updating the hpbest vectors, the hgbest vector; replace the updated particle v

Step 6: Iteration continues from step 4 till the maximum iteration cycles or the conver

band stop filter coefficients (N/2þ1); form complete (Nþ1) coefficients by copying (b

4. Results and discussions

4.1. Analysis of magnitude response of FIR BS filters

The MATLAB simulation has been performed extensively torealize the FIR BS filters of the orders of 20, 28 and 36. Hence, thelengths of the filter coefficients are 21, 29 and 37, respectively.The sampling frequency has been chosen as fs¼1 Hz. Also, for allthe simulations the sampling number is taken as 128. Eachalgorithm is run for 30 times to get the best solutions.

The parameters of the FIR BS filter to be designed are asfollows:

�

tion

se a

dp¼

n li

tion

(5)

mu

ect

gen

ecau

Pass band ripple (dp)¼0.1

� Stop band ripple (ds)¼0.01 � Lower pass band (normalized) cut-off frequency (opl)¼0.30

cycles; filter order N¼20 or 28 or 36; number of filter coefficients to be

nd is of even order); fixing values of C1, C2, Pcr, vcraziness; minimum and

0.1, ds¼0.01; initialization of the velocities of all the particle vectors.

mits; Computation of initial error fitness values of the total population, nP.

of the personal best solution vectors (hpbest), group best solution

and checking against the limits of the filter coefficients; finally, computation

m error fitness value.

ors as initial particle vectors for step 4.

ce of minimum error fitness values; finally, hgbest is the vector of optimal FIR

se the filter has linear phase) before getting the optimal frequency spectrum.

R. Kar et al. / Swarm and Evolutionary Computation 7 (2012) 58–6462

�

TabOpt

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

TabOpt

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

Lower stop band (normalized) cut-off frequency (osl)¼0.35

� Upper pass band (normalized) cut-off frequency (oph)¼0.75 � Upper stop band (normalized) cut-off frequency (osh)¼0.70

The best optimized coefficients for the designed FIR BS filterswith the orders of 20, 28 and 36 have been calculated by PM, RGA,PSO, CLPSO and CRPSO algorithms and are given in Tables 4–6,respectively. Tables 7–9 summarize the comparative results ofdifferent performance parameters obtained using PM, RGA, PSO,CLPSO and CRPSO algorithms for BS filters of orders 20, 28 and 36,respectively.

le 5imized coefficients of FIR BS filter of order 28.

(N) RGA PSO CLPSO CRPSO

(1)¼h(29) 0.0122 0.0062 0.0105 0.0060

(2)¼h(28) 0.0368 0.0350 0.0334 0.0281

(3)¼h(27) �0.0256 �0.0303 �0.0286 �0.0236

(4)¼h(26) �0.0231 �0.0195 �0.0230 �0.0242

(5)¼h(25) �0.0080 �0.0068 �0.0004 0.0005

(6)¼h(24) �0.0344 �0.0287 �0.0238 �0.0228

(7)¼h(23) 0.0613 0.0599 0.0554 0.0515

(8)¼h(22) 0.0466 0.0413 0.0423 0.0408

(9)¼h(21) �0.0512 �0.0558 �0.0496 �0.0538

(10)¼h(20) 0.0003 �0.0006 0.0026 �0.0006

(11)¼h(19) �0.0906 �0.0862 �0.0900 �0.0874

(12)¼h(18) �0.0474 �0.0466 �0.0507 �0.0468

(13)¼h(17) 0.2981 0.3023 0.3006 0.2974

(14)¼h(16) 0.0307 0.0237 0.0271 0.0289

(15) 0.6032 0.6038 0.6038 0.6035

le 6imized coefficients of FIR BS filter of order 36.

(N) RGA PSO CLPSO CRPSO

(1)¼h(37) 0.0007 �0.0011 �0.0046 �0.0050

(2)¼h(36) �0.0321 �0.0262 �0.0311 �0.0250

(3)¼h(35) 0.0044 0.0057 0.0047 0.0015

(4)¼h(34) �0.0033 �0.0025 �0.0081 0.0018

(5)¼h(33) 0.0078 0.0104 0.0061 0.0074

(6)¼h(32) 0.0370 0.0333 0.0278 0.0312

(7)¼h(31) �0.0230 �0.0202 �0.0257 �0.0193

(8)¼h(30) �0.0289 �0.0247 �0.0268 �0.0272

(9)¼h(29) �0.0014 �0.0004 �0.0020 �0.0034

(10)¼h(28) �0.0258 �0.0267 �0.0259 �0.0232

(11)¼h(27) 0.0563 0.0568 0.0533 0.0484

(12)¼h(26) 0.0434 0.0413 0.0342 0.0466

(13)¼h(25) �0.0516 �0.0528 �0.0549 �0.0479

(14)¼h(24) �0.0035 �0.0047 �0.0013 �0.0000

(15)¼h(23) �0.0868 �0.0861 �0.0897 �0.0803

(16)¼h(22) �0.0552 �0.0527 �0.0516 �0.0502

(17)¼h(21) 0.2968 0.2993 0.2934 0.2979

(18)¼h(20) 0.0345 0.0328 0.0271 0.0328

(19) 0.5998 0.5992 0.5939 0.5993

Table 7Comparison summary of the parameters of interest of order 20 for diff

Algorithm Order 20

Stop band attenuation (dB)

Maximum Mean Variance Std. dev

PM 14.18 14.18 0.00016 0.0126

RGA 14.92 15.61 0.8208 0.9059

PSO 15.45 16.12 0.2291 0.4786

CLPSO 16.51 16.55 0.00056 0.0236

CRPSO 17.57 18.03 0.2923 0.5406

Figs. 1–3 show the magnitude (dB) plots for the FIR BS filters oforders 20, 28 and 36, respectively. The statistical test results havebeen calculated by using [37]. The maximum, mean, variance andthe standard deviation of stop band attenuation have beencalculated for the filter orders of 20, 28 and 36 and are shownin Tables 7–9, respectively, for all the above mentioned optimiza-tion algorithms.

The proposed CRPSO based approach for 20th order BS filterdesign results in the highest 17.57 dB stop band attenuation,minimum pass band ripple (normalized)¼0.095, minimum stopband ripple (normalized)¼0.132. The proposed CRPSO basedapproach for 28th order BS filter design results in the highest25.22 dB stop band attenuation, maximum pass band ripple(normalized)¼0.066, maximum stop band ripple (normalized)¼0.0547. The simulation results show that the proposed CRPSObased approach for 36th order BS filter design results in 29.38 dBstop band attenuation, maximum pass band ripple (normal-ized)¼0.061, maximum stop band ripple (normalized)¼0.0339.The novelty of the proposed filter design approach is also justifiedby the comparison made with [24]. The particle swarm optimiza-tion with quantum infusion (PSO-QI) model proposed in [24]reveals no improvement with respect to the PM algorithm,whereas, the proposed filter design technique shows 3.39 dB,6.51 dB, 7.32 dB improvement as compared to PM for the BSfilters of orders 20, 28 and 36, respectively.

From the diagrams and above discussions it is evident thatwith almost same level of the transition width, the proposedCRPSO based filter design approach produces the highest stopband attenuation (dB) and the lowest stop band and pass bandripples (normalized) with a little change in the transition width,compared to those of PM algorithm, RGA, PSO and CLPSO, asshown in Tables 7–9. So, the filters designed by the CRPSO resultin the best responses in the stop band region, Thus, it can befinally inferred that the CRPSO based FIR BS filter design is thebest among the techniques reported in this work.

4.2. Comparative effectiveness and convergence profiles of RGA, PSO,

CLPSO and CRPSO

In order to compare the algorithms in terms of the error fitnessvalue, Fig. 4 shows the convergences of error fitnesses obtainedwhen RGA, PSO, CLPSO and the CRPSO are employed, respectively.The convergence profiles are shown for the BS filter of order 36.Similar plots have also been obtained for the BS filters of orders of20 and 28, which are not shown here. The CRPSO converges tomuch lower error fitness as compared to RGA, PSO and CLPSOwhich yield suboptimal higher values of error fitnesses. As shownin Fig. 4, in case of BS filter of order 36, the RGA converges to theminimum error fitness value of 7.62; the conventional PSOconverges to the minimum error fitness value of 6.72; the CLPSOconverges to the minimum error fitness value of 6.05; whereas,the CRPSO converges to the minimum error fitness value of 5.47.

erent algorithms.

Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

iation

0.196 0.195

0.12 0.179

0.086 0.169

0.07 0.149

0.095 0.132

Table 9Comparison summary of the parameters of interest of order 36 for different algorithms.

Algorithm Order 36

Stop band attenuation Maximum pass bandripple (normalized)

Maximum stop bandripple (normalized)

Maximum Mean Variance Std. deviation

PM 22.06 22.106 0.0018 0.0422 0.079 0.0788

RGA 24.28 34.02 109.398 10.459 0.086 0.0611

PSO 25.13 35.69 67.528 8.217 0.081 0.0554

CLPSO 27.0 35.80 120.489 10.976 0.065 0.0444

CRPSO 29.38 34.89 21.566 4.644 0.061 0.0339

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Frequency

Mag

nitu

de (d

B)

PMRGAPSOCLPSOCRPSO

Fig. 1. Magnitude (dB) plot of the FIR BS filter of order 20.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-50

-40

-30

-20

-10

0

Frequency

Mag

nitu

de (d

B)

PMRGAPSOCLPSOCRPSO

Fig. 2. Magnitude (dB) plot of the FIR BS filter of order 28.

Table 8Comparison summary of the parameters of interest of order 28 for different algorithms.

Algorithm Order 28

Stop band attenuation Maximum passband ripple (normalized)

Maximum stop bandripple (normalized)

Maximum Mean Variance Std. deviation

PM 18.71 18.768 0.004136 0.0643 0.115 0.116

RGA 19.3 24.52 15.82 3.977 0.084 0.109

PSO 20.2 25.09 18.38 4.287 0.08 0.0978

CLPSO 22.13 24.468 3.1666 1.779 0.072 0.0779

CRPSO 25.22 26.464 0.4707 0.686 0.066 0.0547

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

Frequency

Mag

nitu

de (d

B)

PMRGAPSOCLPSOCRPSO

Fig. 3. Magnitude (dB) plot of the FIR BS filter of order 36.

R. Kar et al. / Swarm and Evolutionary Computation 7 (2012) 58–64 63

The above results may be verified from Tables 7–9. Similarobservations hold good for BS filters of orders 20 and 28 asshown in the same tables.

For all BS filters of different orders, the CRPSO algorithmconverges to the least minimum error fitness values in findingthe optimum filter coefficients. With a view to the above fact, itmay finally be inferred that the performance of CRPSO algorithm isthe best among all algorithms. All optimization programs were runin MATLAB 7.5 version on core (TM) 2 duo processor, 3.00 GHzwith 2 GB RAM.

5. Conclusions

In this paper, a novel Craziness based Particle Swarm Optimiza-tion (CRPSO) technique is applied to the solution of the constrained,multimodal, non-differentiable, and highly nonlinear FIR band stopfilter design problem to obtain the optimal filter coefficients. With

0 10 20 30 40 50 60 70 80 90 1004

6

8

10

12

14

16

18

Iteration cycle

J

RGAPSOCLPSOCRPSO

Fig. 4. Convergence profile for RGA, PSO, CLPSO and CRPSO in case of 36th order

BS FIR filter.

R. Kar et al. / Swarm and Evolutionary Computation 7 (2012) 58–6464

almost same level of the transition width, the CRPSO produces thehighest stop band attenuation and the lowest stop band and thepass band ripples as compared to those of PM algorithm, RGA andconventional PSO and CLPSO. It is also evident from the resultsobtained by a large number of trials that the CRPSO is consistentlyfree from the shortcoming of premature convergence exhibited bythe other optimization techniques. Thus, it reveals that the CRPSOmay be used as a good optimizer for the solution of obtaining theoptimal filter coefficients in a practical digital filter design problemin digital signal processing systems.

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