Research Article Design of Digital IIR Filter with ...

Research ArticleDesign of Digital IIR Filter with Conflicting Objectives UsingHybrid Gravitational Search Algorithm

D S Sidhu1 J S Dhillon2 and Dalvir Kaur3

1Department of Electronics and Communication Engineering Giani Zail Singh Punjab Technical University CampusBathinda 151001 India2Department of Electrical and Instrumentation Engineering Sant Longowal Institute of Engineering and TechnologyLongowal 148106 India3Department of Electronics and Communication Engineering Punjab Institute of Technology Kapurthala 144601 India

Correspondence should be addressed to D S Sidhu dssidhuyahoocom

Received 17 May 2015 Revised 14 September 2015 Accepted 15 September 2015

Academic Editor Erik Cuevas

Copyright copy 2015 D S Sidhu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the recent years the digital IIR filter design as a single objective optimization problem using evolutionary algorithms has gainedmuch attention In this paper the digital IIR filter design is treated as a multiobjective problem by minimizing the magnituderesponse error linear phase response error and optimal order simultaneously along with meeting the stability criterion Hybridgravitational search algorithm (HGSA) has been applied to design the digital IIR filter GSA technique is hybridized with binarysuccessive approximation (BSA) based evolutionary search method for exploring the search space locallyThe relative performanceof GSA and hybrid GSA has been evaluated by applying these techniques to standard mathematical test functions The aboveproposed hybrid search techniques have been applied effectively to solve the multiparameter and multiobjective optimizationproblem of low-pass (LP) high-pass (HP) band-pass (BP) and band-stop (BS) digital IIR filter designThe obtained results revealthat the proposed technique performs better than other algorithms applied by other researchers for the design of digital IIR filterwith conflicting objectives

1 Introduction

The electrical signal is the best method to transfer infor-mation to longer distances As the signal becomes complexwe require to process the signal to extricate the informa-tion available in the signal Generally signals are classifiedinto two categories that is continuous-time and discrete-time signals Digital signal processing (DSP) is a numericalmanipulation of discrete-time signal data With the advent ofdigital circuit technology cheaper simple and faster digitalcomputers with huge memory storage capability are able toperform complex real time digital signal processing In therecent period digital signal processing (DSP) is the area ofinterest for engineers and scientists due to their accuracy reli-ability flexibility low cost and small physical size Filteringis an essential part of digital signal processing for differentapplications such as telecommunications speech processingimage processing consumer electronics biomedical systems

industrial applications andmilitary electronics As comparedto the analog filters digital filters are flexible reliable andversatile The sampled values of input signal and the transferfunction of digital filter can be easily stored in the memory[1 2]

Digital filters are classified as finite impulse response(FIR) filters and infinite impulse response (IIR) filters IIRfilter provides better response than FIR filters with the sameorder or same number of coefficients The stability of thedigital IIR filter can be obtained by limiting the parameterspace in a suitable range Multimodal error surface of IIRfilter may force gradient based algorithm to get stuck inlocal minima [3] In order to avoid the above problem andto find the global minima global optimization techniqueslike genetic algorithm (GA) particle swarm optimization(PSO) differential evolution (DE) predator prey optimiza-tion (PPO) artificial bee colony (ABC) and so forth havebeen used by the researchers [3ndash7] to design digital IIR filters

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 282809 16 pageshttpdxdoiorg1011552015282809

2 Mathematical Problems in Engineering

The synergy between exploration and exploitation is athrust area in the optimization process The generalizationof heuristics allows the wide applicability but it limitsthe efficiency of heuristics and simply delivers mediocreperformance The hybridization of general heuristics withproblem-specific heuristics provides better results for morecomplicated problems The technique which uses generalheuristics as global search and a specific heuristic as localsearch is generally called hybrid algorithm With a propertuning hybrid algorithms with a good explorative ability asa population-based global search algorithm also provide agood exploitive performance as a local search algorithm [8]Many researchers have applied various hybrid optimizationtechniques for design of digital IIR and FIR filters Singhet al [9] had applied DE hybridized with pattern searchtechnique for the design of digital IIR filter DE has beenused for global search technique and pattern search methodas local search technique Kaur and Dhillon [10] had pro-posed integrated cat swarm optimization (CSO) and DEfor digital IIR filter design So Kaur and Dhillon discussedhybridization of two global search techniques Hua et al[11] have hybridized DE and PSO for designing 2D FIRfilter Vasundhara et al [12] had proposed hybridization ofadaptive DE and PSO for FIR filter design Both researchershave also considered the hybridization of two differentglobal search techniques Bindiya and Elias [13] had applieddifferent modified metaheuristic algorithms for the designof multiplier-less nonuniform channel filters and comparedtheir results They concluded that gravitational search algo-rithm (GSA) is the fastest and also gives acceptable valuesof the frequency-response characteristics and complexitySaha et al [14] had applied GSA for design and simulationof FIR band-pass and band-stop filters and compared theresults with results obtained by DE PSO RGA and BBOIt has been concluded that GSA converges faster than othertechniques Saha et al [15] had also applied GSAwith waveletmutation for design of IIR filter and concluded that GSAWMprovides better results as compared to other techniques Theabove researchers considered the design problem as a singleobjective optimization problem and applied the techniquesfor minimizing the magnitude response error of digital filteronly with supplementary conditions

Yu and Xinjie [16] had applied cooperative coevolu-tionary genetic algorithm to design the digital IIR filterMagnitude response and phase response are simultaneouslyoptimized along with the minimum filter order Wang et al[17] applied local search operator enhanced multiobjectiveevolutionary algorithm (LS-MOEA) for design of digital IIRfilter withmultiple objectivesWang et al [17] has also appliednondominated sorted based genetic algorithm II (NSGA-II)for the above problem Kaur et al [18] applied real codedgenetic algorithm (RCGA) for design of digital IIR filterwith multiple objectives The literature survey reveals thatmetaheuristicmethods generally suffer from either the explo-ration or exploitation capabilities leading to lack of diversityand stagnation to local minima respectively Involvementof many parameters leads to parameter tuning problemsand becomes time consuming Mostly the aforementioned

methods lack to maintain balance between exploration andexploitation

Any proposed optimizationmethodmay be a good choiceto design digital filters thatmay have exclusive advantages likerobust global search capability little information require-ment ease of implementation parallelism no requirement ofcontinuous and differentiable objective function The intentof this paper is to propose the global search techniquegravitational search algorithm (GSA) to design the digital IIRfilter The global search technique is then hybridized withevolutionary search method for exploring the search spacelocally With the increase in number of filter coefficientsthe number of comparisons increases exponentially in evolu-tionary search technique during exploration of search spaceTo reduce such comparisons a unique binary successiveapproximation (BSA) strategy has been implemented whileexploring the search space locally The opposition-basedlearning strategy is applied to have a start with good initialpopulation In this paper the design of digital IIR filter istreated as a multiobjective problem by simultaneously min-imizing the magnitude response error linear phase responseerror and optimal order along with meeting the stabilitycriterion The proposed hybrid technique is applied to solvethemultiparameter andmultiobjective optimization problemof low-pass high-pass band-pass and band-stop digital IIRfilter design Owing to the imprecise nature of decisionmakerjudgement it is presumed that decision maker may havefuzzy or imprecise goals for each objective [19]HybridGSA isused to simulate trade-off between conflicting objectives thatis magnitude response error and phase response Once thetrade-off has been obtained fuzzy set theory helps decisionmaker to decide the optimal design of IIR digital filter overthe noninferior trade-off curve

This paper is divided into five sections In Section 2digital IIR filter problem is described The gravitationalsearch algorithm (GSA) and binary successive approxima-tion (BSA) based evolutionary search method for optimalIIR filter design are explained in Section 3 The GSA andhybrid GSA have been applied to standard mathematical testfunctions in Section 4 to evaluate the relative performanceIn Section 5 digital IIR filter has been designed by applyingthe proposed hybrid heuristic search technique and obtainedresults are evaluated and compared with results of otherauthors The final conclusion and discussions are outlined inSection 6

2 IIR Filter Design Problem

Digital infinite impulse response (IIR) filter design requiresthe optimization of set of filter coefficients which meetvarious specification of the filter such as pass-band width andgain stop-band width and attenuation edge frequencies ofpass band and stop band peak ripples in pass band and stopband and linear phase response in pass band and transitionband

The cascaded digital IIR filter has several first order andsecond order sections together and is easy to implement

Mathematical Problems in Engineering 3

Putting 119911 = 119890119895120596 the structure of cascading type IIR filter can

be expressed as [20]

119867(120596 119909)

= 119860(

119872

prod

119894=1

1 + 119886119894119890minus119895120596

1 + 119887119894119890minus119895120596

)(

119873

prod

119896=1

1 + 119901119896119890minus119895120596

+ 119902119896119890minus2119895120596

1 + 119903119896119890minus119895120596

+ 119904119896119890minus2119895120596

)

(1)

where

119909 = [119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873 119904119873]119879 (2)

Vector 119909 denotes the set of filter coefficients of first orderand second order sections of dimensions119863times1with119863 = 2119872+

4119873 + 1The coefficients of the transfer function 119867(120596 119909) are

approximated during the design of filter The transfer func-tion119867(120596 119909) is compared this with the ideal transfer function119867119868(120596) The magnitude error function 1198641(119909) is obtained fromthese two values as given below

Minimize 1198641 (119909) =

119896

sum

119894

1003816100381610038161003816119867119889 (120596119894) minus

1003816100381610038161003816119867 (120596119894 119909)

1003816100381610038161003816

1003816100381610038161003816 (3a)

where

119867119889 (120596119894) =

1 for 120596119894 isin pass band

0 for 120596119894 isin stop band(3b)

The ripple magnitude of pass-band 1205751(119909) and of stop-band 1205752(119909) are defined as below

1205751 (119909) = max120596119894

1003816100381610038161003816119867 (120596119894 119909)

1003816100381610038161003816 minus min120596119894

1003816100381610038161003816119867 (120596119894 119909)

1003816100381610038161003816

for 120596119894 isin pass band(4a)

1205752 (119909) = max120596119894

1003816100381610038161003816119867 (120596119894 119909)

1003816100381610038161003816 for 120596119894 isin stop band (4b)

During the design of filter the linear phase response isalso optimized for both pass band and transition band Theexpression for phase is defined as below

0 = tanminus1 imaginary part of numerator

real part of numerator

minus tanminus1 imaginary part of denominator

real part of denominator

(5)

The first order difference in phase is determined as

Δphase = Δ01 Δ02 Δ0119899minus1 (6)

where

Δ0119894 = Δ0119894+1 minus Δ0119894 (119894 = 1 2 119899 minus 1) (7)

119873 is the total number of sampling points in pass bandand transition band The phase response will be linear if allthe elements of Δphase have the equal values [18] So the

H(120596119901)

1 minus 120575p1 1 1 + 120575p2

120583n(H(120596119901))

Hmin(120596119901)

1

0

Hmax(120596119901)

Figure 1Membership function ofmagnitude response during pass-band

second objective function is to minimize variance of phasedifferences and is expressed as below

Minimize 1198642 = var Δ0119899

0119899 isin pass band cup transition band(8)

Consider all these objectives subject to the followingstability constraints

1 + 119887119894 ge 0 (119894 = 1 2 119872) (9a)

1 minus 119887119894 ge 0 (119894 = 1 2 119872) (9b)

1 minus 119904119896 ge 0 (119896 = 1 2 119873) (9c)

1 + 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9d)

1 minus 119903119896 + 119904119896 ge 0 (119896 = 1 2 119873) (9e)

For the design of digital IIR filter with the above mul-tiobjective optimization criterion the fuzzy membershipfunctions are assigned to magnitude response error func-tion and phase response error function along with stabilityconstraints The fuzzy sets are stated by their membershipfunctions Suchmembership functions describe the degree ofmembership in certain fuzzy sets using values from 0 to 1Themembership function value 0 indicates incompatibility withthe sets whereas 1 means full compatibility [19] The memberfunctions are used to convert minimization to maximizationoptimization problem and constraints are converted intoobjective function to be maximized

21 Magnitude Response Error The main objective to designdigital IIR filter is to minimize the magnitude responseerror in a predefined pass-band and stop-band and withinprescribed permissible ripples In the prescribed pass-bandthe signal is allowed to pass and in the prescribed stop-bandthe signal is restricted

Themembership function ofmagnitude response in pass-band is described in Figure 1

4 Mathematical Problems in Engineering

120575s

0

1

H(120596119904)

120583n(H(120596119904))

Hmax(120596119904)

Figure 2 Membership function of magnitude response during stopband

Mathematical representation of membership function ofmagnitude response during pass-band is given as below

119906119899 (119867(120596119901))

=

0 119867(120596119901)le 119867

min(120596119901)

or 119867(120596119901) ge 119867max(120596119901)

119867(120596119901)minus 119867

min(120596119901)

(1 minus 1205751199011) minus 119867min(120596119901)

119867min(120596119901)

lt 119867(120596119901)lt 1 minus 1205751199011

119867max(120596119901)

minus 119867(120596119901)

119867max(120596119901)

minus (1 + 1205751199012)

1 + 1205751199012 lt 119867(120596119901)lt 119867

max(120596119901)

1 1 minus 1205751199011 le 119867(120596119901)le 1 + 1205751199012

(10)

where 120596119901 is the frequency in pass-band 119867(120596119901) is the magni-tude response in pass-band119867min

(120596119901)is theminimummagnitude

in pass band 119867max(120596119901)

is maximum magnitude in the pass-band 1205751199011 and 1205751199012 are the maximum tolerable deviations inmagnitude in pass-band 1205751199012 is kept almost nearer to zero119906119899(119867(120596119901)

) is the membership function of magnitude responsein pass-band 119899 isin pass band

Themembership function ofmagnitude response in stop-band is described in Figure 2

Mathematical expression of membership function ofmagnitude response during stop-band is given as below

120583119899 (119867(120596119904)) =

0 119867(120596119904)ge 119867

max(120596119904)

119867max(120596119904)

minus 119867(120596119904)

119867max(120596119904)

minus 120575119904

120575119904 lt 119867(120596119904)lt 119867

max(120596119904)

1 119867(120596119904)le 120575119904

(11)

where120596119904 is the frequency in stop band119867(120596119904) is themagnituderesponse in stop-band 119867max

119904 is maximum magnitude in thestop-band 120575119904 is the maximum tolerable deviations in magni-tude in stop-band 119906119899(119867(120596119904)) is the membership function ofmagnitude response in stop-band 119899 isin stop band

120583bi1

1

01 + bi1

Figure 3 Membership function of stability constraint

The main objective is to maximize the membershipfunction of magnitude response in both pass-band and stop-band and is represented as below

1205831 = min min 120583119899 (119867(120596119901)) 119899 isin pass band

min 120583119899 (119867(120596119904)) 119899 isin stop band

(12)

The maximum value of 1205831 provides the design of digitalfilter having minimum value of magnitude response error

22 Phase Response Error As described in (8) the phaseresponse error 1198642 is to be minimized This minimiza-tion problem is converted into maximization problem asdescribed as below

1205832 =1

1 + 1198642

(13)

where 1205832 is the objective function of phase response Maxi-mum value of objective function 1205832 provides the minimumvalue of linear phase response error 1198642

23 Membership Function of Stability Constraints The stabil-ity constraints for the design of digital IIR filter are obtainedby using the Jury method [21] on the filter coefficients givenin (9a) Using fuzzy set theory the membership function ofstability constraints given in (9a) is shown in Figure 3

Mathematically membership function of stability con-straint given in (9a) is given below

1205831119894 =

0 if (1 + 119887119894) le 0

1 if (1 + 119887119894) gt 0

(119894 = 1 2 119872) (14)

Similarly membership function for stability constraintsgiven in (9b) to (9e) can be describedThe overall objective tomeet these above five stability constraints is mathematicallygiven below and is to be maximized

1205833

=

1

5

2

sum

119896=1

[

1

119872

(

119872

sum

119894=1

120583119896119894)] +

5

sum

119896=3

[

[

1

119873

(

119873

sum

119895=1

120583119896119895)]

]

(15)

If the value 1205833 is 1 that means all constraints are satisfiedOtherwise it gives the percentage level of satisfaction ofconstraints

Mathematical Problems in Engineering 5

24 Multiobjective Problem Formulation The task of digitalIIR filter design is to find an optimum structure withminimum magnitude response error and minimum linearphase response error while satisfying the stability constraintsBy applying fuzzy set theory multiobjective constrainedproblem is converted intomultiobjective unconstrained opti-mization problem and is stated as below

Maximize [1205831 (119883) 1205832 (119883) 1205833 (119883)]119879 (16)

where 1205831(119883) is the membership function of magnituderesponse error given in (12) 1205832(119883) is the membershipfunction of linear phase response error given in (13) 1205833(119883)

is the membership function of stability constraints given in(15)119883 is a vector decision variable of dimensions119863 times 1 with119863 = 2119872 + 4119873 + 1

The objective is to find the value of filter coefficients beingdecision variables 119883 which maximizes the entire objectivefunctions simultaneously The value of membership functionindicates how much the solution satisfies the 120583119894 objectiveon the scale from 0 to 1 The maximum satisfaction ofmembership function for any filter coefficient combinationis obtained by taking the intersection of the membershipfunctions of participating objectives and is expressed asbelow

fit119895 (119905) = min 1205831119895 1205832119895 1205833119895 (119895 = 1 2 119873119875) (17)

25 Optimal Order of Digital Filter The expression foroptimal order of digital IIR filter is described below

119874 =

119872

sum

119895=0

120572119895 + 2

119873

sum

119896=0

120573119896 (18)

where 120572119895 and 120573119896 are 119895th and 119896th control genes of corre-sponding first order and second order blocks respectivelyand the value of genes will be either 1 or 0 119872 and 119873 arethe maximum number of first order and second order blocksrespectively Maximum order of the digital IIR filter will be119872 + 2119873 The order of the filter has been determined bycontrol genes as shown in Figure 4 The coding method hasbeen taken from Yu and Xinjie [16]The value of control genedetermines whether the particular block will be consideredfor filter design or notThe block will be considered activatedwhen the corresponding control gene is 1 The number ofbinary bits used to generate control genes depends upon thevalue of 119872 and 119873 The decision vector 119909 shown in (2) ismodified as below119909 = [119880119881 119860 1198861 1198871 119886119872 119887119872 1199011 1199021 1199031 1199041 119901119873 119902119873 119903119873

119904119873]119879

(19)

The variable 119880 is a positive integer with maximum valueof (2119872

minus 1) and the variable 119881 is a positive integer withmaximum value of (2119873 minus 1) The variable 119880 and 119881 will alsobe optimized along with the coefficients of the filter

3 Optimization Technique

31 Gravitational Search Algorithm This optimization algo-rithm is based on the law of gravity In this algorithm each

1205721 1205722 1205723 1205731 1205732 1205733 1205734

1 + a2zminus1

1 + b2zminus1

1 + a1zminus1

1 + b1zminus1

1 + a3zminus1

1 + b3zminus1

1 + p1zminus1 + q1z

minus2

1 + r1zminus1 + s1z

minus2

1 + p3zminus1 + q3z

minus2

1 + r3zminus1 + s3z

minus2

1 + p2zminus1 + q2z

minus2

1 + r2zminus1 + s2z

minus2

1 + p4zminus1 + q4z

minus2

1 + r4zminus1 + s4z

minus2

Figure 4 Activation and deactivation of filter coefficients withcontrol genes

candidate of the population is considered as an object (mass)Each object has four specifications that is active gravitationalmass passive gravitational mass inertia mass and positionGravitational and inertia masses are determined by thefitness function and position of the object corresponds to thesolution of the problemAll the objects attract each otherwitha gravitational force Candidates having good solutions haveheavy masses Heavy masses move slowly and attract lightermasses towards good solutionsAt the endof iterationmassesare updated as per their new positionsWith the lapse of timewe consider that all the masses be attracted by the heaviestmass and this mass will represent the optimum solution [22]

In GSA population is considered as an isolated universeof objects (masses) Value of gravitational mass and inertiamass of an object has been assumed equal Each object obeysthe Newtonian laws of gravitation and motion as describedbelow

Law of Gravitation Each object attracts every other objectwith gravitational force The gravitational force between twoobjects is directly proportional to the product of their masses(119872) and inversely proportional to the square of the distance(119877) between them But in GSA only distance 119877 is takeninstead of R2 because it gives better results This is thedeviation of GSA from the Newtonian laws of gravitation

Law of Motion The current velocity of any object is equal tothe fraction of previous velocity of the object and accelerationof the object Acceleration of the object is equal to the forceapplied on the object divided by the inertiamass of the object

Algorithm has been initialized with random populationof 119863 dimensional 119873119875 objects (masses) and each object isdescribes as

119883119894 = (1199091119894 119909

119889119894 119909

119863119894 )

for (119894 = 1 2 119873119875 119889 = 1 2 119863)

(20)

where 119909119889119894 represents the position of 119894th object in the 119889th

dimension

6 Mathematical Problems in Engineering

At a specific iteration ldquo119905rdquo the force acting on 119894th object by119895th object is described as below

119865119889119894119895 (119905) = 119866 (119905)

119872119894 (119905) times 119872119895 (119905)

119877119894119895 (119905) + 120576

(119909119889119895 (119905) minus 119909

119889119894 (119905)) (21)

where 119872119894 and 119872119895 are gravitational masses of 119894th and 119895thobjects respectively 119866(119905) is the gravitational constant atiteration ldquo119905rdquo 120576 a small constant has been added to avoidextra ordinary high value of force between two objects andalmost lies on the same place in the space 119877119894119895 is the Euclidiandistance between 119894th and 119895th objects and described as below

119877119894119895 (119905) =

10038171003817100381710038171003817119883119894 (119905) 119883119895 (119905)

100381710038171003817100381710038172

(22)

To provide stochastic characteristics to the algorithm itis considered that total force acting on the 119894th object in 119889thdimension is a randomly weighted sum of 119889th component ofthe force exerted by all other objects and is given by

119865119889119894 (119905) =

119873119875

sum

119895=1119895 =119894

rand (119895) 119865119889119894119895 (119905) (23)

where rand(119895) is a random variable in the interval [0 1]corresponding to the 119895th object

As per the law ofmotion the acceleration of the 119894th objectin the 119889th direction at iteration ldquo119905rdquo is defined as below

119886119889119894 (119905) =

119865119889119894 (119905)

119872119894 (119905) (24)

The velocity of the object is calculated as sum of fractionof current velocity and acceleration of the object Thereforethe velocity and position of the object is updated as describedbelow

V119889119894 (119905 + 1) = rand (119894) times V119889119894 (119905) + 119886119889119894 (119905)

(25)

119909119889119894 (119905 + 1) = 119909

119889119894 (119905) + V119889119894 (119905 + 1) (26)

where rand(119894) is a random variable in the interval [0 1] Thegravitational constant 119866 has been initialized at the beginningand reduces with successive iterations Gravitational constant119866(119905) at iteration ldquo119905rdquo is described as below

119866 (119905) = 119866119900119890minus120572119905119879

(27)

where 119866119900 is the initial value of gravitational constant 120572 isa predefined constant and 119879 is the maximum number ofiterations

Masses of the objects are updated as follows

119898119894 (119905) =

fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)

119872119894 (119905) =

119898119894 (119905)

sum119873119895=1119898119895 (119905)

for (119894 = 1 2 119873)

(28)

where fit119894(119905) represents the fitness value of the 119894th object atiteration ldquo119905rdquo For minimization problem best(119905) and worst(119905)are described as follows

best (119905) = min119895isin(1119873119875)

fit119895 (119905)

worst (119905) = max119895isin(1119873119875)

fit119895 (119905) (29)

To have good compromise between exploration andexploitation the number of objects can be reduced withsuccessive iterations So it is supposed that only a set of objectswith heavy masses will exert gravitational force on otherobjects But to avoid trapping in local minima algorithmshould use exploration in the beginning By lapse of itera-tions the exploration should fade out and exploitation shouldfade in So only kbest objects exert gravitational force on otherobjects Initially kbest is a taken equal to number of objectsand all objects exert force With successive iterations kbestdecreases linearly in such way that at the last iteration onlyone object has been left and this provides us the optimumresult Therefore (23) can be modified as follows

119865119889119894 (119905) = sum

119895isin119896119887119890119904119905119895 =119894

rand (119895) 119865119889119894119895 (119905) (30)

where kbest is the set of objects with best fitness value andmaximummass

If elements of object velocity V119905119894119889 violate their limits theirvalues are updated as below

V119905119894119889 =

Vmin119889 V119905119894119889 lt Vmin

119889

Vmax119889 V119905119894119889 gt Vmax

119889

V119905119894119889 no violation of limits

(31)

Similarly if elements of object position 119909119905119894119889 violate their

limits their values are updated as below

119909119905119894119889 =

119909min119889 119909

119905119894119889 lt 119909

min119889

119909max119889 119909

119905119894119889 gt 119909

max119889

119909119905119894119889 no violation of limits

(32)

32 Exploratory Move In exploratory move the currentpoint is perturbed in all possible directions for each and everyvariable at a time and the best point is recorded After eachperturbation the present point is changed to the best pointAt the end of all variable perturbations if the point foundis different from the original point the exploratory move iscalled a success otherwise the exploratory move is a failureIn any case the best point arrived at the end of exploratorymove

The evolutionary method is used to search for theoptimal filter coefficients In this method for 119863 number offilter coefficients 2

119863 feasible solutions are generated A 119863

dimensional hypercube of side Δ is formed around the point119909119862119894 represents the coefficients of IIR filter from the current

point of hypercube The better solution is obtained from

Mathematical Problems in Engineering 7

Table 1 Coefficient vector at the corners of hypercube

Hyper cube corners

Possiblecombinations of

3-bit1198622 1198621 1198620

Distance of hypercubefrom centre point

1199091198881198943 1199091198881198942 1199091198881198941

Possible coefficient pattern of the IIR filter at the corner of hypercube

0 000 minusΔ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909

1198881198942 minus Δ 1198942 119909

1198881198941 minus Δ 1198941

1 001 minusΔ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909

1198881198942 minus Δ 1198942 119909

1198881198941 + Δ 1198941

2 010 minusΔ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 minus Δ 1198943 119909

1198881198942 + Δ 1198942 119909

1198881198941 minus Δ 1198941

3 011 minusΔ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 minus Δ 1198943 119909

1198881198942 + Δ 1198942 119909

1198881198941 + Δ 1198941

4 100 +Δ 1198943 minus Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909

1198881198942 minus Δ 1198942 119909

1198881198941 minus Δ 1198941

5 101 +Δ 1198943 minus Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909

1198881198942 minus Δ 1198942 119909

1198881198941 + Δ 1198941

6 110 +Δ 1198943 +Δ 1198942 minus Δ 1198941 1199091198881198943 + Δ 1198943 119909

1198881198942 + Δ 1198942 119909

1198881198941 minus Δ 1198941

7 111 +Δ 1198943 +Δ 1198942 +Δ 1198941 1199091198881198943 + Δ 1198943 119909

1198881198942 + Δ 1198942 119909

1198881198941 + Δ 1198941

3(011) 7(111)

2(010) 6(110)

0(000) 4(100)

5(101)1(001)

xci3 xci2 x

ci1

o

Figure 5 Three-dimensional hypercube representing corners in decimals

4(100)

2(010) 6(110)

1(001) 3(011) 5(101) 7(111)

minus21

minus20

+21

+20minus20 +20

Figure 6 BSA for 3-bit code

the objective function of the IIR filter Another hypercubeis formed around the current better point and the iterativeprocess is continued All the corners of the hypercuberepresented in the 119863 binary bits code are explored for betterresults simultaneously Table 1 shows the coefficient patternfor 3-coefficient digital IIR filter where 3-bit binary codeis used to represent the 8 corners of the three-dimensionalhypercube (Figure 5) The decimal serial numbers of thehypercube are changed into their respective binary codesThe deviation from the current center point is obtained byreplacing 1rsquos of the code with +Δ and 0rsquos with minusΔ As thenumber of coefficients of the IIR filter increased the numberof hypercube corners increases exponentially So the processbecomes time consuming [23]

33 BSA Strategy To reduce the computational time binarysuccessive approximation (BSA) strategy is used to explorethe optimal solution BSA strategy to search for the optimalsolution is explained in Figure 6 where solution proceduremoves towards the optimal solution by comparing twosolutions at a time represented by the two corners of thehypercube [23]

The search process is started by initializing the coefficientvector 119909119862119905119895 giving objective function Ft To performBSA strat-egy by the iterative process 119862119905119895 is initially selected as below

119862119905119894119895 =

1 for (119895 = 1)

0 for (119895 = 2 3 4 119863)

(33)