DESIGN OF TWO-DIMENSIONAL INFINITE IMPULSE RESPONSE RECURSIVE FILTERS USING HYBRID MULTIAGENT PARTICLE SWARM OPTIMIZATION

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DESIGN OF TWO-DIMENSIONAL INFINITE IMPULSE RESPONSERECURSIVE FILTERS USING HYBRID MULTIAGENT PARTICLESWARM OPTIMIZATIONRajesh Kumar a;Anupam Kumar a

a Department of Electrical Engineering, Malaviya National Institute of Technology, Jaipur, India

Online publication date: 12 April 2010

To cite this Article Kumar, Rajesh andKumar, Anupam(2010) 'DESIGN OF TWO-DIMENSIONAL INFINITE IMPULSERESPONSE RECURSIVE FILTERS USING HYBRID MULTIAGENT PARTICLE SWARM OPTIMIZATION', AppliedArtificial Intelligence, 24: 4, 295 — 312To link to this Article: DOI: 10.1080/08839511003715204URL: http://dx.doi.org/10.1080/08839511003715204

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Applied Artificial Intelligence, 24:295–312Copyright © 2010 Taylor & Francis Group, LLCISSN: 0883-9514 print/1087-6545 onlineDOI: 10.1080/08839511003715204

DESIGN OF TWO-DIMENSIONAL INFINITE IMPULSE RESPONSERECURSIVE FILTERS USING HYBRID MULTIAGENT PARTICLESWARM OPTIMIZATION

Rajesh Kumar and Anupam KumarDepartment of Electrical Engineering, Malaviya National Institute of Technology,Jaipur, India

� We incorporate the optimization problem of two-dimensional infinite impulse response(IIR) recursive filters and the optimization methodology of hybrid multiagent particle swarmoptimization (HMAPSO) and then apply the resultant optimized IIR filter in image processingfor justifying HMAPSO robustness over other algorithm and its role in optimizing real-timesituations. The design of the 2-D IIR filter is reduced to a constrained minimization problemwhose robust solution is being achieved by a novel and optimal algorithm HMAPSO. Thisalgorithm integrates the deterministic solution by the multiagent system, the particle swarmoptimization (PSO) algorithm, and bee decision-making process. All agents search parallel inan equally distributed lattice-like structure to save energy and computational time as done bythe bees in their hive selection process. Thus making use of deterministic search, multiagentPSO, and bee, the HMAPSO realizes the purpose of optimization. Experimental results andthe application of the designed filters to focusing the defocused image show that the HMAPSOapproach provides better upshots than the previous design methods.

INTRODUCTION

Over the past few years a lot of research is being done on designing2-D infinite impulse response (IIR) filters. Two different approaches,transformation approach and optimization approach, are developedfor designing digital 2-D filter (Kaczorek 1985; Tzafestas 1986; Luand Antoniou 1992). Transformation approach involves analog-to-digitaltransformation based on a given set of prescribed specifications that donot work well for IIR filters (Mastorakis et al. 2003). In the optimizationapproach the designing problem is formulated as a multiobjective problemsatisfying some constraints having various local minimas (Lu and Antoniou1992; Mastorakis et al. 2003).

Address correspondence to Rajesh Kumar, Associate Professor, Department of ElectricalEngineering, Malaviya National Institute of Technology, Jaipur, India. E-mail: [email protected]

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296 R. Kumar and A. Kumar

The stability of the designed filters plays a vital role for their practicalimplementation, which is governed by the optimization technique used.Various researchers used genetic algorithm-based approaches to solve the2-D filter problem (Kawamata et al. 1994; Tsaia et al. 2009; Lu and Tzeng2009; Wang et al. 2006; Zhao 2005). It is a common observation that in themultiparameter optimization problem, the genetic algorithm (GA)-basedapproach fails when it gets trapped in the local optima of the objectivefunction where the number of the parameters is large, so one can facenumerous local optima in designing digital 2-D IIR filters (Tsaia et al.2009). To obtain better upshots, researchers have applied various methodsto improve GA. Hybrid Taguchi GAs (HTGAs) for one-dimensional (Tsaiet al. 2004), two-dimensional (Tsai et al. 2005), and structure-specifiedfilters (Tsaia et al. 2009) are some such approaches. The plight of GA-based approaches is their casual output and time taken for differentcomputation. Some researchers involve the neural network to obtain anenhanced and stable solution (Mladenov and Mastorakis 2001).

IIR filters, due to their various local minimums, have a multimodalerror surface. Therefore a consistent design method proposed for IIRfilters, based on a global search procedure, must be used to get betterresults. Many global optimization techniques are already used, which mayinclude particle swarm optimization (PSO) or artificial bee colony (ABC)algorithm. Swarm optimization approach is a stochastic, population-basedevolutionary algorithm for obtaining optimization, used for designing 2-D,zero phase, IIR digital filters (Swagatam and Konara 2007). The ABCalgorithm, which simulates the intelligent foraging behavior of the honeybee swarm, is a simple, robust, and very flexible algorithm also used fordesigning digital IIR filters (Nurhan Karabogaa 2009).

Here, we have used a novel multiagent-based hybrid particle swarmoptimization technique, termed HMAPSO (Kumar et al. 2009). Thealgorithm integrates the deterministic multiagent system (MAS), PSO,and bee decision-making approach. The hybrid algorithm comprises twoparts search algorithm and other as decision-making process. A searchin HMAPSO is based on the multiagent approach and PSO, whereas thedecision-making process is based on the bee decision-making processor forits next hive. This algorithm was tested over various cases where HMAPSOshows its robustness and accuracy over GA and PSO (Seeley et al. 2006;Kumar et al. 2009). In this article HMAPSO-based IIR filter results arecompared with the HTGA, quasi-Newton, neural network, and traditionalGA methods.

The reminder of the article is organized as follows. The next sectiondetails the problem formulation of an IIR filter. Next, we describe theHMAPSO algorithm and then elaborated on the experimental results forvarious methods. Finally, we conclude the paper.

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Design of Two-Dimensional IIR Recursive Filters 297

PROBLEM FORMULATION

For design purposes the following 2-D filter is used whose transferfunction is expressed in Eq. (1) as used in previous approaches (Mastorakiset al. 2003; Tsaia et al. 2009):

H (z1, z2) = H0

∑Ki=0

∑Kj=0 aij z

i1z

j2∏K

k=1 (1 + bkz1 + ckz2 + dkz1z2)(1)

Md is the desired amplitude response of the 2-D filter, which is functionof the frequency of w1 and w2, where

(w1,w2 ∈ �0, ��

). The objective of

the system is to find a transfer function H (z1, z2) as given in Eq. (1) suchthat the function H (e jw1 , e jw2) gauges the desired amplitude response ofMd(w1,w2). To obtain the objective, an optimization problem is consideredwhose aim is to minimize, as stated in Eq. (2):

J = J (aij , bk , ck , dk ,H0)

=N1∑

n1=0

N2∑n2=0

[∣∣M (w1,w2)∣∣ − Md (w1,w2)

]p(2)

where M (w1,w2) = H (z1, z2)|z1=e jw1z2=e jw2

and w1 = (�/N1)n1, w2 = (�/N2)n2 p is

an even positive integer (usually p = 2 or p = 4).So Eq. (2) can be written as

J =N1∑

n1=0

N2∑n2=0

[∣∣∣∣M(�n1

N1,�n2

N2

)∣∣∣∣ − Md

(�n1

N1,�n2

N2

)]p

(3)

Hence, the filter design problem is formulated as a minimizationproblem having multiobjective constraints to being satisfied. Because weare dealing with only first-degree factors in the denominator, it is wellknown that the stability constraint can be stated as follows (Kaczorek 1985;Tzafestas 1986; Lu and Antoniou 1992):∣∣bk + ck

∣∣ − 1 < dk , k = 1, 2, � � � ,K

dk < 1 − ∣∣bk + ck∣∣ , k = 1, 2, � � � ,K �

Different authors solve this problem with different approaches.Mladenov and Mastorakis (2001) solved this problem using the neuralnetwork, whereas Mastorakis et al. (2003) applied GA to find the optimalsolution; Tsai et al. (2004, 2005) used the improved GA. PSO is also beingtested and used for designing the 2-D, zero-phase IIR, which proves better

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298 R. Kumar and A. Kumar

than previous methods (Swagatam and Konara 2007). Artificial ABC isalso used for designing low- and high-order digital IIR filters (NurhanKarabogaa 2009). Novel global, robust optimization approaches have beenused, with hybrid features of PSO, MAS, and bee optimization. The resultproves its advantage over other approaches.

HMAPSO ALGORITHM

For the HMAPSO algorithm different agents are sent in the wholesearch area, which is divided into different fragments (Kumar et al.2009). For a global and robust optimal solution the total range of theindependent parameters are divided into smaller volumes, each of whichdetermines the starting point for the exploration for each agent. Theagent then finds its own optimized point by a developed optimizationtechnique, the Nelder–Mead method (Nelder and Mead 1965). Each agentthen passes the information regarding the optimized point by bee waggledance. When all the information of optimized points is obtained, then thebest among these is chosen by a consensus method, as in case of honeybee swarms (Seeley et al. 2006; Kumar et al. 2009).

Particle Search Methodology

For optimization of the any objective function, the Nelder–Mead method is modified (Kumar et al. 2009). Deterministic searchmethodology is used but in a sense similar to swarm local search. Letz = f (x , y) be the function that is to be minimized. For agents this isfood function. To start we assume that agent considers three verticesof a triangle as food points for a two variables problem as z1, z2, andz3. z1 = (x1, y1) represents the initial position of agent z2 = (x2, y2) andz3 = (x3, y3) are the positions of probable food points (i.e., local optimalpoints). The movement of agents from its initial position toward the foodposition (i.e., optimization point) is as follows. Here we considered theproblem as to generate the minima of a function zi = f (xi , yi).

The function zi = f (xi , yi) for i = 1, 2, 3 is evaluated at each of thesethree points. The obtained values of zi are recorded in a way that z1 ≤z2 ≤ z3 with corresponding agents positions and food points from thebest to worst position. The construction process uses the midpoint of theline segment joining the two best food positions z1 and z2, as shown inFigure 1(a).

The value of function decreases as agent moves along z3 to z1 or z3 toz2. Hence, it is feasible that f (x , y) takes a smaller value if agent movestoward z12. For further movement of the agent a test point zt is chosen in

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Design of Two-Dimensional IIR Recursive Filters 299

FIGURE 1 Agents search movements with the proposed optimization algorithm. (a) Starting of themotion in search of solution. (b) Extension in the direction of good optimal point. (c) Contractionof the movement in case optimal point quality is not good. (d) Shrinking of the space towardoptimistic solution.

such a way that it is reflection of the worst food point (i.e., z3), as shownin Figure 1(a). The vector formula for zt is

zt = 2 × z12 − z3 (4)

If the function value at zt is smaller than the function value at z3, thenthe agent has moved in the correct direction toward minimum. Perhapsthe minimum is just a bit further than the point zt . So the line segment isextended further to ze through zt and z12. The point ze is found by movingas additional distance d/2 along the line, as shown in Figure 1(b). If thefunction value at ze is less than the function value at zt , then the agent hasfound a better food point than zt .

ze = 2 × zt − z12 (5)

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300 R. Kumar and A. Kumar

FIGURE 2 Movement of the agents for a given problem.

If the function value at z12 and z3 are the same, another point must betested. Two test points are considered by the agent on the both sides of z12at distance d/2, as shown in Figure 1(c).

The point of smaller value frames a new triangle with other two bestpoints. If the function value at the two test points is not less than thevalue at z3, the points z2 and z3 must be shrunk toward z1, as shown inFigure 1(d). The point z2 is replaced with z12, and z3 is replaced with themidpoint of the line segment joining z1 and z3. Figure 2 shows the pathtrace by the agents and the sequences of triangles �Tk� converging to theoptimal point for the objective function

f (x , y) = x2 − 4x + y2 − y − xy (6)

The starting point in any search lattice plays a vital role in getting robustoptimal solution. Kumar et al. (2009) concluded that the center of thelattice is a good starting point to get a better optimal solution.

Exploration

In MAS, all agents live in an environment (Wooldridge 2002). Anenvironment is organized in a structure, as shown in Figures 3–5. In

FIGURE 3 Domain of the objective function with one independent parameter.

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Design of Two-Dimensional IIR Recursive Filters 301

FIGURE 4 Domain of the objective function with two independent parameters.

the environment each agent is fixed on a lattice point and each circlerepresents an agent; the data in the circle represent the position of agentand the evaluated value of the function. The size and dimension of thelattice depends on the variables and search space.

The value of the objective function depends on p number ofindependent parameters. Let the range of j th parameter ∈ [

Wji ,Wjf

],

where Wji and Wjf represent the initial and final value of the parameter.Thus the complete domain of the objective function can be representedby a set of p number of axis. Each axis is in a different dimension andcontains the total range of one parameter.

FIGURE 5 Domain of the objective function with three independent parameters.

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302 R. Kumar and A. Kumar

The next step is to divide each axis into smaller parts. Each part isknown as a step. Let the j th axis be divided in nj number of step each oflength Sj , where j = 1 to p. This length Sj is known as step size for the j thparameter. The relationship between nj and Sj can be given as

nj = Wjf − Wji

Sj(7)

Hence, each axis is divided into their corresponding branches. If we takeone branch from each axis, then these p number of branches constitute ap dimensional volume.

The total number of such volumes can be calculated as

Number of volumes, Nv =p∏

j=1

nj (8)

The number of volumes indicates the number of agents going out forexploration. One point inside each volume is chosen as the starting pointfor the optimization, which in our approach is the midpoint of thatvolume. The midpoint of total cluster can be calculated as follows;[

Wi1 + Wf 1

2,Wi2 + Wf 2

2, � � � ,

Wip + Wfp

2

](9)

For an objective function having one independent parameter, thecomplete domain is given by a single axis, represented as h1. Here, eachstep gives us one volume.

Let us take the following values:

p = 1, W1i = 1, W1f = 6, S1 = 1

Therefore n1 = 5 and Nv = 5. Thus five agents are sent for exploration.The starting point for each agent is the midpoint of each step, as shownin Figure 3.

For an objective function having two independent parameters, thecomplete domain is given by a set of two axes represented as h1 and h2. Letus take the following values:

p = 2, W1i = 1, W1f = 5, S1 = 1 and

W2i = 1, W2f = 5, S2 = 1

Therefore n1 = 4, n2 = 4, and Nv = 16. Thus 16 agents are sent forexploration, as shown in Figure 4. The starting point of each agent is

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Design of Two-Dimensional IIR Recursive Filters 303

the midpoint of each volume, which in this case are two-dimensionalrectangles.

For an objective function with three independent parameters, thecomplete domain is given by a set of three axes represented as h1, h2, andh3. Let us take the following values:

p = 3, W1i = 1, W1f = 5, S1 = 1, W2i = 1, W2f = 4, S2 = 1 and

W3i = 1, W3f = 4, S3 = 1

Therefore n1 = 4, n2 = 3, n3 = 3 and Nv = 36. Thus 36 agents are sentfor exploration. The starting point for each agent is the midpoint ofthe corresponding volume, which in this case is 3-dimensional cuboid, asshown in Figure 5. Objective functions with more than three independentparameters can also be solved in the similar manner.

Bee Swarm-Based Decision Process

Honey bee swarms have a highly distributed decision-making processthat they use for finding their next hive or a new source of food. Hundredsof bees out of thousands work as scout bees to start a search for the nextpossible site. Upon finding the site, scouts inform other bees by a waggledance (Seeley et al. 2006). Discovered nest sites of sufficient quality arereported on the cluster via the scouts’ waggle dance. Depending on thewaggle dance by scout bees, quiescent bees get activated and decided torecruit or explore for nest site. If an uncommitted bee is not satisfied withany of the scout sites, then she can explore for new sites. When a beeadvertises a site more than once, in every subsequent turn she decreasesthe strength of her dance by about 15 dance circuits. Once the quorumthreshold reaches for any one of the sites, the bee starts piping signals thatelicit heating by the quiescent bees in preparation for flight. There aretwo methods used by bee swarms to find out the best nest site, consensusand quorum (Seeley et al. 1991, 2006). In consensus widespread agreementamong the group is taken into account, whereas in quorum the decision forbest site happens when a site crosses the quorum (threshold) value. In thisarticle the consensus algorithm is used for finding out the optimum solution(i.e., best food site) (Kumar et al. 2009).

Waggle Dance

Here in the proposed algorithm the agents provide their individualoptimal solution to the centralized systems, which then choose thepreferable solution from the searched one. For optimal minimum cases it

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304 R. Kumar and A. Kumar

selects the best optimal solution, which can mathematically stated as

Wdi = min(fi(X )) (10)

where fi(X ) represent the different search value obtained by an agent.Each of these points is recorded in a table known as optimum vector tableX . X is a vector containing p number of elements. These elements containthe value of parameters at that point. So both the optimal solution valueand the corresponding variable values are recorded. This record is knownas Personal Best (i.e., Pbest in PSO). The function value changes accordingto the objective function requirement; that is, if the objective function isto be minimized, then the minimum function is used, and if we have tofind maximize in an objective function, it will switch over to maximizefunction.

ConsensusBee swarms use the consensus method to decide the best obtained or

search value, and here we mimic this event and behavior by comparingthe results obtained. Once exploration and waggle dance (transmission ofdata) is finished, the global optimized point is chosen by comparing thefitness values of all the optimized points in the optimum vector table (i.e.,global best, gbest as in case of PSO). For minimization problems the pointwith the lowest fitness value is selected as the global optimized point. Theglobal optimized point XG is found by

f (XG) = min[f (X1), f (X2), � � � , f (XNv)] (11)

ILLUSTRATIVE EXAMPLE AND COMPARISONS

For comparing the result of the system, a setup is designed, which hasalready been used by various authors (Mastorakis et al. 2003; Tsaia et al.2009; Mladenov and Mastorakis 2001; Swagatam and Konara 2007). Thedesired amplitude response of the 2-D filter is given as

Md(w1,w2) =

1 if

√w2

1 + w22 ≤ 0�08�

0�5 if 0�08� <√w2

1 + w22 ≤ 0�12�

0 otherwise�

Figure 6 shows the desired amplitude response∣∣Md(w1,w2)

∣∣. Thecorresponding minimization constraint optimization problem in Eq. (2) is

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Design of Two-Dimensional IIR Recursive Filters 305

FIGURE 6 Desired amplitude responses |M (x1, x2)| at the design condition of p = 2.

being written as

MinimizeJ =50∑

n1=0

50∑n2=0

[∣∣∣M (�n1

50,�n2

50

)∣∣∣ − Md

(�n1

50,�n2

50

)]2

subject to the constraints given by

∣∣bk + ck∣∣ − 1 < dk , k = 1, 2, � � � ,K

(12)dk < 1 − ∣∣bk − ck

∣∣ , k = 1, 2, � � � ,K

having K = 1, 2 and N1 = 50, N2 = 50.

H (z1, z2) = H0

a00 + a01z2 + a02z22 + a10z1 + a20z21 + a11z1z2+ a12z1z22 + a21z21 z2 + a22z21 z

22

(1 + b1z1 + c1z2 + d1z1z2) (1 + b2z1 + c2z2 + d2z1z2)(13)

With the purpose of illustration, we have taken K = 2 for bettercomparison. Then, H (z1, z2) can be expressed from Eqs. (1) to (13). Toimplement the HMAPSO algorithm the M (w1,w2) is expressed in term of

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306 R. Kumar and A. Kumar

its constraint in Eq. (14a), where

M (w1,w2) = H0

(a00 + a01c01 + a02c02 + a10c10 + a20c20 + a11c11+ a12c12 + a21c21 + a22c22)

D

−j(a01s01 + a02s02 + a10s10 + a20s20 + a11s11

+ a12s12 + a21s21 + a22s22)

D

(14a)

cpq = cpq(w1,w2) = cos(pw1 + qw2)

spq = spq(w1,w2) = sin(pw1 + qw2) (14b)

p, q = 0, 1, 2

And

D = �(1 + b1c10 + c1c01 + d1c11) − j (b1s10 + c1s01 + d1s11)�

× �(1 + b2c10 + c2c01 + d2c11) − j (b2s10 + c2s01 + d2s11)� (14c)

In compact form the M (w1,w2) can be expressed as

M (w1,w2) = H0AR − jAI

(B1R − jB1I )(B2R − jB2I )(15)

where

AR = a00 + a01c01 + a02c02 + a10c10 + a20c20 + a11c11 + a12c12 + a21c21 + a22c22

AI = a01s01 + a02s02 + a10s10 + a20s20 + a11s11 + a12s12 + a21s21 + a22s22

BIR = (1 + b1c10 + c1c01 + d1c11)(16)

B1I = (b1s10 + c1s01 + d1s11)

B2R = (1 + b2c10 + c2c01 + d2c11)

B2I = (b2s10 + c2s01 + d2s11)

Now the magnitude of M (w1,w2) can be expressed as

∣∣M (w1,w2)∣∣ = H0

√A2

R+ A2

I(B2

1R+ B2

1I

)(B2

2R− B2

2I

) (17)

And Eq. (18) for constraint can be stated from Eq. (12) as

−1 + dk < (bk + ck) < (1 + dk)

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Design of Two-Dimensional IIR Recursive Filters 307

−(−1 + dk) < (bk − ck) < (1 − dk) (18)

(1 + dk) > 0 (1 − dk) > 0

where K = 1, 2. Hence, the optimization problem comes as solving theequation optimizing at the cost of following constraint vector: x =(a00, a02, a01, a11, a20, a21, a22, b1, b2, c1, c2, d1, d2,H0)

T �For proving HMAPSO, a better and optimal approach compared the

result with other competitive approaches, which are as follows: HTGA(Tsaia et al. 2009), which emphasizes over convergence of HTGA, andswarm intelligence algorithm, termed MEPSO (Swagatam and Konara2007), which uses PSO for designing 2-D zero-phase recursive filters,are two new methods for optimizing the x vector. The classical DE(Tsai et al. 2005) approach, which use binary crossover for GA, is alsobeing compared. The G3 (generalized generation gap) model (Mladenovand Mastorakis 2001), which incorporates the generic parent centricrecombination scheme (PCX) and steady state, elite preserving, scalable,and computationally fast population alteration model of the GA results,is also compared. Others methods include neural method (Mladenovand Mastorakis 2001), quasi-Newton method (Lu and Tzeng 2009), andGA (Tsai et al. 2005). An optimal upshot is derived using HMAPSOrepresented in Eq. (19a). Its optimized constrained vector can beexpressed as Eq. (19b):

H (z1, z2) = 0�00024

1 + 0�5815z2 + 0�2207z22 + 0�4387z1 + 0�4045z21− 1�4084z1z2 − 0�5720z1z22 − 0�8418z21 z2 + 2�277z21 z

22(

1 − 0�9078z1 − 0�9058z2 + 0�8373z1z2)

× (1 − 0�9075z1 − 0�9101z2 + 0�8406z1z2

)(19a)

x = (1, 0�5815, 0�2207,−1�4084, 0�4045,−0�8418, 2�277,−0�9078,

− 0�9705,−0�9058,−0�9101, 0�8373, 0�8406, 0�00024)T (19b)

For defining the space region, the first search is carried out usingPSO, as explained in the algorithm. Once the exploration area is defined,different agents are sent to explore within their cavity. Each agent comesup with their individual optimal solutions within their search space. Theseindividual solutions are examined using bee swarm decision method asexplained in the algorithm. The algorithm is stated as follows:

1. Initialize the number of parameters, p. Initialize the length of steps (noof axes to be explored), Sj (j = 0 to p).

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308 R. Kumar and A. Kumar

2. Initialize the range for each parameter as[Wij ,Wfj

]where j = 0, 1, � � � , p.

3. Calculate the number of steps in each step:

nj = Wfj − Wij

Sj

4. Calculate the total number of volumes:

Nv =p∏

j=1

nj

5. For each volume, take the starting point of the exploration as themidpoint of the volume

[Wi1+Wf 1

2 ,Wi2+Wf 2

2 , � � � ,WiP +WfP

2

].

6. Explore the volume according to modified Nelder–Mead method.7. Record the value of optimized point obtained corresponding to each

volume in optimum vector table in the following way: [X1,X2, � � � ,XNv ].8. After the exploration is completed, the global optimized point is

calculated in the following manner using the bee decision approach:

F (XG) = min[F (X1), F (X2), � � � , F (XNv)]

Comparison between the optimal result of J with p = 2 uses differentmethods, with HMAPSO is performed, and the results are shown inTable 1. Figure 7 shows different method results. HTGA results are takenfrom Tsaia et al. (2009), and MEPSO, DE, and G3 with PCX results aretaken from Mladenov and Mastorakis (2001), with p = 2. Quasi-Newton,neural network, and GA results are taken from Tsaia et al. (2009) forcomparison with HMAPSO. Similarly, Table 2 shows time comparisonbetween the methods, where the other results are taken from Tsaia et al.(2009). The result shows the robustness and stability of the HMAPSO fordesigning of 2-D IIR filter.

TABLE 1 Computational Results for J by Using Different Methods

Method Best Average Standard deviation

HMAPSO 1.586576 1.586576 0.00HTGA (Tsaia et al. 2009) 2.8664 3.3888 0.2191MEPSO (Mladenov and Mastorakis 2001) 9.0005 9.01665 0.221DE (Mladenov and Mastorakis 2001) 11.907 12.1993 0.221G3 with PCX 10.425 10.0673 0.221Quasi-Newton (Lu and Tzeng 2009) 7.0810 NA NANeural network (Mladenov and Mastorakis 2001) 3.7772 NA NAGA (Tsai et al. 2005) 6.0276 NA NA

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FIGURE 7 Comparison of amplitude responses |M (x1, x2)| by different methods.

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310 R. Kumar and A. Kumar

TABLE 2 Computational Time Results Using Different Agent Sizes

Agent Size Best Time

HMAPSO 1.586576 1.592468HTGA (Tsaia et al. 2009) 2.8664 3.3888Quasi-Newton (Lu and Tzeng 2009) 7.0810 NANeural network (Mladenov and Mastorakis 2001) 3.7772 NAGA (Tsai et al. 2005) 6.0276 NA

FIGURE 8 Processing results of the designed filters for focusing the image.

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Design of Two-Dimensional IIR Recursive Filters 311

To show heftiness of IIR filter so obtained using HMAPSO, it is appliedin image processing. A defocused image is taken from Tsaia et al. (2009)for comparison, and Figure 8 shows the results after applying differentfilters. HMAPSO provides better upshot, which can be used for optimizingIIR filters.

CONCLUSION

Here we enumerate the HMAPSO approach to produce a highlyoptimized, 2-D, IIR digital, structure-specified filter. HMAPSO, dueto its hybrid nature of multiagent approach, particle swarm tactics,and bee-based decision-making approach, is well suited for real-timemultioptimization problems like 2-D IIR filter. IIR filter is first convertedinto a multiobjective optimization problem, which satisfies someconstraints. The objective function is then solved. The performance ofthe proposed method was compared with that of a well-known availablesolution using conventional optimization algorithm, PSO algorithm,GA, and improved GA for the system identification purpose. From thesimulation and experimental results, it was observed that the methodbased on the HMAPSO seems as an alternative approach for designingdigital low- and high-order IIR filters. The results shows that HMAPSOremoves the randomness in the algorithm and provides a better upshotas compared with its counterparts and improves significantly in globaloptimization performance of optimizing real-time optimization problemslike IIR filter.

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