8/8/2019 Sun-Fang-Xu

1/5

IEEE TRANSACTI ONS ON CIRCUI TS AND SYSTEMSII: EXPRES S BRI EFS, VOL. 57, NO. 2, FEBRUARY 2010 141

A Quantum-Behaved Particle Swarm OptimizationWith Diversity-Guided Mutation for the Design

of Two-Dimensional IIR Digital FiltersJun Sun, Member, IEEE, Wei Fang, and Wenbo Xu, Member, IEEE

AbstractThis brief proposes quantum-behaved particleswarm optimization with diversity-guided mutation (QPSO-DGM) to solve the problem of designing the optimal 2-Dzero-phase IIR digital filters. The new method integrates adiversity control strategy into QPSO to guide the particles searchand thus improve the capabilities of exploration. Numerical resultsdemonstrate that the design approach based on QPSO-DGM canobtain better digital IIR filters than the existing methods.

Index TermsDigital IIR filters, heuristics, multidimensional

digital filters, particle swarm optimization (PSO).

I. INTRODUCTION

T WO-DIMENSIONAL (2-D) IIR filter design has attractedgrowing attention due to its extensive applications in thedomain of denoising of digital images, biomedical imaging anddigital mammography, etc. [1][3]. The most popular designmethods, which may result in an unstable filter, are based eitheron appropriate transformation of 1-D filters or on appropriateoptimization techniques such as linear programming, Remezexchange algorithm, and nonlinear programming.

Modern heuristic methods have also been employed for 2-D

IIR filter design problems, such as genetic algorithm (GA)[4], neural network (NN) [5], particle swarm optimization(PSO) [6], and the computer language GENETICA [7]. Thesetechniques were able to find out better solutions than thosementioned in the previous paragraph.

The PSO algorithm, which is motivated by social behavior ofbird flock or fish schooling, is a population-based optimizationtechnique [8]. In PSO, the potential solutions, called particles,fly toward the optima by following their own experiencesand the current global best particles. In this brief, based ona variation of PSO, namely, quantum-behaved PSO (QPSO)[9], we propose QPSO with diversity-guided mutation (QPSO-DGM) for the design of 2-D IIR digital filters. The method istested, and a performance comparison is made with some ofother heuristics.

Manuscript received June 25, 2009; revised October 10, 2009. First pub-lished January 22, 2010; current version published February 26, 2010. Thiswork was supported in part by the Innovative Research Team Project ofJiangnan University under Contract JNIRT0702. This work was recommendedby Associate Editor M. Chakraborty.

The authors are with School of Information Technology, Jiangnan Uni-versity, Wuxi 214122, China (e-mail: sunjun_wx@hotmail.com; wxfangwei@163.com; xwb@jiangnan.edu.cn).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCSII.2009.2038514

The brief is organized as follows. Section II describesthe problem. The proposed QPSO-DGM is presented inSection III. Section IV provides and discusses the experimentalresults on a 2-D filter design problem. Section V offers someconclusions.

II. PROBLEM FORMULATION

For design purposes, we consider the 2-D recursive filterswith the following transfer function:

H(z1, z2) = H0

Ni=0

Nj=0 aijz

i1z

j2N

k=1(1+ qkz1+rkz2+skz1 z2), a00= 1

(1)

where N is the dimension of the filter. Let the frequencies1, 2 [, ], z1 = e

j1 , and z2 = ej2 . The design task

of 2-D filters amounts to finding a transfer function H(z1, z2)as in (1) such that the function M(1, 2) = H(e

j1 , ej2)approximates the desired amplitude response of the 2-Dfilters Md(1, 2). The approximation can be achieved byminimizing [4][7]

J = J(aij , qk, rk, sk, H0)

=

N1k1=0

N2k2=0

M

k1N1

,k2N2

Md

k1N1

,k2N2

p

(2)

where p is a positive even integer (p = 2, 4, or 8), and N1and N2 are positive integers. Thus, the design of 2-D filters isequivalent to the following constrained minimization problem:

Minimize J= J(aij, qk, rk, sk, H0) (3a)

s.t.

|qk+rk|1 < sk, sk < 1|qkr|, k = 1, 2, . . . , N .

(3b)

Here, the prime object is to reduce the difference between thedesired and actual amplitude responses of the filter at N1 N2points. Since the denominator only contains first-degreefactors, we can assert the stability conditions as the constraints(3b) [4][7].

1549-7747/$26.00 2010 IEEE

8/8/2019 Sun-Fang-Xu

2/5

142 IEEE TRANSACTI ONS ON CI RCUI TS AND SYSTEMSII: EXPRESS BRI EF S, VOL. 57, NO. 2, FEBRUARY 2010

III. PROPOSED QPSO-DGM ALGORITHM

In QPSO with m particles in the n-dimensional space, forparticle i (1 i m), the jth (1 j n) component of itsposition at iteration t + 1 is updated by

xt+1i,j

=pti,j Ct

ixt

i,j ln 1/ut

i,j Ct

j= (1/m)

m

i=1

yti,j

(4)

where pti,j = ti,jy

ti,j + (1

ti,j)y

tj , with y

ti,j and y

tj repre-

senting the jth components of the personal best (pbest) positionof the particle and the global best (gbest) position of the pop-ulation. Here, C is called the mean best (mbest) position, andti,j , u

ti,j U(0, 1), where U(0, 1) is the uniform distribution

in [0, 1]. The parameter is called the contractionexpansioncoefficient.

Although QPSO is efficient in solving global optimizationproblems, it may also encounter premature convergence, whichis the result of constant declination of population diversity

during the search. In the proposed QPSO-DGM algorithm, weset a lower bound dl for the diversity, which, as in [10], ismeasured by

dt = [1/ (M |A|)] mi=1

nj=1

xti,j x

tj

2(5)

where xtj = (1/m)m

i=1 xti,j , and |A| is the length of the

longest diagonal in the search space. If dt decreases to belowdl, the following mutation is operated. That is, for each j, let

ztj = ytj + |A| , N(0, 1). (6)

Then set ytj = ztj and y

tg,j = z

tj , where y

tg,j denotes the pbest

position of the current particle with the gbest position (calledthe gbestparticle). N(0, 1) denotes the standard normal distri-bution, and is a user-specified parameter, which should bemuch larger than dl so that dt can promptly increase to abovedl after mutation. We suggest that 10dl. z

t is a temporaryvector. When the mutation is operated, the displacement ofthe gbest particle increases the value of |ytj y

ti,j| and pulls

the position C away from its original position, making theparticles search scope expanded and, in turn, resulting in thegain ofdt. Here, we outline the QPSO-DGM algorithm.

Procedure of QPSO-DGM:Step 0: Initialize particles with random position.Step 1: For t = 1 to the maximum number of iterations,execute the following steps.Step 2: Calculate the mean best position C among theparticles.Step 3: For each particle, compute its fitness value f(xti). Iff(xti) < f(y

t1i ), then y

ti = x

ti.

Step 4: Select the gbestposition yt among particles.Step 5: Measure dt according to (5). If dt < dl, executeStep 6, or else go to Step 7.Step 6: Exert the mutation on each ytj and y

tg,j as (6) and

update the fitness value of the gbest particle.

Step 7: Update each component of the current position by (4)and return to Step 1.

TABLE IPARAMETER CONFIGURATIONS FOR THE COMPETITOR ALGORITHMS

To test QPSO-DGM, we use the same example of the designproblem as in [4][7], where Md(1, 2) is given by

Md(1, 2) =

1,

21 + 22 0.08

0.5, 0.08