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Applied lied Soft Comp Computin uting g 25 (201 (2014) 4) 530– 530–534 534 App

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Applied Soft Computing  j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Swarm intelligence based techniques for digital filter design Archana Sarangi a , Shubhendu Kumar Sarangi a , Sasmita Kumari Padhy a , Siba Prasada Panigrahi b,∗ , Bijay Ketan Panigrahi c a

ITER, SOA University, University, Bhubaneswa Bhubaneswar, r, India CVRCE,, Bhubaneswa CVRCE Bhubaneswar, r, India c IIT IIT,, Delhi,India Delhi,India b

a r t i c l e

i n f o

 Article history: Received Rece ived 15 Septemb September er 2011 Re Recei ceive ved d in revise revised d form form 12 May May 2013 Accep Accepted ted 2 Jun June e 2013 2013 Av Avail ailab able le onl online ine 14 June June 2013 2013

a b s t r a c t

Thi Thiss paper paper deals with the problem of  digital IIR  filter design. Two novel modifications are propos proposed ed to Particle Swarm Optimization and validated through novel application for design of  II IIR R fil filte ter. r. First modification is based on quantu quantum m mechanics and proved to yield a bett better er performance. The second modification isto take care of time of time dependency character of the of the constriction factor. Extensive simulation results validate the superior performance of proposed of  proposed algorithms. © 2013 Elsevier B.V. All rights reserved.

Keywords: Digital Digi tal filte filterr design design Particle Part icle Swa Swarm rm Opti Optimiza mization tion IIR filter

Introduction

Studies by Fang et al. [1] conforms the findings by the [2–13].. This This wo work rk can can be tr trea eate ted d as exte extens nsio ion n of the the researchers of  researchers of [2–13] wo work rkss done done in [1] [1].. For For ease ease of read eading, ing, we have have repr reprod oduc uced ed the wo work rk as first first part part of this this arti articl cle e that that pa pave vess a wa way y for for the the next next part part of the art articl icle. e. The The prob proble lem, m, desi design gn of IIR IIR filte filters rs,, erro errorr surf surfac ace e in mo most st of the the cases is non-q non-quadr uadratic atic and mult multi-mo i-modal dal [2] [2].. In a mul multiti-mod modal al err error or surf surfac ace, e, glob global al mi mini nima ma can can be ac achi hiev eved ed by a gl glob obal al opti optimi miza zati tion on technique. techn ique. Bio-i Bio-inspir nspired ed algor algorithm, ithm, Genet Genetic ic Algor Algorithm ithm (GA) [2] [2],, and its disa hybr hybrids ids [3–5] avebeen use used dck fordigit fordigital al IIRfilterdesig IIRfilterdesign. n.ilit Bec Becaus e of di sadv dvan anta tage gesshof GA (i.e (i.e., ., la lack of good good loca locall sear search ch abil ab ity yause and and or premat pre mature ure conver convergen gence) ce) inv invite ited d Sim Simula ulated ted Ann Anneal ealing ing (SA (SA)) [6] f or use use in the the di digi gita tall IIR IIR fil filte terr desi design gn.. Ag Agai ain, n, stan standa dard rd SA is very very slow slow and als also o req requir uires es rep repeat eated ed eva evalua luatio tions ns of cos costt fun functi ction on to con conver verge ge to the glo global bal min minima ima.. Hen Hence, ce, Dif Differ ferent ential ial Evo Evolut lution ion (DE (DE)) [7] finds nds a sco scope pe in dig digita itall IIRfilte IIRfilterr des design ign.. But But,, theprobl theproblem em wit with h DE alg algori orithm thm is that that it is sens sensit itiv ive e to th the e choi choice ce of its its cont contro roll para parame mete ters rs.. Part Partic icle le Swar Swarm m Op Opti timi miza zati tion on (PSO (PSO)) [8,9] also used in the prob proble lem m of di digi gita tall IIR IIR fil filte terr desi design gn.. PSO PSO can can be ea easi sily ly im impl plem emen ente ted d and and al also so ou outp tper erfo form rmss GA in ma many ny of the the appl applic icat atio ions ns.. But, But, as th the e part article icless in PSO PSO only only sear search ch in a finite nite sam sampl plin ing g spa space, ce, PSO may easi easily ly be trap trapp ped in intto lo loca call opti optima ma.. On One e usef useful ul modifi odifica cattion ion of  PSO PSO is Qu Quan antu tumm-Be Beha have ved d Part Partic icle le Sw Swar arm m Opti Optimi miza zati tion on (QPS (QPSO) O) [10–13]. QPSO SO uses ses the the merit eritss of qua quantum ntum mecha echani nics cs and and PSO. PSO. [10–13] . QP

The The pr prob oble lem m of di digi gita tall IIR IIR filte filterr desi design gn is a mi mini nimi miza zati tion on pr prob oble lem m that that ca can n be solv solved ed by PSO PSO and and QPSO QPSO.. In addi additi tion on,, th the e pa para rame mete ters rs of fil filte terr ar are e real real code coded d as a pa part rtic icle le and and th then en th the e sw swar arm m repr repres esen ents ts all the can candid didate ate sol soluti utions ons.. Fang et al al.. pr prop opos osed ed one one modi modific ficat atio ion n to th the e stan standa dard rd In [1], [1], Fang version of QPSO. This modification, QPSO-M, is proposed by intr introd oduc ucin ing g a ra rand ndom om vect vector or in QP QPSO SO in orde orderr to enha enhanc nce e th the e ra ranndomn domnes esss an and d gl glob obal al sea searc rch h abil abilit ity. y. In th this is pape paperr we pr prop opos ose e a modi mo dific ficat atio ion n to PSO, PSO, wh wher ere, e, th the e co cons nstr tric icti tion on fa fact ctor or is al allo lowe wed d to change cha nge in each each of theitera theiteratio tions ns wit with h a lin linear early ly decrea decreasin sing g K . Im Impor por-tanc tance e an and d util utilit ity y of pr prop opos osed ed modi modific ficat atio ions ns va vali lida date ted d by no nove vell appl applic icat atio ion nsign to de sign gn of IIand R filte fil ter. r. Thre Th ree e exam ex ampl ples es for ford th the eSO. purp pu. rpos ose e of fil filte ter r desi de gndesi ar are e test tested edIIR an d co comp mpar ared ed wi with th PSO PS O an and QP QPSO Re Rest st part part of th the e pa pape perr is orga organi nize zed d as: as: prob proble lem m of filte filterr desi design gn is outl outlin ined ed in sec secti tion on ‘Pro ‘Probl blem em stat statem ement ent’. ’. PSO, PSO, QP QPSO SO and and pr prop opos osed ed mo modi dific ficat atio ions ns ar are e di disc scus usse sed d in sect sectio ion n ‘M ‘Met etho hodo dolo logi gies es’’ foll follow owed ed by sim simula ulatio tion n exampl examples es in sectio section n ‘Simul ‘Simulati ation on result results’. s’. Fin Finall ally, y, the paper pap er is con conclu cluded ded in sect section ion ‘Concl ‘Conclusi usion’ on’.. Problem Prob lem state statement ment

In th this is pape paper, r, we ha have ve used used th the e sa same me syst system em stru struct ctur ure e as th that at reprod oduc uced ed in th this is sect sectio ion. n. He Here re,, th the e mo mode dell of IIR  IIR  used in [1] and repr used fil filte terr is consi conside dere red d to be same same as th that at of th the e auto autore regr gres essi sive ve mo movi ving ng averag ave rage e (AR (ARMA) MA) model. model. The mod model el can be defi definedby nedby the differ differenc ence e equation [3] [3]:: L

M  ∗ Correspo Corresponding nding author. author. Tel.: +91 9437 9437262 262878 878.. E-mail E-ma il address: address: siba panigrahy [email protected] [email protected] m (S.P. Panigrahi).

1568 1568-49 -4946 46/$ /$ – seefront matte matterr © 2013 2013 Els Elsevi evier er B.V. B.V. All rightsreser rightsreserved ved.. http://dx.doi.org http://dx.doi.org/10.1016/j. /10.1016/j.asoc.2013.06 asoc.2013.06.001 .001

 y(k) +

 i=1

bi y(k − i) =

 i=0

ai x(k − i)

(1)

 

531

 A. Sarangi et al. / Applied Soft Computing 25 (2014) 530–534

where  x(k) repr repres esen ents ts in inpu putt to the the fil filte terr and and  y(k) repr repres esen ents ts outoutput put from from the the fil filte ter. r. M (≥ L) is order of the filter, ai   and bi   are are th the e adju adjust stab able le coef coeffic ficie ient nts. s. The The tr tran ansf sfer er fu func ncti tion on of this this mo mode dell ta take kess following follo wing genera generall form form::

H ( z ) =

L a  z −i i=0 i M  b  z −i i=1 i

 

  A( z )  = 1 + B( z ) 1+

(2)

The The prob proble lem m of IIR filter lter desi design gn can can be cons consid ider ered ed as an opti opti-mi miza zati tion on prob proble lem, m, wh wher ere, e, the the mean mean squa square re er erro rorr (MSE (MSE)) is ta take ken n as th the e cost cost fu func ncti tion on.. He Henc nce, e, th the e co cost st fu func ncti tion on is: is:

 J (ω) = E [e2 (k)] = E [{d(k) − y(k)}2 ]

(3)

Here, d(k) is the the desi desire red d resp respon onse se of the the fil filte ter, r, e(k) = d(k) − y(k) is the err error or sig signal nal.. Using Using two set setss of coe coeffic fficien ients, ts, {ai }Li=0  and {bi }M  i=1  we can can get get the the comp compos osit ite e we weig ight ht vect vector or for for the the filte filter. r. This This is de defin fined ed as:

ω = [a0 , . . . , aL , b1 , . . . , bM ]







e2 (k)

(5)

k=1

Here, N  rep repres resent entss the num number ber of sam sample pless use used. d.

This This sect sectio ion n di disc scus usse sess on opti optimi miza zati tion on al algo gori rith thms ms used used in th this is pape paperr fo forr th the e prob proble lem m of di digi gita tall II IIR R fil filte terr de desi sign gn.. QPSO

PSO [14,15] is a glob globa al sear search ch techn echniique que and and ba base sed d on soci social al behavior of animals. In PSO, each particle is a potential solution to a problem. In M -dim - dimen ensi sion onal al spac space, e, pa part rtic icle le posi positi tion on is and velo veloci city ty is de defin fined ed as V i  = defined defi ned as  X i = ( xi1 , . . .,  xid , . . .,  xiM ) and (v i1 , . . . , v id , . . . , v iM ). Part Partic icle less can can re reme memb mber er thei theirr own own pr prev evio ious us bestt pos bes positi ition. on. The veloci velocity ty and pos positi ition on upd updati ation on rul rule e for par partic ticle le i at (k + 1)t 1)th h iterat iteration ion use usess the rel relati ations ons:: v k+1

k  −  xij k );   = w · v ijk  + c 1 · r 1k j  · (P ijk  −  xkij ) + c 2 · r 2k j  · (P  gj

 xkij+1 

r 1k j , r 2k j ∈ U ((0 0, 1)

=  xijk  + v kij+1

(6)

Here, c 1  (cogni  (cognitiv tive) e) and c 2   (soc (socia ial) l) are are two two posi positi tive ve cons consta tant ntss thatt con tha contro trols ls res respec pectiv tively ely the rel relat ative ive pro propor portio tion n of cog cognit nition ion and social soc ial int intera eracti ction. on. Vec Vector tor P i = ( pi1 , . . .,  pid , . . .,  piM ) is the previ reviou ouss bestt pos bes positi ition on (th (the e pos positi ition on giv giving ing the bes bestt fitness fitness val value) ue) of par partic ticle le i, wh whic ich h is call called ed pbest . An And d vect vector or P  g  = ( p g 1 , . . .,  p gd , . . .,  p gM ) is the best best posi positi tion on di disc scov over ered ed by th the e en enti tire re popu popula lati tion on,, wh whic ich h is call called ed  gbest . Par Param amete eterr w is the the iner inerti tia a we weig ight ht and and the the opti optima mall stra strate tegy gy to control it is to initially set to 0.9 and reduce it linearly to 0.4 [14].. QPSO QPSO is deri derive ved d from from fund fundam amen enta tall theo theory ry of part partic icle le swar swarm m [14] havi having ng the the me meri rits ts of quan quantu tum m me mech chan anic ics. s. In the the QP QPSO SO al algo gori rith thm m with M  par partic ticles les in D-di -dimen mensio sional nal spa space, ce, the pos positi ition on of par partic ticle le i at (k + 1) 1)th th iter iterat atio ion n is upda update ted d by by:: k +1  xij  

k =  pij

±˛·

k k MP  j   − xij



 1

  · ln

k  pijk  = ϕijk · P ijk  + (1 − ϕijk ) · P  gj ,

=

1 M 

i=1



 1 P ik1 , . . . , M 



k P iD

 

(9)

i=1

Here, para parameter meter ˛ is cal called led con contra tracti ction– on–exp expans ansion ion (CE) (CE) coefficoefficient. P i   and P  g   ha have ve the sa same me mea meani ning ngss as th tho ose in PSO. SO. MP  is cal calledMean ledMean Bes Bestt Pos Positi ition,which on,which is definedas definedas themean of the pbest  positi pos itions ons of all partic particles les.. Sear Search ch sp spac ace e of QP QPSO SO is wi wide derr th than an th that at of PSO PSO be beca caus use e of in intr trooduct ductio ion n of expo expone nent ntia iall di dist stri ribu buti tion on of posi positi tion ons. s. Al Also so,, unli unlike ke PSO, PSO, wh wher ere e each each part partic icle le co conv nver erge gess to th the e gl glob obal al be best st po posi siti tion on in inde de-pend pe nden entl tly, y, impr imin prov ovem emen t re, has ha,seach been be enpa in intr trod oduc ed by in inse sert rtin ing Mean Meth an Best Best Po Posi siti tion on QP QPSO SO..ent He Here ea ch part rtic icle leuced ca canno nnot t conv co nver erge geg to the e globall bestposition witho globa without ut consid consideringits eringits colleaguesbecause colleaguesbecause that th the e di dist stan ance ce betw betwee een n th the e curr curren entt po posi siti tion on and and MP  deter determines mines the positi pos ition on dis distri tribut bution ion of the par partic ticle le for the nex nextt ite iterat ration ion.. Modifie Mod ified d QPS QPSO O

uijk

[1].. This This sect sectio ion n repr reprod oduc uces es the mod odifi ifica cattio ion n prop roposed osed in [1] Thou Though gh QP QPSO SO finds finds be bett tter er gl glob obal al sear search ch abil abilit ity y th than an PSO, PSO, st stil illl ma may y fal falll int into o prema prematur ture e con conver vergen gence ce [16] sim imil ilar ar to othe otherr evol evolut utio ionnary ary al algo gori rith thms ms in mult multii-mo moda dall opti optimi miza zati tion on li like ke GA and and PSO. PSO. This This resultss in greatloss of perfo result performanc rmance e and yieldsub-optimalsolution yieldsub-optimalsolutions. s. In QP QPSO SO,, in a la larg rger er sear search ch spac space, e, dive divers rsit ity y lo loss ss of th the e wh whol ole e po poppul ulat atio ion n is al also so in inev evit itab able le due due to th the e coll collec ecti tive vene ness ss al alth thou ough gh.. From From eviden entt th that at wh when en MP  j − xij is very very smal small, l, th the e sear search ch Eq. (2), (2), it is evid spac space e wi will ll be smal smalll and and xij ca  cann nnot ot go toa ne new w posi positi tion on in th the e subs subsee-

    

quent ite quent iterat ration ions. s. The exp explor lorati ative ve power power of par partic ticles les is als also o los lostt and th the e evol evolut utio ion n proc proces esss ma may y beco become me stag stagna nate te.. This This case case even even ma may y occu occurr at an earl early y stag stage e wh when en MP  j − xij is zero zero.. Al Also so,, in th the e la latt tter er

Methodologies

ij

MP  =

k ) (MP 1k , . . . , MP M 



(4)

This is ach achiev ieved ed The The obje object ctiv ive e howe howeve verr is to mini minimi mize ze MS MSE E (3). (3). This th thro roug ugh h adju adjust stme ment nt of  ω. Beca Becaus use e of diffi ifficult culty y in reali ealiz zing ing the ensemb ens emble le ope operat ration ion,, the cos costt fun functi ction on (3) can be repla eplace ced d by the time-aver time -averaged aged cost funct function: ion:  1  J (ω) = N 

k



constag stage, e, th the e lo loss ss of dive divers rsit ity y for for MP  j − xij is a problem to be con side sidere red. d. In orde orderr to pr prev even entt th this is unde undesi sira rabl ble e di diffi fficu cult lty, y, we in th this is pape paperr cons constr truc uctt a ra rand ndom om vect vector or.. Thi Thiss ra rand ndom om vec vecto torr wi will ll repl replac ace e certain probabili probability ty CR, according according to the diffe difference rence MP  j − xij with a certain betwee between n two positi positiona onall coordi coordinat nates es tha thatt are rere-ran random domize ized d in the proble pro blem m spa space,and ce,and thevalu thevalue e of thenew pos positi ition on updati updating ng equat equation ion becomes:





ı = mu  xk − mu xs ,

  

xijk+1  =  pkij ± ˛ · ı · ln

 1

ukij

 

(10)

where mu xk and mu xs aretwo ran randompart domparticl icles es gene generat rated ed in the prob proble lem m sp spac ace. e. Beca Becaus use e of in intr trod oduc ucti tion on of th the e ra rand ndom om vect vector or,, th the e particles parti cles may leav leave e from from th the e curr curren entt po posi siti tion on and and ma may y be loc locate ated d in a new new sear search ch doma domain in.. The The mo modi difie fied d ve vers rsio ion n of QP QPSO SO is term termed ed he here re as QP QPSO SO-M -M.. The The proc proced edur ure, e, used used in th this is pa pape perr is outl outlin ined ed as: as: Step Step 1: Ini Initia tializ lize e partic particleswith leswith ran randomposit dompositionand ionand setthe con contro troll parameter CR. Step2 Step2 :For k = 1 to maximum maximum itera iteration,execute tion,execute the follo following wing steps. steps. Step Step 3: Calc Calcul ulat ate e th the e me mean an be best st po posi siti tion on MP  amo among ng the par partic ticles les.. Step 4: For For each each par arti ticl cle, e, com comput pute it itss fitne ness ss va valu lue e  f { xi (k)}. If   f { xi (k)} < f {P i (k)}, th then en P i (k) = xi (k). Step Step 5: Selec Selectt  gbest  position P  g (k) among among partic particles les.. Step Step 6: Ge Gene nera rate te a ra rand ndom om numb number er,, in th the e ra rang nge e [0, [0, 1]. Step7  Step7 : If RN < CR the then n upd updatethe atethe pos positi ition on acc accord ordingto ingto (7)–(9) (7)–(9),, else according accor ding to (8) (8) an and d (1 (10) 0)..

k

,

uij ∈ U ((0 0, 1)

ϕijk ∈ U ((0 0, 1)

(7)

Pro Propos posed ed mod modific ificati ation on wit with h tim timee depend dependant ant con constr strict iction ion fac factor  tor  (PSO-LD)

(8)

ndicat ate e th that at th the e use use of a “con “const stri rict ctio ion n Rece Recent nt wo work rkss in [17,18] indic fact factor or”” ma may y be necessary to insure converg ergence of the PSO. A

 

532

 A. Sarangi et al. / Applied Soft Computing 25 (2014) 530–534

 Table 1 Simulation parameters. parameters.

time-depen timedependent dent and linea linearly-d rly-decreas ecreasing ing K , unli unlike ke th thatof atof a fixed fixed K . Here Here,, we adjust k at each each of the ite iterat ration ionss fol follow lowing ing the rec recurs ursion ion::

PSO

QPSO

Parameter

Value

ω c 1 , c 2

0.9 → 0.4 ˛ 2

 

QPSO-M & PSO-LD

Parameter

Value

Parameter

Value

1 → 0.5

˛ CR

1 → 0.5 0.8

kn  = kmin + (kmax − kmin)

 Table 2 Filterr desi Filte design gn in exam example ple 1 (ran (randomly domly chos chosen en initi initial al posi positions tions). ).

Numbe Nu mberr of hit hitss Global minim Global minimum um −0.312, −0.816

Local minim Local minimum um 0.117, 0.557

27 92 100 100

73 8 0 0



PSO QPSO QPSO-M PSO-LD







 Table 3 Mean Me an va value luess and sta standa ndard rd dev deviat iation ionss of the filt filter er coe coeffic fficien ients ts of 100 ra rando ndom m run runss in exam example ple 1 (with rand randomly omly chos chosen en initi initial al posi positions tions). ). a

−0.6125 −0.7926 −0.8162 −0.8171

0.2084 0.1164 0.0075 0.0045

Here, m and n res respec pectiv tively ely repres represent entss the maximu maximum m number number of it iter erat atio ions ns and and th the e curr curren entt it iter erat atio ion. n. Note Note he here re th that at th the e us use e of a constrictio constr iction n factor factor ensure ensuress compu computati tational onal stability stability of the PSO algorith rithm. m. This This resu result ltss in a stab stabil iliz izin ing g effe effect ct on th the e swar swarm m an and d whic which h theref the reforecalls orecalls forthe use of a low lower er val value ue of the constr constrict ictionfacto ionfactor. r. Simulation Simul ation resu results lts

Perfor Per forman mance ce of abo above ve men mentio tioned ned met method hodolo ologie giess were were stu studie died d thro throug ugh h si simu mula lati tion ons. s. Wi With th QP QPSO SO as refe refere renc nce, e, si simu mula lati tion onss we were re carr carrie ied d for for th thre ree e diff differ eren entt ca case ses. s. On One e hu hund ndre red d Mo Mont nte e Carl Carlo o si simmul ulat atio ions ns we were re unde undert rtak aken en for for ea each ch case case.. Simu Simula lati tion on para parame mete ters rs illustrated illust rated in Tabl Table e 1. Casee 1: second Cas second-or -order der system system and firs first-o t-orde rderr IIR filte filter  r 

CPUtime (s)

b

−0.263 ± −0.2754 ± −0.3192 ± −0.3181 ±

PSO QPSO QPSO-M PSO-LD

m−n m−1

± ± ± ±

0.5222 0.3891 0.0077 0.0011

7.843 2.780 2.308 2.630

simpli simp lifie fied d me meth thod od of in inco corp rpor orat atin ing g a co cons nstr tric icti tion on fa fact ctor or is il illu lusstrat tr ated ed in [17] [17].. In [17] [17],, the the per perfo form rma ance nce of PSO PSO using sing an iner inerttia weight was compared with the PSO performance using a constri strict ctio ion n fa fact ctor or.. It wa wass conc conclu lude ded d th that at the the best best appr approa oach ch is to us use e a cons constr tric icti tion on fact factor or wh whil ile e limi limiti ting ng the the ma maxi ximu mum m velo veloci city ty v max to the dyna ynami micc rang range e of the vari variab able le  xmax   in in each each di dime mens nsio ion. n. It thiss app approa roach ch pro provid vides es a per perfor forma mance nce wa wass al also so show shown n in [17] that thi superi sup erior or to any sim simila ilarr tec techni hnique que rep report orted ed in the litera literatur ture. e. Constricti Const riction on facto factor-bas r-based ed PSO [9] mo moti tiva vate ted d this this wo work rk to propropose a new modification. This is achieved by introducing a

In this case, i.e., when the second-order sy sysstem and firstorder IIR filter, the problem is of falling to local minima. To te test st th the e effe effect ctiv iven enes esss of prop propos osed ed al algo gori rith thms ms,, we ha have ve chos chosen en a pl plan antt same same as th that at di disc scus usse sed d in [6], [6], having numerator numerator coeffi coefficient cientss {0.05, −0.4} 0.4} and den denomi ominat nator or coe coeffic fficien ients ts {1, −1.13 1.1314, 14, 0.25} 0.25}. The obje object ctiv ive e is no now w to find find coef coeffic ficie ient ntss “a” and “b” of 1st order rder filt lter er 1 − rando om Ga Gaus ussi sia an no nois ise e wi with th zero ero mea mean an and d H ( z ) = a/(1+ bz  ). A rand unitvariance was waschosenas chosenas syste system m input input,, x(k). Theerro Theerrorr surfac surface e has aglobalminimumat {a, b} = {−0.312, − 0.816 0.816}} andalocalminimum at {a, b} = {0.11 0.117, 7, 0.576 0.576}. Forall thefive alg algori orithm thms,the s,the sea searchspace rchspace chosenwas (±1). The fixedinitial posit positions ions are chosenrandomlyas: [6].. A com compar pariso ison n among among {0.11 0.117, 7, 0.57 0.576 6}, {0.8, 0.8, 0} and {{0.9, {{0.9, −0.9}} 0.9}} [6] vari variou ouss al algo gori rith thms ms in hi hitt ttin ing g th the e gl glob obal al and and lo loca call mi mini nimu mum m is outoutlined lined in Table Table 2. We fou found nd the these se result resultss by 100 100 ran randomsimul domsimulati ations ons and and wi with th ra rand ndom omly ly chos chosen en in init itia iall posi positi tion ons. s. It is evi evide dent nt from from th the e

 Table 4 Mean Mean va value luess of thefilter coeffic coefficien ients ts of 100 rando random m runs runs in examp example le 1 (with (with four four fixe fixed d pos positi itions ons). ).

Fixed initi initial al posi positions tions



 



 

0.117, 0.576

a

PSO QPSO QPSO-M PSO-LD

0.8, 0

b

0.113 0.114 0.116 0.117

 

0.524 0.519 0.571 0.573

0.9, −0.9

a

b

−0.0361 0.77 0.81 0.83

0.0526 0.027 0 .0 0 9 0.012

CPU tim time e (s)

  0.9, 0.9

a

b

−0.1697 0.9 0.89 0.91

−0.4768 −0.9 −0.9066 −0.9062

a

b

0.263 0.9 0.91 0.901

0.6125 0 .9 0.907 0.903

7.47 2.65 2.298 3.13

 

CPU tim CPU time e (s)

 Table 5 Mean Me an va value luess an and d sta standa ndard rd dev deviat iation ionss of thefiltercoeffi thefiltercoefficie cientsof ntsof 100 ra rando ndom m run runss in exa examp mple le 2 (wi (with th ran random domly ly cho chose sen n ini initia tiall pos positi itions ons). ). a0

PSO QPSO QPSO−M PSO−LD

 

−0.3313 −0.3909 −0.3948 −0.48

± ± ± ±

a1

0.1092 0.0138 0.0134 0.021

 

−0.0586 ± −0.0769 ± −0.0742 ± −0.076 ±

b1

0.12054 0.0162 0.0195 0.02

−0.545 ± −0.2187 ± −0.2230 ± −0.24 ±

 

b2

0.3892 0.0168 0.02 0.024

−0.2354 ± −0.5796 ± −0.5739 ± −0.56 ±

0.3034 0.0148 0.0155 0.019

72.37 48.45 44.071 46.03

 Table 6 Mean Me an va value luess an and d sta standa ndard rd dev deviat iation ionss of thefiltercoeffi thefiltercoefficie cientsof ntsof 100 ra rando ndom m run runss in exa examp mple le 3 (wi (with th ran random domly ly cho chose sen n ini initia tiall pos positi itions ons). ). a0

PSO QPSO QPSO-M PSO-LD

−0.1697 ± −0.2 ± −0.199 ± −0.2 ±

 

a1

0.1051 0.3237 ± 3.523 × 10−3 0.398 ± 1.4 × 10−4 0.399 ± 1.4 × 10−4 0.2 ±

 

a2

 

b1

 

b2

 

b3

0.1364 −0.3622 ± 0.1829 −0.3285 ± 0.4288 0.2493 ± 0.1974 −0.088 ± 4.286 × 10−3 −0.499 ± 4.088 × 10−3 −0.599 ± 7.186 × 10−3 0.25 ± 7.969 × 10−3 −0.199 ± −0.599 ± 2.55 × 10−4 −0.2 ± 2.045 × 10−4 −0.499 ± 1.6 × 10−4 0.249 ± 2.816 × 10−4 −0.5 ± 1.6 × 10−4 −0.6 ± 2.55 × 10−4 2.045 × 10−4 0.25 ± 2.816 × 10−4 −0.22 ±

  0.1618 7.389 × 10−3 2.72 × 10−4 2.72 × 10−4

CPU time (s) time 9.526 4.523 3.998 4.12

 

 A. Sarangi et al. / Applied Soft Computing 25 (2014) 530–534

533

tab able le tha hatt PS PSO O an and d QP QPSO SO ju jum mp to the gl glob obal al min iniimu mum m va vall lley ey in more mo re nu numb mber er of ca case sess an and d co conv nver erge ge to th the e gl glob obal al mi mini nimu mum, m, bu butt they th ey ma may y al also so ju jum mp to the loc ocal al min inim imu um and the hen n co conv nver erge ge to the th e lo loca call mi mini nimu mum. m. Ho Howe weve ver, r, QP QPSO SO-M -M an and d PS PSOO-LD LD ha have ve ab abil ilit ity y to conv co nver erge ge in th the e gl glob obal al mi mini nimu mum m in al almo most st al alll th the e ca case ses. s. In Tables 3 and 4, we demon emonst stra ratte the the mean ean valu values es of filt lter er coef coeffic ficie ient ntss alon along g wi with th th the e stan standa dard rd devi deviat atio ions ns al alon ong g wi with th si simu mu-la lati tion on run run time time fo forr each each of th the e al algo gori rith thms ms.. It is evid evidentfro entfrom m Tab Table le 3 thatQPSO-M couldfind the globa globall minimumwith minimumwith the leaststandar leaststandard d deviat dev iation ionss as com compar pared ed to oth other er fou fourr alg algori orithm thms. s. It is eviden evidentt from from Table Table 4 that hat QPSO QPSO-M -M and and PSOPSO-LD LD can can jump jump out out of any any of the the sett settle led d fixed fixe d ini initia tiall pos positi itions ons and find the glo global bal minimu minimum m whi while le the other other al algo gori rith thms ms are are all all trap trappe ped d in th thes ese e fixed fixed init initia iall al algo gori rith thms ms.. Fig. Fig. 1 illustrat illus trates es the coeffi coefficient cientss learn learning ing curve curvess and conve convergence rgence behav behav-iors ors of the the alg lgor oriithm thms fo forr this this case case of desig esign. n. Th The e re resu sult ltss sho shown are are aver averag aged ed over over 100 100 rand random om runs runs wi with th rand random omly ly chos chosen en init initia iall positions. Ca Case se 2: thirdthird-ord order er sys system tem and and second second-or -order der IIR filte filter  r 

When Wh en th the e plan plantt is a thir thirdd-or orde derr sy syst stem em and and fil filte terr is a seco second nd-order IIR filter, the error surface of the cost function becomes mul ulti ti--mo moda dall beca becaus use e of redu educed ced ord order filt lter er.. We have have cho chosen sen th the e plan plantt havi having ng nume numera rato torr coef coeffic ficie ient ntss {−0.3, − 0.4, − 0.5} 0.5} and denominato denom inatorr coeffi coefficient cientss {1, − 1.2 1.24, 4, 0.5 0.5,, − 0.1 0.1}}. The The prob proble lem m now now redu reduce cess to that that of findi finding ng coef coeffic ficie ient ntss {a0 , a1 , b1 , b2 } of the the filt lter er with tran transfer sfer funct function ion H ( z ) = (a0 + a1 z −1 )/(1+ b1 z −1 + b2 z −2 ). A unif unifor orml mly y di dist stri ribu bute ted d wh whit ite e se sequ quen ence ce in the the rang range e (±0. 0.5) 5) is ta take ken n as th the e inpu inputt  x(k),andth ),andthe e SNR SNR wa wass keptfix keptfixedat edat 30dB 0dB.. Result Res ultss by variou variouss alg algori orithm thmss in cas case e 2 are ill illust ustrat rated ed in Table Table 5. Resu Result ltss prov provid ide e th the e me mean an best best va valu lues es and and stan standa dard rd de devi viat atio ions ns of  the filt filter er coe coeffic fficien ients. ts. Thes These e res result ultss are obt obtain ained ed by ave averag raging ing ove overr 100 100 rand random om runs runs wi with th rand random omly ly ch chos osen en init initia iall po posi siti tion onss li like ke that that depict ctss the the conv conver erge genc nce e be beha havi vior orss for for case case 2 of case 1. Fig. 2(a) depi Tabl ble e 5 that, hat, me mean an be best st fo forr diff differ eren entt algo algori rith thms ms.. It is evid eviden entt from from Ta val valuesprod uesproduce uced d byPSO,QPSO andQPSO andQPSO-Mare -Mare app approx roxima imate,butby te,butby PSOPSO-LD LD is exac exact. t. Conv Conver erge genc nce e sp spee eed d of QP QPSO SO and and QP QPSO SO-M -M is fast faster er th than an th the e othe otherr th thre ree e algo algori rith thms ms as seen seen from from Fi Fig. g. 2(a (a). ). Ca Case se 3: thirdthird-ord order er sys system tem and and thirdthird-ord order er IIR filte filter  r 

When the plant is a third-order system and filter is a thirdorder IIR filter, the error surface of the cost function becomes uniuni-mo moda dall beca becaus use e of same same orde orderr fil filte ter. r. We have have chos chosen en the the pl plan antt having havin g numer numerator ator coeffi coefficient cientss {−0.2 0.2,, 0.4 0.4,, − 0.5 0.5}} and denom denominato inatorr coefficients {1, − 0.6, 0.25 0.25,, − 0.2 0.2}}. The The pr prob oble lem m now now re redu duce cess to that that of findi finding ng coeffi coefficient cientss {a0 , a1 , a2 , b1 , b2 , b3 } of thefilterwithtrans thefilterwithtransfer fer 3 2 1 2 1 − − − − − function H ( z ) = (a0 + a1 z  + a2 z  )/(1+ b1 z  + b2 z  + b3 z  ). We havechos havechosen en sameinpu sameinputt asthatof case1. case1. The The bestsolu bestsoluti tionis onis to be lo loca cate ted d at {−0.2 0.2,, 0.4 0.4,, − 0.5, − 0.6 0.6,, 0.2 0.25, 5, − 0.2} 0.2}. The The me mean an be best st valu values es and and stan standa dard rd devi deviat atio ions ns of fil filte terr coef coeffic ficie ient ntss in case case 3 are are provid pro vided ed in Tabl Table e 6 that hat gi gives ves re resu sult ltss aver averag aged ed over over 100 100 rand random om runs runs wi with th rand random omly ly init initia iall posi positi tion ons. s. Conve Converg rgen ence ce be beha havi vior orss for for Fig. g. 2(b). case case 3 aver averag aged ed ov over er 100 100 rand random om ru runs ns are are pr prod oduc uced ed in Fi As evid eviden entt from from Tabl Table e 6, the the fil filte terr coef coeffic ficie ient ntss foun found d by QP QPSO SO are are exac exactl tly y lo loca cate ted d at the the best best so solu luti tion on and and the the stan standa dard rd de devi viat atio ion n is smal smalle lerr than than that that yi yiel elde ded d by any any othe otherr al algo gori rith thms ms.. QP QPSO SO-M -M and and PSO-L SO-LD D are are the the most and the the seco second nd most ost robu robust st alg lgor orit ithm hmss as comp compar ared ed to two two ot othe herr algo algori rith thms ms.. It is seen seen in Fig. Fig. 2(b) that hat the conver con vergen gence ce spe speeds eds of QPS QPSO, O, QPS QPSO-M O-M and PSO PSO-LD -LD are muc much h fas faster ter th than an thos those e of PSO. PSO. Comp Comput utat atio ion n ti time me us usin ing g MA MATL TLAB AB for for di diff ffer eren entt st stra rate tegi gies fo forSO r thre thare ree exam exster ampl essolv outl ou tlin ed Tab brs, le, 2 . It aippro s sroxi eexima nmatthat-t P SO and an desQP QPSO ar ee fa fast erples to so lve eined tha th aninothe otTa hers but pp ing a pro problem blem,, as QP QPSO SO-M -M and and PSOPSO-L LD gene genera rallly requi equire ress more ore constraint const raintss and addi additiona tionall varia variables, bles, which increa increases ses compu computatio tation n time.

Fig. Fig. 1. (a) Coefficien Coefficientt a; (b) (b) coef coeffic ficie ient nt b le lear arni ning ng cu curv rves es fo forr ex exam ampl ple e 1 and and (c (c)) compariso comp arison n of converge convergence nce behaviorsfor behaviorsfor example example 1.

 

534

 A. Sarangi et al. / Applied Soft Computing 25 (2014) 530–534

has has enha enhance nced d th the e ra rand ndom omne ness ss by mo modi dify fyin ing g th the e upda update te equa equati tion on of QPSO. QPSO-M replaces ces a part of the update equation with a rand random om vect vector or wi with th a cert certai ain n pr prob obab abil ilit ity. y. PSOPSO-LD LD in intr trod oduc uced ed a time-d time-depe epende ndent nt linear linearlyly-dec decrea reasin sing g con constr strict iction ion fac factor tor.. QPSO, QPSO, QP QPSO SO-M -M and and PSOPSO-LD LD we were re used used in th the e desi design gn of digi digita tall IIR IIR filte filters rs for the pur purpos pose e of system system ide identi ntifica ficatio tion. n. Exp Experi erimen mental tal res result ultss hav have e show shown n tha hatt th the e perfo erforrma manc nce e of QP QPSO SO,, QP QPSO SO-M -M an and d PSOPSO-L LD are supe superrio iorr to PSO PSO in the dig igit ital al IIR filt lter er desig esign n pro robl blem em an and d they they wi will ll be an effic efficie ient nt to tool ol for for filte filterr desi design gn pr prob oble lem. m. References

Fig Fig.. 2. Com Compa paris rison on of conver convergen gence ce beh behavi aviorsfor orsfor (a)example (a)example 2 and(b) exampl example e 3.

From From the above bove thr three exam examp ple les, s, QP QPSO SO,, QPSOPSO-M M and and PSO-L SO-LD D have hav e sho shown the their ir str strong onger erodal sea search abili ab ilitie ties s bot both h on themulti-mo -modal dal pro proble blem m wn andon theuni-m theuni-moda lrch one one. . QPS QPSO,QPSOO,QPSO-M M themulti andPSOandPSO-LDoutLDoutperfor per form m PSO in con conver vergen gence ce spe speed, ed, rob robust ustnes nesss and qua qualit litati ativel vely y of  the fina finall sol soluti utions ons [19]. [19]. Conclusion

This This pape paperr intr introd oduc uced ed two two nove novell modifi dificati catio ons to PSO PSO and and its its vari varian antt QP QPSO SO,, and ter termed as QP QPSO SO-M -M and PSOSO-LD. LD. QP QPSO SO-M -M

[1] W. Fan Fang, g, J. Sun Sun,, W. Xu, A new muta mutated ted quan quantumtum-beha behaved ved part particle icle swar swarm m opti opti-mizer miz er fordigit fordigital al IIRfilte IIRfilterr des design ign,, EU EURA RASIPJ. SIPJ. Adv Adv.. Sig Signa nall Pro Proces cess. s. (2 (2009 009), ), Art Articl icle e ID 367 367465 465.. [2] V.J. V.J. Ma Mano noj, j, E. El Elia ias, s, De Desi sign gn of mu mult ltip ipli lier er-l -les esss no nonu nuni nifo form rm fil filte terr ba bank nk tr tran anssmulti mu ltiple plexur xure e us using ing ge genet netic ic alg algori orithm thm,, Sig Signa nall Pro Proces cess. s. 89 (11 (11)) (2 (2009 009)) 22 2274 74– – 2285. [3]  J.-T. Tsai, W.-H. Ho, J.-H. Chou, Design of two-dimensiona two-dimensionall IIR digital structurespeci sp ecifie fied d filt filters ers by usi using ng an imp improv roved ed ge genet netic ic alg algori orithm thm,, Exp Expert ert Sys Syst. t. Ap Appl. pl. 36 (3) (2009 (2009)) 692 6928 8–69 6934 34 (Pa (Part rt 2). [4] C.-W C.-W.. Ts Tsai ai,, C. C.-H -H.. Hu Huan ang, g, C. C.-L -L.. Li Lin, n, St Stru ruct ctur uree-sp spec ecifi ified ed II IIR R fil filte terr an and d co cont ntro roll design desi gn usin using g realstructur realstructured ed gene genetic tic algo algorithm rithm,, App Appl. l. SoftComput.9 (4) (200 (2009) 9) 1285– 1285 –1295. [5] Y.-P. Y.-P. Cha Chang,C. ng,C. Low Low,, S.-Y S.-Y.. Hung Hung,, Inte Integrat gratedfeasibl edfeasible e direc directionmethodandgenetic tionmethodandgenetic algorithmfor algo rithmfor opti optimalplannin malplanning g of harm harmonicfilterswith onicfilterswith unce uncertain rtainty ty condi conditions tions,, ExpertSyst. Exp ertSyst. Ap Appl.36 pl.36 (2)(2009 (2)(2009)) 39 3946 46– –39 3955 55 (Pa (Part rt 2) 2).. [6] S. Chen Chen,, R. Iste Istepani panian,B.L. an,B.L. Luk,DigitalIIR filterdesignusing adap adaptivesimulate tivesimulated d annealin anne aling, g, Digi Digital tal Sign Signal al Proc Process. ess. 11 (3) (2001 (2001)) 241 241– –251. [7] N. Ka Karab raboga oga,, Dig Digita itall IIR filt filter er des design ign usi using ng dif differ ferent ential ial ev evolu olutio tion n alg algori orithm thm,, EURA EU RASIPJ. SIPJ. Ap Appl. pl. Sig SignalProce nalProcess.2005 ss.2005 (8)(2005 (8)(2005)) 12 1269 69– –1276. [8] D.J.Krusienski D.J.Krusienski,, W.K.Jenkin W.K.Jenkins,Adaptiv s,Adaptive e filter filteringvia ingvia par particleswarmoptim ticleswarmoptimizat ization, ion, in: Proc Proceedin eedingsof gsof theConfer theConferenceRecordof enceRecordof theAsiloma theAsilomarr Conf Conferen erenceon ceon Sign Signals, als, System Sys temss an and d Co Compu mputer ters, s, vo vol. l. 1, Nov Novemb ember,2003, er,2003, pp pp.. 57 571 1–575. [9] Y.Gao,Y. Li,H. Qian Qian,, Thedesignof IIRdigitalfilterbasedon chao chaoss par particleswarm ticleswarm optimiza opti mizationalgorit tionalgorithm,in: hm,in: Proce Proceeding edingss of the2nd Inte Internat rnationa ionall Conf Conferen erence ce on Genetic Gene tic and Evolu Evolution tionary ary Comp Computin uting g (WGE (WGEC’08 C’08), ), Jingz Jingzhou, hou, Hub Hubei, ei, Sept Septembe ember, r, 2008, 200 8, pp pp.. 303 303– –306. [10]  J. Sun, B. Feng, W. Xu, Part Particle icle swar swarm m optim optimizat ization ion with par particle ticless havi having ng quan quan-tum be behav havior ior,, in: Pro Procee ceedin dings gs of the Con Congre gress ss on Evo Evolut lution ionary ary Com Comput putati ation on (CEC’ (C EC’04) 04),, vo vol. l. 1, Por Portla tland, nd, OR OR,, USA USA,, Jun June, e, 200 2004, 4, pp pp.. 32 325 5–331. [11]  J. Sun, W. Xu, B. Fen Feng, g, A glo global bal sea search rch str strate ategy gy of qu quant antumum-be behav haved ed pa parti rticle cle swarm sw arm op optim timiza izatio tion, n, in: Pro Procee ceedin dings gs of IEE IEEE E Con Confer ferenc ence e on Cyb Cybern erneti etics cs an and d Intellige Inte lligent nt Syst Systems, ems, 2004 2004,, pp. 111 111– –116. [12]  J.I. Ababneh Ababneh,, M.H. Bataineh, Linear phase FIR filter design using particle swarm optimiza opti mizationand tionand gene geneticalgorith ticalgorithms,DigitalSignalProces ms,DigitalSignalProcess. s. 18(July (4))(2008) (4))(2008).. [13] A. Sar Sarang angi, i, R.K R.K.. Ma Mahap hapatr atra, a, S.P S.P.. Pa Panig nigra rahi, hi, DEP DEPSO SO and PSO PSO-QI -QI in dig digita itall filt filter er design des ign,, Exp ExpertSyst.Appl.38 ertSyst.Appl.38 (2 (2011 011)) 109 10966 66– –10973. [14]  J. Kennedy, R. Eberhart, Particle swarm optimization, in: Proceedings of  IEEE Int Intern ernat ation ional al Co Confe nferen rence ce on Ne Neura urall Net Netwo works rks,, vol vol.. 4, Pe Perth rth,, Au Austr strali alia, a, November– November –Dece December mber,, 1995 1995,, pp. 1942 1942– –1948. [15] Y. Sh Shi, i, R. Ebe Eberha rhart, rt, Mod Modifie ified d pa parti rticle cle sw swarm arm opt optimi imizer zer,, in: Pro Procee ceedin dings gs of the IEEE Conf Conferen erence ce on Evol Evolution utionary ary Comp Computat utation ion (ICE (ICEC’98 C’98), ), Anch Anchorag orage, e, AK, USA USA,, May, Ma y, 199 1998, 8, pp pp.. 69 69– –73. [16] L.D L.D.S.Coelho,A .S.Coelho,A qua quantumpartic ntumparticleswarmoptimiz leswarmoptimizerwith erwith chao chaoticmutationoper ticmutationoper-ator, at or, Ch ChaosSolit aosSolitonsFract onsFractals37 als37 (5)(2008 (5)(2008)) 14 1409 09– –1418. [17] R.C.Eberhar R.C.Eberhart, t, Y. Shi, Comp Comparin aring g inert inertia ia weig weights hts and cons constrict triction ion acto actors rs in par par-ticle swa swarm rm opt optimi imizat zation ion,, in: Pro Procee ceedin dings gs of 200 2000 0 Con Congre gress ss on Evo Evolut lution ionary ary Computat Comp utation, ion, 2000, pp. 84 84– –88. [18] M. Clerc Clerc,, J. Kenn Kennedy, edy, The par particle ticle swa swarm: rm: explo explosion sion,, stab stability ility,, and conve convergen rgence ce in a mu mult ltii-di dime mens nsio iona nall co comp mple lex x sp spac ace, e, IE IEEE EE Tr Tran ans. s. Ev Evol ol.. Co Comp mput ut.. 6 (2 (200 002) 2) 58– 58 –73. [19] S. Ha Hayki ykin, n, Ada Adapti ptive ve Fil Filter ter Th Theor eory, y, 4th ed. ed.,, Pre Prenti ntice ce Ha Hall, ll, Eng Englew lewood ood Cli Cliffs ffs,, NJ NJ,, USA,, 2001. USA

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