Keywords Digital Infinite-Impulse Response (IIR) filter, Digital Finite-Impulse Response (FIR) filter, DE, exploratory move

Transcription

1 Volume 5, Issue 7, July 2015 ISSN: X Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering Research Paper Available online a: A Hybrid Differenial Evoluion Mehod for he Design of Low Pass Digial Fir Filer Pooja Rani *, Balraj Singh Elecronics & Communicaion Engineering, Giani Zail Singh Punjab Technical Universiy Campus, Bahinda, Punjab, India Absrac This paper presens he mehodology for he design of Low pass FIR digial filer using hybrid Differenial Evoluion (DE) mehod. I is an evoluionary algorihm ha exhibis he feaures of boh he basic DE and Hooke- Jeeves exploraory move o aain he opimal soluion. The basic DE and exploraory move have he capabiliy o exploi and explore he search space o design he opimal filer parameers. A mulivariable objecive funcion is opimized o aain he minimum magniude error and pass band and sop band ripples. The designed Low pass digial FIR filer auhenicaes ha he proposed hybrid DE algorihm yields more accurae and efficien soluion and can be effecively applied for he design of High pass, Band pass and Band sop digial filers. Keywords Digial Infinie-Impulse Response (IIR) filer, Digial Finie-Impulse Response (FIR) filer, DE, exploraory move I. INTRODUCTION Opimizaion plays a subsanial role in engineering, science and digial signal and image processing fields. Opimizaion is he process of deermining he condiions ha give he minimun and maximum value of a funcion [1]. Opimizaion deals wih he sudy of hese kinds of problems in which one has o minimize or maximize one or more objecives ha are funcion of some real or ineger variables. Opimizaion can be single objecive or muli objecive. Single objecive opimizaion resuls in opimizing only one objecive. On he oher hand muliobjecive opimizaion deals wih he ask of simulaneously opimizing wo or more objecives wih respec o se of cerain consrains. The design of opimum digial FIR (Finie Impulse Response) filer using opimizaion is a very challenging ask now- a- days. FIR filers are he digial filers having finie impulse response as i requires only finie number of previous samples and presen sample o find he impulse response. FIR filers are non recursive filers having linear phase and always sable [2]. Therefore hese are he mos suiable for phase sensiive applicaions such as radar and audio applicaions. The performance of digial FIR Filers is measured in erms of magniude error and ripple magniude of pass band and sop band. Various opimizaion algorihms such as simulaed annealing, Tabu search, an colony algorihm, immune algorihm, Geneic algorihm, paricle swarm opimizaion and differenial evoluion algorihm are available for digial FIR filer design [3]. Kumar e. al. (2013) has implemened window mehod for he design of band pass FIR filer using various windows such as Blackmann, Hamming, Hanning and Kaiser window [4]. The resuls of Kaiser window proved o be beer among all ohers. Kaur e. al. (2012) has discussed he design of FIR filers using Geneic algorihm in which he number of operaions in he design process are reduced and coefficien calculaion is realized [5]. Geneic algorihm may converge owards local opima insead of global opima if he value of finess funcion is no properly defined. GA is difficul o operae on dynamics ses. I also requires complex geneic operaors of muaion and crossover. The compuaional cos of GA is also very large. Dai e. al. (2010) designed digial IIR filer using Seeker opimizaion algorihm [6]. I uses he concep of simulaing he ask of human search in which he empirical gradien is used o find he search direcion using a simple fuzzy rule. Alhough i is simple o implemen and good a local convergence bu i requires oo many calculaions o obain global minima. Neha e. al. (2014) presened he PSO algorihm for Low pass FIR filer design. This algorihm is used wih consricion facor approach o solve he mulimodal, highly non linear filer design problems. This mehod has he propery of parameer impedance and converges in very less execuion ime [7]. Mondal e. al. (2012) implemened noval PSO o design he FIR digial low pass filer [8] and compared i wih exising PM, RGA and PSO algorihm. NPSO proved o be beer in erms of convergence characerisics, execuion ime, magniude response, minimum sop band ripple and maximum sop band aenuaion. Bu he major shorcoming of PSO is ha he conrol parameers affec he convergence behaviour o a grea exen. So hybrid differenial evoluion can be used o overcome he shorcomings of PSO. Iniially DE algorihm was developed by Sorn and Price in 1995 which is simple, fas and populaion based sochasic algorihm. DE has he capabiliy o handle non linear, non differeniable and mulimodal cos funcion. I is easy o implemen due o need of less conrol parameers. I acquires good convergence speed due o is parallelizabiliy. DE has he capabiliy o explore he search space globally. Bu he basic drawback is ha ou of en muaion sraegies available, only one paricular sraegy is suiable for he desired global opimum soluion. So o overcome his drawback, 2015, IJARCSSE All Righs Reserved Page 895

2 Rani e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering 5(7), basic DE is hybridized wih exploraory move o search he local neighbourhood of global soluion and find beer soluion. The premise of his paper is o implemen a hybrid DE algorihm for he design of Low pass FIR digial filer. This algorihm explores and explois he search space by varying various conrol parameers such as populaion size, crossover rae and muaion facor and opimizes he filer coefficiens o achieve minimum magniude error and pass band and sop band ripples. The remaining paper is arranged in he four secions. The Low pass digial FIR filer design problem is discussed in Secion II. The brief algorihm regarding he Hybrid DE is explained in Secion III. Secion IV depics he performance and resuls of he proposed algorihm. Secion V successfully negoiaes he conclusion. II. PROBLEM FORMULATION FOR FIR FILTER DESIGN Digial filers usually operae on discree-ime signals. FIR Filers have finie impulse response because he filer oupu is compued as a weighed, finie erm sum of pas, presen, and fuure values of he filer inpu. The difference equaion of FIR filer is as below: N 1 y n = b k x n k (1) k=0 where x(n) represens he filer inpu b k represens he filer coefficien y(n) represens he filer oupu N is he number of filer coefficiens (order of he filer) The ransfer funcion of FIR filer is given as: H z = N n=0 h n z n (2) where H z is he impulse response in frequency domain and h n is he impulse response in ime domain. N is he filer order. This paper presens he design of Low pass FIR digial filer which is even symmeric and wih even order. The oal number of filer coefficiens is N+1. Bu due o even symmeric propery of FIR filers, we have o calculae only half number of filer coefficiens i.e. (N/2 + 1). So he dimension of he problem is reduced o half. There are many filer ypes, bu he mos common are low pass, high pass, band pass and band sop. A low pass digial FIR filer allows only low frequency signals (below some specified cu-off) hrough o is oupu, so i can be used o eliminae high frequencies. For a Low pass digial filer he ideal response is defined as: H i e jw 1 for w w = c (3) 0 oherwise The performances of digial FIR filer can be calculaed by using L 1 -norm and L 2 -norm approximaion error of magniude response and ripple magniude of boh pass-band and sop-band. The FIR filer is designed by opimizing some coefficiens. For his purpose, we have o minimize he L p -norm approximaion error funcion. L p -norm is expressed as : k E x = i=0 H d w i H w i, x p 1 p (4) where H d w i is he desired magniude response of he ideal FIR filer and H w i, x is he obained magniude response of he FIR filer. For p=1, magniude error denoes he L 1 -norm error and for p=2 magniude error denoes he L 2 -norm error. The L 1 -norm error is given by he expression: e 1 x = k i=0 H d w i H w i, x The L 2 -norm error is given by he expression: e 2 x = k i=0 H d w i H w i, x The desired magniude response H d w i of FIR filer is given as: 1 for w H d w i = i pass band (7) 0 for w i sop band The ripple magniude of pass band and sop band is expressed as: δ p = max H w i, x min H w i, x for w i pass band (8) δ s = max H w i, x for w i sop band (9) Four objecive funcions for opimizaion are: Z 1 x = Minimize e 1 x Z 2 x = Minimize e 2 x Z 3 x = Minimize δ p Z 4 x = Minimize δ s 2015, IJARCSSE All Righs Reserved Page 896 (6) (5)

3 Rani e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering 5(7), The muli objecive funcion is convered ino single objecive funcion: Minimize f x = w 1 Z 1 x + w 2 Z 2 x + w 3 Z 3 x + w 4 Z 4 x (10) where w 1, w 2, w 3, w 4 are he weighing funcions. III. HYBRID DIFFERENTIAL EVOLUTION In his paper he Low pass digial FIR filer is designed by exploring hybrid DE algorihm. DE is a populaion based evoluionary algorihm. The erm evoluionary comes from he fac ha i uses biological operaors of evoluion like muaion, crossover and selecion along wih he arihmeic operaors. DE is a sochasic echnique as i gives differen resuls everyime he program is run. In his algorihm firs some random populaion is iniialized and hen i is evolved o yield beer resuls for he opimizaion problem. A. Parameer Seup The parameers which have been seleced iniially are populaion size (S), boundary consrains of populaion, muaion facor (f m ), crossover rae (CR) and maximum number of ieraions (T max ). If here are N number of filer coefficiens hen each populaion should be N-dimensional. S number of populaions are generaed each of lengh N. So populaion is represened in he form of marix x ij of size S N. B. Populaion Iniializaion For iniializaion of populaion, firsly random numbers are generaed wih uniform probabiliy disribuion and hese numbers are mapped wihin he given boundary consrains according o he following equaion: min = x j + rand() x max min j x j 11 x ij where j = 1,2,.. N; i = 1,2,.. S where x ij is he j h elemen of i h populaion and represens he generaion. rand () is he uniform random number wih values in he range 0 and 1. C. Muaion In muaion, a se of new vecors is generaed by combining he randomly chosen vecors from he iniial populaion in differen ways. The resulan vecor is known as muan vecor. There are en muaion sraegies available for he combinaion of vecors bu firs five are sudied in his paper which are as below: Z ij 1 = P R1 j + f m x R2 j x R3 j (12) Z ij 2 Z ij 3 Z ij 4 Z ij 5 = x Bj + f m x R1 j = x ij + f B x Bj = x Bj + f m x R1 j x R2 j (13) x ij + f m x R1 j x R2 j (14) (15) + x R2 j = x R5 j + f m x R1 j + x R2 j where i = 1,2,, S; j = 1,2, N x R3 j x R3 j x R4 j x R4 j (16) Z i = [Z i1, Z i2,., Z in ] T represens he muan vecor for he posiion of he i h individual. f m is he muaion facor in he range [0.5,1]. R 1, R 2, R 3, R 4, R 5 are randomly chosen populaion vecor wih uniform disribuion. D. Recombinaion Recombinaion is accomplished by exchanging he parameers of muan vecor wih ha of randomly generaed populaion vecor wih crossover probabiliy of CR. I is mainly used o increase he populaion diversiy. The resulan vecor is called rial vecor or crossover vecor. Recombinaion is carried ou according o following equaion: if r and j CR or j = j rand U ij +1 = Z ij x ij if r and j > CR or j j rand i = 1,2,, S; j = 1,2, N where U +1 i = [U +1 i1, U +1 i2,., U +1 in ] T is he rial vecor. rand(j) is he j h evaluaion of random number in he range [0,1]. j rand is randomly chosen index wihin he range [1,N]. CR is he crossover rae in he range [0, 1]. E. Selecion In selecion process, he rial vecor U ij +1 is checked for survival in he nex generaion. The values of objecive funcions of boh he populaion vecor and rial vecor are compared. If he value of objecive funcion of rial vecor is less han ha of populaion vecor, hen i is seleced for evoluion in he nex generaion, oherwise no. x +1 ij = U ij +1 j = 1,2,. N ; if A U i +1 < A X i X ij j = 1,2,. N ; oherwise +1 where A represens he objecive funcion and x ij represens he nex generaion of populaion. 2015, IJARCSSE All Righs Reserved Page 897 (18) (17)

4 Rani e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering 5(7), F. Exploraory move Hooke and Jeeves paern search mehod has wo kinds of moves: Paern move and Exploraory move. The exploraory move explores he global soluion wih a consan sep in all direcions. An exploraory move is performed in he viciniy of he curren poin sysemaically o find he bes poin around he curren poin [9]. Thus if he objecive funcion wih posiive or negaive perurbaion is less han he global bes objecive funcion, hen ha posiive or negaive perurbed locaion is updaed, oherwise he locaion remains he same. The perurbaion of filer coefficien x i is performed according o following equaion: X n o j i = x i ± i u i ; i = 1,2,.. S; j = 1,2,.. N (19) where u j 1 if i = j i = (20) 0 if i j N represens number of variables. n The value of he objecive funcion A x i is calculaed as below: x o o o i + i u i ; A x i + i u i < A x i X n i = x o o o i i u i ; A x i i u i < A x i (21) o x i ; oherwise If he value of objecive funcion improves in any aemped perurbaions hen exploraory move is a success oherwise failure. The whole process is repeaed o explore all he filer coefficiens unil overall minimum is seleced for nex ieraion. IV. RESULTS AND DISCUSSIONS This secion presens he work performed o implemen he design of low pass digial FIR filer. The hybrid differenial evoluion algorihm has been applied o design low pass digial FIR filer by using various muaion sraegies. The designing of low pass filer is done by seing 200 equally spaced poins wihin frequency domain [0, ]. The prescribed design condiions for he low pass FIR digial filer are given in Table I. Table I: Design condiions for Low Pass digial FIR filer FILTER TYPE PASS BAND STOP BAND MAX VALUE OF H, x Low Pass 0 w o. 2π 0. 3π w π 1 The performance of his algorihm is improved by varying various parameers such as order of he filer, populaion size, muaion facor and crossover rae. The values of he parameers used and ha are opimized are shown in Table II. Table II: DE algorihm Parameers PARAMETER VARIATION OPTIMIZED PARAMETERS Filer Order 20 o Populaion Size 60 o Muaion Facor 0.5 o Crossover Rae 0.1 o Table III: Design resuls of Low pass FIR digial filer for differen orders FILTER ORDER OBJECTIVE FUNCTION MAGNITUDE ERROR 1 MAGNITUDE ERROR 2 PASS BAND PERFORMANCE STOP BAND PERFORMANCE , IJARCSSE All Righs Reserved Page 898

5 Rani e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering 5(7), Fig. 1: Plo of Absolue Magniude Response vs. Normalized Frequency Fig. 2: Plo of Magniude Response in db1 vs. Normalized Frequency Fig. 3: Plo of Phase Response vs. Normalized Frequency Table IV: Coefficiens obained for designing Low Pass digial FIR filer h(n) Filer Coefficiens h(0)=h(44) h(1)=h(43) h(2)=h(42) h(3)=h(41) h(4)=h(40) h(5)=h(39) h(6)=h(38) h(7)=h(37) h(8)=h(36) h(9)=h(35) h(10)=h(34) h(11)=h(33) h(12)=h(32) h(13)=h(31) , IJARCSSE All Righs Reserved Page 899

6 Rani e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering 5(7), h(14)=h(30) h(15)=h(29) h(16)=h(28) h(17)=h(27) h(18)=h(26) h(19)=h(25) h(20)=h(24) h(21)=h(23) h(22) In his firs five muaion sraegies have been applied on various filer orders varying beween The muaion sraegy 2 yields minimum objecive funcion. The muaion sraegy 2 has been implemened by varying he filer order from 20 o 50 and he algorihm runs 100 imes for each order. The design resuls are given in Table III for differen filer orders. From he Table III i is observed ha filer order 44 depics he minimum value of objecive funcion. As he order increases he objecive funcion decreases. The objecive funcion a order 44 is minimum wih value , bu again a order 46 he objecive funcion increases. Then he populaion size is varied from value 60 o 140 on muaion sraegy 2 and order 44. The bes populaion is obained a 130.This populaion is seleced o design low pass FIR filer having objecive funcion value as The muaion facor is varied from 0.5 o 1.0 on populaion size 130, muaion sraegy 2 and order 44. The minimum value of objecive funcion is found a 0.8 muaion facor. The crossover rae value is varied from 0.1 o 0.5 a 0.8 muaion facor. The minimum objecive funcion is obained a 0.25 having value is seleced o design Low pass FIR digial filer. The corresponding filer coefficiens are shown in Table IV. The Fig. 1 shows he graph for Absolue Magniude Response of order 44 for Low pass digial FIR filer. The Fig. 2 shows he graph beween Magniude Response in db and normalized frequency of order 44 o design Low pass digial FIR filer and Fig. 3 shows he graph beween Phase Response and normalized frequency of Low pass digial FIR filer for order 44. The saisical daa including maximum, minimum, average value of objecive funcion and sandard deviaion is shown in Table V. Table V: Maximum, Minimum, Average value of objecive funcion and Sandard Deviaion Filer Type Maximum Value Minimum Value Average Value Sandard deviaion Low Pass From he resuls, i is eviden ha he proposed filer design approach produces minimum objecive funcion, low pass band aenuaion and higher sop band aenuaion. V. CONCLUSION The Hybrid DE is used o design he low pass digial FIR Filer by opimizing he various conrol parameers. The firs five muaion sraegies have been applied for differen filer orders. On he basis of above resuls obained for he design of Low pass digial FIR filer, i is concluded ha i gives beer performance for muaion sraegy 2 wih filer order 44. Simulaion resuls show beer performance of he proposed algorihm in erms of magniude error and maximum pass band and sop band performance. The achieved value of sandard deviaion is which is less han 1. Thus he exploraion search and global search opimizaion yields a powerful ool for he design of FIR filers. The hybrid DE echnique can also be furher used o design high pass, band pass and band sop filers. REFERENCES [1] Yadwinder Kumar, Priyadarshni, Design of Opimum Digial FIR Low Pass Filer Using Hybrid of GA & PSO Opimizaion, Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering, vol. 3, issue 9, pp , [2] Sonika Gupa, Aman Panghal, Performance Analysis of FIR Filer Design by Using Recangular, Hanning and Hamming Windows Mehods, Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering, vol. 2, issue 6, pp , June [3] Balraj Singh, J.S.Dhillon, Y.S.Brar, A Hybrid Differenial Evoluion Mehod for he Design of IIR Digial Filer, ACEEE Inernaional Journal on Signal & Image Processing, vol. 4, no. 1, pp. 1-10, , IJARCSSE All Righs Reserved Page 900

7 Rani e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering 5(7), [4] Mukesh Kumar, Rohi Pael, Rohini Saxena, Saurabh Kumar, Design of Bandpass Finie Impulse Response Filer Using Various Window Mehod, Inernaional Journal of Engineering Research and Applicaions, vol. 3, issue 5, pp , [5] Pradeep Kaur, Simarpree Kaur, Opimizaion of FIR Filers Design using Geneic Algorihm, Inernaional Journal of Emerging Trends & Technology in Compuer Science (IJETTCS), vol. 1, issue 3, pp , [6] Chaohua Dai, Weirong Chen and Yunfang Zhu, Seeker Opimizaion Algorihm for Digial IIR Filer Design, IEEE Transacions on Indusrial Elecronics, vol. 57, no. 5, pp , May [7] Neha, Ajaypal Singh, Design of Linear Phase Low Pass FIR Filer using Paricle Swarm Opimizaion Algorihm, Inernaional Journal of Compuer Applicaions, vol. 98, no. 3, pp , July [8] Sangeea Mondal, S.P.Ghoshal, Rajib Kar, Durbadal Mandal, Novel Paricle Swarm Opimizaion for Low Pass FIR Filer Design, WSEAS Transacions on Signal Processing, vol. 8, issue 3, pp , July [9] G.S Kirga, A.N Surde, Review of Hooke and Jeeves Direc Search Soluion Mehod Analysis Applicable To Mechanical Design Engineering, Inernaional Journal of Innovaions in Engineering Research and Technology (IJIERT), vol. 1, issue 2, pp. 1-14, [10] John G. Proakis and Dimiris G.Manolakis, Digial Signal Processing, Pearson Educaion Limied, fourh ediion, , IJARCSSE All Righs Reserved Page 901