Design of Digital Band Stop FIR Filter using Craziness Based Particle Swarm Optimization (CRPSO) Technique

Transcription

1 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May 05 6 Desgn o Dgtal Band Stop FIR Flter usng Crazness Based Partcle Swarm Optmzaton (CRPSO) Technque Shnam Ran, Darshan Sngh Sdhu, (Department o Electroncs and Communcaton Engneerng (ECE), Gan Zal Sngh PTU Campus, Bathnda, Punjab, Inda) ABSTRACT Ths paper presents the evolutonary optmzaton technque called crazness based partcle swarm optmzaton (CRPSO) or the desgn o a dgtal band stop FIR lter. CRPSO s the much mproved verson o PSO, that proposes a new denton or the velocty vector and swarm updatng and hence the soluton qualty s mproved. It s a populaton based global stochastc search algorthm whch nds near optmal soluton n terms o a set o lter coecents. Perormance o proposed optmzaton technque s compared wth some well establshed algorthm such as partcle swarm optmzaton (PSO). From the smulaton results t has been demonstrated that the proposed algorthm CRPSO can optmze the dgtal FIR lter n terms o mnmzng the magntude errors and mnmzng the rpple magntudes o both pass band and stop band. Keywords- Crazness based partcle swarm optmzaton (CRPSO), Fnte mpulse response (FIR) lter, Magntude response, Partcle swarm optmzaton (PSO). I. INTRODUCTION Dgtal sgnal processng s an area o scence and engneerng that has developed to have a major and ncreasng mpact n many key areas o technology ncludng meda, dgtal televson, dgtal audo, moble phone, bomedcne, CD. The smaller, powerul, aster, and cheaper dgtal computers have developed due to the rapd developments o DSP n ntegrated-crcut technology, start wth medum-scale ntegraton (MSI) and succeedng to large-scale ntegraton (LSI), and now, very-large-scale ntegraton (VLSI) o electronc crcuts. Due to these cheaper and relatvely ast dgtal crcuts have made t possble to construct hghly complcated dgtal systems that are capable o perormng complex dgtal sgnal processng unctons and tasks, whch are too expensve and usually complcated to be perormed by analog crcut or analog sgnal processng systems. To process the analog sgnal, dgtal sgnal processng provdes an alternatve method. To perorm the processng dgtally, an nterace s needed between the analog sgnal and the dgtal sgnal processor. Ths nterace s called an analog-to-dgtal (A/D) converter. The output o the A/D converter s a dgtal sgnal that s gven to the dgtal sgnal processor. The dgtal sgnal processor may be a dgtal computer to perorm the desred operatons on the nput sgnal. Dgtal sgnal processng (DSP) s the mathematcal manpulaton o an normaton sgnal to mody or to mprove t []. The advantages o DSP are ts more ecent, smple to store and use, nexpensve, applcable to very low requency sgnals and easy to construct []. The processng o sgnal manly ncludes lterng whch means to extract the normaton rom the sgnal to remove unwanted part o sgnal and to mprove sgnal qualty. In engneerng applcatons derent lterng operatons are to be perormed or whch derent lters are requred such as low pass, hgh pass, band pass, band reject, all pass, derentator, ntegrator etc. A lter s bascally a network or system that s wdely used n sgnal processng and communcaton crcut systems to remove or at least unwanted parts o the sgnal such as nose. Based on the nput sgnal, lters are classed as: analog lters and dgtal lters. Analog lters are the devce that operates on contnuous-tme sgnals. These lters consst o passve and actve basc elements o electroncs lke operatonal amplers, capactors, resstors etc and these are easy to develop and especally cheap as compared wth dgtal lters. Dgtal lters consst o DSP processors and controller whch plays an mportant role n DSP applcatons such as sgnal analyss and estmaton. In sgnal processng, a dgtal lter s a system that perorms numercal operatons on sampled values o sgnal, dscrete sgnal to reduce or mprove certan aspects o that sgnal. Dgtal lters may be more expensve than analog lters due to ther complex structure, but they make practcal many desgns that are not practcal or mpossble as analog lters. Dgtal lters are classed as nnte mpulse response (IIR) and nte mpulse response (FIR) on the bass o

2 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May ther mpulse response. The mpulse response o IIR lter s nnte, means unt sample response exsts or zero to nnty. These are non lnear and recursve type lters. A recursve type lter has eedback rom output to nput, n general ts output s a uncton o the prevous output samples and the present and past nput samples. The mpulse response o FIR lter s nte. Ths means the mpulse response o FIR lters settles to zero wth n a nte amount o tme. FIR lters are called non-recursve as they have no eedback [3]. The advantage o FIR lter over IIR lters are, FIR lter contan lnear phase by makng the coecents symmetrcal whereas IIR lter has no partcular phase. FIR lter always reman stable and depends only on nput. FIR lters have only zeros and IIR lters consst o both poles and zeros. Conventonally, there are many well known methods or FIR desgn, such as the wndow method, requency samplng method, optmal lter desgn method. The wndowng method smply conssts o truncatng or wndowng a theoretcally lter mpulse response by some sutably chosen wndow uncton. The wndow method s ast, sutable, robust but generally suboptmal. There are varous knds o wndow unctons (Butterworth, Chebyshev, Kaser, and ammng) avalable dependng on the lter speccatons lke rpples n pass band, stop band, maxmum stop band attenuaton and on transton wdth []. Its major dsadvantage s the lack o accurate control o the crtcal requences such as pass band and stop band cut-o requency and the transton wdth. The objectve uncton or the desgn o optmal dgtal lters nvolves accurate control o varous parameters o requency spectrum and s hghly non-unorm, non-lnear, nonderentable and multmodal n nature. These objectve unctons cannot optmze. So, to desgn the optmal dgtal FIR lters, evolutonary optmzaton methods lke GA [5], smulated annealng and artcal bee colony optmzaton have been mplemented whch are qute ecent and havng better control o parameters. Genetc algorthms s ecent to obtan local optmum whle mantanng ts moderate computatonal complexty but t s not very successul n determnng the global mnma n terms o convergence speed and soluton qualty. In ths paper evolutonary optmzaton technque o Partcle Swarm Optmzaton (PSO) or the desgn o dgtal band stop FIR lter s presented. Partcle Swarm Optmzaton (PSO) was developed by Eberhart and t s evolutonary algorthm [6]. The merts o PSO are smple to mplement and ts convergence may be controlled va ew parameters. But t contans lmtatons that are premature convergence and stagnaton problem. To overcome these lmtatons, crazness based partcle swarm optmzaton (CRPSO) technque s used whch s modcaton to PSO and employed or FIR band stop lter desgn. The CRPSO algorthm tres to nd the best coecents that are closely match to the deal requency response and t presents the eectveness, comprehensve set o results and better perormance o the proposed desgned method. Ths paper s arranged as ollows. In secton II, the FIR band stop lter desgn problem s ormulated. Secton III brely dscusses the algorthm o classcal PSO and the CRPSO. Secton IV conssts o the smulaton results that are obtaned or Band stop FIR dgtal lter. Fnally, secton V concludes the paper. II. FIR FILTER DESIGN PROBLEM FIR lter s known as non-recursve lter whch means there s no eedback connecton. The mpulse response o FIR lter s nte. Ths means that the mpulse response sequence o FIR lters has a nte number o non-zero terms. For the realzaton o FIR systems present, past, uture samples o nput are requred. The conventonal desgn o FIR dgtal lter s descrbed by the derence equaton and t s expressed as: M y m b x( m k () ) k 0 k ere b k s the set o lter coecents. M s order o lter. The transer uncton o FIR lter s gven as: z N n0 h n z n, n 0,,..., N () Where z s the requency doman representaton o the mpulse response and s termed as system uncton o the dgtal lter, h n s the tme doman representaton o the mpulse response o the lter, N s the order o the lter. Ths paper presents the FIR wth h n as even symmetrc and the order s even. The length o h n s N+ and the number o coecents are also N +. h n coecents are symmetrcal so that dmenson o the N problem s halved. Thus, number o h(n) coecents are actually optmzed, whch are nally coupled to nd the requred (N+) number o lter coecents. The requency response o a desred FIR lter s gven as:

3 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May d N j jn e hne n0 j ere e d (3) s called the Fourer transorm complex actor. For a BS lter the deal response s dened as: j 0 c c e or () otherwse ω c, ω c are the cuto requency o the deal band stop lter. The perormances o dgtal FIR lter can be calculated by usng L -norm and L -norm approxmaton error o magntude response and rpple magntude o both passband and stop-band. The FIR lter s desgned by optmzng some coecents or parameters so that L p - norm approxmaton error uncton or magntude s to be mnmzed. L p -norm s expressed as [7]: K p Ex d, x (5) 0 Where d (ω ) s the desred magntude response o the deal FIR lter and (ω,x) s the obtaned magntude response o the FIR lter. For p=, magntude error denotes the L - norm error and or p= magntude error denotes the L -norm error. L -norm error, e x o magntude response s stated as below: e K x x d, 0 p (6) And the squared L -norm error, e x o magntude response s stated as below: e K d, 0 x x (7) The desred magntude response d (ω ) o FIR lter s gven as: passband d w or 0 stopband (8) δ p and δ s are the rpple magntude o pass band and stop band whch are to be mnmzed and these are expressed as:, x mn x p max, or ε passband max x or ε stopband (9) s, Four objectve unctons or optmzaton are: x Mnmze e x x Mnmze e x 3 x Mnmze p x Mnmze s The mult objectve uncton s converted nto sngle objectve uncton: Mnmze x x x x x (0) 3 3 ω, ω, ω 3, ω are the weghtng uncton III. EVOLUTIONARY TECNIQUE EMPLOYED. Classcal PSO PSO s a lexble, robust populaton-based stochastc search or optmzaton technque that s based on the ntellgence and movement o swarms. PSO can easly handle non-derental objectve unctons and larger search space unlke tradtonal optmzaton methods. PSO was orgnally nvented by James Kennedy and Russell Eberhart ater beng nspred by the study o the behavor o brd lockng and sh schoolng. From the nspraton o nature, t conssts o a number o partcles that make the swarm and movng n the search space to look or the best soluton. Each partcle s behaved as an ndependent pont n search space whch adjusts ts poston accordng to ther own lyng experence and ther neghbor s lyng experence. In the search space each partcle has some tness value that has been acheved by that partcle so ar. Ths value s known as personal best (pbest). Also, there s another best value that s obtaned so ar by any partcle n the neghborhood o that partcle n the search space. Ths value s known as global best (gbest). So ater ndng the pbest and gbest values, the partcle updates ther velocty and postons by usng these two equatons [8]: V C w V rand C rand gbest X pbest X () X X V ()

4 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May Where rand and rand are random numbers chosen between 0 and, w s the weghtng uncton and wll be updated by progressvely teratons IT w w ( w w ) max max mn max IT wmaxs the maxmum value o weghtng uncton, mn s the mnmum value o weghtng uncton and (k) max IT are the maxmum teratons, V s velocty o a partcle at k th teratons, X (k) s current partcle value at k th teratons, C and C are learnng actors havng value. ere two random parameters rand and rand o () are ndependent. I both are large, the partcle s drven too ar away rom the local optmum. I both are small, the convergence speed o the technque s reduced. So, nstead o takng rand and rand ndependent one sngle random number r s chosen so that when r s large, (- r ) s small and vce versa. Another random parameter r s ntroduced to control the balance o global and local searches. The global search ablty o PSO s mproved wth the help o the ollowng modcatons. Ths moded PSO s termed as mproved partcle swarm optmzaton (IPSO). Wth all modcatons, the velocty can be expressed as ollows [9]: V pbest gbest r X sgn r 3 v ( r ( r ) C X ) C ( r ) r w (3) Where r, r and r 3 are the random parameters unormly taken rom the nterval [0, ] and sgn (r 3 ) s a uncton dened as: r sgn ( r 3 ) where () r For brds lockng or ood, there could be some rare cases that ater the poston o the partcle s changed accordng to (), the drecton o brd s velocty wll be opposte to the sure area o ood due to nerta. So, n that case the drecton o the brd s velocty should be reversed. Sgn(r 3 ) s used to ensure the lyng drecton always towards the promsng regons.. Crazness Based Partcle Swarm Optmzaton (CRPSO) A crazness operator s ntroduced n the proposed technque to ensure that the partcle would have a predened crazness probablty to mantan the dversty o the partcles. Beore updatng ts poston the velocty o the partcle s crazed by [0]: V V P crazness r sgnr v (5) Where r s a random parameter whch s chosen unormly wthn the nterval [0, ], v crazness s a random parameter whch s unormly chosen rom the nterval [ v mn, v max ] and P( r ),sgn( r ) are dened as: P ( r ) where 0 sgn ( r ) where r r P cr P cr Where P cr s a predened probablty o crazness. The steps o CRPSO algorthm are as ollows: r r (6) (7). Intalze the populaton or a swarm o n p vectors, n whch each vector represents a soluton o lter coecent.. Computaton o ntal cost (tness) values o the total populaton. 3. Take the partcle wth the best tness value or mnmum tness value.e. global best (gbest).. Compare the newly calculated tness value wth prevous one and select the one havng better tness value as personal best (pbest). 5. Update velocty o partcles usng Eq.(3) and Eq. (5) and poston o partcles usng Eq. (). 6. Update the pbest and gbest vectors and replace the updated partcle vectors as ntal partcle vectors. 7. Iteraton contnues tll the maxmum teraton cycles or the convergence o mnmum cost values are reached. The desgn am o ths paper s to obtan the optmal combnaton o the lter coecents, so as to acqure the mnmum magntude error and maxmum stop band attenuaton. The values o the parameters used or the CRPSO technque are gven n Table. The desgnng o Band Stop FIR dgtal lter s done by settng 00 equally spaced ponts wthn the requency doman [0, π]. The prescrbed desgn condtons or the desgn o Band Stop FIR lter are gven n Table.

5 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May Flter Type Band stop Parameter Table CRPSO parameters Value Populaton sze 00 Iteraton cycle 00 C, C v mn 0.0 v max.0 P cr v crazness Table Desred Desgn Condtons or Band Stop Flter Pass-band Stop-band Maxmum value o (ω,x) Ths secton presents the smulatons perormed n MATLAB or the desgn o FIR band stop (BS) lter. CRPSO has been appled to desgn FIR BS lter wth order rom 6 to and the results are gven n Table 3. Fg. shows the graph o PSO verses CRPSO and t s plotted between derent orders o lter and objectve uncton. From gure t s observed that objectve uncton o CRPSO s less than PSO or all the orders and at order objectve uncton s mnmum as compared to other orders. So order s selected or desgnng FIR BS lter usng CRPSO. Fg. ndcates the magntude response n db verses normalzed requency or band stop lter o order. The absolute magntude response or band stop lter o order has been shown n Fg.3. Fg. depcts the graph between objectve uncton and no. o teratons. It s concluded that the magntude gets stablzed ater 0 teratons or band stop FIR lter. Table shows the best optmzed lter coecents obtaned or BS lter wth the order o by PSO and CRPSO. Table 5 ndcates that the best value, average value and standard devaton o objectve unctons are much better or CRPSO than PSO whch assured the eectveness o the desgned system. From Table 6 t may be noted that the maxmum stop band attenuaton acheved or BS lter o order usng the CRPSO s db and t s also observed rom Table 6 that the smulaton results obtaned or lter order usng CRPSO are better than the PSO. IV. RESULTS AND DISCUSSIONS Table 3 Desgn Results or Band stop FIR Flter Order L - norm Error L - norm Error Pass-band perormance (rpple magntude) Stop-band perormance (rpple magntude)

6 Objectve uncton Objectve uncton Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May PSO CRPSO Orders o Band stop lter Fg. Graph o PSO verses CRPSO Fg. Magntude (db) plot or the FIR BS Flter o order Fg. 3 Magntude (abs) plot or the FIR BS lter Order No. o teratons Fg. Objectve uncton verses no. o teratons grap Table Optmzed coecents o FIR BS lter o order h(lter coecent) PSO CRPSO h()=h(3) h()=h() h(3)=h() h()=h(0)

7 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May h(5)=h(9) h(6)=h(8) h(7)=h(7) h(8)=h(6) h(9)=h(5) h(0)=h() h()=h(3) h() Table 5 Statstcal Data or FIR BS Flter o order Algorthm Maxmum value o Objectve uncton Mnmum value o Objectve uncton Average o Objectve uncton Standard devaton PSO CRPSO Algorthm Table 6 Summary o CRPSO results wth PSO or FIR BS lter o order Objectve Functon BS lter Maxmum Stop-band attenuaton(db) Maxmum Pass-band rpple (normalzed) Maxmum stop-band rpple (normalzed) PSO CRPSO From the results, t s evdent that the proposed lter desgn approach CRPSO produces mnmum objectve uncton, hgher stop band attenuaton and smaller stop band rpple compared to PSO. V. CONCLUSION Ths paper presents a novel and accurate method or desgnng dgtal Band stop FIR lters by usng CRPSO as a much mproved verson o PSO. Flter o order have been realzed usng CRPSO algorthm. Smulaton results show better perormance o the proposed CRPSO algorthm n terms o objectve uncton, magntude error, and maxmum stop band attenuaton and CRPSO technque can also be used to desgn hgh pass, low pass and band pass lters. REFRENCES [] John G. Proaks and Dmtrs G.Manolaks, Dgtal Sgnal Processng, Pearson,ourth edton, 03.

8 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Volume, Issue 5, May 05 7 [] A.V.Oppenhem, R.W.Schaer and J.R.Buck, Dscrete-Tme-Sgnal-Processng, Prentce all, NJ, Englewood Cls, 999. [3] Parks T W, Mcclellan J. Chebyshev approxmaton or non recursve dgtal lters wth lnear phase [J]. IEEE Transactons on Crcuts Theory, vol. 9, No., pp. 89-9, 97. [] Saurabh Sngh Rajput, Dr. S.S. Bhadaura, Comparson o Band-stop FIR Flter usng Moded ammng Wndow and Other Wndow unctons and Its Applcaton n Flterng a Muttone Sgnal, Internatonal Journal o Advanced Research n Computer Engneerng & Technology (IJARCET) Vol., Issue 8, October 0. [5] ung-chng Lu, Shan-Tang Tzeng, Desgn o arbtrary FIR log lters by genetc algorthm approach, Sgnal Processng, 000, 80, pp [6] J.Kennedy and R.Eberhart, Partcle swarm optmzaton, n Proceedngs o IEEE Internatonal Conerence Neural Network, Perth Australa, no., pp. 9-98, 995. [7] Ranjt Kaur, Manjeet Sngh Patterh, Damanpreet Sngh and J.S. Dhllon, eurstc SearchMethod For Dgtal IIR Flter Desgn, WSEAS Transactons on Sgnal Processng, vol.8, ssue 3, July 0. [] S. K. Saha, R. Kar, D. Mandal, S. P. Ghoshal, "Desgn and Smulaton o FIR Band Pass and Band Stop Flters usng Gravtatonal Search Algorthm," Journal o Memetc Computng, vol. 5, pp. 3 3, Sprnger, 03. [] S. K. Saha, R. Kar, D. Mandal, S. P. Ghoshal, "Optmal Lnear Phase FIR Flter Desgn usng Partcle Swarm Optmzaton wth Constrcton Factor and Inerta Weght Approach wth Wavelet Mutaton", Internatonal Journal o ybrd Intellgent Systems (IJIS), vol., No., pp. 8-96, IOS Press, 0. [3] Yng-png Chen and Pe Jang, Analyss o Partcle Interacton n Partcle Swarm Optmzaton, ELSEVIER, Theoretcal Computer Scence,, 00. [] Subhadeep Chakraborty, Abhrup Patra, Perormance Analyss o IIR Dgtal Band Stop Flter, Internatonal Journal o Advanced Research n Computer Engneerng & Technology (IJARCET), Vol., No 5, May 03. [5] Sangeeta Mondal, Vasundhara,Rajb Kar, Durbadal Mandal, S. P. Ghoshal, Desgn o optmal lnear phase FIR hgh pass lter usng crazness based partcle swarm optmzaton technque, World Academy o Scence, Engneerng and Technology Vol. 5, 0-. [8] A. Mukhopadhyay, R. Kar, D. Mandal, S. Mondal, S. P. Ghoshal, Optmal desgn o Lnear phase FIR band stop lter usng Partcle Swarm Optmzaton wth Improved Inerta Weght Technque, Proc. 9th IEEE JCSSE 0, Bangkok, Thaland, pp , 30 May - 0 Jun 0. [9] S. Mandal, D. Mandal, S. P. Ghoshal and R. Kar, FIR Band Stop Flter Optmzaton by Improved Partcle Swarm Optmzaton, World Congress on Inormaton and Communcaton Technologes (WICT 0), Mumba, Inda, pp , Dec. 0. [0] Suman Kumar Saha, Rajb Kar, Durbadal Mandal, S.P. Ghoshal. IIR Flter desgn wth Crazness based Partcle Swarm Optmzaton Technque, Internatonal Scence Index Vol. 5, 0.