Development and Performance Evaluation of Mismatched Filter using Differential Evolution

Transcription

1 Internatonal Journal of Computer Applcatons ( ) Development and Performance Evaluaton of Msmatched Flter usng Dfferental Evoluton J. B. Seventlne 1 G. V. K. Sharma 2 K. Srdev 3 D. Elzabath Ran 4 K. Raa Raeswar 5 12&3 Dept. of ECE GITAM Insttute of Technology GITAM Unversty Vsahapatnam A.P. INDIA. 4 Dept. of EIE GITAM Insttute of Technology GITAM Unversty Vsahapatnam A.P. INDIA. 5 Dept. of ECE College of Engneerng Andhra Unversty Vsahapatnam A.P. INDIA ABSTRACT Pulse compresson s a technque that plays an mportant role n varous felds le radar sonar and spread spectrum communcaton to acheve the hgh transmt energy of a long pulse whle preservng the range resoluton of a short pulse. In ths paper a new method of desgn of msmatched flter to mnmze the Pea sdelobe rato (PSLR) n the pulse compressed waveform s dscussed. The method used here s the dfferental evoluton optmzaton technque. The performance of the flter s studed wth dfferent codes le bnary barer and chaotc codes.results show that sgnfcant reducton n Pea sdelobe rato s obtaned at the output of the msmatched flter. Keywords Pulse Compresson Autocorrelaton Pea sdelobe rato Dfferental evolutonmsmatched flter. 1. INTRODUCTION In radar pulse compresson technque when a target echo sgnal passes a matched flter the flter output conssts of a man lobe and several sdelobes [12]. These sdelobes can cause false alarms or mas the man lobe of wea target echo sgnals at neghborng range cells.the pea sdelobe s smply the largest sdelobe n the output of the matched flter. Thus the rato of the pea sdelobe to man pea n (PSLR) should be as low as possble so that unwanted clutter gets suppressed.the desgn of radar flters to suppress the range sdelobes has been a subect of consderable nterest n whch lot of research wor has been reported. The least squares nverse flter desgn of msmatched flter was frst proposed by Acroyd and Ghan.Here the sum of squared magntude of the dfference between the nverse flter outputs and deal one s mnmzed [3]. Zoraster has developed a msmatched flter usng lnear programmng technques [4]. Here the flter weghts are desgned n an optmzed manner so that the man lobe ampltude s maxmzed and sdelobe ampltude s mnmzed. For msmatched flter desgn Baden and Cohen used teratvely reweghted least squares to mnmze PSLR.When msmatched flterng s used the man pea wll not be as large as t would have been wth matched flterng.ths loss s referred to as the loss n processng gan or LPG. LPG s expressed n decbels as the rato of the msmatched pea to the matched pea [5].Neural networs usng bac propagaton algorthm developed have also been used for sde lobe reducton [6]. Recently Nunn [7] has suggested that the sgnal/flter optmzaton should start from a sgnal whose own autocorrelaton functon already exhbts low pea or ntegrated sdelobes. Based on hs wor Levanon has developed longer msmatched flters usng optmzaton technques for mnmum ntegrated or pea sdelobes [8].In hs wor the desgn of a msmatched flter usng least squares method to optmze Integrated Sde lobe rato(islr) and desgn of flter usng the matlab functon fmncon to optmze the Pea sdelobe rato has been explaned. The performance of the flter s studed by cross correlatng dfferent bnary codes le barer codes and long bnary codes le MSL codes and bnary chaotc codes wth the optmzed flter coeffcents. In ths paper a n ew method of the desgn of msmatched flter usng dfferental evoluton s dscussed and ts performance study by cross correlatng dfferent codes le bnary barer code and chaotc codes has been presented. The generaton of bnary and ternary chaotc codes s done usng chaotc maps and has been reported n the prevous lterature [9 1]. 2. CLASSICAL DIFFERENTIAL EVOLUTION The dfferental evoluton algorthm (DE) algorthm s a stochastc parallel drect search optmzaton method that s fast and reasonably robust and t was frst proposed by Prce and Storn n 1995[11 12]. The algorthm became qute popular because of easy methods of mplementaton and neglgble parameter tunng [18].It s used to solve problems n varous felds such as control and synchronzaton of chaotc systems [13] parameter dentfcaton [14] and others because of ts consstent and relable performance n nonlnear optmzaton applcatons. The dfferental evoluton combned wth chaotc sequences has also been used to solve the mage enhancement problem [15].The algorthm was named DE due to a specal nd of dfferental operator whch was used to create new offsprng from parent chromosomes nstead of classcal crossover or mutaton. Based on the ndvdual beng perturbed the number of ndvduals used n the mutaton process and the type of crossover used 1 dfferent strateges for DE was proposed by Prce and Storn. In each strategy tral vectors are generated by addng the weghted dfference between other randomly selected members of the populaton. DE/a/b/c s the general conventon used above. DE represents dfferental evoluton a stands for a strng denotng the vector to be perturbed b s the number of dfference vectors consdered for perturbaton of a and c represents the type of crossover beng used (exp: exponental bn: bnomal).the scheme mplemented n ths wor s the DE/rand/1/bn meanng that the target vector s randomly selected and only one dfference vector s used. The recombnaton s controlled by a bnomal decson rule whch s ndcated by the bn acronym. The optmzaton procedure of DE/rand/1/bn s gven by the followng steps and procedures [15][18]. DE also starts wth a populaton of L d dmensonal search varable vectors or parameters le any other evolutonary 56

2 Internatonal Journal of Computer Applcatons ( ) algorthm. The successve generatons n DE s represented by dscrete tme steps le n = n n + 1 etc. The followng notaton for representng the th vector or ndvdual of the populaton at the current generaton (.e. at tme n=n) s used snce the vectors are lely to be changed over dfferent generatons x n [ x 1( n) x2... x d ] (1.1) These vectors are also often called as genes or chromosomes. Step 1: Intalzaton of the varable vectors Intally consder n =.The problem parameters or ndependent varables are ntalzed somewhere n ther feasble numercal range because there may be a certan range wthn whch value of parameter should le for better search results for each search varable. Therefore the th component of the th populaton members may be ntalzed as x () V M rand (1) ( x x ) M x (1.2) where the lower and upper bound of the th parameter of the gven problem s represented as x M and x V respectvely and rand(1) generally exsts between and 1 whch s a unformly dstrbuted random number. Then the ftness value or obectve functon of each ndvdual s evaluated. Step 2: The mutaton operaton An operaton that adds a unform random varable to one or more vector parameters s generally referred as mutaton. The DE reles upon the populaton tself to supply ncrements of the approprate magntude and orentaton. In every teraton of the algorthm or each generaton a donor vector v (n) s created to change each populaton member x (n).the method of creatng ths donor vector vares between the varous DE schemes. In ths wor a specfc mutaton strategy s used whch s explaned here. Three other parameter vectors (say the 1 2 and 3 th vectors( )) are chosen n a random fashon from the current populaton. After that a scalar number f d scales the dfference of any two of the three vectors and the scaled dfference s added to the thrd one to obtan the donor vector v (n).thus the evaluaton for the th component of each vector s expressed as v ( 3 n 1) x f ( x x ) 1 d 2 (1.3) Step 3: The crossover operaton Crossover s appled n the populaton followng the mutaton operaton. There are two types of crossover called bnomal crossover and exponental crossover. In bnomal crossover whch s used n ths wor for each mutant vector v (n+1) tral vector s generated wth z ( z n 1) [ z 1( n 1) z2( n 1)... z d ( n 1)] ( n 1) v x ( n 1) f randb( ) CR otherwse (1.4) where z (n) stands for the th component of the th ndvdual or real-valued vector after crossover operaton randb() s the th evaluaton of a unform random number generaton n the range [ 1] CR s the crossover rate n the range [ 1]. Step 4: Selecton operaton To determne whch one of the target vector x (n) and the tral vector z (n+1) wll survve n the next generaton DE actually nvolves the Darwnan prncple of Survval of the fttest n ts selecton process. So the next step of the algorthm s selecton and ths step ensures that the populaton sze s constant over subsequent generatons. Thus to decde whether or not the vector z (n+1) should be a member of the populaton comprsng the next generaton t s compared to the correspondng vector x (n). Thus f F() denotes the obectve functon then x ( n 1) z ( n 1) x f F( z ( n 1)) F( x ) otherwse (1.5) Step 5:The stoppng crteron The generaton number s ncreased to n = n+ 1 and then we need to proceed to step 3 untl a stoppng crteron s met usually a maxmum number of teratons or generatons n max. 3. PROBLEM FORMULATION The desgn of msmatched flter usng partcle swarm optmzaton technque has been reported earler[17]. The same procedure s adopted here to fnd the optmum weghts of a msmatched flter usng the dfferental evoluton algorthm so that the sdelobe levels n the flter output are reduced. Phase coded pulse compresson s often used to ncrease range resoluton and accuracy when energy requrements dctate a pulse length substantally n excess of the desred resoluton n pulsed radar system. Codes or sequences refer to the phase of the carrer n phase coded pulse compresson. Consder a bnary or ternary sequence gven by S s s... s } (1.6) { 1 N1 where N s the number of samples of the sequence. The bnary sequence has alphabets ±1where as the ternary sequence has alphabets ±1.The output of the matched flter wthout Doppler shft whch s nothng but the aperodc autocorrelaton functon for postve delays s defned n equaton(1.7) r( ) N 1 s s for 1 N 1 (1.7) The flter elements or weghts of the msmatched flter are consdered as H h h... h } (1.8) { 1 M 1 where the weghts are real and M( N) ndcates the flter length. For mathematcal convenence M wll be assumed to be even f N s even and odd f N s odd. The flter output s not necessarly symmetrc. The output of the msmatched flter [17] s gven as G M 1 h s for (N 1) (M 1) (1.9) 57

3 Msmatch flter output n Internatonal Journal of Computer Applcatons ( ) Here we assume s = f < or >N 1.The constrant functon s defned as Maxmse F G M N 2 M 1 h s M N / 2 M 1 Subect to G M N h s 1 2 for (N 1) (M 1) M N/2. (1.1) The man lobe level whch has to be maxmzed s referred to as F. The nequalty condton restrcts all sde-lobes to be less than and equal to one n absolute value whch s gven by 1.1. The Pea sdelobe rato whch s generally referred as PSLR of the msmatched flter output s defned n equaton (1.11).In ths equaton the maxmum value of the flter output or man lobe level whch occurs at =M-N/2 s compared wth maxmum of the absolute value of the flter outputs at M-N/2 whch are consdered as the pea sdelobe levels. By consderng ths constrant functon the PSLR optmzed flter or flter wth reduced sdelobe levels s obtaned max G PSLR 2log G M N / 2 M N 2 (1.11) Here the ndvduals or varable vectors are chosen as msmatched flter mpulse response coeffcents or elements. The mutaton factor f d s chosen as.5 and the crossover rate (CR) s chosen as.3. The ntal populaton of search varable vectors s chosen accordng to step 2. It s well nown that the matched flter weghts are the sequence tself whch may be +1-1 or zero. The msmatched flter weghts are chosen to have a slght devaton from the zero padded matched flter s response whch s of the form H =[Z S Z] where Z s a all zero sequence of length z=m-n/2.so f the length of the sequence (N) s 13 and the flter length(m) s 39 z s 13.When the flter s desgned n ths fashon the pea of the flter output occurs at =M-N/2.For convenence the flter length s consdered as three tmes the length of the sequence. The populaton sze and the number of teratons are vared wth the length of the code. ternary codes of varyng lengths. The PSLR obtaned usng equaton(1.11)s shown n Table 1.1and 1.2. The well nown matched flter response of Barer 13 has an PSLR of but wth the PSLR optmzed flter lower PSLR s observed. The msmatched flter output from =-12 to =38 s plotted n Fg.1.1.As observed from the fgure the pea of the flter output occurs at =13.The matched flter output s not symmetrc about =M-N/2. The flter exhbts a LPG loss of -.33 and the PSLR obtaned s The devaton of the msmatched flter weghts from the matched flter weghts are plotted n Fg.1.2.The populaton sze chosen s 2 and the number of teratons s 2. Fg.1.3 ndcates the msmatched flter output wth a ternary chaotc code of length 21.The PSLR wth matched flter was Wth msmatched flter PSLR as low as s obtaned as observed from the plot. But the LPG s The populaton sze s 5 and the number of teratons s Fg 1.1 Msmatched flter output for Barer code 4. SIMULATION RESULTS The smulaton s done usng Matlab and the performance of the PSLR optmzed msmatched flter s studed for dfferent codes le barer code chaotc bnary code and chaotc 58

4 msmatched flter output n devaton Table 1.1 PSLR of Barer code and chaotc bnary code Internatonal Journal of Computer Applcatons ( ) Code Length of the code PSLR n (From matched flter output) PSLR n (From msmatched flter output) LPG n Barer Chaotc bnary code flter weghts Fg.1.2 The msmatched flter elements devaton from Barer 13 sgnal values flter. The sgnfcant mprovement n PSLR s observed from the table. To understand the reducton n PSLR very clearly a plot of PSLR obtaned wth matched flter and wth msmatched flter for ternary chaotc codes s also plotted n Fg 1.4 Table 1.2 PSLR of ternary chaotc codes from matched flter and msmatched flter Length of sequence PSLR n (From matched flter o/p) PSLR n (From msmatched flter o/p) LPG n K Fg 1.3. Msmatched flter output for ternary chaotc code of length Table 1.2 shows the comparson of PSLR of the ternary chaotc codes obtaned from the autocorrelaton functon output of matched flter reported earler [1] and msmatched 59

5 PSLR n Internatonal Journal of Computer Applcatons ( ) Wth matched flter Wthmsmatched flter Length of the code Fg.1.4.Comparson plot of PSLR wth matched flter and wth msmatched flter for ternary chaotc codes 5. CONCLUSION In ths paper a new method of desgn of msmatched flter to optmze PSLR of bnary and ternary codes s proposed.for the ternary chaotc codes good mprovement n PSLR s observed at lower lengths. The flter can be used for larger length sequences but the length of the flter also proportonally ncreases wth the length of the sequence whch may mae the practcal mplementaton of the flter dffcult. The most sgnfcant reducton n PSLR obtaned wth ternary chaotc code s at length 27. Reducton n PSLR of the msmatched flter output of Barer code s also observed whch s In prevous lterature desgn of msmatched flters usng evolutonary algorthms le partcle swarm optmzaton has been reported. But the advantage of DE algorthm over the other methods s that t presents smple structure convergence speed versatlty and robustness. The performance of the classcal DE s senstve to the choce of control parameters ncludng the desgn of mutaton factor. So ths wor can be extended to obtan better results by usng the other schemes of DE to sequences of any length. 6. REFERENCES [1] SonM.I.Introducton to Radar Systems Tata McGraw Hll Boo Co. New Yor3 rd Ed. 21. [2] Levanon N. Mozeson E.Radar Sgnals Hoboen NJ:John Wley & Sons Inc.24. [3] AcroydM.H.and Ghan.F Optmum msmatched flters for sdelobe suppresson IEEE Transactons on Aerospace and Electronc systemsvol.9pp March [4] ZorasterS. Mnmum pea range sdelobe flter for bnary phase coded waveforms IEEE Transactons on Aerospace and Electronc systemsvol.16pp January 198. [5] BadenJ.M. and M.N Cohen Optmum pea sdelobe flters for bphase pulse compresson Proceedng of the IEEE Internatonal radar Conference pp [6] KwanH.K. and LeeC.K. A neutral networ approach to pulse radar detecton IEEE Trans. Aerosp. electron.syst.29 (1) pp [7] Nunn C. Constraned optmzaton appled to pulse compresson codes and flters. IEEE Int. Radar Conf. pp May 25. [8] LevanonN. Cross Correlaton of long bnary sgnals wth long msmatched flters IEE Proc.-Radar and Sonar navg. July 25. [9] Xn Wu Wexan Lu Le Zhao Jeffrey.S. Fu Chaotc Phase Code for Radar Pulse Compresson. IEEE Radar Conf. 21 pp [1] SeventlneJ.B. Elzabeth Ran D.Raa Raeswar.K Ternary Chaotc Pulse Compresson Sequences Journal of Rado Engneerng Vol.19No:3 pp September 21. [11] Storn R. Prce K. Dfferental evoluton: a smple and Effcent Adaptve scheme for global optmzaton over contnuous spaces Techncal Report TR Internatonal Computer Scence Insttute Bereley USA [12] Storn R.Prce K.V. Dfferental evoluton a smple and effcent heurstc for global optmzaton over contnuous spaces. Journal of Global Optmzaton Vol.11 pp Lu B Wang.L. Jn Y.H. Huang D.X Tang F. Control and Synchronzaton of chaotc systems by dfferental evoluton algorthm Chaos Soltons & Fractals 34(2) Elseverpp [13] ChangW.D. Parameter dentfcaton of Rossler s chaotc system by an evolutonary algorthm ChaosSoltons& Fractals Elsever pp [14] Leandrrodos Santos Coelho Joao Gulherme Sauer and MarceloRude Dfferental evoluton optmzaton combned wth chaotc sequences for mage contrast enhancement Chaos Soltons and Fractals Elsever Volume 42 Issue 1 pp october 29 Swagatam Das Ponnuthura Nagaratnam Suganthan Dfferental Evoluton: A Survey of the State-of-the-Art IEEE Transactons on Evolutonary Computaton Vol. 15 No. 1 pp.4-31february 211 [15] Vas Baghel. Multobectve Optmzaton New Formulaton and Applcaton to Radar Sgnal Processng M.Tech thess29. [16] S. Das A. Abraham and A. Konar Partcle SwarmOptmzaton and Dfferental Evoluton Algorthms:Techncal Analyss Applcatons and HybrdzatonPerspectves Studes n Computatonal Intellgence (SCI) (28). 6