A Multi-modulus Blind Equalization Algorithm Based on Memetic Algorithm Guo Yecai 1, 2, a, Wu Xing 1, Zhang Miaoqing 1

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1 Internatonal Conference on Materals Engneerng and Informaton Technology Applcatons (MEITA 1) A Mult-modulus Blnd Equalzaton Algorthm Based on Memetc Algorthm Guo Yeca 1,, a, Wu Xng 1, Zhang Maoqng 1 1 College of Electronc and Informaton Engneerng, Nanjng Unversty of Informaton Scence and Technology, Nanjng 144, Chna Jangsu Collaboratve Innovaton Center on Atmospherc Envronment and Equpment Technology (CICAEET), Nanjng 144, Chna a emal: guo-yeca@163.com Keywords: Sgnal Processng; Mult-Modulus Algorthm; Memetc Algorthm; Intellgence Optmzaton Algorthm; Optmal Weght Vector Abstract. In vew of slow convergence speed, large steady mean square error (MSE), and exstng blnd phase for the constant modulus blnd equalzaton algorthm(cma), a mult-modulus blnd equalzaton algorthm based on memetc algorthm(ma-mma) s proposed, whch combnes the basc dea of ntellgent optmzaton algorthm and ntroduces the ndvdual own evoluton and socal behavor among ndvduals to the blnd equalzaton technology. In ths proposed algorthm, the recprocal of the cost functon of mult-modulus blnd equalzaton algorthm(mma) s defned as the ftness functon of the memetc algorthm(ma), the ntal optmal weght vector of the MMA s optmzed by usng the global nformaton sharng mechansm and local depth search ablty of the MA. When the ntal optmum weght vector of the MMA s obtaned, the weght vector of the MMA may be updated. The smulaton results wth the hgher-order APSK mult-modulus sgnals show that, compared wth the CMA, the MMA, and the mult-modulus blnd equalzaton algorthm based on genetc algorthm(ga-mma), the proposed MA-MMA has the fastest convergence speed, smallest mean square error(mse), and clearest output constellatons. Introducton In a communcaton system, blnd equalzaton algorthm s ntroduced to the recever n order to effectvely elmnate the nter-symbol nterference(isi) caused by lmted bandwdth and mult-path propagaton. In blnd equalzaton algorthms, constant modulus blnd equalzaton algorthm(cma) can make the output constellaton ponts dstrbute n the crcle wth radus R C (statstcal mode value of nput sgnal) as far as possble, ts cost functon s only related to the receved sequence ampltude, so CMA s very sutable to equalze constant modulus sgnals[1]. However, the dfferent modulus constellaton ponts of hgh order QAM or APSK sgnals are dstrbuted n dfferent radus of the crcle. If CMA s used to equalze hgh order QAM or APSK sgnals, the output constellaton ponts of hgh order QAM or APSK sgnals wll be focused sngle crcle, there wll be large mean square error(mse), and CMA even wll become nvald. In recent years, some mult-modulus blnd equalzaton algorthms(mmas) were proposed to equalze mult-modulus sgnals. The MMAs can make the output constellaton ponts of hgh order QAM or APSK sgnals equalze to dfferent crcles wth dfferent rad[][3]. Compared wth CMA, especally for non-square constellaton and ntensve constellaton, MMA can make full use of the statstcal characterstcs of symbols. However, the MMA and CMA all exst the model error so that the convergence speed and resdual MSE aren t stll very deal. Memetc Algorthm s an optmzaton algorthm wth genetc mechansm and local searchng, can make all ndvduals obtan local optmal value after teraton, and s easy to understand and mplement, as well as has smple operaton [4][]. Accordng to characterstcs of the MA and the MMA, a mult-modulus blnd equalzaton algorthm based on MA s proposed. It can search global optmal group fast by MA, whch s used 1. The authors - Publshed by Atlants Press 13

2 as the ntal optmal weght vector of the MA. The smulaton results show that, the proposed algorthm has the ablty to equalze hgher-order mult-modulus sgnals, ts convergence speed s faster and ts steady MSE s smaller. Mult-modulus Blnd Equalzaton Algorthm The MMA s shown n Fgure 1. In Fg.1, a decson devce s used to determne the decson modulus value R of mult-modulus sgnals. x ( n) s the recevng sequence to equalzer, w ( n ) s the weght vector of the equalzer, zn ( ) s the output of the equalzer, zns ˆ( ) estmated sgnal of zn, ( ) e ( n ) s error sgnal, R J s samplng modulus value. After znpasses ( ) nonlnear estmator wthout memory, the estmated sequence zn ˆ( ) nstead of the expected sgnal can be obtaned. The memory less nonlnear functon of the MMA s defned as zn ˆ( ) = gzn [ ( )] = zn ( )[1 + R zn ( ) ] (1) R J s defned as 4 E[ zn ( ) ] R = J E[ zn ( ) ] () The output of the MMA s gven by T zn ( ) = w ( n) x ( n) (3) The error sgnal s defned as e( n) = zn ˆ( ) zn ( ) = zn ( )[ R zn ( ) ] (4) The cost functon of the MMA s wrtten as J ( n) = E[ R zn ( ) ] () MMA { } The weght vector teratve formula of the MMA s obtaned by stochastc gradent and wrtten as w ( n+ 1) = w ( n) µ e ( n) x ( n) (6) where µ s step sze factor. Accordng to decson modulus value R, MMA can adjust error functon. For 3-APSK sgnal, ts constellatons can be known as the star QAM[6][7] and shown n Fgure. In Fgure, the nput constellaton ponts are dstrbuted n three dfferent crcles wth dfferent rad, two crcles wth the dashed border are decson boundary condton for decson devce. The constellatons are dvded nto three regons correspond to three error functon by two decson crcles. R J R RB R x ( n) w ( n zn ( ) ) g() zn ˆ( ) CMA - + e ( ) n Fg. 1 MMA Fg. 3-APSK In the constellaton, the radus R of crcle are ranked as R 1, R, and R 3 n ascendng order. The radus of decson crcle n ascendng order are R s, R b. The equalzer output zn ( ) s nput to decson devce. In order to obtan R and e ( k ), t s necessary to compare R J wth the decson boundary Rs and R. So we have b 14

3 R 1, RJ < Rs R = R, Rs < RJ < Rb (8) R3, RJ > Rb The error functon of 3-APSK s gven as zn ( )[ R 1 zn ( ) ], RJ < Rs e( n) = zn ( )[ R zn ( ) ], Rs < RJ < Rb (9) zn ( )[ R3 zn ( ) ], RJ > Rb For dfferent modulaton modulus, the decson rules aren t the same. The equalzaton performance of the MMA manly depends on the selecton of decson condton R s and R b. MMA has some advantages such as low computatonal complexty and the fast convergence speed, but ts resdual MSE s large and ts convergence speed s slow. Correlaton Algorthms Genetc algorthm. Genetc algorthm s a non determnstc quas natural algorthm, whch s a smulaton of bologcal genetc and evolutonary process[8]. It starts from a populaton representatve of a potental soluton of possble problem, a new generaton of populaton can be generated by genetc operator of nature genetcs such as code, populaton ntalzaton, and ftness calculaton, as well as the hybrd and mutaton. The whole process smulates the process of bologcal evoluton n natural, the envronmental adaptablty of the newborn generaton s stronger than that of prevous generaton. The decoded best ndvdual n the last generaton can be served as the approxmate optmal soluton of the problem. Memetc algorthm. The word Memetc s derved from meme, whch s generally understood as cultural gene. Therefore, Memetc algorthm s called as a culture genetc algorthm, whch smulates the development of human cvlzaton. Its operaton process s smlar to the genetc algorthm, but ts local search can make all ndvduals n each tme of teraton reach the local optmal. The MA usually can be defned as the combned algorthm of evoluton and personal learnng n a much more dverse background [9]. The followng pece of pseudo-code can make thngs more explct. Begn t := ; X(t) := IntalPop(); whle(stoppng crtera not met )do X (t) := Recombne(X(t)); X (t) := Mutate(X(t)); X (t) := Select(X (t)); EvaluateFtness(X (t)); X(t+1) := SelectNewPop(X(t), X (t)); X (t+1) := LocalSearch(X(t+1)); t := t+1; end end Step1: Code. Bnary code s used. Step: Intalzaton of the populaton. The ntal populaton s generated randomly (where N s the populaton sze) and denoted as X = [ X1, X,, X N ]. Step3: Recombnaton. In accordance wth the recombnaton probablty ( P c ), two ndvduals have been selected randomly to generate two new ndvduals of new generaton of populaton. 1

4 X 1 = ω1x1+ (1 ω1) X X = ωx + (1 ω) X1 (1) where X 1 and X are randomly selected as two parent ndvduals, X 1 and X are the correspondng new offsprng ndvduals after recombnaton calculaton, ω 1 and ω are two random number n [,1]. Step 4: Mutaton operator. After calculatng accordng to certan rules, some ndvduals wth mutaton probablty ( P m ) n new generaton of populaton are selected. The mutate operaton s gven by X + ( Xmax X)( rand gt ), sgn = X = X ( X Xmn )( rand gt ), sgn = 1 (11) where gt = t/ G s the dentfcaton of the populaton evoluton (where t s the current evoluton tmes, G s the maxmum of evoluton tmes). X s the selected mutaton ndvduals, X s mutated ndvduals, rand s a random number n [,1], sgn s set or 1, X max and X mn are the upper and lower lmts of the parameters, respectvely. Step : Evaluate ftness. Evaluate ftness s denoted as ftness( X ). Step 6: Select operator. N excellent ndvduals (hgh ftness value) n the current populaton are selected to make them enter next teraton and some ndvduals wth low ftness value wll be elmnated. Each probablty ( P ) of ndvdual s defned as ftness( X ) P =, = 1,, N ftness( X ) (1) where ftness( X ) s the th ndvdual ftness value. Step 7: local search. The local search of all ndvduals s carred out by usng the MATLAB bult-n optmzaton functon. Step 8: Iteraton end. When the current teraton number s greater than the maxmum teraton tme, the optmal soluton s output, otherwse go to step 3. Based on the mechansm of metaphor n the theory of cultural development, the MA s a collecton of global search and heurstcs local search, t s dfferent from other ntellgent optmzaton algorthms wth specfc mathematcal model, and t s more of a thought or frame. Mult-Modulus Blnd Equalzaton Algorthm Based on MA When the MA s ntroduced to the MMA, mult-modulus blnd equalzaton algorthm based on MA(MA-MMA) s proposed and shown n Fg.3. In Fg.3, s ( n) s the transmtted sgnal, n ( n ) s Gaussan whte nose. MA s( n) h( n) + x ( n) w ( n zn ( ) ) n ( n) Fg.3 MA-MMA Assume that the ntal populaton of randomly generaton s X = [ X1, X,, X N ], each ndvdual X corresponds to one of the weght vector of the MMA. The optmal weght vector of the MMA can be obtaned by local search and global evoluton. In the MA-MMA, the recprocal of 16 MMA

5 cost functon of MMA s defned as the ftness functon of the MA,.e. 1 ftness( X ) = = 1,, N (14) J ( X ) MMA Smulaton and Analyses In order to verfy the performance of the MA-MMA, the smulaton test of the MA-MMA was carred out and compared wth the CMA, MMA, and GA-MMA. In the test, N =, S =, G =, P c =.7, and h = [ ], the receved sgnal-to- nose rato SNR = 3dB, the equalzer had 11 order transversal tap structure, the center tap coeffcent of the CMA and MMA were set to 1, the other tap coeffcent were set to, maxmum teratons of all smulaton ter = 1, Monte Carlo tmes M =. The step-szes of the CMA and MMA were 1 1, the step-szes of the MA-MMA and GA-MMA were 1 6, the smulaton results were shown n Fgure 4. (a) Output of CMA (b) Output of MMA (c) Output of GA-MMA MSE/dB CMA MMA GA-MMA MA-MMA (d) Output of MA-MMA ter/1 3 (e) The convergence curves Fg. 4 Smulaton results of 3-APSK In the same SNR, the teratons and the steady MSE were shown n Table 1. Tab. 1 Convergence teratons and steady MSE Algorthms MSE/dB Iteratons CMA -17 MMA GA-MMA -4 1 MA-MMA -7 1 From Fgure 4 and Table 1, we can come some followng conclusons: (1) the equalzaton performance of the MA-MMA and the GA-MMA are much better than that of the MMA, the convergence speed of the MA-MMA s slghtly faster than that of the GA-MMA, but ts MSE s smallest. () the output constellatons of the MA-MMA s more clearer and more compact than those of the CMA and MMA. Therefore, for hgher-order mult-modulus sgnals, the MA-MMA 17

6 outperforms the CMA, MMA, and GA-MMA n mprovng convergence speed and reducng steady MSE. Conclusons Ths paper proposes a mult-modulus blnd equalzaton algorthm based on Memetc Algorthm. Ths proposed algorthm makes full use of the genetc mechansm and local searchng of the MA to obtan the ntal optmal weght vector, so that the equalzaton performance of the MMA can be mproved. The smulaton results show that, for hgher-order mult-modulus 3-APSK sgnals, MA-MMA has faster convergence speed and smaller steady MSE comparson wth tradtonal CMA, MMA, and GA-MMA. Acknowledgements Project supported the Major Project of Nature Scence Foundaton of Hgher Educaton Insttuton of Jangsu Provnce, Chna(Grant No.13KJA11), Jangsu Scentfc Research Achevements n Industralzaton Project, Chna(JHB 1-9), and Jangsu Provnce, the 1 annual general unversty graduate students practce nnovaton program(sjlx1_398) Reference [1] Gao Yuan, Qu Xnyun. A New Varable Step Sze CMA Blnd Equalzaton Algorthm [C]//IEEE Chnese Control and ecson Conference, Tayuan, Chna, 1, : [] Yang Jan, Jean-Jacques Werner, Guy A. umont. The Mult-modulus Blnd Equalzaton and Its Generalzed Algorthm[J]. IEEE Journal on Selected Areas n Communcatons,, (): [3] Jenq-Tay Yuan. Equalzaton and Carrer Phase Recovery of CMA and MMA n Blnd Adaptve Recevers[J]. IEEE Tran. on Sgnal Processng, 1, 8(6): [4] Swapna ev, evdas G. Jadhav, Shyam S. Pattnak. Memetc Algorthm and Its Applcaton to Functon Optmzaton and Nose removal [C]//IEEE World Congress on Informaton and Communcaton Technologes, 11, [] Ln Geng,, Zhu Wenxng. An effcent Memetc Algorthm for the Max-Bsecton Problem[J]. IEEE Trans. on Computers, 14, 63(6): [6] Yao Ma, Zhang Q.T. versty Recepton of APSK Over Generalzed Fadng Channels[J]. IEEE Trans. on Wreless Communcatons,, 4(4): [7] Chrstan Hager, Alex Alvarado. esgn of APSK Constellatons for Coherent Optcal Channels wth Nonlnear Phase Nose[J]. IEEE Trans. on Communcaton, 13, 61(8): [8] Kalyanmoy eb, Amrt Pratap, Sameer Agarwal et al. A Fast and Eltst Mult-objectve Genetc Algorthm: NSGA-II[J]. IEEE Trans. on Evolutonary Computaton,, 6(): [9] Q. H. Nguyen, Y. S. Ong, N. Krasnogor. A Study on the esgn Issues of Memetc Algorthm[C]//IEEE Congress on Evolutonary Computaton, 7,