Nonlinear Complex Channel Equalization Using A Radial Basis Function Neural Network

Transcription

1 Nonlnear Complex Channel Equalzaton Usng A Radal Bass Functon Neural Network Mclau Ncolae, Corna Botoca, Georgeta Budura Unversty Poltehnca of Tmşoara cornab@etc.utt.ro Abstract: The problem of equalzaton for complex sgnals s presented. It s proposed a compettve method for the estmaton of the centers of a complex radal bass functon neural network. Smulaton results are presented from the pont of vew of mean square error and sgnals space partton. Concludng remarks and further developments are dscussed. Key Words: channel equalzaton, complex radal bass neural network I. INTRODUCTION In modern hgh-speed communcatons networks, the presence of symbol nterference (ISI) s a maor mpedment of data transmsson. Nonlnear actve or passve devces and the transmsson channels themselves ntroduce nonlnear dstortons that affect the sgnals. Especally the sgnals wth a varable envelope modulaton, as for example the quadrature ampltude modulaton (QAM) sgnals, more effcent n transmsson from the spectral pont of vew, are affected, n phase and n ampltude. To compensate the dstortons of QAM and phase shft keyng (PSK) sgnals equalzers for complex sgnals are necessary. These equalzers are extensons of the real equalzers. The problem of equalzaton may be treated as a problem of sgnals classfcaton, so neural networks (NN) are qute promsng canddates because they can produce arbtrarly complex decson regons. Studes performed durng the last decade have establshed the superorty of neural equalzers comparatve to the tradtonal equalzers, n condtons of hgh nonlnear dstortons and rapdly varyng sgnals. Several dfferent neural equalzers archtectures have been developed, mostly combnatons between a conventonal lnear transversal flter (LTE) and a neural network. The LTE elmnates the lnear dstortons, such as ISI, so the NN can be focused on compensatng the nonlneartes. There have been studed the followng structures: a LTE and a multlayer perceptron (MLP) [], [7], a LTE and a radal bass functon network (RBF) [], [3], [4],[8] a LTE and a recurrent neural network [0],[], a functonal lnk equalzer [6] [9], and cellular neural network equalzer []. There have been ntroduced many dfferent nonlnear devces models and channels models, so a untary comparson between all known equalzers s dffcult to be done. Snce MLP networks are sometmes plagued by long tranng tmes and may be trapped at bad local mnma, RBF networks often provde a faster and more robust soluton to the equalzaton problem. In addton, the RBF neural network has a structure smlar to the optmal Bayesan symbol decson equalzer. Note that the Bayesan equalzer does not necessarly yeld a good mnmum mean square error (MSE) performance but does provde the mnmum average bt error rate (BER) achevable for symbol decson and ndrect-modelng equalzer structures. Therefore, the RBF s an deal processng structure to mplement the optmal Bayesan equalzer. The RBF performances are better than the LTE and MLP equalzers [], [4]. In concluson the RBF network s an attractve alternatve to the MLP neural network manly due to ts smple structure and more effcent learnng. Several learnng algorthms have been proposed to update the RBF parameters. However, the most popular algorthm conssts of an unsupervsed learnng rule for the centers of hdden neurons and a supervsed learnng rule for the weghts of the output neurons. The centers are generally updated usng the k-means clusterng algorthm [7] whch conssts of computng the squared dstance between the nput vector and the centers, choosng a mnmum squared dstance, and movng the correspondng center closer to the nput vector. The k mean algorthm has some potental problems: classfcaton depend on the ntals values of the centers of RBF, on the type of chosen dstance, on the number of classes. If a center s napproprate chosen t may never be updated, so t may never represent a class. In [8] s proposed a sequental learnng algorthm referred as complex mnmal resource allocaton network (CMRAN). The studes proved that the equalzer performance s superor to the functonal lnk equalzer of Patra and all [9] and the stochastc gradent RBF equalzer [] of Cha and Kassam. In ths paper we propose a new compettve method to update the RBF centers, whch recompenses the wnnng neuron and penalzes the second wnner, named rval. The algorthm s qute smple and the performances are comparatve to all the others reported equalzers. The classes a automatcally generated at the output of the network. The RBF network has complex centers and connecton weghts, but the nonlnearty of ts hdden nodes s a real-valued functon. The RBF compettve equalzer s able to approxmate an arbtrary nonlnear functon n complex mult-dmensonal space wth a reduced calculus complexty comparatve wth other algorthms.

2 Data source x(n) Communcaton channel model y(n) o(n) Equalzer w(n) Gaussan nose generator Σ - e(n) x(n-d) + Delay Fg. Schematc of a communcaton system II. THE EQUALIZATION PROBLEM The equalzaton problem s tradtonally vewed as an nverse flter problem. Equalzers are desgned to track the tme-varyng channel dstortons by adustng ther coeffcents and mantanng a prescrbed sgnal to nose rato (SNR). Tradeoffs between nose enhancement and channel nverson generally render these technques suboptmal. An alternatve vewpont s to consder the equalzaton problem as a pattern classfcaton problem. The obectve of equalzaton becomes the separaton of the receved symbols n the output sgnal space, whose optmal decson regon boundares are generally hghly nonlnear. Snce neural networks are well known for ther ablty of performng classfcaton tasks by formng complex nonlnear decson boundares, neural equalzers have been recently recevng consderable attenton. Neural equalzers have shown the potental for sgnfcant performance mprovements especally n severely nonlnear dstorted and rapdly varyng sgnals. Fg. represents a model of a communcaton system. If the sgnal x s a 4 QAM, the nput constellaton s gven by: () x = + () x = + x(n) = x R + x I = () (3) x = (4) x = The nput symbols sequence x(n) s passed through the nonlnear communcaton channel model and produces at ts output the sequence y(n). The channel output sgnal s affected by an addtve nose w(n), usually whte Gaussan, and produces a corrupted sgnal o(n). The problem of equalzaton s to determne an estmaton of the nput sgnal x (n) usng the receved sgnal o(n) and the desred delayed sgnal x(n-d). From the NN pont of vew, the equalzer has to classfy the receved sgnal n one of the four possble classes P m,d, accordng to the nput sgnals: or: P m,d (l) = P U m,d Pm,d (l) l 4 = () (l) { y(n) x(n d) = x }, l 4 (3) III. THE COMMUNICATION CHANNEL MODEL Fg. represents a model of the communcaton channel that ntroduces lnear and nonlnear dstortons. The lnear complex part of the channel s often modeled wth a transversal flter FIR whose output s gven by: k = 0 y (n) = a x(n ) (4) where a are the flter coeffcents and k s the order of the flter. Fg. Nonlnear channel model One of models suggested n [] generates the output sgnal accordng to: x(n) FIR ỹ (n) y = ( )x(n) + ( )x(n ) + + ( )x(n ) The order of ths flter s k=3. NL y (n) (5)

3 xˆ ( n d ) Output f RBF ( o) Hdden layer w φ w wh φ K φ φh o ( n) o ( n ) o( n m +) z z Fg.3 The archtecture of the RBF neural network equalzer The nonlnear part of the channel s a very strong one and produces at the output: y(n) 3 = y(n) + 0.[y(n)] [y(n)] (6) Ths sgnal s added wth the Gaussan nose w(n), wth a null mean and a dsperson of σ, and subsequently passed through the neural equalzer: o (n) = y(n) + w(n) (7) IV. THE COMPLEX RADIAL BASIS FUNCTION EQUALIZER An equalzer may be mplemented wth a LTE followed by a neural network, as n Fg.3. If m s the FIR flter order, the receved sgnal o=[o(n) o(n-)...o(n-m+)] s the nput for the RBF network. The number of possble states of the receved QAM sgnal s n e =4 k+m-. The complex RBF s a straghtforward extenson from the real counterpart [8], obtaned by replacng the relevant parameters wth complex values. As depcted n Fg. 3, the RBF network has two layersthe hdden layer and the output layer. The hdden layer s composed of an array of computng neurons, each havng a parameter c, vector called center. Each neuron computes a dstance between ts center and the network nput vector. Ths dstance may be of dfferent types and t s subsequently dvded by a parameter ρ, called wdth, whch s the spread of the correspondng center. The result s passed through a real, nonlnear actvaton functon. φ, ρ ) ( H φ = [ φ(o c ) (o c ), ρ ], n (8) where o s the complex nput vector of n h dmenson, c s the centers vector of the radal bass functons, whch s also a complex vector of n h dmenson, ρ s the center spread parameter, n h s the number of computng nodes. The operator ( ) H =(( ) T T )*, where ( ) s the transposton operator and ( ) * s the complex conugaton operator. The nonlnear output functon s usually the Gaussan functon: χ ρ φ( χ, ρ) = e (9) h

4 The number of hdden neurons n h s gven by the number of possble states of the channel output n e. A number n h greater than n e generates nutle computng. A number n h smaller than n e may degrade the performances of the network. Smlarty wth the Bayesan equalzer mpose that the spread parameter ρ=.σ where σ s the nose dsperson gven by relaton: σ = E o(n) c (0) where E s the mean, the second order momentum. The output layer of the network conssts of eght neurons (two neurons for each class, one for the real part and the other for the magnary part of each class) wth a lnear functon : n = h f (o) φ w () RBF = where w are the complex weghts. Accordng to the relaton ( 9), f RBF becomes: n = (o c ) (o c ) ρ = h f (o) w. e () RBF IV.. COMPETITIVE LEARNING ALGORITHM PENALIZING THE RIVAL The compettve standard algorthm calculates the dstance between the nput vector and the RBF centers vector. The dstance may be of dfferent types, usually the Eucldan norm s used: o(n) c (n) = = o(n) c (n) H o(n m + ) c (n m + ) (3) The neuron wth a mnmum dstance s declared wnner: = argmn o(n) - c (n), =, (4) The wnnng neuron center s moved wth a fracton η towards the nput. The compettve algorthm penalzng the rval [5] determnes not only the wnnng neuron but also the second wnnng neuron r: n h r = argmn o(n) - c (n), =, n h (5) The second wnnng neurons wll move away from the nput ts center wth a rato γ. All the others neurons wll not change ther centers vector. So the learnng law can be syntheszed n the followng relaton: c (n) + η.[o(n) c(n)] f = c (n + ) = c (n) + γ.[o(n) c(n)] f = r (6) c (n) f and r where η and γ are the learnng constants wth real values between 0 and. If the learnng speed η s chosen much greater than γ, the RBF network wll fnd automatcally the number of sgnal output classes. In other words, suppose that the number of classes s unknown and the number n h s greater than the number of the classes. The RBF centers wll converge towards the centers of the nput sgnals clusters. The penalzng compettve algorthm wll move away the rval, n each teraton. If the n h s smaller than the number of the classes, than the network wll oscllate durng tranng, ndcatng that the number of hdden neurons must be ncreased. IV.. LMS ALGORITHM A supervsed algorthm may be used to update the output neurons weghts, for nstance the LMS algorthm gven by the followng relatons: w (n + ) = w (n) + α.e(n). φ(n) (7) where α s the learnng constant. LMS mnmzes the mean square error: N MSE = e (n) (8) N = where N s the number of nput sequences and the complex error e(n) s determned wth: e(n) = x(n d) f (o) (9) RBF V. SIMULATION RESULTS QAM nput sgnals were generated, usng an unform dstrbuton, ndependently for the real part from the magnary part. Smulatons were done usng the channel model presented n secton III. The output channel y(n) had one of 64 possble states. A whte nose w(n) was generated and added to y(n). The FIR flter used n front of the neural equalzer had the order m=. The number of the hdden neurons was chosen n e =64 and of the output neurons 8. The RBF centers were randomly ntalzed to a subset of channel output values. The centers spread was chosen 0.8. The best results were obtaned for the followng learnng constants: η=0.09, γ=0.03 and α=0.0. There were appled N= 000 nput sgnal sequences x(n), x(n)=[x(n) x(n-) x(n-)] to tran the equalzer.

5 Fg.4 Bdmensonal representaton of output channel states wthout nose, nputs of RBF network, ntal and fnal postons of the RBF centers for N= 000 sequences Tranng-Blue Performance s , Goal s 0 am of mnmzng the MSE. Fg.5 depcts the MSE evoluton durng 5000 teratons for σ =0.0 and a delay of d=. The complex space was dvded n ponts usng a samplng pas of δ=0.0 to represent the decson regons of the RBF complex equalzer. Fg. 6 represents the partton sgnals space for a delay of d=, n the worst condtons of nose, σ =0.5, whch had strong nonlnear decsons boundares Epochs Fg.5 Evoluton of mean square error durng 5000 teratons for σ =0.0 The equalzer was tested n dfferent condtons nose, wth a nose dsperson σ =0.0, 0. and 0.5. In all the stuatons the equalzer succeeded to fnd a correct soluton. The compettve algorthm traned the centers of the RBF network. For each sequence t has been calculated the error e(n) and than the MSE. The output weghts were modfed accordng to the relaton (7), n order to mnmze the MSE. In Fg. 4 are represented the output channel states y(n), the corrupted receved sgnal o(n), the ntal and fnal postons of the RBF centers n case of σ =0.0. Ths operaton was repeated a number of tmes, n the Fg.6 The output sgnals space partton

6 VI. CONCLUSIONS The man drawback of the neural network equalzers s the computatonal complexty and the extensve tranng.. Our compettve algorthm, that recompenses the wnner and penalzes the rval to tran the centers of the RBF network, s rather smple and has a fast convergence to a soluton. It generates strong nonlnear regons of decson n the sgnal space So ths algorthm s adequate to the adaptve equalzaton of fast varyng sgnals corrupted wth strong lnear and nonlnear dstorsons.. Because of ts structure smlar to the Bayesan equalzer the performance of the RBF equalzer s superor to the LTE and MLP equalzers. The MSE performance of our equalzer s comparatve to others RBF equalsers reported n lterature, tested n the same condtons. To mprove the performances t mght be ncreased the order of the LTE flter coupled wth the RBF neural network. References [] S.Bouchred, M. Ibnkahla, "Decson Neuronale Applquee a L egalsaton De Canaux Satelltares Mobles", Dx-septeme colloque, Grets, Vannnes, Proceedngs, pp.5-8, 999 [] I. Cha, S.Kassam, "Channel Equalzaton Usng Complex-Valued Radal Bass Functon Networks", IEEE. Journal Select. Areas Commun, vol3, pp.-3, Jan, 995 [3] S. Chen, S. McLaughln, B. Mulgrew, "Complex- Valued Radal Bass Functon Network. Network Archtecture And Learnng Algorthms", Sgnal Processng, No.35, pp. 9-3, 994 [4] S. Chen, et al., "Complex-Valued Radal Bass Functon Network. Part II: Applcaton To Dgtal Communcatons Channel Equalzaton", Sgnal Processng, No. 36, pp ,994 [5] D. Hamad, S. El assad et J. Postare, "Algorthmes d apprentssage compéttf pour la classfcaton automatque", TISVA 98, Ouda, Maroc, Aprl, 998 [6] A. Hussan, J. Soraghan, T. Durran, "A New Adaptve Functonal-lnk Neural Network-based DFE for Overcomng Co-channel Interference", IEEE Transactons Communcatons,, November, pp , 997 [7] M. Ibnkahla, "Applcatons of Neural Networks to Dgtal Communcaton a Survey" IEEE Sgnal Processng Magazne, November, pp.86-5, 997 [8] Deng Janpng, Narasmhan Sundararaan, P. Saratchandran, " Communcaton Channel Equalzaton Usng Complex-Valued Mnmal Radal Bass Functon Neural Network", IEEE Trans. On Neural Networks, Vol 3, No.3, May, pp , 00 [9] J.C.Patra, R.N.Pal, R. Balarsngh, G.Panda, "Nonlnear Channel Equalzaton For Qam Constellaton Usng Artfcal Neural Networks, ", Trans. Syst. Man, Cyb, vol 9,aprl, pp.6-7,999 [0] R. Pars, E. D Claudo, G. Orland, B. Rao, "Fast Adaptve Dgtal Equalzaton by Recurrent Neural Networks", IEEE Trans. Sgnal Process. 45, November, pp , 997 [] R.Perfett " Cellular Neural Networks for Fast Adaptve Equalzaton", Internatonal Journal of Theory and Applcatons, vol.,pp 65-75,993 []"Tme varyng Channel Equalzaton", ChannelEqualzaton/ 00