On Evolutonary Programmng for Channel Equalzaton ADINA BURIAN, ARTO KANTSILA, MIKKO LEHTOKANGAS, JUKKA SAARINEN Dgtal and Computer Systems Laboratory Tampere Unversty of Technology P.O. BOX 553, FIN-33101, Hermankatu 12C FINLAND Abstract: - The use of evolutonary programmng for adaptve equalzaton of bnary data bursts n a baseband dgtal communcaton system s studed. Evolutonary programmng has been found to perform well for blnd channel dentfcaton. In ths paper t s shown that the usage of evolutonary programmng technques are benefcally also for adaptve channel equalzaton. We have also studed the effect of reducng the computatonal complexty of ths technque on the bt error rates performances. Comparsons are made wth mult-layer perceptrons equalzers. Key-Words: - Channel equalzers, ntersymbol nterference, dgtal communcatons, evolutonary programmng, offsprng, populaton Dgtal nput Source encoder Channel encoder Modulator Channel Dgtal output Source decoder Channel decoder Demodulator Fg. 1. Dgtal Communcatons System 1 Introducton We start ths paper by presentng n general terms the adaptve equalzaton concept, the very basc of a dgtal communcaton system and a few adaptve equalzaton methods. In the next secton, we wll present the used approach for adaptve equalzaton wth evolutonary programmng. 1.1 The Dgtal Communcaton System Fgure 1 llustrates the basc elements of a dgtal communcaton system [13]. The source encoder converts effcently the output of an analog or dgtal source nto a sequence of bnary dgts (data compresson). Ths sequence s passed to the channel encoder whch has the purpose to ntroduce n a controlled manner some redundancy n the bnary nformaton sequence. Ths redundancy can be used at the recever to overcome the effect of nose and nterference encountered n the transmsson. The output of the channel s passed to the dgtal modulator, whch maps the coded sequence nto sgnal waveforms. These are transmtted along wth a carrer sgnal at a hgher frequency. The obtaned analog sgnal s passed through the communcaton channel - the physcal medum used to send the sgnal from transmtter to the recever. At the recever, the demodulator converts the contnuous waveforms back to dscrete doman. The obtaned sequence s passed to the channel decoder whch attempts to reconstruct the orgnal sequence usng the redundancy contaned n the receved data. Fnally, the source decoder accepts the output sequence from the channel decoder, and from knowledge of the used source encodng method t tres to reconstruct the orgnal sgnal. The
obtaned sgnal at the output of the source decoder s an approxmaton of the orgnal nput. One of the fundamental problems encountered n dgtal communcaton s the corrupton of the transmtted sgnal by nose. Other undesred effects can occur durng transmsson: ntersymbol nterference (ISI), multpath propagaton, ISI arses n dgtal communcaton systems when the channel mpulse response lasts for more than one symbol nterval. The communcaton channel whch we wll consder throughout ths paper ntroduces ISI and addtve whte Gaussan nose to the transmtted sgnal. The ISI part of the channel s modeled as a fnte mpulse response (FIR) dscretetme flter. To compensate for the ISI, channel equalzaton s needed. The purpose of equalzaton s to reduce the unwanted effects of sgnal dstorton caused by the transmsson channel, so that the transmtted symbols can be correctly nterpreted. Equalzaton of dgtal communcatons channels s usually done by usng a transmtted sequence also known to the recever, durng a preamble perod. The channel response s tme varant and unknown, and adaptve equalzaton methods are employed. In adaptve equalzaton the actual channel pulse response s estmated and an equalzer s automatcally adjusted to equalze the channel [14]. In steady state, the adaptaton of the equalzer s decson drected; ths means that recever decsons are used to generate the error sgnal. Decson drected equalzer adjustment s often not effectve durng ntal acquston, snce the ISI can be so hgh that can ntally cause a very hgh error rate. Blnd equalzers (self-learnng) are used n order to provde the correct dentfcaton of transmtted symbols, when one does not have a tranng perod, or when t s not practcal to use such a strategy (.e. dgtal TV broadcastng or multpont networks, where tranng has to be executed whenever one sngle recever s nserted n the system). The transmsson of tranng sgnals decreases communcatons throughput, although for tme-nvarant channels ths s nsgnfcant because only one tranng s necessary. For tme-varyng channels, the loss of throughput becomes an ssue. Dependng on the degree of channel tme-varance, the repeated transmsson of tranng sequences may leave the communcaton system wth consderable overhead. In blnd channel estmaton, ths overhead could be used for other purposes. The fundamental dea of blnd channel estmaton s to derve the channel mpulse response from the receved sgnal only, wthout access to the channel nput sgnal by means of the tranng sequences. Conventonal approaches to solve the blnd channel dentfcaton problem nclude hgh order statstcs [20], whch explots the hgher order statstcal propertes of the transmtted sgnal, and Bayesan estmaton [15], when the channel s modeled by fnte dmensonal determnstc unknown parameters. Evolutonary Programmng (EP) was also proposed for solvng ths problem [6]. EP was able to dentfy the channel parameters n almost all the stuatons tested n [6]. Channel estmates were not affected sgnfcantly by varatons n the SNR, and the results were comparable wth the ones gven by genetc algorthms. Optmum recevers n dgtal communcaton systems requre the knowledge of the transmsson channel. However, channel estmaton s complcated by the fact that the transmsson channel s frequency selectve, mxed phase, and tme-varant. Present state-of-the-art moble communcaton systems transmt the so-called tranng sequences to assst the recever n estmatng the channel mpulse response. 1.2 Tradtonal Channel Equalzaton Methods The smplest equalzer mechansm s the Lnear Transversal Equalzer (LTE) [9]. The past and current values of the receved sgnal are lnearly weghted by the complex-valued equalzer coeffcents and tuned to produce the estmated output. There are varous crtera for choosng the best equalzer coeffcents, the most used beng the MSE (Mean Square Error) crteron. Several coeffcent adaptaton algorthms have been developed; the best-known are LMS (Least Mean Square) and RLS (Recursve Least Square) algorthms. Another tradtonal equalzer s Decson Feedback Equalzer (DFE). It usually conssts of two FIR flters: a feedforward flter (FFF) that s bascally a LTE, and a feedback flter (FBF). The decsons made on the equalzed sgnal are feed back va the FBF. The FFF s used to remove ISI due to symbols transmtted n the future, whle the FBF s used to cancel ISI due to symbols transmtted n the past. The FFF and FBF n DFE are typcally optmzed usng ether a zero-forcng crteron, or a mnmum MSE crteron, the later beng more prevalent. Both used crterons are equvalent n the lmt of hgh sgnal-to-nose rato (SNR). Optmzaton of DFE under a mnmum symbol error probablty crteron s usually not attempted.
Instead of symbol-by-symbol detecton, the Maxmum Lkelhood Sequence Estmator (MLSE) treats the entre sequence at once and maxmzes the mean tme between error events. It s known that the Vterb Algorthm (VA) can be used to mplement MLSE of the nput sequence n the presence of ISI and addtve nose. The VA s a recursve structure that was orgnally nvented to decode convolutonal codes, but t was also analyzed for channel equalzaton purposes (MLSE/VA). The VA s used to determne the sequence that s closest n dstance to the receved sequence of nosy samples. The evaluaton of the dstance metrc s done by usng path metrcs assocated wth states and branch metrcs assocated wth transtons, n computng the path recursvely. Unfortunately, the complexty of the VA grows exponentally wth the duraton of the channel mpulse response. In applcatons whch have hgh symbol rates and channel mpulse responses whch last for a large number of symbol ntervals, t can be dffcult to mplement the VA and to meet overall system objectves of low power consumpton and low cost. Consequently, suboptmum detecton technques - such as DFE - whch offer good performances, are used. 1.3 Channel Equalzaton Usng Classfcaton Methods Another approach s to consder the equalzaton of the receved sgnal as a classfcaton problem. In [5] the sgnal space was parttoned nto approprate decson regons and a class label was assgned to a partcular regon for all unclassfed vectors belongng to that regon. Top-down approach was used n tree nducton, and splttng was done based on nformaton gan crteron. Overfttng was avoded by utlzng a prunng algorthm. The advantages of ths method are ts smplcty and straghtforwardness and thereby the reducton of computatonal complexty. It was shown that the problem of fndng fnte length DFE flters that mnmze the probablty of symbol error at any SNR subject to a certan separaton condton s a convex optmzaton problem [1]. The problem of determnng DFE flters that mnmze the probablty of symbol error at hgh SNR t was shown to be equvalent to fndng the hyperplane that maxmally separates two gven fnte groups of ponts n a fnte dmensonal Eucldan space. Ths task s equvalent to fndng the optmum separatng hyperplane n support vector machnes (SVMs) [16]. A SVM uses tranng data as an ntegral element of the functon estmaton model as opposed to smply tranng data to estmate parameters of an a pror model usng maxmum lkelhood, whch s the more tradtonal approach. Consequently, SVMs tranng s rather straghtforward, requrng less ad hoc nput from the desgner. Once tranng of the SVM was completed, the equalzaton, or more approprately the nonlnear detecton, s effcent and comparable to Volterra flters and neural networks. 2 Channel Equalzaton usng Evolutonary Programmng Evolutonary Programmng (EP), orgnally conceved by Lawrence J. Fogel n 1960, s a stochastc optmzaton strategy that places emphass on the behavoral lnkage between parents and ther offsprngs [3][4]. The evolutonary process s smulated n the followng manner: an ntal populaton of solutons (N ndvduals) s typcally chosen at random. These parents are measured n ther ndvdual ablty to predct each next event n ther experence, wth respect to whatever payoff functon has been prescrbed (the ftness score - squared error, absolute error). Progeny are created through random mutaton of these parents. Each parent creates a sngle offsprng. The obtaned offsprngs are scored on ther predctve ablty n a smlar manner to ther parents, the ones that are most sutable are probablstcally selected to become the new parents. Par-wse comparsons over the unon of parents and offsprng are conducted. For each comparson, f the ndvdual s ftness s no smaller than the opponent s, t receves a wn. An actual predcton s made when the predctve-fr score demonstrates that a suffcent level of credblty has been acheved or when the avalable computatonal lmt has been exceeded. The dsadvantages of usng such algorthms for channel equalzaton are slow convergence, large error varance and hgh computatonal complexty. The most obvous attempt to use EP for channel equalzaton can be consdered to just use EP for the tranng of a neural network (whch can be MLP) equalzer [2]. In [17] t was concluded that the use of an evolutonary algorthm provdes a more effectve mean of tranng an MLP to perform equalzaton of a non-mnmum phase channel. The evolutonary algorthm would tran the MLP to a more optmal soluton more often than back propagaton (BP). On average, the varance of the bt error rate performance of the MLPs traned wth an evolutonary algorthm was much less than those
traned wth BP. So, evolutonary algorthms offer an effectve alternatve tranng technque for MLPs to perform channel equalzaton. A dfferent approach s presented n [12]. Tradtonally, equalzaton s based on lnear FIR flters, but nfnte mpulse response (IIR) flter s a more general equalzer structure, wth the dsadvantages of a very slow convergence and always beng stucked at a local mnmum. A global optmum soluton based on genetc algorthms (GA) for IIR lattce flter structure was obtaned. In ths paper we propose a methodology n whch an evolutonary programmng-type search s used n combnaton wth the gradent-descent method. The flter coeffcents are evolved n a random manner when the flter s startng to have slow convergence rate. The used flter s a tunable IIR dgtal flter structure. The gradent-descent soluton s used to ntalze the EP method. The scheme of the used EP algorthm s summarzed as follows: 1. Intalzaton: An ntal populaton of N ndvduals s selected randomly from a feasble range n each dmenson, around the soluton gven by the gradent descent method. Each ndvdual s taken as a par of real-valued vectors (s, η ) = S, =1,,N, where S s a random vector, s s the outcome of the random vector the n channel coeffcents. The dstrbuton of ntal trals η s unform. 2. Evaluaton Each s, =1,,N s assgned a ftness score ϕ(s ) that s the mean absolute error (MAE) for the tranng sequence. 3. Creaton of offsprng (mutaton) Generate one offsprng from each ndvdual: each s, =1,,N s altered by addng a Cauchy random varable and assgned to s +N. Cauchy mutaton s used because t performs better than Gaussan mutaton: t has a hgher probablty of makng longer jumps [19]. The mutaton s done accordng to: [ ' N(0,1) + (0,1) ] ' η ( j) = ( j) exp η j (1) ' s ( j) = s ( j) + η ( j) C, (2) where N(0,1) s a normally dstrbuted random number wth zero mean and unt varance, N j (0,1) s the same as N(0,1) but s regenerated for every j, and C j s a Cauchy dstrbuted random varable wth unt scale parameter. The factors τ and τ are (commonly) set 1 to 2 n and ( 2n ) 1, respectvely. Therefore, ndvduals ncludng parents and offsprng exst n a common competng pool. 4. Evaluaton Evaluate the offsprng: to each s +N, =1,,N t s assgned a ftness score. 5. Competton and selecton Each ndvdual n the competng pool must stochastcally strve aganst other members of the pool, based on the functon ϕ(s ). Every ndvdual n the populaton (channel coeffcents n ths case) s compared wth r randomly selected opponents. The ndvduals wth the best functon values are selected to form a survvor set accordng to the decson rule. The better half of the populaton, wth the largest number of wns, s selected to become the new parents for the next generaton. 6. Stoppng rule The process of generatng new trals and selectng those wth best functon values s contnued untl the functon values are not obvously mproved (the obtaned MAE s small enough) or a gven count of maxmum number of generatons G s reached. j
3 Expermental Results In obtanng the results, we used the Global System for Moble Communcatons (GSM) [11]. The nformaton sgnal conssts of bursts of bnary symbols takng values of ether 1 or 1. Each burst begns wth a tranng sequence, whch s 26 bts n length and known to the recever. Therefore t can be used to adapt the equalzer. The latter part of the burst contans 116 bts of data payload, whch s not known to the recever. The total length of the burst s 142 bts, and the burst s oversampled at the transmtter by a factor of 3. The nformaton sgnal s transmtted as two-leveled baseband sgnal. For each transmtted burst, we have frst used the tranng sequence of the burst, and then the equalzaton was performed on the data sequence of the burst wth the obtaned traned soluton. The communcaton channel used n smulatons has both ISI and addtve whte Gaussan nose. The ISI part, resultng from multpath propagaton, s modeled as a FIR dscretetme flter. We can formalze the relatonshp between the channel outputs y and the transmtted bnary sgnal a usng equaton (3), where h are the channel coeffcents and δ represents addtve nose. y n = N = 0 h a n + The obtaned results usng evolutonary programmng were compare wth the correspondng results gven by a cascade-correlaton traned multlayer perceptron neural network equalzer [7]. The evolutonary programmng and neural network learnng are performed for each burst separately. We used a fxed channel mpulse response: h=[h 0 h 1 h 4 ]=[0.5 0.3 0.6 0.7 0.8] T, and sgnal to nose rato (SNR) of 5, 10, and 15 db. A whte Gaussan nose wth zero mean and unt varance was added to the channel output. In all smulatons the maxmum number of generatons G was set to 10000. The results presented are obtaned by averagng over 10 runs. The obtaned bt error rates for the two consdered structures n all the consdered stuatons are presented n Table 1. It appears from ths table that the use of EP mproves the obtaned BER when compared wth MLP equalzer. Computatonal complexty of the neural network depends on the number of nputs (n our case 2), tranng epochs (50) and hdden unts (max. 6). Computatonal complexty of evolutonary programmng depends on the number of evolvng ndvduals. Ths gves a δ n (3) (4) hgher complexty to ths technque. When a small number of evolvng ndvduals N was used, a bgger number of generatons g was needed, but a good soluton was found (see Table 2). In both tables the bt error rate s computed as the decmal logarthm of the error probablty. SNR 5 10 15 BER -1.236-1.638-1.860 NN BER EP -1.237-1.741-1.991 Table 1. Comparatve results between NN and EP wth 100 evolvng ndvduals. N g BER 20 7799-1.443 50 6130-1.652 100 5540-1.741 150 5509-1.726 200 5460-1.742 Table 2. Comparatve results for EP wth dfferent number of evolvng ndvduals for SNR=10 db Future work deas nclude also the evolvng of the used neural network. Eventually, to use an embedded tranng: both cascade correlaton and evolutonary programmng. The purpose of usng such a system s to reduce the searchng space of neural network tranng, and also to assst the neural network tranng n the search of the global mnmum. 4 Conclusons We have studed the use of evolutonary programmng for equalzaton purposes n a tmevaryng communcatons channel, where the channel ntroduces ntersymbol nterference and addtve Gaussan nose to the transmtted sgnal. The usage of evolutonary programmng gves an ncrease n performances (smaller error rates) comparatvely wth mult-layer perceptrons networks, at the cost of an ncreased computatonal complexty. The computatonal complexty s a very mportant factor n moble envronments. Because of that we studed also the possblty of decreasng the number of the evolvng ndvduals, whch reduces the computatonal complexty of evolutonary programmng. For a very small number of evolvng ndvduals, smlar values of the error rates wth MLP networks were obtaned, for a bgger number of generatons requested.
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