Comparison of RLS&LMS Algorithms for OFDM Systems

Transcription

1 Iteratioal Joural of Emergig Techology ad Advaced Egieerig Compariso of RLS&LMS Algorithms for OFDM Systems Vidya S.Bhosale 1, V. N. Ghodke 2 1 PG Studet, 2 Assistat Professor, Departmet of E&TC, Pue Uiversity, Pue, Idia Abstract I this paper adaptive chael estimatio techiques such as least squares (LMS) algorithm ad recursive least squares algorithm (RLS) is plaed for multiple iput multiple output (MIMO) orthogoal frequecy divisio multiplexig (OFDM) systems. These CE methods uses adaptive estimator which are able to update parameters of the estimator cotiuously, so that the kowledge of chael ad oise statistics are ot madatory. This LMS& RLS CE algorithm requires kowledge of the received sigal oly. So by desigig this we are goig to calculate the terms of the sigal to oise (SNR), ad bit error rate (BER) o mat lab software. Simulatio results demostrated that, the RLS CE method has better performace for OFDM systems. I additio, the utilizig of more multiple ateas at the trasmitter ad/or receiver provides a much higher performace compared with fewer ateas. Keywords RLS; LMS; OFDM. I. INTRODUCTION Orthogoal Frequecy Divisio Multiplexig (OFDM) have attracted much attetio as promisig techology i wireless commuicatio systems. OFDM is multi-carrier trasmissio techique i which a sigle high rate data stream is divided ito multiple low rate data streams. The whole spectrum is divided ito several sub carriers &before each OFDM block the cyclic prefix (CP) is iduced.so OFDM systems ca mitigate the effects of multipath &have more spectrum efficiecy. These data streams are the modulated usig subcarriers which are orthogoal to each other. I this way the symbol rate o each sub chael is greatly lowered ad hece the effect of itersymbol iterferece (ISI) due to chael dispersio i time caused by multipath delay spread is reduced. The orthogoality betwee subcarriers ca be maitaied, eve though the sigal passes through a time-dispersive chael by cyclically extedig the OFDM symbols ito guard iterval. The mai advatages of OFDM are its multipath delay spread tolerace ad efficiet spectral usage by allowig overlappig i the frequecy domai. several chael estimatio techiques have bee suggested to reduce iterchael iterferece (ICI) i OFDM system. I this paper least mea square algorithm & has bee proposed to miimize error betwee received Recursive least squares algorithm sigal &estimated sigal &described for OFDM System. These CE methods uses adaptive estimator which are able to update parameter of estimator cotiuously so that kowledge of chael &oise statics are ot ecessary. This LMS/RLS CE algorithm Need kowledge of the received sigal oly. This ca be doe i a digital commuicatio system by periodically trasmittig a traiig sequece that is kow to the receiver. Simulatio results show that the RLS CE method has better performaces compared LMS CE method for OFDM systems. I additio, the utilizig of more multiple ateas at the trasmitter ad/or receiver provides a much higher performace compared with fewer ateas. Furthermore, the RLS CE algorithm provides faster covergece rate compared to LMS CE method. Therefore, i order to combat the chael dyamics, the RLS CE algorithm is better to use for OFDM systems.[7] II. SYSTEM MODEL Data at the trasmitter is first projected to chael ecodig to reduce the probability of error at the receiver due to the chael effects. Usually, covolutio ecodig is preferred. The the bits are mapped to symbols. Usually, the bits are mapped ito the symbols of 16- QAM.The symbol sequece is coverted to parallel format ad IFFT (OFDM modulatio) is applied ad the sequece is oce agai coverted to the serial format. Guard time is provided betwee the OFDM symbols ad the guard time is filled with the cyclic extesio of the OFDM symbol. Widowig is applied to the OFDM symbols to make the fall-off rate of the spectrum steeper. The resultig sequece is coverted to a aalog sigal usig a DAC ad passed o to the RF modulatio stage. The resultig RF modulated sigal is, the, trasmitted to the receiver usig the trasmit ateas. Here, directioal beam formig ca be achieved usig atea array, which allows for efficiet spectrum reuse by providig spatial diversity. At the receiver, first RF demodulatio is performed. The, the sigal is digitized usig a ADC ad timig ad frequecy sychroizatio are performed.. The guard time is removed from each OFDM symbol ad the sequece is coverted to parallel format ad FFT (OFDM demodulatio) is applied. The output is the serialized ad symbol de-mappig is doe to get back the coded bit sequece. Chael decodig is, the, doe to get the user bit sequece.[9] 524

2 Iteratioal Joural of Emergig Techology ad Advaced Egieerig Fig 1:OFDM Block Diagram[8] III. CHANNEL ESTIMATION TECHNIQUES There are may digital sigal processig applicatios i which secod order statistics caot be specified. Such applicatio icludes chael equalizatio echo cacellatio ad oise cacellatio. I these applicatios, filters with adjustable coefficiets called Adaptive filters are employed. A adaptive filter is a filter that self adjusts its trasfer fuctio accordig to a optimizig algorithm. It adapts the performace based o the iput sigal. Such filters icorporate algorithms that allow the filter coefficiets to adapt to the sigal statics. There are differet approaches used i adaptive filterig, which are Adaptive techiques use algorithms, which eable the adaptive filter to adjust its parameters to produce a output that matches the output of a ukow system. This algorithm employs a idividual covergece factor that is updated for each adaptive filter coefficiet at each iteratio. The basic idea behid LMS filter is to approach the optimum filter weights (R-1 P), by updatig the filter weights i a maer to coverge to the optimum filter weight. The algorithm starts by assumig a small weights (zero i most cases), ad at each step, by fidig the gradiet of the mea square error, the weights are updated. That is, if the MSE-gradiet is positive, it implies, the error would keep icreasig positively, if the same weight is used for further iteratios, which meas we eed to reduce the weights. I the same way, if the gradiet is egative, we eed to icrease the weights. [10-14]The output from the chael ca be expressed as: L-1 R( m)= W (m,l ) S (m- l)+ Z (m) (1) l=0 Where S(m-l) is the complex symbol draw from a costellatio s of the lth paths at time m-l, L is the chael legth, Z(m) is the AWGN with zero mea ad variace σ 2.The above equatio ca be rewritte as vector otatio [1]: R(m) = W(m)S(m) - Z(m), (2) The output of the adaptive filter is Y (m) =W est (m) S(m) (3) Where W est( m) is the estimated chael coefficiets at time m. The priori estimated error sigal eeded to update the weights of the adaptive filter is e(m)= R( m) -Y (m) W( m) S( m) +Z (m) W est (m)- S( m) (4) This error sigal is used by the CE to adaptively adjust the weight vector so that the MSE is miimized. Now the cost fuctio j(m) = E[e(m)e * (m)] for the adaptive filter structure is J( m)=e[r( m)r( m)]- E[S]R * W est (m) R m W est (m) E[ S]- W est ( m) W est ( m) E[ S (m) S (m)] S = σ 2 -C( m) W est ( m)- W est ( m) C( m)+d (m) W T ( m) W est ( m) (5) A. LMS ALGORITHM Fig 2.Idea for adaptive CE Methods [7] Least mea squares (LMS) algorithms are class of adaptive filter used to mimic a desired filter by fidig the filter coefficiets that relate to producig the least mea squares of the error sigal (differece betwee the desired ad the actual sigal). It is a stochastic gradiet descet method i that the filter is oly adapted based o the error at the curret time. Where σ 2 is the variace of the received sigal, C(m) = E[S(m)R(m)] is the cross-correlatio vector betwee the tap iput vector S(m) ad the received sigal r(m), ad D (m) =E [S (m) S T (m) is the correlatio matrix of the tap iput sigal S(m). Now takig the gradiet vector with respect to W est ( m) : j(m) =-2C( m) +2D (m) W est (m) =S( m) R * ( m)+2 S( m) S (m) W est ( m) (6) 525

3 Iteratioal Joural of Emergig Techology ad Advaced Egieerig Accordig to the method of steepest descet, if est W (m) is the tap-weight vector at the mth iteratio the the followig recursive equatio may be used to update W est (m) : W est ( m+1)= W est ( m)-1/2 ƞ j( m) =W est ( m) S (m) R( m) West(m)S (m) =W est ( m)+ S( m) e * ( m) (7) Where W est (m+1) deotes the weight vector to be computed at iteratio (m + 1) ad η is the LMS step size which is related to the rate of covergece. The smaller step size meas that a loger referece or traiig sequece is eeded, which would reduce the payload ad hece, the badwidth available for trasmittig data. The term [ S(m)e*(m) ] represets the correctio factor or adjustmet that is applied to the curret estimate of the tap-weight vector. I order to improved system performace, takig ito accout the variatio i the sigal level at the filter iput ad selects a ormalized step size parameter i.e., W est ( m+1)=w est ( m)+ S(m)/S( m)s(m)] e*(m) (8) The iterative procedure is started with a iitial guess W est (0). Therefore, the LMS based CE is least sesitive to the scalig of its iput sigal variatio. Therefore, this algorithm is able to sese the best possible chael coefficiets are chagig. B. RLS CE METHOD The RLS CE requires all the past samples of the iput ad the desired output is available. The objective fuctio of a RLS CE algorithm is defied as a expoetial weighted sum of errors squares: c(m) = λ -m e H (m)e(m) + δλ W H (m)w(m), (9) m=1 Where δ is a positive real umber called regularizatio Parameter, e(m) is the prior estimatio error, ad λ is the Expoetial forgettig factor with 0 < λ < 1. The prior Estimatio error is the differece betwee the desired respose ad estimatio sigal: e(m) = H(m) W H (m) S(m) (10) The objective fuctio is miimized by takig the partial Derivatives with respect to W() ad settig the results equal to zero. δc (m)=0=-2 λ -m S (m) e H ( m)+2δλ W (m) =-2 -m S( m)[ H (m)- W H ( m) S( m)]+ 2 W (m) N W(m)[ -m S (m) S H (m) + I]= -m S (m) H H ( m) R (m) W( m)= R sh ( m) W( m)=r -I s(m) R sh ( m) Where s R (m) is the trasmitted auto-correlatio matrix ad sh R (m) is the cross correlatio matrix i.e., (11) Accordig to the Woodbury idetity, the above sh R (m) ca be writte as For coveiece of computig, let D(m) = Rsh(m) ad The K(m) is referred as a gai matrix. We may rewrite (9) as: So simply (13) to Substitutig (14), (15) ito (11), we obtai the followig RLS CE formula 526

4 BER BER Iteratioal Joural of Emergig Techology ad Advaced Egieerig Therefore, equatio (17) is the recursive RLS CE algorithm to update chael coefficiet. IV. RESULTS & DISCUSSION OFDM Simulatio Parameters parameter Number of carriers 512 Guard iterval 128 guard type Chael Model A. BER PERFORMANCE Specificatio Cyclic Extesio AWGN Chael The BER performace of the LMS estimatio,& RLS estimatio Proposed Algorithm is show i the Figure 3&4. Because of the effect of the oise, the BER performace of LMS estimator is the worst. The RLS CE is better tha LMS estimator.the proposed has the best performace at high SNR, hece it is accurate. After obtaiig the received data it is easy to calculate the bit error rate. Error is calculated as differece betwee received ad trasmitted data. The followig curve shows the bit error rate curve for various sigal to oise ratio values SNR i db Fig 3:BER Curve for LMS Algorithm SNR i db Fig 4:BER Curve for RLS Algorithm V. CONCLUSION The chael is estimated usig least mea square (LMS) & Recursive mea square algorithm is Proposed Method. The performace of the RLS Algorithm is better tha the LMS. At high SNR, the performace of the RLS is much better. Sice the effect of the oise is also small, hece the RLS is much more accurate. It helps to reduce bit error rate (BER). Ackowledgmet The author would like to thak His guide Assistat Professor Vekat N. Ghodke of AISSMS S Istitute of Iformatio Techology for motivatig Author to write a paper o the give topic. He helped author i all the way to complete this Challegig Task. REFERENCES [1] A.Sohail ad M.N.Jafri, Adaptive OFDM over Frequecy Selective ad Fast Fadig Chael Usig Blockwise Bit Loadig Algorithm, IEEE Iteratioal Cofereece o Wireless ad Optical Commuicatio Networks,pp. 1-4, July [2] A.Cyzlwik, Adaptive OFDM for widebad radio chaels, Global Telecommuicatios Coferece, vol 1, pp , Nov 1996 [3] E. Karami, Trackig performace of least squares MIMO chael estimatio algorithm, IEEE Tras. O Wireless Comm. vol. 55, o.11,pp , Nov [4] V. Pohl, P. H. Nguye, V. Jugickel, ad C. Vo Helmolt, How ofte chael estimatio is eeded i MIMO systems, i Proc. IEEE Global Telecommu. Cof., o. 1, pp Dec. 2003, 527

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