Bit Error Rate Prediction of Coded MIMO-OFDM Systems

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1 Bit rror Rate Prediction of Coded MMO-OFDM Sytem Y. Naer member, J.F. Helard Senior member, M. Cruiere member ntitute of leconic and Telecommunication of Renne, UMR CNRS 6464, Renne, France mail : ABSTRACT Bit error rate (BR) rediction over channel realiation ha emerged a an active reearch area. n thi aer, we give analytical ignal to interference and noie ratio (SNR) evaluation of MMO-OFDM ytem uing an iterative receiver. Uing thi analytical SNR exreion, we rooe an accurate BR rediction method baed on ective exonential SNR maing (SM) method. We how by imulation that our method i indeendent of the channel realiation and of the MMO cheme. t i only deendent on the modulation and coding cheme. ndex Term- OFDM, MMO, Sace time block code, Bit error rate, SM techniue.. NTRODUCTON Recently, the combination of multile inut multile-outut (MMO) and orthogonal freuency diviion multilexing (OFDM) techniue i urued a a otential candidate for the future wirele network ince they enure high ecum iciency a well a high diverity gain. However, the erformance of thi combination can be further imroved if adeuate link adatation algorithm are adoted. The choice of the bet algorithm could be baed on the modulation and coding cheme (MCS) but alo on the alication reuirement and channel condition. A a coneuence, an accurate and robut real-time channel rediction i reuired by the higher layer rotocol, in articular bit error rate (BR) rediction []. Accurate BR rediction can facilitate deign, erformance evaluation and arameter tuning of many wirele rotocol and alication. For intance, rate-adative alication and data link rotocol can ue accurate BR rediction to adat their ource and channel coding rate in accordance with the forecated channel condition. n thi aer, we invetigate the ective exonential ignal to interference and noie ratio method (SM) in the MMO-OFDM ytem to redict the BR at the outut of the channel decoder. The redicted BR could therefore be ued by higher layer rotocol in order to adat their anmiion mode. The conibution of thi work i twofold. Firt, a generalized framework i rooed for modelling the ignal to interference and noie ratio (SNR) at each iteration of a ub-otimal iterative receiver in MMO-OFDM ytem. Therefore, we adat the SM techniue, initially validated within 3GPP for OFDM ytem [], to a MMO-OFDM context. We how that the SM techniue i indeendent of the channel fading rofile and deend only on the MCS. Thi aer i uctured a follow. Section decribe the anmiion model in MMO-OFDM ytem. n ection 3 we give analytical exreion of the SNR at the outut of the detector. n ection 4 we decribe the SM techniue and it adatation to the MMO-OFDM ytem. Section 5 validate our model through imulation reult. Concluion are drawn in ection 6.. TRANSMSSON MODL Conider a MMO-OFDM communication ytem uing M T anmit antenna (Tx) and M R receive antenna (Rx). Figure deict the anmitter module. nformation bit b k are firt channel encoded, randomly interleaved, and fed to a uadrature amlitude modulation (QAM) module which aign B bit for each of the comlex contellation oint. Therefore, each grou S=[,, Q ] of Q comlex ymbol i fed to a ace-time block code (STBC) encoder. Let X=[x i,t ] where x i,t (i=,, M T ; t=,, T) be the outut of STBC encoder. The ST coding rate i then R = Q / T. Thi outut i then fed to M T OFDM modulator, each uing N ub-carrier. Then, the ignal ower at the outut of the ST encoder i normalized by M T. b k Source ncoder nterleaver Maer S STBC ncoder OFDM Mod. OFDM Mod. Ant. Ant. M T Figure - MMO-OFDM anmitter. n our anmiion model, we aume that the anmitter and receiver are erfectly ynchronied. Moreover, we aume erfect channel tate information (CS) at the receiver. Since we aume a freuency domain anmiion, the ignal received on the ub-carrier n by the antenna j i a ueroition of the anmitted ignal by the different antenna multilied by the channel coicient h j,i [n] to which white Gauian noie (WGN) i added. t i given by: M T y [ n] = P h [ n] x [ n] + w [ n] () j, t i j, i i, t j, t i= where y j,t [n] i the ignal received on the nth ubcarrier by the jth receiving antenna during the tth OFDM ymbol duration. P i the aigned ower to the ymbol i anmitted by the ith antenna. h j,i [n] i the freuency channel coicient aumed to be contant during T ymbol duration, x i,t [n] i the ignal anmitted by the ith antenna and w j,t [n] i the additive WGN with zero mean and variance N /. n the euel, we will dro the ubcarrier index n for imlicity. By inoducing an euivalent receive M T Y C whoe element are the comlex R maix received ymbol exreed in () we can write the received ignal on the n th ub-carrier on all receiving antenna:

2 Y = HPX + W () where H i the (M R,M T ) channel maix whoe comonent are the coicient h j,i, P i a (M T,M T ) diagonal maix containing the ignal magnitude P i, X i a (M T,T) comlex maix containing the anmitted ymbol x i [t]. W i a (M R,,T) comlex maix correonding to the WGN. Let u now decribe the anmiion link with a general model indeendently of the ST coding cheme. We earate the real and imaginary art of the comlex ymbol inut vector { :,,Q}, of the outut X of the ST encoder, and the received ignal Y. Let,R and, be the real and imaginary art of. The main arameter of the double code are given by it dierion maice U and V correonding (not eual) to the real and imaginary art of X reectively. With thee notation, X i given by: (, R j, ) Q X = U + V (3) We earate the real and imaginary art of S, Y and X and tack them row-wie in vector of dimenion ( Q,), (M R T,) and (M T T,) reectively. We obtain: y x =, R,,,..., Q, R, Q, = y,, y,,..., yt,, yt,,..., ym,, RT y R R R M RT, = x(,),, x(,),,..., x( M, ),, T T x R R ( MT, T ), where hold for maix anoe. Since we ue linear ST coding, the vector x can be written a [3]: (4) x = F. (5) where F ha the dimenion (M T T, Q) and i obtained through the dierion maice of the real and imaginary art of X [3]. A we change the formulation of, x, and y in (4), it can be hown that vector x and y are related through the maix G of dimenion (M R T, M T T) uch that: y = GBx + w (6) The maix B i a (M T T, M T T) diagonal maix whoe comonent are given by: Bi, i = Pi with. T ( ) + i T. =,..., M T Maix G i comoed of block G j,i (j=,,m R ; i=,,.m T ) each having (T,T) element [3]. t comonent are the real and imaginary art of the channel coicient. Now, ubtituting x from (5) in (6), the relation between y and become: (7) y = GB F + w = Ge + w (8) G e i the euivalent channel maix of dimenion (M R T,M T T) between and y. t i aumed to be known erfectly at the receiving ide. Ant. Ant. M R PC detector ( ) ˆ l ɶ ( l ) Demaer (LLR com.) Soft Gray Maer Denterleaver nterleaver SSO Decoder bˆ timated bit Figure - terative receiver ucture Now, the detection roblem i to find the anmitted data given the vector y. The otimal receiver i baed on joint ST detection and channel decoding oeration. However uch receiver i exemely comlex to imlement and reuire large memory for non-orthogonal (NO) STBC code. Thu the ub-otimal olution rooed here conit of an iterative receiver where the ST detector and channel decoder exchange exinic information in an iterative way until the algorithm converge. The iterative detector hown in Figure i comoed of a MMO eualizer, a demaer which i made u of a arallel interference cancellation (PC) module, a log likelihood ratio (LLR) comutation, a oft-inut oft-outut (SSO) decoder, and a oft maer. At the firt iteration, the demaer take the etimated ymbol ŝ, the knowledge of the channel G e and the noie variance, and comute the LLR value of each of the coded bit anmitted er channel ue. The etimated ymbol ŝ are obtained via minimum mean uare error (MMS) filtering according to: ˆ ( σ ) w = g G G + y (9) e e where g of dimenion (M R T, ) i the th column of G e ( Q). ŝ i the etimation of the real art ( odd) or imaginary art ( even) of the comlex ymbol ( Q). The oft Gray maer take the oft LLR outut from the SSO decoder and roduce etimation ɶ of the anmitted ymbol. The etimated ymbol ɶ belong to the contellation oint et. t etimation i baed on the LLR value and the robability value of each bit of a given contellation oint [3]. Once the etimation of the different ymbol ɶ i achieved by the oft maer at the firt iteration, we ue thi etimation for the next iteration roce. From the econd iteration, we erform PC oeration followed by a imle invere filtering (intead of MMS filtering at the firt iteration): yˆ = y G ɶ ˆ ( l ) ( l ) e, = g g g yˆ ( l ) ( l ) () where the uercrit ( ) l refer to the iteration number. Ge, of dimenion (M R T, Q-) i the maix Ge with it th column removed, ɶ of dimenion (Q-, ) i the vector

3 ɶ etimated by the oft maer with it th eny removed. n next ection, we will evaluate the iterative detection roce through SNR and BR analyi. 3. SNR VALUATON Without lo of generality, we aume that we are intereted by the th ymbol. Uing the vector-maix notation of reviou ection, the etimated received ymbol at the firt iteration in (9) could be written in an euivalent form a: n (), = + + ˆ () () () ( σ w ) () = g Ge Ge + g ( σ ) Q () = g Ge Ge + w g ( σ w ) () e e () = g G G + w i the ueful received ignal, () () i the interference ignal between different antenna due to the non-orthogonality of the conidered STBC. We can eaily verify that it i eual to zero for orthogonal STBC cheme. () i the colored noie. The comlex anmitted data ymbol are aumed i.i.d. having zero mean and unit variance (the variance of the real and imaginary art i eual to ½). Due to thi diibution, the SNR exreion can be deduced from () by: () { } () () { } + { } SNR = () The exectation value in () over the random data ymbol are given by: () ( = g G G + σ ) w g Q () { } = ( + σ ) w e e N () ( = g Ge Ge + σ w) g G G g e e (3) At the econd iteration, the etimated ymbol exreed in () become more comlex. t i obtained uing (8) and (9) in () by: () = + + where (4) () () () () ˆ ( σ ) Q () () = g g + Ge Ge w g ɶ g g () = = g g g w (5) For next iteration, it i clear from (5) that the exreion of the etimated received ymbol a well a the etimated SNR become more comlex. Therefore, ome maniulation hould be conidered to give an analytical exreion of the SNR. Baed on the ucture of the iterative receiver, we already know that the outut of the oft Gray maer are comlex ymbol which belong to the contellation oint. Let () () () () () = + = be the total interference t ower at the econd iteration. Then, two cae can be reented at thi tage: f the etimated ymbol Since () ɶ at the outut of the Gray maer i eual to the anmitted ymbol, the ueful ignal () in (5) i uch that = =ɶ () () and the total interference ignal at the econd iteration become: = + = (6) () () () () () () t and () are indeendent and the comlex outut of the Gray maer are zero mean with unit variance, the etimated SNR at the econd iteration i: SNR () where () { } ( ) () () () { + } { ˆ ɶ } i etimated through () and () ˆ outut of the oft Gray maer at the firt iteration. (7) () ɶ i the f the etimated ymbol ɶ at the firt iteration i () different from the anmitted ymbol, the difference between the received ignal at the firt two ucceive iteration yield by ubtituting (5) in (4): Since () () () () () ˆ () from ɶ = + + ɶ (8) ɶ i different from () in thi cae, and the different anmitted ymbol are i.i.d., we can verify due to the exectation oeration that: SNR () { } { ɶ } () () () () () () () () () ( ) () { ˆ ɶ } { ˆ ɶ } (9) t i clear from the lat term of (7) and (9) that the SNR exreion at the econd iteration i imler than that of (5). n thi cae, only the etimated ymbol at each iteration are ued for SNR etimation i.e. we don t have to comute comlex exreion. Alo, we can how that (7) and (9) could be generalized for ucceive iteration. We

4 will now exloit our theoretical SNR model through BR meaurement at the outut of the channel decoder. 4. BR PRDCTON WTH SM TCHNQU n the reviou ection, we derived formula for the etimation of the SNR at each iteration of the detector outut. However, it i deirable to evaluate the ytem level erformance after channel decoding in term of BR. Thi work i motivated by the ractical need of uch meaure for accurate and realitic evaluation of the ytem level erformance but alo for uitable develoment of link adatation algorithm uch a adative modulation and coding, acket cheduling, hybrid-arq, etc [4]. Therefore, an accurate relationhi between the SNR obtained at the outut of the detector and the BR erformance at the outut of the channel decoder mut be identified. Let J denote the acket ize in comlex data ymbol. n general, the data ymbol in the acket are anmitted over different reource element (e.g. ub-carrier) and therefore they may exerience different roagation and interference condition. Thu, the data ymbol may have different SNR value. Let SNR be the vector of J intantaneou SNR received at the outut of the detector. The roblem of determining an accurate BR rediction method come back to looking for a relationhi Pe = f ( SNR ) () where P e denote the bit error robability (BP) and f i the rediction function, which hould be invariant with reect to the fading realization, to the multi-ath channel model and hould be alicable to different MCS in a oft way, i.e., by changing the value of ome generic arameter [5]. n the context of a WGN channel, the SNR become a SNR and it remain contant over the acket. n thi context, a direct relationhi ξ exit between the SNR and the error robability P = ξ ( SNR ) e, WGN () The function ξ i called the maing function. t i obtained through theoretical analyi or link level imulation with a WGN channel. n the general context of a fading channel, where the SNR varie, the rediction function f in () can be written exactly a a comound function of the WGN function ξ and a comreion function r [5]: P = ξ r( SNR) = ξ ( SNR ) e with SNR = r( SNR) () The function r i referred to a the comreion function ince it role i to comre the vector SNR of J comonent into one calar SNR. The calar SNR i called the ective SNR and it i defined a the SNR which would yield the ame error robability in an euivalent WGN channel a the aociated vector SNR in a fading channel. By writing (), we have merely turned the roblem of determining the rediction function f into the roblem of determining the comreion function r. n an OFDM ytem, it wa concluded that the key iue to accurately determine the aroriate BR after channel decoding i to ue the ective SNR in combination with WGN curve. [] rooe the SM techniue which i baed on the Chernoff Union bound [4] to find the ective SNR. The key SM techniue exreion relevant to an OFDM ytem i given by: SNR N SNR[ n] ex N n= λ = λ ln (3) SNR[n] i the SNR obtained over the n th ub-carrier and λ i a uniue arameter which mut be etimated from the ytem level imulation for each MCS. t i etimated once by reliminary imulation for each MCS. When the SNR i comuted, it will be ued for BR rediction at the outut of the channel decoder with a imle look-u table (LUT) a hown in Figure 3. Thi LUT give the BR at the outut of the channel decoder a a function of the SNR for a Gauian channel. t i comuted analytically or by imulation. The uniuene of λ for each MCS i derived from the fact that the ective SNR mut fulfill the aroximate relation ( ) = ( ) BR SNR BR SNR (4) where BR WGN i the BP for the WGN channel which deend only on the MCS. SNR[],...,SNR[N] SM WGN SNR LUT Figure 3- BR rediction through SM BR at the outut of the channel decoder n our tudy, the SM techniue mut be adated to a MMO-OFDM context. ndeed, the etimated received ymbol at each ub-carrier i a ueroition of different ymbol anmitted by the different antenna on that ubcarrier. Therefore, the SM techniue will be alied on the et of Q ymbol anmitted on the M T antenna during T OFDM ymbol. The ective SNR i therefore comuted through: SNR Q N SNR [ n] = λ ln ex (5) NQ = n= λ 5. SMULATON RSULTS n thi ection, we validate by imulation our theoretical model baed on (), (9) and (5). n thi aer, we conider the orthogonal Alamouti code [5] and the Golden code [7] with M T = and M R =. For eual anmitted ower, we aume that the ower of maix B in (6) are eual to db. For uneual anmitted ower, we et P to db and we change P. The conidered imulation arameter are given in Table. The ecal iciencie 4 and 6 [b//hz] are obtained for different ST cheme a hown in Table. The WGN reult are obtained uing Alamouti cheme ince, NO cheme are not icient on WGN channel. Figure 4 (re. Figure 5 ) comare the BR obtained by imulation and the BR redicted with the SM techniue for the Alamouti cheme, conidering a ecal iciency η=4 [b//hz] (re. η=6 [b//hz]) and different value of anmitted ower. Thee figure how the accuracy of the rooed techniue baed on the SNR analytical exreion. Moreover, they how that the

5 arameter λ i contant (λ=.7 for η=4 and λ=.6 for η=6) and it i indeendent of the anmitted ower but deend on the MCS or euivalently on the ecal iciency. The arameter λ i obtained by imulation. t i comuted once for a given MCS. Figure 6 comare the BR obtained by imulation and the BR redicted with the SM techniue for the Golden code cheme, a ecal iciency η=6 [b//hz] and different value of anmitted ower. Since the ecal iciency doe not change with reect to reult of Figure 5, the arameter λ=.6 give an accurate BR rediction and validate our rediction techniue. Table - Simulation Parameter Number of ubcarrier (N c ) 8K mode Guard nterval Rate R c of CC Channel etimation Contellation Secal fficiencie Channel model Secal fficiency 4 amle /, /3, 3/4 uing (33,7) o erfect 6-QAM, 64-QAM, 56-QAM η= 4 and 6 [b//hz] Tyical Urban (TU-6) Table - Different MMO cheme and iciencie ST cheme ST rate R Contellation η=4 Alamouti 64-QAM /3 [bit/sec/hz] Golden 6-QAM / η=6 Alamouti 56-QAM 3/4 [bit/sec/hz] Golden 64-QAM / 6. CONCLUSON n thi aer, we have rooed an analytical SNR evaluation of MMO-OFDM ytem uing iterative receiver. Once the SNR i comuted, we have rooed an adatation of the SM techniue to redict the BR at the outut of the channel decoder. We how by imulation the validation of our rediction method. Our future work conit to ue thi method in cro layer otimization. RFRNCS [] S. Karande, U. Parrikar, K. Mira, and H. Radha, On modeling of 8.b reidue error,css, Mar. 6. [] 3GPP TSG-RAN-, "TR 5.89: feaibility tudy for OFDM for UTRAN enhancement", Mar. 4. [3] Y. Naer, J.-F. Hélard, M. Cruiere, and O. Pauero, fficient MMO-OFDM Scheme for Future Terreial Digital TV with Uneual Received Power, to be aeared in the nt. communication conference, May 8, Beijing, China. [4] ricon, "Sytem level evaluation of OFDM- further conideration", TSG-RAN WG #35, Libon, Portugal, November 3, R-333 [5] A. Tee, S. Yoon, and J. Cleveland, "Link-Sytem nterface Simulation Methodologie," 8. R c Working Grou on Mobile Brodband Wirele Acce, C8.-4/67, June 4. [6] S.M. Alamouti, A imle anmit diverity techniue for wirele communication, J. on Selected Area in Communication, vol. 6, no. 8, , Oct [7] J.-C. Belfiore, G. Rekaya, and. Viterbo, The golden code: a full-rate ace-time code with nonvanihing determinant, Tran. in nformation Theory, vol. 5, no. 4, , Ar. 5. BR Gauian Predicted BR P = db Simulation BR P = db Predicted BR P Simulation BR P b /N Figure 4- Validation of SM techniue, Alamouti cheme, η=4 [b//hz], λ=.7 BR Gauian Predicted BR P = db Simulation BR P = db Predicted BR P Simulation BR P b /N Figure 5- Validation of SM techniue, Alamouti cheme, η=6 [b//hz], λ=.6 BR Gauian Predicted BR P Simulation BR P Predicted BR P Simulation BR P b /N Figure 6- Validation of SM techniue, Golden code cheme, η=6 [b//hz], λ=.6

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