A Reduced Complexity Channel Estimation for OFDM Systems with Precoding and Transmit Diversity in Mobile Wireless Channels Hlaing Minn, Dong In Kim an

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1 A Reduced Complexity Chael Estimatio for OFDM Systems with Precodig ad Trasmit Diversity i Mobile Wireless Chaels Hlaig Mi, Dog I Kim ad Vijay K. Bhargava Departmet of Electrical ad Computer Egieerig, Uiversity of Victoria Victoria, B.C., Caada V8W 3P6. bhargava@ece.uvic.ca Abstract? A precodig approach for reduced complexity chael estimatio for OFDM systems with trasmit diversity is preseted. Two dieret OFDM schemes that employ the precodig are ivestigated: the rst oe is based o space-time codig as i [], the secod is based o a trasmit diversity usig a optimal desig of symbols for chael estimatio. The rst scheme reduces the size of the iverse matrix i chael estimatio to half of the existig method [] at the expese of slightly degraded BER performace. The secod does ot require ay matrix iversio ad hece achieves much complexity savig but has a BER performace degradatio at moderate ad high SNR ad a better BER performace at low SNR. Keywords: OFDM, Precodig, Trasmit diversity, Reduced complexity chael estimatio, Space-time codig, MRC I. Itroductio I order to compesate for the frequecy selective fadig i OFDM, techiques such as error correctig code ad diversity have to be used []-[5]. For OFDM systems, trasmit diversity combied with Reed-Solomo code has bee proposed for clustered OFDM i []. Applicatio of space-time codig [6] i OFDM systems has bee studied i [7] with perfect chael kowledge at the receiver. The chael estimatio for OFDM systems without trasmit diversity has bee studied by may researchers (e.g., [8] - [0]). However, for systems with trasmit diversity, the received sigal is a superpositio of the dieret trasmitted sigals from all trasmit ateas ad, cosequetly, the chael estimatio becomes more complicated. Recetly, the chael estimatio for OFDM This work was supported i part by a Strategic Project Grat from the Natural Scieces ad Egieerig Research Coucil (NSERC) of Caada ad i part by Telus Mobility, Cellular. The work of Dog I Kim was supported by the Research Fud of Uiversity of Seoul, Korea. systems with trasmit diversity usig space-time codig has bee proposed i []. I order to reduce the complexity associated with the matrix iverse operatio, [] also proposed a simplied approach. I this paper, we focus o reduced complexity chael estimatio for OFDM systems with trasmit diversity. We propose a chael estimatio method that has less complexity tha the simplied method of []. The proposed chael estimatio employs precodig o the subcarrier symbols of each trasmit atea ad makes use of the correlatio betwee adjacet subchael frequecy resposes. A optimal criterio for the traiig symbol of the proposed chael estimatio is derived. The proposed chael estimatio is applied i two schemes: the rst scheme uses space-time codig as i [] ad the secod scheme uses a trasmit diversity scheme that satises a optimal desig for the proposed chael estimatio ad maximal ratio combiig (MRC) at the receiver. The rest of the paper is orgaized as follows. I Sectio II, the proposed chael estimatio is preseted. I Sectio III, the mea square error (MSE) of the proposed chael estimatio is derived. The chael estimatio MSE ad BER performaces are evaluated for the proposed schemes ad [] i Sectio IV. Coclusio is give i Sectio V. II. System Descriptio Cosider a OFDM system with two-brach trasmit diversity ad two-brach receive diversity. Let the frequecy-domai subcarrier symbols for i th trasmit atea at time be ft i [; k] : k 0; ; : : : ; K? g for i ; ; with K beig the total umber of subcarriers. The, the DFT output frequecy-domai subcarrier symbols from each re-

2 ceive atea ca be expressed as r j [; k] i H ij [; k] t i [; k] + w j [; k] () w j [; k] is the additive complex Gaussia oise with zero mea ad variace, o the j th receive atea, that is ucorrelated for dieret 's, k's, or j's, ad H ij [; k] is the chael frequecy respose for the k th toe at time, correspodig to the i th trasmit atea ad the j th receive atea. With tolerable leakage, it ca be expressed as [] H[; k] K 0? l0 h[; l] W kl K () W K exp(?jk) ad K 0 is the total umber of WSSUS chael paths. I the proposed chael estimatio scheme, precodig is applied to ft i [; k]g before eterig IDFT block, as show i Fig.. The precoded frequecy-domai subcarrier symbols ft pi [; k]g are give by t pi [; k] (?) ik t i [; k]: (3) The, the frequecy-domai subcarrier symbols from the j th receive atea ca be expressed as r j [; k] i Let us dee the followig: t [; m] r j [; m] r j [; m] w j [; m] w j [; m] H ij [; k] (?) ik t i [; k]+w j [; k]: (4) t [; m]t [; m] +t [; m + ]t [; m + ] t [; m]r j [; m]?t [; m + ]r j [; m + ] t [; m]r j [; m] +t [; m + ]r j [; m + ] t [; m]w j [; m]?t [; m + ]w j [; m + ] t [; m]w j [; m] +t [; m + ]w j [; m + ] x deotes the complex cojugate of x ad m 0; ; : : : ; M? ad M K. If the chael frequecy resposes are such that H ij [; m] H ij [; m + ]; (5) the the followig are obtaied: rj[; m] Hj[; m] t [; m] + wj[; m] (6) rj[; m] Hj[; m] t[; m] + wj[; m]: (7) Assumig that (5) holds, the chael impulse respose estimate ca be obtaied, usig a similar approach to [], by miimizig the followig MSE cost fuctio, C f~ hij [; l]; i ; ; j ; g r ij[; m]? K 0? l0 t i [; m] ( ~h ij [; l] W ml K ti [; m] t [; m]; i t [; m]; i : (8) (9) The chael impulse respose estimate ca the be give by ~h ij [] Q? [] p ij []: (0) Q[] ~h ij [] p ij [] q[; l] p ij [; l] 0 q[; 0] q[;?] : : : q[;?k 0 + ] q[; ] q[; 0] : : : q[;?k 0 + ]... q[; K 0? ] q[; K 0? ] : : : q[; 0] ( ~ h ij [; 0]; ~ h ij [; ]; : : : ; ~ h ij [; K 0? ]) T (p ij [; 0]; p ij [; ]; : : : ; p ij [; K 0? ]) T t [; m] t [; m] W?ml K r ij [; m] t i [; m] W?ml K : Followig [], the complexity ivolved i Q? ca be further reduced ad the chael estimatio performace ca be further improved by usig oly sigificat chael taps. Durig the traiig block, the sigicat P taps ca be idetied by dig the l's with large i jh ij[; l]j for each receive atea. Suppose that J sigicat taps are used ad let fl m : m ; ; : : : ; J; (0 l < l < : : : < l J K 0? )g be the J sigicat chael tap idexes for the receive atea j. The, the simplied chael estimatio is give by h ij [] Q? [] p ij [] () C A

3 Q[] h ij [] p ij [] 0 q[; 0] q[; l? l ] : : : q[; l? l J ] q[; l? l ] q[; 0] : : : q[; l? l J ]..... q[; l J? l] q[; l J? l] : : : q[; 0] ( ~ h ij [; l ]; ~ h ij [; l ]; : : : ; ~ h ij [; l J ]) T (p ij [; l ]; p ij [; l ]; : : : ; p ij [; l J ]) T : C A P [l] is the power delay prole of the chael correspodig to each trasmit ad receive atea pairs. The chael frequecy respose estimate MSE of the proposed method uder optimal coditio ca also be give by MSEf ~ Hij [m]g K 0? l0? W l K K 0 P [l] + K : (4) As metioed i [], a time correlatio of h ij [; l m ] ca be used to further improve the chael estimatio but is ot cosidered i this paper. III. Performace Aalysis I the previous sectio, a reduced complexity chael estimatio for OFDM systems with trasmit diversity is proposed with the assumptio that (5) holds. I this sectio, the performace of the proposed method i a multipath fadig chael ot satisfyig (5) is preseted. For simplicity, the time idex is omitted i the followig aalysis. Dee the followig: Hij[m] He[m] He[m] [l0] Hij[m]? Hij[m + ] Hj[m]? Hj[m] t[m + ] t [m + ] Hj[m]? Hj[m] t [m + ] t[m + ] t[m] He[m] W?ml 0 M [ [0]; []; : : : ; [K0? ]] T The MSE of the chael impulse respose estimate ca be give by MSE o E k h K ~ j? h j k 0 K 0 T race + T race Q?Ho Q? Ef H gq?hoo () From (), a optimal solutio ca be obtaied whe Q is diagoal matrix, i.e., q[; l] 0 for l 6 0. This optimal coditio is achieved whe either of the followig two coditios are satised: (i) t i [; m] t i [; m + ]; i ; ; m 0; ; : : : ; M?, ad (ii) t [; k] t [; k]; k 0; ; : : : ; K?. With optimal coditio, ad assumig jt i j, the MSE ca be give by MSE P K0? l? W l K P [l] + K 0 K (3) Similarly, the MSE of the chael frequecy respose estimate of [] at its optimal coditio ca be show to be equal to MSEf ~ Hij [m]g K 0K: (5) Hece, the proposed method has a larger chael frequecy respose estimatio MSE tha [] due to the assumptio of (5). The extra term i the MSE expressio also idicates the depedece of the proposed chael estimatio MSE performace o the chael power delay prole. We have also validated the MSE expressios by simulatio but the results are ot show i this paper. For systems with L-trasmit diversity, the average SNR at the receiver, assumig jt i j, is deed as SNR PL E i jh ij[; o k]j : (6) From (4), it ca be observed that the chael estimatio MSE maily depeds o which, i tur, idicates that the chael estimatio MSE performace evaluatio by computer simulatio depeds o how to model fefjh ij [; k]j gg. For a SNR value, if fefjh ij [; k]j gg are modeled very small, will be very small ad, cosequetly, the obtaied chael estimatio MSE will be very small. Hece, a suitable simulatio model for chael estimatio performace evaluatio should be adopted. There are two possible approaches: the rst oe is P to model E L i jh ij[; o k]j, ad the sec- od is to model E jh ij [; o k]j. Due to the similar reasoig of the MSE's depedece o chael power gai ad to be comparable with dieret umber of diversity braches, the secod approach of E jh ij [; k]j o is adopted i this paper.

4 IV. Performace Evaluatio by Computer Simulatio IV. Proposed Chael Estimatio with Space-Time Code The proposed chael estimatio is applied i two schemes: the rst scheme uses space-time codig as i []. As show i Fig. (a), at the receiver, the chael estimates are precoded before eterig spacetime Viterbi decodig so that the same space-time decodig as i [] ca be used. I other words, the combied eect of precodig at the trasmitter ad multipath chael respose ca be treated as the eective chael respose for space-time codig/decodig. The precoded chael estimates are give by ~H p;ij [; k] (?) ik ~ Hij [; k]: (7) The, the space-time Viterbi decodig for the system usig proposed chael estimatio uses the metrics r[; k]? ~ Hp [; k] ~ t[; k] (8) k k deotes Euclidea orm, ad r[; k] ~H p [; k] r [; k] r [; k]! ~H p; [; k] Hp; ~ [; k] ~H p; [; k] Hp; ~ [; k] ad ~ t[; k] is the estimated sigal vector, deed as ~ t[; k] ~t [; k] ~t [; k] From (4) - (5), [] has a better chael estimatio MSE at its optimal coditio (i.e., for traiig symbol). Hece, the proposed chael estimatio i the rst scheme will have a larger chael estimatio MSE tha []. By observig that a matrix iversio is ot eeded for the traiig symbol, we may cosider a hybrid scheme for which the chael estimatio of [] is used oly for the traiig symbol i order to yield a better chael estimatio ad the proposed chael estimatio is applied to the data symbols i order to reduce the complexity. IV. Proposed Chael Estimatio with Optimal Sigal Desig To achieve comparable MSE performace, the proposed chael estimatio is applied i the secod! :! scheme, as show i Fig. (b), the trasmit diversity scheme that satises the optimal coditio for the proposed chael estimatio is used, i.e., t [; k] t [; k] before precodig. Similar to the rst scheme, the combied eect of precodig at the trasmitter ad multipath chael respose ca be treated as the eective chael respose for MRC detectio. The MRC output is give by r MRC [; k] r [; k] [; k] + r [; k] [; k] (9) [; k] ( ~ Hp; + ~ Hp; ) [; k] ( ~ Hp; + ~ Hp; ) : The decisio is to choose the sigal poit t from the sigal costellatio poits that miimizes the metric kr MRC [; k]? tk. IV.3 Simulatios, Results ad Discussios The system parameters used i the simulatio are the same as []: 8 subchaels, 4 guardsubchaels o each ed, 60s subchael spacig, 40 s guard iterval, two-ray (TR) ad GSMtypical urba (TU) chael models with delay spread of :06s ad 40 Hz Doppler frequecy, 6-state space-time code with four PSK. The umber of sigicat taps used i all chael estimatio methods is 7. The chael estimatio MSE performaces are show i Fig. for both TR ad GSM-TU chael models. Whe comparig [] ad the rst scheme both usig space-time codig, it is foud that the latter has a slightly worse MSE performace for both chael as expected from (4)-(5). The MSE of hybrid scheme is also plotted, which is almost the same as the rst scheme: just margially better but ot oticeable i the gure. The MSE performaces of the proposed chael estimatio with optimal sigal desig (i.e., the secod scheme) ad [] are similar. At low SNR regio, the secod scheme has a slightly better MSE performace for both chaels. At high SNR regio, [] has a slightly better MSE performace tha the secod scheme for both chaels. The reaso may be explaied as follows. The MSE of [] is cotributed by the oise, the decisio-directed referece data error ad the chael estimatio leakage due to o-sample spaced chael taps, while

5 that of proposed chael estimatio is cotributed by the oise, the decisio-directed referece data error, the chael power delay prole ad the chael estimatio leakage. The eect of oise o MSE of [] is larger tha that of the proposed method with optimal sigal desig because of the o-optimality of data symbols for the chael estimatio of []. At very low SNR regio, oise eect is much more domiat tha the others ad hece [] has a slightly worse MSE. At high SNR regio, the oise eect becomes less domiat tha the others ad hece [] teds to achieve a slightly better MSE. The slightly worse MSE of the secod scheme i both chaels may be attributed to the larger referece data error, as ca be see from its BER performace i Figs. 3 ad 4. Whe comparig [] ad the rst scheme, both usig space-time codig, the latter has a slightly worse (but comparable) BER performace tha the former does, for both chaels. This is due to the slightly worse chael estimatio performace of the latter ad is the price paid for the reduced complexity. It reduces the size of the matrix, the iverse of which has to be foud for chael estimatio at every OFDM symbol, to half of that i []. The BER performace of the hybrid scheme is almost the same as the rst scheme. Although the MSE performaces of the secod scheme ad [] are similar, there are cosiderable diereces i their BER performaces. The secod scheme has a better BER performace at very low SNR regio but [] has a much better BER performace at high SNR regio. The reaso is that, at very low SNR regio, space-time code loses its codig gai ad hece [] has a worse BER performace while, at high SNR regio, o-error correctio capability of the secod scheme causes it to have a worse BER performace. V. Coclusio A ew chael estimatio based o precodig techique has bee proposed for OFDM systems with trasmit diversity, which aims at greatly reducig the complexity i the matrix iversio eeded for every OFDM symbol. I particular, the proposed scheme usig precodig combied with space-time codig, as i [], shows a comparable BER performace to [] while achievig complexity savig, i.e., the matrix iverse size is reduced by half. Also, the other scheme with trasmit diversity based o the optimal desig of precoded symbols, combied with MRC at the receiver, attais much receiver complexity savig at the expese of some BER performace degradatio at moderate ad high SNR but has a better BER performace at low SNR. VI. Refereces [] Y. Li, N. Seshadri ad S. Ariyavisitakul, "Chael estimatio for OFDM systems with trasmitter diversity i mobile wireless chaels," IEEE Joural o Selected Areas i Commuicatios, vol. 7, o. 3, pp , Mar 999. [] L. J. Cimii, Jr., B. Daeshrad, ad N. R. Solleberger, "Clustered OFDM with trasmitter diversity ad codig," IEEE Global Telecommuicatios Cof., Lodo, U.K., pp , Nov [3] L. J. Cimii ad N. R. Solleberger, "OFDM with diversity ad codig for high-bit-rate mobile data applicatios," IEEE Global Telecommuicatios Cof., Phoeix, AZ, pp , Nov [4] A. Wittebe, "A ew badwidth eciet trasmit atea modulatio diversity scheme for liear digital modulatio," IEEE Iteratioal Cof. o Commuicatios, Geeva, Switerlad, pp , Jue 993. [5] J. H. Witers, "The diversity gai of trasmit diversity i wireless systems with Rayleigh fadig," IEEE Iteratioal Cof. o Commuicatios, Chicago, IL, pp.- 5, Jue 994. [6] V. Tarokh, N. Seshadri ad A. R. Calderbak, "Space-time codes for high data rate wireless commuicatio: Performace aalysis ad code costructio," IEEE Tras. o Iformatio Theory, vol. 44, pp , Mar 998. [7] D. Agarwal, V. Tarokh, A. Naguib ad N. Seshadri, "Space-time coded OFDM for high data rate wireless commuicatio over widebad chaels," IEEE Vehicular Techology Cof., Ottawa, Caada, pp. 3-36, May 998. [8] J.-J. va de Beek, O. Edfors, M. Sadell, S. K. Wilso ad P. O. Borjesso, "O chael estimatio i OFDM systems," IEEE Vehicular Techology Cof., Chicago, IL, pp , July 995. [9] O. Edfors, M. Sadell, J.-J. va de Beek, S. K. Wilso ad P. O. Borjesso, "OFDM chael estimatio by sigular value decompositio," IEEE Tras. o Commu. vol. 46, pp , Jul 998. [0] Y. Li, L. J. Cimii, Jr., ad N. R. Solleberger, "Robust chael estimatio for OFDM systems with rapid dispersive fadig chaels," IEEE Tras. o Commu., vol. 46, pp , Jul 998.

6 b[,k] space -time codig t [,k] t [,k] Precodig t [,k] p t [,k] p r [,k] r [,k] space -time decodig ~ H p,ij Precodig ~ H ij ^ b[,k] space -time codig TR chael [] st scheme Hybrid d scheme b[,k] QPSK & Optimal sigal desig t [,k] t [,k] Precodig t [,k] p t [,k] p (a) (b) r [,k] r [,k] chael estimator MRC & Decisio chael estimator ^ b[,k] QPSK ~ H & p,ij Optimal Precodig sigal ~ desig H ij Fig.. Proposed reduced complexity chael estimatio for OFDM systems with trasmit diversity i two schemes: (a) st scheme usig space-time codig, (b) d scheme usig optimal sigal desig ad MRC receptio BER Fig. 3. BER performaces of OFDM systems with trasmit diversity usig dieret chael estimatio methods i TR chael. 4 6 (a) TR chael [] st scheme Hybrid d scheme 4 6 (b) TU chael TU chael [] st scheme Hybrid d scheme 8 8 MSE (db) 0 MSE (db) 0 BER Fig.. Chael estimatio MSE performaces for OFDM systems with trasmit diversity: (a) TR chael ad (b) GSM-TU chael Fig. 4. BER performaces of OFDM systems with trasmit diversity usig dieret chael estimatio methods i GSM-TU chael.

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