Transmitter Precoding for Orthogonal Space-Time Block-Coded OFDM in Transmit-Antenna and Path-Correlated Channels

Transcription

1 ransmitter Precoding for Orthogonal Space-ime Block-Coded OFDM in ransmit-antenna and Path-Correlated Channels Yu Fu, Witold A. Krzymień*, and Chintha ellambura Department of Electrical and Computer Engineering, University of Alberta Edmonton, Alberta, Canada 6G 2V4 * also with RLabs, Edmonton, Alberta, Canada {yufu, wak, chintha@ece.ualberta.ca} Abstract Orthogonal space-time block-coded OSBC orthogonal frequency-division multiplexing OFDM links for frequency-selective multiple-input multiple-output MIMO channels with correlated paths and transmit antennas are considered. In such systems, optimal precoding with only covariance feedback is derived using the minimum pair-wise error probability PEP criterion; linear and non-linear precoders are designed. he proposed precoding only needs the statistical knowledge of the channel at the transmitter, which significantly reduces the feedback requirements. Both linear and non-linear precoders substantially improve the system bit error rate for OSBC OFDM in transmit-antenna and path-correlated channels. he proposed non-linear precoder outperforms the linear precoder. Index erms OFDM, MIMO, optimized precoding, OSBC I. INRODUCION he multiple-input multiple-output MIMO paradigm, employing multiple antenna arrays at both the transmitter and the receiver, has recently emerged as a dominant technology for high-data-rate wireless communications. Using spacetime signal processing techniques, MIMO systems effectively exploit spatial dimension inherent in multiple antennas, and obtain full diversity gains or capacity increases without bandwidth expansion or transmit power increase. Orthogonal space-time block coding OSBC 1], 2] is an important space-time signal processing technique to exploit the total available spatial diversity in MIMO channels. Since OSBC achieves full diversity with low decoding complexity, it is widely used and has been adopted in the third generation cellular standards 3], 4]. Orthogonal frequency-division multiplexing OFDM is a spectrally efficient transmission technique suitable for frequency-selective radio channels. OS- BC, although designed for flat-fading MIMO channels, can be immediately overlayed on OFDM by simply performing coding/decoding on a subcarrier basis. herefore, orthogonal space-time block-coded OSBC 1 OFDM achieves full diversity gain and facilitates the utilization of this gain on frequency-selective MIMO channels. 1 OSBC here stands for orthogonal space-time block-coded and orthogonal space-time block coding, depending on the context. OSBC has been originally designed for uncorrelated Rayleigh fading channels, where the channel gains are distributed as independent and identically distributed i.i.d. zeromean complex Gaussian random variables. However, in practical systems, the MIMO channel may be spatially correlated due to poor scattering and/or insufficient transmit antenna spacing. he temporally-correlated multipath signals can lead to path correlations in each channel between the transmit and receive antenna pair. he path and antenna correlations make the received data streams correlated and lead to difficult stream separation and decoding. If conventional space-time processing techniques are directly used in correlated MIMO channels, the capacity and bit error rate performance can be degraded. If channel state information CSI is available at the transmitter, precoding can exploit spatial diversity, offer higher link capacity, and reduce the complexity of MIMO transmission and reception. ransmitter precoding can increase throughput in spatially-multiplexed OFDM on spatially-correlated frequency-selective MIMO channels 5]. It also offers the flexibility of adapting OSBC to spatially correlated flatfading MIMO channels 6] 8]. Similarly, directly applying OSBC to OFDM in correlated frequency-selective channels leads to substantial increase 9]. Precoding in OS- BC OFDM systems adapts to channel conditions and preprocesses signals at the subcarrier level such that OSBC designed for i.i.d. channels can also be used for correlated frequency-selective MIMO channels. Nevertheless, precoding for error-rate minimization in OSBC OFDM with spatial correlations has not been considered yet. In this paper, we develop linear precoding and non-linear omlinson-harashima precoding HP for OSBC OFDM in transmit-antenna and path-correlated frequency-selective channels to minimize the probability of error. With perfect CSI at the transmitter, a precoded system can achieve a significant capacity gain or reduction. However, the instantaneous and accurate CSI feedback is not realistic since the feedback capacity is usually very limited. Our proposed precoding approach only needs the channel statistical information correlation matrices to be available at the transmitter, i.e., the instantaneous values of the channel gains are not needed /06/$ IEEE

2 Since correlation matrices change at a much slower rate than the channel gains or even do not change at all, the covariance feedback requires much lower capacity. We assume that the receiver has perfect CSI and uses maximum likelihood ML decoding. We derive both linear and non-linear precoding using the minimum pair-wise error probability PEP criterion. he proposed precoding remarkably reduces the system in OSBC OFDM with path and transmit antenna correlations. Moreover, non-linear precoding outperforms linear precoding. II. SYSEM MODEL his section will introduce the system model of an N- subcarrier OFDM system with M transmit antennas and M R receive antennas in the presence of transmit antenna and path correlations. A. Path and ransmit Antenna Correlations We restrict our analysis to the downlink case, where correlations exist between the transmit antennas, and no correlations exist between receive antennas. Between the u-th transmit antenna and v-th receive antenna, a wideband frequencyselective fading channel with L resolvable paths is assumed. he l-th path gain is a zero-mean complex Gaussian random variable Rayleigh fading with variance σl 2, which can be represented by an M R M matrix hl with entries h u,v l, l. We assume that the channel gains remain constant over several OFDM symbol intervals. he channel gain vector is ] h = vec h0...vec hl 1, where vec denotes the vectorization operator 10]. According to the model in 5], the transmit antenna correlation matrix can be represented by R = E h h H ] = R P R I MR, 1 where is the Kronecker product, and R P is the L L path correlation matrix with the {m, n}th entry R P m, n =σ m σ n p m n e jθm,n, 0 <p 1 2 where p is the path correlation coefficient and the θ m,n is the phase of the path correlation between the m-th and the n-th path. If the paths between each transmit-receive antenna pair are uncorrelated, i.e., p =0,theR P = diag ] σ0 2...σL 1 2 is only defined by the power delay profiles. he R is the transmit antenna correlation matrix. From 10], the entries of R are R m, n =J 0 2π m n ζ, 3 where J 0 is zero-order Bessel function of the first kind and ζ = d λ ; λ = c/f c is the wavelength at the center frequency f c, is the angle of arrival spread, and the transmit antennas are spaced by d.asin7],them R LM tap gain matrix can be obtained as ] ] h0 hl 1 = hw R 1/2 ] P R = hw r P r, 4 where h w is an M R M L matrix of i.i.d zero mean complex Gaussian random variables with unit variance; r P = R P and r = R. B. ransmit Antenna and Path Correlations in OFDM At the receiver, the channel on the k-th subcarrier can be represented as L 1 Hk] = hle j 2π N kl. 5 l=0 With the l-th path gain matrix hl satisfying 4, 5 can be written as Hk] =h w r P Fk] r = hw rk], 6 where Fk] = e j 2π N k0...e j 2π kl 1] N is an L- dimensional vector; rk] =r P Fk] r is an M L M matrix. he k-th received signal vector in spatially correlated OFDM channels in which multiple paths are also correlated thus can be given by Yk] =Hk]Xk]+Wk], 7 where Yk] is an M R -dimensional vector and Xk] = X1 k]...x M k] ] is an input data vector; Xu k] denotes an M-ary quadrature amplitude modulation QAM symbol on the k-th subcarrier sent by the u-th transmit antenna. he Wk] is the noise vector where the entries W v k] = M u=1 W u,vk] are additive white Gaussian noise AWGN samples with zero mean and variance σ 2 W, and W u,vk], k, are assumed i.i.d. C. OSBC OFDM Space-time codes improve power efficiency by maximizing spatial diversity. An OSBC matrix is composed of linear combinations of constellation symbols and their conjugates, and encoding therefore only requires linear processing. he M code matrix for orthogonal SBC satisfies P C H C = c t 2 I M, 8 t=1 for all complex codewords c t. he transmission code rate R c is defined as P/, where P represents the number of symbols transmitted over time slots. OSBC can be directly employed in OFDM at a subcarrier level to offer full spatial diversity gain, if there is no correlation between transmit antennas or different paths. For example, the full-rate X1 k] X2 Alamouti-coded OFDM transmits k] X 2 k] X1 onto k] the subcarrier k, i.e., X 1 k] and X 2 k] are transmitted over the 1-st and 2-nd antenna at the first time slot, respectively; the X2 k] and X1 k] are transmitted in the following time slot. Full-rate complex orthogonal designs do not exist for more than two transmit antennas. he system performance of OSBC OFDM can be analyzed using PEP, which is the probability that a transmitted signal vector Xk] is erroneously decoded as a vector ˆXk]. We assume the ML decoder at the receiver uses the Euclidean distance decoding metric ˆXk] = arg min Yk] Xk] Hk]Xk] 2 F ; 9

3 where. F is the Frobenius norm. he PEP on the k-th subcarrier can be upper bounded by 9] P e Xk] ˆXk] exp Hk]Ø 2 F 2, 10 where Ø = Xk] ˆXk] is the codeword difference vector. As in 11], by taking the expectation of 10 over the channel statistics, the average PEP can be bounded by log P e M R log det Qk], 11 where Qk] = ØØH Rk] +I 2 M ; Rk] = r H k]rk] is an M M matrix. For an OSBC structure, ØØ H = di M is a diagonal matrix 2], where d is the distance between codewords in pair. he d min is the minimum distance over all pairs of the codewords and dominates the error probability exponent and hence can be considered an indicator of the system performance. Obviously, the worst PEP primarily depends on Rk], which consists of r P and r. Since the R P and transmit-antenna correlation matrix R are constant, minimizing the worst average PEP is equivalent to maximizing dmin J Q = log det Q min k] = log det 2 Rk]+I M. 12 III. PRECODING FOR OSBC OFDM WIH CORRELAIONS We first show the impact of path correlations on the OSBC OFDM system performance. Linear and non-linear precoders are then proposed to mitigate the performance degradation. A. Impact of Path Correlations he transmit antenna correlations will always degrade the performance in OSBC OFDM systems 9]. We now show the impact of path correlations on the system performance. o analyze the impact of r P, we separately decompose the two correlation matrices using singular value decomposition SVD as follows: r P Fk] =U P Γ P VP H r = U Γ V H 13, where U and V are M M unitary matrices. Γ is the singular value matrix of r ;itisanm M diagonal matrix with real, non-negative entries γ u, u =1,...,M, in descending order γ 1 γ 2 γ M 0. Since r P Fk] is an L 1 vector, U P is an L 1 vector and U H P U P =1; V P =1and Γ P is a rank-one matrix with the only entry γ Pk = F H k]r P Fk]. he matrix rk] in 6 therefore becomes rk] =r P Fk] r = U P γ Pk U Γ V H = γ Pk UP U Γ V H 14. Because U is a unitary matrix, Ũ = U P U is an LM M unitary matrix, i.e., ŨH Ũ = I M. he correlation matrix Rk] can thus be given by Rk] =r H k]rk] =γ 2 P k V Γ H ŨH ŨΓ V H = γ 2 P k V Γ 2 V H. 15 he performance degradation due to path correlations is shown in Fig. 1. here is no correlation between the transmit antennas, i.e., R = I M. We provide three groups of for different values of p at SNR=5 db, 10 db and 15 db, respectively. In each group, when θ m,n = 0, the monotonously increases as p grows. Compared with the of zero phase, the random phase can mitigate the impact of path correlations, especially at the high path-correlation p Zero Phase, SNR=5dB Random Phase, SNR=5dB Zero Phase, SNR=10dB Random Phase, SNR=10dB Zero Phase, SNR=15dB Random Phase, SNR=15dB p Fig. 1. as a function of the path correlation coefficient for different values of SNR for a 2 2 QPSK Alamouti-coded OFDM system. B. Optimal Precoding for OSBC OFDM with Correlations We design optimal precoding on a subcarrier basis to overcome the performance degradation due to path and transmit antenna correlations. Both linear and non-linear H precoding is considered. We first consider linear precoding. With an M M precoding matrix Ek] on the k-th subcarrier, the transmitted signal vector on the k-th subcarrier is Ek]Xk], instead of Xk]. hej Q in 12 becomes dmin Ek]E H k] J Q = log det Rk]+I M. 16 4σ 2 W Our optimal precoding matrix thus can be given by Ek] opt =arg max log det ξzk]rk]+i M, 17 trzk]=m where ξ = dmin, Zk] =Ek]E H k], and tr denotes the 2 trace of a matrix. Substituting 15 into 17 and applying the determinant identity, we have Ek] opt =arg max log det ξγp 2 trzk]=m k Γ V H Zk]V Γ + I M =arg max log det Γ Zk] Γ + I M, tr Zk]=ξM 18 where Γ = γ Pk Γ and Zk] =ξv HZk]V.hewaterfilling solution can be derived from 18 12]. he optimal main-diagonal entries in Zk] opt will then be Z kuu = µ γ 2 uu +, u =1,...,M, 19

4 where a + denotes maxa, 0; the parameter µ is chosen to satisfy tr Zk] = ξm. Hence, the optimal precoding matrix can be obtained by 1 Ek] opt = Zk] opt = ξ V Zk] opt V H. 20 he precoding is designed using the singular values of the transmit antenna correlation matrix and has the waterfilling solution. With the precoding matrix, the effective channel becomes Hk]Ek]. After reception the receiver performs ML decoding on the k-th subcarrier in an OSBC OFDM system. he proposed precoding only needs the correlation matrices r P and r, i.e, only covariance feedback is needed for our precoding design. C. Non-Linear Precoding Fig. 2. M-QAM Mapper ansmitter Channel Receiver W k] ak] Xk] ML Decoder with MOD H D MOD & Slicer _ B - I omlinson-harashima precoding in OSBC OFDM. ^ a k ] M-QAM Demapper In this subsection, we propose a non-linear H precoder. he structure of the proposed precoder is illustrated in Fig. 2. he receiver side consists of a diagonal scaling matrix Pk], an ML decoder and a modulo arithmetic device. he transmitter side includes a modulo arithmetic feedback structure employing the matrix Bk], by which the transmitted symbols Xk] are successively calculated for the data symbols ak] drawn from the initial signal constellation. Without the modulo device, the feedback structure is equivalent to B 1 k], which can be optimally designed as in 20, Bk] opt = E 1 k] opt. he effective channel is Hk]Ek] opt and ML decoding is used at the receiver. he diagonal scaling matrix P is to keep the average transmit power constant. HP employs modulo operation at both the transmitter and the receiver. he modulo 2 M reduction at the transmitter, which is applied separately to the real and imaginary parts of the input, is to restrict the transmitted signals into the boundary of R{Xk]} M, ] M and I{Xk]} M, ] M. If the input sequence ak] is a sequence of i.i.d. samples, the output of the modulo device is also a sequence of i.i.d. random variables, and the real and imaginary parts are independent, i.e., we can assume E Xk]X H k] ] = E s I M, k 13]. At the receiver, the filtered noise vector becomes W = PW, where the k-th entry W k] has individual variance σw 2 k.a slicer, which applies the same modulo operation as that at the transmitter, is used. After the ML decoding and discarding the modulo congruence, the unique estimates of the data symbols âk] can be generated. he details of HP operation are further discussed in 13]. In our proposed linear and non-linear precoder, the transmitter does not require explicit channel gain information and only the channel correlation matrices are delivered to it. Since the correlation matrices may change much slower than the channel response or even may not change at all, the covariance feedback significantly reduces the feedback load. In most applications, transmit antenna spacing can be estimated at the transmitter, i.e., no feedback for R is needed. he feedback requirement can hence be further reduced. Furthermore, due to the non-linear property, HP avoids power efficiency loss present in linear precoding. herefore, low can be expected for the proposed non-linear precoder. IV. SIMULAION RESULS In this section, our simulation results show how the proposed linear and non-linear precoders improve the system performance in OSBC OFDM with path and transmit-antenna correlations. he transmitter knows only the correlation matrices R and R P with ζ = d λ and path correlation coefficient p, respectively; the phase correlation coefficients θ m,n in 2 are assumed zero, m, n. We assume the angle of arrival spread is 12, i.e., 0.2. Perfect channel information is assumed to be available only at the receiver and ML decoding is used. A. Flat-Fading MIMO Channels We first consider the special case with N =1and L =1, where the channel model 6 is reduced to a flat-fading MIMO channel. Only transmit antenna correlations need to be considered. he s of 16-QAM 4 2 systems with ζ =0.25 and 4 4 systems with ζ =0.5 are shown in Fig. 3. he transmission rate R c of the orthogonal SBC matrix is 1/2 as in 2]. he for uncorrelated MIMO channels is shown as reference. Evidently, in an OSBC MIMO system, the transmit-antenna correlations significantly degrade the performance. As transmit-antenna correlation becomes high ζ decreases, the degradation becomes severe. Both the linear and non-linear H precoders mitigate the detrimental impact of the antenna correlations. he H precoder almost completely eliminates the degradation due to correlations. At a of, the linear precoding obtains 0.5 db gain in 4 4 systems and 1 db gain in 4 2 systems; HP achieves gains of 0.8 db and 1.8 db, respectively. B. OSBC OFDM We now consider 64-subcarrier QPSK OSBC OFDM. he vehicular B channel specified by IU-R M ] is used where the channel taps are zero-mean complex Gaussian random processes with variances of 4.9 db, 2.4 db, 15.2 db, 12.4 db, 27.6 db, and 18.4 db relative to the total power. In Fig. 4, 2 2 and 2 4 Alamouti-coded OFDM systems are considered. he paths are uncorrelated, i.e., p =0and ζ = Similarly, both linear and non-linear precoding suppress the increase due to transmit-antenna correlations. Nonlinear H precoding outperforms linear precoding. In Fig. 5, we assume the path correlation coefficient p = 0.9. heζ =0.25 and ζ =0.5 are considered. he is substantially degraded due to path correlations. Both the linear and non-linear precoders mitigate the impact of correlations.

5 Correlated, p=0, HP Correlated, p=0, LP Correlated, p=0, NoP 10 0 =0.25, NoP =0.25, LP =0.25, HP 4 2 MIMO, ζ =0.25 =0.5, NoP =0.5, LP =0.5, HP 4 4 MIMO, ζ = E /N db b E /N db b 0 Fig. 3. with linear precoding LP, HP and no precoding NoP as a function of the SNR for different values of the normalized transmit antenna spacing for 4 2 and QAM 1/2-rate OSBC systems. Fig. 5. with linear precoding LP, HP and no precoding NoP as a function of the SNR for different values of the path correlation coefficient and the normalized transmit antenna spacing for 2 2 QPSK Alamouti-coded OFDM systems. 2 4 OFDM 2 2 OFDM Correlated, p=0, HP Correlated, p=0, LP Correlated, p=0, NoP E b /N 0 db Fig. 4. with linear precoding LP, HP and no precoding NoP as a function of the SNR for 2 2 and 2 4 QPSK Alamouti-coded OFDM systems, ζ =0.25. V. CONCLUSIONS We have derived PEP-optimal linear and non-linear precoding with only covariance feedback for OSBC OFDM systems in the presence of transmit-antenna and path correlations. Not only are the feedback requirements reduced, since our precoding needs only the statistical knowledge of the channel at the transmitter, but also the system is reduced in transmit-antenna and path-correlated channels. he proposed non-linear precoding outperforms linear precoding. ACKNOWLEDGEMEN he authors gratefully acknowledge funding for this work provided by the Natural Sciences and Engineering Research Council NSERC of Canada, Alberta Informatics Circle of Research Excellence icore, RLabs, and Rohit Sharma Professorship. REFERENCES 1] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Select. Areas Commun., vol. 16, no. 8, pp , Oct ] V. arokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE rans. Inform. heory, vol. 45, no. 5, pp , July ] elecommunications Industry Association, IA/EIA Physical Layer Standard for cdma2000 Spread Spectrum Systems, Revision C. IA/EIA/IS ] 3rd Generation Partnership Project, echnical Specification Group Radio Access Network: Physical Channels and Mapping of ransport Channels onto Physical Channels FDD release v.3.2.0, ] E. Yoon, J. Hansen, and A. J. Paulraj, Space-frequency precoding for an OFDM based system exploiting spatial and path correlation, in Proc. IEEE Globecom 04, vol. 1, Dallas, X, Nov. 2004, pp ] Y. Zhao, R. Adve, and. J. Lim, Precoding of orthogonal SBC with channel covariance feedback for minimum error probability, in Proc. IEEE PIMRC 04, vol. 1, Barcelona, Spain, Sept. 2004, pp ] M. Vu and A. J. Paulraj, Linear precoding for MIMO channels with non-zero mean and transmit correlation in orthogonal space-time coded systems, in Proc. IEEE VC 04-Fall, vol. 4, Los Angeles, CA, Sept. 2004, pp ], Linear precoding for MIMO wireless correlated channels with non-zero means: K factor analysis, extension to non-orthogonal SBC, in Proc. IEEE ICASSP 05, vol. 3, Philadelphia, PA, Mar. 2005, pp ] H. Bölcskei, M. Borgmann, and A. J. Paulraj, Impact of the propagation environment on the performance of space-frequency coded MIMO- OFDM, IEEE J. Select. Areas Commun., vol. 21, no. 3, pp , Apr ] D. Shiu, G. J. Foschini, M. J. Gans and J. M. Kahn, Fading correlation and its effect on the capacity of multi-element antenna systems, IEEE rans. Commun., vol. 48, no. 3, pp , Mar ] G. Jöngren, M. Skoglund, and B. Ottersten, Combining beamforming and orthogonal space-time block coding, IEEE rans. Inform. heory, vol. 48, no. 3, pp , Mar ] İ. E. elatar, Capacity of multi-antenna Gaussian channels, Eur. rans. elecomm. E, vol. 10, no. 6, pp , Nov ] R. F. H. Fischer, Precoding and Signal Shaping for Digital ransmission. New York: Wiley, ] International elecommunication Union, Recommendation IU-R M. 1225, Guidelines for Evaluation of Radio ransmission echnologies for IM-2000, Feb