Exact BER analysis for Alamouti's code on arbitrary fading channels with imperfect channel estimation

In this contribution, we investigate the effect of imperfect channel estimation on the bit error rate (BER) performance of multiple-input multiple-output (MIMO) systems employing Alamouti’s code. The propagation channels from both transmit antennas to each of theNr receive antennas are assumed to be affected by (possibly correlated) flat block fading with an arbitrary fading distribution. The transmitted symbols belong to a pulse amplitude modulation (PAM) or quadrature amplitude modulation (QAM) constellation. The mismatched maximumlikelihood (ML) receiver makes use of a least-squares or linear minimum mean-square error (MMSE) channel estimate, obtained from known pilot symbols sent among the data. The resulting average BER for QAM transmission can easily be written as an expectation over8Nr real-valued random variables, but the computing time needed for its numerical evaluation increases dramatically with Nr. We point out that the BER can be expressed in terms of the distribution of the Frobenius norm of the channel matrix, rather than the joint distribution of all channel coefficients. This allows to reduce the BER expression for any number of receive antennas to an expectation over only 6 random variables, which can easily be computed numerically. Moreover, we show that for real-valued constellations and/or realvalued channels, the BER expression reduces to an expectation over less than6 variables. For BER levels of practical interest, the numerical evaluation of the BER is much less time-consuming than a straightforward computer simulation. The presented BER expression is useful not only when the fading distribution is given in closed form, but also when only experimental data (e.g. a histogram) on the fading are available. I. I NTRODUCTION The performance of wireless communication systems is strongly affected by small-scale fading. To combat this det rimental phenomenon, several diversity techniques have been proposed to provide the receiver with independent replicas of the signal. Spatial diversity is achieved by using multip le antennas at the transmitter and/or receiver side. These socalled multiple-input multiple-output (MIMO) communicat ion systems can achieve a maximum diversity order of NtNr (with Nt and Nr denoting the number of transmit and receive antennas, respectively), provided that proper space -time coding is used. In 1998, Alamouti invented a simple coding scheme for data transmission using two transmit antennas [1 ], with the remarkable benefit that the maximum-likelihood (ML ) decoding algorithm reduces to symbol-by-symbol detection , based only on linear processing at the receiver. Tarokh et al. generalized Alamouti’s scheme to an arbitrary number of transmit antennas by introducing the concept of orthogonal space-time block coding [17], [18]. As the orthogonal space time block codes (OSTBCs) achieve full spatial diversity, a nd require only linear processing at the receiver, these codes are a very attractive transmit diversity technique. Under t he assumption of perfect channel knowledge (PCK), the bit error rate (BER) performance of OSTBCs has been studied extensively in e.g. [4], [9], [10] and [11]. In practical wireless applications, however, the receiver has to estimate the channel response, which inevitably results in a performance penalty. Most often, the investigation of the resulting performance has been carried out under the assumption of Rayleigh fading: an analytical expression fo r the BER of OSTBCs in case of minimum mean-square error (MMSE) channel estimation was derived in [8]; in [15], analytical BER expressions as well as the tight Chernoff bou nd were given for orthogonal space-time block coded systems employing M -ary phase-shift keying ( M -PSK) modulation; high-SNR expressions for the pairwise error probability (P EP) were derived in [2] for quite general STBCs with coherent and noncoherent receivers, using an eigenvalue approach; i n [5], an exact closed-form expression for the PEP of both orthogonal and non-orthogonal space-time codes in the case of least-squares channel estimation was obtained by means of characteristic functions. However, from the PEP one can compute only an upper bound on the BER, which in a fading environment does not converge to the true BER at high SNR. In [6], expressions for the exact decoding error probabilit y (DEP) were presented for the case of square OSTBCs on arbitrary fading channels, but the analysis is restricted t o PSK constellations. In this contribution, we provide an exact BER analysis for Alamouti’s code with pulse amplitude modulation (PAM) or quadrature amplitude modulation (QAM) constellations on a rbitrary flat-fading channels with imperfect channel estima on (ICE). In section II we describe the observation model which includes the2Nr arbitrary fading channels. Section III presents a linear pilot-based channel estimation method (with the we llknown least-squares estimation and linear MMSE estimation as special cases), and derives the statistical properties o f the channel estimate. The mismatched ML receiver is briefly outlined in section IV. The main part of our contribution is in section V, where the exact BER expression is reduced to an expectation over (at most) 6 variables, that allows numerical evaluation with a computing time that is independen t of the number ( Nr) of receive antennas. In section VI the results obtained from the numerical evaluation of our BER expression are confirmed by computer simulations, assuming i.i. d. Nakagamim fading channels. Finally, conclusions are drawn in section VII. The main conclusion is that the present ed BER expression can be used when the fading distribution is available in closed form or as an experimentally obtained histogram, and allows for a faster evaluation than by means of straightforward computer simulation, for practical val ues of the operating BER. Throughout this paper, the superscripts T and H represent the vector (matrix) transpose and conjugate transpose , respectively.R{x}, I{x} and E[x] denote the real part, the imaginary part and the expected value of x, respectively, while ‖X‖ refers to the Frobenius norm of X. II. SIGNAL MODEL We consider a MIMO wireless communication system with 2 transmit antennas and Nr receive antennas. The propagation channels between both transmit antennas and each of the receive antennas are affected by flat fading with an arbitrar y distribution. Transmission is organized in frames: in one frame, each transmit antenna sends Kp known pilot symbols andK coded data symbols; the pilot symbols enable channel estimation at the receiver. Within one frame of Nfr = K +Kp symbols, the channel is assumed to be constant (block fading ). The Nr × Nfr received signal matrixRtot is given by Rtot = [Rp,R] = H [Ap,A] + W, (1) where the2×Kp pilot matrix Ap and the2×K data matrix A consist of the pilot symbols and the coded data symbols, respectively, transmitted at each transmit antenna. The pr opagation channel is represented by the Nr × 2 complex random matrix H. We consider an arbitrary joint pdf p(H) of the (possibly correlated) 2Nr complex fading gains. The Nr×Nfr matrix W represents additive spatially and temporally white noise and consists of i.i. d. zero-mean circularly symmetri c complex Gaussian (ZMCSCG) random variables with variance N0. Alamouti’s space-time block code [1], that has been designed for two transmit antennas, transforms two informati on symbolss1(k) and s2(k) into a 2 × 2 coded symbol matrix C(k), given by C(k) = [ s1(k) −s2(k) s2(k) s ∗ 1(k) ] . (2) Hence, assuming that an even number K of information symbols is sent, the transmitted data symbol matrix is given by A = √ Es [C(1), . ,C(K/2)]. Considering a normalized information symbol constellation ( E[|si|2] = 1), it follows that the average energy of the transmitted coded symbols is given by Es: 1 2K E [