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 the Nr 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 over 8Nr 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 than 6 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.