Partially Coherent Multiple Antenna Systems: Design Criterion and Construction Methods for Constellations and Coded Modulation

We consider multiple-antenna communication systems in Rayleigh fading channel, where the transmitter does not know the channel coefficients and the receiver has only an estimate of them. We further assume that the transmitter and receiver know the statistics of the estimation error. We refer to this system as partially coherent system, for which we derive the expressions for the optimal detector and study the code and constellation design problems. Since the Chernoff bound on pairwise error probability of the partially coherent systems appears to be intractable, we use Stein’s Lemma to propose a design criterion based on Kullback-Leibler (KL) distance between conditional distributions. We show that the KL distance is relatively easy to derive and work with, and when used as a design criterion, results in codes and constellations that significantly improve the performance of the partially coherent systems. Using the KL-based design criterion, we construct constellations for multiple-antenna systems which can be demodulated in the presence of channel estimation errors. Such constellations are suitable for scenarios where long preambles or training sequences are either impossible, e.g., fading channels with short coherence intervals; or impractical, e.g., applications that require transmission of short control packets (such as RTS and CTS in IEEE 802.11). The proposed constellations are multi-level, with The material in this paper was presented in part at the Allerton Conference on Communications, Control and Computing, Monticello, IL, October 2002, and at the Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, November 2003. August 25, 2004 DRAFT multi-dimensional spherical constellations at each level. For the constituent spherical subsets of the partially coherent constellations, we use a previously proposed recursive construction, which results in a systematic design, and in some cases, low complexity decoding algorithm. The new constellations provide significant performance improvement over the conventional single-antenna PSK and QAM constellations and multiple-antenna techniques such as Bell Lab’s Space-Time (BLAST) architecture and Orthogonal Transmit Diversity (OTD) schemes, when the estimation variance is comparable to the reciprocal of the signal-to-noise ratio. Next, we consider coded systems, and derive a KL-based design criterion for partially coherent coded modulation. We also propose a construction method for partially coherent coded modulation using the idea of mapping by set partitioning. We show that, in the presence of channel estimation errors, the proposed codes provide substantial performance improvement over the conventional coded modulation techniques, and the gains in this case are even larger than the gains obtained in the case of uncoded systems.