Variable-Length Channel Quantizers for Maximum Diversity and Array Gains

Abstract: We consider a $t \times 1$ multiple-antenna fading channel with quantized channel state information at the transmitter (CSIT). Our goal is to maximize the diversity and array gains that are associated with the symbol error rate (SER) performance of the system. It is well-known that for both beamforming and precoding strategies, finite-rate fixed-length quantizers (FLQs) cannot achieve the full-CSIT diversity and array gains. In this work, for any function $f(P)\in\omega(1)$, we construct variable-length quantizers (VLQs) that can achieve these full-CSIT gains with rates $1+(f(P) \log P)/P$ and $1+f(P)/P^t$ for the beamforming and precoding strategies, respectively, where $P$ is the power constraint of the transmitter. We also show that these rates are the best possible up to $o(1)$ multipliers in their $P$-dependent terms. In particular, although the full-CSIT SER is not achievable at any (even infinite) feedback rate, the full-CSIT diversity and array gains can be achieved with a feedback rate of 1 bit per channel state asymptotically.