Very Low-Rate Variable-Length Channel Quantization for Minimum Outage Probability

We identify a practical vector quantizer design problem where any fixed-length quantizer (FLQ) yields non-zero distortion at any finite rate, while there is a variable-length quantizer (VLQ) that can achieve zero distortion with arbitrarily low rate. The problem arises in a $t \times 1$ multiple-antenna fading channel where we would like to minimize the channel outage probability by employing beamforming via quantized channel state information at the transmitter (CSIT). It is well-known that in such a scenario, finite-rate FLQs cannot achieve the full-CSIT (zero distortion) outage performance. We construct VLQs that can achieve the full-CSIT performance with finite rate. In particular, with $P$ denoting the power constraint of the transmitter, we show that the necessary and sufficient VLQ rate that guarantees the full-CSIT performance is $\Theta(1/P)$. We also discuss several extensions (e.g. to precoding) of this result.