Variable-Length Limited Feedback Beamforming in Multiple-Antenna Fading Channels

We study a multiple-input single-output fading channel, where we would like to minimize the channel outage probability or symbol error rate (SER) by employing beamforming via quantized channel state information at the transmitter (CSIT). We consider a variable-length limited feedback scheme where the quantized CSIT is acquired through feedback binary codewords of possibly different lengths. We design and analyze the performance of the associated variable-length quantizers (VLQs) and compare their performance with the previously studied fixed-length quantizers (FLQs). For the outage probability performance measure, we construct VLQs that can achieve the full-CSIT performance with finite rate. Moreover, as the signal-to-noise ratio P tends to infinity, we show that VLQs can achieve the full-CSIT outage probability performance with asymptotically zero feedback rate. For the SER performance measure, we show that while the SER with full-CSIT is not achievable at any finite feedback rate, the diversity and array gains with full-CSIT can be achieved using VLQs with asymptotically zero feedback rate as P → ∞. Our results show that VLQs can significantly improve upon the traditional FLQs that require infinite feedback rate to achieve the outage probability or the diversity and array gains with full-CSIT.