BER criterion and codebook construction for finite-rate precoded spatial multiplexing with linear receivers

Precoded spatial multiplexing systems with rate-limited feedback have been studied recently based on various precoder selection criteria. Instead of those based on indirect performance indicators, we in this paper propose a new criterion directly based on the exact bit error rate (BER) that is applicable to systems with linear receivers and rectangular/square quadrature-amplitude-modulation constellations. The BER criterion outperforms any other alternative in terms of optimizing the BER performance for an uncoded system with linear receivers. We then develop a precoder codebook construction method based on the generalized Lloyd algorithm from the vector quantization literature. This construction is not directly based on the BER criterion. Hence, it is suboptimal in the BER sense. However, relative to those currently available, our newfound codebooks improve considerably various minimum distances between any pair of codewords of the codebook. Finally, we analyze the BER-optimal precoder in the asymptotic case with infinite-rate feedback that amounts to perfect channel knowledge at the transmitter. The infinite-rate optimal precoder based on the BER criterion is drastically different from the counterparts with other criteria, and it leads to a benchmark performance for finite-rate precoded spatial multiplexing systems. We observe from numerical results that the BER performance of finite-rate feedback with suboptimal codebooks approaches quickly the benchmark performance of infinite-rate feedback. This suggests that i) the number of feedback bits in practical systems need not be large and ii) the room for performance improvement via further codebook optimization shrinks quickly as the codebook size increases.