Performance analysis of RVQ-based limited feedback beamforming codebooks

Codebooks based on Random Vector Quantization (RVQ) are popular in limited feedback beamforming applications over MIMO channels because of their low-complexity design properties. The goal of this work is on understanding the performance of an ensemble of RVQ codebooks as a function of the number of bits of feedback (B), antenna dimensions, and spatial correlation. We analyze the case of correlated MIMO channels and arbitrary choice of B. Towards this goal, we first study the distribution function of weighted norms of isotropically distributed beamforming vectors. From this, we compute the received SNR loss and mutual information loss of a B-bit RVQ scheme relative to a perfect channel information benchmark. Our computation reveals the following: i) The loss terms are a product of two factors. The first factor, which is also common to analysis of i.i.d, channels, decays as B increases at the rate 2-B/(N_t-1) where N_t is the number of transmit antennas; ii) The second factor reflects the condition number of the channel. A channel that minimizes/maximizes the condition number on average also minimizes/maximizes the performance loss, respectively. Such behavior is typical of channels that correspond to rich (i.i.d.) spatial scattering, and poor (rank-l channels) spatial scattering, respectively.