Distributed Transmit Beamforming Over Rician Fading: Design and Analysis of Nonuniform Codebook for Feedback Quantization

Distributed transmit beamforming (DTB) enhances the transmit diversity and energy efficiency in wireless sensor networks. To implement DTB, one key issue is that the fusion center should feed back each sensor the estimated channel phase information (CPI), based on which each sensor can accordingly preprocess the transmit signal phase to coordinate in transmitting a common message to the fusion center. Because of feedback overhead and link rate constraint, codebook-based quantization methods are usually exploited to provide such CPI. However, how to effectively construct the optimal codebook over the general Rician fading channel model is not well understood in the current literature, not to mention its related performance analysis. Motivated by this research gap, in this article, 1) we first develop a low-complexity iterative algorithm to construct the optimal codebook; 2) we then reveal that in the designed codebook, the codewords are not uniformly distributed. More specifically, as the absolute value of the codeword increases, the distance between adjacent codewords increases gradually; 3) based on 1) and 2), we further derive the exact probability density function (PDF) of the residual phase error resulting from feedback quantization; 4) finally, we exploit this PDF and also the Laguerre series approximation to characterize, in closed form, the very tight approximation for the receiving signal-to-noise ratio distribution at the fusion center, which is then exploited for performance analysis of outage probability, diversity order, and ergodic capacity. Numerical results show that our designed codebook can achieve higher performance gains compared with the widely used equidistance phase-shift-keying-based codebook.