Vector Precoding for Wireless MIMO Systems: A Replica Analysis

We apply the replica method to analyze vector precoding, a method to reduce transmit power in antenna array communications. The analysis applies to a very general class of channel matrices. The statistics of the channel matrix enter the transmitted energy per symbol via its R-transform. We find that vector precoding performs much better for complex than for real alphabets. As a byproduct, we find a nonlinear precoding method with polynomial complexity that outperforms NP-hard Tomlinson-Harashima precoding for binary modulation on complex channels if the number of transmit antennas is slightly larger than twice the number of receive antennas.

💡 Research Summary

The paper presents a rigorous statistical‑physics based analysis of vector precoding for wireless multiple‑input multiple‑output (MIMO) systems, using the replica method to evaluate the average transmit energy per symbol in the large‑system limit (number of transmit antennas Nₜ → ∞). The authors start by formulating the precoding problem: a data vector s drawn from a finite alphabet A is transformed by a nonlinear mapping f(·) before transmission through a channel matrix H, yielding the received vector y = H f(s) + z, where z is additive Gaussian noise. The goal of vector precoding is to choose f(·) so that the average transmit power P = E