LSE Precoders for Massive MIMO with Signal Constraints: Fundamental Limits
This paper proposes the nonlinear Least Square Error (LSE) precoders for multiuser MIMO broadcast channels. The output signals of LSE Precoders are limited to be chosen from a predefined set which let these precoders address several constraints such as peak power limitation, constant envelope transmission and discrete constellations. We study the large-system performance of these precoders via the replica method from statistical physics, and derive a closed-form expression for the asymptotic distortion. Our results demonstrate that an LSE precoder with the output peak-to-average power ratio of $3~{\rm dB}$ can track the performance of the Regularized Zero Forcing (RZF) precoder closely. As the peak-to-average power ratio reduces to one, the constant envelope precoder is recovered. The investigations depict that the performance of the RZF precoder is achieved by the constant envelope precoder with $20\%$ of more transmit antennas. For $M$-PSK constellations, our analysis gives a lower-bound on the asymptotic distortion which is tight for moderate antenna-to-user ratios and deviates as the ratio grows. We improve this bound by deriving the replica solution under one-step of replica symmetry breaking. Our numerical investigations for this case show that the bound is tight for antenna-to-user ratios less than $5$.