SINR and Multiuser Efficiency Gap between MIMO Linear Receivers

Due to their low complexity, Minimum Mean Squared Error (MMSE) and Zero-Forcing (ZF) emerge as two appealing MIMO receivers. Although, as the signal-to-noise ratio (SNR) grows large, they provide asymptotically the same achievable rate, a non-vanishing gap between the {color{blue}signal to interference and noise ratio (SINR)} obtained through the two receivers exists, affecting the error and outage probability, as well as the multiuser efficiency. Interestingly, both the SINR and the multiuser efficiency gaps can be compactly expressed as quadratic forms of random matrices, with a kernel that depends solely on the statisticsof the interfering streams. By leveraging cite{Aris_quadratic}, we derive the closed-form distribution of such indefinite quadratic forms with random kernel matrix, which turns out to be proportional to the determinant of a matrix containing the system parameters. Then, specializing our result to different fading conditions, we obtain the closed-form statistics of both the SINR gap and the multiuser efficiency gap. Although the focus of this work is on the finite-size statistics, for completeness we also provide some results on the doubly-massive MIMO case. We validate all our derivations through extensive Monte Carlo simulations.