This paper establishes the exact distribution of the signal-to-interference-plus-noise ratio (SINR) under matched-filter (MF) precoding. Specifically, we derive the exact expressions for the cumulative distribution function (CDF) and the probability density function (PDF) of SINR under MF precoding over Rayleigh fading channels. Based on our exact analysis, we then rigorously prove that the SINR converges to some specific distributions separately in high SNR and in the large MIMO regimes. To simplify the exact result in general cases, we provide a simple approximation based on modelling the interference to follow the Beta distribution. We then shift to the exact analysis of the communication rate, and answer the fundamental question of how the exact rate converges to the well-known asymptotic rate in massive MIMO. After that, we propose a novel approximation for the ergodic rate, which is shown to outperform various existing approximations. Finally, we present some numerical results to demonstrate the accuracy of the derived analytical models.

Index Terms-Matched-filter precoding, and multi-user MIMO

EURECOM's research is partially supported by its industrial members: ORANGE, BMW, SAP, iABG, Norton LifeLock, and by the projects EEMW4FIX (French ANR), CONVERGE (EU H2030), and ERC-PoC Project LIGHT (Grant 101101031).

1 Notations: || • || denotes the norm-2 of a vector, while | • | denotes the magnitude of a complex number. For a matrix A, we use A H , A * and A T to denote its conjugate transpose, conjugate part and non-conjugate transpose respectively. Exp(•), Gamma(•, •), Inv-Gamma(•, •), Beta(•, •) and CN (•, •) denote the exponential distribution, the Gamma distribution, the inverse Gamma distribution, the Beta distribution and the complex Gaussian distribution respectively.