Asymptotic distributions of the signal-to-interference ratios of LMMSE detection in multiuser communications

Abstract

Let $\mathbf{s}_{k}=\frac{1}{\sqrt{N}}(v_{1k},\ldots,v_{Nk})^{T}$, k=1, …, K, where {v_ik, i, k=1, …} are independent and identically distributed random variables with Ev₁₁=0 and Ev₁₁²=1. Let S_k=(s₁, …, s_k−1, s_k+1, …, s_K), P_k=diag (p₁, …, p_k−1, p_k+1, …, p_K) and β_k=p_ks_k^T(S_kP_kS_k^T+σ²I)⁻¹s_k, where p_k≥0 and the β_k is referred to as the signal-to-interference ratio (SIR) of user k with linear minimum mean-square error (LMMSE) detection in wireless communications. The joint distribution of the SIRs for a finite number of users and the empirical distribution of all users’ SIRs are both investigated in this paper when K and N tend to infinity with the limit of their ratio being positive constant. Moreover, the sum of the SIRs of all users, after subtracting a proper value, is shown to have a Gaussian limit.