Central limit theorem for signal-to-interference ratio of reduced rank linear receiver

TL;DR

This paper proves a central limit theorem for the signal-to-interference ratio in reduced rank linear receivers, showing asymptotic Gaussian behavior and improving existing theorems in random matrix theory.

Contribution

It introduces a new CLT for the SIR of multistage Wiener receivers and spectral statistics, extending and refining prior results in random matrix theory.

Findings

Asymptotic Gaussian distribution of SIR and mutual information.

Central limit theorem for spectral statistics of sample covariance matrices.

Improved bounds and results compared to previous theorems.

Abstract

Let $\mathbf{s}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with $\{v_{ik},i,k=1,...\}$ independent and identically distributed complex random variables. Write $\mathbf{S}_k=(\mathbf{s}_1,...,\mathbf {s}_{k-1},\mathbf{s}_{k+1},... ,\mathbf{s}_K),$ $\mathbf{P}_k=\operatorname {diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$, $\mathbf{R}_k=(\mathbf{S}_k\mathbf{P}_k\mathbf{S}_k^*+\sigma ^2\mathbf{I})$ and $\mathbf{A}_{km}=[\mathbf{s}_k,\mathbf{R}_k\mathbf{s}_k,... ,\mathbf{R}_k^{m-1}\mathbf{s}_k]$. Define $\beta_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\ mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k$, referred to as the signal-to-interference ratio (SIR) of user $k$ under the multistage Wiener (MSW) receiver in a wireless communication system. It is proved that the output SIR under the MSW and the mutual information statistic under the matched filter (MF) are both asymptotic Gaussian when $N/K\to c>0$. Moreover, we provide a central limit theorem for linear spectral statistics of eigenvalues and eigenvectors of sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also improve Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605].