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

• Published in: arXiv.org
• Main Authors: Pan, G M, Zhou, W
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 17.06.2008
• Subjects: Communications systems; Complex variables; Covariance matrix; Eigenvalues; Eigenvectors; Interference; Linear receivers; Matched filters; Mathematical analysis; Mathematics - Probability; Matrix methods; Random variables; Theorems; Wireless communications
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0806.2768

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].