A Central Limit Theorem for the SNR at the Wiener Filter Output for Large Dimensional Signals

TL;DR

This paper proves a central limit theorem for the Signal-to-Noise Ratio at the Wiener filter output in large-dimensional signal models, showing it converges to a Gaussian distribution with a specific variance.

Contribution

It establishes a Gaussian approximation for the SNR quadratic form in high-dimensional settings, extending random matrix theory results to practical signal processing metrics.

Findings

The SNR quadratic form converges to a Gaussian distribution as dimensions grow.

The variance of the Gaussian approximation is explicitly derived.

The results apply to wireless communications and array processing scenarios.

Abstract

Consider the quadratic form $\beta = {\bf y}^* ({\bf YY}^* + \rho {\bf I})^{-1} {\bf y}$ where $\rho$ is a positive number, where ${\bf y}$ is a random vector and ${\bf Y}$ is a $N \times K$ random matrix both having independent elements with different variances, and where ${\bf y}$ and ${\bf Y}$ are independent. Such quadratic forms represent the Signal to Noise Ratio at the output of the linear Wiener receiver for multi dimensional signals frequently encountered in wireless communications and in array processing. Using well known results of Random Matrix Theory, the quadratic form $\beta$ can be approximated with a known deterministic real number $\bar\beta_K$ in the asymptotic regime where $K\to\infty$ and $K/N \to \alpha > 0$. This paper addresses the problem of convergence of $\beta$. More specifically, it is shown here that $\sqrt{K}(\beta - \bar\beta_K)$ behaves for large $K$ like a Gaussian random variable which variance is provided.