Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples (0902.4250v1)

Abstract: The detection problem in statistical signal processing can be succinctly formulated: Given m (possibly) signal bearing, n-dimensional signal-plus-noise snapshot vectors (samples) and N statistically independent n-dimensional noise-only snapshot vectors, can one reliably infer the presence of a signal? This problem arises in the context of applications as diverse as radar, sonar, wireless communications, bioinformatics, and machine learning and is the critical first step in the subsequent signal parameter estimation phase. The signal detection problem can be naturally posed in terms of the sample generalized eigenvalues. The sample generalized eigenvalues correspond to the eigenvalues of the matrix formed by "whitening" the signal-plus-noise sample covariance matrix with the noise-only sample covariance matrix. In this article we prove a fundamental asymptotic limit of sample generalized eigenvalue based detection of signals in arbitrarily colored noise when there are relatively few signal bearing and noise-only samples. Numerical simulations highlight the accuracy of our analytical prediction and permit us to extend our heuristic definition of the effective number of identifiable signals in colored noise. We discuss implications of our result for the detection of weak and/or closely spaced signals in sensor array processing, abrupt change detection in sensor networks, and clustering methodologies in machine learning.

• The paper presents a novel random matrix theory-based approach that defines the asymptotic limits for detecting signals using sample generalized eigenvalues.
• It applies a statistical threshold, derived from the Tracy-Widom law, to differentiate signal eigenvalues from noise under limited sample conditions.
• The findings guide practical sensor array design and dimensionality reduction strategies in radar, sonar, and wireless communications.

Fundamental Limits of Signal Detection Using Sample Generalized Eigenvalues

The paper Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples by Raj Rao Nadakuditi and Jack W. Silverstein explores the statistical signal processing problem of detecting signals in noise using sample generalized eigenvalues under conditions of limited signal-bearing and noise-only samples. The authors explore the theoretical limits of detection, particularly in high-dimensional settings, introducing a method based on random matrix theory to reliably detect signals close to critical signal-to-noise ratios (SNRs).

Overview and Contributions

Signal detection is vital in diverse applications such as radar, sonar, and wireless communications. This research examines the detection problem framed in terms of sample generalized eigenvalues, which involve whitening the signal-plus-noise covariance matrix against a noise-only covariance matrix. This approach raises questions about the reliability of signal detection when signal-bearing snapshots and noise-only snapshots are relatively few.

The authors prove the fundamental asymptotic limit of sample generalized eigenvalue detection in arbitrarily colored noise and establish conditions under which reliable detection is unattainable. A core result of the paper is the introduction of an algorithm based on random matrix theory that enhances detection reliability when the eigen-SNR exceeds a critical threshold. The threshold is influenced by dimensions such as the number of noise-only and signal-plus-noise samples.

Key Results and Algorithm

A significant numerical result presented is that when the eigen-SNR is above a determined critical value, reliable detection of the signal is possible using the proposed algorithm. The algorithm evaluates the eigenvalues of the sample covariance matrices to estimate the model order, offering practical insights into signal presence across applications.

The algorithm applies a statistical threshold to distinguish signal eigenvalues from noise eigenvalues based on the Tracy-Widom law, which approximates the distribution of the largest eigenvalue of the multivariate F matrix in high dimensionality. This threshold is calculated as a function of both the number of signal-plus-noise samples and noise-only snapshots, allowing practitioners to gauge the effective number of identifiable signals.

Implications and Future Work

This paper’s findings have theoretical and practical implications. The introduction of this statistical threshold clarifies the conditions under which signals can be reliably identified, thereby influencing sensor array design and deployment strategies in fields like sensor networks and machine learning. The results can guide efforts to optimize system parameters or develop strategies for dimensionality reduction in signal processing.

The authors suggest potential applications beyond direct signal detection, such as in clustering methodologies within machine learning, where understanding the detectable signal limit could impact algorithm design and outcome reliability. The future exploration could focus on extending methods to scenarios where the noise-only covariance matrix is singular, thus broadening the applicability of the findings.

In conclusion, Nadakuditi and Silverstein offer a refined perspective on the limits of signal detection using sample generalized eigenvalues. By grounding their work in a robust theoretical framework, they provide a tool for enhancing detection reliability even in challenging conditions where data is scarce. This contribution holds promise for improving the fidelity of signal detection in practical, real-world scenarios.