Performance of Statistical Tests for Single-Source Detection Using Random Matrix Theory

• Published in: IEEE transactions on information theory Vol. 57; no. 4; pp. 2400 - 2419
• Main Authors: Bianchi, P, Debbah, M, Maida, M, Najim, J
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.04.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Cognitive radio; Computer Science; Context; Convergence; Cooperative spectrum sensing; Covariance matrix; Detection, estimation, filtering, equalization, prediction; Eigenvalues; Eigenvalues and eigenfunctions; Electric noise; Exact sciences and technology; generalized likelihood ratio test; hypothesis testing; Information Theory; Information, signal and communications theory; Jacobian matrices; large deviations; Mathematics; Matrix; Noise; Probability; random matrix theory; ROC curve; Sensors; Signal and communications theory; Signal, noise; Statistics; Statistics Theory; Telecommunications and information theory; Performance evaluation; large deviations; Random matrix; Error probability; hypothesis testing; Theoretical study; Eigenvalue; Covariance matrix; Large deviation; Statistical test; Cooperative spectrum sensing; random matrix theory; Maximum likelihood; ROC curve; Sensor array; generalized likelihood ratio test; Signal detection; cognitive radio; large random matrices; statistical test
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2011.2111710

This paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver. The Generalized Maximum Likelihood Test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace. Using recent results from random matrix theory, a practical way to evaluate the threshold and the p -value of the test is provided in the asymptotic regime where the number K of sensors and the number N of observations per sensor are large but have the same order of magnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Characteristic (ROC) curve. It is, in particular, proved that both Type I and Type II error probabilities converge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are provided for the error exponents. These theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models, and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix.