Performance of Statistical Tests for Source Detection using Random Matrix Theory

This paper introduces a unified framework for the detection o f a 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 st udied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance ma trix and its normalized trace. Using recent results of random matrix theory, a practical way to ev aluate the threshold and the p-value of the test is provided in the asymptotic regime where the numbe r K of sensors and the number N of observations per sensor are large but have the same order of m agnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Char acteristic (ROC) curve. It is in particular proved that both Type I and Type II error probabilities conve rge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are p rovided for the error exponents. These theoretical results rely on a precise description of the lar ge 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 matr ix. This work was partially supported by french programs ANR-07 -MDCO-012-01 ‘Sesame’, and ANR-08-BLAN-0311-03 ‘GranMa’. P. Bianchi and J. Najim are with CNRS and Télécom Paristech , France.{bianchi,najim}@telecom-paristech.fr , M. Debbah is with SUPELEC and holds Alcatel-Lucent/Supéle c Flexible Radio chair, France merouane.debbah@supelec.fr , M. Maida is with Université Paris-Sud, UMR CNRS 8628, Franc e. Mylene.Maida@math.u-psud.fr , October 2, 2009 DRAFT