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

Performance of Statistical Tests for Single Source Detection using Random Matrix Theory