Eigenvalue Based Detection of a Signal in Colored Noise: Finite and Asymptotic Analyses

Signal detection in colored noise with an unknown covariance matrix has a myriad of applications in diverse scientific/engineering fields. The test statistic is the largest generalized eigenvalue (l.g.e.) of the whitened sample covariance matrix, which is constructed via $m$-dimensional $p$ signal-plus-noise samples and $m$-dimensional $n$ noise-only samples. A finite dimensional characterization of this statistic under the alternative hypothesis has hitherto been an open problem. We answer this problem by deriving cumulative distribution function (c.d.f.) of this l.g.e. via the powerful orthogonal polynomial approach, exploiting the deformed Jacobi unitary ensemble (JUE). Two special cases and an asymptotic version of the c.d.f. are also derived. With this new c.d.f., we comprehensively analyze the receiver operating characteristics (ROC) of the detector. Importantly, when the noise-only covariant matrix is nearly rank deficient (i.e., $m=n$), we show that (a) when $m$ and $p$ increase such that $m/p$ is fixed, at each fixed signal-to-noise ratio (SNR), there exists an optimal ROC profile. We also establish a tight approximation of it; and (b) asymptotically, reliable signal detection is always possible (no matter how weak the signal is) if SNR scales with $m$.