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The detection problem in statistical signal processing can be succinctlyformulated: Given m (possibly) signal bearing, n-dimensional signal-plus-noisesnapshot vectors (samples) and N statistically independent n-dimensionalnoise-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 thecritical first step in the subsequent signal parameter estimation phase.  The signal detection problem can be naturally posed in terms of the samplegeneralized eigenvalues. The sample generalized eigenvalues correspond to theeigenvalues of the matrix formed by "whitening" the signal-plus-noise samplecovariance matrix with the noise-only sample covariance matrix. In this articlewe prove a fundamental asymptotic limit of sample generalized eigenvalue baseddetection of signals in arbitrarily colored noise when there are relatively fewsignal bearing and noise-only samples.  Numerical simulations highlight the accuracy of our analytical prediction andpermit us to extend our heuristic definition of the effective number ofidentifiable signals in colored noise. We discuss implications of our resultfor the detection of weak and/or closely spaced signals in sensor arrayprocessing, abrupt change detection in sensor networks, and clusteringmethodologies in machine learning.

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