Fundamental Limit of Sample Eigenvalue based Detection of Signals in Colored Noise using Relatively Few Samples

Sample eigenvalue based estimators are often used for estimating the number of high-dimensional signals in colored noise when an independent estimate of the noise covariance matrix is available. We highlight a fundamental asymptotic limit of sample eigenvalue based detection that brings into sharp focus why in the large system, relatively large sample size limit, underestimation of the model order may be unavoidable for weak/closely spaced signals. We discuss the implication of these results for the detection of two weak, closely spaced signals.