A Sample-Deficient Analysis of the Leading Generalized Eigenvalue for the Detection of Signals in Colored Gaussian Noise
arxiv(2023)
摘要
This paper investigates the signal detection problem in colored Gaussian
noise with an unknown covariance matrix. To be specific, we consider a sample
deficient scenario in which the number of signal bearing samples (n) is
strictly smaller than the dimensionality of the signal space (m). Our test
statistic is the leading generalized eigenvalue of the whitened sample
covariance matrix (a.k.a. F-matrix) which is constructed by whitening the
signal bearing sample covariance matrix with noise-only sample covariance
matrix. The whitening operation along with the observation model induces a
single spiked covariance structure on the F-matrix. Moreover, the sample
deficiency (i.e., m>n) in turn makes this F-matrix rank deficient, thereby
singular. Therefore, a simple exact statistical characterization of the
leading generalized eigenvalue (l.g.e.) of a complex correlated singular
F-matrix with a single spiked associated covariance is of paramount
importance to assess the performance of the detector (i.e., the receiver
operating characteristics (ROC)). To this end, we adopt the powerful orthogonal
polynomial technique in random matrix theory to derive a new finite dimensional
c.d.f. expression for the l.g.e. of this particular F-matrix. It turns out
that when the noise only sample covariance matrix is nearly rank deficient and
the signal-to-noise ratio is O(m), the ROC profile converges to a remarkably
simple limiting profile.
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