Blind Sensing of Signals in Spatially Correlated Noise: Performance and Fundamental Limits

Abstract

This paper addresses a class of blind detection problems characterized by non-random and unknown parameters: the noise covariance matrix, the transmitted data vector, and the channel vector. The data and channel vectors are further constrained to lie within arbitrary, unknown subspaces. The proposed detector employs the largest eigenvalue of a whitened sample covariance matrix as its test statistic. This matrix is constructed from <inline-formula><tex-math notation="LaTeX">$\boldsymbol{m}$</tex-math></inline-formula>-dimensional <inline-formula><tex-math notation="LaTeX">$\boldsymbol{p}$</tex-math></inline-formula> signal-bearing samples and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{m}$</tex-math></inline-formula>-dimensional <inline-formula><tex-math notation="LaTeX">$\boldsymbol{n}$</tex-math></inline-formula> noise-only samples. Under the assumption of Gaussian noise and non-random channel and data vectors, this matrix follows a non-central <inline-formula><tex-math notation="LaTeX">$\boldsymbol{F}$</tex-math></inline-formula>-distribution with a rank-one non-centrality parameter. We derive a closed-form cumulative distribution function (c.d.f.) for the maximum eigenvalue of this matrix by leveraging orthogonal polynomial techniques from random matrix theory. This new c.d.f. enables a precise analysis of the detector’s receiver operating characteristic. We further extend the analysis to a high-dimensional regime where <inline-formula><tex-math notation="LaTeX">$\boldsymbol{m,n,p}\,\mathbf{\to\infty}$</tex-math></inline-formula> with fixed ratios <inline-formula><tex-math notation="LaTeX">$\boldsymbol{m/n}\,\mathbf{\to(0,1)}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{m/p}\,\mathbf{\to(0,1)}$</tex-math></inline-formula>. Notably, we reveal a distinct scaling requirement for detection power. In particular, we determine a critical signal-to-noise ratio (SNR) threshold; for operational SNR values below this threshold, the leading eigenvalue exhibits no statistical power in the high-dimensional regime. In contrast, for the case <inline-formula><tex-math notation="LaTeX">$\boldsymbol{m = n}$</tex-math></inline-formula>, this critical SNR threshold diverges, yet detection with finite power remains possible provided the operational SNR scales at least as <inline-formula><tex-math notation="LaTeX">$\boldsymbol{O(m)}$</tex-math></inline-formula>.