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

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 $\boldsymbol{m}$-dimensional $\boldsymbol{p}$ signal-bearing samples and $\boldsymbol{m}$-dimensional $\boldsymbol{n}$ noise-only samples. Under the assumption of Gaussian noise and non-random channel and data vectors, this matrix follows a non-central $\boldsymbol{F}$-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 $\boldsymbol{m,n,p}\,\mathbf{\to\infty}$ with fixed ratios $\boldsymbol{m/n}\,\mathbf{\to(0,1)}$ and $\boldsymbol{m/p}\,\mathbf{\to(0,1)}$. 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 $\boldsymbol{m = n}$, this critical SNR threshold diverges, yet detection with finite power remains possible provided the operational SNR scales at least as $\boldsymbol{O(m)}$.