Eigenvalues and eigenvectors of covariance matrices for signals closely spaced in frequency

The eigenstructures of common covariance matrices are identified for the general case of M closely spaced signals. It is shown that the largest signal-space eigenvalue is relatively insensitive to signal separation. By contrast, the ith largest eigenvalue is proportional to δω^2(i-1) or δω^4(i-1), where δω is a measure of signal separation. Therefore, matrix conditioning degrades rapidly as signal separation is reduced. It is also shown that the limiting eigenvectors have remarkably simple structures. The results are very general, and apply to planar far-field direction-finding problems involving almost arbitrary scenarios, and also to time-series analysis of sinusoids, exponentials, and other signals