EIGENVALUES AND EIGENVECTORS OF COVARIANCE MATRICES FOR SIGNALS CLOSELY SPACED IN FREQUENCY

There has been considerable recent interest in high-resolution techniques for direction finding and for time-series analysis. The performance of these techniques with regard to resolving closely spaced signals, and estimating the signal parameters depends strongly upon the eigenstructure of an associated covariance matrix. Owing to the difficulty in explicitly solving the characteristic equation, available analytical eigenstructure results appear limited to the case of two closely spaced signals. This paper identifies the eigenstructures of common covariance matrices 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 deltaomega2(i - 1) or deltaomega4(i - 1) where deltaomega is a measure of signal separation. Therefore, matrix conditioning degrades rapidly as signal separation is reduced. Additionally, it is shown that the limiting eigenvectors have remarkably simple structures. For example, in direction-finding applications the limiting eigenvectors are the generic signal vector and its derivatives, appropriately orthonormalized. 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. The identified eigenstructure, together with classical perturbation techniques, provides a powerful tool for analyzing the performance of high-resolution techniques.