Eigenvalues and eigenvectors of covariance matrices for closely-spaced signals in multi-dimensional direction finding

Abstract

The authors characterize the eigenvalues and eigenvectors of covariance matrices that arise in direction finding scenarios with multiple parameters such as azimuth, elevation and, in some applications, also range (multi-D scenarios). They build upon work by H. Lee (1992) for closely-spaced signals with a single directional parameter (1-D). In multi-D, the limiting (small signal spacing) eigenvalues and eigenvectors can be ascertained from a sequence of constant low-rank matrices N/sub k/ expressed in terms of the generic arrival vector, its spatial derivatives, the source configuration, and the source covariances. The limiting eigenvalues are proportional to delta omega /sup 2(k-1)/, where delta omega is the maximum spacing between sources and k epsilon (1,. . .m). It is shown that for a given number of sources m decreases as parameter dimension increases, hence covariance matrix conditioning is improved in multi-D relative to 1-D settings. The results are applicable to analysis of detection and parameter estimation algorithms in multi-D applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>