A VECTOR SPACE APPROACH TO SPATIAL SPECTRUM ESTIMATION (SIGNAL PROCESSING, GEOPHYSICAL APPLICATIONS, RADAR)

Abstract

Array processing for spatial spectrum estimation is reexamined from the vector space viewpoint with the objective of finding a common framework within which the various known superresolution estimators may be compared. Based on the experience with eigenstructure methods, which are ideal in the sense that they asymptotically yield unbiased estimates and have infinite resolving power for point sources, a generic form for an ideal spectrum estimator is proposed. Within this context it is shown that the MUltiple SIgnal Classification (MUSIC) method is an exact realization and the well known superresolution estimators, such as the Maximum Likelihood Method (MLM) of Capon and the Linear Prediction Method (LPM), are approximate realizations of this form. Further, this formulation is shown to suggest ways to modify both MLM and LPM so as to achieve asymptotically ideal performance for point sources. In the case of estimated covariance matrices the compensation for the improved performance is shown to be the requirement of larger number of samples compared to the eigenstructure based methods. The question of how to deploy the array elements for improved performance, in terms of the ability of the array to detect and resolve a larger number of sources than conventionally possible, is addressed. A study related to the statistical properties of the estimator of the unknown angles of arrival is reported. This includes a general result based on Cramer-Rao bound, and specific analyses for various superresolution techniques. A test based on higher powers of the eigenvalues of the sample covariance matrix is derived to estimate the number of point sources present in the data. This test is found to be useful even when the number of array elements is less than the number of point sources and is applied to the augmentation technique where negative eigenvalues are encountered. Other results include determination of the sensitivity of the eigenstructure based techniques on element position uncertainty, and a new spectral estimator combining the properties of the Maximum Likelihood Method and the MUSIC estimator. The latter is potentially useful for dealing with distributed sources. (Abstract shortened with permission of author.