Statistical analysis of a high resolution spatial spectrum estimator utilizing an augmented covariance matrix

In the context of multiple source location with multiple sensors, it was shown [1] that by using an M element array with elements arranged in a minimum redundancy fashion and by augmenting the array output covariance matrix to create a covariance-type matrix of larger dimensions, it is possible to estimate the directions of arrival of as many as M(M - 1)/2 uncorrelated sources. The earlier work assumed perfectly estimated covariances. This paper addresses the statistical properties and related issues of the augmented covariance matrix in the case when the array output sample covariances are estimated from finite data using the maximum likelihood method. Using a matrix factorization technique, the distribution of the sample Bartlett spatial spectrum estimator based on the augmented covariance matrix is shown to be a sum of weighted and dependent χ²-distributed random variables. The degradation in performance in making use of the augmented matrix is here found in terms of the variance of the Bartlett estimator. This is shown to be in the order of twice the variance obtained with a uniform array of actual elements, the latter being equal in number to the dimension of the augmented array. To match the performance of the uniform array of actual elements using the augmented array would then require about twice the number of data samples.