Cramer-Rao bounds and estimation of direction of arrival for narrowband signals

In the presence of two narrowband signal sources, the authors derive simplified explicit expressions for the Cramer-Rao lower bound (CRB) for the covariance matrix of all unknown direction of arrival (DOA) angles. A second order Taylor´s series expansion is employed for the inverse of the data covariance matrix, R. The behavior of the covariance matrix is discussed by quantitative numerical comparisons for different values of correlation and spacing between the sources. By using positive definiteness of the Fisher information matrix, a strict lower limit on the angle of resolution is obtained for a given source correlation. A numerical study is included on the performance of the maximum likelihood (ML) method, which assumes signals radiated from the sources are random. The proposed method is compared with Root-MUSIC (B. D. Rao and K. V. S. Hari, IEEE Trans. vol.ASSP-37, no.12, p.1939-49 of 1989). The observations demonstrate the superiority of the proposed ML method to Root-MUSIC, and its performance agrees well with the theoretically derived bounds.<>