Estimation of phase angles from the cross-spectral matrix

The so-called signal subspace methods in spatial signal processing have been used up to now in the case of steering vectors depending upon very few parameters. For instance, they are used currently for steering vectors of the form (1, e^iθ, e^2iθ, ... , e^i(N-1)θ)^Twhich occur in the case of plane waves and a linear array of N equispaced sensors. We show, by providing a suitable algorithm, that these methods can be extended to the general case of steering vectors of the form (1, e^iθ2, e^iθ3, ... , e^iθN)^Twhere the phase angles θ2,..., θNare treated as free parameters. This kind of situation occurs, for instance, in the case of randomly distorted wavefronts, and/or flexible antennas of unknown geometry, when the fluctuations of the wavefronts and of the antenna are very slow with respect to the time constants used for the estimation of the cross-spectral matrices. As usual in signal subspace methods, the proposed algorithm works very well when the problem presents a strong degree of overdetermination (many more sensors than sources), and the cross-spectral matrix is estimated with good precision.