Signal-Phase Estimation for Product Array Processors

Abstract

In coprime arrays and intensity processing of vector sensor arrays, product array processing can offer advantages in detection or spatial-spectrum estimation over conventional processing of the full array (e.g., in heavy-tailed noise). Product array processing generally focuses on the incoherent response formed by the product of conventional beamforming of subarrays while ignoring the phase of the signal. However, estimating signal phase is important when postdetection estimation is performed coherently over multiple beamforming epochs. In this article, a weak-signal maximum-likelihood (ML) estimator is developed to estimate the phase of a narrowband signal in product array processors from the individual subarray phase estimates, which allows forming the complex envelope of the product beam response. When the subarray signal-to-noise ratios (SNRs) are equal, the estimator simplifies to the angle of a unit-circle average (UCA) of direction vectors, also known as the sample mean direction. A Fourier series representation of the probability density function (PDF) of the subarray phase estimate is used to simplify evaluation of the PDF of the UCA estimator and its circular variance. The UCA estimator is seen to suffer a performance loss relative to the full-array ML phase estimator of no more than 1-dB SNR in independent Gaussian noise. However, increasing SNR or subarray noise correlation reduces the disparity to the point where there is little practical difference. The models of circular variance, including a mixture-based approximation for correlated subarray noise, were seen to very accurately predict phase estimation performance measured from experimental data obtained on a vertical line array in shallow water.