Averaging the Fourier Phase Information in a Signal Ensemble without Calculating Phase

A length-M discrete stochastic process, s(k), and a length-N deterministic sequence, h(k), are convolved to yield the discrete stochastic process, x(k). If we take the length-L discrete Fourier transform of s(k), h(k), and x(k), where L is some integer greater than N+M−2, and pad with zeros as necessary, then (1) for 0 ≤ n < L. Here H(n) is deterministic while X(n) and S(n) are stochastic. Given an ensemble of the process x(k), and sufficient knowledge of the self-statistics of s(k), we wish to recover h(k). Problems such as this arise in the fields of geophysics, radar signal processing, and space object imaging. Although it is easy to estimate the Fourier magnitude of h(k) from (2) estimating the phase of H(n) can be difficult, due to the phase unwrapping problem[1]. The difficulty is worsened by observational noise and by extending the problem to 2-dimensional images.