The estimate of time delay between two signals with random relative phase shift

This paper presents the maximum likelihood estimate of the time delay between two signals with random relative phase shift. The analysis applies for both broad-band and narrowband signals and incorporates a parametric representation of the spread of the phase variation which allows the examination of all cases ranging from uniformly distributed phase shift to known phase shift. A realization of the ML estimator is presented which illustrated the difference between optimum processing for random phase shift and no phase shift. The Cramer-Rao bound is also derived and the performance degradation is discussed.