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 narrow-band 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 illustrates the difference between optimum processing for random phase shift and pure delay. The Cramér-Rao bound is also derived and the performance degradation with respect to the pure delay case is discussed.