The least squares estimation of time delay between two signals with unknown relative phase shift

This correspondence addresses the problem of time delay estimation between two signals with an unknown constant relative phase shift. The method of time delay estimation for the pure delay case of Chan et al. [1] is applied here to derive the least squares estimator (LSE). An explicit expression for the LSE of the delay is obtained which is computationally less complex than the maximum likelihood estimator (MLE) [2]. It is shown that the LSE is unbiased, and that the mean-square error or the LSE is asymptotically equivalent to that of the pertinent MLE.