On time delay estimation involving received signals

The estimation of time delay between two received signals is reexamined for the three basic approaches to this problem: 1) phase data, 2) generalized cross correlation, and 3) parameter estimation. For a simple delay problem with short data lengths, arguments are presented to refute the claim that the parameter estimation approach is superior to the usual cross-correlation method. Special consideration is given to the more complex problem of estimating the time delay when signals are received through transmitting media with unknown transfer functions. Although the phase data and cross-correlation methods would generally yield biased estimates in this case, it is verified that positivity and monotonicity of the media impulse response functions are sufficient conditions to assure an unbiased cross-correlation estimate when the source signal is a white process. The problem is then reformulated as a parameter estimation problem, and comparison is made with the simpler generalized cross-correlation method using time limited data.