Correlator compensation requirements for passive time-delay estimation with moving source or receivers

An analysis is given of the effects of source or receiver motions on the output of a cross correlator used to estimate the source time-delay difference across a baseline. For both narrow-band and wide-band correlators, the need for time-varying correlator delay compensation is quantified for the case of a quadratic signal delay-difference variation. Two useful concepts which emerge are 1) the three-dimensional delay/delay-rate/delay-acceleration mean ambiguity function of the source signal, and 2) the equivalent τ-domain filter, which transforms the source autocorrelation function into the mean output of the mismatched correlator. The required correlator compensation is approximately a quadratic delay modulation matched to the input delay-difference function. For 3 dB peak output loss with a narrow-band signal, the maximum allowable delay-rate mismatch will produce 158° of phase rotation at the centroid frequency f0during the correlation integration time T, while the maximum allowable delay-acceleration mismatch will produce 156° of quadratic phase rotation at f0between the center (t = 0) and each edge (t = ± T/2) of the correlator integration window. For broad-band signals, the mismatch tolerances become about 11 percent tighter.