A fast local maximum likelihood estimator for time delay estimation

A fast algorithm for the local maximum likelihood determination of the difference of arrival time of a common signal at two spatially separated sensors with uncorrelated noise is given. The fast algorithm consists of locally maximizing the cross-correlation function from the two wide-band signals by using Newton´s method for finding the root of an equation. The probability density function of one iteration of Newton´s method is explicitly computed in terms of exponential and error functions. Using a theorem by Rice on the probability density of local maxima of Gaussian processes, the probability density of the local maxima of the cross correlator is obtained. These results are new. When both the signal and the noises have flat power spectral densities, the mean-square error (MSE) of two iterates of Newton´s method is practically equal to the MSE computed from the probability density of the local maxima of the cross correlator (via Rice´s theorem). The above holds if the starting point used in Newton´s method is within a quarter signal resolution binwidth from the true delay and the signal-to-noise ratio (SNR) at the cross-correlator output is 15 dB or higher. The MSE of the local maximum estimator obtained from Rice´s theorem is almost equal to the Cramer-Rao bound even for low SNR, i.e., 5 dB.