EXACT MAXIMUM-LIKELIHOOD TIME-DELAY ESTIMATION FOR SHORT OBSERVATION INTERVALS

This paper presents an exact solution to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T >> tau-c, \d\ made in the derivation of the conventional "asymptotic" maximum likelihood (AML) time delay estimator, where tau-c is the correlation time of the various random processes involved and d is the differential time delay, is relaxed. The resulting "exact" maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators are compared by means of computer simulations for a first-order autoregressive source signal and for values of T, tau-c, and d such that the condition T >> tau-c, \d\ is not satisfied. The results indicate that the AML estimator suffers a dramatic deterioration in performance (large error, bias, and standard deviation) as the ratio d/T increases from 0 to 0.1, making it essentially useless beyond this point. No such effect is observed with the EML estimator which has the best overall performance and whose mean-square error approaches the Cramer-Rao lower bound for large values of the signal-to-noise ratio (SNR). The results also indicate that for large SNR, considerable simplications of the EML estimator are possible without any significant loss in performance.