Exact maximum likelihood time delay estimation

The authors present an exact solution to the problem of maximum-likelihood time-delay estimation over arbitrary observation time T. That is, the standard assumption T≫τ_c +d_max made in the derivation of the asymptotic maximum-likelihood (AML) estimator, where t_c is the correlation time of the various processes involved and d_max the maximum permissible delay, is relaxed. The exact maximum-likelihood (EML) processor is shown to consist of a special finite-time beamformer, followed by a scalar postprocessor based on the eigenvalues and eigenfunctions of a certain integral equation. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities (PSD). The performance of EML and AML are compared by means of computer simulations for a first-order low-pass PSD. The results show that EML can lead to a significant improvement in performances (bias, variance, large errors) when the condition T≫τ_c+d_max is not satisfied