Joint gradient-based time-delay estimation and adaptive minimum mean-squared-error filtering Estimation conjointe du delai et filtrage adaptatif bases sur la methode de gradient du minimum de I'erreur quadratique moyenne

A general estimation ntotlel is defined in which two observations are available; one is a noisy version of the transmitted signal. while the other is a noisy filtered and delayed version of the same trimsniitted siglnl. The time-varying delay and the filter are unknown quantities that must be estimated. A joint estimator is proposed. It is composed of an adaptive dclay element in conjunction with a tnnsversal adaptive filter. The same error signal is used to adjust the delay element and thc filter such that the ~nininiutn mean squared error is attained. Two joint gradient-based adaptation algorithms are studied. The joint steepest-descent (SD) algorithm is first investigated. The possibility of a multitude of stable solutions is established and a condition of convergence is presented. A stochastic implcnicntotio~~ OF thc juint SD algorithm, under the fornl of a joint least-mean-square (LMS) algorithm. is then presented. It is analysed in lertns al' convcrgcncc in tlic nlean and in the mean square of both the delay estimate and the adaptive filter weight vector estimate. The conditions of convcrpence or thc joint L.MS algo~ilhnl are established as n function of the power spectral densities of the observed signals and the minimum tnean squared crror. The joint LMS algorithm is simulated under various conditions and it is shown that the adaptive delay element is very effective in reducing the mean squ;lrod crror lit thc output of 11 long xlaptive filter coping with two asynchronous inputs.