Joint gradient-based time-delay estimation and adaptive minimum mean-squared-error filtering

A general estimation model 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 transmitted signal. 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 delay element in conjunction with a transversal adaptive filter. The same error signal is used to adjust the delay element and the filter such that the minimum 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 implementation of the joint SD algorithm, under the form of a joint least-mean-square (LMS) algorithm, is then presented. It is analysed in terms of convergence in the mean and in the mean square of both the delay estimate and the adaptive filter weight vector estimate. The conditions of convergence of the joint LMS algorithm are established as a function of the power spectral densities of the observed signals and the minimum mean squared error.