Joint time-delay estimation and adaptive recursive least squares filtering

A general estimation model is defined in which two observations are available: a noisy and a noise-filtered and delayed version of the transmitted signal. The delay and the filter must be estimated. The joint estimator is composed of an adaptive delay element operating in conjunction with an adaptive transversal filter. The delay is updated using a form of derivative, with respect to the delay, of the sum of squared errors. The adaptive delay is limited to integer values and is defined as the lag. The lag value is computed and updated so that the optimum least-squares solution is attained. The joint algorithm is obtained by combining the lag update relations with a version of the fast transversal filter RLS algorithm. Simulations show that both stationary and time-varying delays are effectively tracked and that the adaptive filter properly estimates the reference filter impulse response