Least Squares Filtering

A general estimation model is defined in which two observations are available; one being a noisy version of the transmitted signal, while the other is a noisy-filtered and delayed version of the same transmitted signal. The delay and the filter are unknown quantities that must be estimated. An adaptive system, based on the least squares (LS) estimation criterion, is proposed in order to perform a joint estimation of the two unknowns. The joint estimator is conceptually composed of an adaptive delay element operating in conjunction with an adaptive transversal filter. The weighted sum of squared errors is minimized with respect to both the delay and the adaptive filter weight vector. The filter is adapted using a fast version of the recursive least squares (RLS) algorithm, while the delay is updated using a form of derivative, with respect to the delay, of the sum of squared errors. In order to perform this task efficiently, the adaptive delay is limited to integer values and is corrected one sample at a time. The integer delay value is defined as the lag. A series of relations is presented, in order to compute and update the lag value such that the optimum least squares solution is attained. The joint delay estimation and RLS filtering algorithm is obtained by combining the lag update relations with a version of the fast transversal filter RLS algorithm. The simulations of the resulting algorithm show that both stationary and time-varying delays are effectively tracked and that the adaptive filter properly estimates the reference fi lter impulse response.