A generalization of McWhirter's formulas

Spatial recursive least squares (RLS) adaptive filters with exponential forgetting of past data perform not much better than a simple LMS adaptive filter in most applications. A significant improvement can be achieved when a multiple delay is added in the filter path. Delayed filtering with respect to parameter adaptation has the effect of incorporation of a number of “future” samples in the parameter estimation process. Thus the algorithm can adapt in advance on future changes in the data characteristic. This principle leads to a significantly improved tracking capability in nonstationary environments over conventional approaches where adaptation relies merely on past samples of the observed process. The realization of a multiple delay in the filter path is, however, a nontrivial problem in QR-RLS adaptive filters when a fast implicit computation of the residual is desired. A new spatial QR-RLS adaptive filter with fast implicit computation of the residual in the more general case of delayed filtering is presented. The new recursions can be viewed as a generalization of McWhirter’s formulas. Experimental results confirm the superior performance of this algorithm. In a systolic realization these generalized McWhirter recursions can be implemented with only a few layers of an appended array with O ( N ) rotational processors and thus appear as a natural and useful extension of the classical Gentleman and Kung array.