Algorithms and Architectures for Split Recursive Least Squares

In this paper, a new computationally efficient algorithm for recursive least-squares (RLS) filtering is presented. The proposed Split RLS algorithm can perform the approximated RLS with O ( N ) complexity for signals having no special data structure t o be exploited. Our performance analysis shows that the estimation bias will be small when the input data are less correlated. We also show that for highly correlated data, the orthogonal preprocessing scheme can be used t o improve the performance of the Split RLS. The systolic implementation of our algorithm based on the QR-decomposition RLS (QRD-RLS) array requires only O ( N ) hardware complexity and the system latency can be reduced t o O(log, N). A major advantage of the Split RLS is its superior tracking capability over the conventional RLS under non-stationary environments. INTRODUCTION The family of recursive least-squares (RLS) adaptive algorithms are well known for their superiority to the LMS-type algorithms in both convergence rate and misadjustment [l]. However, the O ( N 2 ) computational complexity becomes the major drawback for their applications as well as for their cost-effective implementation. To alleviate the computational burden of the RLS, the family of fast RLS algorithms such as fast transversal filters, RLS lattice filters, and QR-decomposition based lattice filters (QRD-LSL), have been proposed [l]. By exploiting the special structure of the input data matrix, they can perform RLS estimation with O ( N ) complexity. One major disadvantage of the fast RLS algorithms is that they work for data with shifting input only. In many applications like multichannel adaptive array processing and image processing, the fast RLS algorithms cannot be applied because no special matrix structure can be exploited. In this paper, we propose an approximated RLS algorithm based on the projection m e t h o d [2][3][4][5]. Through multiple decomposition of the signal space and making suitable approximations, we can perform RLS for non-structured data with O ( N ) complexity. Thus, both the complexity problem in the conventional RLS and the data constraint in the fast RLS can be resolved. We shall call such RLS estimation the Split RLS. The systolic implementation of the Split RLS based on the QR-decomposition RLS (QRD-RLS) systolic array in [6] is also proposed. The hardware complexity for the resulting RLS array can be reduced to O ( N ) and the system latency is only O(log, N ) . It is noteworthy that since approximation is made while performing the Split RLS, our approach is not to obtain exact least-squares (LS) solutions. The approximation errors will introduce misadjustment (bias) to the LS errors. In order to know under what circumstances the algorithm will produce small and acceptable 0-7803-2123-5194 $4.00