Finite-precision error analysis of QRD-RLS and STAR-RLS adaptive filters

The QR decomposition-based recursive least-squares (RLS) adaptive filtering (QRD-RLS) algorithm is suitable for VLSI implementation since it has good numerical properties and can be mapped onto a systolic array. Recently, a new fine-grain pipelinable STAR-RLS algorithm was developed. The pipelined STAR-RLS algorithm (PSTAR-RLS) is useful for high-speed applications. The stability of QRD-RLS, STAR-RLS, and PSTARRLS has been proved, but the performance of these algorithms in finite-precision arithmetic has not yet been analyzed. The aim of this paper is to determine expressions for the degradation in the performance of these algorithms due to finite precision. By exploiting the steady-state properties of these algorithms, simple expressions are obtained that depend only on known parameters. This analysis can be used to compare the algorithms and to decide the wordlength to be used in an implementation. Since floating- or fixed-point arithmetic representations may be used in practice, both representations are considered in this paper. The results show that the three algorithms have about the same finiteprecision performance, with PSTAR-RLS performing better than STAR-RLS, which does better than QRD-RLS. These algorithms can be implemented with as few as 8 bits for the fractional part, depending on the filter size and the forgetting factor used. The theoretical expressions are found to be in good agreement with the simulation results.