Fixed and floating point error analysis of QRD-RLS and STAR-RLS adaptive filters

The QR decomposition based recursive least-squares (RLS) adaptive filtering (referred to as QRD-RLS) algorithm is suitable for VLSI implementation since it has good numerical properties and can be mapped to a systolic array. Recently, a new fine-grain pipelinable STAR-RLS algorithm was developed based on scaled tangent rotation. The pipelined STAR-RLS algorithm, referred to as PSTAR-RLS, is useful for high-speed applications. The stability of QRD-RLS, STAR-RLS and PSTAR-RLS 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 closed-form expressions are obtained which depend only on known parameters. Since floating-point or fixed-point arithmetic representations may be used in practice, both representations are considered in this paper. The results show that the PSTAR-RLS and STAR-RLS algorithms perform better than the QRD-RLS especially in a floating-point representation. The theoretical expressions are found to be in good agreement with the simulation results