Fixed-Point Implementation of Fast QR-Decomposition Recursive Least-Squares Algorithms (FQRD-RLS): Stability Conditions and Quantization Errors Analysis

The fast QR-decomposition based recursive least-squares (FQRD-RLS) algorithms offer RLS-like convergence and misadjustment at a lower computational cost, and therefore are desirable for implementation on a fixed-point digital signal processor (DSP). Furthermore, the FQRD-RLS algorithms are derived from QR-decomposition based RLS algorithms that are well known for their numerical stability in finite precision. Hence, these algorithms are often assumed numerically stable, although there is no rigorous analysis addressing the stability of such algorithms in finite precision. In this paper, we derive the conditions that guarantee stability of the FQRD-RLS algorithms, and also derive mathematical expressions for the mean-squared quantization error (MSQE) of internal variables of the FQRD-RLS algorithms at steady state. The objective is to quantify the propagation error due to quantization effects. The derived MSQE expressions have been verified by comparisons with fixed-point computer simulations.