Finite-precision analysis: Fast QR-decomposition algorithm

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 fixed point digital signal processors (DSPs). Furthermore, the FQRD-RLS algorithms are derived from QR-decomposition based RLS algorithm that are well-known for their numerical stability in finite-precision, therefore these algorithms are also assumed to be numerically stable. However, no formal proof has been provided till now for the stability of the FQRD-RLS algorithms in finite precision. The objective here is to prove the sufficient condition for stability by deriving the steady-state values of the quantization error of the internal variables of the FQRD-RLS algorithm in presence of a zero mean and unity variance white Gaussian noise. The mean-squared quantization error values of all the variables of the FQRD-RLS algorithm are derived and compared with a fixed-point simulation for verification.