Steady state and convergence characteristics of the fixed point RLS algorithm

The steady-state mean square prediction error is derived for the fixed point recursive-least-squares (RLS) algorithm, both for the exponentially windowed RLS (forgetting factor, γ<1) and the prewindowed growing memory RLS (γ=1) for correlate inputs. It is shown that signal correlation enhances the excess error due to additive noise and roundoff noise in the desired signal prediction computation. However, correlation has no effect on the noise due to roundoff of the weight error update recursion, which is the error term leading to the divergence of the algorithm for γ=1. It is shown that the convergence rate of the algorithm depends on the filter order, on the choice of the forgetting factor, and on the eigenvalue spread of the data. Convergence is slower if the data are highly correlated, i.e., have a large eigenvalue spread