On the sensitivity of transversal RLS algorithms to random perturbations in the filter coefficients

Transversal Recursive Least Squares (RLS) algorithms estimate filter coefficients which minimize the accumulated sum of the square of the error residuals termed the error power. In this paper the sensitivity of this error power to random perturbations about the optimum filter coefficients is investigated. Expressions are derived for the mean and variance of the deviation from the optimum error power. It is shown that for the prewindowed growing memory RLS algorithm (~=1) the mean value of the deviation increases linearly with the number of iterations. The variance of the deviation increases in proportion to the square of the number of iterations. Expressions are also derived for the variance for correlated signals. These expressions show that the variance of the deviation for correlated signals increases compared to uncorrelated white signals by a term related to the sum of the square of the off-diagonal elements of the sample autocorrelation matrix. Expressions are also derived for the mean and variance of the deviation for the exponentially windowed RLS algorithm (A < 1). In this case the deviations are bounded and inversly proportional to 1-~.