Sensitivity Analysis of Transversal RLS Algorithms With Correlated Inputs

In this contribution it is shown that the sensitivity analysis predicts that the mean square excess error is the same for both correlated and white inputs. Correlation increases the variance about the mean square excess error. We also present a stable finite precision RLS algorithm. Many adaptive filtering problems can be cast as a systems identification problem. Consider the desired signal d(n) which is a linear mapping of the sequence x(n) by the weights w*, N-l d(n) = x(n-i) w i* = x '(n) w* i-0 In the Recursive Least Squares algorithm , the weights w (n) are calculated such that the accumulated sum of the error residuals is minimized. The error residual is the difference between the desired response and an estimate of the desired response obtained from w ( n ) . The conventional RLS algorithm can then be stated as follows [4] : w(n) = w(n1) + k(n) e (n) (1 .7) With initialization: w(0) = 0 P(0) = 8-11 (6 << 1) h is the exponential forgetting factor and is equal to 1, for the prewindowed growing memory case where all data is weighted equally . 2. Sensit ivity Analysis of t h e Transversa l RLS A l g o r i t h m In a recent paper [3] it was shown that, one way to analyze finite precision effects is indirectly thrdugh the study of the sensitivity of the RLS algorithm to perturbations in the filter coefficients. That analysis was based on expanding the deviation due to random perturbations in the filter coefficient from the optimum error power, which is minimized by the RLS algorithm, in a Taylor series involving second order panial derivatives. This is necessary. since the ( 1 , 1 ) first order derivatives are zero due to ;he optimum RLS filter formulation. Defining C(n) as the optimum error power that is (1 .2) minimized in the RLS algorithm, we have ISCAS '89 1744 CH2692-2/89/OOUO1744 $1 .oO O 1989 IEEE (2.1) The mean value of the error power deviation due to random perturbations, 6i(n) in the weights w(n), for k=l. was derived as [3] 2 At(") = n N EO, and ox2 is the variance of input samples. N is the order of filter and n is the number of iterations . It should be noted that, the mean of the deviation is the same for equal power white uncorrelated and correlated signals. This is also verified in simulations of fixed point implementation (which will be shown below). However, the variance of the deviation increases for correlated signals. The variance for white Gaussian input samples is [31, 2 2 2 2 oA$n) =n E N o ,