Behavior of the partial correlation coefficients of a least squares lattice filter in the presence of a nonstationary chirp input

This paper studies the performance of the aposteriori recursive least squares lattice lter in the presence of a nonstationary chirp signal. The forward and backward partial correlation (PARCOR) coe cients for a Wiener-Hopf optimal lter are shown to be complex conjugates for the general case of a nonstationary input with constant power. Such an optimal lter is compared to a minimum mean square error based least squares lattice adaptive lter. Expressions are found for the behavior of the rst stage of the adaptive lter based on the least squares algorithm. For the general nth stage, the PARCOR coefcients of the previous stages are assumed to have attained Wiener-Hopf optimal steady state. The PARCOR coe cients of such a least squares adaptive lter are compared with the optimal coe cients for such a nonstationary input. The optimal lattice lter is seen to track a chirp input without any error, and the tracking lag in such an adaptive lter is due to the least squares update procedure. The expression for the least squares based PARCOR coe cients are found to contain two terms: a decaying convergence term due to the weighted estimation procedure; and a tracking component which asymptotically approaches the optimal coe cient value. The rate of convergence is seen to depend inversely on the forgetting factor. The tracking lag of the lter is derived as a function of the rate of nonstationarity and the forgetting factor. It is shown that for a given chirp rate there is threshold adaptation constant below which the total tracking error is negligible. For forgetting factors above this threshold, the error increases nonlinearly. Further, this threshold forgetting factor decreases with increasing chirp rate. Simulations are presented to validate the analysis. 2