Adaptive recovery of a chirped sinusoid in noise. I. Performance of the RLS algorithm

Abstract

The authors study the ability of the exponentially weighted recursive least square (RLS) algorithm to track a complex chirped exponential signal buried in additive white Gaussian noise (power P/sub n/). The signal is a sinusoid whose frequency is drifting at a constant rate Psi . lt is recovered using an M-tap adaptive predictor. Five principal aspects of the study are presented: the methodology of the analysis; proof of the quasi-deterministic nature of the data-covariance estimate R(k); a new analysis of RLS for an inverse system modeling problem; a new analysis of RLS for a deterministic time-varying model for the optimum filter; and an evaluation of the residual output mean-square error (MSE) resulting from the nonoptimality of the adaptive predictor (the misadjustment) in terms of the forgetting rate ( beta ) of the RLS algorithm. It is shown that the misadjustment is dominated by a lag term of order beta /sup -2/ and a noise term of order beta . Thus, a value beta /sub opt/ exists which yields a minimum misadjustment. It is proved that beta /sub opt/=((M+1) rho Psi /sup 2/)/sup 1/3/, and the minimum misadjustment is equal to (3/4)P/sub n/(M+1) beta /sub opt/, where rho is the input signal-to-noise ratio (SNR). >