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

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 _n). The signal is a sinusoid whose frequency is drifting at a constant rate Ψ. 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 (β) of the RLS algorithm. It is shown that the misadjustment is dominated by a lag term of order β^-2 and a noise term of order β. Thus, a value β_opt exists which yields a minimum misadjustment. It is proved that β_opt={(M+1)ρΨ²} ^1/3, and the minimum misadjustment is equal to (3/4)P_n(M+1)β_opt, where ρ is the input signal-to-noise ratio (SNR)