Adaptive recovery of a chirped signal using the RLS algorithm

This paper studies the performance of the recursive least squares (RLS) algorithm in the presence of a general chirped signal and additive white noise. The chirped signal, which is a moving average (MA) signal deterministically shifted in frequency at rate ψ, can be used to model a frequency shift in a received signal. General expressions for the optimum Wiener-Hopf coefficients, one-step recovery and estimation errors, noise and lag misadjustments, and the optimum adaptation constant (β_opt) are found in terms of the parameters of the stationary MA signal. The output misadjustment is shown to be composed of a noise (ξ₀Mβ/2) and lag term (κ/(β²ψ²)), and the optimum adaptation constant is proportional to the chirp rate as ψ^2/3 . The special case of a chirped first-order autoregressive (AR1) process with correlation (α) is used to illustrate the effect the bandwidth (1/α) of the chirped signal on the adaptation parameters. It is shown that unlike for the chirped tone, where the β_opt increases with the filter length (M), the adaptation constant reaches a maximum for M near the inverse of the signal correlation (1/α). Furthermore, there is an optimum filter length for tracking the chirped signal and this length is less than (1/α)