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 /spl psi/, 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 (/spl beta//sub opt/) are found in terms of the parameters of the stationary MA signal. The output misadjustment is shown to be composed of a noise (/spl xi//sub 0/M/spl beta//2) and lag term (/spl kappa//(/spl beta//sup 2//spl psi//sup 2/)), and the optimum adaptation constant is proportional to the chirp rate as /spl psi//sup 2/3/. The special case of a chirped first-order autoregressive (AR1) process with correlation (/spl alpha/) is used to illustrate the effect the bandwidth (1//spl alpha/) of the chirped signal on the adaptation parameters. It is shown that unlike for the chirped tone, where the /spl beta//sub opt/ increases with the filter length (M), the adaptation constant reaches a maximum for M near the inverse of the signal correlation (1//spl alpha/). Furthermore, there is an optimum filter length for tracking the chirped signal and this length is less than (1//spl alpha/).