Parameter estimation for random amplitude chirp signals

We consider the problem of estimating the parameters of a chirp signal observed in multiplicative noise, i.e., whose amplitude is randomly time-varying. Two methods for solving this problem are presented. First, an unstructured nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional (2-D) maximization problem. A theoretical analysis of the NLS estimator is presented, and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance that is very close to the Cramer-Rao bound. The second approach combines the principles behind the high-order ambiguity function (HBF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of the HAF-based estimator is also carried out, and closed-form expressions are derived for the asymptotic variance of the HAF estimators based on the data and on the squared data. Numerical examples attest to the validity of the theoretical analyzes and establish a comparison between the two proposed methods.