Cramer-Rao bounds and parameter estimation for random amplitude phase modulated signals

The problem of estimating the phase parameters of a phase modulated signal in the presence of coloured multiplicative noise (random amplitude modulation) and additive white noise, both Gaussian, is addressed. Closed-form expressions for the exact and large-sample Cramer-Rao bounds (CRB) are derived. It is shown that the CRB is not significantly affected by the colour of the modulating process, especially when the signal-to-noise ratio is high. Hence, maximum likelihood type estimators which ignore the noise colour and optimize a criterion with respect to only the phase parameters are proposed. These estimators are shown to be equivalent to the nonlinear least squares estimators which consist of matching the squared observations with a constant amplitude phase modulated signal when the mean of the multiplicative noise is forced to zero. Closed-form expressions are derived for the efficiency of these estimators, and are verified via simulations