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

The problem of estimating the phase parameters of a phase-modulated signal in the presence of colored multiplicative noise (random amplitude modulatio n) and additive white noise (both Gaussian) is addressed, Closed-form expre ssions for the exact and large-sample Cramer-Rao Bounds (CRB) are derived, It is shown that the CRB is significantly affected by the color of the modu lating process when the signal-to-noise ratio (SNR) or the intrinsic SNR is small. Maximum likelihood type estimators that ignore the noise color and optimize a criterion with respect to only the phase parameters are proposed , These estimators are shown to be equivalent to the nonlinear least square s estimators, which consist of matching the squared observations with a con stant amplitude phase-modulated signal when the mean of the multiplicative noise is forced to zero. Closed-form expressions are derived for the effici ency of these estimators and are verified via simulations.