Fisher information for a complex Gaussian random variable: Beamforming applications for wave propagation in a random medium

We derive a general expression for the Fisher information for a special class of complex Gaussian random variables, whose mean and covariance may be expressed in terms of a specific parameterized linear transformation. The deterministic unknown parameter case is considered. This class is descriptive of signals that propagate in a homogeneous or inhomogeneous medium and also occurs in numerous beamforming applications. The Fisher information derived here encompasses a wide range of previously derived, well-known results for complex Gaussian signals propagating in a homogeneous medium and is also applicable to more recent models for signals propagating in an inhomogenous medium. The conditions for the estimates of the azimuth and elevation to decouple from the estimates of the nuisance parameters are studied for monochromatic plane-wave and spherical-wave propagation in a random medium. For propagation in an anisotropic medium, it is found that, in general, the Cramer-Rao lower bounds (CRLBs) of the entire unknown parameter set will be coupled. For special cases of propagation in an isotropic medium, it is found that for plane waves, the CRLBs of the angles of arrival will be approximately decoupled from the CRLBs of the other unknowns, and for spherical waves, the CRLBs of the angles of arrival will be always be coupled to the CRLBs of all other unknowns, although the couplings may be minimized under certain conditions.