Fisher information for a Gaussian random variable with unequal real and imaginary covariances: Derivation and application to beamforming
Sandra L. Collier, D. Keith Wilson · The Journal of the Acoustical Society of America · 2003
There are many physical situations in which a received signal may be modeled as a complex Gaussian random variable. One such situation occurs when an acoustic wave is strongly scattered as a result of propagation through a random medium such as the atmosphere or ocean. However, there are also many conditions under which this model fails to provide an adequate representation. A specific example is the geometric acoustics regime, in which diffraction and scattering by the medium are both weak and the variance of the phase fluctuations is much larger than the variance of the log-amplitude fluctuations. The reduced wavefunction is the wavefunction of a signal propagating in an inhomogeneous medium normalized by the wavefunction in free space. A statistical model is developed for a reduced wavefunction whose real and imaginary components are Gaussian random variables with unequal covariances. A linear transformation is performed and the probability density function of the received signal at a passive array is calculated. The Fisher information is calculated for special conditions of the transformation that are of interest to acoustic beamforming. The coupling of the Cramer–Rao lower bounds on the parameter estimates (e.g., the angles of arrival, source phase, and medium parameters) is addressed.