Fisher-Rao distances between finite energy signals in noise

This paper proposes to represent finite-energy signals observed in agiven bandwidth as parameters of a probability distribution, and use the information geometrical framework to compute the Fisher-Rao distance between these signals, seen as distributions. The observations are represented by their discrete Fourier transform, which are modeled as complex Gaussian vectors with fixed diagonal covariance matrix and parametrized means. The parameters define the coordinate system of a statistical manifold. This work investigates the possibility of obtaining closed-form expressions for the Fisher-Rao distance. We study two cases: the general case representing any finite energy signal observed in a given bandwidth and a parametrized example of observing an attenuated signal with a known magnitude spectrum and unknown phase spectrum, and we calculate the Fisher-Rao distances for both cases. The finite energy signal manifold corresponds to the manifold of the Gaussian distribution with a known covariance matrix, and the manifold of known magnitude spectrum signals is a submanifold. We derive the expressions for the Christoffel symbols and the tensorial equations of the geodesics. This leads to geodesic equations expressed as second order differential equations. We show that the tensor differential equations can be transformed into matrix equations. These equations depend on the parametric model but simplify to only two vectorial equations, which combine the magnitude and phase of the signal and their gradients with respect to the parameters. We compute closed-form expressions of the Fisher-Rao distances for both studied cases and show that the submanifold is non-geodesic, indicating that the Fisher-Rao distance measured within the submanifold is greater than in the full manifold.