We propose a parametric representation of finite energy signal observations defining a statistical manifold and investigate the possibility of obtaining closed-form expressions for the Fisher-Rao distance. The tensor differential equations defining the geodesics simplify to only two vectorial equations, which combine the magnitude and phase of the signal and their gradients with respect to the parameters. These equations lead to closed-form expressions of the Fisher-Rao distance in certain cases. We study the example of observing an attenuated signal with a known magnitude spectrum and unknown phase spectrum and calculate the Fisher-Rao distance. 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 compute closed-form expressions of the Fisher-Rao distances 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.