What is a natural notion of distance between power spectral density functions?

We introduce a Riemannian metric on the cone of spectral density functions of discrete-time random processes. This is motivated by a problem in prediction theory, and it is analogous to the Fisher information metric on simplices of probability density functions. Interestingly, in either metric, geodesics and geodesic distances can be characterized in closed form. The goal of this paper is to highlight analogies and differences between the proposed differential-geometric structure of spectral density functions and the information geometry of the Fisher metric, and raise the question as to what a natural notion of distance between power spectral density functions is.