  We present several natural notions of distance between spectral density
functions of (discrete-time) random processes. They are motivated by certain
filtering problems. First we quantify the degradation of performance of a
predictor which is designed for a particular spectral density function and then
it is used to predict the values of a random process having a different
spectral density. The logarithm of the ratio between the variance of the error,
over the corresponding minimal (optimal) variance, produces a measure of
distance between the two power spectra with several desirable properties.
Analogous quantities based on smoothing problems produce alternative distances
and suggest a class of measures based on fractions of generalized means of
ratios of power spectral densities. These distance measures endow the manifold
of spectral density functions with a (pseudo) Riemannian metric. We pursue one
of the possible options for a distance measure, characterize the relevant
geodesics, and compute corresponding distances.
