Spectral Representation of Stationary Processes

Abstract

In this chapter we review the spectral representation of stationary processes. This representation theory is useful for at least two reasons. First it leads to concrete representation results of stationary processes in terms of white noise. These representations are basic for filtering and prediction and also for state-space modeling of random signals. Second, spectral representation theory provides a functional calculus for random variables and processes in terms of functions of a complex variable, much in the same spirit as the Fourier transform for deterministic signals. Unfortunately the Fourier transform of a stationary process cannot be defined in a deterministic pathwise sense. For it is well-known that the sample paths of a discrete-time stationary Gaussian process of, say, independent random variables (discrete time white noise) are neither in l 2 nor uniformly bounded with probability one, and hence as functions of time they do not admit a Fourier transform [129].