Stationarizable random processes

Abstract

The familiar notion of inducing stationarity into a cyclostationary process by random translation is extended through characterization of the class of all second-order continuous-parameter processes (with autocorrelation functions that possess a generalized Fourier transform) that are {\em stationarizable} in the wide sense by random translation. This class includes the nested set of proper subclasses: {\em almost cyclostationary} processes, {\em quasi-cyclostationary} processes, and {\em cyclostationary} processes. The random translations that induce stationarity are also characterized. The concept of stationarizability is extended to the concept of asymptotic stationarizability, and the class of {\em asymptotically stationarizable} processes is characterized. These characterizations are employed to derive characterizations of optimum linear and nonlinear time-invariant filters for nonstationary processes. Relative to optimum time-varying filters, these time-invariant filters offer advantages of implementational simplicity and computational efficiency, but at the expense of increased filtering error which in some applications is quite modest. The uses of a random translation for inducing stationarity-of-order-n, for increasing the degree of local stationarity, and for inducing stationarity into discrete-parameter processes are briefly described.