Matched Filtering for Generalized Stationary Processes

The methods for solving optimal filtering problems in the case of the classical stationary processes have been well known since the late 1940s. Practice often gives rise to what is not a classical stationary process but a generalized one, and white noise is one simple example. Hence, it is of interest to describe the system action on the generalized stationary processes, and then to carry over filtering methods to them. For arbitrary generalized stochastic processes this seems to be a challenging problem. In this correspondence, we identify a rather general class of -generalized stationary processes for which the desired extension can be done for matched filters. This class can be considered as a model of colored noise, and it is wide enough to include white noise, positive frequencies white noise, as well as certain generalized processes occurring in practice, namely, when the smoothing effect gives rise to the situation in which the distribution of probabilities may not exist at some time instances. One advantage of the suggested model is that it connects optimal filter design with inverting of integral operators; the methods for the latter can be found in the extensive literature.