# 4

# Linear Filtering of Stochastic Processes

In this chapter we will investigate what the influence will be on the main parameters of a stochastic process when filtered by a linear, time-invariant filter. In doing so we will from time to time change from the time domain to the frequency domain and vice versa. This may even happen during the course of a calculation. From Fourier transform theory we know that both descriptions are dual and of equal value, and basically there is no difference, but a certain calculation may appear to be more tractable or simpler in one domain, and less tractable in the other.

In this chapter we will always assume that the input to the linear, time-invariant filter is a wide-sense stationary process, and the properties of these processes will be invoked several times. It should be stressed that the presented calculations and results may only be applied in the situation of wide-sense stationary input processes. Systems that are non-linear or time-variant are not considered, and the same holds for input processes that do not fulfil the requirements for wide-sense stationarity.

We start by summarizing the fundamentals of linear time-invariant filtering.

## 4.1 BASICS OF LINEAR TIME-INVARIANT FILTERING

In this section we will summarize the theory of continuous linear time-invariant filtering. For the sake of simplicity we consider only single-input single-output (SISO) systems. For a more profound treatment of this theory see references [7] and [10]. The generalization ...