# 3

# Spectra of Stochastic Processes

In Chapter 2 stochastic processes have been considered in the time domain exclusively; i.e. we used such concepts as the autocorrelation function, the cross-correlation function and the covariance function to describe the processes. When dealing with deterministic signals, we have the frequency domain at our disposal as a means to an alternative, dual description. One may wonder whether for stochastic processes a similar duality exists. This question is answered in the affirmative, but the relationship between time domain and frequency domain descriptions is different compared to deterministic signals. Hopping from one domain to the other is facilitated by the well-known Fourier transform and its inverse transform. A complicating factor is that for a random waveform (a sample function of the stochastic process) the Fourier transform generally does not exist.

## 3.1 THE POWER SPECTRUM

Due to the problems with the Fourier transform, a theoretical description of stochastic processes must basically start in the time domain, as given in Chapter 2. In this chapter we will confine ourselves exclusively to wide-sense stationary processes with the autocorrelation function *R*_{XX}(*τ*). Let us assume that it is allowed to apply the Fourier transform to *R*_{XX}(*τ*).

Theorem 3

The Wiener–Khinchin relations are

**Figure 3.1** Interpretation of *S*_{XX}(ω)

The function *S*_{XX}*(ω)*