Introduction

Spectral analysis almost invariably deals with a class of models called stationary stochastic processes. The material in this chapter is a brief review of the theory behind such processes. The reader is referred to Chapter 3 of Priestley (1981), Chapter 10 of Papoulis (1991) or Chapter 1 of Yaglom (1987) for complementary discussions.

Stochastic Processes

Consider the following experiment (see Figure 31): we hook up a resistor to an oscilloscope in such a way that we can examine the voltage variations across the resistor as a function of time. Every time we press a ‘reset’ button on the oscilloscope, it displays the voltage variations for the 1 second interval following the ‘reset.’ Since the voltage variations are presumably caused by such factors as small temperature variations in the resistor, each time we press the ‘reset’ button, we will observe a different display on the oscilloscope. Owing to the complexity of the factors that influence the display, there is no way that we can use the laws of physics to predict what will appear on the oscilloscope. However, if we repeat this experiment over and over, we soon see that, although we view a different display each time we press the ‘reset’ button, the displays resemble each other: there is a characteristic ‘bumpiness’ shared by all the displays.

We can model this experiment by considering a large bowl in which we have placed pictures of all the oscilloscope displays that we could possibly observe. Pushing the ‘reset’ button corresponds to reaching into the bowl and choosing ‘at random’ one of the pictures.