5.1 Introduction

A random process is a random variable with an additional dimension: time. For each measurement or outcome of an experiment there exists a time function instead of a single number. This is also true for all signal and time series measurements with random components or properties. For example, each time there is a large explosion or sudden movement in the Earth's tectonic plates, seismic waves are produced that travel considerable distances. These waves are studied, and Figure 5.1 shows an example of one. So instead of a single data point for each event, there is a record of movement over time. This situation is described by assigning two independent arguments, t and ?, to a random process x( t, ?). This is depicted in Figure 5.2. The variable ? _n indicates the nth outcome of the time function. Each realization random variable, x( t, ? ₀), , x( t, ? _n), is a sample function, and the set of sample functions is an ensemble. The laws of probability and statistics are applied by describing the behavior of all the processes at a specific time, t ₀, as a random variable. This is illustrated in Figure 5.2, and x( t ₀ , ?) is a random variable. The behavior of the process at another time, t ₁, is also a random variable, x( t ₁ , ?). For simplicity, the...