1.8 Introduction to random signals and processes

Before now, you have probably dealt strictly with the theory behind signals and systems, as well as look at some the basiccharacteristics of signals and systems . In doing so you have developed an important foundation; however, most electrical engineers do notget to work in this type of fantasy world. In many cases the signals of interest are very complex due to the randomness ofthe world around them, which leaves them noisy and often corrupted. This often causes the information contained in thesignal to be hidden and distorted. For this reason, it is important to understand these random signals and how to recoverthe necessary information.

Signals: deterministic vs. stochastic

For this study of signals and systems, we will divide signals into two groups: those that have a fixed behavior and thosethat change randomly. As most of you have probably already dealt with the first type, we will focus on introducing you torandom signals. Also, note that we will be dealing strictly with discrete-time signals since they are the signals we dealwith in DSP and most real-world computations, but these same ideas apply to continuous-time signals.

Deterministic signals

Most introductions to signals and systems deal strictly with deterministic signals . Each value of these signals are fixed and can be determined by a mathematicalexpression, rule, or table. Because of this, future values of any deterministic signal can be calculated from pastvalues. For this reason, these signals are relatively easy to analyze as they do not change, and we can make accurateassumptions about their past and future behavior.

Stochastic signals

Unlike deterministic signals, stochastic signals , or random signals , are not so nice. Random signals cannot be characterized by a simple,well-defined mathematical equation and their future values cannot be predicted. Rather, we must use probability andstatistics to analyze their behavior. Also, because of their randomness, average values from a collection of signals are usually studied rather than analyzing one individual signal.

Random process

As mentioned above, in order to study random signals, we want to look at a collection of these signals rather than just oneinstance of that signal. This collection of signals is called a random process .

random process

A family or ensemble of signals that correspond to every possible outcome of a certain signal measurement. Eachsignal in this collection is referred to as a realization or sample function of the process.

A random process is usually denoted by $X(t)$ or $X(n)$ , with $x(t)$ or $x(n)$ used to represent an individual signal or waveform from this process.

In many notes and books, you might see the following notation and terms used to describe different types of randomprocesses. For a discrete random process , sometimes just called a random sequence , $t$ represents time that has a finite number of values. If $t$ can take on any value of time, we have a continuous random process . Often times discrete and continuous refer to the amplitude of the process, and process or sequencerefer to the nature of the time variable. For this study, we often just use random process to refer to a general collection of discrete-time signals, as seen above in .