We should begin by distinguishing between deterministic and random signals. Usually, the signals being considered here will represent some physical quantity such as voltage, current, distance, temperature, and so forth. Thus, they are real variables. Also, time will usually be the independent variable, although this does not necessarily need to be the case. A signal is said to be deterministic if it is exactly predictable for the time span of interest. Examples would be
(a) xðtÞ ¼ 10 sin 2pt (sine wave)
(b) xðtÞ ¼ 1; t 0 0; t < 0 (unit step) (c) xðtÞ ¼ 1 e t; t 0 0; t < 0 (exponential response)
Notice that there is nothing “chancy” about any of these signals. They are described by functions in the usual mathematical sense; that is, specify a numerical value of t and the corresponding value of x is determined. We are usually able to write the functional relationship between x and t explicitly. However, this is not really necessary. All that is needed is to know conceptually that a functional relationship exists.
In contrast with a deterministic signal, a random signal always has some element of chance associated with it. Thus, it is not predictable in a deterministic sense. Examples of random signals are:
(d) X(t)¼ 10 sin(2pt þ u), where u is a random variable uniformly distributed
between 0 and 2p.
(e) X(t)¼ A sin(2pt þ u), where u and A are independent random variables with
known distributions.
(f) X(t)¼ A noiselike signal with no particular deterministic structure—one that
just wanders on aimlessly ad infinitum.
Since all of these signals have some element of chance associated with them, they are random signals. Signals such as (d), (e), and (f) are formally known as random or stochastic processes, and we will use the terms random and stochastic interchange-
ably throughout the remainder of the book.*
Let us now consider the description of signal (f) in more detail. It might be the common audible radio noise that was mentioned in Chapter 1. If we looked at an analog recording of the radio speaker current, it might appear as shown in Fig. 2.1. We might expect such a signal to have some kind of spectral description, because the signal is audible to the human ear. Yet the precise mathematical description of such a signal is remarkably elusive, and it eluded investigators prior to the 1940s (3, 4).
Imagine sampling the noise shown in Fig. 2.1 at a particular point in time, say, t1.
The numerical value obtained would be governed largely by chance, which suggests it might be considered to be a random variable. However, with random variables we must be able to visualize a conceptual statistical experiment in which samples of the random variable are obtained under identical chance circumstances. It would not be proper in this case to sample X by taking successive time samples of the same signal, because, if they were taken very close together, there would be a close statistical connection among nearby samples. Therefore, the conceptual experiment in this case must consist of many “identical” radios, all playing simultaneously, all *
We need to recognize a notational problem here. Denoting the random process as X(t) implies that there is a functional relationship between X and t. This, of course, is not the case because X(t) is governed by chance. For this reason, some authors (1, 2) prefer to use a subscript notation, that is, Xtrather than X(t), to denote a random time signal. Xtthen “looks” like a random variable with time as a parameter, which is precisely what it is. This notation, however, is not without its own problems. Suffice it to say, in most engineering literature, time random processes are denoted with “parentheses t” rather than “subscript t.” We will do likewise, and the reader will simply have to remember that X(t) does not mean function in the usual mathematical sense when X(t) is a random process.
being tuned away from regular stations in different portions of the broadcast band, and all having their volumes turned up to the same sound level. This then leads to the notion of an ensemble of similar noiselike signals as shown in Fig. 2.2.
It can be seen then that a random process is a set of random variables that unfold with time in accordance with some conceptual chance experiment. Each of the noiselike time signals so generated is called a sample realization of the process.
Samples of the individual signals at a particular time t1 would then be sample
realizations of the random variable X(t1). Four of these are illustrated in Fig. 2.2 as
XA(t1), XB(t1), XC(t1), and XD(t1). If we were to sample at a different time, say, t2, we
would obtain samples of a different random variable X(t2), and so forth. Thus, in this
example, an infinite set of random variables is generated by the random process X(t). The radio experiment just described is an example of a continuous-time random process in that time evolves in a continuous manner. In this example, the probability density function describing the amplitude variation also happens to be continuous. However, random processes may also be discrete in either time or amplitude, as will be seen in the following two examples.
Figure 2.2 Ensemble of sample realizations of a random process.
EXAMPLE 2.1
Consider a card player with a deck of standard playing cards numbered from 1 (ace) through 13 (king). The deck is shuffled and the player picks a card at random and observes its number. The card is then replaced, the deck reshuffled, and another card observed. This process is then repeated at unit intervals of time and continued on ad infinitum. The random process so generated would be discrete in both time and “amplitude,” provided we say that the observed number applies only at the precise instant of time it is observed.
The preceding description would, of course, generate only one sample real- ization of the process. In order to obtain an ensemble of sample signals, we need to imagine an ensemble of card players, each having similar decks of cards and each generating a different (but statistically similar) sample realization of the process.
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