B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

9
RANDOM PROCESSES AND
SPECTRAL ANALYSIS
The notion of a random process is a natural extension of the random variable (RV).
Consider, for example, the temperature x of a certain city at noon. The temperature x
is an RV and takes on different values every day. To get the complete statistics of x, we
need to record values of x at noon over many days (a large number of trials). From this data,
we can determine P x (x), the PDF of the RV x (the temperature at noon).
But the temperature is also a function of time. At 1 p.m., for example, the temperature
may have an entirely different distribution from that of the temperature at noon. Still, the
two temperatures may be related, via a joint probability density function. Thus, this random
temperature x is a function of time and can be expressed as x(t). If the random variable is
defined for a time interval t E [t a , tb], then x(t) is a function of time and is random for every
instant t E [t a , tb]-An RV that is a function of time* is called a random process, or stochastic
process. Thus, a random process is a collection of an infinite number of RVs. Communication
signals as well as noises, typically random and varying with time, are well characterized by
random processes. For this reason, random process is the subject of this chapter before we
study the performance analysis of different communication systems.
9.1 FROM RANDOM VARIABLE TO
RANDOM PROCESS
To specify an RV x, we run multiple trials of the experiment and from the outcomes estimate
Px (x). Similarly, to specify the random process x(t), we do the same thing for each time instant
t. To continue with our example of the random process x(t), the temperature of the city, we
need to record daily temperatures for each value of t (for each time of the day). This can be
done by recording temperatures at every instant of the day, which gives a waveform x(t, {i),
where {i indicates the day for which the record was taken. We need to repeat this procedure
every day for a large number of days. The collection of all possible waveforms is known as the
ensemble ( corresponding to the sample space) of the random process x (t). A waveform in this
collection is a sample function (rather than a sample point) of the random process (Fig. 9.1 ).
* Actually, to qualify as a random process, x could be a function of any practical variable, such as distance. In fact, a
random process may also be a function of more than one variable.
456