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

9. 1 From Random Va riable to Random Process 457
Fi g ure 9, 1
Random process
to represent the
tern peratu re of a
city.
'
'
t -
4'
: : t --..
,,
' t -
x(t, n)
Fi g ure 9,2 x ( t, 1
)
Ensemble with a
finite number of
sample functions.
x(t, (n)
'-- ST ___,
lOT
r.7 lJ
L±J
n 17
7T
17 JOT
LJLJ 1-
I tR 61T
I
T
t -
Sample function amplitudes at some instant t = t1 are the values taken by the RV x(t1 ) in
various trials.
We can view a random process in another way. In the case of an RV, the outcome of
each trial of the experiment is a number. We can view a random process also as the outcome
of an experiment, where the outcome of each trial is a waveform (a sample function) that is
a function of t. The number of waveforms in an ensemble may be finite or infinite. In the
case of the random process x(t) (the temperature of a city), the ensemble has infinitely many
waveforms. On the other hand, if we consider the output of a binary signal generator ( over the
period O to lOT), there are at most 2 10 waveforms in this ensemble (Fig. 9.2).
One fine point that needs clarification is that the waveforms (or sample functions) in the
ensemble are not random. They have occurred and are therefore deterministic. Randomness
in this situation is associated not with the waveform but with the uncertainty as to which