We discussed Random Signals briefly and now we return to consider
them in detail. We shall assume that they evolve continuously
with time tt, although they may
equally well evolve with distance (e.g. a random texture in
image processing) or some other parameter.
We can imagine a generalization of our previous ideas about
random experiments so that the outcome of an experiment can be
a 'Random Object', an example of which is a signal waveform
chosen at random from a set of possible signal waveforms, which
we term an Ensemble. This ensemble of random
signals is known as a Random Process.
An example of a Random Process
Xtα
X
t
α
is shown in Figure 1,
where tt is time and
αα is an index to the various
members of the ensemble.
-
tt is assumed to belong to some
set (the time axis).
-
αα is assumed to belong
to some set (the sample
space).
-
If is a continuous
set, such as R or
0
∞
0
, then the process is termed a Continuous
Time random process.
-
If is a discrete set
of time values, such as the integers
Z, the process is termed a
Discrete Time Process or Time
Series.
-
The members of the ensemble can be the result of different
random events, such as different instances of the sound 'ah'
during the course of this lecture. In this case
αα is discrete.
-
Alternatively the ensemble members are often just different
portions of a single random signal. If the signal is a
continuous waveform, then
αα may also be a
continuous variable, indicating the starting point of each
ensemble waveform.
We will often drop the explicit dependence on
αα for notational
convenience, referring simply to
random process
Xt
X
t
.
If we consider the process
Xt
X
t
at one particular time
t=
t1
t
t1
, then we have a random variable
X
t1
X
t1
.
If we consider the process
Xt
X
t
at NN time instants
t1
t2
…
tN
t1
t2
…
tN
, then we have a random vector:
X=(
X
t1
X
t2
…X
tN
)T
X
X
t1
X
t2
…
X
tN
We can study the properties of a random process by considering
the behavior of random variables and random vectors extracted
from the process, using the probability theory derived earlier
in this course.
