Digital Signal Processing Reference
In-Depth Information
2.4 Stochastic Processes
The proper method of including probability theory in the study of infor-
mation signals consists of defining a sample space associated with a set of
functions. This model originates the notion of a stochastic process. Con-
sidering time as the support of the signal, the following definition can be
posed.
DEFINITION 2.4
A stochastic process
X
is a collection or ensemble of
functions engendered by a rule that assigns a function
x
(
t
)
(
t
, ω
i
)
or simply
x
i
(
, called sample of the stochastic process, to each possible outcome in
the sample space.
t
)
Figure 2.4 illustrates in a classical and intuitive way the concept of
stochastic process. From this figure we can observe that
•
For a given ω
i
we have a single time function or sample function
x
i
(
. This function represents a specific realization of the random
signal, which means that, for stochastic processes, the occurrence of
a given signal is a result of a random experiment.
t
)
•
For a given time instant
t
K
, the value to be assigned to
X
depends
on the choice of the sample function, i.e., depends on ω
i
.So
X
(
t
K
)
is a
value corresponding to the outcome of a random experiment, i.e., is
an r.v. Hence each sample function
x
i
(
(
t
K
)
t
)
is a flux of random variables
in time.
As discussed earlier for deterministic signals, time flow can be modeled
in terms of a continuum or an integer set of values, which means that random
signals can also be continuous- or discrete-time. In order to characterize a
x
1
(
t
K
)
Sample
space Ω
x
1
(
t
)
0
ω
1
x
2
(
t
K
)
x
2
(
t
)
ω
2
0
ω
n
x
n
(
t
K
)
x
n
(
t
)
0
t
t
K
FIGURE 2.4
Several sample functions of a random process.
