Graphics Programs Reference
In-Depth Information
t
µ
y
=
A
µ
x
=
44
1
24 24 6
24 34 13
6131
3
A
C
x
A
t
C
y
=
=
13.8. Random Processes
A random variable is by definition a mapping of all possible outcomes of
a random experiment to numbers. When the random variable becomes a func-
tion of both the outcomes of the experiment as well as time, it is called a ran-
dom process and is denoted by . Thus, one can view a random process as
an ensemble of time domain functions that are the outcome of a certain random
experiment, as compared to single real numbers in the case of a random vari-
able.
X
X
()
Since the
cdf
and
pdf
of a random process are time dependent, we will denote
them as
F
X
()
f
X
x
and
()
x
, respectively. The
nth
moment for the random
process
X
()
is
∞
∫
EX
n
x
n
f
X
[
()
]
=
()
d
x
(13.90)
∞
A random process is referred to as stationary to order one if all its s
ta
-
tistical
p
roperties do not change with time. Consequently,
X
()
EX
()
[
]
=
X
,
where
X
is a constant. A random process
X
()
is called stationary to order two
(or wide sense stationary) if
f
X
(
x
1
,
x
2
;
t
1
,
t
2
)
=
f
X
(
x
1
,
x
2
;
t
1
+
∆
t
,
t
2
+
∆
t
)
(13.91)
for all
t
1
t
2
,
and
∆
t
.
Define the statistical autocorrelation function for the random process
X
()
as
ℜ
X
(
t
1
,
t
2
)
=
EXt
()
Xt
()
[
]
(13.92)
The correlation is, in general, a function of . As a con-
sequence of the wide sense stationary definition, the autocorrelation function
depends on the time difference , rather than on absolute time; and
thus, for a wide sense stationary process we have
EXt
()
Xt
()
[
]
(
t
1
,
t
2
)
τ
=
t
2
t
1
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