Graphics Programs Reference
In-Depth Information
∞
∑
j
2
n
π
t
T
--------------
G
n
∗
X
n
e
R
gx
()
=
(13.56)
n
=
∞
When
x
()
g
()
=
, Eq. (13.56) becomes the power autocorrelation function,
∞
∑
∞
∑
j
2
n
π
t
T
j
2
n
π
t
T
--------------
--------------
2
e
2
2
e
R
x
()
=
X
n
=
X
0
+
2
X
n
(13.57)
n
=
∞
n
=
1
The power spectrum and cross-power spectrum density functions are then
computed as the FT of Eqs. (13.57) and (13.56), respectively. More precisely,
∞
∑
2
δω
2
n
π
T
S
x
() 2π
=
X
n
----------
n
=
∞
(13.58)
∞
∑
G
n
∗
X
n
δω
2
n
π
T
S
gx
() 2π
=
----------
n
=
∞
2
The line (or discrete) power spectrum is defined as the plot of
X
n
versus
n
,
2
where the lines are
∆
f
=
1
⁄
T
apart. The DC power is
X
0
, and the total
∞
∑
2
power is
X
n
.
n
=
∞
13.6. Random Variables
Consider an experiment with outcomes defined by a certain sample space.
The rule or functional relationship that maps each point in this sample space
into a real number is called Ðrandom variable.Ñ Random variables are desig-
nated by capital letters (e.g., ), and a particular value of a random vari-
able is denoted by a lowercase letter (e.g.,
XY
…
,,
xy
…
,,
).
The Cumulative Distribution Function (
cdf
) associated with the random vari-
able
X
is denoted as
F
X
()
, and is interpreted as the total probability that the
random variable
X
is less or equal to the value
x
. More precisely,
F
X
()
Pr X
=
{
≤
x
}
(13.59)
The probability that the random variable
X
is in the interval
(
x
1
,
x
2
)
is then
given by
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