Graphics Programs Reference
In-Depth Information
∞
∫
∞
∫
y
()
=
x
()
ht
τ
(
) τ
d
=
h
()
xt
τ
(
)
d
(13.39)
∞
∞
The cross-correlation function between the signals
x
()
and
g
()
is defined
as
∞
∫
x
∗
()
gt
τ
R
xg
()
=
(
+
)
d
(13.40)
∞
Again, at least one of the two signals should be an energy signal for the corre-
lation integral to be finite. The cross-correlation function measures the similar-
ity between the two signals. The peak value of and its spread around
this peak are an indication of how good this similarity is. The cross-correlation
integral can be computed as
R
xg
()
1
X
∗
()
G
()
R
xg
()
F
=
{
}
(13.41)
When
x
()
g
()
=
we get the autocorrelation integral,
∞
∫
x
∗
()
xt
τ
R
x
()
=
(
+
) τ
d
(13.42)
∞
Note that the autocorrelation function is denoted by rather than .
When the signals and are power signals, the correlation integral
becomes infinite and, thus, time averaging must be included. More precisely,
R
x
()
R
xx
()
x
()
g
()
T
⁄
∫
2
1
---
x
∗
()
gt
τ
R
xg
()
=
lim
(
+
)
d
(13.43)
T
→
∞
T
⁄
2
13.5. Energy and Power Spectrum Densities
Consider an energy signal
x
()
. From ParsevalÓs theorem, the total energy
associated with this signal is
∞
∫
∞
∫
1
2π
x
()
2
X
()
2
d
E
=
d
=
------
(13.44)
∞
∞
When is a voltage signal, the amount of energy dissipated by this signal
when applied across a network of resistance
x
()
R
is
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