Graphics Programs Reference
In-Depth Information
a
0
=
X
0
T
⁄
2
1
---
2π
nt
T
∫
------------
a
n
=
x
()
cos
d
(13.34)
T
⁄
2
T
⁄
2
1
---
2π
nt
T
∫
------------
b
n
=
x
()
sin
d
T
⁄
2
The coefficients are all zeros when the signal is an odd function of
time. Alternatively, when the signal is an even function of time, then all
a
n
x
()
b
n
are
equal to zero.
Consider the periodic energy signal defined in Eq. (13.33). The total energy
associated with this signal is then given by
t
0
+
∫
T
∞
∑
a
2
4
a
2
2
b
2
2
1
---
x
()
2
E
=
d
=
-----
+
-----
+
-----
(13.35)
t
0
n
=
1
13.4. Convolution and Correlation Integrals
The convolution
φ
xh
()
between the signals
x
()
and
h
()
is defined by
∞
∫
φ
xh
()
x
()
h
()
=
•
=
x
()
ht
τ
(
)
d
(13.36)
∞
where is a dummy variable, and the operator is used to symbolically
describe the convolution integral. Convolution is commutative, associative,
and distributive. More precisely,
τ
•
x
()
h
()
•
=
h
()
x
()
•
(13.37)
x
()
h
()
g
()
•
•
=
(
x
()
h
()
•
)
•
g
()
=
x
()
h
()
g
()
•
(
•
)
For the convolution integral to be finite at least one of the two signals must be
an energy signal. The convolution between two signals can be computed using
the FT
1
φ
xh
()
F
=
{
X
()
H
()
}
(13.38)
Consider an LTI system with impulse response and input signal . It
follows that the output signal is equal to the convolution between the
input signal and the system impulse response,
h
()
x
()
y
()
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