Graphics Programs Reference
In-Depth Information
11.
Autocorrelation:
∞
∫
X
()
X
∗
()
X
()
2
Fx
()
x
∗
τ
(
t
) τ
d
=
=
(13.22)
∞
12.
ParsevalÓs Theorem: The energy associated with the signal
x
()
is
∞
∫
∞
∫
x
()
2
X
()
2
d
E
=
d
t
=
(13.23)
∞
∞
13.
Moments: The
nth
moment is
∞
∫
n
d
t
n
x
()
d
m
n
=
=
X
()
ω 0
(13.24)
=
ω
n
d
0
13.3. The Fourier Series
A set of functions
S
=
{
ϕ
n
()
n
;
=
1 …
N
,
,
}
is said to be orthogonal over
the interval
(
t
1
,
t
2
)
if and only if
t
2
∫
t
2
∫
0
ij
≠
ϕ
i
∗
()ϕ
j
()ϕ
j
∗
()
d
()
t
d
=
ϕ
i
=
(13.25)
λ
i
i
=
j
t
1
t
1
where the asterisk indicates complex conjugate, and
λ
i
are constants. If
λ
i
=
1
for all , then the set
i
S
is said to be an orthonormal set.
An electrical signal
x
()
can be expressed over the interval
(
t
1
,
t
2
)
as a
weighted sum of a set of orthogonal functions as
N
∑
x
()
≈
X
n
ϕ
n
()
(13.26)
n
=
1
where are, in general, complex constants, and the orthogonal functions
are called basis functions. If the integral-square error over the interval
is equal to zero as
X
n
ϕ
n
()
(
t
1
,
t
2
)
N
approaches infinity, i.e.,
t
2
∫
N
∑
2
lim
x
()
X
n
ϕ
n
()
d
=
0
(13.27)
N
→
∞
t
1
n
=
1
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