Graphics Programs Reference
In-Depth Information
then the set
S
=
{
ϕ
n
()
}
is said to be complete, and Eq. (13.26) becomes an
equality. The constants
X
n
are computed as
t
2
∫
x
()ϕ
n
∗
()
d
t
1
X
n
=
-------------------------------------
(13.28)
t
2
∫
()
2
ϕ
n
d
t
t
1
Let the signal
x
()
be periodic with period
T
, and let the complete orthogo-
nal set
S
be
j
2π
nt
T
--------------
S
=
e
;
n
∞∞
=
,
(13.29)
Then the complex exponential Fourier series of
x
()
is
∞
∑
j
2π
nt
T
--------------
x
()
=
X
n
e
(13.30)
n
=
∞
Using Eq. (13.28) yields
T
⁄
∫
2
2π
nt
T
-----------------
1
---
X
n
=
x
()
e
d
(13.31)
T
⁄
2
The FT of Eq. (13.30) is given by
∞
∑
X
n
δω
2π
n
T
X
() 2π
=
----------
(13.32)
n
=
∞
where
δ⋅
(
)
is delta function. When the signal
x
()
is real we can compute
its trigonometric Fourier series from Eq. (13.30) as
∞
∑
∞
∑
2π
nt
T
2π
nt
T
x
()
a
0
=
+
a
n
cos
------------
+
b
n
sin
------------
(13.33)
n
=
1
n
=
1
Search WWH ::
Custom Search