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Fourier Transforms
Before presenting the concept of Fourier transforms, we’ll discuss the general
definition of an integral transform. In general, an integral transform
is a
transformation that relates a function f(t) to a new function F(s) by an integration
of the form The function κ(s,t) is known as the
kernel of the transformation
.
The use of an integral transform allows us to resolve a function into a given
spectrum of components
. To understand the concept of a spectrum, consider
the Fourier series
representing a periodic function with a period T. This Fourier series can be re-
written as where
for n =1,2, …
∫
⋅⋅=
b
a
dttftssF .)(),()(
κ
()
,sincos)(
1
0
∑
∞
=
⋅+⋅+=
n
nnnn
xbxaatf
ωω
∑
∞
=
+⋅+=
1
0
),cos()(
n
nnn
xAaxf
φϖ
,tan,
122
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=+=
−
n
n
nnnn
a
b
baA
φ