Page 16-44
integration of the form
∫
⋅⋅=
b
a
dttftssF .)(),()( κ
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
()
,sincos)(
1
0
∑
∞
=
⋅+⋅+=
n
nnnn
xbxaatf ωω
representing a periodic function with a period T. This Fourier series can be
re-written as
∑
∞
=
+⋅+=
1
0
),cos()(
n
nnn
xAaxf φϖ where
,tan,
122
=+=
−
n
n
nnnn
a
b
baA φ
for n =1,2, …
The amplitudes A
n
will be referred to as the spectrum of the function and will
be a measure of the magnitude of the component of f(x) with frequency f
n
=
n/T. The basic or fundamental frequency in the Fourier series is f
0
= 1/T, thus,
all other frequencies are multiples of this basic frequency, i.e., f
n
= n⋅f
0
. Also,
we can define an angular frequency, ω
n
= 2nπ/T = 2π⋅f
n
= 2π⋅ n⋅f
0
= n⋅ω
0
,
where ω
0
is the basic or fundamental angular frequency of the Fourier series.
Using the angular frequency notation, the Fourier series expansion is written
as
∑
∞
=
+⋅+=
1
0
).cos()(
n
nnn
xAaxf φω
()
∑
∞
=
⋅+⋅+=
1
0
sincos
n
nnnn
xbxaa ωω
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