Page 16-44
integration of the form
∫
⋅
⋅
=
b
a
dt
t
f
t
s
s
F
.
)
(
)
,
(
)
(
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
(
)
,
sin
cos
)
(
1
0
∑
∞
=
⋅
+
⋅
+
=
n
n
n
n
n
x
b
x
a
a
t
f
ω
ω
representing a periodic function with a period T. This Fourier series can be
re-written as
∑
∞
=
+
⋅
+
=
1
0
),
cos(
)
(
n
n
n
n
x
A
a
x
f
φ
ϖ
where
,
tan
,
1
2
2
=
+
=
−
n
n
n
n
n
n
a
b
b
a
A
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
= 2n
π
/T = 2
π⋅
f
n
= 2
π⋅
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
n
n
n
x
A
a
x
f
φ
ω
(
)
∑
∞
=
⋅
+
⋅
+
=
1
0
sin
cos
n
n
n
n
n
x
b
x
a
a
ω
ω
