Page 16-45
A plot of the values A
n
vs.
ω
n
is the typical representation of a discrete
spectrum for a function. The discrete spectrum will show that the function has
components at angular frequencies
ω
n
which are integer multiples of the
fundamental angular frequency
ω
0
.
Suppose that we are faced with the need to expand a non-periodic function
into sine and cosine components. A non-periodic function can be thought of
as having an infinitely large period. Thus, for a very large value of T, the
fundamental angular frequency,
ω
0
=
2
π
/T, becomes a very small quantity,
say
∆ω
. Also, the angular frequencies corresponding to
ω
n
= n
⋅ω
0
= n
⋅∆ω
,
(n = 1, 2, …,
∞
), now take values closer and closer to each other, suggesting
the need for a continuous spectrum of values.
The non-periodic function can be written, therefore, as
where
and
∫
∞
∞
−
⋅
⋅
⋅
⋅
=
dx
x
x
f
S
)
sin(
)
(
2
1
)
(
ω
π
ω
The continuous spectrum is given by
The functions C(
ω
), S(
ω
), and A(
ω
) are continuous functions of a variable
ω
,
which becomes the transform variable for the Fourier transforms defined
below.
Example 1 – Determine the coefficients C(
ω
), S(
ω
), and the continuous
spectrum A(
ω
), for the function f(x) = exp(-x), for x > 0, and f(x) = 0, x < 0.
∫
∞
⋅
⋅
+
⋅
⋅
=
0
,
)]
sin(
)
(
)
cos(
)
(
[
)
(
ω
ω
ω
ω
ω
d
x
S
x
C
x
f
2
2
)]
(
[
)]
(
[
)
(
ω
ω
ω
S
C
A
+
=
∫
∞
∞
−
⋅
⋅
⋅
⋅
=
,
)
cos(
)
(
2
1
)
(
dx
x
x
f
C
ω
π
ω
