HP 50g Graphing Calculator User Manual
Page 520
Page 16-43
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 = 2π⋅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
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
= 2π/T, becomes a very small quantity, say
Δω. Also, the angular frequencies corresponding to ω
n
= n
⋅ω
0
= n
⋅Δω, (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
∑
∞
=
+
⋅
+
=
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
∫
∞
⋅
⋅
+
⋅
⋅
=
0
,
)]
sin(
)
(
)
cos(
)
(
[
)
(
ω
ω
ω
ω
ω
d
x
S
x
C
x
f
∫
∞
−∞
⋅
⋅
⋅
⋅
=
,
)
cos(
)
(
2
1
)
(
dx
x
x
f
C
ω
π
ω