HP 50G, Руководство пользователя

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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 = 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
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
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