HP 49g+ User Manual Page 519

Graphing calculator.

A plot of the values A
spectrum for a function. The discrete spectrum will show that the function has
components at angular frequencies ω
fundamental angular frequency ω
Suppose that we are faced with the need to expand a non-periodic function
into sine and cosine components.
as having an infinitely large period. Thus, for a very large value of T, the
fundamental angular frequency, ω
say ∆ω.
Also, the angular frequencies corresponding to 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
(
)
[
f
x
C
0
where
(
ω
)
C
and
(
ω
S
The continuous spectrum is given by
A
(
ω
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.
vs. ω
is the
typical representation of a discrete
n
n
which are integer multiples of the
n
.
0
A non-periodic function can be thought of
= 2π/T, becomes a very small quantity,
0
(
ω
)
cos(
ω
)
(
ω
)
x
S
1
(
)
cos(
ω
f
x
2
π
1
)
(
)
sin(
f
x
2
π
2
)
=
[
C
(
ω
)]
+
[
S
(
ω
)]
= n⋅ω
= n⋅∆ω,
n
0
sin(
ω
)]
ω
,
x
d
)
,
x
dx
ω
)
x
dx
2
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