HP 48gII - Manual (Page 523)

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
= 2π/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