HP F2226A - 48GII Graphic Calculator User Manual page 523

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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