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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
∫
∞
∞−
⋅⋅⋅⋅= dxxxfS )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()([)( ωωωωω dxSxCxf
22
)]([)]([)( ωωω SCA +=
∫
∞
∞−
⋅⋅⋅⋅= ,)cos()(
2
1
)( dxxxfC ω
π
ω
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