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Define this expression as a function by using function DEFINE („à). Then,
plot the continuous spectrum, in the range 0 < ω < 10, as:
Definition of Fourier transforms
Different types of Fourier transforms can be defined. The following are the
definitions of the sine, cosine, and full Fourier transforms and their inverses used
in this Chapter.
Fourier sine transform
Inverse sine transform
Fourier cosine transform
Inverse cosine transform
Fourier transform (proper)
Inverse Fourier transform (proper)
Example 1
– Determine the Fourier transform of the function f(t) = exp(- t), for t
>0, and f(t) = 0, for t<0.
∫
∞
⋅⋅⋅⋅==
0
)sin()(
2
)()}({ dtttfFtfs
ω
π
ω
F
∫
∞
−
⋅⋅⋅==
0
1
)sin()()()}({ dttFtfF
s
ωωω
F
∫
∞
⋅⋅⋅⋅==
0
)cos()(
2
)()}({ dtttfFtfc
ω
π
ω
F
∫
∞
−
⋅⋅⋅==
0
1
)cos()()()}({ dttFtfF
c
ωωω
F
∫
∞
−∞
−
⋅⋅⋅== dtetfFtf
ti
ω
π
ω
)(
2
1
)()}({F
∫
∞
−∞
−−
⋅⋅⋅== dteFtfF
ti
ω
ω
π
ω
)(
2
1
)()}({
1
F