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−
−
⋅
+
⋅=
+
⋅=
ω
ω
ωπωπ
ω
i
i
ii
F
1
1
1
1
2
1
1
1
2
1
)(
+
⋅−
+
=
22
11
1
2
1
ω
ω
ω
π
i
which is a complex function.
The absolute value of the real and imaginary parts of the function can be
plotted as shown below
Notes:
The magnitude, or absolute value, of the Fourier transform, |F(ω)|, is the
frequency spectrum of the original function f(t). For the example shown above,
|F(ω)| = 1/[2π(1+ω
2
)]
1/2
. The plot of |F(ω)| vs. ω was shown earlier.
Some functions, such as constant values, sin x, exp(x), x
2
, etc., do not have
Fourier transform. Functions that go to zero sufficiently fast as x goes to
infinity do have Fourier transforms.
Properties of the Fourier transform
Linearity: If a and b are constants, and f and g functions, then F{a⋅f + b⋅g} =
a F{f }+ b F{g}.
Transformation of partial derivatives. Let u = u(x,t). If the Fourier transform
transforms the variable x, then
F{∂u/∂x} = iω F{u},
F{∂
2
u/∂x
2
} = -ω
2
F{u},
F{∂u/∂t} = ∂F{u}/∂t, F{∂
2
u/∂t
2
} = ∂
2
F{u}/∂t
2
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