Fourier transforms
The question immediately arises — is there a transformation analogous to (1.1) and (1.2) that applies to functions that
are not periodic? Let us regard a nonperiodic function as the limit of a periodic one where the period is taken to
infinity. We can write
c
n
=
ω
◦
2
π
integraldisplay
τ/
2
−
τ/
2
f
(
u
)
e
−
jnω
◦
u
du
and substitute into (1.1), yielding
f
(
t
)
=
∞
summationdisplay
n
=
−∞
ω
◦
2
π
integraldisplay
τ/
2
−
τ/
2
f
(
u
)
e
jnω
◦
(
t
−
u
)
du
=
1
2
π
∞
summationdisplay
n
=
−∞
g
(
ω
n
)∆
ω
g
(
ω
n
)
≡
integraldisplay
τ/
2
−
τ/
2
f
(
u
)
e
jω
n
(
t
−
u
)
du
where
ω
n
=
nω
◦
and
∆
ω
=
ω
n
+1
−
ω
n
=
ω
◦
have been defined. Now, in the limit that
τ
→ ∞
, the sum
∑
∞
n
=
−∞
g
(
ω
n
)∆
ω
formally goes over to the integral
integraltext
∞
−∞
g
(
ω
)
dω
,
ω
representing a continuous independent variable. The corresponding
expression for
g
(
ω
)
just above becomes:
g
(
ω
)
=
integraldisplay
∞
−∞
f
(
u
)
e
jω
(
t
−
u
)
du
Substituting into the equation above for
f
(
t
)
yields:
f
(
t
)
=
1
2
π
integraldisplay
∞
−∞
e
jωt
dω
integraldisplay
∞
−∞
f
(
u
)
e
−
jωu
du
Finally, if we change variables and define
F
(
ω
)
as (analogous to (1.2))
F
(
ω
)
=
integraldisplay
∞
−∞
f
(
t
)
e
−
jωt
dt
(1.4)
we are left with (analogous to (1.1))
f
(
t
)
=
1
2
π
integraldisplay
∞
−∞
F
(
ω
)
e
jωt
dω
(1.5)
where the continuous function
F
(
ω
)
replaces the discrete set of amplitudes
c
n
in a Fourier series. Together, these
two formulas define the Fourier transform and its inverse. Note that the placement of the factor of
2
π
is a matter of
convention. The factor appears in the forward transform in some texts, in the inverse in others. In some texts, both
15
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integral transforms receive a factor of the square root of
2
π
in their denominators. What is important is consistency.
Similar remarks hold for the sign convention in the exponents for forward and inverse transforms.
Now
f
(
t
)
and
F
(
ω
)
are said to constitute a continuous Fourier transform pair which relate the time and frequency
domain representations of the function. Note that, by using the Euler theorem, the forward and inverse transformations
can be sinusoidal and cosinusoidal parts which can be used to represent antisymmetric and symmetric functions,
respectively. Fourier transforms are often easy to calculate, and tables of some of the more common Fourier transform
pairs have been tabulated. Fourier transforms are very useful for solving ordinary and partial differential equations,
since calculus operations (differentiation and integration) transform to algebraic operations. The Fourier transform
also exhibits numerous properties that simplify evaluation of the integrals and manipulation of the results, linearity
being among them. Some differentiation, scaling, and shifting properties are listed below, where each line contains a
Fourier transform pair.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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