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Engineering Handbook/Mathematics/Fourier Transformation
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Engineering Handbook
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Mathematics
Fourier Transform
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F
(
j
ω
)
=
F
{
f
(
t
)
}
=
∫
−
∞
∞
f
(
t
)
e
−
j
ω
t
d
t
{\displaystyle F(j\omega )={\mathcal {F}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }f(t)e^{-j\omega t}dt}
Inverse Fourier Transform
edit
F
−
1
{
F
(
j
ω
)
}
=
f
(
t
)
=
1
2
π
∫
−
∞
∞
F
(
j
ω
)
e
j
ω
t
d
ω
{\displaystyle {\mathcal {F}}^{-1}\left\{F(j\omega )\right\}=f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega }
Table of Fourier Transforms
edit
This table contains some of the most commonly encountered Fourier transforms.
Time Domain
Frequency Domain
x
(
t
)
=
F
−
1
{
X
(
ω
)
}
{\displaystyle x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}}
X
(
ω
)
=
F
{
x
(
t
)
}
{\displaystyle X(\omega )={\mathcal {F}}\left\{x(t)\right\}}
1
X
(
j
ω
)
=
∫
−
∞
∞
x
(
t
)
e
−
j
ω
t
d
t
{\displaystyle X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt}
x
(
t
)
=
1
2
π
∫
−
∞
∞
X
(
ω
)
e
j
ω
t
d
ω
{\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega }
2
1
{\displaystyle 1\,}
2
π
δ
(
ω
)
{\displaystyle 2\pi \delta (\omega )\,}
3
−
0.5
+
u
(
t
)
{\displaystyle -0.5+u(t)\,}
1
j
ω
{\displaystyle {\frac {1}{j\omega }}\,}
4
δ
(
t
)
{\displaystyle \delta (t)\,}
1
{\displaystyle 1\,}
5
δ
(
t
−
c
)
{\displaystyle \delta (t-c)\,}
e
−
j
ω
c
{\displaystyle e^{-j\omega c}\,}
6
u
(
t
)
{\displaystyle u(t)\,}
π
δ
(
ω
)
+
1
j
ω
{\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}
7
e
−
b
t
u
(
t
)
(
b
>
0
)
{\displaystyle e^{-bt}u(t)\,(b>0)}
1
j
ω
+
b
{\displaystyle {\frac {1}{j\omega +b}}\,}
8
cos
ω
0
t
{\displaystyle \cos \omega _{0}t\,}
π
[
δ
(
ω
+
ω
0
)
+
δ
(
ω
−
ω
0
)
]
{\displaystyle \pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,}
9
cos
(
ω
0
t
+
θ
)
{\displaystyle \cos(\omega _{0}t+\theta )\,}
π
[
e
−
j
θ
δ
(
ω
+
ω
0
)
+
e
j
θ
δ
(
ω
−
ω
0
)
]
{\displaystyle \pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,}
10
sin
ω
0
t
{\displaystyle \sin \omega _{0}t\,}
j
π
[
δ
(
ω
+
ω
0
)
−
δ
(
ω
−
ω
0
)
]
{\displaystyle j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,}
11
sin
(
ω
0
t
+
θ
)
{\displaystyle \sin(\omega _{0}t+\theta )\,}
j
π
[
e
−
j
θ
δ
(
ω
+
ω
0
)
−
e
j
θ
δ
(
ω
−
ω
0
)
]
{\displaystyle j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,}
12
rect
(
t
τ
)
{\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}
τ
sinc
(
τ
ω
2
π
)
{\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,}
13
τ
sinc
(
τ
t
2
π
)
{\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,}
2
π
rect
(
ω
τ
)
{\displaystyle 2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}
14
(
1
−
2
|
t
|
τ
)
rect
(
t
τ
)
{\displaystyle \left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}
τ
2
sinc
2
(
τ
ω
4
π
)
{\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,}
15
τ
2
sinc
2
(
τ
t
4
π
)
{\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,}
2
π
(
1
−
2
|
ω
|
τ
)
rect
(
ω
τ
)
{\displaystyle 2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}
16
e
−
a
|
t
|
,
ℜ
{
a
}
>
0
{\displaystyle e^{-a|t|},\Re \{a\}>0\,}
2
a
a
2
+
ω
2
{\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}\,}
Notes:
sinc
(
x
)
=
sin
(
π
x
)
/
(
π
x
)
{\displaystyle {\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)}
rect
(
t
τ
)
{\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)}
is the rectangular pulse function of width
τ
{\displaystyle \tau }
u
(
t
)
{\displaystyle u(t)}
is the Heaviside step function
δ
(
t
)
{\displaystyle \delta (t)}
is the Dirac delta function
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![{\displaystyle \pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ba663be89105af0a5374d28953425c223fabf8)
![{\displaystyle \pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d585473a707e4abf1ace17e47a7434ac7020d3c6)
![{\displaystyle j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391da94b975afa006cd1d258fdc30b8cbf0c99a7)
![{\displaystyle j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f00590a519f990c76c17e433f8608d011c34e82f)
is the rectangular pulse function of width 
is the Heaviside step function
is the Dirac delta function
