Signal
Fouriertransformierte
Fouriertransformierte
Hinweise
g
(
t
)
≡
{\displaystyle g(t)\!\equiv \!}
1
2
π
∫
−
∞
∞
G
(
ω
)
e
i
ω
t
d
ω
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}\mathrm {d} \omega \,}
=
∫
−
∞
∞
G
(
f
)
e
i
2
π
f
t
d
f
{\displaystyle =\int _{-\infty }^{\infty }\!\!G(f)e^{i2\pi ft}\mathrm {d} f\,}
G
(
ω
)
≡
{\displaystyle G(\omega )\!\equiv \!}
1
2
π
∫
−
∞
∞
g
(
t
)
e
−
i
ω
t
d
t
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\mathrm {d} t\,}
G
(
f
)
≡
{\displaystyle G(f)\!\equiv }
∫
−
∞
∞
g
(
t
)
e
−
i
2
π
f
t
d
t
{\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\mathrm {d} t\,}
1
a
⋅
g
(
t
)
+
b
⋅
h
(
t
)
{\displaystyle a\cdot g(t)+b\cdot h(t)\,}
a
⋅
G
(
ω
)
+
b
⋅
H
(
ω
)
{\displaystyle a\cdot G(\omega )+b\cdot H(\omega )\,}
a
⋅
G
(
f
)
+
b
⋅
H
(
f
)
{\displaystyle a\cdot G(f)+b\cdot H(f)\,}
Linearität
2
g
(
t
−
a
)
{\displaystyle g(t-a)\,}
e
−
i
a
ω
G
(
ω
)
{\displaystyle e^{-ia\omega }G(\omega )\,}
e
−
i
2
π
a
f
G
(
f
)
{\displaystyle e^{-i2\pi af}G(f)\,}
Zeitverschiebung
3
e
i
a
t
g
(
t
)
{\displaystyle e^{iat}g(t)\,}
G
(
ω
−
a
)
{\displaystyle G(\omega -a)\,}
G
(
f
−
a
2
π
)
{\displaystyle G\left(f-{\frac {a}{2\pi }}\right)\,}
Frequenzverschiebung (Äquivalent zu Nr. 2)
4
g
(
a
t
)
{\displaystyle g(at)\,}
1
|
a
|
G
(
ω
a
)
{\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}
1
|
a
|
G
(
f
a
)
{\displaystyle {\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,}
5
G
(
t
)
{\displaystyle G(t)\,}
g
(
−
ω
)
{\displaystyle g(-\omega )\,}
g
(
−
f
)
{\displaystyle g(-f)\,}
Dualität der Fouriertransformation durch Vertauschung der Variablen
t
{\displaystyle t\,}
ω
{\displaystyle \omega \,}
6
d
n
g
(
t
)
d
t
n
{\displaystyle {\frac {\mathrm {d} ^{n}g(t)}{\mathrm {d} t^{n}}}\,}
(
i
ω
)
n
G
(
ω
)
{\displaystyle (i\omega )^{n}G(\omega )\,}
(
i
2
π
f
)
n
G
(
f
)
{\displaystyle (i2\pi f)^{n}G(f)\,}
7
t
n
g
(
t
)
{\displaystyle t^{n}g(t)\,}
i
n
d
n
G
(
ω
)
d
ω
n
{\displaystyle i^{n}{\frac {\mathrm {d} ^{n}G(\omega )}{\mathrm {d} \omega ^{n}}}\,}
(
i
2
π
)
n
d
n
G
(
f
)
d
f
n
{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {\mathrm {d} ^{n}G(f)}{\mathrm {d} f^{n}}}\,}
Äquivalent zu Nr. 6
8
(
g
∗
h
)
(
t
)
{\displaystyle (g*h)(t)\,}
2
π
G
(
ω
)
H
(
ω
)
{\displaystyle {\sqrt {2\pi }}\,G(\omega )H(\omega )\,}
G
(
f
)
H
(
f
)
{\displaystyle G(f)H(f)\,}
g
∗
h
{\displaystyle g*h\,}
Faltung von
g
{\displaystyle g\,}
h
{\displaystyle h\,}
9
g
(
t
)
h
(
t
)
{\displaystyle g(t)h(t)\,}
(
G
∗
H
)
(
ω
)
2
π
{\displaystyle (G*H)(\omega ) \over {\sqrt {2\pi }}\,}
(
G
∗
H
)
(
f
)
{\displaystyle (G*H)(f)\,}
Äquivalent zu Nr. 8
Quadratisch integrierbare Funktionen
Signal
Fouriertransformierte
Fouriertransformierte
Hinweise
g
(
t
)
≡
{\displaystyle g(t)\!\equiv \!}
1
2
π
∫
−
∞
∞
G
(
ω
)
e
i
ω
t
d
ω
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}\mathrm {d} \omega \,}
=
∫
−
∞
∞
G
(
f
)
e
i
2
π
f
t
d
f
{\displaystyle =\int _{-\infty }^{\infty }\!\!G(f)e^{i2\pi ft}\mathrm {d} f\,}
G
(
ω
)
≡
{\displaystyle G(\omega )\!\equiv \!}
1
2
π
∫
−
∞
∞
g
(
t
)
e
−
i
ω
t
d
t
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\mathrm {d} t\,}
G
(
f
)
≡
{\displaystyle G(f)\!\equiv }
∫
−
∞
∞
g
(
t
)
e
−
i
2
π
f
t
d
t
{\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\mathrm {d} t\,}
10
exp
(
−
a
t
2
2
)
{\displaystyle \exp \left(-{\frac {at^{2}}{2}}\right)\,}
1
a
⋅
exp
(
−
ω
2
2
a
)
{\displaystyle {\frac {1}{\sqrt {a}}}\cdot \exp \left(-{\frac {\omega ^{2}}{2a}}\right)}
2
π
a
exp
(
−
2
π
a
⋅
π
f
2
)
{\displaystyle {\begin{matrix}{\sqrt {\frac {2\pi }{a}}}\end{matrix}}\exp \left(-{\begin{matrix}{\frac {2\pi }{a}}\end{matrix}}\cdot \pi f^{2}\right)}
Die Gaußsche Funktion
exp
(
−
t
2
/
2
)
{\displaystyle \exp(-t^{2}/2)}
R
e
(
a
)
>
0
{\displaystyle \mathrm {Re} (a)>0}
11
r
e
c
t
(
a
t
)
{\displaystyle \mathrm {rect} (at)\,}
1
2
π
|
a
|
⋅
s
i
n
c
(
ω
2
a
)
{\displaystyle {\frac {1}{{\sqrt {2\pi }}|a|}}\cdot \mathrm {sinc} \left({\frac {\omega }{2a}}\right)}
1
|
a
|
⋅
s
i
n
c
(
π
f
a
)
{\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} \left(\pi \ {\frac {f}{a}}\right)}
Die Rechteckfunktion und die sinc-Funktion .
12
s
i
n
c
(
a
t
)
≡
s
i
n
(
a
t
)
a
t
{\displaystyle \mathrm {sinc} (at)\equiv {\frac {\mathrm {sin} (at)}{at}}\,}
1
|
a
|
π
2
⋅
r
e
c
t
(
ω
2
a
)
{\displaystyle {\frac {1}{|a|}}{\sqrt {\frac {\pi }{2}}}\cdot \mathrm {rect} \left({\frac {\omega }{2a}}\right)}
π
|
a
|
⋅
r
e
c
t
(
π
a
f
)
{\displaystyle {\frac {\pi }{|a|}}\cdot \mathrm {rect} \left({\frac {\pi }{a}}f\right)}
Äquivalent zu Nr. 11. Die Rechteckfunktion ist ein idealisierter Tiefpassfilter, und die sinc-Funktion ist die akausale Stoßantwort eines solchen Filters.
13
exp
(
−
a
|
t
|
)
{\displaystyle \exp \left(-a|t|\right)}
2
π
a
ω
2
+
a
2
{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {a}{\omega ^{2}+a^{2}}}}
2
a
(
2
π
f
)
2
+
a
2
{\displaystyle {\frac {2a}{(2\pi f)^{2}+a^{2}}}}
a
>
0
{\displaystyle a>0}
exponentiell abfallenden Funktion ist eine Lorentzkurve .
14
1
t
2
+
a
2
{\displaystyle {\frac {1}{t^{2}+a^{2}}}}
π
2
1
a
exp
(
−
a
|
ω
|
)
{\displaystyle {\sqrt {\frac {\pi }{2}}}{\frac {1}{a}}\exp \left(-a|\omega |\right)}
π
a
exp
(
−
2
π
a
|
f
|
)
{\displaystyle {\frac {\pi }{a}}\exp \left(-2\pi a|f|\right)}
Äquivalent zu Nr. 13.
Distributionen
Signal
Fouriertransformierte
Fouriertransformierte
Hinweise
g
(
t
)
≡
{\displaystyle g(t)\!\equiv \!}
1
2
π
∫
−
∞
∞
G
(
ω
)
e
i
ω
t
d
ω
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}\mathrm {d} \omega \,}
=
∫
−
∞
∞
G
(
f
)
e
i
2
π
f
t
d
f
{\displaystyle =\int _{-\infty }^{\infty }\!\!G(f)e^{i2\pi ft}\mathrm {d} f\,}
G
(
ω
)
≡
{\displaystyle G(\omega )\!\equiv \!}
1
2
π
∫
−
∞
∞
g
(
t
)
e
−
i
ω
t
d
t
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\mathrm {d} t\,}
G
(
f
)
≡
{\displaystyle G(f)\!\equiv }
∫
−
∞
∞
g
(
t
)
e
−
i
2
π
f
t
d
t
{\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\mathrm {d} t\,}
15
1
{\displaystyle 1\,}
2
π
⋅
δ
(
ω
)
{\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )\,}
δ
(
f
)
{\displaystyle \delta (f)\,}
δ
(
ω
)
{\displaystyle \delta (\omega )}
Delta-Distribution .
16
δ
(
t
)
{\displaystyle \delta (t)\,}
1
2
π
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}
1
{\displaystyle 1\,}
Äquivalent zu Nr. 15.
17
e
i
a
t
{\displaystyle e^{iat}\,}
2
π
⋅
δ
(
ω
−
a
)
{\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)\,}
δ
(
f
−
a
2
π
)
{\displaystyle \delta (f-{\frac {a}{2\pi }})\,}
Folgt aus Nr. 3 und 15.
18
cos
(
a
t
)
{\displaystyle \cos(at)\,}
2
π
δ
(
ω
−
a
)
+
δ
(
ω
+
a
)
2
{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!+\!\delta (\omega \!+\!a)}{2}}\,}
δ
(
f
−
a
2
π
)
+
δ
(
f
+
a
2
π
)
2
{\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!+\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\,}
Folgt aus Nr. 1 und 17
19
sin
(
a
t
)
{\displaystyle \sin(at)\,}
2
π
δ
(
ω
−
a
)
−
δ
(
ω
+
a
)
2
i
{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!-\!\delta (\omega \!+\!a)}{2i}}\,}
δ
(
f
−
a
2
π
)
−
δ
(
f
+
a
2
π
)
2
i
{\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!-\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2i}}\,}
20
t
n
{\displaystyle t^{n}\,}
i
n
2
π
δ
(
n
)
(
ω
)
{\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}
(
i
2
π
)
n
δ
(
n
)
(
f
)
{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(f)\,}
Hier ist
n
{\displaystyle n}
Natürliche Zahl .
δ
n
(
ω
)
{\displaystyle \delta ^{n}(\omega )}
n
{\displaystyle n}
Delta-Distribution .
21
1
t
{\displaystyle {\frac {1}{t}}\,}
−
i
π
2
sgn
(
ω
)
{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )\,}
−
i
π
⋅
sgn
(
f
)
{\displaystyle -i\pi \cdot \operatorname {sgn}(f)\,}
22
1
t
n
{\displaystyle {\frac {1}{t^{n}}}\,}
−
i
π
2
⋅
(
−
i
ω
)
n
−
1
(
n
−
1
)
!
sgn
(
ω
)
{\displaystyle -i{\begin{matrix}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(\omega )\,}
−
i
π
(
−
i
2
π
f
)
n
−
1
(
n
−
1
)
!
sgn
(
f
)
{\displaystyle -i\pi {\begin{matrix}{\frac {(-i2\pi f)^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(f)\,}
23
sgn
(
t
)
{\displaystyle \operatorname {sgn}(t)\,}
2
π
⋅
1
i
ω
{\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\ \omega }}\,}
1
i
π
f
{\displaystyle {\frac {1}{i\pi f}}\,}
24
Θ
(
t
)
{\displaystyle \Theta (t)\,}
π
2
(
1
i
π
ω
+
δ
(
ω
)
)
{\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)\,}
1
2
(
1
i
π
f
+
δ
(
f
)
)
{\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi f}}+\delta (f)\right)\,}
Θ
(
t
)
{\displaystyle \Theta (t)}
Heaviside-Funktion ).
25
∑
n
=
−
∞
∞
δ
(
t
−
n
T
)
{\displaystyle \sum _{n=-\infty }^{\infty }\delta (t-nT)\,}
2
π
T
∑
k
=
−
∞
∞
δ
(
ω
−
k
2
π
T
)
{\displaystyle {\begin{matrix}{\frac {\sqrt {2\pi }}{T}}\end{matrix}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\begin{matrix}{\frac {2\pi }{T}}\end{matrix}}\right)\,}
1
T
∑
k
=
−
∞
∞
δ
(
f
−
k
T
)
{\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)\,}
![{\displaystyle ={\frac {x_{0}}{2{\sqrt {2\pi }}}}\left[-{1 \over {\frac {1}{\tau }}-\mathrm {i} (\omega _{s}-\omega )}e^{-t({\frac {1}{\tau }}-\mathrm {i} (\omega _{s}-\omega ))}-{1 \over {\frac {1}{\tau }}+\mathrm {i} (\omega _{s}+\omega )}e^{-t({\frac {1}{\tau }}+\mathrm {i} (\omega _{s}+\omega ))}\right]_{0}^{\infty }}](/media/api/rest_v1/media/math/render/svg/4f78e234649a471a1d81e2c3f01ab90d9932563e)
