Signal
Fourier transform
Fourier transform
Remarks
g
~
(
t
)
{\displaystyle {\tilde {g}}(t)\,}
G
(
ω
)
=
d
e
f
{\displaystyle G(\omega )\!\ {\stackrel {\mathrm {def} }{=}}\ \!}
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}\,dt}
G
(
f
)
=
d
e
f
{\displaystyle G(f)\!\ {\stackrel {\mathrm {def} }{=}}\ }
G
[
k
]
=
1
T
o
∫
0
T
o
g
(
t
)
e
−
i
2
π
k
f
o
t
d
t
{\displaystyle G[k]={\frac {1}{T_{o}}}\int _{0}^{T_{o}}\!\!g(t)e^{-i2\pi kf_{o}t}\,dt}
301
1
{\displaystyle 1\,}
2
π
⋅
δ
(
ω
)
{\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )\,}
δ
(
f
)
{\displaystyle \delta (f)\,}
δ
(
ω
)
{\displaystyle \displaystyle \delta (\omega )}
Dirac delta distribution.
302
δ
(
t
)
{\displaystyle \delta (t)\,}
1
2
π
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}
1
{\displaystyle 1\,}
Dual of rule 301.
303
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 }})\,}
This follows from and 103 and 301.
304
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}}\,}
Follows from rules 101 and 303 using Euler's formula :
cos
(
a
t
)
=
(
e
i
a
t
+
e
−
i
a
t
)
/
2.
{\displaystyle \displaystyle \cos(at)=(e^{iat}+e^{-iat})/2.}
305
sin
(
a
t
)
{\displaystyle \sin(at)\,}
i
2
π
δ
(
ω
+
a
)
−
δ
(
ω
−
a
)
2
{\displaystyle i{\sqrt {2\pi }}{\frac {\delta (\omega \!+\!a)\!-\!\delta (\omega \!-\!a)}{2}}\,}
i
δ
(
f
+
a
2
π
)
−
δ
(
f
−
a
2
π
)
2
{\displaystyle i{\frac {\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!-\!\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\,}
Also from 101 and 303 using
sin
(
a
t
)
=
(
e
i
a
t
−
e
−
i
a
t
)
/
(
2
i
)
.
{\displaystyle \displaystyle \sin(at)=(e^{iat}-e^{-iat})/(2i).}
306
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)\,}
Here,
n
{\displaystyle \displaystyle n}
natural number .
δ
n
(
ω
)
{\displaystyle \displaystyle \delta ^{n}(\omega )}
n
{\displaystyle \displaystyle n}
polynomials .
307
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)\,}
Here
sgn
(
ω
)
{\displaystyle \displaystyle \operatorname {sgn}(\omega )}
sign function ; note that this is consistent with rules 107 and 302.
308
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)\,}
Generalization of rule 307.
309
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}}\,}
The dual of rule 307.
310
u
(
t
)
{\displaystyle u(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)\,}
Here
u
(
t
)
{\displaystyle u(t)}
unit step function ; this follows from rules 101 and 309.
311
e
−
a
t
u
(
t
)
{\displaystyle e^{-at}u(t)\,}
1
2
π
(
a
+
i
ω
)
{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}
1
a
+
i
2
π
f
{\displaystyle {\frac {1}{a+i2\pi f}}}
u
(
t
)
{\displaystyle u(t)}
unit step function and
a
>
0
{\displaystyle a>0}
312
g
[
n
]
=
∑
n
=
−
∞
∞
g
(
n
T
)
δ
(
t
−
n
T
)
{\displaystyle g[n]=\sum _{n=-\infty }^{\infty }g(nT)\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)\,}
The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.
![{\displaystyle G[k]={\frac {1}{T_{o}}}\int _{0}^{T_{o}}\!\!g(t)e^{-i2\pi kf_{o}t}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0be498e8335669534253ad20672ad85321785d0f)
![{\displaystyle g[n]=\sum _{n=-\infty }^{\infty }g(nT)\delta (t-nT)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27d50a65cf37314afd21e33dd79f955f2f669ed5)
