फलन
रूपान्तर
टिप्पणी
1
a
f
(
t
)
+
b
g
(
t
)
{\displaystyle af(t)+bg(t)\,}
a
F
(
ω
)
+
b
G
(
ω
)
{\displaystyle aF(\omega )+bG(\omega )\,}
रैखिकता
2
f
(
t
−
a
)
{\displaystyle f(t-a)\,}
e
−
i
ω
a
F
(
ω
)
{\displaystyle e^{-i\omega a}F(\omega )\,}
विलम्ब (delay)
3
e
i
a
t
f
(
t
)
{\displaystyle e^{iat}f(t)\,}
F
(
ω
−
a
)
{\displaystyle F(\omega -a)\,}
आवृत्ति शिफ्ट
4
f
(
a
t
)
{\displaystyle f(at)\,}
|
a
|
−
1
F
(
ω
a
)
{\displaystyle |a|^{-1}F\left({\frac {\omega }{a}}\right)\,}
यदि
a
{\displaystyle a}
f
(
a
t
)
{\displaystyle f(at)}
|
a
|
−
1
F
(
ω
a
)
{\displaystyle |a|^{-1}F\left({\frac {\omega }{a}}\right)}
5
d
n
f
(
t
)
d
t
n
{\displaystyle {\frac {d^{n}f(t)}{dt^{n}}}\,}
(
i
ω
)
n
F
(
ω
)
{\displaystyle (i\omega )^{n}F(\omega )\,}
Свойство преобразования Фурье от
n
{\displaystyle n}
6
t
n
f
(
t
)
{\displaystyle t^{n}f(t)\,}
i
n
d
n
F
(
ω
)
d
ω
n
{\displaystyle i^{n}{\frac {d^{n}F(\omega )}{d\omega ^{n}}}\,}
7
(
f
∗
g
)
(
t
)
{\displaystyle (f*g)(t)\,}
2
π
F
(
ω
)
G
(
ω
)
{\displaystyle {\sqrt {2\pi }}F(\omega )G(\omega )\,}
फलन
f
∗
g
{\displaystyle f*g}
f
{\displaystyle f}
g
{\displaystyle g}
8
f
(
t
)
g
(
t
)
{\displaystyle f(t)g(t)\,}
(
F
∗
G
)
(
ω
)
2
π
{\displaystyle {\frac {(F*G)(\omega )}{\sqrt {2\pi }}}\,}
9
δ
(
t
)
{\displaystyle \delta (t)\,}
1
2
π
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}
δ
(
t
)
{\displaystyle \delta (t)}
10
1
{\displaystyle 1\,}
2
π
δ
(
ω
)
{\displaystyle {\sqrt {2\pi }}\delta (\omega )\,}
11
t
n
{\displaystyle t^{n}\,}
i
n
2
π
δ
(
n
)
(
ω
)
{\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}
12
e
i
a
t
{\displaystyle e^{iat}\,}
2
π
δ
(
ω
−
a
)
{\displaystyle {\sqrt {2\pi }}\delta (\omega -a)\,}
13
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}}\,}
1 और 12 का उपसाध्य ; आयलर प्रमेय से -
cos
(
a
t
)
=
1
2
(
e
i
a
t
+
e
−
i
a
t
)
{\displaystyle \cos(at)={\frac {1}{2}}\left(e^{iat}+e^{-iat}\right)\,}
14
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}}\,}
15
exp
(
−
a
t
2
)
{\displaystyle \exp(-at^{2})\,}
1
2
a
exp
(
−
ω
2
4
a
)
{\displaystyle {\frac {1}{\sqrt {2a}}}\exp \left({\frac {-\omega ^{2}}{4a}}\right)\,}
इससे स्पष्ट है कि गासियन फलन
exp
(
−
t
2
/
2
)
{\displaystyle \exp(-t^{2}/2)}
16
W
2
π
s
i
n
c
(
W
t
)
{\displaystyle W{\sqrt {\frac {2}{\pi }}}\mathrm {sinc} (Wt)\,}
r
e
c
t
(
ω
2
W
)
{\displaystyle \mathrm {rect} \left({\frac {\omega }{2W}}\right)\,}
रेक्टैगुलर फलन अर्थात्, आदर्श लो-पास-फिल्टर
17
1
t
{\displaystyle {\frac {1}{t}}\,}
−
i
π
2
sgn
(
ω
)
{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )\,}
यहाँ
sgn
(
ω
)
{\displaystyle \operatorname {sgn}(\omega )\,}
sgn फलन (चिह्न फलन) है।
18
1
t
n
{\displaystyle {\frac {1}{t^{n}}}\,}
−
i
π
2
(
−
i
ω
)
n
−
1
(
n
−
1
)
!
sgn
(
ω
)
{\displaystyle -i{\sqrt {\frac {\pi }{2}}}{\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )\,}
17 का सामान्यीकृत रूप
19
sgn
(
t
)
{\displaystyle \operatorname {sgn}(t)\,}
2
π
(
i
ω
)
−
1
{\displaystyle {\sqrt {\frac {2}{\pi }}}(i\omega )^{-1}\,}
17 का द्वैत
20
2
π
θ
(
t
)
{\displaystyle {\sqrt {2\pi }}\theta (t)\,}
1
i
ω
+
π
δ
(
ω
)
{\displaystyle {\frac {1}{i\omega }}+\pi \delta (\omega )\,}
यहाँ
θ
(
t
)
{\displaystyle \theta (t)\,}
हेविसाइड फलन है।
