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Transformasi Fourier nyéta hiji alat matematis anu ngawincik fungsi non-périodik kana fungsi-fungsi sinusoida anu nyusunna. Tranformasi Fourier ogé mangrupa alat pikeun ngarobah fungsi waktu kana wujud fungsi frékuénsi.
Dina matématika , lamun fungsi périodik bisa diwincik kana sajumlah dérét fungsi anu disebut deret Fourier ku rumus
x
(
t
)
=
∑
n
=
−
∞
∞
c
n
e
j
n
ω
o
t
.
{\displaystyle x(t)=\sum _{n=-\infty }^{\infty }c_{n}e^{jn\omega _{o}t}.}
dérét Fourier
Lamun x(t) mangrupa hiji sinyal non-périodik. Mangka transformasi Fourier x(t), anu dilambangkeun ku
F
{\displaystyle {\mathcal {F}}}
X
(
ω
)
=
F
{
x
(
t
)
}
=
∫
−
∞
∞
x
(
t
)
e
−
j
ω
t
d
t
{\displaystyle X(\omega )={\mathcal {F}}\{x(t)\}=\int \limits _{-\infty }^{\infty }x(t)\ e^{-j\omega t}\,dt}
Kabalikan transformasi Fourier
X
(
ω
)
{\displaystyle X(\omega )}
F
′
{\displaystyle {\mathcal {F^{'}}}}
x
(
t
)
=
F
′
{
X
(
ω
)
}
=
1
2
π
∫
−
∞
∞
X
(
ω
)
e
j
ω
t
d
ω
,
{\displaystyle x(t)={\mathcal {F}}^{'}\{X(\omega )\}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )\ e^{j\omega t}\,d\omega ,}
t .di mana
x
(
t
)
j
e
u
n
g
X
(
ω
)
{\displaystyle x(t)jeungX(\omega )}
Urang ngagunakeun perlambang
x
(
t
)
⟺
F
X
(
ω
)
{\displaystyle x(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega )}
x (t ) jeung X (ω) mangrupa pasangan transformasi Fourier.
1. Liniéritas (superposisi):
a
⋅
x
1
(
t
)
+
b
⋅
x
2
(
t
)
⟺
F
a
⋅
X
1
(
ω
)
+
b
⋅
X
2
(
ω
)
{\displaystyle a\cdot x_{1}(t)+b\cdot x_{2}(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad a\cdot X_{1}(\omega )+b\cdot X_{2}(\omega )}
2. Kakalian
x
1
(
t
)
⋅
x
2
(
t
)
{\displaystyle x_{1}(t)\cdot x_{2}(t)\,}
⟺
F
1
2
π
⋅
(
X
1
∗
X
2
)
(
ω
)
{\displaystyle {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{\sqrt {2\pi }}}\cdot (X_{1}*X_{2})(\omega )\,}
(konvensasi normalisasi uniter)
⟺
F
1
2
π
⋅
(
X
1
∗
X
2
)
(
ω
)
{\displaystyle {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2\pi }}\cdot (X_{1}*X_{2})(\omega )\,}
(konvensi non-uniter)
⟺
F
(
F
∗
G
)
(
f
)
{\displaystyle {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad (F*G)(f)}
(frékuénsi biasa)
3. Modulasi:
x
(
t
)
⋅
cos
ω
0
t
⟺
F
1
2
[
X
(
ω
+
ω
0
)
+
X
(
ω
−
ω
0
)
]
,
ω
0
∈
R
f
(
t
)
⋅
sin
ω
0
t
⟺
F
j
2
[
X
(
ω
+
ω
0
)
−
X
(
ω
−
ω
0
)
]
x
(
t
)
⋅
e
j
ω
0
t
⟺
F
X
(
ω
−
ω
0
)
{\displaystyle {\begin{aligned}x(t)\cdot \cos \omega _{0}t&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2}}[X(\omega +\omega _{0})+X(\omega -\omega _{0})],\qquad \omega _{0}\in \mathbb {R} \\f(t)\cdot \sin \omega _{0}t&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {j}{2}}[X(\omega +\omega _{0})-X(\omega -\omega _{0})]\\x(t)\cdot e^{j\omega _{0}t}&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega -\omega _{0})\end{aligned}}\,}
4. Géséran waktu
x
(
t
−
t
0
)
⟺
F
e
−
j
ω
t
0
⋅
X
(
ω
)
{\displaystyle x(t-t_{0})\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad e^{-j\omega t_{0}}\cdot X(\omega )}
5. Géséran frékuénsi:
x
(
t
)
⋅
e
j
ω
o
t
⟺
F
X
(
ω
−
ω
o
)
{\displaystyle x(t)\cdot e^{j\omega _{o}t}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega -\omega _{o})}
6. Skala:
x
(
a
t
)
⟺
F
1
|
a
|
X
(
ω
a
)
,
a
∈
R
,
a
≠
0
{\displaystyle x(at)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{|a|}}X{\biggl (}{\frac {\omega }{a}}{\biggr )},\qquad a\in \mathbb {R} ,a\neq 0}
7. Lawan / kabalikan waktu:
x
(
−
t
)
⟺
F
X
(
−
ω
)
{\displaystyle x(-t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(-\omega )}
8. Dualitas:
X
(
t
)
⟺
F
2
π
x
(
−
ω
)
{\displaystyle X(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad 2\pi x(-\omega )}
9. Diferensiasi waktu:
x
′
(
t
)
=
d
x
(
t
)
d
t
x
(
t
)
⟺
F
j
ω
X
(
ω
)
{\displaystyle x^{'}(t)={\frac {dx(t)}{dt}}x(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad j\omega X(\omega )}
10. Diferensiasi frékuénsi:
(
−
j
t
)
x
(
t
)
⟺
F
X
′
(
ω
)
=
d
X
(
ω
)
d
w
{\displaystyle (-jt)x(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X^{'}(\omega )={\frac {dX(\omega )}{dw}}}
11. Integrasi:
∫
−
∞
t
x
(
τ
)
d
τ
⟺
F
1
j
ω
X
(
ω
)
+
π
X
(
0
)
⋅
δ
(
ω
)
,
{\displaystyle \int _{-\infty }^{t}x(\tau )\,d\tau \quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{j\omega }}X(\omega )+\pi X(0)\cdot \delta (\omega ),\,}
No.
Fungsi waktu
Transfirmasi Fourier (doméin Frékuénsi)
1.
δ
(
t
)
{\displaystyle \delta (t)}
1
2.
δ
(
t
−
t
o
)
{\displaystyle \delta (t-t_{o})}
e
−
j
ω
t
o
{\displaystyle e^{-j\omega t_{o}}}
3.
1
2
π
δ
(
ω
)
{\displaystyle 2\pi \delta (\omega )}
4.
e
j
ω
t
o
{\displaystyle e^{j\omega t_{o}}}
2
π
δ
(
ω
−
ω
o
)
{\displaystyle 2\pi \delta (\omega -\omega _{o})}
5.
cos
(
ω
o
t
)
{\displaystyle \cos(\omega _{o}t)\,}
π
δ
(
ω
−
ω
o
)
+
π
δ
(
ω
+
ω
o
)
{\displaystyle \pi \delta (\omega -\omega _{o})+\pi \delta (\omega +\omega _{o})}
6.
sin
(
ω
o
t
)
{\displaystyle \sin(\omega _{o}t)\,}
−
j
π
δ
(
ω
−
ω
o
)
+
j
π
δ
(
ω
+
ω
o
)
{\displaystyle -j\pi \delta (\omega -\omega _{o})+j\pi \delta (\omega +\omega _{o})}
7.
u
(
t
)
{\displaystyle u(t)}
π
δ
(
ω
)
+
1
j
ω
{\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}
8.
e
−
a
t
u
(
t
)
{\displaystyle e^{-at}u(t)}
1
j
ω
+
a
{\displaystyle {\frac {1}{j\omega +a}}\,}
9.
e
−
a
|
t
|
{\displaystyle e^{-a|t|}}
2
a
ω
2
+
a
2
{\displaystyle {\frac {2a}{\omega ^{2}+a^{2}}}\,}
Всё о Mathcad Archived 2019-10-20 di Wayback Machine Citakan:Ref-ru Fourier Transforms from eFunda - includes tablesDym & McKéan, Fourier Series and Integrals . (For réaders with a background in mathematical analysis .)
K. Yosida, Functional Analysis , Springer-Verlag, 1968. ISBN 3-540-58654-7
L. Hörmander, Linear Partial Differential Operators , Springer-Verlag, 1976. (Somewhat terse.)
A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations , CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
R. G. Wilson, "Fourier Series and Optical Transform Techniques in Contemporary Optics", Wiley, 1995. ISBN 0471303577
R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., Boston, McGraw Hill, 2000. 
![{\displaystyle {\begin{aligned}x(t)\cdot \cos \omega _{0}t&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2}}[X(\omega +\omega _{0})+X(\omega -\omega _{0})],\qquad \omega _{0}\in \mathbb {R} \\f(t)\cdot \sin \omega _{0}t&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {j}{2}}[X(\omega +\omega _{0})-X(\omega -\omega _{0})]\\x(t)\cdot e^{j\omega _{0}t}&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega -\omega _{0})\end{aligned}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7d22764d22d1e60781395392b38083713f813f)
