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
Définisi
édit
Sifat Transformasi Fourier
édit
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 ),\,}
Transformasi Fourier tina sawatara sinyal nu mangfaat
édit
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}}}\,}
Tempo ogé
édit
Всё о Mathcad 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. Tumbu kaluar
édit
![{\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)
