Fourier transform
التعريفات والمعاني
== English ==
=== Alternative forms ===
FT (initialism)
=== Etymology ===
Named after French mathematician and physicist Jean Baptiste Joseph Fourier, who initiated the study of what is now harmonic analysis.
=== Noun ===
Fourier transform (plural Fourier transforms)
(mathematical analysis, harmonic analysis, physics, electrical engineering) A particular integral transform that when applied to a function of time (such as a signal), converts the function to one that plots the original function's frequency composition; the resultant function of such a conversion.
2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 1,
The trigonometric sums of
G
F
{\displaystyle {\mathcal {G}}^{F}}
are thus (up to a scalar) the Fourier transforms of the characteristic functions of the
G
F
{\displaystyle G^{F}\!\!}
-orbits of
G
F
{\displaystyle {\mathcal {G}}^{F}}
.
2012, David Brandwood, Fourier Transforms in Radar and Signal Processing, Artech House, 2nd Edition, page 1,
The Fourier transform is a valuable theoretical technique, used widely in fields such as applied mathematics, statistics, physics, and engineering.
==== Usage notes ====
Like the term transform itself, Fourier transform can mean either the integral operator that converts a function, or the function that is the end product of the conversion process.
The Fourier transform of a function
f
{\displaystyle f}
is traditionally denoted
f
^
{\displaystyle {\hat {f}}}
. Several other notations are also used.
There are also several different conventions used when it comes to defining the Fourier transform and its inverse for an integrable function
f
:
R
→
C
{\displaystyle f:\mathbb {R} \to \mathbb {C} }
. (The two are often defined together to highlight their connectedness.)
One form of this definition pair is:
f
^
(
ξ
)
=
∫
−
∞
∞
f
(
x
)
e
−
2
π
i
x
ξ
d
x
{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx}
f
(
x
)
=
∫
−
∞
∞
f
^
(
ξ
)
e
2
π
i
x
ξ
d
ξ
{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi ix\xi }\,d\xi }
,
where the exponent (including its sign) reflects a convention in electrical engineering to use
f
(
x
)
=
e
2
π
i
ξ
0
x
{\displaystyle f(x)=e^{2\pi i\xi _{0}x}}
for a signal with initial phase 0 and frequency
ξ
0
{\displaystyle \xi _{0}}
.
==== Hypernyms ====
integral transform
==== Derived terms ====
inverse Fourier transform
continuous Fourier transform (CFT)
discrete Fourier transform (DFT)
fast Fourier transform (FFT)
Fourier-transform infrared spectroscopy (FTIR)
==== Translations ====
=== Further reading ===
Fourier analysis on Wikipedia.Wikipedia
Harmonic analysis on Wikipedia.Wikipedia
Fourier series on Wikipedia.Wikipedia
Time–frequency analysis on Wikipedia.Wikipedia
Wavelet transform on Wikipedia.Wikipedia