View source for Gabor transform

An [[Integral transform|integral transform]] introduced by D. Gabor, the Hungarian-born Nobel laureate in physics, who, in his paper [[#References|[a3]]], modified the well-known [[Fourier transform|Fourier transform]] of a function (or a signal) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200101.png" /> by introducing a time-localization window function (also called a time-frequency window). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200102.png" /> denote the Fourier transform

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200103.png" /></td> </tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200104.png" /> denote the Gaussian function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200105.png" /></td> </tr></table>

Then the Gabor transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200106.png" /> is defined by

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where the real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200108.png" /> is used to translate the  "window"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200109.png" />. The Gabor transform localizes the Fourier transform at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001010.png" />. A similar transform can be introduced for Fourier series.

By choosing more general windows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001011.png" />, the transforms are called short-time Fourier transform and the Gabor transform is a special case, based on the Gaussian window. One property of the special choice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001012.png" /> is

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which says that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001014.png" /> of Gabor transforms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001015.png" /> decomposes the Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001017.png" /> exactly.

Gabor transforms (and related topics based on the Gabor transform) are applied in numerous engineering applications, many of them without obvious connection to the traditional field of time-frequency analysis for deterministic signals. Detailed information (including many references) about the use of Gabor transforms in such diverse fields as image analysis, object recognition, optics, filter banks, or signal detection can be found in [[#References|[a4]]], the first book devoted to Gabor transforms and related analysis.

A recent development (starting at 1992) that is more effective for analyzing signals with sharp variations is based on wavelets (see [[#References|[a2]]] or [[Wavelet analysis|Wavelet analysis]]); for the relation between wavelets and the Gabor transform, see [[#References|[a1]]]. The Gabor transform can also be viewed in connection with  "coherent states"  associated with the Weyl–Heisenberg group; see [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ch.K. Chui,   "An introduction to wavelets" , Acad. Press  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Daubechies,   "Ten lectures on wavelets" , SIAM (Soc. Industrial Applied Math.)  (1992)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gabor,   "Theory of communication"  ''J. IEE'' , '''93'''  (1946)  pp. 429–457</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.G. Feichtinger,   Th. Strohmer,   "Gabor analysis and algorithms" , Birkhäuser  (1998)</TD></TR></table>