In applied
mathematics, Gabor atoms, or Gabor functions, are
functions used in the analysis proposed by
Dennis Gabor
Dennis Gabor ( ; hu, Gábor Dénes, ; 5 June 1900 – 9 February 1979) was a Hungarian-British electrical engineer and physicist, most notable for inventing holography, for which he later received the 1971 Nobel Prize in Physics. He obtained ...
in 1946 in which a family of functions is built from translations and modulations of a generating function.
Overview
In 1946,
Dennis Gabor
Dennis Gabor ( ; hu, Gábor Dénes, ; 5 June 1900 – 9 February 1979) was a Hungarian-British electrical engineer and physicist, most notable for inventing holography, for which he later received the 1971 Nobel Prize in Physics. He obtained ...
suggested the idea of using a granular system to produce
sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' b ...
. In his work, Gabor discussed the problems with
Fourier analysis. Although he found the mathematics to be correct, it did not reflect the behaviour of sound in the world, because sounds, such as the sound of a siren, have variable frequencies over time. Another problem was the underlying supposition, as we use sine waves analysis, that the signal under concern has infinite duration even though sounds in real life have limited duration – see
time–frequency analysis
In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains ''simultaneously,'' using various time–frequency representations. Rather than viewing a 1-dimensional signal (a ...
. Gabor applied ideas from
quantum physics to sound, allowing an analogy between sound and quanta. He proposed a mathematical method to reduce Fourier analysis into cells. His research aimed at the information transmission through communication channels. Gabor saw in his atoms a possibility to transmit the same information but using less data. Instead of transmitting the signal itself it would be possible to transmit only the coefficients which represent the same signal using his atoms.
Mathematical definition
The Gabor function is defined by
:
where ''a'' and ''b'' are constants and ''g'' is a fixed function in
''L''2(R), such that , , ''g'', , = 1. Depending on
,
, and
, a Gabor system may be a basis for ''L''
2(R), which is defined by translations and modulations. This is similar to a wavelet system, which may form a basis through dilating and translating a mother wavelet.
When one takes
:
one gets the
Gabor transform The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be tran ...
.
See also
*
Gabor filter
*
Gabor wavelet Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabo ...
*
Fourier analysis
*
Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
*
Morlet wavelet
In mathematics, the Morlet wavelet (or Gabor wavelet)0).
The parameter \sigma in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction \sigma>5 is used to avoid problems with the Morlet wavelet a ...
References
Further reading
*Hans G. Feichtinger, Thomas Strohmer: "Gabor Analysis and Algorithms", Birkhäuser, 1998;
*Hans G. Feichtinger, Thomas Strohmer: "Advances in Gabor Analysis", Birkhäuser, 2003;
*Karlheinz Gröchenig: "Foundations of Time-Frequency Analysis", Birkhäuser, 2001;
External links
NuHAG homepage umerical Harmonic Analysis Group
{{DEFAULTSORT:Gabor Atom
Wavelets
Fourier analysis