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,

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'' by ...

. 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. Gabor applied ideas from

quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

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 :

g_(x) = g(x - a\ell)e^, \quad -\infty < \ell,n < \infty,

where ''a'' and ''b'' are constants and ''g'' is a fixed function in ''L''²(R), such that , , ''g'', , = 1. Depending on

a

b

, and

g

, 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 :

g(t)=A \cdot e^= A \cdot \exp(-\pi t^2)

one gets the Gabor transform.

References

External links

NuHAG homepage
umerical Harmonic Analysis Group {{DEFAULTSORT:Gabor Atom Wavelets Fourier analysis

Overview

Mathematical definition

See also

References

Further reading

External links