Fourier transform


Also found in: Dictionary, Thesaurus, Medical, Acronyms, Wikipedia.

Fourier transform

A mathematical operation by which a function expressed in terms of one variable, x , may be related to a function of a different variable, s , in a manner that finds wide application in physics. The Fourier transform, F(s ), of the function f(x) is given by
F(s) = f(x) exp(–2πixs) dx 
and
f(x) = F(s) exp(2πixs) ds

The variables x and s are often called Fourier pairs. Many such pairs are useful, for example, time and frequency: the Fourier transform of an electrical oscillation in time gives the spectrum, or the power contained in it at different frequencies.

Fourier Transform

 

The Fourier transform of a function f(x) is the function f(x)

If f(x) is even, then g(u) takes the form

(the cosine transform), and if f(x) is odd, then g(u) takes the form

(the sine transform).

Formulas (1), (2), and (3) are invertible, that is, for even functions

for odd functions

and in general

To every operation on functions there corresponds an operation on their Fourier transforms that is frequently simpler than the operation on f(x). For example, the Fourier transform of f’(x) is iug(u). If

then g(u) = g1(u)g2(u). The Fourier transform of f(x + a) is eiuag(u), and the Fourier transform of c1f1(x) is c1g1(u) + c2g2(u).

If

exists, then the integrals in formulas (1) and (6) converge in the mean (seeCONVERGENCE) and

(Plancherel’s theorem). Formula (8) is the extension of Parse-val’s formula to Fourier transforms (seePARSEVAL EQUALITY). In physical terms, formula (8) states that the energy of a certain vibration is the sum of the energies of its harmonic components. The map F:f(x) → g(u) is a unitary operator in the Hubert space of square integrable functions f(x), – ∞ < x < ∞. This operator can also be represented in the form

Under certain conditions on f(x) we have the Poisson formula

which is of use in the theory of theta functions.

If the function f(x) decreases sufficiently rapidly, then it is also possible to define its Fourier transform for certain complex values u = v + iw. For example, if

exists, then the Fourier transform is defined for |w| < a. There is a close connection between the complex Fourier transform and the two-sided Laplace transform

The Fourier transform operator can be extended to classes of functions larger than the class of integrable functions; for example, if (1 + |x|)–1f(x) is integrable, then the Fourier transform of f(x) is given by formula (9). The operator can even be extended to certain classes of generalized functions (generalized functions with a slow rate of growth).

There exist generalizations of the Fourier transform. One of them makes use of various special funtions, such as Bessel functions. This generalization is fully developed in the representation theory of continuous groups. Another generalization is the Fourier-Stieltjes transform, which is frequently applied in, for example, probability theory; it is defined for every bounded nondecreasing function ϕ(x) by means of the Stieltjes integral

and is called the characteristic function of the distribution ϕ. According to the Bochner-Khinchin theorem, for g(u) to be representable by (10) it is necessary and sufficient that

for every choice of u1, . . ., un, ξ1, . . ., ξn.

First introduced in heat conduction theory, the Fourier transform has many applications in mathematics. For example, it is used in the solution of differential, difference, and integral equations and in the theory of special functions. Many branches of theoretical physics also make use of the transform. Thus, the Fourier transform provides the standard apparatus for quantum field theory and is widely applied in the method of Green’s functions for nonequilibrium problems of quantum mechanics and thermodynamics and in scattering theory.

REFERENCES

Sneddon, I. Preobrazovanie Fur’e. Moscow, 1955. (Translated from English.)
Vladimirov, V. S. Obobshchennye funktsii v matematicheskoi fizike. Moscow, 1976.

Fourier transform

[‚fu̇r·ē‚ā ′tranz‚fȯrm]
(mathematics)
For a function ƒ(t), the function F (x) equal to 1/√(2π) times the integral over t from -∞ to ∞ of ƒ(t) exp (itx).

Fourier transform

(mathematics)
A technique for expressing a waveform as a weighted sum of sines and cosines.

Computers generally rely on the version known as discrete Fourier transform.

Named after J. B. Joseph Fourier (1768 -- 1830).

See also wavelet, discrete cosine transform.
References in periodicals archive ?
Analysis on measuring accuracy of Fourier transform profilometry due to different filtering window, Laser Journal, 24(4), 43-45.
Another method of time-frequency analysis is this one of dynamic coefficient of transition K(j[omega], t) (DCT) [1], using Fourier transform technique applied to the pulse response with variable upper limit
We used Fourier transform properties along with its tables in a sensible manner to compute the closed form expressions of these important pulses.
For comparison purposes, the spectra based on Fast Fourier Transform (FFT) were also computed.
In conformity with above definitions and iterating the sums, an arbitrary component of the discrete Fourier transform of the circular convolution of the matrices X and Y is given by the formula
Sometimes the factor 1 /2[pi] is distributed between the two integrals (the Fourier and the Inverse Fourier Transform Integrals) as 1/[square root of (2[pi])] 'Jin.
In this section the tools used are first introduced and then the 2D conformal Fourier transform (2D CFT) algorithm is formulated.
NN] is the inverse Fourier transform operator, F(u) and G(u) are the Fourier transform of the N(x) and g(x) functions.
FORWARD FOURIER TRANSFORM and its inverse furnish a means for data conversion from time to frequency domain and vice versa.
One of the methods used in business today to predict future fluctuations in sales is called The Fast Fourier Transform, in honour of Fourier's work.
Odense, Denmark 4/28/08; Palo Alto, CA 4/28/08 -- Proxeon A/S has entered into an OEM agreement to provide its nanoscale Easy-nLC system to Varian for interface with Varian's 920-MS Fourier Transform triple quadrupole MS.
A recent publication in a biotech journal acknowledges a Top 10 pharmaceutical company has utilized Digilab's sample preparation and fractionation technologies to expose standard and novel diabetes substrates in a Fourier Transform Mass Spectroscopy experimental paradigm (Yates, et al.

Full browser ?