Fourier transform

(redirected from Fourier transformation)
Also found in: Dictionary, Medical.

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 ?
Discrete Fourier Transformation (DFT) is a method transforming the discrete time signal used in many fields of engineering into frequency period [17-20].
A Fourier Transform converts time to frequency and vice versa, Similar to the Cooley-Tukey Fast Fourier Transformation (FFT) algorithm, the planned algorithm can generate the next higher-order DCT from two identical lower-order DCTs.
Let us consider the work of elementary digital filter (EDF) on the basis of the sliding discrete complex Fourier transformation [8, 9].
Therefore, using a Fourier transformation, which is well determined, the required formula for the energetic spectrum in the coastal zone can be derived:
Analysis operators corresponding to generalized frames are everywhere defined bounded operators; however, Fourier transformation on [R.
For data acquisition, Fourier transformation carries out highly sensitive and accurate deformation amplitudes and the exact phase shift.
Proceeding from this, the Fourier transformation of the measured near-field gives the far-field scattering pattern.
It identified various server functions, such as XML mapping, encryption and fast Fourier transformation (FFT), and SERT measures the power consumption and performance of servers while performing those functions at different load levels.
This occurs for example in field configurations that are locally invariant under Fourier transformation, such as linear and spherical harmonics.
First principle is based on mathematical methods like Fast Fourier Transformation (FFT) or wavelet transformation.
No zero-filling or digital filtering was used prior to Fourier transformation.
The spectrogram calculated by Short Time Fourier Transformation is shown in the figure 6.

Full browser ?