Fourier-Analysis
The Fourier Analysis (pronounced fuʁie ), also known as Fourier analysis or classical harmonic analysis, the theory of Fourier series and Fourier integrals. Its origins date back to the 18th century. Named are the Fourier Analysis, Fourier series and Fourier integrals after the French mathematician Jean Baptiste Joseph Fourier, who studied in his Théorie de la chaleur Analytique Fourier series in 1822.
The Fourier analysis is in many science and engineering branches of the utmost practical importance. The applications range from physics ( acoustics, optics, tides, Astrophysics ) over many branches of mathematics ( number theory, statistics, combinatorics and probability theory ), the signal processing and cryptography up to oceanography and economics. Depending on the application branch undergoes decomposition to interpretation. In the acoustics, for example, the frequency transformation of sound in harmonics.
From the perspective of the abstract harmonic analysis of both the Fourier series and the Fourier integrals and the Laplace transform of the Mellin transform or Walsh transform are ( here, the trigonometric functions, the Walsh functions replaced) special cases of a more general ( Fourier ) transformation.
- 5.1 Mathematical Foundations
- 5.2 Fourier series
- 5.3 Aperiodic processes ( Fourier integral )
Variants of the Fourier transform
The various terms in this context are not used consistently in the literature and there are several names for the same process. Thus one uses Fourier transform very often as a synonym of the continuous Fourier transform, and Fourier analysis is often the decomposition into a Fourier series meant, but sometimes the continuous transformation.
According to the characteristics of the function to be tested, there are four variants, as shown in the illustration:
Obtained in all transformations a frequency spectrum depending on the variant discrete ( infinitely sharp lines) or is continuous:
Fourier series
Each continuously differentiable function defined on the interval, can develop into a Fourier series, ie, both sides of the transformation exist. At the fundamental frequency and the angular frequencies where:
It can be developed in a Fourier series more general types of functions, as in sections continuous, bounded functions, or more generally measurable square integrable functions.
Continuous Fourier transform
The continuous Fourier transform is defined by
The inverse transformation is this:
In the literature, there are also other definitions that have a pre-factor instead of just 1 or. This depends on the scaling conventions used in each case. The version used here has the aesthetic advantage that the prefactor in return transformation is the same. In addition, it simplifies the presentation of the set of Parseval:
This condition is, for example, in physics important for conservation of energy by the Fourier transform. Mathematically, the equation means that the Fourier transform is a unitary map, which is fundamental in quantum mechanics, among others.
Sometimes, for example in the signal theory, it is preferably - also energy-conserving - version of the Fourier transform, wherein the - also called spectral - Fourier Transform depends on the frequency, instead of the angular velocity:
The relationship between the two kinds of Fourier transformation is taught.
The inverse transform is then
Since in this case the variable instead of being integrated, eliminating in this form of representation of prefactor.
Discrete Fourier transformation,
There are no restrictions in the application of the transformation and the development of formula. Are positive numbers, and any integer shifts, then a more general variant of the transformation formulas are given. With true and
And
For calculating the discrete Fourier transform, the fast Fourier transform (FFT ) is often used, an algorithm, wherein the number of calculation steps for calculating the Fourier coefficients is considerably smaller than in the case of a direct implementation of the integration.
Fourier synthesis
All transformations contemplated in the Fourier analysis, have the property that there is a corresponding inverse transform. In the engineering sciences, physics and numerical mathematics is also called the decomposing a function into its Fourier spectrum analysis. Thus, the term describes not only this area of functional analysis, but also the process of decomposition of a function. The representation of the output function with the help of the spectrum from the Fourier analysis is referred to as Fourier synthesis. Since this term is common for forming particularly in the applied sciences, this occurs even more in relation to the discrete Fourier transform and fast Fourier transform.
Applications
The Fourier transform is comprised mainly in the engineering sciences, such as signal processing and physics, important application areas. This also special terms and nomenclatures are used:
In technically motivated applications of the relation between the time domain with the original function and the frequency range of the image function is also shown with the following symbols:
In physics, the Fourier transformation of the wave mechanics, the relationship between the time domain and the frequency domain dar. If instead of time signals signals as a function of the location considered, represents the Fourier transform of a link between the local area and the present in the frequency-space spatial frequency or wavenumber dar. in several dimensions the wave numbers described in terms of wave vectors. In crystallography of the real space reciprocal space is called reciprocal frequency space.
In quantum mechanics correspond, up to a proportionality factor, the wave numbers of the momentum of the particle, resulting in a connection with the Heisenberg uncertainty principle results. Since position and momentum space are related by the Fourier transform, leads linking the extensions to a blur. Analogously, also the energy - time uncertainty from the Fourier transformation, in which case the frequency of which corresponds to the proportionality factor of the energy and thus a combination of energy and time is given by the Fourier transform, resulting in a blur.
History
Starting in 1740 discussed mathematicians such as Bernoulli and d' Alembert the opportunity to present periodic functions as trigonometric series. The now well-known series expansion for periodic functions goes back to the French mathematician Fourier. At the beginning of the 19th century, he published his work Théorie de la chaleur Analytique, in which he assumes that any function can be expanded in a trigonometric series. He used this series in particular for solving the heat conduction equation. In this work he also introduced the continuous Fourier transform in the form of a cosine transform. With this, he tried to solve the heat equation on unbounded sets in particular on the real axis.
Peter Gustav Lejeune Dirichlet examined these trigonometric series that are now called Fourier series, continue and could prove first convergence properties. So he was able to show in 1829 that the Fourier series converges pointwise when the output function is Lipschitz continuous. For an exact calculation of the Fourier coefficients Bernhard Riemann led then set its integral term and discovered in 1853, the localization principle. This implies that the convergence or divergence and, where appropriate, the value of the Fourier series of a function in the conduct of in an arbitrarily small neighborhood of is uniquely determined.
Only in 1876 was Paul Du Bois- Reymond is a continuous function whose Fourier series does not converge pointwise. In his statement, however, Fejér in 1904 showed that the Fourier series converges for every continuous function on the arithmetic average. Nikolai Nikolaevich Luzin in 1915 raised the question of whether the Fourier series converges for each function. This was not until 1968 that answered positively by Lennart Carleson and Hunt 1968 generalized the result to functions with. However, the requirement is essential, as the example of an integrable function with everywhere divergent Fourier series, the Kolmogorov 1926 found shows.
Because the Fourier transform and outside the mathematics has a wide range of applications, one is interested in an algorithm with which a computer can calculate the Fourier coefficients with a minimum of effort. Such a process is called Fast Fourier Transform. The best known algorithm comes from James Cooley and John W. Tukey, who published it in 1965. However, an algorithm was developed in 1805 by Carl Friedrich Gauss. He used it to calculate the trajectories of the asteroid ( 2) Pallas and (3) Juno. For the first time a variant of the algorithm by Carl Runge in 1903 or 1905 has been published. In addition, prior to Cooley and Tukey already limited variants of the fast Fourier transform has been published. So has also published such an algorithm, for example, Irving John Good 1960.
Mathematical motivation
Mathematical Foundations
We consider steady, of real time t dependent functions or operations ( eg, as a vector-valued functions) f ( t) that are repeated after a time, that are periodic with period T, f ( t T) = f (t). Joseph Fourier postulated in his work, that f can be composed of periodic, harmonic oscillations, ie sine or cosine functions of different phase and amplitude and well-defined frequency. Consider such a composite function with (N 1) summands:
The individual oscillations have the angular frequency, ie the frequency. So that the first oscillation (fundamental ), the frequency, the next, ...
Because a sine wave is only a phase-shifted cosine, the series representation could be limited to cosine functions. We immediately also get the sine terms, if we use the addition theorems:
Along with we obtain a phase-free representation
In the next step, the sum is to be rewritten by using complex numbers. It is then allowed complex coefficients, and the number is complex-valued. If real functions are considered, it can be recovered as a real part of the sum. Of the Euler's formula or as defined in the trigonometric function with the exponential function
Thus
With the complex coefficients, and for n> 0, we obtain a sum with also negative indices
Fourier series
So we now know the trigonometric sum in different representations. But it was asked to approximate a periodic continuous function by means of such a sum. We find that the complex coefficients, and thus also that of the other representations can be recovered from the sum function.
For this, the above equation is multiplied by, and then integrated on both sides over the interval, i.e., over a period. With transformations to reach the following statement:
It follows
For the integral -th on the right side of the following applies:
It provides only the summand for n = 0 a contribution, the integral is simplified so to
We can now try to replace the trigonometric sum by any continuous periodic function f to determine the coefficients according to the above formulas and compare the trigonometric sums formed with these coefficients with the output function:
With the Dirichlet kernel
Aperiodic processes ( Fourier integral )
Prerequisite for the derived Fourier series is the periodicity over the time interval. Of course, there are also non-periodic functions that satisfy this condition for any finite time interval. As already shown, the - th harmonic has frequency. The difference between the - th harmonic frequency of the previous, that is, the harmonic frequencies are at a distance. For approaches infinity its distance to zero - the sum is in the limit for Riemann integral.
The Fourier integral of the continuous Fourier transform is thus given by
With
From the result, the continuous spectrum has now become. Is known to be precise, the second transform as a Fourier transform, the first, the inverse, the Fourier synthesis.
The second equation can be derived analogously as for the series.
The relationship pair specified applies again and Others for square integrable functions.
Differential equations
The Fourier transform is often used to solve the differential equations. Because the or which are eigenfunctions of differentiation, and transformation converts linear differential equations with constant coefficients into ordinary algebraic equations.
Thus, for example in a linear time-invariant system, the physical frequency conserved and the performance can be solved individually for each frequency. The application of the Fourier transform of the equation yields the frequency response of the system.
Abstract harmonic analysis
The abstract harmonic analysis is the development of Fourier analysis on locally compact topological groups. On these groups can be measure using the hair, comprising the Lebesgue measure as a special case, to define an integral. Central in abstract harmonic analysis is the concept of character, which was introduced by Lev Semenovich Pontryagin. This is a continuous homomorphism of the locally compact abelian group in the sphere. In analogy to linear functionals and dual spaces their totality form the dual group. The term dual group is justified by the duality theorem of Pontryagin. From the perspective of abstract harmonic analysis is then understood as the figure
The Fourier transform. If one chooses and then to give the traditional continuous Fourier transform. Harmonic analysis in the abstract, there is the same as in the classic Fourier analysis for this transformation also an inverse transformation. This abstract also comprises Fourier transform, the Fourier series, and the Laplace transform of the Mellin transform, and other transformations as special cases.