Fourier series
As a Fourier series (after Joseph Fourier ) is defined as the series expansion of a periodic, partially continuous function into a series of functions of sine and cosine functions.
The basic functions of the Fourier series form a prominent example of an orthonormal basis. Within the framework of the theory of Hilbert spaces and developments of any complete orthonormal system are referred to as a Fourier series.
- 3.1 triangular pulse
- 3.2 square pulse
- 3.3 Sägezahnpuls (lowest to highest )
- 3.4 sine pulse
- 4.1 Theorem of Dirichlet
- 4.2 set of Carleson
- 4.3 set of Fejér
History
Already in the 18th century known mathematicians such as Euler, Lagrange or the Bernoulli Fourier series for some features. At the beginning of the 19th century, was claiming Fourier in his work Théorie de la chaleur Analytique that there were such series expansions for all functions. This claim initially met with leading mathematicians such as Cauchy and Abel rejection.
Dirichlet proved that Fourier's assertion is true, at least for Lipschitz continuous functions in 1829. Du Bois -Reymond in 1876 found a continuous function whose Fourier series diverges. In the 20th century it finally came to the realization that there are convergent Fourier series for continuous or piecewise continuous functions when the convergence term is suitable attenuated ( Lennart Carleson ).
As an early pre- geometric approximation by a Fourier series of the epicycle theory can be considered.
Forms of representation
The partial sums of a Fourier series are trigonometric polynomials. How can these Fourier series are represented in three equivalent forms. There are corresponding formulas for determining the coefficients and parameters of the Fourier series expansion of a periodic function for each of these representations.
A Fourier series expansion of a periodic function with period is possible in the following, gradually becoming general cases:
General form
A periodic function with period, which belongs to one of the specified classes, can be represented by a series of sine and cosine functions whose frequencies are integer multiples of the fundamental frequency,
The angular frequency scaled in this case the period of sine and cosine to the corresponding period. In practical application it will frequently terminate the sequence number of a finite number of members. This then gives only an approximation in the form of a trigonometric polynomial,
This finite sum is then called a partial sum of the Fourier series. The resulting trigonometric polynomial, trigonometric polynomials in all the same structure, the one with minimum mean square error for the original function.
The coefficients of the expansion of are:
This represents a shift in the interval, and can be arbitrarily chosen for simplicity.
Simple properties of this development are that
- Applies in all cases, if there just is,
- Applies in all cases, if is odd.
So is straight, then all, and the coefficients can also be calculated over. This is possible because the symmetry of the cosine function and the values of the integral are the same in the two half- intervals. So there are often simplifications. This also applies analogously for odd, ie at.
If the underlying function is unknown or are only given discrete data (eg measured values) before, except from the bases approximated ( Trigonometric interpolation).
Amplitude-phase notation
In the above illustration, the signal is represented by a sine spectrum and a Kosinusspektrums. But there is also a representation by means of phase angle and amplitude of the spectrum is possible, as it may represent a sine and a cosine wave as a phase-shifted cosine, the additive superposition (interference )
It is calculated by:
The calculation of is more complex. If one chooses representatives from so applies with the signum function:
For the more common choice is, however, offers the use of the arc tangent of two arguments:
For this simplifies to:
For the above functions are not defined - in this case, however, in any case. Therefore irrelevant.
Shows in the quadrant in which also the point lies.
Complex Fourier series
One can now interpret each pair of amplitude and displacement as a complex number in polar form. Thus, the two spectra can be converted into one. However, a simplification of even and odd functions as in the real is not possible.
It is
It is often chosen. This results in
The calculation is often easier, since not only the e-function is easy to integrate and would only have a coefficient instead of two has to be calculated. When it is in this representation form a so-called complex amplitude, which also contains the phase information. The discrete and complex amplitude spectrum shows a line spectrum, the distance between two spectral lines corresponding to the reciprocal of the period:
Relationship between real and complex Fourier coefficients
Reell too complex:
Complex to real:
Related to the Fourier transform
Using Fourier series can be only periodic functions and their spectrum describe. In order to describe non-periodic functions spectrally, one performs a crossing of the period. Thus, the frequency resolution becomes arbitrarily fine, resulting in a disappearance of the complex amplitude spectrum. For this reason it performs the complex amplitude of a spectral density, starting from the complex Fourier series for the first discrete arguments:
By formation of the threshold value ( the same time ), so that immediately following the Fourier transformation:
Examples
Triangular pulse
The delta function can be approximated according to the desired phase with sine and cosine terms. The peak value are the Fourier series:
Rectangular pulse
The square wave is defined by
Is the Fourier series to
Using this feature, you recognize that you can be a square wave by an infinite number of harmonics. They contain respectively the odd harmonics with the amplitude decreasing with increasing frequency. Because of a square wave signal is also often taken for testing electronic circuits, as such the frequency response of this circuit is detected.
Generally contain all periodic oscillations with the period of the fundamental oscillation and any pattern within the period only odd harmonics if:
In the right image, the Fourier synthesis of a square wave is shown. The graphs in the first column show that vibration that is added in the corresponding line. The graphs in the second column show all previously unrecognized vibrations, which are then added in the diagrams of the third column to come as close as possible to the signal to be generated. The vibration of the first line is called the fundamental vibration, all others to be added, are harmonics (harmonics). The more of such multiples of the fundamental frequency are taken into account, the more closer to an ideal square wave. Forming at the discontinuous points of the square wave signal by the Fourier synthesis implies a so-called overshoot, which does not disappear even at higher approximation. This phenomenon is called the Gibbs phenomenon, it has a constant and independent of the range of the overshoot of about 9 % of the full on jump. The fourth column shows the amplitude spectrum normalized to the fundamental.
Sägezahnpuls (lowest to highest )
Similarly, point-symmetric functions of sine terms can be approximated. Here you can reach a phase shift by alternating sign:
Sine pulse
Convergence results for Fourier series
One can safely set up a periodic function a Fourier series, but this series does not converge. If this is the case, no additional information is also obtained by this transformation. The series converges, so you have to be clear in what sense there is convergence. In most cases examined to Fourier series on pointwise convergence, uniform convergence and convergence with respect to the norm. The following are some important theorems on the convergence of Fourier series are enumerated.
Set of Dirichlet
Peter Gustav Lejeune Dirichlet proved that the Fourier series of a differentiable, periodic function converges pointwise to the output function. With the proviso that even continuously differentiable, the message can be improved.
Be a continuously differentiable, periodic function, then the Fourier series converges uniformly to of.
Set of Carleson
The set of Carleson is a low-lying result for convergence of a Fourier series.
Let be a square integrable function, then the Fourier series converges with respect to f almost everywhere.
This statement is true even for all rooms and is called in this general form set of Carleson - Hunt. The fact that the statement is false, Kolmogorov 1923 could show by a counterexample. However, Nikolai Nikolaevich Luzin suspected as early as 1915, the validity of the theorem of Carleson, they could not prove. The proof only succeeded Lennart Carleson in 1964.
Set of Fejér
Leopold Fejér proved that the arithmetic mean of the partial sums of the Fourier series of a continuous, periodic function converge uniformly to the function.
Let be a continuous, periodic function and the Fourier series of. With the n-th partial sum of this series will be described. Then the set of states Fejér that the partial sums converge uniformly to. It is therefore
Where the convergence is uniform.
Gibbs phenomenon
In the vicinity of discontinuities the Fourier series converges no longer uniform, but only pointwise. It created there in the partial sums of the series typical over-and undershoots of about 9 % of the step height. This effect has far-reaching effects in the signal processing.
- See also: Discrete Fourier Transform
Generalized Fourier series
Be a Hilbert space with an orthonormal basis. Then you can each element of the Hilbert space by
Represent. This series representation is also called the ( generalized) Fourier series.
Generalizations
Generalizations of the Fourier series, although they can be described as representation in the orthonormal bases, but additionally have a similar certain properties of the Fourier series with respect to symmetry, examines the harmonic analysis. The Pontryagin duality generalizes doing the Fourier series of functions on any locally compact abelian topological groups, the set of Peter -Weyl on compact topological groups.