A white vibration is a realization of a random process in which the spectral concentration of power is the same one for all the frequencies.

One often speaks about Gaussian white vibration , it is about a white vibration which follows a normal Loi of average and variance given.

In synthesis and treatment of the its, one considers only the frequencies ranging between 20Hz and 20kHz since the human ear is sensitive only to this waveband (in fact rather 25Hz-19kHz). The impression obtained is that of a breath.

Its product at the time of the effect of " neige" on a television set not regulated is very an good example of white vibration.

White vibration and analytical solutions of differential equations

In any rigor a white vibration cannot exist because an identical spectral concentration for all the frequencies would lead to a variance, measured by the surface under the curve, infinite (and thus an infinite energy). There thus does not exist, as in the example which precedes, that white vibrations limited to a waveband.

This concept of white vibration is interesting in certain practical problems because, although it cannot exist, it is shown that the answer to a white vibration of a deadened system remains finished. The replacement of an unspecified excitation by a white vibration thus provides, by simplifying considerably calculations, an approximation of as much better than the damping of the system is weaker.

White vibration and simulations

A white vibration of spectral concentration (see spectral Analysis) S₀ sampled with the step T contains frequencies lower than 1/2T (see Théorème of Shannon ). It thus has a finished variance which is written, if the spectral concentration is expressed on a scale in positive frequencies, σ² = S₀/2T.

This white vibration is regarded as a realization of a random process described, in addition to its spectral concentration, by a law of probability (see Random variable).

A white vibration can be generated by a sequence of numbers randomly which corresponds to a density of uniform probability on an interval of width unit. To obtain numbers on an interval of width has, it is enough to multiply the result by A.

Consequence of the Theorem of the central limit, the Gaussian white vibration is particularly useful. To create it, one can use the formula of Rice

$\backslash \; Phi$ is a sequence of uniform variables on an interval of width 2π.

Has is a sequence of variables of Rayleigh whose function of distribution is written, $\backslash \; sigma^2$ being the variance sought for the variable of Gauss:

By equalizing this function of distribution to that of a number randomly noted R, one obtains a realization of the variable of Rayleigh:

From there, one builds a realization of a Gaussian white vibration. One can then obtain a realization of an unspecified Gaussian process by taking his Transformée of Fourier, by multiplying it by the square root of the spectral concentration and by reversing the transform.

White vibration and statistical

In the study of the Time serieses in statistics, it is often useful to also define a process of white vibration in the temporal field (whereas the definitions higher are in the field of the frequencies). According to Hamilton (1994):

Déf a process $\backslash \; epsilon\_t$ is described as noise blan C if:

*$E=0\; \backslash ,$

*$E=\; \backslash \; sigma\; ^2$

*$E\; \backslash \; epsilon\_\; \{\backslash \; tau\}\; =0\; \backslash \; qquad\; \backslash \; forall\; T\; \backslash \; \backslash \; tau$

A process of white vibration is thus by stationary definition.

Déf a process $\backslash \; epsilon\_t$ is described as independent white vibration if:

*$E=0\; \backslash ,$

*$E=\; \backslash \; sigma\; ^2$

*$\backslash \; epsilon\_t\; \backslash ,$ and $\backslash \; epsilon\_\; \{\backslash \; tau\}\; \backslash ,$ is independent $\backslash \; forall\; T\; \backslash \; \backslash \; tau$

Déf Finalement, if in more of the last definitions:

*$\backslash \; epsilon\; \backslash \; sim\; \backslash \; mathcal\; \{NR\}\; (0,1)\; \backslash ,$

The process $\backslash \; epsilon\_t$ is qualified of Gaussian white vibration .

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