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Concept# White noise

Summary

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal. White noise draws its name from white light, although light that appears white generally does not have a flat power spectral density over the visible band.
In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance; a single realization of white noise is a random shock. Depending on the context, one may also require that the samples be independent and have identical probability distribution (in other words independent and identically distributed random variables are the simplest representation of white noise). In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise.
The samples of a white noise signal may be sequential in time, or arranged along one or more spatial dimensions. In , the pixels of a white noise image are typically arranged in a rectangular grid, and are assumed to be independent random variables with uniform probability distribution over some interval. The concept can be defined also for signals spread over more complicated domains, such as a sphere or a torus.
An infinite-bandwidth white noise signal is a purely theoretical construction. The bandwidth of white noise is limited in practice by the mechanism of noise generation, by the transmission medium and by finite observation capabilities. Thus, random signals are considered "white noise" if they are observed to have a flat spectrum over the range of frequencies that are relevant to the context.

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White noise

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal.

Time series

In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart).

Spectral density

The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

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We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.

Garance Hélène Salomé Durr-Legoupil-Nicoud

The StatComp package is a Matlab statistical toolbox developed over the years by Dr. Testa and his students. It has been inspired by M. R. Brown’s paper Magnetohydrodynamic Turbulence: Observation and experiment [2]. It first performed the analysis of the edge magnetic turbulent field in the TCV. It started in 2015 by A. Yantchenko and has been constantly improved and supplemented since then. The last addition to the package was many separate functions for the ”big data” analysis of the results, done by S. Ogier-Collin. The entire code is currently under review for release in the MHD analysis package within the SPC’s General Analysis Toolkit. The present document reports the latest evolution of this package in the perspective of using the charac- terisation plasma turbulence to possibly provide useful information for the optimisation of real-time plasma control and the fusion performance of a tokamak. The mathematical theory of the StatComp analyses and some examples of application are presented in the section 2. The section 3 presents the evolution of the existing functions as well as the addition of the loading function for the electrostatic data from the edge of the plasma, and the multifractality and predictability analyses. These enhancements are put in the perspective of one particular usage: the characterisation of the turbulence in order optimise potentially plasma control. Then, the up-to-date running instructions and interpretation guidelines are detailed in the section 4. The latter are based on the output figures resulting of the analysis of a standard dataset constituted of a white noise sample, three fractional Brownian motions of different known Hurst index, of a linear ramp and of a sample of the solar wind. The section 5 shows the results of the test on four actual shots realised on the TCV tokamak. The varying parameters are the signs of the poloidal magnetic field and of the plasma current. The four shots are each the resultant of a positive or negative poloidal field and a positive or negative plasma current. The shape and position of the plasma in the vacuum vessel are the same for each shot as well as the amplitude of the varied parameters, i.e. the magnetic field and plasma current. The emphasis is made on the presentation and interpretation of the results obtained with the electrostatic data on the low-field side of the plasma. The obtained results are discussed along the limits of the package and its possible improvements in section 6 before concluding in section 7. In the appendix, the structures necessary to the use of the package are detailed and examples of run commands are presented. In order to offer to the reader a frame of reference for reflection, the main parameters and orders of magnitude related to the plasma shots in TCV are given. Some of the mathematical basis of the statistical theory are also elaborated to complete the description of the different tools of the package. Finally, the reduced bibliography of all the sources explicitly mentioned in this report is doubled by a second bibliography presenting a wider selection of relevant sources each accompanied with a brief description of its content and its link to the present study.

2022We investigate the existence and regularity of the local times of the solution to a linear system of stochastic wave equations driven by a Gaussian noise that is fractional in time and colored in space. Using Fourier analytic methods, we establish strong local nondeterminism of the solution and the existence of jointly continuous local times. We also study the differentiability and moduli of continuity of the local times and deduce some sample path properties of the solution.

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