# Phase Congruential White Noise Generator
## 2 White Noise Autocorrelation Matrix
These intensities were calculated using the Euler-Fourier Formulas (38)-(40) with the subsequent application of the elementary transformation \(A_{k}=\sqrt{a_{k}^{2}+b_{k}^{2}}\). All meanings received with the values 0.7 finely match to the first property of the uniform white noise process. In the last part of this listing, the phases _psiFourier_ show coincidence with the congruential phases in program _P070403_. It should also be noted that the application of the Euler-Fourier transform (38)-(40) completely recovers the generation of white noise process received in the class _cDeonYuliCongPhase62A_. Careful analysis of all results above satisfies that the generator _cDeonYuliCongPhase62A_ does indeed provide the equal amplitudes of all internal phase frequencies, and that it ideally satisfies to the first property of the uniform white noise process. Thus, taking into account the better approximation of the matrix \(R\) of autocorrelation coefficients obtained in the program _P070403_ to the same matrix of theoretical white noise, and also taking into consideration an ideal coincidence of the intensities of the internal phase frequencies, it should be recognized that the here proposed congruential phase generator does indeed ensure a sufficiently high quality of generation of the white noise signals, which closely approximate true natural white noise. ## 6 Conclusions

Analysis of the source material shows that the algorithms of the commonly used generators of white noise signals have a low stochasticity of countings in the given observation intervals. Based on this, in this article the instrumental algorithmic tools for generating statistically independent white noise signals have been proposed. The designed techniques allowed for the creation of a new phase signal generator with an improved matrix of autocorrelation coefficients. The mathematical expressions used confirm that at Fourier frequencies a single dimensional phase random variable could be obtained. As a result, the derivative _cDeonYuliCongPhase62A_ phase generator made it possible to create information signals with a better approximation to the uniform white noise process. The simulation outcomes verify that the information signals received have the properties of white noise signals with equal amplitudes at all internal frequencies with uniformly distributed random phases. These results can be used in a huge number of applications where white noise processes are used. All the authors equally contributed to this work. All authors have read and agreed to the published version of the manuscript. The authors have no support or funding to report. Not applicable. The authors are thankful to [PERSON] and [PERSON] (University of Arkansas for Medical Sciences, Little Rock, AR, USA) for the proofreading. The authors declare no conflict of interest. ## References

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