PRELIMINARY DESIGN OF ANALOG DISCRETE UNIFORM NOISE GENERATOR

The paper presents the basic idea of the construction of an analog discrete uniform noise generator. The source of noise is a carbon resistor, the noise is linearly strongly amplified and limited to around zero. The probability density function (PDF) of the carbon resistor thermal noise in that region is square. By narrowing the symmetric allowable gap (interval) around zero, PDF of the noise approaches a uniform distribution. The factor of deviation from the uniform distribution is correctly and precisely defined. This quantity has been shown to be practically negligible. In addition, the paper discusses the application of the proposed ditheter noise, both in the two-bit and in the multi-bit stochastic digital measurement method (SDMM). It has been shown that noise is more suitable for application in multi-bit SDMM, because it is less sensitive to deviations from the uniform distribution. Commercially available track-and-hold circuits provide at least an order of magnitude wider bandwidth of the described generator compared to the standard solution that uses numerical random number generator and a corresponding D/A converter. However, the realization of such a generator requires hard engineering work, and therefore goes beyond the scope of this paper.

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Statistical Justification of Pearson's Criterion for Testing a Complex Hypothesis on the Uniform Distribution

Mechanical Engineering and Computer Science ◽

10.24108/0418.0001392 ◽

2018 ◽

pp. 45-53

Author(s):

T. V. Oblakova

Keyword(s):

Uniform Distribution ◽

Degrees Of Freedom ◽

Likelihood Function ◽

Random Number Generator ◽

Maximum Likelihood Estimates ◽

Chi Square ◽

Main Hypothesis ◽

Pearson Criterion ◽

Complex Hypothesis ◽

Uniform Law

The paper is studying the justification of the Pearson criterion for checking the hypothesis on the uniform distribution of the general totality. If the distribution parameters are unknown, then estimates of the theoretical frequencies are used [1, 2, 3]. In this case the quantile of the chi-square distribution with the number of degrees of freedom, reduced by the number of parameters evaluated, is used to determine the upper threshold of the main hypothesis acceptance [7]. However, in the case of a uniform law, the application of Pearson's criterion does not extend to complex hypotheses, since the likelihood function does not allow differentiation with respect to parameters, which is used in the proof of the theorem mentioned [7, 10, 11].A statistical experiment is proposed in order to study the distribution of Pearson statistics for samples from a uniform law. The essence of the experiment is that at first a statistically significant number of one-type samples from a given uniform distribution is modeled, then for each sample Pearson statistics are calculated, and then the law of distribution of the totality of these statistics is studied. Modeling and processing of samples were performed in the Mathcad 15 package using the built-in random number generator and array processing facilities.In all the experiments carried out, the hypothesis that the Pearson statistics conform to the chi-square law was unambiguously accepted (confidence level 0.95). It is also statistically proved that the number of degrees of freedom in the case of a complex hypothesis need not be corrected. That is, the maximum likelihood estimates of the uniform law parameters implicitly used in calculating Pearson statistics do not affect the number of degrees of freedom, which is thus determined by the number of grouping intervals only.

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