Multiplexing: B

Chapter 10.1007/978-3-030-49995-2_3 used the Central Limit Theorem to determine the number of users that can safely share a common cable or link. We saw that this result is also fundamental to calculate confidence intervals. In this section, we prove this theorem. A key tool is the characteristic function that provides a simple way to study sums of independent random variables.Section 4.1 introduces the characteristic function and calculates it for a Gaussian random variable. Section 4.2 uses that function to prove the Central Limit Theorem. Section 4.3 uses the characteristic function to calculate the moments of a Gaussian random variable. The sum of squares of Gaussian random variables is a common model of noise in communication links. Section 4.4 proves a remarkable property of such a sum. Section 4.5 shows how to use characteristic functions to approximate binomial and geometric random variables. The error function arises in the calculation of the probability of errors in transmission systems and also in decisions based on random observations. Section 4.6 derives useful approximations of that function. Section 4.7 concludes the chapter with a discussion of an adaptive multiple access protocol similar to one used in WiFi networks.

MODEL OF DIGITAL FIBER-OPTICAL MEASUREMENT SYSTEM

A new approach is proposed to obtain a generalized model of distribu­ -ted digital fiber-optic measuring systems of interferometric type using multichannel reception of signals of a fiber-optic inter-mode interferometer to improve the accuracy of measurements. On the basis of this approach, generalized equations for the con-version of fiber-to-digital converters of the geometric coordinates of the points of the measured object are obtained. The equations combine all the private mathema­ ti­ cal models of energy information processes. The approach is based on the representa-tion of the "coordinate of point (move) — code" in the form of an equation of perfect digital-to-analog source code conversion, the processes of which change bit codes are given in the form of logical functions from the input move and points of real multidimensional spatial parameters. The fiber optic line is used in bidirectional optical sig-nal mode in conjunction with the code element element. In this function, the supply of radiation from the measuring units to the points of reading information, the control ele­ -ment, transmitters of modulated radiation are combined in a single fiber. The spatial separation of optical streams is carried out in a block of bidirectional optical communication devices, which is a set of fiber-optic Y-splitters. For multichannel reception, the principle of making a decision on registration of influence on the interferometer is in-troduced: if the module of the output signal exceeds the set level, the signal is fixed. Changes in the measuring signal from external conditions are determined by changes in the parameters of the fiber, the processes of interaction of modes and double re-fraction. Changes in the measurement signal are presented as random variables. Using the central limit theorem for a large number of double sums, the values of the signals at a particular point in time are described by independent random variables, with a normal distribution law and a variance. The beneficial effect is considered regu-lar, and at the time of measurement it is represented by a centered Gaussian random variable with variance. The useful signal component is a Gaussian random variable with standard deviation.

ASYMPTOTIC CRAMÉR TYPE DECOMPOSITION FOR WIENER AND WIGNER INTEGRALS

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

Wind Farm Power Forecasting

Forecasting annual wind power production is useful for the energy industry. Until recently, attention has only been paid to the mean annual wind power energy and statistical uncertainties on this forecasting. Recently, Bensoussan et al. (2012) have pointed that the annual wind power produced by one wind turbine is a Gaussian random variable under a reasonable set of assumptions. Moreover, they can derive both mean and quantiles of annual wind power produced by one wind turbine. The novelty of this work is the obtainment of similar results for estimating the annual wind farm power production. Eventually, we study the relationship between the power production for each turbine of the farm in order to avoid interaction between them.

WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.