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QUASI-DETERMINISTIC SIGNALS
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QUASI-DETERMINISTIC SIGNALS
Veshkurtsev Yury 1
Authors:
1.  A.M. Prokhorov Academy of Engineering (Siberian Branch)
Russian Federation
Type:
Chapter
DOI:
https://doi.org/10.26526/chapter_628a8925285cd4.45558771
Chapter pages from - to
26 - 36
Publisher:
AUS PUBLISHERS
Language:
English
Keywords:
new-generation modems, information, signal, signal modulation and demodulation method, IT equipment, digital communication, radio engineering
Abstract and keywords
Abstract (English):
The monography presents the fundamentals of the theory of construction new-generation modems. Modems are built on the principles of statistical communication theory, based on the use of a random signal (chaos) as a carrier of information. In such a signal, a characteristic function is modulated, which is a fundamental characteristic of a random process. The signal modulation and demodulation method is patented and allows you to create modems with efficiency and noise immunity indicators several orders of magnitude higher than those of the known devices of the same name. New-generation modems immediately improve the technical characteristics of digital IT equipment by several orders of magnitude, since they work without errors in wired and radio channels when receiving one hundred duodecillion of binary symbols. The book is recommended for scientists and specialists in the field of digital communication systems, statistical radio engineering and instrumentation, and may be useful for graduate students, masters and students of relevant specialties.

Keywords:
new-generation modems, information, signal, signal modulation and demodulation method, IT equipment, digital communication, radio engineering
Text
To implement statistical modulation, quasi-deterministic signals are required, which, by definition [4 p. 171] ".... are described by time functions of a given type, containing one or more random parameters , that do not depend on time." From this class of signals, the model and probabilistic characteristics of a quasi-deterministic signal with the arcsine distribution law are quite fully presented in the publication. Information about other quasi-deterministic signals is presented only fragmentarily. 2.1. Signal model with arcsine distribution law Let us consider a signal with a mathematical model of the form , (2.1) where is the constant amplitude of the signal- constant circular frequency of the signal; - a random variable (initial phase angle) with a uniform distribution law within - . . ; instantaneous values of the signal, obeying the distribution law of the arcsine. In the publication, this signal is called quasi-deterministic. We begin the description of probabilistic characteristics with the probability density of instantaneous values of the signal, which, by definition [4], is equal to , (2.2) where dispersion (average power) of the signal. Here and below, the number 1 denotes the one-dimensionality of the function. The graph of function (2.2) is shown in Figure 2.1. Let's take a look at the shape of the graph. It looks like a horseshoe and is centered around zero because the mathematical expectation of the signal is zero. At the edges, the value of the function tends to infinity when the value of the argument is equal to the amplitude of the signal. This results in the probability of occurrence of the signal amplitude as if equal to one, since this is like depicting a probability density of a constant value using a delta function. However, this is possible, and this is explained only by the fact that the signal model (2.1) is some mathematical abstraction. The physical process has amplitude fluctuations [8], which are random. And as Figure 2.1. Signal probability density a result of this, the specified equality a=const is violated. The arcsine law was established in 1939 by the mathematician P. Levy for random walks of a point on a straight line and later adapted for signal (2.1). The characteristic function of the signal in accordance with (1.5) is equal to Θ1(Vm)=∫_(-∞)^∞▒〖W_1 (x) 〗 , (2.3) where J0(∙) is the zero-order Bessel function of the first kind (see Fig. 2.2). For a random variable η with probability density W1(η)=(1/2π) ch.f. will be [4] . (2.4) For a constant signal amplitude ch.f. was defined earlier with the help of expression (1.63), which we repeat with our designations . (2.5) The signal correlation function (2.1) will be (2.6) where W(η) is the probability density of the random phase η; τ is the shift in time. The dependence of the average signal power on time turns out to be harmonic, it is shown in Figure 2.3. Figure 2.2. Ch.f. of signal (separate real and imaginary part) Figure 2.3. Correlation function of a signal with a frequency of 30 Hz Let us proceed to the analysis of the power spectral density (energy spectrum) of the signal (2.1). Let's write down its energy spectrum . (2.7) The spectrum (2.7) turned out to be lined. It contains a spectral component ranging from - to 0 and a spectral component ranging from 0 to , where delta is a function. In the transition to the physical spectrum, i.e. to the spectrum in the region of positive frequencies, we get . (2.8) The energy spectrum as a function of frequency is shown in Figure 2.4. Figure 2.4. Signal power spectral density at 30 Hz To clarify, Figures 2.1–2.4 show estimates of the probabilistic characteristics of the signal (2.1) measured using the virtual instrument “Characteriometer” [3, 9]. Signal (2.1) was obtained from the output of the G3-54 generator. 2.2. Signal model with the Veshkurtsev distribution law Let's repeat the mathematical model of the signal (2.1) u(t)= a sin (ω0t +η), (2.9) where a,η - random variables (amplitude and initial angle of phase shift, respectively); ω0 - constant circular frequency of the signal; u(t) - instantaneous signal values distributed according to the Veshkurtsev law [10]. There is no description of such a signal in the publication. Let the signal amplitude (2.9) be distributed according to the Gauss law (normal law)〖 W〗_1 (x)=1/(σ√2π) e^(-x^2/(2σ^2 )) , (2.10) and the random phase - according to a uniform law within – π . .π, where is the amplitude dispersion. Then the instantaneous values of the signal obey the Veshkurtsev distribution (2.11) where is the cylindrical function of the imaginary argument (the Macdonald function) [11]. The MacDonald function at 𝑥=0 asymptotically tends to infinity on both sides of the y-axis (Fig. 2.5). In this way, Veshkurtsev's law resembles the arcsine law (Fig. 2.1), where the value of the probability density tends to infinity at the edges when the argument value is equal to the signal amplitude. It turns out that Veshkurtsev's law is a kind of copy of the transformed arcsine law, therefore, it is also some kind of mathematical abstraction. A physical process with such a law does not exist, and only digital technology will make it possible to put it into practice in the form of a random process sensor. Figure 2.5. Signal Probability Density Since this law was obtained for the first time, we will agree to call it the Veshkurtsev law in the future by the name of the author, who was the first to write it down analytically and apply it in practice in solving new problems [10,12,13]. Naturally, all the properties of the statistical law prescribed in the theory of probability have been verified by the author and they are fulfilled. Using the Fourier transform of this law, we obtain the ch.f. signal (2.12) where I0(∙) is the Bessel function of the imaginary zero-order argument. A similar transformation of the distribution law (2.10) gives the ch.f. (2.13) of signal amplitude with a Gaussian distribution law. Ch.f. for the random phase of the signal will be . (2.14) Quasi-deterministic signal (2.9) is centered (its mathematical expectation is zero), it has dispersion (average power) , (2.15) where - generalized hypergeometric series [11]; signal amplitude dispersion. Concluding the analysis of the probabilistic characteristics of the quasi-deterministic signal (2.9), we clarify that its instantaneous values are distributed according to the Veshkurtsev law, the amplitude is distributed according to the Gauss law, and the phase is distributed according to the uniform law. The signal correlation function (2.9) has the form (2.16) The energy spectrum of the signal (2.9) coincides with the spectrum (2.7) . (2.17) In the transition to the physical spectrum, i.e. to the spectrum in the region of positive frequencies, we obtain . (2.18) The physical spectrum of the signal with the Veshkurtsev distribution law contains only one spectral component located on the frequency axis at the point with the abscissa , when , and coincides with the origin. 2.3. Signal model with cosine distribution law Let's repeat the mathematical model of the signal (2.1) , (2.19) where - random variables (amplitude and phase shift angle, respectively), each with its own distribution law; ω - constant circular frequency; - instantaneous signal values obeying the distribution law at , where > 0. (2.20) Here and below, the number 1 denotes a one-dimensional function. The statistical law (2.20) is given in the book [ 5 p. 46 ] without a title and additional explanations, there is no information about its use in the literature. Apparently, we are the first to pay attention to this statistical law. For further actions, we take the value in formula (2.20), then , and expression (2.20) takes the form (2.21) at The statistical law (2.21) has all the properties prescribed in the probability theory. We will call it the law of cosine in the future. There are no quantitative parameters in the mathematical description of the law of cosine. It should be noted that this law is centered, the mathematical expectation is zero, and the dıspersion is (2.22) It is always constant and depends only on the bounds of the values of the variable. By this, this law is inferior to the Gauss law (normal law), in which the dispersion and mean square deviation (MSD) are included in the mathematical description of the law. If the instantaneous values of the quasi-deterministic signal (2.19) are distributed according to the cosine law, then the signal amplitude will be distributed according to the law [14] at , (2.23) where J0(∙) - Bessel function of the zero order of the first kind; Г (∙) – gamma – function. Since this law was obtained by us for the first time, we will call it the Bessel law in the future by analogy with the function of the same name included in it. Using the Bessel law, we determine the initial moments of the distribution of the signal amplitude (2.19), while obtaining the initial moment of the first order (expectation) [14] (2.24) and the initial moment of the second order [14] (2.25) where - Lommel function [11 ] ; is the Bessel function of the kth order of the first kind [11]. Turning to the random phase of the signal (2.19), we say that as a result of mathematical calculations, we have obtained a uniform law according to which the phase is distributed within . The characteristic function (ch.f.) of a centered quasi-deterministic signal (2.19) is [14] . (2.26) The work [2] describes the properties of the ch.f., which the function (2.26) satisfies. In particular, one property of the ch.f. concerns the signal distribution law, from which it follows that the ch.f. for a signal with a non-centered distribution law is equal to the ch.f. obtained for a signal with a centered distribution law , multiplied by the exponent , where the expectation of the signal. Let us use this property and write the function (2.26) for a non-centered quasi-deterministic signal (c.q.s.), i.e. a signal that has an expectation. As a result, we will have , (2.27) where are the real and imaginary parts of the ch.f. respectively. In contrast to (2.27), the ch.f. (2.26) has only a real part. Passing to the ch.f. random amplitude of the centered signal (2.19), we have [14] , (2.28) ; ; where - Bessel function of the first order of the first kind; - Bessel function of the second order of the first kind [11]. Expression (2.28) is a particular solution; it is valid for the value , while the general solution for the ch.f. signal amplitude (2.19) is still in the search stage. Since the expectation of the signal amplitude (2.19) is not equal to zero, the ch.f. (2.28) is a complex function. For the random phase of the signal (2.19), the ch.f. known [4 p. 162] and is equal to , (2.29) it is a real function, since the phase distribution law is centered. Concluding the analysis of the probabilistic characteristics of the quasi-deterministic signal (2.19), we clarify that its instantaneous values are distributed according to the cosine law, the amplitude - according to the Bessel law, and the phase - according to the uniform law. The signal correlation function (2.19) will be , (2.30) where W(y) - amplitude probability density (2.23); W(η) - probability density of the random phase η; is the initial moment of the second order (2.25). Let us proceed to the analysis of the power spectral density (energy spectrum) of signals (2.19). Let's write the energy spectrum of the signal . (2.31) The spectrum (2.31) turned out to be lined. It contains a spectral component ranging from - to 0 and a spectral component ranging from 0 to , where where delta is a function. In the transition to the physical spectrum, i.e. to the spectrum in the region of positive frequencies, we obtain (2.32) Like the signal (2.9), the physical spectrum of a signal with a distribution according to the cosine law contains only one spectral component located on the frequency axis at the point with the abscissa , when and coincides with the origin of coordinates. 2.4. Signal model with the Tikhonov distribution law Let's repeat the mathematical model of the signal (2.1) , (2.33) where - random variables (amplitude and phase shift angle, respectively), each with its own distribution law; ω - constant circular frequency; - instantaneous signal values obeying the Tikhonov distribution law within -π≤x≤π. (2.34) The authors of the book [ 5 p. 46] without additional explanations call the statistical law (2.34) the distribution of V.I. Tikhonov, a well-known scientist who was the first to propose it to describe the phase of self-oscillations of a synchronized generator in a phase-locked loop system. Apparently, we are the leader in the use of this statistical law in the formation of a quasi-deterministic signal. Similarly to the cosine law (2.20), in the mathematical model (2.34) there are no quantitative parameters of the distribution law, except for the coefficient D, which determines the shape of the probability density graph, since it enters the Bessel function I0(D). Tikhonov's law is centered, its dispersion is determined [15, p.334] , (2.35) it is constant and depends on the coefficient D, for example, at D=1 the dispersion is equal to , where is the Bessel function of the imaginary argument of the nth order of the first kind. The signal amplitude (2.33) is distributed according to the law described by the probability density of the form [16] . (2.36) Since the statistical law (2.36) was obtained for the first time, we will call it the Bessel-Lommel law by analogy with the known functions included in it. The properties of the law (2.36), prescribed in the theory of probability, have been verified by us and they are fulfilled. The Bessel-Lommel law describes the distribution of the random signal amplitude (2.33) within . The expectation of a random signal amplitude is [16] , (2.37) and the initial moment of the second order of the signal amplitude will be . (2.38) In this case, the dispersion of the signal amplitude will be . The designations in expressions (2.37), (2.38) were explained earlier when describing formulas (2.24), (2.25), (2.28), (2.35). From the analysis of Tikhonov law, it follows that the random phase of the signal (2.33) is distributed according to a uniform law within -π…+π. The characteristic function of the signal (2.33) is the Fourier transform of the probability density (2.34) . (2.39) Properties of ch.f. depend on properties - the Bessel function of the imaginary argument - th order of the first kind. The graph of this Bessel function is shown in the figure in the handbook [17, p.196]. For each value of the parameter, the graph of the function is different, however, for the value function . Thus, it can be argued that the properties of the ch.f. (2.39) are observed. If the signal (2.33) has expectation , then its ch.f. will be . (2.40) Concluding the analysis of the probabilistic characteristics of the quasi-deterministic signal (2.33), let us clarify that its instantaneous values are distributed according to the Tikhonov law, the amplitude - according to the Bessel-Lommel law, and the phase - according to the uniform law. The signal correlation function (2.33) will be [16] , (2.41) where W(y) is the amplitude probability density (2.36); W(η) - probability density of the random phase η; - the initial moment of the second order (2.38). Let us proceed to the analysis of the power spectral density (energy spectrum) of signals (2.33). Let's write the energy spectrum of the signal . (2.42) The spectrum (2.42) turned out to be lined. It contains a spectral component ranging from - to 0 and a spectral component ranging from 0 to , where where delta is a function. In the transition to the physical spectrum, i.e. to the spectrum in the region of positive frequencies, we obtain (2.43) Similarly to what was said earlier, the physical spectrum of a signal with a distribution according to Tikhonov's law contains only one spectral component located on the frequency axis at the point with the abscissa , when when and coincides with the origin of coordinates. Completing the stage of formation of mathematical models of quasi-deterministic signals for statistical modulation, let's say that we recorded three of them for the first time with all probabilistic characteristics, including the characteristic function. A quasi-deterministic signal with an arcsine distribution law already exists practically as a source of physical oscillations and can be used when performing statistical modulation. The remaining quasi-deterministic signals can be implemented in practice only in the form of new computer programs, which will later serve as signal sensors as part of digital technologies. Now it is premature to talk about the creation of new sources of physical oscillations, in our opinion. However, the filling of a separate class of random processes with other quasi-deterministic signals must go on constantly.
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