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

 

 

For many random processes, the ch.f. is defined and calculated. Information about them is systematized in the literature [2,5]. A graph of the ch.f. widely used Gaussian random process is shown in Figure 1.2. It's plotted for the function

(1.12)

for which the coefficient m₁ =0. When σ=1, the value of ch.f. for V_m = 4 differs from θ₁(0) = 1 by a factor of 300. Therefore, ch.f. of normal random process rapidly decreases to zero. This behavior ch.f. is determined by its noise filtering property, which will be discussed below. In addition, the ch.f. has

Figure 1.2. Characteristic function of a Gaussian process

    (m₁=0, σ=1)

 

other properties, for example, its maximum value is equal to one at V_m = 0 and the modulus |θ1(0)| = 1. This implies the measurability and boundedness of the ch.f. if all values .

When solving many problems, it is especially useful to have the property that the ch.f. additive mixture of signal and noise is equal to the product of the characteristic functions of individual terms. This property is easy to

extend to the sum  of independent random processes. Ch.f. of the amount will be

(1.13)

where θ_1i(V_m) is the ch.f. i-th random process. Here we can also talk about the probability density of the sum of independent random processes, which, as you know, is calculated by the convolution formula

(1.14)

where z(t)=u(t)+y(t). Then, using transformation (1.5), we can pass to the characteristic function of the additive mixture z(t). However, these two options for solving the problem do not reject each other. In some cases, it turns out to be easier to find the probability density, in others, the characteristic function. For example, when measuring the probability density of only a signal from a mixture z(t), of course, it is much easier to measure the ch.f. mixtures of z(t) and ch.f. interference y(t) when the signal u(t)=0. Then, it is necessary to find the ch.f. signal from the ratio of the characteristic functions, and to calculate its probability density using the Fourier transform

(1.15)

The solution of such a problem with the help of convolution (1.14) is much more complicated.

         If we divide the characteristic function of the additive mixture by the ch.f. interference, we get the quotient. It contains expressions for the real and imaginary parts of the ch.f. separate signal. Expressions are equal

	(1.16)
	(1.17)

where , ,  is the real part of the ch.f. of signal, interference and additive mixture, respectively; , ,  is the imaginary part of the ch.f. of signal, interference and additive mixture, respectively. When the noise has a symmetric distribution function and mathematical expectation equal to zero, then the calculation algorithms (1.16), (1.17) become simpler:

	(1.18)
	(1.19)
In accordance with (1.2), the ch.f. signal will be equal to
		(1.20)

    If the additive mixture of signals is represented as z(t)= K₀u(t) + U₀, then the ch.f. sum looks like this: 

(1.21)

where  is deterministic signal, constant in time (for example, constant voltage); θ_1z(V_m) is ch.f. process z(t), and θ_1u(V_m) is ch.f. of signal u(t). Where U₀ = 0, expression (1.21) is transformed into the equality

 

.

(1.22)

Consequently, amplification (attenuation) of the signal by a factor of K₀ leads only to a change in the real parameter of the ch.f. the same number of times. This property of the ch.f. is especially important when constructing instruments of a new class with the conditional name characterometers. When the sign of the real parameter of the ch.f. equality (1.22) takes the form

=

(1.23)

i.e. θ_lz(-V_m) and θ_1u(K₀V_m) are complex conjugate functions. If we put K₀ = 1 in expression (1.23), then it is reduced to the known

= =

(1.24)

Let us onsider a complex signal

in which the terms are functions of a real variable. Then

(1.24a)

where ,  are the characteristic functions of the signals ξ(t), u(t), respectively. Thus, ch.f. of complex signal is equal to the product of the ch.f. of the real part of this signal and the expectation of the exponential function, whose exponent with the opposite sign is the imaginary part of the complex signal amplified by Vm times. If a random signal v(t) is subject to the distribution law W₁(x), then 

For example, for a law of the form W1(x)=1/(2π) we have	(1.24б)
	(1.24в)

Multidimensional ch.f. of random process has the following form:

(1.25)

The quantization step ∆V is applicable to any ch.f. parameter. Taking into account the previously adopted notation, we can write

  

(1.26)

Moreover, it is not necessary at all in (1.26) to have a constant quantization step for all parameters of the ch.f. Each parameter (or group of parameters) ch.f. can be discretized by its step.

If the instantaneous values of the random process, taken at the moments of time t_l, t₂,...,t_n, do not depend on each other, then the expression (1.25) takes a different form:

(1.27)

where  is ch.f. of i-th set of instantaneous values of a random process.

  For stationary random processes, expression (1.25) is written in a simpler way:

(1.28)

since ch.f. does not depend on time or depends only on the differences of individual moments of time t₂-t₁, t₃-t₁,...,t_n-t₁ = пτ, i.e.

(1.29)

It is quite clear that the ch.f. of lower dimension is quite simply obtained from the n-dimensional characteristic function, namely:

 

(1.30)

Calculation of n-dimensional ch.f. random process is a difficult task. Therefore, the publication contains few examples of multidimensional ch.f. Nevertheless, the expression for the n-dimensional ch.f. normal random process:

 

(1.31)

where m₁, σ² are the mathematical expectation and variance of the random process; R_ik(τ) is the normalized correlation function, with R_ik(τ)=R_ki(τ), R_ii(0) = 1, R_kk(0)=1.  Examples of other multidimensional ch.f. can be found in the books [2,5]. 

 

1.2. Estimates of the characteristic function of instantaneous values ​​of a random process

 

Formula (1.1) contains the mathematical operation m₁{·} is the statistical mean (or the expectation operator), which involves averaging an infinitely large number of values ​​of the function exp[jV_mξ(t)], depending on the instantaneous values ​​of the random process ξ(t).

It is quite clear that in the instrumental analysis of ch.f. a finite number of instantaneous values ​​of the random process or its parameters (envelope, instantaneous phase, frequency) will be used. The result of calculating the value of ch.f. according to a limited set of sample data[1]⁾  of a random process is called the estimate of the function. The estimate of the ch.f. will be denoted by the previously adopted symbol, and to distinguish it, mark with an asterisk:

	(1.32)
	(1.33)

 

 

where L is the transformation operator for an array of sample data, a set of sample values, or time-limited implementations of a random process; k is the number of implementations or sequences used).

The transformation operator L can be different. In [6], three operators are considered, namely: an ideal integrator with normalization with respect to T

(1.34)

ideal adder normalized by N

(1.35)

ideal adder-integrator with normalization on N and T

(1.36)

where T is the averaging time (the duration of the implementation of the random process); N is the amount of sample data or the set of sample values. Taking into account (1.34) - (1.36), we can write:

	(1.37)               (1.38)
	(1.39)

By definition [6], estimates (1.37) - (l.39) are called k - current, t - current and average, respectively. The application of these estimates is largely determined by the type of random process. Using the classification [4] of random processes, we can specify the following:

- for a stationary ergodic random process, all estimates of the ch.f. are equal to each other;

- for a stationary non-ergodic random process t - current and average estimates of ch.f. are equal to each other;

- for a non-stationary ergodic random process k - current and average estimates of the ch.f. are equal to each other;

- for a non-stationary non-ergodic random process, all estimates of the ch.f. are not equal to each other. In this case, we recommend using average ch.f. estimates, since they converge better than others to a probabilistic characteristic (characteristic function).

     Passing to the estimates of the real and imaginary parts of the ch.f., taking into account (1.37) – (1.39), we write

	(1.40)
	(1.41)
	(1.42)
	(1.43)
	(1.44)
	(1.45)

Our further analysis is connected with the k-current estimate of the ch.f. To simplify the notation, we omit the symbols k, n in formulas (1.37), (1.40), (1.41), and we will only have them in mind. If in estimates (1.37), (1.40), (1.41) the integration operation is performed by a non-ideal integrator, and a real device with an impulse response q(t), then these expressions take the form

                                     (1.46)

 

	(1.47)
	(1.48)

Let us extend the notion of an estimate to an n-dimensional ch.f. Similarly to estimates (1.37)–(1.39), let us write 

	(1.49)
	(1.50)
	(1.51)

Separately for the real and imaginary parts of the ch.f. we have 

	(1.52)
	(1.53)
	(1.54)
	(1.55)
	(1.56)
	(1.57)

For a k-current estimate of a multidimensional ch.f. taking into account (1.46) - (1.48) we can write

	(1.58)
	(1.59)
	(1.60)

In the above formulas, the model of a random process can be any, and the model of the impulse response of a physically realized integrator is presented in Table 1.1.

           In the theory of estimating the probabilistic characteristics of a random process, much attention is paid to the fundamental nature of estimates. Fundamentality is characterized by properties of estimates. They must be wealthy, efficient and unbiased. The properties of estimates (1.40 - 1.45, 1.47, 1.48) are studied in detail and described in the book [2]. It is also shown there that the estimates for the real and imaginary parts of the ch.f. are consistent, efficient, and asymptotically unbiased, i.e. they are fundamental.

Table 1.1. 

Classification of integrators

Integrator type	Characteristic q(T-t)
RC -integrator	βeβ(t-T/2)/2sh(βT/2)
RC -integrator	(l-βT+βt)eβt/T
RC -integrator	2(T-t-βeβ(t-T))/(2e-βT+T2-2)
Measuring instrument with critical damping	β2(T-t)eβT/(eβT-βT-1)

 

         Turning to estimates of the form (1.36 - 1.39, 1.46), we can say that they are also fundamental. Their properties will be the same as those of the estimates (1.40,1.41,1.47,1.48) of the real and imaginary parts of the ch.f. Here it is appropriate to say the following. Estimates (1.40), (1.41), (1.47), (1.48) are assigned the role of some approximators, with the help of which estimates of the probability density, correlation function, initial and central moment functions are constructed. Since the initial estimates (approximators) of the probabilistic characteristics of a random process are consistent, other estimates of the probabilistic characteristics constructed from them by classical methods will also be consistent, which is consistent with the conclusion in [2].

In the instrumental analysis of random processes, it can be difficult to obtain an estimate of a probabilistic characteristic that satisfies all of the above properties at once. For example, it may turn out that even if an effective estimate exists, then the algorithms for calculating it are too complicated and one has to be content with a simpler estimate, the variance of which is somewhat larger. The same can be said about biased estimates, as slightly biased estimates are sometimes used. The final choice of the estimate, as a rule, is dictated by the results of its analysis in terms of simplicity, ease of use in equipment or mathematical statistics, and reliability of properties. Estimates of the probability distribution function, probability density, correlation function, initial and central moment functions of the kth order are given in the book [2].

 

1.3. Estimation of the characteristic function of a discrete quantity

       Let us consider a special case when the process ξ(t) is represented only by instantaneous values ξ_i(t_n). To simplify the formulas, we introduce the notation ξ_i(t_n) = ξ_i. This is a discrete random variable for which the probability density has the form [4]

	(1.61)
where pi is the probability of occurrence of the i-th value of a discrete random variable; N is the total number of discrete values of a random variable; δ (·) is the delta function [4]. In view of (1.61), the ch.f. discrete random variable
	(1.62)

If a discrete random variable ξ_i=U₀= const with probability p=1, then the ch.f. constant value is calculated by the formula

(1.63)

Known distribution laws for a discrete random variable are tabulated in the book [5]. The characteristic functions obtained taking into account the known distribution laws of a discrete quantity are also given there. For example, a discrete random variable distributed according to the Poisson law has a ch.f. kind

(1.64)

where λ – Poisson distribution parameter. This function is widely used when observing the flows of elementary particles, such as electrons.

Formulas (1.7) - (1.11) can be used to calculate the distribution moments of a discrete random variable. Substituting the ch.f. (1.64) into formula (1.7), we obtain

 

,  ,    .

(1.65)

They coincide with the results contained in [4]. Thus, the above material regarding ch.f. random process can be extended to a discrete random variable, the ch.f. which is defined by formula (1.32).

Let us consider a complex number ξi=ui+jvi, which, in accordance with the notation adopted earlier, can be called a complex discrete quantity. Similarly to (1.24a), we obtain

.

(1.66)

Let’s denote  Then we have

                                                                               (1.67)

where p_i is the probability with which the random variable takes on the value y_i. If the probability density of this quantity is

(1.68)

and the probability density of the random variable v is equal to

(1.69)

then we get the equation

Taking into account (1.67), (1.70), we have the ch.f. discrete random variable

(1.71)

When the probability is   , then

,

(1.72)

where N is the number of equiprobable values of the discrete random variable v. For this case, expression (1.71) takes on the form

(1.73)

New formulas of the form (1.73) follow from consideration of other laws of distribution of a discrete random variable.

 

1.4. New property of the characteristic function

 

In the above analysis of the characteristic function, special attention was paid to its properties known from the publication, such as, for example, boundedness, measurability, and others. We have discovered a new property of ch.f., which concerns filtering in the space of noise probabilities with the help of this function.

Expressions (1.16 - 1.19) form the basis of the filtering capacity of the characteristic function. Filtering capacity of ch.f. has an applied value [3,7] and the following physical meaning.

The probability density of a random process and its ch.f. are connected by a pair of Fourier transforms (1.5), (1.15). It turns out that the ch.f. is the spectral density of the probability density (or, in short, the spectral density of the probabilities) of a random process in the domain of probabilities, if we use the terminology of the Fourier transform of signals, in which the domain is called the frequency domain. Ch.f. carries information about the probabilities of occurrence of instantaneous values ​​of a random process depending on the parameter Vm, which we previously proposed to call the multiplicity of values ​​of a random process. This multiplicity can be written as integer and fractional real numbers. For integer multiplicities, V_m= ±1, ±2, …, ±¥, and for fractional multiplicities, V_m takes any other values ​​on the real axis from ¥ to +¥. This is done in the same way as in the frequency domain, when imaging a line spectrum, the spectral lines are located at points with abscissas w,  2w,  3w,…,   nw, where w is the circular frequency of the signal. By analogy with the physical spectrum when using the ch.f. in practice, the multiplicity V_m is taken to be integer and positive. At V_m=0  ch.f. Q₁(0)=1 is the total probability. This total probability is distributed between the probabilities of presence in the signal of instantaneous values ​​with a multiplicity one (Vm=1), with a multiplicity two (V_m=2), with a multiplicity three (V_m=3), etc. For example, for "white" noise with dispersion σ² =1 and ch.f. of the form (1.12) at  m₁ = 0  we have:    p₁ =0,6065 at V_m =1;   p₂ =0,1353 at  V_m =2;   p₃=0,0111 at V_m  =3 etc. to infinity, and for V_m =∞  the probability p_∞ =0. When filtering the additive mixture using expressions (1.16 - 1.19) with V_m =1, all instantaneous noise values ​​that are present with a probability p₁ =0.6065 are “cut out” from it. In this case, after filtering, instantaneous noise values ​​remain in the additive mixture with a total probability of 0.3935, i.e. with a probability less than one. At the filter output, the additive mixture is different, it will be partially “cleaned” of noise. And, as a result of this, the signal-to-noise ratio at the filter output increases.

       You can continue filtering further the "purified" additive mixture, "cutting out" from it at V_m= 2 the instantaneous noise values ​​that are present with a probability p₂ = 0.1353. Then, at the output of the second filter, the signal-to-noise ratio increases even more, and the additive mixture becomes “cleaned” twice. And so it should be continued further, changing only the value of the multiplicity V_m=m∆V. When ∆V=1, it is appropriate to use the well-known term "filter section" and write the definition of a new device in the form m - link virtual filter.

         Simulation of virtual filters using the characteristic function with V_m =1 showed good results. The works [3,7] present quantitative and graphical data obtained using equation (1.19). As an example, Figure 1.3 shows graphs of noise and signal suppression when filtering an additive mixture, where σ₀, σ_f are the SNR at the input and output of the filter, respectively; σs is the SNR of the signal; h_вх is the signal-to-noise ratio at the filter input. The designation N is taken from expressions (1.42,1.43). An analysis of the curves in Figure 1.3 shows that the filter suppresses signal and noise differently. "White" noise with the help of ch.f. is suppressed by a factor of 5053, while the quasi-deterministic signal is attenuated by only a factor of 120. Thus, the signal-to-noise ratio at the output of a single-link virtual filter increased by an average of 30 times.

а)                                                                     b)

Figure 1.3. Filter suppression of noise (a) and signal (b):

1, 4 – N=5; 2 – N=10; 3 – N=50

 

2. QUASI-DETERMINISTIC SIGNALS

 

         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)

     

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(V_m)=  ,                                                           (2.3)

.

(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.