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

The theoretical analysis of modem noise immunity is based on the determination of the real and imaginary parts of the ch.f. additive mixture. Then the values are calculated separately for each of the parts of the ch.f. additive mixture. And finally, these values of the ch.f. are compared with the thresholds set in the sine and cosine channels of the demodulator in order to make decisions in accordance with the observance of the inequalities recorded in the signal discrimination algorithm. Separately, a quantitative analysis of the probability of errors is carried out. In total, thirteen different modems of the new generation are considered together. To determine whether the material in this chapter belongs to the device model, the modem name was used, which includes a cipher of letters and numbers denoting the following: A - arcsine law; K is the law of cosine; B - Veshkurtsev's law; T is Tikhonov's law; 1 - one channel; 2 - two channels; 2-1 - one channel resulting from combining two different demodulator channels using digital logic circuits. Let's correctly write down and decipher, for example, such a name: A2-1 modem is a single-channel modem for receiving signals with distribution according to the arcsine law. 4.1. Noise immunity of modem A when receiving an additive mixture of noise and signal with the distribution of instantaneous values according to the arcsine law Let's recall that a signal with the distribution of instantaneous values according to the arcsine law can be modulated in two ways, described above in Section 3.1. We will consider each of them separately and, on their basis, we will build modems that are different in structure and characteristics. As a result, we get three models of a new generation modem. 4.1.1. Noise immunity of the modem A2 when receiving an additive mixture of noise and a non-centered signal with the distribution of instantaneous values according to the arcsine law The modem contains a modulator (Fig. 3.1), the quasi-deterministic signal at the output of which is shown in Fig. 3g, and a two-channel demodulator (Fig. 3.7). Its name will be: modem A2. The modulation algorithm for a quasi-deterministic signal (2.1) is written in Table 4.1. Table 4.1. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0,18 0 logical "1" 0,18 0,9 The methodology and results of the studies were published in [26]. Let us turn to the analysis of the noise immunity of the demodulator under the action of an additive mixture of a quasi-deterministic signal (2.1) and "white" noise at its input z(t)=u(t)+n(t), (4.1) where n(t) is “white” noise, u(t) is a signal with a=U0, and the probabilistic characteristics are known from section 2.1. Using expressions (3.3, 3.4) and the data in Table 4.1, using formulas (3.13), we calculate the thresholds in the sine and cosine channels of the demodulator. As a result, with the value Vm = 1 and U0 =0,6 we get П1=J0(U0,t)sin(e0) = 0,7116; П2 = J0(U0,t) = 0,912. Further, with the value Vm=1 and s(t) =0 we define for the additive mixture (4.1) . (4.2) When s(t)=0, similarly to (4.2), we calculate for the value Vm=1 for the additive mixture (4.1) (4.3) where W(z) – probability density of instantaneous values of the additive mixture; – signal-to-noise ratio; – the dispersion of the quasi-deterministic signal; – the dispersion of "white" noise. The results (4.2, 4.3) need to be quantified. Tables 4.2, 4.3 present the results of calculations for П1=0,7116; П2=0,912; К1=0,56; К2=0,88, written in the line with the name of the evaluation. Table 4.2. The probability of errors in the cosine channel at a logical "1" Threshold 0,912∙ 0,88 = 0,8 Evaluation 0 0 0,37 0,83 0,9 0,9 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 2,2∙10-5 2∙10-45 2∙10-45 Table 4.3. Probability of errors in the sinus channel at logical "0" Threshold 0,7116∙ 0,56 = 0,4 Evaluation 0 0 0 0 0 0 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 0 0 0 When analyzing the data in tables 4.2, 4.3, we always compare the values of the estimates of ch.f. additive mixture with the thresholds recorded in the first line of the tables. At the same time, we see that the data in Table 4.2 exceeds the threshold, starting from the signal-to-noise ratio from 1 to 100, i.e. in the range of 20 dB. This means that there will be no errors here when receiving a logical "0", so the modem has maximum noise immunity. In the range of signal-to-noise ratios from 0.1 to 1, errors when receiving a logical "0" are possible. However, it can be stated that the noise immunity of the cosine channel of the modem is an order of magnitude better than the data given in the publication. Analyzing the data in Table 4.3, we see ideal results. In the sinus channel of the demodulator, all data is below the set threshold. Therefore, we have the maximum noise immunity when receiving a logical "0" in the range of signal-to-noise ratios from 10-3 to 102 or 50 dB, and the lower limit of the range is minus 30 dB. These data are at least twenty orders of magnitude better than the noise immunity of the device known from the publication. Suppose the additive mixture (4.1) contain a non-centered quasi-deterministic signal at the demodulator input; this corresponds to the condition s(t)=1. Similarly, to (4.2), for the value Vm=1 we define (4.4) or similarly to (4.3) for the value Vm=1 we calculate (4.5) The results (4.4), (4.5) need a quantitative analysis. Tables 4.4, 4.5 show calculation data at П1=0,7116; П2=0,912; К1=0,56; К2=0,88, written in the line with the name of the evaluation. Table 4.4. Probability of errors in the cosine channel at logical "0" Threshold 0,912∙ 0,88 = 0,8 Evaluation 0 0 0,23 0,52 0,57 0,57 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 2∙10-45 2∙10-45 2∙10-45 Table 4.5. The probability of errors in the sinus channel at a logical "1" Threshold 0,7116∙ 0,56 = 0,4 Evaluation 0 0 0,29 0,65 0,71 0,71 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 8∙10-32 2∙10-45 2∙10-45 Similarly, to the analysis of tables 4.2, 4.3, we will study the data of tables 4.4, 4.5. The data in Table 4.4 is below the set threshold. Hence, they correspond to the ideal case. Here we can say that the reception of the logical "1" in the cosine channel of the demodulator occurs without errors, i.e. with ultimate noise immunity, in the range of signal-to-noise power ratios from 10-3 to 102 or in the range of 50dB. These data are at least twenty orders of magnitude better than the noise immunity of the device known from the publication. The data in Table 4.5 are much more modest than the previous ones. In the sinus channel of the demodulator, a logical "1" is received without errors only when the signal-to-noise ratio is from 1 to 100 or in the range of 20 dB. With a signal-to-noise ratio from 0.1 to 1 in the sinus channel of the demodulator, errors are possible when receiving a logical "1". Let's move on from qualitative data analysis to a quantitative assessment of modem noise immunity. In tables 4.2–4.5, the following designations are adopted: Р0 – the probability of errors when receiving a logical "0"; Р1 – the probability of errors when receiving a logical "1"; - is the total probability of device errors. Quantitative assessment of the noise immunity of the modemA2. In expressions (3.11,3.12), instead of the expectation operator, an ideal adder is used. And, as a result of this, we obtain estimates of the real and imaginary parts of the ch.f., which are recorded in tables 4.2 - 4.5. Both estimates are random variables with their own properties and distribution laws. Let us recall that estimates for the real and imaginary parts of the ch.f. are efficient, consistent, and unbiased. This is shown in earlier works, for example [2], in which the effectiveness of estimates is characterized by their variances. In the book [2, p. 95 – 96] the dependence of the variance of estimates (3.11, 3.12) on the dimensionless time is shown , where - the duration of the signal realization; - the width of the energy spectrum of the signal. With a value of S= 100 the variance of the estimate of the real part of the ch.f. , and the variance of the estimate of the imaginary part of the ch.f. . The value S =100 will be obtained when we take with s and . Here the designations are borrowed from expressions (3.11,3.12). The law of distribution of estimates of the real and imaginary parts of the ch.f. depends on the probability density of the additive mixture of signal and noise. Let it be normal in the first approximation, since it is difficult to solve this problem mathematically exactly, and maybe even not possible. According to Professor S.Ya. Vilenkin, who has been solving similar problems for many decades, "..an exact solution is possible only in some cases [27, p.106]". For example, in the same place, the author obtained the exact distribution law for the estimate of the correlation function of a Gaussian signal, and then, after some assumptions, suggested that it be considered approximately normal. Let's follow this example. Looking ahead, we say that when modeling a demodulator (Fig. 3.7), the validity of such a hypothesis was proved in [28]. Next, we proceed similarly to the procedure for discretizing a continuous value by level, with one level equal to the threshold, and the second level is not limited by the threshold, i.e. it is variable without negative consequences for the probability of errors. At the same time, we consider that the center of the distribution law coincides with the value of the ch.f., recorded in tables 4.2 - 4.5, since estimates of ch.f. are not displaced. There is a corridor between the value of the assessment and the threshold, it is different when . If the value of the estimate of the ch.f. goes beyond the corridor boundary, then an error occurs when receiving a logical element. For example, the corridor is 0.23 in table 4.4 with a value =10. We divide the value of the corridor by "sigma", i.e. on estimates of the real or on estimates of the imaginary part of the ch.f. depending on the demodulator channel in question. The mean square value of the estimate in the cosine channel is , so we get the number of "23sigma" separating these two values. Then we apply a rule similar to the “three sigma” rule and calculate the value of the error integral at “L sigma”. In our example L=23. The error probability that interests us will be equal to the difference between unity and the value of the error integral. Unfortunately, in reference books on special functions [17], the values of the error integral are limited to the size L≤10. Therefore, in tables 4.2 - 4.5, the values of the error probability are sometimes overestimated, for example, in table 4.4 at =10 . In fact, the errors will be smaller by many orders of magnitude. For greater clarity and understanding of what was said above, we use the well-known distribution law for the estimate of the real part of the ch.f. [28], the view of which is shown in Figure 4.1. We will calculate the modem error probabilities using a method developed on the basis of the statistical decision theory [29]. The estimate is a random variable, depends on the signal-to-noise ratio and has a dispersion of the real part of the ch.f. and the dispersion of the estimate of the imaginary part of the ch.f. The values of estimates (3.11, 3.12) are distributed according to the Gauss law [28], in which the values recorded in tables 4.2 - 4.5 are the most probable, i.e. expectations. The distribution law and the initial data, where W(A) – the probability density of the ch.f. estimate is shown in Figure 4.1a; m1{A} – the expectation of the estimate of the ch.f. You can also see the interval between the mathematical expectation of the estimate and the threshold in the demodulator. In Figure 4.1a, the thresholds are shown to the left and right of the expected value. This is done because either the left or the right half of the distribution law is involved in determining the logical "0" and logical "1". Let's start calculating the error probabilities in the demodulator when a logical "0" is received. The mathematical expectation of the estimate is equal to the value (4.4). All values of the evaluation (3.11) must exceed the threshold П2k, shown in Figure 4.1a on the left. Let's use the three sigma rule. Let's define the number of sigmas using the relation . The probability of errors when receiving a logical "0" is equal to the area under the curve W(A), lying to the left of the threshold П2k. Fig. 4.1. The probability density of the real a) and imaginary b) parts of the ch.f. It is equal numerically , (4.5d) where erf (∙), erfc (∙) – probability integral (error function). Formula (4.5d) is suitable for calculating errors in the demodulator when a logical "1" is accepted. Only the expectation of the estimate in this case is equal to the value (4.2), and the probability of errors is equal to the area under the curve W(A), which lies to the right of the threshold П2k (colored in black), and will be denoted by . Then the probability of modem errors. When calculating the error probability according to formula (4.5d), data from tables 4.2, 4.4 were substituted in place . A similar description can be repeated for estimating (3.12) the imaginary part of the ch.f. using the data in tables 4.3, 4.5 and figure 4.1b. The total error probability of the sine (curve 1) and cosine (curve 2) demodulator channels is shown in Figure 4.2, and its main values are listed in Table 4.6. For comparison, in the same place from [15, p.473], the probability of errors (curve 3) of ideal phase modulation (PM), calculated in a noisy channel, is given. The choice of PM for comparison is not accidental. It is recognized by all as the most noise-resistant modulation. Table 4.6. Probability of errors of different modems Total sinus channel error probability 0,5 2∙10-21 4∙10-32 1∙10-45 Less than 1∙10-45 Total cosine channel error Probability 0,5 4,9∙10-1 1,1∙10-5 1∙10-45 Less than 1∙10-45 PM error probability 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio 0,1 0,5 1,0 10 100 Comparison of the noise immunity of the new modem with the noise immunity of the known device, in which ideal PM is used, shows its superiority by at least four orders of magnitude and more, up to thirty orders of magnitude when working with weak signals. This causes distrust among modem developers, whose opinion says: "this cannot be, because it can never be." In the cosine channel (curve 2), the new modem has a reference point with a non-zero error probability , i.e. from the limiting noise immunity of the device, if the probability of errors 10-32 is conventionally equated to zero. Its occurrence may be associated with a random, without any justification, choice for modulating the quantitative parameters of the distribution law of a quasi-deterministic signal. Probably, the optimization of these parameters will eliminate the modem reference point. There are no reference points in the sinus channel (curve 1) of the modem. Therefore, even with such data, one can hope for a good future for the new modem with two channels. Figure 4.2. Probability of errors of the two-channel modem A2 As a result, we can say that in the presence of "white" noise in the data transmission channel, the potential noise immunity according to Kotelnikov of the proposed modem is limiting, because with accurate synchronization of both channels of the modem, there are no errors when receiving a telegraph signal. In the sine and cosine channels of the modem, the range of signal-to-noise power ratios is different. In the sine channel it is equal to 30 dB, and in the cosine channel - 25 dB, and the lower limit of the range in the sine channel lies at the level of minus 10 dB, while in the cosine channel it is equal to minus 5 dB. Thus, the sine channel of the modem has better noise immunity than the cosine channel. In Figure 4.2, curves 2 and 3 run parallel in the section . Therefore, the cosine channel at the error probability level of 1∙10-5 and less has an energy gain of 10 dB relative to the ideal PM signal modem. Single-channel modem A2-1. The new modem contains a modulator (Fig. 3.1) and a single-channel demodulator (Fig. 3.8). Its name will be: modem A2-1. The modulation algorithm for a quasi-deterministic signal (2.1) remains the same and is recorded in Table 4.1. At the same time, the above theoretical analysis of modem noise immunity when operating in a noisy channel remains unchanged for the new modem model. However, the new modem model has only one channel and one output. Let's recall that the demodulator (Figure 3.8) combines the advantages of the sine and cosine channels of the demodulator in Figure 3.7. Table 4.3 shows that in the sinus channel of the demodulator, the logical "0" is determined without errors in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB. Table 4.4 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors also in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB, if the probability of errors 2∙10-45 is conventionally equated to zero. Theoretically, when these advantages of both channels are combined, they should get a new modem model with maximum noise immunity in the range of signal-to-noise ratios of 50 dB, with the lower limit of the range equal to minus 30 dB. However, in practice this does not work out, which is confirmed by table 3.1 of truth. The probability of errors in modem A2 - 1 decreases on average by 20 times compared with the probability of errors in the cosine channel of modem A2. Figure 4.3 shows the error probability of different new generation modems, where curve 1 is plotted for a known 4-QAM modulation and curve 3 for known QPSK modulation. Curves 2, 4 – 7 are plotted for the new SSK modulation. Curve 2 shows the error probability of modem A1 with a non-optimal modulation algorithm, and curve 4 - with the optimal modulation algorithm. Curve 5 refers to modem A2 (cosine channel), curve 7 - to modem A2 (sine channel). Curve 6 refers to an A2-1 dual-channel modem with channel bonding connected. Figure 4.3. The probability of errors of different modems of the new generation Modem A2–1 is superior in noise immunity to the cosine channel of modem A2 and modem A1. It has a potential noise immunity in the range of 30 dB and in this indicator exceeds, at least twenty orders of magnitude, modems known from domestic and foreign literature. The A2-1 modem with such characteristics has no analogues and competitors all over the world. To test the theory, statistical modeling was first carried out, and then, for re-verification, simulation modeling of the A2 modem and the A2–1 modem, its results were published [24,30]. The simulation confirmed the results of the theoretical analysis of the noise immunity of the demodulator. There is marked point D in Figure 4.3 where the results of calculations and simulations coincided. 4.1.2. Noise immunity of A1 modem when receiving an additive mixture of noise and a centered signal with the distribution of instantaneous values according to the arcsine law The modem contains a modulator, the quasi-deterministic signal at the output of which is shown in Figure 3.2e and coincides in shape with the classical amplitude keying, and the demodulator is single-channel (Figure 3.9). Its name will be: modem A1. The modulation algorithm for a quasi-deterministic signal (2.1) is written in Table 4.7. Table 4.7. Suboptimal signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "1" 1,125 0 logical "0" 0 0 The methodology and results of the studies were published in [25]. Let us turn to the analysis of the noise immunity of the demodulator, under the action of an additive mixture (4.1) of a centered quasi-deterministic signal (2.1) and "white" noise at its input z(t)=u(t)+n(t), where n(t) – "white" noise, u(t) – a signal with a=U0. Using expressions (3.3, 3.4) and the data in Table 4.7, using formulas (3.13), we calculate the threshold in the cosine channel of the demodulator. As a result, with the value Vm = 1, we get П2 = J0(U0,t) =J0(0) = 1. Further, with the value Vm=1 and s(t) =1 let’s define for the additive mixture (4.1) . (4.5а) where W1(z) – probability density of instantaneous values of the additive mixture (4.1); – signal-to-noise ratio; – is the dispersion of the quasi-deterministic signal. If the signal s(t) =0, then expression (4.5a) takes a different form . (4.5б) The results (4.5a), (4.5b) require a quantitative analysis. Tables 4.8,4.9 present calculation data at П2=1; К2=0,55 ; , written in a line with the name evaluation. Table 4.8. Probability of errors at logical "1" Threshold 1∙ 0,55 = 0,55 Evaluation 0 0 0,004 0.57 0,95 1 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 5∙10-3 2∙10-45 2∙10-45 When analyzing the data in Table 4.8, we always compare the values of the ch.f. additive mixture with the threshold recorded in the first row of the table. At the same time, we see that the data in Table 4.8 exceed the threshold, starting from the signal-to-noise ratio of 1 to 100, i.e. in the range of 20 dB. This means that there will either be no errors here when accepting a logical “1”, or they will be minimal. Let's recall that the inverter is turned on at the demodulator output, therefore, the recipient of information has a logical "0". Table 4.9. Probability of errors at logical "0" Threshold 1∙ 0,55 = 0,55 Evaluation 0 0 0,002 0,29 0,49 0,51 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 2∙10-45 2∙10-45 2∙10-45 2∙10-45 2,2∙10-17 1,5∙10-8 The data in Table 4.9 came out below the set threshold. Therefore, they correspond to the ideal case. This allows us to say that accepting of the logical "0" in the demodulator occurs without errors, i.e. with maximum noise immunity in the range of signal-to-noise power ratios from 10-3 to 102 or in the range of 50 dB. And, as a result of this, simple control commands (“turn on” or “turn off”, “open” or “close”, “raise” or “lower” and others) will be accepted by the modem with a reliability equal to one. If there is an inverter in the demodulator, the accepting of information receives the message in the form of a logical "1". Let's move on from qualitative data analysis to a quantitative assessment of modem noise immunity. In tables 4.8,4.9, the following designations are accepted: P0 - the probability of errors when accepting a logical "0"; P1 - the probability of errors when accepting a logical "1"; total probability of device errors. Quantitative assessment of noise immunity of modem A1. Again, we note that in expression (3.11), an ideal adder is used instead of the expectation operator. And, as a result of this, we obtain an estimate of the real part of the ch.f., which is recorded in tables 4.8,4.9. Repeating verbatim the reasoning stated above in section 4.1.1, we obtain, in relation to the data of tables 4.8, 4.9, the quantitative values of the error probability of the new modem model. The probability of demodulator errors depending on the signal-to-noise ratio with the help of graphs is shown in Figure 4.4, where curves 1 (at K2=0.55), 2 (at K2=0.53) characterize SSK (statistical shift keying) according to the data obtained here, curve 3 is amplitude keying according to the data of [15, p.478] , curve 4 is the ideal PM according to the data of [15, p.473]. The variable coefficient K2 significantly affects the noise immunity of the A1 modem, since up to the value inclusive, curve 1 looks better than curve 4, which characterizes the noise immunity of an ideal PM. The difference between them reaches ten orders of magnitude, and the gain in noise immunity belongs to amplitude manipulation. Here, the SSK competes as a leader, outperforming phase keying even in noise immunity. To detail the error probability, its main values are recorded in Table 4.10. For comparison, in the same place from [15, p.478], the probability of errors of ideal amplitude shift keying (AM), calculated in a noisy channel, is given. When the value of the coefficient K2 =0.55, modem A1 is superior in noise immunity to the known device using the ideal AM in the range of signal-to-noise power ratios of 20 dB. Table 4.10. Probability of errors of different modems New modem error probability (curve 1) 0,5 5∙10-1 2,5∙10-3 1,1∙10-17 7,5∙10-9 Probability of errors AM 0,82 0,62 4,8∙10-1 6∙10-2 3∙10-7 Signal-to-noise ratio 0,1 0,5 1,0 10 100 Comparison of the noise immunity of the A1 modem with the noise immunity of the known device, in which the ideal AM is used, shows its superiority by at least two orders of magnitude or more, up to fifteen orders of magnitude. Thus, new knowledge makes it possible to improve, at least two orders of magnitude, the noise immunity of modems with amplitude shift keying, which have been operating in digital communication systems over the past decades. For this, not much is required at all - to build a demodulator patented in Russia and conduct its full-scale tests. Figure 4.4. Error probability of a single-channel modem A1 First, statistical modeling was carried out, and then, for re-verification, simulation modeling of the A1 modem was carried out, its results were published [24, 30]. The simulation confirmed the results of the theoretical analysis of the noise immunity of the demodulator. It seems that new knowledge allows, in our opinion, to make a big leap towards improving the noise immunity of old-generation modems operating with amplitude-keyed signals. Our conclusion can be treated differently. However, it cannot be left unattended, because in the era of the digital economy, a new generation modem has a great future. Comparison of Figures 4.3, 4.4 shows that a single-channel A1 modem with a non-optimal modulation algorithm (Table 4.7, curve 1 in Fig. 4.4 or curve 2 in Fig. 4.3) is inferior in noise immunity to the same A1 modem with an optimal modulation algorithm (Table 4.11, curve 4 in Figure 4.3). The A1 modem with the optimal modulation algorithm is 10 dB more energy efficient than the known QPSK modulation. Table 4.11. Optimal signal modulation algorithm with Vm=2 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "1" 0,72 0 logical "0" 0 0 The search for optimal signal modulation algorithms is aimed at constructing such a special function in formulas (4.5a, 4.5b), the values of which at and differ from each other by several orders of magnitude. 4.2. Noise immunity of modem B when receiving an additive mixture of noise and signal with the distribution of instantaneous values according to the Veshkurtsev law The statistical law of Veshkurtsev [12] has quantitative parameters that can be changed abruptly, i.e. modulate the signal with the distribution of instantaneous values according to this law. In total, we have considered 12 methods of modulating the characteristic function (ch. f.) of a quasi-deterministic signal with the distribution of its instantaneous values according to the Veshkurtsev law. The variables in this law were the variance and the expectation of the random amplitude of the signal, which, in turn, was distributed according to the Gauss law. The result is 6 direct signal modulation methods and 6 inverse modulation methods. The 10 signal modulation methods in the demodulator require sine and cosine channels to transform its instantaneous values in order to recover the transmitted information. Only 2 modulation methods require one cosine channel in the demodulator to recover the telegraph signal. 4.2.1. Noise immunity of the B1 modem when receiving an additive mixture of noise and signal with the distribution of instantaneous values according to the centered Veshkurtsev law The modem contains a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.9). Its name will be: modem B1. The modulation algorithm for a quasi-deterministic signal (2.9) is written in Table 4.12. Table 4.12. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "1" 1,0 0 logical "0" 0,0009 0 The research methodology and results are published in [10,12,31]. Let us turn to the analysis of the noise immunity of the demodulator, when an additive mixture of a quasi-deterministic signal (2.9) and "white" noise acts at its input z(t)=u(t)+n(t), (4.6) where n(t) – "white" noise, u(t) – signal (2.9). Using expression (2.12) and the data in Table 4.11, using formula (3.13), we calculate the threshold in the demodulator. As a result, with the value Vm = 1, we get П1= 0,7917. Let us represent the functional transformation in the demodulator circuit with the dependence y =cosz at the value Vm = 1 and N>> 1. Let us calculate the expectation m1{y}, since the ch.f. is the expectation of the cosine function for the real part and the sine function for the imaginary part. Let's recall that the imaginary part of the ch.f. is equal to zero. We get with the value Vm =1 (4.7) where W(z) - the probability density of the additive mixture (4.6); σш2 - dispersion of "white" noise. Dispersion of the modulated c.c.s. changes abruptly from σ02 to σ12, the values of which are recorded in Table 4.12. Then, when transmitting a logical "0", we get (4.8) and when transmitting a logical "1" will be (4.9) Having performed the following substitutions in expressions (4.8,4.9) σш2=σ02 /h02, σш2=σ12 / h12 , we get (4.10) (4.11) where h0=σ0 / σш - signal-to-noise ratio when receiving logical "0"; h1= σ1 / σш - signal-to-noise ratio when receiving a logical "1". The results (4.10), (4.11) require a quantitative analysis. Tables 4.13, 4.14 show the calculation data at K1 =1,14, П1 = 0,7917, recorded in the line with the name of the evaluation. In this case the following is taken into account. The modulation algorithm in Table 4.12 contains and . This means that at a constant noise power in the case of transmission of a logical "0" and a logical "1", the ratio . For the given values of the dispersions, we will obtain a value and use it in the calculations. When the modem is operating in a noisy channel, it is impossible to provide a different signal-to-noise ratio at its input when receiving a logical "0" and a logical "1", because the noise power in the channel does not depend on the logical "0" and "1". Therefore, the final conclusions about the probability of modem errors in this case should be taken depending on the values of the ratio . Table 4.13. Probability of errors at logical "1" Threshold 0,7917∙1,14 = 0,9 Evaluation 0,6376 0,9512 1 1 1 1 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1,5∙10-12 2∙10-45 2∙10-45 2∙10-45 2∙10-45 In Table 4.13, the evaluation values A ̂(1,t) exceed the threshold at a signal-to-noise ratio from 0.01 to 100; here, in a noisy channel, there are no errors when accepting a logical "1" in the demodulator. When h12 < 0,01 errors appear in the demodulator in the channel with noise when accepting a logical "1". Thus, the range of signal-to-noise ratios is only 40 dB, with the lower limit of the range being minus 20 dB. Table 4.14. Probability of errors at logical "0" Threshold 0,7917∙1,14 = 0,9 Evaluation 0,505 0,757 0,788 0,792 0,792 0,792 Relation 1,11111 11,1111 111,111 1111,11 11111,1 111111 Probability of errors P0 0 0 0 0 0 0 In Table 4.14, all evaluation values A ̂(1,t) are less than the threshold for any signal-to-noise ratio. This means that in a channel with noise, the demodulator does not have errors when receiving a logical “0” in the range of signal-to-noise ratios of 50 dB. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the B1 modem. In tables 4.13,4.14, the following designations are accepted: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when accepting a logical "1"; - the total probability of device errors. Quantitative assessment of modem noise immunity B1. In expression (3.11), an ideal adder is used instead of the expectation operator. And, as a result of this, we obtain an estimate of the real part of the ch.f., which is written in tables 4.13, 4.14. An estimate is a random variable that has its own properties and distribution law. Repeating verbatim the rationale and methodology for calculating errors in the demodulator, set out in Section 4.1.1, we get the data recorded in Table 4.15. For comparison, the error probability of ideal phase modulation (PM) is given in the same place from [15, p.473], calculated in a noisy channel. Table 4.15. Probability of errors of different modems Probability of modem errors 7,5∙10-13 1∙10-45 Less than 1∙10-45 Less than 1∙10-45 Less than 1∙10-45 PM error probability 1,0 0,9 1,5∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio 0,01 0,1 1,0 10 100 The error probability of the demodulator depending on the signal-to-noise ratio with the help of graphs is shown in Figure 4.5, where curve 1 characterizes the error probability according to the data obtained here, curve 2 - the ideal PM according to the data of [15, c.473]. The variable coefficient K1 significantly affects the probability of modem errors. Thanks to it, you can adjust the amount of demodulator errors. Comparison of the noise immunity of a new modem model with the noise immunity of a well-known device in which ideal PM is used shows its superiority by at least ten orders and even up to thirty orders. These figures are simply fantastic. Perhaps they determine the potential noise immunity of statistical modulation for the near future, which must be achieved. Therefore, in a noisy channel, modem B1 has maximum noise immunity or, in other words, it works without errors when receiving a telegraph message in the range of signal-to-noise ratios from 0.01 to 100, i.e. in the range of 40 dB, starting from a value of minus 20 dB. This indicates that the expectation operator in the mathematical model of ch.f. reliably protects the signal from noise. Modems Figure 4.5. Error probability of a single-channel modem B1 of the new generation can work without errors when the signal-to-noise ratio is much less than one. This modem model can be used in radio and wired, cable, fiber-optic communication channels. This modem has no analogues and competitors all over the world.   4.2.2. Noise immunity of the B2 modem when accepting an additive mixture of noise and signal with the distribution of instantaneous values according to the non-centered Veshkurtsev law The modem contains a modulator (Fig. 3.3) and a two-channel demodulator (Fig. 3.7). Its name will be: modem B2. The modulation algorithm for a quasi-deterministic signal (2.9) is written in Table 4.16. Table 4.16. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0,01 0 logical "1" 0,01 0,6 The research methodology and results are published in [10,13,31]. Let us turn to the analysis of the noise immunity of the demodulator, when an additive mixture of a quasi-deterministic signal (2.9) and "white" noise acts at its input z(t)=u(t)+n(t), (4.12) where n(t) – "white" noise, u(t) – signal (2.9). Using expressions (2.12,3.10) and the data in Table 4.16, using formulas (3.13), we calculate the thresholds in the demodulator. As a result, with the value Vm = 1, we get П1= 0,5646 , П2= 1. Let us calculate the real and imaginary parts of the ch.f. additive mixture (4.12) and is comparable with the thresholds. Then, during the transmission and value in the channels of the demodulator, the threshold devices will accept the values of the real and imaginary parts of the ch.f. additive mixture equal to , , (4.13) where - the probability density of the additive mixture; - signal-to-noise ratio; - dispersion of "white" noise. When transmitting and the value in the channels of the demodulator, the threshold devices will accept the values of the real and imaginary parts of the ch.f. additive mixture, equal , (4.14) . (4.15) Let . The results of calculations by formulas (4.13,4.14,4.15) are summarized in tables 4.17, 4.18,4.19,4.20, recorded in the line with the name of the evaluation. Table 4.17. Probability of errors in the cosine channel of the modem at a logical "1" Threshold 1∙ 0,96 = 0,96 Evaluation 0,0067 0,6065 0,9512 0,99 1 1 Relation h2 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 0,988 2,1∙10-5 1,5∙10-8 1,5∙10-8 Analysis of the data in tables 4.17–4.20 shows that the logical "1" in the cosine channel and the logical "0" in the sine channel are determined correctly, i.e. without errors, at any signal-to-noise ratio in the range from 10-3 to 102 or 50 dB in power. Logical "0" in the cosine channel is determined without errors when the signal-to-noise ratio is from 0.1 to 100, i.e. in the range of 30 dB in power. Table 4.18. Probability of errors in the cosine channel of the modem with a logical "0" Threshold 1∙ 0,96 = 0,96 Evaluation 0,0056 0,5005 0,785 0,817 0,8253 0,8253 Relation h2 0,001 0,01 0,1 1.0 10 100 Probability of errors P0 0 0 2∙10-45 2∙10-45 2∙10-45 2∙10-45 Table 4.19. Probability of errors in the sinus channel of the modem at a logical "1" Threshold 0,532∙ 0,5646 = 0,3 Evaluation 0,0038 0,3424 0,537 0,559 0,5646 0,5646 Relation h2 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 5,8∙10-2 2,8∙10-26 4,3∙10-31 1,5∙10-32 1,5∙10-32 Table 4.20. Probability of errors in the sinus channel of the modem at a logical "0" Threshold 0,532∙ 0,5646 = 0,3 Evaluation 0 0 0 0 0 0 Relation h2 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 3,8∙10-41 3,8∙10-41 3,8∙10-41 3,8∙10-41 3,8∙10-41 3,8∙10-41 At the same time, in the sinus channel, the logical "1" is determined without errors when the signal-to-noise ratio is from 0.01 to 100, i.e. in the range of 40 dB on power. In this modem, the sinus channel prevails, since it has a maximum noise immunity in the range of signal-to-noise ratio of 40 dB when operating in a communication channel with white noise interference, and the lower limit of the range is minus 20 dB. From a qualitative analysis of the data, let's move on to a quantitative assessment of the noise immunity of the B2 modem. In tables 4.17 - 4.20, the following designations are adopted: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when accepting a logical "1"; total probability of device errors. Quantitative assessment of modem noise immunity B2. In expressions (3.11,3.12), instead of the expectation operator, an ideal adder is used. And, as a result of this, we get evaluations of the real and imaginary parts of the ch.f., which are recorded in tables 4.17 - 4.20. Both evaluations are random variables with their own properties and distribution laws. Repeating verbatim the rationale and methodology for calculating errors in the demodulator channels, set out in Section 4.1.1, we obtain the data recorded in Table 4.21. For comparison, the error probability of ideal phase modulation (PM) in the same place from [15, p.473] is given, calculated in a noisy channel. Table 4.21. Probability of errors of different modems Total sinus channel error probability 2,9∙10-2 1,4∙10-26 2,1∙10-31 7,5∙10-33 7,5∙10-33 Total cosine channel error probability 0,5 0,5 1∙10-5 7,5∙10-9 7,5∙10-9 PM error probability 1,0 0,9 1,5∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio 0,01 0,1 1,0 10 100 The dependence of the B2 modem error probability on the signal-to-noise ratio is shown in Figure 4.6, where curves 1,2 refer to the cosine channel of the demodulator; curves 3,4 - to the sinus channel of the demodulator; curve 5 - to the ideal PM. The sine and cosine channels have different noise immunity, and in each channel it strongly depends on the value of the thresholds (3.13). With the help of variable coefficients K1, K2, the modem can be configured. An example of this is shown in Figure 4.6, where curve 2 is obtained with a coefficient K1 = 0.52; curve 4 - with coefficient K1=0.532; curve 1 - with coefficient K2 = 0.98; curve 3 - with coefficient K2 = 0.96. Curves 1, 2 have the expectation of the signal e0 =0.4, and curves 3,4 have the value e0 =0.6, which is recorded in Table 4.16. This means that changing the value of e0 also affects the noise immunity of modem B2. Therefore, it is necessary to create the foundations of the theory of statistical signal modulation algorithms with the Veshkurtsev distribution law in order to obtain good noise immunity of the B2 modem. Unfortunately, this has not been done yet, and the signal modulation algorithm is written approximately. Nevertheless, the probability of errors in the sinus channel of the modem turned out to be tremendously low and lies at the level of 7.5∙10-33 when Figure 4.6. Error probability of a dual-channel modem B2 receiving weak signals, when the signal-to-noise power ratio does not exceed 17 dB, and the lower limit of the ratio is minus 10 dB. According to this indicator, the B2 modem is thirty orders of magnitude superior to a similar device with ideal phase modulation (curve 5). On the other hand, the cosine channel of modem B2 has an error probability of 7.5∙10-9 and competes very little with the well-known device using PM. Therefore, it has no prospects for existence in the future. Single-channel modem B2-1. The new modem contains a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.8). Its name will be: modem B2-1. The modulation algorithm for a quasi-deterministic signal (2.9) is written in Table 4.16. At the same time, the above theoretical analysis of modem noise immunity when operating in a noisy channel remains unchanged for the new modem model. However, the new modem has only one channel and one output, on which the telegraph signal will appear as a result of the execution of the state table 3.1. Let's recall that the demodulator (Figure 3.8) combines the advantages of the sine and cosine channels of the demodulator shown in Figure 3.7. Table 4.20 shows that in the sinus channel of the demodulator, the logical "0" is determined without errors in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB, if the probability of errors 3.8∙10-41 is conventionally equated to zero. Table 4.18 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors also in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB, if the probability of errors 2∙10-45 is conventionally equated to zero. When combining these advantages of both channels together, we get a new modem with maximum noise immunity in the range of signal-to-noise ratios of 50 dB, with the lower limit of the range equal to minus 30 dB. However, this theoretical result cannot be realized in practice. The demodulator in Figure 3.8 improves the noise immunity of the B2 modem's cosine channel by only a factor of 20 on average. The error probability of different modems of the new generation is shown in Figure 4.7, where curve 3 refers to modem B2 - 1, curve 1 - to modem B1. It also shows the error probability of the sine channel (curve 2) and the cosine channel (curve 4) of the modem B2, as well as the known device (curve 5) for accepting signals with phase modulation. The B2-1 modem is superior in noise immunity to the cosine channel of the B2 modem, but inferior to the sine channel of the B2 modem, the B1 modem and the device using phase modulation. Such a modem has no prospects for the future. Figure 4.7. The probability of errors of different modems of the new generation 4.3. Noise immunity of the modem K2 when accepting an additive mixture of noise and signal with the distribution of instantaneous values according to the cosine law The modem contains a modulator (Fig. 3.3) and a two-channel demodulator (Fig. 3.7). Its name will be: modem K2. The modulation algorithm for a quasi-deterministic signal (2.19) is written in Table 4.22. Table 4.22. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0,4674 0 logical "1" 0,4674 0,8 The modem research technique was developed in [14]. Let us turn to the analysis of the noise immunity of the demodulator, when an additive mixture of a quasi-deterministic signal (2.19) and "white" noise acts at its input z(t)=u(t)+n(t), (4.16) where n(t) – "white" noise, u(t) – a signal with a=U0. Using expressions (2.26,2.27) and the data in Table 4.22, using formulas (3.13), we calculate the thresholds in the demodulator. As a result, with the value Vm = 1, we get П1= =0,5634 , П2 =( π)⁄4 =0,7854. For the value Vm=1, we define for the additive mixture (4.16) the real part of the ch.f. , (4.17) where - signal-to-noise ratio. When s(t)=0, similarly to (4.17) we calculate for the value Vm=1 for the additive mixture (4.16) the imaginary part of the ch.f. (4.18) The results (4.17), (4.18) require a quantitative analysis. Tables 4.23, 4.24 present the results of calculations at , , , , written in a line with the name of the evaluation. Table 4.23. Probability of errors in the cosine channel of the modem at a logical "1" Threshold 0,7854∙ 0,764 = 0,6 Evaluation 0 0 0,08 0,624 0,77 0,79 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 6,9∙10-4 2∙10-45 2∙10-45 Table 4.24. Probability of errors in the sinus channel of the modem at a logical "0" Threshold 0,5634∙ 0,53 = 0,3 Evaluation 0 0 0 0 0 0 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 0 0 0 Analysis of the data in Table 4.23 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors in the range of signal-to-noise power ratios from 1 to 100 or from 0 dB to 20 dB. Table 4.24 presents the ideal results, as logical "0" in the sinus channel of the demodulator is determined without errors, i.e. with ultimate noise immunity, at any signal-to-noise power ratio in the range of 50 dB. This allows us to say that simple control commands such as turn on-off, open-close and others will be accepted with a reliability equal to one, in any operating conditions of the K2 modem. Suppose the additive mixture (4.16) at the demodulator input contain non-centered quasi-deterministic signal, this corresponds to the condition s(t)=1. Similarly to (4.17), at the value Vm=1 we define (4.19) or similarly (4.18) with the value Vm=1 we calculate . (4.20) The results (4.19), (4.20) require a quantitative analysis. Tables 4.25, 4.26 show calculation data at , , , , written in a string with the name of the evaluation. Table 4.25. Probability of errors in the sinus channel of the modem at a logical "1" Threshold 0,5634∙ 0,53 = 0,3 Evaluation 0 0 0,05 0,44 0,55 0,56 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 1,1∙10-12 8∙10-32 8,6∙10-35 In case of the selected threshold values according to the data of tables 4.25, 4.26, the distinction of logical "1" from zero in the sinus channel of the demodulator occurs without errors in the range of signal-to-noise ratios from 1 to 100, i.e. in the range equal to 20 dB. In this case, in the cosine channel of the demodulator Table 4.26. Probability of errors in the cosine channel of the modem at a logical "0" Threshold 0,7854∙ 0,764 = 0,6 Evaluation 0 0 0,05 0,43 0,54 0,55 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 2∙10-45 2,2∙10-17 1,5∙10-12 the maximum noise immunity is maintained at a signal-to-noise power ratio of 0.001 and higher up to 100, i.e. in the 50 dB range, for which the lower limit is minus 30 dB. Therefore, simple control commands such as turn on-off, close-open and others will be accepted by the cosine channel with a reliability equal to one, under any operating conditions of the K2 modem. As a result of the analysis of the noise immunity of the K2 modem, we can say that in the presence of "white" noise in the data transmission channel, the noise immunity according to Kotelnikov of the proposed modem is limiting. With accurate synchronization of the operation of both channels of the K2 modem, there are no errors when receiving a telegraph signal in the range of signal-to-noise ratios of 20 dB or more, and the lower limit of the range is 0 dB. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the K2 modem. In tables 4.23-4.26, the following designations are accepted: Р0 – the probability of errors when receiving a logical "0"; Р1 – the probability of errors when receiving a logical "1"; - the total probability of device errors. Quantitative assessment of the noise immunity of the K2 modem. In expressions (3.11,3.12), instead of the expectation operator, an ideal adder is used. And, as a result of this, we obtain estimates of the real and imaginary parts of the ch.f., which are recorded in tables 4.23 - 4.26. Both estimates are random variables with their own properties and distribution laws. Let's recall that evaluations for the real and imaginary parts of the ch.f. are efficient, consistent, and unbiased. This is shown in earlier works, for example [2], in which the effectiveness of estimates is characterized by their variances. Repeating verbatim the reasoning stated above in section 4.1.1, we get, in relation to the data of tables 4.23 - 4.26, the quantitative values of the error probability of the new modem model. The probability of demodulator errors depending on the signal-to-noise ratio with the help of graphs is presented in Figure 4.8. Curve 1 characterizes the error probability of the sine channel, curve 2 characterizes the cosine channel of the K2 modem, curve 3 characterizes the device in which PM is applied. The total error probability of the sine and cosine channels of the K2 modem is shown in Table 4.27. For comparison, in the same place from [15, p. 473] there is presented the error probability of ideal phase modulation (PM) calculated in a noisy channel. Comparison of the noise immunity of a new modem with the noise immunity of a well-known device in which an ideal Table 4.27. Probability of errors of different modems Total sinus channel error probability 0,5 9∙10-3 5,5∙10-13 4∙10-32 4,3∙10-35 Total cosine channel error probability 0,5 0,5 3,5∙10-4 1,1∙10-17 7,5∙10-13 PM error probability 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio 0,1 0,5 1,0 10 100 PM, shows its superiority by thirteen orders and up to thirty orders. Modem K2 in the cosine channel has a reference point with an error probability of a value , different from zero, i.e. from the maximum noise immunity of the device. Its occurrence may be associated with a random, without any justification, choice for modulating the quantitative parameters of the distribution law of a quasi-deterministic signal. It is likely that the optimization of these parameters using the newly constructed theory of statistical modulation will eliminate the modem reference point. There are no fixed points in the sinus channel of the modem. Therefore, even with such data as they are recorded in table 4.27, one can hope for a good future for the K2 modem. Comparison of the noise immunity of the K2 modem with the previously considered modems A2, A2-1 shows that it is lower, since the A2-1 modem under equal conditions has a maximum noise immunity in the range of signal-to-noise ratios of 20 dB. The difference between the two compared modems lies only in the models of quasi-deterministic signals used in them. Apparently, a quasi-deterministic signal with an arcsine distribution law Figure 4.8. Probability of modem K2 errors has an entropy less than an oscillation (2.19) with a cosine distribution law or a random process (2.9) with the Veshkurtsev distribution law, which has no equal among the distribution laws of signals considered in Chapter 2. Single-channel modem K2 -1. Suppose the new modem contain a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.8). The modulation algorithm for a quasi-deterministic signal (2.19) is written in Table 4.22. At the same time, the above analysis of modem noise immunity when operating in a noisy channel remains unchanged for the new modem model. However, the new modem has only one channel and one output, on which the telegraph signal will appear as a result of fulfilling the positions of the truth table 3.1. Let's recall that the demodulator (Figure 3.8) combines the advantages of the sine and cosine channels of the demodulator shown in Figure 3.7. Table 4.24 shows that in the sinus channel of the demodulator, the logical "0" is determined without errors in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB. Table 4.26 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors also in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB. When combining these advantages of both channels together, we get a new modem with maximum noise immunity in the range of signal-to-noise ratios of 50 dB, with the lower limit of the range equal to minus 30 dB. However, this theoretical result cannot be realized. The demodulator in Figure 3.8 improves the noise immunity of the cosine channel of the K2 modem on average only 20 times. The error probability of the K2 modem and the K2-1 modem is shown in Figure 4.9, where curve 1 refers to the sinus channel of the K2 modem; curve 3 - to the cosine channel of modem K2; curve 2 - to modem K2 - 1; curve 4 - to the device in which phase modulation is applied. An analysis of the graphs in Figure 4.9 shows that the K2-1 modem works well with weak signals and has a minimum error probability of 1∙10-19 at a signal-to-noise ratio of 10 dB, and then the error probability begins to increase to a value of 1∙10-14 if the signal is growing. Modem K2–1 is inferior in noise immunity to the sinus channel of modem K2 up to the signal-to-noise ratio of 20 dB. However, both modems are superior in noise immunity to the device for receiving signals with ideal PM (curve 4) in the range of signal-to-noise power ratios . Let's recall that the K2 modem and the K2–1 modem use statistical modulation of a quasi-deterministic signal. Therefore, the K2 modem and the K2-1 modem have no analogues and competitors all over the world. Figure 4.9. The probability of errors of different modems of the new generation 4.4. Modem noise immunity when accepting an additive mixture of noise and signal with the distribution of instantaneous values according to Tikhonov law Tikhonov law (2.34) contains the parameter D, which is included in all the probabilistic characteristics of the signal. Parameter value . Let's look at a few examples. 4.4.1. Noise immunity of the modem T2 when accepting an additive mixture of noise and signal with the distribution of instantaneous values according to Tikhonov law with the parameter D = 1 The modem contains a modulator (Fig. 3.3) and a two-channel demodulator (Fig. 3.7). Its name will be: modem T2. The modulation algorithm for a quasi-deterministic signal (2.33) is recorded in Table 4.28. The method and results of modem studies are published in [32]. Table 4.28. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 1,604 0 logical "1" 1,604 0,8 Let us turn to the analysis of the noise immunity of the demodulator, when an additive mixture of a quasi-deterministic signal (2.33) and "white" noise acts at its input z(t)=u(t)+n(t), (4.21) where n(t) – "white" noise, u(t) – signal (2.33). With the help of expressions (2.26,2.27) and the data in Table. 4.27 using formulas (3.13) we calculate the thresholds in the demodulator. As a result, at a value of Vm = 1 and we get П1= =0,32 , П2 = =0,4464. At the value Vm=1 we define for the additive mixture (4.21) the real part of the ch.f. , (4.22) where - the signal-to-noise ratio. When s(t)=0, similarly to (4.22) we calculate at the value Vm=1 for the additive mixture (4.21) the imaginary part of the ch.f. (4.23) The results (4.22), (4.23) require a quantitative analysis. Tables 4.29, 4.30 present the results of calculations at , , , , written in a line with the name of the evaluation. Table 4.29. Probability of errors in the cosine channel of the modem at a logical "1" Threshold 0,4464∙ 0,448 = 0,2 Evaluation 0 0 0 0,201 0,412 0,446 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 8,9∙10-1 2∙10-45 2∙10-45 Table 4.30. The probability of errors in the sinus channel of modem at a logical "0" Threshold 0,32∙ 0,375 = 0,12 Evaluation 0 0 0 0 0 0 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 1,5∙10-8 1,5∙10-8 1,5∙10-8 1,5∙10-8 1,5∙10-8 1,5∙10-8 Analysis of the data in Table 4.29 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors in the range of signal-to-noise power ratios from 1 to 100 or from 0 dB to 20 dB. Table 4.30 presents the ideal results, as logical "0" in the sinus channel of the demodulator is determined without errors, i.e. with ultimate noise immunity, at any signal-to-noise power ratio. This allows us to say that simple control commands such as turn on-off, open-close and others will be accepted with a reliability equal to one, in any operating conditions of the sinus channel of the modem T2. Suppose the additive mixture (4.21) contain a non-centered quasi-deterministic signal at the demodulator input, this corresponds to the condition s(t)=1. Similarly to (4.22) at the value Vm=1 we define (4.24) or similarly (4.23) with the value Vm=1 we calculate . (4.25) The results (4.24), (4.25) require a quantitative analysis. Tables 4.31, 4.32 show the calculation data at , , , , written in the line with the name of the evaluation. Table 4.31. Probability of errors in the sinus channel of the modem at a logical "1" Threshold 0,32∙ 0,375 = 0,12 Evaluation 0 0 0 0,144 0,296 0,32 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 4∙10-1 1∙10-16 1∙10-21 In case of the selected values of the thresholds according to the data of tables 4.31, 4.32, the discrimination of the logical "1" from zero in the sinus channel of the modem T2 occurs without errors in the range of signal-to-noise ratios from 1 to 100, i.e. in the range equal to 20 dB. In this case, in the cosine channel of the demodulator, the limiting Table 4.32. Probability of errors in the cosine channel of the modem at logic "0" Threshold 0,4464∙ 0,448 = 0,2 Evaluation 0 0 0 0,14 0,28 0,31 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 2,2∙10-17 0,5 0,5 noise immunity is maintained at a signal-to-noise power ratio from 0.001 and above to 1.0, i.e. from minus 30 dB to 0 dB. Therefore, simple control commands such as turn on-off, close-open and others will be received by the cosine channel of the modem T2 with a reliability equal to one in the range of signal-to-noise ratios of 30 dB. If the signal-to-noise ratio is greater than ten, there will be continuous errors in the cosine channel of the demodulator, i.e. the operation of this channel of the modem T2 in the channel with noise becomes impossible. As a result of the analysis of the noise immunity of the modem, we can say that in the presence of "white" noise in the data transmission channel, the noise immunity according to Kotelnikov of the modem T2 turned out to be different in both channels. With accurate synchronization of the operation of the sinus channel of the modem T2, there are no errors when receiving a telegraph signal in the range of signal-to-noise ratios of 20 dB or more, and the lower limit of the range is 0 dB. But the cosine channel of the modem T2 has an error probability of 0.5 and cannot work here. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the modem T2. In tables 4.29-4.32, the following designations are adopted: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when accepting a logical "1"; - the total probability of device errors. Quantitative assessment of the noise immunity of the modem T2. In expressions (3.11,3.12), instead of the expectation operator, an ideal adder is used. And, as a result of this, we get estimates of the real and imaginary parts of the ch.f., recorded in tables 4.29 - 4.32. Both evaluations are random variables with their own properties and distribution laws. Let's recall that evaluations for the real and imaginary parts of the ch.f. are efficient, consistent, and unbiased. This is shown in earlier works, for example [2], in which the effectiveness of estimates is characterized by their variances. Repeating verbatim the reasoning stated above in section 4.1.1, we get, in relation to the data of tables 4.29 - 4.32, the quantitative values of the error probability of the new modem model. The total error probability of the sine and cosine channels of the modem T2 is shown in Table 4.33. For comparison, in the same place from [15, p. 473] there is presented the error probability of ideal phase modulation (PM) calculated in a noisy channel. In addition, for clarity, the probability of modem errors using graphs is shown in Figure 4.10. Table 4.33. Probability of errors of different modems Total sinus channel error probability 0,5 0,5 2∙10-1 7,5∙10-9 7,5∙10-9 Total cosine channel error probability 0,5 0,5 0,45 0,5 0,5 PM error probability 0,5 0,5 1,5∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio 0,01 0,1 1,0 10 100 Figure 4.10. Probability of modem T2p errors In Figure 4.10, curve 1 characterizes the cosine channel of the T2 modem at K2 = 0.448, curve 2 - the cosine channel of the T2 modem at K2 = 0.78 according to [30], curve 3 - the sine channel of the T2 modem at K1 = 0.375, curve 4 - device for accepting PM signals. Comparison of the noise immunity of the modem with the noise immunity of the known device, in which the ideal PM is used, shows the superiority of the sinus channel characteristics in comparison with the prototype within the values , and then it disappears. The gain in noise immunity is only 0.8 dB at an error probability level of 7,5∙10-9. The cosine channel of the modem T2 is practically inoperable at the value of the variable coefficient K2 = 0.448, because for any signal-to-noise ratio, the error probability is 0.5. But with the value of the variable coefficient K2 = 0.78, the cosine channel of the modem T2 becomes better than the sine channel. Therefore, we will consider this modem to be bad and replace its code with another one: modem T2p (the letter p is a bad modem). The modem T2p does not have great prospects for existence in the future. 4.4.2. Noise immunity of modem T2 when receiving an additive mixture of noise and signal with the distribution of instantaneous values according to Tikhonov law with the parameter D = 2 The modem contains a modulator (Fig. 3.3) and a two-channel demodulator (Fig. 3.7). Its cipher will be: modem T2. Let's repeat the analysis of modem noise immunity with the value of the Tikhonov distribution parameter. The modulation algorithm for a quasi-deterministic signal (2.33) is recorded in Table 4.34. The method and results of modem research are published in [33]. Table 4.34. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0,7645 0 logical "1" 0,7645 0,8 The results (4.22 – 4.25) need to be quantified. Tables 4.35 - 4.38 present the results of calculations at the values , , , , written in the line with the name of the evaluation. Table 4.35. Probability of errors in the cosine channel of the modem at a logical "1" Threshold 0,7∙ 0,715 = 0,5 Evaluation 0 0 0,015 0,48 0,67 0,7 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 0,99 2∙10-45 2∙10-45 Table 4.36. Probability of errors in the sinus channel of the modem at a logical "0" Threshold 0,5∙ 0,5 = 0,25 Evaluation 0 0 0 0 0 0 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 8∙10-32 8∙10-32 8∙10-32 8∙10-32 8∙10-32 8∙10-32 Table 4.37. Probability of errors in the sinus channel of the modem at a logical "1" Threshold 0,5∙ 0,5 = 0,25 Evaluation 0 0 0 0,34 0,48 0,5 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 2,2∙10-5 3,7∙10-27 8∙10-32 Analysis of the data in Table 4.35 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors in the range of signal-to-noise ratios from 1 to 100 or from 0 dB to 20 dB. Table 4.36 Table 4.38. Probability of errors in the cosine channel of the modem at a logical "0" Threshold 0,7∙ 0,715 = 0,5 Evaluation 0 0 0,01 0,33 0,47 0,49 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 0 2,2∙10-5 1,6∙10-1 presents the ideal results, as logical "0" in the sinus channel of the demodulator is determined without errors, i.e. with ultimate noise immunity, at any signal-to-noise power ratio. This allows us to say that simple control commands such as turn on-off, open-close and others will be accepted with a reliability equal to one, in any operating conditions of the modem T2. With the selected threshold values according to tables 4.37, 4.38, the distinction between logical "1" and zero in the sinus channel of the demodulator occurs without errors in the range of signal-to-noise ratios from 1 to 100, i.e. in the range equal to 20 dB. In this case, in the cosine channel of the demodulator, the maximum noise immunity is maintained at a signal-to-noise power ratio from 0.001 to 1, i.e. with a signal-to-noise ratio of 30 dB. If the signal-to-noise ratio is greater than one, there will be continuous errors in the cosine channel of the demodulator, i.e. operation of the modem in a noisy channel becomes impossible. And as a result of this, the cosine channel of the modem can be excluded from the structure of the demodulator, and only the sine channel can be left in the modem. Let's move on from qualitative data analysis to a quantitative assessment of modem noise immunity. In tables 4.35 - 4.38, the following designations are adopted: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when accepting a logical "1"; - the total probability of device errors. Quantitative assessment of the noise immunity of the modem T2. In expressions (3.11,3.12), instead of the expectation operator, an ideal adder is used. And, as a result of this, we get evaluations of the real and imaginary parts of the ch.f., which are recorded in tables 4.34 - 4.37. Both evaluations are random variables with their own properties and distribution laws. Repeating verbatim the rationale and methodology for calculating errors in the channels of the demodulator, set out in Section 4.1.1, we get the data recorded in Table 4.39. The total error probability of the sine (curve 1,2,3) and cosine (curve 4,5,6) demodulator channel for different values of the parameter is shown in Figure 4.11. For comparison, in the same place from [15, p. 478] there is presented the error probability (curve 7) of ideal phase modulation (PM), calculated in a noisy channel. The main fragments of this noise immunity are listed in Table 4.39. An analysis of the curves in Figure 4.11 confirms that the noise immunity of the sine and cosine channels of the demodulator is different and depends on the - parameter of Tikhonov law. According to Table 4.39, it is desirable to take a large value of the parameter, and its optimal value can be obtained only as a result of constructing the theory of statistical modulation and conducting additional research. Comparison of the noise immunity of the new modem with the noise immunity of the known device, in which ideal PM is used, shows the superiority of its characteristics by at least ten orders, Figure 4.11. The probability of errors of different models of the modem T2 if we analyze weak signals, when . In case of strong signals, when , the noise immunity of an ideal PM is higher. The cosine and sine channels of the T2 modem are configured using the variable coefficients K1, K2 in different ways. In the cosine channel of the modem, the highest noise immunity at value is obtained at ratio (curve 6), and with increasing value it decreases. This modem T2 channel is suitable for excellent performance with marginal noise immunity with weak signals. In the sinus channel of modem T2, the opposite is done (curve 3). The sinus channel of the modem T2 works equally well with both weak and strong signals. Therefore, we indicate in the modem T2 cipher that it is good. Its new cipher will be: T2x modem, where x is good. Thus, increasing the value of the parameter D of the Tikhonov distribution up two allows raising the thresholds П1с ,П2к at least twice in the sine and cosine channels of the modem T2 demodulator. And as a result of this, probability of errors in the sinus channel of modem T2 Table 4.39. Probability of errors of different modems Sinus channel D=1 Curve 1 0,5 0,5 2∙10-1 8∙10-9 8∙10-9 D=2 Curve 2 0,5 0,5 1∙10-5 2∙10-27 4∙10-32 D=5 Curve 3 0,5 7∙10-3 1∙10-17 5∙10-28 4∙10-32 Cosine channel D=1 Curve 4 0,5 0,5 0,5 9∙10-19 8∙10-9 D=2 Curve 5 0,5 0,5 0,5 1∙10-5 8∙10-2 D=5 Curve 6 0,5 2∙10-2 1∙10-45 5∙10-28 2∙10-23 Ideal PM Curve 7 0,9 3∙10-1 1∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio h2 0,1 0,5 1 10 100 decreases at a value by four orders of magnitude (Table 4.39). By increasing only the thresholds in the demodulator channels, the downward trend in the error probability in the sinus channel of the modem T2 continues up to the signal-to-noise ratio of 10 dB. In the cosine channel of the modem T2, unfortunately, such a trend is not observed. Nevertheless, such a model of the modem T2 is promising and can be included in the class of new generation modems. 4.4.3. Noise immunity of the modem T2 when accepting an additive mixture of noise and signal with the distribution of instantaneous values according to Tikhonov law with the parameter D = 5 The modem contains a modulator (Fig. 3.3) and a two-channel demodulator (Fig. 3.7). Its cipher will be: modem T2. Let's repeat the analysis of the noise immunity of the modem with the value of the Tikhonov distribution parameter. The modulation algorithm for a quasi-deterministic signal (2.33) is recorded in Table 4.40. The method and results of modem research are published in [34]. Table 4.40. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0,228 0 logical "1" 0,228 0,8 The results (4.22 – 4.25) need to be quantified. Tables 4.41 - 4.44 present the results of calculations at , , , , written in a line with the name of the evaluation. Table 4.41. Probability of errors in the cosine channel of the modem at a logical "1" Threshold 0,9∙ 0,78 = 0,7 Evaluation 0 0 0,29 0,81 0,89 0,9 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 2∙10-45 2∙10-50 2∙10-55 Table 4.42. Probability of errors in the sinus channel of the modem at a logical "0" Threshold 0,64∙ 0,625 = 0,4 Evaluation 0 0 0 0 0 0 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 0 0 0 Table 4.43. Probability of errors in the sinus channel of the modem at a logical "1" Threshold 0,64∙ 0,625 = 0,4 Evaluation 0 0 0,21 0,58 0,64 0,65 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P1 1 1 1 2,2∙10-17 1∙10-27 8∙10-32 Table 4.44. Probability of errors in the cosine channel of the modem at a logical "0" Threshold 0,9∙ 0,78 = 0,7 Evaluation 0 0 0,2 0,56 0,62 0,63 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors P0 0 0 0 2∙10-45 1∙10-27 4,2∙10-23 Analysis of the data in Table 4.41 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors in the range of signal-to-noise power ratios from 1 to 100 or from 0 dB to 20 dB. Table 4.42 presents the ideal results, as logical "0" in the sinus channel of the demodulator is determined without errors, i.e. with ultimate noise immunity, at any signal-to-noise power ratio. This allows us to say that simple control commands such as turn on-off, open-close and others will be accepted with a reliability equal to one, in any operating conditions of the modem T2. With the selected threshold values according to the data of tables 4.43, 4.44, the distinction between logical "1" and zero in the sinus channel of the demodulator occurs without errors in the range of signal-to-noise ratios from 1 to 100, i.e. in the range equal to 20 dB. In this case, in the cosine channel of the demodulator, the maximum noise immunity is maintained at a signal-to-noise power ratio from 0.001 to 100, i.e. in the range of 50 dB. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the modem T2. In tables 4.41 - 4.44, the following designations are accepted: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when acceptinga logical "1"; total probability of device errors. Quantitative assessment of the noise immunity of the modem T2. In expressions (3.11,3.12), instead of the expectation operator, an ideal adder is used. And, as a result of this, we obtain estimates of the real and imaginary parts of the ch.f., which are recorded in tables 4.40 - 4.43. Both estimates are random variables with their own properties and distribution laws. Repeating verbatim the rationale and methodology for calculating errors in the channels of the demodulator, set out in Section 4.1.1, we get the data recorded in Table 4.45. The total error probability of the sine (curve 1) and cosine (curve 2) channels of the demodulator is shown in Figure 4.12. For comparison, in the same place from [15, p. 478] shows the error probability (curve 3) of ideal phase modulation (PM), calculated in a noisy channel. Table 4.45. Probability of errors of different modems Total sinus channel error probability 0,5 7,2∙10-3 1,1∙10-17 5∙10-28 4∙10-32 Total cosine channel error probability 0,5 1,7∙10-2 1∙10-45 5∙10-28 2,1∙10-23 PM error probability 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 Signal-to-noise ratio 0,1 0,5 1,0 10 100 Comparison of the noise immunity of the modem T2 with the noise immunity of the known device, in which ideal PM is used, shows the superiority of its characteristics by at least ten orders of magnitude. The cosine and sine channels of the modem T2 are configured using the variable coefficients K1, K2 in different ways. With the value D=5 in the cosine channel of the modem, the error probability itself is obtained with the ratio . Figure 4.12. Probability of modem T2 errors at D=5 (curve 2), and with increasing value , it grows. This modem T2 channel is suitable for excellent performance with marginal noise immunity with weak signals. In the sinus channel of the modem T2, the opposite is done (curve 1). The sinus channel of the modem T2 works equally well with both weak and strong signals. The probability of errors at a point in both modem channels is the same, and then curves 1,2 in Figure 4.12 diverge. Moreover, in the cosine channel of the modem, the error probability increases to the value Р=2.1∙10-23 with a signal-to-noise ratio of 20 dB, while in the sinus channel of the modem, the error probability continues to decrease to the value Р= 4∙10-32 with the same signal-to-noise ratio. Thus, increasing the value of the parameter D of the Tikhonov distribution to five makes it possible to raise the thresholds П1с, П2к at least three times in the sine and cosine channels of the modem T2 demodulator. And as a result of this, with a value by seventeen orders of magnitude (Table 4.39), the probability of errors in the sinus channel of the modem T2 decreases. By increasing only the thresholds in the demodulator channels, the downward trend in the error probability in the sinus channel of the modem T2 continues up to the signal-to-noise ratio of 20 dB. In the cosine channel of the modem T2, unfortunately, such a trend is not observed. Nevertheless, such a model of the modem T2 is promising and occupies a worthy place in the class of new generation modems. Single-channel modem T2-1. Let the new T2-1 modem contain a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.8). The quasi-deterministic signal modulation algorithm (2.33) is written in Table 4.40. At the same time, the above analysis of the modem noise immunity when operating in a noisy channel remains unchanged for the new modem model. However, the new modem has only one channel and one output, on which the telegraph signal will appear as a result of the execution of the truth table 3.1. Let's recall that the demodulator (Figure 3.8) combines the advantages of the sine and cosine channels of the demodulator in Figure 3.7. Table 4.42 shows that in the sinus channel of the demodulator, the logical "0" will be determined without errors in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB. Table 4.44 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors also in the entire range of signal-to-noise power ratios, i.e. in the range of 50 dB. When combining these advantages of both channels together, we get a new modem with maximum noise immunity in the range of signal-to-noise ratios of 50 dB, with the lower limit of the range equal to minus 30 dB. However, in practice this does not work out, which is confirmed by table 3.1 of truth. The probability of errors in the modem T2 - 1 is reduced by an average of 20 times compared with the error probability of the cosine channel of the modem T2. The error probability of the modem T2 and the modem T2-1 is shown in Figure 4.13, where curve 1 refers to the T2-1 modem; curve 2 - to the cosine channel of modem T2; curve 3 - to the sinus channel of modem T2; curve 4 - to the device in which phase modulation is applied. Figure 4.13. Probability of modem errors T2 - 1 An analysis of the graphs in Figure 4.13 shows that the modem T2-1 works well with weak signals and has an error probability of 1∙10-47 in the range of signal-to-noise ratios from 0.1 to 1.0 (minus 10 dB), and then the error probability starts to increase to the value of 1∙10-24 if the signal grows. After a signal-to-noise ratio of 10 dB, the modem T2-1 is equal in noise immunity to the sinus channel of the modem T2. However, both modems are superior in noise immunity to the device for receiving signals with ideal PM (curve 4) in the range of 26 dB, the lower limit of which is minus 10 dB. Starting from 16 dB to 20 dB, the modem T2 and the T2-1 modem together are inferior in terms of noise immunity to a device for receiving signals from PM. Let's recall that the modem T2 and the modem T2-1 use statistical modulation of a quasi-deterministic signal. Therefore, the modem T2 and the modem T2-1, when receiving weak signals, have no analogues and competitors all over the world. They can work with signals 10 times weaker than the noise power. 4.4.4. Noise immunity of the modem T1 when receiving an additive mixture of noise and signal with the distribution of instantaneous values according to Tikhonov law with a variable parameter D The modem contains a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.9). Its name will be: modem T1. Let us repeat the analysis of the noise immunity of the modem, modulating only the distribution parameter of Tikhonov law. The modulation algorithm for a quasi-deterministic signal (2.33) is recorded in Table 4.46. The method and results of modem research are published in [16]. Table 4.46. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the distribution parameter of the Tikhonov law logical "0" 1,604 1 logical "1" 0,228 5 Taking into account the modulation algorithm described above, we calculate the threshold in the demodulator in accordance with expression (3.13). As a result, with the value Vm = 1 and D = 5 we get П1 = 0,89 . At the value Vm=1 we define for the additive mixture (4.21) the real part of the ch.f. , (4.26) where - the signal-to-noise ratio. In expression (4.26) the signal dispersion changes according to the modulation algorithm. Taking this into account, we write down the value of the estimate for the ch.f. at and at , (4.27) where - the signal-to-noise ratio when accepting a logical "0"; - signal-to-noise ratio when accepting a logical "1"; ; . The results (4.27) need to be quantified. Tables 4.47, 4.48 present the results of calculations at , , written in a line with the name of the evaluation. In this case the following is taken into account. The modulation algorithm in Table 4.46 contains and . This means that at a constant noise power in the case of transmission of a logical "0" and a logical "1", the ratio . For the given values of the dispersions, we will get a value and use it in the calculations. When the modem is operating in a noisy channel, it is impossible to provide a different signal-to-noise ratio at its input when receiving a logical "0" and a logical "1", because the noise power in the channel does not depend on the logical "0" and "1". Therefore, the final conclusions about the probability of modem errors in this case should be taken depending on the values of the ratio . Analysis of the data in Table 4.47 shows that the modem T1 has maximum noise immunity when accepting a logical "1" in the range of signal-to-noise ratios from 1 to 100, i.e. in the range of 20 dB. From table 4.48 we see that when accepting a logical "0", the maximum noise immunity Table 4.47. Probability of modem errors with logical "1" Threshold 0,89∙ 0,84 = 0,75 Evaluation 0 0 0,29 0,81 0,89 0,9 Relation 0,001 0,01 0,1 1,0 10 100 Probability of errors Р1 1 1 1 2,2∙10-17 2∙10-51 2∙10-51 Table 4.48. Probability of modem errors at logical "0" Threshold 0,89∙ 0,84 = 0,75 Evaluation 0 0 0,143 0,4 0,442 0,446 Relation 0,007 0,07 0,703 7,035 70,35 703,5 Probability of errors P0 0 0 0 0 2∙10-51 2∙10-51 modem T1 takes place in the range of signal-to-noise ratios from 0.001 to 100, i.e., in the range of 50 dB. Let's remember that in the modem T1 the relation . So, when , relation . Therefore, the final noise immunity characteristics of the modem T1 should be evaluated by errors when accepting a logical "1", i.e. in relation to . From a qualitative analysis of the data, let's move on to a quantitative assessment of the noise immunity of the modem T1. In tables 4.47, 4.48, the following designations are accepted: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when accepting a logical "1"; total probability of device errors. Quantitative assessment of the noise immunity of the modem T1. In expression (3.11), an ideal adder is used instead of the expectation operator. And, as a result of this, we get an estimate of the real part of the ch.f., which is written in tables 4.47, 4.48. Evaluation of ch.f. is a random variable that has its own properties and distribution law. Repeating verbatim the rationale and methodology for calculating errors in the demodulator, set out in Section 4.1.1, we get the data recorded in Table 4.49. The error probability of the modem T1 depending on the signal-to-noise ratio is shown using graphs in Figure 4.14, where curve 1 characterizes statistical modulation (SSK), and curve 2 characterizes QPSK modulation. For details on the error probability of the modem T1, its individual values are recorded in Table 4.49. For comparison, the error probability of a device for receiving signals with ideal phase modulation (QPSK) in the same place from [15, p.473] is given, calculated in a noisy channel. Figure 4.14. Probability of modem T1 errors Table 4.49. Probability of errors of different modems Probability of modem T1 errors 0,5 1,5∙10-8 1,1∙10-17 2∙10-51 Less than 2∙10-51 PM error probability 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 Signal-to-noise ratio 0,1 0,5 1,0 10 100 Comparison of the noise immunity of the new modem T1 using SSK modulation with the noise immunity of the known device, which uses ideal PM, shows its superiority by at least ten orders and up to forty orders of magnitude. SSK modulation provides good noise immunity for weak signals. This effect arises due to the properties of ch.f. ideal to filter weak signals [7]. Therefore, at the error probability of the modem T1 is only 1.5∙10-8. In our opinion, with such data, we can hope for a good future for the modem T1. Thus, the new modulation algorithm for a centered quasi-deterministic signal with the Tikhonov distribution made it possible to significantly improve the noise immunity of the modem T1 and made it possible to use it in a radio channel, since the expectation of the signal is always identically equal to zero. It turns out that a quasi-deterministic signal with the Tikhonov distribution law has advantages over other signals if only its dispersion is modulated. However, the demodulator then becomes single-channel (Fig. 3.9). 4.5. Performance indicators of digital systems with new generation modems The class of quasi-deterministic signals includes oscillations of the form (2.1), (2.9), (2.19), (2.33) proposed by us, whose mathematical models do not contain time functions, but have only random variables. And, as a result of this, the energy spectra (2.8), (2.18), (2.32), (2.43) have one spectral line located at a frequency equal to the carrier frequency of the signal. In case of statistical modulation of signals (2.1), (2.9), (2.19), (2.33), their mathematical model changes and takes a different form (3.5), (3.8), which includes a telegraph signal s(t), which is a sequence of elements in the form of logical "0" and logical "1". The elements of a telegraph signal carry information. As an example, let's consider the signal (3.8). We assume that the telegraph signal has the Poisson distribution law and the correlation function , and it does not depend on the signal (2.9). Therefore, the signal correlation function (3.8), in accordance with its properties, will be equal to the sum of the correlation functions of independent variables [5] , (4.29) Where - the average number of pulse jumps per unit time; - shift in time; - amplitude of impulses. The signal correlation function (2.9) is defined in Chapter 2 and is equal to (2.16). Let us proceed to the analysis of the power spectral density (energy spectrum) of the signal (3.8). Let us write the energy spectrum of the signal [16] (4.30) where [16] (4.31) the spectrum was obtained and described earlier in Section 2.2. The spectrum (4.31) is continuous. The spectrum envelope has the form of a Gaussian curve at any carrier frequency of the numerical axis from -∞ to +∞. In the transition to the physical spectrum, i.e. to the spectrum in the region of positive frequencies, we obtain [16] (4.32) Let us determine the effective width of the energy spectrum of the modulated signal (3.8) at the frequency , after which we have [16] (4.33) where Expression (4.33) needs clarification. The fundamental provisions recorded in the book explain that ". . . the value can be interpreted as the width of a energy spectrum process uniform in the band, equivalent to the given one in terms of average power [4, p.202]”. If the frequency band (4.33) is implemented in a digital communication system, then half of the power of the received signal will be lost. This is not allowed in practice. To preserve the total power of the received signal, one should fulfill the equality , where - the effective bandwidth of the digital system. Then the signal power loss will be 1.54∙10-6 %, i.e. practically zero. Digital systems in the literature are characterized by performance indicators [35], which include noise immunity, spectral and energy efficiency. Let's compare the potential capabilities of a digital system with a new generation modem, for example, a modem T1, and a digital system with a modem for accepting signals with ideal phase modulation, as the most promising among the known types of modulation. The potential noise immunity of a digital system with a modem T1 after calculations (Tables 4.47, 4.48) turns out to be limiting in a channel with “white” noise, i.e. the modem T1 has no errors when receiving data in the range of signal-to-noise ratios of 27 dB, starting from a ratio of minus 3 dB. The potential noise immunity of a modem with ideal phase modulation (PM) is lower, and the error probability is greater by sixteen orders of magnitude. This is shown in Figure 4.14 and using the data in Table 4.49. The spectral efficiency of digital systems with PM according to [36] is 2 [(bit/s)/Hz]. Theoretically, for modems with statistical modulation, it will be 4 [(bit / s) / Hz], if you use the formula [35] (4.34) where Rb =1000 - information transfer rate at binary coding, bit/s; - modem bandwidth at = 1000 Hz. Taking into account the Nyquist bandwidth, the spectral efficiency of a statistic modulation modem can decrease to 2.8-3.6 [(bps)/Hz], but it is still greater than the spectral efficiency of digital PM systems. The energy efficiency of digital systems with PM is calculated by the formula [35] with the same error probability in the channel with "white" noise in the 1 Hz band (4.35) where h2 – the signal-to-noise power ratio. We choose the value of the error probability at the level of 1∙ 10-17 (fig. 4.14). Then for systems with PM at Rb = 1000 bit/s, the energy efficiency will be 0.13 [(bit/s)/Hz∙dB]. For modems with statistical modulation, it will be 1.4 [(bit/s) / Hz∙dB], i.e. it approaches the limit equal to 1/ ln2 =1,44 [35]. Summing up the analysis of the effectiveness of digital systems with new generation modems, we see that these digital systems in all respects are approaching the limiting theoretical values of system efficiency known in the literature, and this confirms their promise. 4.6. Comparison of noise immunity of new generation modems In total, we have considered 13 models of new generation modems. At the first stage of selection, 11 models of modems were left for further analysis, unfortunately, the T2p modem and the T2x modem were left without attention. For clarity, all other modem models are listed in Table 4.50. Let's pay attention, in accordance with the cipher in table 4.50, single-channel and dual-channel modems are recorded, which have sine and cosine channels independent of each other. Table 4.50. Models of new generation modems New generation modems modem А1 modem В1 modem Т1 modem А2 modem В2 modem К2 modem Т2 modem А2–1 modem В2–1 modem К2–1 modem Т2–1 Completing the analysis of the noise immunity of different variants of new generation modems, let us compare the error probabilities of modem A, modem B, modem K, modem T when operating in a channel with noise and signals that have different distribution laws. Such information is prepared separately for wired and radio communication channels. It is shown in Figure 4.15, where a) - radio - and wired communication channels; b) - wired communication channel. In Figure 4.15a, curve 1 refers to modem B1, curve 2 - to modem T1, curve 3 - to modem A1 with the optimal modulation algorithm (Table 4.11), curve 4 - to the modem for receiving signals from PM. The main values of the error probability in Figure 4.15a are recorded in tables 4.15, 4.49. It follows from the analysis of the graphs in Figure 4.15a that in the radio channel, preference should be given to modems for receiving modulated signals with distribution according to Veshkurtsev's law or according to Tikhonov's law. Moreover, Veshkurtsev law has priority out of the two named distribution laws, since the error probability of the modem B1 when accepting weak signals in the range is less by 27 orders of magnitude than that of the modem T1 and the modem A1. In addition, the energy gain for the modem B1 compared to QPSK (curve 4) at the error probability level of 1∙10-45 is 30 dB, for the T1 modem it is only 12 dB, and for the modem A1 it does not exceed 10 dB. Fig. 4.15a. Error probability of single-channel modems in radio and wired communications In Figure 4.15b, curve 1 refers to the sinus channel of modem B2, curve 2 - to modem T2-1, curve 3 - to the sinus channel of modem A2, curve 4 - to the sinus channel of modem K2, curve 5 - to the modem for receiving signals from PM. The main values of the error probability in Figure 4.15b are presented in Table 4.51. It follows from the analysis of the graphs in Figure 4.15b that modem B2 and modem T2–1 have a low probability of errors in the section. There is a big difference between modems in the area. Modem A2 becomes the leader in noise immunity, followed by modem B2, Table 4.51. Probability of errors of different modems Sinus channel modem A2 0,5 0,5 4∙10-32 1∙10-45 Less than 1∙10-45 Sinus channel modem B2 2,9∙10-2 1,4∙10-26 2,1∙10-31 7,5∙10-33 7,5∙10-33 Modem T2 - 1 2,5∙10-2 2,5∙10-2 5∙10-47 2,5∙10-29 1∙10-24 Sinus channel of modem K2 0,5 0,5 5,5∙10-13 4∙10-32 4∙10-35 Relation 0,01 0,1 1,0 10 100 Figure 4.15b. Probability of modem errors in wired communications and then modem K2 goes and modem T2-1 closes the circuit. On the site with modem A2, a modem competes for accepting signals from PM. In this section, the energy gain of the modem A2 compared to the QPSK modem is 10 dB. For weak signals, when , all new-generation modems have no analogues and competitors all over the world, and for the A2 modem, this can be said even with any signal-to-noise ratio. In our opinion, the modem A2 will soon appear in digital systems with quadrature amplitude shift keying (QAM). Let's recall that the modem A2 is proposed for receiving signals with distribution according to the arcsine law. Such a distribution law has a physical process at the output of the generator, for example, electrical oscillations. The rule follows from the history of the development of telegraphy on planet Earth that the decrease in the probability of modem errors by an order of magnitude while maintaining the signal strength occurs after ten years of exploratory scientific research in this direction. If you follow this rule, then the appearance of the modem B1 with an error probability of 1∙10-45 can be expected for more than three hundred years, counting from the present time, in which known PM modems have an error probability of 1∙10-8. This time will come, because technological progress cannot be stopped. In the meantime, the modems in table 4.51 are lined up in a certain queue one after another, the first in which, in our opinion, are modem B2 and modem A2.

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