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

"Non-white" noise has a Gaussian distribution law and, unlike "white" noise, it can have an expectation. For the operation of a new generation modem, such noise is dangerous, because significantly reduces its noise immunity. Let's consider the operation of each modem model separately in a channel with "non-white" noise or simply in a channel with Gaussian noise. 5.1.1. Noise immunity of the modem A2 when accepting an additive mixture of Gaussian 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) and a two-channel demodulator (Fig. 3.7). Its cipher was recorded earlier as modem A2. Let's repeat the modulation algorithm for a quasi-deterministic signal (2.1) using Table 5.1. Table 5.1. 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 analysis technique was developed in [26, 37]. Let's consider the noise immunity of a demodulator under the action of an additive mixture of a quasi-deterministic signal (2.1) and "non-white" Gaussian noise at its input z(t)=u(t)+n(t), (5.1) where n(t) – Gaussian noise, u(t) – a signal with a=U0 . Using expressions (3.3, 3.4) and the data in Table 5.1, 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, at the value Vm=1 we define for the additive mixture (5.1) (5.2) When s(t)=0, then, similarly to (5.2), we calculate for the value Vm=1 for the additive mixture (5.1) (5.3) , where W(z-eш) – the probability density of instantaneous values of the additive mixture; – signal-to-noise ratio; – the dispersion of the quasi-deterministic signal; – the dispersion of the Gaussian noise; eш – the expectation of Gaussian noise; the ratio of mathematical expectations of the signal and Gaussian noise (coefficient). The results (5.2, 5.3) need to be quantified. Tables 5.2, 5.3 present the results of calculations at П1=0,7116; П2=0,912; К1=0,56; К2=0,88, , written in the line with the name of the evaluation. In addition, in tables 5.2, 5.3, the values of the coefficient ρ are recorded in a separate column on the right. Table 5.2. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 0,912 0,88 = 0,8 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,23 0,52 0,57 0,57 1 Evaluation A ̂(1,t) 0 0 0,36 0,82 0,89 0,89 5 Evаluation A ̂(1,t) 0 0 0,37 0,83 0,91 0,91 10 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.3. The values of the estimate of the ch.f. in the sinus channel of the modem Threshold 0,7116 0,56 = 0,4 Coefficient ρ Evaluation B ̂(1,t) 0 0 0,29 0,65 0,71 0,71 1 Evaluation B ̂(1,t) 0 0 0,06 0,15 0,16 0,16 5 Evaluation B ̂(1,t) 0 0 0,03 0,07 0,08 0,08 10 Relation 0,001 0,01 0,1 1,0 10 100 In tables 5.2, 5.3, we compare the values of the ch.f. with the threshold specified in the first line. The values of the estimate depend on the coefficient ρ. Therefore, the noise immunity of the modem when operating in a channel with Gaussian noise will depend on two variables, namely, the signal-to-noise ratio and the coefficient ρ. Analysis of the data in tables 5.2, 5.3 shows that there are errors in the sine and cosine channels of the demodulator when accepting a logical "0". When the value of the coefficient =1, then continuous errors appear in the cosine channel of the modem when accepting a logical "0" for any signal-to-noise ratio. And, conversely, in the sinus channel of the modem, the maximum noise immunity is observed when accepting a logical "0" in the range of the signal-to-noise ratio of 20 dB, the lower limit of which is 0 dB. The situation changes dramatically when the value of the coefficient . Now, in the cosine channel of the demodulator, the modem has the maximum noise immunity when operating in a channel with Gaussian noise when accepting a logical "0" in the range of signal-to-noise ratios of 20 dB, the lower limit of which is 0 dB. In this case, continuous errors are observed in the sinus channel of the demodulator when accepting a logical "0". Therefore, the modem has a maximum noise immunity when accepting a logical "0" in the range of signal-to-noise ratios of 20 dB, depending on the coefficient , the values of which were indicated above. Moreover, the sine and cosine channels of the demodulator work in this case in different ways and there is no algorithm for choosing the preferred channel. Suppose the additive mixture (5.1) contain a non-centered quasi-deterministic signal at the demodulator input, this corresponds to the condition s(t)=1. Similarly to (5.2), at the value Vm=1 we define (5.4) or similarly (5.3) at the value Vm=1 let's calculate (5.5) The results (5.4), (5.5) require a quantitative analysis. Tables 5.4, 5.5 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. In addition, in tables 5.4, 5.5, the values of the coefficient ρ are recorded in a separate column on the right. Table 5.4. The values of the estimate of the ch.f. in the cosine channel of the modem Threshold 0,912 0,88 = 0,8 Coefficient ρ Evaluation A ̂(1,t) 0 0 -0,08 -0,19 -0,21 -0,21 1 Evaluation A ̂(1,t) 0 0 0,17 0,39 0,43 0,43 5 Evaluation A ̂(1,t) 0 0 0,2 0,45 0,5 0,5 10 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.5. The values of the estimate of the ch.f. in the sinus channel of the modem Threshold 0,7116 0,56 = 0,4 Coefficient ρ Evaluation B ̂(1,t) 0 0 0,36 0,81 0,9 0,9 1 Evaluation B ̂(1,t) 0 0 0,33 0,73 0,8 0,8 5 Evaluation B ̂(1,t) 0 0 0,31 0,7 0,76 0,76 10 Relation 0,001 0,01 0,1 1,0 10 100 Analysis of the data in tables 5.4, 5.5 shows that the accepting a logical "1" in the sine and cosine channels of the demodulator occurs without errors at any value of the coefficient in the range of signal-to-noise ratios of 20 dB, the lower limit of which is different for each channel. For the sine channel of the demodulator, it is equal to 0 dB, and for the cosine channel, it is minus 30 dB. It turns out that the expectation of Gaussian noise does not affect the reception of the logical "1" by both channels of the modem. As a result, we can say that in the presence of Gaussian noise in the data transmission channel, the noise immunity according to Kotelnikov of the proposed modem is limiting, depending on the value of the expectation of noise. So, for example, at and accurate synchronization of both channels of the modem, there are no errors when receiving a telegraph signal in the range of signal-to-noise power ratios of 20 dB, the lower limit of which is 0 dB. Let's recall that the expectation of "non-white" Gaussian noise will be present in a wired communication channel and absent in a radio channel. In the radio channel, the effect of "white" and Gaussian noise on the operation of the modem is identical and was discussed earlier in Section 4.1. Therefore, the error probabilities of modem A2 in the channel with Gaussian noise at will be obtained from Table 4.6 and calculated for other values using the method described in Section 4.1.1. The main values of the error probability of modem A2 are recorded in Table 5.6. To visualize the error probability of the modem A2 depending on the signal-to-noise ratio and the value of ρ, the graphs in Figure 5.1 are presented. Curves 1 - 4 characterize the error probability of the sine channel, and curves 5 - 8 characterize the cosine channel of modem A2. Curve 9 shows the error probability of the device for accepting signals with ideal PM according to the work [15, p.473]. In Figure 5.1, curves 1.5 are the same for any signal-to-noise ratio. In addition, curves 7 and 8 also coincide with each other. This means that the expectation of “non-white” Gaussian noise at value does not affect the noise immunity of the cosine channel of modem A2, but positively affects the operation of the sine channel. In the sinus channel of modem A2 (curves 3 and 4), noise immunity increases by 10 dB. This means that the signal modulation algorithm in Table 5.1 is not optimal and can be corrected. Apparently, it would be more correct to write . Then the probability of errors in the sinus channel of modem A2 at will decrease by five orders of magnitude up to the value Р=1,1∙10-5. Moreover, nothing will change in the cosine channel of the modem A2. And it's a completely different matter when the value of . At For both modem channels, the error probability is 0.5 (curves 1.5) for any signal-to-noise ratio. Here, the noise immunity of the A2 modem in the channel with Gaussian noise reaches a minimum, as a result of which it becomes inoperable. To ensure the operation of the A2 modem with high noise immunity in a channel with Gaussian noise, additional measures are required. The contents of these activities are outlined below. Table 5.6. Probability of errors of different modems Р 0,5 0,5 0,4 0,5 0,5 0,9 Curve 1 Р 0,5 5,5∙10-4 4∙10-32 5,5∙10-30 5,5∙10-30 0,45 Curve 2 Р 0,5 1,1∙10-5 1∙10-45 1∙10-45 Less than 1∙10-45 0,09 Curve 3 Р 0,5 0,5 4∙10-32 1∙10-45 Less than 1∙10-45 0 Curve 4 Р 0,5 0,5 0,5 0,5 0,5 0,9 Curve 5 Р 0,5 0,5 2,4∙10-3 2∙10-37 2∙10-37 0,45 Curve 6 Р 0,5 0,5 1,1∙10-5 1∙10-45 Less than 1∙10-45 0,09 Curve 7 Р 0,5 0,5 1,1∙10-5 1∙10-45 Less than 1∙10-45 0 Curve 8 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 9 0,01 0,1 1,0 10 100 еш Figure 5.1. Modem A2 Error Probability in a Channel with “Non-White” Gaussian Noise Single-channel modem A2-1. The new modem contains a modulator (Fig. 3.1) and a single-channel demodulator (Fig. 3.8). The modulation algorithm for a quasi-deterministic signal (2.1) remains the same and is recorded in Table 5.1. At the same time, the above theoretical analysis of the noise immunity of the modem A2 when operating in a channel with "non-white" Gaussian noise remains unchanged for the modem A2 - 1. However, the new model of the modem A2 has only one channel and one output, on which the telegraph signal will appear as a result of the states in table 3.1 of truth. 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. The data in the table. 5.3 show that in the sinus channel of the demodulator, the logical "0" will be determined without errors. There are no errors in the sinus channel in the range of signal-to-noise power ratios equal to 50 dB, while the value of the coefficient . Table 5.4 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors. There are no errors in the cosine channel in the range of signal-to-noise ratios equal to 50 dB, if the probability of errors 2∙10-45 is conditionally equated to zero. Combining these advantages of both channels together, we 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 being minus 30 dB. However, in practice this does not work, which is confirmed by truth table 3.1. 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. The probability of modem errors A2–1 is shown on fig. 5.2, where curve 1 is obtained with a coefficient ρ=1; curve 2 – with coefficient ρ=5; curve 3 – with coefficient ρ=10. In the same place, for comparison, the probability of errors of a known device (curve 5) using ideal phase modulation is shown. Curves 1,2,3,4 coincide in the section and curves 3,4 coincide at any value of . The main values of the error probability of the modem A2-1 in the channel with "non-white" Gaussian noise are listed in Table 5.7. Table 5.7. Probability of errors of different modems Р 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 0,9 Curve 1 Р 2,5∙10-2 2,5∙10-2 1,2∙10-4 1∙10-38 1∙10-38 0,45 Curve 2 Р 2,5∙10-2 2,5∙10-2 5,5∙10-7 5∙10-47 Less than 5∙10-47 0,09 Curve 3 Р 2,5∙10-2 2,5∙10-2 5,5∙10-7 5∙10-47 Less than 5∙10-47 0 Curve 4 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 5 0,01 0,1 1,0 10 100 еш An analysis of the curves in Figure 5.2 shows that the error probability of the A2-1 modem in a channel with “non-white” Gaussian noise strongly depends on the noise expectation value. If the value is , then the probability of modem errors A2 - 1 is minimal and lies at the level of 1∙10-47 and even lower. In this case, the probability of modem errors A2 - 1 increases to a value of 2.5∙10-2 (curve 1), as a result of which the noise immunity is completely lost. To restore the noise immunity of the A2-1 modem, it is necessary to compensate for the expectation of "non-white" Gaussian noise. Recommendations for eliminating the influence of non-white Gaussian noise characteristics on the noise immunity of new generation modems are outlined below. Modem A2-1 in terms of noise immunity surpasses only the cosine channel of modem A2. 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. Figure 5.2. Probability of modem А2–1 errors in a channel with “non-white” Gaussian noise Recommendations for eliminating the influence of the characteristics of "non-white" Gaussian noise. It is not possible to know the signal-to-noise ratio and the status of noise in the communication channel in advance before the communication session, namely: to consider the noise as "white" or "non-white" Gaussian. Unlike “white”, other noise may have an expectation, which negatively affects the noise immunity of the demodulator (Fig. 3.7) of the modem A2 we propose. Therefore, additional measures are required to resolve this issue positively, leaving everything else unchanged. To this end, we recommend the following. We propose to modernize the expressions (3.11, 3.12 ), bringing them to the form , (5.6) , (5.7) where - the estimate of the expectation of the noise. This expectation of noise is measured using the characteristic function in advance for no more than one second in the channel before the communication session, when there is still no signal, and is recorded in memory, and then, during modem operation, is subtracted from each current discrete instantaneous reading of the additive mixture (5.1) . Algorithm for measuring the expectation of a random process using ch.f. already developed [2] , (5.8) where (5.9) the estimate of the imaginary part of the ch.f.; , - quantization step of the ch.f. The previously constructed theory for measuring estimates of probabilistic characteristics of random processes using ch.f. is described in the book [2], where the step is calculated and all the properties of estimates, including estimates (5.8,5.9), are studied. Moreover, a virtual device XN 31.1 beta has been developed, with which you can measure estimates of 15 probabilistic characteristics of a random process for no more than five seconds and thus control the status of noise and other interference. The description of the device and instructions for its use are published in the book [3]. We recommend to include separate files of the program of the device for measuring estimates (5.8,5.9) into the computer program of the modem A2 and thereby eliminate the influence of the numerical characteristics of noise (mathematical expectation) on the noise immunity of digital systems with amplitude shift keying. 5.1.2. Noise immunity of modem A1 when receiving an additive mixture of Gaussian noise and a centered signal with the distribution of instantaneous values according to the arcsine law The modem contains a modulator (Fig. 3.2d) and a single-channel demodulator (Fig. 3.9). Its cipher was recorded earlier as modem A1. We repeat the modulation algorithm for a quasi-deterministic signal (2.1) using Table 5.8. Table 5.8. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0 0 logical "1" 1,125 0 The analysis technique was developed in [37]. Let us turn to the analysis of the noise immunity of the demodulator, when an additive mixture of a centered quasi-deterministic signal (2.1) and “non-white” Gaussian noise acts at its input z(t)=u(t)+n(t), (5.10) where n(t) – Gaussian noise, u(t) – a signal with a=U0 . Using expressions (3.3, 3.4) and the data in Table 5.8, using formulas (3.13), we calculate the threshold in the cosine channel of the demodulator. As a result, at the value Vm = 1 and U0 =1,5 we get П2 = J0(U0,t) =J0(0) = 1. Further, at the value Vm=1 and s(t) =0 we define for the additive mixture (5.10) . (5.11) where W(z-eш) – the probability density of instantaneous values of the additive mixture; – signal-to-noise ratio; – the dispersion of the quasi-deterministic signal; – the variance of the Gaussian noise; eш – the expectation of noise; ρ – the coefficient. The result (5.11) needs to be analyzed quantitatively. Table 5.9 presents the results of calculations at П2=1; К2=0,55 and , , written in the line with the name of the evaluation. In addition, in table 5.9, the values of the coefficient ρ are recorded in a separate column on the right. Table 5.9. The values of the evaluation of the ch.f. in the modem channel Threshold 1 0,55 = 0,55 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,002 0,35 0,59 0,62 1 Evaluation A ̂(1,t) 0 0 0,004 0,51 0,86 0,9 0,5 Evaluation A ̂(1,t) 0 0 0,004 0,57 0,95 0,99 0,1 Relation 0,001 0,01 0,1 1,0 10 100 In table 5.9, the values of the ch.f. compare with the threshold written in the first line. The noise immunity of the modem when operating in a channel with Gaussian noise now depends on two coefficients, namely, on the signal-to-noise ratio and on ρ, i.e. from the expectation of noise. Analysis of the data in Table 5.9 shows that there are errors in the modem A1 demodulator when accepting a logical "0". When the value of the coefficient is , then continuous errors appear in modem A1 when accepting a logical "0" at a signal-to-noise ratio from 10-3 to 10. The situation changes dramatically when the value of the coefficient . Now modem A1 has the maximum noise immunity when operating in a channel with Gaussian noise when accepting a logical "0" in the range of signal-to-noise ratios of 20 dB, the lower limit of which is 0 dB. Therefore, modem A1 has a maximum noise immunity when accepting a logical "0" in the range of signal-to-noise ratios of 20 dB, depending on the expectation of Gaussian noise, the values of which vary from 0 to 0.45. In the range of signal-to-noise ratios from 0.1 to 1, errors are possible when receiving a logical "0". However, it can be stated that the noise immunity of the modem A1 when operating in a channel with Gaussian noise is an order of magnitude better than the data given in the publication. Let the additive mixture (5.10) at the demodulator input contain a centered quasi-deterministic signal with dispersion , which corresponds to the condition s(t)=1. In this case, expression (5.11) will not change. The result (5.11) needs to be analyzed quantitatively. Table 5.10 shows the calculation data at П2=1; К2=0,55 and , , written in the line with the name of the evaluation. In addition, in table 5.10, the values of the coefficient ρ are recorded in a separate column on the right. Similarly to the analysis of table 5.9, we will study the data in table 5.10. The data in Table 5.10 turned out to be below the set threshold at any value of the expectation of Gaussian noise. It turns out that the expectation of Gaussian noise does not affect the accepting the logical "1". Hence, they correspond to the ideal case. Therefore, it can be stated that the accepting the logical "1" in modem A1 occurs without errors (i.e. with maximum noise immunity) in the range of signal-to-noise ratios from 10-3 to 102 or in the range of 50dB. This data is like Table 5.10. The values of the evaluation of the ch.f. in the modem channel Threshold 1 0,55 = 0,55 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,001 0,18 0,3 0,32 1 Evaluation A ̂(1,t) 0 0 0,002 0,26 0,44 0,46 0,5 Evaluation A ̂(1,t) 0 0 0,002 0,29 0,49 0,51 0,1 Relation 0,001 0,01 0,1 1,0 10 100 at least by three orders of magnitude better than the noise immunity of the modem, known from the publication. As a result, we can say that in the presence of "non-white" Gaussian noise in the data transmission channel, the potential noise immunity according to Kotelnikov of the proposed modem A1 is limiting, because with accurate modem synchronization, there are no errors when accepting a telegraph signal in the range of signal-to-noise ratios of 20 dB, and the lower limit of the range is 0 dB. To visualize the demodulator error probability depending on the signal-to-noise ratio and the value of the coefficient ρ, the graphs are presented in Figure 5.3. The main values of the error probability of modem A1 are recorded in Table 5.11. Curves 1,2,3 characterize the noise immunity of the modem A1 according to the data obtained here, curve 4 - devices according to the data of [25], curve 5 - devices for receiving signals with ideal PM according to the data of [15, p.473]. In figure 5.3 curves 3,4 coincide. This means that the expectation of Gaussian noise at value does not affect the noise immunity of the modem A1. And it's a completely different matter when the value is . Here, the noise immunity of modem A1 in a channel with Gaussian noise depends significantly on the signal-to-noise ratio. For strong signals, the noise immunity of the modem A1 increases by an order of magnitude or more, and for weak signals it drops sharply. Let us determine, starting from the value using curve 1, the decrease Table 5.11. Probability of errors of different modems P 0,5 0,5 0,5 7,7∙10-9 2,1∙10-23 0,9 Curve 1 P 0,5 0,5 0,5 1,1∙10-45 2∙10-35 0,45 Curve 2 P 0,5 5∙10-1 2,5∙10-3 1,1∙10-17 7,5∙10-9 0,09 Curve 3 P 0,5 5 ∙10-1 2,5∙10-3 1,1∙10-17 7,5∙10-9 0 Curve 4 P 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 5 0,01 0,1 1 10 100 eш Figure 5.3. Modem A1 error probability in a channel with “non-white” Gaussian noise noise immunity of the modem in the channel with "non-white" Gaussian noise in comparison with its characteristic obtained earlier (curve 4). It turned out to be 5 dB. Therefore, additional measures are needed to eliminate the influence of the expectation of Gaussian noise on the noise immunity of the modem A1. These activities are outlined in Section 5.1.1. 5.2.1. Noise immunity of the modem B2 when receiving an additive mixture of Gaussian noise and a 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 cipher was recorded earlier as modem B2. The modulation algorithm for a quasi-deterministic signal (2.9) is repeated in Table 5.12. Table 5.12. 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 modem B2 under the action of an additive mixture of a quasi-deterministic signal (2.9) and “non-white” Gaussian noise at its input z(t)=u(t)+n(t), (5.12) where n(t) – the Gaussian noise, u(t) – signal (2.9). Using expressions (2.12,3.10) and the data in Table 5.12, using formulas (3.13), we calculate the thresholds in the demodulator. As a result, at the value Vm = 1 we get П1= 0,5646 , П2= 1. Let us calculate the real and imaginary parts of the ch.f. of additive mixture (5.12) and is comparable with the thresholds. Then, during the transmission and value in the channels of the demodulator (Fig. 3.7), the threshold devices will receive the values of the real and imaginary parts of the ch.f. additive mixture, equal to , (5.13) , (5.14) where - the probability density of the additive mixture; - signal-to-noise ratio; - variance of Gaussian noise; - expectation of Gaussian noise; ρ – the coefficient. When transmitting and the value in the channels of the demodulator (Fig. 3.7), the threshold devices will receive the values of the real and imaginary parts of the ch.f. of additive mixture, equal to , (5.15) . (5.16) Suppose ; . The results of calculations by formulas (5.13 - 5.16) are summarized in tables 5.13 - 5.16, recorded in the line with the name of the assessment. In addition, in tables 5.13 - 5.16, the values of the coefficient ρ are recorded in a separate column on the right. An analysis of the data in tables 5.13 - 5.16 shows that modem B2 is very sensitive to the mean of Gaussian noise. For example, logical "0" in the sine channel and in the cosine channel of modem B2 is determined with errors if the expectation of Gaussian noise lies within . But the logical "1" in the cosine channel and in the sine channel of modem B2 is determined correctly, i.e. without errors, for any value of the expectation in the range . An analysis of the data in tables 5.13 - 5.16 shows that modem B2 is very sensitive to the mean of Gaussian noise. For instance, Table 5.13. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 1 0,96 = 0,96 Coefficient ρ Evaluation A ̂(1,t) 0,0055 0,5 0,785 0,817 0,8253 0,8253 1 Evaluation A ̂(1,t) 0,0064 0,58 0,91 0,95 0,9553 0,9553 0,5 Evaluation A ̂(1,t) 0,0067 0,61 0,95 0,99 0,9982 0,9982 0,1 Relation 0,001 0,01 0,1 1,0 10 100 logical "0" in the sine channel and in the cosine channel of modem B2 is determined with errors if the expectation of Gaussian noise lies within . But the logical "1" in the cosine channel and in the sine channel of modem B2 is determined correctly, i.e. without errors, for any value of the expectation in the range . Table 5.14. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 1 0,96 = 0,96 Coefficient ρ Evaluation A ̂(1,t) 0,0024 0,22 0,345 0,359 0,362 0,362 1 Evaluation A ̂(1,t) 0,0042 0,377 0,591 0,615 0,622 0,622 0,5 Evaluation A ̂(1,t) 0,0053 0,479 0,751 0,782 0,79 0,79 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.15. The values of the estimate of the ch.f. in the sinus channel of the modem Threshold 0,532 0,5646= 0,3 Coefficient ρ Evaluation B ̂(1,t) 0,0038 0,342 0,537 0,559 0,565 0,565 1 Evaluation B ̂(1,t) 0,002 0,179 0,281 0,293 0,296 0,296 0,5 Evaluation B ̂(1,t) 0,0004 0,036 0,057 0,059 0,06 0,06 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.16. The values of the estimate of the ch.f. in the sinus channel of the modem Threshold 0,532 0,5646 = 0,3 Coefficient ρ Evaluation B ̂(1,t) 0,0062 0,565 0,887 0,923 0,932 0,932 1 Evaluation B ̂(1,t) 0,0052 0,475 0,745 0,775 0,783 0,783 0,5 Evaluation B ̂(1,t) 0,0041 0,372 0,583 0,607 0,613 0,613 0,1 Relation 0,001 0,01 0,1 1,0 10 100 In this modem, the sinus channel prevails, since it has a maximum noise immunity in a wide range of signal-to-noise power ratios of 50 dB when operating in a channel with Gaussian noise, and the lower limit of the range is minus 30 dB. From a qualitative analysis of the data, let's move on to a quantitative assessment of the noise immunity of the B2 modem. For this, 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 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 obtain estimates of the real and imaginary parts of the ch.f., which are recorded in tables 5.13 - 5.16. Both estimates are random variables with their own properties and distribution laws. Repeating verbatim the rationale and methodology for calculating errors in the demodulator channels (Fig. 3.7), set out in Section 4.1.1, we obtain the data recorded in Table 5.17. For comparison, in the same place from [15, p.473], the error probability of a device for receiving signals with ideal phase modulation (PM), calculated in a noisy channel, is given. The dependence of the modem B2 error probability on the signal-to-noise ratio in the channel with Gaussian noise is shown in Figure 5.4, where curves 1-4 refer to the modem's sine channel; curves 5 - 8 - to the cosine channel of the modem; curve 9 - to the device for receiving signals with ideal PM. Curves 1,5,6 coincide at any value of , and in the range of values, curve 2 joins them. Curves 7,8,9 merge with curves 1,5,6 in the section . An analysis of the curves in Figure 5.4 shows that the modem B2 error probability in a channel with "non-white" Gaussian noise is at the level of 0.5 when the noise expectation is large and lies within . Here, there is no need to talk about any noise immunity of the modem B2, since it becomes unable to receive digital data. To restore the noise immunity of modem B2, it is necessary to compensate for the expectation of Gaussian noise. Recommendations for the implementation of such an activity are developed and presented in section 5.1.1. Following them Figure 5.4. Modem B2 error probability in a channel with “non-white” Gaussian noise Table 5.17. Probability of errors of different modems Р 0,5 0,5 0,4 0,5 0,5 0,6 Curve 1 Р 7,5∙10-9 0,18 0,38 0,42 0,42 0,3 Curve 2 Р 3,5∙10-4 1,1∙10-30 5,5∙10-30 5,5∙10-30 5,5∙10-30 0,06 Curve 3 Р 2,9∙10-2 1,4∙10-26 2,1∙10-31 7,5∙10-33 7,5∙10-33 0 Curve 4 Р 0,5 0,5 0,5 0,5 0,5 0,6 Curve 5 Р 0,5 0,5 0,5 0,25 0,25 0,3 Curve 6 Р 0,5 0,43 3,3∙10-5 3,4∙10-8 3,4∙10-8 0,06 Curve 7 Р 0,5 0,5 1,1∙10-5 7,5∙10-9 7,5∙10-9 0 Curve 8 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 9 0,01 0,1 1,0 10 100 еш it is enough to reduce the expectation of noise to the value (ρ=0,1). Then the noise immunity of modem B2 will be practically the same as that considered earlier in the channel with "white" noise. In Figure 5.4, curves 3,4 and curves 7,8 confirm this. Again, in modem B2, the sine channel prevails over the cosine channel. Up to a signal-to-noise ratio of 16 dB, this channel is better in terms of noise immunity than a device for receiving signals with ideal PM (curve 9), and for the cosine channel of modem B2, this superiority remains only up to a ratio of 10 dB (curves 7,8). Single-channel modem B2-1. The new modem contains a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.8). The modulation algorithm for a quasi-deterministic signal (2.9) is the same as before and is written in Table 5.12. At the same time, the above theoretical analysis of the modem B2 noise immunity in a channel with “non-white” Gaussian noise 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 coincidence of the channel states recorded in 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 5.15 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 there is the inequality еш < 0,06. Table 5.14 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, which is confirmed by truth table 3.1. The probability of errors in modem B2 - 1 is reduced by an average of 20 times compared with the error probability of the cosine channel of modem B2. The error probability of the new generation B2-1 modem is shown in Figure 5.5, where curve 1 is plotted for the value еш =0,6; curve 2 - for the value еш =0,3; curve 3 - for the value еш =0,06; curve 4 - for the value еш =0. It also shows the error probability of a known device (curve 5) for receiving signals with phase modulation. The main values of the B2-1 modem error probability in a channel with Gaussian noise are recorded in Table 5.18. Table 5.18. Probability of errors of different modems Р 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 0,6 Curve 1 Р 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 0,3 Curve 2 Р 2,5∙10-2 2,1∙10-2 1,6∙10-6 1,7∙10-9 1,7∙10-9 0,06 Curve 3 Р 0,5 0,5 1,1∙10-5 7,5∙10-9 7,5∙10-9 0 Curve 4 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 5 0,01 0,1 1,0 10 100 еш Figure 5.5. Modem B2-1error probability in a channel with “non-white” Gaussian noise An analysis of curves 1,2,3 in Figure 5.5 shows that the modem B2-1 error probability is highly dependent on the expected value of non-white Gaussian noise over the 40 dB signal-to-noise ratio range. Only when еш =0 the error probability stabilizes at the level of 7.5∙10-9 in the range of the signal-to-noise ratio . Therefore, the modem B2-1 has no potential noise immunity in a channel with "non-white" Gaussian noise, in which the expectation is present and changing. Recommendations for compensation of the mathematical expectation of "non-white" Gaussian noise in a wired channel are developed and presented in Section 5.1.1. The modem B2-1 is inferior in noise immunity to the sinus channel of the modem B2 by more than twenty orders of magnitude when working with "non-white" Gaussian noise. 5.2.2. Noise immunity of modem B1 when receiving an additive mixture of Gaussian noise and a 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 cipher was recorded earlier as modem B1. The modulation algorithm for a quasi-deterministic signal (2.9) is repeated in Table 5.19. Table 5.19. Signal modulation algorithm with Vm=1 Telegraph signal Signal dispersion value The value of the expectation of the signal logical "0" 0,0009 0 logical "1" 1,0 0 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 "non-white" Gaussian noise with expectation acts at its input z(t)=u(t)+n(t), (5.17) where n(t) –the Gaussian noise, u(t) – signal (2.9) . Using expression (2.12) and the data in Table 5.19, using formula (3.13), we calculate the threshold in the demodulator. As a result, at the value Vm = 1 we get П1= 0,7917. Let us represent the functional transformation in the demodulator circuit (Fig. 3.9) by the dependence y =cosz at the value Vm = 1 and N>> 1. 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. signal (2.9) is equal to zero. We get at the value Vm =1 , (5.18) where W(z-еш) - the probability density of the additive mixture (5.20); σш2 - dispersion of Gaussian noise; еш – the expectation of the Gaussian noise. Dispersion of the modulated c.c.s. varies discretely from σ02 to σ12, the values of which are recorded in Table 5.19. Then, when transmitting a logical "0", we get , (5.19) and when transmitting a logical "1" we will have . (5.20) Having performed the following substitutions in expressions (5.19,5.20) σш2=σ02 /h02, σш2=σ12 / h12 , we get , (5.21) , (5.22) where h0=σ0 / σш - signal-to-noise ratio when receiving logical "0"; h1= σ1 / σш - signal-to-noise ratio when receiving a logical "1"; ρ – coefficient. The results (5.21), (5.22) require a quantitative analysis. Tables 5.20,5.21 show calculation data at Vm =1, σ1=0,03, σ0=1,П1с =0,9, Table 5.20. The values of the evaluation of the ch.f. in the modem channel Threshold 0,7917 ∙ 1,14 = 0,9 Coefficient ρ Evaluation A ̂(1,t) 0,444 0,663 0,7 0,7 0,7 0,7 1 Evaluation A ̂(1,t) 0,59 0,88 0,92 0,92 0,92 0,92 0,5 Evaluation A ̂(1,t) 0,635 0,948 0,997 0,997 0,997 0,997 0,1 0,001 0,01 0,1 1,0 10 100 Table 5.21. The values of the evaluation of the ch.f. in the modem channel Threshold 0,7917 ∙ 1,14 = 0,9 Coefficient ρ Evaluation A ̂(1,t) 0,352 0,527 0,549 0,552 0,552 0,552 1 Evaluation A ̂(1,t) 0,465 0,697 0,726 0,73 0,73 0,73 0,5 Evaluation A ̂(1,t) 0,503 0,755 0,785 0,789 0,789 0,789 0,1 1,11111 11,1111 111,111 1111,11 11111,1 111111 , written in a line with the name of evaluation. In addition, in tables 5.20, 5.21, the values of the coefficient ρ are recorded in a separate column on the right. When the modem is operating in a noisy channel, it is impossible to know the different signal-to-noise ratio at its input when accepting a logical "0" and a logical "1", because the noise power in the channel does not depend on the telegraph signal. In our example, the signal dispersions during the transmission of telegraph signal elements correlate with each other as ,11. In this regard, in a noisy channel, the ratio is at constant noise power. In Table 5.21, all evaluation values A ̂(1,t) are less than the threshold for any signal-to-noise ratio, regardless of the coefficient ρ. This means that in a channel with "non-white" Gaussian noise, modem B1 does not have errors when accepting a logical "0" in the range of signal-to-noise ratios of 50 dB. In Table 5.20, the evaluation values A ̂(1,t) exceed the threshold for a signal-to-noise ratio of 0.1 to 100 when the coefficient is . Here, in a channel with Gaussian noise, modem B1 has no errors when accepting a logical "1". At and the value of the coefficient of the demodulator (Fig. 3.10), errors appear in the channel with Gaussian noise when accepting a logical "1". Thus, at a value the signal-to-noise ratio range is only 30 dB. If the value is , then the range of signal-to-noise ratios for modem B1 increases to 40 dB. It turns out that the expectation of "non-white" Gaussian noise affects the noise immunity of modem B1. The final conclusions about the noise immunity of the modem will be made according to the data in Table 5.20 (the accepted designations are further simplified to the form ). Its analysis shows that the noise immunity of modem B1 will be the limit in the range of signal-to-noise power ratios from 0.01 to 100, i.e. in the range of 40 dB, if the coefficient is . This indicates that the expectation operator in the mathematical model of ch.f. reliably protects the signal from Gaussian noise. New generation modems can work without errors when the signal-to-noise ratio is less than one. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the B1 modem. Let's introduce the following designations: Р0 – the probability of errors when accepting a logical "0"; Р1 – the probability of errors when accepting a logical "1"; - is the total probability of device errors. Quantitative assessment of modem noise immunity. The demodulator (Fig. 3.10) measures the value of the real part of the ch.f. with some error. And, as a result of this, we obtain - an estimate of the real part of the ch.f. Estimated ch.f. is a random variable that has its own properties and distribution law. Repeating verbatim to the conditions of our problem the methodology for calculating errors in the demodulator, written in detail in Section 4.1.1, we obtain the data included in Table 5.22. Curve 1 was obtained at a value of ρ =0,5, curve 2 - at a value of ρ = 0.1, curve 3 - at a value of ρ =0. For comparison, in the same place from [15, p.473], the probability of errors of ideal phase modulation is given (curve 4). Table 5.22. Probability of errors of different modems Curve 1 0,5 0,5 2,3∙10-3 2,3∙10-3 2,3∙10-3 2,3∙10-3 Curve 2 5,7∙10-12 3,8∙10-42 3,8∙10-42 3,8∙10-42 3,8∙10-42 3,8∙10-42 Curve 3 0,5 7,5∙10-13 1∙10-45 Less than 1∙10-45 Less than 1∙10-45 Less than 1∙10-45 Curve 4 1 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 Signal-to-noise ratio 0,01 0,1 0,5 1,0 10 100 To visualize the dependence of the B1 modem error probability on the signal-to-noise ratio, the graphs in Figure 5.6 are presented. The figure shows that curves 1,2,3 differ significantly from each other. This means that - the expectation of "non-white" Gaussian noise has a strong influence on the noise immunity of the considered modem B1. The initial state of noise immunity is characterized by curve 3, obtained when modem B1 operates in a channel with "white" noise, when . And it is a completely different matter when the product , which is calculated with a coefficient ρ =0,1. In this case, the noise immunity of modem B1 in the channel with "non-white" Gaussian noise (curve 2) increases by almost 10 dB compared to its noise immunity (curve 3) in the channel with "white" noise (if the comparison is performed at the error probability level of ). Indeed, in this case, curve 2 then rises by three orders of magnitude, and the error probability decreases to the value . The probability (curve 2) is greater than the probability (curve 3), however, at this stage of research, these two probabilities are equivalent for us. It can even be considered that a small expectation of Gaussian noise ( ) has a positive effect on the operation of the B1 modem. Figure 5.6. B1 modem error probability in a channel with "non-white" Gaussian noise The situation changes with the value of the product . Curve 3 rises sharply and is located parallel to the x-axis at the probability level (curve 1). At the same time, it is not necessary to talk about the maximum noise immunity of the B1 modem. Here, additional measures are needed to compensate . Recommendations for eliminating the influence on the noise immunity of the modem were formulated earlier in section 5.1.1 and recorded in [37]. As a result, the behavior of modem B1 in a channel with "non-white" Gaussian noise is ambiguous. Its noise immunity first increases when the expectation of the Gaussian noise does not exceed 0.1, and then drops noticeably if the expectation of the noise reaches 0.2 and continues to grow. At a coefficien and value we get curve 1. And this is not the end. Further increase in the expectation of noise to a value of 0.8 straightens curve 1 to a straight line running parallel to the x-axis at the error probability level . Again, the expectation of Gaussian noise affects the noise immunity of modem B1 only in a wired communication channel. In the radio channel, the antenna-feeder system at the input of the receiver filters the expectation of noise and thereby eliminates its effect on the noise immunity of modem B1. Comparison of the noise immunity of modem B1 with the same characteristic of a known device in which ideal PM is used (curve 4) shows its superiority by at least 30 dB with an error probability , if the value is . At a value superiority disappears. 5.3.1. Noise immunity of the modem T2 when accepting an additive mixture of Gaussian noise and a 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 was recorded earlier as a T2 modem. Let us repeat the analysis of the noise immunity of the T2 modem in the channel with "non-white" Gaussian noise at the value of the Tikhonov distribution parameter . The modulation algorithm for a quasi-deterministic signal (2.33) in its previous form is written in Table 5.23. The method and results of modem research are published in [34]. Table 5.23. 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 Let us turn to the analysis of the noise immunity of the T2 modem under the action of an additive mixture of a quasi-deterministic signal (2.33) and "non-white" Gaussian noise with expectation at its input z(t)=u(t)+n(t), (5.23) where n(t) – the Gaussian noise, u(t) – signal (2.33). Using expressions (2.26,2.27) and the data in Table 5.23, using formulas (3.13), we calculate the thresholds in the demodulator. As a result, at the value Vm = 1 and we get П1= =0,64 , П2 = =0,9. At the value Vm=1, we define for the additive mixture (5.23) the real part of the ch.f. , (5.24) where - signal-to-noise ratio: expectation of Gaussian noise; ρ – is the coefficient. When s(t)=0, similarly to (5.24) we calculate at the value Vm=1 for the additive mixture (5.23) the imaginary part of the ch.f. . (5.25) The results (5.24), (5.25) require a quantitative analysis. Tables 5.24, 5.25 present the results of calculations at , , , , , , written in a line with the name of the evaluation. In addition, in tables 5.24, 5.25, the values of the coefficient ρ are recorded in a separate column on the right. Table 5.24. The values of the estimate of the ch.f. in the cosine channel of the modem Threshold 0,9 0,78 = 0,7 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,202 0,564 0,62 0,627 1 Evaluation A ̂(1,t) 0 0 0,267 0,746 0,82 0,829 0,5 Evaluation A ̂(1,t) 0 0 0,289 0,807 0,887 0,897 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.25. The values of the evaluation of the ch.f. in the sinus channel of the modem Threshold 0,64 0,625 = 0,4 Coefficient ρ Evaluation B ̂(1,t) 0 0 0,208 0,581 0,638 0,646 1 Evaluation B ̂(1,t) 0 0 0,113 0,315 0,347 0,35 0,5 Evaluation B ̂(1,t) 0 0 0,023 0,065 0,071 0,072 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Analysis of the data in Table 5.24 shows that in the cosine channel of the T2 modem, the logical "1" is determined with errors, the probability of which depends on еш – the mathematical expectation of "non-white" Gaussian noise. When еш =0,8 (ρ =1) there will be continuous errors in the cosine channel. A similar conclusion regarding errors when accepting a logical "0" in the sinus channel of the T2 modem follows after analyzing the data in Table 5.25. Let the additive mixture (5.23) contain a non-centered quasi-deterministic signal at the demodulator input, this corresponds to the condition s(t)=1. Similarly to (5.24) at the value Vm=1 we get (5.26) or similarly (5.25) at the value Vm=1 we calculate . (5.27) Table 5.26. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 0,9 0,78 = 0,7 Coefficient ρ Evaluation A ̂(1,t) 0 0 -0,008 -0,024 -0,026 -0,026 1 Evaluation A ̂(1,t) 0 0 0,105 0,294 0,323 0,326 0,5 Evaluation A ̂(1,t) 0 0 0,185 0,516 0,567 0,573 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.27. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 0,64 0,625 = 0,4 Coefficient ρ Evaluation B ̂(1,t) 0 0 0,29 0,81 0,87 0,9 1 Evaluation B ̂(1,t) 0 0 0,27 0,755 0,83 0,84 0,5 Evaluation B ̂(1,t) 0 0 0,224 0,624 0,686 0,694 0,1 Relation 0,001 0,01 0,1 1,0 10 100 The results (5.26), (5.27) require a quantitative analysis. Tables 5.26, 5.27 show the calculation data at , , , , , , written in a line with the name of the evaluation. In addition, in tables 5.26, 5.27, the values of the coefficient ρ are recorded in a separate column on the right. Table 5.26 presents the results of an ideal discrimination of a logical "0" in the cosine channel of the modem T2 in the signal-to-noise ratio range of 50 dB, for which the lower limit is minus 30 dB. There are no errors here within the values of the expectation of "non-white" Gaussian noise. The data in Table 5.27 shows that a logical "1" in the sinus channel of the modem T2 is determined without errors, regardless of the value of еш in the 20 dB signal-to-noise ratio range, which has a lower limit of zero decibels. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the modem T2. Let's introduce the following designations: 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 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 5.24 - 5.27. Both estimates are random variables with their own properties and distribution laws. Repeating verbatim the justification and the method for calculating errors in the channels of the modem T2, set out in Section 4.1.1, we obtain the data recorded in Table 5.28. For clarity of presentation of the data in Table 5.28, Figure 5.7 shows the dependencies of the modem T2 error probability on the signal-to-noise ratio in a channel with “non-white” Gaussian noise. Curves 1-4 refer to the sine channel of the modem T2, curves 5-8 refer to the cosine channel of the modem T2. For comparison, in the same place from [15, p.473], the error probability of a device for receiving signals with ideal PM is given (curve 9). Curves 1,5 coincide for any value of the signal-to-noise ratio. In Figure 5.7, the graphs are very densely focused on the area . Curves 7 and 8 show good noise immunity of the cosine channel of the modem T2 in a channel with Gaussian noise. Indeed, curves 7,8 diverge at the point , and then curve 7 goes parallel to the abscissa at the level of error probability P = 1∙10-45, and curve 8 rises. It turns out that a small value of the expectation ( ) of Gaussian noise increases the noise immunity of the cosine channel of the modem T2 by 20 dB at the level of error probability Р = 2∙10-23 . The same is observed in the sinus channel of modem T2 (curves 3 and 4). Table 5.28. Probability of errors of different modems Р 0,5 0,5 0,4 0,5 0,5 0,8 Curve 1 Р 0,5 0,5 3∙10-5 6,5∙10-3 9∙10-3 0,4 Curve 2 Р 0,5 0,5 2,5∙10-26 1,9∙10-41 5,5∙10-44 0,08 Curve 3 Р 0,5 0,5 1,1∙10-17 5∙10-28 4∙10-32 0 Curve 4 Р 0,5 0,5 0,5 0,5 0,5 0,8 Curve 5 Р 0,5 0,5 3,9∙10-11 1∙10-45 Less than 1∙10-45 0,4 Curve 6 Р 0,5 0,5 1∙10-45 1∙10-45 Less than 1∙10-45 0,08 Curve 7 Р 0,5 0,5 1∙10-45 5∙10-28 2,1∙10-23 0 Curve 8 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 9 0,01 0,1 1,0 10 100 еш Figure 5.7. Probability of modem T2 errors in a channel with "non-white" Gaussian noise Comparison of the noise immunity of the modem T2 with the noise immunity of the known device (curve 9), in which the 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 behave identically in the channel with Gaussian noise and show high noise immunity. To do this, it is necessary to control the expectation of "non-white" Gaussian noise in the communication channel. The necessary recommendations have already been developed and are presented in section 5.1.1. As a result, such a model of the T2 modem 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 modulation algorithm for a quasi-deterministic signal (2.33) remains the same and is recorded in Table 5.23. At the same time, the above analysis of the noise immunity of the modem T2 in the channel with "non-white" Gaussian noise 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 modem channel transitions to the states recorded in 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 5.25 shows that in the sinus channel of the modem T2, 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 coefficient is . Table 5.26 shows that in the cosine channel of the modem T2, 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 truth table 3.1. The probability of errors in the T2-1 modem decreases on average by a factor of 20 compared to the error probability of the cosine channel of the modem T2. The error probability of the modem T2-1is shown in Figure 5.8, where curve 1 is calculated for the value еш =0,8; curve 4 - for the value еш =0,4; curve 3 - for the value еш =0,08; curve 2 - for the value еш =0. For comparison, in the same place from [15, p.473], the error probability of a device for receiving signals with ideal PM is given (curve 5). The main values of the modem T2-1 error probability are recorded in Table 5.29. Table 5.29. Probability of errors of different modems Р 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 0,8 Curve 1 Р 2,5∙10-2 2,5∙10-2 1,9∙10-12 5∙10-47 Less than 1∙10-47 0,4 Curve 2 Р 2,5∙10-2 2,5∙10-2 5∙10-47 Less than 1∙10-47 Less than 1∙10-47 0,08 Curve 3 Р 0,5 0,5 1∙10-45 5∙10-28 2,1∙10-23 0 Curve 4 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 5 0,01 0,1 1,0 10 100 еш Figure 5.8. Probability of modem T2-1 errors in a channel with “non-white” Gaussian noise An analysis of the graphs in Fig. 5.8 shows that the modem T2-1 works fine in a channel with "non-white" Gaussian noise. It has an error probability of 1∙10-47 in the range of signal-to-noise ratios from 10-1 to 101, i.e. 20 dB with a lower limit of minus 10 dB if the value еш =0,08. When the value еш=0, then the error probability of the modem T2-1starts to increase to the value of 1∙10-23 if the signal grows. It turns out that small values of the expectation of "non-white" Gaussian noise increase the noise immunity of the modem T2-1. Therefore, compensation algorithms (5.6), (5.7) for the expectation of Gaussian noise in the modem T2–1 should be reduced to the form , (5.28) . (5.29) And then the potential noise immunity of the modem T2-1 will become limiting, and the error probability will lie at the level of 1∙10-47 in the range of signal-to-noise ratios of 20 dB, in which the lower limit is equal to minus 10 dB. In this range, the modem T2-1 is superior in noise immunity to a device for receiving signals with ideal PM (curve 5). The T2-1 modem has no analogues and competitors all over the world. It can handle signals with 10 times less noise power. 5.3.2. Noise immunity of the modem T1 when accepting an additive mixture of Gaussian noise and a signal with the distribution of instantaneous values according to Tikhonov centered law The modem contains a modulator (Fig. 3.3) and a single-channel demodulator (Fig. 3.9). Its cipher was recorded earlier as a modem T1. The modulation algorithm of a quasi-deterministic signal (2.33) is repeated in Table 5.30 Table 5.30. 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 The research methodology and results are published in [32 – 34,38]. Let's proceed to the analysis of the noise immunity of the modem T1. Suppose an additive mixture of a centered quasi-deterministic signal (dynamic chaos with Tikhonov's law) and "non-white" Gaussian noise act at the demodulator input (Fig. 3.9) z(t)= +n(t), (5.30) where – a signal (2.33) with the distribution of instantaneous values according to Tikhonov law, - "non-white" Gaussian noise with a characteristic function of the form , - a parameter of the characteristic function (ch.f.); - the dispersion (average power) of the noise. At the value Vm=1 we define for the additive mixture (5.30) the real part of the ch.f. , (5.31) where - signal-to-noise ratio; - coefficient; - parameter of Tikhonov distribution law; - the Bessel function of the imaginary argument of the n-th order of the first kind. In expression (5.31), the signal dispersion changes in accordance with the modulation algorithm written in table 5.30, at parameter , and dispersion ; at parameter , and dispersion , where - a telegraph signal in the form of a sequence of logical "0" and logical "1". The time dependence appeared due to the telegraph signal. Taking this into account, we write down the value of the ch.f. at and at , (5.32) where - the signal-to-noise ratio when accepting a logical "0"; - signal-to-noise ratio when accepting a logical "1". The results (5.32) need to be analyzed quantitatively. Tables 5.31, 5.32 present the calculation data at , , written in a line with the name of the evaluation. In addition, in tables 5.31, 5.32, the values of the coefficient ρ are recorded in a separate column on the right. When the modem is operating in a noisy channel, it is impossible to know the 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 telegraph signal. In our example, the signal dispersions during the transmission of elements of a telegraph message correlate with each other as =7,03508. In this regard, in a noisy channel, the ratio is at a constant average noise power. In addition, the demodulator (Fig. 3.9) with some error measures the value of only the real part of the characteristic function, so the threshold device receives an evaluation of the ch.f. in the form . In Table 5.32, all evaluation values A ̂(1,t) are less than the threshold for any signal-to-noise ratio, regardless of the coefficient ρ. This means that in a channel with "non-white" Gaussian noise, errors when accepting a logical "0" modem T1 does not have a signal-to-noise ratio of 50 dB in the range. In Table 5.31, the evaluation values A ̂(1,t) exceed the threshold for a signal-to-noise ratio of 1 to 100 when the coefficient . Here in the channel Table 5.31. The values of the evaluation of the ch.f. in the modem channel Threshold 0,75 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,2 0,564 0,62 0,627 1 Evaluation A ̂(1,t) 0 0 0,267 0,746 0,82 0,829 0,5 Evaluation A ̂(1,t) 0 0 0,289 0,807 0,887 0,897 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.32. The value of the evaluation of the ch.f. in the modem channel Threshold 0,75 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,1 0,279 0,31 0,31 1 Evaluation A ̂(1,t) 0 0 0,132 0,368 0,407 0,44 0,5 Evaluation A ̂(1,t) 0 0 0,142 0,399 0,441 0,445 0,1 Relation 0,007 0,07 0,703 7,035 70,35 703,5 Gaussian noise there are no errors when accepting a logical "1" modem T1. At and the value of the coefficient the modem has errors in the channel with Gaussian noise when accepting a logical "1". Thus, at a value the signal-to-noise ratio range is only 10 dB. If the value is , then the signal-to-noise ratio range is increased to 20 dB. It turns out that the expectation of “non-white” Gaussian noise affects the noise immunity of the modem T1, and at and value of and the value of the error when accepting a logical “1” is constantly present in the modem for any signal-to-noise ratio. We will draw the final conclusions about the noise immunity of the modem T1 according to the data in Table 5.31 (the accepted designations are further simplified to the form ). Its analysis shows that the noise immunity of the modem will be limiting in the range of signal-to-noise ratios from 1 to 100, i.e. in the range of 20 dB, if the coefficient . This indicates that the expectation operator in the mathematical model of ch.f. reliably protects the signal from noise. Let's move on from a qualitative data analysis to a quantitative assessment of the noise immunity of the modem T1. Let's introduce the following designations: 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 T1. The demodulator (Fig. 3.9) measures the value of the estimate of the real part of the ch.f. And, as a result of this, we obtain - an estimate of the real part of the ch.f. Estimated ch.f. is a random variable that has its own properties and distribution law. Repeating verbatim to the conditions of our problem the methodology for calculating errors in the demodulator, written in detail in Section 4.1.1, we obtain the data included in Table 5.33. Curve 1 was obtained at a value of ρ = 0.5, curve 2 - at a value of ρ = 0.1, curve 3 - at a value of ρ = 0. For comparison, in the same place from [15, p.473], the error probability of a device for receiving signals with ideal PM is given (curve 4). To visualize the dependence of the modem T1 error probability on the signal-to-noise power ratio, the graphs in Figure 5.9 are presented. The figure shows that curves 1,2,3 differ significantly from each other. This means that - the expectation of "non-white" Gaussian noise has a strong influence on the noise immunity of the modem T1 over the entire range of signal-to-noise ratios. Therefore, we select two sections in the figure, namely: the first, where , and the second, where . Let's consider each section separately. Table 5.33. Probability of errors of different modems Curve 1 0,5 2,8∙10-1 2,1∙10-23 2,8∙10-28 Curve 2 0,5 3,8∙10-16 1∙10-45 1∙10-45 Curve 3 0,5 3,8∙10-16 2∙10-51 2∙10-52 Curve 4 0,9 1,5∙10-1 8∙10-6 2∙10-45 Relation h2 0,1 1 10 100 In the first section of the figure, curves 2 and 3 coincide up to the ratio , and then diverge. Curve 3 shows the error probability of the modem T1 when operating in a channel with "white" noise, when . In this case, the noise immunity of the modem turns out to be maximum, and it drops by 1.25 dB if the coefficient , and . This result is obtained by comparing the abscissas of curves 2,3 at the level of error probability . Comparing the abscissas of curves 1,3 at the error probability level , we see a drop in modem noise immunity by 7 dB when the coefficient , . Let's continue the analysis of the figure. Suppose the coefficient and . Then, regardless of the signal-to-noise ratio, curve 1 is transformed into a straight line running parallel to the x-axis at the level . As a result, the noise immunity of the modem T1 reaches a minimum. In the second section of the figure, curves 1,2,3 run almost parallel to each other at different levels of error probability. It ranges from 0.5 at value ( ) to at value . This shows that the expectation of "non-white" Gaussian noise must be reduced. Recommendations on this matter were formulated earlier in section 5.1.1 and published in [37]. It should be noted that the expectation of "non-white" Gaussian noise affects the noise immunity of the modem T1 only in a wired communication channel. In the radio channel, the antenna feeder system at the receiver input filters the expectation of noise and thereby eliminates its effect on the noise immunity of the modem T1. Figure 5.9. Probability of modem T1 errors in a channel with "non-white" Gaussian noise Comparison of the noise immunity of the modem T1 with the noise immunity of the known device, in which the ideal PM is used (curve 4), shows the superiority of the modem T1 by at least 5 dB with the error probability , if the value of . At value of and superiority is lost. 5.4. Noise immunity of the modem K2 when receiving an additive mixture of Gaussian noise and a 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 cipher was recorded earlier as the modem K2. The modulation algorithm for a quasi-deterministic signal (2.19) is repeated in Table 5.34. Table 5.34. 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's proceed to the analysis of the noise immunity of the K2 modem, when an additive mixture of a quasi-deterministic signal (2.19) and "non-white" Gaussian noise acts at its input z(t)=u(t)+n(t), (5.33) where n(t) – the “non-white” Gaussian noise, u(t) – signal (2.19). Using expressions (2.26,2.27) and the data in Table. 5.34 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. At the value Vm=1, we define for the additive mixture (5.33) the real part of the ch.f. , (5.34) where the signal-to-noise ratio; expectation of Gaussian noise; ρ – the coefficient. When s(t)=0, similarly to (5.34) we calculate at the value Vm=1 for the additive mixture (5.33) the imaginary part of the ch.f. . (5.35) The results (5.34), (5.35) require a quantitative analysis. Tables 5.35, 5.36 present the results of calculations at , , , , , written in a line with the name of the evaluation. In addition, in tables 5.35, 5.36, the values of the coefficient ρ are recorded in a separate column on the right. An analysis of the data in Table 5.35 shows that in the cosine channel of the modem K2, 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 with a coefficient value of . When , then the range of signal-to-noise ratios in the cosine channel of the K2 modem narrows to 10 dB. And for the values of the coefficient in the cosine channel of the K2 modem, there will be continuous errors for any signal-to-noise ratio. Table 5.35. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 0,7854 ∙ 0,764 =0,6 Coefficient ρ Evaluation A ̂(1,t) 0 0 0,05 0,388 0,48 0,49 1 Evaluation A ̂(1,t) 0 0 0,072 0,56 0,69 0,71 0,5 Evaluation A ̂(1,t) 0 0 0,079 0,622 0,767 0,787 0,1 Relation 0,001 0,01 0,1 1,0 10 100 In table 5.36, logical "0" in the sinus channel of modem K2 is determined without errors, i.e. with ultimate noise immunity, with signal-to-noise power ratio from 10-3 to 102, i.e. in the range of 50 dB, starting from minus 30 dB, if the coefficient This allows us to say that simple commands Table 5.36. The values of the evaluation of the ch.f. in the sinus channel of the modem Threshold 0,5634 ∙ 0,53 = 0,3 Coefficient ρ Evaluation 0 0 0,063 0,49 0,603 0,62 1 Evaluation 0 0 0,035 0,27 0,335 0,344 0,5 Evaluation 0 0 0,007 0,056 0,069 0,071 0,1 Relation 0,001 0,01 0,1 1,0 10 100 This allows us to say that simple control commands such as turn on-off, open-close and others will be accepted with a certainty equal to one, in any operating conditions of the K2 modem. Then things get worse. When , then the range of signal-to-noise ratios in the sinus channel of modem K2 is reduced to 30 dB with a lower limit equal to minus 30 dB. Here, in the sinus channel of the modem K2, the logical "0" will be determined without errors, and there will be errors outside the specified range. Suppose the additive mixture (5.33) contain a non-centered quasi-deterministic signal at the demodulator input, this corresponds to the condition s(t)=1. Similarly to (5.34) at the value Vm=1 let's define (5.36) or similarly (5.35) at the value Vm=1 let's calculate . (5.37) The results (5.36), (5.37) require a quantitative analysis. Tables 5.37, 5.38 show the calculation data at , , , , , written in a line with the name of the evaluation. In addition, in tables 5.37, 5.38, the values of the coefficient ρ are recorded in a separate column on the right. At the selected threshold values according to Table 5.38, the discrimination of logical "1" from zero in the sinus channel of modem K2 occurs without errors in the range of signal-to-noise ratios from 1 to 100, i.e. in the range, Table 5.37. The values of the evaluation of the ch.f. in the cosine channel of the modem Threshold 0,7854 ∙ 0,764 =0,6 Coefficient ρ Evaluation A ̂(1,t) 0 0 - 0,01 -0,08 - 0,1 - 0,1 1 Evaluation A ̂(1,t) 0 0 0,025 0,197 0,243 0,25 0,5 Evaluation A ̂(1,t) 0 0 0,05 0,393 0,485 0,497 0,1 Relation 0,001 0,01 0,1 1,0 10 100 Table 5.38. The values of the evaluation of the ch.f. in the sinus channel of the modem Threshold 0,5634 ∙ 0,53 = 0,3 Coefficient ρ Evaluation 0 0 0,079 0,619 0,764 0,783 1 Evaluation 0 0 0,076 0,592 0,73 0,75 0,5 Evaluation 0 0 0,062 0,481 0,594 0,61 0,1 Relation 0,001 0,01 0,1 1,0 10 100 equal to 20 dB, at any value of the coefficient ρ. At the same time, in the cosine channel of the K2 modem (Table 5.37), when determining the logical "0", the maximum noise immunity is maintained at a signal-to-noise ratio of 10-3 to 102, i.e. in the range of 50 dB, in which the lower limit is equal to minus 30 dB, and it does not depend on the value of the coefficient ρ. 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 modem K2. As a result of the analysis of the noise immunity of the modem K2, we can say that in the presence of "non-white" Gaussian noise in the data transmission channel, the Kotelnikov noise immunity of the modem K2 depends on the value of the expectation of noise. At accurate synchronization of the operation of both channels of the K2 modem, there are no errors when accepting a telegraph signal in the range of signal-to-noise ratios of 20 dB or more, with the lower limit of the range equal to 0 dB, and the value of the expectation of Gaussian noise . Let's move from qualitative data analysis to quantification noise immunity modem K2. Let's take the following designations: 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 K2. 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 5.35 - 5.38. Both evaluations are random variables with their own properties and distribution laws. Repeating verbatim to the conditions of our problem the method of calculating errors in the demodulator, written in detail in Section 4.1.1, we obtain the data included in Table 5.39. To visualize the error probability of the K2 modem depending on the signal-to-noise ratio and the value of ρ, the graphs in Figure 5.5 are presented. Curves 1 - 4 characterize the error probability of the sine channel, and curves 5 - 8 characterize the cosine channel of the K2 modem. Curve 9 shows the error probability of the device for receiving signals with ideal PM according to the work [15, p.473]. On fig. 5.10 curves 1.5 coincide at any signal-to-noise ratio, and curves 7,8 coincide in the section and in the section . In addition, in the section curves 7 and 8 are so close to each other, that it is sometimes difficult to distinguish them. This means that the expectation of Gaussian noise at value of has almost no effect on the noise immunity of the cosine channel of the K2 modem, but it has a positive effect on the operation of the sine channel. In the sinus channel of modem K2 (curves 3.4), noise immunity increases by 4.26 dB. This means that the signal modulation algorithm in Table. 5.34 is not optimal and can be adjusted. Table 5.39. Probability of errors of different modems Р 0,5 0,5 0,5 0,5 0,5 0,9 Curve 1 Р 0,5 0,5 8∙10-2 5∙10-2 1,9∙10-2 0,45 Curve 2 Р 0,5 0,5 1,1∙10-17 5,5∙10-44 1∙10-45 0,09 Curve 3 Р 0,5 0,5 5,5∙10-13 4∙10-32 4,3∙10-35 0 Curve 4 Р 0,5 0,5 0,5 0,5 0,5 0,9 Curve 5 Р 0,5 0,5 0,5 2,1∙10-37 1∙10-45 0,45 Curve 6 Р 0,5 0,5 1,1∙10-5 1∙10-45 Less than 1∙10-45 0,09 Curve 7 Р 0,5 0,5 9,3∙10-4 1∙10-45 Less than 1∙10-45 0 Curve 8 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 9 0,01 0,1 1,0 10 100 еш Apparently, it would be more correct to write . Then the probability of errors in the sinus channel of the K2 modem at will decrease by four orders of magnitude up to the value Р=1,1∙10-17. Moreover, in the cosine channel of the K2 modem, the probability of errors will decrease only by one order and will be Р =1,1∙10-5 (table 5.39). And it's a completely different matter when the value of . When (ρ =1) in both modem channels, the error probability is 0.5 (curves 1.5) for any signal-to-noise ratio. Here, the noise immunity of the modem K2 in the channel with Gaussian noise reaches a minimum, as a result of which it becomes inoperable. To ensure the operation of the K2 modem with high noise immunity in a channel with Gaussian noise, additional measures are required. The content of these activities is described in Section 5.1.1. Figure 5.10. Error probability of modem K2 in a channel with "non-white" Gaussian noise Comparison of the noise immunity (curves 3,6,7,8) of the K2 modem with the noise immunity (curve 9) of a known device in which ideal PM is used shows its superiority by thirteen orders and up to thirty orders in a channel with "non-white" Gaussian noise. The noise immunity of the K2 modem in the channel with "non-white" Gaussian noise in the section is even better than in the channel with "white" noise. This follows from the comparison of curve 4 with curves 3,6,7,8. Single-channel modem K2 -1. Let the K2 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 5.34. At the same time, the above analysis of the noise immunity of the modem K2 when operating in a channel with "non-white" Gaussian noise remains unchanged for the new model of the K2 modem - 1. However, the new modem has only one channel and one output, on which the telegraph signal will appear as a result of the transition of the modem channels to the states recorded in 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 5.36 shows that in the sinus channel of the demodulator, a logical "0" is determined without errors in the range of signal-to-noise ratios from 10-3 to 10-1, i.e. in the range of 30 dB. Table 5.37 shows that in the cosine channel of the demodulator, the logical "1" is determined without errors 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 truth table 3.1. The probability of errors in the modem K2 - 1 is reduced by an average of 20 times compared with the probability of errors in the cosine channel of the modem K2. The error probability of the modem K2-1 is presented in Table 5.40 and in Figure 5.11, where curve 1 is plotted at (ρ =1); curve 2 – (ρ=0,5); curve 3 – (ρ =0,1); curve 4 - (ρ =0); curve 5 refers to a device in which phase modulation is applied. Curves 3 and 4 coincide in the section . An analysis of the graphs in Figure 5.11 shows that the K2-1 modem works well with large signals. It has an error probability of 1∙10-45 in the range of signal-to-noise ratios from 1 to 100, and Table 5.40. Probability of errors of different modems Р 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 2,5∙10-2 0,9 Curve 1 Р 2,5∙10-2 2,5∙10-2 2,5∙10-2 1∙10-38 5∙10-47 0,45 Curve 2 Р 2,5∙10-2 2,5∙10-2 5,5∙10-7 5∙10-47 Less than 1∙10-47 0,09 Curve 3 Р 2,5∙10-2 2,5∙10-2 4,6∙10-5 5∙10-47 Less than 5∙10-47 0 Curve 4 Р 0,9 3,2∙10-1 1,5∙10-1 8∙10-6 2∙10-45 0 Curve 5 0,01 0,1 1,0 10 100 еш Figure 5.11. Probability of modem K2 errors - 1 in the channel with "non-white" Gaussian noise the probability of errors depends on the value of the expectation of "non-white" Gaussian noise. For example, at the probability of modem errors K2 - 1 remains at the level of 1∙10-45 (curve 3) for any signal-to-noise ratio in the range of 10 dB, starting from 10 dB. This is the best indicator of the K2-1 modem. Curve 1 in the section shows the low noise immunity of modem K2 - 1 with an error probability of 2.5∙10-2. If the expectation of “non-white” Gaussian noise is compensated to the value , then the noise immunity of the K2-1 modem will be restored and will correspond to curve 3. Recommendations for eliminating the expectation of “non-white” Gaussian noise are recorded in section 5.1.1.

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