To confirm the results of theoretical studies, computer simulation of different models of modem A, recorded in Table 4.50, was performed. At the same time, the Matlab application software package was used, which has significant advantages over currently existing mathematical modeling programs. The Matlab package was created for scientific and engineering calculations and is focused on working with data arrays. All these features make the Matlab package very attractive for solving various problems, including modeling devices and systems for transmitting discrete information over communication channels.
The characteristics of the systems under study are entered in an interactive mode, by graphical assembly of the connection diagram of standard elementary links. The elementary links are blocks (or modules) stored in the built-in library of the Simulink environment. The composition of the library can be supplemented by the user's own developments. Any model can have a nested structure, i.e. consist of lower level models [39 – 45]. In this case, the number of nested models can be very large. Further, we will agree to call nested models subsystems.
6.1. Description of the computer model
The computer model includes a modulator (Fig. 3.1), a communication channel and a demodulator (Fig. 3.7). Together they allow us to investigate the noise immunity of all models of modem A (Table 4.50) in a channel with "white" noise.
Modeling and subsequent study of the modem was performed in the Matlab software environment. To build the model, standard modules of the Simulink base library, as well as DSP Blocksets and Communication Blocksets libraries were used. Processed signals are stored in the working space of the environment. The built-in generators of pseudo-random sequences of the Matlab package were used for noise modeling.A general view of the computer model is shown on fig. 6.1 [24]. The model requires adjustment of block parameters taken from Simulink environment libraries. In the computer model, the communication channel contains the Add1 adder block, which is used to form an additive mixture.
Figure 6.1. General view of the computer model
In the modulator, the formation of the telegraph signal s(t) in the form of binary parcels is performed using the Random Integer Generator1 block. This block generates a pseudo-random binary sequence that simulates the transmitted information. The Product1 block multiplies the quasi-deterministic signal u(t) and the pseudo-random binary sequence. To transmit one data bit, N samplings of a quasi-deterministic signal are used. Loading samples of a quasi-deterministic signal into the model is performed using the FromWorkspace1 block.
The demodulator is built in accordance with the block diagram in Figure 3.8, in the model its functions are extended using the Manual Switch key, which allows you to use the modem channels separately or together through a special combination scheme. Let's recall that the demodulator measures estimates of the real and imaginary parts of the Lyapunov characteristic function (ch.f.) of the additive mixture of signal and noise
, (6.1)
, (6.2)
where z(t)=u(t)+n(t) – is an additive mixture of a quasi-deterministic signal u(t) and "white" noise n(t); Δt is the discretization interval, k is the ordinal number of the discrete sample of the additive mixture. The additive mixture is fed to the input of a demodulator having sine and cosine channels. Evaluation (6.2) is measured in the sine channel, estimate (6.1) is measured in the cosine channel. The sine values of the additive mixture of the useful signal and noise are calculated using the Trigonometric Function1 block, and the cosine values are calculated using the Trigonometric Function2 block.
The sum of sines averaged over 100 values in accordance with expression (6.2) is compared with the П1с threshold set in the Constant3 block, and the sum of cosines averaged over 100 values in accordance with expression (6.1) is compared with the P2k threshold set in the Constant4 block. The threshold devices are implemented using the Relational Operator2 block and the Relational Operator3 block. Each block compares the value of the average sum with a threshold. If the average sum is greater than the threshold, the threshold device outputs a logical unit. Otherwise, the threshold device outputs a logical zero. The output of the Relational Operator2 block is the output of the sinus channel (Output 1). The Relational Operator3 block is connected to a logic element "NOT" implemented using the Logical Operator1 block, whose output corresponds to the output of the cosine channel (Output 2). The threshold devices of the sine and cosine channels are combined using a special circuit. It consists of blocks Logical Operator1 ("NOT" logic element) and Logical Operator2 ("AND" logic element). Thus, after combining the sine and cosine channels, the modem signal is generated at the output of the Logical Operator2 block (Output 3). A more detailed description of the computer model in Figure 6.1 is available on the website [46].
6.2. Characteristics of a quasi-deterministic signal
Samplings of the quasi-deterministic signal are loaded into the model from the workspace of the Matlab package, into which they are previously entered from the output of the ADC connected to the source AKIP - 3409/4 of physical processes. The From Workspace1 block is used to load samplings of a quasi-deterministic signal into the model. The settings window of the From Workspace1 block contains the name of the workspace variable of the Matlab package (the Data field), as well as the values of the sampling period of the quadri-deterministic signal (the Sample time field). The signal oscillation frequency is set to 50 kHz, and the sampling period is set to
s.
Verification of the probabilistic characteristics of the quasi-deterministic signal of the generator AKIP - 3409/4 was performed using a virtual instrument XN 31.1 beta [3]. Figure 6.2 shows an estimate of the probability density of a quasi-deterministic signal.
Figure 6.2. Signal probability density evaluation
The estimate of the probability density of the signal repeats the form of the arcsine law, is symmetrical about zero, and has maxima at values equal to the signal amplitude. Using the scale grid in Figure 6.2, we calculate the signal amplitude and get a value of 0.95. Measurements of the initial and central moment functions of the signal using the XN 31.1 beta device showed the following result: an evaluation of the initial moment of the first order (the expectation of the signal); evaluation of the central moment of the second order (signal dispersion). Using the formula, we calculate the signal amplitude, as a result of which we obtain U0 =0,95. The scale grid in Figure 6.2 does not allow you to see the expectation of the signal with a value of 0.0067. However, its presence will adversely affect the modem simulation results.
Other evaluations of the probabilistic characteristics of a quasi-deterministic signal, measured by the XN 31.1 beta virtual instrument, do not differ from the theoretical curves obtained for a signal with a mathematical model of the form (2.1). They are not shown here, because were previously shown in Figures 2.2, 2.3, 2.4.
As a result, we can assume that the physical process of the AKIP-3409/4 source has properties similar to those of a quasi-deterministic signal (2.1) and can be used in modem modeling.
6.3. Noise characteristics
In the computer model in Figure 6.1, "white" noise is formed by the Band - Limited White Noise 1 block. The value of the dispersion of "white" noise is set in the workspace of the Matlab package, so the variable N0 is set in the Noise Power field. The initial value of the pseudo-random number generator base (Seed field) is taken by default. The sampling period of "white" noise corresponds to the sampling period of a quasi-deterministic signal, i.e. value s.
The probabilistic characteristics of the noise were investigated using the XN 31.1 beta virtual instrument. An evaluation of the noise probability density is presented in Figure 6.3.
The shape of the graph in Figure 6.3 repeats the form of a Gaussian curve. Therefore, we can assume that the instantaneous values of the noise are distributed according to the normal law, or, in other words, have a Gaussian distribution. The initial and central moment functions have the following value:
Figure 6.3. Noise probability density evaluation
evaluation of the initial moment of the first order (expectation of noise); second-order central moment evaluation (noise dispersion).
Figure 6.4 shows an estimate of the correlation function of noise, the value of which at the value is equal to . The view of the graph in Figure 6.4 approaches the image of a delta function. When the value of the noise correlation function is . This means that the discrete instantaneous noise values taken over the sampling interval s, are uncorrelated.
An evaluation of the noise power spectral density is shown in Figure 6.5. It has unevenness. If the density value is conditionally taken as average, then the unevenness of the noise power spectral density will be 1.8% up and 3.6% down. The energy bandwidth of the noise is 100 kHz. Therefore, the carrier frequency of the quasi-deterministic signal is chosen to be 50 kHz.
Figure 6.4. Noise correlation function evaluation
Figure 6.5. Noise power spectral density evaluation
At the end of the show, let's give an evaluation of the characteristic function of noise in Figure 6.6. As expected, the imaginary part of the ch.f. equals to zero. It turns out, indeed, that the expectation of noise is zero. Therefore, the previously indicated value can be considered the error of the device and nothing more.
Figure 6.6. Evaluations for the real and imaginary parts of the ch.f.
In conclusion, based on statistical radio engineering [4], let's write down the main properties of "white" noise:
1) instantaneous noise values are distributed according to the normal law;
2) any two adjacent instantaneous noise values are uncorrelated;
3) the noise power spectral density is constant within the energy bandwidth;
4) the expectation of noise is zero.
In our opinion, the noise formed in the computer model (Fig. 6.1) satisfies all these properties, which means that it is “white”.
6.4. Characteristics of estimates of the characteristic function of a quasi-deterministic signal
The modem demodulator in the absence of noise measures the values of the ch.f. evaluations of quasi-deterministic signal according to algorithms (6.1, 6.2). Ch.f. evaluations are random variables that have their own properties and distribution laws, which are taken into account when analyzing the noise immunity of the modem. Since it is theoretically possible to obtain the laws of distribution of evaluations of ch.f. difficult enough, the hypothesis was put forward that the scores are distributed according to the normal law. For the first time, this law was obtained by the empirical method based on the results of demodulator simulation using the computer model Figure 6.1 in [28] and is shown in Figure 6.7.
The cosine channel of the demodulator, which measures the estimate (6.1), is considered first. In total, one thousand evaluation values were processed , each of which was measured in accordance with equation (6.1) at a value of , when the amplitude of the quasi-deterministic signal (2.1) with the arcsine distribution law is 1,5, and . The exact (principal) value of the estimate is 0.5118 and is calculated using the fundamental formula
(6.3)
at , where - the probability density (2.2) of the signal with the arcsine distribution law; - Bessel function of the zeroth order of the first kind. The probability density of estimate is shown in Figure 6.7 (6.1).
Figure 6.7. Estimating the probability density of values
The sinus channel of the demodulator, which measures estimate (6.2), is similarly studied. The evaluation probability density (6.2) repeats the graph in Figure 6.7. Therefore, we can say that the probability density of evaluations of the real and imaginary parts of the ch.f. almost repeats the Gaussian curve. This means that the law of distribution of the estimate of the real part and the evaluation of the imaginary part of the ch.f. is normal. Previously, this was discussed only hypothetically, however, with the help of experimental studies of the demodulator, the hypothesis has now been confirmed [28].
6.5. Studying of modem noise immunity using a computer model
In the computer model in Figure 6.1, white noise is generated by the Band-Limited White Noise1 block. Samples of the quasi-deterministic signal are loaded into the model from the workspace of the Matlab package, into which they are preliminarily entered from the output of the ADC connected to the source AKIP - 3409/1 of physical processes. The oscillation frequency of the quasi-deterministic signal is set to 50 kHz. The sampling period of the signal and "white" noise with a bandwidth of 100 kHz is Δt = 2∙10-6 s. The probabilistic characteristics of both processes in statistical radio engineering are well studied, and the estimates of these characteristics were checked before the study using the XN 1.31 beta virtual instrument [3]. The check confirmed the status of both processes.
To determine the error probability when receiving binary messages, the computer model contains the Subsystem1 subsystem, the structure of which is shown in Figure 6.8. In this subsystem, the number of erroneously received symbols (bits) Nerr is calculated, and the total number of received symbols Ntot. In the computer model in Figure 6.1, the number of transmitted and received binary characters is the same.
Subsystem1 has three inputs: Strob, Tx and Rx. The Strob input receives a control signal from the Relational Operator1 block (Fig. 6.1). The Rx input receives a binary sequence from any modem output (Output 1, Output 2, Output 3). The original binary pseudo-random sequence is fed to the Tx input through the Delay1 delay block. The Add1 block calculates the difference
Figure 6.8. Model subsystem for detecting errors in the transmission of binary characters
between the values of each transmitted bit and the corresponding received bit. The Abs1 block calculates the modulus of the difference values coming from the Add1 block. Thus, to determine the number of erroneously received characters, the original binary pseudo-random sequence is compared with the binary sequence at any output of the modem. The Cumulative Sum1 block of the Subsystem1 subsystem counts and stores the number of erroneously accepted symbols (bits).
The total number of transmitted binary symbols is counted by the single pulses received at the Strob input from the output of the Logical Operator1 block (Fig. 6.1). The total number of received symbols is counted using the Cumulative Sum2 block of Subsystem1.
The study of the noise immunity of the modem was carried out at the value N=100 in expressions (6.1), (6.2). Therefore, at the specified sampling time of the additive mixture, the duration of one symbol (bit) is equal to N∙Δt = 100∙2∙10-6 = = 2∙10-4 s. Then the information transfer rate will be 5000 bps.
Using the model in Figure 6.1, different versions of the modem (single-channel, dual-channel, with a connected channel combining scheme according to the scheme in Figure 3.8) were successively studied. The results of the modem study are recorded in tables 6.1 - 6.6, which include additional explanations and the following designations: P0=P1=Nerr/Ntot – error probability; Pex=0,5∙(P0+P1) – total probability of errors in modem modeling; Pcalc – the calculated probability of modem errors; h2 – the signal-to-noise power ratio; U0 – amplitude and e0 – the expectation of the quasi-deterministic signal.
In tables 6.1 - 6.5, errors are shown separately when accepting logical "0" and logical "1". Errors are different. For example, in a two-channel modem (Table 6.2) in the sinus channel, the errors are equal to zero when accepting logical "0", while in the cosine channel (Table 6.3), on the contrary, when accepting logical "1". This point allows you to get a positive effect on modem errors. When combining demodulator channels
Table 6.1.
The results of the study of a single-channel modem
Modem type: single-channel
Suboptimal modulation algorithm: U0=0; e0=0 at s(t)=0 and U0=1,425; e0=0 at s(t)=1
Transmission of logical "1"
h2 0,1 0,5 1,00 2,00 5,00 10,00 20,00 100
П2к = 0,6325
Cosine channel P1 0 0 0 0 0,00001 0 0 0
Nerr, bites 0 0 0 0 1 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Transmission of logical "0"
П2к = 0,6325
Cosine channel P0 1,0 1,0 0,71499 0,00584 0 0 0 0
Nerr, bites 100000 100000 71499 584 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Table 6.2.
The results of the study of the two-channel modem
Modem type: two-channel
Suboptimal modulation algorithm: U0=0,594; e0=0 at s(t)=0 and U0=0,594; e0=0,9 at s(t)=1
Transmission of logical "0"
h2 0,01 0,10 1,00 2,00 4,00 10,00 20,00 100
П1с =0,4
Sinus channel P0 0 0 0 0 0 0 0 0
Nerr, bites 0 0 0 0 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
П2к = 0,75
Cosine channel P0 1,0 1,0 0,00001 0 0 0 0 0
Nerr, bites 100000 100000 1 0 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Table 6.3.
The results of the study of the two-channel modem
Modem type: two-channel
Suboptimal modulation algorithm: U0=0,594; e0=0 at s(t)=0 and U0=0,594; e0=0,9 at s(t)=1
Transmission of logical "1"
h2 0,01 0,10 1,00 2,00 4,00 10,00 20,00 100
П1с =0,4
Sinus channel P1 1,0 0,09492 0 0 0 0 0 0
Nerr, bites 100000 9492 0 0 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
П2к = 0,75
Cosine channel P1 0 0 0 0 0 0 0 0
Nerr, bites 0 0 0 0 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Table 6.4.
Modem study results with connected channels scheme of union
Modem type: with connected channels scheme of union
Suboptimal modulation algorithm: U0=0,594; e0=0 at s(t)=0 and U0=0,594; e0=0,9 at s(t)=1
Transmission of logical "1"
h2 0,01 0,10 1,00 2,00 4,00 10,00 20,00 100
П1с =0,15
The modem has a common output P1 0,98355 0,01074 0 0 0 0 0 0
Nerr, bites 98355 1074 0 0 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Transmission of logical "0"
П2к = 0,67
The modem has a common output P0 0,01633 0,01549 0 0 0 0 0 0
Nerr, bites 1633 1549 0 0 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Table 6.5.
The results of the study of a single-channel modem
Modem type: single-channel
Optimal modulation algorithm: U0=0; e0=0 at s(t)=0 and U0=1,1999; e0=0 at s(t)=1
Transmission of logical "1"
h2 0,01 0,10 1,00 1,50 4,00 10,00 20,00 100
П2к = 0,2
Cosine channel P1 1,0 0,00218 0,00208 0,00142 0,00015 0 0 0
Nerr, bites 100000 218 208 142 15 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Transmission of logical "0"
П2к = 0,2
Cosine channel P0 1,0 1,0 0,28751 0,00147 0 0 0 0
Nerr, bites 100000 100000 28751 147 0 0 0 0
Ntot, bites 100000 100000 100000 100000 100000 100000 100000 100000
Table 6.6.
Modem study results
h2 0,1 1,00 1,5 4,0 7,0 20,0 Note
σА 0,07 0,06 0,04 0,04 0,04 0,01 experiment
Pcal 0,5 0,3 2,6∙10-3 3,8∙10-11 2,2∙10-13 1∙10-45 Table 6.1
Pex 0,5 0,36 2,9∙10-3 no data 5∙10-6 0 Table 6.1
Pex 0,5 0,14 1,4∙10-3 7,5∙10-5 0 0 Table 6.5
Pex 4,7∙10-2 0 0 0 0 0 Table 6.2,6.3
sine channel
Pex 0,5 1∙10-5 0 0 0 0 Table 6.2,6.3
cosine channel
Pex 1,3∙10-2 0 0 0 0 0 Table 6.4
Pcal 0,5 0,1 4,2∙10-3 2,3∙10-3 1∙10-4 1∙10-10 QPSK
using an additional circuit (Table 6.4), we obtain on average almost 20 times fewer errors at the total output of the device [24].
To compare the simulation results with the theoretical values of the error probability, graphs are plotted in Figure 6.9, which belong to signal modems with different types of modulation. Curve 1 is plotted for 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 a single-channel modem with a non-optimal algorithm, and curve 4 with an optimal algorithm. Curve 5 refers to a two-channel modem (cosine channel); curve 7 - sinus channel. Curve 6 refers to a two-channel modem with channel bonding connected.
The limited amount of computer RAM made it possible to write only 107 discrete values of the signal into the workspace of the Matlab package and then transfer 105 binary elements. The marginal probability of errors in this case was 1∙10-5. The calculated (total) probability of modem errors is much less than the level of 1∙10-15 and it is not possible to check it on the model. Indeed, in the cosine channel of the modem, we managed to check the probability of errors at the ratio . It turned out to be equal to 1∙10-5 (Table 6.2) and coincided with the calculated value of the error probability (point D in Fig. 6.9) [24].
When modeling a modem with one channel, it was found that the mean root square (RMS) of the estimate of the real part of the Lyapunov ch.f. exceeds the previously known theoretical value σА at ratios h2<10. In accordance with the work [2] σА=0,01. The new RMS values of the assessment are recorded in Table 6. 6.
Taking into account these data, the error probabilities of a single-channel modem were calculated for the set threshold П2к=0,6325 and the modulation algorithm recorded in Table 6.1. The order of the calculated and experimental error probabilities coincided with the ratio 0,1≤ h2 ≤1,5.
At the end of the analysis, consider the optimal signal modulation algorithm (Table 4.11) in a modem with one channel, the data on which are recorded in Table 6.5. They are better than the data of the same name in Table 6.1 and confirm the results of the theory. For comparison, the last row of Table 6.6 shows the error probabilities of the QPSK signaling modem. Comparison of the digits of this line with the numbers of other lines of Table 6.6 generates a conclusion not in favor of QPSK modulation. It was theoretically found that the new SSK modulation
Figure 6.9. Probability of errors of different modems
in terms of energy performance, outgoes by 10 dB over QPSK modulation (compare curves 3 and 4).
Thus, the simulation results of the modem do not contradict the theoretical data regarding its performance, and they are better than those of the QPSK signal modem.
The simulation showed the feasibility of the practical implementation of the device and the operability of the new modem, and also provided a test of its characteristics when operating in a channel with "white" noise. The noise immunity of a modem of a single-channel, two-channel modem and a modem with a connected circuit for combining the outputs of the sine and cosine channels of the device is studied. At a signal-to-noise ratio of one or less, the noise immunity of the devices turned out to be different, and at large signal-to-noise ratios, there were no errors when receiving binary symbols. The minimum probability of modem errors is determined at the level of 1∙10-5 with the volume of received binary symbols 105. This volume of symbols is limited by the technical characteristics of the computer and could not be increased more. Nevertheless, in the error probability interval 1.0…1∙10-5, the signal modem with SSK modulation has energy efficiency indicators better than all known devices.
