Modems with amplitude, phase, frequency modulation are widely used in communication technology, but they have low noise immunity when operating in a noisy channel, and amplitude modulation is the most unprotected from interference. Currently, they are trying to increase the noise immunity of modems by combating interference, while inventing various devices and blocks for suppressing interference, which, at times, are much more complicated than the modems themselves.
We offer another direction for improving the theory of modulation and another way of building modems, based on the complication of the mathematical model of the signal and the modulation of its characteristics, in particular the characteristic function. It is defined in the domain of probabilities or the space of probabilities proposed in 1933 by academician A. Kolmogorov when building information theory. At the same time, the probability theory is a mathematical tool for describing all signal transformations in the probability space. For a signal with a mathematical model (2.1, 2.9, 2.19, 2.33), the characteristic function (ch.f.) is strictly defined, i.e. fundamentally. Thus, by introducing random variables into the models (2.1, 2.9, 2.19, 2.33), a transition to the model of the so-called quasi-deterministic signal, which is an element of statistical radio engineering, is achieved. At first glance, replacing a deterministic oscillation model with a quasi-deterministic signal mathematical model is a fairly simple operation, but the modem noise immunity after replacing the oscillation model turns out to be limiting in the sense that there are no errors when receiving data.
3.1. The first method of signal modulation
We will consider a new modulation method [18], in which all signal parameters are “hidden” inside the expectation operator, as a result of which we obtain the function
(3.1)
widely known in mathematics, physics, statistical radio engineering. The mathematician A. Lyapunov proposed this function and published its description in 1901 [19]. In the literature [4], it is called the characteristic function. Applying L. Euler's formula, let’s write
(3.2)
where A(Vm), B(Vm) – real and imaginary parts of the characteristic function;
Vm is the parameter of the characteristic function.
By analogy with cosmonautics, the characteristic function (ch.f.) is a “spacesuit” for a signal, it serves as a fundamental probabilistic characteristic of a signal, for example, a quasi-deterministic oscillation (2.1)
with parameters where – a random phase shift angle with a uniform distribution law within . The physical meaning of ch.f. studied in [2], and it is shown that it is the spectral density of the probabilities of the instantaneous values of the signal (2.1). Ch.f. depends on the probability density of the signal. Consequently, each model of a quasi-deterministic signal has its own fundamental ch.f., which has many positive properties. It is limited, measurable, filters noise, has limiting values , , . Other remarkable properties of it are described in [2]. Based on the advantages of the ch.f., we propose a method for modulating this function.
A ch.f. modulation method in which a constant voltage is multiplied with a telegraph signal s(t), which takes on the value either "1" or "0", after which the product e0s(t) is summed with a centered quasi-deterministic signal (2.1), expectation which is equal to zero, and thus carry out the modulation of the ch.f. of the transformed quasi-deterministic signal according to the law:
for s(t)=0 to obtain functions of the form
A(Vm,t) = J0(VmU0,t), B(Vm,t) = 0 ; (3.3)
for s(t)=1 to obtain functions of the form
A(Vm,t) = J0(VmU0,t) cos (Vm e0), B(Vm,t) = J0(VmU0,t) sin (Vm e0), (3.4)
where J0(∙) is Bessel function of zero order; U0 – the signal amplitude Vm – the ch.f. parameter, and at Vm = 1 function A(1,t) and function B(1,t) change in antiphase. By the way, the dependence of the ch.f. from time to time appeared due to the modulation of the signal, since the modulated signal is a non-stationary process.
In the future, we propose to call the modulation of a new type statistical modulation (SSK - statistical shift keying).
A block diagram of the modulator is shown in Figure 3.1, it contains a multiplier 1 and an adder 2. Timing diagrams explaining its operation are shown in Figure 3.2. The following explanations can be given to the figures. In accordance with the definition of the modulation method, a non-centered quasi-deterministic signal is formed
(3.5)
with ch.f. as [2]
(3.6)
Let the telegraph signal be a sequence of logical zeros and ones (Fig. 3.2a). If s(t)=0, then the ch.f. has only a real part, and its imaginary part is equal to zero [2], i.e.
In this case, with Vm=1, we have A (1, t), B (1, t) in Figure 3.2d, e. When s(t)=1, the ch.f. is equal to (3.6). Then we get
A(Vm,t) = J0(VmU0,t) cos (Vm e0), B(Vm,t) = J0(VmU0,t) sin (Vm e0).
At Vm=1 we have functions
A(1,t) = J0(U0,t) cos (e0 ), B(1,t) = J0(U0,t) sin (e0), (3.7)
which are shown in Figure 3.2d, e. These functions change according to the law of the telegraph signal. Therefore, ch.f. modulated by a telegraph signal, and the functions A(1, t), B(1, t) change in antiphase.
Figure 3.1. Signal modulator circuit
In our opinion, the structure of the modulator turned out to be simple; there are no complex nodes and sources of oscillations in it. Quasi-deterministic signal (2.1) is present at the output of any self-oscillator up to atomic frequency standards. It is known from the review [20] that they and quartz oscillators have short-term phase instability, or, in other words, phase fluctuations, and thus fall under the signal model (2.1). Moreover, the characteristics in Figures 2.1 - 2.4, measured experimentally when studying the signal at the output of a standard generator, confirm this. Constant voltage value by analogy with computers, cell phones can be obtained from the battery. In addition, the amplitude of the high-frequency oscillation in the modulator does not change and, as a result of this, the power amplifier of the transmitting device has the linearity of the input and output characteristics in a narrow range of input signals. All taken together characterizes the modulator only from the positive sides.
Fig. 3.2. Timing waveform diagram in the modulator
However, this modulation method, in which there is a constant component of the signal, is used only in wired communication and is not used in radio communication. Antenna-feeder devices (AFD) in radio communications do not pass the constant component of the signal. For radio communications, a method for amplitude shift keying (AM) of a signal has been developed, which can be used in statistical modulation, since ch.f. of a quasi-deterministic signal will change, similarly to the amplitude of a deterministic oscillation. In this case, the amplitude-time diagram in Figure 3.2c will be different and will take the form shown in Figure 3.2e. The power amplifier of the transmitting device will change the linear mode of operation to non-linear.
At the AM of a centered quasi-deterministic signal (2.1)
Figure 3.2f. Time diagrams of changes in telegraph and AM signals
We obtain only the real part of the ch.f., which changes in antiphase with the telegraph signal and has the form shown in Figure 3.2d. The imaginary part of the ch.f. centered signal (2.1) is always zero. Schemes of modulators with amplitude keying of the signal are described in detail in textbooks [21] and monographs [22].
In passing, we present in more detail the diagram in Figure 3.2c for the telegraph signal shown in Figure 3.2e. As a result, we get a non-centered quasi-deterministic signal with amplitude keying, shown in Figure 3.2g.
Figure 3.2g. Non-centered amplitude-shift keying signal
3.2. The second method of signal modulation
Let a quasi-deterministic signal (2.9) be modulated, for which the random variable a is distributed according to the normal law with quantitative parameters - expectation, - expectation, σ2 - dispersion, and the random variable is distributed uniformly within 0…2π. Then we have
. (3.8)
The characteristic function of A. Lyapunov for the quasi-deterministic signal (2.9), obtained by us earlier using the well-known expression [4, p.263]
(3.9)
and tables [11] , taking into account the distribution law of the random variable a at Vm > 0 , = 0 has the form (2.12). Ch.f. has properties [2], for example, if the law W1(x) corresponds to the ch.f. Θ(Vm), then the law W1(x±e0) corresponds to the ch.f. Θ(Vm)exp( ∓jVme0). Therefore, for a law with expectation ≠ 0 ch.f. (2.12) is transformed into the expression
(3.10)
Ch.f. (3.10) at Vm=const, excluding zero and infinity, depends on the variables ,σ2 . Consequently, by changing the expectation and amplitude dispersion of the quasi-deterministic signal (2.9) with the help of a telegraph signal s(t), one can modulate the ch.f. this signal. Using this method of influencing the amplitude, it is possible to implement twelve variants of the considered method of ch.f. modulation. signal. A new method, the so-called statistical modulation, using a characteristic function, a quasi-deterministic signal and a telegraph message, consists in changing the quantitative parameters of the distribution law of the amplitude of the quasi-deterministic signal in accordance with the change in the telegraph message containing a sequence of logical "0" and logical "1".
A block diagram of the modulator is shown in Figure 3.3 [13], it contains a (IC) interface circuit, a centered quasi-deterministic signal sensor (c.q.s) and a (SB) settings block. In the settings block, the values of the quantitative parameters of the distribution law of the amplitude of the quasi-deterministic signal are stored in the memory, which are written to the signal sensor through the interface circuit. The algorithm for writing parameters includes a telegraph signal s(t), from the logical "0" and "1" of which the values of the settings depend. For example, when a logical “0” arrives, the setting σ02 =1, and when a logical «1» arrives - σ12= 0,0009 is selected, while in both cases the setting e0 =0. Then at the output of the modulator we get a modulated centered quasi-deterministic signal, which is shown in Figure 3.4. In form, the time diagram of the signal resembles a random process that obeys the statistical law of Veshkurtsev, with a probability density of the form (2.11).
Figure 3.3. Modulator circuit
Figure 3.4. Timing diagram of the modulated signal
In the modulator circuit in Figure 3.3, there is no source of physical oscillations, and instead of it, a centered quasi-deterministic signal sensor (c.q.s.) is used. This sensor is built as a computer program using digital technology. The same can be said about other blocks of the modulator block diagram, which are separate files of the general program.
The instantaneous values of the centered quasi-deterministic signal (2.9) vary in the range ±3σ and can reach large values at σ=1, where σ – is the mean square deviation (MSD) of the signal amplitude. This, in turn, places great demands on the linearity of the input and output characteristics of the power amplifier of the transmitting device.
3.3. Combined signal modulator
The first and second methods of signal modulation are implemented individually using their own modulator. However, it is possible to build some combination of two modulators [16], shown in Figure 3.5. The combined modulator circuit includes a multiplier 1, an adder 2, a setpoint block 3, a generator or sensor of a centered quasi-deterministic signal 4. In contrast to the modulator in Figure 3.3, the setpoint block contains only the energy quantitative parameters of the generator (or sensor) oscillation in the form of dispersions σc2, σ02, σ12 signal, which are set using the logical "1" and "0" of the telegraph signal.
Figure 3.5. Modulator circuit
The expectation is introduced using a telegraph signal s(t) through another channel containing a multiplier 1 and an adder 2, which receives a centered quasi-deterministic signal from a sensor (or generator). As a result of these transformations, at the output of the modulator, we obtain a non-centered quasi-deterministic signal shown in Figure 3.6.
а)
b)
Figure 3.6. Timing diagrams of the modulated signal
The amplitude-time dependence of the fluctuation in Figure 3.6a does not contain the expectation ( =0), while in Figure 3.6b the quasi-deterministic signal (2.9) has the expectation =1.
As a result of this, the characteristic function of the signal (2.9) will be modulated, and the variance σ02 = σ12 of the signal (2.9) in both pictures is constant, where σ02, σ12 is the variance of the signal amplitude (2.9) when a logical “0” and a logical “1” arrive at the modulator telegraph signal, respectively.
The amplitude-time dependence of the oscillation at the output of the modulator with a non-centered quasi-deterministic signal (3.5) is shown in Figure 3.2c. Let's recall that the adder 2 of the modulator in this case receives a quasi-deterministic signal (2.1) with a constant dispersion σc2 from generator 4, which in this case replaces the sensor.
3.4. Two-channel signal demodulator
To demodulate the signal, we propose a new method [23], which uses an analog-to-digital signal conversion, multiplication of discrete instantaneous signal values with the parameter Vm, a functional transformation in order to obtain the functions of the sine and cosine products, followed by the accumulation of the values of these functions over a time interval equal to the duration symbol logical "0" and logical "1". After that, using the sine function, the estimate B ̂(V_m,t) of the imaginary part of the ch.f. is calculated, and using the cosine function, the estimate A ̂(V_m,t) of the real part of ch.f., the current values of which are compared with the thresholds, and the decision is made in accordance with the fulfillment of the following inequalities:
if B ̂(V_m,t)< П_1c, then it is considered that the logical ͈ 0 ̎ is accepted;
if B ̂(V_m,t)≥ П_1c, then it is considered that the logical ͈ 1̎ is accepted;
3) if A ̂(V_m,t)≥ П_2к, then it is considered that the logical ͈ 0 ̎ is accepted;
4) if A ̂(V_m,t) < П_2к, then it is considered that the logical ͈ 1 ̎ is accepted.
The block diagram of the demodulator is shown in Figure 3.7. It contains an analog-to-digital converter (ADC) 1, a multiplier 2, functional converters 3,4, accumulating averaging adders 5,6, threshold devices 7,8, an inverter 9. The principle of operation of the demodulator is as follows. The demodulator input receives, for example, a signal (3.8). After conversion to the ADC, the discrete instantaneous values of the signal u_2 (k∆t) are multiplied with the parameter Vm, and the products are converted to obtain the function sin [u_2 (k∆t)V_m] and the function cos [〖V_m u〗_2 (k∆t)]. Accumulating averaging adders 5.6 work simultaneously. The adder 5 accumulates the current values of the sine function, and the adder 6 - the current values of the cosine function. When a synchronization pulse appears at the strobe inputs of the adders, the estimates of the real and imaginary parts of the ch.f. appear at their outputs.
A ̂(V_m,t)=1/N ∑_(k=1)^N▒〖cos [〖V_m u〗_2 (k∆t)]〗, (3.11)
B ̂(V_m,t)=1/N ∑_(k=1)^N▒〖sin [〖V_m u〗_2 (k∆t)]〗, (3.12)
where N - - sample size of instantaneous signal values; ∆t is the signal sampling interval. The properties of the estimates (3.11,3.12) were studied in [2], and it was found that for N>>1 they are asymptotically consistent, effective, and unbiased.
The values of the estimates of the ch.f. (3.11,3.12) with the value Vm=1 are compared in threshold devices 7,8 with the threshold П1c, П2k. For convenience of analysis, the series connection of blocks 3, 5, 7 will be called the sine channel of the demodulator, and the series connection of blocks 4, 6, 8, 9 will be called the cosine channel of the demodulator. Each channel has its own output, hence the demodulator has two outputs. At the output of the cosine channel, the telegraph signal is received inverse with respect to the original. Therefore, the inverter 9 is turned on at the channel output. If the above inequalities with the value Vm=1 are not met, errors occur in the decision regarding the received symbol of the telegraph signal.
Figure 3.7. Two-channel signal demodulator
Thresholds (sine channel), (cosine channel) are set in accordance with the equalities
, , (3.13)
where К1, К2 - variable coefficients; П1 , П2 are thresholds, the values of which will be different depending on the model of the quasi-deterministic signal and are calculated further when analyzing the noise immunity of the modem.
3.5. Single-channel signal demodulator
The sine and cosine channels of the demodulator in Figure 3.7 are not equally affected by interference and, as a result, have different noise immunity. If we combine the advantages of each channel together, we get a single-channel demodulator circuit [24], shown in Figure 3.8.
Figure 3.8. Structural diagram of the demodulator
The block diagram of the demodulator is shown in Figure 3.8. It contains an analog-to-digital converter (ADC) 1, a multiplier 2, functional converters 3.4, accumulating averaging adders 5.6, threshold devices 7.8, an inverter 9, logical AND circuit. Logic circuit 10 combines the outputs of the demodulator channels in the figure 3.7, after which the demodulator has only one output. The demodulator becomes a single-channel device.
Up to logic diagram 10, the single-channel demodulator operates in full accordance with the description of the principle of operation of the circuit in Figure 3.7. Further, the logic circuit 10 is included in the work, the operation of which is explained in Table 3.1.
Table 3.1.
Truth or state table
Sequence number 1 2 3 4
Sinus channel output log. «1» log. «1» log. «0» log. «0»
Cosine channel output log. «1» log. «0» log. «1» log. «0»
Demodulator output log. «1» log. «0» log. «0» log. «0»
Looking ahead, let's say that in the first case, when determining the logical "1", errors are possible, since the sinus channel determines the logical "1" satisfactorily. But the logical "0" sinus channel determines without errors. Therefore, in all subsequent cases, the absence of errors can be expected. The simulation of the circuit in Figure 3.8 confirms what has been said [24]. On average, the single-channel demodulator in Figure 3.8 reduces the error rate by a factor of 20 compared to the cosine channel of the two-channel demodulator. The circuit in Figure 3.8 is not the only one; other options for combining the demodulator channels in Figure 3.7 are possible. Additional studies are required to determine the optimal option for combining two demodulator channels into one channel.
Consider another version of the single-channel demodulator [25], shown in Figure 3.9. In fact, there is actually only one channel in it, namely, this is the previously designated cosine channel. However, the circuit in Figure 3.9 can equally belong to the sine channel of the demodulator if the FC will form a sine function.
Figure 3.9. Single-channel signal demodulator
The circuit in Figure 3.9 includes an ADC - an analog-to-digital converter; P1 - multiplier; FC is the functional converter of the cosine function; AA - accumulative averaging adder; TD is a threshold device, in the output circuit of which an inverter is included, as is done, for example, in the circuit of Fig. 3.7. Therefore, at the output of the demodulator, we will receive an inverse set of logical "0" and "1", which are taken from the output of the control panel to the inverter.
Signal conversion, for example (3.8), in the demodulator proceeds in the following sequence. The quasi-deterministic signal (3.8) is discretized by the ADC, and each discrete instantaneous value of the signal is multiplied with the ch.f. parameter , the product is converted by the functional converter into the value of the function , where ∆t is the sampling interval of the signal. The values of the cosine function are accumulated in the adder, and when a synchronization command is received, they are averaged. The result of averaging enters the threshold device and is compared with the threshold, and the decision is made in accordance with the inequalities:
1) if A ̂(V_m,t)≥ П_1с, then it is considered that the logical ͈ 1 ̎ is accepted;
2) if A ̂(V_m,t) < П_1с, then it is considered that the logical ͈ 0 ̎ is accepted.
The result after averaging the AA adder data is
(3.14)
where N is the sample size of discrete instantaneous values of the signal. In expression (3.14), the expectation operator is replaced by an ideal adder. Studies of the evaluation of the real part of the ch.f. showed [2] that, as N →∞ it is asymptotically consistent, efficient, and unbiased, i.e., evaluation properties tend to fundamental properties. Consequently, the value of the estimate (3.14) will be equal to the value of the ch.f. (2.12), while the threshold will be (3.13)
, where К1 - variable coefficient; П1 – TD device threshold. The coefficient K1 in each modem is different, it depends on the signal modulation algorithm.
