**3. Results**

*3.1. Investigation of the PPSG Based on MAFG When q* = 3

3.1.1. Determining the Repetition Period of the MAFG

At a fixed number of decades, the MAFG repetition period of a pseudo-random sequence of numbers in its output *Tn* and, thus, the repetition period of the pulse sequence in the output of the PPSG, also depended on the number of involved members of Equation (2) and from the initial settings of the registers Rg1–Rg5.

The performed investigations showed that the initial settings of the registers affected the statistical characteristics of the output sequence. The values of these settings obtained asasimulationresult,whenthestatisticalcharacteristicsweresatisfactory,isshownbelow.

Dependence of the repetition period *Tn* on the used number of members from Equation (2) was significant. Some confirmed results are presented in Table 1, which were obtained for such initial states of registers Rg1–Rg5, correspondingly 1, 0, 0, 0, 0.


**Table 1.** Dependence *Tn* on output signal *a* of the logical scheme LS, *q* = 3.

Optimization of equation choosing for the output signal LS was a separate partial task requiring additional research. Its solution would also affect the speed of the generator.

3.1.2. Determination of Statistical Characteristics and the Range of Values of the Control Code

Figure 2 illustrates the investigation results of PPSG statistical characteristics based on the MAFG for *q* = 3.

Here the following notations were used:

• SS\_n—value *χ*2 *c* ;


The results were obtained at the following values of the method parameters for evaluating the quality of the pulse sequence: *nmax* = 1000, *kc* = 10, *<sup>χ</sup>*2*cr* = 25.

The output signal of the logic circuit of the LS was formed by the following expression:

$$a = \left(a\_{\mathbb{O}\_0} \oplus a\_{\mathbb{O}\_1} \oplus a\_{\mathbb{O}\_2} \oplus a\_{\mathbb{O}\_3}\right) \oplus \left(a\_{1\mathbb{O}} \oplus a\_{1\mathbb{I}\_1} \oplus a\_{1\mathbb{I}\_2} \oplus a\_{1\mathbb{J}\_3}\right) \oplus \left(a\_{2\mathbb{J}\_0} \oplus a\_{2\mathbb{I}\_1} \oplus a\_{2\mathbb{J}\_2} \oplus a\_{2\mathbb{J}\_3}\right) \tag{8}$$

as a result of the search for initial states of various variants of registers Rg1–Rg5, it was found that the value of these settings was satisfactory— *G*, 0, 0, 0, 0, correspondingly. That is, the option in which the initial settings depended on the control code. This was for such initial settings for which results are presented in Figure 2.

**Figure 2.** PPSG statistical characteristics based on MAFG (*q* = <sup>3</sup>): (**a**) the value *χ*2*c* ; (**b**) the average value of the last five (current) values *χ*2*c* − *<sup>χ</sup>*2*cav*; (**c**) number of values *<sup>χ</sup>*2*cav* greater than *<sup>χ</sup>*2*cr*. G—control code value.

Thus, the range of the control code values G −(*<sup>G</sup>*<sup>1</sup> ÷ *<sup>G</sup>*2), in which the original pulse sequence corresponded to the Poisson distribution law, in this case (when *q* = 3), was determined by the equation

$$G\_1 = 1, \ G\_2 = 124 \tag{9}$$

In this case, the value *G*1 = 1, determined as a result of simulation, coincided with the value *G*1, defined theoretically by the expression (6):

$$G \ge \frac{10^9 \cdot k\_c \cdot n\_{\text{max}}}{T\_n} = \frac{10^3 \cdot 10 \cdot 10^3}{10^9} = 10^{-2} \tag{10}$$

#### *3.2. Dependence of the Average Value of the Output Signal Frequency on the Control Code*

This section is divided by subheadings. It should provide a concise and precise description of the experimental results and their interpretation, as well as the experimental conclusions that can be drawn.

Figure 3a illustrates the dependence of the average frequency of the PPSG output pulse sequence on the control code *G*, while Figure 3b illustrates a fragment of that dependence.

Here solid lines show the dependences obtained by simulation, and dotted lines show theoretical values, calculated on the basis of Equation (3). Solid and dotted lines in Figure 3a almost coincide. To specify the calculations, it was accepted that *fm* = 1000 Hz. All real dependences were obtained for the condition of formation of the LS output signal correspondingly with logical Equation (8) and explained initial states Rg1–Rg5: *G*, 0, 0, 0, 0, correspondingly.

**Figure 3.** The value of the average frequency of the PPSG output signal based on MAFG (*q* = 3) when Δ*G* = 1: (**a**) the dependence of the average frequency on the control code *G;* (**b**) the fragment of that dependence. G—control code value.

Thus, the dependences of the values of the average frequency of the output pulse sequence of the generator from the control code, obtained as a result of simulation, were close to theoretical. That practically allowed the use of Equation (3) while determining the average frequencies of the PPSG output signal.

#### *3.3. Investigation of the PPSG Based on MAFG When q* = 6 3.3.1.DeterminingtheMAFG RepetitionPeriod

Dependence of the repetition period *Tn* on the number of involved members of Equation (2) is presented in Table 2. The following initial states of the registers Rg1–Rg5, correspondingly 1, 0, 0, 0, 0, were obtained.


**Table 2.** Dependence of *Tn* on the output signal *a* of a logical scheme LS (*q* = <sup>6</sup>).

3.3.2. Determination of Statistical Characteristics and the Range of Values of the Control Code

Investigation results of the PPSG statistical characteristics based on MAFG for *q* = 6 are presented in Figure 4.

**Figure 4.** PPSG statistical characteristics based on MAFG (*q* = <sup>6</sup>): (**a**) the value *χ*2*c* ; (**b**) the average value of the last five (current) values *χ*2*c* − *<sup>χ</sup>*2*cav*; (**c**) number of values *<sup>χ</sup>*2*cav* greater than *<sup>χ</sup>*2*cr*. G—control code value.

The results were obtained for the same values of the parameters of the quality assessing method of the pulse sequence, as in the previous case (for *q* = 3): *nmax* = 1000, *kc* = 10, *<sup>χ</sup>*2*cr* = 25.

Output signal of the logic scheme LS was formed according to the following expression:

$$a = \begin{pmatrix} a\_{0\_0} \oplus a\_{0\_1} \oplus a\_{0\_2} \oplus a\_{0\_3} \end{pmatrix} \oplus \begin{pmatrix} a\_{1\_0} \oplus a\_{1\_1} \oplus a\_{1\_2} \oplus a\_{1\_3} \end{pmatrix} \oplus \begin{pmatrix} a\_{2\_0} \oplus a\_{2\_1} \oplus a\_{2\_2} \oplus a\_{2\_3} \end{pmatrix} \oplus \begin{pmatrix} \begin{pmatrix} a\_{3\_0} \oplus a\_{3\_1} \oplus a\_{3\_2} \oplus a\_{2\_3} \end{pmatrix} \oplus \begin{pmatrix} \begin{pmatrix} a\_{3\_0} \end{pmatrix} \oplus a\_{3\_1} \oplus a\_{3\_2} \end{pmatrix} \oplus \begin{pmatrix} \begin{pmatrix} a\_{2\_0} \end{pmatrix} \oplus a\_{2\_1} \oplus a\_{2\_2} \end{pmatrix} \tag{11}$$

where the initial settings of registers Rg1–Rg5 were correspondingly the following: *G*, 0, 0, 0, 0.

Control code range values *G* −(*<sup>G</sup>*<sup>1</sup> ÷ *<sup>G</sup>*2), in which the output pulse sequence corresponded to the Poisson distribution law, in this case (when *q* = 6), was determined by the following equation.

$$G\_1 = 1, G\_2 = 111010 \tag{12}$$

In this case the value *G*1 = 1, determined as a result of simulation, coincided with the value *G*1, determined theoretically by Expression (6):

$$G \ge \frac{10^q \cdot k\_c \cdot n\_{\text{max}}}{T\_n} = \frac{10^6 \cdot 10 \cdot 10^3}{10^{10}} = 10^0. \tag{13}$$

#### *3.4. Dependence of the OUTPUT Signal Frequency Average Value on the Control Code*

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**Figure 5.** Output signal average frequency of the PPSG based on MAFG (*q* = <sup>6</sup>), when Δ*G* = 1000: (**a**) the dependence of the average frequency on the control code *G;* (**b**) the fragment of that dependence. G—control code value.

Here, similarly to Figure 3, solid lines illustrate the dependences obtained by simulation, and dotted lines illustrate theoretical values, calculated on the basis of Equation (3). Solid and dotted lines in Figure 5a almost coincide. It was accepted that. *fm* = 1000 Hz. All the real dependences were obtained under the condition of LS output signal formation according to the logic of Equation (11) and the above-justified initial states Rg1–Rg5: *G*, 0, 0, 0, 0, correspondingly.

The fundamental difference between the dependencies presented in Figures 3 and 5 is that they were obtained using different values for the control code step changing *G* − Δ*G*: in Figure 3 (for *q* = 3) when Δ*G* = 1; while in Figure 5 (for *q* = 6) when Δ*G* = 1000.

A decrease in value Δ*G* for *q* = 6, led to some ambiguity in establishing the average frequency value of the PPSG output sequence. This is illustrated in Figure 6 where the dependences were similar to the dependences in Figure 5, for Δ*G* = 100.

**Figure 6.** Output signal average frequency of the PPSG based on MAFG (*q* = <sup>6</sup>), when Δ*G* = 100: (**a**) the dependence of the average frequency on the control code *G;* (**b**) the fragment of that dependence. G—control code value.

#### *3.5. Comparing PPSG Characteristics Based on MAFG for q* = 3 *and q* = 6

Increasing the number of decades of the generator could significantly increase the repetition period of the sequence of numbers in the MAFG output and, thus, also the pulse sequence period of the PPSG output. However, this did not lead automatically to an increase in the generator's "distinguishing ability" concerning the established value of the output sequence average frequency *fout*, which was actually setting the ability to specify the changing step *fout* − Δ *fout*.

The performed research showed that "distinguishing ability", at a fixed value for the number of decades *q*, depended on the initial settings of the registers Rg1–Rg5 and on the involved members of Equation (2), which determined the logic of signal generation on the output of the LS scheme. In this case, the statistical characteristics of the original sequence depended on these parameters. Taking into account the above considerations, improving a generator's "distinguishing ability" could be the subject of a separate study.

As far as increasing the number of decades from *q* = 3 to *q* = 6, during the abovementioned conditions, in fact did not lead to an increase in the PPSG's "distinguishing ability" (decreasing Δ *fout*) and expanded the range *fout*. For future work, it would be worth considering the possibility of the practical use of this generator when *q* = 3.

#### *3.6. Using the PPSG Based on the MAFG When q* = 3

Figure 7 shows the structural scheme of the device, in which to expand the range of average values of the output frequency, an additional frequency divider FD was introduced.

**Figure 7.** Structural scheme of the Poisson pulse sequence generator with an extended range of average values in the output frequency.

At the FD output the clock pulse sequence was formed for the PPSG, the frequency of which was determined by the equation

$$f\_m = \frac{f\_\mathcal{S}}{K\_d} \tag{14}$$

where *Kd* is the division factor FD and *fg* is the reference generator frequency.

Some generator parameters are presented in the Table 3, of which one is presented in Figure 7, when *fg*= 1 MHz. The PPSG was implemented based on the MAFG for *q* = 3, while its internal parameters corresponded to the above: the output signal of the logic scheme LS was formed by the expression (8), and the initial states of the registers Rg1–Rg5 were *G*, 0, 0, 0, 0, correspondingly. This allowed us to tell whether the statistical characteristics of the output pulse sequence in the given ranges of values *fout*, corresponded to Poisson's law of distribution.

The construction of PPSGs based on MAFGs, all elements of which, except those of the LS, work in binary-decimal code, improves significantly the quality of the output sequence. This was confirmed by a generalized technique for studying parameters of the PPSG output pulse sequence for compliance with the Poisson distribution law using the Pearson test. Investigations of the proposed solutions illustrated that the dependences of the average frequency values of the generator's output pulse sequence from the control code, obtained as a result of simulation, were close to the theoretical ones. It was shown that the number of decades was enough to choose *q* = 3, because greater numbers of decades did not actually lead to an increased "distinguishing ability" for the PPSG; while scheme realization would be more complicated in that case. In order to expand the output frequency average values the introduction of a division factor into the PPSG structural scheme was proposed, which would be divided by the frequency of the reference generator. The question of the selecting

number of the equation members to calculate the logical variable *a*, the value of which was obtained at the LS output, was rather significant. The number of data members of the equation and approaches to their choice significantly affected the size of the repetition period *Tn*. Further research is needed in this direction in order to improve the initial characteristics of the PPSG and increase its performance.


**Table 3.** PPSG parameters with additional FD.
