THE FORMATION METHOD OF COMPLEX SIGNALS ENSEMBLES BY FREQUENCY FILTRATION OF PSEUDO-RANDOM SEQUENCES WITH LOW INTERACTION IN THE TIME DOMAIN

Context. The problem of forming complex signal ensembles on the basis of frequency band filtering and research of their properties is considered. The object of research is the process of synthesis of signal ensembles based on frequency filtering of pseudo-random sequences of short video pulses with low interaction in the time domain. Objective. It is to form complex signal ensembles with satisfactory values of intercorrelation properties, which are close to the signals with minimal energy interaction. Method. The results of the application of forming complex signal ensembles method by frequency filtering of pseudo-random sequences with low interaction in the time domain are presented. As a result of the spectral band selection of the studied pseudo-random short video pulse sequences due to the use of bandpass filters based on the Chebyshev filter of the first kind, new samples of sequences with spectrum restriction are obtained. By applying intercorrelation analysis to the obtained sequence samples, the values of the maximum emissions of the side lobes of the cross-correlation functions (CCF) for all possible signal pairs are estimated. If the values of the maximum emissions of the side lobes of the CCF signals exceed the limit values, the sequence of the analyzed pair with a smaller value of the number of pulses is removed from the ensemble. In case of satisfactory value – the received signals are accepted for the signal ensemble formation with the minimum power interaction. Thus, a new set of values of the maximum emissions of the side lobes of the CCF is formed. This approach increases the number of signals in ensembles with satisfactory values of statistical characteristics with limited signal spectrum width, and the correlation properties of such sequences approach the signals with minimal energy interaction, which reduces the level of multiple access interference. As a result, complex signal ensembles obtained by frequency filtering should be used in cognitive radio systems with code division multiplexing. Results. Based on the software implementation of the method of forming complex signal ensembles by frequency filtering of pseudo-random sequences with low interaction in the time domain, signals with satisfactory values of statistical characteristics with limited signal spectrum width with intercorrelation properties close to signals with minimal energy interaction and higher ensemble volume were selected. Conclusions. The application of frequency filtering to pseudo-random sequences of short video pulses with a low level of cross-correlation allows to obtain complex signal ensembles, which will be similar in correlation properties to sequences with minimal energy interaction. It will reduce the level of multiple access interference. The analysis revealed that the use of frequency filtering of sequences will slightly worsen the mutual correlation properties of signals, possibly due to suboptimal synthesis of values of maximum emission levels of side lobes of CCF signals, but, nevertheless, it is possible to use such signals in modern cognitive systems radio access multiple access with code division multiplexing.


ABBREVIATIONS
FR is a frequency response; CCF is a cross-correlation function.

NOMENCLATURE α is a coefficient;
В is a signal base; Е і is energy of i-th signal; ΔF is a signal spectrum width; Δf is a filtering band width; σ Rmax is a standard deviation of the maximum emissions of the side lobes CCF; m Rmax is a mathematical expectation of maximum emissions of side lobes CCF; N is a number of signal pairs interacting with each other; n i is a number of pulses in i-th sequence; n j is a number of pulses in j-th sequence; Q i is a duty cycle of i-th pulse sequence; Q j is a duty cycle of j-th pulse sequence; R k is a CCF of the sequence pair with serial number k; rect(·) is a pulse of single amplitude and fixed impulse duration; S i (t) is і-th pulse sequence; T is a signal duration; T i is a pulse follow-up period in i-th sequence; T j is a pulse follow-up period in j-th sequence; τ i is a impulse duration of i-th sequence; τ j is a impulse duration of j-th sequence; U i is a value of the amplitude in the i-th sequence.
INTRODUCTION Current trends in the development of multiple access radio systems and the impact of multiple access interference on them, especially in cognitive radio networks, require a further increase in the number of subscribers, provided that the specified quality of service in wireless networks. The study of the properties of complex signal ensembles based on pseudo-random sequences with minimal energy interaction allows to determine the statistical characteristics of complex signal ensembles obtained by applying the method of forming complex signal ensembles through frequency filtering of pseudo-random sequences with low interaction in the time domain due to which it is possible to significantly increase the volume of the signal ensemble by reducing the values of the maximum emissions of the side lobes of the CCF of such signals, while the level of multiple access interference remains within acceptable values. The received signals should be used in cognitive radio systems based on code division multiplexing, and increasing the number of signals in the ensemble allows you to increase the number of subscribers in such systems, while maintaining a high level of service quality.
The object of study is the process of synthesis of complex signal ensembles formed on the basis of frequency filtering of pseudo-random sequences of short video pulses. Given the limited frequency and time resources in modern cognitive radio systems, the urgent problem is to increase the volume of complex signal ensembles, in order to improve the quality of service and performance in such systems.
The subject of study is the method of forming complex signal ensembles by frequency filtering of pseudo-random sequences with low interaction in the time domain based on samples of pseudo-random sequences with minimal energy interaction. There are known methods of forming ensembles for phase-manipulated, amplitude-manipulated and other signals based on linear and nonlinear sequences [1,2,3,4,5], but the correlation properties of complex signal ensembles obtained by frequency filtering of sequences with low interaction in the time domain were not performed.
The purpose of the work is to develop a method for the synthesis of complex signal ensembles with limited spectrum width, the correlation properties of which are close to the signals with minimal energy interaction.

PROBLEM STATEMENT
Suppose a given sample of pseudo-random sequences with minimal interaction in time domain, with the parameters: The task of forming complex signals ensembles by frequency filtering of pseudo-random sequences with low interaction in time domain is to choose the optimal value of Δf based on R k analysis taking into account the limitations of m Rmax and σ Rmax to match the signals with minimal energy interaction, which in turn increase the number of received signals used to form ensembles.

REVIEW OF THE LITERATURE
Methods of forming complex signals are widely considered in the literature [3,4,7]. The value of the crest factor of such signals is close to 1 even taking into account the passage of the filters of the transmission paths [11,12], and the maximum value of the cross-correlation coefficient of the signal ensemble is of such order that does not significantly reduce noise immunity and does not provide protection against multiple access interference under conditions of a significant dynamic range of received signals. When using recurrent sequences, it is quite difficult to obtain large complex signal ensembles with satisfactory correlation properties [1,3,6,7,10].
Statistical characteristics of signals obtained by frequency filtering of pseudo-random sequences of short video pulses with low energy interaction are considered insufficiently [2] and therefore require more detailed study to increase the number of complex signals in the ensemble.

MATERIALS AND METHODS
To evaluate the complex signals properties, pseudorandom sequences of short video pulses with a low level of cross-correlation were chosen, which are described by the expression [1,2]: the duty cycle of pulse sequence is defined as: to pulses of unit amplitude and fixed pulse duration the following restrictions are put forward [2]: To increase the volume of ensembles using the method of forming complex signal ensembles through frequency filtering of pseudo-random sequences with low interaction in the time domain, a sample of pseudorandom sequences with minimal energy interaction was chosen.
In the analysis process, the frequency range of such sequences is divided into equal bands. The selection of spectrum parts from the common frequency band of complex signals is done using Chebyshev filter of the first kind. The use of such a filter is appropriate where it is necessary to provide with a small-order filter the necessary amplitude-frequency characteristics, in particular, proper suppression of frequencies from the suppression band.
As a result of the application of such filtering, pseudorandom sequences with minimal energy interaction in the frequency domain were subdivided into sequences characterized by constraints with equal intervals of frequency bands. Analysis of such sequences shows that the signals obtained by selection the frequency bands will differ in shape from each other and, moreover, will meet the condition of minimum similarity of signals (4).
The use of complex signals based on pseudo-random sequences with minimal energy interaction in cognitive systems with multiple access based on code division multiplexing provides a minimum level of multiple access interference, which can be estimated by determining the maximum allowable emissions of side lobes of CCF [3].
the signal base is calculated as [3]: The calculation of the emissions values of the side lobes of the CCF, obtained as a result of permutations of the sequences, takes place in accordance with [3]: Due to the different number of pulses in the sequences, and because the energies of the continuous signals will be different to estimate the CCF by expression (6) it is necessary to normalize the signal energy values [1]: The calculation of the CCF is performed for all possible pairs from the complex signal ensemble, checking the CCF of each pair of signals to meet the condition of ensuring a given maximum emission level of side lobes of mutual correlation. If the maximum emission values of the side lobes of the CCF signals exceed the limit values, the sequence of the analyzed pair with a smaller value of the pulses number is removed from the ensemble. In case of satisfactory value -the received signals are accepted for formation of a signals ensemble with the minimum power interaction. Determination of the optimal filter bandwidth is based on the analysis of the dependence of the maximum values of CCF R(τ) which depends on the number of elements in the involved sequences The dependence constructing of the maximum emission levels of the side lobes of the CCF on the filtration band width and the number of pulses in the sequences is based on the method described in [4,5].
Statistical characteristics estimation of signal ensembles based on frequency filtering of short video pulse sequences with minimal energy interaction is performed using the calculation of mathematical expectation of maximum emissions of side lobes CCF signals m Rmax when changing the values of filter bands in the range from 0.1% to 2% of the total spectral width at a constant value of duty cycle and signal duration [12]: To estimate the standard error of the arithmetic mean from the mean value, the calculation of the standard deviation of the maximum emissions of the side lobes CCF σ Rmax relative to the mathematical expectation is performed: Thus, the study based on the proposed method for evaluating the correlation properties of complex signal ensembles by frequency filtering allows obtaining sequences with different waveforms, obtained even from the same sequence. The signals obtained by synthesizing different sequences are weakly correlated when they are located in different frequency bands, in addition, this method is relatively easy to implement and does not require significant computational resources.

EXPERIMENTS
To implement the proposed method of forming complex signal ensembles by frequency filtering of pseudo-random sequences with low interaction in the time domain, a software model was developed in the Matlab environment, which practically confirms the obtained theoretical results.
The original sample consists of 50 pseudo-random sequences with minimal energy interaction, on the basis of which, by band filtering, an ensemble of complex signals is formed.
As a result of searching all pairs of signals, the values of cross-correlation of pseudo-random sequences were calculated. Their estimation was carried out on the basis of the constructed models of each pair of sequences.
The selection of spectrum portions from the frequency band, on the basis of small-kind bandpass filters, which provide the necessary frequency response (FR), as well as the proper suppression of frequencies in the suppression band, was applied to the sampling. The frequency filtering band was selected in the range from 5 kHz to 200 kHz, in increments of 10 kHz. As a result of bandpass filtering of pseudo-random sequences on different frequency bands, signals differing in shape were obtained.
For further evaluation of the CCF of the filtered elements of the sequences, a necessary condition is the normalization of the received signals by energy, as due to the different number of pulses in the sequences and their continuity, the signal energies differ. The next step is to determine the correlation properties of energy-normalized signals and build their models, on the basis of which further calculation of the maximum values of emissions of side lobes CCF signals for all possible pairs from the complex signal ensemble. The result is a dependence model of the maximum values of CCF, which depends on the number of elements in the involved pseudo-random sequences and the width of the filter band. After analyzing in each pair of sequences to meet the condition of ensuring a given level of maximum emissions of the side lobes CCF, only those sequences remain that meet the requirements for the limit values of the signals with minimal energy interaction.
After completing the analysis of the obtained signals that meet the requirements for the limit values of signals with minimal energy interaction, the statistical characteristics of ensembles are evaluated based on the calculation of mathematical expectations of maximum emissions of side lobes CCF signals m Rmax to determine the mean value of the sample of the processed results, the calculation of the standard deviation of the maximum emissions of the side lobes CCF σ Rmax relative to the mathematical expectation, to estimate the standard error of the arithmetic mean.
Based on the analysis of the statistical characteristics calculation, it is possible to ensure the effective formation of complex signals by filtering the spectrum bands of pseudo-random sequences with minimal energy interaction.

RESULTS
The statistical characteristics calculation of the maximum emissions values of the side lobes of the CCF is given in Table 1. To present the results, the following notations were used: m Rmax -mathematical expectation of the maximum emissions of the side lobes of the CCF, σ Rmax -standard deviation. The calculations were performed taking into account the frequency filtering band from 0.1% to 2% of the original spectrum width =of the studied sequences.
Comparing the values from the Table 1, the optimal results of solving the problem of complex signals synthesis by filtering bands of pseudo-random sequences of short video pulses with minimal energy interaction were obtained, provided that the condition fulfillment of required level compliance of maximum emissions the side lobes of the CCF with the limit values and at satisfactory values of the statistical characteristics of the studied signals. Fig. 1 shows the results of the statistical characteristics calculating of the signals obtained by frequency filtering of pseudo-random sequences of short video pulses with minimal energy interaction. Fig. 1a shows the results of the limit values calculation of the side lobes maximum emissions of the CCF, taking into account the frequency filtering band. Obviously, not all calculated pairs satisfy the limit condition, which is shown as a slice. Pairs of sequences whose maximum value exceeds the allowable values must be re-analyzed in order to remove from the sequence signals ensemble that leads to exceeding the limit value. Fig. 1b presents the calculation results of large mathematical expectations of significant maximum emissions of the side lobes of the CCF, taking into account the standard deviation. Fig. 1c shows the dependence of the mathematical expectation of the maximum emissions of CCF side lobes on the frequency band, which shows the correspondence of the maximum values scatter of CCF side lobes to the calculated values, and its value does not exceed B 1 at a filter bandwidth of 0.2%. From the obtained calculations we can conclude that the optimal filter band is equal to 0.2% of the total value of the sequences spectrum.   (Table 1) it is seen that based on the analysis of mathematical expectations of maximum emissions of CCF side lobes, depending on the frequency band, we can solve the problem of forming ensembles of complex signals from filtered frequency bands of the pseudo-random sequences of the short videopulses with minimal energy interaction.
The distribution of the maximum emissions values of the CCF side lobes depending on the number of pulses in the pairs of sequences and the filtration band width in Fig. 1a is limited in the form of a slice of the maximum values of the levels. Those pairs of sequences that exceed these limits are re-analyzed to determine which of the sequences is not suitable for forming an ensemble of signals. Thus, after re-analysis, only those signals remain that satisfy the limit values of the constraints.
The choice of the optimal filtration band width is made on the basis of the analysis of the mathematical expectation dependence of the maximum emissions of the CCF side lobes (Fig. 1b) depending on the filtration band width. In comparison, condition (3) with different values of α was used (Fig. 1c). As a result, the mathematical expectation of the maximum emissions of the CCF side lobes does not exceed B 1 at a filtration bandwidth of 0.2%. It should be noted that with the filtration band expansion, the cross-correlation properties of the signals are improved, and, therefore, satisfy the condition of minimal signals similarity, with the disadvantage of a slight increase in the crest factor of such signals. Also, the application of frequency filtering by means of filters with equal frequency bands to the synthesized pseudo-random sequences with minimal energy interaction makes it possible to obtain a difference in the waveforms obtained even from the same sequence. The signals obtained by synthesizing different sequences are uncorrelated when they are located in different frequency bands.

CONCLUSIONS
Studies of the correlation properties of complex signal ensembles obtained by frequency filtering of pseudorandom sequences with minimal energy interaction allow forming much larger signal ensembles than existing complex signals used in modern radio systems with code division multiplexing.
The scientific novelty of obtained results lies in the development of a method for forming ensembles of complex signals by frequency filtering of pseudo-random sequences with low interaction in the time domain, which have a low level of multiple access interference. This approach simplifies the synthesis of complex signal ensembles based on pseudo-random sequences with minimal energy interaction and increases the number of received signals used to form ensembles with specified emission levels of CCF side lobes.
The practical significance of obtained results is that the possibility of using complex signals ensembles obtained by frequency filtering in cognitive radio systems with multiple access, which are affected by interference from multiple access.
Prospects for further research are to improve the selective capabilities of the proposed method and further modernization, taking into account the permutations of the filtration bands, as well as a deeper study of the ensemble properties of the signals obtained by the proposed method.