PROPERTIES AND FORMATION OF OFDM AND DERIVED SIGNALS

: The article discusses the technology of forming signals used in mobile, information and telecommunication systems, and also provides an analysis of promising technologies that can be used in wireless communication systems of broadband access. It is shown that the widely used modulation scheme with orthogonal frequency division (OFDM) has a number of drawbacks, which can lead to a decrease in system performance. Alternative technologies for generating signals are presented, in particular, a technology based on windowed signal processing (W-OFDM), a technology based on time division (w-OFDM); UFMC technology and others to eliminate the disadvantages of OFDM technology. New points of view are proposed on the use of multi-carrier transmission technology in the form of multiplexing with orthogonal frequency division (in order to increase the security of modern wireless broadband access communication systems from external and internal threats), a class of non-linear discrete cryptographic sequences to form a physical data carrier – signal. Copyright © Research Institute for Intelligent Computer Systems, 2020.


OFDM TECHNOLOGY ADVANTAGES
The main idea of OFDM is to achieve a high transmission rate in the frequency domain, by dividing full signal frequency range into a number of non-overlapping frequency subchannels with lower speeds. In addition, each subchannel (subcarrier) is modulated by a separate symbol, then these channels are multiplexed in frequency domain and data are transmitted in parallel in orthogonal subchannels. Compared to single carrier transmission, this approach provides enhanced resistance to narrowband interference and channel distortion. Specified, in particular, allows for a high level of system flexibility, since modulation parameters, such as constellation size, coding rate, can be independently selected for each subchannel.
The widespread use of the OFDM digital modulation scheme is due to a number of remarkable features of this technology: -resistance to multipath effects; -high noise immunity to narrowband interference; -resistance to intersymbol interference due to the fact that the duration of the symbol in the auxiliary subcarrier is significantly longer compared to the propagation delay than in traditional modulation schemes; -high spectral efficiency in comparison with traditional systems with frequency division of channels due to the large number of subcarriers; -the ability to use different modulation schemes for different subcarriers, which allows you to adapt to the conditions of signal propagation and to different requirements for the quality of received signals; -simple implementation using digital processing methods, etc.

SECURITY INDICATORS EVALUATION
State level of informatization is determined primarily by the development of information communications, as a set of network resources intended for the production and provision of telecommunications, information and other services. The development of wireless communications technologies has been constantly shaped based on studies of waveforms. As an example, we can use the technology of multiplexing signals with orthogonal frequency division multiplexing in modern wireless broadband access systems (WiMAX, Wi-Fi, LTE, etc.). The use of this technology allows increasing the information capacity of the system with a limited bandwidth, data reception and transmission speed, bringing it closer to the channel capacity, increasing the secrecy transmission capacity and noise immunity of signal reception, and as a result, to meet the everincreasing needs of network users in high-speed connections services.
Analytically, the OFDM signal can be represented as [13]: where k is the subcarrier index, ( ) is the signal on the k-subcarrier, is the amplitude component of the sequence of information symbols, N is the number of subcarriers, T is the duration of the information symbol.
The block diagram of an OFDM modulator is presented in Figure 1 [14,15]. In the transmitter, the serial stream of binary symbols s[n] is encoded with an error-correcting code, interleaved further, using inverse multiplexing (demultiplexing), turns into N parallel streams, each of which is matched (complexly) with the output stream s[n] using certain constellation modulations (quadrature modulation QAM, quadrature phase modulation QPSK, etc.). The number of outputs of the demultiplexer is determined by the number of subcarrier frequencies. Next, the modulated 01 ,..., N XX − symbol streams undergo a fast inverse conversion. The received analog signals are used for modulation in accordance with the sine wave and cosine wave (obtained by shifting the sine wave by 90) of the carrier frequency. After modulation, the signals are summed to form a signal s(t), which enters the communication channel. Subcarriers orthogonality makes it possible to select each of them from the common signal at the reception even in the case of partial overlapping of their spectra. Since the subcarriers are located close to each other and even partially overlap, the spectral efficiency of the modulated OFDM signal is high. The parameters of the subcarrier signals are selected in such a way that they are orthogonal to each other, that is, the condition is met for them: where t is the duration of the information symbol, l f and k f are the frequencies of the l-th and k-th subcarriers, respectively. The orthogonality of the carrier signals guarantees the frequency independence of the channels from each other and, therefore, the absence of inter-channel interference. For the fast implementation of this procedure, the inverse fast Fourier transform algorithm is used, that is, the signal values at the input of the IFFT block belong to the frequency domain. At the output of the block IFFT receives the signal value in the time domain. Combining all the values, a complex OFDM signal is obtained. Taking into account the fact that IFFT works effectively with arrays of dimension 2k, the number of subcarriers is chosen with the same multiplicity. For example, in WiMAX wireless communication systems, the number of subcarriers is chosen from 128 to 2048 and can occupy frequency bands from 1.25 MHz to 20 MHz. For each of the subcarriers, a different modulation type is used depending on the requirements and the type of interference in the channel. At the receiving end, the inverse operations are performed, in this case, instead of a digital-to-analog converter, an analogto-digital converter (ADC) is used, instead of a reverse FFT, direct FFT. The structure of the OFDM signal can be quite complex because it consists of many components: -the structure of the time-frequency distribution, given by: initial frequency, frequency grid pitch, number of subcarriers; -time slots specified by: the duration of the symbol, the duration of the guard interval; -type of manipulation: phase (BPSK, QPSK, 8-PSK) or amplitude-phase quadrature modulation (QAM) -discrete sequences that determine the law (rule) of manipulating the phase of the high-frequency carrier, and given the dimension of the signal space; -type of symbol sync; -the presence and type of noise-resistant coding (Reed-Solomon code, Bowes-Choudhury-Hokvingem code, turbo codes, etc.) -the presence and type of data interleaving and so on.
The above features of the OFDM signal structure can be used in the construction of ICT, for which the ensuring requirements of the specified security indicators against the introduction (imposition) of spurious messages, falsification of messages; data integrity, confidentiality, noise immunity of reception, operation secrecy are decisive.
One of the components of information security (along with information secrecy) is imitation protection (ensuring integrity) of information. The mathematical apparatus of the imitation protection system includes a cryptographic algorithm for simulating the encryption of information (this may be an encryption algorithm, an authentication code, or another transformation) and an algorithm for determining the truth of the information received, as well as a key system. In essence, imitational security is a complex service that is provided by such services as integrity, authenticity (truth), as well as the use of various cryptographic protocols with certain properties [14,15]. As studies [16,17] have shown, it is possible to provide the imitability necessary in ICS at the source level of complex signals by increasing the dimension of the signal space, the degree of correlation between them, the complexity of the laws of their construction. In accordance with the above definitions, the theory of authentication by J. Simons [18] can be used to quantify the simulated security. It was Symons who showed that for the quantitative assessment of authenticity one can use the probability Pimp of deception: where ( , ) I C K  is the amount of information on the authentication key K entered into the C cryptogram.
Let us analyze the expression (3). (3) is achieved are referred to as systems that are absolutely resistant to deception.

Systems in which equality
2. To reduce the probability of fraud, it is necessary to increase Taking into account the peculiarities of the OFDM signal structure, imitability ( (4) At the complex signals source level (physical level), the probability of cheating or imposing a false signal is defined as where i l is the length authentication code, the dimension of the signal space Analysis of the data in Table 1 shows that the proposed method for the synthesis of complex nonlinear discrete cryptographic signals allows for the formation of large ensembles of discrete sequences. So, for the period of the sequence N = 63 the number of pairs of CS satisfying the maximum value of the maximum PCCF side lobes -17 is 12214869. For a representative of the class of linear sequencessequences with a three-level cross-correlation function (Gold set), which are optimal from the point of view of cross-correlation functions [23], the number of pairs of signals corresponding to a given boundary is 975. The excess of the volume of a CS over an ensemble composed of M-sequences is more than 10 7 times. For the period of a sequence of 1023 elements, the number of pairs of gearboxes satisfying the limiting value for the side lobes of the crosscorrelation function (CCF) 100 is 5293538, whereas for a representative of the class of linear sequences of M-sequences, the number of pairs that meet this boundary is 435, then there is an excess of the volume of the signal system is more than 105 times. With a slight decrease in the requirements for the limiting value of the maximum lateral peak of PCCF, according to which the selection of signals is carried out (in fact, a reduction in the noise immunity of reception), the performance of the ICS system can be significantly improved. So, for the period of the sequence N = 127, increasing the limit value by 1.2 dB will increase the ensemble volume from M = 11610 (at the border of 17) to 9006648 signals, with a limit value of 27, that is, 776 times. As follows from the data in Table 1, the probability values of imposing in the case of the application of the CS are much less. So, with a period of the sequence L = 1023, it is four orders of magnitude less than using M-sequences and an order of magnitude less than when using sequences with a 3level PCCF. An improvement in the imitation resistance index of the ICS is achieved due to the fact that the CSs have improved ensemble properties in comparison with linear classes of signals, in particular, M-sequences. In Table 2  Analysis of the data given in Table 2, suggests that maximum lateral emissions values of the CS, as well as the statistical characteristics of this class of signals are not inferior to the corresponding characteristics of the signals constructed using Msequences and characteristic discrete signals [7][8][9][10][11][12][21][22][23][24][25][26][27][28][29][30][31]. This, in turn, indicates that the use of a CS provides noise immunity for receiving signals no worse than when applying the decree of signals based on linear formation laws. According to the data in Tables 1-2, it also follows that by varying the limiting values of the side-lobe level of the correlation function, depending on the requirements for the ICS, the tasks of achieving the required values of the noise immunity indicators for signal reception, imitation resistance and secrecy of the ICS, can be solved. Let us make an ICS protection assessment from imposing false messages, for the case when the system uses a dynamic shift mode (according to the law of the control sequence) correspondence: the message bit is a complex signal. In this case, the value of the probability of imposing a false message ( where: 2 k − a number of possible states of the source of the control sequence, which is determined by an ensemble of discrete signals of information carriers; nmessage length provided in bits Table 3 shows the values of the probability of imposing  Data analysis in Table 3 shows that in ICSs, which apply signal multiplexing technologies in orthogonal frequency division of channels, the value of / imp mes P for nonlinear CS is much less than in the case of using linear classes of signals.

DERIVED CRYPTOGRAPHIC SIGNAL SYSTEMS SYNTHESIS
Among the systems of phase-shifted signals, many are based on Walsh systems [1]. It is known that the auto-and cross-correlation functions of the Walsh sequences have large side peaks. To improve signals correlation properties, derived signal systems (DSS) are formed by multiplying Walsh sequences (source sequences) by a signal that has certain properties (producing a signal), in particular, have small side peaks of the autocorrelation function.
The authors formulated a hypothesis about the possibility of using nonlinear cryptographic sequences as generating ones, the theoretical foundations of which are given in [24].
The method of synthesizing derived signal systems based on the use of CS includes the following steps.
1. The selection of M cryptographic sequences of a fixed period N, with the minimum values of the maximum side lobes (Rmax.) ACPF.
2. A set of Walsh codes (matrix N•N) is formed, in which each row corresponds to a separate code.
3. Perform the multiplication of sequences (each of the lines of the Walsh code of the original sequences) on the cryptographic signal, forming N derived orthogonal signals.
4. Carry out a study of the correlation properties of the obtained derived orthogonal signals (in particular, ACPF, ACAF). To study the functions of cross-correlation, they form a matrix of dimension N•N. The number of such matrices: L•N. Table 4 shows cryptographic sequences (M = 14), selected from a set of sequences, by the criterion of the minimum values of the maximum side lobes ACPF (Rmax < 10). The results of the CCPF DSS study based on cryptographic sequences show that the number of pairs of signals for a period of sequences is 64 characters, for which the values Rmax do not exceed 17 (this is the so-called «dense packing» border, achieved in the class of the best, from the point of view of CCF, sequences with a three-level CCPF), 604 pairs (about 30% of the total number of possible combinations of pairs of signals). The number of signals pairs for which values Rmax does not exceed 20 -1577, which is 77% of the total number of signal pairs. With limit Rmax < 25 the maximum number of selected signal pairs is 1984 (96,8%). Such values Rmax have a place for sequences that are most prevalent in modern telecommunications systems M-sequence. Table 5 shows the results of studies of the correlation function statistical characteristics of various signal classes, including DSS, when used as generating cryptographic signals. As correlation function statistical characteristics were used: largest Analysis of the data given in Table 5, indicates that the values of the maximum lateral emissions of CS, as well as the statistical characteristics of this class of signals are not inferior to the corresponding characteristics of signals based on the use of Msequences. As follows from the data presented in the table, the statistical characteristics of the DSS are close to the corresponding characteristics for linear and nonlinear signal classes. The values of the maximum lateral peaks of the DSS cross-correlation functions are smaller than those of the linear Msequences widely used in modern ICS.
Due to the fact that CSs have ensemble properties that are improved compared to other classes of signals, ICS protection indicators against imposing (entering) spurious messages can be improved. At the same time, it should be noted that the use of CS ensures the noise immunity of receiving signals not lower than when applying the above signals based on the linear formation laws.
An assessment of security of ICS, protecting against false messages, is performed when manipulating (extending the range) CS. At the same time, we will assume that the system implements a dynamic mode of operation, which implies, among other things, a change in the correspondence of m message bits -2m complex signals. Change in compliance is made at set time intervals using a control sequence that meets the requirements of randomness. To provide the necessary noise immunity of signal reception, we will use CSs that have good correlation properties [1][2]17].
The probability of imposing a false message is determined by the ability of the reaction station to determine the law of conformity: m bits of the message -2m complex signals, or, in other words, determine the structure (law of formation) of the control sequence establishing the specified correspondence, and is determined from the ratio: where: 2-k-the source control sequence possible states number; nmessage length expressed in bits. Note that the number of possible states of the source control sequence (2 -k ) is determined by an ensemble of discrete sequences, by which the phase of the high-frequency carrier is manipulated to form a phase-shift keyed broadband discrete signal. Table  5 shows the values. Pimp./message for various systems of discrete signals obtained on the basis of Msequences, sequences with a three-level periodic cross-correlation function (CCPF), non-linear characteristic sequences (NLCS) and non-linear cryptographic sequences (NLCCS). The message dimension is set to n = 32. As the sequences period N were selected: 31, 63, 127, 1023.
It must be emphasized that in the calculations Pimp./message (for use in the system NLCCS), sequences were selected whose correlation characteristics are close to optimal boundary values ("dense packing") from CCPF point of view. Such boundary values are achieved in a class of sequences with a three-level cross-correlation function and constitute .max 1,5 side PN  . As it can be seen from the data table, the values Pimp./message for NLCCS are significantly smaller than in the case of using the most widely used in practice linear classes of signals (M-sequences and sequences with three-level CCPF). In the case when the system does not have strict requirements for the Pimp./message (imitation resistance), but it is necessary to provide increased requirements for signal reception noise immunity and to fulfill high requirements in terms of the structural secrecy of the complex signals used, the values of the maximum lateral peaks of the CCPF can be selected as the limiting values "dense packing". In this case, the volume of the system of signals satisfying this boundary will "be less, and accordingly Pimp./message will be higher, but at the same time, the noise immunity of receiving signals will be improved. Thus, when using systems of nonlinear signals, it becomes possible to vary (taking into account the requirements for ICS) indicators of the noise immunity of signal receptionthe system is protected from unauthorized data modification (imitation resistance).
The above estimates suggest that in ICS, in which as a method of information exchange, the dynamic mode of changing the correspondence is implemented, m bits of the message -2m complex signals and apply nonlinear cryptographic signals, provide high levels of system security against unauthorized data modification and the imposition of false information.
In essence, the presented system of providing imitability is [18] a cryptographic system, because it contains all the attributes of such a system: an algorithm for protecting against imposing a false message and hiding the semantic content of a message, which is based on the implementation of the dynamic mode of ICS functioning using cryptographic discrete signals as information carriers; algorithm for indentifying the authenticity of the information received; a key system that implements the functions of generating a control sequence for a change of correspondence: the message bit is a complex signal, as well as the generation of cryptographic discrete signals.

CONCLUSION
This paper presents the technologies for generating signals that are already used in communication and telecommunication systems, and also provides an analysis of promising technologies that may be used in various new systems, including wireless broadband access communication systems, in particular derived cryptographic signals.
Studies of the DSS properties formed on the basis of nonlinear cryptographic sequences show that more than 30% of the total number of combinations of such DSS pairs have side peaks of the correlation function equal to potentially achievable values, for the remaining pairs the maximum side peaks of CF are less than in widely used linear M sequences. In addition, the resulting DSSs have improved structural and ensemble properties compared with orthogonal signals. Currently, mathematical model and software have been developed that implement methods for synthesizing and studying the properties of nonlinear cryptographic signal systems, which are almost available for possible use as part of prototypes and elements of modern digital communication tools, and allow us: to generate nonlinear cryptographic signals for almost any period; determine the values of the minimum and maximum lateral emissions of various correlation functions; compare the obtained values with known, potentially achievable boundaries for the corresponding correlation functions; assign to the implementations of the synthesized sequences, as well as the parameters used for the synthesis of signals, unique identifiers that are necessary for optimal signal processing; calculate the statistical characteristics of the various correlation functions of the synthesized signals; carry out studies of the ensemble characteristics of the synthesized signals.