Annotation of the results obtained in 2023
Within the framework of the assumed research, a novel probabilistic model of a wireless multipath communication channel with fading has been proposed (α–Beaulieu–Xie shadowed (α-BXS)), which is a modification of the Beaulieu–Xie shadowed model and assumes a "cluster" representation of electromagnetic wave propagation. The proposed model belongs to the class of generalized models and takes into account the following cases: – signal propagation in the absence of line–of–sight (No–Line–of–Sight (NLoS)); – signal propagation in the presence of a constant line–of–sight (LoS) component; – signal propagation when the line–of–sight channel experiences random shadowing (Shadowed–Line–of–Sight (SLoS)); – the presence of multiple line–of–sight components; – effects of possible nonlinear distortion (both physical and effective) of the signal envelope in the propagation channel. For the proposed α–Beaulieu–Xie shadowed model, a basic probabilistic description (the probability density function of the instantaneous signal–to–noise ratio, the cumulative distribution function of the instantaneous signal–to–noise ratio) is obtained, and arbitrary–order raw moments, as well as their asymptotic expressions at large values of the average signal–to–noise ratio are derived. For the proposed α-BXS model, the problem of wireless communication system operation quality and reliability description is studied, and the following results are obtained: – an analytical expression for the Outage Probability (OP) and its asymptotic approximations (upper and lower bounds of OP) are obtained; – analytical representation of the Amount of Fading (AoF) and Channel Quality Estimation Index (CQEI) within the α-BXS model is obtained using the derived expressions for the moment functions; – the analytical expression for the Amount of Fading of the α-BXS model is used to analyze the weak hyper–Rayleigh regime of such a channel; – a comprehensive analysis of the reliability of the communication session (evaluated by the value of the OP), depending both on the channel parameters (the severity of total shadowing, the degree of LoS components’ shadowing, the fraction of energy attributable to LoS/NLoS components, and the nonlinearity coefficient α) and on the values of AoF and CQEI; – for the case of quadrature amplitude modulation, an analytical expression for the average bit error probability (ABEP) in the form of a Meyer G–function expansion, as well as an approximation of the analytical solution and its asymptotic expression, which is used to find the diversity gain and the coding gain for the α-BXS channel; – an analytical expression for the ergodic capacity, which takes into account the optimal rate adaptation strategy with constant transmit power (ORA), in the form of a Meyer G–function expansion, as well as its asymptotic behavior under infinitely increasing signal–to–noise ratio is obtained. For the proposed α-BXS model, the problem of the physical layer security is studied, and the following results are obtained: – an analysis of the secrecy outage probability of a wireless communication system operating under multipath and fading conditions is performed, which demonstrated that an increase in the signal’s envelope transformation nonlinearity degree (α) leads to an exponential increase in the secrecy of the communication session; – an analysis of the dependence between the decrease of the communication link secrecy, the relative distance between the agents (the transmitter, the eavesdropper and the legitimate receiver), and the path-loss exponent in the propagation channel, which demonstrated that the reduction of the signal’s attenuation during propagation leads to a decrease of the communication link secrecy; – the existence of a strictly linear dependence (on a logarithmic scale) between the power of LoS and NLoS components, providing a constant level of the secrecy outage probability of a wireless communication session was demonstrated. A novel metric reflecting the potential degree of improvement of link quality (in terms of ABEP) due to improvements in propagation conditions: Error Improvement Margins (EIM) was proposed. Modifications of the assumed metric are considered for the cases of improvement due to reduction of LoS component shadowing (EIML) and due to reduction of shadowing severity (EIMS). Analytical expressions for the proposed modifications of the EIM metric (EIML and EIMS) are derived for the case of fading obeying the fluctuating double Rayleigh channel model with the presence of a shadowed LoS component (fdRLoS). For a wireless communication system utilizing quadrature amplitude modulation, under multipath channel conditions with fading described by the fdRLoS model, the dependence of the finite signal-to-noise ratio diversity gain on the channel parameters is analyzed. It is found that it is possible to achieve a spatial diversity gain greater than the asymptotic one at finite values of the signal-to-noise ratio under conditions of weak shadowing of the strong LoS components. For the fading propagation model corresponding the fluctuating Rayleigh channel with the presence of shadowed LoS and double scattering (fSOSF), an analytical expression for the average bit error probability of the signal with M-level phase modulation in the form of the generalized bi- and trivariate hypergeometric Fox H-functions is obtained for arbitrary values of the model parameters. Both a simplified closed-form expression for the case of an integer value of the shadowing parameter of the LoS component and an asymptotic representation at infinitely increasing signal-to-noise ratio are obtained. For a MIMO system operating under multipath channel conditions with fading described by the complex Nakagami-m model, the following results are derived: – analytical expressions for the statistical higher orders of spectral efficiency; – analytical expressions approximating the derivative function of the moments of instantaneous capacity for each of the spatial substreams, as well as the total capacity of the communication channel in the case of Zero–Forcing processing at the receiver. In the context of the MIMO capacity analysis, the presence of an extremum (maximum) of the capacity reliability depending on the AoF value for small-dimensional systems was identified. A joint analysis of channel capacity and capacity reliability of a MIMO system was proposed and performed for fading complex Nakagami-m fading channel model. Within the framework of the assumed research, the problem of wireless communication systems’ performance description in case of hyper-Rayleigh fading conditions was analyzed, and the following results were obtained: – for NLoS channels (η–μ model) and SLoS (κ–μ model with shadowing), analytical expressions linking the values of AoF, for which both models are in the weak hyper-Rayleigh regime have been obtained, and the behavior of the energy detection probability in such regime has been described; – for the model of a fluctuating double Rayleigh channel with the presence of a shadowed LoS component (fdRLoS), an analytical study of the communication system functioning in the presence of weak/strong/full hyper-Rayleigh is carried out, analytical expressions for the AoF and outage probability are obtained, as well as the bounds of the model parameters values at which it does not demonstrate the effects of hyper-Rayleigh regime are derived.

 

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Annotation of the results obtained in 2022
Within the framework of the assumed research, an analytical review was carried out of the most common special functions used in wireless communication theory for describing the probabilistic characteristics of the performance quality and reliability. The results of the review helped to classify the mathematical approaches (and the corresponding classes of problems) that are used for wireless communication system performance analysis: 1) an approach based on the generalized/confluent multivariate hypergeometric Horn-type functions/ Kampé de Fériet/Lauricella/Appell (problems of signal detection, bit/symbol error probability calculation, moment-generating function derivation); 2) an approach based on the multidimensional Fox H-function (problems of bit/symbol error probability calculation, the outage probability derivation, deriving the probability density/cumulative distribution function); 3) an approach based on the multidimensional Meijer G-function (problems of bit/symbol error probability calculation, derivation of the information-theoretic characteristics: capacity, spectral efficiency, amount of information, higher-order capacity statistics); 4) an approach based on the generalized/confluent univariate hypergeometric Gaussian /Kummer/Tricomi/Meijer functions, as well as "classical" special functions (e.g., Bessel, incomplete gamma-functions, Marcum and Q-Gauss Q-function), which arise when the asymptotic description of the problems noted above is sought. An analytical review was carried out to identify the most widely used modern and promising wireless communication multipath channels models. The results of the review helped to classify models in accordance with the mathematical framework being used: 1) "classical" special functions (e.g., Beaulieu-Xie with shadowing and fluctuating double-Rayleigh with line-of-sight (fdRLoS), 2) generalized/degenerate multidimensional hypergeometric functions (e.g., Fluctuating Beckmann model). Based on the review of channel models and their mathematical representation, an approach was proposed to unify the description of No-Line-of-Sight (NLoS) and Shadowed-Line-of-Sight (SLoS) wireless multipath models. It assumes that the moment-generating function of the instantaneous signal-to-noise ratio admits the Factorized Power-Type (FPT) representation. The connection of the proposed generalized FPT model with a wide range of existing both classical and generalized ones is demonstrated for the following cases: Rayleigh, Nakagami-m, Hoyt, one-sided Gaussian, η-μ, k-μ shadowed, gamma, mixture-gamma, etc. For a wireless communication channel admitting the generalized FPT model, using the contour-integral representation of the Marcum Q-function, analytical expressions in the case of the energy-based spectrum sensing are obtained for the probability of detection, the receiver operating characteristic (ROC) and the area under the receiver operating characteristic curve (AUC) in terms of a confluent Lauricella hypergeometric function ΦD of (N+1) variable (where N – the order of the FPT model) for ROC and the generalized Lauricella hypergeometric function FD of (N+1) variable for AUC. The results were supplied with the simplified computationally efficient approximating expressions, representing ROC/AUC in the form of a confluent Kummer hypergeometric function 1F1/ Gauss hypergeometric function 2F1. The obtained general results are applied to the problem of ROC and AUC analytical derivation for the fading conditions conforming to the recently proposed models: Fluctuating Beckmann and Beaulieu-Xie shadowed. For the Beaulieu-Xie shadowed channel model, in the problem of physical layer security analysis, an analytical expression as well as its computationally efficient approximation are obtained for the secrecy outage probability in terms of the hypergeometric Gaussian functions 2F1. A connection between the generalized Marcum Q-function and the Gauss Q-function was establishing, and the contour-integral representation of the latter was obtained. The derived result makes it possible to expand the possibilities for analytical calculation of wireless signal transmission error rates. For a wireless communication system using digital signal modulation types (MQAM and MPSK), jointly applying an approach based on the moment-generating function and Gauss Q-function contour integral representation, an analytical closed-form expression and its computationally efficient approximation for the averaged bit error probability in case of Nakagami-m fading are obtained in terms of the hypergeometric Appel F1 functions. The same approach applied to the case of the fluctuating double-Rayleigh with line-of-sight (fdRLoS) channel model yielded the result in terms of the extended generalized bivariate Meijer G-functions. For the obtained analytical averaged bit error probability expression of fdRLoS channel model, asymptotic performance was derived for the following cases: 1) infinitely large signal-to-noise ratio, 2) strong dominant components of a multipath signal (infinitely large K-factor), 3) weak dominant components (infinitely small K-factor), 4) weak shading of the line-of-sight channel (infinitely large value of the Nakagami m-parameter), 5) strong shading of the line-of-sight channel (Nakagami m-parameter equal to 0.5). For the fdRLoS channel model, the existence of an irreducible level of the averaged bit error probability was established and demonstrated for: 1) an infinite increasing K-factor, 2) both with weak (infinitely large Nakagami m-parameter) and strong (Nakagami parameter equal to 0.5) shadowing. This effect is observed regardless of the signal-to-noise ratio and the modulation order. Using the contour integral representation for extended generalized bivariate Meijer G-functions, an analytical expression in closed form and asymptotic expressions (under conditions of an infinitely high signal-to-noise ratio) for the ergodic capacity of the fdRLoS channel are obtained when using: 1) optimal rate adaptation strategy with constant transmit power (ORA), 2) optimal simultaneous power and rate adaptation strategy (OPRA). The obtained analytical results are valid for both integer and non-integer values of the Nakagami m-parameter, which expands the existing description for special cases of the fdRLoS model (largely based on the results obtained under the assumption of the integer value of the m-parameter). The connection of analytical descriptions of the ergodic capacity of the fdRLoS channel between the adaptation strategies under consideration (ORA and OPRA) is established.

 

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