Special Integral Functions Used in Wireless Communications Theory

The following sections are included:

• Preface
• Acknowledgements
• Contents

Throughout the whole development of the mathematical theory of communication one of key questions was a question of a potential noise immunity of constructions receiving signals in various types of communication channels. In the first stage the subjects of study were either binary signals (the amplitude shift keying (ASK), phase shift keying (PSK) and frequency shift keying (FSK)) which were commonly used in communication equipment of that time or multidimensional signals with low specific speed — simplex, cubic, etc. If in the first case (for classical channel model with additive white Gaussian noise (AWGN) in case of coherent, quasi-coherent and non-coherent reception) the exact relations were obtained for symbol (bit) error probability, then for multidimensional signals this problem did not always have the exact solution and in many cases scientists used asymptotic values obtained for limiting variants (for example, at great values of the signal-to-noise ratio ≫ 1 or with dimension of space N ≫ 1). Intermediate position was held by multipositional signals (M > 2) for which later with the beginning of their practical usage many scientists in 1960 and 1970th years began to try finding the exact formulas for error probabilities. At the beginning among the various variants of signal constructions multipositional signals of phase shift keying were applied, which belong to the class of signals with superficial-spherical packing of signal points. However, the tendency of increasing the specific speed, when applied the PSK signals, led to the considerable decrease in error probabilities, especially well noticeable at M > 32. The development of theory and the communication practice led to the possibility of usage of signals, more complex in application and accordingly studying, which belong to the volume-spherical signal points packing of signal points. Among the great number of two-dimensional multipositional signal constructions studied recently the greatest practical and theoretical interest is caused by signals of quadrature amplitude modulation (QAM), which possess the worst energy characteristics than other signals (for example, the signals constructed on the basis of a hexagonal lattice, in the limit, at a great number of signals is more effective on 0.14 dB) in certain cases, but in this case they are easier realized as a scheme of a receiver and a transmitter…

The following sections are included:

• Basic Definitions and Properties of Gaussian Q-function
• Approximation of Gaussian Q-function
• The Upper and Lower Bounds of Gaussian Q-function
• Examples of the Problems Solution with Gaussian Q-function

The following sections are included:

• Bivariate Normal Density of Distribution and its Basic Properties
• Integral Representations of Owen T-function and its Basic Algebraic Properties
• The Upper and the Lower Bounds of Owen T-function
• Polynomial and Power Approximations of Owen T-function
• The Calculation of the Hit Probability of Bivariate Gaussian Random Variable into the Area Bounded with a Polygon (Broken Line)
  • The characteristic of the existing technique of calculating hit probability of bivariate Gaussian random variable into the area bounded with a polygon (broken line)
  • The technique of calculating the hit probability of bivariate Gaussian random variable into the area bounded with the polygon (broken line) in the form of algebraic sum of Owen T-function
  • The classification rule of the position of the origin coordinates in relation to the boundaries of the arbitrary polygon (a closed broken line)
• Examples of the Hit Probability Calculation of a Bivariate Gaussian Random Variable of Various Types of Areas
• Basic Relations for the Integral Calculation from the Density of a Trivariate Normal Distribution

The following sections are included:

• The Mathematical Model of a Communication Channel with Determined Parameters and Additive White Gaussian Noise
• Exact Formulas or Probability of Symbol and Bit Errors of M-PSK
  • Classical signal construction M-PSK
  • Hierarchical signal construction of M-PSK
• Exact Formulas of Symbol and Bit Error Probability of M-QAM
  • Classical signal construction M-QAM
  • Hierarchical signal construction M-QAM
• Exact Formulas of Symbol Error Probability M-QAM-“cross”
• Exact Formulas of Symbol Error Probability of M-HEX
• Exact Formulas of Symbol Error Probability of M-APSK
• Exact Relations for Probabilities of Symbol Errors of Receiving the Signal Constructions Applied in Telecommunication Standards

The following sections are included:

• The Mathematical Model of a Communication Channel with the Generalized Fading
• The Special Integral ℋ-function for the Decision of a Problem of the Analysis of a Noise Immunity in the Channel with the Rayleigh, Rice, Rice–Nakagami and Beckmann Generalized Fadings
• The Basic Relations and Algebraic Properties of ℋ-function
• Application of ℋ-function for a Noise Immunity Estimation in the Channel with the Rayleigh, Rice, Rice–Nakagami and Beckmann Generalized Fadings
• The Examples of Calculation of Errors Probability in the Channel with the Rayleigh, Rice and Rice–Nakagami Generalized Fadings

The following sections are included:

• Special Integral 𝒮-function for Calculation of Error Probabilities in the Channel with the Generalized Four-parametric Fadings
• The Basic Relations and Algebraic Properties of 𝒮-function
• The Five-parametric Distribution Law for a Channel with Generalized Fadings
• Application of ℋ-functions for Noise Immunity Estimation under Noncoherent Reception

The following sections are included:

• Diversity Methods in Radio Communication Systems
• Basic Abbreviations, Notations and Assumptions
• Maximal Ratio Combining
  • Computational procedure of error probability in a channel with the diversity reception
  • Four-parametric distribution law for a channel with generalized fadings and diversity reception
  • Special integral ℋ^(L)-functions and 𝒮^(L)-functions for the calculation of error probability in a channel with generalized four-parametric fadings and diversity reception
  • Noise immunity of the coherent diversity reception of multipositional signal constructions at the correlated Rayleigh and Nakagami fadings
• Selection Diversity Combining
  • Noise immunity of the coherent diversity reception of multipositional signal constructions at the correlated Rayleigh, Nakagami and Rice fadings

The material given in this book doesn't, of course, cover all the problems of the theory of potential noise immunity. Nevertheless it is possible to assume that the obtained outcomes will allow to compare with each other various two-dimensional multipositional signal constructions by criterion of minimum of symbol or bit error probabilities. The given formulas of the errors probabilities dependence on the signal-to-noise ratio and the graphs constructed on their basis allow to solve a comparison problem in a formalized way on the basis of exact given within limits suppositions of the model of a communication channel and decision-making rule relations. Many described in the monograph signal constructions are presented in telecommunication standards (DVB, IEEE 802.11, IEEE 802.16 and etc.) and used in equipment…

The following sections are included:

• Appendix A Rational Approximation of Function erfc (x)
• Appendix B The Noise Immunity Analysis at Coherent Reception in a Communication Channel with AWGN and Nonideal Carrier Synchronization
• Appendix C The Noise Immunity Analysis at Coherent Reception in Conditions Joint Influence of Interference Similar to Signal and AWGN
• Appendix D The Graphs of Probabilities Errors for Signals M-QAM at Generalized Gaussian Four-parametric Fadings
• Appendix E Gray Code
• Appendix F Abbreviations of Signal Constructions

The following sections are included:

• Bibliography
• Glossary
• Index