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Special Integral Functions Used in Wireless C mmunications Theory 9168_9789814603218_tp.indd 1 21/4/14 11:43 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Special Integral Functions Used in Wireless C mmunications Theory Nikolay V. Savischenko Saint Petersburg State University of Telecommunications, Russia World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9168_9789814603218_tp.indd 2 21/4/14 11:43 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Savischenko, Nikolay V. Special integral functions used in wireless communications theory / by Nikolay V. Savischenko (Saint Petersburg State University of Telecommunications, Russia). pages cm Includes bibliographical references and index. ISBN 978-9814603218 (hardback : alk. paper) 1. Wireless communication systems--Mathematical models. 2. Calculus, Integral. 3. Signal processing--Mathematics. I. Title. TK5102.83.S28 2014 621.38401'5154--dc23 2014007316 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore April21,2014 11:7 BC:9168–SpecialIntegralFunctions Special˙Function˙eng pagev Preface Teach not thoughts but to think Immanuel Kant1 Oneofthescientificfieldsdevelopingintheframeworkofthemathematical (statistic)communicationtheoryisthe potentialnoiseimmunity theoryon the basis of the fundamental works by V. A. Kotel’nikov2 and D. Middle- ton3. A. J. F. Siegert, P. M. Woodward and I. L. Davies made a consider- able contribution to the foundation and the development of the potential noise immunity theory. Investigations of optimal reception methods and potential noise immunity in the channels with fluctuation noise, indefinite phase and Rician signal fadings were conducted in the work of L. M. Fink (see Ref. [36]). Thesimilartasksrelatingtofour-parametricchannelsmodelweresolved by D. D. Klovskiy who also obtained a number of important results on stochastic models of space-time channels and conducted the analysis of 1“NichtGedanken, sonderndenkenLernen.”(I. Kant). 2Kotel’nikovV.A.(1908–2005)isaremarkablesovietandRussianscientistinthefield of radio engineering, radio communication and radio astronomy. The IEEE president Prof.BruceEisensteinspeaksabouthisroleinthedevelopmentofthetelecommunication theory: “The outstanding hero of the present. His merits are recognized all over the world. Infrontof us isthe giant of radioengineering thought, whohas madethe most significantcontributiontomediacommunicationdevelopment. OvertheyearstheWest haditsShannon; andthe Easthadits Kotel’nikov”. Kotel’nikov’sthesis foraDoctor’s degree“TheTheoryofOptimumNoiseImmunity”wasdefendedinJanuary1947butit wasopenlypublishedasabookonlyin1956andbecameknowntotheworldsciencein 1959whenitwaspublishedinEnglish(seeRefs.[22,63]). 3Middleton David (1920–2008) is an outstanding American scientist in the field of communications theory. Without regard to V. A. Kotel’nikov D. Middleton developed a theory of signals optimal reception on the base of mathematical theory of statistical decisions(seeRefs.[22,74,75]). v April15,2014 12:17 BC:9168–SpecialIntegralFunctions Special˙Function˙eng pagevi vi Special Integral Functions Used inWireless Communications Theory communicationsystemsnoiseimmunitytakingintoaccounttime,frequency and space distributed and concentrated additive noise (see Ref. [57]). In 1960–1980investigationsofpotentialandrealradiocommunicationsystems noiseimmunity onexposureto fluctuation, concentratedandimpulse noise were conducted (L. M. Fink, H. L. Van Trees, J. M. Wozencraft, I. Ja- cobs, D. D. Klovskiy,A. D. Viterbi etc.). The scientific results receivedfor the period covering were collected in a unique reference book on the noise immunity evaluation of the digital transmission systems (see Ref. [61]). In recent decade a number of works on the noise immunity evaluation of signal construction coherent reception in the channel with additive white Gaussian noise (AWGN) and generalized fadings was published in IEEE Transactions of Communication, IEEE Transactions Information Theory (Mohamed-Slim Alouini, Annamalai Annamalai, Norman C. Beaulieu, Vi- jay K. Bhargava, Andrea J. Coldsmith, George K. Karagiannidis, Marvin K. Simon, Chintha Tellambura etc.). It is important to note a cycle of publications of M.-S. Alouini and M. K. Simon as the English investigations on radio communication systems noise immunity with signals (see Refs. [109–111, 113–116] etc.). The noise immunity analysis during the digital information transmission through the channels with constant parameters and additive white Gaussiannoise(AWGN), withgeneralizedfadings(frequency-nonselective) during the five-parametric distribution coefficient of the channel transmission and also in the communication channel with diversity reception and generalized fadings is analyzed in this monograph. In total the famous results of the potential noise immunity theory applied to the multipositional signals which are completely based on the geometric models research results and metric characteristics of the modern signal constructions were generalized. Themonographgivesthe methodstoobtainexactformulasof symbol and bit error probabilities (P — SEP or SER and P — BEP or e b BER) for arbitrary signal M positioning (most often coinciding with 2K), for arbitrary values of signal-to-noise ratio (among them small signal-to- noise ratios) and in some cases for the arbitrary signal N dimensionality. Practically the whole conception is based on such theoretical fundamental principlesasShannontheory,Kotel’nikovtheoryanddiscretegeometrybe- causeitisthegeometricalinterpretationofsignalanddecisionschemethat was considerably productive in the works of Kotel’nikov and Shannon4. 4Shannon ClaudeElwood(1916–2001) isanoutstanding Americanscientist,acreator ofinformationmathematical theory. April15,2014 12:17 BC:9168–SpecialIntegralFunctions Special˙Function˙eng pagevii Preface vii Thetheoreticalfundamentalsoferrorprobabilitiescalculationformulti- positionalsignalinthechannelwithconstantparametersandadditivewhite Gaussian noise in the case of coherent reception are shown in the works of J.M.WozencraftandI.Jacobs(seeRef.[141]),A.D.Viterbi(seeRef.[135]) etc. It is shown that in this case of Bayesian optimality criteria choice the areas of optimal decision acceptance in a common case will have a form of regular polyhedrons and the corresponding noise immunity analysis problem is to estimate the multidimensional integral of probability Gaussian density in these areas. The foundations of mathematical decision of this problemfortwo-dimensionalprobabilityGaussiandensitywereobtainedin the works of D. B. Owen (see Refs. [85–87]), and also L. N. Bol’shev and N. V. Smirnov(see Ref. [118]). The procedure ofintegralcalculationof ar- bitrarytwo-dimensionalnormaldistributiondensityoverarbitrarypolygon on the base of one-fold multiple5 integral— OwenT-functionusing shown in Ref. [118]. But the mathematical inaccuracies and mistakes in these works could cause false conclusions. The critical opinion on the earlier obtained results and their correction shown in this monographwas required. The second main problem considered in the monograph is devoted to noise immunity analysis problem in the channel with generalized fadings and white noise in the case of coherent reception. Numerous publications on this theme emphasizing its actuality give results based on either direct integral calculation and the result representationas infinite rows or on the Laplacetransformationusingoffadingdistributiondensity(Rayleigh,Rice and Nakagami) and the result record as one-fold multiple integral (MGF — Moment Generating Function). The absence of the error probability formulas for arbitrary multipositional signals in the channel with constant parameters and white noise in the case of coherent reception holds the second approach application often used at practice. The conceptions given in the monograph are based on generalization andsystematizationofwell-knownandnewscientific resultsrepresentedin Refs. [104,105] and also obtained by the author in recent time and which were not published in scientific publications by various reasons. In view of themonographsizelimitationtheauthordidnotincludetheresultsdevoted to analysisoftelecommunicationsystemnoise immunity functioning under influenceofmutualandmaliciousinterferencesimilartoasignal(structure 5Anexampleofintegralcalculationoftwo-dimensionalnormaldistributiondensityover triangleonthebaseofNicholson’stwo-foldmultipleintegralusingwasgiveninRef.[1] butthecorrespondingresultswerenotgeneralized. April15,2014 12:17 BC:9168–SpecialIntegralFunctions Special˙Function˙eng pageviii viii Special Integral Functions Used in Wireless Communications Theory targetingandimitatingnoise)(seeRef.[104])andalso6 intheconditionsof error account in the carrier signal phase (see Ref. [104]). The main part of themonographreferstothespecial7integralfunctionsandtheirapplication forthe errorprobabilitycalculationinthe caseofcoherentreceptioninthe channel with generalized fadings and additive white Gaussian noise. The author aimed at some main problems: (1) To formalize the calculation conducting methods of symbol P and bit e P error probabilities in the case of coherent reception on the base of b criteriaminP inthe channelwithadditivewhite Gaussiannoiseusing e with Owen T-function properties application. (2) Toperformsystematicdecisionofthisproblemforthechannelwithgen- eralizedfadingsandadditivewhiteGaussiannoiseusingexactrelations for error probabilities obtained as a result of the first item realization. In this case Rayleigh, Rice, Hoyt, Rice–Nakagami and Beckmann dis- tributions are considered to be fading distribution laws. The problem solvingforthesedistributionlawscauseditssolvingforfour-parametric andfive-parametricGaussianmodelofacommunicationschannel. The mathematicalsolvingoftheseproblemsledtothenecessityofnewspe- cial integral functions introduction: H-function and S-function. (3) To obtain a new solving of the error probability calculation problem for the channel with diversity reception based on generalization of H-function and S-function and application of the approaches used in the previous item. (4) To study properties of new special integral functions which small part is given in the monograph. A part of the monograph material namely Chap. 1 is devoted to the propertiesofGaussianQ-function. Thisspecialfunctionalreadyknownfor 6A part of the material devoted to noise immunity analysis in the conditions of error account inthecarriersignalphaseisgiveninAppendixB. 7Functions appearing during probability theory and mathematical statistics problems solvingaswellasproblemsofmathematicalphysicsareconventionallyspecialareas. In- tegralfunctionsdenotespecialfunctionswhichareexpressedbymeansofintegralsfrom simpleelementaryfunctionsandalsofunctionswhichareobtainedfromthembymeans of a finite number of computational and differential operations (see Ref. [92]). Proba- bility integral and functions connected with it are referredto special integral functions used in communications theory. Elementary functions are functions of one argument whichvalues arederivedbymeans offinitenumber of computational operations. They representfourarithmeticoperations,raisingtothewholeorfractionalpower,takingup trigonometric functions and functions inversed to them, taking logarithms and antilog- arithmsfromanargument,adependent variableandconstant numbers(seeRef.[92]). April22,2014 9:42 BC:9168–SpecialIntegralFunctions Special˙Function˙eng pageix Preface ix abouttwohundredyearsattractsscientificinterestasbefore, andthereare alotofpublicationsdevotedtoitsnewpropertiesnowadays. InChap.1an attempt was made to collect the material devotedto Gaussian Q-function. The less known Owen function (T-function) given in Refs. [85–87, 101] required more attention because namely its using allows to formalize the problemofintegralcalculationofbivariateGaussiandistributiondensityin the arealimitedby apolygonora brokenline. The solvingofthis problem with T-functionusing forms the basis of symbol and bit errorprobabilities calculation in the conditions of coherent reception that is when the areas ofdecisionreceptionintwo-dimensionalspacearerepresentedas polygons. InRef. [118]methods ofintegralcalculationofbivarite Gaussiandensityof probability distribution over an arbitrary polygon had mathematical inaccuracies which were solved in Chap. 2. The symbol and bit error probabilities for two-dimensional signals of phase shift keying (PSK), quadratureamplitude modulation (QAM) (clas- sical and Hierarchical), QAM-“cross” and amplitude phase shift keying (APSK), used in different modern telecommunication standards (DVB-S2, V.29 and etc.) calculation examples are collected in Chap. 3. Chapters 4 and 5 are devoted to the problem considerationof the error probability calculation in the channel with generalized fadings (frequency nonselective or flat fading) and additive Gaussian white noise. As a con- sequence of Introduction and Chap. 2 we can see that symbol (bit) error probabilities can be represented in the form of a linear combination of Owen T-functions (in particular, using properties of T-function the probability can be represented as a linear combination of Owen T-functions, Gaussian Q-functions and product of two Gaussian Q-functions), a formal solution is reduced to a problem of the mathematical expectation defining of Owen T-function with its argument depending on a random coefficient ofμcommunicationchanneltransmissionwhichisdistributedaccordingto four-parametriclawω(μ)inacommoncase. Itisshownthatmathematical expectationcanbedefinedonthebasisofanewspecialintegralH-function intheconditionsofRice–Nakagamidistribution. Theprovedfunctionω(μ) expansion in a row according to Rice–Nakagami distribution enabled to solve the problem of T-function averaging for four(five)-parametric distribution law andin this case a new special integralS-function is introduced. Nikolay V. Savischenko Saint Petersburg, Russia 2014