Random Signal Analysis

323 Pages·2018·3.039 MB·English

by  Congfeng Liu, Jie Yang

JieYang,CongfengLiu
RandomSignalAnalysis
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
Information and
Computer Engineering
|
Volume 2
Alreadypublished intheseries:
Volume1
BeijiaNing,AnalogElectronicCircuit, 2018
ISBN978-3-11-059540-6, e-ISBN978-3-11-059386-0,
e-ISBN(EPUB)978-3-11-059319-8
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
Jie Yang, Congfeng Liu
Random Signal
Analysis
|
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
Authors
Prof.JieYang
Associateprofessor,SchoolofCommunicationandInformation
Ai’anUniversityofPost&Telecommunication
[email protected]
Prof.CongfengLiu
Associateprofessor,ResearchInstituteofElectronicCountermeasure
XidianUniversity
cfl[email protected]
ISBN978-3-11-059536-9
e-ISBN(PDF)978-3-11-059380-8
e-ISBN(EPUB)978-3-11-059297-9
ISSN2570-1614
LibraryofCongressCataloging-in-PublicationData
Names:Liu,Congfeng,author.
Title:Randomsignalanalysis/CongfengLiu.
Description:Berlin;Boston:DeGruyter,[2018]|Series:Informationandcomputerengineering|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018026142(print)|LCCN2018029153(ebook)|ISBN9783110593808(electronic
PortableDocumentFormatpdf)|ISBN9783110595369(alk.paper)|ISBN9783110593808(e-book
pdf)|ISBN9783110592979(e-bookepub)
Subjects:LCSH:Signalprocessing.
Classification:LCCTK5102.9(ebook)|LCCTK5102.9.L5652018(print)|DDC621.382/2–dc23
LCrecordavailableathttps://lccn.loc.gov/2018026142
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2018WalterdeGruyterGmbH,Berlin/Boston
Coverimage:Prill/iStock/GettyImagesPlus
Typesetting:le-texpublishingservicesGmbH,Leipzig
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
Preface
Random signals (stochastic signals) arealsoknown asrandomprocesses (stochas-
ticprocesses).Itisaquantitativedescriptionofthedynamicrelationshipofaseries
ofrandomevents.Randomresearchandotherbranchesofmathematicssuchaspo-
tentialtheory,differentialequations,themechanicsandtheoryofcomplexfunctions,
andsoon,arecloselylinkedinnaturalscience,engineeringscienceandsocialscience
researchineveryfieldofrandomphenomenaisanimportanttool.Randomsignalre-
searchhasbeenwidelyusedinareassuchasweatherforecasting,statisticalphysics,
astrophysics,managementdecision-making,economicmathematics,safetyscience,
populationtheory,reliabilityandmanyfieldssuchascomputerscienceoftenuseran-
domprocesstheorytoestablishmathematicalmodels.
Inthestudyofrandomprocesses,peopleaccidentlycametodescribetheinher-
entlawofnecessityandtodescribetheselawsinprobabilityform,realizingthatthe
inevitableisthecharmofthisdiscipline.
Thetheoreticalbasisofthewholedisciplineofstochasticprocesseswaslaidby
KolmogorovandDub.Thisdisciplinefirstoriginatedfromthestudyofphysics,such
asbyGibbs,Boltzmann,Poincareandothersstudyingstatisticalmechanics,andlater
Einstein,Wiener,LevyandotherswiththepioneeringworkoftheBrownianmove-
ment.Beforeandafter1907,Markovstudiedaseriesofrandomvariableswithspecific
dependencies,whichwerelatercalledMarkovchains.In1923,Wienergavethemathe-
maticaldefinitionofBrown’smovement,andthisprocessisstillanimportantresearch
topictoday.
Thegeneraltheoryofstochasticprocessesisgenerallyconsideredtohavebegun
inthe1930s.In1931,KolmogorovpublishedtheAnalyticalMethodofProbabilityThe-
ory. In 1934,Khintchine published “Thetheory of smooth process,” whichlaidthe
theoreticalbasisoftheMarkovprocessandthestationaryprocess.In1953,Dubpub-
lishedthefamous“randomprocesstheory,”systematicallyandstrictlydescribingthe
basictheoryofrandomprocesses.Atthispoint,thestochasticprocessdevelopedinto
asystematicscientifictheory.
Inourdailylives,becauseofthepresenceofnoiseandinterference,thesignalswe
receiveisnolongeraclearsignal,butarandomprocess;usuallywecallthisarandom
signal.Arandomsignalisakindofsignalthatisprevalentintheobjectiveworld.It
isvery importantfor college students intheinformationtechnology field to havea
deepunderstandingofthestatisticalcharacteristicsandtomasterthecorresponding
processingandanalysismethods.Therefore,randomsignalanalysisisanimportant
basiccourseinthefieldofelectronicinformation.Throughthestudyofthecourse,
studentsaretaughttounderstandthebasicconceptsofrandomsignals,tomasterthe
basictheoryofrandomsignals,statisticalcharacteristicsandanalyticalmethods,to
learn“statisticalsignalprocessing”or“signaldetectionandvaluation,”withother
follow-upcoursesandfuturedevelopmentslayingasolidfoundation.
https://doi.org/10.1515/9783110593808-201
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
VI | Preface
ThebookwaswrittenonthebasisofthetextbookRandomSignalAnalysiscom-
piledbyProfessorZhangQianwufromXidianUniversity,whichabsorbedtheexpe-
riencesofsimilarteachingmaterialsinbrothercollegesanduniversities,andwhich
wasfinalizedafteranumberofdiscussionsintheprojectgroup.Thetextbookcharac-
teristicscanbesummarizedas:
(1)Focusontheconstructionofthewholeknowledgesysteminthefieldofsignal
processing.
Fromthepointofviewoftheknowledgesystem,themathematicalbasisofrandom
signalanalysisis“highermathematics,”“probabilitytheory,”and“linearalgebra,”
andaprofessionalbackgroundfrom“signalsandsystems,”thefollowingcoursesare
“statisticalsignalprocessing,”or“signaldetectionandestimation.”Therefore,itcon-
tinuestostrengthen students’foundationofmathematicalanalysisandtheknown
basicconceptofsignalanalysis,basicprinciplesandbasicanalysisandprocessing
methods,andatthesametimehelpsstudentstounderstandtheapplicationofran-
domsignalanalysismethodstosignaldetectionandparameterestimationwithnoise
inthebackground.Thetextbookemphasizestheknowledgesystemandthestructure
ofsignalprocessinginitsentirety,soastoavoidstudentslearningandunderstanding
randomsignalprocessinginisolation.
(2)Continuousrandomsignalanddiscreterandomsequenceanalysis.
Traditionalrandomsignalanalysismaterialsmostly focusonthedescription,char-
acterizationandanalysisofcontinuousstochasticprocess,oftenignoringtheintro-
ductionofdiscreterandomsequences,sothatthestudentsdoingthiscoursecanonly
carryouttheoreticalanalysisandderivationandcannotusecomputersforsimulation
andemulation.However,makingfulluseofcomputerstoprocessandanalyzerandom
signals,ontheonehand,isbeneficialforstudentstoobtainanintuitiveunderstand-
ing, and, on the other hand is helpful for students to apply their knowledge, truly
combiningtheoreticalresearchandpracticalapplications.Therefore,inthecourseof
compilingthetextbook,theanalysisofthediscreterandomsequencewasalsotaken
intoconsiderationindetailwhilethecontinuousrandomprocessisanalyzed.
(3)Thecombinationoftheoreticalanalysisandexperimentalpractice.
Randomsignalanalysisisapracticalcourse,andmostcurrenttextbooksonlyfocus
on theoretical teaching instead of experimental practice. This textbook will design
thecorrespondingexperimentalcontentforeachchapter,sothatstudentscanunder-
standandgraspbasicconcepts,basicprinciplesandbasicmethodsthroughcomputer
simulationexperiments.
(4)Introductionofthelatestresearchresults.
Randomsignalanalysisofexistingteachingmaterialismainlylimitedtothecharac-
terizationandanalysisofstationaryrandomprocesseslackadescriptionofnonsta-
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
Preface | VII
tionaryrandomprocessesandrelatedanalysisofrandomprocessesafterpassingnon-
linear systems. With the advancement of modern signalprocessing, nonstationary,
aperiodic,non-Gaussianand nonlinear stochastic signalprocessing problemshave
ledtoalotofresearchresults;theseresultsshouldbethebasisofapreliminaryunder-
standingoftoday’sundergraduates.Therefore,thistextbookwilldevoteachapterto
theintroductionoftime-frequencyanalysisandbasicknowledgeofwaveletanalysis.
Thebookisdividedintosixchapters:Chapter0isanintroduction,whichreviews
and summarizesthe basicknowledge ofprobability theory and introduces random
variables and their related digital features and characteristic functions, as well as
otherknowledgepoints.Chapter1introducesthebasicconceptofrandomsignals.It
discussestheirbasiccharacteristicsandmethodstodescribethem,complexstochas-
tic process. The discrete-time stochastic process are also detailed, and the normal
stochastic processand itsspectralanalysisand white noiseprocessareintroduced
as well. Chapter 2 introduces the linear transformationof the stationary stochastic
process,reviewsthelineartransformationandlinearsystem.Moreover,theprocess
ofdifferentialandintegralofrandomprocessisalsointroducedtherein.Thestochas-
ticprocessisanalyzedbycontinuousanddiscrete-timesystems.Whitenoiseisana-
lyzedbyalinearsystemandthemethodofsolvingtheprobabilitydensityafterthe
lineartransformationoftherandomprocess.InChapter3,wediscussthestationary
narrowbandstochasticprocessandfirstintroducetheanalyticalprocessandHilbert
transform,narrowbandstochasticrepresentation,andtheanalyticcomplexstochas-
ticprocess.Wethendiscusstheprobabilitydensityoftheenvelopeandphaseofthe
narrowbandnormalprocessandtheprobabilitydensityofthesquareofitsenvelope.
Chapter4discussesthenonlineartransformationmethodofstationaryrandompro-
cess,includingthedirectmethod,transformationandtheanalysisofthestochastic
process through limiters and the method of calculating the signal-to-noise ratio at
the outputofthenonlinear system arealsodetailed.Thecharacteristicdescription
and analysis method of the nonstationary stochastic process is given in Chapter 5.
First, the definition and description of the nonstationary stochastic process are in-
troduced,andthecorrelationfunctionandpowerspectraldensityarediscussed.Fi-
nally,theanalysismethodofthenonstationarystochasticprocessinmodernsignal
processing, such as Wigner–Ville distribution and wavelet analysis are introduced.
Thebookincorporatesalargenumberofexamplesandillustrations,andattheend
ofeachchapterthereareenoughexercisesforpractice.Italsoprovidessomerefer-
enceanswersattheendoftheChinesebook.Thebookalsoprovidesthederivation
andproofofsomeimportantformulas,theEnglish–Chineseterminology,andother
appendices.
The book was completed by associate Professor Yang Jie and Liu Congfeng.
OngbwaOllomoArmelonas,aninternationalstudentofXidianUniversity,hasmade
great efforts in the translation process of this book. The authors express their ap-
preciationtoFuPanlong,YinChenyang,YunJinwei,LiuChenchong,ShaZhaoqun,
SuJuan,HouJunrongfortranslatingandcorrectingChapters0,1,2,3,4,5andthe
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
VIII | Preface
Appendix,respectively.Thepreparationprocessofthisbookwasencouraged,helped
andsupportedbytheXi’anUniversityofPost&TelecommunicationandXidianUni-
versitycolleagues.Thesciencepublisherswerestrongsupportinthepublicationof
thisbook; PanSisiand other editors dedicated alot ofenergy to the bookand the
authorswishtoexpresstheirheartfeltappreciationofthis.
Limited to the knowledge of editors, the book fallacies and omissions are in-
evitable.Readersareencouragedtooffercriticismandsuggestcorrections.
TheEditors
2017.08
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:50 AM
Contents
0 Introduction|1
0.1 Probabilityspace|1
0.1.1 Randomizedtrials|1
0.1.2 Samplespace|2
0.1.3 Probabilityspace|2
0.2 Conditionalprobabilityspace|3
0.2.1 Conditionalprobability|3
0.2.2 Multiplicationformula|4
0.2.3 Totalprobabilityformula|4
0.2.4 TheBayesianformula|5
0.3 Randomvariables|6
0.3.1 Theconceptofrandomvariables|6
0.3.2 Discreterandomvariables|7
0.3.3 Continuousrandomvariables|7
0.3.4 Multidimensionalrandomvariables|9
0.4 Distributionofrandomvariablefunctions|13
0.4.1 Distributionofdiscreterandomvariablefunctions|14
0.4.2 Distributionofcontinuousrandomvariablefunctions|14
0.5 Numericalcharacteristicsofrandomvariables|15
0.5.1 Mathematicalexpectations|16
0.5.2 Varianceandstandarddeviation|17
0.5.3 Covarianceandcorrelationcoefficients|18
0.5.4 Themomentofrandomvariables|18
0.6 Characteristicfunctionsofrandomvariables|22
0.6.1 Complexrandomvariables|22
0.6.2 Characteristicfunctionsofrandomvariables|23
0.6.3 Propertiesofcharacteristicfunctions|24
0.6.4 Relationshipbetweencharacteristicfunctionsandmoments|25
0.6.5 Characteristicfunctionsofmultidimensionalrandomvariables|26
0.7 Chebyshevinequalityandthelimittheorem|28
0.7.1 Chebyshevinequality|28
0.7.2 Centrallimittheorem|28
1 Randomprocesses|33
1.1 Basicconceptsofrandomprocesses|33
1.1.1 Definitionofrandomprocesses|33
1.1.2 Probabilitydistributionofrandomprocesses|36
1.1.3 Themomentfunctionofrandomprocesses|40
1.1.4 Characteristicfunctionsofrandomprocesses|43
Brought to you by | provisional account
Unauthenticated
Download Date | 1/8/20 9:51 AM