Table Of ContentSpringer Series in Wireless Technology
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codingandmodulation;
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cognitiveradio;
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full-duplexwirelesscommunication;
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model-freedesign;
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multipleaccess;
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resourceallocation;
•
usesofdigitalsignalprocessinginwirelesssystems;
•
wirelessenergytransfer;
•
wirelessnetworks:4G,5Gandbeyondandnext-generationWiFi;adhocwireless
networks, device-to-device networks; heterogeneous mobile networks; wireless
sensornetworks;
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wirelessopticalcommunications.
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· ·
Shih-Chun Lin Tsung-Hui Chang
·
Eduard Jorswieck Pin-Hsun Lin
Information Theory,
Mathematical Optimization,
and Their Crossroads in 6G
System Design
Shih-ChunLin Tsung-HuiChang
DepartmentofElectricalEngineering SchoolofScienceandEngineering
NationalTaiwanUniversity TheChineseUniversityofHongKong
Taipei,Taiwan Shenzhen,China
EduardJorswieck Pin-HsunLin
InstituteforCommunicationsTechnology InstituteforCommunicationsTechnology
TechnischeUniversitätBraunschweig TechnischeUniversitätBraunschweig
Brunswick,Germany Brunswick,Germany
ISSN 2365-4139 ISSN 2365-4147 (electronic)
SpringerSeriesinWirelessTechnology
ISBN 978-981-19-2015-8 ISBN 978-981-19-2016-5 (eBook)
https://doi.org/10.1007/978-981-19-2016-5
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SingaporePteLtd.2023
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Preface
Withthecommercializationof5Gcommunications,industrialandacademiccommu-
nitieshavebeguntoconceivethetechnologyandrequirementsofthenextgeneration
of communications. This monograph provides a solid understanding of the funda-
mentaltoolsandmethodsfrominformationtheoryandmathematicalprogramming
withspecificapplicationsto6Gsystemdesigns.Thefuture6Gsystemswillincorpo-
ratethe5GcommunicationnetworkwithArtificialintelligenceinInternetofThings
(AIoT)andopentheeraof“InternetofIntelligence”,whichwilllinkpeople,things,
and information with smart computing. The smart Internet collects environmental
information from networked sensors, and people or things can obtain processed
and useful information through the smart Internet. It is expected that the informa-
tiontheory,mathematicaloptimization,andtheirintersectionwillplayanimportant
role in the design. Information theory offers fundamental guidelines on optimal
datatransmissionandstatisticalinferenceoverthenetwork,whiletheoptimization
theoryallowssystematicmodellingandalgorithmicmethodstoapproachthemand
performfurtherdataprocessing.Thismonographwillprovideanover-archofthese
twofieldsandtheirinterplay,andalsodemonstratehowtheyareappliedtothe6G
systemdesigns.
Inthefirstpartofthisbook,informationandoptimizationtheoryarediscussed.
Two types of system design approaches will be discussed, that is, centralized and
distributed ones. The former suits 6G systems pursuing maximum performance,
whereperfectnetworkinformationisavailableatnodes,whilethelattersuitsAIoT
applicationswhencoordinationbetweennetworknodesislimitedduetoconstraints
onavailableresources.InChap.1,itstartsfromtheinformationtheoryforcentralized
wireless networks where all nodes have global channel state information (CSI).
Next,inanetworkwherenotallnodesfullycooperateorprovidetheirlocalCSI,the
informationtheoryofthisdistributednetworkwillbepresentedinChap.2.InChap.3,
wediscussthestatisticaldetectiontheories,wherethedistributedpartsofthemalso
servesasabasisofdistributedlearning.Tofurtheroptimizetheschemesdiscussed,
centralizedanddistributedoptimizationtheorieswillthenbeaddressedinChaps.4
and 5, respectively. Not only the convex optimization but also the state-of-the-art
non-convexmethodswillbeincluded.
v
vi Preface
Inthesecondpart,applicationsoftheoriesinthefirstpartandtheirintersection
will be addressed. Note that 3GPP already provided the 5G New Radio, with the
servicesandapplicationsbeingclassifiedasfollows:
•
Enhanced mobile broadband (eMBB): This use case mainly focuses on high
data-ratehuman-centriccommunications.Augmentedrealityandhigh-resolution
videocommunicationarebothapplicationexamples.
•
Massivemachinetypecommunications(mMTC):Thisusecasehasaverylarge
numberofconnecteddevices,andisalsoknownasIoT.Lowcostandlongbattery
lifeareimportantforthesedevices.
•
Ultra-reliableandlowlatencycommunications(URLLC):Thisusecasepromises
toservemultipleautonomousmachineswithhighreliabilityandlowlatency.Its
examplesincludesmartgrid,InternetofVehicles,andwirelesscontrolofindustrial
manufacturing.
Wearguethattheaforementionedusecasesarealsoimportantfor6G.InChap.6,
wefocusontheresourceallocationofhighdata-rate6Gnetwork.Toachievehigh
trafficcapacity,basestation(BS)scanfullycooperateunderthelowrequirementof
mobilityandbecomeacentralizedsupertransmitterknowingperfectCSI.However,
when users desire high mobility, the channel state information at the transmitter
(CSIT)isnotperfectandBSsmayevennotfullycooperate.Thecentralizedinfor-
mationandoptimizationtheorieswillbeimportantforthefirstpartofChap.6,while
the distributed theories are essential for the second part. Next, URLLC should be
extended toallowmultipleusersin6G,andthusnotonlyfiniteblocklength infor-
mationtheorybutalsonon-convexoptimizationareneeded,asdiscussedinChap.7.
TowardsInternetofIntelligence,inChap.8wediscussFederatedLearning(FL)asa
formofAIoT.Inthisscenario,aserverperformsdistributedparameteroptimization
throughquantizedinformationfromcapacity-limitedlinksofedgedevices.Finally,
securityissuesautomaticallyariseindistributedinformationsystems.Twoexamples,
thedistributedByzantinedetectionanddistributedsecrecy,arediscussedinChap.9.
Notations:Inthisbook,ifnotspecificallymentionedineachchapter,(differen-
tial)entropyandmutualinformationaredenotedbyH(·)andI(;),respectively,and
randomvariablesaredenotedinitaliccapitals.Deterministicmatricesaredenoted
in bold-face capitals. For matrix G, Tr(G) and Rk(G) denote the trace and rank;
GT and G† denote the transpose and conjugate transpose, respectively. G−1 and
s
|Gs|aretheinverseanddeterminant(cid:2)ofasquarematrixGs.TheFrobeniusnormof
a matrix G is denoted by (cid:2)G(cid:2) := Tr(GG†) and I denotes the identity matrix
F n
ofdimensionn.Thepartialorderingbetweensymmetricmatricesisdenotedby(cid:3)
and(cid:4),forexample,G (cid:4) G (“largerthanorequalto”betweenmatrices)means
1 2
(G − G ) is a positive semi-definite matrix. The Kronecker product between
1 2
matricesis⊗.Pleasereferto[1]fordetailsofthesematrixoperations.
Preface vii
Randomvectorswillbedenotedbylowercaseboldfontorwithasuperscriptindi-
catingtheirlength(e.g.,aor A[1:T] =(A ,...,A )).Forarandomvectora,(cid:2) or
1 T a
Cov(a)denotesitscovariancematrix.ForcomplexGaussianvectora∼CN(0,(cid:2) ),
a
weassume itiscircularly symmetric such thatitsdistributionis completely deter-
minedbythecovariancematrix(cid:2) .ThesameassumptionisappliedtoarealGaussian
a
vectora∼N(0,(cid:2) ).
a
The indicator function 1{.} is 1 if its input event is true and 0 otherwise. Also
(x)+ = max{x,0}.Theset{1,...,N}isdenotedas[1 : N]or[N],where N isa
positiveinteger.
Taipei,Taiwan Shih-ChunLin
Shenzhen,China Tsung-HuiChang
Brunswick,Germany EduardJorswieck
Brunswick,Germany Pin-HsunLin
Reference
1.HornRA,JohnsonCR(1985)Matrixanalysis.CambridgeUniversityPress,Cambridge,UK
Acknowledgements
PartoftheworksofShih-ChunLinisfundedbyMinistryofScienceandTechnology,
Taiwan,andbyNationalTaiwanUniversityunderGrantNTU-CC-111L894402.S.-
C.LinthankshisstudentsandDr.Yen-FuChouforpreparingthisbook,andmany
colleaguesforhelpfuldiscussions.Especially,Prof.I-HsiangWang,Prof.Yu-Chih
Huang,Prof.Yu-JuiHuang,andProf.AlirezaVahid.TheworkofTsung-HuiChang
issupportedinpartbytheShenzhenScienceandTechnologyProgramunderGrant
JCYJ20190813171003723 and RCJC20210609104448114, in part by the NSFC,
China,underGrant62071409,andinpartbytheGuangdongProvincialKeyLabo-
ratoryofBigDataComputing.Tsung-HuiChangwouldliketoacknowledgeMiss
Songyang Ge and Miss Yanmeng Wang for fruitful discussions and for preparing
the materials for Chaps. 5 and 8, respectively. The discussion with Dr. Chao Shen
at Shenzhen Research Institute of Big Data for Sect. 6.2.6 is also acknowledged.
Part of the work of E. Jorswieck is funded by the Federal Ministry of Education
and Research (BMBF, Germany) through the 6G Research and Innovation Cluster
6G-RICunderGrant16KISK020KandbytheGermanResearchFoundation(DFG)
under grant JO 801/23-1. E. Jorswieck would like to acknowledge the discussions
withthecolleaguesfromthejointGerman-IranianDFG-INSFfundedprojectunder
grant JO 801/24-1, and the groups of Dr. Mokari and Dr. Javan. Furthermore, E.
JorswieckacknowledgesdiscussionswithDr.Matthiesenaboutthematerialonnon-
convexoptimizationinSect.4.2,inparticularonbranch-and-boundandmixedmono-
tonicprogramming,withDr.Zapponeonfractionalprogramming,withDr.Björnson
onmonotonicprogramming.FortheresultsinSect.6.1,E.Jorswieckacknowledges
fruitfuldiscussionswithMr.Rezvani,forSect.6.2.1withMr.ZakeriandMr.Khalili,
forSect.6.2.3withDr.Matthiesen,Dr.HellingsandDr.Utschick.Finally,thediscus-
sionswithDr.ShehabandDr.AlvesoneffectiveenergyefficientURLLC(Sect.7.1)
isacknowledged.
ix
Contents
1 InformationTheoryinCentralizedWirelessNetwork .............. 1
1.1 SingleUserMIMOChannel ................................. 2
1.1.1 Capacity-Achieving Code foraGaussian(SISO)
Channel ............................................ 4
1.1.2 BeamforminginSIMOandMISOChannels ............. 20
1.1.3 MIMOCapacitywithFullCSI ......................... 23
1.1.4 MIMOOFDMforMulti-pathChannels ................. 30
1.2 Multiple Access Channel with Global Channel State
Information ................................................ 34
1.2.1 GaussianMultipleAccessChannel ..................... 34
1.2.2 SuccessiveInterferenceCancellationattheMAC
Receiver ............................................ 37
1.2.3 MIMOMultipleAccessChannel ....................... 39
1.3 BroadcastChannelwithGlobalChannelStateInformation ....... 41
1.3.1 GaussianBroadcastChannel ........................... 41
1.3.2 SuccessiveInterferenceCancellationattheStronger
Receiver ............................................ 44
1.3.3 MIMOGaussianBroadcastChannel .................... 46
1.3.4 DirtyPaperCodingwithFullCSIT ..................... 48
1.4 Conclusion ................................................ 60
References ..................................................... 60
2 InformationTheoryinDistributedWirelessNetwork .............. 63
2.1 CSIFeedbackinSingleUserChannel ......................... 64
2.1.1 ChannelStatistics .................................... 64
2.1.2 CodinginaFastFadingChannel ....................... 66
2.1.3 CodinginaSlowFadingChannel ...................... 72
2.2 IntermittentBroadcastChannelswithDistributedCSI ........... 78
2.2.1 BinaryExpansionModelsofIntermittentGaussian
BroadcastChannels .................................. 79
xi
xii Contents
2.2.2 Hybrid Intermittence State Information
attheTransmitterandReceivers ........................ 81
2.2.3 DoF Regions for Channels with Perfect CSIT
forReceiver1 ....................................... 82
2.2.4 DoFRegionsforChannelsWithoutPerfectCSIT
forReceiver1 ....................................... 88
2.3 IntermittentInterferenceChannelswithDistributedCSI .......... 96
2.3.1 GaussianInterferenceChannels ........................ 96
2.3.2 DoFRegionsofIntermittentGaussianInterference
Channels ........................................... 98
2.3.3 Intermittent Gaussian Interference Channels
withLocalDelayedCSIT ............................. 110
2.4 ComputationOverIntermittentMultipleAccessChannel ......... 112
2.4.1 DoF for Computation Over a Gaussian Multiple
AccessChannel ...................................... 113
2.4.2 DoFBoundsforComputationOvertheTwo-UserCase .... 119
2.4.3 AchievableDoFfortheComputationOverK-User
Case ............................................... 124
2.5 Conclusion ................................................ 128
References ..................................................... 128
3 CentralizedandDistributedDetection ............................ 131
3.1 CentralizedDetectionTheory ................................ 132
3.1.1 CentralizedHypothesisTesting ........................ 132
3.1.2 CentralizedQuickestChangeDetection ................. 135
3.2 DistributedDetectionTheory ................................. 138
3.2.1 DistributedHypothesisTesting ......................... 138
3.2.2 DistributedQuickestChangeDetection .................. 141
3.3 Conclusion ................................................ 142
References ..................................................... 142
4 CentralizedMathematicalOptimization .......................... 145
4.1 ConvexOptimization ....................................... 145
4.1.1 ConvexSetsandFunctions ............................ 146
4.1.2 ConvexOptimizationProblems ........................ 149
4.1.3 LagrangeDualityTheorem ............................ 151
4.1.4 TheKKTConditions ................................. 153
4.1.5 ConcludingRemark .................................. 155
4.2 Non-convexOptimization ................................... 155
4.2.1 SequentialConvexApproximation ...................... 156
4.2.2 Semi-definiteRelaxation .............................. 157
4.2.3 FractionalProgramming .............................. 159
4.2.4 MonotonicProgramming .............................. 162
4.2.5 Branch-and-Bound ................................... 166
4.2.6 MixedMonotonicProgramming ....................... 169
