Capacity Region of Multiple Access Channel with States Known Noncausally at One Encoder and Only Strictly Causally at the Other Encoder

0.36 MB·English

by  Abdellatif Zaidi

1
Capacity Region of Multiple Access
Channel with States Known Noncausally
at One Encoder and Only Strictly Causally
at the Other Encoder
2
1
0
2 Abdellatif Zaidi Pablo Piantanida Shlomo Shamai (Shitz)

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a
J

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Abstract
T
I
.
s Weconsideratwo-userstate-dependentmultiaccesschannelinwhichthestatesofthechan-
c
[ nelareknownnon-causallytooneoftheencodersandonlystrictlycausallytotheotherencoder.

1
Both encoders transmit a common message and, in addition, the encoder that knows the states
v
8 non-causallytransmitsanindividualmessage.Wefindexplicitcharacterizationsofthecapacity
7
2 region of this communication model in both discrete memoryless and memoryless Gaussian
3
. cases.Inparticularthecapacityregionanalysisdemonstratestheutilityoftheknowledgeofthe
1
0 statesonlystrictlycausallyattheencoderthatsendsonlythecommonmessageingeneral.More
2
1 specifically,inthediscretememorylesssettingweshowthatsuchaknowledgeisbeneficialand
:
v increasesthecapacityregioningeneral.IntheGaussiansetting,weshowthatsuchaknowledge
i
X doesnothelp,andthecapacityissameasifthestateswerecompletelyunknownattheencoder
r
a thatsendsonlythecommon message.Furthermore,wealsostudythespecialcaseinwhichthe
The material in this paper was presented in part at the IEEE International Symposium on Information Theory,
Saint-Petersburg,Russia,August2011.ThisworkhasbeensupportedbytheEuropeanCommissionintheframework
oftheFP7NetworkofExcellenceinWirelessCommunications(NEWCOM++).TheworkofS.Shamaihasalsobeen
supported by the CORNET consortium.
Abdellatif Zaidi is with Universite´ Paris-Est Marne La Valle´e, 77454 Marne la Valle´e Cedex 2, France. Email:
[email protected]
Pablo Piantanida is with the Department of Telecommunications, SUPELEC, 91190 Gif-sur-Yvette, France. Email:
[email protected]
ShlomoShamai iswiththeDepartmentofElectricalEngineering,TechnionInstitute ofTechnology,TechnionCity,
Haifa 32000, Israel. Email: [email protected]
January17,2012 DRAFT
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two encoders transmit only the common message and show that the knowledge of the states
onlystrictlycausallyattheencoderthatsendsonlythecommonmessageisnotbeneficialinthis
case, in both discrete memoryless and memoryless Gaussian settings. The analysis also reveals
optimal ways of exploiting the knowledge of the state only strictly causally at the encoder
that sends only the common message when such a knowledge is beneficial. The encoders
collaborateto convey tothe decodera lossy version of the state, in additionto transmitting the
information messages through a generalized Gel’fand-Pinsker binning. Particularly important
in this problem are the questions of 1) optimal ways of performing the state compression and
2) whether or not the compression indices should be decoded uniquely. By developing two
optimal coding schemes that perform this state compression differently, we show that when
used as parts of appropriately tuned encoding and decoding processes, both compression a`-
la noisy network coding, i.e., with no binning, and compression using Wyner-Ziv binning
are optimal. The scheme that uses Wyner-Ziv binning shares elements with Cover and El
Gamal original compress-and-forward,but differs from it mainly in that backwarddecoding is
employed instead of forward decoding and the compression indices are not decoded uniquely.
Finally,byexploringthe propertiesofour outerbound,we showthat,although notrequiredin
general,thecompressionindicescaninfactbedecodeduniquelyessentiallywithoutalteringthe
capacityregion, but atthe expense of largeralphabetssizes for the auxiliaryrandom variables.
I. Introduction
The study ofchannels that arecontrolledbyrandom states has spurredmuch interest, due to itsimportance
frombothinformation-theoreticandcommunicationsaspects.Forexample,state-dependentchannelsmaymodel
communicationinrandomfadingenvironments[1]orinthepresenceofinterferenceimposedbyadjacentusers.
Thechannel states maybeknowninastrictly-causal,causalornoncausalmanner, toalloronlyasubset ofthe
encoders.Foratransmissionoflengthn,letSn =(S ,S ,...,S )denotethestatesequence,withS representingthe
1 2 n i
channelstateaffectingthechannelattimeorblocki.Forthetransmissioninblocki,thestatesequenceisknown
non-causallyifitisknownentirelybeforethebeginningofthetransmission.Itisknowncausallyifitisknownup
toandincludingtimei;anditisknownstrictlycausallyifitisknownonlyuptotimei 1.Thewaythechannel
−
stateinformationisutilizedandinfluencescapacitydependsalsoonwhichoftheencoders(s)anddecoder(s)are
aware of it. In single user channels, the concept of channel state available at only the transmitter dates back to
Shannon[2]forthecausalchannelstatecase,andtoGel’fandandPinsker[3]forthenon-causalchannelstatecase.
Inmultiuserenvironments,agrowingbodyofworkstudiesmulti-userstate-dependentmodels.Recentadvances
in this regard can be found in [4]–[27], and many other works. For acomprehensive review of state-dependent
channelsandrelatedwork,thereadermayreferto[4].
There is a connection between the role of states known strictly causally at an encoder and that of output
feedbackgiventothatencoder.Insingle-userchannels,itisnowwellknownthatstrictlycausalfeedbackdoesnot
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increasethecapacity[28].Inmultiuserchannelsornetworks,however,thesituationchangesdrastically,andoutput
feedbackcanbebeneficial—butitsroleisstillhighlymissunderstood.Onehasasimilarpicturewithstrictlycausal
states at the encoder. In single-user channels, independent and identically distributed states available only in a
strictlycausalmannerattheencoderhavenoeffectonthecapacity.Inmultiuserchannelsornetworks,however,
likefeedback,strictlycausalstatesingeneralincreasethecapacity.
Advances in the study of the effect of strictly causal states in multiuser channels are rather very recent and
concern mainly multiple access scenarios. In [15], Lapidothand Steinberg study a two-encoder multiple access
channelwithindependentmessagesandstatesknowncausallyattheencoders.Theyshowthatthestrictlycausal
statesequencecanbebeneficial,inthesensethatitincreasesthecapacityforthismodel.Thisresultisreminiscent
of Dueck’s proof [29] that feedback can increase the capacity region of some broadcastchannels. In accordance
with[29],themainideaoftheachievabilityresultin[15]isablockMarkovcodingschemeinwhichthetwousers
collaboratetodescribethestatetothedecoderbysendingcooperativelyacompressedversionofit.Asnoticedin
[15],althoughsomenon-zeroratethatotherwisecouldbeusedtotransmitpureinformationisspentindescribing
the state to the decoder, the net effect can be an increase in the capacity. In [16], they show that strictly causal
state information is beneficial even if the channel is controlled by two independent states each known to one
encoderstrictlycausally.Inthiscase,eachencodercanhelptheotherencodertransmitatahigherratebysending
acompressedversionofitsstatetothedecoder.In[18],Li,SimeoneandYenerimprovetheresultsof[15],[16]and
extend themtothecaseofmultipleencoders.Theachievabilityresultsin[18]areinspiredbythenoisynetwork
codingschemeof[30]and,unlike[15],[16],donotuseWyner-Zivbinning[31]forthecompressionofthestate.In
averyrecentcontribution[32],LapidothandSteinbergderiveanewinnerboundonthecapacityregionforthe
caseofasinglestategoverningthemultiaccesschannel.Theyalsoprovethattheinnerboundof[18]forthecase
oftwoindependentstateseachknownstrictlycausallytooneencodercanindeedbestrictlybetterthanprevious
boundsin[15],[16]–aresultwhichisconjecturedpreviouslybyLi,SimeoneandYenerin[18].
A. Studied Model
Inthispaper,whichgeneralizesaformerconferenceversion[33],westudyatwo-userstate-dependentmultiple
accesschannelwiththechannelstatesknownnon-causallyatoneencoderandonlystrictlycausallyattheother
encoder.Thedecoderisnotawareofthechannelstates.AsshowninFigure1,bothencoderstransmitacommon
messageand,inaddition,theencoderthatknowsthestatesnon-causallytransmitsanindividualmessage.This
modelgeneralizes onewhosecapacityregionisestablished in[5]and inwhichtheencoder that sendsonlythe
commonmessagedoesnotknowthestatesatall.Moreprecisely,letW andW denotethecommonmessageandthe
c 1
individualmessagetobetransmittedin,say,nusesofthechannel;andSn =(S ,...,S )denotethestatesequence
1 n
affectingthechannelduringthistime.Attimei,Encoder1knowsthecompletesequenceSn =(S ,...,S ,S,...,S )
1 i 1 i n
−
andsendsX =φ (W ,W ,Sn),andEncoder2knowsonlySi 1 = (S ,...,S )andsendsX =φ (W ,Si 1)–the
1i 1 c 1 − 1 i 1 2i 2,i c −
−
functionsφ andφ aresomeencodingfunctions.Inthispaper,westudythecapacityregionofthisstate-dependent
1 2,i
MACmodel.Asouranalysiswillshow,thisrequires,amongothers,understandingtheroleofthestrictlycausal
partofthestatethatisrevealedtoEncoder2.
January17,2012 DRAFT
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Si−1
Encoder 2 X2n
Wc WMAC Yn Decoder (Wˆc,Wˆ1)
Y|X1,X2,S
Encoder 1 X1n
W1
Sn
Fig.1. State-dependentMACwithdegradedmessagesetsand states knownnoncausallyatthe encoderthat sends
both messages and only strictly causally at the other encoder.
B. Main Contributions
Inthediscretememorylesscase,wecharacterizethecapacityregionforthegeneralfinite-alphabetcasewitha
single-letter expression.Theproofoftheachievabilitypartisbasedonablock-Markovcodingschemeinwhich
thetwoencoderscollaboratetoconveyalossyversionofthe statetothedecoder,inthespiritof[15],[16],[32],
inadditiontoageneralizedGel’fand-Pinskerbinningforthetransmissionoftheinformationmessages[3].From
theangleofthestatecompression,codingschemesthatperformthestatecompressionforourmodeltiewithvery
recentworksoncompressionsincompress-and-forwardtyperelayingnetworks[30],[34]–[36].Wefirstdevelopa
codingschemeinwhichthestatecompressionisperformeda`-laKimetal.noisynetworkcodingschemeandshow
thatitisoptimal,i.e.,achievesanouterboundthatweestablishforthestudiedmodel.Inthiscodingscheme,unlike
[15],[16],[32]whereeveryinformationmessageisdividedintoblocksanddifferentsubmessagesaresentoverthese
blocksandthendecodedoneatatimeusingthesamecodebookasintheoriginalcompress-and-forwardschemeby
CoverandElGamal[37],heretheentirecommonmessageandtheentireindividualmessagearetransmittedover
allblocksusingcodebooksthataregenerated independently, oneforeachblock,andthedecodingisperformed
simultaneouslyusingallblocksasinthe noisynetwork codingschemeof [30].Also,like[30],ateachblockthe
compressionindexofthestateofthepreviousblockissentusingstandardratedistortion,notWyner-Zivbinning.
Atthe end ofthe transmission, the receiver uses the outputs ofall blocksto performsimultaneous decodingof
theinformationcommonandindividualmessages,withoutuniquelydecodingthecompressionindices.Fromthis
angle,ourcodingschemeconnectsmorewith[18],thanwith[15],[16]and[32].
Twoofthemostimportantfeaturesofourcodingschemethatisbasedonnoisynetworkcodingarei)standard
compressionwithoutWyner-Zivbinningandii)non-explicitdecodingofthecompressionindices.Investigating
whetherthesefeaturesarepivotalforoptimalityinourproblem,asarguedin[30]forsomerelatedmodels,wealso
explorebinning-basedcompressions.Weshowthatthecapacityregionofourmodelcanalsobeachievedusing
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analternate codingschemeinwhichthe state compressionisrealizedusingWyner-Zivbinning. Theemployed
optimalalternatecodingschemeshareselementswithCoverandElGamalcompress-and-forward[37],butdiffers
fromitintwoaspects:1)backwarddecodingisutilizedinsteadoftheforwarddecodingof[37],and2)unlike[37],
the compression indices are not decoded uniquely. Decoding backwardly instead of forwardly seems essential
forthe optimalityofthis alternate codingscheme here. Atthislevel, wenote that the finding inthis paper that
backwarddecodingwithnon-uniquedecodingofthecompressionindicesisbeneficial,mayholdmoregenerally
in other scenarios that involve Wyner-Ziv binning. In the fading setting, this is also observed in [38]. Next, by
exploringourouterbound further, weshowthat, althoughnotrequired,onecanmodifythiscodingschemein
amannertogetthecompressionindicesdecodedatthereceiveressentiallywithoutalteringthecapacityregion
but at the expense of larger alphabets sizes of the involved auxiliary random variables. The decoding of the
compressionindicesintroducesanadditionalrateconstraint;butweshowthatthisconstraintissatisfiedbythe
auxiliaryrandomvariablesoftheouterbound.Finally,wenotethatthefindinginthispaperthatinthecontext
of Wyner-Ziv binning backward decoding with non-unique decoding of the compression indices improves the
transmissionratemaybebeneficialinotherscenarios.Inthefadingsetting,thiswasalsoobservedin[38].
Thesingle-letter characterizationofthecapacityregionofourmodelremainsintactifoneallowsfeedbackto
the encoder that sends both messages. Also, the capacity region of our model contains that of the model of [5]
inwhichtheencoderthatsendsonlythecommonmessageisunawareofthechannelstates;andthisshowsthat
revealingthestatesevenonlystrictlycausallytothisencoderpotentiallyincreasesthecapacityregion.Next,by
investigating a discrete memoryless example, we show that this inclusioncan be strict, thus demonstrating the
utilityofconveyingacompressedversionofthestatetothedecodercooperativelybytheencoders.
Wealsospecializeourresultstothecaseinwhichthetwoencoderssendonlythecommonmessage.Werefer
tothecapacityinthiscaseascommon-messagecapacity.Weshowthat,whenoneofthetwoencodersisinformed
noncausally,theknowledgeofthestatesonlystrictlycausallyattheotherencoderdoesnotincreasethecommon-
messagecapacity.Itshouldbenotedthatthisresultisnotadirectconsequenceofthatfeedbackdoesnotincrease
thecapacityinamultiaccesschannelinwhichtheencoderssendonlyacommonmessage;andourconverseproof
isneededhere.
Next,weconsiderthememorylessGaussiansettinginwhichthechannelstateandthenoiseareadditiveand
Gaussian.Weestablishanoperativeouterboundontheachievableratepairs.Then,weshowthatthisouterbound
is achievable, yielding a closed-form expression of the capacity region. The resulting capacity region coincides
withthatofthe modelof[5]inwhichthe encoderthatsendsonlythe commonmessageiscompletelyunaware
ofthestates,thusdemonstratingthat,byoppositiontothediscretememorylesscase,revealingthestatesstrictly
causallyto this encoder isnotbeneficial inthe Gaussiancase,inthe sense that itdoesnotincreasethe capacity
region.
Finally,wenotethatincontrasttotherelatedMACmodelsin[5],[7],ourconverseproofsinthispaperdonot
followdirectlyfromtheconversepartproofofthecapacityformulaforthestandardGel’fand-Pinskerchannel[3].
Thisisbecause,attimei,theencoderthattransmitsonlythecommonmessagesendsinputswhicharefunctionof
notonlythatmessage,butalsotheobservedpaststatesequence.
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C. Outline and Notation
Anoutlineoftheremainderofthispaperisasfollows.SectionIIdescribesinmoredetailthecommunication
modelthatweconsiderinthiswork.SectionIIIprovidesthecapacityregionofthediscretememorylessmodel.In
thissectionwealsoestablishanalternativeouterboundonthecapacityregionthatwillturntobeusefulinthe
Gaussiancase, provideanexample demonstrating the utility ofrevealing the states onlystrictly causallyto the
encoderthatsendsonlythecommonmessage,andderivethecommon-messagecapacity.SectionIVcharacterizes
thecapacityregionaswellasthecommon-messagecapacityoftheGaussianmodel.Finally,SectionVconcludes
thepaper.
Weusethefollowingnotationsthroughoutthepaper.Uppercaselettersareusedtodenoterandomvariables,
e.g., X; lower case letters are used to denote realizations of random variables, e.g., x; and calligraphic letters
designatealphabets,i.e.,X.TheprobabilitydistributionofarandomvariableXisdenotedbyP (x).Sometimes,
X
forconvenience,wewriteitasP .WeusethenotationE []todenotetheexpectationofrandomvariableX.A
X X
·
probabilitydistribution of a random variable Y given X is denoted by P . The set of probabilitydistributions
YX
|
defined on an alphabet X isdenoted by P(X).The cardinalityof a set X isdenoted by X. For convenience, the
| |
lengthnvectorxnwilloccasionallybedenotedinboldfacenotationx.TheGaussiandistributionwithmeanµand
varianceσ2isdenotedbyN(µ,σ2).Forintegersi j,wedefine[i: j]:= i,i+1,...,j .Finally,throughoutthepaper,
≤ { }
logarithmsaretakentobase2,andthecomplementtounityofascalaru [0,1]isdenotedbyu¯,i.e.,u¯ =1 u.
∈ −
II. SystemModelandDefinitions
Weconsiderastationarymemorylessstate-dependentMACW whoseoutputY Yiscontrolledbythe
Y|X1,X2,S ∈
channelinputsX X andX X fromtheencodersandthechannelstateS Swhichisdrawnaccordingto
1 1 2 2
∈ ∈ ∈
amemorylessprobabilitylawQ .Weassumethatthe channel stateSn isknownnon-causallyatEncoder1,i.e.,
S
beforehand,atthebeginningofthetransmissionblock.Encoder2knowsthechannelstatesonlystrictly-causally;
thatis,attimei,itknowsthestatesonlyuptotimei 1,Si 1 =(S ,...,S ).
− 1 i 1
− −
Encoder2wantstosendacommonmessageW andEncoder1wantstosendanindependentindividualmessage
c
W alongwiththe commonmessage W .Weassumethat the commonmessage W andthe individualmessage
1 c c
W are independent random variables drawn uniformly from the sets W = 1, ,M and W = 1, ,M ,
1 c c 1 1
{ ··· } { ··· }
respectively.ThesequencesXnandXnfromtheencodersaresentacrossastate-dependentmultipleaccesschannel
1 2
modeled as a memoryless conditional probabilitydistribution W . The joint probability mass function on
Y|X1,X2,S
W W Sn Xn Xn Ynisgivenby
c× 1× × 1× 2×
n
P(wc,w1,sn,xn1,xn2,yn)=P(wc)P(w1) QS(si)P(x1,i|wc,w1,sn)P(x2,i|wc,si−1)
Yi=1
W (y x ,x ,s). (1)
· Y|X1,X2,S i| 1,i 2,i i
Thereceiverguessesthepair(Wˆ ,Wˆ )fromthechanneloutputYn.
c 1
Definition1: For positive integers n, M and M , an (M ,M ,n,ǫ) code for the multiple access channel with
c 1 c 1
statesknownnoncausallyatoneencoderandonlystrictlycausallyattheotherencoderconsistsofamapping
φ :W W Sn Xn (2)
1 c× 1× −→ 1
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atEncoder1,asequenceofmappings
φ2,i :Wc Si−1 X2, i=1,...,n (3)
× −→
atEncoder2,andadecodermap
ψ:Yn W W (4)
c 1
−→ ×
suchthattheaverageprobabilityoferrorisboundedbyǫ,
Pn=E Pr ψ(Yn),(W ,W )Sn =sn ǫ. (5)
e S c 1 | ≤
h (cid:16) (cid:17)i
Therateofthecommonmessageandtherateoftheindividualmessagearedefinedas
1 1
R = logM and R = logM , (6)
c n c 1 n 1
respectively.
Aratepair(Rc,R1)issaidtobeachievableifforeveryǫ>0thereexistsan(2nRc,2nR1,n,ǫ)codeforthechannel
W . The capacity region of the considered state-dependent MAC is defined as the closure of the set of
Y|X1,X2,S
achievableratepairs.
III. DiscreteMemorylessCase
Inthissection,itisassumedthatthealphabetsS,X ,X arefinite.
1 2
A. Capacity Region
LetPstandforthecollectionofallrandomvariables(S,U,V,X ,X ,Y)suchthatU,V,X andX takevaluesin
1 2 1 2
finitealphabetsU,V,X andX ,respectively,and
1 2
P (s,u,v,x ,x ,y)=P (s,u,v,x ,x )W (yx ,x ,s) (7a)
S,U,V,X1,X2,Y 1 2 S,U,V,X1X2 1 2 Y|X1,X2,S | 1 2
P (s,u,v,x ,x )=Q (s)P (x )P (vs,x )P (u,x s,v,x ) (7b)
S,U,V,X1,X2 1 2 S X2 2 V|S,X2 | 2 U,X1|S,V,X2 1| 2
P (s,u,v,x ,x )=Q (s). (7c)
S,U,V,X1,X2 1 2 S
u,vX,x1,x2
Therelationsin(7)implythat(U,V) (S,X ,X ) YisaMarkovchain,andX isindependentofS.
1 2 2
↔ ↔
DefineCtobethesetofallratepairs(R ,R )suchthat
c 1
R I(U;YV,X ) I(U;SV,X )
1 2 2
≤ | − |
R +R I(U,V,X ;Y) I(U,V,X ;S)
c 1 2 2
≤ −
forsome (S,U,V,X ,X ,Y) P. (8)
1 2
∈
ThefollowingpropositionstatessomepropertiesofC.
Proposition1: (propertiesofcapacityregion)
1. ThesetCisconvex.
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2. ToexhaustC,itisenoughtorestrictVandUtosatisfy
V S X X +1 (9a)
1 2
| |≤| || || |
U S X X +1 S X X . (9b)
1 2 1 2
| |≤ | || || | | || || |
(cid:16) (cid:17)
Proof:TheproofofProposition1appearsinAppendixA.
As stated in the following theorem, the set C characterizes the capacity region of the state-dependent discrete
memorylessMACmodelthatwestudy.
Theorem1: Thecapacityregionofthemultipleaccesschannel withstatesknownonlystrictlycausallyatthe
encoderthatsendsthecommonmessageandnon-causallyattheencoderthatsendsbothmessagesisgivenbyC.
Proof: Anoutlineproofofthecodingschemethat weuseforthedirectpartwillfollow.Theassociatederror
analysisandtheproofoftheconverseappearinAppendixB.
Theorem1continuestoholdifin(7)wereplaceP byP .Also,itshouldbenotedthatsettingV= in
U|S,V,X2 U|S,V ∅
(8),thecapacityregionCreducestotheunionofallrate-pairs(R ,R )satisfying
c 1
R I(U;YX ) I(U;SX )
1 2 2
≤ | − |
R +R I(U,X ;Y) I(U,X ;S) (10)
c 1 2 2
≤ −
forsomemeasureonS U X X Yoftheform
1 2
× × × ×
P =Q P P W . (11)
S,U,X1,X2,Y S X2 U,X1|S,X2 Y|X1,X2,S
LetC denotetheregiondefinedby(10)and(11)intheremainingofthispaper.Ithasbeenshownin[5]thatthe
′
regionC isthecapacityregionoftheMACmodelofFigure1butwiththestatescompletelyunknownatEncoder
′
2,i.e.,whiletheencodingatEncoder1isgivenby(3),theencodingatEncoder2isdefinedbythemapping
φ :W Xn. (12)
2 c −→ 2
ObservingthatC CshowsthattheknowledgeofthestatesonlystrictlycausallyatEncoder2inourmodelin
′
⊆
generalincreasesthecapacityregion.InSectionIII-Bwewillshowthattheinclusioncanbestrict,i.e.,C (C.
′
Furthermore,onecaneasilycheckthatinthecaseofachannelthatdoesnotdependonthestates,i.e.,W =
Y|X1,X2,S
W ,thecapacityregionCreducestotheclosureoftheunionofallrate-pairs(R ,R )satisfying
Y|X1,X2 c 1
R I(X ;YZ,X )
1 1 2
≤ |
R +R I(X ,X ;Y) (13)
c 1 1 2
≤
forsome
P =P P P W . (14)
Z,X1,X2,Y Z X1|Z X2|Z Y|X1,X2
January17,2012 DRAFT
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Also,itis noted that Theorem 1 remainsintact if we allowfeedback to Encoder 1,i.e.,before producingthe ith
channel input symbol, Encoder 1 also observes the past channel output sequence Yi 1. That is, the encoding at
−
Encoder2isstillgivenby(3)andthatatEncoder1isreplacedbyasequenceofmappings φ n ,with
{ 1,i}i=1
φ1,i :Wc W1 Sn Yi−1 X1. (15)
× × × −→
WenowturntotheproofofachievabilityofTheorem1.Thefollowingremarkisusefulforabetterunderstanding
ofthecodingschemethatweusetoestablishtheachievabilityofTheorem1.
Remark1: TheproofofachievabilityofTheorem1isbasedonablock-Markovcodingschemeinwhichalossy
versionofthestateisconveyedtothedecoder,inthespiritof[15],[16],[32],inadditiontoageneralizedGel’fand-
Pinsker binningforthetransmissionoftheinformationmessages[3].However,unlike[15],[16]and[32]where
Wyner-Zivcompression[31]isutilizedforthetransmissionofthelossyversionofthestate,here,inspiredbythe
noisynetworkcodingschemeof[30],ateachblockthecompressionindexofthestateofthepreviousblockissent
using standard ratedistortion, notWyner-Ziv binning. Also,unlike [15], [16] and [32] where every information
message isdividedintoblocksanddifferent submessages aresent over these blocksand then decodedoneata
timeusingthesamecodebookasintheoriginalcompress-and-forwardschemebyCoverandElGamal[37],here
theentirecommonmessageandtheentireindividualmessagearetransmittedoverallblocksusingcodebooksthat
aregeneratedindependently,oneforeachblock,andthedecodingisperformedsimultaneouslyusingallblocksas
in[30].Attheendofthetransmission,thereceiverusestheoutputsofallblockstoperformsimultaneousdecoding
oftheinformationcommonandindividualmessages,withoutuniquelydecodingthecompressionindices. (cid:3)
ProofofAchievability:
ThetransmissiontakesplaceinBblocks.ThecommonmessageW andtheindividualmessageW aresentover
c 1
allblocks.WethushaveB =nBR ,B =nBR ,N=nB,R =B /N=R andR =B /N =R ,whereB is
Wc c W1 1 Wc Wc c W1 W1 1 Wc
thenumberofcommonmessagebits,B isthenumberofindividualmessagebits,N isthenumberofchannel
W1
usesandR andR aretheoverallratesofthecommonandindividualmessages,respectively.
Wc W1
Codebook Generation: Fix a measure P P. Fix ǫ > 0, η > 0, η > 0, ηˆ > 0, δ > 1 and denote
S,U,V,X1,X2,Y ∈ c 1
Mc =2nB[Rc−ηcǫ],M1=2nB[R1−η1ǫ],Mˆ =2n[Rˆ+ηˆǫ]andJ=2n[I(U;S|V,X2)+δǫ].
Werandomlyandindependentlygenerateacodebookforeachblock.
1) For each block i, i = 1,...,B, we generate M Mˆ independent and identically distributed (i.i.d.)codewords
c
x (w ,t)indexedbyw =1,...,R ,t =1,...,Mˆ,eachwithi.i.d.componentsdrawnaccordingtoP .
2,i c ′i c c ′i X2
2) For each block i, for each codeword x (w ,t), we generate Mˆ i.i.d.codewordsv(w ,t,t) indexed by t =
2,i c ′i i c ′i i i
1,...,Mˆ,eachwithi.i.d.componentsdrawnaccordingtoP .
V|X2
3) Foreachblocki,foreachcodewordx (w ,t),foreachcodewordv(w ,t,t),wegenerateacollectionofJM
2,i c ′i i c ′i i 1
i.i.d.codewords u(w ,t,t,w ,j) indexedbyw =1,...,M , j =1,...,J,eachwithi.i.d.componentsdraw
{ i c ′i i 1 i } 1 1 i
accordingtoP .
U|V,X2
Encoding:SupposethatacommonmessageW =w andanindividualmessageW =w aretobetransmitted.
c c 1 1
Aswementionedpreviously,w andw willbesentoverallblocks.Wedenotebys[i]thestateaffectingthechannel
c 1
in block i, i = 1,...,B. For convenience, we let s[0] = and t = t = 1 (a default value). The encoding at the
1 0
∅ −
beginningofblocki,i=1,...,B,isasfollows.
January17,2012 DRAFT
10
Encoder2,whichhaslearnedthestatesequences[i 1],knowst andlooksforacompressionindext [1:Mˆ]
i 2 i 1
− − − ∈
suchthatv (w ,t ,t )isstronglyjointlytypicalwiths[i 1]andx (w ,t ).Ifthereisnosuchindexorthe
i 1 c i 2 i 1 2,i 1 c i 2
− − − − − −
observedstates[i 1]isnottypical,t issetto1andanerrorisdeclared.Ifthereismorethanonesuchindext ,
i 1 i 1
− − −
choosethesmallest.Encoder2thentransmitsthevectorx (w ,t ).
2,i c i 1
−
Encoder 1 obtains x (w ,t ) similarly. It then finds the smallest compression index t [1 : Mˆ] such that
2,i c i 1 i
− ∈
v(w ,t ,t) is strongly jointly typical with s[i] and x (w ,t ). Again, if there is no such index or the ob-
i c i i i 2,i c i 1
− −
served state s[i] is not typical, t is set to 1 and an error is declared. Next, Encoder 1 looks for the smallest j
i i
such that u(w ,t ,t,w ,j) is jointly typical with s[i] given (x (w ,t ),v(w ,t ,t)). Denote this j by j⋆ =
i c i−1 i 1 i 2,i c i−1 i c i−1 i i i
j(s[i],w,t ,t,w ). If such j⋆ is not found, an error is declared and j(s[i],w ,t ,t,w ) is set to j = J. Encoder
c i−1 i 1 i c i−1 i 1 i
1thentransmitsavectorx [i]whichisdrawni.i.d.conditionallygivenu(w ,t ,t,w ,j⋆),s[i],v(w ,t ,t)and
1 i c i−1 i 1 i i c i−1 i
x (w ,t )(usingtheconditionalmeasureP inducedby(7)).
2,i c i−1 X1|U,S,V,X2
Decoding:Attheendofthetransmission,thedecoderhascollectedalltheblocksofchanneloutputsy[1],...,y[B].
Step(a):Thedecoderestimatesmessagew usingallblocksi=1,...,B,i.e.,simultaneousdecoding.Itdeclaresthat
c
wˆ issentifthereexisttB =(t ,...,t ) [1:Mˆ]B,w [1:M ]and jB =(j ,...,j ) [1: J]Bsuchthatx (wˆ ,t ),
c 1 B 1 1 1 B 2,i c i 1
∈ ∈ ∈ −
u(wˆ ,t ,t,w ,j),v(wˆ ,t ,t)andy[i]arejointlytypicalforalli=1,...,B.Onecanshowthatthedecoderobtains
i c i 1 i 1 i i c i 1 i
− −
thecorrectw aslongasnandBarelargeand
c
R +R I(U,V,X ;Y) I(U,V,X ;S). (16)
c 1 2 2
≤ −
Step(b): Next, the decoderestimates message w using againallblocksi = 1,...,B,i.e.,simultaneousdecoding.
1
Itdeclaresthat wˆ issentifthereexisttB = (t ,...,t ) [1 : Mˆ]B, jB = (j ,...,j ) [1 : J]B suchthat x (wˆ ,t ),
1 1 B 1 B 2,i c i 1
∈ ∈ −
u(wˆ ,t ,t,wˆ ,j),v(wˆ ,t ,t)andy[i]arejointlytypicalforalli=1,...,B.Onecanshowthatthedecoderobtains
i c i 1 i 1 i i c i 1 i
− −
thecorrectw aslongasnandBarelargeand
1
R I(U;YV,X ) I(U;SV,X ) (17a)
1 2 2
≤ | − |
R I(U,V,X ;Y) I(U,V,X ;S). (17b)
1 2 2
≤ −
(cid:3)
InthecodingschemeofTheorem1,thestatecompressionisstandard,i.e.,usesnoWyner-Zivbinning,thesame
messageissentineveryblock,andthedecodingofthesentmessageisperformedjointlyusingallblocks.Although
ofnobenefitinthecaseofonerelay,thecombinationofthesethreefeatureswasshowntobeessentialinachieving
rates that are strictlylarger than those offered by schemes based onCover and El Gamalclassic compress-and-
forwardscheme[37]forcertainnetworkswithmultiplerelaysin[30].Thatis,thecodingschemeof[30]outperforms
CoverandElGamalclassiccompress-and-forwardforsomemulti-relaynetworksin[30].Onecanwonderwhether
thesameholdsforourmodel,i.e.,whetherschemesbasedonCoverandElGamalclassiccompress-and-forward,
i.e.,blockMarkovencodingcombinedwithWyner-Zivbinning,fallshortofachievingoptimalityforourmodel.In
thispaper,weshowthatthecapacityregionCasgivenby(8)canbeachievedalternativelywithacodingscheme
thatweobtainbybuildinguponandmodifyingCoverandElGamaloriginalcompress-and-forwardscheme.The
modificationconsistsessentially in1) decodingblock-by-blockbackwardlyinstead of block-by-blockforwardly
and2)non-uniquedecodingofthecompressionindices.(Infact,byinvestigatingmorecloselytheconverseproof
January17,2012 DRAFT