A Rate-Splitting Approach To Robust Multiuser MISO Transmission

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by  Hamdi Joudeh

A RATE-SPLITTING APPROACHTO ROBUST MULTIUSER MISO TRANSMISSION
Hamdi Joudeh∗ andBrunoClerckx∗†
∗ DepartmentofElectrical and ElectronicEngineering,ImperialCollegeLondon,United Kingdom
† School ofElectrical Engineering,KoreaUniversity,Seoul,Korea
Email: {hamdi.joudeh10, b.clerckx}@imperial.ac.uk
6
1 ABSTRACT packing thecommon parts such that Wc = {Wc,1,...,Wc,K} ∈
0 For multiuser MISO systems with bounded uncertainties in the Wc, where Wc = Wc,1 ×...×Wc,K. Wc is encoded using a
2 Channel StateInformation (CSI),we consider twoclassical robust commoncodebooksharedbyallusers,whileW1,...,WK areen-
codedusingprivatecodebooksknownbytheircorrespondingusers.
n designproblems:maximizingtheminimumratesubjecttoatransmit
a power constraint, and power minimization under a rate constraint. TheresultingK+1encodedsymbolstreamsarelinearlyprecoded
J Contrary to conventional strategies, we propose a Rate-Splitting andsimultaneouslytransmitted.Ateachreceiver,thecommonmes-
sageisdecodedfirstbytreatingallprivatesignalsasnoise. Thisis
9 (RS)strategywhereeachmessageisdividedintotwoparts,acom-
followedbydecodingtheprivatemessageafterremovingthecom-
1 monpartandaprivatepart. Allcommonpartsarepackedintoone
super common messageencoded usingasharedcodebook andde- mon message via Successive Interference Cancellation (SIC). The
] codedbyallusers,whileprivatepartsareindependentlyencodedand originalmessagesaredeliveredgiventhateachreceiversuccessfully
T decodesthecommonmessageanditsprivatemessage.
retrievedbytheircorrespondingusers. WeprovethatRS-basedde-
I A special case of the described RS scheme [7,8], where only
signsachievehighermax-minDegreesofFreedom(DoF)compared
.
s to conventional designs (NoRS) for uncertainty regions that scale oneuser messageissplit, wasshown toboost thesum Degreesof
c Freedom (DoF) under CSITerrors that decay withincreased SNR
withSNR.Forthespecial caseof non-scaling uncertainty regions,
[ RS contrasts with NoRS and achieves a non-saturating max-min atarateofO(SNR−α)forsomeconstantα∈ [0,1]. Thisstrategy
1 rate. In the power minimization problem, RS is shown to combat was leveraged to enhance the sum-rate performance under various
v thefeasibilityproblemarisingfrommultiuserinterferenceinNoRS. CSIT assumptions [9,10]. RS was also shown to provide signifi-
5 ArobustdesignofprecodersforRSisproposed, andperformance cantsum-rategainsinthelarge-scalearrayregime[11]. Employing
0 gainsoverNoRSaredemonstratedthroughsimulations. RSinamoregeneralmannertoachievemax-minfairnesswasfirst
1 reportedinourpreviouswork[12],wheretheaverageperformance
5 IndexTerms— MISO-BC,degreesoffreedom, linearprecod- overtheerrordistributionwasconsidered. Thispapergivesamore
0 ing,max-minfairness,quality-of-service,robusttransceiverdesign. completetreatmentofthemax-minproblembyderivingtheasymp-
. totic rate performance (in a DoF sense) of the optimally designed
1
1. INTRODUCTION RSscheme. Inaddition, theapproachisextendedtotacklethein-
0
versepoweroptimizationproblem.Weshouldalsohighlightthatthe
6
ConsideraMulti-User(MU)MultipleInputSingleOutput(MISO) worst-caseoptimizationconsideredinthisworkposesanextrachal-
1
: systemwhereaBaseStation(BS)equippedwithNtantennasserves lenge in comparison to [12] due to the minimization embedded in
v asetofsingle-antennausersK , {1,...,K},withK ≤ Nt. Un- eachworst-caserateexpression. Therestofthepaperisorganized
i der the assumption of erroneous Channel State Information at the as follows. The system model is described in Section 2. In Sec-
X
Transmitter (CSIT) with bounded uncertainty regions, we address tion 3, the problem isformulated and the asymptotic performance
r twotypicaldesignproblems: maximizingtheminimumworst-case isderived. AnoptimizeddesignofprecodersfortheRSstrategyis
a
ratesubjecttoatotaltransmitpowerconstraint(rateproblem),and proposedinSection4.Simulationresultsandanalysisarepresented
minimizingthetransmitpowersubjecttoaworst-caserateconstraint inSection5,andSection6concludesthepaper.
(power problem). Such problems, which are non-convex in gen-
eralandsemi-infinite,wereaddressedinliteratureusingvariousap-
proaches and approximations [1–5]. However, all existing works 2. SYSTEMMODEL
consider aconventional transmissionscheme, i.e. eachmessageis
encodedintoanindependent datastream, thenallstreamsarespa- Forthesystemintroducedintheprevioussection,consideratrans-
tiallymultiplexed. Fortherateproblem,suchdesignsareknownto missiontakingplaceoverablockofchanneluseswherethechannel
yieldasaturatingperformanceathighSNRswhereMUinterference remainsfixed(quasi-static). Thesignalreceivedatthekthuserina
becomesdominant[4–6].Conversely,thiscreatesafeasibilityissue givenchannelusewritesasy =hHx+n ,whereh ∈CNtisthe
k k k k
forthepowerproblem,sinceratesbeyondthesaturationlevelcan- channelvectorfromtheBStothekthuser,x∈CNt isthetransmit
notbeachieved[1,4].Inthiswork,weproposeaRate-Splitting(RS) signal,andn ∼CN(0,σ2 )istheAdditiveWhiteGaussianNoise
k n,k
strategytocombattheseshortcomings. (AWGN)atthereceiver. Thetransmitsignal issubject toanaver-
LetWt,1,...,Wt,Kbemessagesuniformlydrawnfromthesets age power constraint E{xHx} ≤ Pt. Without loss of generality,
Wt,1,...,Wt,K,andintendedforreceivers1,...,K respectively. weassumeequalnoisevariancesacrossusers,i.e. σn2,k = σn2,and
Intheproposedscheme,eachusermessageissplitintoaprivatepart channelentrieswithnormalizedaveragegains. Therefore,thelong
W ∈W andacommonpartW ∈W ,whereW ×W = termSNRisdefinedasSNR , P /σ2. Moreover, σ2 isnon-zero
k k c,k c,k k c,k t n n
W . A super message (the common message) is composed by andremainsfixed.Hence,P →∞impliesSNR→∞.
t,k t
Wc,W1,...,WKareencodedintotheindependentdatastreams correspondstoperfectCSITresultingfromaninfinitelyhighnum-
sc,s1,...,sK respectively. For a given channel use, symbols are beroffeedback bits. Inthefollowing, theexponentsaretruncated
grouped ass , [sc,s1,...,sK]T ∈ CK+1, whereE{ssH} = I. suchthatα ∈ [0,1]whichiscustomaryinDoFanalysisasα = 1
ThesymbolsaremappedtotheBSantennasthroughtheprecoding correspondstoperfectCSITintheDoFsense[6,7].
matrixP,[pc,p1,...,pK],yieldingthetransmitsignalx=Ps, DuetotheCSITuncertainty,theactualratescannotbeconsid-
where the common stream is superimposed on top of the private ered as design metrics at the BS. From the BS’s point of view,
streams. Thepowerconstraintisrewrittenastr PPH ≤ P . The achievable rates also lie in bounded uncertainty regions. We
t
kthaveragereceivepowerisexpressedas consider a robust design where precoders are optimized w.r.t the
(cid:0) (cid:1) worst-case achievable rates defined for the kth user as R¯ ,
c,k
Sc,k Sk Ik minhk∈HkRc,k(hk) and R¯k , minhk∈HkRk(hk), where the de-
Tc,k =z|hHk}p|c|2{+z|hHk}p|k|2{+zXi6=k|hHk}p|i|2+σn2{. (1) pamceihnttidienevngacbiWleescc,ooWmf mt1h,oe.n.r.raa,tteWesiKsondaetfihnkreadtaerasesRRh¯¯iccg,h,R¯li1gm,h.tien.d.j.,{RR¯¯TKc,hje}reKjw=sp1oe.rcsTtti-rvcaeanlssye-,
Ic,k=Tk guarantees successful decoding at the receivers for all admissible
channelswithintheuncertaintyregions. FollowingtheRSstructure
| {z }
Thecommonmessageisfirstdecodedbytreatinginterferencefrom inSection1, thekthuser’sportionof thecommon rateisdenoted
aalslspcr,ikva=tesgigc,nkaylsk,aswnhoeirsee.gTc,hkeisktahsucsaelra’rseeqstuiamliazteer.ofGsicviesnotbhtaatintehde tboytaCl¯kacwhiheevraeblerKka=te1Ci¯skgi=venR¯acs. TR¯hte,kre=forRe¯,kth+eCk¯tkh,ucsoerrr’esspwoonrdsitn-cgastoe
common message is successfully decoded, SIC is used to remove therateatwhiPchtheoriginalmessageW istransmitted.
t,k
the common signal from y before decoding the private message.
b k
Theestimateofs writesass = g (y −hHp s ), whereg
k k k k k c c,k k 3.1. RateOptimizationProblem
isthecorrespondingequalizer. Attheoutputofthekthreceiver,the
commonandprivateMSEs,dbefinedasεc,k , E{|sc,k−sc|2}and UsingtheRSstrategy, therobust rateoptimizationproblem which
ε ,E{|s −s |2}respectively,writeas: achievesmax-minfairnessisposedas
k k k
b εc,k =|gc,k|2Tc,k−2ℜ gc,khHkpc b+1 (2a) R¯tm,R¯ac,x¯c,P R¯t
w(ChSeIrRe)Tkobitsadineεefikdn=etdhr|iognuk(g|12hT).kcWo−mem2aℜsos(cid:8)nugmak(cid:8)nehdpHkedprefkde(cid:9)cicta+CteS(cid:9)1dIadtotwhnelRinekcetirv(a2eibnrs)- RRS(Pt):s.t. RRC¯¯¯kcKk,k+≥=1≥C0¯C¯,Rk¯k∀c≥=k, R∀∈¯R¯ktcK,∈∀Kk∈K (3)
ing. ThisassumptionisjustifiedbynotingthattheeffectsofCSIR Pk
ewUshstieimcrshateiismonopvleoeryrrwoorhpsetliammreeudmatb(ytghtehe,sagimnfl)e,ueip.noecw.eetorhfeleCwvSeeIllTl-okfunnoacwdedrnittaiMivneitnynim[o1ius3me],. whereR¯t isanauxiliaryvariable,tra(cid:0)nPdP¯cH,(cid:1)[≤C¯1P,t....,C¯K]T. Point-
MSE (MMSE) equalizerscg,kivenk as: gMMSE = pHh T−1 and wiseminimizationsinR¯t andR¯c arereplacedwithinequalitycon-
c,k c k c,k straintsin(3),whereequalityholdsatleastforoneuseratoptimal-
gεMkMMMSSEE ==Tp−Hk1hIkTk−an1d, fεrMomMSwEhi=chTth−e1IM.MTShEeskatrheSoIbNtaRinsewdraitse: ity. The constraints C¯k ≥ 0 guarantee non-negative splitting. In
c,k c,k c,k k k k contrast,theNoRSversionoftheproblemisformulatedas
as: γ = S I−1 = (1−εMMSE)/εMMSE andγ = S I−1 =
c,k c,k c,k c,k c,k k k k max R¯
(1−εMkMSE)/εMkMSE. Under Gaussian signalling, the kth max- R¯,Pp
−eimnsluoumgre2a(tεchhMca,iktMevaSalElb)lueasnerardsteRsdkeacro=edgeliovWge2n(,1asi+t: isRγkct)r,kan=s=m−itltloeogdg22(i(n1εMk+aMmγScuE,lk)ti.)caT=sot R(Pt):s.t. tR¯rkP≥pPR¯Hp, ∀k≤∈PKt. (4)
c
fashionatthecommonrateRc ,minj{Rc,j}Kj=1. where R¯ is the rate auxiliary varia(cid:0)ble, and(cid:1)Pp , [p1,...,pK].
Solving(4)isequivalenttosolving(3)overarestricteddomainchar-
3. PROBLEMSTATEMENTANDASYMPTOTIC acterizedbysetting¯c = 0, whichinturnforcesR¯c and kpck2 to
PERFORMANCE zerosatoptimality. Asaresult,wehaveRRS(Pt)≥R(Pt). Next,
welookattheoptimumasymptoticperformanceofthetwoschemes.
For each channel vector h , the BS obtains an erroneous estimate ToanalysetheperformanceasSNRincreases,wedefineapre-
k
h ,fromwhichtheerrorunknowntotheBSisdefinedash ,h − coding scheme for (3) as a family of feasible precoders with one
k k k
h . AsfarastheBSisaware,h isboundedbyanorigin-centered precoderforeachSNRlevel,i.e. P(P ) . Theassociatedpow-
k k t Pt
rsbbepghieorneHwikth,radihuskδ|kh.kH=enchek,h+ekhisk,ckohnfiknke≤dwδkithi.nTthheisueinscreerlteavinatnyt earsskapllcokc2at=edOto(Pthtaecp)raencoddkinpgkkv2ec=(cid:8)torOs(aPreta(cid:9)aks)s,uwmheedretoascc,aalkew∈it[h0,P1t]
aresomescalingexponents. Under agiven precoding scheme, the
inlimitedfeednbacksystemsasquantizationerroorsarebounded. In
kthworst-casecommonandprivateDoFaredefinedas
scenarioswherethenumberboffeeedbackebitsismadetoscalewithin-
creasedSNRtoprovideimprovedCSITquality[6,9],theuncertainty R¯ (P ) R¯ (P )
sretagniotnexsphorinneknststuhcahtqthuaatnδtik2fie=stOhe(PCt−SαIT),qwuhaelirteyαas∈SN[0R,∞gr)owissalcaorgne- d¯c,Ptli→m∞logc2(Ptt) and d¯k ,Ptli→m∞logk2(Ptt) (5)
(αas=sum0eredptroesbeenttsheascaomnsetaancrto(sosrussloewrsl)y,is.ec.alαin,g)lniummPbt→er∞of−felleoodgg(b(δPak2tc))k. wThheerkeththeusdeerp’sensdpelintcoiefsd¯ocnisthdeefiponwederalsevc¯ekl a,relhimigPhtli→gh∞tedlCo¯gki2n((PP(t5t))),.
bits,yieldingnon-decayingCSITerrors.Ontheotherhand,α=∞ where K c¯ = d¯. All definitions extend to the NoRS case
k=1 k c
P
wherethecommonpartisdiscarded,andaprecodingschemefor(4) change withthe transmit power variation during the design proce-
isdenotedby P (P ) . dure,aschannelestimationandfeedbackiscarriedoutpriortothe
p t Pt precoder design. The rate and the power problems are monotoni-
Theorem1. (cid:8)TheNoR(cid:9)Sproblemyieldsanoptimummax-minDoF cally non-decreasing in their arguments, and are related such that
R P(R¯) = R¯ andR P (R¯ ) = R¯ ,whichcanbedemon-
RS RS t t
R(P )
d¯∗ , lim t =α (6) stratedusingthesamestepsin[15,16]. CombiningthiswithThe-
Pt→∞log(Pt) ore(cid:0)m1,it(cid:1)followsthatund(cid:0)ernon-sca(cid:1)lingCSITerrors,R Pt con-
vergestoafinitemaximumvalueasP → ∞, whichisthemaxi-
whiletheoptimummax-minDoFfortheRSproblemisgivenby mumfeasiblerateforP(R¯). Ontheothterhand,R (P )(cid:0)do(cid:1)esnot
RS t
d¯∗ , lim RRS(Pt) = 1+(K−1)α. (7) converge.Hence,anyfiniterateisfeasibleforPRS(R¯t).
RS Pt→∞ log(Pt) K
4. ROBUSTOPTIMIZATION
Proof. Let {a∗}K be the power exponents of (4)’s optimum
k k=1
precoding scheme. We initially assume that the optimum pow- Problems(3) and(8) aresemi-infiniteand appear tobeintractable
ers satisfy a∗,...,a∗ = a∗. We have d¯∗ ≤ min{α,a∗}. intheir current forms. Even finiteinstances of the problems seem
1 K k
This is shown through upper-bounding the worst-case SINR, i.e. to be intractable due to the non-convex coupled sum-rate expres-
γ¯k , minhk∈Hkγk(hk), by selecting hk ∈ Hk such that the lth sions embedded in each user’s total rate. Therefore, we employ
(l∈K\k)interferenceismaximized,anddiscardingsomeinterfer- theRate-WeightedMSE(WMSE)relationshipwhichisparticularly
ence terms[14]. Sincethe point-wise minimum isupper-bounded suitableforproblems featuringsum-rateexpressions [17–19]. The
by any element in the set, and d¯∗ ≤ d¯∗ for all k ∈ K, we have kthuser’saugmented WMSEs(referredtoasWMSEsfor brevity)
k
d¯∗ ≤ α. Next, assume that (3)’s optimum precoding scheme has are defined as: ξc,k , uc,kεc,k −log2(uc,k) and ξk , ukεk −
power exponents a∗ anda∗ for thecommon andprivateprecoders log (u ),withu andu asthecorrespondingweights. Optimiz-
c 2 k c,k k
respectively.UsingR¯ +R¯ ≤R¯ +R¯ ,andthepreviousbound- ing over the equalizers and weights, the Rate-WMSE relationship
c k c,k k
ingtechniques,weobtaind¯∗c+d¯∗k ≤min{1+α−a∗,1}.Sincethe writesas: ξcM,kMSE , minuc,k,gc,kξc,k = 1−Rc,k andξkMMSE ,
max-minDoFisupper-boundedbytheaverageuserDoF,wewrite minuk,gkξk =1−Rk,wheretheoptimumequalizersandweights
d¯∗RS ≤ d¯∗c+PKKk=1d¯∗k ≤ 1+(KK−1)α,wheretheright-mostinequality aregivenby: gc∗,k = gcM,kMSE,gk∗ = gkMMSE,u∗c,k = εMc,kMSE −1,
isobtainedfromd¯∗+d¯∗ ≤1andd¯∗ ≤α. andu∗ = εMMSE −1 [10,12]. Fromthisrelationship, theworst-
c 1 k k k (cid:0) (cid:1)
The upper-bounds are achieved through feasible precoders as caseratesareequivalentlywrittenas
(cid:0) (cid:1)
follows. Best-effortZero-Forcing(ZF)obtainedusingtheavailable R¯ =1− max min ξ h ,g ,u (10a)
channelestimatewithpowersscalingasO(Ptα)isusedforprivate c,k hk∈Hkuc,k,gc,k c,k k c,k c,k
precoders,achievingd¯1,...,d¯K =α.ForRS,thisissuperimposed R¯k =1− max min ξk hk,(cid:0)gk,uk . (cid:1) (10b)
by a random common precoder with power that scales as O(Pt), hk∈Hkuk,gk
achievingd¯c = 1−αwhichissplitequallyamongusers. Relax- Equivalent WMSEproblems areo(cid:0)btained by(cid:1)substituting(10)into
ingtheassumptionthata∗,...,a∗ areequalyieldsthesameresult (3)and(8),wherethedomainsareextendedtoincludetheequaliz-
1 K
usingslightlymoreinvolvedupper-boundingsteps. ersand weightsasoptimization variables. Suchproblems have an
interestingblock-wiseconvexstructurewhichcanbeexploitedus-
ItshouldbenotedthatalthoughtheZFprecodersproposedinthe
ingtheAlternatingOptimization(AO)principle. However,thenew
proofareoptimuminaDoFsense, theymaybefarfromoptimum
problemshaveinfinitelymanyoptimizationvariablesandconstraints
fromaworst-caserateperspectiveatfiniteSNRs.Thismotivatesthe
duetothedependencies oftheoptimumequalizersandweightson
needfor robustlydesigned precoders. Itisevident that d¯∗ ≥ d¯∗
RS perfectCSI.Weresorttotheconservativeapproximationin[14]by
holds for all α ∈ [0,1], and strictly holds for α ∈ [0,1). d¯∗ is
RS swapping the minimization and maximization in (10). Equalizers
lower-boundedby1/K. Hence,anoptimallydesignedRSscheme
andweightsloosetheirdependenciesonperfectCSIandweobtain
isexpectedtoachieveanever-growingmax-minrate.
R =1− min max ξ h ,g ,u (11a)
c,k c,k k c,k c,k
ubc,k,gbc,khk∈Hk
3.2. PowerOptimizationProblem (cid:0) (cid:1)
bRk =1− min max ξk hk,gk,ubk b (11b)
ThepowerproblemwitharateconstraintR¯twritesas ubk,gbkhk∈Hk (cid:0) (cid:1)
min tr PPH whereRcb,k ≤ R¯c,k andRk ≤ R¯k areblowebr-boundsontheworst-
PRS(R¯t):sR¯.ct.,¯c,P RR¯¯kc(cid:0)Kk,k+=1≥C¯C¯kR¯k(cid:1)c≥=, R∀¯R¯ktc,∈∀Kk∈K (8) caraaapntspdeelip(erubrdaoctbb,tekolse,amub(lslkeyc)eiheaaflrondeonsttetnhlhoesetiaencbo1tsbnhtrseiaenucrvtn[ea1cdt4eivr,eteqaSiuWenactltyMiizoseSnerEtssI.VcaoPn.Budlun.2gwte]g)eri,pingaaghrntt(ds11w()bghicin,cktho,bgtahkree)
OntheotherhanPd(,Rt¯h)e:N(oRmsP.tSi.pncouCPtR¯n¯rktk(cid:0)eP≥r≥ppa0PRr¯,tHp,i∀s(cid:1)∀kfko∈r∈mKuK.la.tedas (9) RRS(Pt):s1R1b.−tt−.,RbξmξckKc,(cid:0)bca,kh,xPk,Ch,gbgkb,ubk,=g,Rucbb,Rktk(cid:1),u+cC,bkk ≥≥RRbtc,,∀∀hhkk∈∈HHkk,,kk∈∈KK
b k=1(cid:0) k c (cid:1)
AW0.seT(c9ho)insissiidsaerrreelnesovtrnaic-nsttectdaolivn(e8gr)sCiaoSnndITo(f9w)(i8twh),hδwe12ree,.ht.ha.ev,eδCK2PSIRT=S(qROu¯at(l)1it)y≤, id.ePo.e(sαR¯nt=o)t.  PCtbrk(cid:0)P≥P0Hb,(cid:1)∀≤bk∈PbtKb b (12)
where g , {g ,g | k ∈ K} and u , {u ,u | k ∈ K}.
c,k k c,k k
Extendingthisapproachtothepowerproblemyields 7 NRSoRS 7 NRSoRS
6 6
b b mibn tr PPH b b b Hz)5 Hz)5
PRS(Rt):s1R1b.−ct−.,bcξ,ξkPc(cid:0),,gkbh,kubh,kgk,(cid:0)g,uc,kk(cid:1),u+c(cid:1)C,kk ≥≥RRtc,,∀∀hhkk∈∈HHkk,,kk∈∈KK Rate(bps/1234 Rate(bps/1234
Tbhesembi-infiniteCPbnkeKks≥=s1i(cid:0)s0Cb,ebkl∀im=kbibn∈RbatcKebd.bby(cid:1)refobbrmulatingtheinfinite(s1e3ts) Fig0.51(.a1R0)aαte1=5SpNeR0r2,f0(odδBrm=)25a0n.c310efo3r5K =(b)N0α5t==1003.,51a,5SnδNdR=2δ0(1d0B,.δ)2152p,δ31300=P3t−5δ.α
ofrateconstraintsintoequivalentLinearMatrixInequalities(LMIs)
ucosinnsgtrtaτhikneψt−r0eikubnsλku(k(cid:0)l1tτ2ik)n−a+[n2δd0|ψkbg]g(Ik,kH1k|b32Pa)σspine2sd(cid:1)reo−−wnδlrtokihλtg0Pteek2TSnHp(Iub-abgpkskHr)o≤ced1(cid:23)u+re0.C,bTkλhk−ek≥Rbtht0tota((l11r44abate)) Num.offeasiblechannels124680000000 0.0NR1SoR0S.05 δ 0.1 0.15 TransmitPower111000012fceha.s0:..01NR100SoR0S.0953δ 0.611 0.1359
whilethekthcommonrateconstraintisexpressedas
b (a)Feasiblechannelsoutof100 (b)Averagetransmitpower
u τ +|g |2σ2 −log (u )≤1−R (15a)
c,k c,k c,k n 2 c,k c
Fig.2.Powerminimizationunderarateconstraintof3.3219bps/Hz,
τc,kψ−cb,kλc,k(cid:0) ψIcHb,k −(cid:1)δkP0THgcHb,k(cid:23)0,λc,kb≥0 (15b) forK =Nt =3,andδ1,δ2,δ3 =δ.
0 −δ g P λ I
k c,k c,k uncertainty regions, from which channel estimates are obtained as
whereψH , g hHP −eT andψH b, g hHP−eT. Fora hk =hk−hk,∀k ∈K. Fig. 1showstheconservativeworst-case
k k k pb k c,k c,k k 1 rate performance averaged over 100 channels. The scaling uncer-
detailed description of the procedure, please refer to [2] and [14].
tbaintyinFig.e1bcoincideswiththenon-scalinguncertaintyinFig.1a
Next,wedevelobpbanunifiedAOalgorithmthbatsoblves(12)and(13). at20dBSNR.ItisevidentthatRSoutperformsNoRSforallSNRs,
withalmost30%improvementachievedathighSNRinFig.1a,and
4.1. AlternatingOptimizationAlgorithm 25%improvementachievedatintermediateSNRinFig.1b.Atfirst
glance,theRSratesaturationinFig. 1aseemstocontradictTheo-
Ineachiterationofthealgorithm,gisfirstoptimizedbysolvingthe
rem1. However,thiscanberegardedtothedesign’ssub-optimality
problemsminmaxε h ,g andminmaxε h ,g forall
c,k k c,k k k k
k ∈ K, fogbcr,mkuhlka∈teHdkwith(cid:0)objectiv(cid:1)be funcgbtikonhsk∈τHk +(cid:0)|g |2(cid:1)σ2 and amnidndthtehalotoTsheenoersesmof1thqeuacnotnifiseersvathtieveopatpimpruomximpeartfioornm[a1n4c],eb.earingin
τ +|g |2σ2,andconstraintsb(15b)and(14b),rec,skpectivecl,ykb(wenights Forpoweroptimization,Fig. 2acomparesthefeasibilityofRS
k k n
are fixed and ignored in this step). Such problems are posed as andNoRSover arangeofδs, whereRSachieves animprovement
b
Semidefinite Programs (SDPs) and can be solved efficiently using exceeding100%comparedtoNoRSatδ =0.15.Ontheotherhand,
b
interior-point methods [21]. The resulting conservative MMSEs, Fig. 2b shows the improved power performance. Intuitively, we
εMMSEandεMMSE,areusedtoupdatetheweightsinthenextstepas expectthepowergaptoincreasewithδ. However,sinceinfeasible
k c,k
u =1/εMMSEandu =1/εMMSE.Finally,(P,c)andtheaux- channelsareomittedintheaveragepowercalculation,averagingis
k k c,k c,k
ibliaryratevabriablesareupdatedbysolvingaSDPformulatedbyfix- restrictedtoverywellconditionedchannelsforlargerδs.
ing(g,u)in(12),or(13).Thisprocedureisrepeatedinaniterative
b b b b b
manner until convergence, which is guaranteed since the bounded
6. CONCLUSION
objectivefunctionsbehavemonotonicallyoveriterations. However,
b b
appropriateinitializationofPisrequiredforthepowerproblemto In this contribution, we developed a robust RS transmission strat-
avoidfeasibilityissues.Thisisdonebyperformingrateoptimization egy to address the rate and power design problems in MU-MISO
fordifferentpowerconstraintsinthefirstfewiterationsuntilafea- systemswithCSITuncertainties. ThisbuildsupontheexistingRS
siblesolutionisfound,beforeswitchingtopoweroptimization.The approach used to boost the sum-DoF and sum-rate performances.
conservativeapproximationsguaranteethattheAOalgorithmyields WeanalyticallyprovedthatproperlydesignedRSschemesachieve
feasible (although possibly sub-optimal) solutions for the original superiormax-minDoFperformancescomparedtotheirNoRScoun-
problems. However, global optimality cannot be guaranteed even terparts. Moreover, we showed that RS can be used to tackle the
w.r.t(12)and(13)duetonon-convexity. Despitethissub-optimality, feasibilityproblemappearinginNoRSpowerdesigns.Weproposed
suchalgorithmswereshowntoperformwell[14]. a sub-optimal unified algorithm that solves the robust RS rateand
powerproblemsbasedonaconservativeapproximation. Thesupe-
5. SIMULATIONRESULTS riorperformanceoftheRSalgorithmcomparedtoitsNoRScounter-
partwasdemonstratedthroughsimulations. Intheextendedversion
We consider a system with K,N = 3, and i.i.d channel entries of this work, we seek todevelop anon-conservative robust design
t
drawnfromCN(0,1). Wesetσ2 = 1, yieldingalong-termSNR thatachievesthetheoreticallyanticipatednon-saturatingrateperfor-
n
of P . CSIT errors are uniformly drawn from the corresponding manceundernon-scalingCSITuncertainties.
t
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