Table Of ContentA FRAMEWORKFORDESIGNINGMIMOSYSTEMS WITHDECISION FEEDBACK
EQUALIZATION ORTOMLINSON-HARASHIMA PRECODING
M. BotrosShenoudaandT. N. Davidson
Department ofElectricaland ComputerEngineering
McMasterUniversity,Hamilton,Ontario,L8S4K1, Canada
7 s x y ˆs
0 ABSTRACT P GH Quantizer
0
We consider joint transceiver design for general Multiple-Input
2 Multiple-Output communication systems that implement interfer- B
n ence (pre-)subtraction, such as those based on Decision Feedback
Transmitter DFE Receiver
a Equalization(DFE)orTomlinson-Harashimaprecoding(THP).We
J developaunifiedframeworkforjointtransceiverdesignbyconsider- Fig.1.MIMOtransceiverwithDecisionFeedbackEqualization.
8 ingdesigncriteriathatareexpressedasfunctionsoftheMeanSquare
2 Error (MSE) of the individual data streams. By deriving two in-
Schur-concave functions of the mean square error (MSE) of each
equalities that involve the logarithms of the individual MSEs, we
datastream. Fortheclassofinterference(pre-)subtraction,designs
] obtainoptimaldesignsfortwoclassesofcommunicationobjectives,
T forDFEbasedschemesusinganMMSEcriterionreceiverwerecon-
namelythosethatareSchur-convexandSchur-concavefunctionsof
I theselogarithms. For Schur-convex objectives, theoptimal design sidered in [3, 4], and designs subject to a zero-forcing constraint
. wereconsideredin[5,6]. SomeTHPcounterpartsofthesedesigns
s resultsindatastreamswithequalMSEs.Thisdesignsimultaneously
c werepresentedin[3]and[7],respectively.
minimizesthetotalMSEandmaximizesthemutualinformationfor
[ In this paper, we develop a broadly applicable framework for
the DFE-based model. For Schur-concave objectives, the optimal
jointtransmitterandreceiverdesignforMIMOsystemswithaDFE
2 DFEdesignresultsinlinearequalization andtheoptimalTHPde-
oraTHP.Weconsiderthebroadrangeofdesigncriteriathatcanbe
v signresultsinlinearprecoding. Theproposedframeworkembraces
expressedaseitherSchur-convexorSchur-concavefunctionsofthe
9 awiderangeofdesignobjectivesandcanberegardedasacounter-
logarithmoftheMSEofeachdatastream,andweprovideoptimal
6 partoftheexistingframeworkoflineartransceiverdesign.
transceiverdesignsforthesetwoclasses. Inadditiontoprovidinga
1
1 Index Terms— Decision Feedback Equalization, Tomlinson- generalizationof existingdesignsbased ontheoverall MSE,these
0 Harashimaprecoding,transceiverdesign,MIMOchannels. classesoffunctionsembraceotherdesigncriteriasuchasminimiz-
7 ingthemaximumoftheindividualMSEs,orminimizingaweighted
0 1. INTRODUCTION geometricmeanoftheMSEs.Moreover,fortheDFEmodel,design
/ criteriaexpressedintermsofthesignaltointerference-plus-noisera-
s
c One of the key advantages of Multiple-Input Multiple-Output tio(SINR)andbiterrorrate(BER)ofeachstreamareincludedinthe
: (MIMO)communications schemesisthattheyfacilitatethesimul- setofobjectivescoveredbytheseclasses. Interestingly,theoptimal
v taneous transmission of multiple data streams. Typically, such designforbothSchur-convexandSchur-convexobjectivesyieldsa
Xi schemes involve processing of the data streams at the transmitter diagonalMSEmatrix. ForSchur-convexobjectives,theoptimalde-
(precoding)andprocessingofthereceivedsignals(equalization)to signresultsindatastreamswithequalMSEs. Furthermore,forthe
r
a “match” the transmission to the channel and to mitigate the inter- DFE model, the optimal design for this class simultaneously min-
ference between the received streams at reasonable computational imizes the total MSE and maximizes the mutual information. For
cost. One approach to the design of such a scheme isto focus on Schur-concave objectives, the optimal design results in linear pre-
linear precoding and linear equalization; e.g. [1, 2]. An alterna- codingandequalization. Fromaboarderprospective,theproposed
tive approach that offers some advantages is to allow interference framework can be viewed as a counterpart for the design of DFE-
(pre-)subtraction at either the transmitter or the receiver. This ap- basedandTHP-basedtransceiversoftheunifiedframeworkforthe
proachincludesschemeswithlinearprecodingandDecisionFeed- designoflineartransceiversin[1].
backEqualization(DFE),andschemeswithTomlinson-Harashima
precoding (THP) and linear equalization, and will be the focus of
2. TWOSYSTEMMODELS
thispaper.
Alargenumberofdesignstrategieshavebeenproposedforthe
Weconsider agenericMIMOcommunicationsysteminwhichthe
classoflinearMIMOtransceivers(e.g.,[2]),andauniformframe-
receivedsignalcanbewrittenasy=Hx+n,whereH∈CNr×Nt
workthatencompassesmanyofthesedesignswasproposedin[1].
representsthechannel,thetransmittedvectorxissynthesizedfrom
Thisframeworkconsistsoffunctionsthatcaptureabroadrangeof
avectors ∈ CK ofdatasymbols, andtheadditivenoisehaszero-
communicationobjectives,namelythosethatareSchur-convexand
mean and covariance matrix E {nnH} = R . Wewill consider
n n
ThisworkwassupportedinpartbyNaturalSciencesandEngineering ageneral designapproach thatencompasses several designcriteria
ResearchCouncilofCanada.Theworkofthesecondauthorisalsosupported fortwocommunication systems, namelythosesystemswithlinear
bytheCanadaResearchChairsProgram. precoding at the transmitter and a DFE at the receiver, and those
Transmitter FormoderatetolargevaluesofM thispowerincreaseisnegligible
s Modulo v P x andtheapproximationE{vvH} = Icanbeused. Wewillusethe
− moreaccurateapproximationE{vvH}=σ2I;e.g.,[3,7].
v
B FortheTHPscheme,thereceivedsignalvectorcanbewrittenas
y=HPC−1u+n,andhencethereceiver’sestimateoftheofthe
Receiver modifieddatasymbolsisuˆ = GHHPC−1u+GHn. Following
y
GH Modulo Quantizer linear equalization, the modulo operation is used to eliminate the
effect of the periodic extension of the constellation induced at the
transmitter. Intermsofthemodifieddatasymbols,theerrorsignal
Fig.2.MIMOtransceiverwithTomlinson-Harashimaprecoding e = uˆ −u = GHHPv+GHn−Cvcanbeusedtodefinethe
MeanSquareErrormatrixE=Ev{eeH}:
i
s u v x E=σ2CCH −σ2CPHHHG−σ2GHHPCH
P v v v
−
+σ2GHHPPHHHG+GHR G. (2)
v n
B
For the TH precoding model, the transmitter power constraint is
givenbyE{xxH}=σ2tr(PPH)≤P .
v total
Fig.3.EquivalentlineartransmittermodelforTHP-basedsystem
systems with THP at the transmitter and linear equalization at the 2.3. GeneralModel
receiver. (The linear transceiver is a special case of both systems
Fromequations(1) and(2), weobservethattheMSEmatrixEof
withthefeedbackmatrixB=0;seeFigs1and 2.)
bothsystemshasacommonform:
2.1. DecisionFeedbackEqualization E=σ2CCH−σ2CPHHHG−σ2GHHPCH+GHR G, (3)
y
AsshowninFig.1,thetransmittedvectorisgeneratedbylinearpre- whereR = σ2HPPHHH +R . FortheDFEmodel σ2 = 1
y n
coding, x = Ps, and hence the received vector y = HPs+n. whilefortheTHprecodingmodelσ2 =σ2.Theaveragetransmitter
v
The DFE is implemented using a feedforward matrix GH and a powerconstraintcanberewrittenastr(PPH)≤P /σ2 =P.
total
strictly lower triangular feedback matrix B ∈ C. Assuming cor-
rect previous decisions, the vector of inputs to the quantizer is
ˆs=(GHHP−B)s+GHn.Definingtheerrorsignale=s−ˆs,
3. OPTIMALFEEDFORWARDANDFEEDBACK
andusingtheassumptionEs{ssH}=I,themeansquareerrorma- MATRICES
trixcanbewrittenas:
WeconsiderthejointdesignofthetransceivermatricesG,C,Pin
E=Es{eeH}=CCH−CPHHHG−GHHPCH ordertooptimizesystemdesigncriteriathatareexpressedasfunc-
+GHHPPHHHG+GHR G, (1) tionsoftheMSEoftheindividualdatastreamsE . Wewilladopt
n ii
three-stepdesignapproach.First,anexpressionfortheoptimalfeed-
whereC=I+Bisaunitdiagonallowertriangularmatrix.Theob- forwardmatrixGHwillbefoundasafunctionofCandP.Second,
jectiveistodesigntheG,C,Pfordifferentdesigncriteria,subject usingtheexpressionoftheoptimalG,anexpressionoftheoptimal
tothetransmitterpowerconstraintEs{xxH}=tr(PPH)≤Ptotal. CwillbefoundasafunctionofP. Finally,usingtheobtainedex-
pressionsofGandC,wewilldesigntheoptimalprecoderP.
2.2. Tomlinson-HarashimaPrecoding
AsshowninFig.2,inTHPthetransmitterperformssuccessivein- 3.1. OptimalfeedforwardmatrixGH
terference pre-subtraction and spatial precoding using the strictly
lowertriangularmatrixBandtheprecodingmatrixP,respectively. ForgivenCandP,theMSEoftheithdatastream,E ,isaconvex
ii
Weassume that theelements of s arechosen froma square QAM functionoftheith columnofG,denotedg ,andisindependent of
i
constellation S with cardinality M and that Es{ssH} = I. The othercolumns. Therefore,thecolumnsofGcanbeindependently
Voronoiregionofthisconstellation,V,isasquarewhosesidelength optimizedtominimizetheindividualMSEs.Asimilarpropertywas
isD.Followingpre-subtractionoftheeffectofpreviouslyprecoded observed in[1] for linear transceivers. Settingthe gradient of E
ii
symbols,thetransmitterusesthemodulooperationsothatthesym- with respect to g to zero, we obtain following expression for the
i
bols of v lie within the boundaries of V. The effect of the mod- optimalG:
ulooperationisequivalenttotheadditionofik = irkeD+iikmagD G=σ2R−y1HPCH. (4)
to s , where ire, iimag ∈ Z. Using this observation, we obtain
k k k Since each g independently minimizes the MSE of the ith data
the standard linearized model of the transmitter shown in Fig. 3 i
stream, theexpression of Gin(4)isalsotheoptimal feedforward
(e.g. [7]), in which v = (I + B)−1u = C−1u. As a result
matrixinthesenseofthesumofMSEs,tr(E). Usingthisexpres-
of the modulo operation, the elements of v are almost uncorre-
sion,theMSEmatrixcanbewrittenas:
lated and uniformly distributed over the Voronoi region V [7, Th.
3.1]. Therefore, the symbols of v will have slightly higher aver- E=σ2C(I+σ2PHHHR−1HP)−1CH =CMCH, (5)
n
ageenergythantheinput symbolss. ForasquareQAM,wehave
σ2 = E{|v |2} = M E{|s |2}forallkexceptthefirstone[7]. wherethematrixinversionlemmahasbeenused.
v k M−1 k
3.2. OptimalfeedbackmatrixB Applyingtheabovelemmatothepositivedefinitelowertriangular
matrixL,weobtainoutfirstinequality:
From(5)weobservethattheMSEofeachdatastreamE isconvex
ii
function of the ith row of C = I+B and is independent of the (L211,...,L2KK)≺× (σ12(L),...,σK2 (L)). (10)
otherrows. Therefore,theoptimalCthatminimizestheindividual
MSEs can be obtained by minimizing any convex combination of The second inequality involves the more common notation of
E . Bychoosingthatconvexcombinationtobethesum,ourgoal additivemajorization:
ii
reducestominimizingtr(CMCH)subjecttoCbeingunitdiagonal AdditiveMajorization[10]:Leta, b∈RK.Thevectorbissaidto
lower triangular matrix. UsingtheCholesky decomposition M = majorizea,a ≺ b,if j a ≤ j b ,forj =1,...,K−
LLH, whereL isalower triangular matrixwithpositive diagonal 1and K a = KPib=1 [i] Pi=1 [i]
elements,wecanrewritetheobjectiveastr(CMCH) = kCLk2F, WPeoib=s1erv[ei]thatPifi=el1em[ein]tsofaandbarepositive,thena≺×
where k·kF denotes the Frobenius norm and the product CL is b⇔ln(a)≺ln(b).Consequently,(10)canbewrittenas:
positivedefinitelowertriangularmatrix[8]. Letλ (CL) ≥ ...,≥
1
λK(CL)andσ1(CL)≥...≥σK(CL)denotetheorderedeigen l≺m, (11)
valuesandsingularvalues,respectively,ofthematrixCL.Thenthe
unitdiagonallowertriangularCthatminimizestr(CMCH)canbe where l = (lnL211,...,lnL2KK) and m =
obtainedusingthefollowinglowerbound: (lnσ12(L),...,lnσK2 (L)).
Toderivethesecondinequality,wewillusethefollowingcon-
K sequenceofadditivemajorization: Anyvectora ∈ RK majorizes
kCLk2 = K σ2(CL) ≥ λ2(CL) (6)
F Pi=1 i X i its mean vector a whose elements are all equal to the mean; i.e.,
i=1 a = 1 K a . Thatis,a ≺ a. Now,sinceM = LLH,we
= XK (CL)2ii=XK L2ii, (7) knioKwthlKat=QPKi=il=n11dLe2itii(=Md)eatn(dLoLuHr)se=codndeti(nMeq)u.aAlitsyaisr:esult,wehave
i=1 i=1 Pi=1 i
wheretheboundin(6)isobtainedbyapplyingWeyl’sinequality[9], l≺l, (12)
and(7)followsfromthefactthatCLislowertriangularandCis
unitdiagonal. Theinequalityin(6)issatisfiedwithequalitywhen whereli = K1 lndet(M).
thematrixisnormal[9].SinceourmatrixCLisatriangularmatrix, Theproposeddesignswillbebasedonthefollowingclassesof
it can only be normal if it is diagonal [8, pp 103]. Therefore, the functions[10]: Areal-valuedfunctionf(x)definedonasubsetA
matrixCthatattainsthelowerboundis: ofRK issaidtobeSchur-convexifa ≺ bonA ⇒ f(a) ≤ f(b),
andissaidtobeSchur-concaveifa ≺ bonA ⇒ f(a) ≥ f(b).
C=Diag(L11,...,LKK)L−1. (8) Inparticular,wewillconsidercommunicationobjectivesthatcanbe
expressed as theminimization of afunctions of theMSEsof each
UsingthisoptimalC,theMSEmatrixcanberewrittenas: datastream,g(L2 ,...,L2 )=g(el1,...,elK))=g(el).
11 KK
E=Diag L2 ,...,L2 . (9)
` 11 KK´
WeobservethatforanygivenprecodingmatrixP,theoptimalfeed- 4.2. Schur-convexfunctions
forward and feedback matrices will yield a diagonal MSE matrix,
withtheindividualMSEsbeingE =L2. Examples of objectives that result in g(el) being a Schur-convex
ii ii function of l include: minimization of the maximum of individ-
ual MSEs: g(el) = max eli; minimization of the total MMSE:
i
4. OPTIMALPRECODINGMATRIXP g(el) = eli; and minimization of the (log) determinant MSE
matrix: dPet(iE) = eli, which is also Schur-concave function
Giventheoptimal GandC, thelaststepistodesignaprecoding of l. For the DFE mQoidel, the SINR of the ith stream is given by
matrixPtooptimizedesigncriteriaexpressedasfunctionsofindi- SINR = (1/MSE )−1 = e−li −1. Hence, manyobjectivesin
i i
vidualMSEofeachstream,L2ii.Wewillfirstderivetwoinequalities termsofSINRandBERcanbeexpressedasSchur-convexfunctions
involvingLiithatenableustocharacterizetheoptimalprecoder. ofl. Aswewillshowbelow,theoptimaltransceiverdesignisiden-
ticalforalltheseobjectives.
4.1. Preliminaries Ifg(el)isaSchurconvexfunctionofl,thenfrom(12)wehave
thatg(el)≤g(el)andtheoptimalvalueisobtainedwhenalll are
i
Toderivethefirstinequality,wewillusetheconceptofmultiplica- equaltol = 1 lndet(M);i.e.,E = L2 = K det(M). Since
tivemajorization: i K ii ii p
the objective is an increasing function of the individual MSE, the
MultiplicativeMajorization[9,10]: Leta,b ∈ RK andleta ≥
+ [1] designgoalreducestominimizingdetMsubjecttothepowercon-
...≥a denotetheelementsofaindescendingorder.Thevector
[K] straintandtotheconstraintthatdiagonalelementsoftheCholesky
bis said to multiplicatively majorize a, a ≺× b, if Qji=1a[i] ≤ factorofMareallequal. Wewillstartbycharacterizingthefam-
j b ,forj =1,...,K−1and K a = K b . ily of solutions that minimize det(M) subject to the power con-
Qi=1 [i] Qi=1 [i] Qi=1 [i]
Animportantexampleofthisdefinitionis: straint, then we will show that there is a member of this fam-
ily that yields a Cholesky factor of M with equal diagonal ele-
Lemma1 Weyl [9]: Let A ∈ CK×K and let λ (A) and σ (A) ments. Minimizingdet(M)isequivalenttomaximizingtheGaus-
i i
denotetheeigenvaluesandsingularvaluesofA,respectively.Then sianmutualinformation,andthefamilyofoptimalprecodersisob-
wehave(|λ1(A)|2, ..., |λK(A)|2)≺× (σ12(A), ..., σK2 (A)). tainedusingastandardwater-fillingalgorithm[11]. Inparticular,if
IfAisnormal,then|λi(A)|=σi(A). RH =σ2HHR−n1H=UΛHUH,thefamilyofoptimalprecoders
takestheform:
100
P=U1ΦˆV=U1[Φ 0]V, (13) 10−1
whereU1 ∈CNt×Kˆ containstheeigenvectorsofRHcorrespond- ms10−2
itnivgetdoefithneiteKˆm≤atriKxΦlaragreesotbetiagiennedvafrloumes,thKˆewanatdert-hefildliinaggoanlgaolrpitohsmi- R of K strea10−3
Sr[e1cs1hu]ul,tra-scnhodonwVvesxt∈hfauCtncfKoti×roKDnFoiEfslab,atuhsneeidtoapsryytismmteaamltrssioxdluedsteiioggnrneeiedsoianfcfcfororemreddainotigmon.tolToahsnsiys- Average BE1100−−54 LLDiiFnnEeeaa−rrS−−CMDoeMnt(SvEeE)x
less. To complete the design of P, weneed to select V such that 10−6 THP−Sconvex
theCholeskydecompositionofM = LLH yieldsanLfactorwith
equaldiagonalelements.Using(13): 10−7
0 4 8 12 16 20 24 28
SNR in dB
M = “VH(I+ΦˆTΛH1Φˆ)−1/2”“(I+ΦˆTΛH1Φˆ)−1/2V”
Fig.4.BERsoftheproposedSchur-convexdesignsandtheoptimal
= LLH =RHR=(QR)H(QR), (14) lineartransceivers: minimumMSE(Linear-MMSE),andmaximum
mutualinformation(Linear-Det(E)),forN =N =K =4.
where ΛH1 is the diagonal matrix containing the largest Kˆ eigen t r
valuesofRH,andQisamatrixwithorthonormalcolumns.Hence, design resultsinlinear equalization andtheoptimal THprecoding
findingVisequivalenttofindingaVsuchthatQRdecomposition
designresultsinlinearprecoding.
of(I+ΦˆTΛH1Φˆ)−1/2VhasanR-factorwithequaldiagonal.This
problemwassolvedin[6]and V canbeobtained byapplying the
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