On Achivable Rates in a Multi-Antenna Gaussian Broadcast Channel
GiuseppeCaire
Eurecom,Sophia-Antipolis,France
ShlomoShamai
Technion-32000Haifa,Israel
Abstract | A Gaussian broadcast channel with r
single-antennareceivers and t antennas at the trans-
mitter isconsidered. Both transmitter and receivers
haveperfectknowledgeofthechannel. Despiteappar-
entsimplicity,thismodelisingeneralanon-degraded
broadcast channel, for which the capacity region is
not fully known. We propose a novel transmission
schemebasedon\rankedknowninterference"(RKI).
Inbrief,thetransmitterdecomposesthechannelinto
an ordered (or ranked) set of interference channels
forwhichtheinterferencesignalofthei-thchannelis
generatedasalinearcombinationofthesignalstrans-
mitted in channels j < i. In this way, known tech-
niques of coding for non-causally known interference
canbeappliedtomaketheinterferenceineachchan-
nelharmlesswithoutfurtherpowerpenalty. Weshow
that the proposed scheme is throughputwise asymp-
toticallyoptimalforbothlowandhighSNR.Wepro-
vide a modication of the basic RKI scheme which
achievesoptimalthroughputfor all SNRs inthe spe-
cial case of 2 antenna and 2 users. For independent
Rayleigh fading closed-form throughput expressions
areobtained in variouscases ofinterest and numeri-
calexamplesareprovidedfortheinnite-dimensional
Rayleigh channel.
I. Problem statement
The t1:::r Gaussian boradcast channel (GBC) is de-
scribedbyy = Hx+zwhere the notationt1:::r desig-
nates that the r receivers cannot cooperate. Here, x 2 C t
denotesthevectorofsymbolstransmittedinparallelfromthe
tantennasinany channeluse, H 2C rt
denotes thechan-
nelmatrix,whosei;j-thelementisthecomplexchannelgain
from antenna j to antenna i, y 2 C r
denotes the vector of
receivedsignalsattherreceiversandz2C r
isthenoisevec-
tor, N
C (0;N
0
I). The channelmatrixH is knownto the
transmitterandtoallreceiversandtheinputisconstrainedto
satisfy trace (E[xx H
]) E where E isthe maximumallowed
totaltransmitenergyperchanneluse.
WeareinterestedinthechannelthroughputR gbc
(orrate-
sum),denedasthesumofallachievableindividualuserrates.
We shallconsider the following scenarios: i)H is determin-
istic andxed;ii)His xedduring thetransmissionofeach
code word,but it is randomly selectedaccording to agiven
probabilitydistribution(compositechannel).
II. The \rankedknowninterference" scheme
Let H = GQ be a QR-type decomposition of H where
G2C rm
islowertriangularandQ2C mt
hasorthonormal
rows (we dene m
= minfr;tg). The transmitted signal is
obtainedasx=Q H
u. Letg
i;j
denotethe(i;j)-thelementof
Gandletdi=jgi;ij 2
.Theoriginalchannelisturnedintothe
setofinterferencechannelsy
i
=g
i;i u
i +
P
j<i g
i;j u
j +z
i ,for
i=1;:::;m,whilenoinformationissenttousersm+1;:::;r.
H H H
uiare allgeneratedbythetransmitter: foreachi-thchannel
u
i
isthewantedsignaland P
j<i g
i;j u
j
isaninterferencesignal
non-causallyknowntothetransmitter. Withsomecare[5 ],we
canapplytheresults of[2 ]or(equivalently)themorerecent
results of[3 ], and show thatbyusinganappropriatecoding
strategy,the capacityofeachi-thchannelisequalto thatof
thevirtualinterference-freechannely
i
=u
i +z
i :
Proposition 1. Themaximumachievable throughputof
theRKIschemeisgivenbyR rki max
= P
m
i=1 [log(d
i )]
+ where
isthesolutionof P
i=1 [ 1=d
i ]
+
=
=E=N
0
.
III.Results
We denote by R zf max
and by R
coop max
the maximum
achievable throughputs of a conventional system based on
zero-forcing beamforming [4] and of a system where the re-
ceivers are allowed to cooperate (equivalent to the single-
user multiple-antenna case [1 ]). For H given and constant
wehave thefollowing results: 1) ForanychannelmatrixH,
R rki max
R zf max
;2) Foranychannel matrixH withfull
row-rankr,limA!1 R
coop max
R rki max
=0;3)Forany
channelmatrixH,lim
A!0 R
rki max
R zf max
=1.
Notice that for H of rank r the RKI is asymptoti-
cally throughputwise optimalfor large SNR,since obviously
R
coop max
R gbc
.
A generalized RKIstrategy where x = Q H
Ru, where R
is anoptimizeduppertriangularmatrixis shownto achieve
themaximumthroughputofthe21:::2GBC,foranySNR
andanymatrixH[5 ].
InthecaseofrandomHwithi.i.d. Rayleighfading,dier-
entRKIstrategieswithequal-powerandconstant-rateactive
usersarecomparativelyinvestigatedforalargearray(i.e.,for
r;t!1 and r=t xed). It is demonstrated that theequal-
powerschemeperformsveryclosetothewaterllingRKIand
approachestheupperboundofcooperativerateformoderate
and high SNRs[5 ]. The results essentially hold when H is
availabletothetransmitteronly[5 ].
References
[1] E.Telatar,\Capacityofmulti-antennaGaussianchannels,"Eu-
ropeanTrans.onTelecomm.ETT,Vol.10,No.6,pp.585-596,
Nov.1999.
[2] M.Costa,\Writing ondirtypaper,"IEEETrans.onInform.
Theory,Vol.29,No.3,pp.439-441,May1983.
[3] U. Erez, S. Shamai and R. Zamir, \Capacity and lattice-
strategiesforcancellingknowninterference,"ISITA2000,Hon-
olulu,Hawaii(USA),Nov.2000.
[4] P.Baier,M. Meurer,T.WeberandH. Troeger, \Jointtrans-
mission(JT),analternativerationaleforthedownlinkoftime
divisionCDMAusingmulti-elementtransmitantennas,"IEEE
6th Int.Symp. onSpread-Spectrum Tech.&Appl. (ISSSTA),
NJIT,N.J.(USA),Sept.2000.
[5] G.CaireandS.Shamai,\Onachievableratesinamulti-antenna
Gaussian broadcast channel," submitted to IEEE Trans. on
Inform.Theory,April2001.
