Capaity of MIMO Broadast Channels

2 o

. Thehannel gainof eahpath

φ p

isassumedtobezero-meanomplexGaussiandistributedandallpathshaveunit variane.

2.4 Capaity of MIMO Broadast Channels

Theompleteharaterizationoftheapaityregionofmulti-antennabroadasthannelwas

the foremosttheoretial hallenge in multiuserinformation theoryoverthelast veyears.

TheanalysisofbroadasthannelswasinitiatedbyCover[14℄andtheirapaityisgenerally

knownonlyinspeialases,wherethesignalssenttotheusersanbeorderedaordingto

their `strength'. In ontrastto single-usersystems wherethe apaityis asingle number,

the apaityof amultiuser systemwith

K

^users ^isharaterizedbyaapaity region, i.e.

K

-dimensional rateregion,where eah pointis avetorofratesahievablebyallthe

K

userssimultaneously. Arate vetorisahievableifthere exists aoding shemefor whih

theerrorprobabilityforallusersisarbitrarysmallastheodebloklengthinreases. The

maximumofthesumoftheommuniationratesistheso-alledsum-ratepointandlieson

theboundaryoftheapaityregion. Clearly,sinethe

K

^users^share^the^same^bandwidth,^a

tradeoarisesbetweenthereliableommuniationuserrates: ifonewantstoommuniate

at ahigherrate,theotherusersmayneedto lowertheirrates.

A large lass of broadast hannels, known as `more apable' hannels [15℄, ontains

two important ategories as speial ases: `degraded' and `less noisy' hannels. Roughly

speaking,abroadasthannelisdegradedwhentheusersanbeorderedfromthestrongest

totheweakestinanaturalorder.Forinstane,aSISObroadasthannelisdegraded,sine

theusersanbeorderedaordingtotheir

| H k | ²

^,ând^theâpaity^regionân^beâhieved^by

superposition oding [14℄. However,MIMObroadasthannels aregenerallynon-degraded

asthereisnotanaturalwaytoorderhannelmatries.

2.4.1 Capaity with perfet CSI at the transmitter

Although the haraterization of the general (fading) broadast apaity region is a long

standing problemin multiuserinformation theory,substantialprogresshasbeenmade for

Gaussian MIMO hannels. Despite not being degraded, the Gaussian MIMO BC oers

signiant struture that an be exploited to haraterize its apaity region. The key

theoretial toolfor haraterizing the MIMO BC apaity region with full CSI,the Dirty

Paper Coding (DPC), wasrevealed by the seminalwork of Caire and Shamai (Shitz)[7℄.

doesindeedahievetheapaityofa2-userMISObroadasthannel. Theresultsof[7℄were

extendedandgeneralizedby[1618℄,untilthefullharaterizationofMIMOGaussianBC

apaityregion(foranyompatsetofinputovarianesandnotonlyunderatotalpower

onstraint)byWeingartenetal.[8℄,establishingtheoptimalityofDPCasapaity-ahieving

strategy.

Assuming noise with unit variane and given a set of positive semi-denite matries

P _k ≥ 0, ∀ k

^that^satisfy^the^power^onstraint

Tr n

TheDPC regionis givenbytheonvexhullofalltheahievableratesas

C ^{DP C} =

^onv

andisshowntobeequivalenttotheapaityregionofMIMO broadasthannel[8℄.

Theapaityexpression(2.29)anbesimplied asfollows:

C ^{DP C} = E _H

TheoneptofdirtypaperodingwasintroduedbyCosta[6℄,whoshowedthatforasalar

GaussianhannelwithAWGNandaninterferingGaussiansignalknownnon-ausallyatthe

transmitter(butnotatthereeiver),theapaityisthesameasiftherewasnoadditive

in-terferene,orequivalentlyasifthereeiveralsohadknowledgeoftheinterferene. Inother

words,dirtypaperoding allowsnon-ausally knowninterferenetobe`pre-subtrated'at

thetransmitterwithnoinreaseinthetransmitpower. Assume,withoutlossofgenerality,

thattheenodingproessisperformedin asendingorder. Theenoderrstpiksa

ode-wordfor

i

^-th^reeiver,ând^then^hoosesâôdeword^for^reeiver

(i + 1)

^-th^reeiver^with^full

(non-ausal)knowledgeoftheodewordintendedforreeiver

i

^. ^Thus,^the^enoder^onsiders

theinterferenesignalaused byusers

j < i

^as^knownnon-ausally and subsequently, the

i

^-th^deoder^treats^theinterferenesignalausedbyusers

j > i

^as^additional^noise.

Uplink-Downlinkduality

Themain tool that failitatedthe extensionof theworkin [7℄ andsimplied the problem

of nding the apaity region of MIMO BC was the uplink-downlink duality, introdued

in[1719℄. Theoneptofuplink-downlinkdualityanbeseen,ingeneral,astheequivalene

between the performane of a lass of reeive and transmit strategies when the role of

transmittersand reeiversare reversed. This equivalene has beenobserved in seemingly

dierent ontexts in the literature. For instane, in point-to-point links, the duality is

that the apaityregion oftheMIMO BC,

C ^{DP C}

^with ^power^onstraint

P

^is ^equal^to ^the

apaityregionoftheso-alleddualMIMOMAC,

C ^MAC

^with^sum^power^onstraint

P

C ^{DP C} (P, H _1...K ) = [

T r { ^P ^K ^k=1 ^P ^k ^≤ ^P }

C ^MAC ( P _1...K , H ^T _1...K )

^(2.31)

where theunionistakenoverallmatries

P k ≥ 0 ∀ k

^suh^that

Tr n P K

k=1 P k ≤ P o

The major benet of the uplink-downlink duality is that the apaity region of the

downlinkanbealulatedthroughtheunionofregionsofthedualuplink,whihisonvex

and whose boundary an be alulated using interior-point methods [20℄. An additional

benet is from an optimization theory point of view, sine by exploiting the duality the

dimensionalityoftheoptimizationproblemissigniantlyredued. Inmanypratialases,

the number of transmit antennas in the broadasthannel is greater than the numberof

reeiveantennas of anyof thereeivers. Therefore, insteadofoptimizing over

K

^matries

ofsize

M × M

^,^we^need^to^optimize^over

K

^matries^of^sizes

N × N

^. ^Note^that^the

uplink-downlink duality only holds under a total power onstraint, and extensions of the DPC

optimalityto generalonstraintsettings(e.g. per-antennapoweronstraint)arebased on

themoregeneraloneptofmin-maxduality[8,21℄.

On theoptimal numberof users with non-zeroalloated power

Multiuser information theory advoatesfor transmittingto multiple users simultaneously

by properlydistributingthespatial dimensionsamong thebest groupof usersasameans

to boost the system throughput. A natural question that arises is how many users an

besimultaneously ative, and how the spatial dimensions are distributed among them. Yu

and Rhee[22℄obtainedatheoretialupperbound onthenumberofsimultaneouslyative

usersbyountingthenumberofvariablesandunknownsin thesetofKarush-Kuhn-Tuker

(KKT)optimalityonditionsforthesum-ratemaximizationproblem. Thisboundindiates

that inthedownlinkhannelmaximizingthesumrateentailsshedulingatmost

M ²

^users

simultaneously. In pratie, simulations show that typially the number of ative users

is four times the number of transmit antennas in the high SNR regime using optimum

ovarianematries,and thatshedulingupto

M

^users,^although suboptimal, resultsto a small apaityloss. In[23℄,it wasindependentlyshownthat under ertainonditionsin a

vetordownlink with

K

ûsersândâ ^BS^with ^two^transmitântennas, ^the^numberôf ûsers

thatanbesimultaneouslyservedanbehigherthantwo. Thepoweralloatedtothe

k

^-th

user is no longer a water-lling proedure, but it is found by the KKT onditions. Note

that whenrestritingto linearpreodingtehniques,aswedointhis thesis,thenumberof

servedusersisdiretlylimitedbythenumberofdegreesoffreedomattheBS,i.e.

M

2.4.2 Capaity with no CSI at the transmitter

The Gaussian MIMO BC with no CSIT is still degraded nomatter whether the reeivers

haveCSIRornot,assumingthatthetransmitterorthereeiversareequippedwithmultiple

antennas [7℄. In that ase, the apaity region is ahieved by superposition oding [24℄.

When the users have the same number of antennas, it an be shown that superposition

oding isthe sameastime sharing. In thisase, the sumapaity isthe sameasifthere

simultaneously. TheapaityregionoffadingMIMOBC isanopenproblemoftheoretial

interest. Theapaityregionisnotexpliitlyharaterized,andonly asymptotiallytight

boundsurrentlyexist. ThefadingMISOBCisonsideredin[25℄assumingthedistribution

ofthefadingoeientsisisotropi. Itwasshownthattheapaityregionisequivalentto

thatofthefadingsalarBC,resultinginamultiplexing gainofone. Whenthetransmitter

hasinompleteCSIonthefadingrealization,thepre-logfator(multiplexinggain

Dans le document Multiuser multi-antenna systems with limited feedback (Page 40-43)

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φ p

K

K

K

K

| H k | 2

P k ≥ 0, ∀ k