Interferene-bounded Multiuser Eigenbeamforming with limited feed-

4.4 Exploiting Statistial CSIT in Spatially Correlated Channels

4.4.4 Interferene-bounded Multiuser Eigenbeamforming with limited feed-

lim-ited feedbak

Intheprevioussetions, wedealt with theproblem ofdening aneienttypeof

instan-taneousCQI to beombinedwith long-termstatistial hannelknowledge. Theproposed

oarseML hannel estimate framework is mainly usefulfor the purpose of user seletion.

Althoughpreoding designbasedonthehannel estimatesisfeasible, providing good

per-formaneforsmallanglespreads,itisin generalsensitiveandpronetosignambiguities.

In this paragraph, we exploit the long-term statistial information in a dierent way

fortheproblem ofjointshedulingandbeamformingwith limitedfeedbakandfousona

pratial,low-omplexitysheme. Inbrief,eahuser

k

^has^a^xed,^predenedbeamforming vetor,mathedto theprinipal eigenvetorof itshannelorrelationmatrix

R k

^. ^At^eah

shedulingslot,theusersareallowedtofeedbaktwosalarvalues: thealignmentbetween

thehannelandtheirpredenedbeamformingvetorsandtheirhannelnorms. Inturn,the

shedulerseletsthegroupofusersthatmaximizesthesystemthroughputusinggreedyuser

seletionand byestimating of thereeivedSINR basedoninter-userinterferenebounds.

One theusers to serveare identied, the preoding matrix ontainsthe preferred

beam-formingvetors(prinipaleigenvetors)oftheseletedusers. Theproposedshedulingand

preodingalgorithmisoutlinedin Table4.3.

FeedbakStrategy Weproposethateahuserfeedsbakthefollowingtwosalarvalues:

•

^its^hannel^norm

γ _k ⁽¹⁾ = k

^k k

•

^the^alignment^(angle)^between^itsinstantaneoushannelvetorandapresetnormalized beamformingvetor

w _k

^,^i.e.

γ _k ⁽²⁾ = | ^h ^H ^k ^w ^k |

k h _k k

The intuition behind this feedbak poliy is two-fold: in MIMO BC with partial CSIT

aneientshedulingsetshould ontainuserswithlargeinstantaneoushannelgainsand

mutuallyquasi-orthogonalhannelspatialsignatures,asmeanstoahievebothspatial

mul-tiplexingandmultiuserdiversitygains. TherstsalarCQI

γ _k ⁽¹⁾

^allows^to^selet^users^with

highhannelgainsasameanstobenetfrommultiuserdiversity. Inontrasttoafeedbak

metrioftype

γ k = h ^H _k w _k

2

,large

γ ⁽¹⁾ _k

^learly^identies^the^users^with^the^most^favorable

onditions,whereasthe lattermetrianbelargeevenfor userswith moderategains but

whose vetorhannels are perfetly alignedwith theirbeamforming vetors. The seond

salar metri

γ _k ⁽²⁾

^provides ^a ^measure ^of ^the misalignment between the hannel and the beamformer, and an be interpreted as a measure of the hannel quantization error due

tolimited CSITknowledge. Insingle-usersettings, thequantization erroraets onlythe

reeivedsignaland is translatedto apoweroset. However,in multiuserSDMA settings,

itanbeshown that

γ _k ⁽²⁾

^playsâ^vital^roleⁱⁿ ^theêstimationôf ^theînter-userinterferene.

Therefore,both

γ _k ⁽¹⁾

^and

γ _k ⁽²⁾

ân^beûsedâsâ^means^toêstimate^theînter-userinterferene duetolimitedfeedbak.

User Seletion If a perfetly orthogonal set of beamforming vetorsanbe found, the

abovelimitedfeedbakissuienttoahievethesameasymptotisumrateasthatofDPC.

Table4.3: ResoureAlloationAlgorithmwithStatistial CSIT

ConstrutbeamformingmatrixW

( S )

annotbealulatedexpliitly. Forthat,approximateexpressionsandboundsonthe

inter-user interferenebased onlimited hannel knowledge are of interest. Foruser

k ∈ S

^, ^the

interfereneanbeexpressed as

I k ( S ) = P

i ∈S ,i 6 =k P i |

^H k

i | ² = k

^k k ² I k ( S )

^, ^where

I k ( S )

denotestheinterfereneoverthenormalizedhannelh

k

^. ^Let

I ^UB _k ( S )

^denote^an^upper^bound

I k ( S )

^,â^lower^bound ôn^the^SINRâssumingîs^given^by

usersthatmaximizesalowerbound onthesumrateasfollows

S ^∗ =

^arg

max

Analyti low and upper bounds on the inter-user interferene under linear preoding are

presentedindetailin thefollowinghapter. Atthis point,weproposetousethefollowing

upperbound[90℄

Linear Preoding Lettheeigenvaluedeompositionof thetransmit orrelationmatrix

R k = E { h k h ^H _k } = V k Σ k V ^H _k

^, ^where

Σ k

^is ^a^diagonal ^matrix^with ^the eigenvalues of

R _k

ⁱⁿ ^desending^order^and

V _k

îsâûnitary^matrix^with^the eigenvetorsof

R _k

^. ^As^a^low

omplexity approah, we propose a systemwhere eah user has apreferred beamforming

vetorknown bothby the BS and the mobile terminal. As shown in [91℄,for single-user

MIMOommuniations,givenaertainuser

k

^with^orrelation^matrix^R

k

^the^average^rate^is

maximizedbymathingthebeamformingvetortotheprinipaleigenvetorofitsorrelation

matrix, w

k =

¹ _k

(eigenbeamforming). Hene, wedesign eah user's beamforming vetor inspiredbythissingle-userstrategy. Thismultiusereigenbeamformingtransmissionsheme

anbeseenasanequivalentodebook-basedsystemwhereeahuserhasatrivialodebook

ofsize one. Theodebook ontainsasingleodevetor,i.e. theprinipal eigenvetor,and

isupdatedat verylowrateequalto theoherene timeofthe seond-orderstatistis. We

shouldalsoremarkthatundertheprismofhannelestimationframework,the

interferene-bounded eigenbeamforming an be seen as a method where the transmitter designs the

preoderbasedonaoarsehannelestimategivenby

h ˆ _k = k h _k k cos(∠h k , v ¹ _k )

¹ _k = γ _k ⁽¹⁾ γ _k ⁽²⁾

¹ _k

^(4.27)

4.4.5 Performane Evaluation

For the system evaluation, we assume that the hannel evolves aording to a speular

modelwherethehannelimpulseresponseisasuperpositionofanitenumberofpaths,as

desribedin Setion2.3.1. WeonsiderULA atthetransmitterwith antennaspaing

d = 0.4λ

^,^where

λ = 0.15m

îs^the^wavelength^(here^for^2GHz). ^Weônsiderânarrowband, at-fadingRayleigh(spatiallyorrelated)hannelwhere eah user

k

^has^a^dierent^ovariane

matrix

R k

^. ^Theâssumption ^that ^the^reeivers^do^not^have^the^sameôrrelation ^matrixîs

well motivated by the fat that in broadast hannels, theangle-of-arrivalis dierent for

eah user beause theyare notphysially o-loated. Theovarianematrix is omputed

usingtheassumption ofGaussian distributed satteringwithangular spread

σ θ

^(standard

deviation of the distribution) and is averagedover 60 time slots. Note that the angular

spreadorrespondstoanangularspreadsetorof

2σ θ

^degrees. ^Unless^otherwise^stated,^the

BSisequippedwith

M = 2

^antennas ^and^the^transmit

SNR

^is^set^to

10dB

MLEstimationwith BeamGain Information

Weompare the sumrate ahieved using theoarse ML hannel estimate with BGI with

thatofoptimalMMSEbeamformingwithfullCSITandwitharandombeamforming-based

shedulingapproah[9℄.

Figures4.3and4.4showtheperformaneomparisonasafuntionoftheanglespread

and the number of users, respetively. One the group of seleted users

S

^is ^identied

basedoneahuser'soarsehannelestimate,thetransmitterobtainsfullCSITonly forthe

seleted

M

ûsersând^designs^the^MMSE^preoding^matrixôfûser^set

S

^. ^In^RBF^approah,

the users are seleted based on the maximum SINR [9℄. We observe that the sheduler,

despiteusingonly oarseonlyhannelestimate, isableto identify abetter groupofusers

thanRBFforallanglespreads. Whentheangle spreadisloseto zero,ourmethod loses

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 7.5

8 8.5 9 9.5 10

Angle spread, σ _θ /π

Sum Rate [bits/s/Hz]

MMSE Precoding with full CSIT Constr. ML channel estimation with BGI Orth.Basis Expansion method with BGI Random Beamforming

Figure 4.3: Sum rateperformane versusanglespreadofproposedML estimationmethod

for

M = 2

^,^and

K = 50

ûsers. ^Full^CSITîsôbtained^for^the^seletedûsersâtâ^seond^step.

10 20 30 40 50 60 70 80 90 100

6 6.5 7 7.5 8 8.5 9 9.5 10

Number of users

Sum Rate [bits/s/Hz]

MMSE Precoding with full CSIT Constr. ML channel estimation with BGI Orth. Basis Expansion method with BGI Random Beamforming

Figure 4.4: Sum rate performane versusthe numberof users of ML hannel estimation

methodfor

M = 2

^,^and

σ θ = 0.2π

^. ^Full^CSIT^for^the^seletedûsersîsôbtained^for^preoder

design.

estimation methods exhibitexatlythesameperformaneastheyare equivalentsolutions

forthesameoptimization problem,dieringonlyin termsofomputationalomplexity.

In Figure 4.5, we evaluate the performane of the hannel estimation methods when

user seletionandbeamformingdesign areperformedin one stepbasedon oarsehannel

estimates. Evidently,MMSEpreodingdesignbasedontheestimatedhannelisrobustonly

in highly orrelated hannels, forwhih thehannel estimateis loserto thereal hannel.

Nevertheless, bothestimation methods show - with no additional feedbak - asigniant

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 5

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Angle spread, σ _θ /π

Sum Rate [bits/s/Hz]

MMSE Precoding with full CSIT Constr. ML channel estimation with BGI Orth. Basis Expansion method with BGI Random Beamforming

Figure4.5: SumrateperformaneversusanglespreadofproposedMLestimationframework

for

M = 2

^,^and

K = 50

ûsers. ^Partial^CSITîsêmployed^for^preoding^design.

20 30 40 50 60 70 80 90 100

4 5 6 7 8 9 10

Number of users

Sum Rate [bps/Hz]

MMSE Precoding with full CSIT Greedy User Selection with norm feedback ML channel estimation with norm feedback Random Beamforming

Figure4.6: Sumrateasafuntionofthenumberofusersforvarioususerseletionshemes

with

M = 2

^,^antenna^spaing

d = 0.5λ

^and

σ θ = 0.1π

MLEstimationwith NormFeedbak

InFigures4.6-4.8weevaluatetheMLhannelestimationframework withnormfeedbak

andgreedyuserseletionalgorithm(Table4.2)asafuntion of

K

^,^antenna^spaing

d

^,^and

the angle spread

σ θ

^, respetively. As a benhmark, wealso plot the sumrate of MMSE beamformingwithfullCSITandRBF.Inallmethods, onethegroupofuserstobe

shed-uled is identied, the BS obtains full CSIT for the seleted users in order to design the

MMSEpreodingmatrix. Ourmethods showalear gainoverRBF forangle spreadless

than35degreesmakingitpratialapproahforellularoutdoorsystems,astypial

mea-0 0.5 1 1.5 2 2.5 1

2 3 4 5 6 7 8 9

Antenna Spacing (d/λ)

Sum Rate [bps/Hz]

MMSE Precoding with full CSIT Greedy User Selection with norm feedback ML channel estimation with norm feedback Random Beamforming

Figure 4.7: Sum rate asa funtion of antenna spaing for various userseletion shemes

with

M = 2

σ θ = 0.1π

^and

K = 50

^users.

0 0.1 0.2 0.3 0.4 0.5

4 5 6 7 8 9 10

σ _θ /π

Sum Rate [bps/Hz]

MMSE Precoding with full CSIT Greedy User Selection with norm feedback ML channel estimation with norm feedback Random Beamforming

Figure 4.8: Sumrateasafuntion ofangle spreadforvarious userseletionshemeswith

M = 2

^,^antenna^spaing

d = 0.5λ

^and

K = 50

^users.

theBS[89℄. Interestingly,theantennaspainganbeoptimizedanditisfoundthat about

0.4λ

^givesôptimal ^results,âsît ^gives^the^best ^tradeo ^between^resolutionândsuppression of spatial aliasing. Note that asmall antennaspaing redues transmitantennadiversity,

howevermultiuserdiversityanompensateforthatduring thephaseofsheduling.

Interferene-bounded Eigenbeamforming

We evaluate now the performane of interferene-bounded multiuser eigenbeamforming

(Interf.-bounded MU EigenBF). Figures 4.9 and 4.10 show the ahieved sum rate of our

Foromparison,wealsoplottheperformaneof optimalMUeigenbeamformingwith

per-fet CSIT and that of interferene-bounded multiuser eigenbeamforming using full CSIT

foruserseletion. Asweansee, the performaneof theproposedlow-omplexity sheme

exeeds that of RBF but depends on the level of antenna orrelation, i.e. angle spread

σ θ

^. Êxpetedly, ^gainsâre ^more^pronouned^for ângle^spread^less^than⁴⁵^degrees^(outdoor

ellularnetworks).

20 40 60 80 100 120 140 160 180 200

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5

Number of users

Sum Rate [bits/s/Hz]

Optimal MU EigenBF with full CSIT Interf.−bounded MU EigenBF with Full CSIT Interf.−bounded MU EigenBF − sum rate lower bound Interf.−bounded MU EigenBF − effective sum rate Random Beamforming

Figure4.9: Sumrateasafuntionofthenumberofusersfor

M = 2

^,^and

σ θ = 0.1π

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5

Angle spread, σ _θ /π

Sum Rate [bits/s/Hz]

Optimal MU EigenBF with full CSIT Interf.−bounded MU EigenBF with Full CSIT Interf.−bounded MU EigenBF − sum rate lower bound Interf.−bounded MU EigenBF − effective sum rate Random Beamforming

Figure 4.10: Sumrate as afuntionof angle spread for

M = 2

^, ^antenna^spaing

d = 0.4λ

^and

K = 100

^users.

4.5 Conlusions

In this hapter, we showed that the redundany that arises in temporally and spatially

orrelated hannels an be exploited in order to inrease the system throughput by

opti-mizing the SDMA shedulingdeisions. In the rstpart, motivated by thefat that the

performaneofrandombeamformingdegradesseverelywithlownumberofusers,weshow

howexploitinghanneltimeorrelationweanalleviatethisproblematminimalost. The

proposedmemory-basedopportunistibeamformingprovidesawaytolosethegapto

op-timalityforarbitrarynumberofuserswhenthehanneloherenetimeislarge,e.g. inlow

mobility (indoor) settings. In theseond part, we investigated spatially-orrelatedMISO

hannels and showed how statistial hannel knowledge anbe eiently ombined with

instantaneous salarhannel feedbak for thepurpose of sheduling andlinear preoding.

Speially,itwasdemonstratedthat,inSDMA systemswithhannel-awaresheduling,it

issuienttofeedbakasinglesalarCSITparameter-eitherthehannelnormorbeam

gaininformation-in ordertoahievenearoptimalsum-rateperformane. Wederivednew

shedulingmetristhat havetheadvantageofaommodatingstatistialhannel

informa-tionandlimitedinstantaneoushannelfeedbak. AMLhannelestimationframeworkhas

beenestablishedthat issuitableforresourealloationinwide-areamulti-antennaellular

systems. Finally, alowomplexitypreoding/shedulingalgorithm, basedon

interferene-bounded SDMA eigenbeamforming forspatially orrelated MISO hannels. All the above

shemes exhibit performane lose to that of omplete CSI when the multipath angular

spread per user at the BS is small enough, making these approahes suitable to wireless

systems withelevatedBS suhasoutdoorellularnetworks, in whih the elevation of the

BSabovethelutterdereasestheanglespreadofthemultipath.

APPENDIX

4.A Proof of Proposition 4.1

To provethis statement, wean equivalently showthat for the set of i.i.d. random

uni-tary matries,

Q 1 , . . . , Q _|U| ⊂ U

max

1 ≤ i ≤|U| X i

^onverges ^to

R ^∗

^for

|U|

^suiently ^large

and xed number of users

K

^. ^Thus, ^we ^want ^to ^show ^that

∀ ǫ, δ > 0

∃ |U|

^suh ^that

Pr

max 1 ≤ i ≤|U| ≤ R ^∗ − ǫ ≤ δ

Asthesequeneof unitarymatries

{ Q _i } ^|U| 1

âreî.i.d. ^r.v.'s, ând

{ X i } ^|U| 1

âreâlso î.i.d. ^for

i

^,^using^order^statistis^we^have^that

Pr

1 ≤ max i ≤|U| X i ≤ R ^∗ − ǫ

= [F X ( R ^∗ − ǫ)] ^|U|

^(4.28)

Forahannelwithmemory

L

^,îtîsêvidently^meaningful^to^have

|U| ≥ L

^. ^As

0 ≤ F X (x) ≤ 1

asymptotiallyfor

L → ∞

^,^we^have^that

Pr

1 ≤ max i ≤|U| X i ≤ R ^∗ − ǫ

→ 0

^(4.29)

Limited Feedbak Broadast

Channels based on Codebooks

5.1 Introdution

Inthe previoushapters,weinvestigatedlimited feedbakapproahesthat anbemainly

ategorizedasdimensionredutionorprojetiontehniques(f. Setion2.8.2). The

major-ityofourproposedsolutionswerebuilton-althoughnotlimitedto-arandombeamforming

ontext. Conventional RBF [9℄wasmainly employedasapre-shedulingtehnique, while

therandom beamformerwereoptimized during thepreoding designphase. A limitation

of RBF is that the resolution of CDI is xed to

B D = log 2 M

^, ^thus ^the ^sheme ^annot

be extended for the ase where additionalCDI bits anbe utilized. On the other hand,

reent ndingssuggestthat CDIis ofpartiular importane in limitedfeedbakmultiuser

multi-antenna systems, espeially in the high power regime. As it wasshown in [10℄, if

hannelinversion(zero-foring)isemployedastransmissionstrategy,thefeedbakloadper

usermustinreaseapproximatelylinearlywith

M

^and^the^transmit^power⁽ⁱⁿ^dB)ⁱⁿ^order

toahievethefullmultiplexinggain. Moreover,upto now,weonsideredshemeswherea

randomunitarypreoderisrstgeneratedwithno apriori CSITandtheBSollets

low-rate(salar)CQIfromeahuserasameanstoseletagroupofgoodusersandpotentially

re-designthepreodingmatrixfortheseletedgroup.

In this hapter, we take on a quantization-based approah (f. 2.8.1). The preoding

matrixis not pre-designed(before thefeedbak phase), but is generated based onpartial

CSIT obtained by all ative users. In other words, eah user rst reports some form of

quantized CSIT, whih in turn is used at the BS foruser seletionand preoding design.

Severallimitedfeedbakapproahes,imposingabandwidthonstraintonthefeedbak

han-nel have been studied in point-to-point MIMO systems [5456,58℄. In this ontext, eah

userfeeds baknitepreision(quantized)CSIT onitshanneldiretion byquantizingits

normalizedhannelvetortothelosestvetorontainedinapredeterminedodebook. In

this hapter, weonsider amulti-antennabroadasthannel with

K ≥ M

^users, ⁱⁿ ^whih

eah useris allowedto report feedbak bak to theBS viaanite ratefeedbak hannel.

ThisCSITonsistsof

B D

^-bit^quantizedinformationonitshannelvetordiretion,referred to asCDI, omplemented with additional instantaneous CQI. CDIinformation is mainly

employedforthepurposesofpreodingdesign,whileCQIservesasameanstointelligently

selet

M

^spatially^separableûsers^with^large^hannel^gains. ^Thisâpproahân^be^seenâsân

extensionofRBFtoaodebookontaining

N D > M

beamformingvetors(notneessarily orthonormal). Ithastheabilitytotunethefeedbakloadperuser,providingmore

exibil-ityinrealistiniteratefeedbaksenarios,inwhihthefewfeedbakbitsneedtobesplit

betweenhanneldiretionaland hannelqualityinformation. Ourmodel isonthelinesof

workin [62℄whihextendedthenite feedbakratemodel [10,52℄fortheaseof

K ≥ M

Astransmissionstrategy,severalbeamformingmethodshavebeeninvestigatedinthe

liter-ature, inludingorthogonalunitarybeamforming[92℄,transmitmathed-ltering [93℄,and

zero-foringbeamforming[64,9395℄. Notethatintheaboveontributions,thehannelgain

feedbakisonsidered unquantizedforanalytialsimpliity.

A major partof this hapter fouses on the following question: "What type of salar

CQIinformationneedstobeonveyedinordertoahievelose-to-optimumperformane?"

ReentresultsshowthatifthesalarCQIontainsinformationonlyonthehannelnorm,

the sum rate growth is independent of the average SNR and the number of ative users

K

^[62,^64℄. ^Therefore, ^the ^system^beomes interferene-limited forhigh SNR,and fails to

ahieve the optimum sum rate growth, even when the number of users goes to innity

(no multiuser diversity gain). This is due to the fat that an estimate on the inter-user

interferene isneeded, and thusadditional knowledge in theform of hannel quantization

error is neessaryin orderto ahievebothmultiplexing and multiuser diversitygains and

approahtheapaitywithperfetCSIT.

TheproblemofeientCQIdesignforsum-ratemaximizationwithshedulingandlinear

preodingintheaboveniteratefeedbaksettingisaddressedhere. Ourmainontributions

anbesummarizedasfollows:

•

^We^propose ^several ^salar^feedbak ^metris^based ^on ^inter-user interferenebounds, whihenapsulateinformationonthehannelgain,thehanneldiretion,aswellason

thequantizationerror. These metrisanbeinterpretedasestimatesofthereeived

SINR,whih is generallyunknownto the individualusers that haveknowledge only

ontheirownhannels.

•

^Weêmploy^these^metrisⁱⁿ â^systemêmploying^linear^ZFbeamformingonthe quan-tized hannel diretions and greedy user seletion. For that, we extendthe greedy

sheduling algorithm of [11℄ for the limited feedbak ase. This algorithm has the

advantages that it does notdepend on any apriori dened systemparameter (suh

as quantized hannels' orthogonality [62℄) and is able to swith from multiuser to

single-usertransmission.

•

Ûsing^theâbove^preoding^setting,^we^deriveûpper^boundsôn^theinstantaneous mul-tiuser interferene that allowsus to analytially predit the worst ase interferene

andaSINRlowerboundinasystememployingzero-foringonthequantizedhannel

•

^The^system^throughputîsânalyzedândîtsâsymptotiôptimalityⁱⁿ^termsôfâpaity

growth(i.e.

M log log K

⁾^is^shown^for

K → ∞

^. ^Sum ^rate^upper^bounds^for^the^high

SNR regimearealsoderived.

•

^Sheduling^metris^suitable^for^swithing^thetransmissionmodefrommultiuser(SDMA) tosingle-user(TDMA)areproposed,basedonarenedfeedbakstrategy. Weshow

that expetedly single-usermode ispreferredastheaverage SNR inreases,whereas

multiusermodeisfavoredwhenthenumberofusersinreases.

Dans le document Multiuser multi-antenna systems with limited feedback (Page 105-117)

Interferene-bounded Multiuser Eigenbeamforming with limited feed-

4.4 Exploiting Statistial CSIT in Spatially Correlated Channels

4.4.4 Interferene-bounded Multiuser Eigenbeamforming with limited feed-

k

R k

•

γ k (1) = k

k k

•

w k

γ k (2) = | h H k w k |

k h k k

γ k (1)

γ k = h H k w k

2

γ (1) k

γ k (2)

γ k (2)

γ k (1)

γ k (2)

( S )

k ∈ S

I k ( S ) = P

i ∈S ,i 6 =k P i |

H k

i | 2 = k

k k 2 I k ( S )

I k ( S )

k

I UB k ( S )

I k ( S )

S ∗ =

max

R k = E { h k h H k } = V k Σ k V H k

Σ k

R k

V k

R k

k

k

k =

1 k

h ˆ k = k h k k cos(∠h k , v 1 k )

1 k = γ k (1) γ k (2)

1 k

d = 0.4λ

λ = 0.15m

k

R k

σ θ

2σ θ

M = 2

SNR

10dB

S

M

S

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 7.5

8 8.5 9 9.5 10

Angle spread, σ θ /π

Sum Rate [bits/s/Hz]

MMSE Precoding with full CSIT Constr. ML channel estimation with BGI Orth.Basis Expansion method with BGI Random Beamforming

M = 2

K = 50

10 20 30 40 50 60 70 80 90 100

6 6.5 7 7.5 8 8.5 9 9.5 10

Number of users

Sum Rate [bits/s/Hz]

MMSE Precoding with full CSIT Constr. ML channel estimation with BGI Orth. Basis Expansion method with BGI Random Beamforming

M = 2

σ θ = 0.2π

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 5

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Angle spread, σ θ /π

Sum Rate [bits/s/Hz]

MMSE Precoding with full CSIT Constr. ML channel estimation with BGI Orth. Basis Expansion method with BGI Random Beamforming

M = 2

K = 50

20 30 40 50 60 70 80 90 100

4 5 6 7 8 9 10

γ _k ⁽¹⁾ = k

^k k

w _k

γ _k ⁽²⁾ = | ^h ^H ^k ^w ^k |

k h _k k

γ _k ⁽¹⁾

γ k = h ^H _k w _k

γ ⁽¹⁾ _k

γ _k ⁽²⁾

γ _k ⁽²⁾

γ _k ⁽¹⁾

γ _k ⁽²⁾

^H k

i | ² = k

^k k ² I k ( S )

I ^UB _k ( S )

S ^∗ =

R k = E { h k h ^H _k } = V k Σ k V ^H _k

R _k

V _k

R _k

¹ _k

h ˆ _k = k h _k k cos(∠h k , v ¹ _k )

¹ _k = γ _k ⁽¹⁾ γ _k ⁽²⁾

¹ _k

Angle spread, σ _θ /π

Angle spread, σ _θ /π

σ _θ /π

Angle spread, σ _θ /π

Q 1 , . . . , Q _|U| ⊂ U

R ^∗

max 1 ≤ i ≤|U| ≤ R ^∗ − ǫ ≤ δ

{ Q _i } ^|U| 1

{ X i } ^|U| 1

1 ≤ max i ≤|U| X i ≤ R ^∗ − ǫ

= [F X ( R ^∗ − ǫ)] ^|U|

1 ≤ max i ≤|U| X i ≤ R ^∗ − ǫ