Random Opportunisti Beamforming

2.9 Linear Preoding and Sheduling with Limited Feedbak

2.9.3 Random Opportunisti Beamforming

h ˆ ^H _k h ¯ k

2

under RVQ, the interested

readerisreferredto[10,61℄. Nevertheless,performaneanalysisofmultiuserMIMOshemes

employing RVQ, despite its simpliity, does not often result in simple alulations and

integralswithlosed-formsolution. Insuhase,thefollowingodebookdesignframework

mightbeof interest.

ApproximateCell VetorQuantization

A geometrialframework forvetorquantizationwaspresentedin [56℄. Therein,in order

to evaluatetheareaofno-outageregions,theauthorsdenedspherialaps onthesurfae

of the hypersphere,whih yields agood approximationfor the area of no-outage regions.

Assuming that eah odewordis isotropially distributed in

C ^M

, the unit norm sphere

U

wherearandomvetor

h ¯ _k

^lies^ispartitionedinto

N D

`quantizationregions'(deisionregions)

{ H ¯ ⁱ ; i = 1, . . . , N d }

^, ^where

H ¯ ⁱ = { h ¯ _k ∈ U : | h ¯ ^H _k v _i | ² ≥ | h ¯ ^H _k v _j | ² , ∀ j 6 = i, 1 ≤ j ≤ N D }

If the hannel

h ¯ _k ∈ H ¯ ⁱ

^, ^the ^reeiver

k

^feeds ^bak ^the ^index

i

^. Approximate ell vetor quantization results assuming that eah quantizationell is a Voronoi region of spherial

apwiththesurfaearea

1/N D

ôf^the^total^surfaeâreaôf^theûnit^sphere^[62℄. ^Sine

h ¯ _k

^is

uniformlydistributedover

U

^,^we^have^that

Pr { h ¯ _k ∈ H ¯ ⁱ } ≈ 1/N D , ∀ i

^,^and^the(approximate) quantizationellisgivenby[55,56,63,64℄

H ˜ ⁱ = { h ¯ _k ∈ U : 1 − | h ¯ ^H _k v _i | ² ≤ δ } , ∀ i, k

for

δ = (1/N D ) ^M−1 ¹ = 2 ⁻ ^B ^D ^/(M ⁻ ¹⁾

^. Âlthough^generally ^thereâre ôverlapsⁱⁿ ^the

approxi-mate quantizationells,thisapproximationisshownthroughnumerialresultstobequite

aurateevenforsmall

N D

^[63℄. ^Furthermore,itanbeshownthatACVQyieldsanaurate lowerbound tothequantizationerrorforanyvetorquantizationodebook[55,56℄.

2.9.3 Random Opportunisti Beamforming

Ifweonsiderthateahuserisallowedtouseonly

B D = log ₂ M

^bits^for^CDIquantization, the optimal hoie for a randomly generated odebook is one that ontains orthonormal

vetors. Therefore, the above vetor quantization-basedtehniques anbe viewed as

ex-tensiontoapopular,alternativelow-ratefeedbaksheme,oinedasrandomopportunisti

beamforming(RBF).

InRBF,

1 ≤ B ≤ M

^mutuallyôrthogonal^random^beamsâre^generatedât^the

transmit-ter. Thesingle-beamRBF(

B = 1

⁾^was^proposedⁱⁿ^[53℄,^while^multi-beam^RBF⁽

B = M

⁾^is

proposedin [53℄andanalyzedin[9℄. Aunitarypreodingmatrix

Q

^is^generated^randomly

aordingtoanisotropidistribution. Its

M

^olumns^(vetors)

q _m ∈ C ^M ^× ¹

aninterpreted as randomorthonormal beams. An isotropially distributed (i.d.) unitary matrix anbe

generated byrstgeneratinga

M × M

^random^matrix

X

^whose^elements^areindependent irularly symmetri omplex normal

CN (0, 1)

^, ^and ^then ^perform ^the ^QR deomposition

X = QR

^,^where

R

îsûpper^triangularând

Q

îs ânî.d. ûnitary^matrix. Ât^time^slot

t

^the

transmittedsignalisgivenby

x (t) =

B

X

m=1

q _m (t)s m (t)

^(2.59)

where

s m (t)

îsâ^salar^signalîntended^for^theûser^servedôn^beam

m

^. ^The^SINR^of^user

k

nds the beam

b k

^that ^provides ^the ^maximum ^SINR, ^i.e.

b k =

^arg

max

1 ≤ m ≤B

SINR

k,m

^, ^and

feeds bak the value of SINR

k,b k

ⁱⁿ ^addition ^to ^the orresponding beam index

b k

^. ^An

underlyingassumption hereis thattheusersknowtheirownhannel oeients. Inturn,

thetransmitterassignseahbeam

m

^to ^the^user

k m

^with^the^highestorrespondingSINR, i.e.

k m =

^arg

max

1 ≤ k ≤ K

SINR

k,m

^. ^Sine^theûsers^haveî.i.d. ^hannels,^the^CDFôf^the^SINRôf

aseleteduser(after sheduling)

F s (x)

^is^given^by^[9℄:

F s (x) = (F SIN R (x)) ^K = 1 − e ⁻ ^x ^B ^σ ² ^/P (1 + x) ^B− ¹

! K

(2.61)

Theahievablesumrate(assumingGaussiansignaling)isgivenby

R ^RBF ≈ E

wheretheapproximationisusedsinethereisaprobabilitythatusermaybethestrongest

userformorethanonebeam.

Asymptotisum-rateanalysisshowedthat,forxed

M

P

^and

K → ∞

^,^the^average^sum

rateofRBFsalesas

M log log K

^,^whihîs^the^sameâs^the^saling^lawôf^theâpaity^when

perfetCSIisavailable. Thisisduetothefatthatthe

max

1 ≤ k ≤ K

SINR

k,m

^behaves^like

log K

whihisthebehaviorofthenumerator(maximumof

K

^i.i.d.

χ ² ₍₂₎

^r.v.'s),^as^theinterferene termsbeomearbitrarysmall. Inotherwords,inthelarge

K

^regime,^RBF^with^partial^CSIT

doesnotsuer anyapaitylossduetointer-userinterferenedespiterelyingonimperfet

(salar) CSIT. The intuition behind that sheme is that for large

K

^, ^there ^exists ^almost

surelyauserwell-alignedto eahbeam, aswellaswith verylittleinterferene from other

beams. Thus, we have

M

^data ^streams ^being transmitted simultaneously in orthogonal spatialdiretions and asaresult,full spatial multiplexinggainis exploited. Furthermore,

the authors in [9℄ show that if

M = O(log K)

^, ^then ^a^linear ^apaity ^saling ^with

M

^is

guaranteed, and fairnessis ahievedasa byprodut. Here, theterm fairness impliesthat

theprobabilityofhoosinguserswithunequalSNRsisequalized.

A limitationof [9℄isthat it isoptimalfor averylarge,typiallyunrealisti numberof

users. Theperformaneisquiklydegradingwithdereasingnumberofusers. Furthermore,

thisdegradationisampliedwhenthenumberoftransmitantennasinreases. Thereasonis

intuitive: asthenumberofativeusersdereasesand

M

înreases,ît^beomes^moreând^more

unlikelythat

M

^randomly^generated,equipoweredbeamswillmathwellthevetorhannels of any set of

M

ûsersⁱⁿ ^theêll. În ^Chapters ³ând ⁴^we^propose^several enhanements in order to restore robustness and inrease the sum-rate performane of RBF in sparse

networks. Moreover,RBF is highly sub-optimal at high SNR,i.e.

lim

P →∞

R ^RBF

log P = 0

^, ^as^it

beomesinterferenedominated. Asinterferenesaleswith

P

ândânnot^beêliminated^due

Scheduled users Step 1

The BS sends M random orthogonal beams

Step 2

Users feed back their maxSINR

Step 3

The BS assigns each beam to the user with the maxSINR for this beam

Figure2.4: ShematiofRandomOpportunistiBeamforming.

to partial hannelknowledgeofxed rate,the multiplexinggainof

M

^annot^be^ahieved

at highSNR.

Enhaned Multiuser Random

Beamforming

3.1 Introdution

In this hapter, we onsider the downlink of a wireless system with a

M

^-antenna ^base

stationand

K

single-antennausers. Alimitedfeedbak-basedshedulingandbeamforming senarioisstudiedthatbuildsuponthemulti-beamRBFframework[9℄presentedindetail

in Setion 2.9.3. The popularity of RBF has been spurred by the fat that it yields the

sameapaitysaling,intermsofmultiplexingandmultiuserdiversitygain,astheoptimal

fullCSIT-basedpreodingsheme. Theoptimalapaitysalingof

M log log K

^is^ahieved

whenthenumberofusers

K

îsârbitrary^large,^withônly^little^feedbak^from^theûsers,î.e.

in the form of individual SINR. RBF-based approahes havein fat evolvedin atopi of

researhin itsownrightandmanypossiblestrategiesanbepointedout[6568℄.

The intuition behind the RBF onept is that although the beamsare generated

ran-domlyandwithoutanyaprioriCSIT,forlarge

K

^,^the^seleted^groupôfûsersêxhibit^large

hannelgainsaswellasgoodspatialseparability,andtheprobabilitythattherandombeam

diretionisnearlymathedtoertainusersisinreased. However,amajordrawbakofthis

tehniqueisthatitsperformaneisquiklydegradingwithdereasing

K

^. Furthermore,this degradationis ampliedwhen thenumberoftransmitantennas inreases. As thenumber

ofativeusersdereasesand

M

înreases,ît^beomes^moreând^moreûnlikely^that

M

ran-domly generated, equipowered beams will losely math the vetor hannels of any set of

M

ûsers. ^This ^situation ôuld êasily ^be ^faed âs^tra îs ^normally ^bursty ^with ^frequent

silentperiodsindata-aessnetworks,thustheshedulermaynotountonalargenumber

ofsimultaneouslyative usersat alltimes. Another limitation ofRBF is that it beomes

interferenedominatedathighSNR,anditsmultiplexing gainvanishessineinterferene

-whihsaleswithSNR -annot beeliminatedwithxed-ratepartialCSIT.

Intherstpartofthishapter,weprovideanalytisum-rateexpressionsforonventional

random beamforming [9℄and deriveapaitysaling laws at high SNR. Mainimpliation

ofourresultsisthat inertainasymptotiregimes,itisbeneial toreduethenumberof

ativebeams,i.e. thebeamsalloatednon-zeropower.

Intheseondpartofthishapter,weinvestigatesolutionstoirumventthelimitations

of RBF for

K

^dereasing. ^We întrodue â ^new ^lass ôf ^random ûnitary beamforming-inspiredshemesthatexhibitsrobustnessinellswith-pratiallyrelevant-lowtomoderate

numberofusers(sparsenetworks),whilepreservingthelimitedfeedbakandlow-omplexity

advantagesofRBF.Onerstkeyideaisbasedonsplittingthedesignbetweenthesheduling

andthenalbeamomputation(or"userserving")stages,thustakingprotfromthefat

the numberof usersto beserved at eah shedulingslot ismuh lessthan thenumberof

ativeusers(i.e.,

B ≤ M << K

^). ^In^the^sheduling^phase,^a^nite^feedbak^rate^sheduling

shemeispresentedexploitingtheoneptofRBF.WeusetheSINRreportedbyallusers,

whihismeasuredupontheinitialpreodingmatrixasabasisonwhihtofurtherimprove

the designofthe nalbeamsthat will beused to servetheseletedusers. Ingeneral, the

initial preoderanbedesignedbasedonanyapriorihannelknowledge;howeverherewe

assumethattherst-stagebeamsaregeneratedatrandomasin [9℄sinenoaprioriCSIT

isassumed. Onethegroupof

B (1 ≤ B ≤ M )

^users^ispre-seletedusingtheSINRfeedbak ontherandombeams,additionalCSITmayberequestedtoonlytheseletedusergroupin

order to designthenal preoder. More speially, wemakethefollowing proposalsand

ontributions:

•

^Theseond-stagepreodingmatrixmayrequirevariablelevelsofadditionalCSIT feed-baktobeomputed,dependingon designtargets, andthenal beamswill improve

overtherandom beamforming used in [9℄. In partiular, while weexpet littlegain

over[9℄forlarge

K

^,^signiant^throughput^gain^appears^for^sparse^networksⁱⁿ ^whih

theinitialrandombeamformermaynotprovidesatisfatorySINRforall

M

^users.

•

Îf^we^restritôurselves^to^theâse^that^theînitial ^beam^diretions^do^not^hange,^we

propose then to adapt the power and the numberof ative beams aordingto the

numberofusers, theaverageSNR andthe numberoftransmit antennas asameans

tomaximizethesystemthroughput.

•

Înône^variantôf^the^proposed^designs,^we^studyâ^powerâlloation^shemeâross^the

B

^(initially equipowered) random beamsshowing substantial apaity improvement over[9℄ for a wide range of values of

K

^. ^The ^sheme ^requires

B ≤ M

real-valued salar values to be fed bak from eah of the

B

pre-seleted users. For a 2-beam system,the global optimalbeampowersolutionis provided in losed-form, whereas

forthegeneral

B

^-beamâse,^solutions^basedônîterativeâlgorithmsâre^proposedând

numeriallysimulated.

•

Înânother^proposed ^robust^variantôf^RBF, ^noâdditional^CSIT^feedbakîs ^required

during the seond stage. Instead, weexploit the SINR information obtained under

therandombeamsintherststage inorderto notonlyperformshedulingbutalso

to rene thebeamforming matrix itself. An on/o beam powerontrol is proposed

as a low-omplexity solution, yielding a dual-mode sheme swithing from TDMA

areativewithequalpower. Thethroughputgainsover[9℄areshowntobesubstantial

forhighSNRand low

K

^values.

Dans le document Multiuser multi-antenna systems with limited feedback (Page 58-63)

2.9 Linear Preoding and Sheduling with Limited Feedbak

2.9.3 Random Opportunisti Beamforming

h ˆ H k h ¯ k

2

C M

U

h ¯ k

N D

{ H ¯ i ; i = 1, . . . , N d }

H ¯ i = { h ¯ k ∈ U : | h ¯ H k v i | 2 ≥ | h ¯ H k v j | 2 , ∀ j 6 = i, 1 ≤ j ≤ N D }

h ¯ k ∈ H ¯ i

k

i

1/N D

h ¯ k

U

Pr { h ¯ k ∈ H ¯ i } ≈ 1/N D , ∀ i

H ˜ i = { h ¯ k ∈ U : 1 − | h ¯ H k v i | 2 ≤ δ } , ∀ i, k

δ = (1/N D ) M−1 1 = 2 − B D /(M − 1)

N D

B D = log 2 M

1 ≤ B ≤ M

B = 1

B = M

Q

M

q m ∈ C M × 1

M × M

X

CN (0, 1)

X = QR

R

Q

t

x (t) =

B

X

m=1

q m (t)s m (t)

s m (t)

m

k

b k

b k =

max

1 ≤ m ≤B

k,m

k,b k

b k

m

k m

k m =

max

1 ≤ k ≤ K

k,m

F s (x)

F s (x) = (F SIN R (x)) K = 1 − e − x B σ 2 /P (1 + x) B− 1

! K

R RBF ≈ E

M

P

K → ∞

M log log K

max

1 ≤ k ≤ K

k,m

log K

K

χ 2 (2)

K

K

M

M = O(log K)

M

M

M

M

lim

P →∞

h ˆ ^H _k h ¯ k

C ^M

h ¯ _k

{ H ¯ ⁱ ; i = 1, . . . , N d }

H ¯ ⁱ = { h ¯ _k ∈ U : | h ¯ ^H _k v _i | ² ≥ | h ¯ ^H _k v _j | ² , ∀ j 6 = i, 1 ≤ j ≤ N D }

h ¯ _k ∈ H ¯ ⁱ

h ¯ _k

Pr { h ¯ _k ∈ H ¯ ⁱ } ≈ 1/N D , ∀ i

H ˜ ⁱ = { h ¯ _k ∈ U : 1 − | h ¯ ^H _k v _i | ² ≤ δ } , ∀ i, k

δ = (1/N D ) ^M−1 ¹ = 2 ⁻ ^B ^D ^/(M ⁻ ¹⁾

B D = log ₂ M

q _m ∈ C ^M ^× ¹

q _m (t)s m (t)

F s (x) = (F SIN R (x)) ^K = 1 − e ⁻ ^x ^B ^σ ² ^/P (1 + x) ^B− ¹

R ^RBF ≈ E

χ ² ₍₂₎

R ^RBF