Exploiting redundany in time-orrelated hannels

4.2.1 User Seletion in time-orrelated hannels

Considerthat the hannel exhibits orrelationfrom oneshedulingtime slot to theother.

Evidently, in suh ongurations, the sheduling deisions exhibit in turn some form of

orrelationoversuessiveintervals. Inotherwords,hannelorrelationinthetimedomain

reates temporal redundany, whih an be exploited asmeans to either redue feedbak

rate or inreasethesystemthroughput. If thehannel varies slowly,then learlythebest

user in termsof hannel qualityat urrent time slot

τ

^{ishighly likelyto be thebest user}

at the subsequent time slots

τ + T c

^{. Therefore, the fat that previously seleted users}

arehighly likelyto remaingoodanbefurtherexploitedduring theuserseletionproess.

Temporalorrelationhasbeenexploitedinpaketswithdesign,eitherbyusingamaximum

weightmathingalgorithm[81℄orbyarandomizedalgorithmexploitingtemporalorrelation

of queue states [82℄. In [83℄, the authors proposed arandomized sheduler that exploits

temporalorrelationsinslowfadinghannels.

4.2.2 Beamformingand Sheduling exploitingtemporalorrelation

Sine shedulingand linearpreoding is theleitmotiv of this dissertation,weaddress the

problemhowtoexploittemporalorrelationandenhanethesum-rateperformaneof

low-novelSDMA sheduling/preoding sheme, oined asMemory-based Opportunisti

Beam-forming (MOBF). Theshemebuildsonmulti-beamrandombeamforming[9℄presentedin

Setion 2.9.3, and exploits memory in thehannel asameans to ll the gap to sum-rate

optimality.

Inanutshell, MOBFreplaestherandomseletionofpreodingmatrieswitha

ombi-nationofrandomandpastfeedbak-aidedbeamformingmatriesthat arekeptinmemory.

Theshemean beseenassuessiverenementof thepreodingmatrixinside the

oher-enetimeofthehannel. Whentheoherenetimeofthehannelishigh(e.g. largeDoppler

spread),MOBFapproahesthesumapaityofoptimalunitarypreoderwithperfetCSIT.

Forunorrelatedi.i.d. hannels, theperformaneof theproposedshemeremainssuperior

to that of [9℄ at the expense of moderate additional feedbak (two SINR valuesper user

insteadofone).

Interestingly,theshemeanbeseentoalsorelatetoreentusefulresults[84℄,presented

toimprovethedelayperformaneofthesingle-beamopportunistibeamforming[53℄. In[84℄

shedulingislimitedtooneuserandtemporalhannelorrelationisexploitedthroughthe

useof a xed set of beamsdetermined in advaned. This sheme does notautomatially

reah theperformaneof afull CSIT senario,sinethe temporal orrelationwasusedto

restore fairness and users with long waiting times are prioritized. Another sheme that

exploits temporal orrelation in orthogonal frequeny division multiple aess (OFDMA)

systemshavebeenproposedin [85℄.

4.2.3 Memory-based Opportunisti Beamforming

Asstated before, MOBFbuilds onrandom beamforming (f. Setion 2.9.3),in whih the

transmittergeneratesat eahtime slot

t

^a

B × B ( B ≤ M )

^{unitarypreoding matrix}

Q (t)

randomly, as a means to redue the feedbak burden and omplexity requirements, i.e.

W (t) = Q (t) = [ q ₁ (t) . . . q _B (t)]

^{. In}onventionalRBF[9℄,anewrandomunitarypreoding isgenerated and used forserving theseleted usersat eah time slot. Hene,anykind of

strutureinthephysialhannelisnotexploited. Memory-basedOpportunisti

Beamform-ing attempts to exploit memory in the hannel by making at eah time slot animproved

seletionof theunitarypreodingmatrixbasedonpastCQIinformation. Temporal

orre-lation is exploited by memorizingthe previous best shedulingdeision(s), i.e. thegroup

of seleted users

S

^{for arandom preoder}

Q (t)

^{, and omparing it with the next random}

mathings

Q(t + i)

^for

i = 1, . . . , T c

^.

Speially,weonsiderthattheBShasaodebook(set)of`preferred'unitarymatriesof

size

U

^:

Q = { Q ₁ , Q ₂ , . . . , Q _U }

^(4.1)

with

Q ⊆ U (M, M )

^,where

U (M, M )

^{denotestheunitarygroupofdegree}

M

^{,i.e. thegroup}

of

M × M

^{unitarymatriesdeningtheomplexStiefelmanifold. Thenotionof}`preferred' is used in the sense of (relative) maximizationof the sum rate among pastused random

beamformingmatries.

Ateahtimeinstant

t

^{,theunitarymatrixofthepreferredset,denoted}

Q ˜

^{anddenedasthe}

preoderthathasprovidedthehighestsumrateinprevioustimeslots,isappliedanditssum

rateismeasured(updated)underurrenthannelonditions. Theahievablesumrateof

Q ˜

attimeslot

t + 1

^{isomparedwiththat ofanew,randomlygeneratedunitarymatrix}

Q _r

^,

andthebeamformingmatrixthatoersthehighestsumrateisseletedfortransmission. In

thephaseofupdatingtheodebook

Q

^{,thesumratevalueof}

Q ˜

^{intheodebookisupdated,}

and thenewlygeneratedrandompreoder

Q _r

^{isaddedinto theodebookifandonlyifits}

sumrateishigherthanthesumrateof theodebook matrixwiththeminimumsumrate.

Let

S ^t

^{denote the set of seleted users at eah sheduling window}

t

^and

H ( S ^t )

^{be the}

orrespondingsubmatrixof

H = [h ^T ₁ . . . h ^T _K ] ^T

^{. With}

R (Q, S ^t )

^{wedenotethesumratewhen}

unitary beamformingmatrix

Q

^{isused forservingthe usersbelongingto}

S ^t

^{. The stepsof}

theproposedalgorithmareoutlinedinTable4.1.

Table4.1: Memory-basedOpportunistiBeamformingAlgorithm

Memory-basedOpportunisti Beamforming(MOBF)Algorithm

First phase: (`best'unitarymatrixseletion)

Step 0Initializeodebook

Q = { Q ₁ , Q ₂ , . . . , Q _U }

^,

eahwithsumrate

R (Q i ), i = 1, . . . , U

Ateahtimeslot

t

^,

Step 1Generateanewrandompreoder

Q r

Step 2Selet

Q ˜ ∈ Q

^:

Q ˜ =

^arg

max

Q i ∈Q R (Q i )

Step 3Apply

Q ˜

^{,olletupdatedfeedbakfromtheusers}

andalulate

R ( Q, ˜ S ^t+1 )

Step 4If

R ( Q, ˜ S ^t+1 ) > R (Q r , S ^t )

^,

Q ^∗ → Q ˜

^,else

Q ^∗ → Q _r

Seond phase: (Updateofodebook

Q

⁾

Step 5Updatethevalue

R ( Q) ˜

^intheset

Q

Step 6If

[ R (Q r ) > R (Q min )]

^,

Q _min → Q _r

^,where

Q _min =

^arg

min

Q i ∈Q R (Q i )

⁾

Some ommentsare in order: The algorithm outlined in Table 4.1 presents a general

frameworkformemory-based,randomizedshedulingin slowtime-varyinghannels. First,

in pratie, at eah time slot

t

^theset

Q

^{ontains only onepreoder matrix(}

U = 1

^{), i.e.}

the onethat has provided the highestsystem throughputup to the urrent time instant.

Seondly,althoughMOBFisbasedonRBFforpreodinganduserseletion,ourproposed

shemeisnotonlyrestritedtosuhsystems. Theideaofmemory-basedpreodinganbe

also applied to systemswhere the usersutilizeaodebook toquantize theirhannelsand

feed bak quantized CDI. If the hannel is strongly orrelated, the above oneptan be

used to redue the feedbakloadby dereasingthefeedbak reporting rate. At eah slot,

additional CDI is then fed bak only if it is suiently dierent than the one previously

reported. Alternatively,ifweenforeCDIreportingateahtimeinstant,usersmayhavethe

PerformaneAnalysis

Theunderlying ideabehindthe sum-rateanalysis of MOBFis the following: theproess

of memorizing at eah sheduling slot the sum-ratemaximizing preoding matrixan be

seenasarandomsearh ofbeamformingongurationsin thespaeof orthogonalunitary

preoders. Evidently, the performane ofsuh sheme depends on thedistribution of the

sumrateonditionedto aertainhannel realization

H ( S )

^{forthe seletedgroupofusers}

S

^{. Tosimplify ouranalysis,wexthehanneloftheseleteduserstoaertain}realization

H

^{andweanalyzethepropertiesof}

X i = R ( Q _i , H )

^{,whihrepresentsthesumrateprovided}

by random unitary matries

Q _i

^{for a given hannel}

H

^{. Therefore,}

{ X i } ^∞ i=1

^{is a random}

proesswhosedistributiondependsontheunderlyingrandomvariable

Q _i

^{. Forxedhannel}

realization,

X i

^{is i.i.d. for}

i

^{withassoiated PDF}

f X ( · )

^andCDF

F X ( · )

^{. Ifthe hannel is}

quasi-stati,memory-basedbeamforming aimsat ndingthe unitarybeamformingmatrix

Q ^∗

^{from thefeasible setof unitarymatries}

U

^{that maximizes thesumrate. This anbe}

mathematiallywritten as:

Q ^∗ =

^arg

max

Q i ∈U R (Q i ) =

^arg

max

1 ≤ i ≤|U| X i

^(4.2)

Note that this optimizationreturns oneoutof possibly many globalmaximizers

Q ^∗

^sine

theglobal maximizer isnot unique, i.e.

R (Q ^∗ ) = R (Q ^∗ Q ^{′ H} )

^{, for any}

Q ^′ ∈ U

^{. However,}

themaximumvalueofthesumrate,

X ^∗ = R ( Q ^∗ , H )

^{, isuniqueovertheset}

U

^.

Assumingthat the set ofunitary matries

U

^{isnite with ardinality}

|U|

^{, then for}

|U|

i.i.d. randomunitarymatries

{ Q _i } ^|U| i=1

^{,theahievablesumrate}

X ^∗

^isgivenby

X ^∗ = max

1 ≤ i ≤|U| X i = Z ∞

0

xdF _X ^|U| (x)

^(4.3)

Forasymptotiallylarge

|U|

^{, the}distribution of

max

1 ≤ i ≤|U| X i

^{onverges-afterpropershifting}

andsaling-toalimitingdistribution(l.d.) ofGumbel,FréhetorWeibulltype. However,

astheexatform of the CDF

F X (x)

^{is diultto obtain, theexatl.d. is diultto be}

inferred. Hene,weresortto thefollowingresultin ordertoderivetheasymptoti(in

|U|

⁾

onvergeneofouralgorithm.

Proposition 4.1: Consider ahannel with memory

L = ^T _T ^c _s

^{, where}

T c

^{is the hannel}

oherene time, and

T s

^{is the slot duration. For}

L → ∞

^{, the sum rate of} memory-based beamforming

R ^MOBF

^{onvergesto the apaity of optimum unitary} beamforming

R ^∗

^{for a}

givenhannel

H

^:

R ^MOBF → R ^∗ = max

Q ∈U R ( Q _i , H )

^(4.4)

Proof. Theproofisgivenin Appendix4.A.

The above result implies that the maximum of the sum rate oered by using various

preoders

Q _i

^onvergesasymptotiallytotheoptimumapaityofunitarybeamforming

R ^∗

^.

As aresult, the orrespondingunitary preoding matrix, denoted

Q ^∗

^{, whih orresponds}

tothematrixthat maximizesthesumrateonvergestooneofthepossiblymanyoptimum

unitarypreoders. Therefore,ifthehannel isquasi-stati(verylarge

L

^{),theodebook of}

MOBFwillontainanoptimalbeamformingmatrix,i.e. aunitarymatrixthat maximizes

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