Deoupled Feedbak Optimization

In this setion, instead of determining jointly the optimal feedbak bit split, we follow a

werst ndtheoptimal numberofCDI bitsrequired toguaranteefull multiplexing gain,

implyingthat thefeedbakloadalloatedtoCQIis

B Q = (B tot − B D )

^{,andoptimizingthe}

2 ^{B Q}

quantizationlevelsbyusing(5.38). Thisapproahismotivatedbyresultsshowingthat lakofaurateCDIfeedbakinthehighSNRregimeresultsinlossofmultiplexinggain.As

thelossofthepre-logfatorof

M

^ismoredetrimentalontheahievablesum-ratethanthe lossinmultiuserdiversity,webelievethatforsuhkindsoffeedbakrateoptimizations,an

eientruleofthumbistoguaranteeappropriateCDIfeedbakratetoahievelose-to-full

spatialmultiplexinggain.

To illustrate this feedbak optimization tehnique, we apply this deoupled approah

tofeedbakoptimization of strategy1. Basedon theasymptoti growth of(5.9) forlarge

K

^{givenin [64℄,wederive thesalingof CDIfeedbakload, whih in turn determinesthe}

remaining CQI feedbak bits. We dene the power gap (per user) between the SINR of

theabovesheme,

SINR I

^{, and that of} zero-foring with perfet CSI,

SINR ZF

^{asthe ratio}

SINR I

SINR ZF = α

^{. Note that this power gap is translated to a rate gap. In order to ahieve}

fullmultiplexing gainfornite

K

^{, thenumberofCDIbits}

B D

^perreeiver

k

^{should sale}

aordingto:

B D = (M − 1) log ₂ (P/M ) − (1 − b) log ₂ K + c

^(5.39)

where

c = log ₂ (K/ K ⁱ )

^{is a onstant apturing the multiuser diversity redution at eah}

step

i

^{ofthegreedy-SUSalgorithmduetothe}

ǫ

-orthogonalityonstraintbetweensheduled users. As

b < 1

^{, havingmore usersin theell, asmallernumberof feedbak bits}

B D

^per

useris requiredin order to ahievefull multiplexing gain. Forexample, in asystem with

M = 4

^antennas,

K

^{=30users andSNR = 10dB, whena3-dBSINR gap isonsidered,}

eahuserneedstofeedbakatleast

B D

^=9bits.

Salingof CDI feedbak bits at highSNRregime

Inthehigh SNR regime,therole of CDIis moreritialdue to theeet ofquantization

error[10℄. For

P → ∞

^andxed

K

^{,basedonasymptotiresultsof [64℄,weanshowthat}

thefeedbakloadshouldsaleas

B D = (M − 1) log ₂ P − log ₂ K

^(5.40)

Forinstane,forasystemwith

M

^{=4antennas,SNR=20dBand}

K

^=60users,

B D

⁼¹⁴

bitsarerequiredto guaranteefull multiplexinggain. Expetedly,thefeedbakload

B D

^at

highSNRislargerthanthatof(5.39). Thus,itismorebeneialtoalloatemorefeedbak

bitsonthequantizationofhanneldiretion(

B D

^{)athighSNR,andassignlessbitsforCQI}

(

B Q

^).

5.7 Performane Evaluation

In order to assess the sum-rate performane of the proposed shemes, simulations have

been performed under the followingonditions:

M = 2

^{transmit antennas,}orthogonality onstraint

ǫ = 0.4

^{andodebooksgeneratedusingrandomvetor}quantization(RVQ)[10,61℄.

The ahieved sum rate is ompared with two alternative transmission tehniques for the

MIMOdownlink,randombeamforming[9℄andzero-foringbeamformingwithperfetCSI

0 5 10 15 20 25 30 35 40 45 50 0

5 10 15 20 25 30 35

SNR

Sum Rate [bits/s/Hz]

Zero Forcing, Full CSIT Zero Forcing, Greedy−SUS w/ metric I Zero Forcing, Greedy−SUS w/ metric II Zero Forcing, Greedy−US w/ metric III Zero Forcing, Greedy−US w/ metric IV Random Beamforming

Figure 5.2: Sumrate versusthe averageSNR for

B D = 4

^bits,

M = 2

^{transmit antennas}

and

K = 30

^users.

0 5 10 15 20 25 30 35 40 45 50

4 6 8 10 12 14 16

Users, K

Sum Rate [bits/s/Hz]

Zero Forcing − Full CSIT Zero Forcing − Greedy−SUS w/ metric I Zero Forcing − Greedy−SUS w/ metric II Zero Forcing − Greedy−US w/ metric IV Random Beamforming

Figure5.3: Sumrateasafuntionofthenumberofusersfor

B D = 4

^bits,

M = 2

^transmit

antennasandSNR =20dB.

UnquantizedCQI

In Figure5.2 we omparethesumrates oftheproposed CQImetrisasafuntion of the

averageSNR, for

K

^=30usersand

B D

^{=4bits peruserforCDI}quantization. Strategy 1 (metri I)andstrategy2(metri II) oersimilarthroughput,exhibitinghoweverthesame

bounded behaviorathigh SNR, wherethe systemapaityonvergesto aonstant value.

GivenaxednumberofCDIbits

B D

^{,thesystembeomes}unavoidablyinterferene-limited at high SNR and the rate urves atten out. This is due to the fat that the auray

aswellasdue to that Greedy-SUSfores thesystemto shedule always

M

^{users. On the}

ontrary,thesheme usingstrategy 3(withfeedbakof twosalarvalues)provideshigher

exibilityby transmittingto

M ≤ M

^{users, thus keepingalinearsum-rategrowthin the}

interferene-limitedregionandonvergingtoTDMAfor

P → ∞

^(where

M = 1

^isoptimal).

In Figure 5.3 we plot the sum rate asa funtion of

K

^{for average SNR = 20dB and}

odebookof size

B D

^{=4bits. It anbeseenthat allsalarmetrisaneientlybenet}

from the multiuser diversity gain. The gap with respet to the full CSIT ase an be

dereased by inreasing the feedbak load

B D

^{. However, the slightly dierent saling of}

strategy4(shedulingmetriIV) is due to thefat that the userseletionbasedon

sum-rateestimates deides that

M < M

^{beamsoughtto beused. Sine the alulations are}

performed using inomplete CSIT, erroneous or loose estimations an sometimes lead to

sub-optimal deisionsin termsof thenumberof usersto besheduled. Furthermore,in a

systemwith xed orthogonality fator

ǫ

^{, the auray of the lowerbound (}

γ ^II

^{) does not}

improveas

K

^{inreases. Onthe otherhand,theupperbound (}

γ ^I

^{)beomesmorerealisti}

duetoahigherprobabilityofndingorthogonalquantizedhannels,heneyieldingslightly

betterperformane.

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35

SNR [dB]

Sum Rate [bits/s/Hz]

0

Zero Forcing w/ Full CSIT

metric I w/ Greedy SUS

K = 300 K = 50

K = 1000

0

Figure5.4: Sumrateperformaneasafuntionof theaverageSNR forinreasingvalueof

thenumberofusers,with

B D = 4

^{bitsoffeedbakperuserand}

M = 2

^{transmitantennas.}

We study now theperformane of strategy 1 with dierent number of users and CDI

feedbakbitsinordertoobtainaninsightontheCQIfeedbakmetridesignandtheresults

ofourasymptotianalysis. Figures5.4and5.5showasum-rateomparisonasafuntionof

theaverageSNR,illustratingthemultiplexinggainahievedbystrategy1. Inbothgures,it

anbeseenthatgivenaxednumberoffeedbakbits

B D

^{,thesystembeomes}unavoidably interferene-limited at high SNR and therate urvesatten out. Given axed odebook

size, Figure 5.4 shows the performane improvement of the proposed CQI metrisas the

numberofativeusersinreases. Indeed,itanbeseenthattheperformanegapbetween

theshemewithperfetCSITandstrategy1withpartialCSITisnarrowerfor

K

^inreasing.

Althoughtheshemeenterstheinterferenelimitedregimeforlargevaluesof

P

^,thelarger

0 5 10 15 20 25 30 35 40 0

5 10 15 20 25 30

SNR [dB]

Sum Rate [bits/s/Hz]

Zero Forcing w/ Full CSIT

Random Beamforming

0

Metric I w/ Greedy−SUS BD = 9 BD = 7 BD = 4

Figure5.5: Sumrateasafuntion oftheaverageSNRforinreasingodebooksize,

M = 2

transmitantennas,and

K = 50

^users.

0 100 200 300 400 500 600 700 800 900 1000

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

Number of users

Sum Rate [bits/s/Hz]

Zero Forcing w/ Full CSIT

Random Beamforming Metric I w/ Greedy−SUS

BD = 4

BD = 7

BD = 9

Figure5.6: Sumrateperformaneasafuntionofthenumberofusersforinreasing

ode-booksize,

M = 2

^{transmitantennas,andSNR=10dB.}

Forxednumberofativeusersintheell(Fig.5.5),byinreasingthenumberofodebook

bits, strategy1onvergesto ZFBF with perfet CSIT, whileproviding onsiderablegains

with respet to RBF. Note also that inreasing the number of bits for hannel diretion

quantizationat highSNR ismorebeneialthanat lowSNR.Thesumrateasafuntion

of thenumber of users

K

^{is shown in Figure 5.6. As the size of the odebook inreases,}

the performane of sheme I approahes that of the sheme with perfet CSIT, showing

the expeted saling with thenumber of users. This is due to the fat that metri I an

EetofCQI quantization

InordertoevaluatetheeetofCQIquantization,weonsiderasysteminwhiheahuser

hasin total

10

^{bits availablefor feedbakreporting. Asum-rateomparisonasafuntion}

ofthenumberofusersforSNR=20dBisshownin Figure5.7. Weuse

B D

^{bitsforfeeding}

baktheindex of thequantizedhannel andthe remaining

B Q = (10 − B D )

^{bitsfor CQI}

quantization. ForStrategy4(twosalarvaluesoffeedbak),2bitsareusedforquantization

ofthe hannel norm(

γ ⁽¹⁾

^{)and 3bits forthe alignment(}

γ ⁽²⁾

^{). The random}beamforming shemeuses

B D = 1

^{bitsinordertospeifythehosen}transmittedbeam(

B D = ⌈ log 2 M ⌉

⁾

andtheremaining(

9

^bits)forSINRquantization. Asimplequantizationtehniquehasbeen usedthat minimizesthemeansquareddistortion (maxLloydalgorithm). Forthisamount

ofavailable feedbak, itan be seenthat forthe simulatedrange of

K

^,

6

^{bitsare enough}

to apture alarge portion of multiuser diversity and preservethe saling (ase

B D = 4

^).

Notealso that the performane issimilar to that of Figure 5.3, in whih the CQImetris

areonsideredunquantized.

0 5 10 15 20 25 30 35 40 45 50

2 4 6 8 10 12 14 16

Users, K

Sum Rate [bits/s/Hz]

Zero Forcing − Full CSIT Zero Forcing − Greedy−SUS w/ metric I Zero Forcing − Greedy−SUS w/ metric II Zero Forcing − Greedy−US w/ metric IV Random Beamforming

Figure5.7: Sum rateversusthenumberof usersforwith SNR =20dB,

M = 2

^transmit

antennas and 10-bittotal feedbakbits.

B D = 5

^{bits are usedfor odebook indexingand}

(

B Q = 10 − B D

^bits)forCQIquantization. FormetriIV,2bitsareusedforquantization ofthehannelnormand3bitsforthealignment.

Finitesumrate feedbak onstraint

Weevaluate nowthesumrate performane ofstrategy1under anitesumrate feedbak

onstraint. Thetotalnumberofavailablefeedbakbitsis

B tot

^{=7bits. CQI}quantization is performed through Max-Lloyd's algorithm. One both the input quantization levels

q i

andoutputrepresentativelevels

γ q i

^{arefound, thequantizersets}

γ q i = q i

^,

0 ≤ i ≤ N Q − 1

inordertoavoidinformationoutageevents.

Figures 5.8 and 5.9 show the sum rate asa funtion of the number of users for SNR

=10 dB and SNR = 20 dB respetivelyfor dierent CDI and CQI feedbak alloations.

0 50 100 150 200 250 300 350 400 450 500

in asystemwith lownumberof ativeusers. On theother hand, asthe numberof users

inreases,itbeomesmorebeneialtoalloatebitsonCQIquantizationinstead. Theblak

urve

B D

^{=1bit orrespondsto theRBF for}

M = 2

^{transmitantennas [9℄. In asystem}

with optimalquantization,i.e. mathedtothePDF ofthemaximumCQIvalueamong

K

users,theamountof neessaryquantizationlevelsisreduedasthenumberofusersin the

ell inreases. Thus, fewer amounts of feedbak bits are needed for CQI quantization in

orderto apturethemultiuserdiversity.

0 100 200 300 400 500 600 700 800 900 1000

In Figure 5.10,the envelope ofthe urvesin the twoprevious guresis shown, whih

orrespondsto asystemthat hoosesthebest

B D /B Q

^{balanefor eah averageSNR and}

K

^{pair. Inthis gure,weompare howthis best pairof(}

B D

^,

B Q

^{)hanges asthesystem}

0 100 200 300 400 500 600 700 800 900 1000 5

6 7 8 9 10 11 12 13 14

Number of users

Sum Rate [bits/s/Hz]

Optimal Tradeoff − SNR = 10 dB Optimal Tradeoff − SNR = 20 dB BD = 6 BD = 5

BD = 4 BD = 3 BD = 1

BD = 6

BD = 5 BD = 4 BD = 3 BD = 1

SNR = 20 dB

SNR = 10 dB

Figure5.10: Sum ratevs. numberofusersin asystemwithoptimal

B D /B Q

^{balaning for}

dierentSNRvalues.

averageSNRinreases. Bothurvesaredividedindierentregions,aordingtotheoptimal

(

B D

^,

B Q

^{)pairin eahregion. Itanbeseenthattheoptimalthresholdforswithingfrom}

B D → B D − 1

^bits(andthus

B Q → B Q + 1

^{)isshiftedtotherightforhigheraverageSNR}

values(upperurve). Thismeansthat asthe average SNR inreases,morebitsshould be

alloatedonhanneldiretioninformation. Summarizing,givenapairofaverageSNRand

K

^{values,thereexistsanoptimalompromiseof}

B D

^and

B Q

^,giventhat

B tot

⁼

B D

⁺

B Q

^.

5.8 Conlusion

Inthishapter,westudymulti-antennabroadasthannels,inwhiheahuserreportsbak

totheBSquantizedCDIandreal-valuedsalarCQIthroughalimitedratefeedbakhannel.

We proposed various salar CQI feedbak and sheduling metris that, if ombined with

eientjointshedulingandzero-foringbeamforming,anahieveasigniantfrationof

theapaity of thefull CSI ase bymeans ofmultiuser diversity. These metrisare built

uponinter-userinterfereneboundsandinorporatesinformationonbothhannelgainand

quantizationerrorasameanstoestimatesatisfatorilythereeivedSINR.Anovelfeedbak

strategyisalso identied, whih allowsforadaptiveswithing betweenmultiuser(SDMA)

andsingle-usertransmission(TDMA)modewasalso identied asameans to ompensate

forthesum-rateeilingeetathighSNR.Ourshemeisshowntoahievelinearsum-rate

growthintheinterferene-limitedregionbydynamiallyadaptingthenumberofsheduled

users. Under apratially relevant xedfeedbakrate onstraint per user,weformulated

the problem of optimal feedbak balaning in order to exploit spatial multiplexing and

multiuser diversitygains. A low-omplexity optimization approah has been suggestedin

ordertoidentify theneessaryCDIandCQIfeedbakloadsaling,revealinganinteresting

interplaybetweenthenumberofusers,theaverageSNR andthenumberoffeedbakbits.

Table5.1: GreedySemi-orthogonalUserSeletionwithLimitedFeedbak

Step0set

S = ∅

^,

Q ⁰ = 1, . . . , K

For

i = 1, 2, . . . , M

^repeat

Step1

k i =

^arg

max

k ∈Q ⁱ⁻¹

CQI

k

Step2

S = S ∪ k i

Step3

Q ⁱ = n

k ∈ Q ^{i − 1} | | h ˆ _k h ˆ ^H _k

i | ≤ ǫ o

Table5.2: GreedyUserSeletionAlgorithmwithLimitedFeedbak

Step0Initialization: Set

S ⁰ = ∅

^,

R ( S ⁰ ) = 0

^,and

Q ⁰ = 1, . . . , K

Step 1

k 1 =

^arg

max

k ∈Q ⁰

CQI

k

Set

S ¹ = S ⁰ ∪ { k 1 }

While

i < M

^repeat

i ← i + 1

Step 2

k i =

^arg

max

k ∈ ( Q ⁰ −S i−1 ) R ( S ⁱ − 1 ∪ { k } )

Step 3 Set

S ⁱ = S ⁱ − 1 ∪ { k i }

if

R ( S ⁱ ) ≤ R ( S ⁱ − 1 )

Step 4 nishalgorithmand

i ← i − 1

Step 5 Set

S = S ⁱ

^and

M = i

APPENDIX

5.A Proof of Theorem 5.1

Beforeproeedingto theproofofTheorem5.1,werststatethefollowingresult.

Lemma5.3: LetU

k ∈ C ^{M × (M − 1)}

bean orthonormalbasis spanning the nullspae ofw

k

^.

Proof. Dene theorthonormalbasisZ

k

^of

C ^M

obtainedbystakingtheolumn vetorsof U

k

^andw

k

^{: Z}

k = [

^U

k

^w

k ]

^{. SineZ}

k

^Z

H

Then,bydenitionofZ

k

^{weanseparatethepowerofh}

k

^asfollows

dening

ω _k ² = cos ² θ k

^,theinterfereneoverthenormalizedhannelforuser

k

^andindexset

S

^,denotedas

I k ( S )

^{, anbeexpressedas}

Thenormalizedhannelh

k

^{anbeexpressedasalinearombinationof}orthonormalbasis vetors. Using Lemma 5.3, all possible unit-norm h

k

^{vetors with}

h k

^w

k

Substituting(5.45)into (5.44)weget

I k ( S ) = ω _k ²

^w

^H _k Ψ k ( S )

^w

k

Sinethersttermin(5.46)isperfetlyknown,theupperboundon

I k ( S )

^{isfoundbyjoint}

maximizationofthesummands(a)and(b)withrespetto

α k

^,B

k

^ande

k

^{. Weuseasimpler}

(a) Dening A

k ( S ) =

^U

^H _k Ψ _k ( S )

^U

k

^{for larity of} exposition, the seond term an be

where the operator

λ max {·}

^{returns the largest} eigenvalue. The maximum in (5.47) is obtainedwhenthevetorB

k

^e

k

^{equals theprinipal eigenvetorofthematrixA}

k ( S )

^.

(b)Deningq

k =

^B

^H _k

^U

^H _k Ψ _k ( S )

^w

k e ^{jα k}

^{andnotingthatthematrix}

Ψ _k ( S )

^{isHermitianby}

onstrution,theboundonthethird termin(5.46)anbewrittenasfollows

max

k

^q

^k k

^{, whih satises theunit-norm onstraint,}

yieldingthemodiedboundin (5.48). Thesolutionisgivenby

max

Finally, inorporatinginto (5.46) the bounds obtained in (5.47) and (5.49) weobtain the

desiredbound.

5.B Proof of Lemma 5.1

By noting that

ǫ ZF

^{orresponds to the maximum possible amplitude of the o-diagonal}

terms of

H

ˆ k

H

ˆ

H k

− 1

and under the non restritive assumption

ǫ < _M ¹ _{− 1}

^{, the bound on}

ǫ ZF

^{is found by bounding theamplitude of theo-diagonal terms in the Neumannseries}

P _∞

,where

offdiag( · )

^{takestheo-diagonalpartsettingtheelementsin}

thediagonaltozero. Byrepresentingthenon-normalizedzero-foringbeamformingvetors

as the sum of h

ˆ k

^{and its orthogonal omplement w}

˜ k

^{, i.e.}

w k = ˆ

^h

k + ˜

^w

k

^{and bounding}

theamplitudeof thediagonaltermsofI

+ P ∞

n=1 offdiag

, weobtainthedesired

boundonthehannelalignment

cos θ k

^.

5.C Proof of Theorem 5.2

By using the denition of eah user'sSINR

k

^,

cos θ k

^{and equal power alloation, we have}

that

We aim to nd an upperbound on the multiuser interferenegivenby Theorem 5.1 that

takesintoaounttheworst-aseorthogonality

ǫ ZF

^{. Expressingtheworst-ase}interferene

reeived by the

k

^{-th user in terms of}

cos θ k

^and

ǫ ZF

^{, the following bounds an be easily}

Hene, by substituting these values in equation (5.15), we obtainthe upper bound

I k = cos ² θ k (M − 1)ǫ ² _ZF + sin ² θ k [1 + (M − 2)ǫ ZF ] + 2 sin θ k cos θ k (M − 1)ǫ ZF ≤ sin ² θ k

^{. By}

substituting

ǫ ZF = ϑ

^and

cos θ k = | ^{cos φ k − √ ϑ} |

1+ϑ

^(i.e. inequalities (5.17) and (5.18), respe-tively beome equalities), where

ϑ = _{1 − (M} ^ǫ _{− 1)ǫ}

^{in the previous} expression, we have the upperbound givenby(5.21). Usingthis bound ontheSINR

k

^{expressionderivedin(5.50),}

weobtaintheSINRbound inequation(5.20).

5.D Proof of Lemma 5.2

Beforeproeedingtotheproof,werststatesomepreliminary alulationsthat areuseful

inthederivationoftheCDFof

γ _k ^I

^{. Tosimplifythenotation,wedenetherandomvariable}

ν := k h k k ²

^{whih is Gamma} distributed with parameter

M

^{and mean}

E {k h i k ² } = M

^;

hene,itsPDFis givenby

f ν (x) = x ^{M − 1}

Γ(M ) e ^{− x}

^(5.52)

where

Γ(M ) = (M − 1)!

^{istheompletegammafuntion.}

In[62℄,itisshownthatundertheACVQframework,theinterferene

Y = k h _k k ² sin ² φ k

followsahi-square

χ ² _{(2M − 2)}

distributionwith

(2M − 2)

^{degreesoffreedomweightedby}

δ

^,

i.e.

Y ∼ δχ ² _{(2M − 2)}

^{. Similarly,the} distribution ofthe reeived signal

X = k h _k k ² cos ² φ k = k h _k k ² (1 − sin ² φ k )

^{isthesumoftwo}independentweightedhi-squaredistributions

χ ² ₍₂₎ + (1 − δ)χ ² _{(2M − 2)}

^.

Denethefollowinghangesofvariables

ψ := sin ² φ k u := ¹ _δ ν(1 − ψ)

ν := k h _k k ² v := ¹ _δ νψ

^(5.53)

Then,themetriinequation(5.9)anbeexpressedas

γ = u

ran-dom variables for i.i.d. hannels, thejoint PDF of

u

^and

v

^{is obtainedfrom}

f uv (u, v) =

1

Hene,wegetthejointdensity

f uv (u, v) = δ

Γ(M − 1) e ^{− δ(u+v)} v ^{M − 2}

^(5.57)

TheCDFoftheCQImetriI isfoundbysolvingtheintegral

F γ (x) = ZZ

u,v ∈ D x

f uv (u, v) du dv

^(5.58)

Theboundedregion

D x

^inthe

uv

^{-planerepresentstheregionwheretheinequality}

_v+ ^u M P δ ≤ x

holds. In addition, sine the domain of

ψ

^is

D ψ = [0, δ]

^{, we also obtain the} inequalities

v

u+v ≥ 0

^,

_u+v ^v ≤ δ

^andthus

u ≥ ^{1 −} δ ^δ v

^{. Hene,}

F γ (x)

^isobtainedbyintegrating

f uv (u, v)

^over

therst quadrantof the

uv

^{-plane, intheregion dened by}

u ≤ x v + _{P δ} ^M

and

u ≥ ^{1 −} δ ^δ v

^.

Dependingontheslopesoftheselinearboundaries,theintegralin(5.58)isarriedoutover

dierentregions

value of

v

^{in whih the linearboundaries interset,}

v = _{P(1 −} ^Mx _{δ − δx)}

^{. Solving the integrals}

above,weobtaintheCDFoftheSINRmetri.

5.E Proof of Theorem 5.3

Let

γ _k ^I _i

^{denotetheupper bound ontheahievedSINRof user}

k i

^{(i.e. theuserseletedat}

the

i

^{-thiteration,for}

i = 1, 2, . . . , M

^{. FromTheorem1in[64℄,wehavethat}

manipu-lations itanbeshownthatforlarge

K

^,wehave

Pr

Sine

log( · )

^{is aninreasingfuntion,wehavethat}

Pr { log ₂

Hene,

Bysubstituting

u _{K 1}

^and

u _{K i}

^{intheaboveequation,weonludethattheLHSandtheRHS}

oftheinequalitiesbothonvergetooneas

K → ∞

^,therefore

K lim →∞

R

log _{2 M} ^P log K = 1

^(5.62)

with probabilityone. Assuming equal power alloation and that

M

^{perfetly orthogonal}

usersanbefound, as

Pr {|S| = M } ^K → ^→∞ 1

^{,wehavethattheproposedshemeahievesa}

sumrateof

M log _{2 M} ^P log K

.

An upperboundon

R ^opt

^{isgivenin [44℄,where}

Pr

where the RHS of the inequality inside the

Pr

^{goes to zero for}

K → ∞

^{. As a result, for}

large

K

^,wehavethat

0 ≤ log ₂ (1 + γ _k ^I _i ) − R ^opt

M , i = 1, . . . , M

withprobabilityone,whihresultsto(5.30)for

K → ∞

^,as

R ^opt

^{isanupperboundonthe}

sumrateofourproposedsheme.

5.F Proof of Theorem 5.4

For

P → ∞

^,wehave

Theexpetedsumrateforauserset

S

^{(ofardinality}

M

^)isgivenby

R ≤ E

=

where (a)followsfrombinomialexpansionandto get(b)theNörlund-Rieintegral

repre-sentationisapplied[105℄. Combining(5.64)with

K ⁱ ≤ K

^{,weget(5.31).}

Feedbak Redution using

Ranking-based Feedbak

6.1 Introdution

Intheprevioushapters,weinvestigated severalshedulingandlinearbeamforming

teh-niquesand tried to identify low-ratefeedbak measuresthat providethe transmitter with

suientyetpartialhannelknowledge,asameanstoahievenearoptimalsystem

through-put. Itwasshownthatifsomeformofimpliithannelknowledge(e.g. hannelorrelation)

is exploited, it suient to feed bak one or two salar feedbak parameters in order to

ahievesatisfatoryperformane. Inthishapter,wetakeadierentapproahtothe

prob-lemoffeedbakredutionandaimatndingarepresentationoffeedbakmetristhatallows

forfurther ompression. Usingthepromising two-step shedulingandpreodingapproah

proposed in Chapter 3, we point out that the hannel information to be onveyedto the

sheduleranbefurtherdereased. Astherst-stagehannelinformationismainlyutilized

for thepurposes of user seletion and notfor beamdesign or ratealloation, wepropose

anew type offeedbak representation,oinedasranking-basedfeedbak. In thisapproah,

eahuser-insteadofreportingaquantizedversionofCSITfeedbak-alulatesandfeeds

baktheranking,anintegerbetween1and

W + 1

^,ofitsinstantaneousCSITamongasetof

baktheranking,anintegerbetween1and

W + 1

^,ofitsinstantaneousCSITamongasetof

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