Lower bound on instantaneous reeived SINR

Intheprevioussetion,westudiedboundsontheaveragereeivedSINRandidentied an

eientCQI metri. However, from apratialpoint of view,metri I hasthelimitation

that is not ahievable (upper bound), sine in general the beamforming vetors are not

perfetly orthogonal, espeially in networks with lowto moderate numberof users. As a

result, metri I may be useful foruser seletionpurposes; howeveritannot beemployed

forrateadaptation. Ifthesystemmathestheodingrateandmodulationorderbasedon

the

γ _k ^I

^{value (f. eq. (5.9)), the link will suer from signiant outage events sine CQI}

metriIoverestimatesthereeivedSINR.Toirumventthat,theBSisrequiredtoaskfor

additionalfeedbakfrom theseletedusersto performratealloation. Thisseond stepof

feedbakmaybedetrimental in termsofsignalingoverhead andprotooldelays,and itis

ratherimpratialin fasttime-varyinghannels.

Inordertoavoidtheneedforthisseondstepandtoguaranteeoutage-freetransmissions,

weaimat ndingafeedbakmetrithat anbeeientlyutilized forbothshedulingand

ratealloationsimultaneously. Forthat,weproposetofeedbakalowerboundontheSINR

rather than an upper bound. Additionally, we derive bounds on the instantaneous SINR

•

^{alowerbound onthereeivedsignalpower.}

•

^{anupperboundontheatualmultiuser}interferene.

NotealsothattheSINRestimatedby(5.9)doesnottakeintoaountthefatthataspei

preoder is used for transmission sheme. Therefore, it neglets the eet of preoding

andthat of theorrespondingmisalignment betweenthe quantizedhannel diretion and

thebeamforming vetor. In this paragraph, assuming that ZFBF is employed, we derive

interfereneboundsthatinorporatethepowerlossintroduedbythemisalignmentbetween

theinstantaneoushannelandtheZFbeamformers.

Notation: Thefollowing orthogonalityonstraints, whih that areused extensivelybelow,

anbeimposed:

•

^Theorthogonalitybetweenthequantized hannelandthezero-foringbeamformeris denedas:

ξ ≤

h ˆ _k w _k

^.

•

^{The worst-ase} orthogonality between two zero-foring beamformers is dened as

ǫ ZF = max

i,j ∈S

w ^H _i w _j

^.

Lower bound on reeived signalpower

The quantity

|

h

¯ k

^w

k | ² = cos ² (∠(¯

^h

k ,

^w

k ))

^{that appears in the numerator of (5.4) an be}

bounded as follows: using theinequality

∠(¯

^h

k ,

^w

k ) ≤ ∠(¯

^h

k , h ˆ _k ) + ∠(ˆ h _k ,

^w

k )

^{, andthefat}

thatthefuntion

cos x

^ismonotoniallydereasingin

x

^{fortheintervalofinterest,wehave}

cos ² (∠(¯

^h

k ,

^w

k )) ≥ cos ² (∠(¯

^h

k , h ˆ _k )+∠(ˆ h _k ,

^w

k ))

^{. Therefore,thereeivedsignalpowerofuser}

k

^,denotedas

S k

^{,anbelowerboundedas}

S k = P |

^h

^k

^w

^k | ² ≥ P k h _k k ² cos ² (φ k + ∠(ˆ h _k ,

^w

k )) =

^(5.10)

NotethatiftheBSisabletondperfetlyorthogonaluserhannels,thequantizedhannel

diretion

h ˆ k

^and zero-foringbeamformingvetor

w k

^{oinide, and hene}

∠(ˆ h k , w k ) = 0

^,

yieldingthefollowingsimpleexpressionforthelowerboundin(5.10):

S k = P k h _k k ² cos ² φ k

^.

Astheabovelowerboundannotbealulatedexpliitlyatthereeiverside,weareobliged

tousetheorthogonalityonstraint

ξ

^{,whihresultsinthefollowinglowerbound:}

S _k ^LB1 ≥ P k h _k k ² cos ² (φ k + arccos(ξ))

^(5.11)

Adierentlowerboundonthereeivedsignalanbederivedasfollows:

cos ² θ k = |

h

¯ k

^w

k | ² =

Sinethereeiversannotalulate theabovelowerbound (astheydonothaveaess

inthequantity

| h ˆ _k

^w

_k |

^{),thereeivedsignalpowerneedsto befurtherboundedas:}

S _k ^LB2 ≥ P k h _k k ²

ξ ² cos ² φ k − ξ p

1 − ξ ² | sin(2φ k ) |

(5.12)

Bounds on Instantaneous MultiuserInterferene

Theinter-userinterfereneofthe

k

^{-thuseranbeexpressedas:}

I k ( S ) = P k h k k ² X

where I

k ( S )

^{denotes themultiuser}interferene experiened bythe

k

^{-thuser overthe}

nor-malized hannel. Sine the zero-foring beamformers satisfy the orthogonality onstraint

h ˆ _k w _j = 0, ∀ j 6 = k

^,wehavethat

In order to bound the inter-user interferene, weneed to bound either theterm I

⊥

spanning thenullspae of

w _k

^{. Wereallthat theZF}beamformersareonsideredas unit-normvetors.

Theorem5.1: Givenasetofnormalizedbeamformingvetors

{ w _k } , k ∈ S

^{,thenormalized}

interfereneterm I

k ( S )

^{isupperboundedby}

I

k ( S ) ≤ cos ² θ k α k ( S ) + sin ² θ k β k ( S ) + 2 sin θ k cos θ k ω k ( S )

^(5.15)

Proof. TheproofisgiveninAppendix5.A.

For notationsimpliation, wedropbelowthedependene on

S

^.

Lemma5.1: The worst-aseorthogonality of asetof

M

zero-foring beamforming vetors andthe alignmentwith thenormalizedhannel(

cos θ k

^{)are boundedasafuntion of}

cos φ k

for

ǫ < _M ¹ _{− 1}

^asfollows:

Proof. TheproofisgiveninAppendix5.B.

Itis worthnotingthattheabovelemma providesanotherlowerbound onthereeived

signalpower,givenby:

Basedonthisresult,wehave:

Theorem5.2: Given agroup of

ǫ

-orthogonal users with ardinality

|S| = M

^{, the reeived}

SINRinasystememploying zero-foringbeamforming islower boundedby

SINR

k ≥ P k h _k k ² cos ² θ k

P k h _k k ² I ^UB1 _k + M

(5.20)

where

I ^UB1 _k = (M − 1) (ϑ cos θ k + sin θ k ) ² − (M − 2)(1 − ϑ) sin ² θ k

^(5.21)

with

cos θ k = | ^{cos φ k − √ ϑ} |

1+ϑ

^and

ϑ = _{1 − (M} ^ǫ _{− 1)ǫ}

^.

Proof. Theproofisgivenin Appendix5.C.

An additional inter-user interferene upper bound

I ^UB2 _k

^{an be derived by trying to}

upperbound thetermI

⊥ k ( S )

^.

CQIfeedbak metriII

MotivatedbytheabovelowerboundontheinstantaneousSINR(f. Theorem2),wepropose

thateahuserfeedsbaktotheBSthefollowingsalarmetri

γ _k ^II = S _k ^LBx I ^UBx _k + M

(5.22)

wherethewildard`x'an bereplaedby1,2or3forthereeivedsignal(LB) and1or2

forthe interferene(UB). In thenumerialresults setion, weonly simulate thefollowing

metri:

γ ^II _k = S _k ^LB3 I ^UB1 _k + M =

P

(1+ϑ) ² k h _k k ² (cos φ k − √ ϑ) ² P k h k k ² I ^UB1 _k + M

(5.23)

Inordertoalulate(5.23),thereeiverhastoknowtheorthogonalitysystemparameters

ǫ

^and

ξ

^{and to assumethat}

M = M

^{exatlyuserswill be sheduled. Thebasidierene}

between(5.9)and(5.23)isontheestimationoftheinter-userinterfereneandthereeived

signalpower. In(

5.9

^)theinterfereneisreplaed byanupperbound onitsaveragevalue, i.e.

E n

P

j ∈S\{ k } P

M k h k ² | h ¯ _k w _j | ² o

≤ M ^P k h k ² sin ² φ k

^{, where for CQI metri II an upper}

bound ontheinstantaneousmultiuserinterferene(f. eq. 5.21)isusedinstead.

CQImetriIanbeviewedasanestimationofreeivedSINRassumingthatthe

quan-tizedhannel

h ˆ _k

^{and the} zero-foringbeamformer

w _k

^oinide,i.e.

∠(ˆ h _k , w _k ) = 0

^{. This}

assumptionbeomesvalidforlargenumberofusers

K

^{. Therefore,in metriI,the}

approxi-mation

cos ² (∠(¯

^h

k ,

^w

k )) ≈ cos ² (∠(¯

^h

k , h ˆ _k ))

^{isused,whereasinCQImetriIIthepowerloss}

introdued bytheangle shiftdueto themisalignment ofh

ˆ k

^{and w}

k

^{is taken intoaount}

(usongLemma5.1).

Evidently,thetwoproposed metrisoinidefor

ǫ = 0

^sinewehave

cos ² θ k = cos ² φ k

^,

thus

I ¯ UB k = sin ² θ k = sin ² φ k

^and

(cosφ k − √ ϑ) ²

(1+ϑ) ² = cos ² φ k

^{. Hene, metri II (5.23) takes}

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