ML oarse Channel Estimation with CQI Feedbak

4.4 Exploiting Statistial CSIT in Spatially Correlated Channels

4.4.3 ML oarse Channel Estimation with CQI Feedbak

islarge. Asuboptimal,yeteient,greedyuserseletionshemeanbeusedinstead,similar

to the approahin [11℄. Here weextendthis sheme for MMSE linearbeamformingwith

long-term spatial information and instantaneous salar CQI feedbak

γ k

^. ^The ^proposed

greedyuserseletionalgorithmisgiveninTable4.2. Inthisalgorithm,usersareaddedone

by one to the set. The omplexity anbe redued by onsidering only usersexeeding a

threshold

γ th

^. ^Theûser ^with ^the ^highest ^CQI îs êxamined ât êah ^time, ând ît îs âdded

to thesetof sheduledusers

S

ônlyîfît^resultsⁱⁿ^sum-rateînrease. ^We^should ^note^that

theoverallperformaneofgreedyuserseletiondepends heavilyonwhetherthepreoding

matrixanbereproessedeahtimeauserisadded,whihinturndependsontheformof

thehannelfeedbakavailableatthetransmitter.

4.4.3 ML oarse Channel Estimation with CQI Feedbak

Asstatedbefore,theorrelationmatrixprovidesusefulinformationaboutthespatial

han-nel harateristis, espeially if it is ill-onditioned, however it does notreveal any

infor-mation aboutthe qualityof the urrenthannel realization. Inorder to exploit multiuser

diversity,theshedulerrequiresproperlydesignedinstantaneouslow-ratefeedbak

γ k

^,^whih

Table4.2: GreedyUserSeletionwithStatistialCSIT

GreedyUserSeletionwithstatistialCSIT

Ateahtimeslot

t

1. Initialize

S = ∅

^and

G = ∅

2. Selettheusersthat exeedthethreshold

γ th

G = {∀ k ∈ { 1, . . . , K }| γ k ≥ γ th }

3. Selettheuserwith thehighestCQIvalue

k max =

^arg

max

1 ≤ k ≤ K γ k

S ← S ∪ { k max } , G ← G \ S

4. Repeat

k ^∗ =

^arg

min

k ∈G Tr (

Ψ( ˆ S ) + βI − 1 2 )

S ← S ∪ { k ^∗ } , G ← G \ S

until

|S| = M

5. Returnuserset

S

anbeameasureofthequalityoftheurrenthannel. Inthissetion,werestritourselves

toRayleighfadingorrelatedhannels,i.e.

h ¯ k = 0

^,^and^we^propose^a^simple^frameworkⁱⁿ

whihlong-termstatistialhannelknowledgeisombinedwithshort-termpartialCSITas

ameanstoprovideaoarsehannelestimateateahslot.

MLEstimationwith BeamGain Information

Weadoptherethefeedbakstrategy1andonsiderthateahuser

k

^feeds^bak^the^squared

magnitudeofthe hannelwith abeamforming vetor

z k ∈ C ^M ^× ¹

, i.e.

γ k = h ^H _k z k

2

. The

beamforming vetors an be interpreted as pilot signals during the training phase or as

the preferredbeamformer in a two-stage preoding and sheduling approah (see Setion

3.4). Thisbeamformeranbehosenrandomlyoritanbeoptimizedbasedonlong-term

statistialinformation.

Optimized training vetors As the orrelation matrix of eah user is known at the

transmitter side, the training vetors

z _k

ân ^be ôptimized. ^Briey ^speaking, ân êient

trainingodebookanontain

N = N p + N l + N r

^vetors,^where ^the^indies^p,l,r^indiate

prinipal,loal, and random,respetivelyasexplained below. Theodebook onstrution

1)Basedoneahuser'sstatistialCSIT,theodebookwillontain

N p

^prinipaleigenvetors of theovarianematrix

R k

⁽

N p = 1

^for^MISO^hannels).

2)Inthisstep,weselet

N l

^vetorsⁱⁿ^the^loalâreaôfêah^prinipal^statistial^diretion

v _k

asameanstoaountforthosehannelrealizationsthatsteertheprinipalsingularvetor

in aloalityoftheprinipalstatistial diretion. Theloal areaoftheprinipalstatistial

diretion is dened by a one around

v _k

^and ^is haraterized by the angle between the trainingandtheprinipalstatistialvetors.

3)Duringthethirdstep,wegenerate

N r

^vetors^thatâreôutside^theône^dened ⁱⁿ^step²

andaountforthehannelrealizationsinwhihthediretionoftheprinipalrightsingular

vetor(orvetorhannel)isfarfromthestatistial(mean)hanneldiretion. Thesevetors

anbehosenrandomly orastheones that oversoptimallytheremainingspae, outside

theone,whihisrelatedtotheGrassmannianlinepakingproblem. Intheidealase,the

size of

N r

^should^beâdapted ^basedôn^the^strengthôforrelation,asitgivesameasure on thefrequenythatthesedeviationsour.

Random training vetors Forsimpliity, werather adopt alow-omplexity approah

and onsider a randomopportunisti beamformingsetting [9℄. In this setting, we assume

that thevetors

z _k

^are isotropiallydistributed and hosenrandomly, i.e.

z _k = q _m

^where

{ q m } ^M m=1

âre ^the ôlumns ôf ^the ûnitary ^matrix

Q

^. ^We ^ombine ^the information ex-trated from the orrelation matrix with a salar instantaneous feedbak in the form of

γ k = | h ^H _k ˜ q k | ²

^,^where^the^vetor

z k = ˜ q k

^is^hosen^by^user

k

^as

˜

q k =

^arg

max

m=1,...,M | h ^H _k q m | ²

^(4.12)

Clearly,thistypeofsalarCQIprovidesajointinstantaneous measureofthequalityofthe

urrenthannel realizationand its diretionof thehannelinstantaneously. Although the

amount of spatial information enapsulated into this metri annot be deomposed from

the hannel gain information, it is partiularly useful for userswith strong hannels, i.e.

users that are verylikely to be sheduled. It an bealso shown that the hoie of

q ˜ _k

^is

equivalentto seletingthebeamoverwhih user

k

^experiene ^the^highest^reeived^SINR

k

in [9℄. Assume that user

k

^has^its ^maximum^SINR ^on ^beam

i

^out^of ^the

j ∈ { 1, . . . , M }

Sine

f (x)

^is^always^monotonous^positive^for

x ∈ (0, c)

^, ^we^have^that

i =

^arg

max

We propose aML estimation framework that ombines long-term statistial knowledge

andinstantaneousCSITprovidedbythefeedbakmetri

γ k

^. ^This^feedbak^allows^us^to^pik

userswhose hannels span spatially separated ones of multipath and havegood hannel

gains. This so-alled Constrained Maximum Likelihood (CML) hannel estimate is the

onethat maximizesthelog-likelihoodfuntion of thePDF (4.5) onditionedto thesalar

feedbakonstraint

γ k = | h ^H _k q ˜ k | ²

h ˆ _k =

^arg

max f (h | γ k )

^(4.15)

Thisresultstothefollowingoptimization problem:

max

k

h ^H _k R k h k

s.t. | h ^H _k q ˜ _k | ² = γ k

(4.16)

Itanbeeasilyshownthat(4.16)isequivalenttosolvingthefollowinggeneralizedeigenvalue

problem(GEV):

R _k h _k = λ Φ _k h _k

^,^where

Φ _k = ˜ q _k q ˜ ^H _k

^. ^The^maximumgeneralizedeigenvalue oftheHermitianmatrixpair(

R _k , Φ _k

^),^with

Φ _k > 0

^is^dened ^as

λ max (R k , Φ _k ) = sup { λ |

^det

(λΦ k − R _k ) = 0 } = sup

q 6 =0

h ^H _k R _k h _k h ^H _k Φ k h k

(4.17)

Thesolutionof(4.16),in theviewofthegeneralizedRayleigh-Ritzquotient,isgivenby

h ˆ _k =

^arg

max

h k

h ^H _k R _k h _k h ^H _k Φ _k h _k

(4.18)

whihorrespondstothedominantgeneralizedeigenvetor,denotedas

u _k

^,^assoiated^with

thelargestpositivegeneralizedeigenvalueoftheHermitianmatrixpair(

R k , Φ k

^). ^Therefore,

theMLhannelestimateisgivenby

h ˆ _k =

√ γ k

| q ˜ ^H _k u _k | u _k

^(4.19)

OrthogonalBasis expansion

ThesolutionoftheCMLestimateasageneralizedeigenvalueproblemrequiresthe

om-putation of the prinipal generalized eigenvetor at eah time slot, thus it may exhibit

remarkableomputational omplexity in pratie. In order to failitate thealulation of

the oarse estimate, we derive an equivalent hannel estimation framework in whih the

hannel of the

k

^-th ûser îs êxpressed âs â^linear ômbination ôf ôrthogonal ^vetors.

Al-thoughanyorthogonalbasisanbeused, intheaseofrandomtrainingvetorsitismore

naturaltohoosethebeamformingvetors

{ q _m } ^M i=1

^as^ourorthonormalbasis. Inthatase, thehannelvetoranbeexpressedas

h ^H _k =

M

X

m=1

α m q ^H _m

^(4.20)

where

α m

âre^the^(omplex) ^weightsôf^theôrthogonalêxpansion.

Consider, without loss of generality, that

q ₁

^orresponds ^to ^the ^best ^beam ^hosen ^by

user

k

^. Substituting (4.20) into (4.16), and solvingtheoptimization problem (4.16)using Lagrangemultipliers,weobtainthattheoptimalweights

b _opt = [α 2 , · · · , α M ] ^T

^equal^to

b _opt = − α 1 A ⁻ ¹ c

^(4.21)

where

and

α 1 = √ γ k

^so^that ^theinstantaneousCQIfeedbakonstraintissatised.

Observingthesimilarityinthestrutureofmatrix

A

^with^that^of

Q ^T R ⁻ _k ¹ Q ^∗

^,^the

omputa-tional omplexityof thematrixinversionof

A

ân^be^further^redued^throughûseôf^blok

matrixdeomposition. Denote

F = Q ^T R ⁻ _k ¹ Q ^∗

^,^then

Theinverse

A ⁻ ¹

ân^beêasily ôbtainedûsing^theêquation:

S _A ⁻ ¹

ML ChannelEstimation with Channel NormFeedbak

ConsidernowthattheinstantaneousCQImetritakesontheformofthehannelnorm,i.e.

γ k = k

^k k ²

^. ^Clearly,^the^above^diretion independentCQI feedbakprovide less instanta-neous spatialinformation than

γ k =

h ^H _k z _k

2

. However,foruserswith largehannel gain,

thusforusersthataremorelikelytobeseleted,hannelnormfeedbakprovidesomeform

of additionalspatial information(espeially in Riean hannels). Moreover,the largerthe

hannel gain,the moreaurate this hannel diretional information. There is howevera

dierenebetweenRayleighandRieanhannels. IntheRayleighase,thesignambiguity

onthediretionannotbeeliminated,whereasin Rieanhannels(non-zeromean)thereis

additionalCDIonthesignoflargehannelrealizations.

SimilarlytoSetion4.4.3,weformulateaoarseMLhannelestimateassumingthatthe

hannel norm of user

k

^is ^known, ^whih ^results ⁱⁿ ^the ^following^onstrained optimization problem:

max h k

h ^H _k R _k h _k

s.t. k h _k k ² = γ k

(4.22)

Thesolutionof (4.22)isgivenby

h ˆ _k = √ γ k u _k

^(4.23)

where

u _k

îs ^theêigenvetorâssoiated^with ^the^largest êigenvalueôf

R _k

^and

γ k

^is ^hosen

Dans le document Multiuser multi-antenna systems with limited feedback (Page 100-105)

ML oarse Channel Estimation with CQI Feedbak

4.4 Exploiting Statistial CSIT in Spatially Correlated Channels

4.4.3 ML oarse Channel Estimation with CQI Feedbak

γ k

γ th

S

γ k

t

S = ∅

G = ∅

γ th

G = {∀ k ∈ { 1, . . . , K }| γ k ≥ γ th }

k max =

max

1 ≤ k ≤ K γ k

S ← S ∪ { k max } , G ← G \ S

k ∗ =

min

k ∈G Tr (

Ψ( ˆ S ) + βI − 1 2 )

S ← S ∪ { k ∗ } , G ← G \ S

|S| = M

S

h ¯ k = 0

k

z k ∈ C M × 1

γ k = h H k z k

2

z k

N = N p + N l + N r

N p

R k

N p = 1

N l

v k

v k

N r

N r

z k

z k = q m

{ q m } M m=1

Q

γ k = | h H k ˜ q k | 2

z k = ˜ q k

k

˜

q k =

max

m=1,...,M | h H k q m | 2

q ˜ k

k

k

k

i

j ∈ { 1, . . . , M }

f (x)

x ∈ (0, c)

i =

max

γ k

γ k = | h H k q ˜ k | 2

h ˆ k =

max f (h | γ k )

max

k

h H k R k h k

s.t. | h H k q ˜ k | 2 = γ k

R k h k = λ Φ k h k

Φ k = ˜ q k q ˜ H k

R k , Φ k

Φ k > 0

λ max (R k , Φ k ) = sup { λ |

(λΦ k − R k ) = 0 } = sup

q 6 =0

h H k R k h k h H k Φ k h k

h ˆ k =

max

h k

h H k R k h k h H k Φ k h k

u k

k ^∗ =

S ← S ∪ { k ^∗ } , G ← G \ S

z k ∈ C ^M ^× ¹

γ k = h ^H _k z k

z _k

v _k

v _k

z _k

z _k = q _m

{ q m } ^M m=1

γ k = | h ^H _k ˜ q k | ²

m=1,...,M | h ^H _k q m | ²

q ˜ _k

γ k = | h ^H _k q ˜ k | ²

h ˆ _k =

h ^H _k R k h k

s.t. | h ^H _k q ˜ _k | ² = γ k

R _k h _k = λ Φ _k h _k

Φ _k = ˜ q _k q ˜ ^H _k

R _k , Φ _k

Φ _k > 0

λ max (R k , Φ _k ) = sup { λ |

(λΦ k − R _k ) = 0 } = sup

h ^H _k R _k h _k h ^H _k Φ k h k

h ˆ _k =

h ^H _k R _k h _k h ^H _k Φ _k h _k

u _k

h ˆ _k =

| q ˜ ^H _k u _k | u _k

{ q _m } ^M i=1

h ^H _k =

α m q ^H _m

q ₁

b _opt = [α 2 , · · · , α M ] ^T

b _opt = − α 1 A ⁻ ¹ c

Q ^T R ⁻ _k ¹ Q ^∗

F = Q ^T R ⁻ _k ¹ Q ^∗

A ⁻ ¹

S _A ⁻ ¹

^k k ²

h ^H _k z _k

h ^H _k R _k h _k

s.t. k h _k k ² = γ k

h ˆ _k = √ γ k u _k

u _k

R _k