Semi-blind methods for FIR multichannel estimation

SEMI{BLIND METHODS FOR

FIR MULTICHANNEL

ESTIMATION

Elisabeth de Carvalho

StanfordUniversity

DavidPackardElectricalEngineering

StarLab

350SerraMall

Stanford,CA94305-9515,USA

Tel: +1650-724-3633 Fax: +1650-723-9251

email: carvalho@dsl.stanford.edu

Dirk Slock

InstitutEURECOM

2229routedesCr^etes,B.P.193,

06904SophiaAntipolisCedex, FRANCE

Tel: +33493002626 Fax: +33493002627

email: slock@eurecom.fr

CorrespondingAuthor: Elisabeth de Carvalho

Abstract

The purpose of semi-blind equalization is to exploit the information used by

blindmethodsaswellastheinformationcomingfromknownsymbols. Semi{blind

techniquesrobustifytheblind andtrainingproblems,andoerbetterperformance

than these two methods. We rst present identiability conditions and perfor-

manceboundsforsemi{blindestimation. Semi{blindmethodsareabletoestimate

any channel, even when the position of the known symbols in the burst is arbi-

trary. Performance bounds for semi{blind multichannel estimation are provided

through the analysis of Cramer-Rao bounds (CRBs) and a comparison of semi{

blind techniques with blind and training sequence based techniques is presented.

Threecategoriesofsemi{blindmethodsareconsidered. Optimalsemi{blindmeth-

odstakeintoaccountallblindinformationandalltheinformationcomingfromthe

known symbols,evenifnotgrouped. MainlyMaximum{Likelihood(ML)methods

willbeconsidered. Suboptimalsemi{blindsolutionsarealsoinvestigatedwhenthe

knownsymbolsaregroupedinatrainingsequence: thesuboptimalsemi{blindcri-

teriaappearasalinearcombinationofatrainingsequencecriterionandablindML

criterion. Thirdly,wepresentmethods thatcombine agivenblindcriterionwitha

trainingsequencebasedcriterion.

7.1 Introduction

7.1.1 Training Sequence based Methods and Blind Methods

Traditionalequalizationtechniquesare basedontraining. Thesendertransmitsa

trainingsequence(TS)knownatthereceiverwhichisusedtoestimatethechannel

coeÆcients orto directly estimatethe equalizer. Most of theactual mobile com-

municationstandards include atraining sequenceto estimate the channel, like in

GSM [1]. Inmost cases,training methods appear as robust methods but present

somedisadvantages.Firstly,bandwidtheÆciencydecreasesasanon{negligiblepart

ofthedataburstmaybeoccupied: inGSM,forexample,20%ofthebitsinaburst

are used for training. Furthermore, in certain communication systems, training

sequences are not available or exploitable, such as when explicit synchronization

betweenthereceiverandthetransmitterisnotpossible.

Blindequalizationtechniquesallowtheestimationofthechannelortheequalizer

basedonlyonthereceivedsignalwithoutanytrainingsymbols. Theintroductionof

multichannels,orSIMOmodelswhere asingleinputsymbolstream istransmitted

throughmultiple symbol ratelinear channels, has given riseto aplethora of new

blind estimation techniquesthat do notrequirehigherorder{statistics. Themost

popular second{order statistics (SOS) based estimation techniques suer from a

lackofrobustness: channels mustsatisfydiversityconditionsandmanyblindSOS

methods fail when the channel length is overestimated. Furthermore, the blind

techniquesleaveanindeterminacyinthechannelorthesymbols,ascaleorconstant

phaseora discretephase factor. Thissuggests that SOS blind techniques should

Training Sequence based Estimation

estimate the channel

Blind Estimation

All symbols considered as unknown All channel outputs used Training sequence used to Channel

Output Burst

Training Sequence Tail Bits Tail Bits

BURST

Combines training sequence and blind

informations SEMI-BLIND ESTIMATION

Figure 7.1. Semi{BlindPrinciple: exampleofaGSMburst.

mayalsobetruefor TSbasedmethods, especiallywhen thesequenceistooshort

foracertainchannellength. Semi{blindtechniquesprovideasolutiontoovercome

theseproblems.

7.1.2 Semi{Blind Principle

Inthis chapter,weassumeatransmissionbyburst, i.e. thedata isorganizedand

transmittedbyburst, andwefurthermoreassumethat knownsymbolsarepresent

in each burst in the form of a trainingsequence aimed at estimating the channel

orsimplyintheformofsomeknownsymbolsusedforsynchronizationorasguard

intervals,likein theGSMorthe DECTbursts. Inthistypicalcase,whenusinga

trainingorablindtechniquetoestimatethechannel,informationgetslost. Train-

ing sequence methods base the parameter estimation only on the received signal

containing known symbols and all the other observations, containing (some) un-

knownsymbols,areignored.Blindmethodsarebasedonthewholereceivedsignal,

containingknownandunknownsymbols,possiblyusinghypothesesonthestatistics

of the input symbols, likethe fact that they are i.i.d. for example, but no use is

madeoftheknowledgeofsomeinputsymbols. Thepurposeofsemi{blindmethods

is to combine both training sequence and blind information (see gure 7.1) and

exploitthepositiveaspectsofbothtechniques asstatedin section7.1.1.

Semi{blindtechniques,becausetheyincorporatetheinformationofknownsym-

bols,avoidthepossiblepitfallsofblindmethodsandwithonlyafewknownsymbols,

any channel, singleormultiple, becomesidentiable. Furthermore,exploiting the

blindinformationin additiontotheknownsymbolsallowstheestimationoflonger

channelimpulseresponsesthanispossiblewithacertaintrainingsequencelength,

afeature thatisofinterestfortheapplicationofmobilecommunicationsinmoun-

wewillcallGaussianmethods),oneknownsymbolissuÆcienttomakeanychannel

identiable. Inaddition,it allowstheuseofshortertrainingsequencesforagiven

channel length and desired estimation quality, compared to atraining approach.

Apartfromtheserobustnessconsiderations,semi{blindtechniquesappearalsovery

interesting from aperformance point of view, astheir performance is superior to

thatoftrainingsequenceorblindtechniquesseparately. Semi{blindtechniquesare

particularlypromisingwhenTSandblindmethodsfailseparately:thecombination

ofbothcanbesuccessfulinsuchcases.

In section 7.2, we describe the multichannel model. In section 7.4, we give

identiabilityconditionsforFIRmultichannelestimationforthedeterministicand

Gaussian classesof methods. In section7.5, theCRBs are usedasaperformance

measure for semi{blind estimation, which is compared to TS and blind channel

estimation. Semi-blind estimation is shown to be superior to either TS or blind

methods. In section 7.7, we describe optimal semi{blind methods which are able

to take into account all blind information and all the information coming from

the known symbols, even if not grouped. These methods are mainly based on

ML.Insection7.9,wepresentsuboptimaldeterministicsemi{blindmethodswhich

can be constructedwhen theknown symbols are grouped in atraining sequence.

Thesuboptimal semi{blindcriteriaappear as alinearcombination ofa TSbased

criterionandablind MLcriterion. Theproposed MLbasedmethods aresolvedin

aniterativequadraticfashion. Insection7.10,welinearlycombineablindcriterion

with a TS criterion: particularly theSubchannel Response Matching (SRM) and

Subspace Fitting (SF) blind criteria are considered. We provide a performance

study of the resulting semi{blind deterministic quadratic criteria in section 7.11.

At last,in section7.12, wegivean exampleof the Gaussian method: semi{blind

covariancematching.

Throughoutthis chapter,weshallusethefollowingnotation:

(:)

,(:) T

, (:) H

conjugate,transpose,conjugatetranspose

(:) +

Moore{PenrosePseudo{Inverse

tr(A),det(A) traceanddeterminantofmatrixA

vec(A) [A

T

i;1 A

T

i;2 A

T

i;n ]

T

Kroneckerproduct

^

, o

estimateofparameter,truevalueofparameter

E

X

mathematicalExpectationw.r.t.therandomquantityX

Re(:), Im(:) realandimaginarypart

I Identitymatrixwithadequatedimension

w.r.t. withrespectto

7.2 Problem Formulation

We consider here linear modulation (nonlinear modulations such as GMSK can

belinearizedwith goodapproximation[2], [3])overalinearchannelwith additive

H

1 (z)

a(k)

y

1 (k)

H

2 (z)

y

2 (k) v

2 (k) v

1 (k)

Figure7.2. Multichannelmodel: examplewith2subchannels.

the transmittedsymbolswith an overallchannel impulseresponse,which is itself

the convolution of the transmit shaping lter, the propagation channel and the

receiverlter,plusadditivenoise. Theoverallchannelimpulseresponseismodeled

asFIR which for multipath propagation in mobile communications appears to be

well justied. In mobile communicationsterminology, thesingle-usercase will be

considered.

We describe here the FIR multichannel model used throughout the chapter.

Thismultichannelmodelappliesto dierentcases: oversamplingw.r.t.thesymbol

rateofasinglereceivedsignal[4],[5],[6]ortheseparationintothereal(in{phase)

and imaginary(quadrature) component of the demodulated received signal ifthe

symbolconstellationisreal[7],[8]. Inthecontextofmobiledigitalcommunications,

athirdpossibilityappearsin theformofmultiplereceivedsignalsfrom anarrayof

sensors. Thesethreesourcesformultiplechannelscanalsobecombined.

Inthe multichannel model, asequence of symbolsa(k) is receivedthrough m

channelsoflengthN andwithcoeÆcientsh (i)(seegure7.2):

y(k)= N 1

X

i=0

h(i)a(k i)+v(k); (7.2.1)

v(k)isanadditiveindependentwhiteGaussiannoise,r

vv

(k i)= Ev(k)v(i) H

=

2

v I

m Æ

k i

. Assume wereceiveM samples,concatenatedin thevectorY

M (k):

Y

M (k)=T

M (h)A

M+N 1 (k)+V

M

(k) (7.2.2)

Y

M

(k) = [y T

(k)y T

(k M+1)]

T

, similarly for V

M

(k), and the input burst is

A

M+N 1

(k)= [a(k)a(k M N+2)]

T

. T

M

(h) is a block Toeplitzmatrix with

M blockrowsand

H 0

m(M 1)

asrstblockrow:

H=[h(0) h(N 1)] andh= h

h T

(0)h T

(N 1) i

T

: (7.2.3)

Thechannel transferfunction isH(z)= P

N 1

i=0 h (i)z

i

=[H

1

(z)H

m (z)]

T

, where

H

i

(z)isthetransferfunctionofthei th

subchannel.

ThechannellengthisassumedtobeN whichimpliesh (0)6=0andh(N 1)6=0

simplify thenotationin(7.2.2) withk=M 1to

Y =T(h)A+V: (7.2.4)

Semi{BlindModel Thevectorofinputsymbolscanbewrittenas:A=P

A

K

A

U

whereA

K

aretheM

K

knownsymbolsandA

U theM

U

=M+N 1 M

K

unknown

symbols. Theknown symbolscanbedispersed in the burstand P designatesthe

appropriatepermutation matrix. Forblind estimation A=A

U

, whileA =A

K

=

A

TS

forTSbasedestimation. Wecansplitthecorrespondingpartsin thechannel

outputasT(h)A=T

K (h)A

K +T

U (h)A

U .

Irreducible, Reducible, and Minimum-phase Channels A channel iscalled irre-

ducibleifitssubchannelsH

i

(z)havenozerosincommon,andreducibleotherwise.

Areduciblechannelcanbedecomposedas:

H(z)=H

I (z)H

c

(z); (7.2.5)

where H

I

(z),of lengthN

I

, is irreducible and H

c

(z), of lengthN

c

=N N

I +1,

is a monochannel for which we assume H

c

(1) = h

c

(0) = 1 (monic). A channel

is called minimum{phase if all its zeros lie inside the unit circle. Hence H(z) is

minimum{phaseifandonlyifH

c

(z)isminimum{phase.

Commutativity of Convolution We shall exploit thecommutativity property of

convolution:

T(h)A=Ah (7.2.6)

where: A=A

1 I

m ,

A

1

= 2

6

6

6

6

4

a(M 1) a(M 2) a(M N)

a(M 2) .

. .

. .

. .

.

.

.

.

. .

. .

. .

. .

.

.

a(0) a( N+1)

3

7

7

7

7

5

: (7.2.7)

Minimum Zero-Forcing (ZF) Equalizer Length, Eective Number of Channels

TheBezoutidentitystatesthatforanFIR irreduciblechannel, FIRZFequalizers

exist[9]. TheminimumlengthforsuchaFIRZFequalizeris

M =minfM : T

M

(h)hasfullcolumn rankg : (7.2.8)

Onemay note that T

M

(h) has full column rankfor M M. In [10],it is shown

thatifthemN elementsofHareconsideredrandom,morepreciselyindependently

distributed withacontinuousdistribution,then

M =

N 1

withprobability1, (7.2.9)

and M =1for N =1. In this case,the channel is irreduciblew.p. 1. Onecould

consider other (perhapsmorerealistic) channel models. Consider, for example,a

multipath channel with K pathsin which themultichannel aspect comes from m

antennas. Without elaborating thedetails, itis possibleto introduce aneective

numberofchannelsm

e

=rank(H)whichinthis casewouldequal(w.p.1)

m

e

= minfm;N;Kg : (7.2.10)

With a reduced eective number of channels, the value of M increases to M =

N 1

m

e 1

w.p. 1. Note that in the rst probabilistic channel model leading to

(7.2.9),ifm>N,theninfactm

e

=N,butthisdoesnotchangethevalueofM =1.

Anothertypeofchannelmodelarisesinthecaseofahillyterrain. Inthatcase,two

ormorerandomnon-zeroportionsofchannelimpulseresponsearedisconnectedby

delays. Ifthesedelaysaresubstantial,thenforthepurposeofdeterminingM,the

problem can be approached as amulti{user problem by interpretingthe dierent

chunks of the channel as channels corresponding to dierent users. Multi{user

resultsforM [9]couldthenbeapplied.

Ingeneral,foranirreduciblechannel,MN 1[11]inwhichtheupperbound

wouldcorrespondtom

e

=2. Notethatm

e

=1correspondstoareduciblechannel

(inwhichcaseM=1).

7.3 Classication of Semi{Blind Methods

Semi{blindmethodscanbeclassied(roughly)accordingtotheamountofapriori

knowledgeontheunknowninputsymbolsthatgetsexploited,seegure7.3:

1. Noinformationexploited: deterministicmethods.

These methodsaredirectlybasedonthestructureofthereceivedsignaland

particularlyonthestructureoftheconvolutionmatrixT(h). Amongtheblind

deterministic methods, onecannd theSubspaceFitting (SF) method [12],

theSubchannelResponseMatching[13]method,thedeterministicMaximum

Likelihoodmethod [14],[5]and also the leastsquares smoothing method or

two{sidedlinearpredictionapproach[15],[16]. Semi{blindextensionsofthese

blind methodsalreadyexistasstatedlaterinthis chapter.

2. Second{orderstatistics: Gaussianmethods.

These methods exploit the second{order moments of thedata. The (blind)

predictionmethod[5],[17]orthe(blind)covariancematchingmethod[18]be-

longto thiscategory,butalsothesemi{blindGaussianMaximumLikelihood

(GML) approach [19] which treatsthe unknown input symbols asGaussian

random variables. A semi{blind covariancematching approach will be pre-

sentedlaterin thischapter.

3. Higher{orderstatistics.

Increasing A Priori Knowledge Exploited Increasing Non Convexity Distribution

Joint Part of

STOCHASTIC Distribution

Joint

GAUSSIAN Statistics Second-Order

HIGH ORDER

STATISTICS ALPHABET FINITE Distribution

Joint Part of

DETERMINISTIC No Information

Figure7.3. Classicationof(semi)blindchannelidenticationmethodsaccording

totheassumedAPrioriknowledgeontheunknowninputsymbols.

4. FiniteAlphabet oftheinputsymbols.

Thesemethodsexploitthenitealphabetnatureoftheinputsymbols. Among

these approaches,wendtheMLmethods [21].

5. Complete distributionoftheinputsymbols.

Inthesemethods,thetruedistributionoftheinputsymbolsisexploited;for

example,forBPSK,weexploitthetruediscretedistributiontheinputsymbols

(i.e. theirnitealphabetplustheirprobabilities). ThestochasticML(SML),

whichwaspresentedinitssemi{blindversionin[22],belongstothiscategory.

Themoreinformationontheunknowninputsymbolsgetsexploited,thebetter

the channel estimation. However, at the sametime, the associated methods are

typicallymorecostly,withcostfunctions presentinglocalminima. Inthischapter,

we mainlyfocusondeterministicand Gaussian methodswhich canbesolved ina

simplewaywithsometimesquadraticcostfunctions/closed-formsolutions.

7.4 Identiability Conditions for Semi{Blind Channel Estimation

Inthissection,wedeneidentiabilityconditionsforsemi{blinddeterministicand

Gaussian channel estimation. Werstrecall resultson blind estimationand then

extend them to semi{blind estimation. One remarkable property is that, in the

deterministiccase, semi{blindtechniques canestimatethechanneleven whenthe

known symbols are arbitrarily dispersed. The following results are detailed and

provedin[23].

7.4.1 Identiability Denition

Let be the parameter to be estimated and Y the observations. We denote by

f(Yj)theprobabilitydensityfunctionofY. Inaregularcase(i.e.in anonblind

case),iscalledidentiableif[24]:

8Y; f(Yj)=f(Yj 0

) ) =

0

: (7.4.1)

This denition has to be adapted in the blind identication case because blind

inthedeterministicmodelandjj=1intheGaussianmodel(isrealinthecase

ofrealsymbols). Theidentiabilitycondition(7.4.1)willbefortoequal 0

upto

theblindindeterminacy.

ForbothdeterministicandGaussianmodels,f(Yj)isaGaussiandistribution:

identiability in this casemeans identiability from themean and the covariance

ofY.

7.4.2 TS Based Channel Identiability

WerecallheretheidentiabilityconditionsforTSbasedchannelestimation. From

(7.2.6), T(h)A = Ah. h is determined uniquely if and only if A hasfull column

rank,whichcorrespondsto conditions(i) (ii)below.

Necessary and suÆcient conditions [TS]The m{channel H(z) isidentiable

byTS estimationif andonlyif

(i) BurstLengthMN ornumberof known symbols M

K

2N 1.

(ii) Number ofinputsymbol modes 1

N.

TheburstlengthM isthelengthofY,expressedin symbolperiods.

7.4.3 Identiability in the Deterministic Model

Inthedeterministicmodel,Y N(T(h)A;

2

v

I)and =[A T

U h

T

] T

. Identiability

of is based on the mean only; the covariancematrix only contains information

about 2

v

, the estimationof whichhence getsdecoupled from theestimation of.

A

U

andhareidentiableif:

T(h)A=T(h 0

)A 0

)

(

A

U

=A 0

U

andh=h 0

forsemi{blindandTSbasedestimation

A= 1

A

0

andh=h 0

forblindestimation

(7.4.2)

withcomplex,foracomplexinputconstellation,andreal,forarealinputconstel-

lation. Identiabilityishencedenedfromthenoise{freedataT(h)A(the mean).

Blind ChannelIdentiability

SuÆcient conditions [DetB] In the deterministic model, the m{channel H(z)

andthe inputsymbolsA areblindly identiable if

(i) H(z)isirreducible.

(ii) BurstlengthMN+2M.

1

(iii) Number ofinputsymbol modesN+M.

These conditionsexpress the fact that oneshould haveenough data with the

rightpropertiesto beableto completelydescribethesignalornoisesubspace(i.e.

thecolumnspaceofT(h)oritsorthogonalcomplement). Theseconditionsappear

to be suÆcient for all the deterministic methods listed in section 7.3 except for

SRM[14].

Semi{Blind ChannelIdentiability

We distinguish here between the case of grouped known symbols (training se-

quence case) andthe caseof arbitrarily dispersed knownsymbols. Inbothcases,

the channel can be identied for the same number of (non-zero) known symbol-

s. When the known symbols are all equal to zero, the channel cannot be iden-

tied when they are grouped. However, when they are dispersed in the burst,

the channel can be identied (up to a scale factor). We will consider the gen-

eral case of a reducible channel H(z) = H

I (z)H

c

(z): M

I

is dened as M

I

=

minfM : T

M (h

I

)hasfullcolumnrankg.

Grouped knownSymbols

SuÆcient conditions[DetSB] In the deterministic model, the m{channel H(z)

andthe unknowninputsymbolsA

U

aresemi{blindlyidentiable if

(i) BurstlengthMmax(N

I +2M

I

;N

c N

I +1)

(ii) Number of excitation modes of the input symbols: atleast N

I +M

I

that are

notzerosofH(z)(or henceof H

c (z)).

(iii) Grouped known symbols: number M

K

2N

c

1 (which is also a necessary

condition), withnumberof excitation modesN

c .

For an irreducible channel, 1 known symbol is suÆcient. For amonochannel,

2N 1 grouped known symbols are suÆcient. If 2N 1 grouped known symbols

containingN independentmodesareavailable,condition(ii)becomessuperuous.

ForareduciblechannelH(z)=H

I (z)H

c (z),H

I

(z)canbeidentied blindlywhile

H

c

(z) can be identied by TS. In general, at least as many known symbols are

needed as the number of (continuous) parameters that cannot be determined by

blindestimation.

Arbitrarily Dispersed Known Symbols In [26], we prove that the regularity of

the Fisher Information Matrix (FIM) is equivalent to local identiability for the

deterministicmodel(andalsotheGaussian model). Bystudyingthe regularityof

theFIM,thetreatmentofthearbitrarilydispersedknownsymbolscase(alsotreated

in[27]tosomeextent)becomestractable. Strictlyspeaking,wecanproveonlylocal

identiability. Ingeneral,FIMregularityimpliesanite numberofsolutionsin A

and h [28], however,for asuÆcient burst length [27] (larger than the ones given

Non-ZeroKnownSymbols

Theorem 1: The channelH(z)islocallyidentiable with probability 1 2

if

(i) Burstlengthmax(N

I +2M

I

;N

c N

I +1).

(ii) Numberof excitation modesN+M

I .

(iii) Numberof knownsymbols 2N

c

1,which isalso anecessarycondi-

tion.

Note that no further conditionsonthe excitation modes ofthe known sym-

bols are required. Furthermore, amonochannel can be identied if 2N 1

arbitrarilydispersed knownsymbolsareavailable.

KnownSymbolsequalto Zero

When the known symbols are all equalto zero, the channel canat best be

identied up to a scale factor. Indeed, T(h)A = T(h 0

)A 0

, with h 0

= h,

A 0

=A= andA

K

=A 0

K

. Wehaveshownin [26] that the channel issemi{

blindly locally identiable up to a scale factor if and only if the FIM is 1-

singular. The position of the known symbols cannot be totally arbitrary

though. Infact,wehavethefollowingtheorem:

Theorem 2: Thechannel H(z)islocally identiablewithprobability 1upto

ascalefactor if

(i) BurstlengthN

I +2M

I .

(ii) Numberof excitation modesN+M

I .

(iii) Numberofknownsymbols (zeros)2N

c

2,whichisalsoanecessary

condition.

(iv) Theknownsymbols(zeros)are\suÆciently"dispersed: thereareatleast

N

c

1knownsymbolsthatdonotbelongtoagroupofN

c

ormoreknown

symbols.

If only some known symbols are equal to zero, Theorem 1 can be applied

providedcondition(iv)ofTheorem2isadded.

Semi{Blind RobustnesstoChannel LengthOverestimation

A major disadvantageof the deterministic blind methods is their non robustness

to channel length overestimation. Semi{blind estimation overcomesthis problem.

Consideragainareduciblechannel: H(z)=H

I (z)H

c (z).

SuÆcientconditions[DetSBR]Inthedeterministicmodel,them{channelH(z)

andthe unknowninputsymbols A

U

are semi{blindlyidentiable when theassumed

channellengthN 0

isoverestimatedif

2

(i) BurstlengthMmax(N

I +2M

I

;2(N 0

N

I

+1) N).

(ii) Number ofinputsymbol excitation modes: atleastN

I +M

I

thatarenot zeros

ofH

c (z).

(iii) Knownsymbols: M

K 2(N

0

N

I

)+1,grouped.

Number ofknown symbolmodesN 0

N

I +1.

Theseresultsarealsovalidwithprobabilityoneforarbitrarilydispersedknown

symbols.

7.4.4 Identiability in the Gaussian Model

IntheGaussianmodel,Y N(T

K (h)A

K

; 2

a T

U (h)T

H

U

(h)+ 2

v

I)and=[h T

2

v ]

T

.

2

a

isthevarianceoftheinputsymbols. Recallthat identiabilityisidentiability

from the mean and covariance matrix, so identiability in the Gaussian model

impliesidentiabilityinanystochastic model,sincesuchamodelcanbedescribed

intermsofthemeanandthecovarianceplushigher{ordermoments.

Blind ChannelIdentiability

In theblind case, m

Y

()=0, soidentiability is basedon thecovariancematrix

only. In the Gaussian model, the channel and the noise variance are said to be

identiableif:

C

YY (h;

2

v )=C

YY (h

0

; 2

v 0

))h 0

=e j'

hand 2

v 0

= 2

v

: (7.4.3)

Whenthesignalsarereal, thephasefactorisasign; whentheyarecomplex, itis

aunitarycomplexnumber.

Unlikeinthedeterministiccase,zeroscanbeidentied: itisonlynotpossibleto

determineifthezerosareminimumormaximum{phase(discretevaluedambiguity).

Soifitisknownthatthechannelisminimum-phase,thechannelcanbeidentied.

Irreducible Channel

SuÆcient conditions [GaussB1] In the Gaussian model, the m{channel H(z)

isidentiable blindly uptoaphase factorif

(i) H(z)isirreducible.

(ii) BurstlengthMM+1

TheseconditionsaresuÆcientconditionsfortheprediction,covariancematching

and GML methods. Note that not allthe non{zero correlations (lag0to N 1)

Reduciblechannel LetH(z)beareduciblechannel: H(z)=H

I (z)H

c (z).

SuÆcient conditions [GaussB2] In the Gaussian model, the m{channel H(z)

isidentiable blindly uptoaphase factorif

(i) H

c

(z) isminimum{phase.

(ii) Mmax(M

I +1;N

c N

I +1).

Inthemonochannel case,thenoisevariance 2

v

cannot beestimatedandhence

neither h. However, if we consider 2

v

as known, the channel can be identied

byspectralfactorization. ThesuÆcientconditionsare forthe monochannel to be

minimum-phaseand theburstto beatleastoflengthN.

Semi{Blind ChannelIdentiability

In the semi{blind case, identiability is based on the mean and the covariance

matrix.

Identiability for any channel Inthesemi{blind case,theGaussian modelhas

theadvantageofallowingidenticationfromthemeanonly. m

Y

()=T

K (h)A

K

=

A

K

h: if A

K

has full column rank, h canbe identied. The dierence with the

trainingsequencecaseisthatintheidenticationofhfromm

Y ()=T

K (h)A

K ,the

zerosdueto themeanofA

U

alsogiveinformation, which lowerstherequirements

on the number of known symbols. For one non-zero known symbol a(k) (with

0kM N,i.e.notlocatedattheedges),A

K

=a(k)I

Nm

. TheGaussianmodel

appearsthusmorerobustthanthedeterministicmodelasitallowsidenticationof

anychannel,reducibleornot,multiormonochannel,underthefollowingconditions:

SuÆcient conditions [GausSB1] In the Gaussian model, the m-channel H(z)

issemi{blindly identiable if

(i) BurstlengthMN.

(ii) At least one non-zero known symbol a(k) not located at the edges (0 k

M N).

Identiabilityforan Irreducible Channel

SuÆcient conditions[GausSB2] In the Gaussian model, the m-channel H(z)

issemi{blindly identiable if

(i) H(z)isirreducible.

(ii) Atleast1non-zeroknown symbol (locatedanywhere)appears.

Theseresultsonidentiabilityindicatethesuperiorityofthesemi{blindmethods

overTSandblindmethods,aswellasthesuperiorityofGaussianmethodsoverthe

0 10 20 30 40 50 60 70 80 90 100 10 ⁻¹

10 ⁰ 10 ¹

Number of known symbols Semi-Blind

all symbols known

DeterministicCRBsforBPSKandH

well

0 10 20 30 40 50 60 70 80 90 100

10 ⁻¹ 10 ⁰ 10 ¹

Number of known symbols 25

all symbols known Gain: 3

Gain: 5 Gain: 3

Semi-Blind Training Sequence

Gain: 20

DeterministicCRBsforBPSKandHwell

Figure7.4. CRBsfordeterministicsemi{blindchannelestimation(left);compar-

isonbetweendeterministicsemi{blindandTSchannel estimation(right).

7.5 Performance Measure: Cramer{Rao Bounds

We comparehere the performance of semi{blind channel estimation to blind and

training based estimation through the Cramer{Rao bounds for the channel with

theinputsymbolsbeingconsideredasnuisanceparameters. Thesecomparisonsare

illustrated by curves showing the trace of the CRBs w.r.t. the numberof known

(orunknown)symbolsintheinputburstforacomplexinputconstellation,QPSK,

andarealone,BPSK.Theknowninputsymbolsarerandomlychosenandgrouped

at the beginning of the burst. The SNR, dened as

2

a khk

2

m 2

v

(average SNR per

subchannel),is10dB;theburstlengthisM=100.

Three dierenttypes of channels are tested: an irreducible channel H

well , an

ill{conditionedchannelwithnearlya(common)zeroH

ill

,areduciblechannelwith

irreduciblepartH

I

andreduciblepartH

c .

The results are mainly shown in the deterministic case; the Gaussian curves

present similar shapes[26]. In gure 7.4 (left), we show the CRBs of semi{blind

channel estimation for a xed number of input symbols and variable number of

knownsymbols. Weseeadramaticimprovementoftheperformancewithveryfew

knownsymbols.

In gure 7.4 (right), we compare the performance of semi{blind and training

modes. For 10 known symbols, wehave a gainof performance of 20 brought by

semi{blind estimation. For the same estimation quality, 10 known symbols for

semi{blindestimationrequire50knownsymbolsforTSestimation. For25known

symbols,wehaveaperformancegainofafactor3;onerequires 70knownsymbols

0 10 20 30 40 50 60 70 80 90 100 10 ⁻²

10 ⁻¹ 10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵ 10 ⁶

Number of known symbols Semi-Blind

Constrained Semi-Blind Blind

DeterministicCRBsforBPSKandH

ill

Figure7.5. Comparisonbetweendeterministicblindandsemi{blindchannelesti-

mationforH

ill .

Deterministic blind channel estimation leaves a scale factor indeterminacy in

thechannel. Onecommonway toadjust this scalefactoristo impose constraints

onthe parameters; dierentconstraintscanbeconsidered. Here, weconsider the

blindCRBcomputedunderanormconstraintkhk 2

=kh o

k 2

andaphaseconstraint

Im(h H

h o

)=0. Thischoice ofconstraintsisrstmotivated bythecommonuseof

thenorm constraint. Furthermore, theresultingconstrained CRB is equalto the

pseudo{inverse of the Fisher information matrix for the channel (with the input

symbolsconsidered asnuisanceparameters) [30]. Thisis aparticular constrained

CRBasityieldsthelowestvalueforthetraceoftheCRBamongallsetsofaminimal

numberofindependentconstraints. Thepreviousnormand phaseconstraintsgive

the sameconstrained CRB as the linearconstrainth H

h o

= h oH

h o

: in general,a

setofconstraintisequivalentto anothersetoflinearconstraints,in thesense that

theygivethesameconstrainedCRB,see[30]forfurtherdetails.

Ingure7.5,wecomparesemi{blindandblindmodesevaluatedunderthenorm

andphaseconstraintsforanill{conditionedchannel. Thesuperiorityofthesemi{

blindmodecan benoticed especiallyforasmallnumberofknownsymbols.

In gure 7.6, the case of a reducible channel is shown. The CRB is drawn

w.r.t. the number of unknown symbols in the burst, for a xed number, 10, of

known symbols. Inthedeterministiccase,theblind partbrings asymptoticallyno

information to the estimation of the zerosof the channel. In the Gaussian case,

however,theblindpartbringsinformationtotheestimationofH .

0 20 40 60 80 100 120 140 160 180 200 10 ⁻²

10 ⁻¹

Number of unknown symbols Deterministic CRBs for BPSK - 10 known symbols

h

I

hc

0 10 20 30 40 50 60 70 80 90 100

10 ⁻³ 10 ⁻² 10 ⁻¹

Number of unknown symbols Gaussian CRBs for QPSK - 10 known symbols

h

c hI

Figure7.6. CRBsfordeterministic(left)andGaussian(right)semi{blindestima-

tionofareduciblechannelw.r.t.thenumberofunknownsymbols. 10symbolsare

known.

7.6 Performance Optimization Issues

Values ofthe knownsymbols (Deterministically)whiteinputsymbolsequences,

in the sense that A H

A = M

2

a

I, optimize the performance of training sequence

basedestimation. Optimization of thesemi{blind CRB w.r.t.the known symbols

leadsto channel dependentresults; weexpect howeverthat such whitesequences,

eveniftheydonotstrictlyoptimizethesemi{blindperformance, wouldbeamong

thebest choices.

Distribution of the known symbols over the burst Should theknown symbols

begrouped or separated? The answer seemsagain to depend on thechannel. In

thissection wewillcall \minimum{phasemultichannel",amultichannelforwhich

all the subchannels are minimum{phase, the energy is then concentrated in the

rst coeÆcients of the multichannel; a \maximum{phase multichannel"will have

maximum{phasesubchannels.

WedidteststocomparethedeterministicCRBs foraminimumandmaximum

phase channel H

min

and H

max

and a randomly chosen channel H

rand

. In the

tables below, we show the trace of the CRBs for the three channels for a xed

sequenceof 10knownsymbols,randomlychosenfrom aQPSKconstellation (top)

or equalto 0(bottom), groupedinthemiddle oftheburst oruniformly dispersed

allovertheburst. TheburstlengthisM=100. TheCRBsareaveragedover1000

realizationsoftheunknownsymbolsinthecaseofQPSK.

KnownSymbols H

min H

max H

rand

grouped 0.36 0.79 0.22

KnownSymbols H

min H

max H

rand

grouped 3.23 9.99 0.78

separated 0.53 1.36 0.16

Whentheknownsymbolsarechosenrandomly,fortheminimumandmaximum{

phasechannels,performance isbetter whenthe knownsymbolsare groupedthan

uniformly separated in the burst. For the random channel, both choices seeme-

quivalent. Whentheknownsymbolsareallequalto0,itishoweverbettertohave

themdispersed allovertheburstin allcases.

Positionofthetrainingsequenceintheburst Again,theanswerdependsonthe

characteristics of the channel. What could bedone is study the CRBs w.r.t. the

positionofthetrainingsequenceforastochasticchannelmodel(suchase.g.COST

207 models [31]). Our simulation experience tends to indicate that the training

sequencepositionshould besuchthat T

K

(h) hasmaximalenergy. Inotherwords,

puttingthe trainingsequencein themiddleof theburstis neverabadchoice, re-

gardlessofthechannel.

7.7 Optimal Semi{Blind Methods

Optimalsemi{blindalgorithmsshouldfulll acertainnumberofconditions:

Theyshould exploit alltheinformationcomingfromtheknownandtheun-

knownsymbolsintheburst,andespeciallytheobservationscontainingknown

andunknownsymbolsatthesametime. ThiscouldbeadiÆculttask,asthe

classicaltrainingsequencebasedestimationcannotdoitandblindestimation

doesnotdoit(andconsiderstheknownsymbolsasunknown).

They should work when the known symbols are arbitrarily dispersed in the

burst.

Semi{blind identiability conditions should also be respected: for example,

themethodsshouldworkforonlyoneknownsymbolforirreduciblechannels,

somethingthatisnotsystematicallysatisedbythesuboptimalmethods.

With asuÆcientnumberofknownsymbols,theoptimalsemi{blindmethods

should beabletoidentifyanychannel,particularlymonochannels.

Optimalsemi{blindmethodsaremethodsthatnaturallyincorporatetheknowl-

edge of symbols. Maximum{Likelihood methods fulll this condition. Methods

estimatingdirectlytheinputsymbolslike[32]arealsogoodcandidatesforoptimal

DeterministicMaximumLikelihood(DML) InDML,boththechannelhandthe

unknownsymbolsA

U

aretobeestimated. TheDML criterioncanbewritten as:

min

h;A

U

kY T(h)Ak 2

= min

h;A

U

kY T

K (h)A

K T

U (h)A

U k

2

(7.7.1)

Thiscriterioncanbeoptimizedusingalternatingminimizations betweenhandA

U

(see [33], [34] for the blind version of the algorithm and [35] for the semi{blind

version). This algorithm oers the advantage of decreasing the cost function at

eachiteration. TheblindversionconvergestotheMLminimumasymptotically(in

SNRand/ornumberofdata)[34]. However,thisalgorithmrequiresalargenumber

ofiterationswhichrendersitlesspractical.

When solving criterion (7.7.1) w.r.t. A

U

and replacing the expression of A

U

foundinthecriterion,wegettheDMLcriterionforh:

min

h

(Y T

K (h)A

K )

H

P

?

TU(h) ( Y T

K (h)A

K

) (7.7.2)

P

X

is the orthogonal projectiononto the column space of X and P

?

X

= I P

X

istheprojectiononto itsorthogonalcomplement. This criterioncanbeoptimized

bythemethod ofscoring[36],whichrepresentsacomputationallyheavysolution.

Insection7.9,thecriterion(7.7.2) willbesimpliedin thecaseofgroupedknown

symbolsandlowcomplexitysolutionswill beproposed.

GaussianMaximumLikelihood(GML) IntheGaussianapproach,theparameters

tobeestimatedarethechannelandthenoisevariance. TheGMLcriterionhasthe

form:

min

h;

2

v

lndetC

YY

+(Y T

K (h)A

K )

H

C 1

YY

(Y T

K (h)A

K

) (7.7.3)

Again,thiscriterioncanbeoptimizedbythemethodofscoring[36].

MaximumLikelihoodwithnitealphabetconstraintsontheinputsymbols(FA{

ML) TheFA{MLcriterionissimilar tothe DMLcriterion exceptthat thenite

alphabet (denotedA

p

)constraintontheinputsymbolsisimposed.

min

h;AU2Ap

kY T

K (h)A

K T

U (h)A

U k

2

(7.7.4)

FA{ML can be solved by alternatingminimizations betweenh and A

U

, with A

U

constrainedtothenite alphabet. Themostproblematicestimationisthat ofthe

symbolsbecauseoftheFAconstraint. In[21] wheretheblind versionof(7.7.4) is

optimized, theFA constraintisrst ignoredand then theestimatesare projected

onto the nearest discrete value of the nite alphabet. In [37], this technique is

extendedto the semi{blindcase. In[38], theblind versionof(7.7.4) is optimized

Stochastic Maximum Likelihood (SML) SML considers the input symbols as

random variables. Theirtrue distribution is taken into account: the symbolsare

assumed zero mean, i.i.d., equiprobable, and with values of the nite alphabet.

f(Yjh)= X

A2Ap

f(YjA;h)f(A) X

A2Ap

f(YjA;h),sotheSMLcriterionis:

min

h;

2

v 1

2

v X

AU2Ap exp

1

2

v kY T

K (h)A

K T

U (h)A

U k

2

(7.7.5)

DirectoptimizationoftheSMLcriterionrepresentsacostlysolution. TheExpectation{

Maximization (EM) [39] algorithm can be used to solve SML using the Hidden

MarkovModel(HMM) framework: see [40],foradescriptionofdierentmethods.

TheEMalgorithmwillconvergetotheSMLsolutiongivenagoodinitialization. A

semi{blindSMLisformulatedin[22].

When the known symbols are dispersed, the \blind problem part" looses its

structure asT

U

(h)hasnoparticular structuralproperties. Asaconsequence,fast

algorithms cannot be built. So for complexity reasons but also for performance

reasons(seesection7.6),whenthechoiceispossible,itispreferabletohavegrouped

knownsymbols.

Suboptimalsemi{blindcriteriacanbeconstructedwhentheknownsymbolsare

grouped. Therstgroupofproposedsemi{blindmethodsisagainbasedonML.In

thatcase,aMLbasedsemi{blindcriterioncanbewrittenas:

Semi{blindCriterion=

1

Trainingsequencecriterion+

2

Blindcriterion

The weights

1 and

2

are the optimal weights in the ML sense: they are not

arbitraryand are deduced from the semi{blindML problem. Such methods were

initiatedin[19]and[41].

Themostinterestingsemi{blindcriteriaaretheones basedonablindcriterion

thatisquadratic;thesemi-blindcriterionoftheformaboveisthenalsoquadratic.

Wenowmainly focusondeterministic MLmethods. Wedescribeamethod to

optimizeblindDMLinaniterativequadraticfashionandthencombineblindDML

tothree dierentTSbasedcriteria.

7.8 Blind DML

A low cost solution to solve blind DML is basedon alinear parameterization of

the noise subspace [9], [14]. For a given H(z), there exists an H

?

(z) such that

H

?

(z)H(z)=0and T(h

?

)T(h)=0where T(h

?

)is theconvolutionmatrixbuilt

from H

?

(z). Forexample, form =2(2 subchannels), H

?

(z)=[ H (z) H (z)];

form>2dierentchoicesarepossible,wewillconsider [42]:

H

?

(z)= 2

6

6

4 H

2 (z) H

1

(z) 0 0

0 H

3 (z) H

2 (z)

.

.

.

.

.

.

.

.

. .

.

.

0

H

m

(z) 0 0 H

1 (z)

3

7

7

5

: (7.8.1)

TheDMLcriterionforhcanthenbewritten as:

min

khk=1 Y

H

T H

(h

?

)

T(h

?

)T H

(h

?

)

| {z }

R(h) +

T(h

?

)Y : (7.8.2)

We only specify here the norm constraint on the channel; a phase constraint is

necessaryin thecomplexchannelcase[30]. Thechannelisassumedirreducible.

IterativeQuadraticML(IQML)isaniterativealgorithmwhich,ateachiteration

considersthedenominatorR(h)=Rasconstant,evaluatedatthechannelestimate

from theprevious iteration. The criterion becomes quadratic in this way. Using

thecommutativityofconvolution,wecanwriteT(h

?

)Y =Yh;theIQMLcriterion

canthenbewrittenas:

min

khk=1 h

H

Y H

R +

Yh: (7.8.3)

Inthe noiselesscase,Yh o

=0: thetruechannelh o

nullsthe(quadratic)criterion

(regardless ofinitialization) and thesolutionis h o

. At high SNR,arstiteration

givesaconsistent estimateof hand asecond iterationgivestheML solution. At

low SNR however, IQML results in a biased estimation of the channel and its

performance is poor. Indeed, asymptotically (M ! 1) the IQML cost function

becomesequivalenttoitsexpectedvaluebythelawoflargenumbers:

tr n

T H

(h

?

)R +

T(h

?

)EYY H

o

=

tr n

T H

(h

?

)R +

T(h

?

)XX H

o

+ 2

v tr

T H

(h

?

)R +

T(h

?

) :

(7.8.4)

h = h o

nulls the rst term but is not in general the minimal eigenvector of the

secondtermandhenceofthesum.

7.8.1 Denoised IQML (DIQML)

Therstapproach[41]weproposeremovesthenoisefromtheIQMLcriterion. We

subtractfromtheIQMLcriterionanestimateofthenoisecontribution( c

2

v

denotes

aconsistentestimateofthenoisevariance):

min

khk=1 tr

n

P

T H

(h

?

)

YY H

c

2

v I

o

, min

khk=1 0

B

@ Y

H

P

T H

(h

?

) Y

c

2

v tr

P

T H

(h

?

)

| {z }

constant 1

C

A

, min

khk=1 n

h H

Y H

R +

(h)Yh c

2

v trfT(h

?

)R +

(h)T H

(h

?

)g o

:

(7.8.5) is solved in the IQML way: we consider R(h) = R as constant at each

iterationandtheproblem becomesquadratic:

min

khk=1 h

H n

Y H

R +

Y c

2

v D

o

h (7.8.6)

whereh H

Dh=trfT H

(h

?

)R +

T(h

?

)g.

The choice for c

2

v

turns out to be crucial. For a nite amount of data, the

centralmatrixQ=Y H

R +

Y c

2

v

Disindeniteif c

2

v

isnotproperlychosen: inthat

case,DIQMLwillnotimproveIQML.Inordertohaveawell{denedminimization

problem at each iteration, we choose the c

2

v

which renders Q(h) positive with 1

singularity. Theproblembecomes:

min

khk=1;

h H

Y H

R +

Y D h: (7.8.7)

withthenon{negativityconstraintforthecentralmatrix. is theminimalgener-

alizedeigenvalueofY H

R +

Y andDandhistheassociatedgeneralizedeigenvector.

Asymptoticallyinthenumberofdata,arstiterationgivesaconsistentestimateof

handaseconditerationgivestheglobalminimizerwhatevertheinitialization[43].

Also,! 2

v

. TheperformanceofIQMLisinferiorto thatofDML.Asimilarap-

proachwasdeveloppedindependently[44]withalessjudiciouschoiceforthenoise

varianceestimate;somerelatedworkcanalsobefoundin[45]. Thenextproposed

algorithm,PQML,willgivethesameperformanceasDML.

7.8.2 Pseudo Quadratic ML (PQML)

PQML [41] isan iterative algorithmwhich at each iteration tries to nullthetrue

gradient of DML.This gradient can be written asP(h)h where P(h) is ideallya

positive denite matrix. At each iteration P(h) = P is considered as constant

(evaluated from the previousiteration). Thetrue (unconstrained) DML gradient

canthenalsobeinterpretedasthe(unconstrained)gradientofthefollowingpseudo-

quadraticcriterion:

min

khk=1 h

H

P h: (7.8.8)

For the DML problem, the matrix P(h) can be written as P(h) = Y H

R +

Y

B H

(h)B(h). When M ! 1, B H

(h)B(h) ! 2

v

D+signalterm. Evaluated at a

consistenth,thesignaltermbecomesnegligible,andtheeect ofB H

(h)B(h) isto

removethenoisecontributionfromtheIQMLHessianbutin a(statistically)more

eÆcientwaythanDIQML does.

Again, for anite amount of data, the matrix P(h) will beindenite. So by

analogywith DIQML,weintroduceascalarthat willrenderthematrixP(h)=

Y H

R +

Y B

H

(h)B(h)positivewithonesingularity. Thecriterionbecomes:

min h H

Y H

R +

Y B

H

B h: (7.8.9)

withnon{negativityconstraintonthecentralmatrix. Asymptotically, withacon-

sistent initialization, PQML convergesto its minimum at the rst iteration and

providesthesameasymptoticperformanceasDML.Also,!1inthiscase. Both

DIQML andPQMLrequireacomplexitythatislinearintheburstlengthM,and

sowill thesemi{blindcriteriapresentedbelow.

7.9 Three Suboptimal DML based Semi{Blind Criteria

7.9.1 Split of the Data

Theoutputburstcanbedecomposedinto3parts,seegure7.7(top):

1. Theobservationscontainingonlyknownsymbols.

2. TheN 1overlapobservationscontainingknownandunknownsymbols.

3. Theobservationscontainingonlyunknownsymbols.

Theproposed semi{blindcriteriawill consider adecomposition of thedata into2

parts,withtheoverlapzoneassimilatedtothetrainingpartortotheblindpartof

thesemi{blindcriteria.

7.9.2 Least Squares{DML

Intherstsemi{blindapproach,theoverlapzoneisincorporatedintotheblindpart

ofthesemi{blindcriterion. ThedataissplitasY =[Y T

TS Y

T

B ]

T

,seegure7.7:

Y

TS

=T

TS (h)A

TS +V

TS

groupsalltheobservationscontainingonlyknown

symbols.

Y

B

=T

B (h)A

B +V

B

groupsalltheobservationscontainingunknownsymbols

andespeciallytheoverlapobservations,wherewedonotexploittheknowledge

of the known symbols, which will hence be treated as unknown. So, some

information is lost. This lossofinformation canbecritical, especially when

thetrainingsequenceis veryshort,oflessthanN symbols.

WeapplytheDMLprincipleto:

Y =

Y

TS

Y

B

N

T

TS (h)A

TS

T

B (h)A

B

; 2

v I

(7.9.1)

As Y

TS andY

B

are decoupledin terms ofnoisecomponents, theDML criterion

forY isthesumoftheDMLcriteriaforY

TS andY

B :

min

h;A

B

kY

TS T

TS (h)A

TS k

2

+kY

B T

B (h)A

B k

2

: (7.9.2)

Thiscriterioncanbeoptimizedbyalternatingminimizationsw.r.t.handA

B . We

canalsosolvew.r.t.A andsubstitutethesolutiontogetthefollowingsemi{blind

Unknown Symbols Only

Known Symbols Unknown Symbols

Only Symbols

Symbols Known +

Unknown Known

Overlap Zone

Y

B Y

TS

Figure7.7. Output Burst: splitofthedataforLS{DML.

DMLcriterionforh:

min

h n

kY

TS T

TS (h)A

TS k

2

+Y H

B P

T H

B (h

?

) Y

B o

: (7.9.3)

Thiscriterioncanalsobeoptimizedinthesemi{blindPQMLway:

min

h

kY

TS T

TS (h)A

TS k

2

+h H

Y H

B R

+

B Y

B B

H

B B

B

h (7.9.4)

where is chosenas theminimal generalizedeigenvalueofY H

B R

+

B Y

B andB

H

B B

B .

We will call (7.9.4) Least{Squares PQML (LS{PQML). LS{PQML represents a

suboptimalwayof solvingthesemi{blindproblem andthesemi{blindidentiabil-

ity condition for thenumber of known symbolsrequired no longer holds exactly.

For irreducible channels, the criterion requires at least N known symbols to be

well{dened: P

B

(h) = Y H

B R

+

B Y

B

^

B H

B B

B

is indeed positive semi{denite with

1 singularity, and with N known symbols, A H

TS A

TS

has rank 1, which is suÆ-

cient to allowP

B

(h) to bepositivedenite. Forareducible channel with N

c 1

zeros, asymptotically P

B

(h) ! X H

B R

+

X

B has N

c

singularities, and N +N

c 1

known symbols are necessaryto have a well{conditioned problem. Furthermore,

for monochannels the semi{blind criterion reduces to its TS part since the blind

part does notadd any information (in fact, the blind criterion is notdened for

monochannels). LS{PQMLaswellasthefollowingalgorithmsneedaninitialization

andcan beinitializedbyTS orbyoneofthealgorithmsofsection7.10.

7.9.3 Alternating Quadratic DML (AQ{DML)

Here,theoverlapzonewill be incorporatedintothe trainingpartof thecriterion.

ThedataissplitasY =[Y T

Y T

] T

,see gure7.8:

Unknown Symbols Only

Unknown Symbols Symbols

Known and Unknown Only

Known Symbols

Symbols Known +

Unknown

Y

B Y

AQ

=Y

WLS

Figure7.8. OutputBurst: splitofthedataforAQ{DMLand WLS-DML.

Y

AQ

=T

AQ (h)A

AQ +V

AQ

= T 0

K (h)A

TS +T

0

U (h)A

0

U +V

AQ

groups allthe

observationscontainingtheknownsymbolsA

TS

plustheoverlapobservations.

TheunknownsymbolsA 0

U in Y

AQ

areconsideredasdeterministic.

Y

B

=T

B (h)A

U +V

B

groupsallthe observationscontainingonlyunknown

symbolswhichareconsideredasdeterministicunknownquantities.

DMLis appliedto[Y T

AQ Y

T

B ]

T

,whichgives:

min

h;AB;A 0

U

kY

AQ T

0

K (h)A

TS T

0

U (h)A

0

U k

2

+kY

B T

B (h)A

B k

2

: (7.9.5)

Semi{blindAQMLproceedsas:

1. Initialization

^

h (0)

2. Iteration(i+1):

AQMLonY

AQ

, initializedby

^

h (i)

.

Criterion min

h;A 0

U kY

AQ T

0

K (h)A

TS T

0

U (h)A

0

U k

2

issolvedbyalternating

minimizations w.r.t. A 0

U

and h. We keep only the estimate of A 0

U , to

formthenewestimateof b

A

AQ :

b

A (i+1)

AQ

=[A T

K A

0 (i+ 1)

U T

] T

.

Solvethesemi{blindcriteriontoget

^

h (i+1)

(withA 0

U

frozen).

Wecanperformalternatingminimizationsw.r.t.A

B

andh,startingfrom

^

h (i)

,basedonthecriterion:

min

h;A n

kY

AQ T

AQ (h)A

(i+1)

AQ k

2

+kY

B

T(h)A

B k

2 o

: (7.9.6)

WecanalternativelysolvethePQMLbasedcriterion:

min

h n

kY

AQ T

AQ (h)A

(i+ 1)

AQ k

2

+h H

Y H

B R

+

B (

^

h (i)

)Y

B B

H

B (

^

h (i)

)B

B (

^

h (i)

)

h o

:

(7.9.7)

7.9.4 Weighted{Least{Squares{PQML (WLS{PQML)

WLS{PQML is based on the same decomposition as for AQ{PQML. It mixes a

deterministicandaGaussianpointofview: theunknownsymbolsA 0

U

intheoverlap

zone are modeled asi.i.d. Gaussian random variables of mean0 and variance 2

a .

Wedenote Y

WLS

=T

WLS (h)A

WLS +V

WLS

=T 0

K (h)A

TS +T

0

U (h)A

0

U +V

WLS .

GMLisappliedtoY

WLS

andDMLto Y

B :

8

>

<

>

: Y =

Y

WLS

Y

B

N

T 0

K (h)A

TS

T

B (h)A

B

;

C

YWLSYWLS 0

0

2

v I

;

C

YWLSYWLS

= 2

a T

0

U (h)T

0H

U

(h)+ 2

v I

(7.9.8)

ThemixedMLcriterionis:

min

h;A

B

; 2

v n

lndetC

Y

WLS Y

WLS +(Y

WLS T

0

K (h)A

TS )

H

C 1

YWLSYWLS (Y

WLS T

0

K (h)A

TS )

+lndet 2

v I+

1

2

v kY

B T

B (h)A

B k

2

:

(7.9.9)

Weconsider 2

v

asknown(inpractice,itwillbeestimatedseparately). C

YWLSYWLS

isconsidered asconstant(computedusing thechannel estimatefromtheprevious

iteration). Thecriterionthenbecomes:

min

h;AB

kY

WLS T

WLS (h)A

WLS k

2

C 1

Y

WLS Y

WLS +

1

2

v kY

B T

B (h)A

B k

2

: (7.9.10)

Solvedin thePQMLfashion, thecriterionbecomes:

min

h

kY

WLS T

WLS (h)A

WLS k

2

C 1

Y

WLS Y

WLS +

1

2

v h

H

Y H

B R

+

Y

B B

H

B B

B

h

(7.9.11)

The approximation of considering C

Y

WLS Y

WLS

as constant is justied in [41] by

usingasemi{blindPQMLstrategy.

AQ{PQML and WLS{PQMLoutperform LS{PQML because the information

comingfrom theknownsymbolsintheoverlapzoneisused.

Foranirreduciblechannel,AQ{PQMLandWLS{PQMLaredenedwithonly

1knownsymbol. ForareduciblechannelwithN

c

1zeros,N

c

knownsymbolsare

suÆcienttohaveawell{denedproblem.

0 1 2 3 4 5 10 ⁻²

10 ⁻¹ 10 ⁰

M=100 − 10dB − 7 known symbols − Channel Length=4

Iteration

NMSE

LS−PQML AQ−PQML WLS−PQML alterning min.

ML perf

Figure7.9. Semi{blindalgorithms.

7.9.5 Simulations

In gure7.9, we show theNMSE given by allthe PQML based semi{blindalgo-

rithms as well as the optimal semi{blind DML in (7.7.1) solved by AQML for a

randomlychosenchannel(N =4,m=2) and1000Monte-Carlorunsofthe noise

andinputsymbols;7symbolsareknown(whichisthelowerlimitforTSidentiabil-

ity). ThePQMLbasedsemi{blindcriterionand AQMLareinitializedbyTS.The

semi{blindalgorithms improve dramaticallyTS performance. WLS{PQMListhe

best of thealgorithms with performanceclose to the theoretical MLperformance

(thiswasconrmed by othersimulations). Wealsonotice theslowconvergenceof

AQML(which in fact requires many iterationsfor A 0

U

per indicatediteration for

h).

To illustrate the lack of robustness of blind methods, we show in gure 7.10

simulations with 5000 Monte{Carlo runs for channel, noise and input symbols:

the particularly poor performance of blind estimation can be noticed. This does

notmean that blind PQMLis aweakalgorithm. Amongthe 5000realizations of

thechannels,somegaveill{conditionedchannelsresultinginpoorperformanceand

yieldingpooraveragedperformance. Thesemi{blindalgorithmdoesnotsuerfrom

thisproblem.

7.10 Semi{Blind Criteria as a Combination of a Blind and a TS

Based Criteria

Somesemi{blindalgorithmshavebeenproposedthat linearlycombineaTSanda

blindcriterion: in[46]asemi{blindcriterionisproposedbasedonthe(unweighted)

SubspaceFitting(SF)criterion;in[47],[48],anotheroneisbasedontheblindCMA

0 1 2 3 4 5 10 ⁻³

10 ⁻² 10 ⁻¹ 10 ⁰

M=100 − 20dB − 9 known symbols − Channel Length=5

Iteration

NMSE

WLS−PQML Blind PQML

Figure7.10. Comparisonbetweensemi{blindPQMLandblindPQML.

7.10.1 Semi{Blind SRM Example

Semi{BlindSubchannel Response Matching (SRM) [49],[13], [14], [25] illustrates

thefact that onehasto becarefulwhen buildingasemi{blind criterionthat way.

Blind SRMcan beseenasanon{weightedversionofIQML:min

h h

H

Y H

Yh. Con-

siderthefollowingsemi{blindcostfunction:

h H

Y H

B Y

B h+kY

TS T

TS (h)A

TS k

2

: (7.10.1)

This criterion is based on the decomposition of gure 7.7. An intuitive way to

weighbothTSand blindparts isto associatethem withthenumberofdatathey

are built from, as suggested in [46] for semi{blind subspace tting. In the SRM

case,theoptimalwouldthen beequalto 1.

In gure 7.11, we showthe NMSE for the channel averagedover100 Monte{

Carlorealizationsofchannel(withi.i.d.coeÆcients),noiseandinputsymbols. The

NMSE for (7.10.1) is plotted w.r.t. the value of in dotted lines. For = 1,

semi{blindSRMgivesworseperformancethanTSestimation.

Theblind SRMcriterion givesunbiasedestimatesonly underaconstantnorm

constraintfor thechannel. As the semi{blind criterionis optimized without con-

straints,theblindSRMpartgivesbiasedestimateswhich renderstheperformance

of thesemi{blind algorithm poor. In [50],the criterion asis 7.10.1wasproposed

withoutbeingdenoisedorproperlyscaled.

Forthechannelestimatestobeunbiased,thetermY H

B Y

B

needstobedenoised.

Weremove

min (Y

H

B Y

B

)I from Y H

B Y

B

(where

min (Y

H

B Y

B

)denotestheminimum

eigenvalueofY H

B Y

B

). TheresultingmatrixY H

B Y

B

min (Y

H

B Y

B

)I hasexactlyone

singularity.

Once the criterion is denoised, the choice for the weight remains unsolved.

of the semi{blind SRM channel estimate w.r.t. . However, since an analytical

optimization appears impossible, one would have to resort to search techniques

whichwouldrepresentanincreasein complexity.

Inthenextsection,weconstructsemi{blindSRMasanapproximationofsemi{

blindDIQML:inthisway,theblindSRMpartwillbeautomaticallydenoisedand

scaled.

Semi{Blind SRMas an Approximationof DIQML

Weknowthatthesemi{blindMLcriteriongivestheoptimalweightingbetweenthe

blindandtrainingsequenceparts:

h H

Y H

B R

+

(h)Y

B h+kY

TS T

TS (h)A

TS k

2

: (7.10.2)

Wenowneglecttheo-diagonaltermsinR(h): R(h)D(h). Form=2channels,

the diagonal elements are constant and equal to khk 2

. Form > 2, the diagonal

contains thesquared norm of pairs of subchannels. For example, for m =3, the

rstthree diagonalelements are: kH

1 k

2

+kH

2 ]k

2

, kH

2 k

2

+kH

3 k

2

,kH

1 k

2

+kH

3 k

2

andthesethreevaluesarerepeatedalongthediagonalM times.

Withthisapproximation,(7.10.2)becomes:

kY

TS T

TS (h)A

TS k

2

+h H

Y H

B D

+

(h)Y

B

h (7.10.3)

andweoptimizethiscriterionintheDIQMLfashioninordertodenoiseit;D(h)=D

isconsideredasconstant. Thesemi{blindcriterionbecomes:

min

h n

kY

TS T

TS (h)A

TS k

2

+h H

Y H

B D

1

Y

B c

2

v D

v

h o

(7.10.4)

whereh H

D

v h=tr

T H

B (h

?

)D 1

T

B (h

?

) ,and c

2

v

istheminimalgeneralizedeigen-

valueofY H

B D

1

Y

B and D

v .

Thenorm of the dierentsubchannels, used to computeD, can be estimated

fromthedenoisedsamplesecond{ordermomentr

yy (0)=

2

a T

1 (h)T

H

1 (h): r^

yy (0)

c

2

v I =

1

M P

M 1

k =0 y(k)y

H

(k) c

2

v

I. Aswill beseenin thesimulations,atlowSNR,

theweightontheblindpartshould infact besmallerthanthetruevalueD o

ofD

indicates. So insteadof estimating theenergy of each channel from thedenoised

r

yy

(0), weusethenoisyversion. TheresultingDwillbelargerthanD o

.

Ingeneral,thedierentchannelstendtohavethesameenergy,sothatDcanin

turnbeapproximatedbyamultipleofanidentitymatrix,themultiplebeing [

khk 2

m

2 .

Whenm=2,thisapproximationisexact. With Dbeingamultipleofidentity, c

2

v

istheminimaleigenvalueofY H

B Y

B

,andthesemi{blindcriterionbecomes:

min

h (

2

m 1

[

khk 2

h H

Y H

B Y

B

min (Y

H

B Y

B )I

h+kY

TS T

TS (h)A

TS k

2 )

:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10 ⁻²

10 ⁻¹ 10 ⁰

Semi−Blind SRM − M=100 − 10 dB

Valeur de α

NMSE

10 known symbols

10 known symbols 20 known symbols

20 known symbols

*

DenoisedSemi-BlindSRM(7:10:5) NonDenoisedSemi-BlindSRM(7:10:1) Training

Figure 7.11. Semi{blind subspacetting built as alinear combination of blind

SRMandtrainingsequencebasedcriteria.

An alternative to this semi{blind SRM criterion is to use the decomposition of

gure7.8andtouseWLSorAQMLtohandlethetrainingsequencepart.

Ingure7.11insolidlines,weshowtheperformanceofthecorrectedsemi{blind

criterion. The scalar scales the blind part in (7.10.5). The value = 1gives

approximatelytheoptimalperformanceandwenoticethatinfacttheperformance

isroughlyconstantintheneighborhoodof=1.

In gure 7.12, semi{blind SRM is used to initialize the dierent DML based

semi{blindalgorithmsin thecasewhere thetrainingsequence istooshort toesti-

matethechannel.

7.10.2 Subspace Fitting Example

Blind SubspaceFitting

Considerthesample covariancematrixof thereceived signalY

L

of lengthL and

itsexpectedvalue(w.r.t.thenoiseonly,as Aisdeterministic):

R

Y

L Y

L

=T

L (h)

"

M L 1

X

k =0 A

L (k)A

H

L (k)

#

T H

L

(h)+ 2

v

I : (7.10.6)

Provided the blind deterministic identiabilityconditions [DetB] arefullled, the

matrix P

M L 1

k =0 A

L (k)A

H

L

(k)isnon-singular,so thespacespannedbythematrix

T

L (h)

h

P

M L 1

k =0 A

L (k)A

H

L (k)

i

T H

L

(h) is the signal subspace. R

YLYL

admits L+

N 1(the dimensionof thesignalsubspace) eigenvectorsbelongingto the signal

subspace,andLm (L+N 1)eigenvectorsbelongingtothenoisesubspaceand

allassociatedtotheeigenvalue 2

v

. TheeigendecompositionofR

Y

L Y

L is:

R

YLYL

=V

S

S V

H

+V

N

N V

H

(7.10.7)

0 1 2 3 4 5 10 ⁻²

10 ⁻¹ 10 ⁰

M=100 − 10dB − 5 known symbols − Channel Length=4

Iteration

NMSE

LS−PQML AQ−PQML WLS−PQML alternating min.

ML Perf

Figure 7.12. DML based semi{blind algorithms initialized by semi{blind SRM.

Caseofatooshort trainingsequencetoestimatethechannelviaTSalone.

where the columns of V

S

span the signal subspace and the columns of V

N the

noise subspace,

N

= 2

v I. Let

b

V

S and

b

V

N

beestimates of the signaland noise

eigenvectorsobtainedfrom thesamplecovariancematrix. SignalSubspaceFitting

(SSF) tries to t the column space of T(h) to that of b

V

S

through the quadratic

criterion:

min

khk 2

=1 kP

b

VN T(h)k

2

, min

khk 2

=1 h

H

S H

Sh : (7.10.8)

(seeciteAbedMeraim:Subsp97,foradescriptionofthestructureofS).

Semi{Blind SubspaceFitting

Considernowthefollowingsemi{blindcostfunction:

M

U h

H

S H

B S

B h+kY

TS T

TS (h)A

TS k

2

: (7.10.9)

Weadopt herethedecomposition ofgure7.7. In[46],M

U

waschosenequalto

thenumberofdataonwhich theblind criterionisbased,i.e. =1.

Ingure7.13(left),weplottheNMSEofchannelestimationw.r.t.fordierent

sizes L, for SNR=10dB and M

K

= 10 known symbols. For L = N, the semi{

blind criterion is relatively insensitive to the value of . For L larger than N

however,thecriterionisvisibly verysensitiveto thevalueof. Thechoice=1

givesperformanceworse thanthatfor trainingsequence basedestimationfor L>

N. These simulationssuggest that thelinearly combinedsemi{blind algorithm is