Deterministic quadratic semi-blind FIR multichannel estimation : algorithms and performance

ESTIMATION: ALGORITHMS AND PERFORMANCE

Elisabeth de Carvalho

Stanford University

Stanford, CA 94305-9515,USA

carvalho@dsl.stanford.edu

Dirk Slock

Institut Eurecom, B.P. 193

06904 Sophia AntipolisCedex, FRANCE

slock@eurecom.fr

ABSTRACT

Thepurposeofsemi-blindchannelidenticationmethodsis

toexploitthe informationused by blindmethodsandthe

informationcomingfromknownsymbols. Themainfocus

ofthispaperisthestudyofdeterministic quadraticsemi{

blind algorithms which are of particular interest because

oftheir lowcomputationalcomplexity. Theassociatedcri-

teriaare formedas a linearcombinationof ablind and a

trainingsequence based criterion. Through the examples

of Subchannel Response Matching and Subspace Fitting

basedsemi{blindcriteria,westudyhowtoconstructprop-

erlysuchsemi{blindcriteriaandhowtochoosetheweights

ofthelinearcombination. Weprovideaperformancestudy

forthesealgorithmsandgivetheoreticalconditionsforthe

semi{blindperformancetobeindependentoftheweights.

1. INTRODUCTION

Dierent ways of building semi{blind criteria are possi-

ble. Optimal semi{blind methods can take into account

theknowledge ofsymbolsevenarbitrarilydispersedinthe

burst: in[1 ],weproposedoptimalmethodsbasedonMaxi-

mumLikelihood(ML).Thissymbolcongurationisingen-

eral undesirable as the associated semi{blind criteria will

requirecomputationallydemandingalgorithmsbecausethe

structureoftheblindproblemislost.

For grouped knownsymbols (training sequence) com-

putionallylowsolutionscanbebuiltbecausethestructure

of the blind problem is kept. By neglecting some infor-

mationabout the knownor unknownsymbols, ML easily

allows oneto construct semi{blindcriteria that are a lin-

ear combinationof a blindand atrainingsequence based

criterion,withoptimalweightsintheMLsense.

A third way to buildsemi{blind criteria is to linearly

combine agivenblindcriterion and aTS basedcriterion.

Westudyherethe SubchannelResponse Matching(SRM)

andSubspaceFitting(SF)basedsemi{blindcriteriawhich

are of particular interest because they are quadratic and

representclosedformedsolutions;semi{blindSRMinpar-

ticularhasalowcomplexity. WeshowthattheblindSRM

criterionneedsrsttobedenoisedandthencorrectlycom-

binedtothe TScriterion. In[2 ], theSRMcase istreated

butthe criterionis neither denoised nor correctlyweight-

ed. The SF case is treated in [3 ,4]. In [4 ], the optimal

for the performance, whichrepresentsanincreaseincom-

plexity. Here,weprovideasimplesolutiontondthecor-

rectweights.

At last, we provide a performance studyof determin-

istic quadratic semi{blindcriteria. We especially charac-

terizetheirperformanceinthecaseofanasymptoticnum-

berofunknownsymbolsM

U

and nite numberofknown

symbols MK. In [4 ], a similar study was conducted for

M

K

M

U

,butwith M

K

!1,whichdoesnotallowto

draw the conclusionswe get. Here, we give theoretically

foundedconditionsforthesemi{blindcriteriaperformance

tobeindependentoftheweightsofthelinearcombination.

2. PROBLEM FORMULATION

Consider a sequence of symbols a(k) received throughm

channelsoflengthNwithcoeÆcientsh (i):

y(k)= N 1

X

i=0

h(i)a(k i)+v(k); (1)

v(k) isanadditive independent whiteGaussiannoiseand

r

vv

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

= 2

v I

m Æ

k i

. Assumewe receive

M samples,concatenatedinthevectorY

M (k):

Y

M

(k)=TM(h)AM(k)+V

M

(k) (2)

Y

M

(k) =[y T

(k M+1)y T

(k)]

T

, similarly for V

M (k),

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

T

. (:) T

denotestrans-

pose and (:) H

hermitian transpose. The channel trans-

fer functionis H(z)= P

N 1

i=0 h (i)z

i

=[H T

1 (z)H

T

m (z)]

T

.

T

M

(h)isablockToeplitzmatrixlledoutwiththechannel

coeÆcientsgroupedinh=[h T

(N 1)h T

(0)]

T

. Weshall

simplifythenotationin(2)withk=M 1to:

Y = T(h)A+V : (3)

The space spanned by the columnsof T(h)will be called

signal subspace and itsorthogonal complement, the noise

subspace. WeassumethatatrainingsequenceA

TS located

at thebeginningoftheburstispresentintheinputburst.

Y

TS

=TTS(h)AT

S +VT

S

=ATSh+V

TS

is theportion

oftheoutputburstcontainingonlyknownsymbols. Y

B

=

TB(h)AB+VB istherest ofthe outputburst(itcontains

theunknownsymbolsbutalsosomeknownsymbolswhich

3. SEMI{BLINDSRM

SRM [5] is based on a linear parameterization H

?

(z) of

the noise subspace which satises T(h

?

)T(h) = 0 where

T(h

?

) is the convolution matrix built from H

?

(z) and

spans the entire noise subspace. For m = 2, H

?

(z) =

[ H

2 (z) H

1

(z)]. Form>2[6 ],

H

?

(z)= 2

6

6

6

4 H

2 (z) H

1

(z) 0 0

0 H3(z) H2(z) .

.

.

.

.

.

.

.

. .

.

.

0

H

m

(z) 0 0 H

1 (z)

3

7

7

7

5 :

(4)

Inthenoisefreecase,T(h

?

)Y (=T(h

?

)T(h)A)=0. Us-

ingthecommutativityofconvolutionT(h

?

)Y =Yh,where

YisastructuredmatrixlledoutwiththeelementsofY:

thechannelcoeÆcientscanbeidentieduniquelyfromthis

equationas the minimal left eigenvectorof Y. Whenthe

receivedsignalis noisy,hisobtained bysolvingtheleast{

squaresquadraticcriterionmin

khk=1 kYhk

2

.

Considerthefollowing semi{blindcostfunction:

h H

Y H

B Y

B h+kY

TS T

TS (h)A

TS k

2

: (5)

NotethatotherTSbasedcriteriacanalsobeconsidered[7 ].

AnintuitivewaytoweighbothTSandblindpartsistoas-

sociatethemwiththenumberofdatatheyarebuilt from,

assuggested in[3 ]for the semi{blindsubspace tting. In

theSRMcase,theoptimalwouldthenbeequalto1. In

gure1,weshowtheNMSEforthechannelaveragedover

100Monte{Carlorealizationsofthechannel(withi.i.d.co-

eÆcients)thenoise andthe inputsymbols. TheNMSEis

plotted w.r.t. the value of in dotted lines. For = 1,

semi{blindSRMgivesworseperformancethanTSestima-

tion.

Theblind SRMcriteriongivesunbiasedestimates on-

ly under a norm constraint for the channel [6 ]. As the

semi{blindcriterion is optimized withoutconstraints, the

blindSRM part gives biased estimates which rendersthe

performance of the semi{blind algorithm poor. For the

criterionto beunbiased, the termY H

B

YB needsto be de-

noised. Asymptoticallyinthe numberof data, Y H

B Y

B

!

X H

B

XB+EV H

B

VB, where XB is built from the noise free

signal T

B (h)A

B and V

B

from the noise V

B

; EV H

B V

B

!

2

v

where is a constant. The minimal eigenvalue of

Y H

B

YB,min(Y H

B

YB),isanestimateof 2

v

andthequan-

tityY H

B Y

B

min (Y

H

B Y

B

)Iis thedenoisedSRMHessian.

Oncethecriterionis denoised,the choicefor the constan-

t remains unsolved. To nd it, we refer to semi{blind

DeterministicMLfor h[7 ]. TheblindDMLforhis:

min

h Y

H

P

?

T(h)

Y (6)

where PX is the orthogonal projection onthe columns of

X and P

?

X

= I P

X

is the projection onthe orthogonal

complementofX. Byusingthelinearparameterizationof

thenoisesubspace,(6)becomes:

min

h Y

H

T H

(h

?

)

T(h

?

)T H

(h

?

)

+

T(h

?

)Y

,minh H

Y H

R +

(h)Yh

(7)

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(9) NonDenoisedSemi-BlindSRM(5) Training

Figure1: Semi{blindSRMbuiltasalinearcombinationof

blindSRMandTSbasedcriteria.

NotethatSRMappearsasanon{weightedversionofDML.

In[7 ],thefollowingsemi{blindcriterionbasedonDMLwas

proposed:

h H

Y H

B R

+

B

(h)YBh+kY

TS

TTS(h)ATSk 2

: (8)

Semi{blind DML gives the optimal weights between the

blindandTSpart: webuildnowsemi{blindSRMasanap-

proximationofDML.WeapproximateRB(h)asamultiple

oftheidentitymatrixwithmultipleequaltothemeanofthe

diagonalelements,i.e.

m

2 khk

2

.Thenormofthechannelcan

beestimatedthankstoanestimateofthedenoisedsecond{

ordermomentofadatasampletr(ryy(0))= 2

a khk

2

. After

denoising,thesemi{blindSRMcriterionis:

min

h f

2

m 1

[

khk 2

h H

Y H

B Y

B

min (Y

H

B Y

B )

h+

kY

TS

TTS(h)ATSk 2

g:

(9)

It canbe shownthat replacing khk 2

by a consistent esti-

mate does not change the asymptotic performance. This

algorithmcouldalsobeinterpretedasanapproximationof

thesemi{blindDenoisedIQMLalgorithm presentedin[7 ].

Ingure1,insolidlines,we show theperformance ofthe

correctedsemi{blindcriterion. Thescalarscalestheblind

partin(9). Thevalue =1 givesapproximately theop-

timalperformanceand intheneighborhoodof=1,the

performancehardlydependsonthevalueof.

4. SEMI{BLINDSUBSPACEFITTING

ConsiderthereceivedsignalcovariancematrixoflengthL:

R

Y

L Y

L

= 2

a T

L (h)T

H

L (h)+

2

v

I. LetR

Y

L Y

L

=V

S

S V

H

S +

VNNV H

N

betheeigendecomposition. SgroupstheL+N

1(dimensionofthesignalsubspace)largesteigenvaluesof

R

Y

L Y

L and

N

= 2

v

Ithesmallestones. ThecolumnsofV

S

spanthe signalsubspaceand thecolumnsofVN the noise

subspace. Signal Subspace Fitting (SSF) ts the column

spaceofT(h)tothatofVSthroughthequadraticcriterion:

min

khk 2

=1 kP

b

V

N T(h)k

2

, min

khk 2

=1 h

H

S H

Sh (10)

where b

V

N

is anestimateofV

N

obtained from thesample

covariancematrix.

0 0.5 1 1.5 2 2.5 3 10

⁻²

10

⁻¹

10

⁰

Value of α M=100 − 2 Subchannels − 10dB − 10 known Symbols

L=2N L=7N

L=5N L=4N

L=N L=7N

L=5N L=4N

L=3N L=3N

L=2N

Semi-blindSF(11)

Denoisedsemi-blindSF

0 0.5 1 1.5 2 2.5 3

10

⁻²

10

⁻¹

10

⁰

Value of α M=100 − 2 Subchannels − 10dB − 10 known Symbols

L=7N

L=N

Semi-blindSF(12)

Figure2: Semi{blindSF built as a linear combinationof

blindSFandTSbasedcriteria.

Considernowthefollowing semi{blindcostfunction:

MUh H

S H

B

SBh+kY

TS

TTS(h)ATSk 2

: (11)

In[3 ], M

U

was chosen equal to the numberof data on

whichthe blindcriterionis based,i.e. = 1. Ingure 2

(left,dashedlines), we plotthe NMSEof channel estima-

tionw.r.t.fordierentsizesL,forSNR=10dB,10known

symbols. For L = N (thesolid line and dotted line are

superposed), the semi{blindcriterionis relatively insensi-

tiveto thevalueof . ForL larger thanN however, itis

visiblyverysensitiveto itsvalue. Thechoice=1gives

performanceworsethanthatoftrainingsequencebasedes-

timation for L > N. These simulationssuggest that the

linearlycombined semi{blindalgorithmis sensitive tothe

dimensionofthenoisesubspacewhichvarieswhenLvaries.

Weproposeto\denoise"theblindpartoftheSFcrite-

rion,i.e.weforcethesmallesteigenvalueofS H

B S

B

to0via

S H

B S

B

min (S

H

B S

B

)I. Note that theoretically we do not

remove any noise contribution here, but cleanthe matrix

S H

B S

B

. This strategy will be laterdiscussed in section 5.

Theperformance of the denoised semi{blind SF criterion

w.r.t. for 500 Monte{Carlo realizations of the channel,

noiseandinputsymbolsareshowningure2(left)insolid

lines. We notice the signicant eectof the denoising on

thesemi{blindalgorithm performance, butthis algorithm

stilldoesnotgivesuÆcientlygoodperformancefor=1.

We propose here to scale the blind part of the semi{

blindSFcriterionbythedimensionN ofthenoisesubspace.

Theresultingcriterionis:

min

h

MU

N h

H

S H

B SB

^

minI

h+kY

TS

TTS(h)ATSk 2

:

(12)

Ingure2(right),weshowtheperformanceof(12)inthe

caseofrandomchannelswith2subchannels: thesemi{blind

SFcriterion(12)givessatisfactoryresultsfor=1.

5. PERFORMANCE STUDY

Considerthegeneralsemi{blindquadraticcriterion:

min

h n

MUh H

b

QBh+kY

TS

TTS(h)ATSk 2

o

: (13)

b

Q

B

= 2

m [

khk 2

1

M

U

Y H

B Y

B

min (Y

H

B Y

B )I

for semi{blind

SRMand b

QB= 1

N h

S H

B SB

^

min(S H

B SB)I

i

forsemi{blind

SF.WhenthechannelhasNc 1zeros(e.g.duetochan-

nellengthoverestimation), b

QB tendsasymptoticallytoQB

which has Nc null eigenvalues. Let b

QB = c

W1 b

1 c

W H

1 +

c

W2 b

2 c

W H

2

betheeigencompositionof b

QB, b

2!0asymp-

totically. Notethatthe smallest eigenvalueof b

QB is 0for

the semi{blindcriteria proposed. Onecan show that the

elementsof b

2 (not exactlyequalto 0)areoforder1=MU

for bothSRMandSF.

Thesolutionof(13) is:

^

h=

M

U b

Q

B +A

H

TS A

TS

1

A H

TS Y

TS

(14)

We studythe performance ofthe semi{blind criterion

(13) fortheasymptoticcases:

MU and MK innite, with condition p

M

U

M

K

! 0,

whichaccountsforthefactthattheTSpartofthecri-

terionshouldnotbenegligiblew.r.t.theblindpart[8 ].

M

U

innite,M

K nite.

Asimilaranalysisexistsin[4 ]forSFwherethelastasymp-

toticconditionisMKMU,withMK innitehowever.

5.1. MU and MK innite

Inthatcase,itcanbeshownasin[8]that:

C

hh

= M

U Q

B +A

H

TS A

TS

1

2

M 2

U E(

b

Q

B h

o

h oH

b

Q H

B )

+ 2

v A

H

TS A

TS

M

U Q

B +A

H

TS A

TS

1

:

(15)

WedonotprovidehereexpressionsforE(

b

Q

B h

o

h oH

b

Q H

B )for

lack of space. The performance in that case depends on

the value of . In orderto optimize the performance, it

wouldbenecessarytondtheoptimal,whichshouldbe

avoided,becauseoftheadditional computationalcost.

5.2. MU innite,MK nite

LetR 1=2

R H =2

betheCholeskydecompositionofA H

TS ATS.

MU b

QB+A H

TS AT

S

1

=

R H =2

M

U R

1=2

b

Q

B R

H =2

+I

1

R 1=2

: (16)

LettheeigendecompositionofR 1=2

b

QBR H =2

= b

Q 0

B be:

b

Q 0

B

= c

W 0

b

0

c

W 0H

= c

W 0

1 b

0

1 c

W 0H

1 +

c

W 0

2 b

0

2 c

W 0H

2

: (17)

b

0

2

!0asymptoticallyinthenumberofdata.

MUR 1=2

b

QBR H =2

+I

1

=

c

W 0

1

MU b

0

1 +I

1

c

W 0H

1 +

c

W 0

2

MU b

0

2 +I

1

c

W 0H

2

=

c

W 0

2

M

U b

0

2 +I

1

c

W 0H

2

(18)

at rst orderin1=

p

M

U . When

b

0

2

6=0, as the non{zero

element of b

0

2

are of order1=M

U ,M

U b

0

2

is of the same

orderasI,andtheperformancedependson.When b

0

2

=

0, c

W 0

2

MU b

0

2 +I

1

c

W 0H

2

= c

W 0

2 c

W 0H

2

and we can show

that the performance are independent of , and at rst

orderin1=M

U

,thecovariancematrixoftheestimationerror

h=

^

h h o

(h o

isthetruevalueofthethechannel)is:

Chh=

2

W2

W H

A H

ATSW2

1

W H

(19)

0 1 2 3 4 5 6 7 8 9 10 10 ⁻²

10 ⁻¹ 10 ⁰

M=100 − 2 Subchannels − 10dB − 10 known symbols

Value of α

D=3 D=4 D=5

D=2 D=1

Figure3:Semi{blindSRM:D=1{5eigenvaluesforcedto0.

Thisexpression can be interpreted as the performance of

the estimationof the zeros of the channelby trainingse-

quence with perfect knowledge of the irreducible part of

thechannel.

For b

0

2

=0,whenconsideringexpression(15),withMK

nite,we nd expression(19): the asymptotic expression

(15)inMK andMUisvalidwhenMK isnite.Sothereis

a continuity between both expressions (15) and (19) and

expression (15) is valid in any case and should be used

to characterize the performance of the semi{blind criteri-

a.ThisisnottruewhentheNcsmallesteigenvaluesof b

QB

arenot exactly equalto 0: thereis adiscontinuityinthe

expressions, andboth analysesdonot coincide. Ingener-

al,inthis last case, itmay notbe obviousto know which

analysisbetweenM

K

inniteorM

K

niteisappropriate.

Inpractice the performance for small MK is notcon-

stant w.r.t. evenwhen b

0

2

=0. For arandomly chosen

channel,ingeneral,thematrix b

Q

B

exhibitsmorethanN

c

\small" eigenvalues (theextra smalleigenvalues are suÆ-

cientlysmall to inuence the performance). We take the

exampleof semi{blindSRM. Toforcen smallesteigenval-

uesto0,weconsiderthequantityY H

B

YB nI,wheren

isthe largest ofthe n smallest eigenvalues ofY H

B Y

B ,and

forcethe negative eigenvaluesto 0. For randomly chosen

channels of length 5 (soa priori irreducible channels) we

force1to5eigenvaluestozero: ingure3,weshowthere-

sultingNMSEfor100runsofthechannel,noiseandinput

symbols.Foranumberof3to5eigenvaluesforcedto0,we

seethat the performance are not dependent onthe value

of. TheproposedDML,SRMand SFbasedalgorithms

forcetozero only1eigenvalue andas alreadystatedhave

theirperformancedependenton. Howeverthealgorithms

wereconstructed(weighted)suchthattheoptimalisap-

proximatelyequalto 1: thissolutionispreferable because

itis lesscomplex. This analysis shows us the importance

oftheeigenvaluesof b

QB inthebehaviorofthesemi{blind

algorithms.

5.3. OptimalWeighted

BlindSRMandSFcriteriaareoftheformminhh H

U H

B UBh.

Theoptimally weighted version is min

h h

H

U H

B W

+

B (h)U

B h

withW +

B

(h)=EUBh o

h oH

U H

B

andgivesbetterperformance

than the non{weighted version. When M

K and M

U are

weightedsemi{blindcriterionis:

min

h

h H

U H

B W

+

B UBh+

1

2

v kY

TS

TTS(h)ATSk 2

(20)

Fortheseasymptoticconditions,ndingtherightscalefac-

tortooptimizetheperformanceofthenon{weightedpart

of the criterionis diÆcult as w mentionedin section 5.1,

howeverndingtherightweightingmatrixiseasier.

In fact, semi{blind DML (6) is built as anoptimally

weightedcombinationoftheblindandTScriteriaandSRM

canbeseenasanapproximationofthisweightedcriterion.

WehavenottestedtheweightedversionoftheSFcriterion:

the introduction of N may perhaps be explained by the

blindweightingmatrix.

6. CONCLUSION

Inthispaper,wehavepresentedtwoquadraticsemi{blind

channel estimation criteria basedona linearcombination

ofa blind and aTS criterion. We have seen that it may

bediÆculttoconstructasemi{blindcriterionthisway. A

performanceanalysishasshowntheimportanceofthesmall

eigenvaluesoftheHessianoftheblindpartofthecriterion.

Wehavestudiedconditionsforthesemi{blindperformance

tobeindependentfromtheweightsassociatedtotheblind

andTSparts.

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