ArraySignalProcessing

�Attaractedalotofattentionbecauseofthe\StarWar"projects

duringReaganAdministration.

�Thepracticalityofthemanyarrayprocessingalgorithmshas

beenunderattack.

-Forantennaarrays,forexample,mutualcoupling,phase

centers,etc.,aremakingassumeddatamodelsinerror(sometimes

insigni�canterror).

�Manyarrayprocessingalgorithmsaretheoreticallybeautifuland

canbemodi�edtosolveotherproblems,suchascode-timing

estimationinCDMAcommunications.

106

d

dsinθ θ

� Consider a uniform linear array shown above.

- The incident wave is a plane wave.

- It is CW (continuous wave), i.e., s(t) = ej!0t.

- The s(t� �) = ej!0te�j!0� = s(t)e�j!0�

107

-Let�=dsin(�)

c

,thene

�j!0�=e�

j!0d

c

sin�4=ej

-Whend=�2;

=��sin�:

-Thenthereceivedsignalsatdi�erentsensorsare

s(t);s(t)ej ;s(t)ej2 ;���;s(t)ej(M�1) ;

whereM

isthenumberofsensors.

-Ata�xedtimet0acrossthearray,thesignalisasinusoidwith

complexamplitudes(t0 )andfrequency !

�Foranarrowbandsignals(t);i.e.,bandwidthofs(t)<<

1

(M�1)�;

theabovemodelisstillapproximatelycorrect.

108

Lety(t)= 26664

y0 (t)

...

yM

�1 (t) 37775

;denotethesensoroutputvectorattimet,

y(t)= 26666664

1ej ...

ej(M�1) 37777775

s(t)+

e(t)

|{z}noisevector

�ForK

signalsfromK

di�erentangles�1 ;���;�k;

y(t)= 26666664

1

���

1

ej 1

���

ej K

...

...

ej(M�1) 1

���

ej(M�1) K 37777775 26664

s1 (t)

...sK(t) 37775

+e(t)

109

UsingFFTforComplexSpectralEstimation

�TheFFTmethodisanon-parametricspectralestimationmethod.

�ComputationallyE�cient.

�Robust.

�HighSidelobes.

�PoorResolution.

Considersn

=�0 ej2�f0n;

n=0;���;N�1

whichisacomplexsinusoidwithcomplexamplitude�0and

frequencyf0 .

sn

canbewrittenassn

=unwR(n)

where

un

=�0 ej2�f0n;

�1

<n<1

wR(n)= 8<:1;

n=0;���;N�1

0;

otherwise

110

wR(n)

DTFT

!

−π π

ω

0

WR

2π

(ω)

N

WR(!) =sin�!N

2

�ej!N�1

2

sin(!2 )

:

WR(2�1

N) = 0:

The Fourier transform (DTFT) of sn is the convolution of DTFT of

un and WR(!).

ej2�f0n !

2πf −π π

ω

0

111

1N

iscalledtheResolutionLimitoftheFFTmethod.

Let�S(!)denotethenormalizedDTFTofsn,

�S(!)=

1N PN�1

n=0sne�j!n:

-�S(!)peaksat!=2�f0 .

-Thecomplexheightat!=2�f0of�S(!)gives�0 .

�TheFFTmethod:

Givenxn;

n=0;���;N�1,thecomplexspectraofxn

is

calculatedas

�X(!)=

1N PN�1

n=0xne�j!n,

whichcanbedonee�cientlyviaFFT.

112

ParameterEstimationofSinusoidsinNoise

yn

=xn+en;

wherexn

=

KXk

=1�kej!kn;n=0;1;���;N�1

�k=complexamplitudeofthekthsinusoid.

�k

=

j�k jej�k

j�k j

=

magnitude

�k

=

phase

!k

=

freq.ofthekth

sinusoid:

!k2[��;�)or[0;2�)

en

=

additivenoise,complexGaussian,zero-mean.

113

Lety= 26666664

y0

y1...

yN�1 37777775

;

a(!k )= 26666664

1ej!k

...

ej(N�1)!k 37777775

�= 26664

�1...

�K 37775

;

e= 26664

e0...

eN�1 37775

:

LetA=[a(!1 )

a(!2 )

a(!K)]:

114

Then

y=A�=e

�Forarbitrarycolorednoise,

e�N(0;Q):

�Forwhitenoise,

e�N(0;�2I):

�ThisisaspecialcaseofcolorednoisewithQ=�2I:

Let!=[!1���!K] T:

Then�and!aredeterministicunknownstobeestimatedfromy.

115

Maximum

LikelihoodEstimator

f(yj�;!)=

1�jQj e

�(y�A�)H

Q�

1(y�A�):

�ForthecaseofunknownQ,MLestimationisill-de�ned.(The

numberofunknownsislargerthanthenumberofmeasurements.)

�ConsiderbelowQ=�2Iwith�2unknown.Thenmaximizingthe

likelihoodfunctionisequivalentto

^!=

argmax

!

�yHA(AHA)�1AHy �

^�= �AHA ��1AHy ���!

=^!

z

116

Remarks:

1.)Themaximizationtoobtain^!isdi�culttoimplement.

2.)Thesearchmaynot�ndtheglobalmaximum.(Therearemany

localmaximumsinthecostfunction).

g1=yHA(AHA)�1AHy:

3.)Thesearchismultidimensionalandhencecomputationally

expensive.

4.)WhenQ=�2I;(z)givesMLestimatesthatareasymptotic(for

largeN)statisticallye�cient(accordingtothegeneralML

estimationtheory).

117

5.)

Let�=[�1

���

�K] T:

Let�=[j�1 j

�1

!1���

j�Kj

�K

!K]:

ForlargeN

andQ=�2I,

CRB(�)� 26666666664

�2

2N

0

0

0

2�2

Nj�1j 2

�

3�2

N2j�1j 2

0

0

�

3�2

N2j�1j 2

6�2

N3j�1j 2

0

...

�1!

�K 37777777775

Note:CRB(�k )�5�2

2N

118

SpecialCases:

1.K

=1.

^!=

argmax

!

�yHA(AHA)�1AHy �:

A= 26666664

1ej!...

ej(N�1)! 37777775

;

AHA=N:

119

AHy

=

[1

e�j!

e�j(N�1)!] 26664

y0...

yN�1 37775

=

N�1

Xn=0yne�j!n

=

DTFTofyn

)

^!=

argmax

!

�����1N

N�1

Xn=0yne�j!n ����� 2

|{z}

Periodogram

.

1.)^!correspondstothehighestpeakoftheperiodogram,which

canbecomputede�cientlyviaFFT,withzeropadding.

2.)^�=

1N PN�1

n=0yne�j!n ���

!=^!:

fEasilyobtainedviaNormalizedFFTg

120

�! = infi6=k j! � !kj >2�

N:

Since V ar(^!k � !k) /

1N3

) ^!k � !k /

1N3

2

) infi 6=k j^!i � ^!kj >2�

N:

At FFT points, ~!i =2�

N

i; i = 0; 1; � � � ; N � 1; we can resolve all

K sine waves by evaluating

g1 = yHA(AHA)�1AHy:

g

ω

1

π2

N

121

AnyK

ofthese~!gives

AHA=NI;

I=

Identitymatrix:

)

g1=

KXk

=1

1N �����N�1

Xn=0yne�j~!kn ����� 2

|{z}

~gk

Tomaximizeg1 ,wemaximizeeachterm~gk ,whichisachievedby

theK

di�erent~!'sthatmaximizetheperiodogram,whichalso

correspondstotheK

largestpeaksofthemagnitudeofthe

normalizedFFT!

122

Remarks:

�^!k

estimatesobtainedbyusingtheK

largestpeaksofthe

normalizedFFThaveaccuracy^!

k�!k/2�N

.

�^�k

arethecomplexamplitudescorrespondingtotheK

largest

peaksofthenormalizedFFT.

^�k=

1N

N�1

Xn=oyne�j^!kn

�OurnormalizedFFTorperiodogramisdoingquitewellforthis

case.(ThiswasintroducedbySchusteracenturyago!)

123

Backtothegeneralcase:

^!=

argmax

!

�yHA(AHA)�1AHy �

Remarks:

1.Manypapershavebeenpublishedontheabovemaximization

problem.

1.)AlternatingProjection/MaximizationbyZiskindandWax

(IEEETran.SP,Oct.88)

2.)AlternatingNotch-Periodogramalgorithm(ANPA)by

HwangandChen(IEEETrans.SP,Feb.93)

...

124

The idea of alternating maximization (AM) :

1.) Iteratively update one unknown with all other unknowns �xed.

2.) Under mild conditions, guaranteed to converge to at least a

local maximum.

3.) Multidimensional search becomes a sequence of one-dimensional

search problem.

ω

ω

2

1

125

�Theideaofalternatingprojection(AP)istoimplementAMina

fastway.

Let

PA

=A(AHA)�1AH

-PA

istheprojectionoperatorontothecolumnspaceofA.

-PAPA

=PA

)

theeigenvaluesofPA

are1or0.

�LetP?A

=(I�PA)

-P?A

istheorthogonalprojectorontothenullspaceofAH:

-P?AP?A

=P?A.

�APusesthefollowingproperty:

P[B;C]=PB

+P(I�PB

)C:

126

�Let~AbeformedfromAwithoutitsi thcolumn.Letaibethe

i thcolumnofA.

Letus�x!1 ;���;!i�1 ;!i+1 ;���;!K

andupdate!i .

PA

=P[~A

ai ]=P~A

+P(I�P~A)ai

^!i

=

argmax

!i

�yHPAy �

=

argmax

!i

�yHP(I�P~A)ai y �

ThestepsofAPare:

1.)K

=1,

obtain^!1 :

2.)K

=2,

use^!1in1.)toobtain^!2 .

...K.)K

=K,

use^!1 ;���;^!K�1toobtain^!K:

K+1.)Alternativelyupdatethe!'s,tillconvergence.

127

�TheideaofANPAistoimplementAPe�ciently.

yHPP?~Aai y

=

yHP?~Aai aHiP?~Ay

aHiP?~AP?~Aai

=

��� aHiP?~Ay ��� 2

��� ��� P?~Aai ��� ��� 2

:

(NotethatwehaveusedP?~A

= �P?~A �H

:)

�Letb(z)=b0+b1 z�1+���+bKz�K

(b0=1)havezeroes

ej!1;ej!2;���;ej!K

:

128

LetB= 266666666666664

b�K...

...

0

b�0

...

...

b�K

0

...

b�0 377777777777775

N�(N�K):

Notethat

BHai= 26664

bK

���

b0

...

...

0

0

bK

���

b0 37775 26666664

1ej!i

...

ej(N�1)!i 37777775

129

SincebK

+bK�1 ej!i

+���+b0 ej!i K

=ejK!i(b

0+���+bK�1 e�j(K�1)!i

+bke�jK!i )=0

)

BHai=0,

fori=1;���;K:

)

BHai=0

SincebothAandBhavefullcolumnranksK

andN�K,Aand

B'scolumnsspanK-dimand(N�K)-dimorthogonalsubspaces,

respectively.

)

P?A

=PB

�Let~Bcorrespondto~A,thenP~B

=P?~A:

)

^!i=

argmax

!i

�j aHi

P~Byj2

jj P~Bai jj2 �.

130

�

Letyv

=P~By:

��aHiyv �� 2= �����

N�1

Xn=0yv(n)e�j!in ����� 2:

LetP~B

= 266666666664q0

q1

q2

���

q�1

...

...

...

...

...

...

q2

...

...

...

q1

...

...

q�1

q0 377777777775

;

jjP~Bai jj 2=

(N�1)

Xl=�(N�1) q

l ej!i l

131

�ThecomputationaladvantagesofANPAare:

+withyv

andP~B,FFTcanbeusedtosearchfor^!i

+P~B

canbecalculatede�cientlysince~BisbandedToeplitz.

-~BH

~BisbandedHermitianandToeplitzandpositive-de�nite.

-~BH

~B=GGH

(CholeskyDecomposition.)

-Gisbandedlowertriangularmatrixandcanbecomputed

iteratively.

-G�1~BH

canbecomputediteratively.

132

Remarks on NLS Method:

NLS methods minimize

g = (y �A�)H(y �A�): (�)

� NLS is ML when Q = �2I.

� NLS is no longer ML when Q 6= �2I:

� When the noise is circularly symmetric Gaussian with unknown

Q, but Q is described by a �nite number of unknowns (e.g. AR,

MA, or ARMA noise) and under mild conditions, NLS

asymptotically (for large N) achieves the CRB, and hence

asymptotically statistically e�cient.

CRB

NLS

MSE

N

133

TheRELAXAlgorithm

�Mostoftheexistingalgorithms�rstconcentrateout�andobtain

^!=

argmax

!

hyHA �AHA ��1AHy i:

SeeforexampleAPandANPA.^�isthenobtainedviaaleast

squaresmethodbyusing^!.

-Yetthisapproachinsteadofsimplyingtheproblem,could

complicatetheproblem.

-Someofthe^!k 'scouldbeveryclosetoeachother,whichmakes

�AHA ��1

ill-conditionedand^�verypoor.

134

�TheRELAXalgorithmisarelaxationbasedoptimization

algorithmthatminimizestheoriginalNLScostfunction(�).

�Theonlydi�erencebetweenAP,ANPAandRELAXis

howtheoptimizationoftheNLScostisimplemented.

�RELAXiscomputationallymoree�cientthanANPA,and

requiresonlyasequenceofFFT's.

�ThecomputationalsimplicitymakesRELAXarobustalgorithm.

�Forverycloselyspacedsinusoids,RELAXconvergesveryslowly.

RELAXperformswellfor�!>

�N

.

135

TheStepsoftheRELAX

Algorithm:

Step(1):K

=1.Obtain^f1and^�1 ,vianormalizedFFT,fromy.

Step(2):K

=2.a)

Use^f1and^�1obtainedinStep(1)to

compute

y2=y�^�a(^!1 ):

Determine^f2and^�2fromy2viathenormalizedFFT.

b)

Computey1=y�^�2 a(^!2 ):

Redetermine^f1and^�1fromy1 .

Iteratethetwosubstepsa)andb)untilconvergence.(NLScostno

longerdecreases).

136

Step(3):K

=3.a)

Use n^fi ;^�i o

i=1;2tocompute

y3=y�^�1 a(^!1 )�^�2 a(^!2 ):

Determine^f3and^�3fromy3 .

b)

Use n^fi ;^�i o

i=2;3tocompute

y1=y�^�2 a(^!2 )�^�3 a(^!3 ):

Redetermine^f1and^�1fromy1 .

c)Use n^fi ;^�i o

i=1;3tocompute

y2=y�^�1 a(^!1 )�^�3 a(^!3 ):

Redetermine^f2and^�2fromy2 .

Iteratethethreesubstepstillconvergence.

ContinuetillK=desirednumberofsinusoids.

137

RemarksonRELAX

�OnecanchecktherelativechangeoftheNLScostbetweentwo

consecutiveiterationsineachstepofRELAX.

�WhennoiterationperformedineachstepofRELAX,RELAX

becomesCLEAN.CLEANwasproposedin1974inaAstrophysics

journalandisusedinmanyapplications.

�WhenoneiterationisusedRELAXbecomesMCLEAN.

�RELAXisnicknamedSUPERCLEAN.

�Undermildconditions,RELAXisguaranteedtoconverge

(usuallyinafewsteps)toatleastalocalminimum.

138

HighResolutionMethods

�StatisticalPerformanceclosetoMLestimatororCRBespecially

forlargeN

orhighSNR.

�AvoidMultidimensionalsearchoverparameterspace.

�DONOTdependonresolutioncondition.

�Allprovideconsistentestimates.

�AllgivesimilarperformanceespeciallyforlargeN.

�Methodofchoiceisdeterminedbythecomputationsrequired.

�Robustnessagainstnon-whitenoise?unknown.

139

MODE(MethodofDirectionEstimation)

�MODEandWSF(WeightedSubspaceFitting)originally

proposedforangleestimationinarrayprocessing.

�MODEandIQML(InterativeQuadraticMaximumLikelihood)

arethesameforsinusoidalparameterestimationorangle

estimationwithonesnapshot.

�MODEisanapproximate(forhighSNR)MLestimatorwhen

Q=�2I.(MODEmayneverachievetheCRBwhenQisarbitrary

unknown.)

�RecalltheMLestimatorwhenQ=�2I.

^!

=

argmax

!

�yHA(AHA)�1AHy �

=

argmin

!

�yHP?Ay �:

140

�TheideaofMODEistore-parameterizetheabovecostfunction

viathepolynomialcoe�cients

b=[bo

b1

���bK];wherebK

+bK�1 ej!i

+���+bo ejK!i

=0

Let 26666666664

b�K

0

...

...

b�0

b�K

...

...

0

b�0 37777777775

N�(N�K):

�SincePB

=P?A:

^b=

argmin

b

�yHPBy �

^b=

argmin

b

tr �(BHB)�1BHyyHB �:

141

TheStepsofMODE:

Step1:Let(BHB)=Iandobtainaninitialestimateofb,whichis

usedtoform

^B:

Step2:Obtain^bvia

^b=

argmin

b

tr h(^BH

^B)�1BHyyHB i:

(�)

Step3:Solvefortherootsof

KXk

=0

^bkzK�k=0

Thephasesoftherootsgivef^!k g.

142

Remarks:�(*)isaquadraticoptimizationproblem,andhence

easytoimplement.

�Step1ofMODEprovidesconsistentestimateofbas�!

0.

Note:

BHyyHB=BHeeHB

�!

�!0 0

(BHA=0!)

For�<<1,

^b(i)�b=�(f001)�1f01 ;

wheref1=

tr �BHyyHB �;

andthei thelementoff01is

[f01 ]i=@f1

@bi:

[f01 ]i=

tr �(B0i )HyyHB+BHyyHB0i �

whereB0i

=@B

@bi

143

)

[f01 ]i

=

tr �(B0i )HyeHB+BHeyHB0i �:=O(�):

[f001]ij

=

tr h(B00ij )HyyHB+(B0i )H

yyHB0j

+

B0Hj

yyHB0i+BHyyHB00ij �

=

O(1))

^b(i)�b=O(�)

)

Theinitialestimateofbisconsistent

�Using �^BH

^B ��1

formedfromtheconsistentestimateof^b(i)

insteadof �BHB ��1

inStep2ofMODE

doesnota�ect

theasymptotic(�<<1)accuracyof^b:

144

Letf

=

tr �(BHB)�1BHyyHB �

=

tr �PByyH �

[PB] 0i= �By �H �BH �0iP?B

+ n(By)H �BH �0i P?B oH

;

whereBy= �BHB ��1BH

^b(i)�b=�(f00)�1f0;

f0i=2Re �[PB] 0iyyH =

O(�)

Letf2=

tr ��(BHB)�1+� �BHyyHB ;

�=O(�)

[f2 ] 0i

=

f0i+

tr ��BHiyyHB+�BHyyHBi

=

f0i+O(�2)

�

f0i=O(�)

145

Similarly,

f00ij=O(1):

[f2 ] 00ij

:=f00ij=O(1)

Hence

^b�b=�(f00)�1f0��(f002)�1f02

)

minimizingfisequivalenttominimizing(*)for�<<1

�ThesecondstepofMODEmaybeiteratedseveraltimes

toachievebetteraccuracy.Althoughtheconvergenceis

notguaranteed,numericalexamplesshowthatiteratingStep2

convergesmostofthetimes.

�MODEisderivedasanapproximateMLestimatorforwhite

noise.IQMLisanapproximateoptimizationmethod.Theyare

di�erentformulti-snapshotarrayprocessingproblem.

146

TheImplementationdetailsofMODE

LetW

=I

(Step1)or

W

= �^BH

^B ��1

(Step2):

^b=

argmin

b

tr hWBHyyHB i

BHy= 26664

bK

���

b0

...

...

0

0

bK

���

b0 37775 26666664

y0

y1...

yN�1 37777775

147

= 26666664yK

���

y0

yK+1

���

y1

...yN�1

���

yN�K�1 37777775 26666664

b0

b1...b

K 377777754=Yb

^b=

argmin

b

��� ��� W12

Yb ��� ��� 2

�Ifnoconstraintonb,thenb=0isthesolution.

�Toobtainmeaningfulsolution,wecouldconstraineither

b0=1

orjjbjj=1

148

�Whenimposingjjbjj=1,thesolutionistheeigenvectorof

YHWYcorrespondingtoitssmallesteigenvalue.Thisconstraint

seemstogivebetteraccuracy.

�Sinceej!k

arethezerosofthepolynomialformedfromb,andare

ontheunitcircle,wecanfurtherconstrainb.

�Tofurtherguaranteethatthezerosofthepolynomialareonthe

unitcircle,theconstraintsonbwillbetoocomplicatedandthe

slight,ifany,accuracyimprovementisnotworthit.

�ConjugateSymmetrydoesnotguaranteethatthezerosareon

theunitcircle,butitiseasytoinclude.Itisanecessarybutnot

su�cientconditionforzerostobeontheunitcircle.

bk=b�K

�k;

k=0;1;���;K

�Withconjugatesymmetry,thenumberofunknownsisabout

halved.

149

�LetI= 26664

1

0

...

0

1 37775J= 26664

0

1

...

1

0 37775

Let�= h�b

0

�b1

����b

bK2c

~b0

~b1

���~b

bK

�

12

c iT

;

where�bi=

Re(bi );

~bi=

Im(bi );

bK2c=

theintegerpartofK2

:

ForK

odd,

b= 24I

jI

J

�jJ 35�

4=G�:

150

ForK

even,

b= 2664

I

0

jI

0

1

0

J

0

�jJ 3775

�

4=G�:

)

^�=

argmin

�

��� ��� W12

YG� ��� ��� 2

151

Angle and Waveform Estimation with a Sensor Array

� Consider auniform linear array (ULA) (this case is closely related

to the sinusoidal frequency and complex amplitude estimation).

kth

Signal

θ k

d

N Sensors

y(t) =2

66666641 � � � 1

ej!1 � � � ej!K

...

ej(N�1)!1 � � � ej(N�1)!K

37777775

26666664�1(t)

�2(t)...

�K(t)3

7777775+ e(t)

N = number of sensors ( Note the notation similar

K = number of signals. to the sinusoids case )

152

�Weconsidertheestimationof!k;

Since!k=�!0d

c

sin�k ,from^!k ,wecandetermine�k

easily.(!k

=

spatialfrequency.)

�d��2

willavoidambiguities.

�y(t)issampledatt=t1 ;���;tM

whereM

isthenumberof

snapshots.

��(tm)= 26664

�1 (tm)

...

�K(tm) 37775

;

m=1;2;���;M

��(tm)isthesignalwaveformvector.

�Whenmodeledasdeterministicunknown,�(t1 );���;�(tM

)

contain2KM

unknowns.|

DeterministicDatamodel

153

-When�(t1 );���;�(tm)ismodeledasstationarystochastic

processeswithcomplexGaussiandistribution,zero-mean,

covariancematrixS,thedatamodelisreferredtoasthe

StochasticDataModel.(no.ofunknownsinS/K2:)

-

CRBSTO

�

CRBDET

-DeterministicML(DML)methodsneverachievesCRBDET

as

M

!

1

sinceno.ofunknownsin�(tm);

m=1;���;M,is

2KM

!

1.

-StochasticML(SML)methodsasymptotically(M

!

1

)

achievesCRBSTO

.

-

MSEofDML�MSEofSML

154

-Thewaveforms�k (t);

k=1;���;K;maybeuncorrelatedwith

eachother,maybecorrelatedorevencoherentwitheachother(

thishappensinamultipathenvironment).

+ForstochasticdatamodelsSisdiagonalforuncorrelated

signalsandissingularforcoherentsignals.

+Fordeterministicdatamodels,1M PMm

=1�(tm)H�(tm)is

singularforcoherentsignals.ItbecomesdiagonalasM

!

1

for

uncorrelatedsignals.

155

DeterministicDataModels

y(tm)=A�(tm)+e(tm);

orsimply

y(m)=A�(m)+e(m),or

ym

=A�m

+em;

whereAisthedirectionmatrixandcontainsthespatialfrequency

information,�m

istheunknownwaveformvector,andem

isthe

additivenoise.

�Weassume

E �e(m1 )eH(m2 ) =Q�m1;m2 ;

where�m1;m2istheKroneckerdelta.

�Further

em

�N(0;Q)

156

Maximum

LikelihoodEstimator

f(y1 ;���;yM

j�1 ;���;�M

;!)=�Mm

=1

1�jQj e

�(ym

�A�m

)H

Q�

1(y�A�m

)

�ConsiderbelowQ=�2I.Thenmaximizingtheabovelikelihood

functionisequivalenttominimizing

g=

MXm=1

(ym

�A�m)H

(ym

�A�m)

�IfQ6=�2Iorem

isnon-Gaussian,minimizinggistheNLS

method.

�Mostexistingestimators�rstconcentrateout�m.

157

�Bysetting

@g

@�m

=0,weget

^�m

= �AHA ��1AHym ��!=^!

and

^!=

argmax

!

"MXm

=1yHmA �AHA ��1AHym #

Let

g1

=

1M

MXm=1yHmA �AHA ��1AHym

=

tr "A �AHA ��1AH

MXm=1 �y

myHm �

M

#

=

tr hPA

^R i;

where

^R=

1M PMm

=1ymyHm:

�^Risknownasthesamplecovariancematrix.

158

AP/ANPA

�InAP,weusethesameidea:

^!i=

argmax

!i

tr hPP?~Aai^R i

�InANPA,

tr hPP?~Aai^R i=

tr hP?~Aai aHiP?~A

^R i

aHiP?~AP?~Aai

:

=

PMm

=1 ��� aHiP?~Aym ��� 2

��� ��� p?~A ai ��� ��� 2

-FFTscanbeusedtospeedupthecomputations.

-ThesavingsoncomputingP?~A

willnolongerhaveasigni�cant

impactontheoverallcomputationforlargeM.

159

RELAX

�TheextensionofRELAXtothemultiplesnapshotcaseis

straightforward.

�Let

y(k)

m

=ym

�

KXi=

1;i6=k

[ ^�m]i a(^!i );

where[ ^�m]idenotesthei th

elementof^�m,[ ^�m]iand^!i ;i6=k,

areassumedtobegivenorpreviouslyestimated.

�Considerthecostfunctiong2 .

g2=

MXm=1 ny(k)

m

�a(!k )[�m]k oH n

y(k)

m

�a(!k )[�m]k o

�Weminimizeg2withrespectto!k

and[�m]k;

m=1;���;M:

160

g2

=

MXm=1 n

[�m]k� �aH(!k )a(!k ) ��1aH(!k )y(k)

m oH

:

aH(!k )a(!k ) n[�m]k� �aH(!k )a(!k ) ��1aH(!k )y(k)

m o

+

MXm=1 �y(k)

m hy(k)

m iH

�y(k)

m

a(!k ) �aH(!k )a(!k ) ��1aH(!k )ym �

�

[ ^�m]k=

aH

(!k)y(k)

m

N

���!k=^!k

;

m=1;2;���;M

^!k

=

argmin

!k

MXm=1 ���� ���� �I�a(!k )aH(!k )

N

�y(k)

m ���� ���� 2

=

argmax

!k

PMm

=1 ��� aH(!k )y(k)

m ��� 2

N

161

�^!k

isthelocationofthedominantpeakofthesumofthe

periodogram

j aH

(!k)y(k)

m

j2

N

overm=1;2;���;M:

�[ ^�m]karethecorrespondingheights.

162

StepsoftheMulti-SnapshotRELAX

�Step1:LetK

=1.Estimate!1and[�m]1from

y(1)

m

;

m=1;2;���;M.

�Step2:LetK

=2.a)Compute ny(2)

m oMm

=1byusing^!1and

f[ ^�m]1 gMm

=1:Obtain^!2andf[ ^�m]2 gMm

=1from ny(2)

m oMm

=1 .

b)Re-Determine^!1andf[ ^�m]1 gMm

=1from ny(1)

m oMm

=1obtained

byusing^!2andf[ ^�m]2 gMm

=1 .

Iteratetheabovesub-steps.

...ContinuetillK

=desirednumberofsignals.

163

IQML

^!

=

argmax

!

tr hPA

^R i

=

argmin

!

tr hP?A

^R i

)

^b

=

argmin

b

tr hPB

^R i

MLestimate

�IQMLisanapproximateMLmethodandisiterative.

^bj+1=

argmin

b

tr �B �^BHj

^Bj �

�1

BH

^R �;

where^Bjisobtainedfrom^bj,whichisestimatedfromthej th

iteration.

�IQMLisnotconsistentasM

!

1

andalmostalwaysprovides

biasedestimate.

�IQMLconvergesmostofthetime,butnottheoretically

164

guaranteed.

165

MLEstimationforUnknownQ

�WhenthenoisecovariancematrixQisunknown,MLestimation

ofanglesandwaveformsisill-de�ned.

�Thelikelihoodfunctionisproportionalto:

C=�lnjQj�

tr (Q�11M

MXm=1

(ym

�A�m)(ym

�A�m)H )

�Let@Q

@Qij

=Q0ij ;

Qij=(ij) thelementofQ.

@lnjQj

@Qij

=

tr �Q�1Q0ij �

@Q�1

@Qij

=�Q�1

@Q

@QijQ�1=�Q�1Q0ij Q�1:

166

�

@C

@Qij

=�tr �Q�1Q0ij �+tr �Q�1Q0ij Q�1(:::) ;

where(...)=

1M PMm

=1(ym

�A�m)(ym

�A�m)H

:

�

tr �Q�1Q0ij �= �Q�1 �

ji:

�Q�T

+ �Q�1(:::)Q�1 �T ���

Q=^Q

=0

)

^Q=

1M PMm

=1(ym

�A�m)(ym

�A�m)H

�ThecostfunctionwithQconcentratedouthastheform

F= �����1M

MXm=1

(ym

�A�m)(ym

�A�m)H �����:

�Since(...)isatleastpositivesemide�niteF�0.

167

�f^!k gandf^�mgaredeterminedbyminimizingF.However,there

aretoomanyunknownparameters(toomanydegreesoffreedom).

Therearemanychoicesoff�mgtomakeF=0,theultimate

minimum.

�Forexample,let

^�m

= �AHA ��1AHym:

ThenF

=

�����1M

MXm=1P?AymyHmP?A �����

=

����� P?A "1M

MXm=1ymyHm #P?A �����

=

0;

Since ��P?A ��=0duetoitsrankN�K.

�HenceMLestimationisill-de�nedhere.

168

MLAngleEstimationforSignalswithKnownWaveforms

�Formanyapplications,thesignalwaveforms

f�mgMm

=1areallknown,exceptforunknowngains.

�Forexample,incommunications,knowntrainingsignalsare

oftenusedforsynchronization.

�Forthiscase,MLestimationofanglesis

NOLONGERill-de�nedforunknownQ.

�ThisideahasbeenthesourceoftheAPESalgorithm,which

turnsouttobeanon-parametricmethod.

�Theideais�ndingmoreandmoreapplicationsin

communicationssuchascode-timingestimationinDS-CDMA.

169

�Knownwaveformsbutunknowngains:

�m

=�sm

where�=diagf 1 ; 2 ;���; Kgwithf!k gKk

=1denotingtheK

unknowncomplexgainsoftheK

signals,andfsmgMm

=1denoting

theknownwaveforms.

�ThelikelihoodfunctionforunknownQisproportionalto:

�lnjQj�

tr (Q�11M

MXm=1

(ym

�Csm)(ym

�Csm)H );

where

C=A�:

�ConcentratingoutQyields:

^Q=

1M

MXm=1

[ym

�Csm][ym

�Csm] H:

170

�Thenthecostfunctionbecomes:

F= �����1M

MXm=1

[ym

�Csm][ym

�Csm] H �����:

�Let

^Rys=

1M

MXm=1ymsHm

=^RHs

y

^Ryy=

1M

MXm=1ymyHm

^Rss=

1M

MXm=1smsHm:

171

ThenH

=

1M

MXm=1

[ym

�Csm][ym

�Csm] H

=

^Ryy�C^Rsy�^RHs

y CH

+C^Rss CH

=

hC�^RHs

y^R�1

ss i^Rss hC�^RHs

y^R�1

ss i

H

+

^Ryy�^RHs

y^R�1

ss

^Rsy:

�

^C=^RHs

y^R�1

ss

minimizesH.

�Sincedet(.)andtr(.)arenondecreasingfunctionsofpositive

de�nitematrices,theabove^CminimizesF.

�Notethat^CisanestimateofA�withoutgivingconsiderationto

itsstructures,andhencenotparsimonious.

172

�Using^C,weget

^Q=^Ryy�^RHs

y^R�1

ss

^Rsy:

�Notethattheabove^QisnotMLsincethe^Cusedisnot

parsimonious.Weusethis^QtoobtainMLestimatesoff!k gKk

=1 .

�MLestimationofQisnotourconcern!

173

�TousethestructureofC,considerFagain:

F

=

jHj

=

���^Ryy�^C^Rsy+(C�^C)^Rss (C�^C)H ���

=

���^Q hI+^Q�1(C�^C)^Rss (C�^C)H i ���

=

���^Q ��� ���� I+^Q�1 �C�^C �^Rss �C�^C �H ����

:

�ObtainingtheMLestimatesoff!k gKk

=1andf k gKk

=1via

minimizingFrequiresthemultidimensionalsearchoverthe

parameterspace.

�ForlargeM,minimizingFisequivalenttominimizing

F2=

tr �Rss �C�^C �H

^Q�1 �C�^C � �;

where

Rss=limM

!1

1M PMm

=1smsHm:

174

�Foruncorrelatedsignals,Rssisdiagonal.Hencetheminimization

ofF2isdecoupledtominimizing

F3=[ ^ck� k a(!k )] H^Q�1[ ^ck� k a(!k )]:

�HenceweobtainthedecoupledML(DEML)estimatorfor

estimatingf!k gandf k g:

^ k=

aH

(!k)^Q�

1^ck

aH

(!k)^Q�

1a(!k) ���!k=^!k

^!k=

argmax

!k

j aH

(!k)^Q�

1^ck j2

aH

(!k)^Q�

1a(!k) :

�Evenforanarbitraryarray,DEMLdecomposesmultidimensional

searchintoK

separateone-dimensionalsearchproblems,whichis

non-iterative.

�ForULA,wecanavoideven1-Dsearch.

175

�ForULA,toavoid1-Dsearches,weconsiderapolynomialwith

coe�cients

bk= 24b0k

b1k 35;

k=1;���;K

as

b0k+b1k=b0k �z�ej!k �;

b0k6=0:

BHk

= 26664b1k

b0k

...

...

0

0

b1k

b0k 37775

;

N�(N�1)

BHka(!k )=0

) h^Q12

Bk iH

^Q�

12

a(!k )=0:

)

Let~ak=^Q�

12

a(!k ):

176

P?~a

k

=P^Q12

Bk

=^Q12

Bk (BHk

^QBk )�1BHk

^Q12

�

^!k

=

argmax

!k

^cHk

^Q�1a(!k )aH(!k )^Q�1^ck

aH(!k )^Q�1a(!k )

=

argmax

!k

^cHk

^Q�12

P~ak

^Q�12

^ck

=

argmin

!k

^cHk

^Q�12

P?~a

k

^Q�12

^ck

�

^bk=

argmin

bk

tr hBk (BHk

^QBk )�1BHk^ck ^cHk i

177

�Step1,letBHk

^QBk=Itoobtainaconsistentestimateofbk .

Step2,form

^BHk

^Q^Bk

fromthe^b(i)

k

obtainedinStep1.

^bk=

argmin

bk

tr ��^BHk

^Q^Bk �

�1

BHk^ck ^cHkBk �:

�Theabovestepswillnota�ecttheasymptaticaccuracyofDEML.

�Further,impose:^b

0k=^b�1

k

andRe2(b

0k )+Im2(b

0k )=1toobtain

meaningfulresultwithzeroontheunitcircle.

178

PropertiesofDEML

Comparingknownandunknownwaveformsforuncorrelated

signals,wehave:

1.DEMLcomputetionallymuchsimpler.

2.DEMLgivesbetteraccuracyfortheangleestimatesthanthe

bestestimatorforunknownwaveforms.

3.Asthesmallestangleseparationapproacheszero,DEMLdoes

notdegrade.Forunknownwaveforms,theaccuracydegradestill

failure.

4.DEMLhasnoconstraintsonthemaximum#ofincidentsignals

aslongasM

islargeenough.Forunknownwaveforms,K

<N.

5.DEMLhandlesunknownQwithoutdi�culty.Forunknown

waveforms,MLnotpossible.

179