The M-DSO-ESPRIT Method for Maximum Likelihood DoA Estimation

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UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

THE M-DSO-ESPRIT METHOD FOR MAXIMUM LIKELIHOOD

DOA ESTIMATION

L. Lizzi, F. Viani, M. Benedetti, P. Rocca, and A. Massa

January 2008

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The

M

− DSO − ESP RIT

Method for Maximum Likeli-hood

DoA

Estimation

L. Lizzi, F. Viani,M. Benedetti, P. Ro a, and A. Massa, EM A ademy Fellow

Department of Information and Communi ation Te hnology University of Trento,Via Sommarive 14,38050 Trento - Italy Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it,

{leonardo.lizzi,federi o.viani,manuel.benedetti,paolo.ro a}dit.unitn.it Web-site: http://www.eledia.ing.unitn.it

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The

M

− DSO − ESP RIT

Method for Maximum Likeli-hood

DoA

Estimation

L. Lizzi, F. Viani,M. Benedetti, P. Ro a, and A. Massa

Abstra t

The estimation of the dire tions-of-arrival (

DoA

s) of multiple signals is a topi of great relevan e in smart antenna synthesis and signal pro essing appli ations. In this paper, a memory-based method is proposed to ompute the maximum likeli-hood (

M L

)

DoA

estimates. Su h a on eptually-simple te hnique is based on the data-supportedoptimization(

DSO

)andtheestimationofsignalparameters via ro-tational invarian e te hnique (

ESP RI T

), but fully exploits a memory me hanism forimprovingtheestimationa ura yespe iallywhendealingwith riti als enarios hara terizedbylowsignal-to-noiseratios(

SN R

)or/andsmallnumberofsnapshots. Simulationresultsassessthepotentialities andlimitationsoftheproposedapproa h thatfavorably ompares withstate-of-the-art methods.

Key-words: Dire tion-of-ArrivalEstimation,Data-SupportedOptimization,Smart An-tennas, Wireless Communi ations

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Smartantennas area hallengingresear h topi inele tromagneti sand wireless ommu-ni ations. The main reason for the growing interest about su h an issue when dealing withmulti-users ommuni ationsystems,mainlyliesintheneedofadaptivelyfa ingwith unknown time-varyings enarios [1℄[2℄[3℄. In general,smartsystems onsistof anarray of radiating elements able to steer the main lobe beam towards the desired signal [4℄[5℄[6℄ and tolo atesuitable nulls of the radiation patternin the dire tions of the interferen es [7℄[8℄[9℄. A ordingly, a relevant step for building a smart re eiver is on erned with the estimation of the dire tions of arrival (

DoA

s) of the re eived signals. Towards this end, variouste hniqueshavebeen developed not onlyforwireless ommuni ations, butalsoin various appli ations ranging from radar [10℄[11℄[12℄ to sonar [13℄ and spee h pro essing [14℄.

The maximumlikelihood(

M L

) estimatorhas beenlargelyused indire tionnding prob-lemsbe auseofits apabilityofrea hing,asymptoti allyandunderregularity onditions, the Cramer-Rao Bound (

CRB

) [15℄. Unfortunately, it is hara terized by an intrinsi omplexity arising from the multi-modal nature of the likelihood fun tion (

LF

) and by the high omputational load of the involved multivariate nonlinear maximization prob-lem[16℄[17℄[18℄. Therefore, othersub-optimal approa hes have been proposedinorder to rea h asuitable trade-obetween estimation a ura y and omplexity.

Con erning learning-by-examples (

LBE

) te hniques, some methods based on the use of radial-basis fun tions (

RBF

) [19℄[20℄and support ve tor ma hines [21℄[22℄[23℄ have been e iently applied tosingle- and multiple-sour e dire tionnding,as well.

Unlike

LBE

te hniquesthatneedsofalearningphasefortrainingthe underlyingnetwork ar hite ture,super-resolutionapproa hesdire tlypro essthere eivedsignalswithoutany o-linepre-pro essing or training. In su h a framework, the multiplesignal lassi ation method(

M U SIC

)[24℄isaneigenstru ture-baseddire tionndingte hniquethatemploys the noise-subspa e eigenve tors of the data orrelation matrix for determining a null spe trum, whoseminimaare iteratively omputedtoyield the

DoA

estimates. Although it asymptoti ally onverges to the

CRB

[25℄ for an in reasing number of snapshots, the

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far as uniform linear arrays (

U LA

s) are on erned, the so- alled

ROOT

− MUSIC

[27℄ version an be protably used by solving a polynomial rooting problem,thus improving the omputationalperforman es of the

M U SIC

algorithm [28℄.

Likewise

M U SIC

,theestimationofsignalparametersviarotationalinvarian ete hnique (

ESP RIT

) [29℄ is a ve tor subspa e-based methodology that, instead of identifying the spe tral peaks, dire tly determines the

DoA

s by exploiting the rotational invarian e of the underlying signal subspa e indu ed by the translational invarian e of the sensors array. Sin e the

ESP RIT

omplexitystri tly depends onthe numberof sensors,faithful estimates and a redu ed omputational burden an be a hieved dealing with a limited number of array elements, while the size of the orrelation matrix be omes greaterthan those from

M U SIC

or

ROOT

− MUSIC

when largearrays are onsidered[26℄.

In order to redu e the omputational osts as well as the risk of being trapped in false maxima of the

LF

, the data-supported optimization (

DSO

) an be suitably employed with

ESP RIT

[18℄[30℄. Su h a pro edure onsists in partitioning the data sample in a large numberof "elemental sets"for allowing asimpler omputationof the estimates for ea helementaldataset. The so-obtainedvalues onstitute adata-supportedgrid (

DSG

) over whi h the

LF

fun tion ismaximized.

Althougheigenstru ture-based approa hes onstitute the state-of-the-artin

DoA

estima-tion and demonstrated their optimality (as the best ompromise between a ura y and omputationalload) indealingwith alimitednumberofin omingsignals and/orlimited numberof re eivers, unsatisfa tory performan es (i.e., with onspi uous dieren es om-pared to

M L

) o ur in the so- alled threshold region, namely, when the signal-to-noise ratio is low, oralternatively,when the numberof snapshots issmall.

In order to deal with these situations, this paper presents an hybrid approa h alled memory-based

ESP RIT

-like (

M

− DSO − ESP RIT

). Following the guideline of the

DSO

− ESP RIT

[31℄, the proposed method onsiders an

ESP RIT

-basedestimator for omputing the

DSG

and a memory me hanism for enhan ing the estimation a ura y thanks tothereallo ationoftheinformationa quiredattheprevioussteps,whi hisused as a-priori knowledge for su essive estimates.

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i allyformulated. The

M

− DSO − ESP RIT

methodisdes ribed inSe t. 3by fo using on its innovative features. Se tion 4 is devoted at presenting a set of sele ted numeri al results in order to point out potentialities and limitationsof the proposed approa h also in omparisonwithstate-of-the-artsuper-resolutionte hniques. Finally,some on lusions are drawn (Se t. 5).

2 Mathemati al Formulation

Letus onsidera

U LA

of

M

equally-spa edsensorsand

L

(

L < M

)un orrelated narrow-band signals impingingat ea h time instant

t

s

= t

0 + s △ t

,

s

= 1, ..., S

, on the antenna with plane wavefronts from dierent dire tions,

Θ (t

s

) = {θ

l

(t

s

) ; l = 1, ..., L}

(Fig. 1). Moreover, let us indi ate with

y

(t

s

) = {y

m

(t

s

) ; m = 1, ..., M}

T

the snapshot (i.e., the olle tionofdata samplesat

t

s

) olle tedbythe

M

sensorsatea htimeinstant

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. Under theassumptionthatthenumberofavailablesnapshotsisequalto

N

(

N

≥ L

),there eiver output

Y

(t

s

)

an be expressed, a ording tothe matrix notation[17℄[32℄, asfollows

Y

(t

s

) = A [Θ (t

s

)] X (t

s

) + E (t

s

) .

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Inparti ular,

Y

(t

s

) =

y

(t

s−N+n

) ; n = 1, ..., N

isa omplexmatrixof

M

× N

elements (i.e.,

Y

∈ C

M ×N

),

X

∈ C

L×N

is the matrix of signal waveforms, and

A

∈ C

M ×L

is the steering matrix given by

A

[Θ (t

s

)] = {a [θ

l

(t

s

)] ; l = 1, ..., L}

(2)

being

a

[θ

l

(t

s

)] =

n

e

j(m−1)

2π

λ

dsin[θ

l

(t

s

)]

_{; m = 1, ..., M}

o

T

the steeringve tor of the array to-wards the dire tion

θ

l

(t

s

)

. Moreover,

E

∈ C

M ×N

is related to the noise modeled by means of a stationary and ergodi omplex-valued Gaussian pro ess of zero-mean har-a terized by anassigned signal-to-noiseratio (

SN R

). Furthermore, the noise samples at the re eivers,

e

m

(t

s

)

,

m

= 1, ..., M

,

s

≥ 1

, are assumed to be statisti allyindependent.

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Under these hypotheses, the maximum likelihood lo alizationof the

L

sour es at

t

s

[i.e., the estimationof

Θ (t

b

s

) =

n

b

θ

l

(t

s

) ; l = 1, ..., L

o

℄ isobtained as [17℄[32℄

b

Θ (t

s

) = arg max

Θ(t

s

)

[f

M L

{Θ (t

s

)}]

(3)

where

f

M L

is the likelihoodfun tion given by

f

M L

{Θ (t

s

)} = tr

P

[Θ (t

s

)] R (t

s

)

.

(4)

and

tr

{·}

indi atesthe tra e of the matrix. Moreover,

P

[Θ (t

s

)] = A [Θ (t

s

)]

A

H

[Θ (t

s

)] A [Θ (t

s

)]

−

1 A

H

[Θ (t

s

)]

and

R

(t

s

) =

1 N

N

X

n=1

y

(t

s−N

+n

) y

H

(t

s−N+n

)

are the proje tion and the sample ovarian e matrix [17℄, respe tively.

Althoughthe

M L

lo alizationmethodallowsonetoobtaintheoptimalestimate,itusually requirestheevaluationofthe

LF

in orresponden e withea hpossible ombinationofthe

DoA

s. Su h anevent results ina omputationally-expensive pro edure, espe ially when dealing with multiplesour es. Consequently, asuitable strategy aimedat optimizingthe trade-o between lo alization a ura y and omputational load ould be advantageous. Towards this purpose, a new estimation te hnique isproposed in the following se tion.

3 The M-DSO-ESPRIT Method

Unlike the optimal

M L

approa h, the

M

− DSO − ESP RIT

method resorts to the evaluation of the

LF

in alimited set of ombinationsof

DoA

and it onsiders a memory me hanisminordertofullyexploitthe a quired-knowledge(orexperien e)fromprevious estimates. More indetail, the following multi-steppro edure is arried out atea h time-step

t

s

,

s

≥ 1

:

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The

M

−DSO−ESP RIT

operatesamemoryenhan ementofthe olle teddata. Towards this end, anew memory-enhan ed output matrix

D

(t

s

)

D

(t

s

) = {d

b

(t

s

) ; b = 1, ..., B} ∈ C

M ×B

is dened from the standard output matrix

Y

(t

s

)

as des ribed in the following. At the rst time-step (

s

= 1

), the pro ess is initializedby assuming

D

(t

1 ) = Y (t

1 )

, that is

d

_b

(t

1 ) = y (t

s−N+b

)

b

∈ [1, B] ;

B

= N

otherwise (

s >

1

)

d

_b

(t

s

) =











y

(t

s−N+b

)

b

∈ [1, N]

a

h

θ

b

b−N

(t

s−1

)

i

b

∈ [N + 1, B]

B

= N + L.

3.2 Data-Spa e Re-Sampling

The so-dened data spa e is then re-sampled. Starting from the matrix

D

(t

s

)

and on-sideringthe whole set of olumn ombinationstoobtainamatrix of dimension

M

× L

, it is possible todene

K

[being

K

=

B!

L!(B−L)!

℄ matri es

W

(k)

_{∈ C}

M ×L

W

(k)

(t

s

) =

n

w

(k)

_l

= d

_c

(k)

l

; l = 1, ..., L

o

k

= 1, ..., K

(5)

where the index

c

(k)

l

∈ [1, B]

identies the olumn of

D

(t

s

)

orresponding to the

l

-th element of

W

(k)

_(t

s

)

, a ording to the iterative generation pro edure detailed in Tab. I. Moreover, in order to avoid wrong/unne essary su essive omputations, the matri es whose ondition numbers

η

(k)

s

are greater than a xed stability threshold (i.e., the ill- onditioned matri es) are omitted.

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As in[26℄[31℄, an

ESP RIT

-likealgorithmis used forgenerating the data-supportedgrid points. Letus onsider the matrix

Φ

(k)

_{∈ C}

L×L

given by

Φ

(k)

(t

s

) =

h

V

(k)

(t

s

)

i

H

V

(k)

(t

s

)

−

₁

h

V

(k)

(t

s

)

i

H

U

(k)

(t

s

)

(6) where

V

(k)

and

U

(k)

are two matri es obtained from

W

(k)

by eliminating the rst and lastrow,respe tively. Then,the

ESP RIT

-likeestimates(i.e.,the

K

data-supportedgrid points)turn out to be

b

Θ

(k)

(t

s

) =

n

ˆ

θ

(k)

_l

(t

s

) ; l = 1, ..., L

o

k

= 1, ..., K

(7) where

θ

ˆ

(k)

l

= arcsin

n

λ

2πd

arg

µ

(k)

_l

o

and

n

µ

(k)

_l

; l = 1, ..., L

o

aretheeigenvaluesof

Φ

(k)

[29℄.

3.4

M L DoA

Estimation

Finally,the

M L

estimates of the

DoA

sof the

L

signals are omputedbymaximizingthe

LF

overthe

K

-sized data-supported grid:

b

Θ (t

s

) = arg max

b

Θ

(k)

(t

s

)

h

f

M L

n

b

Θ

(k)

(t

s

)

oi

.

(8) 4 Numeri al Assessment

The numeri al assessment has been arried out by omparing the performan es of the

M

− DSO − ESP RIT

with thoseof otherstate-of-the-art approa hes, su h as

ROOT

−

M U SIC

[27℄,

ESP RIT

[29℄,

DSO

− ESP RIT

[31℄inordertopointoutitspotentialities and urrentlimitationsaswellastherangeof onvenientappli ability. For ompleteness, the asymptoti performan es a hievable by an unbiased estimator of the parameters

θ

l

(i.e., the so- alledCramer-Rao bound (

CRB

) [33℄) are reported, aswell.

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index ofe ien y. Moreover, be auseofthestatisti alnatureofthe s enariosundertest, its value averaged over

Q

= 100

independent realizations of ea h simulation has been omputed

RM SE

=

1 Q

Q

X

q=1

1 S

S

X

s=1

v

u

t 1

L

X

l=1

θ

l

(q)

(t

s

) − ˆ

θ

(q)

l

(t

s

)

2

(9)

where the index

q

denotes the

q

-threalization (

q

= 1, ..., Q

)of asimulation,

S

being the total numberof time-instants(

S

= 100

).

As far as the referen e antenna ar hite ture is on erned, a linear array of

M

= 20

omnidire tionalsensors

λ

2

-spa edhas been adopted.

The rsttest asedeals withstationary s enarios where

L

un orrelatedsignalsimpinge from random, but xed, dire tions [i.e.,

θ

l

(t

s

) = θ

l

,

∀s

℄. Under this assumption, a set of experiments has been performed in order to show the ee t of the size of the snapshot window(

N

),ofthesignal-to-noiseratio,andoftheangularseparation(i.e.,

△θ

l

= θ

l+1

−θ

l

,

l

= 1, ..., L − 1

) between the

DoA

sof the sour es onthe method performan es evaluated in terms of angulara ura y (i.e.,

RM SE

value) and omputational osts.

In therst experiment,the s enarioundertestis hara terizedbyheavy noisy onditions (

SN R

= 2 dB

) and

L

sour es oming fromrandom angulardire tions equally-spa ed by

△θ

l

= 10

o

,

l

= 1, ..., L − 1

. Underthe assumptionthat

N

= L

(i.e.,the minimumnumber of snapshots for a given onguration of sour es), Figure 2 shows the behavior of the estimation error versus the number of sour es

L

. Con erning the estimation a ura y, even though the resolution error is an in reasing quantity, the

M

− DSO − ESP RIT

outperforms the other dire tionnding methods and it results loser to the

CRB

when

L

≤ 4

pointingout anon-negligiblerobustnesstothe noise inlo alizingmultiplesour es. In order to quantify the omputational ost, the amount of oating point operations needed atea h time-instant

t

s

(i.e.,

Ω

)is analyzed (Fig. 3). As expe ted, be ause of the memory enhan ement and the dependen e of the dimension of the output matrix

D

(t

s

)

on

L

, the omputational burden required by

M

− DSO − ESP RIT

grows with

L

, but not in a linear fashion sin e the ltering pro edure at the Data-Spa e Re-Sampling step avoids the pro essing of ill- onditioned matri es. Moreover, whatever the number

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of sour es, the arising

Ω

value is greater than that of the

DSO

− ESP RIT

, but of the same orderinmagnitudeofthe standard

ESP RIT

. Furthermore,the

ROOT

− MUSIC

te hnique results mu h more expensive insu h a situation.

The se ondexperiment isaimedatevaluatingthe behaviorofthe

M

− DSO − ESP RIT

in orresponden e with a variation of the number of snapshots. Towards this purpose, the s enarioisthe sameof the rstexperiment,but thenumberofsour eshas been xed to

L

= 2

(Figs. 4 and 5) and

L

= 4

(Figs. 6 and 7), respe tively. Figures4 and 6 show the resulted

RM SE

error as a fun tion of

N

. As expe ted the statisti al performan es of the proposed estimatorimprove as

N

in reases sin e the numberof

DSO

grid points grows thusallowingthe samplingof a larger portionof the entire

DoA

parameter spa e. Asymptoti ally, the estimation a ura y of the ompared methods are quite similar to one another, but the

M

− DSO − ESP RIT

onrms its ee tiveness when operatingin the threshold region when

N

is small.

On the other hand, unlike

ESP RIT

and

ROOT

− MUSIC

, the omputational burden required at ea h iteration by the

DSO

-based te hniques depends on

N

. As a matter of fa t, su h approa hes usually ompute a number of

DoA

estimates equal to the number of ombinations between the available temporal snapshots. Therefore, in reasing the number of snapshots

N

involves a longer response time that ould make the advantages in terms of resolution a ura y fruitless. Fortunately, the memory me hanism of the

M

−DSO−ESP RIT

positivelya tsallowinganevident(seeFig. 7-

L

= 4

)improvement overthe

DSO

−ESP RIT

inthe riti al(froma omputationalpointofview)region(i.e.,

N

large).

The thirdexperimentdealswith as enario hara terizedby abetter signal-to-noiseratio (

SN R

= 20 dB

)inordertostudythebehaviorofthememory-enhan ed

DSO

−ESP RIT

in a region outside (or only partially overlapped, when

N

is small) the threshold region. As expe ted, the

M

− DSO − ESP RIT

appears to satisfa tory perform for a limited number of sour es (

L

≤ 3

- Fig. 8). As a matter of fa t, by keeping

L

= 2

and varying

N

(Fig. 9), it asymptoti ally guarantees similar results to those of the other methods, while the

M

− DSO − ESP RIT

signi antly over omes the standard

DSO

− ESP RIT

implementation forsmallvalues of

N

(

RM SE

DSO

₋

ESP RI T

RM SE

M

₋

DSO

₋

ESP RI T

k

L=N =2

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howthesour eseparationae tsthe dire tionndinga ura yoftheproposedapproa h. Figure10showstheroot-mean-squarederrorvaluesversus

△θ

l

when

SN R

= 2 dB

,

L

= 2

and

N

= L

. Asit anbeobserved,the

M

−DSO−ESP RIT

performsquitewelland lose tothe

CRB

when

△θ

l

>

8 o

. Moreover,ane ien ydegradationveriesin orresponden e with smallerseparations (

△θ

l

<

8 o

),althougha better resolution, omparedto the other te hniques, is always a hieved. Similar on lusions on the omparative assessment hold true by varying the numberof snapshots and keeping onstant the angularseparation to

△θ

l

= 2

o

(Fig. 11).

Inordertoqualitativelysummarizetheperforman eofthe

M

−DSO−ESP RIT

interms of both estimation apabilities and omputational osts when dealing with stationary onditions, Table II pi torially resumes the behavior of the approa h versus the number of sour es

L

(

1

: many,

0

: few), the number of snapshots

N

(

1

: many -

N > L

,

0

: few

-N

= L

),theangularsour eseparation

∆θ

(

1

: large,

0

: small),andthe

SN R

(

1

: lowlevel of noise,

0

: high level of noise). A ording to the indi ations drawn from the numeri al experiments,TableII pointsoutand onrms thatthe approa hisanee tivealternative to standard estimators espe iallyin the threshold region dened by a low

SN R

or/and a small numberof snapshots.

The se ondtest ase is on ernedwitha omplexenvironment hara terizedbythe time-varian eof the randomly-distributed

DoA

s of the impingingsignals. Figure12(a) pi to-rially des ribes the s enariounder test hara terized by

L

= 3

sour es having a variable dire tionofin iden e[

θ

l

(t

s

) ∈ [−40

o

_,

₅₀

o

_]

;

l

= 1, ..., L

;

s

= 1, ..., S

;

S

= 60

℄. Asfarasthe estimationpro essis on erned,

N

= 3

snapshotshavebeen onsideredanddierentnoisy onditions have been simulated (i.e.,

SN R

= 2 dB

,

SN R

= 10 dB

, and

SN R

= 20 dB

). The a hieved performan es intermsof

RM SE

areshown inFig. 12anddetailedinTab. III. Asexpe ted, whendealingwith the aseof

SN R

= 2 dB

, the

M

− DSO − ESP RIT

usuallya hievesthebesta ura yintheestimates[Fig. 12(b)℄as onrmedby omparing themean(overthe omplete temporalwindowof

S

= 60

time-steps)valuesofthe

RM SE

inTab. III(

ς

M −DSO−ESP RIT

RM SE

= 4.45

SN R=2 dB

vs.

ς

ROOT −M U SIC

RM SE

= 8.59

SN R=2 dB

,

ς

ESP RIT

RM SE

= 12.29

SN R

, and

ς

DSO−ESP RIT

RM SE

= 12.21

(14)

10 dB

, the reliability of

M

− DSO − ESP RIT

is signi antly better than those of

ESP RIT

and

DSO−ESP RIT

[Fig. 12( )-Tab. III℄(

ς

M −DSO−ESP RIT

RM SE

= 2.94

SN R=10 dB

vs.

ς

ESP RIT

RM SE

= 6.63

SN R=10 dB

,

ς

DSO−ESP RIT

RM SE

= 6.59

SN R=10 dB

)andit alsoover omes the performan e of

ROOT

− MUSIC

(

ς

ROOT −M U SIC

RM SE

= 3.59

SN R=10 dB

). Su h an event is mostly due tothe fast rea tion of the

M

− DSO − EP SRIT

to the s enariovariations in orresponden ewiththetime-steptransitions(i.e.,

t

s

= 10, 50

). Finally[

SN R

= 20 dB

- Fig. 12(d)℄,despite the lower noise level and although there isa slight improvement in the estimation a ura y with the in reasing of

SN R

(

ς

M −DSO−ESP RIT

RM SE

SN R=10 dB

= 2.94

vs.

ς

M −DSO−ESP RIT

RM SE

SN R=20 dB

= 2.82

), the

M

− DSO − ESP RIT

does not rea h

the ee tiveness of the

ROOT

− MUSIC

approa h (

ς

ROOT −M U SIC

RM SE

SN R=20 dB

= 1.96

). On the other hand, it should be noti ed that, whatever the noise level, the memory-enhan edversionofthe

DSO

− ESP RIT

alwaysover omes thestandardimplementation (

ς

DSO

−

ESP RI T

RM SE

ς

M

−

DSO

−

ESP RI T

RM SE

k

SN R=2 dB

∼

_{= 2.74}

,

ς

DSO

−

ESP RI T

RM SE

ς

M

−

DSO

−

ESP RI T

RM SE

k

SN R=10 dB

∼

_{= 2.24}

,and

ς

DSO

−

ESP RI T

RM SE

ς

M

−

DSO

−

ESP RI T

RM SE

k

SN R=20 dB

∼

=

1.19

). 5 Con lusions

Inthis paper,a

DoA

estimationmethodhasbeenproposedinordertodeal with omplex s enarios belongingto the so- alled threshold region, thus improving the ee tiveness of dire tion nding te hniques and extending their range of appli ability. Starting from a simple and omputationally-e ient data supported optimization for the solution of the maximum likelihoodestimationproblem,amemory me hanism has been introdu ed. The approa h, alled

M

− DSO − ESP RIT

,in reases the numberofthe data-supported samples,over whi h thelikelihoodfun tion ismaximized,by extendingthe dataset with the maximum likelihoodestimates of the previous time-steps.

Con erning the main features of the

M

− DSO − ESP RIT

, they an besummarized as follows

•

ee tive exploitationof the informationa quired during the estimation pro ess;

(15)

methodsare on erned, the obtained resultsdemonstrated the e ien y of the proposed approa h in terms of both estimation a ura y and omputational burden, espe ially in those s enarios where

•

the environmental onditions heavily ae t the signal-to-noise ratio;

•

the numberof olle table snapshots is limited.

On the other hand, it annot be negle ted that the

M

− DSO − ESP RIT

guarantees a eptableperforman esalsointhepresen eoflowlevelsofnoiseorwhenlongertemporal windows are onsidered, thus indi atingthe exibilityof the method.

Future works will be devoted at experimentally validating the memory-based

DSO

−

ESP RIT

te hnique aswellasimprovingitsee tiveness infa ingwith fasttime-varying s enarios.

A knowledgments

This work has been supported in Italy by the Progettazione di un Livello Fisi o 'Intel-ligente' per Reti Mobili ad Elevata Ri ongurabilità, Progetto di Ri er a di Interesse Nazionale - MIUR Proje t COFIN 2005099984.

(16)

[1℄ Alexiou,A.,andM.Haardt,Smartantennate hnologiesforfuturewirelesssystems: trends and hallenges, IEEE Commun. Mag., Vol. 42,No. 9,90-97, 2004.

[2℄ Fakoukakis, F. E., S. G.Diamantis, A. P. Orfanides, and G.A. Kyria ou,

"Develop-ment of an adaptive and a swit hed beam smart antenna system for wireless

om-muni ations,"JEMWA,Vol.20, No.3, 399-408,2006.

[3℄ Wu,W.,B.Z.Wang,andS.Sun, "Patternre ongurablemi rostrippat hantenna,"

JEMWA, Vol. 19,No. 1,107-113,2005.

[4℄ Ragheb, M. K., S.H. Elramly, and S. Mostafa, "The ee t of varying the input pa-rameters onthe performan e apabilitiesofthreedierentdoa estimationalgorithms using smart antennas," Pro . 20th NationalRadio S i. Conf., Mar h2003, 1-9.

[5℄ Varlamos, P. K., and C. N. Capsalis, "Ele troni beam steeringusing swit hed

par-asiti smart antenna arrays,"PIER, Vol. 36,101-119,2002.

[6℄ Wu, W., and Y. H. Bi, "Swit hed-beam planar fra tal antenna," JEMWA, Vol. 20,

No. 3, 409-415,2006.

[7℄ Mouhamadou, M.,P. Vaudon,and M.Rammal,"Smartantenna array patterns

syn-thesis: null steering and multi-user beamforming by phase ontrol," PIER, Vol. 60,

95-106, 2006.

[8℄ Mukhopadhyay, M., B. K. Sarkar, and A. Chakrabarty, "Augmentation of anti-jam

Gps system using smart antenna with a simple DoA estimation algorithm," PIER,

Vol.67, 231-249,2007.

[9℄ Babayigit, B., A. Akdagli, and K. Guney, "A lonal sele tion algorithm for null

synthesizing of linear antenna arrays by amplitude ontrol," JEMWA, Vol. 20, No.

8, 1007-1020, 2006.

[10℄ Gavan,J.,andJ.S.Ishay,"Hypothesisofnaturalradartra kingand ommuni ation

(17)

radar systems," JEMWA, Vol.11, No.2, 215-224,1997.

[12℄ Paine,A.S., FastMUSICforlarge2-Delementdigitisedphasedarrayradar, Pro . Int. Radar Conf. 2003,September2003, 200-205.

[13℄ Owsley, N. L., Sonar array pro essing, in Array Signal Pro essing, S.Haykin, Ed. Englewood Clis,NJ: Prenti e-Hall,1984, 115-193.

[14℄ Nishiura, T., and S. Nakamura, Talker lo alization based on the ombination of DOA estimation and statisti alsound sour e identi ation with mi rophone array, IEEE Workshop Statisti al Signal Pro essing 2003, O tober2003, 597-600.

[15℄ S hweppe, F. C., Sensor array data pro essing for multiple signal sour es, IEEE Trans. Inform. Theory, Vol.14,294-305, 1968.

[16℄ Stoi a, P., R. L. Moses, B. Friedlander, and T. Söderström, Maximum likelihood estimation of the parameters of multiple sinusoids fromnoisymeasurements, IEEE Trans. A oust., Spee h,Signal Pro essing, Vol.37,378-392, 1989.

[17℄ Ziskind, I., and M. Wax, "Maximum likelihood lo alization of multiple sour es by alternating proje tion," IEEE Trans. A oust., Spee h., Signal Pro ess., Vol. ASSP-36, 1553-1560,1988.

[18℄ Stoi a, P., and A. B. Gershman, "Maximum-likelihood

DoA

estimation by data-supported grid sear h," IEEE Signal Pro essingLett.,Vol.6, No.10, 273-275,1999.

[19℄ El Zooghby, A. H., C. G. Christodoulou, and M. Georgiopoulos, "Performan e of radial-basis fun tion network for dire tion arrival estimation with antenna arrays, IEEE Trans.Antennas Propagat., Vol.45, 1611-1617,1997.

[20℄ El Zooghby,A. H., C. G.Christodoulou, and M.Georgiopoulos, "Aneural network-basedsmartantennaformultiplesour etra king, IEEETrans.AntennasPropagat., Vol.48, 768-776,2000.

(18)

estimation based ona support ve torregression, IEEE Trans. Antennas Propagat., Vol.53, No.7, 2161-2168, 2005.

[22℄ Donelli, M., R. Azaro, L. Lizzi, F. Viani, and A. Massa, A SVM-based multi-resolution pro edure for the estimation of the DOAs of interfering signals in a om-muni ationsystem, Pro .EuropeanConferen eAntennasPropagat.(EuCAP),Ni e, Fran e, 363898,November2006.

[23℄ Pastorino,M., and A. Randazzo, "The SVM-based smart antenna for estimation of the dire tions of arrivalof ele tromagneti waves, IEEE Trans. Intrum.Meas., Vol. 55, No.6, 1918-1925, 2006.

[24℄ S hmidt, R. O., "Multipleemitter lo ation and signal parameter estimation,"IEEE Trans. Antennas Propagat., Vol. 34,No. 3,276-280,1986.

[25℄ Godara, L. C., "Appli ation of antenna arrays to mobile ommuni ations, part II: beam-forming and dire tion-of-arrival onsiderations," IEEE Pro ., Vol. 85, No. 8, 1195-1245, 1997.

[26℄ Al-Ardi,E.M.,R.M.Shubair,andM.E.Al-Mualla,"Investigationofhighresolution DoAestimationalgorithmsforoptimalperforman eofsmartantennasystems,"Pro . 4th Int. Conf. 3G Mobile Comm.Te hnol., June 2003, 460-464.

[27℄ Barabell, A., "Improving the resolution of eigenstru tured based dire tion nding algorithms," IEEE Pro . ICASSP'83,Vol.8, April1983, 336-339.

[28℄ Rao, B. D., and K. V. S. Hari, "Performan e analysis of root-musi ," IEEE Trans. A oust., Spee h, Signal Pro ess.,Vol.37,No. 12, 1939-1949, 1989.

[29℄ Roy, R.,and T. Kailath, "ESPRIT - Estimationof signal parameters via rotational invarian ete hniques,"IEEE Trans.A oust., Spee h,SignalPro ess.,Vol.37,No.7, 984-995, 1989.

[30℄ Hawkins, D. M., D. Bradu, and G.V. Kass, Lo ation ofseveral outliersin multiple regression data usingelementalsets, The hnometri s, Vol. 26,197-208,1984.

(19)

hoodDoA estimation,"IEEE Pro . SAMSP'00, Mar h2000, 337-341.

[32℄ Stoi a,P.,andK.C.Sharman,"Maximumlikelihoodmethodsfordire tion-of-arrival estimation,"IEEE Trans.A oust., Spee h, SignalPro ess.,Vol.ASSP-38,1132-1143, 1990.

[33℄ Stoi a, P.,and A. Nehorai, Musi , maximum likelihood,and Cramer-Rao bound, IEEE Trans.A oust., Spee h, SignalPro ess., Vol. 37,No. 5,720-741,1989.

(20)

•

Figure 1. Problem Geometry.

•

Figure 2. Stationary S enario (

SN R

= 2 dB

,

N

= L

,

△θ

l

= 10

o

,

θ

1 = 10

o

)

-RM SE

values versus numberof sour es

L

.

• SN R

= 2 dB

,

N

= L

,

△θ

l

= 10

o

,

θ

1 = 10

o

) -Computational ost

Ω

versus number of sour es

L

.

• SN R

= 2 dB

,

L

= 2

,

△θ

l

= 10

o

,

θ

1 = 10

o

)

-RM SE

values versus numberof snapshots

N

.

• SN R

= 2 dB

,

L

= 2

,

△θ

l

= 10

o

,

θ

1 = 10

o

Ω

versus number of snapshots

N

.

• SN R

= 2 dB

,

L

= 4

,

△θ

l

= 10

o

,

θ

1 = 10

o

)

-RM SE

N

.

• SN R

= 2 dB

,

L

= 4

,

△θ

l

= 10

o

,

θ

1 = 10

o

Ω

versus number of snapshots

N

.

• SN R

= 20 dB

,

N

= L

,

△θ

l

= 10

o

,

θ

1 = 10

o

)

-RM SE

values versus numberof sour es

L

.

• SN R

= 20 dB

,

L

= 2

,

△θ

l

= 10

o

,

θ

1 = 10

o

)

-RM SE

N

.

• SN R

= 2 dB

,

L

= 2

,

N

= L

,

θ

1 = 10

o

) -

RM SE

values versus sour e separation distan e

∆θ

.

• SN R

= 2 dB

,

L

= 2

,

△θ

l

= 2

o

,

θ

1 = 10

o

)

-RM SE

N

.

•

Figure 12. Time-Varying S enario (

L

= 3

,

N

= L

,

θ

l

(t

s

) ∈ [−40

o

_,

₅₀

o

_]

,

S

= 60

). Referen e onguration (a).

RM SE

atea h

t

s

,

s

= 1, ..., S

when(b)

SN R

= 2 dB

, ( )

SN R

= 10 dB

,and (d)

SN R

= 20 dB

.

(21)

•

Table I. Pro edure fordening the matri es

W

(k)

_{∈ C}

M ×L

,

k

= 1, ..., K

.

•

Table II. Stationary S enario - Summary of the

M

− DSO − ESP RIT

perfor-man es.

•

Table III. Time-Varying S enario (

L

= 3

,

N

= L

,

θ

l

(t

s

) ∈ [−40

o

_,

₅₀

o

_]

,

S

= 60

). Time-averaged values of the

RM SE

.

(22)

Estimation

DOA

1 y (t)

2 y (t)

3 y (t)

M

y (t)

2 θ

x (t, )

L

θ

x (t, )

1 θ

x (t, )

Sources

x

2 x

₁

x

_L

x

₁

x

₂

x

_L

s

t

_s+1

x

L

x

1 x

2 t

_s−1

Sensors

θ

t

(23)

10

3

10

2

10

1

10

0

10 -1

1

2

3

4

5

6 RMSE [deg]

Number of Sources, L

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(24)

10

1

10

0

10 -1

10 -2

1

2

3

4

5

6 Ω

[x 10

6 ]

Number of Sources, L

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

(25)

10

2

10

1

10

0

10 -1

10 -2

2

4

6

8

10

12

14

16

18

20 RMSE [deg]

Number of Snapshots, N

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(26)

10

2

10

1

10

0

10 -1

10 -2

2

4

6

8

10

12

14

16

18

20 Ω

[x 10

6 ]

Number of Snapshots, N

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

(27)

10

2

10

1

10

0

10 -1

10 -2

4

6

8

10

12

14

16

18

20 RMSE [deg]

Number of Snapshots, N

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(28)

10

3

10

2

10

1

10

0

10 -1

4

6

8

10

12

14

16

18

20 Ω

[x 10

6 ]

Number of Snapshots, N

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

(29)

10

3

10

2

10

1

10

0

10 -1

10 -2

10 -3

1

2

3

4

5

6 RMSE [deg]

Number of Sources, L

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(30)

10

1

10

0

10 -1

10 -2

10 -3

2

4

6

8

10

12

14

16

18

20 RMSE [deg]

Number of Snapshots, N

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(31)

10

2

10

1

10

0

10 -1

2

4

6

8

10

12

14

16

18

20 RMSE [deg]

Source separation,

∆θ

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(32)

10

2

10

1

10

0

10 -1

2

4

6

8

10

12

14

16

18

20 RMSE [deg]

Number of Snapshots, N

M-DSO-ESPRIT

DSO-ESPRIT

ROOT-MUSIC

ESPRIT

CRB

(33)

-40

-30

-20

-10

0

10

20

30

40

50

60

50

40

30

20

10

0 θ

[deg]

Time Instant, t

0

10

20

30

40

50

60

50

40

30

20

10

0 RMSE

Time Instant, t

DSO-ESPRIT

ESPRIT

MUSIC

M-DSO-ESPRIT

(a) (b)

0

5

10

15

20

25

30

35

40

45

60

50

40

30

20

10

0 RMSE

Time Instant, t

DSO-ESPRIT

ESPRIT

MUSIC

M-DSO-ESPRIT

0

5

10

15

20

25

30

35

40

45

60

50

40

30

20

10

0 RMSE

Time Instant, t

DSO-ESPRIT

ESPRIT

MUSIC

M-DSO-ESPRIT

12 -L. Lizzi et al. , The M-DSO-ESPRIT metho d for maxim um lik eliho o d ... 31