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
The
M
− DSO − ESP RIT
Method for Maximum Likeli-hoodDoA
EstimationL. 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
The
M
− DSO − ESP RIT
Method for Maximum Likeli-hoodDoA
EstimationL. 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
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 theDoA
estimates. Although it asymptoti ally onverges to theCRB
[25℄ for an in reasing number of snapshots, thefar as uniform linear arrays (
U LA
s) are on erned, the so- alledROOT
− MUSIC
[27℄ version an be protably used by solving a polynomial rooting problem,thus improving the omputationalperforman es of theM 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 theDoA
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 theESP 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 fromM U SIC
orROOT
− 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 withESP 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 theLF
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 toM 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 theDSO
− ESP RIT
[31℄, the proposed method onsiders anESP RIT
-basedestimator for omputing theDSG
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.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
ofM
equally-spa edsensorsandL
(L < M
)un orrelated narrow-band signals impingingat ea h time instantt
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 withy
(t
s
) = {y
m
(t
s
) ; m = 1, ..., M}
T
the snapshot (i.e., the olle tionofdata samplesat
t
s
) olle tedbytheM
sensorsatea htimeinstant(1)
. Under theassumptionthatthenumberofavailablesnapshotsisequalto
N
(N
≥ L
),there eiver outputY
(t
s
)
an be expressed, a ording tothe matrix notation[17℄[32℄, asfollowsY
(t
s
) = A [Θ (t
s
)] X (t
s
) + E (t
s
) .
(1)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.(1)
Under these hypotheses, the maximum likelihood lo alizationof the
L
sour es att
s
[i.e., the estimationofΘ (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 byf
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
)]
andR
(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 requirestheevaluationoftheLF
in orresponden e withea hpossible ombinationoftheDoA
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, theM
− DSO − ESP RIT
method resorts to the evaluation of theLF
in alimited set of ombinationsofDoA
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-stept
s
,s
≥ 1
:The
M
−DSO−ESP RIT
operatesamemoryenhan ementofthe olle teddata. Towards this end, anew memory-enhan ed output matrixD
(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 assumingD
(t
1
) = Y (t
1
)
, that isd
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-SamplingThe 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 dimensionM
× L
, it is possible todeneK
[beingK
=
B!
L!(B−L)!
℄ matri esW
(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 ofD
(t
s
)
orresponding to thel
-th element ofW
(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 numbersη
(k)
s
are greater than a xed stability threshold (i.e., the ill- onditioned matri es) are omitted.As in[26℄[31℄, an
ESP RIT
-likealgorithmis used forgenerating the data-supportedgrid points. Letus onsider the matrixΦ
(k)
∈ C
L×L
given byΦ
(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) whereV
(k)
andU
(k)
are two matri es obtained from
W
(k)
by eliminating the rst and lastrow,respe tively. Then,the
ESP RIT
-likeestimates(i.e.,theK
data-supportedgrid points)turn out to beb
Θ
(k)
(t
s
) =
n
ˆ
θ
(k)
l
(t
s
) ; l = 1, ..., L
o
k
= 1, ..., K
(7) whereθ
ˆ
(k)
l
= arcsin
n
λ
2πd
arg
µ
(k)
l
o
andn
µ
(k)
l
; l = 1, ..., L
o
aretheeigenvaluesofΦ
(k)
[29℄.
3.4
M L DoA
EstimationFinally,the
M L
estimates of theDoA
sof theL
signals are omputedbymaximizingtheLF
overtheK
-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 AssessmentThe 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 asROOT
−
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 parametersθ
l
(i.e., the so- alledCramer-Rao bound (CRB
) [33℄) are reported, aswell.index ofe ien y. Moreover, be auseofthestatisti alnatureofthe s enariosundertest, its value averaged over
Q
= 100
independent realizations of ea h simulation has been omputedRM SE
=
1
Q
Q
X
q=1
1
S
S
X
s=1
v
u
u
t 1
L
L
X
l=1
θ
l
(q)
(t
s
) − ˆ
θ
(q)
l
(t
s
)
2
(9)where the index
q
denotes theq
-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 tionalsensorsλ
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 theDoA
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
) andL
sour es oming fromrandom angulardire tions equally-spa ed by△θ
l
= 10
o
,l
= 1, ..., L − 1
. Underthe assumptionthatN
= 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 esL
. Con erning the estimation a ura y, even though the resolution error is an in reasing quantity, theM
− DSO − ESP RIT
outperforms the other dire tionnding methods and it results loser to theCRB
whenL
≤ 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-instantt
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 matrixD
(t
s
)
onL
, the omputational burden required byM
− DSO − ESP RIT
grows withL
, 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 numberof sour es, the arising
Ω
value is greater than that of theDSO
− ESP RIT
, but of the same orderinmagnitudeofthe standardESP RIT
. Furthermore,theROOT
− 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 toL
= 2
(Figs. 4 and 5) andL
= 4
(Figs. 6 and 7), respe tively. Figures4 and 6 show the resultedRM SE
error as a fun tion ofN
. As expe ted the statisti al performan es of the proposed estimatorimprove asN
in reases sin e the numberofDSO
grid points grows thusallowingthe samplingof a larger portionof the entireDoA
parameter spa e. Asymptoti ally, the estimation a ura y of the ompared methods are quite similar to one another, but theM
− DSO − ESP RIT
onrms its ee tiveness when operatingin the threshold region whenN
is small.On the other hand, unlike
ESP RIT
andROOT
− MUSIC
, the omputational burden required at ea h iteration by theDSO
-based te hniques depends onN
. As a matter of fa t, su h approa hes usually ompute a number ofDoA
estimates equal to the number of ombinations between the available temporal snapshots. Therefore, in reasing the number of snapshotsN
involves a longer response time that ould make the advantages in terms of resolution a ura y fruitless. Fortunately, the memory me hanism of theM
−DSO−ESP RIT
positivelya tsallowinganevident(seeFig. 7-L
= 4
)improvement overtheDSO
−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 edDSO
−ESP RIT
in a region outside (or only partially overlapped, whenN
is small) the threshold region. As expe ted, theM
− 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 keepingL
= 2
and varyingN
(Fig. 9), it asymptoti ally guarantees similar results to those of the other methods, while theM
− DSO − ESP RIT
signi antly over omes the standardDSO
− ESP RIT
implementation forsmallvalues ofN
(RM SE
DSO
−
ESP RI T
RM SE
M
−
DSO
−
ESP RI T
k
L=N =2
howthesour eseparationae tsthe dire tionndinga ura yoftheproposedapproa h. Figure10showstheroot-mean-squarederrorvaluesversus
△θ
l
whenSN R
= 2 dB
,L
= 2
andN
= L
. Asit anbeobserved,theM
−DSO−ESP RIT
performsquitewelland lose totheCRB
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 esL
(1
: many,0
: few), the number of snapshotsN
(1
: many -N > L
,0
: few-N
= L
),theangularsour eseparation∆θ
(1
: large,0
: small),andtheSN 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 lowSN 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 byL
= 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
, andSN R
= 20 dB
). The a hieved performan es intermsofRM SE
areshown inFig. 12anddetailedinTab. III. Asexpe ted, whendealingwith the aseofSN R
= 2 dB
, theM
− DSO − ESP RIT
usuallya hievesthebesta ura yintheestimates[Fig. 12(b)℄as onrmedby omparing themean(overthe omplete temporalwindowofS
= 60
time-steps)valuesoftheRM 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
, andς
DSO−ESP RIT
RM SE
= 12.21
10 dB
, the reliability ofM
− DSO − ESP RIT
is signi antly better than those ofESP RIT
andDSO−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 ofROOT
− 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 theM
− 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 ofSN 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
), theM
− DSO − ESP RIT
does not rea hthe 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 edversionoftheDSO
− 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
,andς
DSO
−
ESP RI T
RM SE
ς
M
−
DSO
−
ESP RI T
RM SE
k
SN R=20 dB
∼
=
1.19
). 5 Con lusionsInthis 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, alledM
− 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;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.
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•
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 esL
.•
Figure 3. Stationary S enario (SN R
= 2 dB
,N
= L
,△θ
l
= 10
o
,
θ
1
= 10
o
) -Computational ost
Ω
versus number of sour esL
.•
Figure 4. Stationary S enario (SN R
= 2 dB
,L
= 2
,△θ
l
= 10
o
,
θ
1
= 10
o
)
-RM SE
values versus numberof snapshotsN
.•
Figure 5. Stationary S enario (SN R
= 2 dB
,L
= 2
,△θ
l
= 10
o
,
θ
1
= 10
o
) -Computational ost
Ω
versus number of snapshotsN
.•
Figure 6. Stationary S enario (SN R
= 2 dB
,L
= 4
,△θ
l
= 10
o
,
θ
1
= 10
o
)
-RM SE
values versus numberof snapshotsN
.•
Figure 7. Stationary S enario (SN R
= 2 dB
,L
= 4
,△θ
l
= 10
o
,
θ
1
= 10
o
) -Computational ost
Ω
versus number of snapshotsN
.•
Figure 8. Stationary S enario (SN R
= 20 dB
,N
= L
,△θ
l
= 10
o
,
θ
1
= 10
o
)
-RM SE
values versus numberof sour esL
.•
Figure 9. Stationary S enario (SN R
= 20 dB
,L
= 2
,△θ
l
= 10
o
,
θ
1
= 10
o
)
-RM SE
values versus numberof snapshotsN
.•
Figure 10. Stationary S enario (SN R
= 2 dB
,L
= 2
,N
= L
,θ
1
= 10
o
) -
RM SE
values versus sour e separation distan e∆θ
.•
Figure 11. Stationary S enario (SN R
= 2 dB
,L
= 2
,△θ
l
= 2
o
,
θ
1
= 10
o
)
-RM SE
values versus numberof snapshotsN
.•
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 ht
s
,s
= 1, ..., S
when(b)SN R
= 2 dB
, ( )SN R
= 10 dB
,and (d)SN R
= 20 dB
.•
Table I. Pro edure fordening the matri esW
(k)
∈ C
M ×L
,
k
= 1, ..., K
.•
Table II. Stationary S enario - Summary of theM
− 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 theRM SE
.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
t
s+1
x
L
x
1
x
2
t
s−1
Sensors
θ
t
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
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
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
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
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
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
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
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
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
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
-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
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 ... 31k
= 1
c
(k)
0
= 0
fori
= 1, ..., L
c
(k)
i
= c
(k)
i−1
+ 1
endgo
=
TRUE fork
= 1, ..., K
forl
= 1, ..., L
w
(k)
l
= s
c
(k)
l
endj
= L
while