passive geolocalization of radio transmitters: algorithm and performance in narrowband context

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Signal Processing

Signal Processing 92 (2012) 841–852

0165-16

doi:10.1

n Corr

avenue

E-m

Jonatha

ferreol@

(A. Fer

larzaba# E

journal homepage: www.elsevier.com/locate/sigpro

Passive geolocalization of radio transmitters: Algorithm andperformance in narrowband context

Jonathan Bosse a,b,#,n, Anne Ferreol a,b, Cecile Germond a, Pascal Larzabal b

a THALES Communications, 160 Boulevard de Valmy, 92704 Colombes, Franceb SATIE, ENS Cachan, CNRS, Universud 61 avenue du president Wilson, 94230 Cachan, France

a r t i c l e i n f o

Article history:

Received 27 September 2010

Received in revised form

12 July 2011

Accepted 9 September 2011Available online 24 September 2011

Keywords:

Emitter geolocalization

Array processing

AOA estimation

Cramer–Rao bounds

84/$ - see front matter & 2011 Elsevier B.V. A

016/j.sigpro.2011.09.008

esponding author at: SATIE, ENS Cachan, C

du president Wilson, 94230 Cachan, France.

ail addresses: [email protected],

[email protected] (J. Bosse),

satie.ens-cachan.fr, [email protected]

reol), [email protected] (C

[email protected] (P. Larzabal).

URASIP member.

a b s t r a c t

Passive localization commonly consists of a two steps strategy. In the first step,

intermediate parameters, often called measurements (such as angles of arrival (AOA),

times of arrival (TOA), etc.) are measured on several base stations equipped with sensor

arrays. In a second step, the transmitted intermediate parameters are then used to

estimate the position at a central processing unit. Such approach is suboptimal. To

overcome this limitation, one step algorithms were recently proposed. They exploit

simultaneously all received signals of all base stations seen as a global array in order to

provide the source positions directly. In this paper, we propose an original one step

algorithm called global MUSIC approach (GMA). The GMA offers better performance in

narrowband context. Moreover it does not require the use of filter banks in the

wideband signal context, contrary to the recently proposed direct position determina-

tion (DPD). GMA appears to outperform the DPD in wideband context. We also

investigate in this paper by means of a Cramer–Rao bound analysis the potential gain

achievable by a one step approach compared to a conventional two steps approach.

Finally, numerical results illustrate the improvement of the proposed method compared

to existing techniques in terms of location error and robustness to the time-bandwidth

product.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

In radio communication the geolocalization problem ofradio transmitters has always received great attention. Bymeans of multiple sensors base stations (seen as sub-arrays), forming what we call here a global array, theproblem lies in estimating the position of multiple radio

ll rights reserved.

NRS, Universud 61

roup.com

. Germond),

emitters in far field context. Traditionally [1,2], thelocalization algorithm consists of two steps illustrated inFig. 1.

In the first step, parameters (often called measure-ments) are independently estimated at each base station.Parameter estimation has always been an intensiveresearch theme in the array processing community. Dur-ing the last two decades it leads to a well-suited theore-tical framework (see [3,4]). Among the available studiedproblems the angle of arrival (AOA) estimation has beenwidely discussed [5]. Using the narrowband assumptionon a multi-sensor base station the AOA information iscontained in the phase difference between the sensors. Itappears to be particularly convenient for passive localiza-tion since it does not require any prior information on thetransmitted signals.

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Fig. 1. Conventional two steps position determination scheme.

Fig. 2. One step position determination scheme.

J. Bosse et al. / Signal Processing 92 (2012) 841–852842

In the second step, the geolocalization is computedbased on the set of measurements available from step 1[6,7] and transmitted to a central processing unit (CPU).Previous studies aimed to quantify and reduce the biasand the variance of this second step [8,9].

However, such two steps approach suffers from limita-tions [1]. First, solving the problem by means of a multiplestep strategy is often suboptimal. As the first step does nottake into account the fact that the received signals at thestations come from the same emitters this step is sub-optimal. Secondly, in passive multi-emitter context, theproblem is ambiguous since one has to identify among allavailable measurements, which subset of measurementscharacterizes each source. This problem is sometimescalled ‘‘data association’’ [10]. Lastly, the performanceand the number of resolvable emitters at each station areknown to be locally limited by the number of sensors ofthe considered station. These are the reasons why anapproach based on a one step strategy, gathering thereceived signals of all available stations, that directlyprovides the emitter location in LOS context, as illustratedin Fig. 2, appears of great interest. Such an approachassumes that each station is able to transfer all the signalsto a central processing unit.

Fewer works provide one step procedures that directlyestimate the position of emitters using the signal col-lected by all multi-sensor stations. Only [11–14] investi-gate recently such one step procedures. Studies [14,15]deal with the global navigation satellite system (GNSS),another [13] focuses on moving targets proposing alocation algorithm based on frequency Doppler shiftsexploitation. To the best of our knowledge, only Weissand Amar [11,12] treat the passive localization problem ina one step approach called direct position determination(DPD) exploiting the information of the steering vector(angular response) of each sensor. Based on filter banks,they proposed a unique criterion gathering all signals ofall stations firstly in the presence of a single emitter [11]and then proposed a MUSIC [16] approach of the problemin order to treat the multiple emitters case [12]. Theirapproach relies on an incoherent sum of criteria obtainedfor each frequency of the filter bank. It suggests that thisstrategy is not optimal. They proved [17] the theoretical

gain of such techniques in terms of the number ofresolvable emitters. They also provide a performanceanalysis [12] under known waveforms or random Gaus-sian signal assumptions. To the best of our knowledge, theperformance study for deterministic signals was neverprovided for direct position estimation. We fill the lack inthis paper.

Solving the problem by means of a one step strategyleads to estimate the emitters location but it also impliesto estimate the complex gain of the signal received in allstations, seen as a nuisance parameter [11], in order toaccount for the possible path attenuation, which is dif-ferent for each station. It underlines the need of investi-gating the theoretical gain of such a one step strategycompared to a conventional two steps strategy.

In this paper, we consider a global array composed ofseveral multi-sensor base stations, each one able totransfer the data to a CPU, and we focus on passive (ornon-cooperative) localization problem of multiple trans-mitted signals in LOS context, with no prior informationon the signals. Based on preliminary work [18], wepropose a one step location algorithm called global MUSICapproach (GMA) that does not require the use of filterbanks in wideband signal context. We explore moresoundly the performance improvement in terms of loca-tion error of such a one step approach using deterministicCramer–Rao bounds (CRBs) comparisons. In Section 2, weformulate the problem and study the case of narrowbandsignals for the global array. Then in Section 3 the MUSICapproach called GMA is derived. Section 3 also providesthe gradient and the Hessian required for a practicalimplementation based on a Newton algorithm. The Cra-mer–Rao bound expression of the conventional two stepsand the one step approach for deterministic signals aregiven in Section 4. Extension of the GMA to the widebandsignal case is discussed in Section 5. Simulations illustratein Section 6 the gain of the proposed approach comparedto existing techniques.

2. Signal model

We focus on the problem of locating multiple LOSemitters on L separated base stations each composed of Nl

(1r lrL) sensors as illustrated on Fig. 3. Let us denoteN¼

PlNl the total number of sensors. Let xlðtÞ be the

Fig. 3. Propagation from an emitter located at pm to multiple multi-

sensor base station located in Sl.

J. Bosse et al. / Signal Processing 92 (2012) 841–852 843

observation vector of the l-th station and let us considerM emitters, whose unknown signals are sm(t) (1rmrMÞ.Assuming classically that the impinging signals are nar-rowband for the base station, we have then:

xlðtÞ ¼XM

m ¼ 1

rl,malðylðpmÞÞe�j2pf 0tlðpmÞsmðt�tlðpmÞÞþnlðtÞ,

ð1Þ

where f0 is the carrier frequency, pm denotes the D� 1position of the m-th emitter (if D¼2, pm ¼ ½xm,ym�

T and ifD¼3, pm ¼ ½xm,ym,zm�

T , where ð�ÞT denotes the transposeoperator). rl,m is an unknown complex parameter stand-ing for the channel attenuation. We denote alðylðpÞÞ thesteering vector of the array at station l depending on theAOA yl, seen here as a function of the position of thesource p. Since we assume that the wavefront is planar ateach station the only parameter that can be estimated at aparticular station is the AOA. This is why a multiple basestation system is required in order to localize an emitter.For notational convenience we will write in the sequelalðpÞ9alðylðpÞÞ usually obtained by a calibration process inpractice and whose theoretical expression on n-th sensoris

ðalðpÞÞn ¼ e�j2pf 0dl,nðpÞ, ð2Þ

where dl,nðpÞ is the time delay between the n-th sensor ofthe base station l and the center of phases of the station.We define tlðpmÞ as the relative time delay of the m-thsignal on the l-th station, the origin being arbitrarilychosen on the first station. So, we define

tlðpmÞ9Jpm�pðlÞJ�Jpm�pð1ÞJ

c, ð3Þ

where pðlÞ ð1r lrLÞ is the position vector of the l-thstation. The additive noise vector nlðtÞ is white Gaussianwith a covariance matrix equal to s2INl

, where IN is theN�N identity matrix. We also assume that all nlðtÞ

ð1r lrL,1rtrTÞ are statistically independent. All sig-nals are supposed temporally independent.

Note that there is an ambiguity in (1) between thephase of rl,m and e�j2pf 0tlðpmÞ, so that both cannot beestimated separately. Since in practice the steering vec-tors of the stations are not calibrated with a commonphase reference and since the impinging signals at eachstation are not supposed to have the same amplitude andphase, we will include the term e�j2pf 0tlðpmÞ into rl,m.

We build the following stacked observation vector ofthe global array:

xðtÞ ¼

x1ðtÞ

^

xLðtÞ

264

375: ð4Þ

We assume here that the impinging signals (complexenvelops) are narrowband for the global array. That is tosay the time-bandwidth product (denoted tm � Bm) isclose to zero. Here tm stands for the time delay ofpropagation across the sensor network (i.e. time delay ofarrival between the two most separated sensors) and theemitter bandwidth Bm. We have

Bm �maxi,j

9tiðpmÞ�tjðpmÞ951: ð5Þ

We can write the signals sm(t) as smðtÞ ¼ amðtÞej2pf mt ,where amðtÞ is the complex envelop, f0 the carrier fre-quency and fm an eventual carrier frequency residual.Assuming (5) amðt�tlðpmÞÞ � amðtÞ so that we can write

smðt�tlðpmÞÞ � smðtÞe�j2pf mtlðpmÞ: ð6Þ

The term e�j2pf mtlðpmÞ can be also included into rl,m. Let usremark that since the base stations are not assumed to lieclose to each other, this assumption is more restrictivethan the classical narrowband assumption performed ateach station that only requires

Bm �maxi,j

9dl,iðpmÞ�dl,jðpmÞ951: ð7Þ

The global observation (4) writes

xðtÞ ¼XM

m ¼ 1

uðpm,wm,/mÞsmðtÞþnðtÞ, ð8Þ

where we introduce the following stacked steeringvector:

uðpm,wm,/mÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

l

Nl

Nc2

l,m

rc1,mejf1,m a1ðpmÞ

c2,mejf2,m a2ðpmÞ

^

cL,mejfL,m aLðpmÞ

266664

377775, ð9Þ

where

wm ¼ ½c1,m � � � cL,m�T ð10Þ

and

/m ¼ ½f1,m � � � fL,m�T ð11Þ

are unknown real deterministic parameter vectors suchthat

rl,m ¼cl,mejfl,mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

l

Nl

Nc2

l,m

r : ð12Þ

The steering vector uðpm,wm,/mÞ has now a constantnorm equal to the number of sensors N as encounteredin classical AOA estimation problem. We choose withoutloss of generality c1,m ¼ 1 and f1,m ¼ 0 (relative nuisanceamplitudes and phases origin). We also define

nðtÞ ¼ ½nT1ðtÞ � � � nT

L ðtÞ�T : ð13Þ


The covariance matrix of the additive noise nðtÞ is s2IN ,assuming that the noise vectors are spatially white. Wedefine the following signal to noise ratio for the m-thsource:

SNRm ¼ 10 log10

E½9smðtÞ92�

s2

!: ð14Þ

We can write

xðtÞ ¼XM

m ¼ 1

uðpm,wm,/mÞsmðtÞþnðtÞ, ð15Þ

or, using a more compact expression:

xðtÞ ¼ AðP,W,UÞsðtÞþnðtÞ, ð16Þ

where

P¼ ½p1 � � � pM �, ð17Þ

W¼ ½w1 � � � wM�, ð18Þ

U¼ ½/1 � � � /M�, ð19Þ

AðP,W,UÞ ¼ ½uðp1,w1,/1Þ � � � uðpM ,wM ,/MÞ�, ð20Þ

sðtÞ ¼ ½s1ðtÞ � � � sMðtÞ�T : ð21Þ

3. A global MUSIC approach (GMA)

3.1. Algorithm

In this section we propose a one step approach calledglobal MUSIC approach (GMA), in contrast to a conven-tional two steps approach where all measurements areindependently estimated on each station. As it can beseen from Eq. (15) the number of parameters to beestimated quickly grows with the number of stations. Inorder to avoid optimizations in a possible huge dimen-sion, a MUSIC [16] approach is preferred compared to amaximum likelihood. The MUSIC algorithm exploits thestructure of the following covariance matrix:

R¼ E½R�, ð22Þ

where E½�� stands for the mathematical expectation and

R ¼1

T

XT

t ¼ 1

xðtÞxHðtÞ: ð23Þ

The signal model (15) is well suited to the use of a MUSICalgorithm applied on the estimated covariance matrix R ,knowing the steering vector uðp,w,/Þ.

According to (16) under the deterministic signalassumption:

R¼AðP,W,UÞRsAHðP,W,UÞþs2I, ð24Þ

Rs ¼1

T

XT

t ¼ 1

sðtÞsHðtÞ: ð25Þ

Following the MUSIC approach, assuming that the size ofthe signal subspace is K¼M in this context, the para-meters fP,W,Ug are estimated by finding the M zeros of

the following objective function:

Cðp,w,/Þ ¼uHðp,w,/ÞPnuðp,w,/Þ

uHðp,w,/Þuðp,w,/Þ, ð26Þ

where Pn ¼ I�UsUH

s is the estimated noise projector, andthe N�K matrix Us is the signal subspace, consisting ofthe K eigenvectors of the matrix corresponding to the K

largest eigenvalues of R . The optimization of (26) wouldlead to a 2ðL�1ÞþD dimensional search, whereas w and /

are nuisance parameters. Note that

uðpm,wm,/mÞ ¼UðpmÞqðwm,/mÞ, ð27Þ

where

UðpmÞ ¼ diagða1ðpmÞ, . . . ,aLðpmÞÞ, ð28Þ

qðwm,/mÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

l

Nl

Nc2

l,m

r1

c2,mejf2,m

^

cL,mejfL,m

266664

377775, ð29Þ

and diagðvÞ is a diagonal matrix built with the entries inthe vector v. Since finding the minimum of the quadraticform ratio (26) is equivalent to searching the minimaleigenvalue of the matrix ½UH

ðpÞUðpÞ��1UHðpÞPnUðpÞ [19],

criterion (26) can be simplified by computing the follow-ing criterion:

C2ðpÞ ¼ lminf½UHðpÞUðpÞ��1UH

ðpÞPnUðpÞg, ð30Þ

where lminfAg is the minimal eigenvalue of the matrix A.For the sake of computational cost and to avoid the fullnumerical optimization of (30) the following equivalentcriterion is preferred [20]:

CrðpÞ ¼9UHðpÞPnUðpÞ9

9UHðpÞUðpÞ9

, ð31Þ

where 9A9 stands for the determinant of the matrix A.Finally, the estimated positions are deduced from the M

zeros of the cost function (31).

3.2. Practical implementation

A first possible implementation could be a grid searchperformed with the criterion CrðpÞ. The positions pm

correspond to the M lowest minima. But we can alsooptimize (31) by using a Newton algorithm, since theobjective function (31) is particularly well suited to obtainclosed-form expressions for the gradient and the Hessiancontrary to (30).

The Newton algorithm can be initialized with p0m

ð1rmrMÞ, obtained by a conventional triangulation,that can be performed, for example, through two AOAestimation algorithms on two stations. In order to miti-gate the ambiguities, among the M2 possible positions(deduced from the two sets of the M estimated AOAs ateach station), the M most appropriate positions are thoseproviding the M lowest values when inserted in thecriterion (31). A grid search can also be employed.

At each step i we obtain

piþ1m ¼ pi

m�H�1ðpi

mÞ=ðpimÞ, ð32Þ


where Hð�Þ and =ð�Þ are the Hessian and the gradientrespectively of the criterion (31). Since UH

ðpÞUðpÞ doesnot depend on p, well known mathematical formulasprovide [3]:

½=ðpÞ�k ¼9M9

9UHðpÞUðpÞ9

Tr½M�1Mk�, ð33Þ

½HðpÞ�kl ¼9M9

9UHðpÞUðpÞ9

ðTr½M�1Ml�Tr½M�1Mk�

�Tr½M�1MkM�1Ml�þTr½M�1Mkl�Þ, ð34Þ

where

MðpÞ ¼UHðpÞPnUðpÞ, ð35Þ

and with Mk ¼ @M=@pk and Mkl ¼ @2M=@pk@pl.

4. Deterministic Cramer–Rao bounds

By means of the Cramer Rao bound we provide in thissection new insights in the theoretical gain provided bythe global one step approach in narrowband context forthe global array compared to a conventional two stepsapproach. In this approach all signals of all stations haveto be transmitted to a CPU. Due to this stronger practicalrequirement, it appears of fundamental interest to quan-tify when and in what manner the global approach out-performs the conventional one. We here provide theexpression of the deterministic CRB of the model (16)called global CRB (GCRB), assuming the noise to beGaussian circular distributed. Then we also provide thedeterministic CRB for a conventional approach calledCCRB based on observation vectors received on eachstation separately seen as sub-blocks of the vector xðtÞ.

4.1. CRB of the global observation (GCRB) in the

narrowband context

Let us denote n the following unknown parametervector in the global approach, where the term s2 isdiscarded from the unknown parameter g:

n¼ ½gT sTr ð1Þ sT

c ð1Þ . . . sTr ðTÞ sT

c ðTÞ�, ð36Þ

where

g¼ ½P1 . . . PD W2 . . . WL U2 . . . UL�T ð37Þ

and srðtÞ and scðtÞ are respectively the real and imaginarypart of sðtÞ. Pd denotes the d-th row of the D�M matrix P,Wl the l-th row of the ðL�1Þ �M matrix W and Ul the l-throw of the ðL�1Þ �M matrix U. The log-likelihood of theproblem writes, up to an additive irrelevant constant

Lðx9nÞ ¼ �1

s2

XT

t ¼ 1

½xðtÞ�AðgÞsðtÞ�H½xðtÞ�AðgÞsðtÞ�, ð38Þ

where x¼ ½xT ð1Þ . . . xT ðTÞ�T . The Fisher informationmatrix (FIM) is [3]

JðnÞ ¼ E@Lðx9nÞ@n

� �@Lðx9nÞ@n

� �T( )

: ð39Þ

We consider the noise as Gaussian, circular. Using [21] asin [22] we finally get the block of the CRB from the invert

of J, only function of the parameter vector g:

GCRB�1ðgÞ ¼

2

s2

XT

t ¼ 1

RefSHðtÞDHPADSðtÞg, ð40Þ

PA ¼ I�AðAHAÞ�1AH , ð41Þ

SðtÞ ¼ IðDþ2ðL�1ÞÞ � diagðsðtÞÞ, ð42Þ

D¼ ½DP DW DU�, ð43Þ

DP ¼ ½DP1. . . DPD

�, ð44Þ

DPd¼

@uðp1,w1,/1Þ

@pd,1. . .

@uðpM ,wM ,/MÞ

@pd,M

� �, ð45Þ

DW ¼ ½Dw2. . . DwL

�, ð46Þ

Dwl¼

@uðp1,w1,/1Þ

@cl,1

. . .@uðpM ,wM ,/MÞ

@cl,M

" #, ð47Þ

DU ¼ ½D/2. . . D/L

�, ð48Þ

D/l¼

@uðp1,w1,/1Þ

@fl,1

. . .@uðpM ,wM ,/MÞ

@fl,M

" #, ð49Þ

where � denotes the Kronecker product. The result (40)can be straightforwardly expressed in the more compactform

GCRB�1ðgÞ ¼

2T

s2RefDHPAD� ðR 0sÞ

Tg, ð50Þ

where

R 0s ¼ 1K�K �1

T

XT

t ¼ 1

sðtÞsHðtÞ, ð51Þ

with � denoting the Hadamard product, andK ¼Dþ2ðL�1Þ and 1K�K being the matrix of dimensionK�K with all elements equal to 1. The expressions for thederivatives of the steering vectors can be found inAppendix A.

Finding an analytical expression of the GCRB depend-ing only on p is complicated in the general case, never-theless we provide in the sequel the calculation in theparticular case of one source and two stations.

4.2. CRB of the conventional observation (CCRB)

Conventional methods does not exploit the extendedstaked vector, they do not exploit any relation betweenthe signals received on each station. Consequently, theyrely on the following model, seen as a set of reparame-trized sub-blocks of the global observation xðtÞ consideredindependently of each other:

x1ðtÞ ¼XM

m ¼ 1

a1ðy1ðpmÞÞs1,mðtÞþn1ðtÞ,

^

xLðtÞ ¼XM

m ¼ 1

aLðyLðpmÞÞsL,mðtÞþnLðtÞ,

8>>>>>>><>>>>>>>:

ð52Þ


where

sl,mðtÞ ¼

ffiffiffiffiNp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPlc

2l,mNl

q clejfm smðtÞ: ð53Þ

The noise vectors nlðtÞ ð1r lrLÞ are assumed Gaussiancircular. The unknown parameters are

ptot ¼ ½P1 . . . PD�T , ð54Þ

s¼ ½sT1 . . . sT

L �T , ð55Þ

sl ¼ ½sTl ð1Þ . . . sT

l ðTÞ�T , ð56Þ

slðtÞ ¼ ½sTl,1ðtÞ . . . sT

l,MðtÞ�T , ð57Þ

so that the unknown parameter vector is

g¼ ½pTtot sT

r sTc �

T : ð58Þ

Where, again, sr and sc denotes the real and imaginarypart of s, respectively. Let us denote JðgÞ the correspond-ing Fisher matrix. We now consider the following variablechange:

g : g¼

ptot

sr

sc

0B@

1CA/n¼

htot

sr

sc

0B@

1CA, ð59Þ

where

htot ¼ ½hT1 . . . hT

L �T , ð60Þ

hl ¼ ½yl,1 . . . yl,M�T , ð61Þ

with yl,m ¼ ylðpmÞ. Consequently, the reparametrizedmodel is

x1ðtÞ ¼XM

m ¼ 1

a1ðy1,mÞs1,mðtÞþn1ðtÞ,

^

xLðtÞ ¼XM

m ¼ 1

aLðyL,mÞsL,mðtÞþnLðtÞ:

8>>>>>>><>>>>>>>:

ð62Þ

We denote JðnÞ the corresponding FIM. We have [23]

JðgÞ ¼@nT

@gJðnÞ

@n

@gT: ð63Þ

Let us denote

JðnÞ ¼Jyy Jys

Jsy Jss

" #: ð64Þ

Since

@nT

@g¼

@pTtot

@htot0

0 I2LMT

" #, ð65Þ

using the matrix inversion lemma [3], we have

CCRB�1ðptotÞ ¼

@hTtot

@ptot

Jyy@htot

@pTtot

�@hT

tot

@ptot

JysJ�1ss Jsy

@htot

@pTtot

, ð66Þ

CCRB�1ðptotÞ ¼

@hTtot

@ptot

ðJyy�JysJ�1ss JsyÞ

@htot

@pTtot

: ð67Þ

And since Jyy�JysJ�1ss Jsy ¼ CRB�1

ðhtotÞ, thanks to the matrixinversion lemma again, we have

CCRB�1ðptotÞ ¼

@hTtot

@ptot

CRB�1ðhtotÞ

@htot

@pTtot

, ð68Þ

with

CRB�1ðhtotÞ ¼

CRB�1ðh1Þ 0 0

0 & 0

0 0 CRB�1ðhLÞ

264

375: ð69Þ

The term CRB�1ðhlÞ is the deterministic CRB on AOA [24]

on the l-th station:

CRB�1ðhlÞ ¼

2T

s2RefDHPlD� R

T

s,lg, ð70Þ

Rs,l ¼1

T

XT

t ¼ 1

sHl ðtÞslðtÞ ð71Þ

D¼@alðyl,1Þ

@yl,1. . .

@alðyl,MÞ

@yl,M

� �, ð72Þ

Pl ¼ I�AlðAHl AlÞ

�1AHl , ð73Þ

Al ¼ ½alðyl,1Þ . . . alðyl,MÞ�: ð74Þ

Since the global approach exploits more information fromthe observations by taking into account correlationbetween the impinging signals on each station, it isexpected that the GCRB is smaller than the CCRB. Never-theless, each bound reflects different technological meansso that their comparison allows us to quantify thepotential gain of a global one step approach comparedto a conventional approach.

4.3. Analytic calculation for one source and two stations

We now examine the special case of two stations(L¼2) and one source (M¼1) for the planar geolocaliza-tion problem (D¼2). We have

xðtÞ ¼ uðp,c,fÞsðtÞþnðtÞ: ð75Þ

In this section, since there is only one source, the index m

is omitted and the parameters c2;1 and f2;1 are denotedas c and f instead since there is no possible ambiguity.Here we have

uðp,c,fÞ ¼ffiffiffiffiNp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN1þN2c

2q a1ðpÞ

cejfa2ðpÞ

" #: ð76Þ

Calling y1 and y2 the angles of arrival on the first and thesecond station respectively, we can write the observationson both stations used by a conventional approach as

x1ðtÞ ¼ a1ðy1Þs1ðtÞþn1ðtÞ, ð77Þ

x2ðtÞ ¼ a2ðy2Þs2ðtÞþn2ðtÞ, ð78Þ

where

s1ðtÞ ¼

ffiffiffiffiNp


2q sðtÞ, ð79Þ

s2ðtÞ ¼

ffiffiffiffiNp


2q cejfsðtÞ: ð80Þ


4.3.1. CCRB for two stations and one source

The CCRB reads

CCRB�1ðpÞ ¼G½CRB�1

ðy1,y2Þ�GT , ð81Þ

where

CRB�1ðy1,y2Þ ¼

CRB�1ðy1Þ 0

0 CRB�1ðy2Þ

" #, ð82Þ

G¼

@y1@x

@y2@x

@y1@y

@y2@y

24

35: ð83Þ

We have

CRB�1ðyiÞ ¼

2T

s2Re

@ai

@yi

H

PðaiÞ@ai

@yiRsi

( ), ð84Þ

with

Rsi¼

1

T

XT

t ¼ 1

9siðtÞ92, ð85Þ

and for any vector b

PðbÞ ¼ I�bðbHbÞ�1bH: ð86Þ

We finally get

ðCCRB�1Þij ¼

2T

s2Re

@a1

@pi

H

Pða1Þ@a1

@pj

Rs1

(þ@a2

@pi

H

Pða2Þ@a2

@pj

Rs2

):

ð87Þ

4.3.2. GCRB for two stations and one source

We prove in Appendix B that the CRB of the globalapproach is written as

ðGCRB�1Þij ¼

2TRs

s2Re

@u1

@pi

H

Pðu1Þ@u1

@pj

(þ@u2

@pi

H

Pðu2Þ@u2

@pj

),

ð88Þ

where

u1 ¼

ffiffiffiffiNp


2q a1,

u2 ¼

ffiffiffiffiNp


2q cejfa2:

We can easily deduce that PðuiÞ ¼PðaiÞ, @u1=@pi ¼ ðffiffiffiffiNp

=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN1þN2c

2q

Þ@a1=@pi and @u2=@pi ¼ ðffiffiffiffiNp

=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN1þN2c

2q

Þ

cejf@a2=@pi. Since by definition Rs1¼ ðN=ðN1þN2c

2ÞÞRs

and Rs2¼ ðN=ðN1þN2c

2ÞÞc2Rs we finally get

ðGCRB�1Þij ¼

2T

s2Re

@a1

@pi

H

Pða1Þ@a1

@pj

Rs1

(þ@a2

@pi

H

Pða2Þ@a2

@pj

Rs2

),

ð89Þ

which proves

GCRB�1ðpÞ ¼ CCRB�1

ðpÞ: ð90Þ

This means that in the narrowband signals case, in thespecial case of one source with two base stations noasymptotic gain of performance can be achieved with theproposed one step approach compared to a conventional

approach. Identity (90) is valid for one source and is dueto the particular form of the noise projector and to thefact that the FIM of the narrowband one step approachcontains some terms equal to zero. This study appears tobe more complicated in multiple sources context. This isthe reason why in the sequel we will calculate by meansof simulations the GCRB under some multi-emitter con-text in order to examine the gain of the one step narrow-band approach.

5. Wideband signal case

When the narrowband assumption is not verified onthe global array we can still write, according to (1), (4)and (9):

xðtÞ ¼XM

m ¼ 1

XL

l ¼ 1

0N1þ��þNl�1

alðpmÞ

0Nlþ 1þ��þNL

264

375smðt�tlÞþnðtÞ, ð91Þ

xðtÞ ¼XM

m ¼ 1

XL

l ¼ 1

uðpm,cl,mel,fl,melÞslmðtÞþnðtÞ, ð92Þ

where slmðtÞ ¼ smðt�tlÞ and el is the l-th column of theL� L identity matrix. We have rl,m ¼cl,mejfl,m . We define0K as a K � 1 vector composed of zeros.

In wideband signal context the model (92) is equiva-lent to (16) except that the size of the signal subspace K ofthe covariance matrix R is now equal to ML. The noiseprojector is Pn ¼ I�UsU

H

s where the columns of Us nowconsist of the K¼ML eigenvectors corresponding to the K

largest eigenvalues of R . Then, we can still optimize thecriterion (31) in order to estimate the position of theemitters. Eq. (91) shows that a wideband emitter can beseen as multiple components at same location and differ-ent nuisance parameters. The difference between thenarrowband and the wideband signal case lies in the sizeof the signal subspace embedded in R . In practice we donot know if the signals are narrowband or wideband onthe global array and both case can occur simultaneouslyin multiple emitter context. This is why, even if thenumber of sources is known, since the signal subspacecan be larger than the number of sources, we propose toestimate the signal subspace size. We use here a simpleempirical thresholding on the eigenvalues of R . Of coursethis stage can be refined with a more elaborate test but itis out of the scope of this paper. Note that [12] proposesto treat the case of wideband signals on the sensornetwork by increasing the size of the filter banks. As aconsequence, the number of independent snapshotsdecreases at the output of each frequency of the filter.Moreover, since the DPD criterion relies on an incoherentsum at the output of the filter banks, it appears to besuboptimal. In the proposed approach the number ofindependant snapshots is preserved.

6. Simulations

In this section we consider four base stations A, B, C

and D whose coordinates are ðxA,yAÞ, ðxB,yBÞ, ðxC ,yCÞ andðxD,yDÞ, respectively. Each base station is equipped with a

−10 −5 0 5 10101

102

103

SNR [dB]R

MSE

[m]

Ψ = [1 1]

CMA (2 steps)GMA (1 step)CCRB=GCRB

−10 −5 0 5 10SNR [dB]

RM

SE [m

]Ψ = [1 2]

CMA (2 steps)GMA (1 step)CCRB = GCRB

101

102

103

104

Ψ = [1 5]


three sensor uniform circular array whose radius is 0.5wavelength (NA ¼NB ¼NC ¼ND ¼ 3). The signals aredeterministic (generated by a Gaussian random process).The carrier frequency is f 0 ¼ 100 MHz. The scenariodescribing the position of the emitters and the basestations is illustrated in Fig. 4. The performance is studiedby means of the root mean square error (RMSE) indistance:

RMSE¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

K

XK

k ¼ 1

ðx�xkÞ2þðy�ykÞ

2

vuut , ð93Þ

where K is the number of Monte-Carlo runs and ðxk,ykÞ

denotes the k-th estimation of the true position (x, y) ofthe emitter. The empirical RMSE is compared to thecorresponding CRB in distance CRB¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCRBðxÞþCRBðyÞ

p.

We here compare the proposed global MUSICapproach (GMA) to a conventional two steps MUSICapproach (CMA). The GMA is computed thanks to theNewton method initialized by a grid search. The CMArelies on MUSIC algorithms implemented at each station,estimating the AOAs. When there are two stations thelocation is achieved by a simple geometrical relation.When there are more than two stations the location isthen obtained by a classical least-square approach [7]. Wecompare these algorithms to their corresponding Cramer–Rao bound called GCRB and CCRB, respectively.

6.1. Single narrowband emitter case

In Fig. 5 we compare the RMSE for the GMA with theCMA in the case of one source in S1, with only two basestations A and B. As we saw in Section 4 the CRB are equal.Since the stations are not supposed to receive the samepower of signal we consider three cases in Fig. 5: W¼ ½1 1�that is to say the stations observed signals of the samepower, W¼ ½1 2� corresponding to a difference of 6 dBbetween the two impinging signals and W¼ ½1 5� corre-sponding to a difference of 14 dB on both impingingsignals. In all cases we can see that the GMA outperformsthe conventional approach at low SNR. Moreover whenthe signals impinging on station have different

Fig. 4. S1ð300;300Þ and S2ð�300,�300Þ and four base stations A, B, C and

D in ð�1500;0Þ, ð1500;0Þ, ð0;1500Þ and ð0;1500Þ, respectively.

amplitudes the gain becomes more obvious. If there ismore than two stations, as illustrated in Fig. 6 the CRBsappear to be still equal and the proposed GMA still

−10 −5 0 5 10101

102

103

104

SNR [dB]

RM

SE [m

]

CMA (2 steps)GMA (1 step)CCRB=GCRB

Fig. 5. RMSE of one source in S1 in the presence of two stations A and B,

with different values of the attenuation between the two stations W,

T¼300, number of Monte-Carlo runs¼100.

−15 −10 −5 0 5100

101

102

103

SNR [dB]

RM

SE [m

]

CMA (2 steps)GMA (1 step)CCRBGCRB

Fig. 6. RMSE of one source in S1 in the presence of a second source in S2

with the four stations A, B, C and D, T¼300, number of Monte-Carlo

runs¼100.

−5 0 5 10 15101

102

103

SNR [dB]

RM

SE [m

]

CMA (2 steps)GMA (1 step)CCRBGCRB

Fig. 7. RMSE of one source in S1 in the presence of a second source in S2

with two stations A and B, T¼300, number of Monte-Carlo runs¼100.

−15 −10 −5 0 5 10 15100

101

102

103

SNR [dB]

RM

SE [m

]CMA (2 steps)GMA (1 step)CCRBGCRB

Fig. 8. RMSE of one source in S1 in the presence of a second source in

S2 with four base stations (A,B,C,D), T¼300, number of Monte-Carlo

runs¼100.


outperforms the conventional two steps approach CCMAat low SNR.

6.2. Multiple narrowband emitter case

Now we consider the presence of two sources S1 and S2

(as the scenario is symmetric we will only show the RMSEof S1) in Fig. 7. We first observe that the GCRB lies underthe CCRB demonstrating the potential gain of an algorithmthat exploits all information of the stations together. Wesee that both algorithms converge toward their corre-sponding CRB, suggesting the relevancy of the consideredalgorithms with regard to the corresponding models. Aswe can see the GMA still outperforms the CMA (a 5 dBgain for a 100 m error). The GCRB still appears to be arelevant approximation of the asymptotic variance ofthe GMA.

In Fig. 8 we compare the RMSE of the source S1 in thepresence of the source S2 using the four base stations. As wecan see the GMA still outperforms the CMA, for example fora 20 m error the gain is approximatively 10 dB.

6.3. Performance comparison between one step and two

steps approach for the narrowband signal case

In [25] thanks to the EXIP theorem [26], it is demon-strated that the performance of a two steps approach arealways inferior or equal to the performance of a one stepapproach. At best a two steps approach can be asympto-tically equivalent to the one step approach. The drasticperformance improvement of the one step approachcompared to the conventional two steps lies in thenumber of sources. This is confirmed by the simulations.Indeed: if there is one source both approaches areasymptotically equivalent whereas in multiple sourcescase, the one step approach appears to be always better.This is consistent with the relationship between the twocorresponding CRBs. For one source we first demonstratethat with two stations both CRB are equal. For an arbitrarynumber of stations (four in the simulations) this equality

appears to be still verified. It underlines the fact thatimprovements for the one step approach can still appearat low SNR in single emitter context. For an arbitrarynumber of sources, the provided simulations underlinethe fact that the CRB of the one step approach is smallerthan the conventional two steps CRB. This is why the onestep approach appears to be always better in such amultiple source context.

6.4. Wideband signal case

It is of interest to compare the GMA performance whenthe time-bandwidth product (t� B) is growing, that is tosay when the narrowband assumption cannot be verifiedon the global array. In Fig. 9 we vary the bandwidth of thesignal without changing the emitter location, so that thetime-bandwidth changes too. The GMA is equivalent to aparticular case of the DPD technique when the numberof FFT-bins J is equal to one in narrowband signal case.

10

20

30

40

50

60

τ × B

τ × B

RM

SE [m

]SNR = 5dB

GMADPD, J=1DPD, J=2DPD, J=4

10−4 10−2 100 102

10−4 10−2 100 102

0

1

2

3

4

5

Estim

ated

sign

al su

bspa

ce si

ze

Fig. 9. (a) RMSE of one source in ð300;500Þ with four base stations

(A,B,C,D), SNR¼5 dB, T¼300, number of Monte-Carlo runs¼200.

(b) Mean estimated signal subspace size used for the GMA algorithm.


When the time-bandwidth product is growing the receivedsignals at each station become uncorrelated. As a conse-quence, the size of the signal subspace grows too as shownin Fig. 9(b) and should be estimated. Fig. 9(a) showsthat the proposed methods outperforms the DPDwhen the time-bandwidth product becomes large. Wesee in Fig. 9(b) that the performance breakdown and thedistinction between the GMA and the DPD with J¼1 occurswhen the estimated signal subspace goes from one to four.When t� B is high emitters are uncorrelated on eachstations so that the size of the signal subspace becomesequal to ML. Choosing a larger filter banks for the DPDreduces the number of independent snapshots at eachfrequency and remains outperformed by the proposedapproach.

7. Conclusion

Whereas conventional location algorithms rely on a twosteps strategy, in this paper we investigate the potential gainoffered by a one step approach. By means of the Cramer–Rao

bounds we underline the fact that in a narrowband contexta one step approach will outperform the conventionalapproach in multiple emitter context. Moreover, we providean algorithm (called GMA) that is shown to perform close tothe corresponding Cramer–Rao bound. In a narrowbandcontext, it outperforms the conventional two steps approach.In a wideband context for the global array, the proposedmethod outperforms the recently proposed DPD technique.Future work might focus on the wideband signal case inorder to propose a performance study.

Acknowledgments

This work was partially supported by the network ofexcellence in wireless communication NEWCOMþþ underthe Contract no. 216715 funded by the European commis-sion FP7-ICT-2007-1.

Appendix A. Steering vectors differential

The expression of the differential of the steeringvectors are

@uðpm,wm,/mÞ

@cl,m

¼�Nlcl,mP

l0Nl0c2l0 ,m

Vcðcl,mÞuðpm,wm,/mÞ, ðA:1Þ

where

Vcðcl,mÞ ¼ diagðvCðcl,mÞÞ, ðA:2Þ

½vCðcl,mÞ�i ¼

1, ial,

Pl0al

Nl0c2l0 ,m

Nlc2l,m

, i¼ l,

8>><>>: ðA:3Þ

@uðpm,wm,/mÞ

@cl,m

¼ Vfðfl,mÞuðpm,wm,/mÞ, ðA:4Þ

where

Vfðfl,mÞ ¼ diagðvfðfl,mÞÞ, ðA:5Þ

½vfðfl,mÞ�i ¼0, ial,

j, i¼ l:

(ðA:6Þ

Appendix B. Analytic calculation of the GCRB for onesource and two base stations

Let us first denote

u¼

ffiffiffiffiNp


2q a1ðpÞ

cejfa2ðpÞ

" #¼

u1

u2

" #ðB:1Þ

and

PðuÞ ¼P1 �C

�CH P2

" #, ðB:2Þ

where Pi ¼ INi�uiu

Hi =N and C¼ u1uH

2 =N. To go furtherwe need the simple following results obtained by


straightforward calculations:

P1u1 ¼N2c

2

N1þN2c2

u1, ðB:3Þ

P2u2 ¼N1

N1þN2c2

u2, ðB:4Þ

CHu1 ¼N1

N1þN2c2

u2, ðB:5Þ

Cu2 ¼N2c

2

N1þN2c2

u1, ðB:6Þ

uH1 P1u1 ¼ uH

2 P2u2 ¼NN1N2c

2

ðN1þN2c2Þ2

, ðB:7Þ

the unknown parameter vector is

n¼ ½RefsgT ImfsgT pT c f�: ðB:8Þ

Denoting

g¼ ½pT c f�, ðB:9Þ

according to (50) the FIM is

ðJðgÞÞij ¼2T

s2Re

@u

@Zi

H

PðuÞ@u

@Zj

Rs

( ): ðB:10Þ

Let us consider

JðgÞ ¼2TRs

s2

Jxx Jxy Jxc Jxf

Jxy Jyy Jyc Jyf

Jxc Jyc Jcc JcfJxf Jyf Jcf Jff

266664

377775: ðB:11Þ

Let us first remark that the derivative of u with respect top1 ¼ x and p2 ¼ y can be written as follows:@u=@pi ¼ diagðvpi

Þu where

vpi¼�j2pf 0

@d1@pi

@d2@pi

24

35, ðB:12Þ

where di ¼ ½di,1ðpÞ . . . di,NiðpÞ�. We can also write @u=@c¼

Vcu and @u=@f¼Vfu where

Vc ¼1

N1þN2c2

�cN2 0

0 N1c

" #, ðB:13Þ

Vf ¼0 0

0 j

" #: ðB:14Þ

Straightforward calculation provides

PðuÞ@u

@c¼PðuÞVcu

¼

ffiffiffiffiNp

ðN1þN2c2Þ3=2

�N2cðN2c2þN1Þu1

N1ðN2cþ N1c Þu2

24

35:

We can deduce

@u

@f

H

PðuÞ@u

@c¼�j

ffiffiffiffiNp

ðN1þN2c2Þ3=2

N1 N2cþN1

c

� �Ju2J

2:

ðB:15Þ

We conclude

Re@u

@f

H

PðuÞ@u

@c

( )¼ 0: ðB:16Þ

In the same manner we observe that

Re@u

@pi

H

PðuÞ@u

@c

( )¼ 0: ðB:17Þ

So the FIM is now

JðgÞ ¼2TRs

s2

Jxx Jxy 0 Jxf

Jxy Jyy 0 Jyf

0 0 Jcc 0

Jxf Jyf 0 Jff

266664

377775: ðB:18Þ

Thanks to a well-known matrix inversion lemma we have

ðGCRBðpÞ�1Þij ¼

2TRs

s2Jpipj�

JpifJpjf

Jff

!: ðB:19Þ

Straightforward calculations lead to

Jff ¼@u

@f

H

PðuÞ@u

@f, ðB:20Þ

¼ uH2 P2u2, ðB:21Þ

¼NN1N2c

2

ðN1þN2c2Þ2

, ðB:22Þ

Jpif ¼ Re@u

@pi

H

PðuÞ@u

@f

( ), ðB:23Þ

¼ Re@u

@pi

H P1 �C

�CH P2

" #Vfu

( ), ðB:24Þ

¼1

N1þN2c2

Im N2c2@u1

@pi

H

u1�N1@u2

@pi

H

u2

( ): ðB:25Þ

So it leads to

JpifJpjf

Jff¼

N2c2

NN1Im

@u1

@pi

H

u1

( )Im

@u1

@pj

H

u1

( )

�1

NIm

@u1

@pi

H

u1

( )Im

@u2

@pj

H

u2

( )

�1

NIm

@u1

@pj

H

u1

( )Im

@u2

@pi

H

u2

( )

þN1

NN2c2

Im@u2

@pi

H

u2

( )Im

@u2

@pj

H

u2

( ): ðB:26Þ


We have then

Jpipj¼ Re

@uH

@pi

PðuÞ@u

@pj

( )

¼ Re@uH

1

@pi

I�u1uH

1

N

� �@u1

@pj

( )�

1

NRe

@u1

@pi

H

u1uH2

@u2

@pj

( )

�1

NRe

@u2

@pi

H

u2uH1

@u1

@pj

( )þRe

@uH2

@pi

I�u2uH

2

N

� �@u2

@pj

( ):

ðB:27Þ

Using the fact that for any complex z1 and z2 we haveRefz1z2g ¼ Refz1gRefz2g�Imfz1gImfz2g and the fact that

Re@ui

@pj

H

ui

( )¼ 0, ðB:28Þ

we finally get

ðGCRBðpÞ�1Þij ¼

2TRs

s2Re

@u1

@pi

H

Pðu1Þ@u1

@pj

( )

þRe@u2

@pi

H

Pðu2Þ@u2

@pj

( )!: ðB:29Þ

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