Localization of Multiple Emitters by Spatial Sparsity Methodsin the Presence of Fading Channels

Joseph S. Picard and Anthony J. Weiss Fellow, IEEE

Abstract—The problem of multiple emitters geolocation usingsensor arrays is addressed, in the case of fading channels. Asparsity-based covariance-matrix fitting method is described. Theprocedure consists of finding a sparse representation of the sam-ple covariance matrices obtained at the arrays, by representingeach matrix by an over-complete basis. Sparsity is encouragedby an �1-norm based penalty function. The penalty function isminimized by semi-definite programming. The proposed methodprovides useful insight and it does not require the identificationof the signal and noise subspaces. Therefore, the method doesnot rely on a good estimate of the number of emitters. Some ofthe approach properties are super-resolution, robustness to noise,robustness to emitter correlation, no sensitivity to initializationand no need for synchronizing the arrays. Special emphasis isgiven to uncorrelated sources and uniform linear arrays.

Index Terms—Geolocation, spatial spectrum, sparsity, convexoptimization, semi-definite programming, �1 norm, covariance-matrix fitting, time of arrival (TOA), angle of arrival (AOA),grid-based positioning.

I. INTRODUCTION

The geolocation problem consists of estimating the locationof radio transmitters using signals collected at multiple basestations, each equipped with a sensor array and placed at aknown location. This problem has received considerable atten-tion by the signal processing, communications, and underwateracoustics communities. Applications include the localizationof cellular phones, spectrum monitoring, and law enforcement.See [1]-[3] for comprehensive reviews of signal angle-of-arrival (AOA) estimation using antenna array processing andrelated issues. See Van Trees [4] for a complete book on arrayprocessing.

Cellular phone localization, radar systems, and underwa-ter acoustics frequently invoke AOA/TOA based positioning.The traditional AOA/TOA approach that is composed of twoseparate steps: (1) AOA/TOA independent estimates and (2)triangulation based on the results of the first step. It is usuallyassumed that AOA/TOA estimates are accurate or degradedby small errors only, whereas large errors or biased measure-ments are overlooked. However, in a recent publication [5] anovel framework was proposed in order to achieve accurategeolocation in the presence of outliers in the measurementsset. All these techniques can be classified as decentralizedprocessing methods, since the observed signals are exploitedindependently at each sensor array.

This research was supported by the Israel Science Foundation (grant No.218/08), by the Institute for Future Technologies Research named for theMedvedi, Shwartzman and Gensler Families, by the Center for Absorption inScience, Israel, and by the Weinstein Research Institute for Signal Processing.

J. S. Picard and A. J. Weiss are with School of Electrical Engineering,Systems Department, Tel Aviv University, Tel Aviv 69978, Israel (phone:+97236408605; fax: +97236405027; e-mail: {picard,ajw}@eng.tau.ac.il).

Centralized processing methods solve the localization prob-lem using the data collected at all sensors at all base stationstogether. As indicated by Wax and Kailath [6], measuringAOA/TOA at each base station separately and independentlyis suboptimal since this approach ignores the constraint thatthe measurements must correspond to the same source posi-tion. Therefore centralized method outperforms decentralizedmethods. The Direct Position Determination (DPD) algorithmby Weiss [7] is an efficient centralized location estimator withseveral side benefits.

In the context of geolocation of multiple transmitters in atwo-dimensional field, the spatial spectrum is defined as asurface, generated by the estimation method, of some scorefunction versus the x and y coordinate axis. The estimatedlocations are associated with peaks in the score function. Thesepeaks are hopefully centered at the actual transmitter location,but two peaks may merge into a single peak if the sources arenot separated enough. Moreover, in the presence of noise andlimited number of observations, even super-resolution methodssuffer from wide lobes in the spatial spectrum, instead ofobtaining thin peaks.

Denote by Q the number of sources at different locations.Ideally, the spatial spectrum should consist of Q peaks, eachpeak located at an actual transmitter location. Consider a N -points grid that uniformly samples the field of interest. Arrangethe corresponding samples of the spatial spectrum in a N × 1vector. Provided that N � Q, this vector is sparse since(ideally) most of its entries are close to zero and only Q entriesare considerably different from zero. Therefore, transmitterlocation estimates can be obtained by encouraging sparsity ofthe spatial spectrum. An alternative naive approach is possibleif the number of sources, Q, is known. Simply examine allpossible selections of Q out of N location samples. This is aNP-hard approach. For Q = 4 transmitters and N = 20×20 =400 location candidates, more than 109 selections must beconsidered, which is obviously not practical. An approachinspired by the theory of sparse representation of signals offersa good alternative. We therefore give a short overview of thistopic.

A. Sparse Signal Representation

Consider the linear set of equations b = Ap whereb is a vector that has a representation p in the span ofA, a.k.a. dictionary. In some applications the sparsest p isdesired. The representation is sparse if the dictionary is over-complete, i.e., A has more columns than rows. In this case,b = Ap is an under-determined set of equations that cannotbe solved uniquely. In recent publications, it has been shown

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that if the sparsest representation is sufficiently sparse, thennot only it is unique, but it can also be easily obtained bylinear programming [8]-[10]. Conditions ensuring uniquenessof the sparsest solution were established in [11]. The problemhas been extended to a wider variety of bases in [12]-[13]including dictionaries built by concatenation of non-unitarymatrices [14]-[15]. These approaches are all based on �1-normminimization replacing the practically intractable �0-normminimization. The validity of finding sparse representation by�1-norm relaxation has been studied in [16]. The theory ofsparse signal representation has been used to solve the NP-hard linear decoding problem in [17].

B. Sparsity Methods in Source Localization

Sparsity methods have been invoked in transmitter local-ization only very recently. The above-mentioned method [5]successfully mitigates outliers within a measurements set andaccurately estimates transmitter coordinates, by the mean ofsparsity concepts. Indeed, for AOA, time-of-arrival (TOA) andtime-difference-of-arrival (TDOA) measurements, the outliererrors were represented by a sparse vector. Then, using meth-ods inspired from the theory of sparse signal representation,outliers were identified and removed from the data set.

The other few works that use sparsity in transmitter lo-calization are all dedicated to AOA estimation exclusively.In [18] the spectrum sparsity results from the recursiveweighted minimum-norm algorithm FOCUSS. The author of[19] proposes a low-complexity approach in the beamspacedomain, which he solves by linear programming. Provided thatsources are uncorrelated and that the number of observationsis sufficiently large, the method consists of finding a sparserepresentation for beamformer outputs using an over-completebasis of beamformer outputs evaluated at equally spacedangles. In the context of acoustic sources mapping, a sparsity-constrained deconvolution approach was proposed in [20] inorder to improve the classical delay-and-sum approach thatsuffers from low resolution and high sidelobes. Furthermore,the method described in [21] uses the singular value decompo-sition (SVD) of the data matrix, in order to summarize multipletime samples. The resulting optimization problem is solved bysecond-order cone programming. Finally, in covariance-matrixfitting approaches [22]-[23], AOA estimation was realized byfinding a sparse representation of the sample covariance ma-trix, using an over complete basis obtained from the manifoldsamples. These approaches minimize a convex cost functionbuilt of an �1-norm penalty term and a data fidelity term. Thisdata fidelity term involves a Frobenius norm in [22], whereasin [23] an �1-norm was chosen in order to obtain a solutionby linear programming.

C. Contribution

We extend the covariance-matrix fitting approach initiallydedicated to AOA estimation [23] to the problem of two-dimensional geolocation in the presence of fading channels.The objective is to obtain a method with similar advanta-geous properties: robustness to noise, robustness to correlatedsources, no need for accurate initialization, no requirement for

knowing the number of sources and no heavy computationssuch eigen-decomposition or SVD of the data. We show thatstraightforward vectorization of the sample covariance matrixyields a formulation that satisfies these requirements.

The problem is defined in Section II and its solution bysemi-definite programming is described for simple cases inSection III. We examine there the case of correlated sourcesand the case of uncorrelated sources. In section IV, wepropose extensions of the algorithm to general cases, and someenhancements are also discussed. Finally, numerical examplesare given in section V.

D. Notations

In the sequel, the k-th entry of a vector r is denoted {r}k.For a matrix R the (k, �)-th entry is denoted {R}k,�. Thek-th column and the �-th row of R are denoted {R}:,k and{R}�,: respectively. Furthermore, 1N and 0N are the N × 1vectors of ones and zeros. Similarly, 1N,M and 0N,M are theN × M matrices of ones and zeros, respectively. Finally, forany matrix R denote by vec {R} the vertical concatenation ofthe columns of R.

II. PROBLEM FORMULATION

Consider L base stations intercepting the signals transmit-ted by Q possibly correlated sources. Each base station isequipped with an antenna array consisting of M elements.The bandwidth of the signal is small compared to the inverseof the propagation time over the array aperture. Denote byψq the actual two-dimensional cartesian coordinates of the q-th source. Let T be the total observation time. In the timedomain, the complex envelopes of the signals observed by the�-th base station array are given by

ρ�(t) =Q∑

q=1

β�q(t)a�(ψq)sq

(t − τ�(ψq)

)+ η�(t) (1)

0 ≤ t ≤ T

where ρ�(t) is a time-dependent M × 1 vector, β�q(t) is anunknown complex scalar representing the channel attenuationbetween the q-th transmitter and the �-th base station, a�(ψq)is the �-th array response to a signal transmitted from ψq,and sq(t − τ�(ψq)) is the q-th signal waveform, delayed byτ�(ψq). The vector η�(t) represents noise and interferenceobserved by the array. The observed signal is partitionedinto K sections. Since the channels attenuation phase andamplitude are very sensitive to movement of the sources,the sensors and the scatterers, the unknown scalar β�q(t)fluctuates with t. However, it is assumed that each time sectionhas a duration T/K sufficiently short (in comparison withthe coherence time) to consider β�q(t) as constant over thecorresponding time section. Thus, for the k-th section there isa scalar β�q[k] such that β�q(t) = β�q[k] for (k−1)T

K ≤ t < kTK .

Each section is Fourier transformed, which yields

ρ�[j, k] =Q∑

q=1

β�q[k]a�(ψq)sq[j, k]e−iωjτ�(ψq) + η�[j, k] (2)

j = 1, ..., J k = 1, ...,K

63

where ρ�[j, k] is the Fourier coefficient of the k-th section ofthe observed signal corresponding to the baseband frequencyωj � 2π(j−1)

T/K , and s[j, k] is the j-th Fourier coefficient of thek-th section of the signal. Finally, η�[j, k] represents the j-thFourier coefficient of the k-th section of the noise waveform.

Time delay information is taken into consideration only forJ ≥ 2 and results in a slight enhancement of the localizationaccuracy. If L = 1 then the single base station can onlyachieve angle-of-arrival (AOA) estimation since time-delayinformation is not exploitable, and the choice J = 1 isrecommended. For notation convenience and better insight,we focus in a first step on the special case J = 1. Theultimate version of the proposed algorithm, described in alater section, supports J ≥ 1 and therefore exploits also timedelay information. Since for now J = 1, any dependence onj completely disappears in (2),

ρ�[k] =Q∑

q=1

β�q[k]a�(ψq)sq[k] + η�[k] k = 1, ...,K (3)

Define the matrices A� and the column vectors s�[k] by

A� = [a�(ψ1), ..., a�(ψQ)]{s�[k]}q = β�q[k]sq[k] (4)

With these definitions, equation (3) yields trivially

ρ�[k] = A�s�[k] + η�[k] k = 1, ...,K (5)

Geolocation consists of estimating the number Q of sourcesand their locations ψq using the observations ρ�[k]. Note thatthe model described in (5) takes into consideration eventualfluctuations of the radio channels, through the dependance ofB�[k] on k.

III. GEOLOCATION BASED ON SPATIAL SPARSITY

The area of interest is sampled using a two-dimensional gridcomposed of N points. Each of the N grid points is labeledwith an index, and let πn denote the coordinates of the n-th grid point. The set Π = {π1, ..., πN} of the N locationsyields different sets of steering vectors at the L sensor arrays.Recall that ψq is the vector of the actual coordinates of theq-th source. For better insight and simpler notation, we makefirst the assumption that the different transmitters locations areon the grid, which implies ψq ∈ Π for q = 1, ..., Q. Therefore,equation (5) yields immediately

ρ�[k] = A�s�[k] + η�[k] k = 1, ...,K (6)

where

A� � [a�(π1), ..., a�(πN )]

{s�[k]}n �{{s�[k]}q if ψq = πn

0 otherwise

The matrix A� is actually the �-th array manifold matrix.Furthermore, the sparse vector s�[k] has exactly Q non-zeroentries, and the indices of non-zero entries correspond to thelocations of the transmitters: the n-th entry of s�[k] differsfrom 0 if and only if one of the sources lies at the n-th grid-point location πn.

Definition 1: Two vectors a and b have same support ifthe indices of non-zero entries in a and b are the same.Clearly, the vectors s�[k] have same support for any (�, k).Denote R� and Σ� the covariance matrices for the signalsρ�[k] and η�[k]. Then, equation (6) yields

R� = A�D�AH� + Σ� (7)

whereD� � E

{s�[k]s�[k]H

}(8)

Note that at most Q2 out of the N2 entries of D� differ fromzero. Furthermore, the vectors D�1N have same support forany � and the indices of their non-zero entries point out thelocations of the multiple emitters on the grid. Define now

r� = vec {R�} n� = vec {Σ�} A� =(A∗

� ⊗ A�

)(9)

Then, equation (7) gives

r� = A�vec{D�

}+ n� (10)

The problem of location estimation of multiple emitters isequivalent to finding the sparse Hermitian matrices D� thatsatisfies r� = A�vec {D�} + n� and such that the vectorsvec {D�} have same support for any �.

A. General Case: Possibly Correlated Sources

In order to find the sparse matrices D�, an intuitive approachwould consist of minimizing

∑L�=1 ‖D�‖1 since sparsity is

encouraged by �1-norm minimization. Denote D the N ×NLcomplex matrix obtained by horizontal concatenation of all thematrices D� for � ∈ {1, ..., L}, i.e.

D = [D1, ...,DL] (11)

In order to improve the localization accuracy, we take intoconsideration the fact that the observations at the differentbase stations result from the same sources. In other words,non-zero columns of D must have same support. Minimizing∑L

�=1 ‖D�‖1 would only encourage sparsity of D, withoutencouraging same support for its columns. Note that D is amatrix of zeros with only few non-null rows, and the indicesof these rows indicate emitters locations. More specifically,these rows are the only rows of D with non-zero Frobeniusnorm. Therefore, it is recommended to minimize the �1-normof the vector d built from the Frobenius norm of the rows ofD according to {d}n = ‖{D}n,:‖2. The �1-norm encouragessparsity of d and the Frobenius norm makes the non-zerocolumns of D have same support. Since d has non-negativeentries, its �1-norm reduces to a sum, and the emitters locationscan be estimated by solving the convex optimization problemin D,

minD�∈CN×N

N∑n=1

‖{D}n,:‖2

s.t.

⎧⎪⎨⎪⎩

r� = A�vec {D�} + n�

D = [D1, ...,DL]D� hermitian

(12)

Such optimization problem can be solved by second order coneprogramming.

64

B. Common Case: Uncorrelated Sources

In many practical localization problems the sources areuncorrelated. In these conditions the signal covariance matrixis a diagonal positive matrix, thus the matrix D� defined in(8) is a N ×N sparse diagonal real positive matrix with onlyQ non-zero entries. The N2 × 1 vector vec

{D�

}consists of

N(N − 1) entries corresponding the off diagonal entries ofD� and therefore are known to be zero. In order to reduce theproblem dimension, these entries are removed from vec

{D�

}together with the corresponding columns of A� without loss ofinformation. This can be accomplished by inserting a selectionmatrix S0. Since the diagonal entries of D� have indices 1,N + 2, 2N + 3,...,N2 in the vector vec

{D�

}, we construct

the N ×N2 selection matrix S0 that keeps only these entries.The matrix S0 is therefore defined by

S0 = [e1, eN+2, e2N+3, ..., eN2 ]T (13)

where en is a N2 × 1 vector of zeros except the n-th entrywhich is 1. Inserting S0 in (10) gives

r� = A�ST0 S0vec

{D�

}+ n�

= A�ST0 d� + n� (14)

where d� � D�1N . Then, taking into consideration (14) theproblem stated in (12) is adapted to uncorrelated sources asfollows,

mind�∈RN×1

N∑n=1

‖{D}n,:‖2

s.t.

⎧⎪⎨⎪⎩

r� = A�ST0 d� + n�

D = [d1, ...,d�]{d�}n ≥ 0 for any n ∈ {1, ..., N}

(15)

Once this optimization problem is solved by second ordercone programming, sources locations estimates are obtainedby identifying large entries of the N × 1 vector of which then-th entry is ‖{D}n,:‖2. The indices of these large entries haveunique correspondence to locations on the grid that samplesthe field of view.

IV. ALGORITHM EXTENSIONS AND IMPROVEMENTS

A. Avoiding Data Redundancy in Uniform Linear Arrays

The matrices R� are all Hermitian, thus consist of N(N−1)2

conjugate pair entries. In an effort for avoiding data redun-dancy, entries of R� located above the main diagonal areremoved, and corresponding entries in vec {R�} too. To thatend, we pre-multiply (14) by an appropriate selection matrixS� constructed from the M2×M2 identity matrix by removingall rows with index k = mM + t for m = 1, ...,M − 1 andt = 1, ...,m.

In certain cases the selection matrix S� can be furthermodified to remove uninformative or redundant rows of A�

that may result from specific array manifolds. For example,we have already shown [23] that when localizing uncorrelatedsources using uniform linear arrays, S� should be defined by

S� = 1√M

[S(1), ..., S(M)

](16)

where S(m) is an M ×M upper triangular matrix with entriesdefined by{

S(m)}

p,q=

{1

M−p+1 if q − p = m − 10 otherwise

for p, q and m in {1, ...,M}. In these conditions, the definitiongiven in (9) for r�, n� and A� should be modified into

r� = S�vec {R�} n� = S�vec {Σ�} A� = S�

(A∗

� ⊗ A�

)and equation (10) still holds. As a result, it appears that theproblem complexity can be reduced by considering effects ofthe arrays geometry on the data model.

B. Finite Observation Time and Off-Grid Source Locations

In practice, the actual source locations generally do notcoincide with the grid points that samples the area of interest.Furthermore, the covariance matrix R� is unknown exactly,but can be approximated by the sample covariance matrixR� = 1

K

∑Kk=1 ρ�[k]ρ�[k]H . Thus, in the example of uncorre-

lated sources, the vector r� is replaced in (14) by the vector r�

obtained from R�. Similarly, if n� is not known precisely it canbe replaced by an estimated n�. The difference between actualand assumed values for source locations and data yields theapproximation r� ≈ A�ST

0 d�+n�. We adapt (15) by replacingthe equality constraint r� = A�[j]ST

0 d� + n� with a residualterm in the cost function, as follows.

mind�∈RN×1

N∑n=1

‖{D}n,:‖2 +μ

2

L∑�=1

res�

s.t.

⎧⎪⎨⎪⎩

res� = ‖A�d� + n� − r�‖22

D = [d1, ...,dL]{d�}n ≥ 0 for any n ∈ {1, ..., N}

(17)

The n-th grid sample of the spatial spectrum is still ‖{D}n,:‖2

although the cost function has been modified. The �1-norm,which reduces to a sum on n in the cost function, still ensuressparsity of the spatial spectrum.

C. Time delay data using multiple Fourier coefficients J ≥ 2The case of multiple frequency coefficients is now consid-

ered, i.e. J ≥ 2. Back to (2) a version of (6) for J ≥ 2is derived in a similar way as for J = 1. Few immediatemanipulations give

ρ�[j, k] = A�[j]B�[k]s[j, k] + η�[j, k]j = 1, ..., Jk = 1, ...,K

(18)

where

A�[j] � [a�(π1)e−iωjτ�(π1), ..., a�(πN )e−iωjτ�(πN )]

{B�[k]}n,m �{{B�[k]}q,q if ψq = πn and m = n

0 otherwise

{s[j, k]}n �{

sq[j, k] if ψq = πn

0 otherwise

Note that the covariance matrix obtained at a single basestation does not contain any time delay information. Therefore,

65

in order to include time delay information, it is necessaryto consider the cross covariance matrix between the signalscollected at the i-th and the �-the base stations, given by

Ri,�[j] = E{ρi[j]ρ�[j]H} (19)

Similarly to (9) and (10), and under the assumption of flatspectrum sources, simple vectorization of the cross covariancematrix yields

ri,�[j] = Ai,�[j]vec{Di,�

}+ ni,�[j] (20)

where

ri,�[j] � vec {Ri,�[j]} (21)

ni,�[j] � vec {Σi,�[j]} (22)

Ai,�[j] �(A�[j]∗ ⊗ Ai[j]

)(23)

Di,� � E{Bi[k]s[j, k]s[j, k]HBH

� [k]}

(24)

and Σi,�[j] is the cross covariance matrix of the noise. As in(17) let ri,�[j] and ni,�[j] be the available approximate versionsfor ri,�[j] and ni,�[j], then a convex optimization problem thatsimultaneously yields to a sparse representation and achievescovariance matrix fitting, is

mindi,�∈CN×1

N∑n=1

‖{D}n,:‖2 (25)

+J∑

j=1

L∑�=1

(μ

2res�,�[j] +

λ

2

�−1∑i=1

resi,�[j]

)

s.t.

⎧⎪⎨⎪⎩

resi,�[j] = ‖Ai,�[j]di,� + ni,�[j] − ri,�[j]‖22

D : horizontal concatenation of all the di,�

di,� : complex vector, and positive real-valued if � = i

where λ and μ are user-defined parameters. Choosing λ = μenables different weighting for angle of arrival informationcontained in R�,�[j] and time delay information contained inRi,� when i = �. for example, one can check analytically thatif the standard deviation of the channel attenuations processesβ�q[k] is large in comparison with their mean value, thenit is preferable to choose λ > μ, otherwise the time delayinformation would not be fully exploited.

V. NUMERICAL EXAMPLES

The distance unit is set to 10 km. The positioning system isbuilt of L = 2 base stations, each of them being an arrayof M = 8 sensors with an intersensors spacing of half awavelength. These base stations are located in the (x, y) planeat coordinates [0, 0] and [1, 0], i.e. at [0, 0] and [10, 0] km. Twosources transmit from locations [0.46, 0.80] and [0.54, 0.80].The sources separation is null along the y-axis, and along thex-axis it is 0.54 − 0.46 = 0.08, i.e. 800 m. Assume someside information indicate the eventual presence of sources inan area of interest that spreads from x = 0.4 to x = 0.6, andfrom y = 0.7 to y = 0.9. The transmitters, whose numberis unknown to the receivers, are therefore searched withinthis area of interest using a grid of N = 21 × 21 = 441location candidates. This grid yields a 100 m resolution along

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X axis

Y a

xis

Fig. 1. Base stations (black squares), transmitters (black circles) and the21×21 points grid that samples the area of interest in which the transmittersare searched.

both x and y axes. The locations of the base stations andthe transmitters are represented on Fig. 1, together with theconsidered grid.

The number of observations is K = 1000, and the signalssq[j, k] are independent, complex, zero mean, unit variancegaussian numbers. The channel attenuation coefficients β�,q[k]are randomized according to β�,q[k] = β�,q[0]+ γ�,q[k], wherethe coefficients β�,q[0] are complex phasors (complex scalarswith unit norm and phase uniformly randomized in [0, 2π])and γ�,q[k] has a zero-mean complex normal distribution.The constant complex phasors are aimed to avoid zero meanchannels, in which time delay information would disappear.The number of Fourier coefficient is J = 2. Finally, the noisevector η�[j, k] has a complex, zero-mean multivariate normaldistribution.

Geolocation is achieved by solving the convex optimizationproblem introduced in (17) by semi-definite programmingsolvers, namely SeDuMi combined with Yalmip. The parame-ter λ is set to 10. For a single experiment, the resulting spatialspectrum is given on Fig. 2 after normalization w.r.t. its peakvalue. Clearly, two peaks are visible: one at [0.46, 0.81] andthe other at [0.54, 0.79], meaning that for both of them thepositioning error is 0.01 distance unit, i.e. 100 m, which isactually the grid resolution. Results would have been better ifthe number of observations was larger.

The proposed method could be a first phase solution forsimultaneous and robust estimation of the number of transmit-ters and their locations, without any need for side information.Then, in a second phase, other algorithms could be invokedand exploit the reliable information obtained in the first phase,and finally yield accurate location estimates.

66

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.60.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

X axis

Y a

xis

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. The spatial spectrum (associated with the a grid represented on Fig.1) and the actual emitters locations (black circles).

VI. CONCLUSION

The proposed positioning method finds a sparse spatialspectrum that best fits the sample covariance matrices ofthe observed data. Two versions of the method have beendiscussed, one for possibly correlated sources and one foruncorrelated sources. The estimation is obtained by semi-definite programming. In the case of ideal uniform linear arraythe computational load is significantly reduced by squeezingthe data set without loss of information. Further, the methodis not sensitive to source correlation, does not require edu-cated initialization, information on the number of sources, orsynchronization of the arrays.

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