IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 6, JUNE 2005 457

Simple Robust Bearing-Range Source’s LocalizationWith Curved WavefrontsE. Boyer, A. Ferréol, and P. Larzabal, Member, IEEE

Abstract—In array processing, the far-field assumption ofplanar wavefronts is widely used by direction of arrival (DOA)estimators but not always satisfied. In this letter, we introduce anew method for bearing-range estimation, which extends classicalsubspace-based bearing estimators to a curved wavefront con-text. The bearing estimation is provided by a one–dimensionalprocedure, and ranges are simply analytically deduced from thebearing estimates. The proposed approach is illustrated by theintroduction of the music curved wavefront (MCW) algorithm.

Index Terms—Curved wavefronts, direct of arrival (DOA) esti-mation, range estimation, subspace-based methods.

I. INTRODUCTION

PASSIVE source localization using an array of sensors is animportant topic that is raised in many different fields, such

as sonar, radar, and communication. In a far-field context, thewaves impinging on the array are planar.

Numerous classical algorithms have been developed underthat underlying far-field hypothesis, such as beamforming [1],CAPON’s type methods [2], and MUSIC [3], [4]. Nevertheless,in a context of curved wavefronts, the wavefront curvature in-duces a performances degradation of previous classical algo-rithms [5], [6]. Various solutions have been proposed in orderto adapt classical methods to this context. In the case of nar-rowband sources, Huang and Barkat [7] proposed a two-dimen-sional bearing-range MUSIC estimator. Starer and Nehorai [8]reduced the computational complexity of the bearing-range esti-mation by the use of path following, but this method is limited touniform linear arrays (ULAs). Sahin and Miller [9] partitionedthe receiver array into subarrays, where the scattered fields areassumed to be locally planar, and a classical MUSIC algorithmis then used to estimate the DOAs, the range being estimated bya triangulation method. Weiss and Friedlander [10] reduced thecomputational cost of bearing-range estimation by replacing thetwo-dimensional (2-D) minimization of the 2-D MUSIC costfunction by a 2-D polynomial rooting (see [11] for the extensionof the method to the three-dimensional case). For the broadbandsources case, see [12]–[14] and the references therein.

Manuscript received October 25, 2004; revised January 13, 2005. This workhas been partially supported by NEWCOM. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. MounirGhogho.

E. Boyer is with the SATIE, ENS Cachan, UMR 8029, 94235 Cachan Cedex,France (e-mail: boyer@satie.ens-cachan.fr; e.boyer@tele2.fr).

A. Ferréol is with the THALES communication, 92700 Colombes, France(e-mail: Anne.FERREOL@fr.thalesgroup.com).

P. Larzabal is with the SATIE, ENS Cachan, UMR 8029, 94235 CachanCedex, France, and also with the IUT de Cachan, Université Paris Sud, BP140,94234 Cachan Cedex, France (e-mail: larzabal@satie.ens-cachan.fr).

Digital Object Identifier 10.1109/LSP.2005.847885

In this letter, we introduce a new method for bearing-rangeestimation. The localization of sources is provided by a two-step approach. In a first step, the range parameter is consideredas a nuisance parameter, leading to a concentration of the costfunction with respect to DOA: Bearing estimates are providedby a 1-D minimization of the cost function. The range parameteris simply analytically deduced from the bearing estimates in asecond step. The concentration of the 2-D MUSIC cost functionis based on a Taylor series expansion of the steering vector. Theproposed approach is illustrated by the introduction of the MCWalgorithm.

This letter is organized as follows: Section II presents themodel. In Section III, the MCW algorithm is introduced, and itsperformances are compared with the Cramer-Rao bound (CRB)and the 2-D MUSIC estimator [7] in Section IV. We briefly con-clude in Section V.

II. SIGNAL MODEL

Let us consider an array of sensors, and let be thevector of the complex envelopes of the received narrow-

band signals at the output of the antennas.We assume that the signals are on the same plane as the array.Each antenna , located at in polar coordinates, isassumed to receive a linear mixture of point sources lo-cated at , where is the DOA, and

is the distance from the origin ( see Fig. 1). Under theseassumptions, the observation vector can be written

(1)

where is the steering vector of the source, is the complex envelope, and is the com-

plex temporally white Gaussian noise vector independent of thesources signals.

The steering vector can be given by variousexpressions, depending on the physical context. In this letter,without loss of generality, we will focus on punctual sensors innear field acoustic sources or electromagnetic sources beyondthe Fresnel area. Taking into account the attenuation over thepropagation, the steering vector of a point source located at

can be written

(2)where denotes the transpose operator, is the imaginaryunit, and the wavelength.

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458 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 6, JUNE 2005

Fig. 1. Position of the problem and notations.

Equation (1) can also be written

(3)

with

(4)

(5)

III. MCW ALGORITHM

A. Bearing Estimation

The MCW bearing-only estimation amounts for concen-trating the classical 2-D MUSIC cost functionwith respect to , where

(6)

denotes the conjugate transpose operator,is the orthogonal projector onto the noise subspace spannedby the columns of , being theth eigenvalue and eigenvector of the covariance matrix

.In this section, the distance of a source is, therefore, con-

sidered as a nuisance parameter. The MCW algorithm relies ona factorization of the steering vector . This factorizationis based on a -order Taylor series expansion of , withrespect to the variable , about

(7)

where denotes the th derivative of with respectto . Equation (7) can be factorized as

(8)

with

(9)

(10)

This factorization is the key point at the basement of the pro-posed algorithm. Although this factorization could be favorablyused in various classes of DOA estimators, we now focus on theextension of the MUSIC algorithm to curved wavefronts.

After some simple algebraic manipulations described in [15],we get

(11)

with

(12)

and

(13)

(14)

where is the smallest eigenvalue of matrix .In simulations, a third-order Taylor expansion of will

be used. With the notations introduced in Fig. 1, we have

(15)

and

(16)

(17)

(18)

with

(19)

B. Range Estimation

The range of the th source can be obtained fromthe estimate . Using [15] and according to (12), the

estimate of the vector cor-responding to the eigenvector associated to the eigenvalue

is the generalized eigenvector ofthe matrix associated to the eigenvalue

[16]. An estimate of can, there-fore, be analytically deduced from the estimates by therelation

(20)

BOYER et al.: SIMPLE ROBUST BEARING-RANGE SOURCE’S LOCALIZATION 459

Fig. 2. Comparison of MCW and MUSIC inverse cost functions for twosources.

Fig. 3. One-source case: DOA standard deviation versus D (� = 40 ,SNR = 40 dB, � � D � 30�, and T = 500 snapshots).

IV. SIMULATIONS

For simulations, we consider a circular antenna composed ofsensors of polar coordinates (

, ). The beamwidth at 3 dB is 28 degrees. Theproposed MCW algorithm is implemented with a third-orderTaylor expansion of .

In the first simulation, we consider two sources and ,with ( , ), ( , ). Thesignal-to-noise ratio (SNR) of both sources is fixed at a valueof 40 dB. The two sources separation is about 1/6 beamwidth.The inverse of the MCW and MUSIC cost functions are plottedin Fig. 2. As expected, the MCW algorithm exhibits two peakscorrectly located. On the contrary, the classical MUSIC algo-rithm fails in estimating and , due to an important far-fielderror model.

Fig. 4. One-source case: range standard deviation versus D (� = 40 ,SNR = 40 dB, 3� � D � 30�, and T = 500 snapshots).

Fig. 5. Two-source case: DOA standard deviation of source 1 versus DOAseparation (with � = �20 , D = 4�, SNR = 20 dB, � = �20 +��,D = 6�, SNR = 10 dB, and T = 500 snapshots).

In the next simulations, we compare the performances of theMCW algorithm, respectively, to the 2-D MUSIC algorithm (in-troduced in [7]) and to the CRB, which provides a lower boundon the covariance matrix of any unbiased estimator [17]. Indeed,as in several other estimators introduced in the literature [7], [8],[10], the MCW algorithm is based on the classical 2-D MUSICalgorithm, and consequently, its statistical performances are,at most, those of the 2-D MUSIC estimator. Simulations will,therefore, provide a quantification of the possible degradationof performances due to the concentration of the 2-D MUSICcost function with respect to DOA. Simulations have been con-ducted over 1000 Monte Carlo realizations and withsnapshots.

We first consider one source with DOA ,dB, and variable range . The standard

deviation of estimated DOA and range are, respectively,plotted in Figs. 3 and 4 as a function of . For both and ,

460 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 6, JUNE 2005

Fig. 6. Two-source case: range standard deviation of source 2 versus DOAseparation (with � = �20 , D = 4�, SNR = 20 dB, � = �20 +��,D = 6�, SNR = 10 dB, and T = 500 snapshots).

the MCW algorithm, as the 2-D MUSIC algorithm, achieve theCRB.

Second, we consider two sources and , with ,, , and . The SNR of both

sources is fixed at a value of 10 dB. The standard deviation errorof estimated DOA and range are, respectively, plotted inFigs. 5 and 6 as a function of the DOA separation of the twosources. Let us note that when is less than 28 degrees, thesources are located in the same lobe. Simulations show that, de-spite the simplicity of the algorithm, its performances are closeto those of the 2-D MUSIC algorithm.

V. CONCLUSION

In this letter, we introduced a new method for bearing-rangeestimation. This method extends classical subspace-basedbearing estimators to curved wavefronts and replaces an ex-haustive 2-D minimization of the 2-D MUSIC cost functionby a 1-D procedure for DOA estimation, with ranges being

analytically deduced from bearing estimates. The proposed ap-proach is illustrated by the introduction of the MCW algorithm.Simulations confirm the good behavior of the method.

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