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Digital Signal Processing 95 (2019) 102579Contents lists available at ScienceDirectDigital Signal localization of near-field sources using a realistic signalpropagation model and optimally-placed sensorsJean-Pierre Delmas a, , Houcem Gazzah b , Mohammed Nabil El Korso cabcTelecom SudParis, Departement CITI, CNRS UMR 5157, Evry, FranceDepartment of Electrical and Computer Engineering, University of Sharjah, 27272, United Arab EmiratesUniversité Paris Nanterre, LEME laboratory EA 4416, Francea r t i c l ei n f oArticle history:Available online 3 September 2019Keywords:Cramer Rao boundsLinear antenna arraysPower profileDirection-of-arrival and range estimationNear-field source localizationArray optimizationa b s t r a c tIn this paper we analyze the estimation of the angle and the range of a narrow-band source located in thenear-field of an arbitrary centro-symmetric linear array (CSLA). This analysis deals with the Cramer Raobound (CRB) on both angle and range, obtained thanks to an exact expression of the source-to-sensordelay and a realistic (range-dependent) model of source-to-sensor attenuation, ultimately achievingtwo objectives. On the first hand, closed-form approximate expressions of the CRB are developed andcompared to those obtained assuming (unrealistically) that sensors perceive the same power despitebeing at different distances from the source. While the impact on angle estimation is negligible, rangeCRB significantly decreases if one incorporates the more appropriate range-dependent power model(except for sources at broadsides). An important consequence is that localization algorithms takingthis range-dependent modelization of the apparent source power into account in their signal modelingshould have much better range performance. On the second hand, the obtained CRBs are used todesign nonuniform CSLA taking into account the ambiguities, with improved angle and range estimation,comparatively to uniform linear arrays (ULA). Finally, we show that our optimized CSLA for a singlesource also brings some benefits for two closely-spaced sources. 2019 Elsevier Inc. All rights reserved.1. IntroductionCRBs are usually used to benchmark parameter estimation algorithms. Furthermore, if interpretable expressions are obtained, theycan be used to optimize the system design, for instance, to minimize the variance of the estimated parameters (see e.g., [1,2]). Inthe particular context of source localization, much effort has beenmade to the far-field case for decades (see e.g., [3–5] and references therein) where the distance of the source to the array islarge compared with the array aperture, and hence the propagating waves are considered to be plane waves at the sensor arrayand only the source direction of arrival (DOA) can be estimated.It is possible to estimate the range (distance from the sourceto the array) if this distance is not too large compared to thearray aperture. This near-field situation occurs in many practicalapplications such as sonar [6], speaker localization [7], electronicsurveillance [8], object detection [9], collision avoidance radar [10],robot navigation [11], seismic exploration [12], biomedical imag-*Corresponding author.E-mail addresses: (J.-P. Delmas), (H. Gazzah), (M.N. El 51-2004/ 2019 Elsevier Inc. All rights [13], [14], seismic exploration [15], etc. In this near-field case,wavefronts are spherical and received power varies from sensorto sensor. However, to reduce the complexity of the localizationalgorithms, an approximate propagation model relevant to the socalled Fresnel zone has been used. This latter makes use of thesecond-order Taylor expansion of the time delay parameter, withconstant amplitude gain however. Numerous methods have usedthese approximations, such as a polynomial rooting approach [16],an high-order ESPRIT algorithm [17], a weighted linear predictionmethod [18], an ESPRIT/MUSIC procedure exploiting subarrays [19],a two-stage MUSIC algorithm [20], a least-square procedure [21], aprediction and oblique projection operator method [22] and manyother approaches. Furthermore, these approximations facilitate theCRB derivations (see e.g., [23]).Only lately the exact time delay and range-dependent modelization of the apparent source power (called also power profile) have been used [24], but only to derive a complicated noninterpretable approximate expression of the near-field CRB for theULA case. We consider here arbitrary CSLA made of pairs of sensorssymmetrically located along the two sides of the linear antennaarray. Such class of nonuniform linear arrays are chosen for theirattractive features proved in [25] for constant amplitude gains. Thisincludes lower DOA and range CRBs and faster convergence to the

2J.-P. Delmas et al. / Digital Signal Processing 95 (2019) 102579lower far-field DOA CRB. Furthermore, thanks to the decoupling between the DOA and range parameters to the second-order w.r.t. theinverse of the range in the Fisher information matrix, the derivation of closed-form approximate expressions of the CRB is greatlysimplified. Note that we use a definition of the near-field that isfamiliar in the signal processing literature, designating the regionwhere range estimation makes sense (to be distinguished from thereactive and radiative region, as understood in electromagnetism[26, ch.2].In this paper, we first develop interpretable and accurateclosed-form approximate expressions of the CRB for both sourceangle and range. They are compared to those unaware of dependence of received power on source range. These expressionsare proved to depend only on three geometric parameters only:the second, fourth and sixth-order moments of the positions ofthe sensors forming the arbitrary but centro-symmetric linear array (CSLA). The obtained expressions tend to prove that the CRBon the angle is generally barely impacted by the power profile.In contrast, the CRB on the range is strongly reduced, except atbroadside directions, i.e. almost everywhere. Second, thanks tothese closed-form expressions we design nonuniform CSLA withimproved range estimation (by as much as 60%) with identical CRBon the angle with respect to ULAs. This design also incorporatesgeometric constraints to account for the array ambiguity problem.Specifically, these constraints lead to a constrained max-min problem. We use its equivalent to a global polynomial maximizationunder, both polynomial equalities and inequalities which can beefficiently solved using the Matlab GloptiPoly utility [36]. Finally,we show that our optimized CSLA for a single source also bringssome benefits for both DOA and range estimation in the context oftwo closely-spaced sources.The paper is organized as follows. Section 2 introduces the datamodel. After giving the general expression of the deterministic andstochastic CRB concentrated on the localization parameters, we develop an interpretable closed-form approximate expressions of theCRB on both angle and range in Section 3. Section 4 is dedicatedto analytical comparisons of these CRBs to the CRBs not takingthe power profile into account. These closed-form expressions areused in Section 5 to design nonuniform CSLA with improved rangeestimation and immunity against array ambiguities. Finally, a conclusion is given in Section 6.2. Data modelWe consider a linear (possibly nonuniform) antenna array madeof P sensors C 1 , · · · , C P depicted in Fig. 1, located along a straightline at coordinates x1 , · · · , x P , respectively. Without loss of generality, we assume the array centroid to be at the origin O of thisaxis. This choice allows for more compact expressions of the CRBcompared to [23,24].A source is located at point S, at a range r from the plane origin O , and forming an angle θ w.r.t. the axis perpendicular to thearray. This single source is emitting a narrow band signal of wavelength λ with no multipath so that the complex baseband snapshotcollected by the sensor p at time index t readsy p (t ) g p e i τ p s(t ) n p (t ),(1)where s(t ) and n p (t ) represent, respectively, the source signal collected at the origin and the ambient additive noise collected bysensor p. The exact expression of the phase τ p is defined asτ p 2π ( S O SC p )/λ. Using the law of cosine, it is rewritten asτ p 2πwithr λ1 βp(2)Fig. 1. Source in the near-field impinging on an arbitrary linear array.defβp 1 2xprsin θ x2pr2(3).Note that, because we fix the phase and amplitude references atthe centroid of the array, our definition of the tuple (θ, r ) is different from the one in [24], which fixes the phase reference at thefirst sensor. Regarding the gain g p at sensor p, we assume a spherical wavefront and a specific range-dependent power profile, wherethe signal magnitude is inversely proportional to the distance fromthe source [26, Chap.2]:gp SOSC p1 βp(4).Thus the sensed power is variable from sensor to sensor. Again,because we use the origin as reference, our definition of g p is different from the one used [24], for which the gain is not definedwith respect to a reference: g p SC1 1 .pr βpWe collect N snapshots { y p (t )} p 1,., P ;t t1 ,.,t N , to estimateboth angle θ and range r. Estimation accuracy is evaluated in termsof the CRB, developed under the following commonly used assumptions about signal and noise [27]:(i) n p (t ) and s(t ) are independent,(ii) {n p (t )} p 1,., P ;t t1 ,.,t N are independent, zero-mean circularGaussian distributed with variance σn2 ,(iii) {s(t )}t t1 ,.,t N are assumed to be either deterministic unknownparameters (the so-called conditional or deterministic model) withσs2 N1 nN 1 s(tn ) 2 , or independent zero-mean circular Gaussian distributed with variance σs2 (the so-called unconditional orstochastic model).3. Expressions of the CRB3.1. Theoretical general background on CRB for near-field sourcesWe focus on a single near-field source whose location is characterized by the parameter of interestα [θ, r ]T .(5)When the sensed power is constant across all sensors, stochasticand deterministic matrix-valued CRBs (concentrated on the parameter of interest) are equal, up to a multiplicative term dependingonly on the SNR σs2 /σn2 of the source and the number P of sensors [25]. This contrasts with the case where the power profile istaken into account for which the multiplicative term depends onα: CRBsto (α ) 1 σn2CRBdet (α ), a(α ) 2 σs2(6)where a(α ) is the steering vector of components g p e i τ p , p 1, ., P . Obviously, the expression of the stochastic CRB can nolonger be decoupled in power and geometric terms. Instead, theseCRBs are given by the following expressions:CRBsto (α ) c σsto (α )F 1 (α ) and CRBdet (α ) c σdet (α )F 1 (α ), (7)where both CRBs appear to be inversely proportional to the matrix

J.-P. Delmas et al. / Digital Signal Processing 95 (2019) 102579 F(α ) Re a(α ) 2 D H (α )D(α ) D H (α )a(α )a H (α )D(α ) ,(8)through the constantsdefc σsto (α ) 2 2σn2 (σn2 a(α ) 2 σs2 )def a(α ) σndetandc(α) σ2N σs42N σs2where D(α ) is defined as a(α ) a(α ). We note that whenever θ , rP σs2 σn2 , we havec σsto (α ) c σdet (α ).(9)The above condition means that the source is more powerful thanambient noise, which is more relevant to near-field sources. Wemaintain this assumption and realize that the stochastic CRB reduces to the deterministic CRB, on which we focus our attentionfrom now on. In this case, the elements of the 2 2 matrix F(α )given in (8) reads [F]i , j P g 2p p 1Pp 1g p ,i g p Pτ2 p ,i g pτ2 p, j g p , i 1, 2,(10)p 1def gdef gwhere g p ,1 θp , g p ,2 rp ,3.3. Taylor expansion of CRB(θ) and CRB(r ) and key geometricparametersIn the process of deriving the new CRB expressions, we identifythe following three key geometric parameters (S 2 , κ , η ) wheredefκ P S4S 22anddefη P 2 S6(14)S 23(15)we prove in the Appendix A.2, the following second-order expansions:CRB(θ) def τ pτ p ,1 α 1), the derivation of (11), (12) and (13)Pr Pwithfrom (10) would give even more complicated expressions difficultto exploit. S2 a(α ) 2 P 1 2 (1 4 sin2 θ) o(r 2 ) ,g p, j g p p 1 p 1 P αSOSC pgp play an important role in the array processing performance as wellthe antenna design. Using results (11)-(13) and (7) in which wereplaceg p ,i g p , j τ p ,i τ p , j g 2p p 1defwhere lim 0 o( )/ 0 with x1 , .x P and λ fixed and λπ 2λπ .These expressions are to be compared to the expressions [25, rels.(7-9)] which can be retrieved from (10) by setting g p ,i 0, i 1, 2 to account for constant gain. Compared to [25, rels. (7-9)],(11), (12) and (13) are much more intricate because they include the powerprofile. Note that for more general profiles (i.e., P3 θ anddef τ p rτ p ,2 .c λ2πP S 2 cos2 θ( 2κ S 2 λ2π ) 1 P3.2. Taylor expansion of the matrix F(α )sin2 θg (sin2 θ,Pr2S 2 , λ2π , κ ) o(r 2 ) ,(16)To highlight the impact of the power profile on the localizationperformance, we use, similarly to [25], a Taylor expansion of thematrix F(α ). To this end, we consider arbitrary CSLA, where foreach sensor placed at coordinate x p , there exists a symmetric sensor placed at coordinate x p . As a consequence, sums of the form Pdefr 2 cos2 θ [F(α )]1,1 Pr4S2 4c λ2π S 22 4Pλ2π 1 P2k 12k, k 0, 1, . are zero, while constants S 2k p 1 x pp 1 x p ,characterizing the array geometry, will appear in the followingTaylor expansion of the matrix F(α ). After tedious algebraic manipulations (main steps are shown in Appendix A.1), we obtain:λ2πCRB(r )S2sin2 θ (κ 1) cos4 θh(sin2 θ, S 2 , λ2π , κ , η)r4r sin θ cos θ o(r 4), 2P λπ S 2 (5P S 4 3S 22 ) cos2 θr4 o(r 4 ),λ2π [F(α )]2,2 24P λπ S 2 sin θ ( P S 4 2defc σn22N σs2P(18) 4P S 2 λ2π [(6 7κ ) sin2 θ (5κ 3)] (19)2(12)S 22 ) cos4 θ4r 4λπ [ P S 4 (1 15 sin θ 24 sin4 θ) S 22 (1 7 sin2 θ 12 sin4 θ )]and where h(sin θ, S 2 , λ2π , κ , η) is an intricate function given inthe Appendix A.2. We note that its expression is not useful to givea good numerical approximation of CRB(r ) as it will be shown inFig. 3, where the approximate value deduced from (17) and (23) isvery close to the approximate value deduced from only the ratioof the dominant terms of (17) and (23) given by (27).22r62 (17)g (sin2 θ, S 2 , λ2π , κ ) 4P 2 λ4π S 22 κ (13κ 9) cos4 θ2[F(α )]1,2 ) ,where(11)2 λπ o(r 2andr2P λ2π S 2 2P S 4 (6 sin2 θ 1) S 22 (1 5 sin2 θ)2r2 46P S 6 ( 5 57 sin θ 99 sin θ 47 sin θ)4.1. Far-field vs near-field performance8r 6S 2 S 4 (5 49 sin2 θ 83 sin4 θ 39 sin6 θ)8r 6 o(r 6 ),4. Analysis of the CRB(13)We recall the deterministic far-field DOA CRB for arbitrary linear arrays [28, rel. (7)]

4J.-P. Delmas et al. / Digital Signal Processing 95 (2019) 102579λ21CRBFF (θ) N 8π2Sσn2,σs2cos2 θ2(20)allowing one to rewrite (16) as follows ( 2κPS 2 λ2π ) CRB(θ) CRBFF (θ) 1 o(r 2 ) ,sin2 θg (sin2 θ,Pr2S 2 , λ2π , κ )(21)which is consistent with (20) for r tending to infinity where theeffect of the power profile disappears.4.2. Constant gain vs range and angle-dependent gainOn the one side we have the near-field angle (16) and range(17) CRBs. On the other side we have the near-field angle CRBCG (θ)and range CRBCG (r ) assuming constant gain (not taking in accountthe power profile, i.e., g p 1). The former are given by [25, rels.(15-16)]:CRBCG (θ) c λ2π1 P S 2 cos2 θ 2 o(r ) ,CRBCG (r )r4 κ S2 1 1 P r2 4κFig. 2. Exact and approximate CRB(θ)/CRBCG (θ) as a function of θ . 2κ 1sin θ(22) 4c λ2π21 k(sin θ, κ , η)S 22 (κ 1) cos4 θS2P r2 o(r 2 ) ,(23)wheredefk(sin2 θ, κ , η) (2 18κ 2 3κ 23η) sin2 θ 3(η κ ).κ 1(24)c λ2πP S 2 cos2 θObviously, the dominant termsof (16) and (22) areequal, implying that the near-field DOA CRB is barely affected bythe power profile for ranges that are not too small. To go further,we look into the second-order term (in 1/r 2 ) in (16) and (22). Forexample for θ 0, we get:CRB(θ) θ 0CRBCG (θ) θ 0κ S2where1 r2κ S2P λ2π o(r 2 ),(25) λπ is in practice positive. Indeed κ 1 [25] implies 2 λ2π 1P pP 1 x2p 4λπ 2 and the condition 1P pP 1 x2p 2Pκ S2that 1 Pλ2is in practice satisfied given the non-ambiguity and apertureconstraints.For example, for a ULA with half-wavelength spacing and P 2Q ,4π 21PPx2p λ2 (4Q 2 1)48p 1 λ24π 2from Q 1.(26)Consequently CRB(θ) is slightly larger than CRBCG (θ) for broadsidedirections (i.e., θ 0).A similar comparison of the near-field range CRB is expressedby the following ratio of the dominant terms of (17) and (23) forarbitrary angle θCRB(r )CRBCG (r ) 1 4P λ2πsin2 θS2(κ 1) cos4 θ 1(1 o(r 1 )).(27)Fig. 3. Exact and approximate CRB(r )/CRBCG (r ) as a function of θ .From the above, the dominant term of CRB(r ) is always smallerthan the dominant term of CRBCG (r ), except for θ 0, for whichthey are equal. In particular for array end-fire directions (i.e., θ π /2), CRB(r ) is much lower than CRBCG (r ). Consequently,taking into account this power profile allows one to achieve better range estimation without deteriorating angle estimation. Thisis explained by a larger sensitivity of the gain to the range withrespect to the angle, for the end-fire directions. Furthermore, forthese directions the time delay profile is less sensitive than thepower profile for the range.These results are confirmed in Figs. 2 and 3 which show respectively the ratios CRB(θ)/CRBCG (θ) and CRB(r )/CRBCG (r ), as a function of the angle θ [0, π /2). There, we assume a ULA of 6 sensorswith half-wavelength inter-sensors spacing for a source at ranger 10λ. We see, in particular, that CRB(θ)/CRBCG (θ) [0.83, 1.03],whereas CRB(r )/CRBCG (r ) strongly decreases in [0, π /2), takingvalues 1, 0.3 and 0.005 for θ 0, 60 and 80 , respectively. Thesefigures also show a good agreement between the approximate expressions of the CRBs deduced from (16), (17) and (22), (23) andthe exact ones (deduced from the exact expression of the matrix F(7), (8) and (10). Furthermore, we see in Fig. 3 that the dominantterms of (17) and (23) given in the ratio (27) show also a goodapproximation of CRB(r )/CRBCG (r ).

J.-P. Delmas et al. / Digital Signal Processing 95 (2019) 102579λ2π4P S24P5sin2 θ (κULA 1) cos4 θ2λπS2sin2 θ (κ 1) cos4 θ 1.(29)We note that 4P λ2π / S 21 for P 4. Thus for values of θ not inthe vicinity of π /2, we have the following approximationR P (κ ) Fig. 4. CRB(r )/r 2 and CRBCG (r )/r 2 with N 1000 andσs2 /σn2 20 dB.On the other hand, Fig. 4 shows the relative CRBs on the rangeCRB(r )/r 2 and CRBCG (r )/r 2 , assuming the same ULA used abovewith which we collect N 1000 snapshots and σs2 /σn2 20 dB.We see clearly in this figure that taking into consideration thepower profile allows to enlarge the domain of possible range estimation but not for broadside directions.As a result, the localization algorithms will benefit from incorporating the power profile into the parameterization of the steering vector. This can be achieved in two ways which are outsidethe scope of this paper. In the first one, the localization algorithmswould use the exact parameterization (2)–(4) of the steering vector. In the second one, they can use the traditional constant gainquadratic wavefront approximation model to take advantage of thelow computational algorithms, but with some correction methodstaking the exact parameterization model [29].5. Near-field array optimizationWhen the source is in the array far-field, DOA estimation performance, as expressed in (20), depends fully on the geometricparameter S 2 . But when the source is in the array near-field, additional parameters appear in (16) and (17), including the geometricparameters κ (for DOA estimation) and both κ and η (for range estimation). By focusing on the dominant terms in (1

J.-P. Delmas et al. / Digital Signal Processing 95 (2019) 102579. lower far-field DOA CRB. Furthermore, thanks to the decoupling be-tween the DOA and range parameters to the second-order w.r.t. the inverse of the range in the Fisher information matrix, the deriva-tion of closed-form approximate expressions of the CRB is greatly simplified.

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