Wenmeng Xiong

José Picheral

Sylvie Marcos

PERFORMANCE ANALYSIS OF MUSIC IN THE PRESENCE OF MODELING ERRORS DUE TO THE SPATIAL DISTRIBUTIONS OF SOURCES

Keywords: array signal processing, distributed sources, model error, performance, MUSIC

In this paper, the direction of arrival (DOA) localization of spatially distributed sources impinging on a sensor array is considered. The performance of the well known MUSIC estimator is studied in the presence of model errors due to angular dispersion of sources. Taking into account the coherently distributed source model proposed in [1], we establish closed-form expressions of the DOA estimation error and mean square error (MSE) due to both the model errors and the effects of a finite number of snapshots. The analytical results are validated by numerical simulations and allow to analyze the performance of MUSIC for coherently distributed sources.

INTRODUCTION

The effects of model errors on DOA estimation has been widely studied in the literature with the sources assumed to be far-field point transmitters or reflectors [START_REF] Weiss | Effects od modeling errors on the resolution threshold of the MUSIC algorithm[END_REF]. In many applications, for instance, aero-acoustic sources imaging [START_REF] Chu | Robust Bayesian super-resolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF], ocean waves parameter estimation [START_REF] Young | Cross-shore decay of cliff top ground motions driven by local ocean swell and infragravity waves[END_REF], bio-medical source localization [START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF] or also mobile communication [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF], the physical sources can no longer be considered as point sources and a spatially distributed model of the sources could be more appropriate.

The models for spatially distributed sources have been classified into two types, namely incoherently distributed (ID) sources and coherently distributed (CD) sources. On one hand, for ID sources, signals coming from different points of the same distributed source can be considered uncorrelated [START_REF] Zoubir | Efficient subspace-based estimator for localization of multiple incoherently distributed sources[END_REF] [START_REF] Hu | An ESPRITbased approach for 2-d localization of incoherently distibuted sources in massive MIMO SYSTEMS[END_REF]. On the other hand, in the scenario of CD sources, the received signal components are delayed and scaled replicas from different points of the same one [START_REF] Souden | A two-stage approach to estimate the angles of arrival and the angular spreads of locally scattered sources[END_REF] [START_REF] Zoubir | Efficient DSPE algorithm for estimating the angular parameters of coherently distributed sources[END_REF].

In the literature, different techniques propose to estimate the nominal DOA of distributed sources but they generally require that the shape of the dispersion is known which may not be true in practice. Even if it is the case, a mismatch on the angular spread dispersion between the real signals and the models will bring estimation errors on the DOA estimates.

In previous works, the model errors have been considered, in the case of punctual sources, as uncertainties on the array elements positions or patterns, or inter-element coupling, and are described as random variables [START_REF] Swindlehurst | A performance analysis of subspace method in the presence of model errors : Case of MUSIC[END_REF] [START_REF] Kangas | Finite sample and modeling error effects on ESPRIT and MUSIC diretion estimators[END_REF] [13] [START_REF] Marcos | Les méthodes à haute résolution, traitement d'antenne et analyse spectrale[END_REF]. For ID sources, the performance analyses have been investigated in the presence of the model error as a mismatch between the angular dispersion of the actual sources and the MUSIC estimator in [START_REF] Astély | The effects of local scattering on direction of arrival estimation with music[END_REF], and the ultimate asymptotic performance, that is to say, the lower bound of the estimation covariance has been investigated in [START_REF] Raich | Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms[END_REF]. We here focus on the CD sources model proposed in [START_REF] Valaee | Parametric localization of distributed sources[END_REF], which is well adapted to the applications such as the aero-acoustic sources imaging [START_REF] Chu | Robust Bayesian super-resolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF]. The MUSIC estimator has been extended for a model with an angular dispersion, and the model error refers to a mismatch between the angular dispersion of the actual sources and that of the extended-MUSIC estimator in our scenario.

In this paper, we first propose a new approach to obtain an accurate approximated analytical expression of the DOA estimation error and MSE. Then, the finite sample effects due to the estimation of the received data covariance matrix will be investigated. Finally, assuming that the shape of the angular distribution is known, an analytical expression of the DOA estimation error as a polynomial function of the angular spread dispersion error is proposed. This expression allows to see explicitly the influence of the model error on the DOA estimation performance, and could be useful in a future work to optimize the antenna parameters in order to reduce the DOA estimation error due to the angular spread of the sources.

The organization of this paper is as follows. The signal model and 1D-DSPE are given in section 2. In section 3, the sensitivity of the estimator is theoretically analyzed. Numerical simulations are presented in section 4 to validate the analytical expressions of the previous section. Finally, conclusions are given is section 5.

SIGNAL MODEL AND 1D ESTIMATOR FOR DISTRIBUTED SOURCES

Let us consider q spatially CD far-field sources impinging on an array of M sensors. The q sources and the M signals received at the array at moment t are denoted by s(t) = [s1(t), . . . , sq(t)] T and y(t) = [y1(t), . . . , yM (t)] T , respectively, with:

y(t) = C(θ)s(t) + n(t), (1)
where n(t) ∈ C M ×1 represents the complex Gaussian distributed additive noise, C(θ) = [c h 1 (θ1), . . . , c hq (θq)] ∈ C M ×q is the array steering matrix composed of q steering vectors c hi (θ) that can be written as proposed in [START_REF] Valaee | Parametric localization of distributed sources[END_REF] by:

c hi (θ) = π 2 -π 2 a(θ + φ)hi(φ)dφ, (2)
where i = 1 . . . q, and a(θ) is the steering vector for a point source which arrives from the DOA θ. In the most general case, the steering vector a(θ) is also a function of the array geometry, the sensor gains, the form of the wavefront, and other possible parameters which are supposed to be known. The function h(φ) is introduced to describe the angular spread distribution (for instance, Uniform and Gaussian distributions) and it can be parameterized by an angular dispersion ∆ which is omitted in the notation. The source signals and the additive noise are considered to be centered Gaussian independent random variables. Assuming that signals and noises are uncorrelated and the sources are uncorrelated with each other, the correlation matrix is given by:

Ry = E[yy H ] = CRsC H + σ 2 b I, (3)
where E[.] is the expectation operator, Rs and σ 2 b are the source covariance matrix and the noise variance, respectively.

The DSPE method proposed in [START_REF] Valaee | Parametric localization of distributed sources[END_REF] is a 2D MUSIC-like estimator allowing to jointly estimate the two parameters θ and ∆ by maximizing the criterion

1 c H h (θ,∆)V b 2
, where V b is the matrix of the M -q eigenvectors associated to the smallest eigenvalue of Ry and which spans the noise subspace. However, the form of h(φ) may not be known in practice. In the following, we will consider that the shape of h(φ) and ∆ are imperfectly known and that the only parameter to estimate is θ. We will then investigate the effect of an imperfect knowledge of h(φ) and ∆, it is to say of c h (φ), on the 1D-DSPE algorithm therefore denoted by:

θi = arg max θ 1 c H h (θ)V b 2 (4)

PERFORMANCE ANALYSIS

General case

The data covariance matrix Ry0 = E[y 0 y H 0 ] is estimated from N snapshots {y 0 (tn)}n=1,...,N by Ry0 ≈ 1 N N n=1 y 0 (tn)y H 0 (tn). It follows that the DOA estimator of θi is :

θi = arg max θ 1 c H h (θ) Vb0 2 , ( 5
)
where Vb0 is associated to the eigendecomposition of:

Ry0 = Vs0 Λs0 VH s0 + Vb0 Λb0 VH b0 , (6)
and c h (θ) is given by (2) but with a form of the angular spread distribution h(φ) which may be different from the true distribution of the actual sources h0i(φ). Note that the parameters related to the true values are indexed by 0 to distinguish them from parameters used by the estimator. The theoretical covariance matrix of the signal related to the actual steering matrix C h0 (θ) is thus defined as Ry0 = Vs0Λs0V H s0 + σ 2 b V b0 V H b0 , and we write Ry0 = R yh + ∆R yh , where R yh would correspond to the covariance matrix of the signal related to a steering vector c h (θ) and where

R yh = V sh Λ sh V H sh + σ 2 b V bh V H bh .
We introduce the error on the steering vector model ∆c = c h (θi) -c h0i (θi). We also note ∆V bh = V b0 -V bh where ∆V bh is an error on the noise eigenmatrix due to the model mismatch between c h and c h0 . Let us also introduce the error due to the finite number of snapshots N on the estimation of the data covariance matrix ∆Ry0 = Ry0 -Ry0 and the error on the noise corresponding eigenmatrix ∆V b0 = Vb0 -V b0 . The error due to the finite number of snapshots on the noise subspace projector is

∆Π b0 = Vb0 VH b0 -V b0 V H b0 , with Π b0 = V b0 V H b0 .
According to [START_REF] Weiss | Effects od modeling errors on the resolution threshold of the MUSIC algorithm[END_REF], the relation of ∆Π b0 and ∆R b0 is:

∆Π b0 = -Π b0 ∆Ry0Q -Q∆Ry0Π b0 , (7)
where

Q = Vs0(Λs0 -σ 2 b I) -1
Vs0, I is a q × q identity matrix. According to [START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF], the DOA estimation θi satisfies that the first order derivative of the denominator of ( 5) equals zero which yields:

2Re{ċ H h ( θi) Vb0 VH b0 c h ( θi)} = 0, (8)
where ċh ( θi) = ∂c(θ) ∂θ | θi . Assuming that, θi is not far from θi, we introduce the second order Taylor series approximations of c h (θ) and ċh (θ) in ( 8), and exploiting [START_REF] Zoubir | Efficient subspace-based estimator for localization of multiple incoherently distributed sources[END_REF] to substitute Vb0 Vb0 yields:

A(θi)∆θ 2 i + B(θi)∆θi + C(θi) = 0, (9)
where the terms of order greater than 2 in ∆θi have been neglected, and the scalar A, B, C are defined with B = B1 + B2 and C = C1 + C2:

A(θi) =Re 1 2 ċh (θi) H V b0 V H b0 ch (θi) + ch (θi) H V b0 V H b0 ċh (θi) + 1 2 ... c h (θi) H V b0 V H b0 c h (θi) , B1(θi) =Re ċh (θi) H V b0 V H b0 ċh (θi) + ch (θi) H V b0 V H b0 c h (θi) , E[∆θ 2 i ] = B1(θi) 2 -2A(θi)C1(θi) -B1(θi) B1(θi) 2 -4A(θi)C1(θi) 2A(θi) 2 + σ 2 b N 1 4A(θi) 2 - B1(θi) 4A(θi) 2 B1(θi) 2 -4A(θi)C1(θi) ϕ(θi) + χ(θi) 2A(θi) B1(θi) 2 -4A(θi)C1(θi) , (11)
where:

ϕ(θi) =Re 4ċ(θi) H V b0 V H b0 ċ(θi) ċ(θi) H QRy0Qċ(θi) + 4c(θi) H V b0 V H b0 ċ(θi)c(θi)QRy0Qc(θi) + c(θi) H V b0 V H b0 c(θi)c(θi)QRy0Qc(θi) , χ(θi) =Re 2ċ(θi) H V b0 V H b0 ċ(θi)ċ(θi)QRy0Qc(θi) + c(θi) H V b0 V H b0 ċ(θi)c(θi)QRy0Qc(θi) . ( 12
) B2(θi) =Re 2ċ h (θi) H V b0 V H b0 ∆Ry0Qċ h (θi) +c h (θi) H V b0 V H b0 ∆Ry0Qc h (θi) , C1(θi) =Re ċh (θi) H V b0 V H b0 c h (θi) , C2(θi) =Re ċh (θi) H V b0 V H b0 ∆Ry0Qc h (θi) .
The expression of ∆θi can be obtained by solving the 2 nd order equation ( 9), and taking into account the convexity of the criterion to select the right root, so that:

∆θi = -B(θi) + B(θi) 2 -4A(θi)C(θi) 2A(θi) . (10)
In addition, according to the formulas in [START_REF] Marcos | Les méthodes à haute résolution, traitement d'antenne et analyse spectrale[END_REF], the MSE can be expressed by [START_REF] Swindlehurst | A performance analysis of subspace method in the presence of model errors : Case of MUSIC[END_REF], as shown at the bottom of the previous page, exploiting the fact that B2(θi) and C2(θi) are random terms related to ∆Ry0, while A(θi), B1(θi), C1(θi) are deterministic, and using E[∆Ry0] = 0.

It is interesting to see that the MSE is composed of two terms, one depends only on the model error, the other depends on the model error but with a factor σ 2 /N . It can be expected that the second term will be negligible when N increases.

First order approximation

In this subsection, we discuss the situation that the estimation error ∆θi is small enough, so that the second order terms in ∆θi can be negligible with respect to the first order terms. Keeping only the first order terms in [START_REF] Souden | A two-stage approach to estimate the angles of arrival and the angular spreads of locally scattered sources[END_REF], replacing V b0 by V bh + ∆V bh , neglecting the second order terms in ∆V b0 ∆θi and ∆V bh ∆θi, and using R yh V b0 = V b0 Λ bh , the estimation error can be simplified as:

∆θi = Re{ċ H h (θi)V bh V H bh ∆c} ċH h (θi)V bh V H bh ċh (θi) + Re{ċ H h (θi)V bh V H bh ∆Ry0Qc h (θi)} ċH h (θi)V bh V H bh ċh (θi) . ( 13
)
The MSE can be given by:

E[∆θ 2 i ] = Re{ċ H h (θi)V bh V H bh ∆c} ċH h (θi)V bh V H bh ċh (θi) 2 + σ 2 b 2N • Re{ċ H h (θi)V bh V H bh ċh (θi)c H h (θi)Ry0Qc h (θi)} (ċ H h (θi)V bh V H bh ċh (θi)) 2 . ( 14
)
As already noted for [START_REF] Swindlehurst | A performance analysis of subspace method in the presence of model errors : Case of MUSIC[END_REF], the second term decreases when N increases, which is not the case for the first terms of ( 13) and ( 14). Note that this simplification allows to express the first terms of ( 13) and ( 14) as a function of the model error ∆c.

Estimation error as an explicit function of the model error

In this subsection, we assume that the covariance matrix is perfectly known and we only study the deterministic terms of (10) so that B2 and C2 are null. We assume that in this case, the shape of h related to the actual signals sources is known, where all the sources have the same shape and the same angular spread dispersion ∆0. The model error is therefore caused by the error on ∆ or by the fact that the sources are assumed to be points when the classical MUSIC estimator is used. Assuming again that ∆ is not far from ∆0, and noting δ = ∆-∆0, we can introduce the second order Taylor series approximations in δ:

c h (θi) ≈ c h0 (θi) + δg h01 (θi) + 1 2 δ 2 g h02 (θi), (15)
where

g h01 (θi) = ∂c h0 (θ i ) ∂∆ |∆ 0 , g h02 (θi) = ∂ 2 c h0 (θ i ) ∂∆ 2
|∆ 0 . We can pay attention that g h01 and g h02 reveals the sensibility of our model to the variation of the angular dispersion of the actual signal. The estimation error can be thus given by:

∆θi = -(f4 + f5δ + f6δ 2 ) + x + yδ + zδ 2 + wδ 3 + kδ 4 2(f1 + f2δ + f3δ 2 ) , (16)
where fi(i = 1, . . . , 5) and x, . . . , k can be expressed in function of V b0 , derivatives of g and c. Details can be found in [START_REF] Xiong | Performance analysis of MUSIC for spatially distributed sources[END_REF].

Ignoring the second terms in ∆θi in ( 9) and extending Taylor series in δ to third order terms for an accurate approximation, the estimation error can be further expressed as a polynomial in δ:

∆θi = α(θi, ∆0)δ + β(θi, ∆0)δ 2 + γ(θi, ∆0)δ 3 , (17)
where α, β and γ are functions depending on the source signals and sensor parameters (see [START_REF] Xiong | Performance analysis of MUSIC for spatially distributed sources[END_REF] for details).

NUMERICAL SIMULATIONS

In this section, numerical examples are presented to illustrate the validity of the analytical results of the estimation performances established in section 3. In all simulations, a uniform linear array composed of M = 10 sensors spaced by d = λ/2 is considered, SN R = 10dB, and N = 1000 snapshots. Figure 1 exhibits the estimation error or bias on θ for two Uniform distributed sources at θ1 = 21 • and θ2 = 39 • , respectively, with an angular dispersion ∆0 = 10 • . The model error due to the dispersion parameter ∆ has been varied to study its effect on the DOA's estimation accuracy. As expected the bias is null when the exact model parameter ∆ = ∆0 = 10 • is used. We can observe that the estimation error is smaller when the parameter ∆ is smaller than ∆0, than when it is larger than ∆0. Focusing on the validity of the expression derived in the previous section, we can notice that for θ2, the DOA estimation error ∆θ2 obtained in [START_REF] Zoubir | Efficient DSPE algorithm for estimating the angular parameters of coherently distributed sources[END_REF] and the one obtained in [START_REF] Ferréol | Statistical analysis of the MUSIC algorithm in the presence of modeling errors, taking into account the resolution probability[END_REF] both match the simulation results. However, for θ1, which has an estimation error much bigger than θ2 when the model error increases, the result obtained in [START_REF] Zoubir | Efficient DSPE algorithm for estimating the angular parameters of coherently distributed sources[END_REF] with the second order approximation outperforms the one in [START_REF] Ferréol | Statistical analysis of the MUSIC algorithm in the presence of modeling errors, taking into account the resolution probability[END_REF] with the first order approximation.

In Figure 2, the RMSE of θ evaluated with 100 Monte Carlo simulations is plotted versus the snapshot number, with the model parameter set to ∆ = 15 • . The MSE decreases as well as N increases, and then converges to a non zero value whose expression is given in [START_REF] Swindlehurst | A performance analysis of subspace method in the presence of model errors : Case of MUSIC[END_REF]. This reveals that when there are two perturbations, the finite number of snapshots effect dominates in the case of a small number of snapshots whereas the model error effect dominates when the snapshots number is large. Again we can see that as in Figure 1, the result with the second order approximation outperforms the one with the first order approximation.

Figure 3 illustrates the approximation obtained in ( 16) and [START_REF] Xiong | Performance analysis of MUSIC for spatially distributed sources[END_REF], in order to express the estimation error as an explicit function of the model error, in the same previous scenario. We can observe that, in this case, [START_REF] Raich | Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms[END_REF] fits better the simulation results, while Figure 4 shows the results of the absolute value of the DOA estimation error when the model error due to the dispersion parameter and the angle between the two sources both vary, with θm = 1 2 (θ1 + θ2) = 30 • . The green stars mark the region where the sources are not resolved, that is to say, the two sources are so close that the criterion 1D-DSPE gives the false appearance that there is only one source in the middle. These results allow to highlight two behaviors of the estimators: firstly, the closer the sources are, the more the model error impacts on the estimation accuracy; secondly, when the model error is small enough, high resolution is achieved. Taking into account that a distributed estimator with ∆ = 0 • means the classical estimator MUSIC with a point source model, and ∆ = 10 • is the 1D-DSPE without model error, the advantage of a distributed estimator with respect to a classical MU-SIC is highlighted. In addition to the fact that the theoretical results correspond to the simulation results, we can observe that again the result obtained in [START_REF] Zoubir | Efficient DSPE algorithm for estimating the angular parameters of coherently distributed sources[END_REF] works better than the result obtained in [START_REF] Ferréol | Statistical analysis of the MUSIC algorithm in the presence of modeling errors, taking into account the resolution probability[END_REF], even in the case where there are resolution problems.

CONCLUSION

In this paper, we have investigated the effects of both the angular dispersion of the source and the finite number of snapshots on the behavior of the DOA MUSIC-based estimator. New analytical expressions of the DOA estimation error and MSE as a function of these two perturbations have been given. Particularly, in the special case when the theoretical covariance matrix is available, expressions as an explicit function of the model error is proposed, which gives an easier way to analyze the influence of a model error, or to optimize the array configurations to reduce the DOA estimation error. Simulations which are carried out are in adequacy with the proposed theoretical results. The performance of MUSIC for coherently distributed sources can thus be analyzed.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: DOA estimation error vs. angular dispersion model error (2 sources with Uniform angular dispersion, ∆0 = 10 • , θ1 = 21 • , θ2 = 39 • )

Figure 4 :

 4 Figure 4: Absolute value of estimation error |∆θ1| vs. source angular separation |θ2 -θ1| and model angular dispersion ∆ (2 sources with Uniform angular dispersion, ∆0 = 10 • , 100 Monte-Carlo simulations)