Xiong Wenmeng

Picheral José

Marcos Sylvie

ARRAY GEOMETRY IMPACT ON MUSIC IN THE PRESENCE OF SPATIALLY DISTRIBUTED SOURCES

Keywords: array signal processing, distributed sources, angular dispersion, array geometry, performance, MUSIC

The MUltiple SIgnal Classification (MUSIC) estimator has been widely studied for a long time for its high resolution capability in the domain of the direction of arrival (DOA) estimation, with the sources assumed to be point. However, when the actual sources are spatially distributed with angular dispersion, the performance of the conventional MUSIC is degraded. In this paper, the impact of the array geometry on the DOA estimation of spatially distributed sources impinging on a sensor array is considered. Taking into account a coherently distributed source model, we establish closed-form expressions of the MUSIC-based DOA estimation error as a function of the positions of the array sensors in the presence of model errors due to the angular dispersion of the signal sources. The impact of the array geometry is studied and particular array designs are proposed to make DOA estimation more robust to source dispersion. The analytical results are validated by numerical simulations.

Introduction

The DOA estimation based on snapshots received on a sensor array has been widely studied with plenty of methods [START_REF] Krim | Two decades of array signal processing research: the parametric approach[END_REF]. Among these methods, the MUltiple SIgnal Classification (MUSIC) [START_REF] Stoica | Music, maximum likelihood, and cramer-rao bound[END_REF] is famous for its high resolution in the case of point sources. However, in many applications, such as acoustic source imaging [START_REF] Chu | Robust Bayesian super-resolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF] and mobile channel communication [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF], where angular dispersion of the sources up to 10 • may occur, the physical sources can no longer be considered as points. In this case the performance of the DOA estimation obtained by the conventional MUSIC are degraded, and a spatially distributed model of the sources would 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 [START_REF] Valaee | Parametric localization of distributed sources[END_REF]. On one hand, for ID sources, signals coming from different points of the same distributed source can be considered uncorrelated. 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] Valaee | Parametric localization of distributed sources[END_REF]. For CD sources, the performances of MUSIC with discretely distributed sources and continuously distributed sources have been investigated in [START_REF] Astély | The effects of local scattering on direction of arrival estimation with music[END_REF], and [START_REF] Xiong | Performance analysis of music in the presence of modeling errors due to the spatial distributions of sources[END_REF], respectively. As expected, due to the angular dispersion the mismatch between the steering vector model of MUSIC and the actual steering vectors of the sources causes estimation errors. Plenty methods such as the joint estimation of both the DOA and the angular dispersion parameter [START_REF] Valaee | Parametric localization of distributed sources[END_REF][9] [START_REF] Zoubir | Efficient subspace-based estimator for localization of multiple incoherently distributed sources[END_REF] have been proposed to solve these problems. However, these methods require a high computational burden or a knowledge of the shape of the source angular distribution for the model, which motivates us to rather keep the conventional point-source MUSIC for the DOA estimation and seek other ways to improve the performances as, for example, the optimization of the array geometry.

The array geometry effect on the DOA estimation has been studied in plenty of publications and in different contexts. The uniform linear array (ULA) is the simplest. However, a more complex geometry can lead to better performance. Optimal array geometries have been designed to reach isotropic and/or optimal performance based on the Cramér-Rao bound (CRB) criterion (eg : [START_REF] Baysal | On the geometry of isotropic arrays[END_REF][START_REF] Gazzah | Optimum ambiguity-free directional and omnidirectional planar antenna arrays for doa estimation[END_REF][START_REF] Gazzah | Cramer-rao bounds for antenna array design[END_REF]), the lower bound of the mean square error (MSE) can be uniform for all the possible DOAs, or the inferior bound of the MSE of the DOA in the elevational and horizontal direction can be decoupled. More recently, based on the spatial aliasing phenomenon, a class of non-uniform array geometries composed of two or more uniform linear arrays (ULAs) with different inter-element spacing has been used to reduce the computational burden of the Maximum Likelihood (ML) estimator [START_REF] Shin | Reduced-complexity maximum likelihood direction-of-arrival estimation based on spatial aliasing[END_REF]. Also, many techniques have been applied in the sparsity array design or large-scale broadband array to reduce the number of elements in an array, to offer a lower cost, power consumption, and heat dissipation (eg : [START_REF] Oliveri | Bayesian compressive sampling for pattern synthesis with maximally sparse nonuniform linear arrays[END_REF] [16] [START_REF] Kassis | Advantages of nonuniform arrays using root-music[END_REF]).

In our work, we focus on the impact of the array geometry on the performance of the MUSIC estimator with the CD source model, in the presence of errors due to modeling mismatch of source dispersion. Let us first briefly review our work in [START_REF] Xiong | Performance analysis of music in the presence of modeling errors due to the spatial distributions of sources[END_REF] and [START_REF] Xiong | Antenna array design for music in the presence of spatially distributed sources[END_REF]. In [START_REF] Xiong | Performance analysis of music in the presence of modeling errors due to the spatial distributions of sources[END_REF], the point source MUSIC has been extended into CD-MUSIC (coherent source MUSIC), the first order Talyor approximation DOA estimation bias is proposed in the case that the steering vectors in the model and of the actual sources are mismatched. In [START_REF] Xiong | Antenna array design for music in the presence of spatially distributed sources[END_REF] we make use of the theoretical estimation bias expression in [START_REF] Xiong | Performance analysis of music in the presence of modeling errors due to the spatial distributions of sources[END_REF], and express the DOA estimation bias as a function of the sensor positions. Impacts of particular array geometries on the estimator performances are studied in the scenario of single source and multiple sources, respectively. In this paper, besides the results in the conference papers, our contributions consist of: firstly, a general condition for canceling the DOA estimation bias in the case of one source is derived; secondly, the condition for canceling the cross terms of the DOA and the angular dispersion parameter in Cramer-Rao bound in the case of one source is derived; thirdly, for the results concerning UCA in the case of two source in [START_REF] Xiong | Antenna array design for music in the presence of spatially distributed sources[END_REF], the calculation details are given in 6.1.

The organization of this paper is as follows. The signal model and a brief recall of MUSIC are given in section 2. The impact of array geometry on the performance of MUSIC and on the crossed terms of the CRB are studied in section 4. Particular geometry designs are studied in section 5. UCA for the case of two sources are studied in section 6. Finally, conclusions are given in section 7.

Signal model

Let us consider q spatially CD far-field narrow-band sources impinging on an array of M sensors. The sources arrive from the DOA θ 1 , ..., θ q , and the position of the m -th sensor is given by the polar coordinates ρ m and α m . Without loss of generality, the signals and the sensors are assumed to be in the same plane, as shown in Figure 1. The q source signals and the M signals received by the array at moment t are denoted by s(t) = [s 1 (t), . . . , s q (t)] T and y(t) = [y 1 (t), . . . , y M (t)] T , respectively. In the case of CD sources, it is common to exploit the model proposed in [START_REF] Valaee | Parametric localization of distributed sources[END_REF] in order to express the received signal:

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 (θ 1 ), . . . , c hq (θ q )] ∈ C M ×q is the array steering matrix composed of q steering vectors c h i (θ) that can be written by:

c h i (θ i ) = π 2 -π 2 a(θ i + φ)h i (φ)dφ, (2)
where i = 1 . . . q, and a(θ) is the steering vector for a point source, which can be given by:

a(θ i ) = e -j2π ρ 1 λ cos(θ i -α 1 ) , . . . , e -j2π ρ M λ cos(θ i -α M ) T , (3)
where λ is the wavelength, and [•] T is the transpose operation.

The functions h i (φ) are introduced in (2) to describe the angular spread distribution (for instance, Uniform and Gaussian distributions). The source signals and the additive noise are considered to be complex 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:

R = E[yy H ] = CR s C H + σ 2 b I, (4)
where E[.] is the expectation operator, R s and σ 2 b are the source covariance matrix and the noise variance, respectively.

Under the hypothesis that q < M and R s and C are not rank deficient, it is well known that the decomposition of R into eigenvalues λ m and eigenvectors e m is as follows :

R = M m=1 λ m e m e m = UΛ s U H + σ 2 b VV H , (5)
where U = [e 1 , . . . , e q ] spans the signal subspace defined by the columns of C, V = [e q+1 , . . . , e M ] spans the noise subspace defined as the orthogonal complement of U, and Λ s = diag{λ 1 , . . . , λ q }.

MUSIC estimator and performances

The MUSIC [START_REF] Stoica | Music, maximum likelihood, and cramer-rao bound[END_REF] method makes use of the orthogonal property of the subspaces spanned by C(θ) and V to estimate the DOAs θ i . In practice it is difficult to know exactly the angular dispersion of the actual sources, consequently, the steering vector model of the point source a(θ) is used here instead of c h i (θ) to estimate the value of θ:

θi = argmax θ 1 a H (θ)V 2 . ( 6
)
Here, we assume that the number of snapshots is large enough such that the estimation error of the noise subspace can be neglected and the DOA estimation error comes mainly from the model error, it is to say, the mismatch of the angular dispersion parameter between the model of the MUSIC estimator a(θ) and the actual source c h i (θ i ). We recall here the standard analysis in order to express the estimation error ∆θ i = θi -θ i , θi should satisfy that the first derivative of denominator in ( 6) is null:

∂a(θ) H VV H a(θ) ∂θ θi = 0, (7)
which gives:

2Re{ ȧ( θi ) H VV H a( θi )} = 0, (8)
where ȧ( θi ) = ∂a(θ) ∂θ θi .

Assuming that, θi is not far away from θ i , we make the first order approximation of Taylor:

a( θi ) ≈ a(θ i ) + ∆θ i ȧ(θ i ), (9)
and:

ȧ( θi ) ≈ ȧ(θ i ) + ∆θ i ä(θ i ), (10)
where ä(θ i ) = ∂ ȧ(θ)

∂θ θ i .
Note ∆c(θ i ) = a(θ i ) -c(θ i ) the error on the steering vector due to the mismatch between the estimator model and the actual signals. Note also ∆V = Ṽ -V the error on the noise eigenvectors matrix, where Ṽ is the noise eigenvectors matrix which would appear if the steering vectors a(θ i ) would replace c(θ i ) in [START_REF] Xiong | Antenna array design for music in the presence of spatially distributed sources[END_REF], where i = 1, . . . , q. Introducing ( 9) and ( 10) into [START_REF] Xiong | Performance analysis of music in the presence of modeling errors due to the spatial distributions of sources[END_REF], and keeping only the first order of ∆θ i , the DOA estimation error for source θ i can be expressed as:

∆θ i = Re{ ȧH i Πc i } ȧH i Π ȧi , (11)
where Π = I -Ṽ ṼH is the noise subspace projector of the steering vector model of the estimator, which would correspond to the case of point sources, I is the M × M identity matrix. Note that θ i is omitted in ȧ(θ i ), a(θ i ) and c(θ i ) which are now noted ȧi , a i and c i , respectively.

Array geometry analyses for one source

In this section, we firstly investigate the array geometry condition for MUSIC to be robust to the source angular dispersion. Then, we link and compare our results with the existing results based on the CRB for an isotropic array in the case of point source. At last, we derive the CRB for joint estimation of the DOA and the angular distribution in the scenario of a distributed source.

Array geometry for MUSIC

In this subsection, we study the estimation error of the point source MUSIC estimator defined in [START_REF] Valaee | Parametric localization of distributed sources[END_REF] in the case where there is only one source arriving from the DOA θ. Notice that in this case

Π = I -1 M a 1 a H 1 .
From [START_REF] Baysal | On the geometry of isotropic arrays[END_REF] the DOA estimation error can then be given by:

∆θ = Re{ ȧH c -1 M ȧH aa H c} ȧH ȧ -1 M | ȧH a| 2 , ( 12
)
where c 1 and a 1 have been replaced by c and a. In the following, we wish to express the estimation error as an explicit function of the sensor positions, so as to study the impact of the sensor array geometry on the performance of MUSIC. For the m -th sensor as illustrated in Figure 1, let us define:

ϕ m = 2π ρ m λ cos(θ -α m ), φm = -2π ρ m λ sin(θ -α m ), φm = -2π ρ m λ cos(θ -α m ), u m = cos(ϕ m (θ + φ) -ϕ m (θ))h(φ)dφ, v m = sin(ϕ m (θ + φ) -ϕ m (θ))h(φ)dφ. (13)
Assuming that the angular dispersion of the distributed source is small enough with a symmetrical distribution, we introduce a third order Taylor approximation in φ of ϕ m (θ + φ) -ϕ m (θ) so as the last equation in [START_REF] Gazzah | Cramer-rao bounds for antenna array design[END_REF] yields:

v m ≈ 1 2 φm σ 2 , (14)
where:

σ 2 φ 2 h(φ)dφ, (15)
remark that σ is the angular dispersion of the source signal.

Introducing the notations ( 13) in [START_REF] Gazzah | Optimum ambiguity-free directional and omnidirectional planar antenna arrays for doa estimation[END_REF], ∆θ can be given by:

∆θ = φ • v -φ • v φ2 -( φ) 2 , ( 16
)
where • is operator to calculate the "average" of a deterministic series such that x = 1 M M m=1 x m . Eq [START_REF] El-Makadema | Geometry design optimization of large-scale broadband antenna array systems[END_REF] makes it possible to design the geometry in order to minimize ∆θ.

Let us introduce ( 14) in ( 16), ∆θ can be given by:

∆θ = 1 2 σ 2 φ • φ -φ • φ φ2 -( φ) 2 . ( 17
)
We can observe that this expression of the DOA estimation error is proportional to the square of the source angular dispersion σ 2 .

From [START_REF] Kassis | Advantages of nonuniform arrays using root-music[END_REF], the estimation error as a function of x m and y m can be derived as:

∆θ = 1 4 σ 2 • sin(2θ) x 2 -(x) 2 -y 2 + (y) 2 -cos(2θ)(xy -x y) sin 2 θ x 2 -(x) 2 + cos 2 θ y 2 -(y) 2 -sin(2θ) [xy -x y] . (18)
The sufficient and necessary condition for ∆θ to be zero for any value of θ is given by:

x 2 -(x) 2 = y 2 -(y) 2 , xy = x y. ( 19
)
If the condition ( 19) is verified then the modelling mismatch introduced in (6) by using point source steering vector does not introduce an estimation error. Despite the fact that the sensor coordinates are deterministic, one can give a statistical interpretation, xy = x y means that the sensors' the abscissa and ordinates are uncorrelated, and x 2 -(x) 2 = y 2 -(y) 2 means that the standard deviation on abscissa and ordinate are the same. Geometrically, this means that the sensor position should not exhibit a specific direction and that the spread of the sensor position is the same in all the directions.

Notice that ( 19) can be exploited to improve an existing array by adding (or removing) one sensor such that the condition ( 19) is verified. Figure 2 illustrates an example of the optimal geometry of the sensor array obtained by minimizing numerically the criterion 2π 0 |∆θ|dθ, where ∆θ is given in [START_REF] Stoica | Performance study of conditional and unconditional direction-of-arrival estimation. Acoustics, Speech and Signal Processing[END_REF], in this example we have a source with a uniform angular distribution 10 • . The numerical "greedy research" method proposed in [START_REF] Gazzah | Optimum ambiguity-free directional and omnidirectional planar antenna arrays for doa estimation[END_REF] is used here to avoid the ambiguity problem. In this case, condition ( 19) is verified, since:

x 2 -(x) 2 = y 2 -(y) 2 ≈ 0.49, xy = x y ≈ 0.21. -2 -1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 x (λ/2) y (λ/2)

Link with the Cramer-Rao bound

Firstly, one can note that the condition to obtain an isotropic array in the scenario of point source proposed in [START_REF] Baysal | On the geometry of isotropic arrays[END_REF] is identical to [START_REF] Goldberg | Inherent limitations in the localization of a coherently scattered source[END_REF]. Note that [START_REF] Goldberg | Inherent limitations in the localization of a coherently scattered source[END_REF] has been established in the presence of a model error due to the sources dispersion and an infinite number of snapshots to estimate the covariance matrix whereas the expression in [START_REF] Baysal | On the geometry of isotropic arrays[END_REF] corresponds to the point sources case and the error is due to a finite number of snapshots.

Secondly, we investigate the CRB for the joint estimator of the DOA and the source angular dispersion. We now consider, the scenario with one distributed source, where the shape of angular distribution h i is known but the angular dispersion parameter σ is unknown and has to be jointly estimated with the DOA. An example of such an estimator is the 2D MUSIC-like estimator named DSPE [START_REF] Valaee | Parametric localization of distributed sources[END_REF]. Under the unconditional-model assumption (UMA) [START_REF] Stoica | Performance study of conditional and unconditional direction-of-arrival estimation. Acoustics, Speech and Signal Processing[END_REF], the CRB matrix is composed of the derivatives of the elements in the parameter vector Φ 0 = [θ, σ, σ 2 s , σ 2 b ]. We will compute the CRB in this case, and express the condition such that the lower bound on the covariance of the estimation of θ and σ is null. When this condition is fullfilled, the consistent estimator is known to be robust to the model error due to the source dispersion since cov( θ, σ) → N →∞ 0, where θ = θ -θ, σ = σ -σ. In other words, error on σ does not impact on the θ estimation.

It is well known that for point sources, the variances of parameters of signal and noise power are decoupled of the variance of the DOA. Similar results can be found for distributed sources when the angular dispersion is small [START_REF] Goldberg | Inherent limitations in the localization of a coherently scattered source[END_REF]. Therefore, without loss of generality, we assume that σ 2 s and σ 2 b are known. The CRB expression for distributed sources has been derived in [START_REF] Goldberg | Inherent limitations in the localization of a coherently scattered source[END_REF]:

CRB(θ, σ|σ 2 s , σ 2 b ) = 1 + βSN R 2N βSN R Re{C H Ψ ΠC Ψ } + Ω -1 , (20)
where

β = c H c, Ω = 2bb T β(1+βSN R) , b = Re{C H Ψ c}, C Ψ = ∂c ∂θ , ∂c
∂σ and SN R is the value of the signal to noise ratio.

Exploiting the approximation proposed in ( 14), we develop the CRB expression as an explicit function of the parameters:

CRB θθ = f σσ η [f θθ f σσ -|f θσ | 2 ] , (21)
CRB σσ = 4 σ 2 σ η • f θθ f θθ f σσ -|f θσ | 2 , ( 22
)
CRB θσ = 2 σ σ η • f θσ f θθ f σσ -|f θσ | 2 , (23)
where:

σ σ φ 2 ∂h(φ) ∂σ dφ = ∂σ 2 ∂σ = 2σ, η = 2N σ 4 s σ 2 b (σ 2 s (M + σ 2 Real{ä H a}) + σ 2 b ) , f θθ =M ȧH ȧ -| ȧH a| 2 + σ 2 Real{ ... a H ȧM + ȧH ȧä H a -a H ȧ( ... a H a + ȧH ä)}, f θσ =Real{M ȧH ä -ȧH aa H ä}, f σσ =M äH ä + Real{a H äa H ä} - 2M σ 2 s Real{ä H a} 2 σ 2 b + σ 2 s (M + σ 2 Real{ä H a}) . ( 24
)
Taking into account the notations in [START_REF] Gazzah | Cramer-rao bounds for antenna array design[END_REF], f θσ can be rewritten as:

f θσ = M 2 φ • φ -φ • φ . ( 25
)
As discussed previously, in order to design array geometry for which consistent DOA estimators are robust to mismodelling, the condition is f θσ = 0. From (25) it yields φ • φ = φ • φ, we can see that the results of ( 19), (25), and the results in [START_REF] Stoica | Performance study of conditional and unconditional direction-of-arrival estimation. Acoustics, Speech and Signal Processing[END_REF] are identical. Notice that [START_REF] Goldberg | Inherent limitations in the localization of a coherently scattered source[END_REF] yields the condition for MUSIC to be robust to the source spatial dispersion whereas (25) corresponds to a CRB criterion for a model of source dispersion and that the condition in [START_REF] Baysal | On the geometry of isotropic arrays[END_REF] has been derived through the CRB calculus for point sources. It follows that the optimal design criterion obtained for MUSIC in the case of spatially distributed sources yields to the same solution as those obtained by CRB both in the distributed and point cases.

Particular array geometries design

In this section, we investigate particular array geometries with pretty simple structures which satisfy the optimal condition proposed in [START_REF] Goldberg | Inherent limitations in the localization of a coherently scattered source[END_REF] in the case of one source.

Uniform circular array

Inspired by [START_REF] Kassis | Advantages of nonuniform arrays using root-music[END_REF], we find that for a UCA with an even number of symmetrical elements, where ρ m = ρ, and α m = 2π (m-1) M with M even, φ and φ • φ can be zero:

φ = - 1 M 2π ρ λ M m=1 sin(θ -α m ) = - 1 M 2π ρ λ   M 2 m=1 sin(θ -α m ) + M 2 m=1 sin(θ -α m + π)   = 0, (26)
and similarly :

φ • φ = (2π ρ λ ) 2 M M m=1 sin(θ -α m ) cos(θ -α m ) 1 2 sin(2θ-2αm) = 0. ( 27
)
This result illustrates that a symmetrical with even elements UCA can be robust to the model error due to the source dispersion, in the scenario of one source. In addition, a sensor array composed of a combination of UCAs with such a structure with a same center but different radii, can also be proved, with a similar demonstration, to be robust to the model error. The V-shape array (VA) is another type of particular geometry which has been widely studied, thanks to its easiness to be parameterized. It has been proved in [START_REF] Gazzah | Optimum ambiguity-free directional and omnidirectional planar antenna arrays for doa estimation[END_REF] that with the array angle fixed to a specific value, a VA with any size can have an isotropic behavior, and outperforms the UCA in terms of CRB.

V-shape array

Let us assume that, without loss of generality, the two identical branches are symmetrical with respect to the y axis, with no sensor placed at the origin, as depicted in Figure 3, and that, the inter-sensor distance is fixed to d. The coordinates of the sensors can be expressed as:

ρ m = md, 1 m M 2 (m -M 2 )d, M 2 m M , (28)
and,

α m = α, 1 m M 2 π -α, M 2 m M . ( 29
)
Considering the above described V-shape geometry, ( 17) can be derived as:

∆θ = 1 2 σ 2 sin θ cos θ (4 + 4M ) cos 2 α + (2 -M ) sin 2 α (4 + 4M ) cos 2 α sin 2 θ + (2 -M ) sin 2 α cos 2 θ , (30)
which shows that the DOA estimation error is independent of d. In addition, we can see that ∆θ can be zero, under the condition that the numerator of (30) vanishes, which yields:

tan 2 α = 4M + 4 M -2 . ( 31
)
This expression reveals that the array angular α can be fixed to a special value to be robust to the model error due to the source dispersion, which depends roughly on the array size, and is robust to other parameters. For example, in the case M = 10 with 5 sensors on each branch, (31) yields α ≈ 66.9 • and β ≈ 46.2 • .

Linear array

Notice that the scenario of α m = 0 corresponds to the linear array (LA). After some straightforward calculus from ( 13)- [START_REF] Kassis | Advantages of nonuniform arrays using root-music[END_REF] with α m = 0, the DOA estimation error is derived as:

∆θ = 1 2 σ 2 cot θ, (32)
which reveals that in the case of LA, the DOA estimation error does not depend on the sensor positions but only on the DOA and the square of the angular dispersion of the source σ 2 . Taking into account that the model error can not be modified, the sensor array should be rotated to reduce the value of cot θ, so as to get a smaller estimation error, if θ is roughly known. Besides, this result is identical to the conclusion that we have for point sources: it is better to have the DOA perpendicular to the array than in the axis of the array.

Simulation results

In this subsection, we validate the theoretical results through numerical experiments in the scenario of one source. In all experiments, one uniformly distributed source with a support within θ -∆ 0 2 , θ + ∆ 0 LA simulation LA theory eq(32) VA simulation VA theory eq(30) V iso simulation V iso theory eq(30) In Figure 4, we consider a V-shaped array and compare the theoretical value of the optimal β to the value obtained by simulation: β sim min MUSIC is the value of β which minimizes the estimation error of MUSIC for 100 trials. This result can be similarly obtained by exploiting the cross term of the CRB as proposed in section 4.1. To illustrate this, we also plot the value of β corresponding to the minimum |Cov(θσ)| = |E[( θ -θ)(σ -σ)]| obtained by the joint estimator DSPE [START_REF] Valaee | Parametric localization of distributed sources[END_REF]. Let us recall the definition of the DSPE, assuming that the distribution shape of the source is known, the DOA and the angular dispersion parameter can be estimated as:

{ θ, σ} = argmax θ,σ 1 c H h (θ)V 2 .
(

) 33
As one may expect, for a V-shape array, the optimal geometry for an 1D estimator is robust to the DOA of the source, in addition, this optimal geometry makes it possible to decouple the estimation of the DOA and the angular dispersion parameter for a 2D estimator.

In Figure 5, the DOA estimation error of MUSIC versus the angular dispersion of the signal source with θ = 45 • is presented, in the case of VA with β = 100 • and VA with β = 53.13 • (V iso ). The V-shape array V iso proposed in [START_REF] Gazzah | Cramer-rao bounds for antenna array design[END_REF] satisfies the isotropic condition for the CRB in the scenario of a point source and also corresponds to the solution of (31) when M tends to infinity or at least becomes large. The ∆θ with array V iso is the smallest among the three arrays, because β = 53.13 • is close to β th min in Figure 4. We can observe the validation of expression (30) and (32) which makes it possible to consider the DOA estimation error as an explicit convex quadratic function of the angular dispersion of the signal source. Furthermore, regardless of a more complicated geometry, the two VAs outperforms LA, which reveals the interest to optimize the array geometry to have a better performance.

6. Two sources in the case of a UCA

Theoretical results

In this subsection, we focus on the above-mentioned case of a UCA with an even number of symmetrically positioned sensors, which is easier to manipulate mathematically comparing to other geometries.

Assuming two sources arriving from θ 1 and θ 2 , with θ 2 > θ 1 . For the m -th sensor, let us introduce the notation:

∆ϕ m = ϕ m (θ 2 ) -ϕ m (θ 1 ) = ϕ m,2 -ϕ m,1 . (34)
Notice that with a UCA:

Re{ ȧH i a i } = - M m=1 2π ρ λ sin(θ -α m ) = 0, (35)
and:

Re{ ȧH i c i } = M m=1 φm,i u m,i =   M/2 m=1 Φ(θ i ) + M/2 m=1 Φ(θ i + π)   = 0, (36)
where

Φ(θ i ) = φm (θ i ) [cos(ϕ m (θ i + φ)) cos(ϕ m (θ i ))+ sin(ϕ m (θ i + φ)) sin(ϕ m (θ i ))] h(φ)dφ.
Introducing (35), (36), and the steering matrix A = [a 1 , a 2 ] in [START_REF] Baysal | On the geometry of isotropic arrays[END_REF] gives:

∆θ 1 = Re{-M ȧH 1 a 2 a H 2 c 1 + ȧH 1 a 2 a H 1 c 1 a H 2 a 1 } ȧH 1 ȧ1 (M 2 -|a H 2 a 1 | 2 ) -M | ȧH 1 a 2 | 2 . (37)
We introduce the following notations:

u (i) = 1 M M m=1 u m,i , v (i) = 1 M M m=1 v m,i , ϕ (i) = 1 M M m=i ϕ m,i
, where u m,i , v m,i , and ϕ m,i are simplified for u m (θ i ), v m (θ i ), ϕ m (θ i ), respectively. i = 1, 2 in the case of two sources. Introducing the notations of ( 13) and (34) in (37), with the results in appendix A, ∆θ 1 can be derived as:

∆θ 1 = - 1 2 σ 2 18 φ2 (1) -3δ 2 φ2 (1) + φ4 (1) ( φ2 (1) -φ2 (1) ) 2 δ -9( φ2 (1) ) 3 + 12 φ2 (1) • φ4 (1) + φ2 (1) -δ 2 ( φ2 (1) + φ4 (1) ) 2 , ( 38
)
where δ = θ 2 -θ 1 . We can see that (38) is proportional to the square of the angular dispersion σ 2 . The value of the estimation error is inversely proportional to the value of δ for small δ, which illustrates that the estimation error decreases as the sources move away from each other. In order to explore the peak separation, let us consider the sign of ∆θ 1 . Firstly, Π is Hermitian, so the denominator ȧH i Π ȧi in ( 11) is non-negative, it is enough to focus on the numerator in (11) for θ 1 which yields 1 2 σ 2 δ 3 (18 φ2

(1) -3δ 2 φ2

(1) + φ4

-φ2 (1) ) 2 (see (A.5) in appendix) after some straightforward calculus. Secondly with a small δ, 18 φ2

(1) -3δ 2 φ2

(1) + φ4

(1) can be approximated by 18 φ2

(1) which is positive, and obviously ( φ2

(1) -φ2 (1)
) 2 is positive. Therefore, in the case that θ 1 < θ 2 , ∆θ 1 is opposite to the sign of δ.

For θ 2 , after similar calculus in appendix B, we have:

∆θ 2 = -∆θ 1 . ( 39
)
∆θ 1 is negative and ∆θ 2 is positive, on the opposite of the well known result for point source, the lower DOA is underestimated and the other one is overestimates, in consequence, the two peaks move away. This implies that when the two sources are close to each other, MUSIC is always able to give the estimation of two DOAs separately, despite that the value of the estimation error increases as it is illustrated in Figure 6.

Numerical results

In this section, we validate the theoretical results through numerical experiments in the scenario of two sources. In all experiments, two uniformly sources with the angular dispersion ∆ 0 = 10 • are considered. Signals received on the sensor array are simulated with N = 1000 snapshots and 100 Monte-Carlo experiments, with SN R = 10dB. The array configurations are the same for the UCA, VA and LA as in subsection 5.4.

Figure 6 illustrates the peak separation of criterion MUSIC in the case of 2 sources. A UCA is used for the DOA estimation of two uniformly distributed sources coming from 29.5 • and 30.5 • with angular dispersion 10 • , assuming that the correlation matrix is perfectly known. Two peaks can be detected, but for arguments which are different from the actual DOAs. The analyses of the signs of the two sources in section 6.1 can be verified in this figure .  In Figure 7, we compare the performance of MUSIC with the UCA and with the LA versus the DOA separation, in the scenario of two sources. The DOA separation varies with θ c = 1 2 (θ 1 + θ 2 ) 12 fixed to 60 • , assuming that θ 2 > θ 1 . We can see that when the two sources are too close to each other, the estimation error can not be ignored despite the array geometry, due to the interference between the sources. For LA, ∆θ 1 decreases as the DOA separation increases and as θ 1 tends to zero, by contrast, ∆θ 2 decreases as the interference phenomenon fades, but increases again along with the DOA separation, which coincides with the well-known conclusion that the ULA has a better performance for broadside sources. For UCA, both ∆θ 1 and ∆θ 2 tend to zero quickly as the DOA separation increases, which is in adequacy with the scenario of one source where the DOA estimation error is null and outperforms its LA counterpart. In addition, it is interesting to see the validity of the approximated expression (38) and (39) for describing the trend and the sign of the estimation error when the sources are close to each other.

In Figure 8, we compare the performance of MUSIC with the UCA and a two-level nested array proposed in [START_REF] Pal | Nested arrays: A novel approach to array processing with enhanced degrees of freedom[END_REF] [START_REF] Han | Nested array processing for distributed sources[END_REF]. The nested array is composed of M = 8 sensors, the four sensors in the first level are spaced by d 1 = λ/2, and the last four sensors in the second level are spaced by d 1 = 3λ/2. Therefore, the nested array has a same array aperture as the UCA. The advantage of UCA with respect to the nested array is mainly illustrated when the angular separation between the two sources becomes larger, in this case, MUSIC with UCA has a better performance than nested array, especially for θ 2 .

In Figure 9, ∆θ 1 and ∆θ 2 is plotted versus the angular extension of the sources, two sources of θ 1 = 55 • and θ 2 = 65 • are used in this experiment. The goal is to validate the expression (38) as an explicit function of the angular extension ∆ 0 (recall that for uniform source σ 2 = ∆ 2 0 12 ).

Conclusion

In this paper, we have studied the impact of the array geometry on the performance of the MUSIC DOA estimation in the presence of spatially distributed sources. We have found that the DOA estimation error can be reduced and even canceled for particular array geometries in the case of one source or in the case where the DOA separation between two sources is large enough. Additionally, the impact of the array geometry on the crossed terms of CRB has been studied, and the condition for decoupling the estimation of the DOA from the angular dispersion parameter has been found the same as the one for canceling the DOA estimation error of MUSIC. The separation property of MUSIC in the scenario of two distributed sources has been explored with a UCA, and motivates the further investigation of the array geometry design in the scenario of two sources.

Appendix A.

Introducing ( 13) and (34) in (37), ∆θ 1 is given as:

φ2

(1) -φ2

(1) • cos 2 (∆ϕ) -φ(1) sin(∆ϕ) We take the third order of Taylor approximation in δ to develop (B.1) rather than the first order, because the angular separation between the sources can be up to 50 • -60 • . Hence, noting that δ = θ 2 -θ 1 , the approximated sin(∆ϕ m ) and cos(∆ϕ m ) can be given by: sin(∆ϕ m ) ≈∆ϕ m -1 6 ∆ϕ

sensor s o u r c eFigure 1 :

 1 Figure 1: Planar array and source DOAs

Figure 2 :

 2 Figure 2: one example of a nine-element optimal array for MUSIC

Figure 3 :

 3 Figure 3: Illustration of V-shape arrays

2 2 0 12 .Figure 4 :

 2124 Figure 4: Value of β corresponds to the minimum value of: 1)the theoretical expression of |∆θ| in eq(30), 2)the |∆θ| obtained by the simulation with MUSIC 3)the covariance of θ and σ of DSPE (simulation) , (VA with M = 10 and d = λ/2, one source with uniform angular dispersion, ∆0 = 10 • )

Figure 5 :

 5 Figure 5: DOA estimation error vs. the source angular extension ∆0 (LA with M = 20 and d = λ/4, VA with M = 20 and d = λ/4 and β = 100 • , Viso with M = 20 and d = λ/4 and β = 53.13 • , one source with uniform angular dispersion with θ = 45 • , ∆0 = 2 √ 3σ)

Figure 6 :

 6 Figure 6: Pseudo-spectrum of MUSIC for two distributed sources with a UCA

Figure 7 :

 7 Figure 7: DOA estimation error vs. DOA separation (UCA with M = 20, LA with = 20 and d = λ/4, two sources with uniform angular dispersion, ∆0 = 10 • , θc = 1 2 (θ1 + θ2) = 60 • )

Figure 8 :

 8 Figure 8: DOA estimation error vs. DOA separation (UCA with M = 20, nested array with M = 8 and d1 = λ/2, d2 = 3λ/2, two sources with uniform angular dispersion, ∆0 = 10 • , θc = 1 2 (θ1 + θ2) = 60 • )

Figure 9 :

 9 Figure 9: DOA estimation error vs. the source angular extension ∆0 (UCA with M = 20, two sources with uniform angular dispersion, θ1 = 55 • , θ2 = 65 • )

∆θ 1 = 2 . 2 -

 122 Re{j φ(1) e -j∆ϕ • e j∆ϕ (u (1) -jv (1) ) -j φ(1) e -j∆ϕ • (u (1) -jv (1) ) • e j∆ϕ } φ2 (1) (1 -e -j∆ϕ 2 ) -φ(1) e -j∆ϕ j∆ϕm , e j∆ϕ (u (1) -jv (1) ) = 1 M M m=1 e j∆ϕm (u m,1 -jv m,1 ).Noting that e -j∆ϕm = cos(∆ϕ m ) -j sin(∆ϕ m ), we have:α m ) cos 2π ρ λ cos(θ 1 -α m ) -cos 2π ρ λ cos(θ 2 -α m ) sin 2π ρ λ cos(θ 1 -α m ) -2 -α m + π) cos 2π ρ λ cos(θ 1 -α m + π) -cos 2π ρ λ cos(θ 2 -α m + π) sin 2π ρ λ cos(θ 1 -α m + π) } = 0. (A.2)Similarly, v (i) and φ(i) • cos(∆ϕ) equal to 0. Considering the terms which equal to 0, (A.1) yields:∆θ 1 = φ(1) • sin(∆ϕ) cos(∆ϕ) • u (1) -cos(∆ϕ) • u (1) + sin(∆ϕ) • v (1)

2 .

 2

  cos 2 (θ 1 + α m )δ 2 -cos(θ 1 + α m ) sin(θ 1 + α m )δ 3 .

						3
						m
	=2π	ρ λ		cos(θ 1 + α m )(δ -	1 6	δ 3 ) -	1 2	sin(θ 1 + α m )δ 2 -	1 6	(2π	ρ λ	) 3 cos 3 (θ 1 + α m )δ 3 ,
	cos(∆ϕ m ) ≈1 -	1 2	∆ϕ 2 m
	=1 -	1 2	(2π	ρ λ	) 2 (A.4)
	Introducing (A.4) in (B.1), ∆θ 1 results in:
	∆θ 1 =	δ 4 -9( φ2 (1) ) 3 + 12 φ2 (1) • φ4 (1) + φ2 (1) -δ 2 ( φ2 (1) + φ4 (1) ) 2 δ 3 18 φ2 (1) -3δ 2 φ2 (1) + φ4 (1) φ2 (1) • u (1) -φ2 (1) • u (1)	(A.5)
				=		18 φ2 (1) -3δ 2 φ2 (1) + φ4 (1)	φ2 (1) • u (1) -φ2 (1) • u (1)	.	(A.6)
					δ -9( φ2 (1) ) 3 + 12 φ2 (1) • φ4 (1) + φ2 (1) -δ 2 ( φ2 (1) + φ4 (1) ) 2

Taking into consideration that u m,1 ≈ cos(ϕ m (θ 1 + φ) -ϕ m (θ 1 ))h(φ)dφ ≈ cos(φ φm (θ 1 ) + 1 2 φ 2 φm (θ 1 ) + 1 6 φ 2 ... ϕ m (θ 1 ))h(φ)dφ ≈ 1 -1 2 σ 2 φ2 m (θ 1 )

, ∆θ 1 results in (38) .

Appendix B.

Similarly to (A.1), ∆θ 2 can be given as:

Recalling that θ 2 = θ 1 + δ, the notations in θ 2 can be approximated as functions of notations in θ 1 and δ:

+ sin 2π ρ m λ (cos(θ 1 + φ) -cos θ 1 ) sin 2π ρ m λ (-δ 2 φ sin θ 1 -δφ 2 sin θ 1 + 2δφ cos θ 1 ) h(φ)dφ


Wenmeng Xiong

José Picheral

José Picheral •

Sylvie Marcos

Robustness of the coherently distributed MUSIC algorithm to the imperfect knowledge of the spatial distribution of the sources

Keywords: array signal processing, distributed source, CD MUSIC, performance analyses

Robustness of the coherently distributed MUSIC algorithm to the imperfect knowledge of the spatial distribution of the sources

Introduction

DOA estimation issues such as the effects of model errors, the resolution of closely spaced sources, and the geometry of antennas have been widely studied in the past, with the sources assumed to be far-field point transmitters or reflectors [START_REF] Ciuonzo | Performance analysis of time-reversal MU-SIC[END_REF][START_REF] Ma | Ica-based direction-of-arrival estimation of uncorrelated and coherent signals with uniform linear array[END_REF][START_REF] Liu | A novel direction-of-arrival estimation method for wideband signals[END_REF][START_REF] Tao | Directionfinding of coherent signals based on cylindrical vectorhydrophones array[END_REF][START_REF] El Markhi | An improved cyclic beamforming method for signal doa estimation[END_REF]. Indeed, in most scenarios the point-source assumption is correct, but in some cases, as for instance the localization of acoustics [START_REF] Chu | Robust Bayesian super-resolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF] [START_REF] Brooks | A deconvolution approach for the mapping of acoustic source(DAMAS) determined from phased microphone arrays[END_REF] or bio-medical [START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF] sources, ocean waves [START_REF] Adam P Young | Cross-shore decay of cliff top ground motions driven by local ocean swell and infragravity waves[END_REF], or for mobile channel communication [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF], a spatially distributed model of the sources is 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 the first hand, for ID sources, signals coming from different points of the same distributed source can be considered uncorrelated, therefore the rank of the noise-free correlation matrix does not equal the number of signals [START_REF] Besson | Decoupled estimation of DOA and angular spread of a spatially distributed source[END_REF] [12] [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 from different points of the same signal components are delayed and scaled replicas, therefore the rank of the noise-free correlation matrix equals the number of CD sources [START_REF] Valaee | Parametric localization of distributed sources[END_REF]. Most of the works deal with the ID model since in wireless communication, the multi-path dispersion effects are modeled as discrete ID scatterers of one source [START_REF] Raich | Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms[END_REF]. In our paper, we focus on other applications such as acoustic localization where the sources are rather modeled as continuous CD ones, this work has been partially presented in [START_REF] Xiong | Performance analysis of MUSIC in the presence of modeling errors due to the spatial distributions of sources[END_REF].

Most of the localisation methods that have been proposed to take into account the distributed sources model are based on the joint [START_REF] Valaee | Parametric localization of distributed sources[END_REF] [START_REF] Trump | Estimation of nominal direction of arrival and angular spread using an array of sensors[END_REF] or iterative [START_REF] Zoubir | Efficient subspace-based estimator for localization of multiple incoherently distributed sources[END_REF] [18] 2D estimation of the DOA and the angular dispersion parameters. These approaches present two main drawbacks: they are time consuming due to multi-parameters estimation and error modeling is introduced due to the imperfect knowledge of the angular distribution shapes of the sources. Otherwise, the standard MUSIC DOA estimator under point source hypothesis can be used despite the angular dispersion of actual sources, in this case an estimation error of the DOA arises due to the mismatch between the model and the sources. In this paper, we introduce an estimator called CD-MUSIC which can be considered as trade off between a 2D estimator and the standard MUSIC based on the point source hypothesis. This DOA estimator exploits a generalized steering vector based on a CD source model, nevertheless it requires to know the angular distribution shape and dispersion parameters of the sources.

Previous works dealing with modeling error due to spatially distributed sources can be found in [START_REF] Raich | Bearing estimation for a distributed source via the conventional beamformer[END_REF] for Beamforming and [START_REF] Astély | The effects of local scattering on direction of arrival estimation with MUSIC[END_REF] for MUSIC. Those works are mainly based on an 1D model for telecommunication applications with a small spread dispersion, and a first order Taylor approximation in the DOA estimation error is used to obtain theoretical expressions for the performances.

The motivations of the present paper are to study the sensitivity of MUSIC to error modeling due to coherently distributed sources (since the standard point source MUSIC estimator is a particular case of CD-MUSIC, theoretical performance analysis provided for CD-MUSIC is also valid for standard MUSIC). Secondly, we will provide tools to determine when point source assumption is acceptable for MUSIC, when CD source model should be considered, or when more complex 2D estimators should be used. In this paper, the study is focused on applications such as acoustic imaging and source localisation [START_REF] Chu | Robust Bayesian super-resolution approach via sparsity enforcing a priori for near-field aeroacoustic source imaging[END_REF] where due to the experiment configuration, the CD model is more appropriate, large spread dispersion is expected and hypothesis of Gaussian shape for the angular distribution is not always valid.

Modeling errors can originate from the following configurations: i) the shape (uniform, or Gaussian or others) of the angular distribution is badly known; ii) the shape of the angular spread distribution is known but with an improper spread dispersion parameter value (for example, a uniform distribution with an improper angular width, or a Gaussian distribution with an improper standard deviation, etc). To analyze the performances, we derive expressions with a second order Taylor approximation in the DOA estimation error rather than the first order techniques proposed in [START_REF] Raich | Bearing estimation for a distributed source via the conventional beamformer[END_REF] and [START_REF] Astély | The effects of local scattering on direction of arrival estimation with MUSIC[END_REF]. The result is a little more complex, but follows better the simulation especially when two sources are close and the angular dispersion parameter is over-estimated. In addition, we also consider the effect of the finite number of snapshots which was done in [21] [22] in a similar way in the scenario of point source. Here, we show that when the two problems arise simutaneously, the finite snapshots effect can be explicitly factored in the theoretical expressions for the performance analyses. Simulation results validate the theoretical results and illus-trate the limits of the point source MUSIC for a large modeling error.

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

Signal model and CD-MUSIC

Notations

In this paper, we use lower case boldface letters to denote vectors and upper case boldface letters to denote matrices. In addition, we use the following notations throughout the paper: (•) T : transpose; (•) H : Hermitian transpose; Re{•}: real part; E[•]: expectation operator; I: identity matrix.

Signal model

Let us consider q spatially coherently distributed farfield sources impinging on an array of M sensors. The q sources and M signals received on the array at moment t are denoted by s(t) = [s 1 (t), . . . , s q (t)] T and y(t) = [y 1 (t), . . . , y M (t)] T , respectively. In the scenario of distributed sources, y(t) is given by:

y(t) = C(θ)s(t) + n(t), (1)
where n(t) ∈ C M ×1 represents additive Gaussian or uniform noise σ 2 b , C(θ) = [c h1 (θ 1 ), . . . , c hq (θ q )] ∈ 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]:

c hi (θ i ) = π 2 -π 2 a(θ i + φ)h i (φ)dφ, (2)
where a(θ) ∈ C M ×1 is the steering vector for a point source which arrives from the DOA θ. In the general case, a(θ) can be given by a(θ i ) = e -j2π v λ t1(θi) , . . . , e -j2π v λ tm(θi) , . . . , e -j2π v λ t M (θi) T , where t m (θ i ) is the propagation time for source i to the m th sensor, v is the propagation speed of the signal and λ is the wavelength.

The function h i (φ) is introduced to describe the angular spread distribution and it can be parameterized by an angular dispersion ∆ i which is omitted in the notation. For instance, ∆ i stands for the support for a uniform distribution and for the standard deviation for a Gaussian distribution.

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 signal sources are uncorrelated with each other, the correlation matrix is given by: R = E[yy H ] = CR s C H + σ 2 b I, where R s is the source covariance matrix.

CD-MUSIC

Under the hypothesis that q < M and R s and C are not rank deficient, it is well known that the decomposition of R into eigenvalues λ m and eigenvectors e m is given by: R =

M m=1 λ m e m e H m = UΛ s U H + σ 2 b VV H
, where U = [e 1 , . . . , e q ] spans the signal subspace defined by the columns of C and V = [e q+1 , . . . , e M ] spans the noise subspace defined as the orthogonal complement of U, Λ s = diag{λ 1 , . . . , λ q }.

In this paper, we focus on a MUSIC based criterion which makes use of the orthogonal property of span{C} = span{U} and span{V} to estimate the DOA. We assume that the sources distribution shapes and the sources angular dispersion parameter ∆ i are known in the model, then there is only one parameter θ i for the i th source to estimate:

θi = argmax θ 1 c H h (θ)V 2 , (3)
which is named CD-MUSIC in the following. Note that the standard point source MUSIC estimator belongs to this class of estimators, since the point source case is obtained for ∆ = 0 and h i (φ) in (2) being the Dirac delta function. In the following of this paper, the goal is to evaluate the robustness of this estimator to model errors.

Model error definitions

The criterion (3) can be impacted by two types of errors. First, the actual angular spread distribution h of the source, may not be known in practice, in this case the function h used by the estimator will be different from the actual h, that is to say, the shape of h is badly known, or the shape of h is known but with an error on the dispersion parameter ∆. Secondly, the covariance matrix R is estimated from a finite number of snapshots, therefore the estimated covariance matrix R and its noise subspace V are different from the actual R and V. Taking into consideration the two types of errors, the algorithm that we study is:

θi = arg max θ 1 c H h (θ) V 2 . ( 4
)
Let us introduce the definitions related to the errors. Assuming that the estimator ( 4) is based on an angular distribution h, the model error on the steering vector is defined as ∆c(θ i ) = c h(θ i ) -c hi (θ i ). Similarly the model error on the covariance matrix can be de-

fined as ∆ R = R -R, where R = CR s CH + σ 2 b I, C = [c h(θ 1 ), . . . , c h(θ q )],
mator R = 1 N N n=1 y(t n )y H (t n ), the errors are de- fined by ∆ R = R -R and ∆ V = V -V, where R = Û Λs ÛH + σ 2 b V VH . Let us also define Π = Ṽ ṼH , Π = V VH
, and Π = VV H .

Performance analysis

In this section, we will investigate the effects of an imperfect knowledge of h(φ), and the finite number of snapshots on the CD-MUSIC algorithm.

General case

According to (4), for the i -th source, the DOA estimation θi satisfies that the first order derivative of the denominator of (4) equals zero so that :

2Re{ ċH h ( θi ) Πc h( θi )} = 0, (5)
where ċh ( θi ) = ∂c(θ) ∂θ | θi . Assuming that, the estimation error (∆θ i = θi -θ i ) is small, we introduce the second order Taylor series approximations of c h( θi ) and ċh ( θi ):

c h( θi ) ≈ c h(θ i ) + ∆θ i ċh (θ i ) + 1 2 ∆θ 2 i ch (θ i ), (6) ċh
( θi ) ≈ ċh (θ i ) + ∆θ i ch (θ i ) + 1 2 ∆θ 2 i ... c h(θ i ), (7)
where ch

(θ i ) = ∂ 2 c(θ) ∂θ 2 | θi , and ... c h(θ i ) = ∂ 3 c(θ) ∂θ 3 | θi .
In order to derive the expressions of the DOA estimation error, we make use of the first order approximation in [START_REF] Weiss | Effects of modeling errors on the resolution threshold of the MUSIC algorithm[END_REF], which says that:

Π -Π = -Π∆ RQ - Q∆ RΠ, where Q = U(Λ s -σ 2 b I) -1 U H , I is the q × q identity matrix.
Introducing ( 6) and ( 7) in [START_REF] El Markhi | An improved cyclic beamforming method for signal doa estimation[END_REF], and the expression of Π yields:

A(θ i )∆θ 2 i + B(θ i )∆θ i + C(θ i ) = 0, (8)
where the terms of order greater than 2 in ∆θ i have been neglected, and the scalar

A(θ i ), B(θ i ), C(θ i ) are defined below with B(θ i ) = B 1 (θ i )+B 2 (θ i ) and C(θ i ) = C 1 (θ i ) + C 2 (θ i ).
In the following, we omit the θ i in the notations for simplicity:

A = Re 1 2 ċH h Πc h + cH h Π ċh + 1 2 ... c H h Πc h , B 1 = Re ċH h Π ċh + cH h Πc h , B 2 = Re 2 ċH h Π∆ RQ ċh + cH h Π∆ RQc h , C 1 = Re ċH h Πc h , C 2 = Re ċH h Π∆ RQc h .
The expression of ∆θ i can be obtained by solving the 2 nd order equation [START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF].The CD-MUSIC criterion chooses the minimum value of the denominator as the estimation result, so θi should satisfy:

∂ 2 c H h (θ) Πch(θ) ∂θ 2
| θi > 0, which makes it possible to choose the solution of ( 8):

∆θ i = -B + √ B 2 -4AC 2A . ( 9
)
It follows that the estimation bias and the MSE are given by:

E [∆θ i ] = -B 1 + B 2 1 -4AC 1 2A , (10)
E[∆θ 2 i ] = B 2 1 -2AC 1 -B 1 B 2 1 -4AC 1 2A 2 + σ 2 b N 1 4A 2 - B 1 4A 2 B 2 1 -4AC 1 ϕ + χ 2A B 2 1 -4AC 1 . ( 11
)
The calculations and the definitions of ϕ and χ can be found in Appendix A.

It is interesting to see that the MSE is composed of two terms: one only depends on the model error, the other depends on the model error but with a factor σ 2 b /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 where the estimation error ∆θ i is small enough, so that the second order terms in ∆θ i can be negligible with respect to the first order terms. Keeping only the first order terms in (8) yields:

B∆θ i + C = 0. ( 12
)
Introducing the property Πc h = Π∆c, and neglecting the second order terms in ∆θ i ∆c and ∆θ i ∆ R, [START_REF] Zoubir | Efficient subspace-based estimator for localization of multiple incoherently distributed sources[END_REF] becomes:

∆θ i ċH h Π ċh = Re{ ċH h Π∆c + ċH h Π∆ RQc h}. ( 13
)
Replacing Π by Π -∆ Π, and neglecting the second order terms in ∆ Π∆θ i , ∆ Π∆c and ∆ Π∆ R, the expression of the estimation error of DOA can be simplified as:

∆θ i = Re{ ċH h Π∆c} ċH h Π ċh + Re{ ċH h Π∆ RQc h} ċH h Π ċh . ( 14
)
As ∆ R follows a Wishart distribution, E[∆ R] = 0 [START_REF] John | Expectations of useful complex wishart forms[END_REF], it follows that the estimation bias is derived as:

E[∆θ i ] = Re{ ċH h Π∆c} ċH h Π ċh , ( 15
)
and the MSE can be given by:

E[∆θ 2 i ] = Re{ ċH h Π∆c} ċH h Π ċh 2 + σ 2 b 2N • Re{ ċH h Π ċh c H h QRQc h} ( ċH h Π ċh ) 2 . ( 16
)
Note that this first order approximation makes it possible to express [START_REF] Raich | Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms[END_REF] and the first term of ( 16) as an explicit function of the model error ∆c.

To obtain ( 15) and ( 16), the second order of Talyor approximation in ∆θ i has been ignored. Therefore, [START_REF] Raich | Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms[END_REF] and ( 16) are valid in the case that ∆θ i is small enough, while [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF] and [START_REF] Besson | Decoupled estimation of DOA and angular spread of a spatially distributed source[END_REF] are also valid for a larger ∆θ i .

Assuming that the source angular distribution is symmetric and the source dispersion is small enough, and the array aperture is not too large, we introduce a second order Taylor approximation in φ such that:

c hi (θ i ) = π 2 -π 2 a(θ i + φ)h i (φ)dφ ≈ a(θ i ) + 1 2 ä(θ i )σ 2 h , (17)
c hi (θ i ) = π 2 -π 2 a(θ i + φ) hi (φ)dφ ≈ a(θ i ) + 1 2 ä(θ i )σ 2 h, (18)
where σ 2 h = φ 2 h(φ)dφ, σ 2 h = φ 2h (φ)dφ is the angular spread of the source and the model, respectively.

For the sake of simplicity, assume again that we are in the case of one source. Taking into account that Π = Ic h(c H h c h) -1 c H h , and introducing ( 17) and ( 18) in (15), the estimation bias can be given by:

E[∆θ i ] = 1 2 (σ 2 h -σ 2 h )(a H ä ȧH a -M ȧH ä) M ȧH ȧ -ȧH aa H ȧ , ( 19
)
where a is simplified for a

(θ i ), ȧ = ∂a ∂θ , ä = ∂ 2 a ∂θ 2 , ... a = ∂ 3 a ∂θ 3 .
Let us recall that a(θ i ) = e -jτ1 , . . . , e -jτm , . . . , e -jτ M , where τ m = 2π v λ t m (θ i ). The estimation bias can be given as:

E[∆θ i ] (20) = 1 2 (σ 2 h -σ 2 h )( M m=1 τm M m=1 τm -M M m=1 τm τm ) M M m=1 τ 2 m -( M m=1 τm ) 2 , ( 21
)
where τm = ∂τm ∂θ , τm = ∂ 2 τm ∂θ 2 . An important result is that, at first order, the estimation bias only depends on the difference of the angular spread of the source and the model and not on their shapes.

Assuming a linear array is used, τ m = 2πd m sin(θ), where d m is the distance between the m th sensor and the origin, and the estimation bias can be given as:

E[∆θ i ] = 1 2 (σ 2 h -σ 2 h ) tan(θ). ( 22
)
From ( 19) and ( 22) we can see that the sign of the estimation bias depends on the difference between the angular spread of the source and the model. Note that in the case of a uniform distribution of the spatial dispersion, σ 2 h = ∆ 2 /12 while for the Gaussian distribution σ 2 h = ∆ 2 .

Numerical results

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, source signals s(t) and noise n(t) are complex Gaussian independent random variables, with zero as mean value and 1 as source power. A uniform linear array (ULA) is composed of M = 10 omni-directional sensors spaced by half-wavelength. In this case the steering vector pour point source i can be given as a(θ i ) = 1, e -jπsin(θi) , . . . , e -jπ(M -1)sin(θi) T . The steering vector for uniformly distributed and Gaussian source can be given as

c hi (θ i ) = 1 ∆ ∆ 2 -∆ 2 a(θ i + φ)dφ, and c hi (θ i ) = 1 √ 2π∆ +∞ -∞ a(θ i + φ) exp{-φ 2 2∆
2 }dφ, respectively. In addition, SN R = 10dB, and N = 1000 snapshots. Each Monte-Carlo simulation is a realization of source signals and additive noise. Different analytical results are compared to simulation results. We first present the validation of our theoretical results and then we analyze the performance.

Validation of theoretical results

The validation of estimation bias formulas [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF] and ( 15) is illustrated in figure 1, with two uniform distributed sources (θ 1 = 33 • , θ 2 = 47 • ), and with a same angular dispersion ∆ = 10 • . Figure 1(a) shows the error for an underestimation of the dispersion parameter ∆ while Figure 1(b) shows error for an overestimation. As expected, the estimation bias is null when the exact model parameter ∆ = ∆ = 10 • is used. Comparing the simulation results, one can observe that the DOA estimation bias is smaller when the parameter ∆ is smaller than ∆ (Figure 1(a)), than when it is larger than ∆ (Figure 1(b)). Focusing on the validity of the expressions derived in the previous section, we can notice that for both θ 1 and θ 2 , the DOA estimation bias obtained in [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF] outperforms the one obtained in [START_REF] Raich | Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms[END_REF], where the advantage of ( 10) is much more evident for θ 1 , which has a larger estimation error.

Figure 2 illustrates the validation of the results obtained in [START_REF] Kangas | Finite sample and modeling error effects on ESPRIT and MUSIC diretion estimators[END_REF] as a function of the difference of the angular spread of the source and model, in order to express the estimation error as an explicit function of the model error. In this scenario one source is considered at θ 0 = 45 • with uniform distribution and ∆ varies from 0 • to 15 • , the model is uniform in figure 2(a) and Gaussian in figure 2(b). The origin is placed at the center of the array. We can see that for such an array configu- ration, the sign of the estimation bias depends on the difference between angular dispersion of the source and the model. In particular, when the angular spread of the source equals to the angular spread of the model, the estimation bias vanishes.

Robustness to shape mismatch

These simulations investigate the impact of the model error due to a mismatch of the shape of the angular distribution. To localize a source with an actual uniform distribution, we compare the performance of three estimators: i) MUSIC, assuming a point source model (figure 3(a), 3(b)), ii) CD-MUSIC assuming a uniform angular dispersion (figure 3(a)), or iii) a Gaussian angular dispersion (figure 3(b)). We consider the case where two uniform distributed sources arrive from θ 1 = 28 • and θ 2 = 32 • , with an angular dispersion ∆ = 10 • , assuming that the covariance matrix is known. Estimation error from simulation and theoretical equations [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF] are plotted versus the model dispersion parameter ∆.

In figure 3(a), it is shown that the model error due to a point source model (point model MUSIC) provides a larger estimation error than the model error due to a mismatch on parameter ∆, except when ∆ is overestimated.

As expected, the error due to distributed sources can be canceled with CD-MUSIC if the source dispersion parameters are known (in this case : uniform distribution and ∆ = ∆ = 10 • ). Moreover, if the dispersion is only roughly known (i.e. ∆ ≈ ∆ = 10 • one can see that CD-MUSIC still outperforms standard MUSIC unless dispersion is too overestimated.

When the model error is due to the use of a Gaussian shape instead of a uniform one (figure 3(b)), we can note that the CD-MUSIC error can be also reduced near to zero for the value ∆ ≈ 3.5 • , which approximately corresponds to the case σ 2 h = σ 2 h, when ∆ = 10 • is the spread of the rectangular model and ∆ = 3.5 • is the standard deviation of the Gaussian model. The results reveal the robustness of CD-MUSIC to the spatial distribution of the sources. Even if we take an angular distribution shape for the model which is different from that of the source signal, it is possible to have a DOA estimation bias near to zero providing ∆ is well chosen. The only case where standard MU-SIC outperforms CD-MUSIC is when the dispersion is badly known and it is overestimated with respect to the actual value. Also note that in this scenario the theoretical expression fits the simulation results except when resolution problems arise (mainly when the parameter ∆ is over-estimated).

Angular separation

In order to explore the resolution capabilities of CD-MUSIC in presence of model errors, figure 4 shows the absolute value of the DOA estimation bias when the model error due to the dispersion parameter ( ∆) and the angle between the two sources (θ 2 -θ 1 ) both vary. In this scenario, we consider two sources with uniform angular dispersion (∆ = 10 • ) and with θ m = 1 2 (θ 1 + θ 2 ) = 30 • . The stars mark the region where the sources are not resolved, that is to say, the two sources are so close that the CD-MUSIC criterion has only one peak in the middle. In order to highlight the behavior of the estimator, figure 5 plots the estimation error in two extreme cases : for close sources (θ 2 -θ 1 = 4 • -figure From these results one can observe that firstly, CD-MUSIC achieves high resolution performance for CD sources under the constraint of small modeling error (i.e. ∆ ≈ ∆ = 10 • ). In contrary, standard point source MUSIC estimator (i.e. ∆ = 0) can not provide reliable estimate for close CD sources. Secondly, when CD sources are well separated, CD-MUSIC is robust to model error on the angular dispersion parameter unless it is overestimated ( ∆ > ∆). Focussing on point source MUSIC (∆), one can notice that it provides accurate estimates in this case. In conclusion, unless for high resolution purpose, error modeling due to CD sources can be neglected.

Performance versus the number of snapshots

In figure 6, the RMSE (root mean square error) of the estimator is plotted versus the snapshots number. A uniform source and a Gaussian source arrive from θ 1 = 26 • and θ 2 = 34 • , respectively. We compare the performance of CD-MUSIC (with uniform distributed model in two cases: with a large error on dispersion parameter ∆ = 5 • and with a smaller error ∆ = 9 • ), MU-SIC for point source, and DSPE (with uniform source model). DSPE proposed in [START_REF] Valaee | Parametric localization of distributed sources[END_REF] estimates jointly the DOA and the source angular dispersion parameter, using a 2D MUSIC based approach but it is more timeconsuming. The RMSE for CD-MUSIC and MUSIC decrease as well as N increases, and then converge to non zero values given by ( 16). 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 number of snapshots is large. Also, in the case where the finite number of snapshots effect dominates, the 1D estimators have better performances; in contrary, when the model error effect dominates, the 2D estimator has better performance.

The advantage of CD-MUSIC in the case that the information of angular distribution of the actual sources is roughly known is illustrated by these simulations. When the model error is small (case ∆ = 9 • ) CD-MUSIC outperforms conventional MUSIC but also the more complex estimator DSPE for N < 100 or N < 1000 according to the type of sources. When the model error is larger (case ∆ = 5 • ) CD-MUSIC is almost equivalent to MUSIC except for a small number of snapshots.

Note that for CD-MUSIC and MUSIC, when N is too small, the theoretical results given by ( 16) do not suit the simulation results, because a first order of approximation of ∆ R is not enough when ∆ R is too large.

Conclusion

This paper has investigated the effects of both the angular dispersion of the source and the finite number of snapshots on the behavior of the CD-MUSIC. New analytical expressions of the DOA estimation bias and MSE as a function of these two perturbations have been given based on the second order Taylor approximation in order to provide accurate expressions even for a large error.

Numerical simulations are in adequacy with the proposed theoretical results. We have shown that, CD-MUSIC can be viewed as a trade-off between the standard MUSIC and 2D estimator DSPE. In the case that the angular dispersions of the sources are roughly known, it outperforms the standard MUSIC and the performance can be quite similar to DSPE (or even better for a small number of samples) while DSPE has the main drawback to require the maximization of a 2D criterion. We also have shown that, at the first order and in the one source case, the DOA estimation bias only depends on the difference of the angular spreads between the model and signal spatial distributions and not on their shapes. These results also make it possible to determine when we can use the standard MUSIC and which error shall be expected, and when, on the contrary, it will be necessary to use a more computationally complex 2D method such as DSPE.

Appendix A

Based on (9), we study the case that the number of snapshots N is big enough, so that ∆ R, B 2 , C 2 is small, we can make the approximation :

B 2 -4AC ≈ B 2 1 -4AC 1 (1 + 1 2 2B 1 B 2 + B 2 2 -4AC 2 B 2 1 -4AC 1
).

(

) 23
Introducing ( 23) in [START_REF] Adam P Young | Cross-shore decay of cliff top ground motions driven by local ocean swell and infragravity waves[END_REF], and keeping only first order terms in ∆ R, the expression of DOA estimation error can be given by:

∆θ i ≈ -(B 1 + B 2 ) + B 2 1 -4AC 1 (1 + 1 2 2B1B2-4AC2 B 2 1 -4AC1 ) 2A . ( 24
)
∆ R is a Wishart distribution [START_REF] John | Expectations of useful complex wishart forms[END_REF] matrix with the property E[∆ R] = 0, so that E[B 2 ] = E[C 2 ] = 0. It follows that the DOA estimation bias can be derived as [START_REF] Pedersen | A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments[END_REF]. Similarly, the DOA estimation MSE can be given by:

E[∆θ 2 i ] = B 2 1 -2AC 1 -B 1 B 2 1 -4AC 1 2A 2 + (1/2A 2 - B 1 2A 2 B 2 1 -4AC 1 )E[B 2 2 ] + 1 A B 2 1 -4AC 1 E[B 2 C 2 ]. ( 25
)
Using the same method in [START_REF] Swindlehurst | A performance analysis of subspace method in the presence of model errors : Case of MUSIC[END_REF], we obtain E[B

σ 2 bṼ

 2 and ∆ Ṽ = Ṽ -V, where the subspace decomposition of R is given by R = Ũ Λs ŨH + ṼH . Assuming again that the covariance matrix is estimated from N snapshots by the empirical esti-

Fig. 1

 1 Fig. 1 Estimation bias vs. the angular dispersion model error. (2 sources)

Fig. 2

 2 Fig. 2 DOA estimation error vs. difference of the angular spread between signal and model. (1 source)

5 ∆

 5 (degree) estimation error (degree) simu : Uni model CD MUSIC simu : Point MUSIC theo : Uni model CD MUSIC theo : Point MUSIC (a) Uniform distribution and point source models

5 ∆

 5 (degree) estimation error (degree) simu : Gaus model CD MUSIC simu : Point MUSIC theo : Gaus model CD MUSIC theo : Point MUSIC (b) Gaussian distribution and point source models

Fig. 3

 3 Fig. 3 DOA estimation error |∆θ 2 | vs. angular dispersion of the model (2 sources)

Fig. 4 |E

 4 Fig. 4 |E [∆θ1] | vs. the source angular separation |θ 2 -θ 1 | and model angular dispersion ∆ (2 sources)

Fig. 5

 5 Fig. 5 |E [∆θ1] | vs. model angular dispersion ∆ (2 sources)

10

 10

  Fig. 6 DOA RMSE vs. number of snapshots (2 sources) 100 Monte-Carlo simulations)

  Re 4 ċh (θ i ) H Π ċh (θ i ) ċh (θ i ) H QRQ ċh (θ i ) + 4c h (θ i ) H Π ċh (θ i )c h (θ i )QRQc h (θ i ) + ch (θ i ) H Πc h (θ i )c h (θ i )QRQc h (θ i ) , χ Re 2 ċh (θ i ) H Π ċh (θ i ) ċh (θ i )QRQc h (θ i ) +c h (θ i ) H Π ċh (θ i )c h (θ i )QRQc h (θ i ) .

	and E[B 2 C 2 ] =	σ 2 b 2N χ, where:	2 2 ] =	σ 2 b 2N ϕ

ϕ

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *


Laure-Anne Parpaleix

Keywords: innovation management, economic growth, capital investment

Innovation and growth potential: managing investment in middle market companies

In recent years, economic growth has turned into a key challenge for both public politics and private actors thus leading to extensive discussions about growth determinants. Innovation early stood out as one of the main growth driver. Nevertheless, despite of prolific empirical economic studies, correlation tests between innovation and growth remains inconclusive. Based on recent management researches findings, my research aims at investigating new factors to explain these empirical discrepancies as well as the link between investment and innovation policies. It will be conducted in partnership with Bpifrance, a state-owned organism backing national public policies by assisting companies through different types of investments. Bpifrance provide an appropriate field of research as its main mission is to ensure firms forthcoming competitiveness through investment, growth and innovation. Beyond a better understanding of the growth trajectories, I expect to develop new management devices for investors such as Bpifrance, whose aim is to identify how to better contribute to the firms' and the national economic growth. My research focuses on middle market companies, a single category hiding a large diversity of innovation and growth patterns. As the focus on middle market companies is fairly new, little is known about their specific innovative capabilities and growth potential, thus making it hard for public actors to tailor appropriate policies. Both theoretical and managerial findings are expected among which a better understanding of the determinants of growth and the role of innovation capabilities in firms' growth trajectories; original growth and investments models with parameters linked to the description of innovation capabilities; and devices to steer the adjusted investment strategies.

Introduction

Economic growth which is a key challenge for public politics and private actors has turned into an intensive research topic in both economy and management. Many policies aim at ensuring a financing continuum for every key phase of business development and route investment in favor of strategic projects (e.g. export, innovation, and buyout). Sustaining growth is often at the heart of these policies. Therefore, search for a better understanding of growth determinants has been extensively discussed in numerous theoretical and empirical studies [START_REF] Coad | Empirical investigations into the characteristics and determinants of the growth of firms[END_REF]). Yet, although a wide list of potential growth process inputs has been established, correlations based on static and in retrospect input-output economic models remains inconclusive (Gupta, Guha et al. 2013). Besides, growth dynamic mechanism still need for further investigation (e.g. continuous innovation provides non persistent and unpredictable growth (Coad, Cowling et al. 2014)). Because of this failure in understanding growth drivers, investors lack of managing strategies allowing them to guide their investing policies.

Among the potential growth factor, innovation has gradually become increasingly explored. Although the study of the impact of innovation on economic growth has produced a voluminous and diverse literature [START_REF] Cameron | Innovation and Growth: a survey of the empirical evidence[END_REF]), a precise relationship has yet to be unequivocally established [START_REF] Demirel | Survey of the literature on innovation and economic performance[END_REF]. Besides, characterizing the relationship between firm innovativeness and growth raises the stake of identifying accurate and appropriate measurements for both growth and innovation. My thesis major challenges lie in exploring the relationship between innovation and growth as well as the link between investment and innovation policies.

Relevance of my dissertation

My PhD is carried out in partnership with Bpifrance, a state-owned organism backing national public policies by assisting companies through different types of investments.

First, Bpifrance provide an appropriate field of research as its main mission is to ensure firms forthcoming competitiveness through investment, growth and innovation. One of Bpifrance current strategic challenge is the support of French firms' growth strategies through a unique state-owned venture capital fund. Bpifrance doctrine states that it should "behave like a prudent investor operating under market conditions to serve public interest". Therefore, the venture capital fund has both to meet financial performances and to carry out a public service mission. A broad consensus suggests that innovative capabilities are one of the main sources of firm's economic growth. In line with this thesis, Bpifrance investors challenge lies in identifying firms with high-growth potential thanks to an analysis of their innovative capabilities. Thus, the aim is to provide investors with management strategies and devices helping them to invest in innovation capabilities.

Secondly, Bpifrance work in close partnership with middle market companies. This firm category is of particular interests for my thesis as it is a quite new one so the relationship between innovation and growth has been little examined and stays even more mysterious. One of Bpifrance venture capital fund; it is directed towards middle market companies. In 2008 a French law created this new business category to reach more precise analysis on these firms considered as valuable assets for French economy. It gathers companies employing between 250 and 5 000 workers with an annual turnover below 1.5 billion euro. In-between SMEs and large firms, French middle market companies' category, as German Mittelstand, gathers long lasting firms with entrenched innovation capabilities that should drive value generation and employment opportunities. However, this single category hides a large diversity of innovation and growth patterns. These various potential growth dynamics could explain why middle market companies growth lack of steadiness and persistence. Growth process remains tricky to characterize and monitor with current statistic tools used for SMEs and large firms, hence the need to adjust current tools.

Therefore, the purpose of my research is to answer to the following questions raised by the above assessment:

 Are innovation capabilities an accurate indicator of growth potential? How to characterize both innovation capabilities and growth (i.e. indicators, measurement methodology)?  How can investment reach greater impact on firm growth? How to shape investments tools (i.e. innovative firms' selection mechanisms, financial vehicles) and steering process accordingly?

Theoretical framework

Despite abundant literature on the relationship between innovation and growth, economic researches remain inconclusive.

Since Adam Smith first theory on macro-economic growth [START_REF] Smith | An Inquiry into the Nature and Causes of the Wealth of Nations[END_REF], there has been extensive academic research on its causes and consequences. Various disciplines contribute to model economic growth including economy (macro, micro and political economy), management [START_REF] Teece | Explicating dynamic capabilities: the nature and microfoundations of (sustainable) entreprise performance[END_REF], finance [START_REF] Timmons | Venture capital's role in financing innovation for economic growth[END_REF], law [START_REF] Levine | Law, Finance, and Economic Growth[END_REF]) and psychology (entrepreneurship literature). Nevertheless, economic growth remains an intellectual puzzle.

Historically, economists were the first to identify and explore technological progress as a driver of economic growth through macro-economic frameworks. Over time, various theories refined causal links between technological progress and growth rate at different level of analysis. Gradually, a shift happened from theories developed at an aggregate scale to a firm and an individual scale. Recent researches in management focused on the relationship between innovation and growth at the firm level.

In 1776, Adam Smith laid the founding principles of the classical growth model. Growth of the labor force and capital accumulation is measured by the increase of a production function with three variables (Y = f( N, L, K)): land (N), labor (L) and capital (K). Later, David Ricardo (Ricardo 1817) identified technical progress as an additional variable (Y = f( N, L, K, S)). Classical economists were interested in political economy and theories of distribution. Thus they developed macro-economic growth theories to better understand world economy and nations comparative advantages. Later, neoclassical theorists, Solow and Swan [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF]) modeled growth as an aggregate production function (Y (t) = F [K (t), L (t), A (t)]) using two factors of production: labor (L) and capital (K) adjusted by the total factor productivity, broad notion related to technological progress (A). This dynamic model assumes that economy admits one representative firm characterized by a unique aggregate production function (Y) contributing to the path towards a constant growth rate in the long run. Technological progress is considered as an exogenous and undefined variable which is often refers as the "black-box problem". Since the 60's, literature on endogenous technical change further developed neoclassical models of economic growth [START_REF] Schneider | What's New and What's Old in New Growth Theory: Endogenous Technology, Microfoundation, and Growth Rate Predictions[END_REF]).

Arrow's model [START_REF] Arrow | The economic implications of learning by doing[END_REF]) of learning-by-doing focuses on introducing common knowledge as a positive externality in the production function. In the 80's, the new endogenous growth model of Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF]) and Lucas further developed the analysis of innovative activity trying to understand the economic forces underlying technological progress. Technological progress, previously defined as a productivity factor, is described as a number of ideas, or the stock of knowledge accumulated up until time, later referred as human capital. In their theory, technological change is no more automatic as in the neoclassical theories but it arises because of intentional actions.

These three major economic growth models (classical, Solow and endogenous) refine the causal links between technological progress and growth. However, these theories remain at a macro-economic level and define a unique representative firm to depict capital accumulation.

Through the process of creative destruction, Schumpeter was one of the first economists to assign a leading role in economic growth to individual firms through the innovative entrepreneur. While previous economists assumed an aggregate production function for a unique final good, in his model, Schumpeter supposes heterogeneous firms in interaction that drive aggregate economic growth by their ability to introduce innovation to the market. [START_REF] Nelson | An evolutionary theory of economic change[END_REF] built on Schumpeter's ideas and borrow from biology the concept of natural selection to shape the framework of an evolutionary theory of firm behavior. As opposed to neoclassical models, although firms are motivated by profit, they are not concerned with maximizing it and no hypothetical state of industry equilibrium is assumed. They study firm's innovation dynamics and are interested in the differing patterns across industry. Their contributions led to subsequent work on cross-industry comparisons of industrial dynamics. Nelson and Winter search economic growth determinants at the atomistic scale of people in the firm. Therefore, their approach has been a source of inspiration for later work in psychology and other social sciences.

Some researchers have focused on refining mechanisms underlining creative capacity at firm. Knowledge spillover theory on ideas diffusion leading to collective growth is part of a flourishing economic literature [START_REF] Audretsch | R&D Spillovers and the Geography of Innovation and Production[END_REF] on the impact of R&D side effects and eco-systems on firm's growth.

This theoretical framework review highlights two parallel phenomenon regarding the growth process analysis. First, economic theories have gradually shifted from macroeconomics to microeconomics. Both Schumpeter and the evolutionally theories introduce individuality in the innovation-growth mechanism while older ones (classical, Solow and endogenous models) assumed collective macroeconomic progress. Secondly, innovation has been progressively recognized as the main source of growth.

While theoretically, economic approach assumes that innovation fosters firm performance, quantitative researches remain inconclusive [START_REF] Demirel | Survey of the literature on innovation and economic performance[END_REF]. Most of quantitative studies focus on correlation tests either test if growth followed a random walk (literature on Gibrat's Law [START_REF] Mansfield | Entry, Gibrat's Law, innovation and the growth of firm[END_REF])) or to evaluate the impact of endogenous (i.e.

patent granted, R&D expenditure, new product) and exogenous indicators (i.e. spillovers, clusters). Although some studies highlight a positive correlation between innovative activities and growth, its conclusions often cannot be duplicated. Besides, others (e.g. (Brouwer, Kleinknecht et al. 1993)) demonstrate that innovation and growth remains unrelated. Establishing a correlation between innovation capabilities and growth potential raises stakes in identifying accurate and appropriate measurements for both growth and innovation. Management literature should contribute to overcome these issues:

 Heterogeneity of growth measure is a first obstacle [START_REF] Delmar | Measuring Growth: Methodological Considerations and Empirical Results[END_REF], Delmar, Davidsson et al. 2003). It reflects both a technical issue (i.e. time period, firm size, type of industry) and a debate about the nature of targeted growth (i.e. sales, employment, market share, goodwill).

 Similarly, measuring innovation is one of central methodological issues. The question is: how to find accurate measures of innovation and examine its impact on firm's performance? Innovation is often measured in retrospect through R&D spending or number of patents. Both are quantitative indicators failing in differentiating how disruptive innovations are.

New approaches for research on relationships between investment and innovation

Theories on public action often assume a correlation between R&D investments, innovation outcomes and growth. However current economic studies, aiming at defining in retrospect a correlation between inputs and outputs, are not sufficient to demonstrate such a relationship. The difficulty to demonstrate the impact of R&D investment on companies' innovation and growth is known as "the R&D paradox" (Le Masson, Weil et al. 2010). An analysis of top 700 international companies' investments in industry demonstrated no correlation between R&D expenditures and firms growth rates of turnover. Other studies confirm those results whatever timescales, performance indicators or sectors used [START_REF] Jaruzelski | Money Isn't Everything[END_REF]. For example, among the most innovative companies, some, like Apple in 2010, have reached high-growth rates spending less in R&D than the average firm in their field. Therefore, taking into account the intensity of R&D investment only seems too restrictive. Beyond the initial financial inputs, appropriate design management and governance models seem necessary to support firms' innovation capabilities and assess innovation potential beforehand.

One hypothesis is that recent researches on innovative design regimes (Hatchuel and Weil 2003) would help reviewing traditional economic indicators and develop new ones tailored for measuring beforehand firms innovation dynamics thus helping to guide future growth and investments strategies.

Effects of growth support policies through investments can now be reviewed in the light of recent literature on design theories and innovation capabilities. For example, literature discusses the coupling between investment and innovation by questioning how a firm capital structure influences its innovative projects performance. Several studies advocate that venture capital funded R&D generates substantial increase in innovative outputs compared to corporate R&D without debating on causes of this productivity differential. Compared to corporate R&D or relationship lending, venture capital main asset relies in its ability to combine an initial financial input and support throughout the project lifetime. Capital venture, by interfering in firms at critical stages of development affects their design management and governance models. Then, different hypothesis could be investigated to explain R&D investment performance differential, for example whether it is due to strict project selection mechanisms ex-ante, to closer ex-post monitoring or to specific innovation management methods [START_REF] Engel | Accelerating Corporate Innovation: Lessons from the Venture Capital Model[END_REF].

Methodological approaches

My research will be conducted along multiple paths:  Contextualization through historical cases of public policies strategies that have supported firm growth. The aim is to size a public operator's specific assets for sustaining firms' growth compared to external private partners or internal corporate capabilities.

 Design of new growth models to better tackle the prolific economic and management literatures in order to detect new hypothesis based, among others, on recent researches about innovation capability.

 Characterization of innovation capabilities through multiple bodies of literatures and empirical cases. Both economic literature and management literature on dynamic capabilities theories (O'Connor 2008) argue on relationships between innovation and growth. Besides, cases study of financed firms, notably through Bpifrance records, in the light of design theories (e.g. Concept-Knowledge theory [START_REF] Hatchuel | Théories et modèles de la conception. Cours d'ingénierie de la conception[END_REF])), will contribute to model innovation capabilities and suggest news indicators. Qualitative studies enable the focus on surprising cases that are essential to single innovation parameters out.

 Understanding of the impact of investments on innovation and growth to adjust investments tools and steering processes through finance, innovation management and public action policies literatures. At the same time, I will conduct empirical analysis of investors' reasoning (notably when assessing firms' innovation capabilities, forecasting potential growth or shaping strategic guidance).

 In parallel of the above mentioned qualitative analysis, quantitative tests based on multiple databases provided by Bpifrance, completed if necessary by external sets of data (INSEE, CASD Community innovation survey and social data statements) will contribute to refine hypothesis on the coupling between investment, innovation and growth.

Expected empirical findings and implications for future research and for practitioners

On one hand, I expect the following theoretical implications:  A better understanding of the determinants of growth and the role of innovation capabilities in firms' growth trajectories.

 Original model with parameters linked to the description of innovation capabilities.

 Clarification of diverging firms growth objectives which are not always in line with public interest and its potential impact on innovation and investments strategy.

On the other hand, I aim at managerial implications regarding tools to steer the investment strategies (i.e. innovative firms' selection mechanisms, financial vehicles). Besides, I also expect findings regarding public policies as this research will question the hypothesis that the best strategy to serve public interest, as a state-owned venture capital fund, is maximizing a few individual firms' growth. I wish to examine potential alternative investment tools serving general interest.

Questions to focus upon at the Doctoral Colloquium

First issue that could be discussed at the doctoral colloquium is investors' theory for dealing with firms innovation capabilities. Investors are used to create well balanced risk portfolios but lack of criteria to assess the potential of financed project based on their innovation capabilities. Can findings on dynamic capabilities be adapted to determine firm's innovation capabilities from an investor external point of view?

A second question to focus upon tackles Bpifrance investment fund innovative mission of "behaving like a prudent investor operating under market conditions to serve public interest".

Here to invest and how to tailor innovative investment strategies that both generate financial returns and serve public interests?

Depending on my findings up until the doctoral colloquium, a last question to focus upon could be the role of innovation capabilities in firms' growth trajectories in light with growth models that I am currently defining based both on literature and empirical findings.