R G Novikov
email: novikov@cmap.polytechnique.fr

Non-abelian Radon transform and its applications

Considerations of the non-abelian Radon transform were started in [Manakov, Zakharov, 1981] in the framework of the theory of solitons in dimension 2+1. On the other hand, the problem of inversion of transforms of such a type arises in different tomographies, including emission tomographies, polarization tomographies, and vector field tomography. In this article we give a short review of old and recent results on this subject.

Introduction

We consider the transport equation

θ∂ x ψ + A(x, θ)ψ = 0, x ∈ R d , θ ∈ S d-1 , (1)
where

θ∂ x = d ∑ j=1
θ j ∂/∂x j and A is a sufficiently regular function on R d ×S d-1 with sufficient decay as |x| → ∞. We assume that A and ψ take values in M (n, C) that is in n × n complex matrices.

For equation (1) we consider the "scattering" matrix S:

S(x, θ) = lim s→+∞ ψ + (x + sθ, θ), (x, θ) ∈ T S d-1 , ( 2
)
where

T S d-1 = {(x, θ) ∈ R d × S d-1 : xθ = 0} (3)
and ψ + (x, θ) is the solution of (1) such that lim s→-∞

ψ + (x + sθ, θ) = I, x ∈ R d , θ ∈ S d-1 , ( 4
)
where I is the identity matrix.

We interpret T S d-1 as the set of all rays in R d . As a ray γ we understand a straight line with fixed orientation. If γ = (x, θ) ∈ T S d-1 , then γ = {y ∈ R d : y = x + tθ, t ∈ R} (up to orientation) and θ gives the orientation of γ.

We say that S is the non-abelian Radon transform along oriented straight lines (or the non-abelian X-ray transform) of A.

We consider the following inverse problem for d ≥ 2: Problem 1. Given S, find A.

Note that S does not determine A uniquely, in general. One of the reasons is that S is a function on T S d-1 , whereas A is a function on

R d × S d-1 and dim R d × S d-1 = 2d -1 > dim T S d-1 = 2d -2.
In particular, for Problem 1 there is a gauge type non-uniqueness, that is S is invariant with respect to the gauge transforms A → A ′ , A ′ (x, θ) = g -1 (x, θ)A(x, θ)g(x, θ) + g -1 (x, θ)θ∂ x g(x, θ),

(5

)
where g is a sufficiently regular GL(n, C)-valued function on R d × S d-1 and g → I sufficiently fast as |x| → ∞.

In addition, in particular, for Problem 1 there are Boman type non-uniqueness (see [Bo], [GN]) and non-uniqueness related with solitons (see [N1]).

Equation (1), the "scattering" matrix S and Problem 1 arise, for example, in different tomographies (see Sections 2-6, 8), in differential geometry (see Section 7) and in the theory of the Yang-Mills fields (see Section 9). In Sections 2-9 we give a short review of old and recent results on this subject.

Classical X-ray transmission tomography

Problem 1 arises as a problem of the classical X-ray transmission tomography in the framework of the following reduction:

n = 1, A(x, θ) = a(x), x ∈ R d , θ ∈ S d-1 , (6) S(γ) = exp[-P a(γ)], P a(γ) = ∫ R a(x + sθ)ds, γ = (x, θ) ∈ T S d-1 , ( 7
)
where a is the X-ray attenuation coefficient of the medium, P is the classical Radon transform along straight lines (classical X-ray transform), S(γ) describes the X-ray photograph along γ.

In this case, for d ≥ 2,

S T S 1 (Y ) uniquely determines a Y , ( 8
)
where Y is an arbitrary two-dimensional plane in R d , T S 1 (Y ) is the set of all oriented straight lines in Y . In addition, this determination can be implemented via the Radon inversion formula for P in dimension d = 2; see [R]. In connection with this formula see also Remark 2 in Subsection 2.2. For more information on the classical X-ray transmission tomography and on the classical X-ray transform, see, e.g., [GGG], [Na] and references therein.

Single-photon emission computed tomography (SPECT)

In SPECT one considers a body containing radioactive isotopes emitting photons. The emission data p in SPECT consist in the radiation measured outside the body by a family of detectors during some fixed time (where expected p is described by P a f defined below). The basic problem of SPECT consists in finding the distribution f of these isotopes in the body from the emission data p and some a priori information concerning the body. Usually this a priori information consists in the photon attenuation coefficient a in the points of body, where this coefficient is found in advance by the methods of the classical X-ray transmission tomography (mentioned in Section 2).

Problem 1 arises as a problem of SPECT in the framework of the following reduction [N1]:

n = 2, A 11 = a(x), A 12 = f (x), A 21 = 0, A 22 = 0, x ∈ R d , ( 9
)
S 11 = exp [-P 0 a], S 12 = -P a f, S 21 = 0, S 22 = 1, ( 10
)
where a is the photon attenuation coefficient of the medium, f is the density of radioactive isotopes, P 0 = P is defined in (7), P a is the attenuated Radon transform along oriented straight lines (attenuated ray transform), P a f describes the expected emission data,

P a f (γ) = ∫ R exp[-Da(x + sθ, θ)]f (x + sθ)ds, γ = (x, θ) ∈ T S d-1 , (11) Da(x, θ) = +∞ ∫ 0 a(x + sθ)ds, x ∈ R d , θ ∈ S d-1 , ( 12
)
where D is the divergent beam transform. In this case (as well as for the case of the classical X-ray transmission tomography),

for d ≥ 2, S T S 1 (Y ) uniquely determines a Y and f Y , ( 13
)
where Y is an arbitrary two-dimensional plane in R d , T S 1 (Y ) is the set of all oriented straight lines in Y . In addition, this determination can be implemented via the following inversion formula [N2]:

f = P -1 a g, where g = P a f, ( 14
)
P -1 a g(x) = 1 4π ∫ S θ ⊥ ∂ x ( exp [-Da(x, -θ)]g θ (θ ⊥ x) ) dθ, gθ (s) = exp (A θ (s)) cos (B θ (s))H(exp (A θ ) cos (B θ )g θ )(s)+ exp (A θ (s)) sin (B θ (s))H(exp (A θ ) sin (B θ )g θ )(s), A θ (s) = (1/2)P 0 a(sθ ⊥ , θ), B θ (s) = HA θ (s), g θ (s) = g(sθ ⊥ , θ), ( 15
)
Hu(s) = 1 π p.v. ∫ R u(t) s -t dt, ( 16
) x ∈ R 2 , θ ⊥ = (-θ 2 , θ 1 ) for θ = (θ 1 , θ 2 ) ∈ S 1 , s ∈ R.
Remark 1. The assumptions on a and f in ( 13)-( 15) can be specified as follows:

a, f are real -valued, a, f ∈ L ∞ (R 2 ), a, f = O(|x| -σ ) as |x| → ∞ for some σ > 1, ( 17
)
where Y is identified with R 2 in (13).

Remark 2. For a ≡ 0, formulas ( 14), ( 15) are reduced to the classical Radon inversion formula for P defined in (7) for d = 2.

For more information on SPECT and for more results on the attenuated ray transform P a we refer to [Na], [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF], [START_REF] Kunyansky | A new SPECT reconstruction algorithm based on the Novikov's explicit inversion formula[END_REF], [N2], [N3], [START_REF] Guillement | Optimized analytic reconstruction for SPECT[END_REF], [START_REF] Guillement | Inversion of weighted Radon transforms via finite Fourier series weight approximations[END_REF] and references therein.

Tomographies related with weighted Radon transforms

We consider the weighted Radon transforms P W (along oriented straight lines) defined by the formula

P W f (x, θ) = ∫ R W (x + sθ, θ)f (x + sθ)ds, (x, θ) ∈ T S d-1 , ( 18
)
where

W = W (x, θ) is the weight, f = f (x) is a test function.
The assumptions on W can be specified as follows:

W ∈ L ∞ (R d × S d-1 ), W = W , 0 < c 0 ≤ W ≤ c 1 , ( 19
) lim s→±∞ W (x + sθ, θ) = w ± (x, θ), (x, θ) ∈ T S d-1 .
If W = 1, then P W is reduced to the classical X-ray transform P defined in (7).

If W (x, θ) = exp ( -Da(x, θ) ) , ( 20
)
where Da is defined by ( 12), then P W is reduced to the classical attenuated ray transform P a defined by ( 11), ( 12). Transforms P W with some other weights also arise in applications. For example, such transforms arise in positron emission tomography, optical tomography, fluorescence tomography; see [Na], [Ba], [MP].

The transforms P W f arise in the framework of the following reduction of the nonabelian Radon transform S:

n = 2, A 11 = θ∂ x ln W (x, θ), A 12 = f (x), A 21 = 0, A 22 = 0, ( 21
)
S 11 = w - w + , S 12 = - 1 w + P W f, S 21 = 0, S 22 = 1. ( 22
)
In connection with P W and with the reduction ( 21), ( 22) we consider the following version of Problem 1, where we assume that W is known.

Problem 2. Given P W f and W , find f . General uniqueness and reconstruction results on Problem 2 were given, in particular, in [LB], [Be], [MQ], [F], [BQ], [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF], [N7], [START_REF] Guillement | Inversion of weighted Radon transforms via finite Fourier series weight approximations[END_REF], [I].

For some W exact and simultaneously explicit formulas for solving Problem 2 are also known, see [R], [N1], [BS], [Gi], [N6] and references therein.

Note that Problem 2 is nonoverdetermined for d = 2 and is overdetermined for d ≥ 3. Indeed,

P W f is a function on T S d-1 , whereas f is a function on R d and dim T S d-1 = 2d -2, dim R d = d, 2d -2 = d for d = 2, 2d -2 > d for d ≥ 3.
Nevertheless, Problem 2 is not uniquely solvable, in general, even for d ≥ 3. An example of non-uniqueness for Problem 2 for d = 2 was constructed in [Bo]. In

this example W ∈ C ∞ (R 2 × S 1 ), f ∈ C ∞ 0 (R 2
). An example of non-uniqueness for Problem 2 for d ≥ 3 was constructed in [GN]. In this example

W ∈ C α (R d × S d-1 ) for some α > 0, f ∈ C ∞ 0 (R d ).
In these examples assumptions ( 19) are also fulfilled. The notation C ∞ 0 stands for infinitely smooth compactly supported functions.

For more information on the theory and applications of the transforms P W we refer to [LB], [Be], [MQ], [F], [Na], [BQ], [Bo], [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF], [N7], [START_REF] Guillement | Inversion of weighted Radon transforms via finite Fourier series weight approximations[END_REF], [I], [GN] and references therein.

Neutron polarization tomography (NPT)

In NPT one considers a medium with spatially varying magnetic field. The polarization data consist in changes of the polarization (spin) between incoming and outcoming neutrons. The basic problem of NPT consists in finding the magnetic field from the polarization data. See, e.g., [DMKHSB], [LDS] and references therein.

Problem 1 arises as a problem of NPT in the framework of the following reduction:

n = 3, A 11 = A 22 = A 33 = 0, A 12 = -A 21 = -g B 3 (x), A 13 = -A 31 = g B 2 (x), A 23 = -A 32 = -g B 1 (x), ( 23
)
where B = (B 1 , B 2 , B 3 ) is the magnetic field, g is the gyromagnetic ratio of the neutron; in addition, S for equation (1) with A given by ( 23) describes the polarization data (but, in general, S can not be given explicitly in this case).

In this case S on T S 2 uniquely determines B on R 3 as a corollary of items (1), (2) of Theorem 6.1 of [N1]. In addition, the related 3D -reconstruction is based on local 2Dreconstructions based on solving Riemann conjugation problems (going back to [MZ]) and on the layer by layer reconstruction approach. The final 3D uniqueness and reconstruction results are global.

For the related 2D global uniqueness, see [E].

Electromagnetic polarization tomography (EPT)

In EPT one considers a medium with zero conductivity, unit magnetic permeability, and small anisotropic perturbation of some known (for example, uniform) dielectric permeability. The polarization data consist in changes of the polarization between incoming and outcoming monochromatic electromagnetic waves. The basic problem of EPT consists in finding the anisotropic perturbation of the dielectric permeability from the polarization data. See [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF], [NS], [START_REF] Sharafutdinov | The problem of polarization tomography[END_REF], [N5] and references therein.

Problem 1 arises as a problem of EPT (with uniform background dielectric permeability) in the framework of the following reduction (see [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF], [NS]):

n = 3, A(x, θ) = -π θ f (x)π θ , x ∈ R d , θ ∈ S d-1 , ( 24
)
where

π θ ∈ M (3, R), π θ,ij = δ ij -θ i θ j , f takes values in M (3, C
) and describes the anisotropic perturbation of the dielectric permeability tensor; by some physical arguments f must be skew-Hermition, f ij = -fji ; in addition, S for equation (1) with A given by ( 24) describes the polarization data (but, in general, S can not be given explicitly in this case).

In this case S on T S 2 does not determine f on R 3 uniquely, in general, (in spite of the fact that dim T S 2 = 4 > dim R 3 = 3), in particular, if

f 11 = f 22 = f 33 ≡ 0, ( 25
)
f 12 (x) = ∂u(x)/∂x 3 , f 13 (x) = -∂u(x)/∂x 2 , f 23 (x) = ∂u(x)/∂x 1 , f 21 = -f 12 , f 31 = -f 13 , f 32 = -f 23 ,
where u is a real smooth compactly supported function, then S ≡ I on T S 2 ; see [NS].

On the other hand, a very natural additional physical assumption is that f is an imaginary-valued symmetric matrix: f = -f , f ij = f ji . According to [N4], in this case

S on Λ uniquely determines f, at least, if f is sufficiently small, ( 26
)
where Λ is an appropriate 3d subset of T S 2 , for example,

Λ = ∪ 6 i=1 Γ ω i , Γ ω i = {γ = (x, θ) ∈ T S 2 : θω i = 0}, ( 27
)

ω 1 = e 1 , ω 2 = e 2 , ω 3 = e 3 , ω 4 = (e 1 + e 2 )/ √ 2, ω 5 = (e 1 + e 3 )/ √ 2, ω 6 = (e 2 + e 3 )/ √ 2,
where e 1 , e 2 , e 3 is the basis in R 3 . In addition, this determination is based on a convergent iterative reconstruction algorithm.

For more information on EPT and for more results on related non-abelian ray transforms we refer to [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF], [NS], [START_REF] Sharafutdinov | The problem of polarization tomography[END_REF], [N5] and references therein.

Inverse connection problem Let

A(x, θ) = a 0 (x) + d ∑ j=1 θ j a j (x), x ∈ R d , θ = (θ 1 , . . . , θ d ) ∈ S d-1 , ( 28
)
where a j are sufficiently regular M (n, C)-valued functions on R d with sufficient decay as |x| → ∞, j = 0, 1, . . . , d. Then Problem 1 arises in differential geometry.

In particular, for a 0 ≡ 0 equation (1) with A given by ( 28) describes the parallel transport of the fibre in the trivial vector bundle with the base R d and the fibre C n and with the connection a = (a 1 , . . . , a d ) along the Euclidean geodesics in R d ; in addition, S(γ) for fixed γ ∈ T S d-1 is the operator of this parallel transport along γ (from -∞ to +∞ on γ); see [START_REF] Sharafutdinov | On an inverse problem of determining a connection on a vector bundle[END_REF], [N1].

Besides, for a 0 ̸ ≡ 0 equation (1) with A given by ( 28) describes the parallel transport of the fibre in the trivial vector bundle with the base R d+1

1,d and the fibre C n and with the connection a = (a 0 , a 1 , . . . , a d ) (independent of time) along the light rays in the Minkowski space R d+1 1,d ; in addition, S(γ) for fixed γ = (x, θ) ∈ T S d-1 is the operator of this parallel transport along the light rays

l(γ, τ ) = {(t, y) ∈ R d+1 : t = 2 -1/2 s + τ, y = 2 -1/2 sθ + x, s ∈ R}, τ ∈ R,
with the orientation given by the vector 2 -1/2 (1, θ) (from -∞ to +∞ on l(γ, τ ) for an arbitrary τ ∈ R); see [N1].

In these cases Problem 1 is an inverse connection problem. The determination in this problem is considered modulo gauge transforms

a = (a 0 , a 1 , . . . , a d ) → a ′ = (a ′ 0 , a ′ 1 , . . . , a ′ d ), a ′ 0 = g -1 a 0 g, a ′ i = g -1 a i g + g -1 ∂ i g, ∂ i g(x) = ∂g(x) ∂x i , i = 1, . . . , d, (29)
where g is a sufficiently regular GL(n, C)valued function on R d and g → I sufficiently fast as |x| → ∞.

Global uniqueness and reconstruction results on this inverse connection problem in dimension d ≥ 3 were given for the first time in [N1]. The related reconstruction is based on local 2D-reconstructions based on solving Riemann conjugation problems (going back to [MZ]) and on the layer by layer reconstruction approach.

In addition, counter examples to the global uniqueness for the aforementioned inverse connection problem for a 0 ≡ 0 in dimension d = 2 were also given for the first time in [N1].

These counter examples use the soliton solutions constructed in [Wa], [V] for equation (38) mentioned below.

In addition, for the global uniqueness in dimension d = 2 for the case of compactly supported a = (a 0 , a 1 , . . . , a d ), see [E].

Note that [N1] was stimulated by [START_REF] Sharafutdinov | On an inverse problem of determining a connection on a vector bundle[END_REF], where [START_REF] Sharafutdinov | On an inverse problem of determining a connection on a vector bundle[END_REF] was preceded by [We].

For more information on the inverse connection problem we refer to [MZ], [START_REF] Sharafutdinov | On an inverse problem of determining a connection on a vector bundle[END_REF], [N1], [E], [N4], [P], [GPSU] and references therein.

Vector field tomography

The inverse connection problem of Section 7 arises as a problem of the vector ultrasonic tomography in the framework of the following reduction:

n = 2, a 0 = ( a(x) 0 0 0 ) , a j = ( 0 u j (x) 0 0 ) , x ∈ R d , j = 1, . . . , d, (30)
S 11 = exp[-P 0 a], S 12 = exp[-P a u], S 21 = 0, S 22 = 1,

where P 0 a is defined as in ( 7), (10),

P a u(γ) = ∫ R exp[-Da(x + sθ, θ)]θu(x + sθ)ds, γ = (x, θ) ∈ T S d-1 , (32) θu = d ∑ j=1
θ j u j , Da is defined as in ( 12), a is the attenuation coefficient, u = (u 1 , . . . , u d ) is the flow velocity, P a u is the attenuated vectorial Radon transform of u along oriented straight lines. The transform P a u for a = 0 is the standard vectorial Radon transform of u and is related to time-of-flight measurements or to Doppler measurements; P a u for a ̸ ≡ 0 is related to the attenuated Doppler measurements; see [Sch] and references therein.

In connection with mathematics of vector field tomography we refer to [GGG], [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF], [N1], [START_REF] Sharafutdinov | Slice-by-slice reconstruction algorithm for vector tomography with incomplete data[END_REF], [KB], [Sch] and references therein.

Theory of the Yang-Mills fields

A. The inverse connection problem of Section 7 for a 0 ≡ 0 arises, in particular, in the framework of studies on inverse problems for the Schrödinger equation

d ∑ j=1 - ( ∂ ∂x j + a j (x) ) 2 ψ + v(x)ψ = Eψ (33)
in the Yang-Mills field a = (a 1 , . . . , a d ) at high energies E (i.e., for E → +∞); see [N1] and references therein. The reason of this consists in the fact that for ψ of the form

ψ = e isθx (µ 0 (x, θ) + O(s -1 )), x ∈ R d , θ ∈ S d-1 , s = √ E → +∞, (34)
equation ( 33) in its leading part is reduced to equation (1) with µ 0 in place of ψ, where A is given by ( 28) with a 0 ≡ 0. B. The inverse connection problem of Section 7 for d = 2 arises, in particular, in the framework of integrating the self-dual Yang-Mills equations; see [MZ], [Wa], [V], [N1] and references therein.

Actually, Problem 1 for

A(x, θ) = a 0 (x) + θ 1 a 1 (x) + θ 2 a 2 (x), x = (x 1 , x 2 ) ∈ R 2 , θ = (θ 1 , θ 2 ) ∈ S 1 , ( 35
)
with M (n, C)-valued a 0 , a 1 , a 2 (and some linear relation between a 1 and a 2 ) was considered for the first time in [MZ] in the framework of integration by the inverse scattering method of the evolution equation (χ

-1 χ t ) t = (χ -1 χ z ) z , ( 36
)
where t, z, z in (36) denote partial derivatives with respect to t, z = x 1 + ix 2 , z = x 1 -ix 2 and where χ is SU (n)-valued function. Equation ( 36) is a (2+1)-dimensional reduction of the self-duel Yang-Mills equations in 2+2 dimensions.

To our knowledge, the terminology "non-abelian Radon transform" was introduced namely in [MZ] where it was used for S in (2) corresponding to the aforementioned A of (35).

The inverse scattering transform in [MZ] is based on Riemann conjugation problems. Related analysis was significantly developed, in particular, in [N1].

In addition, Problem 1 for A(x, θ) = θ 2 a 2 (x), x = (x 1 , x 2 ) ∈ R 2 , θ = (θ 1 , θ 2 ) ∈ S 1 , (37)

with M (n, C)-valued a 2 arises in the framework of the inverse scattering method for the equation (J

-1 J x 1 ) x 1 -(J -1 J x 2 ) t = 0, ( 38
)
where t, x 1 , x 2 in (38) denote partial derivatives with respect to t, x 1 , x 2 and where J is SU(n)-valued function; see [Wa], [V], [N1], at least, for n = 2. Equation ( 38) is also a (2+1)-dimensional reduction of the self-duel Yang-Mills equations in 2+2 dimensions. This reduction is different from (36).

The aforementioned counter examples to the global uniqueness for the inverse connection problem of Section 7 for a 0 ≡ 0 in dimension d = 2 were constructed in [N1] using results of [Wa] and subsequent results of [V] concerning soliton solutions for equation (38).


Abdeldjalil Aissa

El Bey
email: elbey@tsi.enst.fr

Hicham Bousbia-Salah

Karim Abed-Meraim

Yves Grenier
email: grenier@tsi.enst.fr

Yves Grenier Audio

A Aïssa-El-Bey

AUDIO SOURCE SEPARATION USING SPARSITY

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

This paper deals with blind source separation (BSS). The blind context means that neither the sources nor the mixing matrix are known. The goal of BSS is to recover the sources up to scaling and permutation by, only, using the mixtures. Blind source separation (BSS) has applications in several areas, such as communication, speech / audio processing, and biomedical engineering [START_REF] Cichocki | Adaptive Blind Signal and Image Processing[END_REF]. A fundamental and necessary assumption of BSS is that the sources are statistically independent and thus are often separated using higher-order statistical information [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF]. If some information about the sources is available at hand, such as temporal coherency [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF], source nonstationarity [START_REF] Belouchrani | Blind source separation based on time-frequency signal representations[END_REF], or source cyclostationarity [START_REF] Abed-Meraim | Blind source separation using second order cyclostationary statistics[END_REF], then one can remain in the secondorder statistical scenario. In the case of non-stationary signals (including audio signals), certain solutions using time-frequency analysis of the observations exist [START_REF] Yilmaz | Blind separation of speech mixtures via time-frequency masking[END_REF]. Other solutions use the statistical independence of the sources assuming a local stationarity to solve the BSS problem [START_REF] Pham | Blind separation of instantaneous mixtures of non stationary sources[END_REF]. This is a strong assumption that is not always verified [START_REF] Smith | An analysis of the limitations of blind signal separation application with speech[END_REF]. To avoid this problem, we propose a new approach that handles the general linear instantaneous model (possibly noisy) by using the sparsity assumption of the sources in the time domain. The use of sparsity to handle this model, has arisen in several papers in the area of source separation [START_REF] Cichocki | Adaptive Blind Signal and Image Processing[END_REF][START_REF] Zibulevsky | Sparse source separation with relative Newton method[END_REF]. We first present a sparsity contrast function for BSS. Then, in order to achieve BSS, we optimize the considered contrast function using an iterative algorithm based on the gradient technique.

In the following section, we discuss the data model that formulates our problem. Next, we detail the different steps of the proposed algorithm. In Section 4, some simulations are undertaken to validate our algorithm and to show the usefulness of the proposed method.

DATA MODEL

Assume that N audio signals impinge on an array of M ≥ N sensors. The measured array output is a weighted superposition of the signals, corrupted by additive noise, i.e.

x(t) = As(t) + w(t) t = 0, . . . , T -1 (1)
where

s(t) = [s 1 (t), • • • , s N (t)] T is the N × 1 sparse source vector, w(t) = [w 1 (t), • • • , w M (t)]
T is the M × 1 gaussian complex noise vector, A is the M × N full column rank mixing matrix (i.e., M ≥ N ), and the superscript T denotes the transpose operator. The purpose of blind source separation is to find a separating matrix, i.e. a N × M matrix such that s(t) = Bx(t) is an estimate of the source signals.

Before proceeding, note that complete blind identification of separating matrix B (or the equivalently mixing matrix A) is impossible in this context, because the exchange of a fixed scalar between the source signal and the corresponding column of A leaves the observations unaffected. Also note that the numbering of the signals is immaterial. It follows that the best that can be done is to determine B up to a permutation and scalar shifts of its columns, i.e., B is a separating matrix iff:

Bx(t) = PΛs(t) (2)
where P is a permutation matrix and Λ a non-singular diagonal matrix.

ITERATIVE SPARSE ALGORITHM

In this section, we propose an iterative algorithm for the separation of sparse audio signals ISBS for Iterative Sparse Blind Separation. As well known, audio signals are characterized by their sparsity property in the time domain [START_REF] Cichocki | Adaptive Blind Signal and Image Processing[END_REF][START_REF] Zibulevsky | Sparse source separation with relative Newton method[END_REF] which is measured by their p norm where 0 ≤ p ≤ 1. This norm represents how the "energy" is concentrated on a small number of coefficients. Based on this, one can define the following sparsity contrast function,

G p (s) = 1 N N i=1 [J p (s i )] 1 p (3)
where

J p (s i ) = 1 T T -1 t=0 |s i (t)| p (4)
The algorithm finds a separating matrix B such as,

B = arg min B {G p (B)} (5)
where

G p (B) def = G p (z) (6)
and z(t) = Bx(t) represents the estimated sources. The approach we choose to solve ( 5) is inspired from [START_REF] Pham | Blind separation of mixture of independent sources through a quasi-maximum likelihood approach[END_REF].

It is a block technique based on the processing of T received samples and consists in searching the minimum of the sample version of [START_REF] Abed-Meraim | Blind source separation using second order cyclostationary statistics[END_REF]. Solutions are obtained iteratively in the form:

B (k+1) = (I + (k) )B (k) (7) z (k+1) (t) = (I + (k) )z (k) (t) (8)
where I denotes the identity matrix. At iteration k, a matrix (k) is determined from a local linearization of G p (Bx(t)). It is an approximate Newton technique with the benefit that (k) can be very simply computed (no Hessian inversion) under the additional assumption that B (k) is close to a separating matrix. This procedure is illustrated in the following steps: At the (k + 1) th iteration, the proposed criterion (4) can be developed as follows:

J p (z (k+1) i ) = 1 T T -1 t=0 z (k) i (t) + N j=1 (k) ij z (k) j (t) p = 1 T T -1 t=0 |z (k) i (t)| p 1 + N j=1 (k) ij z (k) j (t) z (k) i (t) p
Under the assumption that B (k) is close to a separating matrix, we have

| (k) ij | 1
and thus, a first order approximation of J p (z

(k+1) i
) is given by:

J p (z (k+1) i ) ≈ 1 T T -1 t=0 |z (k) i (t)| p 1 + p N j=1 e( (k) ij ) e z (k) j (t) z (k) i (t) -m( (k) ij ) m z (k) j (t) z (k) i (t) (9)
Table 1: Iterative Sparse Blind Separation (ISBS) algorithm 1. Initialize B (1) randomly (z (1) (t) = B (1) x(t)).

For

k = 1, • • • , K, compute R (k) by (12).
3. Update the separation matrix B (k+1) by (15). 4. Update the source estimate (16). thus,

J p (z (k+1) i ) ≈ 1 T T -1 t=0 |z (k) i (t)| p + p N j=1 e( (k) ij ) e |z (k) i (t)| p-1 e -φ (k) i (t) z (k) j (t) -m( (k) ij ) m |z (k) i (t)| p-1 e -φ (k) i (t) z (k) j (t) (10
) where e(x) and m(x) denote the real and imaginary parts of x and φ

(k) i (t) is the argument of the complex num- ber z (k) i (t).
Using equation (3), minimization of the above criterion [START_REF] Pham | Blind separation of mixture of independent sources through a quasi-maximum likelihood approach[END_REF] is similar to minimization of G p (z (k+1) ). Equation ( 3) can be rewritten in more compact form as:

G p I + (k) = G p (I) + e T r (k) R (k)H (11)
where (•) denotes the conjugate of (•) and the ij th entry of matrix R (k) is given by: R

(k) ij = 1 T T -1 t=0 |z (k) i (t)| p-1 e -φ (k) i (t) z (k) j (t) (12)
and T r is the matrix trace operator. Using a gradient technique, (k) can be written as:

(k) = -µR (k) (13)
where µ > 0 is the gradient step. Replacing (13) into (11) leads to,

G p I + (k) = G p (I) -µ R (k) 2 (14)
So µ controls the decrement of the criterion. Hence, at the (k + 1) th iteration, we have

B (k+1) = (I -µR (k) )B (k) (15) z (k+1) = (I -µR (k) )z (k) ( 16
)
This algorithm is summarized in Table 1.

SIMULATION RESULTS

We present here some numerical simulations to evaluate the performance of our algorithm. We consider an array of M = 5 sensors with half wavelength spacing receiving two audio signals in the presence of stationary complex temporally white noise of covariance σ 2 I (σ 2 being the noise power). 10000 samples are used with a sampling frequency of 8Khz. The sources arrive from the directions θ 1 = 30 and θ 2 = 45 degree.

In order to evaluate the performance, the separation quality is measured using two different criteria, the first one is the mean rejection level criterion [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF] defined as:

Iperf def = p =q E |(BA) pq | 2 ρ q E (|(BA) pp | 2 ) ρ p ( 17
)
where

ρ i = E(|s i (t)| 2 )
is the i th source power evaluated here as

1 T T -1 t=0 |s i (t)| 2 .
The second is the normalized mean square error (NMSE) of the sources defined as:

N M SE i def = 1 N r Nr r=1 min α α s i,r -s i 2 s i 2 (18) N M SE i = 1 N r Nr r=1 1 - s i,r s H i s i,r s i 2 (19) N M SE = 1 N N i=1 N M SE i . ( 20
)
where s i def = [s i (0), . . . , s i (T -1)] and s i,r is defined similarly and represents the r th estimate of source s i , α is a scalar factor that compensate for the scale indeterminacy of the BSS problem and N r is the number of Monte-Carlo runs. Both criteria are estimated over N r = 200 runs. Figure 1 represents the two original sources (s 1 (t), s 2 (t)) and the recovered ones (z 1 (t), z 2 (t)) by the proposed algorithm in a noiseless case. In Figure 2, the mean rejection level is plotted versus the SNR for the proposed algorithm and the algorithm SOBI [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF] which is considered as one of the most performing in separating audio sources. We used SOBI with 6 correlation matrices of respective delays τ = 1, . . . , 6. It is clearly shown that our algorithm (ISBS) performs better in terms of the mean rejection level especially for high SNR. One can observe in Figure 3, that we reach the same conclusion for the N M SE. Figure 4 compares the mean rejection level for ISBS and SOBI when the number of sensors increases. For 2 sensors, both algorithms perform equally. When the number of sensors is greater, ISBS has a much lower mean rejection level than SOBI. Figure 5 shows the mean rejection level against the sample size for ISBS and SOBI. When the sample size is small, SOBI outperforms the proposed algorithm ISBS. Whereas, ISBS has a much lower mean rejection level when the sample size is larger. It can be explained by the size of the samples: if it increases, the signals present more sparsity, which gives an advantage to ISBS.

DISCUSSION

The proposed algorithm outperforms in terms of mean rejection level and N M SE other algorithms that deal with separation from instantaneous mixtures using source independency. It is mostly dedicated to sparse sources in the time domain. Among its other advantages, the algorithm ISBS shows a low computational complexity and thus can be easily implemented. Furthermore, its flexibility allows us to extend the method the adaptive case. Nevertheless, the proposed algorithm presents a relative weakness due to the well known disadvantages of the use of gradient techniques such as, the choice of the step gradient µ that the speed convergence depends on and the problem of local minima.

CONCLUSION

This paper presents a blind source separation method for sparse sources in the time domain. A sparse contrast func- tion is introduced and an iterative algorithm based on gradient technique is proposed to minimize it and perform BSS. Numerical simulations have been performed to evidence the usefulness of the method. They showed good performance in terms of mean rejection level and N M SE compared to other separation technics (SOBI).

Figure 1 :Figure 2 :

 12 Figure 1: Blind source separation example for 2 audio sources and 5 sensors: up the two original source signals and bottom the two estimated sources by our algorithm.

Figure 3 :

 3 Figure 3: NMSE versus the SNR for 2 audio sources and 5 sensors: comparaison between SOBI and the proposed algorithm.

Figure 4 :

 4 Figure 4: Mean Rejection Level versus the number of sensors M for 2 audio sources for SNR= 10dB and 30dB.

Figure 5 :

 5 Figure 5: Mean Rejection Level versus the sample size T for 2 audio sources for SNR=30dB.


Charlotte Campana

Romain Léger

Rodolphe Sonnier

Laurent Ferry

Romain Leger
email: romain.leger@mines-ales.fr

Patrick Ienny

Effect of post curing temperature on mechanical properties of a flax fiber reinforced epoxy composite

Keywords: A. Biocomposite, B. Post-curing, C. Mechanical properties, D. Flax fibers 1

INTRODUCTION

Nowadays, composites reinforced with synthetics fibers such as carbon or glass fibers (CFRP or GFRP) are commonly used in various industrial fields from automotive to aerospace in order to reduce the weight of the final pieces [START_REF] Benzarti | Understanding the durability of advanced fibre-reinforced polymer (FRP) composites for structural applications[END_REF].

But glass and carbon fibers present some drawbacks (energetically expensive, made of nonrenewable resources, or difficult to recycle) [START_REF] Pickering | Recycling technologies for thermoset composite materials-current status. 2nd[END_REF][START_REF] Reynolds | An introduction to composites recycling[END_REF].

An interesting alternative to glass fibers is natural fibers such as flax, hemp, or wood fibers.

Indeed, natural fibers are carbon neutral, come from a renewable source and can easily be biodegraded [START_REF] Joshi | Are natural fiber composites environmentally superior to glass fiber reinforced composites? Compos Part A[END_REF]. They also have a lower density leading to specific properties similar to those of glass fibers [START_REF] Wambua | Natural fibres: Can they replace glass in fibre reinforced plastics?[END_REF][START_REF] Mohanty | Sustainable Bio-Composites from renewable resources: Opportunities and challenges in the green materials world[END_REF]. But natural fibers also present some drawbacks: a large variability of mechanical and physico-chemical properties depending on the period of harvesting, the stem location and the extraction method, their low durability because of their hydrophilicity, their sensitivity to high temperature leading to processing difficulties especially during post-curing of thermoset composites [START_REF] Haag | Influence of flax fibre variety and year-to-year variability on composite properties[END_REF][START_REF] Charlet | Characteristics of Hermès flax fibres as a function of their location in the stem and properties of the derived unidirectional composites[END_REF][START_REF] Baley | Influence of the absorbed water on the tensile strength of flax fibers[END_REF][START_REF] Alix | Effect of chemical treatments on water sorption and mechanical properties of flax fibres[END_REF][START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF][START_REF] Gassan | Thermal degradation of flax and jute fibers[END_REF][START_REF] Placet | Characterization of the thermo-mechanical behaviour of Hemp fibres intended for the manufacturing of high performance composites[END_REF]. This sensibility to high temperature raises several questions in the industrial field: Can the same processing protocol be used for biocomposites? What happens if high temperatures are applied during process? Indeed, GFRP are generally cured at a "low" temperature (between 60 and 100°C) and post cured at higher temperature around 150°C or more to complete the curing and reach the highest crosslinking rate and glass transition temperature possible [START_REF] Cook | Ageing and yielding in model epoxy thermosets[END_REF][START_REF] Kumar | Effect of post-curing on thermal and mechanical behavior of GFRP composites[END_REF]. A post-curing is also done to improve the modulus and strength of both the polymer and the composite and reduce the residual stresses. However, a post-cure can also lead to the thermo-oxidation of the resin. Such curing and post-curing conditions could degrade natural fibers. Natural fibers mechanical properties are mainly dependent on their water content [START_REF] Baley | Influence of the absorbed water on the tensile strength of flax fibers[END_REF][START_REF] Masseteau | An evaluation of the effects of moisture content on the modulus of elasticity of a unidirectional flax fiber composite[END_REF][START_REF] Placet | Influence of environmental relative humidity on the tensile and rotational behaviour of hemp fibres[END_REF][START_REF] Thuault | Effects of the hygrothermal environment on the mechanical properties of flax fibres[END_REF]. Post-curing at high temperature is likely to change the fiber water content and thus modify their mechanical behavior. Müssig and Haag mentioned that exposure of flax fibers at 120°C leads to loss of moisture and degradation of waxes [START_REF] Müssig | The use of flax fibres as reinforcements in composites[END_REF]. Placet showed that the mechanical properties (rigidity and fatigue behavior) of natural fibers are affected by thermal treatment beyond 150°C [START_REF] Placet | Characterization of the thermo-mechanical behaviour of Hemp fibres intended for the manufacturing of high performance composites[END_REF]. It was assumed to be related to the degradation of the cellular walls. Gassan and Bledski showed that a thermal exposure between 170 and 210°C leads to a significant drop in tenacity up to 70% for 2 hours at 210°C [START_REF] Gassan | Thermal degradation of flax and jute fibers[END_REF]. The degradation of mechanical properties could also occur at lower temperature during drying for instance [START_REF] Baley | Influence of drying on the mechanical behaviour of flax fibres and their unidirectional composites[END_REF]. On the contrary Xue and al.

reported that a thermal exposure between 170 and 180°C does not impact significantly the tensile properties of kenaf bast fibers if the exposure lasts less than one hour [START_REF] Xue | Temperature and loading rate effects on tensile properties of kenaf bast fiber bundles and composites[END_REF].

It appears from this quick overview that achieving a complete curing of biocomposites without damaging fibers is very challenging. But, most of the time, in order to substitute glass fibers by natural fibers in structural composites, a traditional curing and post-curing at high temperature, recommended in the resin datasheet, is applied and leads to composite with degraded mechanical properties. The main objective of this study is to assess the impact of such traditional processing upon the properties of a biocomposite based on epoxy resin and unidirectional flax fabrics. The motivation is to identify which component (resin, reinforcement or interphase) is the most sensitive to post-curing at high temperature thus determining the final functional properties of the materials.

EXPERIMENTAL

Materials

The epoxy resin (DER 332) was provided by Dow Chemicals (Midland, USA) with an epoxy equivalent weight of 170g/eq. Isophorone diamine (IPDA) from Sigma Aldrich (Saint-Louis, USA) with a functionality of 4 was used as hardener. The mixing of resin and hardener was carried out at a stoichiometric epoxy/amine ratio (80%wt DER 332 and 20%wt IPDA).

Quasi-unidirectional flax fabrics UD 360 have been supplied by Fibres Recherche Développement (Troyes, France). Its areal weight is 360g/m² (weft: 330g/m 2 ; warp: 30g/m 2 ) and its thickness is 0.4mm.

Preparation of composites

The composites were manufactured using a vacuum infusion process in controlled atmosphere (50%RH and 23°C). The resin was heated at 40°C to obtain a viscosity (of 120P/s) compatible with the vacuum infusion process and then mixed with the hardener during 3 minutes at 400rpm before the infusion. Four plies of fibers (300 x 300mm 2 ) oriented in the weft direction were infused at a constant pressure of 100 mbar during 30 minutes. The composite was then cured at 80°C for 24h (NoPC). Three post-curing cycles were studied: 2 hours at 100°C (PC100), 120°C (PC120) or 150°C (PC150). Resulting composites contain 30%vol of fibers.

Composite plates (300 x 300 x 3mm 3 ) were then cut into various size samples and stored at 23°C and 50%RH according to ISO 291 and ASTM D618.

Characterization

Differential Scanning Calorimetry (DSC)

A PYRIS Diamond DSC from Perkin Elmer was used to assess the evolution of the glass transition temperature and the crosslinking rate of the composites. Samples between 10 and 15mg were tested in aluminum pans with holes. The purge gas was nitrogen (at 20L/min). The heating program consists of a first step of 3 minutes at 20°C followed by a heating step at 10°C/min from 20 to 200°C. The samples were then cooled at 10°C/min to 20°C and heated again following the same heating program.

To determine the crosslinking rate, uncured resin samples were analyzed following the usual thermal program. The DSC curves (Fig. 1) showed an exothermic peak leading to the total curing enthalpy of the system (ΔH total , crosslinking rate of 100%). For the cured samples (NoPC, PC100, PC120 and PC150), the Heat Flow Vs. Temperature curves (Fig. 2) showed an endothermic peak ( ΔH peak , relaxation enthalpy) due to the cure at 80°C (NoPC) or the postcuring at 100 and 120°C (PC100 and PC120). The DSC curves for PC150 (Fig. 2) showed no endothermic peak but a clear glass transition temperature. A simple cross-multiplication allowed us to calculate the composite crosslinking rate following the equation 1. The total enthalpy of curing (ΔH total ) and the various relaxation enthalpies are shown in Table 1.

(Eq. 1)

Thermo-Gravimetric Analysis (TGA)

A Setaram (SETSYS evolution model) TGA device was used to assess the weight loss during different heating programs. Sample between 10 and 15mg were tested in ceramic crucibles.

The purge gas was nitrogen (at 20L/min). The basic heating program consists of a first step of 20 minutes at 20°C followed by a heating step at 10°C/min from 20 to 900°C. The curing/post-curing cycles were also tested on the fibers: 24 hours at 80°C followed by 2 hours at 100°C, 120°C or 150°C to assess to possible weight loss during the processes. According to NF EN ISO 527-4 standard, Young modulus is obtained by linear regression between a strain of 0.05 and 0.25%. However, the NF EN ISO 527-4 standard is not adapted to bio-composites because of their bilinear responses in the range rather than the linear behavior assumed in the standard [START_REF] Baley | Influence of the absorbed water on the tensile strength of flax fibers[END_REF]. This bilinear behavior can be attributed to a reorientation of the cellular fibrils. There is a transition in the slope between 0.2 and 0.3% called a "knee-point" leading to incorrect modulus calculations (Fig. 3a) [START_REF] Liang | Quasi-static behaviour and damage assessment of flax/epoxy composites[END_REF][START_REF] Bensadoun | Fatigue behaviour assessment of flax-epoxy composites[END_REF][START_REF] Berges | Influence of moisture uptake on the static, cyclic and dynamic behaviour of unidirectional flax fibre-reinforced epoxy laminates[END_REF]. To overcome this issue, the modulus was calculated according to NF EN ISO 527-4 standard but also at higher strain between 0.5 and 0.8% leading to a stabilized modulus (Fig. 3b).

Flax fabrics

Tensile tests were carried out on flax fabrics using a Zwick testing machine (TH010 model) with a loading cell of 10kN. The tests were performed at a constant displacement rate of 1 mm/min. Samples were 50mm large with a length ranging from 10 to 200mm to assess different modes of breaking (neat breaking and disentanglement). Series of 3 specimens per length were tested. Some fabrics were previously heated 24h at 80°C (F-No PC) and then 2h at various temperatures (F-PC100, F-PC120 and F-PC150) in order to simulate the curing process used to prepare the corresponding composites. The stiffness of the sample was determined using a linear regression on the linear part of the load-strain curves obtained.

Interlaminar Shear Strength (ILSS) tests

Following the NF EN ISO 14130 standard, ILSS tests were conducted on a Zwick testingmachine (TH010 model) equipped with a 2.5kN load cell; the constant displacement rate was 1mm/min. The distance between the bending supports L was 15mm. The length l, width b and thickness h of specimens were respectively 30, 15 and 3mm. The interlaminar shear strength was calculated from equation 2 and the maximal tensile stress from equation 3, where F is the force applied to break the samples.

(Eq. 2)

(Eq. 3)

Scanning electron microscopy (SEM)

Images of fractured cross-sections or of polished sections were acquired with an environmental scanning electron microscope (FEI Quanta 200). Samples were metallized with carbon in high vacuum to stabilize them during the analysis. The micrographs were then obtained under high vacuum at a voltage of 15kV, a working distance of 10mm and a 100 to 5000 magnification.

RESULTS AND DISCUSSION

Influence of post-curing on composite properties

Evolution of the glass transition temperature and density

Table 2 shows that an increase of the post-curing temperature leads to an increase in the glass transition temperature measured during the first heating ramp from 115°C for NoPC to 167°C

for PC 150. This phenomenon is due to the increase of the crosslinking rate of the composite.

Indeed, the higher the process temperature is, the higher the crosslinking rate and the glass transition temperature will be. The post-curing of 2 hours at 150°C leads to a fully crosslinked resin with a maximum glass transition temperature of 167°C. For all other post-curing conditions, the crosslinking is incomplete and a second heating ramp allows increasing further the glass transition temperature. NoPC samples have a density of 1.276g/cm 3 . When the post-curing temperature increases, the composite density decreases progressively with a maximum decrease of 0.82% for PC150

(d PC100 = 1.270g/cm 3 , d PC120 = 1.266g/cm 3 and d PC150 = 1.262g/cm 3 ). Crosslinking rate, glass transition temperature and density are all related to the curing/post-curing temperature. The higher the glass transition temperature is, the lower the composite density will be (because of the resin density) [START_REF] Gillham | Anomalous behavior of Cured Epoxy Resins : Density at room temperature vs. Time and Temperature of cure[END_REF][START_REF] Enns | Effect of the extent of cure on the modulus, glass transition, water absorptio, and density of an amine-cured epoxy[END_REF][START_REF] Fisch | Influence of structure and curing conditions on the density, degree of cure, and glass transition temperature during the curing of epoxide resins[END_REF]. The post-curing increases the crosslinking rate leading to less free ends. A re-arrangement of the molecular chains causes a variation of the samples specific volume (free volume + occupied volume) and thus a decrease of the density.

Tensile strength tests

The tensile strength of the non-post-cured (NoPC) and post-cured (PC100, PC120 and PC150) composites is shown on Figure 4a.

A decrease of the tensile strength from 252MPa for NoPC to 136MPa for PC150 is observed when the post-curing temperature increases. The higher the temperature of post-curing is, the lower the maximal stress is with a maximal decrease of 46% for the highest temperature. The decrease in tensile strength is particularly significant when post-curing temperature increases from 120 to 150°C with a relative variation of 40% between these two temperatures. A decrease of tensile strength despite a higher crosslinking rate evidences that one (or both) of the composite components is modified during post-curing.

Figure 4 -a) Tensile strength (MPa), elongation at break (%) and Young Modulus (GPa) of the composites post-cured at 100, 120 and 150°C and b) comparison between the Young modulus (GPa) and the stabilized modulus (GPa)

As shown in Figure 4a, the elongation at break (%) follows the same tendency. A significant decrease is observed for PC150 with an elongation at break of 0.74% (versus 1.64%, 1.45% and 1.33% for NoPC, PC100 and PC120 respectively). This may be explained by the modification of the resin which is more brittle when crosslinking rate is higher (see 3.2.1).

Bensadoum and al. studied epoxy composites reinforced with quasi-UD and UD flax fibers with a tensile modulus (calculated between 0.3 and 0.5% strain) of 15.9GPa and 20GPa

respectively. Those composites had a fiber volume fraction of 40% and were manufactured using Resin Transfer Moulding [START_REF] Bensadoun | Fatigue behaviour assessment of flax-epoxy composites[END_REF]. NoPC samples have a stabilized modulus of 12.9GPa (between 0.5% and 0.8% strain) for a composite with a fiber volume fraction of 30%. Using the rule of mixtures, this modulus would be of 16.4GPa for a composite with a fiber volume fraction of 40%.

Baley and al. studied a flax/epoxy composite with a fiber volume fraction of 40% and manufactured by compression molding after wet impregnation of the fibers. The studied composites had an axial modulus (between 0.05 and 0.25% strain) of 22.5±1.51GPa [START_REF] Baley | Influence of drying on the mechanical behaviour of flax fibres and their unidirectional composites[END_REF].

NoPC samples have a modulus between 0.05 and 0.25% strain of 17.9GPa for a fiber volume fraction of 30%. Using the rule of mixture, this composite would have a modulus of 23.06GPa

for a fiber volume fraction of 40%.

Many factors can affect the mechanical properties of a composite (type of flax and epoxy resin used, manufacturing process, …). However, the values obtained for NoPC are comparable to the ones obtained by Bensadoum's and Baley's composites allowing to position our values in an acceptable range of modulus for an epoxy resin reinforced with flax fibers.

Figure 4a also shows the Young modulus for all composites according to NF EN ISO 527-4 standard. Despite the high standard deviations, the Young's modulus tends to increase when the temperature of post-curing increases from 17.9 ± 0.68 GPa for NoPC to 22.2 ± 2.18 GPa for PC150. This trend is also followed by the stabilized modulus which is 28% lower and more reproducible (Figure 4b). We observe a decrease of the elongation at break and the failure tensile strength while the modulus increases meaning that the composite is more brittle.

Influence of post-curing on the composite components

In order to understand the behavior modifications of the composite when a post-cure is carried out, each component of the composite (resin, fibers and interface) was studied separately.

Post curing of the epoxy resin

Evolution of the glass transition temperature

As shown in Table 3, 24 hours of curing at 80°C leads to a resin exhibiting a 92% crosslinking rate and a T g of 121°C. The resin follows the same trend as the composite. When the post-curing temperature increases, an increase of T g is also observed with a maximal value of 165°C for the resin post-cured 2 hours at 150°C. This trend was expected since the aim of post-curing is to obtain a 100% crosslinking rate. After post-curing the molecular mobility within the resin is reduced, leading to a higher T g . It has been reported that the composite T g (Table 2) may be slightly lower than the resin T g (Table 3) due to the disruption of the resin network by the incorporation of fibers that could either absorb the amine hardener in the interface region or generate non cellulosic materials that migrate into the resin [START_REF] Fernández | Role of flax cell wall components on the microstructure and transverse mechanical behaviour of flax fabrics reinforced epoxy biocomposites[END_REF]. But an opposite phenomenon is observed when a post-cure is carried out at 100, 120 or 150°C because the composite T g is higher than the resin T g . This could be explained by a change in specific heat or thermal conductivity when fibers are introduced [START_REF] Crowson | The elastic properties in bulk and shear of a glass bead-reinforced epoxy resin composite[END_REF][START_REF] Nicolais | The Glass Transition Temperature of Poly(phenylene oxide): Annealing and Filler Effects[END_REF].

Tensile tests

A decrease of the Young modulus (Table 4) is observed when a post-cure is realized on the samples with a maximal drop of 15% for R-PC150.This decrease could be explained by a change of resin density that varied from 1.276g/cm3 for NoPC to 1.262g/cm3 for PC150.

However, no significant modification of the modulus was observed when the post-curing temperature increases (or the phenomenon is concealed by the high standard deviation of the results). Table 4 also shows the same trend (high standard deviation around an average strain of 3%) regarding the resin elongation at break after post-curing. The maximal tensile strength at break does not significantly change for R-PC100 and R-PC120 but decreases by 20% when the post-curing temperature reaches 150°C while the opposite trend was expected [START_REF] Aruniit | Preliminary Study of the Influence of Post Curing Parameters to the Particle Reinforced Composite's Mechanical and Physical Properties[END_REF], [START_REF] Moussa | Long-term physical and mechanical properties of cold curing structural epoxy adhesives[END_REF]. For R-PC150, another phenomenon is altering the mechanical properties. Carbas and al. [START_REF] Carbas | Effect of Cure Temperature on the Glass Transition Temperature and Mechanical Properties of Epoxy Adhesives[END_REF] showed that when the curing temperature is above or near the resin T g , the mechanical properties (Modulus and failure tensile strength) of the epoxy resin at room temperature decrease because of a thermal degradation or an oxidative crosslinking (crosslinking within the epoxy polymer).

A slight change of color (from transparent to light yellow) is also evidenced and assigned to an early oxidation of the epoxy resin because the post-curing temperature is approaching the R-PC150 glass transition temperature (T g,R-PC150 = 165°C, Table 3) [START_REF] Tandon | Modeling of oxidative development in PMR-15 resin[END_REF][START_REF] Colin | A new method for predicting the thermal oxidation of thermoset matrices: Application to an amine crosslinked epoxy[END_REF].

Flax fabrics

The second composite component to be studied is the flax fabric used to reinforce the epoxy resin. Flax fibers contain generally between 7 and 10% of moisture in standard conditions (23°C and 50%RH) and this moisture is removed during post-curing at high temperature [START_REF] Baley | Influence of the absorbed water on the tensile strength of flax fibers[END_REF][START_REF] Gassan | Thermal degradation of flax and jute fibers[END_REF][START_REF] Bourmaud | Importance of fiber preparation to optimize the surface and mechanical properties of unitary flax fiber[END_REF][START_REF] Jalaludin | Moisture adsorption isotherms of wood studied using dynamic vapour sorption apparatus[END_REF]. TGA tests (Figure 5) show that the present flax fibers contain only 5% of moisture and that this moisture evaporates when the temperature exceeds 86°C. This was the reason for choosing 80°C as curing temperature. During post-curing at 100, 120 or 150°C, moisture is removed from the fibers and may be extracted from the composites. The higher the temperature is, the more damaging this phenomenon becomes. For instance, when temperature is over 120°C some components of cell walls like waxes are degraded [START_REF] Müssig | The use of flax fibres as reinforcements in composites[END_REF].

Tensile tests were carried out onto fabrics by changing the sample length in order to modify the breaking mechanisms. The evolution of the maximal load until breaking for the different samples is shown in Figures 6 and7. First observations concern the maximal load that decreases when the sample length increases due to a change of the main mode of breaking when longer samples are tested [START_REF] Charlet | Mechanical properties of interfaces within a flax bundle -Part I: Experimental analysis[END_REF][START_REF] Zhu | Recent development of flax fibres and their reinforced composites based on different polymeric matrices[END_REF][START_REF] Charlet | Mechanical characterization and modeling of interfacial lamella within a flax bundle[END_REF]. Bledzki and Gassan also showed that the flax fiber tensile strength is dependent on the length of the fiber itself. The longer it is, the more inhomogeneous it will be, weakening its structure [START_REF] Bledzki | Composites reinforced with cellulose based fibres[END_REF]. Indeed, for short fibers, a neat breaking is observed at the center of the specimen but when the length of the specimen increases, disentanglement is also noted. Hence the behavior of samples depends on a combination of neat breaking and disentanglement. The latter tends to become predominant for longer specimen. Figure 8 shows the evolution of the maximal load normalized in comparison to raw specimens to see the effect of temperature on the fabrics. It shows that the disentanglement is all the more so retarded by the post-curing temperature when sample length increases because the higher the temperature is, the higher the maximal load will be (compared to other similar length samples). Figure 6 and Figure 8 also highlight that for shorter samples (10mm), the higher the temperature of post-curing is, the lower the maximum supported load is, while for longer samples the opposite trend is observed. Indeed, the post-curing process will remove the moisture contained in the fibers. The loss of water will create a modification of the cohesion between the cellulose microfibrils leading to a decrease of the maximal load supported by the fibers [START_REF] Baley | Influence of the absorbed water on the tensile strength of flax fibers[END_REF]. Thus, the maximal load supported will decrease.

Apparent stiffness of fabrics is shown in Figure 9. The apparent stiffness of raw flax fabrics tends to decrease when the specimen length increases. This is due to the occurrence of disentanglement that reduces the global stiffness. The 24 hours curing process at 80°C generates a 10% rise of the fabric apparent stiffness (compare NoPC and raw fabrics) for 10mm samples. Then, the higher the post-curing temperature is, the higher the increase of apparent stiffness will be with a 19% increase for F-PC150. This was expected since heating a single fiber results in an increase of its stiffness because of a reorganization of the fibrils [START_REF] Baley | Influence of drying on the mechanical behaviour of flax fibres and their unidirectional composites[END_REF].

But, for sample length equal to or higher than 50mm, the impact of post-curing on apparent stiffness is limited (Fig. 6, Fig. 9). It seems that disentanglement conceals the impact of the post-curing treatments on fibers.

Characterization of the interface (SEM+ILSS)

The interfacial zone, where interactions between flax fibers and epoxy resin allow the stress transfer from resin to reinforcement, may also be considered as a specific component of the composite that could be affected by the thermal treatments. Indeed, a modification of the interface between fibers and resin may explain a decrease of the tensile strength of the composite. In order to assess the quality of the interface, ILSS tests and SEM fractographies were carried out. We choose to use ILSS tests in order to determine if there was a modification of the interface quality after the various post-curing treatments. Indeed, ILSS allows to determine on a relative basis the tendencies of the bond strength in a given system where only the bonding level is changing. Thus, in our case, ILSS may help determining if the interfacial bonding properties could be responsible of the modification of the composite mechanical properties [START_REF] Narkis | Review of methods for characterization of interfacial fiber-matrix interactions[END_REF]. The ILSS of a composite reinforced with fibers depends on the amount of fibers (synthetic or natural), the processing techniques or even the temperature of measurement. But ultimately, it is mainly dominated by the resin properties and the fibermatrix interfacial strength [START_REF] Kumar | Effect of post-curing on thermal and mechanical behavior of GFRP composites[END_REF][START_REF] Ahmed | Tensile, flexural and interlaminar shear properties of woven jute and jute-glass fabric reinforced polyester composites[END_REF][START_REF] Wu | Effect of matrix modification on interlaminar shear strength of glass fibre reinforced epoxy composites at cryogenic temperature[END_REF][START_REF] Chandrasekaran | Role of processing on interlaminar shear strength enhancement of epoxy/glass fiber/multi-walled carbon nanotube hybrid composites[END_REF]. Synthetic fibers have a rather common geometrical factor whereas natural fibers have a complex geometrical factor (composed of sub-elements)

which will improve the fiber-matrix interfacial strength and therefore, have a positive effect on the ILSS values. There is no sign that the interface was modified by the post-curing treatment either on the fracture images or on the polished section. In order to confirm the SEM observations by quantitative results, ILSS tests were conducted.

Those tests allow to quantify the interfacial interactions and to observe a possible change when the post-curing temperature increases. In this test, the sample undergoes important shear stresses that may cause the failure of the sample. First, it can be observed in Table 5, that the maximal tensile stress in ILSS samples is lower than the tensile strength determined during the tensile tests in Figure 4 with the exception of PC150. Those results allow concluding that the specimens broke under shear stress for NoPC, PC100 and PC120 while no conclusion can be drawn from PC150 results. According to Table 5, it would seem that the post-curing does not affect the composite ILSS whereas the tensile strength decreases significantly.

However, the ILSS for NoPC are 49% higher than those obtain by H. Bos in her study for an epoxy reinforced with flax fabrics composite (15.4 ± 0.4MPa) [START_REF] Bos | The potential of flax fibres as reinforcement for composite materials[END_REF] but similar to Meredith's composite (high strength epoxy resin reinforced with flax fabric) that has an ILSS of 23.3MPa [START_REF] Meredith | On the static and dynamic properties of flax and Cordenka epoxy composites[END_REF]. Epoxy composites reinforced with other natural fibers such as jute fibers exhibit an ILSS 53% superior with an ILSS of 43 ±9 MPa [START_REF] Doan | Jute fibre/epoxy composites: Surface properties and interfacial adhesion[END_REF]. The same flax fabrics (UD360) impregnated with an unsaturated polyester resin lead to a composite with a 40% lower ILSS of 13.73 ± 0.85MPa [START_REF] Testoni | In situ long-term durability analysis of biocomposites in the marine environment[END_REF].

CONCLUSIONS

The objective of this study was to identify the component (fibers, resin or interface) responsible for the progressive change of the composite mechanical properties when a postcuring is carried out at 100, 120 or 150°C. After reproducing the post-curing conditions on each component, static mechanical tests were carried out to assess if an alteration occurred within a 2-hour thermal treatment.

The composite glass transition temperature and the crosslinking rate increase when a postcure is carried out. The higher the temperature of post-curing, the higher the Tg/crosslinking rate will be (115°C/92% for NoPC and 167°C/100% for PC150). But post-curing leads to a modification of composite mechanical properties, especially at high temperature (150°C).

Both tensile strength and elongation at break decrease when the post-curing temperature increases whereas the stabilized modulus does not significantly vary.

Since the interfacial adhesion in NoPC is weak, these alterations cannot be assigned to an interfacial damaging. No significant change of the resin behavior was observed for R-PC100 and R-PC120 but at 150°C, only 15°C below the glass transition temperature, an early oxidation is the first sign of resin damage.

On the contrary, a strong modification of the properties of the fibers when they undergo a thermal treatment similar to post curing was found. Indeed, an increase of the stiffness was highlighted as well as a decrease of the maximal load supported by the fabric when heating at high temperature. However, the impact of heating may be masked by fibers disentanglement when specimen length is high enough. Therefore, tests on short specimens are advised.

It appears that combining a well crosslinked resin with natural fibers to obtain a composite with the highest mechanical properties, is hardly possible. A good compromise for a flax fiber reinforced epoxy composite, would be to perform a 24-hour curing at 80°C followed by a 2 hours post-curing at 120°C. The resulting composite would have a Tg of 145°C and a limited drop of about 14% of the tensile strength compared with an equivalent non-post-cured composite. It would seem that today's epoxy resins are not completely adapted for processing natural fiber reinforced epoxy composites with high (and optimal) mechanical properties. A solution to improve the competitiveness of natural fibers in structural composites would be to associate them with resin that could be cured (rapidly) at low temperature.

Figure 1 :Figure 2 :

 12 Figure 1 : DSC Curve for the system DGEBA -IPDA (cured 100%)

Figure 3 -

 3 Figure 3 -Strain-Tensile strength curves a) and Tangent Modulus b) for NoPC and PC150

Figure 5 -

 5 Figure 5 -TGA of the Flax Fibers

Figure 6 -Figure 7 -Figure 8 -

 678 Figure 6 -Load Vs. Displacement of Raw, NoPC and PC150 series for different sample lengths a)

Figure 9 -

 9 Figure 9 -Stiffness of the different materials Versus Sample length (mm)

Figure 10 -

 10 Figure 10 -a) No PC x1000, b) NoPC x5000, c) NoPC -polished section x800, d) PC150 x1000, e) PC150 x5000, f) PC150 -polished section x800

Table 1 -Total reaction enthalpy for the system DGEBA + IPDA and post relaxation enthalpies for NoPC, PC100, PC120 and PC150

 1

	100% cured (DSC)	NoPC	PC100	PC120	PC150

Table 2 -Glass transition temperature of the various composites

 2

	Sample	NoPC	PC100	PC120	PC150
	1st temperature ramp (20°C -200°C)	115°C	127°C	145°C	167°C
	2nd temperature ramp	158°C	161°C	162°C	167°C
	(20°C -200°C)

Table 3 -Glass transition temperatures and crosslinking rate of the resin

 3

	Sample	R-NoPC	R-PC100	R-PC120	R-PC150
	Tg (°C)	121°C	123°C	139°C	165°C
	(1st heating ramp)
	Tg (°C)	165°C	162°C	163°C	165°C
	(2nd heating ramp)
	Cross-linking rate (%)	92	94	97	100

Table 4 -Young Modulus (GPa), Maximal stress (MPa) and strain (%) of the epoxy resin post-cured at 100, 120 and 150°C

 4

	Sample	R-NoPC	R-PC100	R-PC120	R-PC150
	Modulus (GPa)	3.4 ± 0.52	2.7 ± 0.27	2.9 ± 0.06	2.9 ± 0.37
	according to ISO 527
	Maximal tensile	68 ± 8.1	69 ± 5.2	66 ± 3.2	54 ± 3.8
	strength (MPa)
	Elongation at break (%)	2.67 ± 0.54	3.56 ± 0.56	3.33 ± 0.04	3.12 ± 0.43

Table 5 -Maximal tensile strength and maximal shear stress for the various composites

 5

	Sample	σ max, tensile stress (MPa)	Tensile strength (MPa)	ILSS (MPa)
	No PC	205 ± 15	252	23.02 ± 1.62
	PC 100	200 ± 10	232	22.48 ± 1.10
	PC 120	215 ± 12	218	22.03 ± 1.30
	PC 150	195 ± 11	136	>21.8 ± 1.17


A Aïssa-El-Bey

K Abed-Meraim

Y Grenier
email: grenier@tsi.enst.fr

On the identifiability testing in blind source separation using resampling technique

This paper focuses on the second order identifiability problem of blind source separation and its testing. We present first necessary and sufficient conditions for the identifiability and partial identifiability using a finite set of correlation matrices. These conditions depend on the autocorrelation fonction of the unknown sources. However, it is shown here that they can be tested directly from the observation through the decorrelator output. This issue is of prime importance to decide whether the sources have been well separated or else if further treatments are needed. We then propose an identifiability testing based on resampling (jackknife) technique, that is validated by simulation results.

Introduction

Blind source separation (BSS) of instantaneous mixtures has attracted so far a lot of attention due to its many potential applications [START_REF]Blind estimation using higher-order statistics[END_REF] and its mathematical tractability that lead to several nice and simple BSS solutions [START_REF]Blind estimation using higher-order statistics[END_REF][START_REF] Pham | Blind source separation of instantaneous mixtures of nonstationary sources[END_REF][START_REF] Belouchrani | Blind source separation using second order statistics[END_REF][START_REF] Cardoso | A.Blind beamforming for non-Gaussian signals[END_REF]. The underlaying model is given by: x(t) = y(t) + w(t) = As(t) + w(t)

where s(t) = [s 1 (t), • • • , s m (t)] T is the m × 1 complex source vector, w(t) = [w 1 (t), • • • , w n (t)] T is the n×1 complex noise vector, A is the n×m full column rank mixing matrix (i.e., n ≥ m), and the superscript T denotes the transpose operator. The source signal vector s(t), is assumed to be a multivariate stationary complex stochastic process. In this paper we consider only the second order BSS methods and hence the component processes s i (t), 1 ≤ i ≤ m are assumed to be temporally coherent and mutually uncorrelated, with zero mean and second order moments:

S(τ ) def = E (s(t + τ )s (t)) = diag[ρ 1 (τ ), • • • , ρ m (τ )]
where ρ i (τ ) def = E(s i (t + τ )s * i (t)), the expectation operator is E, and the superscripts * and denote the conjugate of a complex number and the complex conjugate transpose of a vector, respectively. The additive noise w(t) is modeled as a white stationary zero-mean complex random process. In that case, the source separation is achieved by decorrelating the signals at different time lags. This is made possible under certain identifiability conditions that have been developed in [3] and recalled briefly in this paper. Although the previous conditions are expressed in terms of the autocorrelation coefficient of the unknown source signals, we propose here a solution to test them directly out of the received data using the jackknife (resampling) technique.

Second Order Identifiability

In [START_REF] Tong | Indeterminacy and identifiability of blind identification[END_REF], Tong et al. have shown that the sources are blindly separable based on (the whole set) of second order statistics only if they have different spectral density functions. In practice we achieve the BSS using only a finite set of correlation matrices. Therefore, the preview identifiability result was generalized to that case in [START_REF] Belouchrani | Blind source separation using second order statistics[END_REF]3] leading to the necessary and sufficient identifiability conditions given by the following theorem:

Theorem 1. Let τ 1 < τ 2 < • • • < τ K be K ≥ 1 time lags, and define ρ i = [ρ i (τ 1 ), ρ i (τ 2 ), • • • , ρ i (τ K )] and ρi = [ (ρ i ), (ρ i )]
where (x) and (x) denote the real part and imaginary part of x, respectively. Taking advantage of the indetermination, we assume without loss of generality that the sources are scaled such that ρ i = ρi = 1, for all i1 . Then, BSS can be achieved using the output correlation matrices at time lags

τ 1 , τ 2 , • • • , τ K if and only if for all 1 ≤ i = j ≤ m:
ρi and ρj are (pairwise) linearly independent

Interestingly, we can see from condition (1) that BSS can be achieved from only

one correlation matrix R x (k) def = E(x(t + k)x (t)
) provided that the vectors [ (ρ i (k)), (ρ i (k)] and [ (ρ j (k)), (ρ j (k)] are pairwise linearly independent for all i = j. Note also that, from (1), BSS can be achieved if at most one temporally white source signal exists. In contrast, recall that when using higher order statistics, BSS can only be achieved if at most one Gaussian source signal exists. Under the condition of Theorem 1, the BSS can be achieved by decorrelation according to the following result:

Theorem 2. Let τ 1 , τ 2 , • • • , τ K be K time lags and z(t) = [z 1 (t), • • • , z m (t)] T be an m × 1 vector given by z(t) = Bx(t). Define r ij (k) def = E(z i (t + k)z * j (t)).
If the identifiability condition holds, then B is a separating matrix (i.e. By(t) = PΛs(t) for a given permutation matrix P and a non-singular diagonal matrix Λ) if and only if

r ij (k) = 0 and τ K k=τ1 |r ii (k)| > 0 (2)
for all

1 ≤ i = j ≤ m and k = τ 1 , τ 2 , • • • , τ K .
Note that, if one of the time lags is zero, the result of Theorem 2 holds only under the noiseless assumption. In that case, we can replace the condition

τ K k=τ1 |r ii (k)| > 0 by r ii (0) > 0, for i = 1, • • • , m.
On the other hand, if all the time lags are non-zero and if the noise is temporally white (but can be spatially colored with unknown spatial covariance matrix) then the above result holds without the noiseless assumption. Based on Theorem 2, we can define different objective (contrast) functions for signal decorrelation. In [START_REF] Kawamoto | Blind separation of sources using temporal correlation of the observed signal[END_REF], the following criterion2 was used

G(z) = τ K k=τ1 log |diag(R z (k))| -log |R z (k)| (3)
where diag(A) is the diagonal matrix obtained by zeroing the off diagonal entries of A. Another criterion used in [START_REF] Abed-Meraim | A general framework for blind source separation using second order statistics[END_REF] is

G(z) = τ K k=τ1 1≤i<j≤m |r ij (k)| 2 + m i=1 | τ K k=τ1 |r ii (k)| -1| 2 (4)
Equations ( 3) and ( 4) are non-negative functions which are zero if and only if

R z (k) = E(z(n + k)z (n)) are diagonal for k = τ 1 , • • • , τ K or equivalently if (2) is met.

Partial Identifiability

It is generally believed that when the identifiability conditions are not met, the BSS cannot be achieved. This is only half of the truth as it is possible to partially separate the sources in the sense that we can extract those which satisfy the identifiability conditions. More precisely, the sources can be separated in blocks each of them containing a mixture of sources that are not separable using the considered set of statistics. For example, consider a mixture of 3 sources such that ρ1 = ρ2 while ρ1 and ρ3 are linearly independent. In that case, source s 3 can be extracted while sources s 1 and s 2 cannot. In other words, by decorrelating the observed signal at the considered time lags, one obtain 3 signals one of them being s 3 (up to a scalar constant) and the two others are linear mixtures of s 1 and s 2 . This result can be mathematically formulated as follows: assume there are d distinct groups of sources each of them containing d i source signals with same (up to a sign) correlation vector ρi

, i = 1, • • • , d (clearly, m = d 1 + • • • + d d ).
The correlation vectors ρ1 , • • • , ρd are pairwise linearly independent. We write

s(t) = [s T 1 (t), • • • , s T d (t)]
T where each sub-vector s i (t) contains the d i source signals with correlation vector ρi .

Theorem 3. Let z(t) = Bx(t) be an m × 1 random vector satisfying equation (2) for all 1 ≤ i = j ≤ m and k = τ 1 , • • • , τ K . Then, there exists a permutation matrix P such that z(t)

def = Pz(t) = [z T 1 (t), • • • , z T d (t)] T where z i (t) = U i s i (t), U i being a d i × d i non-singular matrix.
Moreover, sources belonging to the same group, i.e., having same (up to a sign) correlation vector ρi can not be separated using only the correlation matrices R

x (k), k = τ 1 , • • • , τ K .
This result (see [3])shows that when some of the sources have same (up to a sign) correlation vectors then the best that can be done is to separate them per blocks and this can be achieved by decorrelation. However, this result would be useless if we cannot check the linear dependency of the correlation vectors ρi and partition the signals per groups (as shown above) according to their correlation vectors. This leads us to the important problem of testing the identifiability condition that is discussed next.

Testing of identifiability condition

Theoretical result

The necessary and sufficient identifiability condition (1) depends on the correlation coefficients of the source signals. The latter being unknown, it is therefore impossible to a priori check whether the sources are 'separable' or not from a given set of output correlation matrices. However, it is possible to check a posteriori whether the sources have been 'separated' or not. We have the following result [3]:

Theorem 4. Let τ 1 < τ 2 < • • • < τ K be K distinct time lags and z(t) = Bx(t).
Assume that B is a matrix such that z(t) satisfies3 equation (2) for all 1 ≤ i = j ≤ m and k = τ 1 , • • • , τ K . Then there exists a permutation matrix P such that for

k = τ 1 , • • • , τ K . E(z(t + k)z (t)) = P T S(k)P
In other words the entries of z(t) def = Pz(t) have the same correlation coefficients as those of s(t) at time lags

τ 1 , • • • , τ K , i.e. E(z i (t + k)z * i (t)) = ρ i (k) for k = τ 1 , • • • , τ K and i = 1, • • • , m.
From Theorem 4, the existence of condition (1) can be checked by using the approximate correlation coefficients r ii (k 2) holds, it does not mean that the source signals have been separated. Three situations may happen:

) def = E(z i (t + k)z i (t)). It is important to point out that even if equation (
1. For all pairs (i, j), ρi and ρj (computed from z(t)) are pairwise linearly independent. Then we are sure that the sources have been separated and that z(t) = s(t) up to the inherent indeterminacies of the BSS problem.

In fact, the testing of the identifiability condition is equivalent to pairwise testing the angles between ρi and ρj for all 1 ≤ i = j ≤ m. The larger the angle between ρi and ρj , the better the quality of source separation (see performance analysis in [START_REF] Belouchrani | Blind source separation using second order statistics[END_REF]). 2. For all pairs (i, j), ρi and ρj are linearly dependent. Thus the sources haven't been separated and z(t) is still a linear combination of s(t). 3. A few pairs (i, j) out of all pairs satisfy ρi and ρj linearly dependent. Therefore the sources have been separated in blocks.

Now, having only one signal realization at hand, we propose to use a resampling technique to evaluate the statistics needed for the testing.

Testing using resampling techniques

Note that in practice the source correlation coefficients are calculated from noisy finite sample data. Due to the joint effects of noise and finite sample size, it is impossible to obtain the exact source correlation coefficients to test the identifiability condition. The identifiability condition should be tested using a certain threshold α, i.e., decide that ρi and ρj are linearly independent if || ρi ρT j |-1| > α. To find α we use the fact that the estimation error of ρi ρT j is asymptotically gaussian 4 and hence one can build the confidence interval of such a variable according to its variance. This algorithm can be summarized as follows:

1. Estimate a demixing matrix B and z(t) def = Bx(t) using an existing second order decorrelation algorithm (e.g. SOBI [START_REF] Belouchrani | Blind source separation using second order statistics[END_REF]). 2. For each component z i (t), estimate the corresponding normalized vector ρ i . 3. Calculate the scalar product R(i, j) = | ρi ρT j | for each pair (i, j). 4. Estimate σ(i,j) the standard deviation of R(i, j) using resampling technique (see Section 5). 5. Choose α (i,j) according to the confidence interval. e.g. to have a confidence interval equal to 99.7% we choose α (i,j) = 3σ (i,j) , and compare | R(i, j) -1| to α (i,j) to test whether sources i and j have been separated or not.

Resampling techniques: The jackknife

In many signal processing applications one is interested in forming estimates of a certain number of unknown parameters of a random process, using a set of sample values. Further, one is interested in finding the sampling distribution of the estimators, so that the respective means, variances, and cumulants can be calculated, or in making some kind of probability statements with respect to the unknown true values of the parameters.

The bootstrap [START_REF] Zoubir | The bootstrap and its application in signal processing[END_REF] was introduced by Efron [START_REF] Efron | The jackknife, the bootstrap and other resampling plans[END_REF] as an approach to calculate confidence intervals for parameters in circumstances where standard methods cannot be applied. The bootstrap has subsequently been used to solve many other problems that would be too complicated for traditional statistical analysis. In simple words, the bootstrap does with a computer what the experimenter would do in practice, i.e. if it were possible: he or she would repeat the experiment. With the bootstrap, the observations are randomly reassigned, and the estimates recomputed. These assignments and recomputations are done hundreds or thousands of times and treated as repeated experiments.

The jackknife [START_REF] Miller | The jackknife -A review[END_REF] is another resampling technique for estimating the standard deviation. As an alternative to the bootstrap, the jackknife method can be thought of as drawing n samples of size n -1 each without replacement from the original sample of size n [START_REF] Miller | The jackknife -A review[END_REF]. Suppose we are given the sample X = {X 1 , X 2 , . . . , X n } and an estimate, θ, from X . The jackknife method is based on the sample delete-one observation at a time,

X (i) = {X 1 , X 2 , . . . , X i-1 , X i+1 , . . . , X n }
for i = 1, 2, . . . , n, called the jackknife sample. This i th jackknife sample consists of the data set with the i th observation removed. For each i th jackknife sample, we calculate the i th jackknife estimate, θ(i) of ϑ, i = 1, 2, . . . , n. The jackknife estimate of the standard deviation of θ is defined by

σ = n -1 n n i=1   θ(i) - 1 n n j=1 θ(j)   2
The jackknife is computationally less expensive if n is less than the number of replicates used by the bootstrap for standard deviation estimation because it requires computation of θ only for the n jackknife data sets. For example, if L = 25 resamples are necessary for standard deviation estimation with the bootstrap, and the sample size is n = 10, then clearly the jackknife would be computationally less expensive than the bootstrap. In order to test the separability of the estimated signals, we have used a jackknife method to estimate the variance of the scalar product quantities R(i, j) for i, j = 1, 2, . . . , m. This is done according to the following steps:

1. From each signal z i = [z i (0), . . . , z i (T -1)] T , generate T vectors such as z

(j) i = [z i (0), . . . , z i (j -1), z i (j + 1), . . . , z i (T - 1
)] T and j = 0, 1, . . . , T -1. 2. For each vector z (j) i , estimate the corresponding vector ρ

(j) i . 3. Estimate R such as its (i, j) th entry is R(i, j) = 1 T T -1 k=0 ρ (k) i , ρ (k) j ρ (k) i ρ (k) j
where •, • denotes the scalar product and • is the euclidian norm.

4. Estimate the standard deviation of R(i, j) by

σ(i,j) = T -1 T T -1 k=0 ρ (k) i , ρ (k) j ρ (k) i ρ (k) j - 1 T T -1 l=0 ρ (l) i , ρ (l) j ρ (l) i ρ (l) j 2 6

Discussion

Some useful comments are provided here to get more insight onto the considered testing method and its potential applications and extensions.

-The asymptotic performance analysis of SOBI derived in [START_REF] Belouchrani | Blind source separation using second order statistics[END_REF], shows that the separation performance of two sources s i and s j depends on the angle between their respective correlation vectors ρ i and ρ j . Hence, measuring this angle gives a hint on the interference rejection level of the two considered sources.

As a consequence, one can use the measure of this angle not only to test the separability of the two sources but also to guarantee a target (minimum) separation quality. Choosing the threshold α (i,j) accordingly is an important issue currently under investigation. -The testing method can be incorporated into a two stage separation procedure where the first stage consists in a second order decorrelation method (e.g. SOBI). The second stage would be an HOS-based separation method applied only when the testing indicates a failure of separation at the first step. -In many practical situations, one might be interested by only one or few source signals. This is the case for example in the interference mitigation problem in [START_REF] Belouchrani | Interference mitigation in spread spectrum communications using blind source separation[END_REF] or in the power plants monitoring applications [START_REF] D'urso | Blind identification methods applied to Electricite de France's civil works and power plants monitoring[END_REF]. In this situation, the partial identifiability result is of high interest as it proves that the desired source signal can still be extracted even if a complete source separation cannot be achieved. -We believe that similar testing procedure can be used for HOS-based BSS methods, at least for those like JADE [START_REF] Cardoso | A.Blind beamforming for non-Gaussian signals[END_REF], that are based on 4 th order decorrelation. This would be the focus of future research work.

Simulation results

We present in this section some simulation results to illustrate the performance of our testing method. In the simulated environment we consider uniform linear array with n = 2 sensors receiving the signals from m = 2 unit-power first order autoregressive sources (with coefficients a 1 = 0.95e j0.5 and a 2 = 0.5e j0.7 ) in the presence of stationary complex temporally white noise. The considered sources are separable according to the identifiability result, i.e. their respective correlation vectors ρ 1 and ρ 2 are linearly independent. The time lags (delays) implicitly involved are τ 0 , • • • , τ 9 (i.e., K = 10). The signal to noise ratio (SNR) is defined as SNR = -10 log 10 σ 2 n , where σ 2 n is the noise variance. We use SOBI algorithm [START_REF] Belouchrani | Blind source separation using second order statistics[END_REF] to obtain the decorrelated sources. The statistics in the curves are evaluated over 2000 Monte-Carlo runs. We present first in figure 1(a) a simulation example where we compare the rate of success of the testing procedure (success means that we decide the 2 sources have been separated) to detect the sources separability for different sample sizes versus the SNR in dB. The confidence interval is fixed to β = 99.7%. One can observe from this figure that the performance of the testing procedure degrades significantly for a small sample size due to the increased estimation errors and the fact that we use the asymptotic normality of considered statistics. In figure 1(b), we present a simulation example where we compare the rate of success according to the sample size for different confidence intervals. The SNR is set to 25dB. Clearly, the lower the confidence interval is, the higher is the rate of success of the testing procedure. Also, as observed in figure 1, the rate of success increases rapidly when increasing the sample size. In figure 2(a), we present a simulation example where we plot the rate of success versus the confidence interval β for different sample sizes and for SNR=25dB. This plot shows somehow the evolution of the rate of success w.r.t. the 'false alarm rate' and confirms the results of the two previous figures. The simulation example presented in figure 2(b) assumes two source signals with parameters a 1 = 0.5e j0.5 and a 2 = 0.5e j(0.5+δθ) , where δθ represents the spectral overlap of the two sources. The number of sensors is n = 5, the sample size is T = 1000 and the SNR=30dB. Figure 2(b) shows the rate of success versus the spectral shift δθ. As we can see, small values of δθ lead to high rates of 'nonseparability' decision by our testing procedure. Indeed, when δθ is close to zero the two vectors ρ 1 and ρ 2 are close to 'linear dependency'. That means that the separation quality of the two sources is poor in that case which explains the observed testing results. In the last figure, we assume there exist three sources. The first two sources are complex white gaussian processes (hence ρ 1 = ρ 2 ) and the third one is an autoregressive signal with coefficient a 3 = 0.95e j0. 5 . The plots in figure 2(c) compares the average values of scalar products for ρ i and ρ j (i, j = 1, 2, 3) with their corresponding threshold values 1 -α (i,j) versus the SNR. The sample size is fixed to T = 500 and the number of sensors is n = 3. This example illustrates the situation where two of the sources (here sources 1 and 2) cannot be separated (this is confirmed by the testing result) while the third one is extracted correctly (the plots show clearly that R(1, 3) < 1 -α (1,3) and R(2, 3) < 1 -α (2,3) ).

Conclusion

This paper introduces a new method for testing the second order identifiability condition of the blind source separation problem. In simple words, this testing allows us to 'blindly' check, out of the observation, whether the unknown sources have been correctly separated or not. To evaluate the statistics needed for the testing procedure we used the jackknife (resampling) technique. The simulation results illustrate and assess the effectiveness of this testing procedure at least for moderate and large sample sizes.

Fig. 1 .

 1 Fig. 1. (a)Rate of success versus SNR for 2 autoregressive sources and 2 sensors and β = 99.7%: comparison of the performance of our testing algorithm for different sample sizes T ; (b)Rate of success versus sample size T for 2 autoregressive sources and 2 sensors and SNR=25dB: comparison of the performance of our algorithm for different confidence interval β.

Fig. 2 .

 2 Fig. 2. (a)Rate of false alarm versus confidence interval β for 2 autoregressive sources and 2 sensors and SNR=25dB: comparison of the performance of our algorithm for different sample size T ; (b)Rate of success versus spectral shift δθ for 2 autoregressive sources and 5 sensors and SNR=25dB; (c)Average values of the |R(i, j)| and thresholds 1-α (i,j) versus SNR for 3 sources and 3 sensors : 2 sources are complex white gaussian processes and the third one is an autoregressive signal.

We implicitly assume here that ρ i = 0, otherwise the source signal could not be detected (and a fortiori could not be estimated) from the considered set of correlation matrices. This hypothesis will be held in the sequel.

In that paper, only the case where τ1 = 0 was considered.

Because of the inherent indetermination of the BSS problem, we assume without loss of generality that the exact and estimated sources are similarly scaled, i.e., ρi = 1.

More precisely, one can prove that the estimation error √ T δ( ρi ρT j ) is asymptotically, i.e. for large sample size T , gaussian with zero mean and finite variance.