LAUM Gwénaël Gabard
email: gwenael.gabard@univ-lemans.fr

Generalised Acoustic Impedance for Viscous Fluids

Keywords: acoustic impedance, surface traction, causality, passivity

A general class of acoustic impedance conditions is considered where a linear dependence is assumed between the fluid velocity and the surface traction. In addition to the classical impedance relating pressure and normal velocity, this generalised definition of the impedance includes effects such as the friction along the surface as well as coupling between the normal velocity and the tangential force on the fluid. After providing a generalised definition based on impedance and admittance matrices, various fundamental principles are considered to ensure the impedance is physically admissible. These principles include the reality, causality, stability and passivity of the surface. To illustrate the influence of the additional parameters introduced by the generalised impedance, a number of results are presented for the reflection of a plane wave on a lined surface, with or without a mean flow.

Introduction

In classical acoustics, i.e. for an inviscid fluid, the surface impedance defines a linear relation between the normal fluid velocity and pressure. In the frequency domain, this boundary condition is sufficient to obtain a well-posed problem. In the time domain, the impedance function Z(ω) should satisfy a number of admissibility constraints, see [START_REF] Rienstra | Impedance models in time domain, including the extended Helmholtz resonator model[END_REF] and [START_REF] Monteghetti | Analysis and discretization of time-domain impedance boundary conditions in aeroacoustics[END_REF]. When considering a viscous fluid, more boundary conditions are needed. In addition to the condition on the normal velocity, a no-slip condition is generally applied, stating that the tangential velocity vanishes on the surface. An alternative is to have a zero shear stress along the surface.

Standard impedance models, such as those from Rienstra [START_REF] Rienstra | Impedance models in time domain, including the extended Helmholtz resonator model[END_REF] or Tam & Auriault [START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF], do not provide any information on the behaviour of the tangential component of velocity at the surface. When applied to viscous flows, and in the absence of more information, these impedance models are generally complemented with the no-slip condition [START_REF] Olivetti | Direct numerical simulation of turbulent flow with an impedance condition[END_REF][START_REF] Scalo | Compressible turbulent channel flow with impedance boundary conditions[END_REF][START_REF] Sebastian | Numerical simulation of a turbulent channel flow with an acoustic liner[END_REF]. Prompted by inconsistencies observed in the impedance educed from measurements with flow [START_REF] Renou | Failure of the Ingard-Myers boundary condition for a lined duct: An experimental investigation[END_REF], it has been recently suggested that additional parameters should be introduced to fully characterize an acoustic treatment under a grazing flow. One such parameter is the momentum transfer impedance relating the tangential force on the surface and the normal velocity [START_REF] Schulz | Modeling of liner impedance with grazing shear flow using a new momentum transfer boundary condition[END_REF]. Another option is the introduction of a friction coefficient [START_REF] Aurégan | On the use of a stress-impedance model to describe sound propagation in a lined duct with grazing flow[END_REF]. Another mechanism under consideration is the coupling with the hydrodynamic field [START_REF] Dai | Acoustic of a perforated liner with grazing flow: Floquet-bloch periodical approach versus impedance continuous approach[END_REF].

In this paper we consider a general class of impedance boundary conditions where a linear dependence is posited between the surface traction and the fluid velocity. In Section 2, these boundary conditions are formulated both in the time domain and the frequency domain. They provide a unified framework to describe a large group of boundary conditions, including the momentum transfer impedance. In Section 3, a number of conditions on the generalised impedance are derived in order to obtain an admissible physical model, namely the reality, causality, stability and passivity of the impedance. The two-dimensional case is considered in more details and results for the reflection of a plane wave are presented.

Generalised Acoustic Impedance

We consider a compressible, viscous fluid in contact with a surface Γ. The surface traction (i.e. the surface force) induced on the surface by the fluid is given by t = -σ • n, where σ is the Cauchy stress tensor and n is the unit normal vector pointing into the surface. For a Newtonian fluid, we will use the constitutive relation σ = -pI + τ, where p is the pressure, I is the identity tensor and τ is the tensor of viscous stresses. It follows that t = pn -τ • n.

Time domain

To generalise the concept of acoustic impedance, we assume a linear dependence between the surface traction t and the fluid velocity u on the surface:

P[u, x, n](t) = Q[t, x, n](t) , (1)
where P and Q are two linear differential operators. x is the position along the surface Γ. We will assume that the operators P and Q only involves u and t as well as their time derivatives but do not involve their spatial derivatives.

In addition, if the surface is uniform, the operators P and Q are independent of the position x. We will use this assumption to simplify the notations, but it is straightforward to extend the present framework to cases where the surface properties vary with position along the surface.

If the operator Q is invertible it is possible to write the surface traction explicitly in terms of the velocity:

t(t) = Z[u, n](t) , ( 2
)
where Z is the impedance operator. Alternatively, if P is invertible, it is possible to write the fluid velocity in terms of the surface traction:

u(t) = Y[t, n](t) , (3)
in which Y is the admittance operator.

If we assume that the impedance and admittance operators are linear, continuous and time invariant they can be written as convolution operators [START_REF] Monteghetti | Analysis and discretization of time-domain impedance boundary conditions in aeroacoustics[END_REF]:

t(t) = +∞ -∞ Z(τ)u(t -τ) dτ , u(t) = +∞ -∞ Y(τ)t(t -τ) dτ , (4)
where Z and Y are the time-domain impedance and admittance functions. These are square matrices.

In addition to this generalised definition of the impedance boundary condition, one has to specify a boundary condition for temperature. The usual adiabatic or isothermal conditions can be used for this purpose.

Frequency domain

It is useful to move to the frequency domain by introducing the Fourier transform and its inverse:

f (ω) = +∞ -∞ f (t)e -iωt dt , f (t) = 1 2π C f (ω)e +iωt dω , (5)
where C is the integration contour for the inverse Fourier transform, which remains to be defined. Quantities expressed in the frequency domain are denoted with a hat. The convolution operators in Equation ( 4) relating surface traction and velocity in the time domain become simple multiplications in the frequency domain:

t(ω) = Ẑ(ω) û(ω) , û(ω) = Ŷ(ω) t(ω) . (6)
If both the impedance and the admittance operators are defined, then the associated transfer functions are related by Ẑ = Ŷ-1 .

Admissible impedance

A number of constraints can be imposed on the impedance and admittance matrices in order to satisfy basic physical requirements. This section discusses the reality, causality, stability and passivity of the generalised surface impedance.

Reality

The impedance and admittance functions Z(t) and Y(t) should be real valued. It follows that Ẑ(-ω * ) = Ẑ(ω) * and Ŷ(-ω * ) = Ŷ(ω) * , in which * denotes the complex conjugate. As a consequence, if an entry of Ẑ or Ŷ has a pole at ω then there should be a matching pole at -ω * and the residues for these two poles should be complex conjugates. Additionally if an entry of Ẑ or Ŷ has a zero at ω then there is a matching zero at -ω * . Note that, if square roots are involved in Ẑ and Ŷ, then the associated branch cuts should be chosen carefully.

Causality

The impedance and admittance functions should be causal, i.e. Z(t) and Y(t) should be identically zero for t < 0. This implies that the integration contour C in the inverse Fourier transform (5) should be placed below all the poles of the entries of the matrices Ẑ(ω) and Ŷ(ω). For t > 0, the contour C is closed above so that all the poles are contained within it. For t < 0, the contour is closed below and doesn't contain any pole.

Stability

Causality does not preclude the presence of instabilities since it is possible to have a causal and exponentially growing impedance and admittance functions (provided that the growth rate remains finite). This is the case if there is a pole in the lower half ω plane and the integration contour C is placed below this pole.

The impedance is stable when all the poles of Ẑ(ω) are in the upper ω plane or on the real axis. The function Ẑ(ω) is therefore analytic in the lower complex half plane. In this case the integration contour C can be placed infinitely close to the real axis from below. The impedance function then satisfies the Kramers-Kronig relations [START_REF] Lucarini | Kramers-Kronig relations in optical materials research[END_REF]Section 4.3]:

Re[Z(ω)] = 1 π - +∞ -∞ Im[Z(ω )] ω -ω dω , Im[Z(ω)] = -1 π - +∞ -∞ Re[Z(ω )] ω -ω dω , (7)
in which the integrals are to be understood as Cauchy principal values. If the admittance is stable then all the poles of Ŷ(ω) are in the upper ω plane or on the real axis. Similar relations to equation [START_REF] Renou | Failure of the Ingard-Myers boundary condition for a lined duct: An experimental investigation[END_REF] hold for the admittance function Ŷ(ω).

Passivity

The rate of work per unit area done by the surface traction is given by

W = t • u = -u • σ • n = pu • n -u • τ • n .
This corresponds to the flux of power into the surface due to the mechanical work done by the fluid on the surface. A passive surface receives more energy than it provides to the fluid. At any time T , the net amount of energy into the surface should therefore be non-negative:

T -∞ W(t) dt ≥ 0 , ∀T . (8)
The requirement of passivity implies the stability of the surface impedance. If the response function Z or Y grows exponentially then the condition above cannot be satisfied for all T and the surface can therefore not be passive.

In the frequency domain, the corresponding quantity is the normal intensity I. Following Fahy [START_REF] Fahy | Sound Intensity[END_REF] we introduce the complex-valued intensity, which, in the present case, takes the form

I = 1 2 ûT t , (9)
in which T denotes the Hermitian transpose (i.e. transpose and complex conjugation). The real part of I corresponds to the active part of the intensity, i.e. the usual time-averaged energy flux. Conversely, the imaginary part corresponds to the reactive part of the intensity vector. The reactive intensity corresponds to the unsteady energy flow that averages to a zero net flux over a period of oscillation. For more information on the interpretation of the reactive part of the intensity, the reader is referred to Fahy [START_REF] Fahy | Sound Intensity[END_REF] and Bruneau [START_REF] Bruneau | Fundamentals of acoustics[END_REF].

The frequency-dependent impedance and admittance matrices Ẑ and Ŷ are complex-valued but not necessarily symmetric nor Hermitian. However, these two matrices can be uniquely split between an Hermitian part and a skew-Hermitian part using the Toeplitz decomposition [START_REF] Horn | Matrix analysis[END_REF]:

Ẑ = R + i X , with R = 1 2 ( Ẑ + ẐT ) and X = 1 2i ( Ẑ -ẐT ) , (10)
Ŷ = Ĝ + i B , with Ĝ = 1 2 ( Ŷ + ŶT ) and B = 1 2i ( Ŷ -ŶT ) . (11)
The matrices R, X, Ĝ and B are Hermitian (hence i X and i B are skew-Hermitian). The Toeplitz decomposition can be understood as the extension to complex matrices of the separation between real and imaginary parts of complex numbers [START_REF] Horn | Matrix analysis[END_REF]. Therefore, the matrices R and X can be interpreted as generalised definitions of the resistance and the reactance of the surface, respectively. Similarly, Ĝ and B provide generalised definitions of the conductance and susceptance.

Using the relation t = Ẑ û together with the splitting of the impedance matrix in terms of resistance and reactance in equation ( 10), we get the following expression for the complex intensity:

I = 1 2 ûT R û + i 2 ûT X û . (12)
Since R and X are Hermitian, the quadratic forms ûT R û and ûT X û are real. The first term (resp. second term) in this expression is real valued (resp. purely imaginary) and therefore corresponds to the active intensity (resp. reactive intensity). This shows that the Toeplitz decomposition (10) leads directly to the separation of the active and reactive parts of the intensity. Instead if we use the relation û = Ŷt and equation [START_REF] Lucarini | Kramers-Kronig relations in optical materials research[END_REF] we get

I = 1 2 tT Ĝt - i 2 tT Bt . ( 13
)
Again we can identify the first term (real valued) as the active intensity and the second term (purely imaginary) as the reactive intensity.

To have a passive surface, the time-average flux of energy should be non-negative, i.e. Re(I) ≥ 0. This implies that we should have ûT R û ≥ 0 for any û, as well as tT Ĝt ≥ 0 for any t. The resistance matrix R and the conductance matrix Ĝ should therefore be positive semidefinite. In other words the matrices Ẑ and Ŷ should have positive semidefinite Hermitian parts. It is useful to recall the result from Mathias [START_REF] Mathias | Matrices with positive definite Hermitian part: Inequalities and linear systems[END_REF]: if a complex square matrix has a positive definite Hermitian part, then its inverse also has a positive definite Hermitian part. Therefore if R is positive definite, then Ĝ is also positive definite (the reverse is also true).

If the resistance matrix R is positive semidefinite then we have the following results, obtained using Corollary 7.1.5 from Horn & Johnson [START_REF] Horn | Matrix analysis[END_REF]:

Rii ≥ 0 , ∀i , and | Ri j | ≤ Rii R j j , ∀i, j with i j . ( 14
)
Similar properties hold for Ĝ. As a consequence we have the following results for a passive surface:

Re( Ẑii ) ≥ 0 , ∀i , and

| Ẑi j + Ẑ * ji | ≤ 2 Re( Ẑii )Re( Ẑ j j ) , ∀i, j with i j , (15)
Re( Ŷii ) ≥ 0 , ∀i , and

| Ŷi j + Ŷ * ji | ≤ 2 Re( Ŷii )Re( Ŷ j j ) , ∀i, j with i j . ( 16
)
The diagonal terms in Ẑ and Ŷ have non-negative real parts. In addition the magnitudes of Ẑi j + Ẑ * ji and Ŷi j + Ŷ * ji should be bounded. This is the main result of the present paper. It provides a generalisation of the requirement that the classical, scalar impedance has a real part.

It is worth noting that the property that R and Ĝ are positive semidefinite for a passive surface applies irrespective of the coordinate system used to define the entries of these matrices. In other words, the properties in equations ( 15) and ( 16) apply in any orthonormal coordinate system. This is shown by introducing the unitary matrix U corresponding to a rotation of the coordinate system. The fluid velocity and the surface traction represented in this new coordinate system are denoted by ǔ = U û and ť = U t. The definitions of the impedance and admittance relations become ť = Ž ǔ and ǔ = Yť , in which Ž = U ẐU T and Y = U ŶU T . It is straightforward to show that the Hermitian parts of Ž and Y are also positive semidefinite, see Observation 7.1.8 in [START_REF] Horn | Matrix analysis[END_REF].

Since the resistance matrix is Hermitian, it can be diagonalised as R = Q Λ QT where Q is a unitary matrix and Λ is a diagonal matrix of real-valued eigenvalues. As a result we have the following expression for the active intensity:

Re(I) = n λ n |ũ n | 2 , ( 17
)
where we have introduced ũ = Q û which corresponds to a change of coordinate system associated with the unitary matrix Q. For a passive surface the eigenvalues λ n are non-negative. A similar result can be obtained for the conductance matrix Ĝ.

Two-dimensional case

To illustrate the proposed formulation, we can consider the two-dimensional case by writing tn tt = Ẑnn Ẑnt Ẑtn Ẑtt ûn ût , and tn tt

= p -τnn -τ nt , (18)
where the subscripts n and t denote the normal and tangential components, respectively. While the term Ẑnn is related to the traditional acoustic impedance, the term Ẑtt represents the friction generated when the fluid elements slide along the surface. The momentum transfer impedance considered by Schultz et al [START_REF] Schulz | Modeling of liner impedance with grazing shear flow using a new momentum transfer boundary condition[END_REF] is represented by the cross term Ẑtn which describes the tangential force generated by a normal velocity. Conversely, the term Ẑnt describes the normal force generated by a tangential velocity. The friction coefficient considered by Aurégan [START_REF] Aurégan | On the use of a stress-impedance model to describe sound propagation in a lined duct with grazing flow[END_REF] is represented by Ẑtt which relates the tangential velocity and the tangential surface traction. It is expected that the cross terms Ẑnt and Ẑtn will be involved for instance in a perforated plate where the holes are drilled at an angle from the normal to the plate.

As indicated above, the resistance matrix of a passive surface should be positive semidefinite and it follows that

Re( Ẑnn ) ≥ 0 , Re( Ẑtt ) ≥ 0 , | Ẑtn + Ẑ * nt | ≤ 2 Re( Ẑnn )Re( Ẑtt ) . ( 19
)
We therefore recover the traditional condition on the real part of the classical impedance Ẑnn . A similar constraint applies for the tangential component Ẑtt . We have also an inequality satisfied by the off-diagonal terms of the impedance matrix. This indicates that if either Ẑnn or Ẑtt is purely imaginary (for instance for a purely reactive acoustic treatment) then Ẑtn = -Ẑ * nt . The standard impedance condition is recovered by neglecting viscosity and setting Ẑnt = Ẑtn = Ẑtt = 0. In this case we have p = Ẑnn ûn . Note however that it is not always possible to define the impedance and admittance matrices. In this case it is preferable to use the more general framework of equation ( 1). An example is when the normal velocity is set to zero and the slip condition (zero tangential force) is applied. This can be written 0 0 0 1

t n t t = 1 0 0 0 u n u t . (20)
In this particular case it is not possible to define the impedance matrix nor the admittance matrix.

Example

To illustrate the impact of the generalised impedance condition, we consider the reflection of an acoustic plane wave off a flat surface where the generalised impedance condition is imposed, see Figure 1.

Numerical model

The problem is two-dimensional with the surface defined by y = 0 and the waves propagating in the region y > 0. The incident acoustic plane wave has an angular frequency ω and an angle of incidence θ. There is a mean flow over the surface. The boundary layer of the mean flow is confined to the region 0 < y < δ where δ is the boundary layer thickness. Above the boundary layer (y > δ) the mean flow is uniform with velocity U 0 along the x axis. In the boundary layer we have a parallel shear flow along the x axis, with a velocity profile denoted by u 0 (y). The mean density ρ 0 and sound speed c 0 are considered uniform. The fluid is viscous and heat conductive, with shear viscosity µ, bulk viscosity µ b and heat conductivity κ.

The linear propagation of small perturbations of the mean flow is governed by the linearised Navier-Stokes equations (linearised conservation of mass, momentum and total energy). In two dimensions these are written as follows

∂ ∂t (Mq) + ∂ ∂x F x q -G xx ∂q ∂x -G xy ∂q ∂y + ∂ ∂y F y q -G yx ∂q ∂x -G yy ∂q ∂y = 0 , (21)
where the vector of field variables q = (ρ , u x , u y , T ) contains the perturbations of density, velocity and temperature.

The coefficient matrices M, F x , F y , G xx , G xy , G yx and G yy are given in Appendix A. Since we consider a parallel shear flow, these matrices depend only on the transverse coordinate y.

Above the boundary layer (y > δ), the incoming acoustic wave is written

q -(x, y, t) = q-exp{iωt -ik x x -ik - y y} ,
with the wavenumbers k x = k -cos(θ) and k - y = -k -sin(θ). The vector qand the wavenumber k -are solutions of the dispersion relation associated with equation (21). Appendix B details the numerical solution of this dispersion relation.

Fourier transforms are applied in time and the x coordinate, which is equivalent to seeking solutions of the form q(x, y, t) = q(y) exp{iωtik x x} .

Since the coefficient matrices are independent of x the governing equations become

iωM q -ik x F x q + ik x G xx q -G xy d q dy + d dy F y q + ik x G yx q -G yy d q dy = 0 . ( 22
)
On the surface y = 0 the generalised impedance condition ( 6) is applied, together with the adiabatic boundary condition dT /dy = 0 for the temperature.

On the upper boundary of the computational domain, y = H, the solution q is written as the sum of the incident acoustic wave with angle θ and a combination of reflected waves:

q(x, y, t) =         q-exp{-ik - y y} + 3 n=1 Ân q+ n exp{-ik + yn y}         exp{iωt -ik x x} , for y > δ , (23)
where Ân represent the unknown amplitudes of the reflected waves. Appendix C explains how the wavenumbers k + yn and vectors q+ n are obtained from the dispersion relation of the linearised Navier-Stokes equations. The one-dimensional governing equations, together with the boundary conditions, are solved using a spectral Galerkin method. The details of the implementation are given in Appendix D.

Results without mean flow

We first consider the case without mean flow by setting u 0 = 0. The fluid is air whose properties are ρ 0 = 1.2 kg/m 3 , T 0 = 293 K, c p = 1012 J/K/kg, γ = 1.4, µ = 1.85 10 -5 kg/m/s and κ = 0.026 W/m/K. The incident plane wave has a frequency of 1 kHz. We vary its angle of incidence θ and calculate the reflection coefficient of the acoustic wave. For reference, with these parameters, the thickness of the unsteady viscous boundary layer is δ v = 70 µm and the thickness of the unsteady thermal boundary layer is δ t = 83 µm. Comparatively, the acoustic wavelength is approximately 344 mm.

Figure 2a shows the magnitude of this coefficient as a function of θ, for Ẑnn = (5i)ρ 0 c 0 , Ẑtn = Ẑnt = 0 and for various values of Ẑtt . We therefore have a fixed standard standard impedance and vary the friction coefficient of the surface. It is clear that in this case the friction coefficient has a negligible effect on the acoustic wave (there is however an effect on the amplitude of the vorticity wave present in the vicinity of the surface y = 0). Also shown in Figure 2a is the reflection coefficient obtained with the standard impedance condition (i.e. for an inviscid model). All the results shown in this graph are identical to those obtained with the standard impedance.

Figure 2b shows the results when Ẑtn = -Ẑnt = ρ 0 c 0 /10. In this case the friction coefficient has a significant impact on the acoustic attenuation by the surface. When the friction coefficient is large, one recovers the solution in Figure 2a. It is interesting to note that when the off-diagonal terms of the impedance matrix are non-zero, there is an asymmetry in the reflection coefficient (i.e. between θ < 90 • and θ > 90 • ). It can be seen that the results with the generalised impedance differ significantly from that obtained with the standard impedance condition.

If we now fix both Ẑnn = (5i)ρ 0 c 0 and Ẑtt = 0 and vary Ẑtn = -Ẑnt we get the results in Figure 3a. Increasing Ẑtn results in a significant increase of the reflection coefficient for most incidence angles. When the friction coefficient is set to a non-zero value Ẑtt = 0.005ρ 0 c 0 , we can see in Figure 3b that the cross terms Ẑtn and Ẑnt have a more limited impact on the reflection coefficient.

Results with flow

We use a sine function to describe the velocity profile inside the boundary layer:

u 0 (y) =        Mc 0 sin π 2 y δ for 0 < y < δ Mc 0 for y > δ .
The Mach number is M = 0.5. Other profiles have been considered (polynomial profiles from linear to quintic). And the same trends as those discussed below have been observed. Results are presented in Figures 4 and5 and the trends are similar to those observed for the no flow case. When Ẑtn = Ẑnt = 0 varying Ẑtt has very little influence on the reflection coefficient (see Figure 4a). However, when the cross terms are non-zero, with Ẑtn = Ẑnt = ρ 0 c 0 /10 in this case, the friction coefficient Ẑtt has a significant impact on the reflection coefficient (see Figure 4b). The overall trend is that the reflection coefficient increases when Ẑtt increases. Again these results match closely those obtained with the standard definition of the impedance.

Similarly to the no-flow case, varying the cross terms Ẑtn and Ẑnt leads to large variations of the reflection coefficient, see Figure 5. These variations are more significant when the friction coefficient Ẑtt is small. In this case, there is a significant difference between the results of the generalised impedance and those of the standard impedance.

Summary and perspectives

We have considered a generalised definition of the acoustic impedance where a linear dependence is assumed between the fluid velocity and the force on the surface. It is possible to define admissibility conditions such as the reality, causality, stability and passivity of the surface. This last condition leads to a generalisation of the requirement that the resistance should be positive. The reflection of a plane wave by a lined surface was then used as a way to illustrate the influence of the additional parameters introduced by the generalised impedance. It was found that the cross terms Ẑnt and Ẑtn and the friction coefficient Ẑtt can have a significant influence on the reflection coefficient.

The present paper provides the theoretical framework for a generalised definition of the impedance. Experimental investigations will be necessary to attempt to measure the admittance or impedance matrix. The questions to address are (i) how one can measure these matrices and (ii) to what extent are these effects significant in practical cases.

A number of models are available to describe the classical impedance of acoustic treatments, in particular perforated liners, for instance [START_REF] Guess | Calculation of perforated plate liner parameters from specified acoustic resistance and reactance[END_REF] and [START_REF] Maa | Potential of microperforated panel absorber[END_REF]. How the additional terms Ẑtt , Ẑnt and Ẑtn could be modelled remains to be investigated. A possible approach would be to use homogenisation methods such as those developed in [START_REF] Rohan | Homogenization of the acoustic transmission through a perforated layer[END_REF] and [START_REF] Marigo | Homogenization models for thin rigid structured surfaces and films[END_REF].

Finally detailed computational modelling (using direct numerical simulations or large eddy simulations) would be useful to study the impact of the generalised impedance on the acoustic field and the grazing flow.

Appendix A. Coefficient matrices

The coefficient matrices that appear in the linearised Navier-Stokes equations (21) are defined as follows:

M =               1 0 0 0 u 0 ρ 0 0 0 0 0 ρ 0 0 e 0 ρ 0 u 0 0 ρ 0 C v               , (A.1) F x =               u 0 ρ 0 0 0 u 2 0 + RT 0 2ρ 0 u 0 0 ρ 0 R 0 0 ρ 0 u 0 0 (e 0 + RT 0 )u 0 ρ 0 e 0 + p 0 + ρ 0 u 2 0 -µu 0 p 0 + ρ 0 u 0 C p               , F y =               0 0 ρ 0 0 0 0 ρ 0 u 0 0 RT 0 0 0 ρ 0 R 0 -µu 0 p 0 + ρ 0 e 0 0               , (A.2) G xx =                0 0 0 0 0 0 µ b + 4 3 µ 0 0 0 µ 0 0 0 (µ b + 4 3 µ)u 0 κ                , G xy =                0 0 0 0 0 0 µ b -2 3 µ 0 0 µ 0 0 0 0 (µ b -2 3 µ)u 0 0                , (A.3) G yx =               0 0 0 0 0 0 µ 0 0 µ b -2 3 µ 0 0 0 0 µu 0 0               , G yy =               0 0 0 0 0 µ 0 0 0 0 µ b + 4 3 µ 0 0 µu 0 0 κ               , (A.4)
in which C p and C v are the heat capacity of the fluid, at constant pressure and volume, respectively. We have also introduced R = C p -C v and e 0 = C v T 0 + u 2 0 /2. H 0 iωw T M qik x w T F x q + ik x G xx q -G xy d q dy dw T dy F y q + ik x G yx q -G yy d q dy dy + w(H) T f (H) -w(0) T f (0) = 0 , (D.1) in which f = F y q + ik x G yx q -G yy d q dy represents the flux of mass, momentum and energy along the y direction. It is precisely these fluxes at y = 0 and H that have to be modified to enforce the boundary conditions in the variational formulation.

For the generalised impedance condition at y = 0, the surface traction corresponds to the second and third components of the flux vector t. The natural boundary condition is therefore the impedance condition written in [START_REF] Rohan | Homogenization of the acoustic transmission through a perforated layer[END_REF] which provides the surface traction in terms of the velocity vector. We can write tn tt = 0 0 1 0 0 1 0 0 f , ûn ût = -û y -û x = -0 0 1 0 0 1 0 0 q , It follows that the generalised impedance condition can be enforced by replacing f (0) in equation (D.1) by

             
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

              f (0) -               0 0 0 1 1 0 0 0              
Ẑ 0 0 1 0 0 1 0 0 q(0) .

In addition, in the calculation of f (0) in this last expression, the last entry of the matrix G yy (which is κ) is set to zero to enforce the adiabatic boundary condition on temperature.

For the radiation condition at the upper boundary y = H the flux term f (H) is written in terms of the incoming and reflected waves: f (H) = (F y + ik x G yx + ik - y G yy ) q-exp{-ik - y H} + 3 n=1

Ân (F y + ik x G yx + ik + yn G yy ) q+ n exp{-ik + yn H} .

The first term goes to the right-hand side of the system since this is the forcing term due to the incident wave. The introduction of the wave amplitudes Ân requires additional equations to close the problem. This is done by writing equation ( 23) at y = H q(y = H) = q-exp{-ik - y H} + It was verified a posteriori that these two boundary conditions are indeed satisfied by the numerical solutions.

Figure 1 :

 1 Figure 1: Diagram of a plane wave reflected by a flat surface in the presence of a grazing mean flow.

Figure 2 :

 2 Figure 2: Reflection coefficient without mean flow as a function of θ, for Ẑnn = (5i)ρ 0 c 0 and for various values of Ẑtt . (a) Ẑtn = Ẑnt = 0; (b) Ẑtn = -Ẑnt = ρ 0 c 0 /10.

Figure 3 :Figure 4 :

 34 Figure 3: Reflection coefficient without mean flow as a function of θ, for Ẑnn = (5i)ρ 0 c 0 and for various values of Ẑtn = -Ẑnt . (a) Ẑtt = 0; (b) Ẑtt /ρ 0 c 0 = 0.005.

Figure 5 :

 5 Figure 5: Reflection coefficient with mean flow as a function of θ, for Ẑnn = (5i)ρ 0 c 0 and for various values of Ẑtn = -Ẑnt . (a) Ẑtt = 0; (b) Ẑtt /ρ 0 c 0 = 0.005.

Appendix B. Dispersion relation

The dispersion relation of the linearised Navier-Stokes equations (21) is obtained by seeking plane wave solutions of the form: q(x, y, t) = q exp{iωtikx cos θiky sin θ} , in which k is the wavenumber and θ the plane wave direction. Introducing this ansatz in Equation (21) yields the dispersion relation

For a given angle θ, the dispersion relation is a quadratic eigenvalue problem for the wavenumber k (with eigenvector q). This problem can be easily solved numerically to obtain 8 eigenvalue-eigenvector pairs. Three of these have wavenumbers with positive real parts and correspond to the acoustic, vortical and entropy waves propagating in the θ direction. Three other eigenvalues correspond to the waves propagating in the opposite direction. The remaining solutions have zero phase velocity ω/k (to machine precision) and do not correspond to physical waves. The acoustic wave propagating in the θ direction is easily identified as having the phase speed with the largest real part. This provides the value of k -as well as the corresponding eigenvector qintroduced in Section 5.1.

Appendix C. Reflected plane waves

To define the plane waves that are reflected into the uniform mean flow (y > δ) it is also necessary to solve the dispersion relation of the linearised Navier-Stokes equation. Instead of fixing the direction θ, the wavenumber k x is prescribed by the incident plane wave. We therefore seek solutions of the form q(x, y, t) = q exp{iωtik x xik y y} , in which k x = k -cos(θ) is fixed. This leads to the following expression

This represents a quadratic eigenvalue problem for the wavenumber k y and the associated eigenvector q. Again one finds 8 eigenvalue-eigenvector pairs, three of which correspond to the acoustic, vortical and entropy waves propagating in the positive y direction. The remaining two solutions have zero phase velocity ω/k (to machine precision) and do not correspond to physical waves (they are artefacts of the mathematics). These are the waves that are used to define the wavenumbers k + yn and the vectors q+ n introduced in Equation ( 23).

Appendix D. Spectral Galerkin method

The linearised Navier-Stokes equations ( 22) are solved using a spectral Galerkin method. A variational formulation is obtained by projecting (22) onto a vector w of test functions and integrating over the computational domain 0 < y < H. After integration by parts we can write:


Thibault Cantat
email: thibault.cantat@cea.fr

Catalytic strategies for the

Catalytic strategies for the reduction of C-O bonds and their utilization in the conversion of CO_2 and other small molecules

Thibault Cantat


p. 9

Liquid-liquid phase separation A universal phenomenon

Schematic representation of a bulk experimental methodology (i) Coacervates are formed by mixing oppositely charged polyelectrolytes.

(ii-iii) A molecule of interest is added, and the mixture is centrifuged. Quantification in the dilute aqueous phase (iv) and in the dense coacervate phase (v)

p. 10

Chemical determinants of phase separation  Immediately after mixing the proteins self-assemble into a large number of small entities.

 Some of which form clusters of irregular shape (aggregation of small entities is faster than self organisation of the cluster ) (A)

 The clusters self-organize into larger entities with more regular shape (BE).

 As the number of entities is reduced, the collision and fusion steps are casual.

 It seems that the coacervates densify with time (F).

Heteroprotein complex coacervation : In between

Early observations under confocal microscopy p. 17

  (i) If a very strong attraction  precipitates (ii) If too weak interactions  no phaseseparation (iii) Optimal strength of the interactions  Coacervates (iv) If multiple polyelectrolyte species, possible to form multiphasic coacervates The strength of electrostatic interactions govern whether coacervates can be formed or not p. 11 Schematic phase diagram of coexisting phases for associative phase separation and schematic associative phase separation Chemical determinants of phase separation: phase diagram p. 12 Multicomponent, multiphase droplets:  Surface tension  Macromolecular density between mixed polymers,  Sensitivity to salt ions Control formation of multiphase liquid droplets ….



  HeteroproteinInternal dynamic of heteroprotein coacervation: exchange between dilute and dense phase . 1. α-LA + LYS-FITC; 2. addition of of LYS-RBITC. Evolution from 0 to 75 min Heteroprotein complex coacervation : Dynamic process between dilute and dense phase 2 µm … Similar of what happens in mono-protein LLPS system: the case of g -gliadin p. 19 Heteroprotein complex coacervation : properties of dense phase  Highly hydrated, viscoelastic network (see R. Hachfi Soussi talk)  Bicontinuous phase with some water riche phase in co-existence with material rich phase  Evidences from FRAP, 1 H NMR and simulation  Heterogeneity: coexistence of dynamic different species Change in the proportion of these structures could explain the variation of the β-LG/LF molar ratio in the coacervate phase isoforms B and A (with one more negative charge)Heteroprotein complex coacervation : Specificity over other macromolecular systems (1. surface charge)LYS and NAP with similar charge but: Homogeneous charge distribution for LYS, relatively patchy charge distribution for NAP.


Gaëlle Laperrière

Valentin Pelloin

Antoine Caubrière

Salima Mdhaffar

Nathalie Camelin

Sahar Ghannay

Bassam Jabaian

Yannick Estève

The Spoken Language Understanding MEDIA Benchmark Dataset in the Era of Deep Learning: data updates, training and evaluation tools

Keywords: Spoken Language Understanding, Benchmark dataset, Deep learning tools

With the emergence of neural end-to-end approaches for spoken language understanding (SLU), a growing number of studies have been presented during these last three years on this topic. The major part of these works addresses the spoken language understanding domain through a simple task like speech intent detection. In this context, new benchmark datasets have also been produced and shared with the community related to this task. In this paper, we focus on the French MEDIA SLU dataset, distributed since 2005 and used as a benchmark dataset for a large number of research works. This dataset has been shown as being the most challenging one among those accessible to the research community. Distributed by ELRA, this corpus is free for academic research since 2020. Unfortunately, the MEDIA dataset is not really used beyond the French research community. To facilitate its use, a complete recipe, including data preparation, training and evaluation scripts, has been built and integrated to SpeechBrain, an already popular open-source and all-in-one conversational AI toolkit based on PyTorch. This recipe is presented in this paper. In addition, based on the feedback of some researchers who have worked on this dataset for several years, some corrections have been brought to the initial manual annotation: the new version of the data will also be integrated into the ELRA catalogue, as the original one. More, a significant amount of data collected during the construction of the MEDIA corpus in the 2000s was never used until now: we present the first results reached on this subset -also included in the MEDIA SpeechBrain recipe -, that will be used for now as the MEDIA test2. Last, we discuss evaluation issues.

Introduction

Spoken Language Understanding (SLU) refers to natural language processing tasks related to semantic extraction from the speech signal. It is "a field in the intersection of speech processing, natural language processing by leveraging technologies from machine learning and artificial intelligence" [START_REF] Tur | Spoken language understanding: Systems for extracting semantic information from speech[END_REF]. For instance, in a human-machine spoken interaction system, such a task aims to convert a user input into a semantic representation according to the user's intention and the target application domain. While treating an SLU task, three questions need to be answered: 1) how is represented the semantic of the domain? 2) Which method is the most adapted to automatically extract the semantic from speech and to project it into the targeted semantic representation? 3) How to evaluate the resulting system? The semantic representation is dependent on the targeted task of the software application. In most available corpora, this semantic representation can be constructed from semantic concepts supported by words or sequences of words. For example, in ATIS (Air Travel Information System), a very popular corpus [START_REF] Hemphill | The atis spoken language systems pilot corpus[END_REF], the task is dedicated to air travel planning scenarios, and semantic labels to retrieve are dedicated to this task. The semantic representation is also represented by 17 intents (aircraft, airport, distance, flight, meal...) that correspond to semantic frames, and by semantic slots associated with each frame (depart date.relative, arrive date.relative). Other corpora exist for SLU [START_REF] Coucke | Snips voice platform: an embedded spoken language understanding system for private-by-design voice interfaces[END_REF][START_REF] Shah | Building a conversational agent overnight with dialogue self-play[END_REF][START_REF] Lugosch | Timers and such: A practical benchmark for spoken language understanding with numbers[END_REF], and each one proposes its own semantic representation, usually based on a list of specific labels like intents or concepts.

Different approaches have been proposed to process SLU tasks. During the last two decades, the main approaches were based on the use of machine learning algorithms. In the 2000s, such approaches were based on different kinds of generative and discriminative algorithms [START_REF] Raymond | Generative and Discriminative Algorithms for Spoken Language Understanding[END_REF]. Until the emergence of deep learning, conditional random fields (CRF) was the most popular tool -since it reached the best results -used to process SLU tasks redefined as word labelling tasks [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF]. The most recent approaches are now based on neural network architectures: recurrent neural architectures [START_REF] Mesnil | Investigation of recurrent-neural-network architectures and learning methods for spoken language understanding[END_REF][START_REF] Kurata | Leveraging sentence-level information with encoder lstm for semantic slot filling[END_REF][START_REF] Dupont | Labeldependencies aware recurrent neural networks[END_REF], encoder-decoder neural networks with attention mechanisms [START_REF] Simonnet | ASR error management for improving spoken language understanding[END_REF][START_REF] Li | A self-attentive model with gate mechanism for spoken language understanding[END_REF]. The latest approaches are based on language representation models pre-trained through self-supervised learning, such as BERT [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF]. Such models have been shown to achieve state-of-the-art results in different SLU tasks [START_REF] Korpusik | A comparison of deep learning methods for language understanding[END_REF][START_REF] Ghannay | Neural networks approaches focused on French spoken language understanding: application to the MEDIA evaluation task[END_REF]. All the SLU models cited above are based on a cascade approach: first an ASR (Automatic speech recognition) system is used to automatically transcribe the user utterance, and then a NLU (Natural Language Understanding) model takes the automatic transcription as input to extract semantic tags. The intermediate transcription may contain recognition errors, and the NLU module has to deal with these errors. End-to-end approaches were proposed in order to skip the use of an intermediate speech transcription, and to avoid ASR errors propagation. In addition, end-to-end approaches permit to optimize the entire model to the final task, while cascade approaches need to optimize each module on a sub-task. An end-to-end approach is based on the use of a single system directly optimized to extract semantic concepts from the speech. Such SLU end-to-end systems can be trained to generate both recognized words and semantic tags [START_REF] Ghannay | End-toend named entity and semantic concept extraction from speech[END_REF][START_REF] Desot | Towards end-to-end spoken intent recognition in smart home[END_REF][START_REF] Dinarelli | A data efficient end-to-end spoken language understanding architecture[END_REF][START_REF] Evain | Task agnostic and task spe-cific self-supervised learning from speech with lebenchmark[END_REF].

Once cascade or end-to-end systems are built, the crucial issue of the evaluation appeared. This evaluation depends on the complexity of the semantic representation. In an intent detection task, accuracy, precision and recall are sufficient to evaluate the performances. In a more complex semantic task, like in slot filling, at least two aspects have to be considered : the semantic label and its value. The Concept Error Rate (CER) and the Concept Value Error Rate (CVER) are introduced in that case.

In this paper, we focus on the French MEDIA SLU dataset, known as being the most challenging one among the ones accessible to the research community [START_REF] Béchet | Benchmarking benchmarks: introducing new automatic indicators for benchmarking spoken language understanding corpora[END_REF]. Unfortunately, the MEDIA dataset is not really used beyond the French research community. To facilitate its use, we present a complete recipe of an end-toend neural architecture, including data preparation, training and evaluation scripts, that has been built and integrated to SpeechBrain, an already popular open-source and all-inone conversational AI toolkit based on PyTorch. By integrating this recipe to SpeechBrain, we expect to make the MEDIA benchmark more accessible to researchers, and to make the source code persistent through a community maintenance. In addition, based on the feedback of some researchers who have worked on this dataset for several years, we also brought manual corrections to the initial manual annotations. Last, a significant amount of data collected during the construction of the MEDIA corpus in the 2000s was never used until now: we present the first results reached on this subset -also included in the MEDIA SpeechBrain recipe -that will be used for now as the MEDIA test2.

The Original MEDIA Benchmark

The French MEDIA benchmark [START_REF] Bonneau-Maynard | Semantic annotation of the french media dialog corpus[END_REF] was created as a part of the Technolangue project of the French government in 2002. It is dedicated to semantic extraction from speech in a context of human-machine dialogues for a hotel room reservation task with touristic information. It aims among others to set up an infrastructure for the production and dissemination of language resources, and the evaluation of written and oral language technologies. The MEDIA data is distributed by ELRA as the ME-DIA Evaluation Package 1 2 .

Data

The MEDIA corpus is composed of telephone dialogue recordings with their manual transcriptions and their se-1 http://catalog.elra.info/en-us/ repository/browse/ELRA-E0024/ 2 International Standard Language Resource Number: 699-856-029-354-6 mantic annotations. It was recorded using a Wizard-of-Oz (WoZ) method [START_REF] Green | The rapid development of user interfaces: Experience with the wizard of oz method[END_REF][START_REF] Dahlbäck | Wizard of oz studies-why and how[END_REF]: a human (the "Wizard"), pretends to be a computer, while the user is made to believe that he is interacting with an intelligent machine. This results in 1258 official recorded dialogues, from 250 different speakers. The semantic annotations are only available for users turns. The original dataset is split into three official parts (train, dev and test) as described in tables 1, 2 and 3. As shown in these tables, a consequent part of the data included in the MEDIA package has not been used during the official campaign in 2005. The reason is this data was finalized after the end of the campaign. As a consequence, even if this data is present in the archive distributed by ELRA, it is not listed in the official data files and is hidden among the subdirectories that structure the MEDIA archive file system. On our knowledge, this data was never used in research work until now. Some complex linguistic phenomena, like co-references, are also managed thanks to this mechanism. By combining attributes and specifiers, the total number of possible attribute/specifier pairs is 1121. These attributes are supported by words or sequence of words, now called word support. For each occurrence of an attribute in the semantic annotation, two other pieces of information are provided in addition to the attribute name: the mode and the normalized value of this occurrence. Four modes are possible: affirmative '+', negative '-', interrogative '?' or optional '˜' .

Data

The following sentence (translated from French) is an utterance extracted from the MEDIA dataset: "I would like to book one double room in Paris up to one hundred and thirty euros". It will be annotated as a sequence of quadruplets (word support, mode, attribute name, normalized value) as (I would like to book, +, reservation, reservation), (one, +, room-number, 1), (double room, +, room-type, double room), (up to, +, comparative-payment, less than), (one hundred and thirty, +, amount-payment, 130), (euros, +, currency-payment, euro).

The study proposed by [START_REF] Béchet | Benchmarking benchmarks: introducing new automatic indicators for benchmarking spoken language understanding corpora[END_REF], revealed why the MEDIA task can be considered as the most challenging SLU benchmark available, in comparison to other well-known benchmarks such as ATIS [START_REF] Dahl | Expanding the scope of the atis task: The atis-3 corpus[END_REF], SNIPS [START_REF] Coucke | Snips voice platform: an embedded spoken language understanding system for private-by-design voice interfaces[END_REF], and M2M [START_REF] Shah | Building a conversational agent overnight with dialogue self-play[END_REF].

Evaluation

Many metrics can be used to evaluate SLU systems. The evaluation metric adopted during the official MEDIA evaluation campaign in 2005 [START_REF] Bonneau-Maynard | Results of the French evalda-media evaluation campaign for literal understanding[END_REF] was called the understanding error rate. It consists on aligning -thanks to the Levenshtein distance -the hypothesis semantic representation to the reference one, and to compare them in terms of deletion, insertion, and substitution. This scoring considers as units the triplets (mode, attribute name, normalized value) presented above. The computation of the understanding error rate is the same as the one used to compute the word error rate (WER) for automatic speech recognition, by considering each triplet (mode, attribute name, normalized value) as a word.

Based on these semantic elements, different scoring processes have been used in the official MEDIA evaluation campaign: the full scoring takes into account the whole set of attributes (1121 possibilities), while the relax scoring does not consider the specifiers (83 possibilities). In addition, the 'mode' can also be reduced to the binary choice 'negative/affirmative' instead of retaining the four initial possibilities.

Issues

Evolution in the use of metrics As far as we know, after the original MEDIA evaluation campaign in 2005, only one study continued using the understanding error rate in relax or full scoring scenarios and 2 or 4 modes [START_REF] Lehuen | A robust semantic parser designed for spoken dialog sytems[END_REF]. In other works related to the MEDIA benchmark, a simplification of the evaluation task has been done. [START_REF] Raymond | Generative and Discriminative Algorithms for Spoken Language Understanding[END_REF] introduced the concept error rate (CER) as a scoring metric for MEDIA. The CER is similar to the understanding error rate, by limiting the reference/hypothesis alignment to the attribute names only, now called as concept. This metric then became the de facto metric in MEDIA and has been used in several research works, as in [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF][START_REF] Dinarelli | A data efficient end-to-end spoken language understanding architecture[END_REF][START_REF] Ghannay | End-toend named entity and semantic concept extraction from speech[END_REF]. [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF] jointly evaluate the recognized concepts and the normalized value of each concept occurrence. In [START_REF] Simonnet | ASR error management for improving spoken language understanding[END_REF][START_REF] Simonnet | Simulating ASR errors for training SLU systems[END_REF] By using these rules, the CVER should be equal to 0% when evaluating the reference. However, we obtain a CVER of 4.7% on the dev and 5.7% on the test corpus. All these errors are substitutions, i.e. support words that are not or badly normalized. [START_REF] Pelloin | End2end acoustic to semantic transduction[END_REF] has introduced a way of obtaining automatically the normalized value directly in the output sequence. They used an end-to-end encoder-decoder system with Attention Mechanism to output the sequence of concept/value pairs from the audio signal. They obtained very good results but similarly to [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF], the same rule-based system designed to normalize concept values obtained better results, even if this normalization is not robust to speech recognition errors and could be perfectible. However, we think that the automatic value normalization process is still a very important issue that could be addressed by researchers in order to propose new approaches that do not use human expertise. Such data-driven solutions could speed up and reduce the cost of the deployment of other human-machine applications in new domains.

In order to get an additional information about the values brought by word sequences supporting the concepts, but which is independent to the value normalization process, we propose in this paper a new evaluation measure: the u-CVER, which takes into account the unnormalized values, instead of the normalized ones.

Error correction in manual annotation During the MEDIA project, a complete semantic annotation scheme have been proposed to the human annotators. As natural language is subject to interpretation and humans are errorprone, some semantic annotation errors remain, leading to take them into account during training and also leading to false errors during the evaluation. Furthermore, we detected some problems in the audio segmentation. For example, there are very long segments at the end of the dialogues when the user has already hung up the phone. We propose to correct some of these errors detected during several analyses of the corpus with the proposed new updated version of MEDIA.

Data preparation

The manual transcription in the ME-DIA corpus is very precise and takes into account some details of pronunciation. For instance, disfluences like false starts or truncated words are annotated by using parentheses or asterisks, e.g.: "an (ho)tel", that means that the "ho" was not pronounced. Such annotations are not processed in the same way by the different authors, and this can make the comparisons between published experimental results hard to interpret. For the sake of transparency, we share our script for data preparation in the SpeechBrain recipe we have implemented.

Integrating the unused data As seen in subsection 2.1., an important part of collected and manual annotated data was never used before now. We suggest to integrate this data in our recipe -including data preparation -and share in this paper the first experimental results got on it.

Data Updates

Corrections have been brought to the initial manual annotations: the new version of the data will be integrated into the ELRA catalogue, as the original one. We started from the already distributed files of the MEDIA dataset to generate the new ones.

Correction of Manual Annotation

First, a simple normalization has been done on the transcription itself. We removed multiple spaces, corrected apostrophe and hyphen connections to their words, and added uppercase to nouns when forgotten. We also corrected some spelling of words and some erroneous semantic labelling. The audio channel -left or right -in which the user's voice has been recorded was not well indicated. It will now be indicated next to the recording ID. The ID of some users did not always respect the expected format: we corrected this also.

The MEDIA test2 Dataset

As presented in section 2.1., we have discovered an available but unused manually annotated data in the MEDIA corpus distributed by ELRA. We have decided to use it to create a new test corpus, named test2 detailed in Tables 1, 2 and 3, on the lines entitled "unused data". The test2 is even bigger than the original test, making it greatly interesting and useful. This corpus is far similar to the original test one. Only a "Full labelling" (attributes+specifiers) manual annotation of the semantic concepts has been realized in test2, we automatically generated the "Relax labelling" (attributes only) by removing the specifier parts of the concepts (following the annotation guide). Only one concept was never seen in the other sets: "personneprénom", standing for "person-firstname". This semantic concept, appearing only once, brought the discussed CVER evaluation problem to the surface.

Data Statistics

In this section, we present the new MEDIA statistics for all datasets, including test2. While Table 4 Taking a closer look at the data, we observe that the most common word is "oui", standing for "yes". It is used 3.8% of the time in total. The most common semantic concepts are "response" and "command-task" (when the user asks to book a room). They appear respectively 18.4% and 6.6% of the time in total.

The MEDIA SpeechBrain Recipe

A complete recipe for the MEDIA corpus has been built and integrated to SpeechBrain 3 , an open-source conversational AI toolkit based on PyTorch. SpeechBrain is a user-friendly toolkit proposing multiple recipes, ready to train. By integrating our recipe to SpeechBrain, we expect to make the MEDIA benchmark more accessible to researchers, and to make the source code persistent through a community maintenance.

The MEDIA recipe is available for running either an ASR task or a SLU task (the latest being the former without considering semantic concepts in the output of the system). It is based on end-to-end architectures, and can be trained by fine-tuning a wav2vec 2.0 model [START_REF] Baevski | wav2vec 2.0: A framework for self-supervised learning of speech representations[END_REF]. wav2vec 2.0 learns speech representations through self-supervision learning from large amounts of speech data, using Convolutional Neural Networks (CNN) and a masked Transformer. After pre-training, this model can then be fine-tuned through supervised learning on labelled data for ASR or SLU task, depending on the final task.

The recipe includes data preparation, training and evaluation scripts. They are detailed in the next subsections.

Data Preparation

To use the MEDIA SpeechBrain recipe, remind that it is necessary to get the original MEDIA data, distributed by ELRA, beforehand. By running an experiment, the data preparation scripts (if asked in the launching command) will create SpeechBrain compatible csv files to train the model. These csv files contain information like the utterance id, the manual annotation, the audio file pathname. Note that some options are available in the recipe in order to sort or remove utterances according to their duration. Indeed, experiments done by many users of the SpeechBrain toolkit have shown that the ascending sorting can be more efficient for ASR or SLU. All special characters, except chevrons, hyphens and apostrophes have been removed from the csv files. The chevrons are used to mark the beginning of word support, indicating the semantic concept, and the ending of the word support is marked with an ending chevron. For example, "<task-command> I want to book > hum <room-type> a double room >". For the apostrophe, it has been decided to bond it to the preceding word, but not to the following one. For instance, "d'Avignon" would now be written "d' Avignon". The only exception is for the word "c'est" (i.e. it is), which is far too common in French to divide it and since the word "c'" does not exist in another form.

Further processing have been made to the MEDIA dataset to be optimized for experiments. Among the removed symbols, asterisks were used to specify very close words in the speech. For example, a user who would say "how are you" very quickly could be transcribed "how* are* you" in the original MEDIA because of a strong assimilation during the co-articulation of these words. It will be transcribed "how are you" after the data preparation process of our recipe. Concerning the truncated words, round brackets and their content have been replaced by asterisks in order to still have an indication about the not pronounced part of the word but also that this word does not exists in the French vocabulary. It also prevents the annotator from misinterpreting any possible word that the user wanted to pronounce. As a result with an example, "exam(ple) " written in the original MEDIA corpus is written "exam* " in the new csv files.

The last modification brought to the transcriptions during the data preparation process was removing hyphens between numbers written in letters. In French, "soixante-dix" (meaning 70) is as much understandable as "soixante dix". Those characters only increase the final vocabulary, but not really adding sense to the transcription.

In the original MEDIA xml files, some synchronization tags are present, in addition to time code related to speaker turns. These synchronization tags are helpful to split a speaker turn into utterances. They have been marked by the human annotators of the original MEDIA corpus. A processing script enables to take some into account in order to reduce the length of utterances.

All the synchronisation tags narrowing the time limits, or cutting the transcription without splitting a semantic annotation labelling a word support, were used.

Neural Architecture

The recipe is based on the use of wav2vec 2.0 models. We used LeBenchmark models [START_REF] Evain | Task agnostic and task spe-cific self-supervised learning from speech with lebenchmark[END_REF] such as the Wav2Vec2-FR-3k large. This model was pretrained through self-supervised learning on 3k hours of speech (mainly read speech and broadcast news) in French language. The LeBenchmark models are freely shared with the community. On the top of the Wav2Vec2-FR-3k large, we added 3 dense layers of 512 neurons, with the LeakyReLU as activation function. These layers are themselves followed by one fully-connected layer and a final softmax layer. The weights of these four additional layers are randomly initialized, while the other weights are initialized by using the pretrained weights in the wav2vec 2.0 part of the neural architecture. As input, the neural network receives a wav audio file sampled at 16 kHz, and the output are characters that cover the alphabet needed to spell all the words of the MEDIA training data, and additional characters that manage the opening and closing tags used to recognize the concepts. After processing through the softmax layer, the outputs are generated thanks to a simple greedy decoder.

Training Process

The training is done in supervised manner from the semantically labelled MEDIA training data. This can be considered mainly as a fine-tuning of the wav2vec 2.0 model on the final downstream task. We also proposed in the recipe an alternative that consists of first fine-tuning the wav2vec 2.0 model on an external audio data, in order to achieve an ASR task. To make the experiments reproducible and accessible, we used the Com-monVoice French dataset4 (version 6.1), collected by the Mozilla Foundation. The train set consists of 425.5 hours of speech, with around 24 hours for the validation and test sets. After removing and re-initializing the weight matrix of the last layer, we finish by fine-tuning the model in the same way as the first solution, on the MEDIA semantically labelled training data.

The optimizer for the wav2vec 2.0 model is Adam, with a learning-rate of 0.0001, and AdaDelta for other layers, with a learning-rate of 1 and momentum of 0.95. Utterances were sorted in ascending order. We only did 30 epochs for the results presented in section 4.5.. Two kinds of models are considered in the following section:

• The models fine-tuned directly on the MEDIA training data to process the targeted SLU task. These models are named media-base.

• The models that are first fine-tuned on the Common-Voice data for an ASR task, then fine-tuned on the MEDIA training data to process the targeted SLU task. These models are named media-comvoice.

Evaluation

We integrated in our recipe several evaluation metrics.

As proposed in the original MEDIA campaign, we align the reference and the hypothesis to compute the number of errors in terms of deletions, insertions and substitutions. Several components are then considered in the reference/hypothesis. The recipe proposes three types of metrics to evaluate the performance of the neural models:

• the character error rate ChER was already integrated in SpeechBrain. It considers all characters in the output, considering also the semantic concept but as only one single character.

• the concept error rate CER, as proposed by [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF], compares the sequence of concepts only. Each token is considered as one unit, not each character like in ChER. Lets take again the example "<task-command> I want to book > hum <room-type> a double room >". The evaluated hypothesis will be "task-command room-type".

• we introduce a new measure: the unnormalized concept value error rate u-CVER. The words appearing in a word support for a concept are concatenated in one single word and also concatenated to the concept. The previous example will be presented as "task-command I want to book roomtype a double room". As a consequence, if any of the characters in the word support prediction or the concept tag prediction differs from the reference, then the entire prediction of that concept is counted as an error.

First Experimental Results

An experiment with the recipe has been launched to obtained the models described in Section 4.2.. Notice that for each kind of scoring (Full or Relax), a specific model was trained (as the semantic lexicon considered for the output is different). Thus, for "Full" or "Relax" scorings, two different media-base models were trained, the same for mediacomvoice.

The results on the new Media, considering both Full and Relax scorings, are presented in Table 7 for the two kinds of neural models present in our MEDIA SpeechBrain recipe.

As the rules based system [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF] we used is only available for the Relax scoring, we do not compute the r-CVER for the FULL scoring. We can see that the recipe reaches a little under end-toend system's state-of-the-art results with 16.3% of CER and 23.7% of r-CVER on the test. This proves the recipe is operational and simply needs tuning to enhance the results. As expected, the proposed metrics u-CVER is stricter than the r-CVER.

In Table 8, the results obtained by our best systems on the new test2 dataset are presented. These systems are based on the media-comvoice models. We can also observe that the already analysed test corpora are quite similar, and that the test2 is well suited to the task. Furthermore, while the original test dataset has been used by researchers for over fifteen years, the results of the test2 appear to show that the models applied to the original test corpus have not been implicitly "overfitted" during this time.

In a future use of a model trained on the MEDIA dataset, we need to be able to evaluate the results obtained by using a better normalization process for concept value than the current rule-based one.

Conclusion

In this paper, we present some updates to the MEDIA benchmark dataset for spoken language understanding. This update will also be integrated into the ELRA catalogue, as the original dataset. We expect to facilitate the use of this very challenging dataset, which became free for academic research in 2020. We also present a complete recipe, including data preparation, training and evaluation scripts, built and integrated to SpeechBrain, an already popular open-source and all-in-one conversational AI toolkit based on PyTorch. Last, a significant amount of data collected during the construction of the MEDIA corpus in the

Table 2 :

 2 Original MEDIA time statistics of the user with mean and median duration of their utterances.

		Nb.	Nb.	Nb.
		Utterances	Turns	Dialogues
	train	13.7 k	13.0 k	727
	dev	1.4 k	1.3 k	79
	test	3.8 k	3.5 k	208
	unused	4.0 k	3.8 k	244
	Table 1: Original MEDIA dataset distribution considering
	only the user's utterances.
	Data	Nb. Hours	Mean Duration	Median Duration
	train	16h56m	4.69s	3.12s
	dev	01h40m	4.77s	2.79s
	test	04h47m	4.89s	3.34s
	unused	05h35m	5.30s	3.86s
	Data	Nb. Hours	Mean Duration	Median Duration
	train	42h10m	209s	194s
	dev	03h37m	165s	158s
	test	11h34m	200s	190s
	unused	14h30m	214s	196s

Table 3 :

 3 Original MEDIA time statistics on global recordings (user, WoZ and blanks) with mean and median of the recordings.

The semantic dictionary defined for the MEDIA project includes 83 basic attributes -including 73 database attributes, 4 modifiers, and 6 general attributes -and 19 specifiers

[START_REF] Bonneau-Maynard | Results of the French evalda-media evaluation campaign for literal understanding[END_REF]

: room-number, hotel-name, location are examples of database attributes, comparative, relative-distance are examples of modifiers, proposition-connector and attribute-connector are examples of general attributes, and address, travel are examples of specifiers, that specializes the attribute role in a dialogue.

  , the authors named this metric the concept-value error rate (CVER), also used in addition to the CER. The CVER is an extension of CER, which considers the correctness of the concept/value pair. Following[START_REF] Simonnet | ASR error management for improving spoken language understanding[END_REF] and[START_REF] Simonnet | Simulating ASR errors for training SLU systems[END_REF] works, the joint CER and CVER metrics have been adopted in several recent studies for MEDIA benchmark[START_REF] Caubrière | Curriculum-based transfer learning for an effective end-to-end spoken language understanding and domain portability[END_REF][START_REF] Ghannay | Where are we in semantic concept extraction for spoken language understanding[END_REF][START_REF] Pelloin | End2end acoustic to semantic transduction[END_REF]

	Concept

value normalization To normalize word supports into values,

[START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF]

proposed three possible ways: 1) hand-crafted rules obtained with the training data, needing a human expert effort; 2) stochastic approaches based on Deep Belief Network or Conditional Random Field; 3) a combination of stochastic approaches and human rules. They concluded that the use of human rules outperforms the results obtained with just stochastic approaches. The main reason was the numerous possible values for some concepts with open values like date, payment-amount or name-client, in conjunction with the small size of the training data. As a result, most of latest works considering the values during the evaluation used the script based on human rules.

Table 4 :

 4 summarizes statistics about words and truncated words (cf. Section 2.3. -data preparation) considering only the user's utterances, Table5presents statistics on concept occurrences and lexicon. Notice that the number of concept occurrences is the same in Full and Relax scoring. Indeed, the number of word supports remains unchanged, only the lexicon of the considered concept labels is different. New MEDIA number of words and truncated words, considering occurrences and lexicon in user's turns.

		Occurrences		Lexicon
	Data	Nb.	Nb.	Nb.	Nb.
		Words	Trunc.	Words	Trunc.
	train	92.6 k	820	2.3 k	372
	dev	10.5 k	134	0.8 k	89
	test	26.0 k	227	1.4 k	146
	test2	28.0 k	159	1.3 k	107
	Concept	Occurrences Full and Relax	Full	Lexicon Relax
	train	31.7 k		144		73
	dev	3.3 k		104		63
	test	8.8 k		125		71
	test2	9.4 k		129		71

Table 5 :

 5 New MEDIA number of occurrences of concepts and size of the concept lexicon, considering Full and Relax scorings.

Table 6 :

 6 Table6makes an update on actual hours of recordings and speech (for users only, not WoZ), thanks to the segmentation we applied from the manual annotation. Statistics of the user's utterances with mean and median duration after processing the new segmentation script.

	Data	Nb. Hours	Mean Duration	Median Duration
	train	10h52m	2.85s	1.69s
	dev	01h13m	3.23s	1.91s
	test	03h01m	2.88s	1.70s
	test2	03h16m	2.94s	1.93s
	total	18h22m	2.90s	1.75s

https://github.com/speechbrain/ speechbrain/tree/develop/recipes/MEDIA

https://commonvoice.mozilla.org/fr/datasets For the sake of comparison, in this paper we also present results with the CVER, when the CVER is computed by the human-rules system as proposed in[START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF]. In order to prevent ambiguity, we rename this metric r-CVER (r-for 'rule-based'). Here, the previous example will be evaluated considering the sequence "taskcommand booking room-type double". Last, while the "Full scoring" was still not used in the last published papers related to the MEDIA corpus, we reintroduce this scoring since we consider it makes the challenging MEDIA benchmark even more challenging. Thus, our recipe includes both "Relax" and "Full" scorings.

2000s was never used until now: we present the first results reached on this subset -also included in the MEDIA SpeechBrain recipe -that can be used for now as the MEDIA test2. We expect a growing community will use our recipe to start working on the MEDIA corpus. While the research community is more and more interested by SLU problems and benchmarks, MEDIA stays one of the most challenging corpus, even in the era of deep learning. For this reason, this corpus constitutes a relevant dataset to investigate new solutions that can have a real impact on such human/machine application.

Acknowledgements

This paper was partially funded by the European Commission through the SELMA project under grant number 957017 and by the AISSPER project supported by the French National Research Agency (ANR) under contract ANR-19-CE23-0004-01.