Christophe Langrenne
email: christophe.langrenne@cnam.fr

Alexandre Garcia

Marc Bonnet

Solving the hypersingular boundary

come

I. Introduction

The boundary element method (BEM) based on the Helmholtz integral equation (HIE) has long been applied for the analysis of acoustic radiation or scattering problems (direct problems) and source characterization (inverse problems). Its mathematical properties are extensively discussed 1;2 . When considering exterior Neumann problems, the solution of this formulation is non-unique for frequencies corresponding to eigenvalues of the associated interior Dirichlet problems [START_REF] Schenck | Improved integral formulation for acoustic radiation problems[END_REF] . This non-uniqueness is purely a drawback of the mathematical formulation of the problem and has no physical significance. To overcome this non-uniqueness problem within a direct BEM approach, two major formulations can be used: the Combined Helmholtz Integral Equation Formulation (CHIEF) proposed by Schenck [START_REF] Schenck | Improved integral formulation for acoustic radiation problems[END_REF] and the Coupled Helmholtz Integral (CHI) presented by Burton and Miller 4 .

CHIEF relies on an overdetermined system of equations generated using both surface and interior collocation points. As the latter points must not lie on a nodal surface of an interior standing wave, the selection of their number and location remains a problem. Wu improved the CHIEF method using a small interior region rather than a single point [START_REF] Wu | A weighted residual formulation for the CHIEF method in acoustics[END_REF] .

The CHI method uses a linear combination of the HIE and its normal derivative. It is proved [START_REF] Lin | A proof for the Burton and Miller integral equation approach for the Helmholtz equation[END_REF] that such a linear combination provides unique solutions at all frequencies, provided the imaginary part of the coupling factor is non-zero. Moreover, numerical experiments [START_REF] Zheng | Is the Burton-Miller Formulation really free of fictitious eigenfrequencies ?[END_REF] show that the eigenvalues of the relevant integral operator are moved away from the real axis by the CHI, and optimal selection of the coupling parameter is studied in [START_REF] Marburg | The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter[END_REF] . The normal derivative of the HIE features a hypersingular integral operator, which makes this approach more difficult to implement than the usual HIE. Burton and Miller 4 considered two methods for handling the hypersingular kernel. The first uses a double surface integral to reduce the order of the hypersingularity, which increases the numerical quadrature work. The second consists in expressing the hypersingular kernel using tangential derivatives on the surface. This approach is applied using planar elements 9;10 or quadratic quadrilateral isoparametric elements [START_REF] Mathews | Numerical techniques for three dimensional steady-state fluid-structure interaction[END_REF] . Another tangential formulation is used by Wu in an isoparametric element environment [START_REF] Wu | On the numerical implementation of Cauchy principal value integral to insure a unique solution for acoustic radiation and scattering[END_REF] . Cunefare modified the Burton and Miller formulation by taking interior additional collocation points, avoiding any singularity [START_REF] Cunefare | A boundary element method for acoustic radiation valid for all wavenumbers[END_REF] .

An alternative way of dealing with the hypersingular operator arising in the CHI exploits known auxiliary solutions to the Laplace equation, leading to identities involving the static Green's function that allow to regularize the hypersingular integrals. Collocation BEM codes for Neumann acoustic problems are traditionally based on C 0 boundary elements (where nodal unknowns are defined at geometrical nodes), a setting adopted in this work as it fits our current implementation. However, later investigations on discontinuous, i.e. non-conforming, interpolations [START_REF] Marburg | Influence of element types on numeric error for acoustic boundary elements[END_REF] show considerable merit for this alternative treatment. The use of C 0 boundary elements in hypersingular operators presents a problem because the unit normal is usually not well-defined at nodes, due to the jump in the derivatives of the interpolating functions at element boundaries. Analytical formulations require collocation points interior to the elements for the unit normal to be well-defined there (whereas the HIE is normally collocated at element nodes). The two systems cannot, therefore, be combined directly as in the original Burton and Miller formulation. To perform this combination, interior pressure and velocity data must be interpolated from corresponding quantities at the element nodes. The choice of these interior points is crucial for the accuracy of the solution [START_REF] Chen | A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem[END_REF] . When an interior point is close to an element edge, the evaluation of the nearly singular integrals arising in adjacent elements requires a special treatment 16;17;18 . On the other hand, results deteriorate as interior points are chosen closer to element centers. Francis makes this choice of using the element centers as the only collocation points for the normal derivative integral formulation [START_REF] Francis | A gradient formulation of the Helmholtz integral equation for acoustic radiation and scattering[END_REF] . To avoid this problem, most implementations use piecewise-constant basis functions to allow using element centers as interior collocation points 20;21 . Marburg and Amini consider 9-node conforming elements and perform a partial combination whereby the HIE and its normal derivative are combined only when the collocation point is the element center node, a formulation that is however only valid for this type of element [START_REF] Marburg | Cat's eye radiation with boundary elements: comparative study on treatment of irregular frequencies[END_REF] .

Few authors have considered formulations involving hypersingular operators that are applicable with usual C 0 conforming elements. Chien et al. propose a special treatment of the hypersingular kernel from known Laplace solutions [START_REF] Chien | An effective method for solving the hypersingular integral equations in 3-D acoustics[END_REF] . Liu and Rizzo compare several kinds of elements (conforming, non-conforming and Overhauser C 1 continuous elements) and conclude that conforming elements give good results even though they do not satisfy the Lyapunov condition at the nodes [START_REF] Liu | A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems[END_REF] . An excellent survey of these methods is presented by Rego Silva [START_REF] Rego | Acoustic and elastic wave scattering using boundary elements[END_REF] . In another paper [START_REF] Rego Silva | A new family of continious/discontinious family of threedimensional boundary elements whith application to acoustic wave propagation[END_REF] , Rego Silva and Wrobel present a new family of continuous/discontinuous three-dimensional boundary elements. Yan et al. introduce a new concept of discretized operator matrix to compute the operator matrix corresponding to the second derivative of the static Green's function [START_REF] Yan | Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship[END_REF] . This approach is only valid for smooth surfaces and numerical examples are performed on spheroid or prolate spheroid geometries.

The motivation for this paper is to propose a robust and efficient computational scheme for the CHI method valid for smooth as well as piecewise-smooth surfaces, based on an indirect procedure exploiting discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. Our proposed approach is presented for C 0 -conforming elements, since they can be used here even though the BEM-discretized surface does not satisfy the Lyapunov condition [START_REF] Liu | A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems[END_REF] at the nodes; it is however equally applicable to non-conforming (i.e. discontinuous) boundary element approximations. Only weakly singular integrals and corresponding standard quadrature rules are needed for the numerical implementation. We compare the Burton and Miller method and the Schenck method on several configurations: sphere (smooth, convex), peanut (smooth, non-convex), cube (non-smooth, convex) and cat's eye (non-smooth, non-convex). Additionally, a regularized version α = ik/(k 2 + λ) of the classical BM coupling factor α = i/k is proposed.

II. The Burton and Miller formulation

For an unbounded acoustic medium (with wave velocity c) occupying the exterior of a closed surface Γ ⊂ R 3 , we consider the acoustic Neumann exterior problem whereby the normal derivative of the acoustic pressure p is given: ∂ n p = g on Γ with g known. The HIE formulation for this problem reads

κ + (P )p(P ) = Γ ∂G k (P, Q) ∂ n Q p(Q) - ∂p(Q) ∂ n Q G k (P, Q) dQ, ( 1
)
where P is the field point, Q the source point on Γ, and n Q the outward-directed normal to Γ at Q. The coefficient κ + (P ) is given by κ

+ (P ) = 1 -κ -(P ), with κ -(P ) = - Γ ∂G 0 (P, Q) ∂ n Q dQ. (2)
In particular, κ + (P ) = 1 for P outside of Γ, κ + (P ) = 0 for P inside Γ, and κ + (P ) = κ -(P ) = 1/2 at any smooth point P of Γ. G k is the radiating free-space Green's function for the Helmholtz equation, given by

G k (P, Q) = e ik|P -Q| 4π|P -Q| , (3)
where k = ω/c. The time-harmonic factor e -iωt is, as usual, implicit in (3) and all other field quantities. Finally, for later reference, G 0 is the free-space Green's function for the Laplace equation:

G 0 (P, Q) = 1 4π|P -Q| . (4)
Equation ( 1) is first used for P ∈ Γ as an integral equation for the unknown surface pressure p, and then provides an integral representation for the pressure at any evaluation point P outside Γ. However, it is well-known that (1) as an integral equation for p is singular at certain discrete values k D of k, namely the characteristic (or resonance) wave numbers for which the interior Dirichlet problem is not uniquely solvable. As a result, (1) does not have a unique solution for these frequencies. To restore solution uniqueness, Burton and Miller 4 introduced a second equation by differentiating (1) in the normal direction at P :

κ + (P ) ∂p(P ) ∂ n P = ∂ ∂ n P Γ ∂G k (P, Q) ∂ n Q p(Q) -G k (P, Q) ∂p(Q) ∂ n Q dQ. ( 5
)
As an integral equation for the unknown surface pressure p, Eq. ( 5) is singular for those discrete values k N of k for which the interior Neumann problem is not uniquely solvable. Importantly, one always has k D = k N . Burton and Miller show that the following combination of integral equations ( 1) and ( 5) yields a unique solution p for all frequencies κ + (P ) p(P ) + α ∂p(P )

∂ n P = Γ ∂G k (P, Q) ∂ n Q p(Q) - ∂p(Q) ∂ n Q G k (P, Q) dQ + α ∂ ∂ n P Γ ∂G k (P, Q) ∂ n Q p(Q) -G k (P, Q) ∂p(Q) ∂ n Q dQ, (6)
where the coupling parameter α is such that Im(α) = 0 when k is real. However, equation ( 6) introduces an integral operator whose kernel function -3 singularity. In the next section, we present a method which can be used to reformulate (5), and hence (6), for any type of boundary element.

∂ n P ∂ n Q G k (P, Q) has a |P -Q|

III. Reformulation of the normal derivative equation A. Partially regularized equation

We start from the following partially regularized reformulation of (5):

κ + (P ) ∂p(P ) ∂ n P = Γ ∂ 2 G k (P, Q) -G 0 (P, Q) ∂ n P ∂ n Q p(Q) dQ + ∂ ∂ n P Γ ∂G 0 (P, Q) ∂ n Q p(Q) dQ - Γ ∂ G k (P, Q) -G 0 (P, Q) ∂ n P ∂p(Q) ∂ n Q dQ - Γ ∂G 0 (P, Q) ∂ n P ∂p(Q) ∂ n Q dQ. ( 7
)
In this equation, the kernels

∂ n P ∂ n Q G k (P, Q) -G 0 (P, Q) and ∂ n P G k (P, Q) -G 0 (P, Q) are
weakly singular and nonsingular, respectively, for P, Q ∈ Γ. Equation ( 7) defines a partially regularized formulation, because the hypersingularity singularity is transferred to derivative of the static Green's function G 0 . When multiple values of ω are considered, e.g. for computing transient responses by Fourier synthesis, the static singular terms need only be computed once as they are frequency-independent. To do so, we use matrix operations, like Yan et al. [START_REF] Yan | Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship[END_REF] , rather than an analytical formulation of the corresponding integrals.

B. Discretization

Upon introducing a conforming isoparametric boundary element discretization 28;29 involving N nodes, Eq. ( 7) takes the form:

K + ∂ n P p = [N k -N 0 ]p + N 0 p -[M k -M 0 ]∂ n P p -M 0 ∂ n P p, (8)
where the matrices

[N k -N 0 ], [M k -M 0 ], N 0 , M 0 and K + are defined by [N k -N 0 ] ij = e∈E j Γ ref ∂ 2 G k (P i , Q e (ξ)) -G 0 (P i , Q e (ξ)) ∂ n P ∂ n ξ N j (ξ)J e (ξ) dξ [M k -M 0 ] ij = e∈E j Γ ref ∂ G k (P i , Q e (ξ)) -G 0 (P i , Q e (ξ)) ∂ n P N j (ξ)J e (ξ) dξ [N 0 ] ij = e∈E j Γ ref ∂ 2 G 0 (P i , Q e (ξ)) ∂ n P ∂ n ξ N j (ξ)J e (ξ) dξ (1 ≤ i, j ≤ N ) (9) [M 0 ] ij = e∈E j Γ ref ∂G 0 (P i , Q e (ξ)) ∂ n P N j (ξ)J e (ξ) dξ K + ij = κ + (P i )δ ij . In these equations, Γ ref is the reference element used in the isoparametric formulation, ξ ∈ R 2 is the parametric coordinate on Γ ref , the mapping ξ ∈ Γ ref → Q e (ξ) ∈ Γ h
defines the geometry of the e-th element of the approximate surface Γ h , J e is the Jacobian of that mapping, and N j is the shape function associated with node P j , used for interpolating both the geometry and the acoustic quantities. Any C 0 -conforming element may be used; in practice, linear or quadratic shape functions are most often chosen. Moreover, E j is the list of all elements that make up the support of the shape function N j on Γ h . The evaluation of the normal n P at a node P depending on which element P is considered to belong to, a reasonable approach consists in defining n P by averaging normals relative to all adjacent elements at P .

C. Evaluation of N 0 and M 0

We now propose an indirect approach for evaluating matrices N 0 and M 0 . For this purpose, we note that any p, ∂ n p solving the interior Laplace problem verify the (discretized) integral equations

K -p = -D 0 p + S 0 ∂ n p (10a) K -∂ n p = -N 0 p + M 0 ∂ n p (10b)
where S 0 and D 0 are the matrices associated with the single-and double-layer integral operators for the Laplace equation:

[S 0 ] ij = e∈E j Γ ref G 0 (P i , Q e (ξ))N j (ξ)J(ξ)dξ [D 0 ] ij = e∈E j Γ ref ∂G 0 (P i , Q e (ξ)) ∂ n ξ N j (ξ)J(ξ)dξ
Solving (10a) for ∂ n p and inserting the result ∂ n p = S -1 0 D 0 + K -p into (10b), we obtain

N 0 p = M 0 -K -S -1 0 D 0 + K -p (11)
Langrenne, Garcia and Bonnet, JASA, p. 7

This equation holds for any p, so the operator N 0 is given by

N 0 = M 0 -K -S -1 0 D 0 + K - (12)
By similar reasoning on the Laplace exterior problem, we find

N 0 = M 0 + K + S -1 0 D 0 -K + (13)
Equations ( 12) and ( 13) imply equality of their right-hand sides. Upon rearrangement and using that K + + K -= I, the resulting equation yields the following expression for M 0 :

M 0 = S -1 0 D 0 S 0 -K + S -1 0 K + S 0 + K -S -1 0 K -S 0 . (14)
It is important to note that M 0 and N 0 are evaluated in terms of quantities which involve only weakly singular integrals, namely K + , K -, D 0 and S 0 . These terms only have to be computed once, using a standard BEM code with k = 0. The matrix M 0 is computed first, then N 0 is evaluated by using ( 14) in either (12) or (13). Performing either derivation and recalling again that K + + K -= I yields

N 0 = S -1 0 (D 0 + K -)(D 0 -K + ) = S -1 0 (D 0 -K + )(D 0 + K -) (15)
In particular, when Γ is a smooth surface (in which case K -= K + = I/2), we find

N 0 = S -1 0 D 2 0 - 1 4 I , (16)
which is the formulation given by Yan et al [START_REF] Yan | Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship[END_REF] . Then equations ( 14) and ( 12) (or ( 13)) can be seen as a generalization of their formulation. Using the foregoing approach, the discretized form of the Burton-Miller integral equation ( 6) for the acoustic Neumann exterior problem is then

D k -K + + α [N k -N 0 ] + N 0 p = S k + α K + + [M k -M 0 ] + M 0 g. ( 17
)
Setting up the system (17) requires only nonsingular and weakly singular integrations. The frequency-independent terms M 0 and N 0 only have to be computed once, before being added to the (weakly singular) terms [M k -M 0 ] and [N k -N 0 ] for each frequency of interest. Equations ( 14) and ( 15) are in fact discretized versions of the Calderón identities linking the integral operators associated with Laplace problems (see e.g. Nedelec 1 , Theorem 3.1.3). The Calderón identities are useful in many aspects of boundary integral formulations, such as preconditioning [START_REF] Steinbach | The construction of some efficient preconditioners in the boundary element method[END_REF] .

IV. Partial combination for piecewise-smooth objects

When the surface Γ is only piecewise-smooth (due to edges and corners), we can only effect a partial combination of the integral equations ( 1) and ( 5) because the normal derivative of the pressure is not clearly defined at edge or corner collocation points. This partial combination has been tested successfully by several authors 19;22;31 , who report that forming such combinations at one-third of the original number of collocation points is sufficient to make the resulting system of equations well-conditioned.

The indirect formulation of Sec. III..C. can still be applied when Γ is piecewise-smooth. Assume that the boundary element approximation of Γ still involves N nodes, which are also pressure nodes. Then, since the normal is not uniquely defined at edge or corner nodes, those nodes support correspondingly multiple values of ∂ n p, so that the mesh involves M > N nodal values of ∂ n p. There are Q < N nodes not located on corners or edges, at which nodal values of ∂ n p are uniquely defined. The integral equation ( 1) for the exterior problem in discrete form is then

K + p = D k p -S k ∂ n p, (18)
where S k is a rectangular N × M matrix while K + and D k are square N × N matrices. Then, the discretized version of equation ( 5) restricted to the Q collocation nodes P where ∂ n p is uniquely defined is

K + v ∂ n p = [N k -N 0 ] + N 0 p -[M k -M 0 ] + M 0 ∂ n p, (19)
where K + v is the rectangular Q × M matrix formed by expanding K + as a M × M square matrix and retaining only the Q lines corresponding to the nodes at which the normal is uniquely defined.

[N k -N 0 ], N 0 are Q × N matrices and [M k -M 0 ], M 0 are Q × M matrices.
Adapting the procedure of Section III..C. to the relevant Laplace integral formulations yields

M 0 = S † 0 D 0 S 0 -K + v S † 0 K + S 0 + K - v S † 0 K -S 0 (20)
(where S † 0 = S T 0 (S 0 S T 0 ) -1 is the pseudo-inverse of the rectangular N × M real-valued matrix S 0 ) and

N 0 = (M 0 + K + v )S † 0 (D 0 -K + ), (21)
or (equivalently)

N 0 = (M 0 -K - v )S † 0 (D 0 + K -). (22)
Now, we can combine the N equations ( 18) and the Q equations ( 19) in Burton-Miller fashion. This corresponds to setting α = 0 for the N -Q edge or corner collocation nodes. The proposed indirect approach, while mainly presented for use with C 0 -conforming boundary elements, is also applicable to discretizations using non-conforming elements. In that context, the partial combination technique of Sec. 4 is not needed if the same approximation spaces are used for p and ∂ n p.

V. Numerical implementation

Tests aiming at fine-tuning the implementation of the formulation proposed in Sections III., IV. have been performed. For brevity, they are not presented in detail, and only the conclusions are reported. The tests were done for triangular elements, but quadrangular elements could also easily be implemented.

Implementing the numerical integrations was easy. For the nonsingular integrals, Lyness's quadrature rule for triangles is used 32 , while the Gauss-Radau quadrature rule (which concentrates quadrature points near the singularity in an appropriate way) is used for the weakly singular integrals [START_REF] Lachat | Effective numerical treatment of boundary integral equations: a formulation for three-dimensions elastostatics[END_REF] . The convergence of the integral evaluations with respect to the number of quadrature points is fast. In our tests, 19 points (Lyness's quadrature rule) are sufficient for regular integration, while 25 points (5 × 5) for each subdivision of a singular element were found adequate for weakly singular integrations. These amounts can be reduced when the collocation point is far from the integration element 34 , but we do not carry out this optimization here. A few element subdivision patterns have also been tested, but this proved not to be critical, so we finally subdivide each singular element into two triangles even if the singular point is at a vertex.

Finally, the normal n P in Eqs. 9 is the average of the element normals at P for all elements sharing P . frequencies lower than the first fictitious eigenfrequency, for which α = i/k is high, we consider the following modified coupling factor for the BM formulation, whose low-frequency behavior is regularized by means of the parameter λ > 0 (with λ = 0 restoring α = i/k):

α = ik k 2 + λ . (25)
For Cases 1,2 involving smooth surfaces, we set λ = π and λ = 4.55 respectively and corresponding to the first fictitious eigenfrequencies of the HIE formulation. For non-smooth surfaces, λ can be set to larger values. For example, we use λ = 15 for Case 3 (cube) and λ = 30 for Case 4 (cat's eye). The value chosen for λ is not critical, provided that the condition number of the resulting matrix does not exceed that arising from the choice α = i/k; the results shown in Figure 2 (right) fulfil this requirement.

C. Comparison of the Schenck and Burton-Miller methods

For Cases 1, 2 and 3, the CHIEF points are the 27 points of a 3 × 3 × 3 regular cubic grid centered at the origin, the grid spacing being 0.25m (which guarantees that all CHIEF points are interior). For Case 4 (cat's eye), the above set of points is translated by (-0.35 0.35 0)m, ensuring that all points are interior. Figure 3 present the results for all four configurations. The present Burton-Miller formulation gives the best results at all frequencies for Case 1, and results similar to those of CHIEF at all frequencies for Case 2. For Cases 3 and 4, corresponding to surfaces with edges and corners, the CHIEF results are better than those produced by the present BM method at low frequencies, but become unsatisfactory at high frequencies for Case 3, possibly due to the (arbitrarily chosen) CHIEF points becoming inadequate in that case. The results given by the BM formulation are acceptably accurate in all cases, and their accuracy has, unlike those produced by CHIEF, a smooth, non-fluctuating dependence on the frequency.

D. Comparison of linear and quadratic shape functions

To see how the solution accuracy is improved by using quadratic shape functions (T6), we compare the relative errors obtained with corresponding results based on T3 shape functions with the same number of nodes (i.e. replacing each 6-noded quadratic triangular element by four 3-noded linear triangular element). Note that, for the same number of nodal unknowns, using T3 elements entails four times as many evaluations of element integrals. Figure 4 shows the improvement in solution accuracy (for both the CHIEF and Burton-Miller formulations) achieved by using T6 elements. This observation is consistent with the inferior performance of linear conforming elements reported in [START_REF] Marburg | Influence of element types on numeric error for acoustic boundary elements[END_REF] .

VII. Conclusion

Yan et al. introduced the concept of discretized operator matrix to compute the hypersingular integral involved in the composite Helmholtz integral equation proposed by Burton and Miller. They used double surface integrals, leading to a formulation that is valid only for smooth geometries. In our paper, the Laplace equations for the interior and exterior problems are used to compute the hypersingular operator N 0 , but also the singular operator M 0 . The method is valid for any kind of geometry and for higher-order element implementations and can be seen as a generalization of the Yan et al. formulation. The first advantage is the numerical implementation, which is very simple because it only requires a polar coordinate change and classical quadrature rules. The second advantage of the method is that the collocation points coincide with the nodal points of the mesh. The HIE and its normal derivative are combined directly at collocation nodes, unlike other methods which need interior element collocation points. When the surface is not smooth, a partial combination is used and is sufficient to give accurate results.

The CHIEF method and the Burton and Miller method are compared for four different configurations, including two involving non-smooth surfaces. CHIEF is found to give better results at low frequencies, where good interior points can easily be found. At higher frequencies, the Burton and Miller method is more reliable because finding good sets of interior points for the CHIEF method becomes difficult. The condition number produced by our BM approach is moderate at all considered frequencies, and consistently lower than that of CHIEF in the upper range of the considered frequencies.

Replacing the usual BM coupling factor α = i/k with a version α = ik/(k 2 + λ) that is regularized at low frequencies improves the condition number at low frequencies without much influence on solution accuracy. The regularization parameter λ can typically be set to the square of the value of the wave number corresponding to the first fictitious eigenfrequency of the HIE. The main additional computational cost of our BM approach lies in having to invert the Laplace single-layer operator (using the singular value decomposition when its pseudo-inverse is needed). However, this is only needed once for a given geometrical configuration, as this part of the computation is independent on the frequency.

Figure 1 Figure 2 :

 12 Figure 1: Test configurations

Figure 3 :

 3 Figure 3: Relative solution errors of the different formulations.

Figure 4 :

 4 Figure 4: Comparison between T3 and T6 triangularizations for the sphere (left) and peanut (right) configurations (λ = 15).

VI. Numerical simulations

In this section, the method is used to solve wave problems with Neumann boundary conditions on the surface of the structure, which is quite common in practice. The gradient pressure or velocity can be measured directly with accelerometers or vibrometric lasers or calculated with an FEM code. For our simulations, we place a monopole of unit strength at a source point X in the interior region surrounded by Γ, which generates the acoustic field G k (X, •) in the entire space. The corresponding normal derivative ∂ n G k (X, •) then defines the Neumann data for the exterior problems solved thereafter (i.e. g = ∂ n G k (X, •)). The computed pressure field p on Γ h can then be compared to its analytical counterpart G k (X, •) on Γ by means of the following error function:

where Q i are the N nodes on Γ and p i are the computed nodal pressures.

A. Case configurations

We consider four different geometrical configurations, thereafter referred to as Case 1 to 4. All meshes are made of quadratic six-node (T6) triangles and created with GMSH [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF] (http://geuz.org/gmsh/).

In Case 1, Γ is a unit sphere, which is both smooth and convex. The mesh is composed of 2622 nodes and 1310 elements. The monopole is located at X = (0 0 0.2).

In Case 2, Γ is a peanut-shaped surface, defined by the parametric representation

z = cos 2θ + 1.5 -sin 2 2θ cos θ

This surface is smooth but not convex (Fig. 1b). The mesh is composed of 2294 nodes and 1146 elements. The monopole is located at X = (0.2 0 0.01). In Case 3, Γ is a unit cube, which is piecewise smooth and convex (Fig. 1c). The mesh is composed of 2402 nodes and 1200 elements. The monopole is located at X = (0.2 0.1 0.3).

Finally, in Case 4, Γ is a cat's eye (unit sphere with one octant removed), which is piecewise smooth and non-convex (Fig. 1d). This is a severe, often used, test case. The mesh is composed of 2338 nodes and 1168 elements. The monopole is located at X = (-0.5 0 0).

B. Condition number

Figure 2 plots the condition number of the governing matrix arising from BEM discretization of the HIE, CHIEF and BM formulations against the frequency for Cases 1 and 4. For the HIE formulation, the condition number clearly shows peaks at frequencies such that k = mπ (m = 1, 2 . . .). Secondary peaks can also be seen; these appear due to the use of quadratic shape functions, a phenomenon previously observed by Marburg and Amini [START_REF] Marburg | Cat's eye radiation with boundary elements: comparative study on treatment of irregular frequencies[END_REF] . The CHIEF method, here used with 27 points for all frequencies (see Sec. C. for details on their arrangement), yields a condition number curve that is smooth only at low frequencies, evidencing the need for increasing the number of interior points with the frequency.

The condition number curve for the BM formulation with the classical value α = i/k of the coupling factor is smooth at all frequencies. However, condition numbers are higher than those from the HIE formulation at low frequencies. Noting that the HIE gives good results at


P G Grinevich

R G Novikov

Generalized analytic functions, Moutard-type transforms and holomorphic maps *

Keywords: generalized analytic functions, spinors, Moutard transforms. We study

. In particular, we suggest an interpretation of generalized analytic functions as spinor fields and show that in the framework of this approach Moutard-type transforms for the aforementioned functions commute with holomorphic changes of variables.

∂ z ψ = u ψ in D, (1)
∂ z ψ + = -ū ψ+ in D, ( 2
)
where D is an open domain in C, u = u(z) is a given function in D, ∂ z = ∂/∂ z; see [START_REF] Vekua | Generalized Analytic Functions[END_REF]. Here and below, the notation f = f (z) does not mean that f is holomorphic. * The main part of the work was fulfilled during the visit of the first author to the IHES, France in November 2015. The first author was partially supported by the Russian Foundation for Basic Research, grant 13-01-12469 ofi-m2 and by the program "Fundamental problems of nonlinear dynamics", RAS.

† Landau Institute of Theoretical Physics, Chernogolovka, 142432, Russia; Lomonosov Moscow State University, Moscow, 119991, Russia; e-mail: pgg@landau.ac.ru ‡ CNRS (UMR 7641), Centre de Mathématiques Appliquées, École Polytechnique, 91128, Palaiseau, France; IEPT RAS, 117997, Moscow, Russia; e-mail: novikov@cmap.polytechnique.fr 1 A new progress in the theory of generalized analytic functions was obtained very recently in [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF] by showing that Moutard-type transforms can be applied to the pair of equations ( 1), [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF]. Note that ideas of Moutard-type transforms were developed and successfully used in the differential geometry, in the soliton theory in dimension 2+1, and in the spectral theory in dimension 2, see [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF] for further references. In particular, our work [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF] was essentially stimulated by recent articles by I.A. Taimanov [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF] on the Moutard-type transforms for the Dirac operators in the framework of the soliton theory in the dimension 2 + 1. On the other hand, we were strongly motivated by some open problems of two-dimensional inverse scattering at fixed energy, where equation ( 1) arises as the ∂-equation in spectral parameter.

A simple Moutard-type transform M = M u,f,f + for the pair of conjugate equations ( 1), ( 2) is given by the formulas (see [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF]):

ũ = Mu = u + f f + ω f,f + , (3) ψ
= Mψ = ψ - ω ψ,f + ω f,f + f, ψ+ = Mψ + = ψ + - ω f,ψ + ω f,f + f + , (4)
where f and f + are some fixed solutions of equations ( 1) and ( 2), respectively, ψ and ψ + are arbitrary solutions of ( 1) and ( 2), respectively, and ω ψ,ψ + = ω ψ,ψ + (z) denotes imaginary-valued function defined by:

∂ z ω ψ,ψ + = ψψ + , ∂ z ω ψ,ψ + = -ψψ + in D, ( 5
)
where this definition is self-consistent, at least, for simply connected D, whereas a pure imaginary integration constant may depend on concrete situation. The point is that the functions ψ, ψ+ defined in (4) satisfy the conjugate pair of Moutard-transformed equations:

∂ z ψ = ũ ψ in D, ( 6
)
∂ z ψ+ = -ũ ψ+ in D. ( 7
)
In addition, we have also the following new important result:

Proposition 1 For a simple Moutard transform (3),( 4) the following formula holds:

ω ψ, ψ+ = ω ψ,ψ + ω f,f + -ω ψ,f + ω f,ψ + ω f,f + + c ψ, ψ+ , ( 8
)
where c ψ, ψ+ is an imaginary constant.

In order to apply the Moutard-type transforms to studies of generalizedanalytic functions with contour poles, we need to study, in particular, compositions of the former transforms and holomorphic maps. Consider a holomorphic bijection W :

W : D → D * , z → ζ(z), (9)
W -1 : D * → D, ζ → z(ζ),
where D is the domain in ( 1), ( 2).

If we treat ψ(z), ψ + (z) as scalar fields in equations ( 1), ( 2), then the conjugate property of these equations is not invariant with respect to holomorphic bijections . In the next theorem we give the proper transformation formulas for the conjugate pair of equations ( 1), ( 2) with respect to holomorphic bijections:

Theorem 1 Let W be a holomorphic bijection as in ( 9). Let

u * (ζ) = u(z(ζ)) √ ∂z ∂ζ ∂ z ∂ ζ = u(z(ζ)) ∂z ∂ζ , (10)
ψ * (ζ) = ψ(z(ζ)) √ ∂z ∂ζ , ψ + * (ζ) = ψ + (z(ζ)) √ ∂z ∂ζ , ( 11
)
where u(z), ψ(z), ψ + (z) are the same that in equations ( 1), [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF]. Then:

∂ζψ * = u * ψ * in D * , ( 12
) ∂ζψ + * = -ū * ψ+ * in D * . ( 13
)
In addition,

ω ψ * ,ψ + * (ζ) = ω ψ,ψ + (z(ζ)), ( 14
)
where ω is defined according to (5).

Remark 1 Formulas (10), ( 11) have the following natural interpretation: ψ(z), ψ + (z) can be treated as spinors, i.e. differential forms of the type

( 1 2 , 0
) , and u can be treated as differential form of the type

( 1 2 , 1 2
)

. The corresponding forms can be written as:

u = u(z) √ dzdz, ψ = ψ(z) √ dz, ψ + = ψ + (z) √ dz. ( 15
)
It is very natural because the generalized analytic function equation ( 1) can be viewed as a special reduction of the two-dimensional Dirac system, see, for example [START_REF] Grinevich | Moutard transform for generalized analytic functions[END_REF].

Theorem 1 implies that W in (9) generates a map of the conjugate pair of equations ( 1), (2) into the conjugate pair of equations ( 12), (13). We also denote the latter map by W . Using this interpretation of W we obtain the following result:

Theorem 2

The following formula holds:

M u * ,f * ,f + * • W = W • M u,f,f + , ( 16
)
where M u,f,f + and M u * ,f * ,f + * are defined according to formulas ( 3), ( 4), and u * , f * , f + * , ω ψ * ,ψ + * are defined according to (10), ( 11), ( 14). Proposition 1 and Theorems 1 and 2 can be proved by direct calculations.

In the framework of the Moutard transform approach, using Theorem 2 we reduce local studies of generalized analytic functions with contour pole at a real-analytic curve to the case of contour pole at a straight line. These studies will be continued in a subsequent paper.


S D Nsele

J G Tartarin
email: tartarin@laas.fr

L Escotte

S Piotrowicz

S Delage

InAlN/GaN HEMT Technology for Robust HF Receivers: an Overview of the HF and LF Noise Performances

Keywords: Low Frequency Noise, High Frequency Noise, X and K band, InAlN/GaN HEMT

From the first developments of Nitride technologies using AlGaN/GaN heterostructures for designing high power, high frequency HEMT devices, we now assist to the emergence of new declination with InAlN/GaN heterostructures. Considering the expected better interface quality of this last technology as a consequence of the better lattice match, and better electrical properties, these HEMT devices attract much interest for high frequency applications (transmitter for power, receiver for low noise and robustness versus jamming). Different InAlN/GaN technological developments have been studied considering their frequency and noise parameters for low noise amplifiers in Ka-band. The paper addresses two issues related to noise in InAlN/GaN HEMT devices; one concerns the study of the HF noise performance for different technological processes and for the optimized technology, whereas the second focuses on the gate and drain current LFN spectra of competing technological versions featuring good HF dynamic and noise performance.

INTRODUCTION

Nitride technologies have registered a rapid growth during the last decade, especially on high power and high frequency segment markets. However, they are still far from exploiting their full potential. If AlGaN/GaN technologies are ready for market, they are regularly improving their reliability figures of merit (pushing the junction temperature and the mean time to failure for high power applications). One of the reasons why AlGaN/GaN devices need strong efforts concerns the electrothermo-mechanical interactions that generate defects hampering to reach the ultimate potential of these technologies. An alternative has been found by using InAlN/GaN heterostructure featuring almost 18% of indium, which content is relevant with lattice match of the two materials. Moreover, the higher mobility and sheet density in the 2DEG is convenient for high power and high frequency applications. This work is part of the Genghis Khan project from the French National Agency for Research; the objective is to develop this technology for HF applications, through the design of a Kaband transceivers. Part of the project, besides the technological issues, concerns the declination of an optimized low noise process for robust HF receivers: this paper addresses two aspects of the noise studies through the analysis of HF and LF noise parameters. The first section presents the studies dedicated to high frequency noise on different technological processes, to find out the optimum declinations of the technologies for high frequency low noise applications. The second paragraph proposes investigations on low frequency noise for the selected technologies to detect some potential weaknesses of the transistor (traps), and to identify its technological maturity (1/f noise level).

II. HIGH FREQUENCY NOISE PARAMETERS

HF noise parameters are measured using different subbands for an overall possible frequency range from 4 GHz to 40 GHz. Devices under test have been characterized on 8 different processes featuring different gate length, gate width and gate-source gate-drain spacing, indium content between 18 and 21%, with different doping concentrations and passivation types. Electrical small signal models have been developed, and frequency dispersion effects have been evidenced over a large frequency range, up to few GHz; these rapid traps (or short term memory effects) impacting the GHz domain can strongly limit the performances of the devices for HF applications [START_REF] Nsele | BroadBand Frequency Dispersion Modeling of the Output Conductance and Transconductance in AlInN/GaN HEMTs[END_REF]. Medium and long term memory effects will also be evidenced in the section dedicated to LFN measurements. These trappingdetrapping HF mechanisms can also be illustrated in figure 1, where the minimum noise figure of merit F min increases for decreasing frequencies below 8 GHz. This behavior has been observed by Danneville et al. on GaAs FETs [START_REF] Danneville | Influence of the gate leakage current on the noise performance of MESFETs and MODFETs[END_REF], and was attributed to shot noise. Fast traps affecting the transconductance and output conductance signatures versus frequency (as revealed for our devices), here combined with shot noise could explain this trend on F min . The device under test from figure 1 features a transition frequency F t =40 GHz and a maximum oscillation frequency F max =70 GHz. The value of F min =1.8 dB at 20 GHz (with associated gain G a =8.8 dB) is still above that of AlGaN/GaN HEMT [3][4] or InAn/AlN/GaN transistors [START_REF] Saunier | InAlN barrier scaled devices for very high fT and low-voltage RF applications[END_REF], featuring noise figures close to 1 dB at 26 GHz [START_REF] Saunier | InAlN barrier scaled devices for very high fT and low-voltage RF applications[END_REF] or at 36 GHz [START_REF] Medjoub | Sub 1-dB minimum-noise-figure performance of GaN-on-Si transistors up to 40 GHz[END_REF] due to their low L G ≤0.1µm. The reduction of the drain current largely affects the performances of F min , as observed between HEMT and MOS-HEMT devices in figure 2. A reduction on F min from 3.1 dB (HEMT) to 1.2 dB (MOS-HEMT) is obtained for 0.7 µm gate length devices, at 4 GHz. However, MOS-HEMT devices feature slightly lower F t and F max , if compared with another competing technology with carbon doping (HEMT device featuring F t =55 GHz and F max =135 GHz): these two technologies can be used to design a 30 GHz LNA. Devices featuring 0.15µm or 0.25µm gate length have been studied for X-band and K-band applications. The optimum biasing point versus V GS (or I DSS /I DS ) and V DS is mainly attributed to the elevated transconductance gain g m condition, to the reduction of diffusion noise at low I DS and to the management of self-heating effects under high dissipated power. An illustration of this optimum bias versus V DS is given in figure 3. F 50 and four noise parameters measurements have been investigated and compared for X-band and Ka-band receiver's applications. F 50 measurements are convenient for fast and systematic measurements as a first order analysis for biasing and sizing the device on each process declination, and also for a cross analysis between processes (with or without carbon doping, passivation type, indium content, gate length…). Then accurate characterization using the 4 noise parameters is performed (F min , equivalent noise resistance R n and complex optimum coefficient Γ opt ). Figure 3 presents the evolution of F 50 versus 1/L G at 18 GHz, for devices from the 8 different wafers under test: devices are biased at the minimum noise conditions. The influence of the dynamic performances (through the reduction of L G ) is noticeable on noise improvement. Figure 5 reports on the four noise parameters for a 0.15µm gate length device (plotted between 8-40 GHz). This device corresponds to the process and to the bias and size that will be implemented for the design of the robust LNA at 30 GHz, and presents a minimum noise figure of F min =2 dB at 30 GHz.

III. LOW FREQUENCY NOISE

In this paper, Low Frequency Noise (LFN) is used as a diagnostic tool to assess the quality of the processed MOS-HEMT evidencing the best HF performances from the previous study, and also on HEMT devices where Carbon has been used for doping. Gate and drain LFN current spectral density analysis of a transistor featuring 0.15µm of gate length are discussed. Measurements are performed between 1 Hz and 100 kHz, in two sub-bands for a better low frequency resolution. The two different processes that have been characterized feature almost the same characteristics in spite of differences (with or without carbon doping, passivation technique). The S IG spectra presents different generationrecombination centers (GR) superimposed with the 1/f flicker noise. Moreover, some Lorentzian contributions can be attributed to random telegraph signal (RTS) measured in time domain on these transistors [START_REF] Nsele | Lowfrequency noise in reverse-biased Schottky barriers on InAlN/AlN/GaN heterostructures[END_REF]. A distributed model is used to account for the 1/f like behavior, based on the plot of S IG ×frequency versus frequency for an equally weighted extraction of the noise parameters. The contribution of the GR centers and of the distributed 1/f models are presented in figure 6, as well as the total fit and measurement of S IG . The evolution of the LFN spectra and of the fitting models is plotted in figure 7, for variable reverse V GS biasing (identified by V R ). Time constants attributed to GR centers are extracted between 1 µs and 150 ms. One GR center depends on the static bias voltage V GS (thermal conditions are stable when the drain is open); the logarithmic representation of its associated time constant evolves with the root square value of the electrical field according to Poole-Frenkel effect (the electrical field is extracted from a model using C-V measurements). Moreover, figure 8 plots the evolution of S IG ×frequency versus I GS for three different frequencies. This plot features a linear model S IG ≈I GS α , with α=3.3 or α=1.5 respectively for low and high levels of I GS , and for different frequencies (10 Hz, 1 kHz, 100 kHz). This normalized representation establishes quite the same behavior versus I GS . The different α values are attributed to different leakage mechanisms at low and high reverse V GS . Electrical modelling of I GS (V GS ) has evidenced different contributions to the reverse leakage current, with essentially Poole-Frenkel contribution before V R =1V (i.e. I GS below 30nA, as delimited in figure 8), and Fowler-Nordheim for higher reverse values of V R (so for gate currents I GS >30nA). This weight of each Poole-Frenkel or Fowler-Nordheim effect is correlated to the change in the slope on figure 8. The α=1.5 value is close to already published results on AlGaN/GaN HEMT [START_REF] Tartarin | Gate defects in AlGaN/GaN HEMTs revealed by low frequency noise measurement[END_REF]. As S IG spectra evidence numerous GR centers, the slope of α=3.3 can be the consequence of these trapping-detrapping processes as stated previously by the time constant dependence with the root square of the electrical field for currents below 30nA (Poole-Frenkel contribution): indeed these GR centers are more pronounced at low V R in figure 7 (i.e. low I GS in figure 8) and trap-assisted mechanisms can be associated with this α=3.3 value.

Noise spectral densities have also been measured on the drain access, when the gate is short circuited. Figure 9 presents the variation of the normalized spectra S ID /I D when the gate voltage changes from open channel to pinch-off conditions. Only HEMT devices with carbon doping are presented and discussed for the drain current LFN measurements; indeed, these devices feature a non-monotonic atypical behavior of S ID plot versus I DS as shown in figure 10. Generally, S ID features an increase with I DS , but for the device under test, a maximum is reached between pinch-off and open bias conditions in figure 10 (at V GS =-1.5 V, i.e. I DS =35mA). As a large number of GR centers have been found to be located at the InAlN/GaN interface, we suppose that the action of the gate bias on the conduction band can be the cause of this inflexion on S ID . The maximum on S ID also corresponds to the bias point where g m is at its maximum, but even considering the carrier number fluctuation (and surface density of slow trap centers) the spectral density should not decrease. We formulate the next hypothesis for this decrease of S ID when V GS increases from V th to 0V. The conduction energy band bends towards the GaN layer when V GS increases; then the carriers are more distributed in the 2DEG volume, and less at the InAlN/GaN interface where traps are located. This is also convenient with the monotonic decrease of S ID /I D when the gate voltage V GS shifts from V th towards 0 V. When V GS is near the pinch-off conditions, the conduction energy band confines the electrons close to this interface, and the normalized S ID /I D noise density is maximum. Charges at the interface between InAlN/GaN are evidenced by S ID and S ID /I D behaviors versus the biasing conditions of the device under test, as well as trapping effects on S IG , with largely distributed effects [START_REF] Nsele | BroadBand Frequency Dispersion Modeling of the Output Conductance and Transconductance in AlInN/GaN HEMTs[END_REF].

IV. CONCLUSION

Wide bandgap InAlN/GaN MOS-HEMT and HEMT devices have been investigated in this paper: High frequency noise and Low Frequency Noise measurements have been performed on different technological processes. HF noise has been used to determine the optimum process for the design of a low noise amplifier in Ka-band: the effects of sizing, biasing and process variations (passivation types, doping levels) have been pointed out. F min =1.9 dB have been measured at 30 GHz on MOS-HEMT devices. Low frequency noise performances have evidenced numerous GR centers on the gate current spectral density, some attributed to RTS noise. Moreover, the evolution of the drain current spectral density does present a maximum value versus I DS , instead of a monotonic increase as expected, and as usually found in the literature. The bending of the conduction energy band with the gate-source voltage could explain a redistribution of the carriers more in the 2DEG, and less at the InAlN/GaN interface where traps have been localized. This is convenient with the behavior of the normalized S ID /I D which decreases with increasing I DS .

Fig. 1 .

 1 Fig. 1. Minimum noise figure Fmin and Associated Gain Ga versus frequency for an InAlN/GaN MOS-HEMT (0.15x2x75µm², VDS=5V and VGS=-1V).

Fig. 2 .

 2 Fig. 2. Minimum noise figure Fmin versus frequency for different InAlN/GaN HEMT and MOS-HEMT devices (measurements are given between 4-12 GHz).

Fig. 3 .

 3 Fig. 3. Minimum noise figure Fmin versus drain voltage VDS for frequencies of 12GHz and 26GHz. Right axis represents the transconductance gain gm.

Fig. 4 .

 4 Fig. 4. F50 noise factor versus 1/LG : devices are biased at their optimum noise conditions (@18 GHz).

Fig. 5 .

 5 Fig. 5. Noise parameters Fmin, Rn and Γopt (magnitude and argument) and associated gain Ga versus frequency (8-40 GHz) for the selected device (MOS-HEMT device feat. 21% In content, LG=0.15µm, 2x75µm gate width, VDS=6V and IDS=20mA).

Fig. 6 .

 6 Fig. 6. Gate current spectral density SIG versus frequency (open drain,VGS=-1V), and associated model based on 2 distributed 1/f models (1-10Hz and 400-1000Hz) [6] and 3 GR centers (80Hz, 1.7kHz, and above 200kHz).

Fig. 7 .

 7 Fig. 7. Gate current spectral density SIG versus frequency (open drain), for variable reverse gate bias VR.

Fig. 8 .

 8 Fig. 8. SIG×frequency versus IGS (at 1 Hz, 1 kHz, 100 kHz).

Fig. 9 .

 9 Fig. 9. Drain current spectral density SID versus frequency (gate short circuited), for variable VGS (@VDS=6V).

Fig. 10 .

 10 Fig. 10. Drain current spectral density SID and SID/ID versus IDS (@1 kHz, and for VDS=6V).

ACKNOWLEDGMENT

This work was supported by the Genghis Khan Project in the Framework of the French Research National Agency. The authors also acknowledge all the other partners of the project (Thales Communication, EGIDE, United Monolithic Semiconductors and Institut Lavoisier) for fruitful exchanges and discussion.


F Gherardi
email: gherardi@dbag.unifi.it

C Souty-Grosset

J Reynolds
email: jrynolds@tcd.ie

UNDERSTANDING AND MANAGING BIODIVERSITY IN RELATION TO NATIVE CRAYFISH POPULATIONS IN EUROPE

Keywords: biodiversity, native crayfish, conservation, management. COMPRÉHENSION ET GESTION biodiversité, écrevisses autochtones, conservation, gestion

The loss or diminution of European crayfish populations because of both habitat deterioration and competition with alien crayfish -also responsible for the dissemination of the crayfish plague -would reduce the biodiversity at the species level. The topic "What is meant by biodiversity?" in the context of native freshwater crayfish in Europe was discussed during the Kilkenny CRAYNET meeting in order to make the point about the varied meanings of biodiversity from genes and individuals to population levels.

INTRODUCTION

Biodiversity is expressed as a core concern of a number of EC networks, united through the EC 5th Framework Programme under the structure "BIOTA Cluster", a European scientific response to the Convention on Biological Diversity and aiming to determine and promote strategic approaches to biodiversity conservation and management in Europe. CRAYNET is thus linked to BIOTA Cluster and among other linked projects, Bio Platform hal-00123403, version 1 -9 Jan 2007 Author manuscript, published in "Bulletin Francais De La Peche Et De La Pisciculture (2003) 370-371" is a European platform for biodiversity aiming to improve the effectiveness and relevance of European diversity research, to contribute to European Research Area for Biodiversity and to promote the dissemination of current best practices and information regarding the scientific understanding of biodiversity conservation.

At the round table during the CRAYNET meeting in Kilkenny four general questions -based on a Bio platform e-conference entitled "Biotic resources in a changing world: science for a better governance" -have been addressed:

1. What do we mean by biodiversity? 2. How dependent are we on biodiversity?

3. What are the associated risks of such dependence? 4. How can we apply our knowledge to address this and to ameliorate the situation?

The present article is an attempt to integrate a number of points that have emerged from the E-conference discussions about flora and fauna but specifically discussed here in terms of crayfish situation in Europe.

BIODIVERSITY AS A CONCEPT AND A MEASURABLE ENTITY

Several authors (e.g. [START_REF] Gaston | Biodiversity[END_REF] have emphasised the multiplicity and range of meanings that have been associated with the term "biological diversity" and its 1986 contraction "biodiversity". This confused semantics is an obvious consequence of the increasing use -and sometimes abuse -of the word within the scientific literature, but it is also the inevitable effect of its adoption by a growing number of spheres besides academia, such as conservation, law, media, politics, science administration. Although the word "biodiversity" has already gained wide currency in the absence of a clear and unique meaning, greater precision will be required of its users in order that policy and programs could be more efficiently defined in the future. Our belief is that a demolition of the Tower of Babel in conservation biology would provide benefits to all concerned with this endeavour and obviously to its beneficiaries.

Biodiversity as a concept refers to the variety of life forms (plants, animals and microorganisms), the genes they contain, and the communities they form within ecosystems (see, e.g., The Convention on Biological Diversity, United Nations Environment Programme 1992). In terms of crayfish, this may relate to population genetic variability, to sympatric crayfish species, and to the macroinvertebrate community containing these species. However, given that the concept of biodiversity must reflect the goals of conservation, more useful than a definition would be "a characterisation of biodiversity that identifies the major components at several levels of organisation", by analysing in the area under concern the "composition" of biodiversity (i.e. species lists and measures of species diversity and genetic diversity), its "structure" (i.e. the physical organisation or pattern of a system, from habitat to landscape complexities), and its "function" (evaluated through ecological and evolutionary processes) [START_REF] Noss R | Indicators for Monitoring Biodiversity: A Hierarchical Approach[END_REF]. Obviously, the pool of living diversity is dynamic. It increases when new genetic variation is produced through the process of evolution and a new species is created or a novel ecosystem is formed, while it decreases when the genetic variation within a species decreases, a species becomes extinct or an ecosystem complex is degraded or lost.

Biodiversity must be viewed also as a measurable entity. Management requires measurements, and measures of biodiversity only become possible when some quantitative value can be ascribed to them and when these values can be compared [START_REF] Groombridge | Global biodiversity[END_REF]. In the context of European crayfish, biodiversity under this connotation may be measured at three levels: (1) genetic diversity, occurring within and between populations of crayfish species as well as between species; (2) species diversity, quantified as the variety of living species in an ecological unit, such as crayfish in stream banks; and (3) ecosystem diversity, relating to the variety of habitats, biotic communities and ecological processes, as well as the tremendous diversity present within ecosystems in terms of habitat differences and of the variety of ecological processes.

From this perspective, species can be weighed on the basis of their ecological role and grouped into functional categories, provided that the ecological role of a species can have a direct effect on the community structure and therefore on overall biological diversity. The ecological role of freshwater crayfish as keystone species in the community appears fundamental.

The quantitative assessment of diversity at the ecosystem, habitat, or community level remains more problematic than that at organismal level. A number of factors may make such measurement and assessment of diversity at this level poorly defined. Many different "units" of diversity are involved at the supra-species level that include patterns of habitats in the community, relative abundance of species, age structure of populations, patterns of communities in the landscape, trophic structure, and patch dynamics. Predator-prey relationships (including the impact of man) are important in the case of crayfish. Further confounding influences are the impact of historical or recent translocations of species, restocking of depleted areas, etc. The management of freshwater crayfish has been going on for a long time, while their community dynamics have only recently been studied. However, by conserving biodiversity at the ecosystem level, not only are the constituent species preserved, but also the ecosystem functions and services protected. These include nutrient and pollutant cycling, climate control, as well as non-consumptive recreation, scientific and aesthetic values (see, e.g., [START_REF] Norton | Scale and Biodiversity Policy: A Hierarchical Approach[END_REF].

Given the complexities of defining biodiversity at community or ecosystem level, a range of different approaches to measuring ecosystem diversity exist. Any number of community attributes is components of biodiversity and may deserve monitoring for specific objectives. There are several generic measures of community level diversity, including biogeographically realms or provinces, based on the distribution of species, and eco-regions or eco-zones, based on physical attributes such as soils and climate. All these apply to freshwater crayfish in Europe.

WE DEPEND ON BIODIVERSITY

In Europe, it seems impossible to find a habitat, landscape, eco-region, or any other large-scale biological system in pristine condition. In practice all "natural systems" on this continent have been altered to various degrees by direct or indirect human influence. In many cases human-related transformations have contributed to a higher biodiversity, although this cannot be equated to higher conservation value. Thus, small scale patchwork agriculture will have higher diversity than a steppe, and an exploited forest will be more diversified than a closed canopy climax stand. This situation is particularly relevant to crayfish in Europe. Therefore, the human component must always be taken into account when evaluating biodiversity.

In high numbers, however, humans also trigger biodiversity losses and affect the quantity and quality of services provided by the environment, adapting themselves to changes in environmental services by modifying their behaviour. This is well exemplified by human pressures on crayfish, including poaching, over-fishing, and degrading of water quality.

Today humans are dependent on fundamental biological systems and processes for their sustenance, health, well-being, and enjoyment of life. Humanity derives all of its food and many medicines and industrial products from the wild and domesticated components of biological diversity. Biotic resources also serve recreation and tourism, and underpin the ecosystems, which provide us with many services. Thus, biodiversity is vital for humans, including the variety of game species, timber and medicinal plants, and functional elements of biodiversity such as regulation of climate, carbon sequestration, decomposition, maintenance of hydrological cycles, pollination etc. But to look at the economic argument another way, can we say that the only species worth saving are those of economic importance?

In pragmatic terms, a species may be a source of products useful to people -thus, conservation has an economic basis [START_REF] Costanza R | The value of the world's ecosystem services and natural capital[END_REF]. Total economic value is the sum of "use values" (consumption, amenities, options, exploration) and "non-use values" -either existence (e.g. fees paid to see heritage items) or bequest (donations, conservation for future generations). Thus, benefits arising from the conservation of components of biological diversity can be (1) ecosystem services (e.g. water resources), (2) biological resources (e.g. crayfish as food), and (3) social benefits (in research; education and monitoring; recreation and cultural value).

From a biological perspective, diversity is vital, because it allows the human population and other species to adapt to environmental changes. Generally, the conservation of components of biological diversity, such as freshwater crayfish, produces benefits arising from the protection of the aquatic environment and water resources, as well as from the maintenance of either food sources or cultural heritage. Concerning Astacus astacus, T. Taugbøl (Norway) highlighted in Kilkenny and at the Halden CRAYNET meeting that the species is highly valued both from a recreational and economic point of view and pointed out that exploitation is a prerequisite for its conservation. As the greatest threat is the spread of crayfish plague following the introduction of alien crayfish, if local people are allowed to catch and benefit from this noble crayfish, this is the best protection against illegal stocking of aliens. (Concrete examples from Norway will be provided in the next BFPP volume devoted to European native crayfish with a special focus on Astacus astacus: linking socioeconomics and conservation.). Social benefits also follow, in terms of research, education, and recreation.

THERE ARE RISKS ASSOCIATED TO OUR DEPENDENCY ON BIODIVERSITY

Biodiversity thus provides a huge variety of goods and services, from food and water to chemicals and tourist attractions. Human society depends on these biological resources that are threatened by anthropogenic pressures. Freshwater crayfish have both a defined monetary and cultural value in Europe, particularly in the native genus Astacus, but also in the case of the established aliens Pacifastacus leniusculus and Procambarus clarkii. The interrelationship between biodiversity and natural resources, in the broad sense, acquires a broader interest under the perspective of global change dynamics, both at physical and biosphere levels and at the socio-economic domain. For example, climate change is expected to cause negative impacts upon fisheries, a major biotic resource, and the use and trade of species or genes in a global economy may generate conflicts over Intellectual Property Rights, a major issue of international politics.

In terms of European crayfish, the clearest case may be made for use values of Astacus species, whose commercial harvest and associated value-added tourism and heritage features may be important supports to the local economy in northern Europe. However, Austropotamobius species may be increasingly considered as having non-use values. In fact, socio-economic and traditional cultural values cannot be clearly separated in terms of freshwater crayfish. For example, the traditional Scandinavian festivals in August provide considerable tourist revenue to crayfish regions, which in its turn emphasises the importance of management of these valuable stocks (see the Norwegian case, above). In contrast, no such value is added in relation to maritime lobster fisheries.

Less tangible resources are the sights, sounds and smells of plants, animals and landscapes. Catching sight of a crayfish in a brook is such an element. Thus, while the economic benefits of such resources may be considerable, the value of biological diversity is not restricted to these. The enormous diversity of life in itself is of crucial value, demonstrably in some cases giving greater resilience to ecosystems and organisms.

Biodiversity has important social and cultural values. To argue that biodiversity should be conserved for its cultural value has a good deal of merit, if you can persuade people that each species has the equivalent complexity of a great city, and that in itself this complexity forms part of our cultural heritage. For crayfish this is still possible because our grandfathers fished the native species, although Austropotamobius species are now protected from exploitation. But we suspect that this is an argument that has limited appeal for many people on the planet, for whom a bug in a tree has nothing in common with their view of cultural heritage. While this will not be further developed here, biodiversity has a social value. The crux of the argument is that humans are today the main agent of change, and our views of what we should do about that change depends entirely on our social perspectives. Human exploitation, or indifference, has led to countless extinctions of species and alteration to ecosystems, and native crayfish are among those species most threatened. The problem comes when society must decide to invest energy into conservation and that energy does not come free of cost. This cost implies trade-offs and compromises -what are the political realities of sacrificing something?

HOW CAN WE APPLY OUR KNOWLEDGE TO CONSERVE BIODIVERSITY?

A better formulation might be: how best to synthesise our knowledge to understand the benefits associated with our dependency on biodiversity?

There is considerable knowledge about the structural and compositional aspects of biological diversity, but there is less information about its functional elements and especially about the relationship between structure/composition and function. This holds true for many aspects of crayfish biology. Ongoing research will increase our knowledge, through synthesis of scientific information, and understanding of its social applications, thus including both scientists and other stakeholders.

The notion of ecosystems services represents a key way to synthesise our knowledge to assess biodiversity. Determining how ecosystem services relate to change in species composition and ecosystem processes requires integrating population and community ecology, ecosystems dynamics, and economics.

In relation to biodiversity in general, scientific research is needed to identify and delineate the network of eco-regions in the scale of the planet and to determine what is indispensable for securing their further existence, as well as to understand:

• How biodiversity supplies the resources on which we depend.

• The feedbacks between biodiversity simplification and climate change.

• The ecology of extinction.

• How European societies manage the conflicts that arise when efforts to conserve biodiversity restrict economic and other human activities.

• In relation to further ecological studies on crayfish, we need to consider the following and through CRAYNET to develop appropriate protocols for management actions, which can be tested and further refined:

• What is an appropriate time span and season for the studies?

• Can we compile a series of indicators for scooping a study?

• How do we deal with obscure or taxonomically difficult groups that are nonetheless part of biodiversity?

• What do we do with all the historical records that do not conform to the new data?

• How are we to decide on standardised protocols for making numerical biodiversity assessments?

• Can we agree on the relative importance of the biodiversity indicated for the groups studied?

• Can we link this sort of approach to the politico-socio-economic background of the decision makers? A first step in management is to provide an inventory and description of local knowledge of biodiversity and sustainable use of its components. We then need to understand the status and trends of local knowledge in Europe (are local crayfish threatened?) and to understand the extent to which European indigenous people or local communities restrict access to local knowledge or treat it as a common good. We need to understand the local arguments (which combine both knowledge and values) and to enrich the debate with our scientific knowledge (biological as well as economic, social or cultural). We should then propose structures that help in finding a decision (these can be participatory approaches, decision support with an open value basis, such as nonutilitarian multi-criteria decision aid) and, finally, help in implementing these decisions.

CONCLUSIONS

There is a need for a standardised approach in biodiversity assessments. The Natura 2000 Network, as a universal system to maintain biodiversity in biogeographically regions, may serve as a basis for comparison of actual biodiversity levels throughout Europe. As white-clawed crayfish are listed for protection under European Legislation such as the Fauna and Habitats directive (93/43/EEC), a number of Special Areas of Conservation (SACs) have been proposed for freshwater crayfish in every European country. For example the Irish government has selected 14 sites as SACs for Austropotamobius pallipes on the basis of the best available published and unpublished information [START_REF] Reynolds | Knowledge-based management of European native crayfish, Crayfish special volume 4[END_REF]. In the U.K., the Joint Nature Conservation Committee (JNCC:http://www.jncc.gov.uk/ ProtectedSites/SACselection/species.asp?FeatureIntCodeS 1092) notes that there is a relatively low proportion of the total UK A. pallipes population contained within the SAC due to the fact that many recorded populations are in areas vulnerable to crayfish plague. In the UK and some other European regions, national legislation or local by-laws allow more stringent or flexible management and control. Thus, although alien crayfish species are widely established, the UK has designated "No-Go" areas. MAFF (1994) has circulated a consultation letter regarding a change in the use of the Import of Live Fish Act 1980: paragraph 13 of this document proposes that "the keeping and release of non-native crayfish will be forbidden in certain specified areas (no-go areas) except under licence" (in [START_REF] Holdich | Conservation and management of crayfish in Britain[END_REF]. This is a good example of how to make the best of a bad situation.

For every habitat type recognised under Natura 2000, it would be necessary to make an assessment of potential biodiversity level (under hypothetical natural conditions) in an agreed standard way. Such a standard should be universal enough to be applied to the whole range of European habitats and not too complicated in order to be used for further routine monitoring.

In agreement with ideas from the E-conference, we conclude that biodiversity should not be conceptually fragmented into "bits to conserve" and "bits to abandon". Since the maintenance of biodiversity through protection of selected components of the biosphere is limited in its effectiveness, the introduction of a new concept, the eco-region as the basic unit for biodiversity conservation is suggested as a suitable way to conserve biodiversity within functional units that encompass all levels of biological organisation from genes to the landscape, and include all biological resources at the regional level. The ecoregion unit is seen as functionally homogenous in the ecological sense, and one where many or all levels of biological organisation are presented. European legislation is starting to recognise such units, e.g. of relevance to crayfish is the catchment as a basic water quality management unit under the Water framework Directive. However, management still needs to understand the benefits and impacts of restocking and translocation of aquatic organisms, including the addition of parasites, pathogens and commensal species, e.g. the spread of crayfish plague across Europe (see the hazard analysis of pathogens of freshwater crayfish exotic to Europe, which is the first step of import risk analysis in EDGERTON, 2002a and2002b), e.g. the spread of crayfish plague across Europe.

The debate on potential and risks of biotic resources may be based around attempts to compare the various values and benefits of preserving biodiversity on a moral ground, and rejection of the idea of conserving only those elements of biodiversity that humans identify as procuring tangible biological resources.


Séraphin Dieudonné Nsele

Charles Robin

Jean-Guy G Tartarin
email: tartarin@laas.fr

Laurent Escotte

S Piotrowicz

O Jardel

S Delage

Ka-band low noise amplifiers based on InAlN/GaN technologies

Keywords: High Frequency Noise, Ka band, InAlN/GaN HEMT, LNA. I

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

If AlGaN/GaN heterostructure are now being under development (in spite of reliability studies that are still going on to push the limits of temperature-power-lifetime budgets), new solutions using InAlN/GaN are in the study phase: these HEMT devices take benefit of theoretically better interface between materials (using In content of 18% to be lattice matched on the GaN material), a better breakdown voltage and a better current density in the 2DEG. Different InAlN/GaN technological development have been studied considering their frequency and noise parameters for low noise amplifier design in Ka-band. Then different versions of hybrid and MMIC LNAs have been designed. A single stage LNA has been developed for RF stress tests, whereas a 3-stages LNA has been designed for optimized performances (Noise Figure andGain). MMIC LNAs have also been designed. In the first section, the technology and dynamic performance of the transistor are presented for the optimized technology. HF noise parameters are proposed in the second paragraph, with also the evolution of the HF noise figure under variable signal pump conditions. Then the design and the performances of the hybrid LNA are discussed in paragraph three. Simulations of MMIC LNA are also presented and compared to hybrid versions and to the state of the art in Ka-band.

II. INALN/GAN MOS-HEMT TECHNOLOGY

The process of InAlN/GaN HEMT and MOS-HEMT devices are developed on SiC substrate by MOCVD technique (3 inches wafer); devices are optimized considering the transition frequency F t and maximum oscillation frequency F max , the output power and the low noise performances for Kaband applications. More details on the technology can be found in [START_REF] Piotrowicz | 12W/mm with 0.15µm InAlN/GaN HEMTs on SiC Technology for K and Ka-Bands Applications[END_REF]. F t =40 GHz and F max =110 GHz have been achieved for the MOS-HEMT used in this study. Power densities of 3.5W/mm and power added efficiencies of 40% have been measured on 0.15µm gate length devices (6x50µm gate width) at 30 GHz for power amplifier design to target performances of 38 dBm with 18 dB of minimum gain in Ka-band over more than 20% frequency bandwidth. The optimum low noise conditions have been found for a 0.15µm gate length device featuring 2x75 µm gate width, biased at V DS =6V and I DS =20mA. State of the art results have been published on InAn/AlN/GaN transistors [START_REF] Saunier | InAlN barrier scaled devices for very high fT and low-voltage RF applications[END_REF], featuring noise figures close to 1 dB at 26 GHz [START_REF] Rudolph | A high survivable 3-7 GHz GaN Low-Noise Amplifiers[END_REF] or at 36 GHz [START_REF] Micovic | GaN MMIC technology for microwave and millimeterwave applications[END_REF] due to their low L G =50 nm technology: power density at 2.6 W/mm for 40% PAE is also achieved at 30 GHz by Saunier et al.

Small-signal model of the transistors have been developed, and the low-noise amplifiers designs in hybrid and MMIC have been studied both using the models and the measurements to assess the impact of the yield on the devices on the final design (more than 20 devices under test for the design of the hybrid amplifiers). The sensitivity of the model on the design procedure is accounted for and topologies can be optimized more securely.

III. HF NOISE CHARACTERIZATION OF ALINN/GAN HEMT

Intensive HF characterization is achieved for determining the optimum sizing and biasing of the devices. First order characterization consists in measuring F 50 noise figure, and then the four noise parameters are evaluated to get accurate appreciation of the noise behavior of investigated devices (minimum noise figure of merit F min , equivalent noise resistance R n and complex optimum coefficient Γ opt ). A plot of these parameters is given in figure 1 between 8 GHz and 40 GHz. It can be noticed the good concordance between the two sub-bands (before and after 25 GHz). Only the phase of Γ opt features a small discontinuity (almost 20°), due to the change in the position of the probes on the contact pads. The device presents a minimum noise figure of F min =2 dB at 30 GHz, with an associated gain of 7.5 dB. Moreover, as the proposed LNAs are supposed to be functional during severe jamming, the integrity of the device must be assessed under high input powers from high EM exposure (as a foreign aggression), but the low-noise characteristics should also be maintained for keeping operational the robust communication system. The nonlinear noise figure is presented in figure 2 to evidence the impact of an RF pump signal on the F 50 degradation of the device (thus of the matched amplifier): for practical reasons, the RF pump frequency is set at F RF =10 GHz, and the measurement of F 50 is performed at 7.5 GHz, (which is obviously not within the frequency band under concern for the LNA). From the P out versus P in plot, the compression point at 1 dB is evaluated at 2.5 dBm at the input of the device (@ 10 GHz). The noise figure F 50 is measured (at 7.5 GHz) versus the pump signal power level P RF ranging from -30 dBm to +14 dBm. The noise figure F 50 evolves when the input RF signal level increases, even below the input compression point. This increase is correlated with the decrease of the gain of the transistor, but occurs before the P 1dB compression point: this non-linear behavior is attributed to the noise contribution to the effective power at the input of the device, as the noise spectral density mixes with the RF pump signal. This study allows to accurately determine the convenient sizing and biasing of the devices, in order to avoid any saturation risk under jamming signals applied at the input of the LNA.

A. Hybrid LNAs

The first designs are developed using alumina substrate and flip-chip reports of the HEMT devices. The flip-chip report is convenient to reduce the length and associated losses, as well as the inductive contribution of the wire bonding, especially at 30 GHz. Moreover, the connection by bumps can act as thermal bridges to dissipate the heat out of the active zone of the transistor. The small wavelength at 30 GHz makes possible the use of distributed matching and design techniques, whereas lumped elements are not available at this frequency. Unfortunately, instabilities may occur at 15 GHz and 7.4 GHz (for an operating frequency at 30 GHz, which is modulo two or four that of the frequencies to stabilize): distributed solutions to stabilize the circuit at these frequencies are forbidden, and a solution has been developed with MIM grounded capacitors and short lines to reduce the gain at these frequencies and get unconditionally stable amplifiers. For the 3-stages LNA, biasing circuits also contribute to the input, output and inter-stage matching networks. The inter-stage networks are also designed to optimize the noise figure, but no degree of freedom can be achieved from the serial feedback inductance brought by the flip-chip report solution (with L S =60pH): no optimization can be realized on the noise optimal reflection coefficient which still remains close to the instabilities zones. Figure 3 is a photograph of the 1-stage hybrid Ka-band LNA. Noise figure have been measured over 4 different low-noise amplifiers, with NF measurements between 3.3 dB and 3.9 dB at 30 GHz, which is 0.3 dB above the simulated value. A photograph of the 3-stages MIC LNA is depicted in figure 4. Unfortunately, an oscillation at 7.9 GHz cannot be overcome on the 5 LNAs under test: from retro-simulation analysis, a high-pass filter must be placed at a specific part of the circuit to remove this instability, but no implementation is possible due to the small size of the estimated zone. However, the quite good agreement between simulated and measured NF on the 1-stage LNA is promising for forecasting that the 3-stage version should demonstrate NF close to simulations (NF=2.9dB at 30 GHz, and NF<3.3dB between 28.5 GHz and 30.5 GHz)

B. MMIC LNAs

Monolithic circuits have also been designed still based on the electrical and HF noise model of the InAlN/GaN MOS-HEMT device (same biasing conditions), with the GH-25 design kit developed by UMS. Different versions have been designed, with lumped matching networks and with distributed lines. The electrical model of the transistor is tailored to fit to the pattern of the MMIC design by de-embedding the tapers, lines and RF pads from the measured/modelled chip. The circuits have not been realized at this time, and only simulations are presented: however during the design steps, sensitivity and yield analysis have been considered to avoid the problems previously evidenced on hybrid amplifiers. Figure 5 represents the layout of a 3-stage MMIC LNA which makes use of L-C networks for matching the different input-output of the transistors (version #A). Layout of the MMIC Ka-band Low-Noise Amplifier (3-stages version #A mainly using L-C matching networks; InAlN/GaN MOS-HEMT are biased at VDS=6V and IDS=20mA). Size of the chip is 2.6x0.8mm².

Another 3-stage MMIC LNA (version #B) is presented in figure 6, based on the use of distributed lines. The coupling limit is set at -30dB between lines and pads of the layout.

C. MIC and MMIC Noise Figures: this work and literature

Noise figure of LNA circuits simulated and measured are reported in figure 7. The minimum gain is 20 dB, for a frequency bandwidth set at a reflection coefficient magnitude lower than -10 dB. From the comparison between the measured and simulated 1-stage MIC low-noise amplifier, a small degradation is noticed on NF (+0.3 dB). It is expected that measurements on 3-stages versions of the LNAs would also present a good concordance between simulations and measurements. Survivability tests have been realized at 30 GHz, but are not presented in this paper: the single-stage hybrid LNA has been exposed to RF step stresses (with RF power levels higher than 20 dBm) with no destruction of the circuit. Self-biasing and non-linear effects have been noticed under elevated RF power. As the technology is not yet mature, charges are activated under high RF power and change the intrinsic operating conditions of the active device. Charges can remove after a long recovery period (up to few minutes). The technology is still under development to improve these trapping-detrapping processes.

V. CONCLUSION

The InAlN/GaN MOS-HEMT technology has been studied for the development of Low-Noise Amplifiers in Ka-band for robust receivers dedicated to SATCOM applications. The HF linear and non-linear Noise Figures have been presented for the optimum sizing and biasing conditions. State of the art Noise Figures are achieved for a single stage MIC LNA (NF=3.3dB @ 30GHz), as well as for MMIC 3-stage LNA (NF<3 dB over more than 14% bandwidth). These new wide bandgap technologies are suitable to remove the limiter usually placed before the LNA in conventional architectures, even if this technology still have to be improved to satisfy the survivability tests under elevated RF power (step stress or CW stress).

Fig. 1 .

 1 Fig. 1. Noise parameters Fmin, Rn and Γopt (magnitude and argument) and associated gain Ga versus frequency (8-40 GHz). The selected device MOS-HEMT device (feat. 21% In content, LG=0.15µm, 2x75µm gate width, VDS=6V and IDS=20mA) is next used during the design of the hybrid and MMIC LNAs.

Fig. 2 .

 2 Fig. 2. Noise figure F50 and gain of InAlN/GaN MOS-HEMT (0.15x2x75µm²) under variable RF pump signal below and above the input 1-dB compression point (RF signal @ FRF=10 GHz, Fmeasurement=7.5 GHz) and for a biasing at VDS=6V and VGS=-1.5V (IDS=20mA).

  IV. KA-BAND LOW-NOISE AMPLIFIERS DESIGN Two technologies are investigated for the design of the LNAs at 30 GHz for SATCOM applications. Hybrid (MIC) realizations are designed using alumina and flip-chipped transistors, whereas Microwave Monolithic Integrated Circuits (MMIC) are based on GH-25 design-kit developed by UMS (United Monolithic Semiconductors).

Fig. 3 .

 3 Fig. 3. Photograph of the MIC Ka-band Low-Noise Amplifier (1-stage version, flip-chipped mounted InAlN/GaN MOS-HEMT is biased at VDS=6V and IDS=20mA). Size of the circuit is 9x7.5 mm².

Fig. 5 .

 5 Fig. 5.Layout of the MMIC Ka-band Low-Noise Amplifier (3-stages version #A mainly using L-C matching networks; InAlN/GaN MOS-HEMT are biased at VDS=6V and IDS=20mA). Size of the chip is 2.6x0.8mm².

Fig. 6 .

 6 Fig. 6. Layout of the MMIC Ka-band Low-Noise Amplifier (3-stages version #B with distributed matching networks; InAlN/GaN MOS-HEMT are biased at VDS=6V and IDS=20mA). Size of the chip is 3.5x1mm². For the MMIC LNA circuits design, much more flexibility is allowed in comparison with hybrid circuits (adjustment of serial feedback, possibility to mix lumped elements and lines). For this reason, very competitive noise figure are achieved: version #A and version #B feature noise figure of 3 dB and 2.6 dB respectively at 30 GHz. The simulated performances over 26-34 GHz bandwidth show a minimum small signal gain of 20 dB, and a noise figure lower than 3.5 dB for version #B MMIC LNA. The lower quality factor of inductance (version#A) in comparison with microstrip lines (version #B) results in a larger bandwidth (+50%), and in a higher noise figure (+0.4 dB) for the same small signal gain (above 20dB for each considered bandwidth).

Fig. 7 .

 7 Fig. 7. Noise figure versus frequency for different design of Hybrid and MMIC LNA based on InAlN/GaN MOS-HEMT technologies (this work). Dashed lines are for the simulations of hybrid-MIC (1-and 3-stages) and MMIC (3-stages, versions #A and #B), and solid line is for the measurement of 1-stage LNA.

Fig. 8 .

 8 Fig.8. State-of-the art in GaN LNA technologies, references are largely inspired by[START_REF] Rudolph | Highly rugged 30 GHz GaN low-noise amplifiers[END_REF]. Simulation of hybrid-MIC (dots) and MMIC version #B (dashed lines) 3-stages LNA, and measurement of 1-stage LNA (solid line) are plotted between 28 GHz and 31-32.5 GHz.

  Some designed LNA are reported in figure 8, with other reported results from literature based of GaN technologies [3]-[13]. Few LNAs have been designed in Ka-band, and this work presents state of the art figure of merit with NF measured at 3.3 dB (MIC LNA) or simulated at 2.6 dB or 3 dB (two versions of MMIC LNAs) at 30 GHz, in spite of a technology not yet mature.

ACKNOWLEDGMENT

This work was supported by the Genghis Khan Project in the Framework of the French Research National Agency. The authors also acknowledge all the other partners of the project (Thales Communication, EGIDE, United Monolithic Semiconductors and Institut Lavoisier) for fruitful exchanges and discussion.