Mathieu Laurière
email: mathieu.lauriere@upmc.fr

Olivier Pironneau
email: olivier.pironneau@upmc.fr

Math ématiques/Mathematics (Analyse Num érique/Numerical Analysis) Dynamic Programming for Mean-field type Control

For mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF ρ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function V (τ, ρ τ ) where the dependency on ρ τ is functional as in P.L. Lions' analysis of mean-filed games (2007). We derive HJB equations and apply them to two examples, a portfolio optimization and a systemic risk model.

Programmation dynamique pour les problèmes de contrôle à champs moyen

. Les résultats sont appliqués à un problème d'optimisation de portefeuille et à un problème de risque systémique.

Introduction

Stochastic control is an old topic [START_REF] Fleming | Controlled Markov Process and Viscosity Solutions[END_REF][START_REF] Øksendal | Applied Stochastic Control of Jump Diffusions[END_REF][START_REF] Touzi | Optimal Stochastic Control, Stochastic Target Problems and Backard SDE[END_REF][START_REF] Zhou | Stochastic Control[END_REF] which has a renewed interest in economy and finance due to mean-field games [START_REF] Lasry | Mean-field games[END_REF][START_REF] Garnier | Large deviations for a mean field model of systemic risk[END_REF][START_REF] Shen | Lqudty Generated By Heterogeneous Belefs and Costly Estmatons[END_REF]. They lead, among other things, to stochastic control problems which involve statistics of the Markov process like means and variance. Optimality conditions for these are derived either by stochastic calculus of variation [START_REF] Pironneau | A Maximum Principle for SDEs of Mean-Field Type[END_REF] or by stochastic dynamic programming in the quadratic case [START_REF] Bensoussan | Control and Nash Games with Mean Field Effect[END_REF][START_REF] Bensoussan | Mean-field Games and Mean-field Type Control[END_REF], but not in the general case for the fundamental reason that Bellman's principle does not apply in its original form on the stochastic trajectories of say X t if those depend upon statistics of X t like its mean value. As noticed earlier in [START_REF] Lions | Mean-field games[END_REF] and in [START_REF] Bensoussan | The Master Equation in Mean-Field Theory[END_REF] 1 , there seems to be no such restriction if one works with the probability measure of X t and use the Fokker-Planck equation.

In this note we apply the dynamic programming argument to the value functional V (τ, ρ τ (•)) where ρ τ is the PDF of X τ . Of course this is at the cost of several regularity assumptions, in particular it requires the existence of PDF at all times.

Once the problem is reformulated with the Fokker-Planck equation, it becomes a somewhat standard exercise to find the optimality necessary conditions by a calculus of variations. So the note begins likewise. Then a similar result is obtained by using dynamic programming and the connection with the previous approach and with stochastic dynamic programming is established, with the advantage that sufficient conditions for optimality are obtained. Finally we apply the method to two mean-field type control problems stated in [START_REF] Pironneau | A Maximum Principle for SDEs of Mean-Field Type[END_REF] and [START_REF] Garnier | Large deviations for a mean field model of systemic risk[END_REF].

The Problem

Let d, s, r ∈ N + . Consider a stochastic differential equation dX t = u(X t , t)dt + σ(X t , t, u(X t , t))dW t , (2.1)
where T > 0, u : R d × (0, T ) → R d , σ : R d × (0, T ) × R d → R d×d and W t is a d-vector of independent Brownian motions. We make the usual assumptions for X t to exist once X 0 is known [START_REF] Touzi | Optimal Stochastic Control, Stochastic Target Problems and Backard SDE[END_REF].

Let H : R d ×(0, T )×R d ×R r → R, h : R d ×(0, T )×R d → R r , G : R d ×R s → R, g : R d → R s . Assume also that ρ 0 is positive with unit measure on R d . Let V d ⊂ R d , U d = {u ∈ (L ∞ (R d × R)) d : u(x, t) ∈ V d ∀x,
t} and consider the problem

min u∈U d J := T 0 E[ H(X t , t, u(X t , t), E[ h(X t , t, u(X t , t))])]dt + E[G(X T , E[g(X T )])]
subject to (2.1) and such that ρ 0 is the PDF of X 0 (2.2)

Andersson et al [START_REF] Pironneau | A Maximum Principle for SDEs of Mean-Field Type[END_REF] analyzed this problem using stochastic calculus of variations, claiming rightly that dynamic programming is not possible unless h = 0, g = 0. Yet denoting

Q = R d × (0, T ) and µ ij = 1 2
k σ ik σ jk , with sufficient regularity, namely if X t has a PDF ρ t (for weaker hypotheses see [START_REF] Neufeld | Nonlinear Lévy processes and their Characteristics[END_REF]), the problem is equivalent to

min u∈U d J = Q H(x, t, u(x, t), ρ t (x), χ(t))ρ t (x)dxdt + R d G(x, ξ)ρ |T dx where χ(t) = R d h(x, t, u(x, t), ρ t (x))ρ t (x)dx, ξ = R d g(x)ρ T (x)dx and ρ t s.t. ∂ t ρ + ∇ • (uρ) -∇ • ∇ • (µρ) = 0, ρ| 0 = ρ 0 (x), x ∈ R d (2.3)
where H = H, h = h if these are not functions of ρ t (x). HYPOTHESIS 1.

-Assume that all data are continuously differentiable with respect to u and ρ and have additional regularity so that the solution to the Fokker-Planck equation is unique and uniformly continuously differentiable with respect to u and µ.

Calculus of Variations

PROPOSITION 1. -Let A : B = trace(A T B). A control u is optimal for (2.3) only if R d H ′ u + h ′ u R d H ′ χ ρdx + ∇ρ * -µ ′ u : (∇∇ρ * ) (v -u)ρdx ≥ 0 ∀t, ∀v ∈ U d with (3.4) ∂ t ρ * + u∇ρ * + µ : ∇∇ρ * = -H ′ ρ ρ + H + (h ′ ρ ρ + h) R d H ′ χ ρdx , ρ * |T = g R d G ′ ξ ρ |T dx + G. (3.5) Proof . -Let us regularize problem (2.3) by replacing R d by Ω := (-L, L) d with L ≫ 1. Now Q = Ω × (0, T ).
We add to the Fokker-Planck equation (2.3) the boundary conditions: ρ(x, t) = 0, ∀x ∈ ∂Ω, t ∈ (0, T ). Consider an admissible variation λδu, i.e. u + λδu ∈ U d for all λ ∈ (0, 1). Such a variation induces a variation λδρ of ρ given by

∂ t δρ + ∇ • (uδρ + ρδu + λδuδρ) -∇ • ∇ • (µδρ + µ ′ u δu(ρ + λδρ)) = 0, δρ| t=0 = 0, (3.6) where µ ′
u is evaluated at x, t, u + θδu for some θ ∈ (0, λ). By hypothesis the solution of the Fokker-Planck eq. in (2.3) depends continuously on the data u, µ, so (3.6) with λ = 0 defines δρ. Also

δχ = Ω [(h ′ u δu + h ′ ρ δρ)ρ + hδρ], δJ = Q [(H ′ u δu + H ′ ρ δρ + H ′ χ δχ)ρ + Hδρ] + Ω [G ′ ξ δξρ |T + Gδρ |T ] 2 = Q (H ′ u + h ′ u Ω [H ′ χ ρ])ρδu + Q (H ′ ρ ρ + H + (h ′ ρ ρ + h) Ω [H ′ χ ρ])δρ + Ω [( Ω [G ′ ξ ρ |T ]g + G)δρ |T ]
The adjoint state ρ * is given by (3.5) and ρ * |∂Ω = 0. Then, multiplied by δρ and integrated on Q (3.5) gives

Q δρ H ′ ρ ρ + H + (h ′ ρ ρ + h) Ω [H ′ χ ρ] = - Q δρ [∂ t ρ * + u∇ρ * + µ : ∇∇ρ * )] = Q [ρ * (∂ t δρ + ∇ • (uδρ) -∇ • ∇ • (µδρ))] - Ω ρ * δρ| T 0 = - Q [ρ * ∇ • (ρδu -∇ • (µ ′ u δuρ))] - Ω [(g Ω [G ′ ξ ρ |T ] + G)δρ| T ] Hence δJ = Q (H ′ u + h ′ u Ω [H ′ χ ρ])ρδu - Q [ρ * ∇ • (ρδu -∇ • (µ ′ u δuρ))] = Q H ′ u + h ′ u Ω [H ′ χ ρ] + ∇ρ * -(∇∇ρ * )µ ′ u ρδu (3.7)

Dynamic Programming

For notational clarity consider the more general case where H, G are functionals of ρ t (•). For any τ ∈ [0, T ] and any ρ τ ≥ 0 with unit measure on R d , let

V (τ ; ρ τ ) = min u∈U d J(τ ; ρ τ , u) := T τ R d H(x, t, u(x, t); ρ t )ρ t (x)dxdt + R d G(x; ρ |T )ρ |T dx (4.8)
subject to (2.3), i.e. such that ρ t is the PDF of X t given by (2.1) starting with ρ τ at time τ Note that the second parameter in V is a function of x, yet it is not V (τ, ρ τ (x)) but V (τ ; ρ τ (•)). We now prove the following version of Bellman's principle of optimality : PROPOSITION 2.

-If the problem is regular, then for any τ ∈ [0, T ] and any positive ρ τ with unit measure on R d , we have :

V (τ ; ρ τ ) = min u∈U d τ +δτ τ R d H(x, t, u(x, t); ρ t )ρ t (x)dxdt + V (τ + δτ ; ρ τ +δτ ) (4.9)
subject to ρ t given by (2.3) on [τ, τ + δτ ] initialized by ρ τ at time τ

Proof . -Denote the infimum of the right-hand side by V (τ ; ρ τ ). For any ǫ > 0, there exists an u ∈ U d such that, if ρ t is the solution of (2.3) with control u :

V (τ ; ρ τ ) + ǫ > J(τ ; ρ τ , u) = T τ R d H(x, t, u(x, t); ρ t )ρ t (x)dxdt + R d G(x; ρ |T )ρ |T dx = τ +δτ τ R d Hρ t + T τ +δτ R d Hρ t + R d Gρ |T ≥ τ +δτ τ R d Hρ t + V (τ + δτ ; ρ τ +δτ ) ≥ V (τ ; ρ τ )
Conversely, given u ∈ U d and ǫ > 0, ∃ũ ∈ U d which coincides with u on R d × [τ, τ + δτ ], such that:

J(τ + δτ ; ρτ+δτ , ũ) ≤ V (τ + δτ ; ρτ+δτ ) + ǫ
where ρt is the solution of (2.3) at t with control ũ starting with ρ τ at time τ . Hence :

V (τ ; ρ τ ) = V (τ ; ρτ ) ≤ J(τ ; ρτ , ũ) = T τ R d H(x, t, ũ(x, t); ρt )ρ t (x)dxdt + R d G(x; ρ|T )ρ |T dx = τ +δτ τ R d
H(x, t, u(x, t); ρt )ρ t (x)dxdt + J(τ + δτ ; ρτ+δτ , ũ)

≤ τ +δτ τ R d H(x, t, u(x, t); ρt )ρ t (x)dxdt + V (τ + δτ ; ρτ+δτ ) + ǫ
We obtain the conclusion by letting ǫ → 0 and by taking the infimum over u ∈ U d .

PROPOSITION 3. -(HJB minimum principle).

There exists {x, τ, ρ τ (•)} → V ′ ∈ R such that :

0 = min v∈V d R d H(x, τ, v(x); ρ τ ) + H ′ ρ (x, τ, v(x); ρ τ ) • ρ τ +∂ τ V ′ + µ(x, τ, v(x)) : ∇ x ∇ x V ′ + v(x) • ∇ x V ′ ρ τ dx (4.10)
where

H ′ ρ • ν = lim λ→0 [H(x, τ ; ρ + λν) -H(x, τ, ρ)]/λ. Proof .
-A first order approximation of the time derivative in the Fokker-Planck equation gives

δρ τ := ρ τ +δτ -ρ τ = δτ [∇ • ∇ • (µ τ ρ τ ) -∇ • (u τ ρ)] + o(δτ ) (4.11)
When everything is differentiable and smooth,

V (τ + δτ ; ρ τ +δτ ) = V (τ ; ρ τ ) + ∂ τ V (τ ; ρ τ )δτ + V ′ ρ (τ ; ρ τ ) • δρ τ + o(δτ ) (4.12)
Using (4.12) and the mean value theorem for the time integral, (4.9) yields

V (τ ; ρ τ ) = min u∈U d δτ R d Hρ τ dx + V (τ ; ρ τ ) + ∂ τ V (τ ; ρ τ )δτ + V ′ ρ (τ ; ρ τ ) • δρ τ + o(δτ )
The terms V (τ ; ρ τ ) cancel; divided by δτ and combined with (4.11) and letting δτ → 0, (4.13) gives

0 = min u∈U d R d Hρ τ dx + ∂ τ V (τ ; ρ τ ) + V ′ ρ (τ ; ρ τ ) • [∇ • ∇ • (µ τ ρ τ ) -∇ • (u τ ρ)] (4.13)
To finalize the proof we need to relate V to V ′ ρ and to its Riesz representative V ′ : PROPOSITION 4. -For any τ ∈ [0, T ] and any initial PDF ρ τ , let û and ρ denote respectively the optimal control and the corresponding solution of (2.3). Then:

R d V ′ (τ ; ρ τ )ρ τ dx = V ′ ρ (τ ; ρ τ ) • ρ τ = V (τ ; ρ τ ) + T τ R d H ′ ρ (x, t, û(x, t); ρt ) • ρt ρt (x)dxdt + R d G ′ ρ (x; ρT ) • ρT ρT (x)dx (4.14)
Proof . -Notice that Fokker-Planck implies ρ t = G(t -τ ) * ρ τ where G is a semi-group operator. Let (û t ) t∈[0,T ] be the optimal control and (ρ t ) t∈[0,T ] the corresponding solution of (2.3). Then :

V (τ ; ρτ ) = T τ R d H(x, t, û(x, t); G(t -τ ) * ρτ )G(t -τ ) * ρτ dxdt + R d G(x; ρ|T )ρ |T dx
This can be differentiated with respect to ρ by computing lim λ→0

1 λ V (τ ; ρτ + λν) -V (τ ; ρτ ) , for a given function x → ν(x) : V ′ ρ (τ ; ρτ ) • ν = T τ R d H(x, t, û(x, t); G(t -τ ) * ρτ )G(t -τ ) * νdxdt + R d G(x; ρ|T )G(T -τ ) * νdx + T τ R d H ′ ρ (x, t, û(x, t); G(t -τ ) * ρτ ) • [G(t -τ ) * ν] G(t -τ ) * ρτ dxdt + R d G ′ ρ (x; G(T -τ ) * ρτ ) • [G(T -τ ) * ν] G(T -τ ) * ρτ dx
Taking ν = ρτ leads to (4.14).

End of proof of Proposition 3 Differentiating (4.14) w.r. to τ leads to

∂ τ V (τ ; ρ τ ) = ∂ τ V ′ ρ (τ ; ρ τ ) • ρ τ + R d H ′ ρ (x, τ, ûτ (x); ρ τ ) • ρ τ ρ τ (x)dx
where ûτ is the optimal control at time τ . Now, let us go back to (4.13), which we rewrite:

0 = min uτ R d H(x, τ, u τ (x); ρ τ ) + H ′ ρ (x, τ, u τ (x); ρ τ ) • ρ τ ρ τ (x)dx + ∂ τ V ′ ρ (τ ; ρ τ ) • ρ τ + V ′ ρ (τ ; ρ τ ) • [∇ • ∇ • (µ τ ρ τ ) -∇ • (u τ ρ τ )] (4.15)
By integrating by parts the last term, Proposition 1 is proved.

REMARK 1. -Notice that (4.14) and (4.8) implies :

R d V ′ |T ρT dx = V (T, ρT ) = R d G + g R d ∂ ξ Gρ T dx ρT dx, ξ = R d g(x, ρT )ρ T dx (4.16) REMARK 2. -By taking ρ τ = δ(x -x 0 ) the usual HJB principle is found if h = g = 0. PROPOSITION 5. -(Hamilton-Jacobi-Bellman equation) When V d = R d , at the optimal solution û ∇ u H + ∇ u H ′ • ρτ + ∇ x V ′ + ∂ u µ : ∇ x ∇ x V ′ = 0 (4.17) R d (H + H ′ • ρτ + ∂ τ V ′ + μ : ∇ x ∇ x V ′ + û • ∇ x V ′ )ρ τ dx = 0 (4.18) REMARK 3. -When H = H(x, t, u(x, t), ρ t (x), χ(t)) with χ(t) = R d h(x, t, u(x, t), ρ t (x))ρ t (x)dx, H ′ ρ (x, τ, u(x, τ ); ρ τ ) • ρ τ = ρ τ ∂ ρ H + R d ∂ χ Hρ τ dx (h + ρ τ ∂ ρ h). ( 4
v = M v = m Figure 1 -The control is v = M when x < x 0 , affine when x 0 < v < x 1 , and v = m when x > x 1 with x 0 = -M σ 2 b + ( 1 2γ + E T x)e -a(T -t) , x 1 = -mσ 2 b + ( 1 2γ + E T x)e -a(T -t)
Then for the optimal û and ρ (4.18) yields

∂ τ V ′ + μ : ∇ x ∇ x V ′ + û • ∇ x V ′ = -H + ρ∂ ρ H + R d ∂ χ H ρdx (h + ρ∂ ρ h) . ( 4

.20)

The link with Section 3. is established: (3.5) and (4.20)coincide with V ′ = ρ * .

Portfolio Optimization

Following [START_REF] Pironneau | A Maximum Principle for SDEs of Mean-Field Type[END_REF], a portfolio of value x t made of a risky asset and a riskless one is optimally managed at t if the quantity v invested at t in the risky asset minimizes, with ρ |0 given,

J = 1 2 Ω (γx 2 -x)ρ |T dx - γ 2 Ω xρ |T dx 2 , ∂ t ρ + ∂ x [(ax + bv)ρ] -∂ xx [ σ 2 v 2 2 ρ] = 0 (5.21)
where a is the interest rate b is a minus the drift of the risky asset and σ is its volatility. We assume that v is a feedback function x, t → v(x, t) there are bounds on v, at each time m ≤ v ≤ M . Thus d = 1 and

H = 0, h = 0, G = 1 2 (γx 2 -x -γx Ω xρ |T dx), u = ax + bv, µ = σ 2 v 2 2 (5.22)
The problem deviates slightly from framework (2.3) but the methodology is the same and gives:

σ 2 v 2 2 ∂Figure 2 -

 22 Figure 2 -Left: x, t → ρ(x, t); x ∈ (-15, 15) is horizontal, t ∈ (0, 2) is from front to back with origin x = -15, t = 0 on the lower left corner. Right: x, t → g(x, t).

Figure 3 -

 3 Figure 3 -ρ * (left) and ρ * (right) versus x (horizontal) and t (vertical).

Figure 4 -

 4 Figure 4 -Iteration history: values of J (top curve) and grad u J| 2 versus iteration count.

In turn, it implies q + q(2a + 2bA + A 2 σ 2 ) = 0, q(T ) = γ 2 , s(T ) = 0, r(T ) = -1 2 -γ R xρ |T and ṙ + (a + bA)r + 2qBb + 2σ 2 ABq = 0, ṡ + rBb + σ 2 B 2 q = 0, s(T ) = 0, b∂ x ρ * + σ 2 v∂ xx ρ * = 2(b + σ 2 A)qx + br + 2σ 2 Bq (5.25)

dτ is required for this regime.

-The advantage here compared with [START_REF] Pironneau | A Maximum Principle for SDEs of Mean-Field Type[END_REF] is that we do not need to guess the shape of the control nor of the adjoint state, once it is assumed polynomial. The analysis also handles constraints.

Numerical Solution of a Systemic Risk Problem

In [START_REF] Garnier | Large deviations for a mean field model of systemic risk[END_REF] it is shown that the rare event probability that the state of a system of N banks, depending on the mean situation of all, transits from a stable situation ρ 0 to a critical one ρ T at time T is given finding the minimum in g of J with

, this is also

Now we notice that ρ = ρ + t T (ρ T -ρ|T ) satisfies the conditions at 0 and T and the PDE with

This means that the problem is in the form analyzed above with state variable {ρ, ρ} and control ũ; naturally the adjoint state has also two components: {ρ * , ρ * }.

Based on the variation of J with respect to u we have used 100 iterations of a gradient method with fixed step size, ω = 0.3. The parameters of the problem are T = 2, h = 0.1, θ = 1, σ 2 = 2 3 θ and ρ 0 , ρ T are as in [START_REF] Garnier | Large deviations for a mean field model of systemic risk[END_REF]. The numerical method for the PDEs is a centered space-time finite element method of degree 1 on a mesh of 150 points and 40 time steps.


Philippe Angot
email: philippe.angot@univ-amu.fr

Thomas Auphan
email: [thomas.auphan@univ-amu.fr

Olivier Guès
email: olivier.gues]@univ-amu.fr

Asymptotic-preserving methods for an anisotropic model of electrical potential in a tokamak

Keywords: Evolution problem, nonlinear anisotropic model, asymptotic-preserving method, numerical tests MSC2010: 00B25, 41A60, 65M30

come

Introduction

The fusion reaction can be performed using a tokamak, a machine whose shape is toroidal. The plasma is confined and warmed in the core of the tokamak to produce the fusion reaction. This technique is expected to maintain the fusion reaction during a long time (more than five minutes, for the ITER project).

One of the main challenges for this objective is to control the wall-plasma interactions. Indeed, the magnetic confinement is not perfect and the plasma is in contact with the wall. In a tokamak such as TORE SUPRA, an obstacle called the limiter, is settled at the bottom of the machine. Due to the strong magnetic confinement, the plasma transport essentially occurs along the magnetic field lines. Thus, the parallel resistivity η is very small (typically, η = 10 -6 ), generating a strong anisotropy in the model. The area where the magnetic lines are interrupted by the limiter is called the scrape-off layer. The numerical simulation of the edge plasma transport allows us to better understand the interactions with the wall.

Anisotropic model of the electrical potential

In this paper, we focus on a 2D model of the electrical potential of the edge plasma φ η in a tokamak with a limiter configuration. A schematic representation of the domain is given in Fig. 1. The x axis corresponds to the curvilinear coordinates along a magnetic field line and the y axis is the radial direction. In the following equations, the curvature terms have been neglected. As the magnetic field lines above the limiter set are closed, periodic boundary conditions are imposed at x = ±0.5. The dimensionless problem for the electrical potential reads:

Limiter Limiter Periodic BC Periodic BC -0.5 0 1 -L L 0.5 Σ Σ l Σ Σ x y Plasma Ω Wall Center Fig. 1 Schematic representation of the 2D domain.                          -∂ t ∂ 2 y φ η - 1 η ∂ 2 x φ η + ν∂ 4 y φ η = S in ]0, T [×Ω ∂ y φ η|t=0 = ∂ y φ ini in Ω ∂ y φ η|Σ = 0 and ∂ 3 y φ η|Σ = 0 on ]0, T [×Σ ∂ x φ η|x=-L = η 1 -e Λ -φ η|x=-L on ]0, T [×]0, l[×{-L} ∂ x φ η|x=L = -η 1 -e Λ -φ η|x=L on ]0, T [×]0, l[×{L}, (1)
where ν corresponds to the ionic viscosity in the perpendicular direction and Λ stands for the reference potential inside the limiter. The initial condition is ∂ y φ η|t=0 = ∂ y φ ini . Negulescu et al. [START_REF] Negulescu | Existence and uniqueness of the electric potential profile in the edge of tokamak plasmas when constrained by the plasma-wall boundary physics[END_REF] proved that, for a fixed value of η > 0, the problem (1) admits a unique weak solution, under suitable hypotheses on the data φ ini and S.

The boundary conditions at the limiter interface x = ±L, are nonlinear. Setting directly η = 0 in the system (1) (after multiplying the first equation by η) leads to an under-determined problem since there are only homogeneous Neumann boundary conditions at the limiter surface x = ±L. Thus, when η is small the numerical resolution of the problem (1) becomes stiff. This issue can be avoided by reformulating the problem (1) thanks to asymptotic-preserving methods.

The micro macro asymptotic-preserving method

We study the Asymptotic-Preserving (AP) method introduced by Degond et al. [START_REF] Degond | An asymptotic-preserving method for highly anisotropic elliptic equations based on a micromacro decomposition[END_REF] for a linear anisotropic elliptic problem. It consists in a decomposition of the solution φ η as φ η = p η + ηq η where ∂ x p η = 0 and q η|x=-L = 0. Then, it yields the problem below where the unknowns are (φ η , q η ):

                                               -∂ t ∂ 2 y φ η -∂ 2 x q η + ν∂ 4 y φ η = S in ]0, T [×Ω ∂ 2 x φ η = η∂ 2 x q η in ]0, T [×Ω ∂ x φ η|x=-L = η∂ x q η|x=-L on ]0, T [×]0, l[×{-L} ∂ x φ η|x=L = η∂ x q η|x=L on ]0, T [×]0, l[×{L} ∂ x φ η|x=-0.5 = η∂ x q η|x=-0.5 on ]0, T [×]l, 1[×{-0.5} ∂ x φ η|x=0.5 = η∂ x q η|x=0.5 on ]0, T [×]l, 1[×{0.5} ∂ y φ η|t=0 = ∂ y φ ini in Ω ∂ y φ η|Σ = 0 and ∂ 3 y φ |Σ = 0 on ]0, T [×Σ ∂ x q η|x=-L = 1 -e Λ -φ η|x=-L on ]0, T [×]0, l[×{-L} ∂ x q η|x=L = -1 -e Λ -φ η|x=L on ]0, T [×]0, l[×{L}, (2)
One important advantage of this AP method is that it can be easily implemented even if the mesh is not aligned with the directions (Ox) and (Oy). The main drawback is the need to compute two unknowns (φ η and q η ) on the 2D domain though only φ η is interesting for the physics.

Let us give the theoretical result which ensures that the modified problem is wellposed for η = 0, and that φ η converges towards φ 0 . First, we provide the definitions of the spaces used for the variational formulation of the problem (2). Definition 1. Let us define the following Hilbert spaces:

• V = f ∈ H 1 (Ω ), ∂ 2 y f ∈ L 2 (Ω ), f periodic on {-0.5, 0.5}×]l, 1[, ∂ y f = 0 on Σ with the scalar product: f , u V = Ω ∂ x f ∂ x u dydx + Ω ∂ 2 y f ∂ 2 y u dydx + 2 l 0 f |x=L u |x=L dy. • Q = f ∈ L 2 (Ω ), ∂ x f ∈ L 2 (Ω ), f |x=-L = 0 on ]0, 1[ , with the scalar product: f , u Q = Ω ∂ x f ∂ x u dydx.
Definition 2. The space A is the set of functions φ such that:

• φ ∈ L 2 (0, T ;V ). • ∂ y φ ∈ L ∞ (0, T ; L 2 (Ω )). • ∂ y φ ∈ L 2 0, T ; { f ∈ H 1 (Ω ), ∂ 2 y f ∈ L 2 (Ω ), f |Σ = 0} . • ∂ 2 y φ ∈ L ∞ (0, T ; L 2 (Ω )). • ∂ t φ ∈ L 2 (0, T ;V ). • ∂ y ∂ t φ ∈ L ∞ (0, T ; L 2 (Ω )).
The weak solution φ η of ( 2) is then searched in the space A . Assumption 3.1 Assume that S and φ ini verify:

1. S, ∂ y S, ∂ 2 y S, ∂ t S, ∂ 2 t S ∈ L 2 (]0, T [×Ω ), S L ∞ (]0,T [×Ω ) ≤ C s and S |t=T L ∞ (Ω ) ≤ C s with C s sufficiently small. 2. φ ini ∈ H 4 (Ω ).
3. φ ini does not depend on x. [START_REF] Negulescu | Existence and uniqueness of the electric potential profile in the edge of tokamak plasmas when constrained by the plasma-wall boundary physics[END_REF].

Ω S |t=0 dydx = ν Ω ∂ 4 y φ ini dydx + 2 l 0 1 -e Λ -φ ini|x=L dy.
The two last hypotheses are compatibility conditions for the initial and boundary conditions with the source term.

We can now write the theorem which asserts the convergence of φ η to φ 0 when η goes to 0: Theorem 1. With the assumption 3.1, the weak formulation of (2): find

(φ η , q η ) ∈ A × L 2 (0, T ; Q) verifying                                        ∀ξ ∈ H 1 (]0, T [), ∀u ∈ V ∩ H 2 (Ω ), ∀w ∈ Q, Ω ∂ y φ η|t=T ∂ y u dydxξ (T ) - T 0 Ω ∂ y φ η|t=T ∂ y u dydx ξ ′ dt + T 0 Ω ∂ x q η ∂ x u dydx ξ dt + ν T 0 Ω ∂ 2 y φ η ∂ 2 y u dydx ξ dt + T 0 l 0 1 -e Λ -φ η|x=-L u |x=-L dy ξ dt + T 0 l 0 1 -e Λ -φ η|x=L u |x=L dy ξ dt = Ω ∂ y φ ini ∂ y u dydx ξ (0) + T 0 Ω S u dydx ξ dt η T 0 Ω ∂ x q η ∂ x w dydx ξ dt = T 0 Ω ∂ x φ η ∂ x w dydx ξ dt,
(3) admits a unique solution. Besides, (φ η , q η ) converges weakly in L 2 (]0, T [×Ω ) 2 , towards (φ 0 , q 0 ) ∈ A × L 2 (0, T ; Q) the solution of (3) when η equals 0.

Finally, the following error estimate holds:

φ η -φ 0 L 1 (0,T ;L 2 (Ω )) ≤ c(T, Ω , φ 0 , S,Λ ) √ η,
where c(T, Ω , φ 0 , S,Λ ) > 0 does not depend on η.

Theorem 1 provides an error estimate for the norm in L 1 (0, T ; L 2 (Ω )), but not for the L 2 (]0, T [×Ω ) norm. This point can be subject to further improvements. This result is shown in [START_REF] Angot | Analysis of asymptotic preserving methods for nonlinear anisotropic models of electrical potential in plasma[END_REF][START_REF] Auphan | Analyse de modèles pour ITER ; Traitement des conditions aux limites de systèmes modélisant le plasma de bord dans un tokamak[END_REF]. The proof of the existence and uniqueness of φ 0 follows the same steps of [START_REF] Negulescu | Existence and uniqueness of the electric potential profile in the edge of tokamak plasmas when constrained by the plasma-wall boundary physics[END_REF], based on a fixed point method. The existence and uniqueness of q 0 and the convergence of (φ η , q η ) when η goes to 0 are shown by extending to a nonlinear case the proof provided in [START_REF] Degond | An asymptotic-preserving method for highly anisotropic elliptic equations based on a micromacro decomposition[END_REF] for a linear elliptic problem.

Numerical experiments

In this section, some numerical tests are presented for the system (2). The space discretisation is done by the centred finite difference scheme. The time resolution uses Euler semi-implicit method.

At first glance, a directional splitting method seems to be interesting. But, the discrete problems obtained in the directions x and y are not invertible. The problem is thus discretised implicitly, except for the nonlinear term. At each time step, a linear system has to be solved to compute the approximations of φ η and q η .

Let us consider a rectangular mesh of the space domain Ω with a constant mesh step δ x (for the direction (Ox)) and δ y (for the direction (Oy)). The time step writes δt. The scalar quantities φ n i, j , q n i, j stands respectively for the approximations of φ η (nδt, -0.5 + iδ x, jδ y) and q η (nδt, -0.5 + iδ x, jδ y). The boundary condition at x = -L is discretised as:

q n+1 I 1 +1, j -q n+1 I 1 -1, j 2δ x -φ n+1 I 1 , j = 1 -e Λ -φ n I 1 , j -φ n I 1 , j ,
where I 1 is the index such that -0.5 + I 1 δ x = -L.

For the boundary condition at x = L, the same technique is used. This time linearisation enables us to have an invertible matrix which is the same at each time step.

The mesh convergence test is performed using a configuration where the limiter goes up to the top of the computational domain, i.e. l = 1. This does not change the results proven for l < 1. For L = 0.4, the chosen manufacturated solution is

φ η (t, x, y) = η t π 2 cos(πy) cos(1.25πx) -ln 1 - 1.25t 2 π cos(πy) + Λ . (4)
Let us note that the source term S associated to the manufactured solution (4) depends on η but is not singular when η goes to 0. This differs from the hypotheses made for Theorem 1.

The plot of the approximated solution is shown in Fig. 2. Studying the L 2 error in Fig. 3, we observe that the numerical scheme is of second-order accuracy in space.

In Fig. 4, we observe that the condition number obtained with the AP method is high but it is bounded independently from η. This is not the case for the matrix obtained for the resolution of (1) without the asymptotic-preserving method. In order to avoid the issues due to the bad conditioning, we choose a LU method to solve the linear problem at each time step, which is faster than a GMRES solver with PETSc library. Finding an efficient preconditioner in order to use iterative methods is a future enhancement of this work.

For the convergence when η tends to 0, the same domain is considered (l = 1, L = 0.4) but another source term is chosen :

S(t, x, y) = 40t cos(2π y) sin π 2L x , φ ini (x, y) = Λ = 0 (5)
This configuration (5) with l = 1 leads to φ 0 (t, x, y) = 0, which enables us to compute numerically φ η -φ 0 L 1 (0,T ;L 2 (Ω )) and φ η -φ 0 L 2 (0,T ;L 2 (Ω )) . For these two norms, we observe a convergence in O(η), see Fig. 5. This suggests that the estimate of Theorem 1 might be improved.

Conclusion

The high anisotropy of the 2D model for the edge plasma electrical potential in a tokamak leads to an ill-conditioned matrix for the numerical approximation using classical methods. The micro-macro decomposition induced by Degond et al. [START_REF] Degond | An asymptotic-preserving method for highly anisotropic elliptic equations based on a micromacro decomposition[END_REF] for a linear anisotropic elliptic problem is studied and analysed for the nonlinear evolution problem of the electrical potential. This method yields a weak formulation which is not degenerated when the parallel resistivity η tends to 0. Moreover, we have the estimate φ η -φ 0 L 1 (0,T,L 2 (Ω )) = O ( √ η) , which can probably be improved, as suggested by the numerical results.

Fig. 2

 2 Fig.2Approximate fields of φ η and q η for δ x = δ y = 0.003125, δt = 0.0001 and η = 0.001. The reference solution is given by (4). Recall that the limiter area corresponds to x ≤ -0.4 and x ≥ 0.4: the values of φ η do not have any physical sense in this zone.

Fig. 4

 4 Fig. 3 φ approx η -φ η L 2 (Ω ) at t = 1 as a function of the space step δ x = δ y for different values of the time step and η = 0.001. The reference solution is given by (4).

Acknowledgements: This work has been funded by the ANR ESPOIR (Edge Simulation of the Physics Of ITER Relevant turbulent transport)and the Fédération nationale de Recherche Fusion par Confinement Magnétique (FR-FCM). We thank Eric Serre, Frédéric Schwander, Guillaume Chiavassa, Philippe Ghendrih and Patrick Tamain for fruitful discussions.


Zhaowu Zhan

Guillaume Villemaud

Florin Hutu

Jean-Marie Gorce

Full-Duplex Dual-Band Radio Dedicated to Flexible Radio Communications

Keywords: Full-Duplex wireless, Dual-Band radio, flxible radio communications radio Full-Duplex, radio bi-bande, communications radio flexibles

This report presents the design of Full-Duplex Dual-Band radio which allows simultaneous transmission and reception in two separate spectrum fragments by combining the Full-Duplex radios and Dual-Band RF front-end. Our design employs two antennas, each one being used for simultaneous transmitting and receiving. To overcome the self-interference (SI) induced by the Full-Duplex radios, simple band pass filters (BPF) are utilized to suppress the inner SI and active analog self-interference cancellation (AASIC) method is used to cancel the inter SI. The two radio frequency (RF) signals corrupted individually by the residual SI are down-converted by the double I/Q structure. Then, the two output baseband signals are processed by the corresponding baseband receivers. We theoretically analyze and evaluate by simulation this Full-Duplex Dual-Band radio operating on the standard WiFi OFDM PHY in two separate spectrum fragments.

Legacy Radio Terminal

Many applications can benefit from the capability of simultaneously transmit and receive on each separate spectrum fragment with one common RF front-end. By this capability, the radio terminal can simultaneously transmit, simultaneously receive or simultaneously transmit and receive on both of the two separate spectrum fragments. For instance, one Full-Duplex Dual-Band radio terminal can communicate simultaneously with two radio terminals as shown in Fig. 1. With respect to the Full-Duplex Dual-Band AP (Access Point: Femto Base Station or WiFi Router), it can accommodate two different clients simultaneously on two different spectrum channels operating on in-band Full-Duplex model. That means the Full-Duplex Dual-Band radio enhance the capacity of an AP by four times compared to the conventional AP which is restricted to the use of only one channel at a time operating on Half-Duplex model. Similarly, the radio terminal such as a mobile phone equipped with the Full-Duplex Dual-Band radio can communicate with the Femto base-station for voice transmission and reception while connecting with laptop for media or files downloading and uploading. Therefore, Full-Duplex Dual-Band radio could provide the users much more flexible radio connection and higher aggregate throughput than the current radio terminals operating on Half-Duplex model in a single spectrum channel at a time.

Such a capability could also be exploited to enhance the radio sharing and coexistence. Portable radio terminals such as smartphones have to accommodate a growing list of wireless standards such as WiFi, Zigbee, Bluetooth, 3G and 4G. Current practical implementation is to use separate radio and antenna to meet the different standard requirement, which is costive and also put high requirements on the space of portable consumer devices. Instead, Full-Duplex Dual-Band radio could enable the radio terminal to accommodate two different wireless standards with one common RF front-end.

In this report, we first studied and demonstrated the Full-Duplex Dual-Band radio via operating on the standard WiFi OFDM PHY in two separate spectrum fragments. The studied radio architecture can be used to enhance the current WiFi system but it is not just limited to that. For example, the radio can operate on the two standards of WiFi and UMTS, WiFi and Zigbee, or 3G and WiFi, and so on.

The rest of the report is organized as follows: Section 2 presents the details of the proposed Full-Duplex Dual-Band OFDM radio transceiver, including Full-Duplex radios and Dual-Band RF front-end. In Section 3, we calculate the power of residual SI to qualify the impact of residual SI caused by Full-Duplex radio on the proposed radio system. Then, the simulation results and discussion are given in Section 4. Finally, the conclusion is drawn in Section 5.

2 Full-Duplex Dual-Band Radio Transceiver In this Section, we first present the general block diagram of the Full-Duplex Dual-Band radio system as shown in Fig. 2 and then the details of the radio transceiver as shown in Fig. 3. As we can see, this radio node employs two antennas which are compatible with future multi-antenna systems. Each antenna is used for simultaneous transmission and reception via a circulator [START_REF] Bharadia | Full Duplex Radios[END_REF] [5] separating the input and output signals, which builds the one-antenna Full-Duplex. From the perspective of both of the two antennas, they are utilized for simultaneously transmitting and receiving, which builds the two-antenna Full-Duplex. Therefore, each antenna of the radio node-B (local radio node) will not only receive the RF signals from the radio node-A (distant radio node) but also the RF SI signal from its own transmission. Then, the signals received by the i th antenna of the local radio node plus the SI signal leaked from the circulator can be represented by

1 f 2 1 f f  2 f 2 1 f f  Full-Duplex Dual-Band RF Front-End Baseband Transceivers Full-Duplex Dual-Band RF Front-End 1 f 2 1 f f  2 f 2 1 f f  Baseband Transceivers Full-Duplex Dual-band Radio Node -A Full-Duplex
S rf,i (t) = S rf,ij (t) + S rf,ii (t) + S rf,siij (t) + aS rf,siii (t), {i, j} ∈ A (1)
where S rf,ij (t) and S rf,ii (t) are the RF signals emitted by the j th antenna and i th antenna of the distant radio node respectively and received by the i th antenna of the local radio node, S rf,siij denotes the inter RF SI signal transmitted from the j th antenna and received by the i th antenna of the local radio node, S rf,siii is the inner RF SI signal transmitted and received by the same antenna and circulator, a being the inner-SI factor including multi-path reflection factor and inner leakage factor of the circulator and

Inria   1 f 2 1 f f  DAC Self-Interference Craft Link QAM Mod. Digital Baseband Transmit System 2 f RF Adder 2 f 2 1 f f  Self-Interference Craft Link RF Adder
SRF(t) Yrf,1(t) Yrf,2(t) Srf,sic21(t) Srf,sic12(t) Srf,1(t) Srf,2(t) 1 f 2 f Channel Estimation Channel Estimation Digital Baseband Receive System Digital Baseband Receive System Output Bits1 Output Bits2 Xsi,1[k] Xsi,2[k] Full-Duplex Dual-Band RF Front-End
A ∆ = {i ∈ [1, 2], j ∈ [1, 2], i + j = 3}.

Full-Duplex Radios

Although the Full-Duplex can improve spectral efficiency and also provide the benefits beyond radio link capacity, it brings to the proposed radio system a strong SI which could saturate the radio receivers. From (1), it can be seen that the receptions of the desired signals from the distant radio node are challenged by the strong inner SI caused by the one-antenna Full-Duplex and inter SI induced by the two-antenna Full-Duplex. These SIs could be hundreds of thousands times of the desired signal. Therefore, suppressing or reducing the strength of the SI signal to a tolerable level is the key point for designing a real Full-Duplex Dual-Band radio. In the proposed radio transceiver, the SI includes inner SI and inter SI. In order to mitigate these two types of SI, we implement two self-interference cancellations (SIC) as shown in Fig. 3. For suppressing the inner SI, we utilize normal BPFs to build the first SIC. While with respect to inter SI, the active analog SIC (AASIC) method [START_REF] Zhaowu Zhan | Design and Evaluation of a Wideband Full-Duplex OFDM System Based on AASIC[END_REF] is used to implement the second SIC.

The First Self-Interference Cancellation

The first SIC is to carry out the suppression of inner SI induced by the one-antenna Full-Duplex. In fact, the inner SI aS rf,siii includes the reflection-path SI transmitted and received by the same

iFFT Modulation DAC Tx RF Front-End Rx RF Front-End Rx Combiner iFFT Modulation DAC Tx RF Front-End ADC FFT Modulation Self-Interference Channel Estimation Tx Self-Interference Craft Link ] [k X i ] [ ~, k H i si ] [ ] [ , k H k X i si i
Figure 4: The second self-interference cancellation model antenna and the SI leaked from the imperfect circulator. BPF-i (perfect BPF is assumed) with center frequency f j is used here to implement the first SIC which eliminates the inner SI signal aS rf,siii (t) and also filters out the distant radio signal S rf,ii (t) with carrier frequency f i . That means the first SIC eliminates completely the SI caused by the one-antenna Full-Duplex. Then, the RF signals with carrier frequency f j including signal of interest S rf,ij (t) and the over-the-air SI signal S rf,siij (t) can be represented by

Y rf,i (t) = S rf,ij (t) + S rf,siij (t), {i, j} ∈ A (2)

The Second Self-Interference Cancellation

After eliminating the inner SI, the inter SI induced by the two-antenna Full-Duplex is still strong enough to saturate the radio receiver. In order to cancel the inter SI signal S rf,siij (t) in (2), [START_REF] Jain | Practical Real-Time Full Duplex Wireless[END_REF] [7] proposed active SI cancellation (ASIC) and [START_REF] Everett | Passive Self-Interference Suppression for Full-Duplex Infrastructure Nodes[END_REF] focus on passive SI suppression (PSIS). As it has been introduced in Section I, we first study the Full-Duplex Dual-Band radio operating on the standard WiFi OFDM PHY. Therefore, we can simplify the two-antenna Full-Duplex model and SIC model as shown in Fig. 4. We craft the RF cancellation signal in another radio link as presented in [START_REF] Zhaowu Zhan | Design and Evaluation of a Wideband Full-Duplex OFDM System Based on AASIC[END_REF] and implement the SIC at the RF stage. Then, the SIC model used in this report can be expressed as

r RF,rsi,j (t) = P si L si IDFT {X i [k]} e j2πfit * h si,i (t) -P ′ si L ′ si IDFT X i [k] H si,i [k] e j2πfit (3)
where r RF,rsi,j (t) denotes the received residual RF SI, from the i th antenna of the local radio node, in the j th receiver chain, P si and P ′ si represent the transmit power of the transmitter and the cancellation link respectively, L si and L ′ si denote the path loss of the SI channel and the cancellation link respectively, X i [k] is the symbol to be transmitted from the i th antenna, h si,i (t) denotes the impulse response of the wireless SI channel from the i th antenna to the j th antenna, H si,i [k] represents the estimated coefficients in the frequency domain of the h si,i (t), f i denotes the carrier frequency of the i th transmitter chain and * represents time domain convolution.

IDFT{χ m [k]} ∆ = 1 Tu N 2 k=-N 2 χ m [k]e j2π∆f kt ,
in which T u is the duration of one OFDM symbol, N denotes the total number of sub-carriers and ∆f denotes the sub-carrier space. After the second SIC, the remaining received signal in the i th receiver chain can be expressed as

S RF,i (t) = S rf,ij (t) + r RF,rsi,i (t) (4)
Therefore, the RF signal S RF,i (t) to be input to the Dual-Band radio consists of the desired signal from the distant radio node and the residual inter SI which can be regarded as another receiver noise superposed on the desired signal. Taking the received RF signal S RF,1 (t) in the 1 st receiver chain as an example, this RF signal includes the desired signal S rf,12 (t) with carrier frequency f 2 from the 2 nd antenna of distant radio node and the residual SI r RF,rsi,1 (t) with the same carrier frequency as S rf,12 (t). In order to qualify the impact of the residual SI on the reception of the signal of interest, we will calculate the power of the residual SI in Section III.

Dual-Band RF Front-End

After the two SICs mitigating the SI induced by the Full-Duplex radios, the two received useful RF signals superposed by the residual SI will be down-converted to baseband signal by using Dual-Band radio with double I/Q structure. The received RF signal S RF,i (t) is with carrier frequency f j , so we define S ′ RF,1 (t) ∆ = S RF,2 (t) and S ′ RF,2 (t) ∆ = S RF,1 (t). Then, the Dual-Band RF front-end is used to separate the RF signals S ′ RF,1 (t) with carrier frequency f 1 and S ′ RF,2 (t) with carrier frequency f 2 . In order to facilitate the theoretical study of the Dual-Band RF frontend, we use

s i = I i + Q i , {i ∈ [1, 2],  2 =
-1} denote the equivalent baseband complex symbols carried by the frequency f i . Then, the received RF signal-1 S ′ RF,1 (t) and RF signal-2 S ′ RF,2 (t) can be expressed as

S ′ RF,1 (t) = I 1 (t)cos(2πf 1 t) -Q 1 (t)sin(2πf 1 t) S ′ RF,2 (t) = I 2 (t)cos(2πf 2 t) -Q 2 (t)sin(2πf 2 t) (5)
Note that the signal model in ( 5) is reasonably valid even in a noisy environment, because I i (t) can represent the sum of the real part of the data symbol and noise (including receiver thermal noise and residual SI) and Q i (t) can represent the sum of the image part of the data symbol and noise (including receiver thermal noise and residual SI) respectively.

As shown in Fig. 5, after the received RF signal S RF (t) = S ′ RF,1 (t) + S ′ RF,2 (t) going through the first frequency translation stage with carrier frequency f LO = f1+f2 2 , and the pass band signals are filtered out. Then, the I/Q branch of intermediate frequency (IF) signals can be obtained as

S IF,I (t) = I 1 (t) + I 2 (t) 2 cos(2πf IF t) + -Q 1 (t) + Q 2 (t) 2 sin(2πf IF t) (6)
S IF,Q (t) = I 1 (t) -I 2 (t) 2 sin(2πf IF t) + Q 1 (t) + Q 2 (t) 2 cos(2πf IF t) (7)
where

f IF = f1-f2 2 = f 1 -f LO = f LO -f 2 .
s IQ (t) = Q 1 (t) -Q 2 (t) 4 ; s QI (t) = Q 1 (t) + Q 2 (t) 4 ; s QQ (t) = -I 1 (t) + I 2 (t) 4 . (8)
After the analog-to-digital converter (ADC) and digital-image-rejection, the received baseband complex symbols s a and s b corresponding to the symbols s 1 and s 2 which include the baseband symbols from the distant radio node are given by

s a = (s II -s QQ ) + (s QI + s IQ ) = s 1 2 ( 9
)
s b = (s II + s QQ ) + (s QI -s IQ ) = s 2 2 (10)
Therefore, the two RF signals S ′ RF,1 (t) and S ′ RF,2 (t) are down-converted and separated completely by using the Dual-Band RF front-end.

From the above studies, it can be seen that the Full-Duplex Dual-Band radio could downconvert and separate two different types of RF signals which are in two separate spectrum fragments. The two output baseband symbols s a and s b including the baseband symbols from the distant radio node and additive noise (receiver thermal noise and residual SI) are further processed by the corresponding baseband receivers.

Residual Self-Interference

In the report, the proposed Full-Duplex Dual-Band radio operates on the standard baseband WiFi OFDM PHY. Therefore, the available AASIC [START_REF] Zhaowu Zhan | Design and Evaluation of a Wideband Full-Duplex OFDM System Based on AASIC[END_REF] method which is suitable for the OFDM SIC is used to cancel the inter SI induced by the two-antenna Full-Duplex as shown in Fig. 4. The radio node knows the transmit symbols

X i [k] with E{|X i [k]| 2 } = 1,
so it can implement the SIC as described in (3) if it can obtain the channel state information (CSI) of the over-the-air SI channel. In the standard WiFi, the long training sequence consisting of two known OFDM symbols in the preamble is used for channel estimation. In order to avoid the impact of the signal from the distant radio node on the SI channel estimation, we redesign the data frame in order that the long training symbols from different radio nodes take different time slots.

Inria

Perfect frequency and timing synchronization and no RF impairment are assumed. Therefore, at the receiver side, after the down-conversion of the RF signal received and removal of the cyclic prefix, the baseband frequency domain signal can be obtained by FFT of the time domain baseband signal. During the time of SI channel estimation, each radio node just receives the signal it transmits. The demodulated long training OFDM symbol in the i th receiver chain can be expressed as:

R si,i [k] = P si L si T j [k]H si,j [k] + Z[k], {i, j} ∈ A (11)
where

T j [k], k ∈ [1, N nz ] with E{|T j [k]| 2 } = 1 denotes the known BPSK symbol sequences, H si,j [k]
is the SI channel from the j th antenna to the i th antenna and Z m [k] represents the frequency domain receiver thermal noise following the normal distribution N (0, δ 2 n ). The CSI of the SI channel is estimated via dividing the demodulated long training symbols received by the pilot pattern with the known BPSK symbol sequences. Therefore, the estimated channel transfer function after normalization is

H si,j [k] = R si,i [k] √ P si √ L si T j [k] = H si,j [k] + Z m [k]/( P si L si T j [k]) (12)
The second term of the right side of the ( 12) is the channel estimation error of the k th subcarrier channel due to the receiver thermal noise.

With the estimated CSI of the SI channel available, the SIC is carried out as (3). After the interference cancellation, the residual baseband SI signal r BB,rsi,j (t) can be represented by (13) when the mixers in the RF front-end are perfect.

r BB,rsi,j (t) = P si L si IDFT {X i [k]} * h si,i (t) -P ′ si L ′ si IDFT X i [k] H si,i [k] = IDFT P si L si X i [k]H si,i [k] -IDFT P ′ si L ′ si X i [k] H si,i [k] (13)
According to the Parseval's theorem [START_REF] Glenn | Advanced Digital Signal Processing: Theory and Applications[END_REF], the power of the time domain signal can be calculated in the frequency domain. The frequency domain residual SI can be represented by

R BB,rsi,j [k] = P si L si X i [k]H si,i [k] -P ′ si L ′ si X i [k] H si,i [k] (14) When √ L si √ P si = L ′ si P ′
si and after replacing the H si,i [k] in ( 14) by ( 12), the residual SI can be expressed as

R BB,rsi,j [k] = - X i [k] T i [k] Z ′ m [k] (15)
Because of the independence between X i [k] and T i [k] in (15), the power of the residual SI in the j th receiver chain is

P rsi,j = E[R BB,rsi,j [k] • R * BB,rsi,j [k]] = δ 2 n ( 16
)
That means we can reduce the SI to noise level and the residual SI could be regarded as another receiver thermal noise.

Simulation Results and Discussion

Simulation Results

In this report, we take the Full-Duplex Dual-Band radio operating on WiFi OFDM PHY in two separate spectrum fragments as the first study case. The two over-the-air SI channels are assumed as single path one delay channel which rely on the fact that the strength of the direct path dominates the power of the SI channel [START_REF] Everett | Passive Self-Interference Suppression for Full-Duplex Infrastructure Nodes[END_REF].

The system parameters are chosen as in Table 1 according to the IEEE 802.11g WLAN standard. We evaluate and demonstrate the Full-Duplex Dual-Band based radio system by employing the co-simulation of ADS (Advanced Design System, Agilent Tech.) and Matlab. The RF frontend and most of the baseband processing system are modeled by ADS and part of the digital signal processing are carried out by Matlab.

The Bit Error Rate (BER) is used to evaluate and compare the performance of IEEE 802.11g, Dual-Band IEEE 802.11g and Full-Duplex Dual-Band IEEE 802.11g as shown in Fig. 6. The BER curves of Dual-Band WiFi-1 and WiFi-2 overlap with that of conventional WiFi system when the Dual-Band RF front-end is ideal, i.e. without any RF impairment, which can be explained and demonstrated by Section II B. While with respect to the Full-Duplex Dual-Band WiFi system, there is around 3 dB gap to the conventional WiFi system. This 3 dB gap in each of the two radio branches is caused by the residual SI due to the thermal noise as presented in Section III. This implies that the proposed radio could simultaneously receive two different signals in two separate spectrum fragment at the cost of tolerable SINR loss. When the radio operating on the standard WiFi OFDM PHY, the SINR loss is just 3 dB.

Discussion

Full-Duplex is defined as a radio which can simultaneously transmit and receive in the same frequency band, which capability can enable the radio terminal to carry out spectrum channel sensing while working [START_REF] Choi | Simultaneous Transmission and Reception: Algorithm, Design and System Level Performance[END_REF]. Dual-Band RF front-end enable the radio systems to simultaneously process two different types of signals. Such a capacity could provide the radio terminal a candidate spectrum channel for immediate channel handover. Therefore, the potential capability of carrying out the spectrum channel sensing and handover could render the proposed Full-Duplex Dual-Band radio a viable candidate radio technique for the cognitive radios.

Conclusion

Spectrum fragmentation will be compounded in 5G driven by the scarcity of spectrum [START_REF] Hong | Applications of Self-Interference Cancellation in 5G and Beyond[END_REF], which requires flexible radio transceiver to support flexible utilization of spectrum fragment in the future wireless network. In this report, a flexible radio: Full-Duplex Dual-Band radio was

FullFigure 1 :

 1 Figure 1: The second self-interference cancellation model

Figure 2 :

 2 Figure 2: Block diagram of the Full-Duplex Dual-Band wireless radios

Figure 3 :

 3 Figure 3: Full-Duplex Dual-Band radio transceiver

Figure 5 :

 5 Figure 5: Architecture of Dual-Band RF front-end

Inria

Figure 6 : 3 2 7 3

 637 Figure 6: The BER of different radio transceiver with different Eb/No

  The baseband signals s II (t), s IQ (t), s QI (t) and s QQ (t) obtained by down-converting IF signals S IF,I (t) and S IF,Q (t) with frequency f IF can be expressed as

	s II (t) =	I 1 (t) + I 2 (t) 4	;

Table 1 :

 1

	System Parameters
	Parameter	Value
	Bandwidth	20MHz
	Carrier frequency f 1	2.4GHz
	Carrier frequency f 2	2.0GHz
	Total number of subcarriers	52
	Number of data subcarriers	48
	Number of pilot subcarriers	4
	Bit rate	36Mbps
	IFFT/FFT period	3.2us
	GI duration	0.8us

RR n➦ 8558