Setting: a more precise definition for some r 0 > 0, let W be the polygonal set

W = {x ∈ R 2 : x • e i ≤ r 0 , ∀i = 1, . . . , N } Then Ω \ W = N ∪ i=1 S i
, where S i is the half-strip

S i = x i e i + x ⊥ i e ⊥ i , x i > r 0 , |x ⊥ i | < 1 W Ω S2 S1 S5 O S4 S3
Y. Achdou Dimension reduction Ω ǫ = ǫΩ "tends" to a network G as ǫ → 0 ℓ ǫ y ǫ (t; x, α), α(t) e -λt dt subject to ẏǫ (t; x, α) = α(t), a.a. t > 0, y ǫ (0; x, α) = x, with α : R + → A measurable, under the state constraint

G = {O} ∪ N i=1 J i , J i = {x i e i , x i > 0}.
y ǫ (t) ∈ Ω ǫ , ∀t ≥ 0. Controlability : A is a compact subset of R 2 such that B(0, r) ⊂ A for some r > 0.
The function ℓ ǫ : Ω ǫ → R is bounded and continuous

ℓ ǫ (x, a) =          ℓ i x i -ǫr 0 , x ⊥ i ǫ , a in ǫS i ℓ 0 x ǫ , a in ǫW ∩ Ω ǫ ℓ i : [0, +∞) × [-1, 1] × A → R ℓ 0 : (W ∩ Ω) × A → R ℓ 2 (x 2 -ǫr 0 , x ⊥ 2 ǫ , a) ℓ 1 (x 1 -ǫr 0 , x ⊥ 1 ǫ , a) ℓ 5 (x 5 -ǫr 0 , x ⊥ 5 ǫ , a) ℓ 0 ( x ǫ , a) ℓ 4 (x 4 -ǫr 0 , x ⊥ 4 ǫ , a)
ℓ i and ℓ 0 match properly in order to ensure the continuity of ℓ ǫ

Y. Achdou Dimension reduction

The function ℓ ǫ : Ω ǫ → R is bounded and continuous

ℓ ǫ (x, a) =          ℓ i x i -ǫr 0 , x ⊥ i ǫ , a in ǫS i ℓ 0 x ǫ , a in ǫW ∩ Ω ǫ

Remark

In ǫS i , ℓ ǫ varies slowly w.r.t. x i and fast w.r.t. x ⊥ i .

It is not restrictive to assume that ℓ 0 ≥ 0.

Y. Achdou Dimension reduction

The value function u ǫ u ǫ is bounded uniformly with respect to ǫ, continuous, and is the unique viscosity solution of

λu ǫ (x) + H ǫ (x, Du ǫ ) ≥ 0 in Ω ǫ , λu ǫ (x) + H ǫ (x, Du ǫ ) ≤ 0 in Ω ǫ ,
where

H ǫ (x, p) = max a∈A -p • a -ℓ ǫ (x, a) .

Questions

Asymptotic behavior of u ǫ as ǫ → 0?

Can we define an effective problem in G?

Y. Achdou Dimension reduction

Background and additional new difficulties

Existing results:

Singularly perturbed control problems in thin domains around smooth manifolds: Bensoussan, Alvarez-Bardi, Arztein-Gaitsgory, Gaitsgory-Leisarowitz, Terrone,... Comparison for viscosity solutions on networks: YA-Oudet-Tchou and Imbert-Monneau Very recent results of Lions-Souganidis on homogenization of HJB equations with defects

Several additional new difficuties:

Identification of the effective problem Will the effective problem keep track of ℓ 0 near O? How?

In the perturbed test-function method of Evans, we need to construct correctors in unbounded domains.

Y. Achdou Dimension reduction

Theorem Under a further technical assumption, u ǫ converges locally uniformly to the bounded viscosity solution u :

G → R of      λu(x) + H i (x i , du dx i (x)) = 0 x ∈ J i , λu(O) + max E, H(O, du dx 1 (0), . . . , du dx N (0)) = 0, Y. Achdou Dimension reduction

Theorem

Under a further technical assumption, u ǫ converges locally uniformly to the bounded viscosity solution u :

G → R of      λu(x) + H i (x i , du dx i (x)) = 0 x ∈ J i , λu(O) + max E, H(O, du dx 1 (0), . . . , du dx N (0)) = 0, with H i (x i , p i ) = sup µ∈Z i [-1,1]×A -p i a i -ℓ i (x i , y, a) dµ(y, a)
Z i is a compact and convex set of Radon probability measures on [-1, 1] × A Z i : limiting relaxed controls.

Y. Achdou Dimension reduction

Theorem Under a further technical assumption, u ǫ converges locally uniformly to the bounded viscosity solution u :

G → R of      λu(x) + H i (x i , du dx i (x)) = 0 x ∈ J i , λu(O) + max E, H(O, du dx 1 (0), . . . , du dx N (0)) = 0,
with -E : an effective cost at the junction

Y. Achdou Dimension reduction

Theorem Under a further technical assumption, u ǫ converges locally uniformly to the bounded viscosity solution u :

G → R of      λu(x) + H i (x i , du dx i (x)) = 0 x ∈ J i , λu(O) + max E, H(O, du dx 1 (0), . . . , du dx N (0)) = 0, with H(O, p 1 , . . . , p N ) := max i=1,...,N H + i (0, p i ), H + i (0, p i ) = sup µ∈Z + i [-1,1]×A
-p i a i -ℓ i (0, y, a) dµ(y, a) ,

Z + i = µ ∈ Z i s.t.
[-1,1]×A a i dµ(y, a) ≥ 0 .

Y. Achdou Dimension reduction

Hamilton-Jacobi on the network Hamilton-Jacobi on the network

(1/2) What is the meaning of      λu(x) + H i (x i , du dx i (x)) = 0 x ∈ J i , λu(O) + max E, H(O, du dx 1 (0), . . . , du dx N (0)) = 0 ? (1
(2/2)
An usc function u : G → R is a sub-solution of (1) if for any x ∈ G and φ ∈ R(G) s.t. u -φ has a local maximum in x, then

λu(x) + H i (x, dφ dx i (x)) ≤ 0 if x ∈ J i , λu(O) + max E, H( dφ dx 1 (O), . . . , dφ dx N (O)) ≤ 0.
A lsc function u : G → R is a super-solution of (1) if for any x ∈ G and φ ∈ R(G) s.t. u -φ has a local minimum in x, then

λu(x) + H i (x, dφ dx i (x)) ≥ 0 if x ∈ J i , λu(O) + max E, H( dφ dx 1 (O), . . . , dφ dx N (O)) ≥ 0.

Y. Achdou Dimension reduction

The effective Hamiltonians away from the junctions

Notation: for x ≥ 0, y ∈ [-1, 1], p ∈ R 2 , H i (x, y, p) = max a∈A (-p • a -ℓ i (x, y, a))
Theorem (Alvarez-Bardi 2000)

∀x i ≥ 0, ∀p i ∈ R, there is a unique H i (x i , p i ) s.t. the cell problem H i (x i , y, p i e i + D y χ i (y)e ⊥ i ) ≥ H i (x i , p i ) y ∈ [-1, 1], H i (x i , y, p i e i + D y χ i (y))e ⊥ i ≤ H i (x i , p i ) y ∈ (-1, 1) has a viscosity solution χ i ∈ Lip([-1, 1]).

Y. Achdou Dimension reduction

For s > 0, the occupational measure µ s generated by (y(t), α(t)) is the Radon probability measure defined on [-1, 1] × A by

µ s = 1 s s 0 δ (y(t),α(t)) dt
where δ (y(t),α(t)) is the Dirac mass concentrated at (y(t), α(t)).

Z(s; i, y 0 ): set of the occupational measures generated by the trajectories (y(t), α(t)) up to time s.

Y. Achdou Dimension reduction

For s > 0, the occupational measure µ s generated by (y(t), α(t)) is the Radon probability measure defined on [-1, 1] × A by

µ s = 1 s s 0 δ (y(t),α(t)) dt
where δ (y(t),α(t)) is the Dirac mass concentrated at (y(t), α(t)).

Z(s; i, y 0 ): set of the occupational measures generated by the trajectories (y(t), α(t)) up to time s.

Theorem (Gaitsgory and Leisarowitz (1999))

There exists a set

Z i ⊂ P([-1, 1] × A) s.t ∀y 0 ∈ [-1, 1], lim s→∞ π H (Z(s; i, y 0 ), Z i ) = 0, (π H : Prokhoroff distance).
Z i is convex and compact for the weak-* topology.

Y. Achdou Dimension reduction

By using results from Terrone-2011 and Alvarez-Bardi-2000

Z i ⊂ µ ∈ P([-1, 1] × A), [-1,1]×A a ⊥ i dµ(y, a) = 0
Z i coincides with the set of limiting relaxed controls, i.e.

H i (x i , p i ) = sup µ∈Z i -p i [-1,1]×A a i dµ(y, a) - [-1,1]×A ℓ i (x i , y, a)dµ(y, a)
Interpretation Away from the junction, the effective Hamiltonian corresponds to a control problem with controls in Z i .

Y. Achdou Dimension reduction

The effective Hamiltonian at the junction

Definition

The effective Hamiltonian at the junction is max(E,

H(O, •))
where E is a suitable constant to be defined and

H(O, p 1 , . . . , p N ) := max i=1,...,N H + i (0, p i ) H + i (0, p i ) = sup µ∈Z + i -p i [-1,1]×A a i dµ(y, a) - [-1,1]×A ℓ i (0, y, a)dµ(y, a)
and The graphs of p → H i (0, p) and of p → H

Z + i = µ ∈ Z i s.t. [-1,1]×A a i dµ(y, a) ≥ 0 Y. Achdou Dimension reduction Z 0 i := µ ∈ Z i s.t. [-1,1]×A a i dµ(y, a) = 0 argminH i (0, •) = p i , p i 0 -22 4 -3 -11 3
+ i (0, p) p < p i ⇒ H i (0, p) = H + i (0, p) is achieved for µ ∈ Z + i \Z 0 i p i < p < p i ⇒ H i (0, p) = H + i (0, p) is achieved for µ ∈ Z 0 i p > p i ⇒ H i (0, p) > H + i (0, p) = min H i (0, •) Y. Achdou Dimension reduction
The constant E Zoom near the junction point O

We extend the function ℓ 0 to the whole domain Ω by setting

ℓ 0 (z, a) = ℓ i (0, z ⊥ i , a), if z i ≥ r 0 , |z ⊥ i | ≤ 1.

Ergodic constants in bounded subdomains

Define Ω R := Ω ∩ W R and the ergodic constant E R , which is the unique number s.t.

H 0 (z, Dw R (z)) ≥ E R in Ω R , H 0 (z, Dw R (z)) ≤ E R in Ω R ,
has a viscosity solution w R .

Ω R

Y. Achdou Dimension reduction

The constant E Lemma ∃C > 0 s.t.

|E R | ≤ C ∀R, Dw R ∞ ≤ C ∀R, R ≤ R ′ ⇒ E R ≤ E R ′ .

Lemma

Under the assumptions, ∀i = 1, . . . , N, min

p i H i (0, p i ) ≤ E.
Y. Achdou Dimension reduction

  → R is an admissible test-function if φ is continuous on G for any j ∈ 1, . . . , N , φ| Jj ∈ C 1 (J j ) R(G): set of the admissible test-functions Y. AchdouDimension reduction

  s.t. ∀z 0 ∈ K 0 , ∀α ∈ A z 0 , ∀T > 0, if z(s) := z 0 + s 0 α(θ)dθ satisfies z(T ) ∈ K 0 , then T 0 ℓ 0 (z(s), α(s))ds ≥ -ET -C.

Proof of the convergence result

The main step is to prove the following theorem Theorem Consider the relaxed semilimits, of u ǫ : for all x ∈ G,

Then, under the assumptions, u is a bounded supersolution of

and u is a bounded subsolution of

then

is achieved by i ∈ I(p) ⊂ {1, . . . , N }. If j / ∈ I(p), then the optimal trajectories starting from z ∈ S j should leave S j : this leads to the fact that the corrector should not be bounded when |z| → ∞, z ∈ S j . Hence, we construct the correctors in subdomain obtained by truncating the half-strips S j if j / ∈ I(p).

Y. Achdou Dimension reduction

The corrector (when H(O, p 1 , . . . , p N ) > E)

The truncated domain Ω p

Theorem

For p ∈ R N : H(O, p) > E. Let ψ p be a smooth function such that Dψ p = p i e i in S i . Under the assumptions, ∃χ p , a bounded and Lipschitz viscosity solution of

Y. Achdou Dimension reduction

The further assumption


I Augustyniak

P Knapkiewicz

K Sareła

J Dziuban

E Debourg

P Pons

M Olszacki

Micromechanical high-doses radiation sensor with bossed membrane and interferometry optical read-out

Keywords: MEMS radiation sensor, high doses of radiation, bossed membrane, optical read-out, polyethylyne degradation, membrane deflection;

The silicon-glass MEMS high-doses radiation sensor with in situ detection, so far not possible in the field of detection of doses above 10 kGy, has been presented. The sensor consists of a chamber filled with the high density polyethylene (HDPE) and a silicon bossed membrane. The radiolysis product of HDPE increases the pressure inside the chamber causing the deflection of the membrane, which is proportional to the pressure, thus to radiation dose. The sensor has been irradiated with high energy electron beam and shows good detectability for 10-40 kGy. The deflection of the membrane has been detected by optical interferometer.

Introduction

Measurement of high radiation doses (>10 kGy) by cheap and easy-to-use sensor became urgent and critical in nuclear power plants and other installations where high radiation doses occur [1]. The other issue is the unsolved problem of the in situ measurements of a high-dose radiation levels [START_REF] Ravotti | Radiation Monitoring in Mixed Environments at CERN: From the IRRAD6 Facility to the LHC Experiments[END_REF][START_REF] Camanzi | The race for new radiation monitors[END_REF]. According to the today's state of art, high-doses of radiation (above 10 kGy) may be only estimated post factum by a family of passive thermo-, radio-and photoluminescence indicators or hydrogen pressure dosimeters [START_REF] Van De Voorde | Effect of radiation on materials and components -Megarad dosimetry[END_REF]. In situ measurements have been up-to-date obtained only for low or medium doses (below 10 kGy) by solid-state MOS based sensors. So, the problem of highdose radiation measurements is still open and important. The idea of high-doses MEMS radiation sensor, working in so called destructive mode, has been presented by us in work [START_REF] Augustyniak | MEMS high-doses radiation sensor[END_REF]. In this solution, the pressure generated by H2, being the effect of radiolysis of HDPE trapped inside the small MEMS chamber with thin silicon membrane, causes mechanical destruction of this membrane. Detection of levels of radiation is done by simple observation of the sensor (by naked eye as well). In this paper, we focus on "proportional" mode in which deflection of a membrane is proportional to the hydrogen pressure, thus to a radiation dose. Moreover, the presented sensor is appointed with a bossed membrane which dimensions are adapted for cooperation with a wireless radio frequency detection system [START_REF] Jatlaoui | Working Principle Description of the Wireless Passive EM Transduction Pressure Sensor[END_REF].

Construction and fabrication

As mentioned earlier the sensor utilizes phenomenon of degradation of some polymers exposed to ionizing radiation. Irradiation releases from the polymer gaseous hydrogen. The pressure of hydrogen deflects thin silicon membrane (Fig. 1). Deflection of the membrane is proportional to the irradiation dose. The construction of the sensor contains hermetically sealed chamber filled up with a small portion of the HDPE (~ 2 mg), a silicon bossed membrane (5×5×0.08 mm3), a connecting channel, a joining hole and a channel under the membrane (Fig. 2a). The sensor is made of silicon and glass in a form of anodically bonded sandwich with dimensions of 25×11×1.925 mm3 (Fig. 2b). The fabrication process utilizes wet etching in tetramethylammonium hydroxide (TMAH) in (100) silicon wafer, in order to make a connecting channel, deep wet etching in potassium hydroxide (KOH) to make a bossed membrane, a chamber and a joining hole in the same procedure. The hydrogen fluoride (HF) etching of a glass wafer has been used to make a channel under the membrane. The back side glass and silicon wafers are anodically bonded in 450°C, at 600 V. HDPE is precisely placed inside the chamber and the second anodic bonding of top side glass cover is done in pure nitrogen in 300°C, at 1200 V (Fig. 3). The second bonding is provided in a decreased temperature process and is the key process of the fabrication.

Results

Deflection of membrane induced pressure changed from 0 to approximately 190 kPa has been measured by interferometric method. After that, calculations were done (FEM modeling) to confirm the experimental data. We obtained good agreement here, as shown in the Fig. 4a. Following, true structure of sensors (with HDPE inside) were irradiated by high energy beam of electrons with variable doses (5-40 kGy) and membrane deflection of irradiated sensors, was measured by interferometric method (Fig. 4b).

Additionally, a pressure value existing in the HDPE chamber after irradiation has been measured by indirect method (Fig. 5a). Test structure has been closed inside special chamber with window, precise pressure sensor and regulator.

Compressed air has been precisely applied into the chamber till silicon membrane had become flat. Flat membrane manifest the differential pressure is equal to zero. This non-destructive method has been applied to measure pressure inside tests structures after each of four irradiation sets. This experiment indicated very high pressure of hydrogen existing inside irradiated sensor (Fig. 5b), much higher (about ten times) than those related in the literature [START_REF] Lawton | Gases liberated during the high voltage electron irradiation of polyethylene[END_REF]. Experiments indicated that hydrogen pressure generated under irradiation is high enough to ensure proper wireless interrogation as described in [START_REF] Jatlaoui | Working Principle Description of the Wireless Passive EM Transduction Pressure Sensor[END_REF].

Summary

The silicon-glass MEMS high-doses radiation sensor has been presented. The sensor consists of a chamber filled with HDPE and a silicon bossed membrane. The radiolysis product of HDPE, mostly the hydrogen, increases the pressure inside the chamber causing the deflection of the membrane, which is proportional to the pressure, thus to radiation dose. The sensor has been irradiated with high energy electron beam.

The sensor shows good proportional detection of doses of radiation in the range 10-40 kGy. The pressure inside the chamber of the sensor is unexpectedly high (up to 300 kPa). It shows that radiolysis of HDPE in micro-scaled MEMS sensor is much more effective in comparison to macro-scaled solutions.

To achieve the range of the sensor for higher doses of radiation, it is necessary to design a sensor with a thicker membrane. This solution opens the way toward creation of family of high dose radiation MEMS sensors for in situ, real-time measurements in harsh and dangerous conditions in which no another sensor could be applied.

Fig. 1 :

 1 Fig. 1: The proportional mode of operation of the sensordeflection of membrane is proportional to hydrogen pressure, thus to radiation dose.

Fig. 2 :

 2 Fig. 2: The MEMS high doses radiation sensor: (a) cross-sectional view; (b) sensor at a glance.

Fig. 3 :

 3 Fig. 3: The fabrication of the sensor -main technological steps.

Fig. 4 :

 4 Fig. 4: Sensor response for (5×5×0.08 mm3) membrane: (a) deflection as a function of pressure inside the chamber; (b) deflection as a function of radiation dose.

Fig. 5 :

 5 Fig. 5: Differential method for measurement of pressure inside the sensor after irradiation: (a) scheme of the measurement set-up; (b) pressure inside the sensor as a function of dose of radiation.

Acknowledgements

The presented work was done in the framework of FP7-MNT-ERA.NET-DOSIMEMS project funded by EU via Foundation for Polish Science and by Midi-Pyrénées Region.


Patrick Pons

Hervé Aubert

Manos Tentzeris

Overview of Electromagnetic Transducers with Radar Interrogation for Passive Wireless Sensors Applications

Keywords: wireless sensor, passive sensor, chipless sensor, electromagnetic transduction, radar interrogation

Passive wireless sensors with electromagnetic transduction are chipless wireless sensors that do not require electronic circuits or power. They are then good candidate for buried sensors or harsh environment applications. We validated at LAAS several transducers concepts with high sensitivity (pressure, stress, temperature, radiation) compatible with radar interrogation up to several ten of meters.

I. INTRODUCTION

All the European prospective (ARTEMIS, EPoSS, ENIAC, H2020) show that "Intelligent Cyber-physical Systems" will play a key role in many sectors (Energy, Transport, Security, Telecommunications, Health Care, Environment, ....) in the next decade. Also called "Internet of Object" or "Ambient Intelligence", this issue puts the smartness in the heart of the "Objects" that will be able to interact with their environment and to communicate to bring new features and new services. In this context Wireless Sensors Networks (WSN) appear as a key technology for a lot of applications.

Wireless sensor consists of two separate units (the sensing unit and the reader) that communicate most of the time using RF waves. Like wired sensors, wireless sensors requirements concern their accuracy, selectivity, response time, reliability, ... and also their size and cost. The specificity of the wireless link leads to additional constraints such as energy autonomy, interrogation distance, identification and data rate. All these specifications may be very different because they are applications dependent and lead to different trade off (technology, design, frequency, …).

In the early 2000s , the availability of miniature MEMS sensors, low power communication modules, highperformance miniature batteries and standard communications protocols ( Zigbee , Bluetooth, … ) opened the way for the development of "active" wireless sensors and this kind of products are currently available on the market. The "active" terminology refers here to the use of a RF transmitter (inside the sensing unit) to communicate with the reader. The main drawback of this kind of sensors concerns the energy autonomy that is limited on one hand by the amount of energy stored in the battery and on other hand by the sensing unit consumption. Significant research efforts have been then undertaken in the world, since the early 2000s, on energy harvesting (solar, mechanical, thermal, RF, ….), on high capacity storage units, on new nuclear energy efficient microsource (electrochemical , thermoelectric , thermionic , ... ) but also on circuits with very low power consumption. However this kind of solutions is not adequate for all the application; that is why "passive" wireless sensors was studied in parallel.

For "passive" sensors, the RF transmitter is removed from the sensing unit and communication with reader is based on backscattering solutions that allows to remove the battery. Two kinds of solutions are developed for long range interrogation (>> 1m). The first one use more or less complex electronics components integrated in the sensing unit (digital modulation of radar cross section [START_REF] Sanchez | RFID and sensor integration standards: State and future prospects[END_REF] or intermodulation techniques [START_REF] Viikari | Passive wireless sensor platform utilizing a mechanical resonator[END_REF]). The second one is called "chipless" as all electronics components have been removed. The oldest example is given by SAW (Surface Acoustic Wave) sensor for which the operating principle is based on the generation of an acoustic wave on the surface of a piezoelectric material through an antenna [START_REF] Pohl | A Review of Wireless SAW Sensors[END_REF]. However all these solutions provide interrogation distance typically lower than 10m that can be too low for several applications.

In order to overcome this restriction, new kind of electromagnetic transducers are developed since the mid-2000s. The operating principle is based on the modification of the properties of high-frequency (>> 1 GHz) passive electromagnetic device by the quantity to be measured. A wide range of sensing properties can be addressed with this principle with the use of a great choice of materials. Another advantage is the use of high frequency that allows to reduce the size elements (antenna, transducer), to increase the antenna directivity, to enhanced immunity to multipath and to increase the available bandwidth. In this paper we present an overview on these studies done at LAAS since 2005.

II. TRANSDUCERS

Electromagnetic transducers are mainly composed of a variable RF impedance or RF resonator connected to an antenna that can be interrogated through backscattered wave analysis. Several concepts are possible to realize a sensor with this kind of device by changing the size, the RF properties of a material constituting the device, the coupling between different parts of the device (using displacement of fluid or mechanical part). At LAAS we investigated the last solution that provides very high sensitivity and we developed transducers working generally at frequency of few ten of GHz.

Membrane displacement has been used at LAAS for pressure [START_REF] Jatlaoui | Radio Frequency Pressure Transducer . European Microwave Week (EUMC2007)[END_REF][START_REF] Jatlaoui | Working Principle Description of the Wireless Passive EM Transduction Pressure Sensor[END_REF] temperature [START_REF] Thai | A Novel Passive Wireless Ultrasensitive Temperature RF Transducer for Remote Sensing[END_REF][START_REF] Thai | Design and Development of a Novel Passive Wireless Ultrasensitive RF Temperature Transducer for Remote Sensing[END_REF], stress [START_REF] Thai | Design of a Highly Sensitive Wireless Passive RF Strain Transducer[END_REF][START_REF] Thai | Novel Design of a Highly Sensitive RF Strain Transducer for Passive and Remote Sensing in Two Dimensions[END_REF] and radiation [START_REF] Augustyniak | MEMS high-doses radiation sensor[END_REF] transducers. For pressure transducer a high resistivity silicon membrane is fabricated above a planar half-wavelength resonator deposited inside a few microns depth glass cavity. The membrane deflection leads to a modification of coupling between the silicon and the resonator evanescent wave that modifies the resonant frequency of the device. For resonator-membrane air gap lower than 10µm, the frequency shift is linear and can achieve sensitivity up to 5%/µm depending on the membrane thickness. The temperature transducer is based on split ring resonators in which each ring is closed by a variable capacitor formed by the metal ring and a bimorph micro-cantilever. The temperature shift will induce a cantilever deflection and then a capacitance modification that changes the resonant frequency of the device. For cantilever-ring distance lower than 10µm frequency shift up to 8%/µm can be obtained. The stress transducer is based on a patch antenna loaded with an open loop in two orthogonal sides in order to sense the stress independently in the two plane directions. Each loop is closed by a variable capacitor formed by the fixed part of the loop and the free end of a metallic cantilever. A substrate deformation causes a displacement of the anchoring point of the beam leading to a modification of the capacitor area and then changes the resonant frequency of the device. Sensitivity around 4.5ppm/με can be obtained that are 4 times greater than existing strain transducers of the same class. The radiation transducer principle is based on polymer (High Density PolyEthylene) outgasing inside a micro-chamber etched in silicon wafer. The generated over-pressure leads to a silicon membrane deflection that modifies the resonant frequency of the RF resonator (see section on pressure transducer). The objective is to develop a wireless sensor for very high doses (10kGy -10 MGy). Tests structured have been designed and fabricated to validate the technology and the HDPE outgazing. Membrane deflections around 0.1µm/kGy have been obtained for doses up to 40kGy. This membrane deflection will induce relative resonance frequency shift around 0.8%/µm (8%/100kGy) for 10µm initial air gap and 400µm thick silicon membrane.

Temperature transducer principle using thermal dilatation of metallic [START_REF] Traille | A Novel Wireless Passive Temperature Sensor Utilizing Microfluidic Principles in Millimeter-Wave Frequencies IEEE Sensors[END_REF] and dielectric [START_REF] Bouaziz | A New millimeter-wave micro-fluidic Temperature sensor for wireless passive radar interrogation[END_REF][START_REF] Bouaziz | Novel Micro-fluidic Structures for Wireless Passive Temperature Telemetry Medical Systems Using Radar Interrogation Techniques in Ka-band[END_REF] fluids in micro-channel have been validated at LAAS. The first one is based on λ/2 dipole antenna array with progressive short circuiting of the two dipole strands by the metallic liquid when the temperature increase, leading to a drastically change of the antenna array Radar Cross Section (RCS) for each short-circuit. RCS variation around 12sBsm has been obtained for the full scale. The second one is based on coplanar gap capacitor whose value is modified by the progressively filling of gap capacitor with a dielectric fluid, leading to a modification of the scattering parameter S11. S11 shift around 8dB between empty and full channel has been obtained.

III. WIRELESS INTERROGATION

FMCW (Frequency Modulated Continuous Wave) Radar has been chosen at LAAS for the wireless interrogation of the electromagnetic transducers. This kind of Radar is simpler than UWB (Ultra Wide Band) Radar and is more compatible with high interrogation distance. FMCW radar is used to measure the RCS variation of the target that is done generally by the transducer connected to an antenna. The signal back-scattered by the target is mixed with the transmitted signal to obtained the beat frequency spectrum. By adding a delay line between the transducer and the antenna it is possible to separate the antenna echo from the transducer echo whose level varies with quantity to measure [5 & 14]. This technique has been successfully applied to the wireless interrogation of the different transducers described previously and has been validated for interrogation distance up to 20m with 20mW input signal and 14dBi antennas. These results have been obtained without complicated signal treatment showing that interrogation distance of several ten of meters may be achievable with efficient noise filtering. The main drawbacks of this technique based on echo level are the low accuracy (around 10% of the full scale) and the sensitivity to

TABLE I .

 I LIST OF FIRST PUBLICATIONS ON DIFFERENT PASSIVE SENSORS WITH ELECTROMAGNETIC TRANSDUCTION

ACKNOWLEDGMENT

Pressure transducer and radar development has been partly supported by the French Project of 'Pôle de compétivité' entitled 'Système Autonome Communicant En Réseau' (SACER). Studies on temperature and stress transducers have been performed in collaboration with Georgia Institute of Technology (USA) Radiation sensor study is done in the framework of FP7-MNT-ERA.NET-DOSIMEMS, funded by Midi-Pyrénées Region and done in collaboration with National Centre for Nuclear Research (Otwock, Poland, M. Olszacki) and Wroclaw University of Technology (Wrocław, Poland, J. Dziuban, P. Knapkiewicz). All these studies have been performed with the assistance of M. Jatlaoui, F. Chebila, T. Thai, S. Bouaziz, A. Traille, A. Rifai, E. Debourg, I. Augustyniak.

parasitic signal attenuation that involves the use of reference sensor. But these drawbacks can be overcome with FCMW radar able to track the frequency for which the RCS is maximum. Specific delay lines are also used to create low frequency bar code for sensors identification [START_REF] Aubert | Wireless Sensing and Identification based on RADAR Cross Sections Variability Measurement of Passive Electromagnetic Sensors[END_REF][START_REF] Jatlaoui | Original Identification technique of passive EM Sensors using Loaded Transmission Delay Lines[END_REF]. In this case, the beat frequency difference between the antenna echo and the transducer echo will provide the identification. We evaluated to 30 the number of different sensors that can be identified in a given direction by the same radar.


I Augustyniak
email: izabela.augustyniak@pwr.wroc.pl

P Knapkiewicz

J Dziuban

M Olszacki
email: michal.olszacki@ncbj.gov.pl

A Tchkalov
email: tchkalov@gmail.com

P Pons
email: ppons@laas.fr

Threshold sensor for high-doses of radiation

Keywords: MEMS sensor, radiaition sensor, modeling of silicon membrane I

MEMS threshold sensor for high-doses of radiation (above 10 kGy) has been design and development. In the sensor, small portion of high density polyethylene (HDPE) is trapped inside silicon-glass MEMS hermetically sealed structure. When the sensor is exposed to ionizing radiation, HDPE degrades and releases atomic hydrogen. The resulting pressure of hydrogen destroys thin silicon membrane of the sensor. The proper parameters of the silicon membrane of the sensor have been selected by simulations in ANSYS software.

INTRODUCTION

High doses of radiation (>10 kGy) can be found in nuclear power plants, storage of spent nuclear fuel, nuclear waste disposal site as well as after nuclear accidents. High radiation doses are also observed in experimental infrastructures for example LHC where level of radiation is low but long term doses are high, above 10 kGy.

Measurement of high radiation doses may be estimated post factum by family of passive thermo-and photoluminescence indicators or hydrogen pressure dosimeters. In situ measurements have been up-to-day obtained only for low or medium doses by solid-state MOS based sensors.

MEMS sensor for measurements of high-doses of radiation, above 10 kGy has been shown. In the sensor, small portion of high density polyethylene (HDPE) is trapped inside siliconglass MEMS hermetically sealed structure (Fig. 1a). When the sensor is exposed to ionizing radiation, HDPE degrades and releases atomic hydrogen proportional to radiation dose. This radiolysis phenomenon of polymers has been described in details in previous work [START_REF] Olszacki | Measurement of the high gamma radiation dose using the MEMS based dosimeter and radiolisys effect[END_REF]. The pressure inside the chamber increase and membrane of known mechanical properties is destroyed under the influence of the growing pressure corresponds to the absorbed radiation dose (Fig. 1b).

The sensors have been irradiated by highly energetic (6 MeV) electron beam with doses above 10 kGy (in National Centre for Nuclear Research, Otwock, Poland). Blow-up effects proportional to irradiation doses have been observed.

Fig. 1

 1 Fig. 1 Principle of operation: (a) sensor before irradiation; (b) sensor after irradiation

ACKNOWLEDGMENT This research was co-financed by European Union under the MNT-ERA.NET, project DOSIMEMS and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme -Project EduMEMS no. 269295.


Giancarlo Vincenzi

George Deligeorgis

Fabio Coccetti

Patrick Pons

Open-Thru de-embedding for Graphene RF devices

Keywords: Calibration, frequency measurements, graphene, integrated circuit modeling, thin films

The performances of graphene RF devices rely heavily on the precision of the de-embedding techniques adopted, and the small size of the lines increase variability issues in the RF standards and test devices. In this work, a robust and high precision de-embedding technique adapted to the needs of microwave graphene devices is presented. Two standards and only one step after SOLT calibration are needed.

I. INTRODUCTION

After nine years from its isolation, graphene is emerging as a valuable candidate in the fields of High-Frequency electronics, as active element, and Flexible and Transparent electronics, as interconnect material [START_REF] Novoselov | A roadmap for graphene[END_REF]. In both fields the high parasitic impedances of graphene/metal contact and interconnects are a critical issue, which greatly limits extrinsic transistor Figures of Merit (FoMs) as f T and f max . De-embedding allows to correct those numbers and boosts intrinsic FOMs up to the double [START_REF] Guo | Record maximum oscillation frequency in C-face epitaxial graphene transistors[END_REF], a dramatic improvement compared to III-V semiconductor devices. The error in de-embeddeding can lead to an error in intrinsic FoMs of the same order of the improvement. A precise and robust de-embedding procedure is critical in graphene RF electronics research.

Multiline Thru-Reflect-Line (TRL) [START_REF] Marks | A multiline method of network analyzer calibration[END_REF] is often regarded as the most accurate calibration algorithm as it does not require the fabrication of high-precision resistors. Neverthless, its validity at lower gigahertz frequencies is limited by the need of impractically long reference lines, especially if integrated in low-κ substrates. Another widely used de-embedding algorithm is the Open-Short (OS) [START_REF] Koolen | An improved de-embedding technique for on-wafer high-frequency characterization[END_REF]. It performs very well at low frequency, but being based on a lumped element circuit model, it requires the lengths of the pads and signal-to-ground interconnects to be smaller than λ/20. For this reason, it should be used after both SOLT calibration and TRL de-embedding, for a total of three steps [START_REF] Yau | Device and IC Characterization Above 100 GHz[END_REF]. Thru-only algorithms decompose the Thru in π-shaped or T-shaped [START_REF] Kobrinsky | Experimental validation of crosstalk simulations for on-chip interconnects using S-parameters[END_REF] networks, but these methods are frequency-limited in the same way as the OS is.

The study of novel materials for RF as metallic nanowires [START_REF] Kim | Contactless Approaches for RF Characterization of Metallic Nanowires[END_REF], CNT [START_REF] Tselev | Microwave impedance spectroscopy of dense carbon nanotube bundles[END_REF] and graphene [START_REF] Skulason | 110 GHz measurement of large-area graphene integrated in low-loss microwave structures[END_REF] is performed from DC to many tens of GHz. Moreover, the impedance to be measured ranges from hundreds of Ω to many kΩ, offering a poor match to 50Ω RF instrumentation and with significant variations in impedance from device to device. The calibration is done up to the probe tips on a standard calibration substrate, while the de-embedding of the interconnects and pads is done through the TRL [START_REF] Kim | Contactless Approaches for RF Characterization of Metallic Nanowires[END_REF], through the OS [START_REF] Tselev | Microwave impedance spectroscopy of dense carbon nanotube bundles[END_REF], or even neglected [START_REF] Skulason | 110 GHz measurement of large-area graphene integrated in low-loss microwave structures[END_REF]. In the case of active devices, the OS technique is generally adopted [START_REF] Guo | Record maximum oscillation frequency in C-face epitaxial graphene transistors[END_REF].

In this work, the Open-Thru (OT), a broadband deembedding procedure adapted to the characterization of the high-frequency impedance of graphene and thin films is introduced. Its use allows for a more precise extraction of the contact and surface impedance of graphene, together with a more precise extraction of active device small-signal parameters and FOMs. In Section II the method will be detailed; in Section III the de-embedding will be tested on a few graphene passive interconnects and compared with the widely adopted OS; in Section IV a high-frequency model of graphene will be extracted; and in Section V the robustness of this method will be measured.

II. METHOD

The OT is an extension of the Thru-only method presented in [START_REF] Kolding | On-wafer calibration techniques for giga-hertz CMOS measurements[END_REF], where the matrix related to each access fixture is first derived by the manipulation of the S-parameters of the Thru, then converted into ABCD chain parameters. In this work a different formulation of the Thru-only is detailed, where the matrix is derived already in the ABCD domain. The final deembedding of the data is the same for the two formulations.

A. Thru-Only de-embedding

In the Thru-Only method, the uncorrected DUT is thought as the chain of a left fixture, the DUT to be de-embedded, and the right fixture. The ABCD parameters of the Thru standard are square-rooted, inverted and left-and right-multiplied to the ABCD parameters of the DUT, as clearly described in [START_REF] Novoselov | A roadmap for graphene[END_REF].

D T = √ T -1 D raw √ T -1 , (1)
where D raw are the measured ABCD parameters of the DUT, T those of the Thru standard and D T are the de-embedded data. The square-root R of a matrix M = a b c d is instead computed as follows:

R = 1 t a + s b c d + s (2) with s = ± √ δ (3)
and

t = ± √ τ -2s , (4)
and τ and δ are respectively the trace and the determinant of M. Two solutions exist for the square root of the matrix of a generic delay line (two more exist and are identical to the first pair). However, only one of them is a physical solution, the

± 2.5% ± 5 µm ± 5% ± 5% ± 5% R S1 R S2 R P L P Central value 1 Ω 1 Ω 0.5 Ω 1 pH Dispersion ± 5% ± 5% ± 5% ± 5%
other one gives anti-symmetric delay in forward and reverse transmission (<S 21 and <S 12 are rotated of π in Smith chart representation).

A general requirement of the Thru-only method is that the Thru device has to be reciprocal and symmetrical, which is not always the case because of some small and yet unavoidable calibration errors after the SOLT step and/or probe landing misplacement. This requirement in S-parameter notation translates in a bi-symmetrical matrix, i.e. simultaneously symmetrical around the main diagonal (S 21 = S 12 , reciprocal device) and the secondary one (S 11 = S 22 , symmetrical device). However, every matrix can easily be separated in a symmetrical and anti-symmetrical component with simple average and difference operations. The small imperfections left by the SOLT calibration can be subtracted away and incorporated in the de-embedding error, and the modified Thru reference structure can be assumed as the chain of its square roots. Hence, the T matrix in (1) must be the ABCD transform of the bi-symmetrized S-parameters of the Thru.

B. Open-Thru de-embedding

The Thru-only can be expanded to include the Open standard. Both the DUT and the Open standard are first corrected with the Thru-only method, then the Y-matrix of the corrected Open is subtracted from the DUT's one, as expressed by ( 5):

D de-emb = D T -O T , (5)
where D T and O T are the Y-matrixes respectively of the DUT and Open corrected with (1) in ABCD domain and D de-emb is the final de-embedded data.

III. SIMULATION AND COMPARISON

The test device is a circuital model drawn in ADS Schematic. The DUT to be corrected is shown in Fig. 1. It contains the intrinsic section DUT int , two 300 µm slightly mismatched lossy transmission lines at the inputs, and parasitic components C S , C P 1 , C P 2 , R S1 , R S2 , R P and L P . The central values of the capacitors are taken from the model of a series discontinuity as described in § 9.5 of [START_REF] Simons | Coplanar Waveguide Circuits, Components, and Sys-tems[END_REF], and the inductor and resistances from [START_REF] Yau | Device and IC Characterization Above 100 GHz[END_REF].

The Short and the Open standards are built according to the specifications in [START_REF] Cheng | MOSFET Modeling for RF IC Design[END_REF], and the Thru contains all the elements with the exception of the short-to-ground impedance.

The de-embedding method presented here will be applied to the measurements of a graphene-loaded CPW line of reduced dimension fabricated on a flexible Polyimide (PI) substrate. between the two signal lines. The CPW ground-to-ground width has been chosen of 25 µm to maintain the compatibility with in-lab mechanically exfoliated graphene. This results in line gaps of 1.5 µm for 50Ω, very close to the resolution limit of the lithographic process used (1 µm) and yielding a variation of ±0.13 µm, measured with low-field SEM. This results in a variation in the Z 0 of the lines of the 2.4% from nominal value and up to the 4.8% of the capacitance between CPW signal and ground electrodes.

The Material Under Test (MUT) is placed in series connection

A study of the robustness of the de-embedding method against geometrical variations can be easily performed introducing a 5% Gaussian dispersion on circuit parameters, in all standards and uncorrected DUT, and simulated with a Monte-Carlo analysis. The central values and the relative dispersions are detailed in Table I. The resulting S-parameters are shown in Figures 2 and3. The output variation of each method is comprised between the symbols (triangles and circles) of the respective color. The errors of OS and OT techniques are listed in Table II The OS is more accurate at low frequency because it includes the short-toground impedance, but at 18 GHz the interconnects are λ/20 and the error grows quickly. The OT performs better than OS at 8.5 GHz and higher frequencies. The error spread in phases is better for OS and is three times better for OT in magnitudes.

An additional study reducing the interconnect length to 30± 5 µm has been carried out. The results are in Table II, showing again better performance of OT for magnitudes and of OS for phases. In this case OT is more accurate than OS after ∼60 GHz.

IV. GRAPHENE INTERCONNECTS MEASUREMENTS

Graphene-loaded CPW lines have been fabricated. CVD monolayer graphene provided by Graphene Supermarket R has been deposited on pre-fabricated Au CPW lines included in a PI flexible substrate, shown in Fig. 4, and then patterned in strips of 20µm of width, shown in Fig. 5. The electrode separation is 2µm for the first device and 20µm for the second. The two graphene strips are electrically short at all tested frequencies. The lines have been measured with an Agilent PNA-X up to 67 GHz calibrated at -20dBm of power, so that graphene is in linear regime [START_REF] Tselev | Microwave impedance spectroscopy of dense carbon nanotube bundles[END_REF]. The fabricated access lines are longer than 180µm on each side, so the OT method has been used. One possible model for graphene at RF frequencies is the lumped element circuit in Fig. 6. It is composed by the sheet resistance R sh , the contact resistance R C and the contact capacitance C C . R C originates from the cascade of transport of electrons in graphene under the metal and their tunneling into the metal [START_REF] Xia | The origins and limits of metal-graphene junction resistance[END_REF]. C C is mainly used to model the charge accumulation at the interface in case of non-ideal contact. However, in the case of graphene, a fraction of C C is predicted to be originated from its intrinsic physics [START_REF] Xia | The origins and limits of metal-graphene junction resistance[END_REF]. Hence, C C and R C should be incorporated in graphene's behavior.

The sum of the resistors in Fig. 6 is fixed to the DC resistance R dc , measured with a multimeter. Fig. 7 shows the de-embedded data and simulations, shows excellent agreement on all frequencies as in DC, proving that the fixture resistance has been correctly removed.

V. ROBUSTNESS

The 2 µm graphene device has been de-embedded also with the OS method in order to evaluate the sensitivity to non-ideal standards. The OT and OS algorithms both use the same Open standard, of which various copies on the same sample have been measured. The de-embedded data have been averaged over all Open standards, and the errors associated to standard variation have been compared between the two methods. This is shown for the transmission magnitudes in Fig. 8 and for transmission phases in Fig. 9. In both cases the OT method resulted in smaller error due to imperfections in standards compared to OS, i.e. 36% smaller for magnitudes and 20% smaller for phases, averaged over all frequencies. The increased robustness of this algorithm might reduce the error associated to the DUT modeling or, conversely, the number of reiterated measurements necessary for the same accuracy. This would increase the standard devices' life and the time employed in measurements campaigns.

The accuracy of this method can be further enhanced by also taking into account the Short standard, making it an improved Open-Short-Thru de-embedding protocol, which would allow the treatment of devices with a low impedance path to ground. However, depending on the impedance of the DUT and on to the resemblance of the electrostatics between DUT and standards terminations, it's not trivial to determine if all the standards in such OST method should be employed. A dedicated study would be necessary to further explore those topics.

In conclusion, the Open-Thru method has been proven as more accurate than the Open-Short for long interconnects, and more robust against imperfections in standards, both in Monte-Carlo simulations and in measurements of micrometric graphene strips. Non-ideal standards are frequently encountered in case of graphene and other carbonaceous thin films RF devices, whose small dimensions are close to lithographic reproducibility limits. The high-precision de-embedding method allowed for accurate modeling of the graphene interconnects under test.

Fig. 1 .

 1 Fig.1. The circuital model of the extrinsic DUT. In the middle, framed, is the intrinsic part of the device that must be de-embedded.

  , averaged between 0 and 110 GHz. They are evaluated as |err(|S 11 | dB )|+|err(|S 21 | dB )| for magnitudes and |err(S 11deg )| + |err(S 21deg )| for phases.

Fig. 2 .

 2 Fig. 2. Simulations of the Reflection magnitude. Comparison between Open-Short (blue dashed line and blue circles), Open-Thru (green dot-dashed line and triangles) and the target DUTint (solid red line). Inset: zoom at 0-10 GHz.

Fig. 3 .

 3 Fig. 3. Same as Fig. 2, simulation of the the Transmission magnitude.

Fig. 4 .

 4 Fig. 4. The flexible PI substrate with included CPW lines (a), and the deposition of graphene on the same (b).

Fig. 5 .

 5 Fig. 5. A 20µm-wide graphene strip deposited on the signal line of the CPW on plastic substrate.

Fig. 6 .

 6 Fig. 6. Graphene RF model.

Fig. 7 .

 7 Fig. 7. De-embedded and simulated S-parameters for two graphene devices of 2 µm (blue triangles) and 20 µm (green circles) of length. Transmission magnitude on left axis, reflection on the right.

TABLE I PARAMETERS

 I OF THE CIRCUIT IN FIG. 1.

		TL Z 0	TL length	C S	C P 1	C P 2
	Central value	47.5 Ω	300 µm	50 fF	5 fF	5 fF
	Dispersion

TABLE II OPEN

 II -SHORT AND OPEN-THRU DE-EMBEDDING ERRORS.

	Line length	Method	Magnitude [dB]	Phase [deg]
	300 µm	Open-Short Open-Thru	4.49 ± 0.25 0.08 ± 0.07	2.28 ± 2.23 1.26 ± 2.83
	30 µm	Open-Short Open-Thru	0.13 ± 0.07 0.08 ± 0.06	1.50 ± 1.56 1.84 ± 3.06

  Table III resumes the extracted parameters. R sh and C C differences are consistent with measured R dc and can be explained with non-ideal contacts.

TABLE III EXTRACTED

 III CONTACT AND SHEET RESISTANCES .

	2 µm	240	1021	1.64	6.40
	20 µm	421	251	2.02	0.78

Length R dc [Ω] R sh [Ω] R C [Ω•mm] C C [pF/mm]


Abderrezak Rachedi
email: rachedi@u-pem.fr

Hakim Badis
email: badis@u-pem.fr

Abderrahim Benslimane
email: abderrahim.benslimane@univ-avignon.fr

How MIMO cross-layer design enables QoS while detecting non-cooperative nodes in wireless multi-hop networks

Keywords: Multi-hop wireless networks, Non-cooperative nodes, MIMO, Monitoring mechanism, Conflict Graph, QoS

How MIMO cross-layer design enables QoS while detecting non-cooperative nodes in wireless multi-hop networks

Abderrezak Rachedi, Hakim Badis,

Introduction

Wireless Multi-hop Networks (WMNs) are a set of nodes based on the cooperation aspect to form and manage a network. The nodes in WMNs act as router and terminal at the same time and a lack of cooperation between them implies the absence of any network. That is why it is important to deal with the non cooperative or selfish nodes problem. The problem of selfish nodes is that they keep their energy to transmit and route their own packets. In other words, the selfish nodes refuse to route and forward the packets of other nodes. The question is why do nodes act selfishly and ignore the cooperation aspect? To answer this question, we investigate the motivation of nodes to adopt this misbehavior. First of all, the resources in WMNs are limited in terms of energy, bandwidth, etc. Then, the nodes try to increase their lifetime duration by reducing their energy consumption and the cost of the transmission operation is important in terms of energy. Secondly, when the nodes route and forward the packets of other nodes, this increases the delay of their own packets transmission and reduces their own average throughput. Thus, this operation may be perceived by nodes as punishment and not as global network interest. In order to deal with this problem, many researchers focus on the monitoring mechanisms in order to detect the selfish nodes and to punish them [START_REF] Marti | Mitigating routing misebehaviour in mobile ad-hoc networks[END_REF][START_REF] Rachedi | Toward a cross-layer monitoring process for mobile ad hoc networks[END_REF][START_REF] Rachedi | Cross-layer approach to improve the monitoring process for mobile ad hoc networks based on ieee 802[END_REF][START_REF] Michiardi | Core: a collaborative reputation mechanism to enforce node cooperation in mobile ad hoc networks[END_REF][START_REF] Tang | Selfish misbehavior detection in 802.11 based wireless networks: An adaptive approach based on Markov decision process[END_REF]. However, all the proposed mechanisms are based on the classical SISO (Single-Input Single-Output) technology to monitor the communication channel and to detect the non forwarding nodes (selfish behavior). The most cited mechanism is Watchdog [START_REF] Marti | Mitigating routing misebehaviour in mobile ad-hoc networks[END_REF]. Many proposed solutions are based on it, but they suffer from a high false alarm rate. The main problem of these mechanisms is related to the interference at the monitor (detector) node which makes the results of its observations wrong. These mechanisms are mainly considering the IEEE 802.11b/g as MAC protocol, so as far as we know, we are the first authors using Multiple-Input Multiple-Output (MIMO) system with IEEE 802.11n to remove the problem of false observation at the monitor nodes [START_REF] Rachedi | MIMODog: How to solve the problem of selfish misbehavior detection mechanism in MANETs using MIMO Technology[END_REF].

In this paper, we extend the study of our proposed MAC protocol called MIMODog [START_REF] Rachedi | MIMODog: How to solve the problem of selfish misbehavior detection mechanism in MANETs using MIMO Technology[END_REF] based on Multi-Input Multi-Output (MIMO) technology, and particularly a SPACE-MAC protocol [START_REF] Park | Space-mac: Enabling spatial reuse using mimo channel-aware mac[END_REF] to significantly reduce the potential interferences at the monitor (detector) nodes and to enhance the accuracy of the monitoring results. The most significant added value consists in i) the modeling of MIMODog to evaluate the impact of the monitoring process on the network performance, ii) the proposition of a monitoring capacity analysis based on graph theory particularly conflict graph, iii) an asymptotic study proposed to investigate lower and upper bounds of the number of monitor nodes.

The contributions of this paper can be summarized in five points:

• We illustrate that the monitoring problem persists in WMNs, even when we use MIMO technology;

• We propose a new MIMO MAC protocol called MI-MODog to reduce the interferences at the monitor (detector) nodes without negatively impacting the total network capacity;

• We present and discuss a different impact on the monitoring process with: DCF MAC, SPACE-MAC and MI-MODog. MIMODog significantly enhances the monitoring process without affecting the network capacity;

• We propose a monitoring capacity analysis using conflict graph from graph theory for different MAC protocols: DCF MAC, SPACE-MAC and MIMODog;

• We investigate lower and upper bounds of the number of monitor nodes based on asymptotic study for MIMODog. Moreover, the obtained numerical results illustrate that the proposed solution overcomes the drawbacks of classical monitoring mechanisms.

This paper is organized as follows: an overview of cooperation models based on monitoring mechanisms and the SPACE-MAC protocol is given in section 2. Section 3 illustrates the DCF and SPACE-MAC protocols vulnerabilities in the monitoring process. A new MAC protocol adapted to the monitoring process with more details on its design and its implementation is proposed in section 6. In addition, the monitoring capacity and its impact on wireless multi-hop networks is investigated in section 5. Moreover, the theoretical model in order to assess the asymptotic bound related to the monitor nodes number is presented in section 6. The numerical results are given and analysed. The final section concludes the paper and presents our future works.

Related Work

In this section, we present the existing works related to the non-cooperative nodes monitoring and the detection models in Wireless Multi-hop Networks (WMNs), and we briefly overview the SPACE-MAC protocol [START_REF] Park | Space-mac: Enabling spatial reuse using mimo channel-aware mac[END_REF].

Non-cooperative nodes monitoring and detection Models

Two kinds of non-cooperative (selfish) nodes can be distinguished: active and passive selfish nodes. The active selfish nodes, also called greedy nodes, try to get more resources by maliciously manipulating protocols' parameters at MAC and routing layers. For instance, at the MAC layer with IEEE 802.11 DCF mode the active selfish nodes can manipulate the backoff parameters in order to quickly access the channel at the cost of their neighbour nodes [START_REF] Raya | Domino: Detecting mac layer greedy behavior in ieee 802.11 hotspots[END_REF]. However, the aim of passive selfish nodes is to optimize their benefits in terms of QoS (like throughput and delay) and minimize their costs like the energy consumption without maliciously manipulating protocols' parameters. The typical action of passive selfish nodes is to refuse to forward the packets of other nodes.

In this paper, we focus on the selfish behaviour of the potential cooperative nodes, particularly passive selfish nodes. Cooperation is an important parameter in wireless multi-hop networks, because without any packet forwarding, the ad hoc network cannot exist and the wireless coverage extension is not possible.

In literature, two main solutions were proposed to overcome the problem of selfish nodes. The first one is based on the reputation mechanisms which consist in assessing a node's contribution like forwarding and routing functionalities [START_REF] Michiardi | Core: a collaborative reputation mechanism to enforce node cooperation in mobile ad hoc networks[END_REF][START_REF] Marti | Mitigating routing misebehaviour in mobile ad-hoc networks[END_REF][START_REF] Buchegger | Performance analysis of the cofidant protocol[END_REF][START_REF] Bansal | Observation-based cooperation enforcement in ad hoc networks[END_REF][START_REF] Chen | A reputation system for wireless mesh networks using network coding[END_REF][START_REF] Braga | RETENTION: A reactive trust-based mechanism to detect and punish malicious nodes in ad hoc grid environments[END_REF]. The reputation model called CONFIDANT is proposed to share the reputation metric and alarm messages in order to detect and punish the misbehaving nodes [START_REF] Buchegger | Performance analysis of the cofidant protocol[END_REF]. Another model called CORE is proposed to implement the reputation function by using the monitoring technique. Each node computes the reputation value for every neighbour and refuses to provide services to misbehaving nodes when their reputation is lower than a certain threshold.

The second one is based on the economics mechanisms also called price or credit-based mechanisms. In these models the nodes are paid to offer message forwarding services and pay to receive forwarding services. The proposed incentive models for cooperation are using the concept of virtual cash. For instance, many incentive models based on game theory are proposed [START_REF] Chen | Conflicts and incentives in wireless cooperative relaying: A distributed market pricing framework[END_REF][START_REF] Rachedi | A Secure Mechanism Design-Based and Game Theoretical Model for MANETs[END_REF]. The nodes are rewarded for forwarding packets by trading virtual cash with source and next hop nodes [START_REF] Chen | Conflicts and incentives in wireless cooperative relaying: A distributed market pricing framework[END_REF]. Buttyan and Hubeaux [START_REF] Buttyan | Enforcing service availability in mobile ad-hoc wan[END_REF] proposed nuglets as credits to manage forwarding transactions. The source node pays relay intermediate nodes by storing nuglets in the packet head. The intermediate nodes acquire the nuglets when they forward the packets. In [START_REF] Li | Analysis of a hybrid reputation management system for mobile ad-hoc networks[END_REF] a hybrid model using the reputation and the price-based mechanism was proposed to overcome the issue of selfish nodes. However, all these solutions are based on the classical monitoring mechanism like Watchdog [START_REF] Michiardi | Core: a collaborative reputation mechanism to enforce node cooperation in mobile ad hoc networks[END_REF]. The Watchdog mechanism consists in the overhearing of the channel activities in order to detect the non-forwarding (selfish) nodes, but it suffers from a high false positive rate. The main issue of the false observation is related to the collision problem at the monitor (detector) node. Despite, many models are proposed to enhance the false observation [START_REF] Tang | Selfish misbehavior detection in 802.11 based wireless networks: An adaptive approach based on Markov decision process[END_REF], but the problem persists. All these models consider the IEEE 802.11b/g as MAC protocol, so as far as we know, we are the first authors using Multiple-Input Multiple-Output (MIMO) system with IEEE 802.11n to remove the problem of false observation at the monitor nodes.

SPACE-MAC protocol: Spatial Reuse Using MIMO

Channel-Aware MAC The SPACE-MAC is a Media Access Control protocol for networks with smart antennas which uses antenna weights to schedule simultaneous transmissions on a single collision domain. Antenna weights are exchanged via control packets (RTS and CTS 1 ) [START_REF] Park | Space-mac: Enabling spatial reuse using mimo channel-aware mac[END_REF].

The main contribution of SPACE-MAC is the fully distributed MAC protocol that exploits the physical layer characteristics and cross-layer techniques to enable spatial reuse in scatter-rich multi-path environments. The main advantage of SPACE-MAC is that it enables multiple data streams at the same time in the same collision area, thereby increasing the overall capacity of the network. The channel control overhead introduced by channel estimation and beam coordination is minimal and effectively countered by the gain provided by the increase in the capacity of MIMO channels.

In SPACE-MAC, the first station that gains access to the channel determines the silence period. All other stations must remain idle following their transmission until the silence period is completed. In SPACE-MAC, the silence period is required because any station currently involved in the transmission is unaware of any other transmission that began during its data packet or acknowledgement packet transmission phase. Additionally, any station that wishes to transmit must not interfere with this ongoing transmission and must not transmit if it cannot complete its entire packet exchange sequence before the end of the silence period.

Based on the RTS/CTS handshake of the existing transmission and for each new communication in the same geographical vicinity, the new sender/receiver nodes will select their weights so that the signal from any existing communication node is nullified. This problem can be formulated as a quadratic optimization and reduced to an unconstrained optimization problem using the null space method which in turn is an eigenvalue problem. Any new additional transmission is only possible if both nodes of a same pair have enough degrees of freedom. For an M antenna system it can null out at most M -1 stations depending on the environment. M is also known as the Degrees of Freedom (DOF). Every time a node nulls out another node, it consumes a DOF.

MAC protocols vulnerabilities in monitoring process

In this section, we highlight vulnerabilities in the monitoring process in different cases: classical IEEE 802.11 DCF (with SISO system), SPACE-MAC (with MIMO system), and MI-MODog (with MIMO system). In this study, we focus on the origin of the mis-observation of the monitor (detector) nodes.

Case of 802.11 DCF MAC with single antenna

The monitoring process acts on the different network protocol layers (eg. MAC, Routing). In this work, we focus on the network layer for the monitoring. The monitor node supervises the packet forwarding activities of its neighbor nodes and their packet integrity. Let us consider a small network illustrated in Figure 1.

Case of SPACE-MAC with multiple antennas

In this subsection, we will show that the monitor nodes cannot recover collided packets using the standard SPACE-MAC protocol.

Let us consider the latest network (Figure 1). We assume that all nodes are silent at the beginning, i.e., there is no ongoing communication and each node has 4 antennas. Node B wants to forward A's packets to C and D wants to communicate with E. Node B transmits an RTS using the default weight vector, [1 1 1 1]/ √ 4, or a random vector. The vector is normalized to have an equal signal power regardless of the number of antennas. The weight vector used to transmit the RTS will be used to transmit the following data packet and to receive the corresponding CTS and ACK. Once node C receives the RTS, it responds with a CTS packet using the current weight vector. The weight vector used to transmit the CTS will be used to receive the following data packet and to send an ACK. The receiver estimates the SIMO (Single-Input Multi-Output) channel vector h BC = w H B H BC , where w B is the weight vector of node B and H BC is M × M MIMO channel matrix with elements h i j and the superscript H denotes an hermitian operation. In fact, as there is no ongoing communication, nodes C (receiver) and A (monitor) can switch their weight vectors to w C = h t BC and w A = h t BA which maximize the combined channel and array gain. When a node other than the designated receiver and the neighbor monitor receives the RTS, say node K, it estimates the effective channel H and adjusts the weight vector so that the signal from the RTS sender is nullified (i.e., h BK C K = 0) for the duration of time specified in the RTS duration field. When a node other than the sender of the RTS (B) receives the CTS, say node L, it estimates the effective channel and stores the weight vector for the duration specified in the CTS duration field. After the RTS/CTS handshaking, node B sends, C receives and A supervises a data frame respectively using the weight vectors w B , w C and w A chosen as described above. Now let us say node D wants to initiate a transmission toward E. Since node D is not currently aware of the antenna weight used by node B (node D cannot overhear B's RTS and C's CTS), it cannot adjust its weight vectors meeting these conditions: w H D H DA w A = 0 (D's signals cannot be nullified by A). Consequently a collision will occur at node A. The example shown in Figure 3 describes such a process problem.

MIMODog protocol

In order to avoid any interference at the monitor node, each new transmitting node must be aware not only of the weight vectors of the existing transmissions in the cover area, but also of the weight vectors used by the monitor nodes. To deal with this issue, we propose a new MIMO MAC protocol called MI-MODog. The basic idea is that the monitor nodes simulate a real reception by sending CTS packet control before starting their monitoring process. We use the previous example to illustrate our MIMO MAC protocol functioning.

Basic protocol operation

When monitor node A hears an RTS packet from its forwarding node B:

1. it estimates the SIMO channel vector h BA = w H B H BA and switches its weight vector to w A = h t BA to well receive B's packets for monitoring; 2. it sends a CTS packet after a SIFS time using a weight vector ŵA meeting this condition: ŵH A H AB w B = 0 (the A's CTS signal is nullified at B to avoid collisions with C's CTS and to ensure that node B will not change its behaviour if it is malicious). The A's CTS contains the weight vector w A and transmits using ŵA . The goal of this operation is to make all future transmitters in the neighborhood believe that node A will receive packets and that its weight vector w A should be considered.

Once it receives the CTS packet from A, each node should estimate the effective channel from A. Now, the transmission of D should ensure that the reception of A is not disturbed. So, it picks W D meeting w H D H DA w A = 0 before transmitting its RTS. The process is graphically explained in Figure 4.

RTS/CTS Control Packet Format

In order to selectively tune in or tune out a particular transmission, the stations have to be aware of the antenna weights that are used by transmitting stations. This requires a mechanism to convey the antenna weights to all neighboring stations. MIMODog uses RTS and CTS control packets to convey antenna weights. The proposed format for RTS and CTS control packets is shown in Figure 5. A separate s byte field is inserted in the payload of the RTS and CTS packets and stores M antenna element weights currently in use. A linear function f is used to obtain the value of s. For example, as each antenna weight can be a complex number, we can store them as a pair of real numbers occupying 4 bytes (per one complex number) and so f (M) = 4M. RTS and CTS packets are also used to perform a channel estimation using pilot symbols embedded in the physical (PHY) preamble.

Monitoring capacity analysis

In this section, we investigate the monitoring capacity and its impact on wireless multi-hop networks. The monitoring capac-ity for cooperation can be evaluated by the ability of a monitor (detector) node to listen (overhear) its neighbors activities under interference consideration. In this regard, we focus on:

• the computation of the available capacity on the link between the monitor-forwarding nodes;

• the necessary conditions to correctly perform monitoring.

Preliminary and definitions

The link capacity in a single-hop network can be defined as the physical transmission bit rate of the source, determined by: the Shannon limit, the fixed modulation scheme and the bit error rate. In a wireless multi-hop network (WMN) context, several links share the same transmission medium, and so the link capacity decreases when more simultaneous transmissions occur. The sum of all active data flow throughputs in the same interference area gives the consumed capacity. Consequently, the available link capacity is the difference between the link capacity (in a single-hop network) and the consumed capacity (Figure 6):

Available Link Capacity = Link Capacity-Consumed Capacity.

(

) 1
Figure 6: Link Capacity.

Table 1 summarizes the notation used in the next sections.

A WMN can be seen as an undirected stochastic geometric graph, called connectivity graph G. For this purpose and as explained in [START_REF] Badis | An efficient bandwidth guaranteed routing for ad hoc networks using ieee 802.11 with interference consideration[END_REF], we use the following interference model: any link within distance H from (i, j) is a potential interfering link. This rule is called the distance-H interference model.

Case of 802.11 DCF MAC with single antenna

Based on LCG, the LCG-nodes in a maximal clique represent the maximal set of mutually contending wireless links, along which only one flow may transit at any given time, consequently only one LCG-node in a clique may be active. Accordingly, the sum of the rates of LCG-nodes in each maximal clique cannot exceed the capacity of the channel; these conditions are the clique constraints. Since the network must satisfy the capacity constraints for all cliques, we can write the clique constraints in a matrix form. We represent a set of flow rates as the column vector F of size n, where n is the number of links in the network G and Fi is the average flow rate assigned to link i. Let C i be a column vector of size n with all entries equal to the channel capacity C i . Hence we have,

∀i Q i .F ≤ C i . (2)
where Q i is an incidence matrix, which is of order qxn. Here, q is the number of maximal cliques that this link i belongs to, and n is the total number of links. The union of the clique matrices across all the links gives the global clique matrix Q.

Note that the flow rate F i assigned to link i in an interval of time ]tτ, t], can be written as:

Fi = 1 τ t t-τ F i (r)dr, (3)
where F i (r) is the instantaneous flow rate utilization on link i at time r.

The clique constraints provide a necessary condition for a realizable schedule to exist, since there cannot be a feasible schedule over links that form a violated clique constraint. One might hope that these constraints would also be sufficient. Unfortunately, that is only true for a special sub-class of graphs called Perfect Graphs [START_REF] Lovasz | A characterization of perfect graph[END_REF]. In prior work [START_REF] Jia | Bandwidth Guaranteed routing for Ad hoc Networks with Interference Consideration[END_REF], the authors have proved the sufficiency condition using clique constraints scaled by 0.46 (using a virtual conflict graph). In our work [START_REF] Badis | An efficient bandwidth guaranteed routing for ad hoc networks using ieee 802.11 with interference consideration[END_REF], we have generalized the notion of scaling factor, β, according to the used interference model. Clique constraints become:

∀i Q i . F ≤ β.C i . ( 4
)
As the CG-node can be a part of multiple cliques, it considers all the cliques that it belongs to, and takes the worst case available capacity over all the cliques. The available capacity on a CG-node , i, is

Γ i = min{(C i × β) -Q i F}. ( 5
)
Γ i is the available capacity on link i, taking into account active flows on i, as well as interference from neighboring links. d,h)} be denoted by { F1 , F2 , F3 , F4 , F5 , F6 , F7 }. Then, the available capacity on the link between the monitor-forwarding nodes, 2 = (b, c), is:

Γ 2 = min              (C 2 × β) -( F1 + F2 + F3 ) (C 2 × β) -( F1 + F2 + F4 + F5 ) (C 2 × β) -( F2 + F3 + F6 + F7 )
In the same way, from Figure 8.b:

Γ 2 = (C 2 × β) -( F1 + F2 + F3 + F4 + F5 + F6 + F7 ).
When the forwarding node forwards the monitor node's traffic, say θ, two conditions are necessary for a successful listening by the monitor node:

• The traffic from/to the monitor node should be null. Otherwise, the monitor node cannot hear the forwarding node's activities. The capacity of the link between monitorforwarding nodes becomes:

       Γ M-F = min{(C M-F × β) -Q M-F F} F M-F = 0 (6)
• The available capacity on the link between the monitorforwarding nodes is more than the amount of the monitored traffic:

Γ M-F ≥ θ. (7)
However, the previous conditions are not sufficient because a feasible schedule isn't guaranteed.

Case of SPACE-MAC protocol with MIMO system

The SPACE-MAC protocol operates in M-dimensional space, where M is the degrees of freedom (DoF) of the MIMO channel. In this case, the interference relationship between different links in the network are conserved in the same DoF. However, the inter-DoF interference is null. Consequently, the capacity of any MIMO link in the network is the sum of all capacities on that link per DOF. Figure 9 illustrates the superposition but independence of the link conflict graphs of the network topology shown in Figure 7. Each DoF has its own LCG. All the LGCs are similar if we apply the same interference model. Let Γ j i be the available capacity on link i under DoF i. The total available capacity on link i, Γ i , is given by the following formula:

       Γ i = M j=1 Γ j i Γ j i = min{(C i × β j ) -Q i F j } (8)
We assume that at any time slot t and under any DoF, a node can either transmit or receive, but not both. In other words, if a node transmits at time slot t and under DoF m, it cannot transmit or receive at the same time slot under the remaining DoFs. To hear the forwarding node, the monitor node should use the same weight vector used by the forwarding node. This means that the monitor-forwarding pair uses the same DoF. According to the DoF used by transmitter/receiver nodes located in the interference area of the monitor node, we distinguish two cases:

• If no transmitter/receiver node uses the same DoF as that used for monitoring, the monitor node can correctly hear the forwarding node's activity. In this case, the link capac-ity of the monitor-forwarding nodes is:

             Γ M-F = M j=1 Γ j M-F Γ j M-F = C M-F if j is the used DoF for monitoring M-F = min{(C M-F × β) -Q M-F F} otherwise (9)
• If there is at least one transmitter/receiver node using the same DoF as that used for monitoring, in this case and following the same reasoning as in section 5.2, two conditions are necessary for a successful listening by the monitor node:

-The traffic from/to the monitor node should be null.

The link capacity of the monitor-forwarding nodes becomes:

       Γ M-F = min{(C M-F × β) -Q M-F F} F M-F = 0 (10)
-The available capacity on the link between the monitor-forwarding nodes is more than the amount of the monitored traffic:

Γ M-F ≥ θ. (11)

Case of MIMODog protocol with MIMO systems

The main goal of the MIMODog protocol is to avoid transmissions and monitoring under the same DoF within the interference area of the monitor node. Consequently, the link capacity of the monitor-relay nodes is similar to that of the SPACE-MAC protocol illustrated in equation 9. The resulting capacity is due to the reservation of the DoF for the monitoring process which demonstrates the advantage of MIMODog protocol. Indeed, to enable a successful monitoring regardless of the nodes activities in the interference area, a capacity reservation is necessary. The SPACE-MAC protocol is a DoF reservation mechanism on the link between a source node and its 1-hop relay node. However, the MIMODog protocol introduces an additional DoF reservation on the predecessor link.

Theoretical results of monitoring process

In this section, we present the obtained theoretical results related to the monitoring and its impact on the network capacity.

Average monitor nodes' number

Let G = (V, E) be a geometric graph with node set v and edge set E, where |V| = n and there is an edge between two nodes u and v with a probability P( uv ). This probability expresses the relationship between the power of the received signal and the distance. If we apply the the log-normal shadowing model, P( uv ) can be given as [START_REF]Degree distribution and hopcount in wireless ad-hoc networks[END_REF]:

P( u -v ) = 1 2 1 -erf 3.07 ln( u -v ) ξ , ξ σ η (12)
where:

             u -v is the normalized distance σ
is the standard deviation of shadowing η is the path-loss exponent

In our work, we consider ξ = 0. In other words, we consider only the case where two nodes with distances less than the normalized distance 1 are connected. The mean number of hops is given by [START_REF]Degree distribution and hopcount in wireless ad-hoc networks[END_REF]:

       hopcount = ln(n) node-degree node -degree = 2(n-1) m(m-1) m i=1 m j=i+1 P( x i -x j ) ( 13
)
where m is the number of small squares containing at most one node.

Any traffic between two nodes in the network has at mean hopcount intermediate (forwarding) nodes and so hopcount potential monitor nodes. According to the number of sourcedestination pair, say K, the average monitor nodes' number, N MN, is

N MN = k × hopcount. ( 14
)
Regardless of the used protocol (IEEE802.11 DCF, Space-MAC and MIMODog), equation 14 remains valid. Using equation 14 and under different values of the number of nodes and degrees of adjacency, we obtain the results shown in figure 14.

In order to study the number of monitor (detector) nodes whatever the monitoring mechanism model used (SISO, MIMO, and MIMODog), we plot in figure 10 the number of monitor nodes (N MN) according to the network density and degrees of neighborhood with different numbers of network traffic flows. We remark that the N MN increases not only with the network density, but also with the number of network traffic flows. However, the N MN significantly decreases when the degree of neighborhood increases. These results are coherent because when the degree of neighborhood is significant the number of potential relayer nodes also increases.

The impact of monitoring on the capacity

We assume that n nodes are randomly located on the surface of a square flat torus unit area. We use this geometric topology to avoid edge effects, which otherwise complicates the analysis. Each node selects a destination randomly to which is sends λ(n) bits/s. We use slotted time for transmission.

The average length of of each source-destination pair in terms of links is hopcount + 1. The average required capacity over the entire network for λ(n) successful delivery is at least as follows:

• for 802.11 DCF MAC with single antenna: (hopcount + 1)nλ(n);

• for SPACE-MAC with M antennas: (hopcount+1)nλ(n)

M

. This is because each node can use all its DoFs for spatial multiplexing, and so the total required capacity is divided by the number of DoFs;

• for MIMODog protocol with M antennas: (hopcount+2)nλ(n) M . As with the SPACE-MAC, the total required capacity is divided by the number of DoFs. However, to allow monitoring without interference, MIMODog protocol extends the interference area of the forwarding node by adding the interference area of the monitor node (see section 4). Consequently, the used protocol interference model becomes distance-(H+1) from the forwarding node, rather than distance-H. In this case, the nodes in the interference area of the monitor node assume the impact of the distance-(H+1) as an increase in the number of hops.

In figure 11, we plot the average consumed capacity in the network according to the network density and the average hop count with different models of monitoring mechanism: SISO (M = 1), SPACE-MAC, and MIMODog (the number of antennas is set to 2). We remark that the required capacity in the case of MIMODog is more important than the case of SPACE-MAC particularly when the average hop count increases. However, the worst results are obtained in the case of SISO compared to both models based on MIMO technology.

In order to study the impact of the number of antennas (M), we plot in figure 12 the consumed capacity with different monitoring models. We remark that the gap between the consumed capacity in the case of MIMODog and SPACE-MAC remains constant when the number of antennas increases.

These results illustrate that the monitoring mechanism is costly if we want to reach 0% of misobservation, because MI-MODog consumes DoF resources to ensure the monitoring process.

Asymptotic study of the number of monitor nodes

In this subsection, we focus on the MIMODog asymptotic study of the lower and upper bound of monitor (detector) nodes' number.

We consider a random multi-hop MIMO ad hoc network with n nodes, where each node, equipped with M antennas, is randomly located in a unit square area. Each node acts as a source node and transmits data to a randomly chosen destination node. The per-node throughput λ(n) is defined as the minimum data rate that can be sent from each source to its destination via multi-hop routing. The maximum data rate that a single data stream can support is W. We assume that a node's transmitter is limited to a transmission range r(n). When a source node cannot transmit data to its destination node in one hop, multi-hop routing is needed to relay the data. Each node also has an interference range (1 + ∆)r(n), where ∆ is non negative-constant.

Lower bound of the number of monitor nodes

In [START_REF] Gupta | The capacity of wireless networks[END_REF], Gupta and Kumar showed that a capacity lower bound for a single-antenna ad hoc network is Ω( W √ n ln n ) by constructing a feasible routing and scheduling scheme. Thus, by adopting the same routing and scheduling scheme in our MI-MODodog ad hoc networks as in [START_REF] Gupta | The capacity of wireless networks[END_REF], a number of monitor nodes lower bound of Ω( M √ n ln n ) can be obtained.

Upper bound of the number of monitor nodes

As shown in [START_REF] Jiang | On the asymptotic capacity of multi-hop mimo ad hoc networks[END_REF], we partition the unit square of the network into small squares, with the size of each small square being cleverly chosen so that the maximum data rate that can be received by the nodes inside the small square can be accurately computed. This method enables to estimate the number of monitor nodes. Proof. Based on [START_REF] Jiang | On the asymptotic capacity of multi-hop mimo ad hoc networks[END_REF], for a square with a side length 1/⌈ √ 2 ∆.r(n) ⌉ (as shown in Figure 13), the maximum number of total data streams that can be received by the nodes inside the square at any time slot for any routing scheme is not greater than M regardless of the number of receiving nodes inside the square. Using our MIMODog, the presence of a monitor node in the • when the number of nodes increases, the upper bound of monitor nodes decreases. This is explained by the fact that the network is more and more dense with a high multihop connectivity and so MIMO interference cancellation is limited.

Conclusion

In this paper, we propose a new scheme called MIMODog based on SPACE-MAC protocol for exploiting adaptive antenna arrays in wireless multi-hop networks using a multiparty propagation channel to efficiently detect the selfish (noncooperative) nodes. The proposed scheme nullifies the beam to competing nodes to enable concurrent transmissions and monitor in the same collision domain. Moreover, we highlight and discuss the different negative impacts on the monitoring process particularly the origin of the false alarm (misobservation) with: DCF MAC, SPACE-MAC and MIMODog. We have shown that the required capacity in the case of MIMODog is more important (but remains close) than the case of SPACE-MAC, due to the allocation of some DoF resources for monitoring. In addition, we have investigated the monitoring process capacity by using conflict graph model. The number of monitor nodes scaling laws for MIMO ad hoc networks with M antennas is evaluated by using asymptotic study. We have shown that the number of monitor nodes is at most M -1 times larger based on MIMODog than with SISO 802.11 DCF MAC.

In our future works, we plan to evaluate the proposed solution by extensive simulations with different parameters like the density of selfish nodes, mobility models, and traffic models.

  1 RTS: Request to Send / CTS: Clear to Send

Figure 1 :

 1 Figure 1: An ad hoc network scenario

Figure 2

 2 depicts the monitoring problem based on 802.11 DCF MAC.

Figure 2 :

 2 Figure 2: Monitoring problem based on 802.11 DCF MAC protocol

Figure 3 :

 3 Figure 3: Monitoring problem based on SPACE-MAC protocol

Figure 4 :

 4 Figure 4: Monitoring mechanism with MIMODog

Figure 5 :

 5 Figure 5: Access Control Packets

Figure 7

 7 shows an example of a connectivity graph. Node b acts as monitor node and node c as forwarding node. We use the conflict graph to model the interference relationships between links and called it the Links Conflict Graph LCG. Every link in connectivity graph G is represented by a node in conflict graph LCG. Two nodes in G are connected by an edge if the nodes corresponding to links in G cannot have simultaneous transmissions according to the protocol's interference model.

Figure 7 :

 7 Figure 7: An example of a connectivity graph.

Figures 8 .Figure 8 :

 88 Figures 8.a and 8.b show distance-2 and distance-4 LCGs for the network topology presented in Figure 7. The link between the monitor-forwarding nodes, 2 = (a, c), belongs to three cliques 2 in Figure 8.a and to one clique in Figure 8.b. We note that, as distance-H of the interference model increases, the number of cliques decreases.

  For example, in Figure 8.a, let the allocated flow on each LCG-node {1 = (a, b), 2 = (b, c), 3 = (c, d), 4 = (a, e), 5 = (a, f ), 6 = (d, g), 7 = (

Figure 9 :

 9 Figure 9: LCGs for network topology.

Figure 10 :

 10 Figure 10: The number of monitor nodes (N MN) versus network density with different degrees of neighborhood.

Figure 11 :

 11 Figure 11: The average consumed capacity in the network according to the network density and the average of hop count with λ(n) = 4 Mbits/s and M = 2.

Lemma 1 . 2 ∆

 12 For a square with a side length 1/⌈ √ .r(n) ⌉, there are at most M -1 times larger monitor nodes based on MIMODog than with SISO 802.11 DCF MAC.

Figure 14 :

 14 Figure 14: The upper bounds of the number of the monitor nodes versus the number of nodes

Table 1 :

 1 Notation

	Symbol	Definition
	G	the connectivity graph
	G-node	a node in the connectivity graph
	G-link	a link in the connectivity graph
	LCG	the links conflict graph
	LCG-node a node in the link conflict graph
	LCG-link a link in the link conflict graph
	C i	the capacity on link i within a single-hop network
	F i (t)	the instantaneous flow rate utilization on link i at time t
	β	the scaling factor
	Q i	the incidence matrix of link i
	Γ i	the available capacity on link i
	Γ M-F	the available capacity on the link between the monitor-forwarding nodes

A clique in the conflict graph is a set of vertices that mutually conflict with each other.

square area consumes exactly one DOF. Let P j (t, i) be the probability that node i is a monitor at time t in a square j. The number of monitor nodes in square j is given by n i=1 P j (t, i). Consequently, the new maximum number of total data streams that can be received by nodes inside a square j is not greater than Mn i=1 P j (t, i) (M ≥ n i=1 P j (t, i)). In the same square and using SISO systems, the maximum number of total data streams that can be received by nodes inside the square is 1. Only one monitor node can be functioning properly. So, there are at most M -1 times fewer monitor nodes with SISO 802.11 DCF than with MIMODog.

Based on Lemma 1, we can now compute the maximum number of monitor nodes that can be supported in the unit square network by taking the sum of the number of monitor nodes among all small squares. Theorem 1. For a random multi-hop MIMO ad hoc network, a number of monitor nodes upper bound for all possible routing and scheduling schemes is O( M √ n ln n ) with a high probability when n → ∞.

Proof. The proof is similar to the proof of Theorem 1 in [START_REF] Jiang | On the asymptotic capacity of multi-hop mimo ad hoc networks[END_REF]. We can easily obtain this equation:

where N MN is the number of monitor nodes and D is the average length of source-destination lines.

Combining the lower and upper bounds of the number of monitor nodes, we can see that the number of monitor nodes in a random multi-hop MIMO ad hoc network with n nodes is Θ( M √ n ln n ).

Numerical results

By running 1000 instances, we obtain the average length of source-destination lines D = 0.52 (see [START_REF] Jiang | On the asymptotic capacity of multi-hop mimo ad hoc networks[END_REF]). We set ∆ = 1. Using equation 1 and under different values of M = 1, 2, 3, 4 we obtain the results shown in figure 14. With M = 1 antenna, MIMODog is exactly the 802.11 DCF MAC. We can extract 2 elements :

• when the number of used antennas increases, the number of monitor nodes increases,