M.-L Gallin-Martel

Y Grondin

N Gac

Y Carcagno

L Gallin-Martel

D Grondin

M Marton

J.-F Muraz

O Rossetto

F Vezzu

two prototype modules in coincidence of a

Keywords: Positron emission tomography (PET), Medical imaging equipment, Liquid Xenon PACS : 87.57.uk

Small animal imaging aims at observing disease mechanisms or drugs functions. One of the powerful tools used for this purpose is Positron Emission Tomography (PET). Coupled with specific radiotracers injected in the organism, it enables to image molecular biological processes in vivo. One of the challenges of small animal imaging is to obtain high resolution images because of the small dimensions of mouse or rat organ structures. Several PET scanners have been developed in the last decade such as MicroPET II [1] which achieves a resolution less than 0.6 mm RMS. However these PETs are made of several longitudinal slices of radially-oriented crystals. Our detector that is extensively described in [2] uses the scintillation properties of the Liquid Xenon in an axial geometry to provide a three dimensional gamma reconstruction free of parallax error. Several groups are studying using liquid xenon for PET imaging but their approach is slightly different [3,4]. Our detector is sensitive only to the scintillation process and it relies on specular reflections within the light guides.

In [2], only performances of a single LXe prototype module in terms of energy, time and spatial resolution capabilities were estimated. Since then, a new experimental set-up with two LXe modules in coincidence has been built. Present paper deals with its characterization. The first image reconstruction of a 22 Na point source located in the centre of the field of view is performed using experimental data sets. The reconstructed image resolutions in the axial, radial and tangential directions are measured. Finally, methods to enhance the image quality are discussed.

The experimental set-up for the test of the two modules in coincidence

The experimental set-up used for the test of both prototype modules in coincidence is described on Fig. 1. The Xenon is grade 4.8 (i.e. a purity better than 20 ppm), which is pure enough for our detector concept [2,5]. As in [2], Xenon is liquefied in a compressor then transferred to a container inside the cryostat. The temperature inside the cryostat is kept around 165 K via a liquid nitrogen heat exchanger. The temperature fluctuation is less than a few tenths of a degree. Both Xenon containers (LXe modules) are stainless steel cylinders 50 mm long and 40 mm in diameter, closed at each end with a 3 mm thick suprasil window. Each container can house various types of cells (see Fig. 1). An experimental test bench has been built to allow the x and y measurement and furthermore to evaluate the z-axis localization, the energy and time resolutions.

In [2] the prototype module was equipped with a matrix of 40 UV light guides each featuring a 5 x 2 mm 2 cross-section. In present paper both prototypes modules were equipped with a matrix of 16 UV light guides each featuring a 5 x 5 mm 2 cross-section By doing so, better energy (σ E /E) and timing (σ t ) resolutions can be achieved but to the detriment of spatial (σ x, σ y and σ z ) resolutions.

A 22 Na source is mounted on a small carriage moving along the z-axis, in the cryostat. The zaxis of the two LXe modules and the one of the 22 Na small carriage are parallels and coplanars. Such an experimental set-up permits us to mimic a 3D image reconstruction of the Finally, it works as if 8 pairs of LXe modules were in coincidence.

The VUV photons are collected with a PSPMT at each LXe module end as in [2]. Hamamatsu provided us with four prototype tubes, belonging to the R8520-06-C12 series [START_REF]8 Rue du Saule Trapu[END_REF]. They are square high resolution position sensitive photomultiplier tubes which are characterized by their compactness and an excellent spatial resolution. Their QE is around 25 % (against 20 % in [2]). These PSPMTs exhibit a 6(X) + 6(Y) cross plate anode. The light pulse position on the photocathode is derived from a barycentre calculation on the X and Y anode signals.

A dedicated read-out electronics was designed to process the PSPMT signals (see Fig. 2). 3 Experimental results

Modules'light guide modelling

It has been demonstrated in the experimental study of a single PET LXe prototype module [2] that the amount of light collected at a light guide end decays exponentially with the axial z coordinate of the photon interaction point. This exponential model is described by :

ref z e I z I λ - = 0 ) ( (1)
The z reconstructed coordinate can be found using the exponential model and the effect of Xenon impurities can be easily introduced in the equations that describe the model [2].

Besides, the main advantage of this model is to qualify the light propagation medium by a single λ ref parameter. The attenuation length λ ref depends on the light guide geometry and its reflectivity.

The number of photoelectrons collected at each module end (module 1 or module 2) is given by:

⋅ - ⋅ = - ⋅ - ⋅ ⋅ = 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 0 2 , 1 2 , 1 2 , 1 2 , 1 0 2 , 1 2 , 1 exp exp 2 . ) ( exp exp 2 ) ( ref ref r ref ref l z l N A z N z l N A z N λ λ λ λ (2)
Where :

• the origin of the z 1,2 axial coordinates is taken at the centre of each module (1 or 2),

• A 1,2 is a constant depending on LA and on the PSPMT quantum efficiency for a given module (1 or 2),

• 2 , 1 0
N is the amount of scintillation photons for a given module (1 or 2),

• l is the module half length.

Unlike in [2], there is no collimation in the z direction so the proposed method for the energy calculation can no longer be used. However, thanks to the exponential model built for the light guide modeling, the reconstructed value of the energy E rec is proportional to

l r N N : l r rec N N E ∝ (3)
Where

• N r is the number of photoelectrons collected at the right end of a given module,

• N l is the number of photoelectrons collected at the left end of the same module.

The l r N N distribution for module 1 is given in Fig. 3 for the z source = 0 position. In the present analysis, the axial z 1,2 reconstruction relies on the dynode signal asymmetry measurement. The photopeak selection in the l r N N distribution obtained for each module permits to level down the background in the z 1,2 reconstruction.

Finally, the reconstructed axial coordinate z rec of the source position is given by:

2 1 1 2 2 1 rec rec rec rec rec rec rec y y z y z y z - - = (4)
where:

• z rec1 (respectively z rec2) is the reconstructed axial coordinate of the 511 keV photon emitted by the source in module 1 (respectively in module 2),

• y rec1 (respectively y rec2 ) is the reconstructed transaxial y coordinate (assimilated with the y coordinate of the guide centre) of the 511 keV photon emitted by the source in module 1 (respectively in module 2).

Calibration

PSPMT and light guides calibration for each module were carried out using Xenon scintillation light as in [2]. The 16 light guides separation is shown on Fig. 4. All of the four guides distributed along the x and y axis can be isolated. As in [2], events reconstructed between two guides are induced by -rays undergoing multiple scattering in several guides.

Such events can not be processed by PSPMTs. Consequently, a Gaussian fit is performed around each peak in the barycentre distributions illustrated by Fig. 4. Only events within a 1.5

σ range around each peak are used for further analysis. A simulation carried out with the GATE toolkit [2,[START_REF] Jan | [END_REF][START_REF] Jan | GATE User's Guide[END_REF][START_REF] Jan | [END_REF] shows that the fraction of the events within the 511 keV peak that are cut out is about 50 %. This efficiency loss is due to the localisation method based on barycentre calculation. A part of these events could be properly processed by a pixelized photodetector [START_REF] Braem | [END_REF].

The PSPMTs mapping consists in the determination of a coefficient table that normalizes the right and left responses for each guide when the gamma interaction is located at the module centre.

The light guide attenuation length λ ref1,2 for each module can be derived from Eq. ( 5):

2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 ln ref l r z C N N λ = × (5)
Where the C 1,2 coefficient is a measurement of the asymmetry between the left and right end of each guide. The λ ref1,2 and C 1,2 coefficient have been calculated using the same calibration method than the one described in [2].The data analysis showed that for each module the 16 λ ref values exhibit a 30 mm mean value and a 1.0 mm standard deviation.

The axial resolution with the two modules in coincidence

The distribution of the reconstructed z rec position for a source located at z source = 0 is shown in Fig. 5. A Gaussian fit of this distribution for various values of z source provides the axial resolution of the two modules experimental set-up. Fig. 6 illustrates the RMS value of the z reconstruction versus the source z position while the Fig. 7 presents the reconstructed position z rec as a function of the source position z source . The average value for σ zrec is around 2.0 ± 0.1 mm.

The energy resolution and reconstruction with the two modules in coincidence

A Gaussian fit of the photopeak leads (see Fig. 3 for the z source = 0 position) to the σ/ resolution where σ and are respectively the RMS and the mean value of the fitted distribution. The mean value for the energy resolution is about 10.4 % for the whole module in experimental data analysis (see Fig. 8).

Present results can be compared with values obtained in [11] where both light and charge collection are operated. In [11], detectors use ultra pure xenon, the charge collection makes on line purification necessary on the contrary to our experiment [2]. Moreover in [11], the photodetectors are immersed in the LXe. In these conditions, the energy resolution reached is at the level of 8 % at zero field. Consequently, present results and [11] exhibit a factor of 1.3 in the energy resolution. But it must be stressed that with our concept of detector the zcoordinate determination relies on UV photon absorption [2] inducing a degradation of the energy resolution.

Time resolution with the two modules in coincidence

The coincidence time resolution has been measured using the TDC signal. Fig. 9 summarizes the coincidence time resolution measurement along the modules by step of 5 mm. A mean value of about 300 ps has been found.

Summary of the performances and discussion

A two LXe modules in coincidence experimental set-up was built, the optimization of its performances relies on three relevant parameters:

• the axial resolution,

• the energy resolution and reconstruction,

• the time resolution.

In present experiment, a z resolution of 2.0 ± 0.1 mm for a guide to guide coincidence can be reached. Immersing the PSMT in LXe would be the next step towards improving the axial resolution . Actually, the objective consists in optimizing the detector and raising the resolution at the level of the expected value of 1.5 mm found in [START_REF] Jan | [END_REF] where a complete simulation of the detector was done assuming that the PSPMTs were immersed in LXe.

A 10.4 % RMS energy resolution was found in average on the modules' length. In [2], it was demonstrated that the energy calculated using the formula l r N N exhibits a dependence on the z-coordinate, the variation of energy is about 10 % on the whole module. This is a limit of the exponential model. As a consequence, in the present analysis the 10.4 % RMS value found for the energy resolution which relies on the shape of the l r N N distribution is a pessimistic estimation given by the exponential model.

In [12] a time-of-flight positron emission tomography using liquid xenon scintillation is studied. To improve the energy resolution the PSPMT signals have to be corrected according to the location of the interaction point in the detector. The energy resolution after correction is about 11 % RMS. In the present work, our concept of detector favors specular reflections of the UV photons emitted at the interaction point in the light guides and then focus the light onto the PSPMTs. At the difference of [12], the energy resolution is found to be free of the localization of the interaction point no correction is needed. In Fig. 8, the shape of the distribution is rather flat. However, the energy resolution obtained with our detector is very comparable to the [12] corrected one since it is about 11 % RMS. In addition, it should be noticed that in our case the z reconstruction relies on UV photon absorption which degrades the energy resolution.

A 300 ps RMS time resolution in average on the modules' length was measured. It is 1.8 times better than in [2]. But in [2] only a single module was under test and coincidences were made with a LYSO crystal with a poorer resolution. In present experiment, such a resolution has permitted us to level down the random coincidences in the guide to guide experimental data analysis. In [12], the average value of timing resolution is 235 ps RMS. To optimize the time resolution, as to optimize the energy resolution, it is necessary to minimize the amount of UV photons absorbed before they reach a photo detector. In our case, an improvement in the UV light detection (for example by immersing the PSPMTs in the LXe) improves all the resolutions ( E , t and z ). Another way to increase the number of photoelectrons is to increase the λ ref1,2 to minimize the UV photon absoption in the detector. But in this last case, the z resolution will be degraded. Consequently, the optimization of the detector is a matter of compromise.

Image reconstruction

Sampling and image reconstruction

The experimental data were used to quantify the performance of PET LXe on reconstructed images of 22 Na point source located in the centre of the field of view. The image reconstruction of the source located at z = 0 (centre of the modules) was performed with our implementation [13,14] of the iterative ML-EM algorithm [15]. We first created 2D sinograms by performing a virtual rotation of the projections obtained from the 8 independent runs as explained in section 2. Consequently, a full set of projections is obtained as for a complete scanner. A 2D sinogram is constructed by filling a table with the Radon variables (s,φ) identifying the LORs in a transverse plane (see Fig. 10). From the coordinates (x 1 ,y 1 ,z 1 ) and (x 2 ,y 2 ,z 2 ) of the coincident gamma photons in the referential (x,y,z) of the experimental test bench, we determine (s,φ) by the following expressions:

s = det(OP 1 ,OP 2 ) || P 1 ,P 2 || ; cos(φ) = (y 1 -y 2 ) || P 1 P 2 || ; sin(φ) = (x 1 -x 2 ) || P 1 P 2 || (6)
where:

• P 1 and P 2 are the geometrical normal projections of the coincident gamma photons on a transverse plane,

• det(OP 1, OP 2 ) is the determinant of the two vectors OP 1 and OP2,

• ||P 1 P 2 || is the vector P 1 P 2 norm.

Each 2D sinogram corresponds to the source projections in a transverse plane identified by the z-coordinate calculated by the equation (4). A sampling path of 1 mm in the z direction has been used to divide the axial field of view in 50 slices, indexed from 1 to 50 in order to have discrete positive values for the image reconstruction step. Hence, a stack of 50 sinograms is obtained. As far as the Radon variables are concerned, a regular sampling that satisfies the Nyquist criteria has been applied. The φ variable is sampled according to the total number of light guides composing a module layer in the case of a complete scanner. The number of projections is therefore 64 (16 modules, 4 light guides in a layer). Due to the field of view size, the number of sampled LORs in one projection is a fourth of 64, it means 16 [START_REF] Townsend | The theory and practice of a 3D PET[END_REF]. Therefore in Radon space (s,φ), the sinogram is of size 16 x 64 x 50 in s, φ and z respectively as shown on Fig. 11. However a common practice in image reconstruction is to interleave the lines of sinograms, which results in dividing by two the number of projections and in doubling the sampling of parallel LORs [START_REF] Townsend | The theory and practice of a 3D PET[END_REF]. The size of the sinogram is therefore 32 x 32 x 50 in s, φ and z respectively.

Axial, radial and tangential resolutions

Fig. 12 and Fig. 13 show the reconstructed image of the source zoomed in axial and transverse views respectively. The axial image horizontal axis corresponds to the z coordinate of the experiment. A linear profile through the point source in this direction shows that the source is well located by the reconstruction process (Fig. 12) at the slices of index 25 and 26.

It corresponds to the centre of the module discretized in 50 divisions. With a 1 mm axial voxel dimension, an axial resolution z of 2.20 ± 0.03 mm is calculated.

Fig. 13 shows the radial and tangential linear profiles through the transverse image. With a 2.5 x 2.5 mm 2 transverse voxel dimension, radial and tangential resolutions of 1.27 ± 0.02 mm and 1.13 ± 0.01 mm are calculated respectively.

Discussion

Several approximations have been made to fill out the sinogram in the Radon space from the coordinates (x 1 ,y 1 ,z 1 ) and (x 2 ,y 2 ,z 2 ) of the two coincident gamma photons. Indeed, our classical implementation of ML-EM algorithm required sinograms in Radon space. It implies (i) interleaving lines of the sinogram, (ii) an arc correction [START_REF] Townsend | The theory and practice of a 3D PET[END_REF], (iii) integration of LORs from several layers into one layer of detectors and (iv) missing data between blocks as illustrated in Fig. 11. The first one is a classical approximation [START_REF] Townsend | The theory and practice of a 3D PET[END_REF], the second and the third approximations correspond to interpolations of the raw data and are therefore quite smooth approximations. The last one is the biggest approximation as illustrated on the sinogram in Fig. 14. Several methods exist to compensate the missing data as exposed in [1] and in [START_REF] Karp | [END_REF].

The geometry of the acquisition could be directly modeled in iterative reconstruction methods as it has been done in [1]. It would allow a better quality of reconstruction.

Our acquired data have a quite low signal to noise (only 600 000 events for the image reconstructed). Regularized methods would be therefore appropriate to enhance image quality [18]. The goal of regularization is to add a constraint of smoothness on the reconstructed object in order to be less sensitive to the noise in the data.

The reconstructed image resolutions are at the level of 2.2 mm RMS, 1.3 mm RMS and 1.1 mm RMS respectively in the axial, radial and tangential views. The 2.2 mm axial resolution deduced from the image reconstruction is in nice agreement with the one found thanks to the exponential model and exposed in section 3.3. However, the radial and tangential resolutions are rather optimistic. Indeed, the light guides having a side length of 5 mm, the standard deviation of a flat distribution is given by 5mm/ 12 = 1.44 mm, which is theoretically the best result that it is possible to achieve. This positive bias, which is observed between our measured resolutions and the theoretical limit, is assumed to be caused by the particular position of the source located in the centre of the field of view. It has been reported that spatial resolution measurements should be done with sources off-centred [19], otherwise, the high density of LORs in the centre of the field of view could affect the reconstructed source image intensity. In our case, it has not been possible to test this configuration due to the incomplete modules ring.

The reconstructed image resolution for a MicroPET II according to [20] ranges from 0.35 mm RMS at the centre of the field of view to 0.63 mm RMS in the radial direction, 0.50 mm RMS in the tangential direction and 0.60 mm RMS in the axial direction at 1 cm offset from the centre of the field of view. Improvements are therefore needed to make the PET LXe more competitive. This should be done mainly by modifying the experimental set-up. Better radial and transaxial resolutions can be achieved by using light guide matrix with a higher number of guides each featuring a smaller cross-section (present one is 5 x 5 mm 2 ) in the transaxial plane of the camera. A better axial resolution is as well reachable if in the next set-up version the PSPMTs are immersed in the LXe (see section 3.6). These two improvements must be envisaged in the future.

Conclusion

Two modules of a parallax error free PET using LXe in an axial geometry have been built.

Each module is optically subdivided into 16, 5 x 5 mm 2 , MgF 2 -coated aluminium UV light guides read on each end by PSPMTs. The light guide modelling has been already extensively discussed in [2]. An experimental test bench has been built. The two modules are read in coincidence to evaluate the axial localization (z coordinate), the energy and time resolutions.

Experimental data have been analyzed using the exponential model. In data analysis, the resolution on the z coordinate has been found to be 2.0 mm once the photopeak is isolated, the energy resolution in average is 10.4 % on the data and time resolution is about 300 ps. The Improvements are needed to make the PET LXe more competitive when compared to

MicroPET II. The aim of present work was to demonstrate the feasibility of this new concept of axial geometry that would provide a three-dimensional gamma reconstruction free of parallax error. The next step would be now to improve all the resolutions to make possible, with the proposed LXe PET, to image the distribution of molecular probes at sub-millimetre resolution in laboratory animals. To reach this goal two solutions are to be envisaged simultaneously : using for example 2 x 2 mm 2 light guides matrices [2] in replacement to the present 5 x 5 mm 2 ones to improve the spatial resolutions and immersing the PSPMT in the liquid xenon to improve the energy and time resolutions.

List of Figures.

1. The experimental set-up used for the test of the two prototype modules in coincidence.

  Experimental results and first 22Na source image reconstruction by two prototype modules in coincidence of a liquid xenon positron emission tomograph for small animal imaging M.L. Gallin-Martel, Yannick Grondin, Nicolas Gac, Y. Carcagno, L. Gallin-Martel, D. Grondin, M. Marton, J.F. Muraz, O. Rossetto, F. Vezzu To cite this version: M.L. Gallin-Martel, Yannick Grondin, Nicolas Gac, Y. Carcagno, L. Gallin-Martel, et al.. Experimental results and first 22Na source image reconstruction by two prototype modules in coincidence of a liquid xenon positron emission tomograph for small animal imaging. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2012, 682, pp.66-74. 10.1016/j.nima.2012.04.006. hal-00703776 1 Introduction

  This multi channel module relies on 52 Charge to Digital Converters (QDC) for the charge measurement of anodes and dynodes signals. Each QDC is composed of a gated integrator coupled to a 12 bit Analog to Digital Converter (ADC). The electronics for the experimental trigger comprises two Constant Fraction Discriminators (CFD) and coincidence module (coinc). Each CFD is fed by the analog sum of the dynode signals of the corresponding cell. A Time to Digital Converter (TDC) computes the time difference between the events in the two cells. The coincidence module uses CFD output signal to trigger the QDCs, the TDC and the PC-based data acquisition.

  experimental data were used to quantify the performance of PET LXe on the reconstructed image of 22 Na point source located in the centre of the field of view. Resolutions of about 2.2 mm RMS, 1.3 mm RMS and 1.1 mm RMS have been found respectively in the axial, radial and transaxial views of the reconstructed image. A complete PET LXe composed of 16 modules was fully simulated in[START_REF] Jan | [END_REF], the detector sensitivity was found to be equal to 1 % in the central region in the energy range [250keV, 650 keV].

4 . 5 . 8 . 9 .Fig. 1 . 1 Fig. 2 .

 4589112 Fig. 1. The experimental set-up used for the test of the two prototype modules in coincidence.

2 1Fig. 3 .

 23 Fig. 3. Spectrum of

Fig. 4 .

 4 Fig. 4. The x and y barycentre distributions computed by the data acquisition system and

Fig. 5 .

 5 Fig. 5. Distribution of the reconstructed Zrec position for a source located at Zsource = 0experimental data.

Fig. 6 . 3 DATAFig. 7 .Fig 8 .

 6378 Fig. 6. RMS value of z rec versus z source position.

Fig. 9 .

 9 Fig. 9.Time resolution scanned over the two modules by 5 mm steps.

Fig. 10 .

 10 Fig. 10. Definition of the Radon variables (s, φ).

Fig. 11 .

 11 Fig. 11. Multi-layer and Radon representation of LORs.

Fig. 12 .

 12 Fig.12. The reconstructed source showed in the axial view (top) and the axial profile (bottom).

Fig. 13 .

 13 Fig. 13. The reconstructed source shown in the transverse view (top) and the radial and tangential profiles (bottom).

Fig. 14

 14 Fig. 14 The central sinogram of the experimental source. The missing data correspond to the lines of low gray level values segmenting the vertical belt of data. Moreover, we notice a shift in the sinogram meaning that the source is slightly off-centered.

Acknowledgments

Financial support from Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), Région Rhône-Alpes and Université Joseph Fourier is acknowledged. The authors would like to thank Pr. Laurent Desbat from TIMC -IMAG, Techniques de l'Ingénierie Médicale et de la Complexité -Informatique, Mathématiques et Applications de Grenoble -Unité Mixte de Recherche CNRS Université Joseph Fourier UMR 5525, Pr. Michel Desvignes and Dr. Stéphane Mancini from GIPSA-LAB : Grenoble Images Signal Parole et Automatique -Unité Mixte de Recherche CNRS Université Joseph Fourier Université Stendhal -UMR 5216, for their contribution in setting up the image reconstruction software.

Dr. Sébastien Jan from SHFJ -Service Hospitalier Frédéric Joliot -CEA Orsay-is thanked for many interesting conversations and bringing his help in using the GATE simulation toolkit.

The authors are grateful to André Menthe from the LPSC laboratory for his implication in the technical support of the mechanical part of the experimental test bench. André Braem and his team at CERN are thanked for providing the LPSC laboratory with new samples of MgF 2coated aluminum leaves used for making the light guides.


Emmanuel Amador
email: emmanuel.amador@insa-rennes.fr

Christophe Lemoine

Philippe Besnier

OPTIMIZATION OF IMMUNITY TESTINGS IN A MODE TUNED REVERBERATION CHAMBER WITH MONTE CARLO SIMULATIONS

Keywords: Reverberation chamber, immunity testing, Monte Carlo simulations, probability of failure, optimization

In this article, we show that an estimation of the probability of failure of a device under test can provide an accurate information about its susceptibility. Monte Carlo simulations give statistical results on the quality of the estimation and a virtual equipment under test is tested numerically with this approach. We propose a modus operandi that reduces the duration of the testing and increases its accuracy. Measurements on a real device confirm our predictions.

INTRODUCTION

Mode-tuned reverberation chambers (RCs) are used for EMC certifications purposes to perform immunity testings on an equipment under test (EUT). During an immunity testing, the EUT is placed in the working volume of the RC. For every position of the paddle, and for every frequency tested, the power injected in the RC is gradually increased until a failure is detected on the EUT. By increasing the injected power, the level of the E-field of the impinging waves on the EUT is increasing. The rotation of the stirrer ensures that the E-field is homogeneous. It means that the angles of arrival of the waves on the device are uniformly distributed, the polarization is uniformly random and the statistical magnitude of the impinging E-field is uniform.

Most of the immunity testings are performed according to the IEC standard [START_REF]Reverberation Chamber Test Methods. International Electrotechnical Commission (IEC) Standard[END_REF]. Yet, few studies about a possible optimization of the modus operandi are available. In [START_REF] Aurand | Reverberation chamber immunity testing: A novel methodology to avoid accidental DUT damage[END_REF], the authors propose a new method for measuring immunity that increases the protection of the EUT against accidental damage. In this article, we try to reduce the duration of the testing and increase the accuracy of the estimation of the susceptibility of an EUT. We choose to use the probability of failure of a device during a testing to predict its level of susceptibility.

After a brief presentation of this probabilistic approach, Monte Carlo (MC) simulations give a good overview of its accuracy and allow to derive a modus operandi. A simulation of a measurement with a numerical model shows that such an approach can give an accurate measurement of the susceptibility and can reduce the duration of the testing. Finally a measurement is performed in an RC with a real EUT. This work should be regarded as a side result from [START_REF] Amador | Modèles de compréhension par la théorie des images des phénomènes transitoires et du régime permanent en chambre réverbérante électromagnétique[END_REF].

STATISTICS OF THE TESTING

Duration

An immunity testing is performed over N stirrer positions. The EUT is tested in a bandwidth ∆f that contains N f r frequencies. For each paddle position, we test N f r frequencies and N p injected power levels. The duration of the testing T can be written :

T ≈ N (∆T + N f r N p T p ) + 2π θ , (1)
where θ is the angular speed (in rad.s -1 ) of the stirrer, T p is the duration of the testing for one power step and ∆T is the time needed for the stabilization of the mechanical stirrer when the step by step motor stops. This article aims at optimizing the duration of a testing. If N f r cannot be reduced, the duration of the testing can be decreased by reducing N or N p . The probabilistic approach we propose in this article allows to find a relation between the number of power steps tested N p and the number of stirrer positions N . Because the magnitude of the electric field exceeds the susceptibility level of the EUT to get access to the probability of failure, this approach is reserved to EUTs that are not prone to be damaged very easily.

Statistics

In this part, we consider that every stirrer position is independent. It means that we use an ideal RC. A rectangular component of the electric field follows a Rayleigh distribution [START_REF] Lehman | The statistical properties of electromagnetic fields with application to radiation and scattering[END_REF][START_REF] Kostas | Statistical model for a mode-stirred chamber. Electromagnetic Compatibility[END_REF]. The probability density function of a random variable X following a Rayleigh distribution with parameter σ is:

f X (x) = x σ 2 e -x 2 /2σ 2 , with x ≥ 0. (2)
The EUT in our simulations is defined by a susceptibility level E s . The probability for one paddle position to have a received power greater than the level of susceptibility E s is given by:

p f = p(X > E s ) = 1 -CDF(E s ) = e -E 2 s /2σ 2 . ( 3
)
An immunity testing is done by choosing N paddle positions. We can estimate the probability of failure p f over N stirrer position by using:

p f = N f /N, (4)
where N f is the number of positions for which a failure of the EUT is detected. Then from (3) we can estimate the level of susceptibility E s :

E s = σ 2 ln(1/ p f ). ( 5
)
Because it is more convenient, we can express E s with the mean value of a rectangular component of the E-field

E m = σ π/2: E s = 2E m ln(1/ p f ) π . (6)
Figure1-(a) shows MC simulations of the ratio E s /E m versus the p f for different values of N . This figure shows that this probabilistic approach allows to measure susceptibilities between from 0.1E m to almost 2.5E m (for p f between 1 % and 99 %). Figure 1-( 1 shows that increasing N allows to increase the accuracy of the immunity estimation and increases the range of value tested and thus allows to reduce the number of power steps tested N p . the maximum error is not symmetrical with p f = 0.5, as shown in figure 1-(b), the error is smaller for values of p f near 0 than with 1p f . Consequently, in order to have a consistent accuracy, we will use the following rule of thumb in the next simulations and measurements and consider only values of p f that verify:

3 N ≤ p f ≤ 0.7 (7)
We can derive from ( 7) and ( 6) the number of test levels N p needed to achieve a good estimation of the susceptibility. For a given number of stirrer positions N , the maximum level of immunity tested and the minimum level are both given by:

E smax ≈ 2E m ln(N/3) π (8) E smin ≈ 2E m ln(1/0.7) π (9)
We note E mi the i-th mean value of the E-field in the chamber corresponding to the i-th level of power injected in the chamber. The successive levels tested E mi can be derived from the following geometric sequence:

E mi+1 ≈ E mi ln(N/3) ln(1/0.7) , (10)
and thus:

E mi (N ) = E m0 ln(N/3) ln(1/0.7) i E mi (N ) = E m0 (α(N )) i , (11)
where E m0 is the first level tested. An alternative and more efficient approach would be to choose the maximum mean field possible with one's equipment and compute the levels downward by using 1/α(N ).

NUMERICAL MODEL OF A MODE-TUNED CHAMBER FOR RADIATED IMMUNITY TESTINGS

Reverberation chamber and stirrer simulation

During an immunity testing in an RC, the EUT receives a random amount of power for a given position of the stirrer. The power received varies continuously when the stirrer is rotating. In order to simulate the power profile received by an EUT during a stirrer rotation, we first use the plane wave integral method [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF] to simulate a random E-field in an RC. The E-field is generated with a large number of plane waves. We use only one rectangular component of the E-field in order to simulate a linearly polarized EUT. In order to simulate the E-field profile received during a stirrer rotation, the E-field is collected along a circle of radius R. In [START_REF] Hill | Spatial-correlation functions of fields and energy density in a reverberation chamber. Electromagnetic Compatibility[END_REF], the correlation of the E-field in a plane wave spectrum is studied. The authors show that positions separated by approximately λ/2 are uncorrelated. If we want N i uncorrelated measurements, the radius R of the circle should be around:

R ≈ λ 2 N i 2π (12)
By knowing the value N i for different frequencies in a given RC, we can define the radius R that would allow to generate E-field profiles that fit approximately the E-field profiles of a real RC. The E-field received (E r ) is computed over M = 9000 positions on the circle. Figure 3 shows normalized by the mean E-field profiles as a function of the angular position θ on the circle of radius R for f = 100 and 500 MHz.

The immunity testing is simulated by using the E-field profiles generated. The number of stirrer positions is N and we choose randomly a first stirrer position among the M computed positions. The remaining N -1 positions are determined on the E-field profile.

Simulation of the EUT

The EUT is simulated by an arbitrary susceptibility level in a given frequency range E s (f ). During the testing, for a given frequency, if the electric field received E r is greater than E s (f ), the testing detects a failure. For a given frequency, we use an arbitrary susceptibility level to simulate an EUT. From 100 MHz to 1 GHz, the susceptibility level of the EUT varies arbitrary between 15 V.m -1 and 100 V.m -1 .

Results

We perform simulations with different numbers of stirrer positions and we use the geometric sequence (11) to determine the level tested. The first level E m0 is set to 10 V.m -1 .

Figure 4 shows the results of the testing with N = 10 stirrer position. According to (11), α ≈ 1.8 (a 5 dB step in terms of power). The mean value of the magnitude of the E-field are successively 10, 18, 32, 58 and 104 V.m -1 . We need N p = 5 power levels to get a complete picture of the immunity of the EUT. This figure shows that the error in the estimation of the susceptibility is around 50 % as expected from MC simulations presented in Fig. 1. The susceptibility is detected for 153 frequencies among the 180. Most of the frequencies for which a susceptibility is not detected are at low frequencies. The main reason is not that the susceptibility level of the EUT is high at these frequencies, but that the probability to obtain a high field at low frequencies with N = 10 stirrer positions is smaller than at high frequencies as expected from the profile of the magnitude of the E-field at 100 MHz presented in Fig. 3. If the testing is stopped for a given frequency once the level E mi allows to compute a probability of failure that verify the condition [START_REF] Hill | Spatial-correlation functions of fields and energy density in a reverberation chamber. Electromagnetic Compatibility[END_REF], the duration of the testing according to (1) with T p = 1 s, ∆T = 3 s and θ = 2π/60 rad.s -1 is around 6800 s, almost 38 s per frequency.

Figure 5 shows the results of the testing with N = 30 stirrer position. According to (11), α ≈ 2.5 (a 8 dB step in terms of power). The mean value of the magnitude of the E-field are successively 10, 25, 62 V.m -1 . We need N p = 3 power levels to get a complete picture of the immunity of the EUT. This figure shows that with N = 30 stirrer positions, the error is reduced and is around 20 %. The susceptibility is detected for 167 frequencies among the 180. The duration of the testing is around 13700 s, al- most 77 s per frequency. Finally, with N = 150, α ≈ 3.3 (a 10 dB step), the error is greatly reduced and a susceptibility is detected for 178 frequencies among 180. The duration of the testing is around 53000 s, almost 300 s per frequency.

EXPERIMENTS

Description of the EUT

In this section, we use a real EUT to perform measurements. This EUT consists of an electronic board that contains an operational amplifier (op-amp) as a comparator placed in a metallic enclosure. A 5 cm long monopole external antenna is connected to the circuit as shown in A schematic of the electronic circuit is given in figure 8. The antenna is associated with an envelope detector for filtering the high frequency disturbances received by the antenna, and also rectifying the signal. Without any disturbance, since V + > V -the op-amp delivers V s = 9 V. With disturbances leading to V -> V + , the op-amp provides V s = -9 V indicating a default. The signal V s is recorded with a digital oscilloscope and a home made program that controls all the experimental setup. The program returns either the value 0 in the case of no susceptibility, or the value 1 if a susceptibility is detected. The measurements are performed between 850 MHz and 1500 MHz. At these frequencies, the behavior of our chamber is ideal and measurements have shown that the rectangular components of the E-field follow a Rayleigh distribution. We choose to use N = 150 stirrer positions and the power injected in the chamber is increased gradually allowing to reach a magnitude of 110 V.m -1 for the mean value of the rectangular components of the E-field. Susceptibility measurements were performed with the same setup in a GTEM cell.

Measurements

Susceptibility measurements in an RC are based on a statistical estimation of the maximum level an EUT may have received over a number N of independent stirrer positions [START_REF]Reverberation Chamber Test Methods. International Electrotechnical Commission (IEC) Standard[END_REF]. The maximum value of a rectangular component of the E-field is a N -order statistics [START_REF] Lehman | Characterization of the maximum test level in a reverberation chamber[END_REF][START_REF] Harima | Statistical characteristics of maximum E-field distribution in a reverberation chamber[END_REF][START_REF] Orjubin | Maximum field inside a reverberation chamber modeled by the generalized extreme value distribution. Electromagnetic Compatibility[END_REF]. It is derived from the size N of stirrer positions and the mean value E m of the E-field in the chamber for a given injected power. The maximum power estimated grows with N . It means that an intrinsic quantity like the susceptibility level of an EUT is a function of the way the measurement is performed. Moreover, the uncertainty of the estimation of the maximum of a rectangular E-field component is large and decreases slowly with N .

Figure 9 offers a comparison of susceptibility measurements in an RC and in a GTEM cell. The susceptibility measured in a GTEM cell is given by the red continuous line. This curve is the result of five measurements at different positions in the testing volume of the GTEM cell. Because the coupling of the device with an external E-field is assured by the short vertical antenna, the GTEM cell should provide an accurate measurement of the susceptibility of the device. For every frequency, we keep the minimum value of the susceptibility measured among the five measurements. For technical reason we could not obtain an E-field greater than 80 V.m -1 in the GTEM. The measurement with N = 150 stirrer positions based on the standard maximum estimation is given by the blue discontinuous line. This estimation exhibits that are caused by the statistical estimation of the maximum of the E-field. The error can reach a factor of 4 (12 dB in terms of power) for some frequencies. The estimation based on our approach N = 10, 30, 150, with E m0 = 3 V.m -1 is given by the different markers. The number of tested levels N p varies from 7 to 3. The susceptibilities measured with our method do not exhibit oscillations and are in good agreement with measure-ments performed in the GTEM cell. Moreover, even if the maximum mean value of the E-field injected reaches 110 V.m -1 , with N = 150 positions, susceptibilities up to 190 V.m -1 are detected.

N

Max. approach Probabilistic approach 10 2000 s 1600 s 30 6000 s 2500 s 150 30000 s 10000 s Table 1. Estimation of the duration of the testings with different values of N .

Table 1 gives the duration of the testing using (1) for an approach based on the maximum and our probabilistic approach. For the measurements based on an estimation of the maximum, the successive power levels injected are separated by 3 dB. Starting from 3 V.m -1 , the number of power level tested varies from 6 to 11 before detecting a failure. The probabilistic approach uses a geometric sequence of power levels according to (11). Starting with E m0 = 3 V.m -1 , with N = 10, the number of power level tested for the different frequencies varies from 4 to 7. With N = 30 the number of power levels tested varies from 2 to 4. Finally with N = 150, the number of power level tested is between 1 and 3. Unlike the method based on an estimation of the maximum, the duration of the testing does not grow linearly with the number of stirrer positions. Moreover, the accuracy of the estimation is enhanced. This experiment on a real EUT shows that the duration of a testing with a probabilistic approach can reduce the duration of a testing and increase its accuracy.

CONCLUSION

In this article we present a new approach for measuring susceptibilities of EUTs in RCs. This approach based on an estimation of the probability of failure during a testing uses more information to predict the susceptibility of the device and therefore is more accurate than an estimation based on the statistical maximum E-field. Moreover, using a probabilistic approach allows to choose the steps of injected power for the testing in accordance with the number of stirrer positions. This allows to reduce the duration of the testing. Simulations with a plane wave spectrum and a virtual EUT show that the accuracy of this approach is very good. Measurements on a real device show that the values of the susceptibility are in agreement with measurements performed in a GTEM cell and are more accurate than measurements based on an estimation of the maximum.

More investigations are needed from a statistical point of view to access analytically the statistics of the testing instead of using MC simulations.

Figure 2 Figure 1 .

 21 Figure 2 gives an idea of the maximum error (95 % CI) for every combination of N and p. This figure shows that

Figure 2 .

 2 Figure 2. Maximum error: max(( E s E s )/Es) for every value of N and p f . MC simulations with 10 4 experiments.

Figure 3 .

 3 Figure 3. Ratio of the magnitude of the E-field received E r by the mean value of the E-field E m vs. the position on the circle, simulated for f = 100 MHz and 500 MHz.

N = 10 , 1 Figure 4 .

 1014 Figure 4. Numerical simulation of a testing with N = 10 stirrer positions. α = 1.8 and N p = 5 power levels are needed starting from E m0 = 10 V.m -1 .

N 1 Figure 5 .N 1 Figure 6 .

 1516 Figure 5. Numerical simulation of a testing with N = 30 stirrer positions. α = 2.5 and N p = 3 power levels are needed.

Figure 7 .

 7 Figure 7. External view of the equipment under test.

Figure 8 .

 8 Figure 8. Schematic view of the equipment under test and its electronic board.

7 N 4 N 3 Figure 9 .

 7439 Figure 9. Comparison of measurements performed in a GTEM cell, with the standard method based on the maximum with N = 150 stirrer positions and with our approach and N = 10, 30 and 150 stirrer positions.

  . When N is increasing, the error is reduced and the range of values of p f is broadened. Consequently, the range of values of E s is widened. With N = 30, for p f values between 0.1 and 0.8, the error is around ±20 % and the range of values tested for E s are between 0.5E m and 1.7E m . With N = 150, for p f values between 0.05 and 0.9, the error is around ±10 % and the range of values tested for E s are between 0.3E m and 2.2E m . Figure

	b) presents the error of
	the estimation E s -E s /E s and gives the confidence intervals (CI) for different values of N . With N = 10,
	if we consider values of p f between 0.3 and 0.7, 95 % of the measurements are within a ±50 % error interval. Such an interval corresponds approximately to a ±3 dB error in terms of power, an error margin that is generally
	accepted for EMC purposes. According to figure 1-(a)
	it allows to measure values of E s between 0.7E m and
	1.3E

m

ACKNOWLEDGMENT

This work was supported by the French Ministry of Defence DGA (Direction Générale de l'Armement), with a Ph.D. grant delivered to Emmanuel Amador.


Emmanuel Amador
email: emmanuel.amador@insa-rennes.fr

Christophe Lemoine

Philippe Besnier

A Probabilistic Approach to Susceptibility Measurement in a Reverberation Chamber

In this article, we propose an alternative approach for measuring the susceptibility level of an equipment under test in a reverberation chamber. Instead of using a statistical estimation of the maximum, our estimation is based on an estimation of the probability of failure of the equipment under test. We show through Monte Carlo simulations that the probability of failure of an equipment under test allows to predict with a good accuracy its susceptibility level. An experimental validation with a simple equipment under test shows that the susceptibility levels measured in a reverberation chamber are consistent with measurements performed in a GTEM cell. c Es and 95% CI for Es = 2 V/m, (r ⇡ 4%) c Es and 95% CI for Es = 0.3 V/m, (r ⇡ 92%)

I. INTRODUCTION

Susceptibility measurements in a reverberation chamber (RC) are based on a statistical estimation of the maximum level an equipment under test (EUT) may have received over a number N of independent stirrer positions [START_REF]Reverberation Chamber Test Methods. International Electrotechnical Commission (IEC) Standard[END_REF]. A lot of work [START_REF] Lehman | Characterization of the maximum test level in a reverberation chamber[END_REF], [START_REF] Koepke | New electric field expressions for EMC testing in a reverberation chamber[END_REF], [START_REF] Harima | Statistical characteristics of maximum E-field distribution in a reverberation chamber[END_REF], [START_REF] Hoijer | Maximum power available to stress onto the critical component in the equipment under test when performing a radiated susceptibility test in the reverberation chamber[END_REF], [START_REF] Hoijer | On "maximum power available to stress onto the critical component in the equipment under test when performing a radiated susceptibility test in the reverberation chamber[END_REF], [START_REF] Orjubin | Maximum field inside a reverberation chamber modeled by the generalized extreme value distribution[END_REF] has been done in the community to characterize statistically the estimation of the maximum value of a rectangular E-field component or the maximum power in an RC. Extreme value theory is used to determine the probability density function of the maximum power [START_REF] Lehman | Characterization of the maximum test level in a reverberation chamber[END_REF] or the maximum of a rectangular component of the E-field [START_REF] Harima | Statistical characteristics of maximum E-field distribution in a reverberation chamber[END_REF] in the chamber.

Let X a random variable with a density probability function f X (x) and a cumulative distribution function F X (x). The cumulative distribution function of the maximum value over N independent samples F N is given by:

F N (x) = F X (x) N (1)
and thus the density probability function of the maximum is given by:

f N (x) = N F X (x) N -1 f X (x). (2)
The maximum value of the power or the maximum value of a rectangular component of the E-field is a N -order statistics.

It is derived from the size N of the sample measured (e.g the number of independent stirrer positions or the number of independent frequency used during the measurement) and the power injected in the RC. It means that by using an estimation of the maximum, an intrinsic quantity like the susceptibility level of an EUT is a function of the number of stirrer positions used during the test. Moreover, the uncertainty of the estimation of the maximum of a rectangular E-field component is large and decreases slowly with the size of the sample N as shown in continuous line in figure 1. In this article we propose a new approach based on the probability 4 of failure of the EUT. This approach uses an estimation of the mean value of a component of the E-field during the measurement and an estimation of the probability of failure of the EUT during the testing. As shown in blue in figure 1, the uncertainty of an estimation of the mean value of a rectangular E-field component decreases more rapidly. After presenting the theoretical background of our approach, we show that it can be used to measure with a good accuracy the susceptibility level of an EUT. The approach is confirmed by an experimental validation and the values obtained are consistent with measurements performed in a guided wave setup like a GTEM cell.

N [ V/m] E( E m) 95% C I of E( E m) E( E M ) 95% C I of E( E M )

II. MEASUREMENT OF THE SUSCEPTIBILITY BASED ON THE PROBABILITY OF FAILURE OF AN EUT

In this section, we propose a method that allows to derive the susceptibility of the EUT from the probability of failure measured in an RC. The statistical distribution followed by a rectangular E-field component in an overmoded and wellstirred RC is a Rayleigh distribution. The mean value of a Rayleigh distribution is σ π 2 [START_REF] Papoulis | Probability, Random Variables, and Stochastic Processes[END_REF]. The cumulative probability function of a Rayleigh distribution with scale parameter σ (shown in figure 2) is given by [START_REF] Papoulis | Probability, Random Variables, and Stochastic Processes[END_REF]:

F (x) = 1 -e -x 2 /2σ 2 , with x ≥ 0. (3)
When the mean value of the E-field E m in the chamber equals the susceptibility level E s of the EUT, the probability of failure r m can be derived by posing:

r m = 1 -F (E s ) = 1 -F σ π 2 = e -π/4 ≈ 46 % (4)
If a default is detected for r m = 46 % of the stirrer positions, the susceptibility level E s of the EUT is directly given by the mean value E m of a rectangular component of the E-field in the chamber.

Let define the susceptibility level E s of an object in an RC as the mean value of a rectangular component for which e -π/4 ≈ 46 % of the stirrer positions provoke a failure on the EUT.

We will derive a general relation that allows to deduce the susceptibility level E s from a measurement with an arbitrary mean value of a rectangular component of the E-field E m and a corresponding probability of failure of the EUT r. If r > 0.46, the mean value of a rectangular E-field component E m is greater than the susceptibility level E s of the EUT, and if r < 0.46, we can deduce that E m < E s . The knowledge of the probability of failure r, the mean value of the E-field E m in the chamber, and the theoretical CDF of an E-field component allow to predict the susceptibility level E s . From (4), we can write:

r = 1 -F (E s ) = e -E 2 s /2σ 2 (5)
We can write the two following equations:

E s = σ 2 ln(1/r) E m = σ π/2. ( 6
)
and thus: We can extract the level of susceptibility of an EUT from its probability of failure, roughly given by r = N f /N and from an estimation of the mean value the magnitude of a rectangular component of the electric field during the measurement E m1 .

E s = 2E m ln(1/r) π (7)

III. MONTE CARLO SIMULATIONS OF OUR APPROACH

In this section we study the statistical properties of our approach by using Monte Carlo simulations. The goal is to have an indication of the quality of the estimator E s of the level of susceptibility E s . From equation [START_REF] Orjubin | Maximum field inside a reverberation chamber modeled by the generalized extreme value distribution[END_REF], we can note that the estimator E s is a function of two estimators, E m and r. As these two estimators are estimated by computing an average over a sample of size N , the confidence interval (CI) should decrease with the sample size N .

A. Effect of the sample size N on the confidence interval

Figure 3 shows the estimation of the susceptibility level for three different levels of susceptibility E s as a function of the number of stirrer positions N . In these simulations, the mean value of the E-field is set to 1 V.m -1 . First in red, the susceptibility level equals the mean value of the Efield. As pointed out in (4), 46 % of the stirrer positions provoke a failure on the EUT. The statistics of E s is similar to the statistics of E m presented in figure 1. We can note that the uncertainty of the estimation varies with the level of susceptibility measured. This is mostly due to the estimation of the probability of failure r. If E s = 0.3E m , as depicted in blue in figure 3, the probability of failure equals 92 %. The CI remains large when the number of stirrer positions increases and the mean value of E s converges to E s when N > 20. If E s = 2E m , as depicted in green in figure 3, the probability of failure equals 4 %. The CI remains significant when the number of stirrer positions increases and the mean value of E s converges to E s if N > 40.

These Monte Carlo simulations show that a probabilistic measurement of the susceptibility gives good results as long as the values of r or 1-r are not too small. Unlike an estimation based on the maximum value of the E-field, this estimation based on its mean value does not increase with N .

B. Effect of the probability of failure r on the confidence interval

Figure 4 shows the mean value and the 95 % CI of E s /E m when N = 30. We can note that with N = 30 the mean value of the estimator is correct for a probability of failure r between 0.05 and 0.95. For a correct estimation of the susceptiblity level, one can use the following rule of thumb 2 N ≤ r ≤ 1-2 N . Figure 4 shows that the absolute width of the CI interval of E s /E m is not varying with r. It means that relative error is increasing when the susceptibility level E s is decreasing. Figure 5 shows the CI of the relative estimator E s /E s for N = 30 stirrer positions and N = 100. If N = 30, the relative error of 95 % of the Monte Carlo experiments does not exceed ±20 % as long as r ≤ 0.6. If N = 100, r should not exceed 0.8.

These Monte Carlo simulations show that the estimation of a susceptibility level based on the probability of failure of an EUT is possible and may be a good alternative to measurements based on the maximum value. The uncertainty of the estimation is relatively well controlled and decreases more rapidly when N is increasing than the estimation based on the maximum.

IV. EXPERIMENTAL VALIDATION

A. Experimental setup

In order to validate our approach, we designed an EUT. This EUT consists of an electronic board with an operational amplifier (op-amp) acting as a comparator. The board is placed in a metallic enclosure. A 5 cm long monopole external antenna is connected to the circuit as shown in figure 6. A schematic of the electronic circuit is given in figure 7. The antenna is associated with an envelope detector for filtering the high frequency and for rectifying the signal. Without any disturbance, since V + > V -the op-amp delivers V s = 9 V. With disturbances leading to V -> V + , the op-amp provides V s = -9 V indicating a default. The signal V s is recorded with a digital oscilloscope and a home made program that Fig. 8. Measurements of the susceptibility in an RC with our probabilistic approach for various injected power (or Em) as calculated from (7) (in red), with the method based on an estimation of the maximum value of the E-field (in blue) and measurements of the susceptibility performed in a GTEM cell (in black). controls all the experimental setup. The program returns either the value 0 in the case of no susceptibility, or the value 1 if a susceptibility is detected. The measurements are performed between 850 MHz and 1500 MHz. At these frequencies, the behavior of our chamber is ideal and measurements have shown that the rectangular E-field components follow a Rayleigh distribution. We choose to use N = 150 stirrer positions and the power injected in the chamber is increased gradually allowing to reach a magnitude of 200 V.m -1 for the rectangular components of the E-field. Susceptibility measurements were performed with the same setup in a GTEM cell at different positions in the test volume. Since the levels obtained from one position to another were similar, we choose to keep only the minimum value of the susceptibility among the different positions.

B. Results

Figure 8 shows the different susceptibility levels measured in the RC and in the GTEM cell of our laboratory. Our setup did not allow to get an E-field magnitude greater than 100 V.m -1 in the GTEM cell. It explains the lack of susceptibility detection below 1 GHz. The measurements of the susceptibility based on an estimation of the maximum magnitude of the E-field in the reverberation chamber with N = 150 stirrer positions are given by the blue curve. In comparison with the measurements made in the GTEM cell, the susceptibility measured is generally higher and for some frequencies, the levels measured are superior by a factor of 4. This susceptibility measurement exhibits oscillations that cannot be explained physically given the length of the antenna. The oscillations are mainly due to the statistical uncertainty in the estimation of the maximum value as explained in section I and presented in figure 1. The red curves in figure 8 show the susceptibility of the EUT by using our approach with only N = 30 stirrer positions. These curves correspond to the different levels of power injected in the chamber. We estimate the susceptibility if and only if the estimated probability of failure r is between 2/N and 1 -2/N . These curves show that our probabilistic approach of the susceptibility levels give results that are consistent with the levels measured in the GTEM cell. If the EUT is not prone to be damaged easily, this approach can reduce the duration of the testing by determining almost immediately the susceptibility level of the EUT instead of increasing gradually the level of power injected in the chamber.

V. CONCLUSION This article gives an alternative approach for measuring the susceptibility in an RC. This approach based on an estimation of the probability of failure r of an EUT allows to predict with accuracy the level of susceptibility of an EUT. We based our investigation on a rectangular component of the electric field in an ideal reverberation chamber. Thus we used a Rayleigh distribution to build our approach. The transposition of this method with other distributions is straightforward, we can use Weibull distributions to perform susceptibility measurements at lower frequencies or exponential distributions if the quantity measured is the power received. The various Monte Carlo simulations show that the CI of this method decreases with the number of stirrer positions. The measurements on an EUT show that the susceptibility levels obtained with this probabilistic approach are consistent with measurements made in a GTEM cell and that the statistical dispersion of the values is reduced in comparison with measurements based on a statistical estimation of the maximum.

Fig. 1 .

 1 Fig. 1. Monte Carlo simulations (10 5 experiments), showing the expected value and the 95 % confidence interval (CI) of both the mean value of the magnitude of a rectangular E-field component Em and the maximum magnitude of rectangular E-field component E M for different values of N . The mean value of the rectangular component is set to 1 V.m -1 in these simulations.

Fig. 2 .

 2 Fig. 2. Cumulative density function of a Rayleigh distribution. Above the susceptibility level Es, failures are detected with a probability r.

Fig. 3 .

 3 Fig. 3. Mean value and 95 % CI of the estimator Es for different values of susceptibility Es and for different values of N . The mean value of a rectangular component Em is set to 1 V.m -1 . Monte Carlo simulations with 10 4 experiments for each value of N .

Fig. 4 .Fig. 5 .

 45 Fig. 4. Mean value and 95 % CI of the estimator Es normalized by Em as a function of the probability of failure r with N = 30 stirrer positions. Monte Carlo simulations with 10 5 experiments for each value of r.

Fig. 6 .

 6 Fig. 6. External view of the equipment under test.

Fig. 7 .

 7 Fig. 7. Schematic view of the equipment under test and its electronic board.

  re m e nt in a GT EM c e ll S u sc e p tib ility b ase d on th e m ax (N = 150) Prob ab ilistic su sc e p tib ility (N = 30)

If the quality factor of the chamber with the EUT is known, Em can be derived directly from it.

ACKNOWLEDGMENT

This work was supported by the French Ministry of Defence DGA (Direction Générale de l'Armement), with a Ph.D. grant delivered to Emmanuel Amador.