domain: phys.phys.phys-optics info sdv

text:
Claire Lefort

Mathieu Chalvidal

Alexis Parente

Véronique Blanquet

Henri Massias

Laetitia Magnol

Emilie Chouzenoux

Claire Lefort¹

Mathieu Chalvidal²

Alexis Parenté

Deep-3D-multiphoton imaging of whole biological structures with a computational assistance and the pipeline FAMOUS

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

claire.lefort@cnrs.fr

We have developed a new instrumental and computational pipeline based on multiphoton microscopy [1] imaging thick biological samples on whole organs without slicing or labeling procedure. This strategy delivers a new viewpoint on the tested biological structures [2]. For the post-processing strategy, we are developing a specific regularized image restoration algorithm [3,4]. The first step is to led an accurate 3D estimation of the point-spread function (PSF) of the instrument all over the entire depth of the structures [5][6]. This latter approach is named FIGARO. An advanced model fitting algorithm measure the depth-variant distortions of the image resulting from the combination between the heterogeneities of the structure resulting into scattering and absorption phenomena and distortions associated to the instrument. The resulting approach is a pipeline called FAMOUS whose performance is evaluated in case of a heterogeneous medium constituted by a whole mouse muscle. The whole EDL (Extansum Digitorum Longus) is resected from tendon to tendon, is unsliced and is imaged with a multiphoton microscope thanks to the emission of a second harmonic generation (SHG) signal, emitted from the myosin assembly of muscle. The images of the PSF are recorded simultaneously in the second recording channel of the multiphoton microscope. Thanks to the pipeline FAMOUS, optical artefacts resulting from the acquisition chain are evaluated, including medium heterogeneities in the 3 dimensions. We are keeping a specific attention to the axial structuration of myosin whose structuration is lost in blur and noise due to the optical resolution lost especially in case of a deep 3D imaging. The restored images reveale for the first time an axial level of structuration unexpected until now. Striated skeletal muscles are organs with a highly ordered structure resulting in an optically heterogeneous medium. The deep imaging of SHG from myosin in a whole mouse muscle, unsliced, lose the axial information of the striated structuration. Figure 1 presents the raw image (A) and the result of the pipeline FAMOUS (B). Figure 1 reveals the high impact of FAMOUS pipeline in the image quality, here revealing the fibrillar organization of the myosin of a wild mouse muscle.

Figure 1 :

 1 Figure 1: 3D image of a mouse muscle structure. Left. Raw acquisition of MPM. Right. Restored image with FAMOUS computational and instrumental pipeline.


Titouan Gendron
email: titouan.gendron@univ-ubs.fr

Emmanuel Boutillon
email: emmanuel.boutillon@univ-ubs.fr

Abdel Charbel
email: charbel.abdelnour-@imt-atlantique.fr

Nour

David Gnaedig
email: david.gnaedig-@turboconcept.com

Revisiting augmented decoding techniques for LTE Turbo Codes

Keywords: Turbo Codes, candidate decoding, Flip and Check, Forced Symbol Methods

This paper revisits several methods used in iterative decoders to improve the performance of the LTE Turbo code for small information blocks. By considering the iterative decoder as a chaotic system, some explanations are given on the unexpected behavior of several post-processing algorithms. A "dither-inspired" algorithm, called blind candidate decoding, is proposed. This algorithm improves error correction by performing several decoding attempts from a set of perturbed input. Combined with the use of the CRC as an outer code, a performance improvement of up to 1 dB in signal to noise ratio can be obtained for a frame error rate of 10 -3 . Moreover, a stopping criterion based on the Euclidean distance between the output and the input is proposed to mitigate the critical problem of undetected errors.

I. INTRODUCTION

The Internet of Things (IoT) generally requires the use of particular error correcting codes, where error correction performance at low Signal to Noise Ratio (SNR) is crucial due to the transmitter's limited energy supply. The SNR range where the Frame Error Rate (FER) decreases rapidly is called the waterfall region, in opposition with the error floor region limited to high SNRs. Turbo Codes [START_REF] Berrou | Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1[END_REF] are one of the error correcting codes that can be used for IoT application [START_REF] Chen | Narrow band internet of things[END_REF]. This paper focuses on the Long Term Evolution (LTE) Forward Error Correction (FEC) code. It includes two binary Recursive Systematic Convolutional (RSC) codes separated by an interleaver, a rate matching system and a Cyclic Redundancy Check (CRC) for outer error detection [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA)[END_REF].

Several methods have been proposed in the literature to achieve better decoding performance with a turbo decoder without modifying the design of the code at the transmitter side. Some of them rely on the CRC to achieve lower FER, by computing several possible hard decisions that can be validated by the CRC. The most intuitive method is the Flip and Check (FC) algorithm [START_REF] Tonnellier | Lowering the error floor of Turbo Codes with CRC verification[END_REF]. If the hard decision of the decoder doesn't pass the CRC check, the FC algorithm consists in first selecting the q F C less reliable bits, then generating all 2 q F C candidate CRC codewords by flipping decoded values of these bits, and finally evaluating the CRC check on all candidates, keeping the one that passes the CRC. This method can improve the performance of the LTE decoder in the waterfall region, but it increases the number of undetected errors (wrong codewords that satisfy the CRC), leading to a higher FER floor.

List decoding [START_REF] Narayanan | List decoding of turbo codes[END_REF] is another method that can be used to improve the performance of codes, generating a list of possible codewords and choosing the one that satisfies the CRC constraints. Unlike FC, List decoding computes several hard decisions by modifying the turbo decoding algorithm. This modification generates several sets of extrinsic outputs by each component decoder using a reliability classifier algorithm like the List Viterbi Algorithm [START_REF] Seshadri | List viterbi decoding algorithms with applications[END_REF] or the 2 k method [START_REF] Nill | List and soft symbol output viterbi algorithms: extensions and comparisons[END_REF]. Processing these multiple candidate extrinsic sets for the same number of decoding iterations would lead to an increased latency unless multiple list Soft Input Soft Output (SISO) decoders are introduced with their associated added complexity.

Candidate decoding, consists in making several hypotheses on the received information and trying to decode them, leading to a plurality of standard decoding processes. Such algorithm was first introduced in [START_REF] Ould-Cheikh-Mouhamedou | A method for lowering Turbo Code error flare using Correction Impulses and Repeated Decoding[END_REF] and later named Forced Symbol Method (FSM) [START_REF] Ould-Cheikh-Mouhamedou | Improving the Error Rate performance of Turbo Codes using the Forced Symbol Method[END_REF]. By selecting different hypotheses on the set of the q b less reliable bits, it provides 2 q b candidates that span all corresponding possibilities. Each decoding process assumes these q b bits as reliable, i.e. their input Log Likelihood Ratio (LLR) is set to plus or minus infinity. A standard decoding process is required to choose these q b bits, and the number of candidates must be a power of 2. Another type of candidate decoding have also been proposed in [START_REF] Leduc-Primeau | Dithered Belief Propagation Decoding[END_REF] for Low Density Parity Check (LDPC) codes: The Randomized Initial State (RIS). Unlike FSM, the RIS algorithm consists in several decoding attempts with randomly perturbed input LLRs. To the best of the authors' knowledge, this idea has not yet been applied for turbo decoders. Inspired by these different variants of the candidate decoding algorithm and the FC algorithm, we propose in this work a way to combine different methods to obtain a performance gain for short blocks LTE decoders.

The rest of the paper is organized as follows : Section II analyzes candidate decoding. Section III presents a new candidate decoding algorithm. Section IV presents a study on the association of the new algorithm with the Flip and Check algorithm and introduces a criterion based on the Euclidean distance to avoid undetected errors. Section V provides the performance of the considered algorithms through different examples. Section VI concludes the paper.

II. SYSTEM DESCRIPTION AND ANALYSIS OF CANDIDATE DECODING

A. System Description

The LTE Turbo encoder applies two component RSC encoders separated by an interleaver, each generating K redundancy bits from the K information bits. The decoder is then made of two component decoders separated by the interleaver. Each component decoder generates extrinsic output from the received information and its a-priori input. The extrinsic output of a component decoder is interleaved to be used as a-priori input by the other component decoder at the next iteration.

B. Analysis of Candidate Decoding

The Forced Symbol Method (FSM) [START_REF] Ould-Cheikh-Mouhamedou | Improving the Error Rate performance of Turbo Codes using the Forced Symbol Method[END_REF] is a candidate decoding algorithm consisting in making several decoding attempts from different initial states. After an unsuccessful decoding attempt, a metric is used to select the q b bits that are more likely erroneous, and 2 q b different input information sets are built, corresponding to all their possible binary combinations. In these input information sets, the q b least reliable bits are saturated, i.e. their LLR values are set to the maximum or minimum (quantized) value of the decoder.

At a first glance, it seemed likely that the choice of the q b bits would be crucial for this algorithm and would play a major role in the performance of the candidate decoding. For example, the selection of q b least reliable bits in [START_REF] Ould-Cheikh-Mouhamedou | Improving the Error Rate performance of Turbo Codes using the Forced Symbol Method[END_REF] is based on a metric that calls upon the LLR difference after decoding (see [START_REF] Ould-Cheikh-Mouhamedou | Improving the Error Rate performance of Turbo Codes using the Forced Symbol Method[END_REF] for details). In order to evaluate the impact of the bit selection method, we have compared their impact using a LTE decoder with floating point precision for different cases of block lengths and code rates (K = {40, 256}, R = {1/3, 9/10}), and for different strategies for the choice of the q b selected bits. Compared methods allow or not the selection of redundancy and punctured bits separately. The list of the methods used to select the q b bits is the following:

• Select randomly the q b bits.

• Select the q b least reliable received bits (i.e. with the smallest absolute LLR value). • Use the metric described in [START_REF] Ould-Cheikh-Mouhamedou | Improving the Error Rate performance of Turbo Codes using the Forced Symbol Method[END_REF]. After simulating all cases, surprisingly all methods were equivalent in terms of performance. In order to verify if the performance gain of the candidate decoding is due only to the multiple decoding attempts and not to the selection method of the candidates, we performed the following experiment: A rate R = 1/3, K = 40 all-zero codeword with an LTE turbo code is transmitted over an Additive White Gaussian Noise (AWGN) channel at a signal to noise ratio E b N0 of 2.75 dB until an error event occurs after 8 decoding iterations. Using this erroneous frame, we performed several decoding attempts after modifying only the a priori LLR L 12 and L 18 of the 12 th and the 18 th systematic bits (the LTE rate matching system punctures these two bits but it can be any couple of bits). In Fig. 1.a), the couple (L 12 , L 18 ) takes values in [-2.0, 2.0] with a step of 0.002, thus generating 1000×1000 distinct combinations. For each combination, the decoding process is executed and if the decoded codeword gives the two decoded bits ( b12 , b18 ) equal to (0, 0), the corresponding pixel in Fig. 1.a) is colored in white. Similarly, if ( b12 , b18 ) = (0, 1), (1, 0) and (1, 1), the corresponding pixel is colored in blue, red and magenta, respectively. In Fig. 1, we observe a chaotic, and to some extent, a fractal frontier between the different areas. We have two explanations to this observation. The first one is due to the nature of iterative processing where each iteration is a non linear function using the result of the previous iteration, thus amplifying the small variations between the initial LLRs. This effect is known as the "butterfly effect" [START_REF] Ghys | The butterfly effect[END_REF] in the chaos theory. The second explanation can be our incapacity to fully grasp the subtle geometry of a highly dimensional vector space. These hypotheses have yet to be verified, which could be an interesting research topic.

The chaotic behavior of LDPC codes has already been studied in [START_REF] Sibel | Evidence of chaos in the Belief Propagation for LDPC codes[END_REF] through the analysis of LDPC Belief Propagation decoding algorithm as a dynamic system. The methods used in [START_REF] Sibel | Evidence of chaos in the Belief Propagation for LDPC codes[END_REF] could be used to study the chaotic nature of iterative decoding using a different approach than the one presented above. In any case, we can see that there is no clear frontier between a decision or another, and that it is possible that a candidate is successfully decoded even if the hypothesis drives away from the original codeword, and vice versa. Then each candidate decoding attempt can be seen as a pick of a random variable with the following 3 possible realizations with unknown probability distribution:

• The candidate leads to the right codeword • The candidate leads to a wrong codeword • The candidate doesn't lead to a codeword. Increasing the number of decoding attempts increases simultaneously the probability to successfully decode on one side, and the undetected error probability on the other side.

With these results in mind, we propose a new candidate decoding algorithm that randomly samples candidates in the vicinity of the received information.

III. PROPOSED BLIND CANDIDATE DECODING

In 2012, [START_REF] Leduc-Primeau | Dithered Belief Propagation Decoding[END_REF] made the same assessment on LDPC codes and presented a candidate decoding algorithm called Random Initial State (RIS). This algorithm consists in performing several decoding attempts with different initial states randomly generated. [START_REF] Leduc-Primeau | Dithered Belief Propagation Decoding[END_REF] showed that such algorithm increases the convergence probability, and presented a method to reduce the undetected error probability. This method consists in storing every codeword found by the candidate algorithm and in keeping only the closest one to the received LLR.

The proposed candidate decoding algorithm, called Blind Candidate Decoding (BCD), is similar to RIS and consists in generating candidates by introducing random perturbations to the received information. This method implies the creation of a set of candidates within an Euclidean sphere around the original received information. For a given number of iterations and number of candidates, we can estimate by Monte Carlo simulation the optimal distance d c between each candidate and its corresponding received information that maximizes the convergence probability of the whole process.

Let n c be the number of random candidates generated for the BCD algorithm. Consider a (N , K) LTE turbo code, let Y 0 be the original received LLR input vector. This latter is a N size vector and Y 0 (k), k ∈ 0, K -1 are the LLR values of the systematic bits, while Y 0 (k), k ∈ K, N -1 represent the LLRs of the redundancy bits. Note that if the k th bit is punctured, Y 0 (k) = 0. Let G γ be a vector of size K filled with equally spaced values centered on 0:

G γ (k) = γ 2k K -1 -1, k ∈ 0, K -1 , ( 1
)
where γ is a parameter that controls the value of the norm

d γ = γ 2 E K of the vector G γ .
The constant E K is given by

E K = K 3 2 K -1 + 1 . (2)
One possible set of candidates can be obtained by the addition of a random perturbation of magnitude G γ to the LLRs of the systematic part of Y 0 , such that each systematic bit is randomly affected by an additive noise, keeping each candidate codeword within a constant value of the Euclidean distance d γ to the original received information vector. Algorithm 1 gives the details of the proposed method. Note that in this algorithm, the distribution type of the G γ vector isn't important as long as it is symmetrically centered around 0. The optimal distance d γ for a given

Algorithm 1: Blind Candidate Decoding input : Y 0 , γ output: hard decision Ĉ = Decode(Y 0 ); if CRC check( Ĉ) then RETURN Ĉ end G γ = γ 2k K-1 -1, k ∈ 0, K -1 ; for i ← 0 to n c -1 do Π = Random Permutation( 0, K -1 ); for k ← 0 to K -1 do Y (k) = Y 0 (k) + G γ (Π(k)); end Ĉ = Decode(Y ); if CRC check( Ĉ) then RETURN Ĉ end end
set of parameters and a block length can be determined by Monte-Carlo simulation at the targeted SNR. The number of candidates denoted by n c and the number I C of decoding iterations per candidates can also be adjusted, given the constraints on throughput and latency. The values of the parameter γ and the couple (n c , I C ) can be optimized separately since the most suitable value for γ remains the best for different couples of (n c , I C ).

The BCD algorithm is more versatile than the FSM [9] algorithm because no prior classical decoding is necessary to build the set of candidates, and the number of candidates is not required to be a power of 2. With more degrees of freedom, it is possible to finely optimize the algorithm parameters for a given architecture. In general, BCD outperforms FSM in most practical cases. More detailed results are presented in the fifth part of this paper.

IV. IMPROVED FLIP AND CHECK ALGORITHM AND THE PROPOSED EUCLIDEAN DISTANCE-BASED CRITERION

For further performance gain, any candidate decoding can be used with a post processing method such as the Flip and Check (FC) algorithm. This latter was first introduced in [START_REF] Tonnellier | Lowering the error floor of Turbo Codes with CRC verification[END_REF] to lower the error floor of the LTE code, but it can also be used to improve performance in the waterfall region. It consists in trying several hard decision codeword candidates with CRC during each decoding iteration. To do so, the q F C bits with the lowest a-posteriori probabilities are chosen, then the 2 q F C possible hard decisions on these bits are tested. This method was already used in [START_REF] Weithoffer | Enhanced decoding for high-rate LTE Turbo-Codes with short block lengths[END_REF] for candidate decoding, with the particularity to choose the q F C candidates from the critical positions of bits. The main drawback of this algorithm is that, combined with a candidate decoding algorithm, it increases significantly the undetected error rate i.e. the probability to have a wrong codeword satisfy the CRC. If no other stopping criterion than the CRC is used on a BCD FC decoder, an error floor will appear around F ER = 10 -3 . This floor is due to the high proportion of wrong codewords that satisfy the CRC. When the FC is used on classic decoding algorithms, the first iterations are usually not CRC checked to avoid this issue. In order to avoid undetected errors, we propose another criterion based on the Euclidean distances. In case of an undetected error, the CRC is verified on a codeword containing erroneous hard decisions of systematic bits (i.e. decoded bits). However, the encoded codeword obtained by re-encoding these systematic hard decisions should be far in terms of Euclidean distance from the received codeword LLR information. Indeed, the redundancy bits obtained after reencoding have a high probability to diverge from the original coded message. It is also possible to predict a-priori the distribution of the distances between the received information and the original coded message, knowing the SNR.

In Fig. 2, the distribution of the Euclidean distances between the encoded hard decision and the received information is plotted. We can see a clear separation between the Euclidean distance of the successfully decoded codewords and the Euclidean distance of codewords with undetected errors. Hence, undetected errors can be avoided if we only take the hard decision on information bits that gives an encoded codeword that is close enough to the received information, i.e. if the Euclidian distance is below a certain threshold.

V. PERFORMANCE COMPARISON AND RESULTS

In this section, a performance comparison between several candidate decoding algorithms using the standard LTE decoder is presented. It has been performed for 4 cases listed in Table I and Table II. For the results presented in this paper, the code rate values do not take into account the 24 redundancy bits of the CRC. To fairly compare FSM and BCD with or without FC, the total number of iterations I max is kept unchanged: I max = 64 or I max = 256. Let q b be the number of tested bits by the FSM, and n c the number of tested candidates by the BCD. Several couples (q b , I C ) for FSM and (n c , I C ) for BCD are possible as long as, respectively, I C 2 q b ≤ I max , and

I C n c ≤ I max .
A first interesting result is the selection of the best parameters for each type of decoding algorithm. They are chosen by identifying a sample of undecoded codewords after 8 iterations of the reference decoder. Then these codewords are used for another decoding run with all possible combinations of the parameters (γ ∈ {0.05; 0.1; . . . ; 1}). Chosen parameters are those which successfully decode the largest proportion of these codewords. The results are shown in Table I. For the cases with high code rates, a limited number of iterations is enough to have a large proportion of these candidates decoded, as shown in Table I. Indeed, only one iteration is generally enough to see if the candidate method will converge or not for short blocks. In that case, a criterion on the Euclidean distance is required to avoid a too high undetected error rate, even without FC. The other interesting information is that the distance d c of the BCD candidates can be larger for low coding rates.

When applying the FC algorithm, the same parameters can be re-used since the specifically optimized parameters for FC have values quite close the selected parameters without FC. Moreover, the performance gained by specifically optimizing the parameters for FC is negligible. For the performance comparison of Fig 3,[START_REF] Ould-Cheikh-Mouhamedou | A method for lowering Turbo Code error flare using Correction Impulses and Repeated Decoding[END_REF] bits are flipped by the FC algorithm for each iteration.

  Fig. 1.a) shows no clear border between decision areas. It is still the case when a zoom is performed, as shown in Fig. 2.b) where (L 12 , L 18 ) takes there values in [-0.25, 0.25].

L 12 ( 2 (Fig. 1 .

 1221 Fig. 1. Representation of the convergence of a LTE code in function of the added perturbation on the 12th and the 18th bit

Fig. 2 .

 2 Fig. 2. Distribution of the Euclidean distances between the re-encoded hard and the received information for K = 40, R = 9/10, E b /N 0 = 4.0 of 2850 successfully decoded frames (valid) and 2850 unsuccessfully decoded frames (erroneous) satisfying the CRC.

TABLE I TABLE OF

 IOF

		OPTIMUM PARAMETERS
	Case	Imax	FSM q b I C	nc	BCD I C	dc
	K = 40	64	3	8	16	4	10.1
	R = 1/3	256	5	8	32	8	14.0
	K = 40	64	5	2	64	1	4.24
	R = 9/10	256	7	2	256	1	5.92
	K = 256	64	3	8	8	8	17.4
	R = 1/3	256	3	32	16	16	17.4
	K = 256	63	5	2	32	2	7.74
	R = 9/10	256	7	2	128	2	10.5

The performance gain of the different algorithms, for each simulation case, in comparison to 8 iterations of the conventional turbo decoding is presented in Table II. We can see that the BCD algorithm is almost always better than the FSM, especially when combined with FC. Candidate decoding is globally more efficient when used on short block sizes and high coding rates. The performance of the K = 40, R = 1/3 case is provided in Fig. 3. In this case, the decoding process of a frame ends if the hard decision of the decoder meets the CRC or if a codeword below the chosen threshold for the Euclidean distance is found by the FC. The value of this threshold is chosen to be larger than the one obtained by 99.99% of the valid codewords. No undetected error occurred for the curves in Fig. 3. Moreover, 300 erroneous frames were detected for each point of the curves. Hence, we can reasonably predict that the undetected error rate is expected to be, at least, 2.5 decades below. We can also mention that, for FER below 10 -3 , the average number of iterations is noticeably lower with BCD than FSM, and FC can reduce it even further.

VI. CONCLUSION

In this paper, a new perspective on candidate decoding for Turbo Codes was presented through potential synergies with the chaotic nature of the non-optimal iterative decoding.

A new candidate decoding algorithm denoted by Blind Candidate Decoding is proposed. In addition to an improved performance at low SNR values, it reveals to be less complex than reference state of the art alternatives.

Coupling candidate decoding with the Flip and Check algorithm can improve performance at the price of unacceptable undetected error rate. As a solution, a new stopping criterion based on the values of the Euclidean distance between the received information and the encoded hard decision of the decoder was introduced. Indeed, it was shown that a distinct separation exists between the Euclidean distance values for valid and erronous codewords that can be simply exploited through a comparison with a well defined threshold. The value of this threshold can be computed for a given block length, code rate and SNR. This comparison reveals to be quite efficient to avoid undetected errors.

Future work includes a deeper analysis of the behaviour of iterative decoding using tools from the complex dynamic systems theory and the chaos theory.


A new design of an air core transformer for Electric Vehicle On-Board Charger

Keywords: «Passive component», «transformer», «coreless», «electric vehicle», «

on board charger».

Introduction

The autonomy of electric vehicles is a key challenge for electrical vehicle and each part of the electrical power architecture must be optimized by reducing its size and weight [START_REF] Khaligh | Global Trends in High-Power On-Board Chargers for Electric Vehicles[END_REF]. For this purpose, active devices development such as GaN and SiC allows to increase the power converter's switching frequency which lead to increase the power density by reducing the size of passive devices [START_REF] Taurou | Utilisation des transistors GaN dans les chargeurs de véhicule électrique[END_REF]. However, these passive components still present the biggest part of the converter in term of volume [START_REF] Meyer | Skin and Proximity Effects for Coreless Transformers[END_REF], They also limit the efficiency at high frequency because of the eddy effect and the copper losses [START_REF] Kazimierczuk | Winding Resistance and Power Loss of Inductors with Litz and Solid-Round Wires[END_REF]. Thus, investigate the passive components and specifically the coreless magnetics components is relevant on-board battery charger development.

Air core transformer

Generally, air core transformers are used in very limited application for inductive charge, superconductor, medical scanner or magnetic sensors [START_REF] Kim | 3D Printed Flexible Coreless Transformers[END_REF]. Without magnetic material, turns of coil emits a magnetic field in its environment. For isolated converter, it is necessary to canalize the magnetic field in order to get a good coupling coefficient. As presented by eq [START_REF] Khaligh | Global Trends in High-Power On-Board Chargers for Electric Vehicles[END_REF], the coupling coefficient expression in the transformer depends on transmitted magnetic flux from its primary to its secondary. Thus, drive the magnetic field becomes crucial for the efficiency of our converter. Moreover, as shown by eq [START_REF] Taurou | Utilisation des transistors GaN dans les chargeurs de véhicule électrique[END_REF], the inductance value of the transformer depends on the core magnetic permeability. A weak permeability as air permeability induces a weak inductance value. For power converter design, the voltage is commonly imposed, and a lower inductance leads to a higher current as expressed by the following eq (3) and eq [START_REF] Kazimierczuk | Winding Resistance and Power Loss of Inductors with Litz and Solid-Round Wires[END_REF]. The use of air core transformer is relevant for high frequency or low voltage applications. On the other hand, low permeability of the air core induces more magnetic field leakage. The only solution is to minimize the magnetic field outside of the transformer by appropriate adjustment of coil's turns.

𝑘 = 𝜑 12 𝑖 1 (1)
Where 𝜑 12 and 𝑖 1 are, respectively, magnetic flux and primary current.

𝐿 = 𝑛 2 𝜇𝑆 𝑒 𝑙 𝑒 ( 2
)
Where 𝑛 is the turns ratio, 𝜇 is the permeability, Se is the equivalent section and le is the equivalent length.

𝑢 = 𝑈𝑠𝑖𝑛(𝜔𝑡) = 𝐿 𝑑𝑖 𝑑𝑡

(3)

𝑖 = - 𝑈 𝜔𝐿 cos (𝜔𝑡) (4)

Field canalization by superposition theorem

For improving the design of an air core transformer, it is important to avoid magnetic field. Superposition theorem could be a good solution. The result of the contribution of all coil turns is the sum of results of each coil turns contribution.

With this method, magnetic field produced by one coil turn is balanced by another coil turn as in fig. 1a. But as shown by simulation with Finite Element Method by FEMM, magnetic field is still spreading. We can use symmetry properties and add other turn to balance the magnetic field as fig. 1b. The best solution is to add coil turns until reaching continuous turns around a revolution symmetric axis of the transformer, meaning that the toroidal geometry is the suitable choice in this case as shown by Fig. 1c.

Moreover, realization of this kind of transformer is manual and difficult because there are few empty spaces. It's necessary to pull in and pull out wires like a sewing technique and requires manual interventions. The equation [START_REF] Kim | 3D Printed Flexible Coreless Transformers[END_REF] shows the method to compute self-inductance in a toroid transformer [START_REF] Sadarnac | Du composant magnétique à l'électronique de puissance -Analyse, modélisation, conception, dimensionnement des transformateurs, inductances, convertisseurs -Cours et exercices corrigés[END_REF] 𝐿

= 𝑛𝜑 𝑖 = 𝑛²𝐻 2𝜋 𝑙𝑛 ( 𝑅 𝑜𝑢𝑡 𝑅 𝑖𝑛 ) (5)
In our case, we choose the outside radius 𝑅 𝑜𝑢𝑡 twice longer than the inside radius 𝑅 𝑖𝑛 and the height 𝐻, the equation ( 5) becomes the equation ( 6)

𝐿 = 𝑛²𝑅 2𝜋 𝑙𝑛(2) (6)

Proposed air core transformer

Because of the difficulties related to the toroidal transformer manufacturing, we design a transformer achievable in holding only once the wires with as main. Objective to keep the lowest leakage magnetic field. Second goal is to obtain a good coupling coefficient, in order to easily transmit the power. Usually, solenoids are used, because of manufacturing issues but the magnetic field is not totally canalized. Indeed, its main part spreads above and below the solenoid. That's why the idea was to complete this geometry by adding turns in opposite direction. As expected, it reduces drastically the leakage magnetic field as represented in figure 4(b). To go further, it's possible to add opposite turns also at each side of the transformer as in figure 4(c). We compute magnetic field at 5cm from the center of the transformer by Finite Element method. Moreover, volume occupied by each geometry and selfinductance of the first coil in each case by the Biot & Savart law was calculated. It shows that the magnetic field is finally well canalized, and the field lines looks like if there was a magnetic core. The outside magnetic field is drastically reduced by more than 300 times.

Comparison and possible applications of two transformers

We created two prototypes of air core transformers according to the best toroid selection and to the new geometry proposed in order to compare them. The toroid transformer was designed to reach 7,8µH following the eq (6) and a Finite element method computed a self-inductance of 11µH. The toroid transformer support was manufactured, and the cylindrical transformer support was 3D printed in order to be ventilated. The model for the 3D printing is shown Fig. 3a. Both realized prototypes can be viewed Fig. 3b We compute the results in table 2. First, we note that toroidal transformer presents worst results than expected. This is due to the winding complexity, where the exact number of turns wasn't reached. Concerning the comparison, the new geometry is twice smaller, and its self-inductance L is twice bigger. That can be resumed as a four times better self-inductance by volume unit. The leakage inductance Lf is fewer in the cylindric transformer but the coupling coefficient stays equivalent (respectively 77% and 81% on average). We can notice that the resistance R is high for both transformers at frequencies exceeding 100kHz because of the quality of the Litz wire used which were initially design for 100kHz maximum operating point.

Conclusion

This article proposes a study and a comparison of different geometry of coreless transformer. It presents an easier method to produce a transformer keeping an equivalent coupling coefficient as a toroidal transformer. Moreover, it proposes a prototype with better value for power density twice smaller. However, we notice the difficulty of simulation and design realization because any analytics techniques are yet developed. Next step will be to compare the magnetic field emitted around each transformer and to optimize the coil turns position for the cylindrical transformer.

(a) (b) (c)

Fig. 1 :Fig. 2 :

 12 Fig. 1: compensation of magnetic radiation by superposition method: (a) two coil turns, (b) four coil turns, (c) toroidal geometry

5 Fig. 3 :Fig. 3 :

 533 Fig. 3: Air core transformer: (a) model (b) cylindric prototype (c) toroidal prototype

Table I : Results and magnetic radiation for air core transformers

 I

	Simple solenoid transformer	Solenoid transformer with turn added	Transformer developed
		above and below
	16,79 µH	16,77µH	14,22µH
	7,12mT	0,13mT	0,02mT
	0,02L	0,05L	0,10L

Table II : Characteristics of two air core transformers

 II

		100 kHz			500 kHz		1 MHz			2 MHz		Volume	Weight
		L	Lf	R	L	Lf	R	L	Lf	R	L	Lf	R
		(µH)	(µH)	(mOhms)
	Proposed	4,8	1,13	23	4,89 1,09	135 4,9	1,09 449 5,04 1,05 1410 0,48	0,88
	toroid
	Proposed	10,35	1,9	26	10,3 1,87	160 10,26 1,89 545 10,1 1,8	1700 0,26	0,84
	cylindric


Loïc Desgeorges

Jean-Philippe Georges

Thierry Divoux

Detection of Security and Safety Threats related to the Control of a SDN Architecture

Keywords: Software-Defined Networking, Safety, Security, Multi-Controllers, Observability

Software Defined Networking is a concept within the networking field which proposed a centralized control considering the control and data planes. To overcome the safety and security threats, solutions might be divided into two categories: enforcing the controller to make it more robust or the architecture using a multi-controller approach. This work aims to pave the way for a multi-controller architecture without East-West interface to avoid the spreading of an attack. There is one nominal controller in charge of the control while the second observes the traffic at the Southbound interface to detect anomalies of control. A detection method is introduced theoretically and relies on Intrusion Detection System theory, more precisely the specificationbased. Here, the specification is a template determined through a projection function of the control logic. The template is compared to the activity of the command observed such that any deviation generates an alarm. The method is finally explained in use cases.

INTRODUCTION

Software-Defined Networking (SDN), [START_REF] Mckeown | Software-defined networking[END_REF], has been introduced to provide a structured software environment to deal with various application requirements and dynamic networks, see [START_REF] Kreutz | Softwaredefined networking: A comprehensive survey[END_REF]. It provides an architecture within the data plane is separated from the control part in a centralized control architecture manner. SDN simplifies network management and facilitates network evolution. The centralized control has two main issues: scalability and robustness. To overcome it, a variant of this architecture has been proposed: a distributed control. A multi-controller architecture permits to balance the load between the controllers while it provides an active redundancy, see [START_REF] Li | A survey of network update in sdn[END_REF]. On top of that, it has some challenges in terms of consistency, reliability, load balancing and security as developed in [START_REF] Hu | Multi-controller based software-defined networking: A survey[END_REF].

Indeed, each plane of the SDN architecture has its own weakness, see Abd [START_REF] Abd Elazim | Software defined networking: attacks and countermeasures[END_REF]. The controller is the heart of the network and has a global view of the network. As a consequence, a security or safety threat of the control plane has consequences on the entire network. Some mechanisms have been proposed in the literature to improve the security and/or safety of the control plane. These methods might be divided into two categories: the one which proposes a more robust controller to the threat considered as in [START_REF] Porras | A security enforcement kernel for openflow networks[END_REF]. Merely, this architecture presents a single point of failure. To consider the safety issue, it is mandatory to consider a multi-controller architecture as in [START_REF] Fonseca | A replication component for resilient openflowbased networking[END_REF]. Thus, the second category proposes to combine several controllers. The role of the second controller may differ according to the method. [START_REF] Qi | An intensive security architecture with multi-controller for sdn[END_REF] proposed a decision-making security architecture for the control layer. In this architecture, each commands is the result of a vote between all controllers. Similarly, [START_REF] Liu | Design of the multi-level security network switch system which restricts covert channel[END_REF] proposed to set a filter which analyses the command sent by the controller and validates it. However, the information is collected by communication between the controllers through an East-West interface. Regarding a security point of view, this interface is opportune to the spread of attacks. To deal with this issue, [START_REF] Lam | Securing distributed sdn with ibc[END_REF] proposed to introduce a Private Key Generator to encrypt the communication between the controllers. In the same way, [START_REF] Shang | Distributed controllers multi-granularity security communication mechanism for software-defined networking[END_REF] proposed to use an indirect way of communication named "interdomain agent flow". Additionally, they proposed a multigranularity approach of the security and safety challenges of the controller. However, little attention has been paid for the consideration of both security and safety threats.

The objective of the paper is to pave the way to a safe and secure multi-controller architecture without East-West interface. In this architecture, one controller is in charge of the nominal control of the network while the second is in charge of the detection of a safety or security issue of the first controller and take the lead in case of anomalies. In this work, a novel detection method relying on a deterministic algorithm is introduced. To develop the logic, attention has been paid to Intrusion Detection System (IDS) theory. Compared to [START_REF] Giotis | Combining openflow and sflow for an effective and scalable anomaly detection and mitigation mechanism on sdn environments[END_REF] or [START_REF] Tang | Deep learning approach for network intrusion detection in software defined networking[END_REF], our method is based on the estimation of the intern variables of the nominal controller only by observation of the activity of the command and not on the information from the nominal controller or from traffic on the data plane (since both might be corrupted). We considered also the combination of anomaly and specification based approaches as in [START_REF] Sekar | Specificationbased anomaly detection: a new approach for detecting network intrusions[END_REF] which proposes to determine a model of the system's recurrent behavior based on a specification and then learn some statistical properties to detect anomaly. Similarly we defined a specification of the command, formalized as a template as introduced in [START_REF] Pandalai | Template languages for fault monitoring of timed discrete event processes[END_REF] and completed on-line based on the estimation of the controller's interns variables in order to verify that there are no anomaly in the control.

THE DETECTION PROBLEM

The multi-controller architecture proposed is represented in Fig. 1. Without East-West interface, the detection method will not be based on the information we get from the other controller but on its capture activity and some a priori knowledge of the control logic.

Principle

To develop the detection logic, attention has been paid to IDS. According to [START_REF] Liao | Intrusion detection system: A comprehensive review[END_REF], IDS works might be divided into two approaches: focusing on the attack behavior or on the unfaulty behavior of the system. The first approach is named signature-based detection and is focused on the attack signature. Thus, this approach concerns just the detection of attacks known and expected.

The second one is composed of Anomaly and Specificationbased approaches. They differ in their knowledge base of the system behavior but both aims to compare the unfaulty behavior known to the running behavior. Basically, anomaly detection techniques work on determining a model of the unfaulty behavior of the system. Regarding specification-based, the behavior is determined by a specification directly extracted from documentation.

In our approach, the specification formalism is a template, as introduced in Pandalai and Holloway (2000), which represents the action/reaction of the command. This specification evolved according to intern variables of the controller. Nevertheless, without East-West interface there is no access to these variables. Consequently, we proposed an observation of the controller activity to estimate its interns variables and then set up the specification's template to compare it with the activity of the command as represented in Fig. 2. Again, it is not the relevance of control that is judged but its consistency. If the controller is attacked before the observation has started then the malicious behavior will be considered as the reference and a non-malicious packet inconsistent will be declared faulty.

Formalisation

This section aims to formalize the activity of the command. OpenFlow, developed by ONF (June 2012), is the protocol which standardized the Southbound interface's communication and is considered in this study.

Events in the network From a controller's point of view, there are two kinds of events in the network: the incoming messages Σ In , the requests from the switches, and the outgoing messages Σ Out , the commands for the switches:

Σ = Σ In ∪ Σ Out
The first set, Σ In , corresponds to the "Packet In" which is the packet received by the controller:

∀pin ∈ Σ In there is pin = (Swp, b, src, dest) with:

• Swp = (p, S) ∈ N × N: the in-port p of the switch S.

• b ∈ N: an identifier named Buffer IDentifier which is tagged to the original packet by the switch. • src: the MAC source address of the packet.

• dest: the MAC destination address of the packet.

The second type of events is related to the commands sent by the controller. There are two types of commands. First, "Packet Out" Σ P O which is single uses: the switch does not retain the information and will have to ask again to the controller what to do. Secondly, "Flow Mod" Σ F M OD which is permanent: the switch adds this command to its flow table. Thus:

Σ Out = Σ P O ∪ Σ F M OD with: ∀pout ∈ Σ P O , pout = (act, b, S):
• act ∈ Actions: the action ordered defined nextly.

• b ∈ N: the buffer ID of the packet.

• S ∈ N: the switch destination of the command.

∀f mod ∈ Σ F M OD , f mod = (act, b, S, src, dest, idle):
• act, b and S are similar to pout.

• src: the MAC source address of the packet.

• dst: the MAC destination address of the packet.

• idle ∈ R + : the storage time of the order by the switch.

Actions The set of actions, Actions, can be divided into two sets {F orward ∪ Set}. Here, layer 2 forwarding applications will be considered so just F orward is formalised. It corresponds to actions sent by the controller to a switch for the forwarding of a packet above a given output port.

So, an action can be modelled as a vector within each component is a binary value, associated with a port of the switch, representing the transmission or not.

Actions = N j=1 b j (b j ) ∈ B
Bias In case of an attack or a failure, the control algorithm returns a biased command. This bias may have several origins in the architecture as explained in [START_REF] Mubarakali | A survey: Security threats and countermeasures in software defined networking[END_REF]. It might be a bias b P in of the Packet In from the data plane, due to a man in the middle attack for example, or a bias b Cgn of the consign from the application plane, due to an application eviction attack as presented in [START_REF] Shin | Rosemary: A robust, secure, and high-performance network operating system[END_REF] or a bias b Cmd of the command from the control plane. As the method does not focus on isolating the fault, only b Cmd will be considered.

Let us formalised a biased command pout ∈ Σ Out as:

pout = pout + b Cmd
With:

• pout ∈ Σ Out : the biased packet.

• pout ∈ Σ Out : the original packet.

• b Cmd ∈ Σ Out : the bias.

• + is the operator defined as:

∀i ∈ [1, lengh(pout)] pout [i] = pout[i] + b Cmd [i]
The challenge consists therefore in detecting the additive bias, i.e. b Cmd .

PROJECTION AND DETECTION

Projection

We assumed that the logic (limited here to layer 2 forwarding applications) embedded in the controller is known. The capabilities of the attacker is to provoke a bias specifically on the controller. Also, the reason of the detected anomaly is not isolated. Regarding the behavior of the controller, whatever the request of a switch, the same number of commands is waited from the controller and the estimation of the computation time is not developed. Moreover, at the starting of the observation of the control, we assume that the controller is not processing a request. We then introduce the projection function which aims to determine the specification template of command algorithms such as Bellman Ford routing, packet filtering, etc. Template Inspired by [START_REF] Pandalai | Template languages for fault monitoring of timed discrete event processes[END_REF], we define a template as an object which gathers a request from a switch and the answered command. Consider T the set of templates and temp ∈ T defined as:

Causality in the algorithm

• temp = {pin, ∪ n i=1 (pout i , [t i , t i ])} • pin ∈ Σ In : the triggered event. It might be empty in case of spontaneous command. • ∪ n i=1 (pout i , [t i , t i ]) ∈ (Σ Out , R 2
) n : the set of command, pout i , expected to answer the event pin. Each command is expected to occur during an interval of time [t i , t i ].

Indeed, a command cannot be waited indefinitely. The function Est is introduced to return this interval of time but is not developed in this work. Est is defined as follows:

Est : I × (Σ In ∪ Σ Out ) → R 2+ (istr, L i ) → it i
With:

• istr ∈ I: the instruction considered.

• L i ∈ Σ In ∪ Σ Out : the line of the instruction to reach.

• it i ∈ R 2+ : the interval of time within the line is reached.

Projection's tool The projection permits to obtain the template for an algorithm. As an algorithm is modelled as a set of instructions, first let us introduce the function P roj Istr as:

P roj Istr : I × (I × Σ) R 2+
→ T (istr, Est) → {pin, {(pout, Est(istr, pout))}} With:

• istr = (pin, op, P OU T ): an instruction.

• {(pout, Est(istr, pout))} ∈ (P OU T × R 2+ ) n : the set of expected commands Moreover, to simplify the reading we assumed that for each instruction the same number n pout of commands is waited.

P roj Algo : Algo → T a → pin, ∪ npout i=1 (pout i , it) With: • it = [t 1 , t 2 ]: fixed interval of time • t 1 = min istr∈Algo min pout∈istr[3] Est(istr, pout):
the lower bound of the intervals of time.

• t 2 = max istr∈Algo max pout∈istr[3] Est(istr, pout):
the upper bound of the intervals of time.

For a trigger event, pin, a set of commands pout i is expected. Besides, the function P roj Algo permits to accomplish the step Projection of the Fig. 2. Intended changes of the algorithms of the nominal controller would have to be notified to the observer through its north interface.

Detection

As mentioned and developed in the section 2, the principle of the method is to compare the trace of the messages send/received by the controller to the instances of the template. The process is synthesized in Fig. 3 and developed here after. In this work, the intern variable to estimate will be the MAC Table M T N of the controller in case of a routing command defined as M T N = {(S, M T )} with:

• S: the switch considered.

• M T = {(p, M AC)}: the MAC Table of the switch S with p the port to join the MAC Address M AC.

Fig. 3. Representation of the algorithms process

The input of the algorithm is the activity of the command, so the first step is to treat those packets by Algo. 1 which launches the algorithm related to the observed packet. It corresponds to the junction of Fig. 3.

Algorithm 1 Analysis Input: Packet p, MAC Table Estimated M T N and O 1 if p ∈ Σ In then 2 Instance(p, M T N, O) 3 else 4 M atch(p, M T N, O)
In case of a Packet In, Algo. 2 is launched and the template is instantiated. The expected commands are completed based on the knowledge stored about the controller's intern variables. It corresponds to launch the instance of the template as represented in Fig. 3. The set of the template instantiations at a date t is noted as O(t). We make the hypothesis that initially there are no command expected:

O(t = 0) = ∅. Algorithm 2 Activation Input: Packet In pin, MAC Table Estimated M T N and O 5 i = 1 , Know = 0 , M T N Check = M T N [1] 6 T emp[1] = pin , T emp[2][2] = it 7 while M T N Check [1]! = pin[1][2] do 8 M T N Check = M T N [i + 1] , i = i + 1 9 M T = M T N Check [2] 10 for Dest ∈ M T [2] do 11 if Dest = pin[3] then 12 T emp[2][1] = M T [1] , Know = 1 13 O = O ∪ T emp , T imer(T emp) 14 if Know = 0 then 15 T emp[2][1] = ∅,O = O ∪ T emp,T imer(T emp)
As soon as the template is instantiated a timer is launched through Algo. 3 to verify that the command is in time. Basically, it consists into waiting the last action before removing the instance from O.

On top of that, when a command is observed, the consistency of the command is checked through Algo. 4 as represented in Fig. 3. For each instance, there is a verification that the command was the one expected by this instance. In addition, there is a formal verification of the safety of the path installed by the controller, which is not developed in Algo. 4 to ease the readability. It checks that there are no forwarding loops, no-dead node and the destination reaching. In such a case, the instance is deleted from O.

Else, the command was not expected and assumed to be inconsistent. Moreover, when a command is observed it is necessary to update the estimation of the controller's intern variables through Algo. 5.

Algorithm 4 Consistency

Input: Packet Out pout, MAC Table Estimated M T N and O 23 f ault = T rue , Learn = 0 24 for T emp ∈ O do 25 if T emp[2][1] = pout then 26 f ault = F alse , O = O \ T emp 27 else if T emp[2][1] = ∅ and pout[3] = T emp[1][1][2] then 28 f ault = F alse , O = O \ T emp , Learn = 1 29 if Learn = 1 then 30 Learn(M T N, pout) 31 return f ault Algorithm 5 Learn Input: Packet Out pout and MAC Table Estimated M T N 32 M T N Check = M T N [1] , newLigne = 1 , i = 1 33 while M T N Check [1]! = pout[3] do 34 M T N Check = M T N [i + 1] , i = i + 1 35 M T = M T N Check [2] 36 for Dest ∈ M T [2] do 37 if Dest = pout[5] then 38 newLigne = 0 39 if newLigne = 1 then 40 M L new = (pout[5], pout[1]) 41 M T [2] = M T [2] ∪ M L new
In these algorithms, the MAC Table Estimated M T N has been considered as the intern variable of the controller. But it can be changed by other intern variables on the condition that Algo. 5 is updated in consequences.

USE CASES

The aim of this section is to illustrate the method presented just above in use cases.

Scenario

The topology represented in Fig. 4 is simulated using Mininet. An ONOS (ONF (2021)) controller is loaded with the MAC learning forwarding application. We name M T N the MAC Table Estimated of the controller defined as in section 3. The algorithm is structured in two instructions : istr 1 retransmits on a particular port the packet and istr 2 floods the packet:

istr 1 = pin 1 ∪ op 1 ∪ pout 1 and istr 2 = pin 2 ∪ op 2 ∪ pout 2
The computation of the time constraints is not further discussed here for the demonstration, we fixed Est(istr i , pout i ) = [0.0005, 0.02] for i = 1, 2. Let's apply P roj Istr : The scenario applied is a ping from h1 to h2. The detection method will be put into practice in three cases: without anomalies, with an attack and with a failure. The frames, captured using wireshark, of the scenario without anomalies is represented in Fig. 5. To ease the readability, the value of the instances of the template and the evolution of O are detailed in Fig. 5. It will be the same for the case of attack and the case of failure. Secondly, at the observation of requests already seen, as pin 2199 or pin 2413 , the template instantiation is based on the knowledge stored previously in M T N , according to the condition line 11 of Algo. 2. Thereafter, commands pout 2200 and pout 2415 are observed and are similar to the one learned pout 2182 . As a consequence, the condition line 25 of Algo. 4 is fulfilled so no anomaly is detected.

Case of an attack

Let's consider the dynamic flow tunneling attack developed in [START_REF] Porras | A security enforcement kernel for openflow networks[END_REF]. To model such an attack, a bias b Cmd of a command pout ∈ Σ Out which leads to pout ∈ Σ Out is introduced:

∀i ∈ [1, lengh(pout)] pout [i] = pout[i] pout [1] = 10 if pout[3] = of 4, pout [1] = pout[1] else
The attack is simulated by adding on the controller an extra code modifying commands such that all traffic which passes through the switch 4 is in practice retransmitted on the port 10. In this section, two pings are considered: a first which permits the detector to learn the unfaulty behavior of the command and a second under attack. The corresponding frames are represented in Fig. 6. The aim of what follows is to show how an anomaly is detected. First, there is the learning phase with the observation of pin 149 and pout 151 as with pin 152 and pout 153 . Based on those exchange of packets, the estimation of M T N has been completed through Algo. 5. As a consequence, in the observation of pin 327 the template is instantiated through Algo. 2 with the expectation of the same command as pout 139 . But pout 329 is different of the command contained in the instance of O. Thus, the condition line 25 of Algo. 4 is not fulfilled and a fault is declared as shown in Fig. 6. In this example, the bias considered was a modification of the port of transmission but any other bias of the command which can be formalized as in section 3 would be detected similarly.

Case of a failure

The case of link failure between the switch 4 and the controller is considered it corresponds to a command pout ∈ Σ Out biased pout ∈ Σ Out as follows:

if pout[3] = of 4: pout = ∅ and else: pout = pout

To set up it, the code of ONOS is modified in order to not send the packets in direction to the switch 4. The observed frames are represented in Fig. 7.

At the observation of pin 589 from the switch 4 the template is instantiated in Algo. 2 and in parallel a timer is launched

CONCLUSION

To conclude, a secure and safe detection method, of both security and safety issue, has been proposed theoretically.

In this objective, a multi-controller architecture is proposed, without East-West interface, within a controller is in charge of the control of the network while the second observes the activity of the command and compares it to the template specification. This template is determined using a projection function and corresponds to the expected behavior of the command. The detection process consists into instantiating this template by the observation of the controller behavior to verify the consistency of its control which permits to detect anomalies. To pursue in this objective, this work would have to be experimented in the future and extended to non-deterministic algorithms using machine learning techniques. Moreover, techniques to take the lead, after the detection of a fault, by the second controller will have to be developed.

Fig. 1 .

 1 Fig. 1. Architecture proposed (c0: nominal controller, D: the detector, h 1 and h 2 : two hosts, s 1 and s n : the switches and t f : the time of the failure or the beginning of the attack)

Fig. 2 .

 2 Fig. 2. Representation of the steps of the detection method

  An hypothesis of this work is that the command algorithms embedded in the controller are known. The action/reaction causality of the algorithm is located in the instructions. The output of the algorithm, which corresponds to a Packet Out, is returned if the condition of the instruction, which corresponds to a specific Packet In, has occurred. Consequently, we decided to model the algorithms Algo through the instructions I. Algo = I m and I = Σ In × Op × Σ n Out An instruction istr ∈ I is defined as: • istr = (pin, op, P OU T ) • cond ∈ (Σ In ) B : a boolean equation which is verified by a set of Packet In pin ∈ Σ In . • op ∈ Op: a set of operations • P OU T = ∪ n i=1 pout i ∈ Σ n Out : the set of output corresponding to the commands.

  An instance T emp 16 f ault = F alse , sleep(min(T emp[2][2])) 17 if T emp / ∈ O then 18 f ault = T rue 19 sleep(max(T emp[2][2]) -min(T emp[2][2])) 20 if T emp ∈ O then 21 f ault = T rue 22 return f ault

P

  Fig. 4. The topology of the network

Fig. 5 .

 5 Fig. 5. Packets exchanged during a ping without anomalies The evolution of O(t) concerning the switch 4 is detailed in Fig. 5. Firstly, the Packet In pin 2158 is observed. The main algorithm Algo. 1 launches the instantiation through Algo. 2. The condition line 11 is not fulfilled as pin 2158 has not yet been observed, so any specific command is expected. The Packet Out pout 2170 is observed in the direction of the switch 4. Similarly, Algo. 1 launches the verification through Algo. 4. As an instance expects ∅ for the switch 4, the condition line 27 is fulfilled and the instance will be deleted from O. Moreover, Algo. 5 is launched to register the command in M T N as represented in Fig. 5.

Fig. 6 .

 6 Fig. 6. Packets exchanged during two pings. The first is the learning phase and the second is under the attack

Fig. 7 .

 7 Fig. 7. Packets exchanged during a ping. Case of a failure. by Algo. 3. After the waiting of 0.02 seconds a fault is declared in Algo. 3 as represented in Fig. 7.

ACKNOWLEDGEMENTS

This work was supported partly by the French PIA project "Lorraine Université d'Excellence", referenceANR-15-IDEX-04-LUE.


Corinne Galy-Lacaux

Claire Delon

Fabien Solmon
email: fsolmon@ictp.it

Marcellin Adon
email: adonatma@yahoo.fr

Véronique Yoboué
email: yobouev@hotmail.com

Jonas Mphepya
email: jmphepya@environment.gov.za

Jacobus J Pienaar

Babakar Diop
email: mbbdiop@yahoo.fr

Luc

Luc Sigha

Laouali Dungall
email: laoualid@yahoo.fr

Aristide Akpo

Eric Mougin
email: eric.mougin@get.obs-mip.fr

Eric Gardrat

Pierre Castera

Dry and Wet Atmospheric Nitrogen Deposition in West Central Africa

Keywords: African ecosystems, Atmospheric deposition budget, Nitrogen, Wet and dry deposition

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The international program DEBITS (Deposition of Biogeochemically Important Trace Species) started in 1990 as part of the IGAC/IGBP (International Global Atmospheric Chemistry/International Geosphere-Biosphere Programme) " Core Project ". It is to study wet-and dry-atmospheric deposition in tropical regions [START_REF] Lacaux | Acid wet deposition in the tropics: two case studies using DEBITS measurements[END_REF]. The DEBITS network includes 25 measuring stations well distributed within the tropical belt. DEBITS activities have been positively reviewed and are thus continuing within the new IGAC structure or DEBITS II [START_REF] Pienaar | Proposal of a new IGAC II task: DEBITS II (Deposition of Biogeochemically Important Trace Species[END_REF]Bates et al. 2006). For tropical Africa, the IDAF (IGAC/DEBITS/AFRICA) Project started in 1994. Since IDAF has been recognized by the Institut National des Sciences de l'Univers (INSU) and the Centre National de la Recherche Scientifique (CNRS) as a part of the Environmental Research Observatory (ORE) network. ORE/IDAF has the mission of establishing a long-term measuring network to study the atmospheric composition and wet-and dry-atmospheric processes and fluxes.

Regional Wet and Dry Nitrogen Deposition

The objective is to present a first estimation of the atmospheric nitrogen (N) deposition budget in West and Central Africa based on experimental measurements. To estimate atmospheric nitrogen (N) deposition, including both wet and dry processes, we compiled the IDAF N data (gas, particles, rain) obtained from the network presented above. We studied a transect going from dry savanna to humid savanna and forest. Presenting the different component of the N atmospheric deposition on these sites, i.e., dry deposition in gaseous and particulate forms associated with wet deposition, this study allows the relative contribution of dry and wet deposition processes to the total N deposition to be given.

Wet and Dry Deposition

Wet Deposition

An automatic precipitation collector specially designed for the IDAF network has been installed in all stations. A local operator collects water from each rainfall event in a Greiner tube (50 ml). Preserving the rainwater samples from contamination is an important issue since microbial input could modify its chemical composition. Samples are refrigerated at 4°C and preserved with 15 mg of thymol biocide or stored in a deep freeze environment. Ion Analytical, and Ionic Chromatography procedures are given in Galy- [START_REF] Galy-Lacaux | Precipitation chemistry in the Sahelian Savanna of Niger, Africa[END_REF].

The laboratory of Aerologie participate since 1996 to the international intercomparison program organized annually by WMO. According to prior results and through intercomparison tests organized by WMO, analytical precision is estimated to be 5 % or better for all ions, within the uncertainties on all measured values presented here. Combining all the uncertainties of measurements and calculations, the uncertainty of the wet deposition fluxes is estimated to be about 10 %.

To calculate wet N deposition in African dry savannas, we have compiled the annual volume weighed mean concentrations of nitrate and ammonium from the precipitation collected at 5 IDAF stations. The computation of nitrate and ammonium wet deposition has been done according to a mean annual rainfall for the studied period of each sites. The mean rainfall depth registered in Banizoumbou (Niger) and Katibougou (Mali) representative of dry savannas is 632 mm, in Lamto (Côte D'Ivoire) for wet savanna 1208 mm and in Zoétélé Cameroon for equatorial forests 1567 mm. The chemical composition of rain shows a strong gradient of nitrate content. In the dry savannas, biogenic emissions of NO x from soils have been identified as the major contributor to the nitrate content of rain, while the high ammonium content is related to ammonia emission from animals [START_REF] Serça | Comparison of nitrogen monoxide emissions from several African tropical ecosystems and influence of season and fire[END_REF][START_REF] Galy-Lacaux | Precipitation chemistry in the Sahelian Savanna of Niger, Africa[END_REF][START_REF] Galy-Lacaux | Long term precipitation chemistry and wet deposition in a remote dry savanna site in Africa (Niger)[END_REF][START_REF] Delon | Atmospheric nitrogen budget in Sahelian dry savannas[END_REF]. Wet deposition of nitrate is estimated to be around 1 kg N-NO 3 -ha -1 year -1 and 1.7 kg N-NH 4 + ha -1 year -1 in dry savannas, 1.4 kg N-NO 3 -ha -1 year -1 and 3 kg N-NH 4 + ha -1 year -1 in wet savannas and 2 kg N-NO 3 -ha -1 year -1 and 3 kg N-NH 4 + ha -1 year -1 in forests. The positive gradient of atmospheric nitrate and ammonium wet deposition fluxes is strongly dependent of the rainfall amounts gradient recorded along the studied ecosystems transect.

Dry Deposition

Considering the difficulties to measure directly dry deposition, the DEBITS program has adopted a strategy to infer indirect dry deposition measurements. Dry deposition is estimated on one hand from measurements of gaseous and particulate species based on continuous measurements of gaseous concentrations through passive gas sampling (NO 2 , NH 3 , HNO 3 ), and on bulk air sampling (ammonium and nitrate particulate content). On the other hand, realistic dry deposition velocity according to the site and the species needs to be calculated in order to estimate dry deposition fluxes.

Dry deposition of nitrogen from particles was calculated according to particulate ammonium (pNH 4

+

) and nitrate (pNO 3 -) concentrations determined in the water soluble content of the aerosols. We calculated annual mean concentrations values for the period 1998-2000. In order to estimate nitrogenous dry deposition fluxes of particles, we used a dry deposition velocity value of 1-2 mm s -1 [START_REF] Whelpdale | A global overview of atmospheric acid deposition fluxes[END_REF]. The calculation shows that total dry deposition of (pNH 4 + ) and (pNO 3 -) are of the same order of magnitude for all types of ecosystems with relatively low values. As an example, mean particle concentrations have been measured in Banizoumbou and Katibougou, and are very low: pNH 4 + = 0.31 ± 0.02 ppb and pNO 3 -= 0.16 ± 0.03 ppb in Banizoumbou, pNH 4 + = 0.17 ± 0.06 ppb and pNO 3 -= 0.23 ± 0.06 ppb in Katibougou. The other ecosystems present comparable particulate nitrate and ammonium concentrations in aerosols. The comparison of these concentrations in aerosol with gaseous NH 3 concentrations in the two dry savannas sites (2.9-10.4 ppb in Banizoumbou, 1.8-6.9 ppb in Katibougou) leads to the conclusion that particulate deposition is negligible. The mean annual deposition fluxes (pNH 4 + + pNO 3 -) in the semiarid, wet savanna and forested ecosystems are around 0.3, 0.4 and 0.06 kg N ha -1 year -1 , respectively. One should note that particulate dry deposition of nitrogen is smaller by an order of magnitude than the wet deposition.

Gaseous dry deposition of nitrogen has been calculated as the sum of dry deposition fluxes of ammonia (NH 3 ), nitric acid (HNO 3 ) and nitrogen oxide (NO 2 ). Gaseous measurement (NH 3 , HNO 3 , NO 2 ) are monthly integrated samples using passive sampling techniques following the work of [START_REF] Ferm | New measurement technique for air pollutants[END_REF]. This technique has been tested in different tropical and subtropical region [START_REF] Ferm | Measurements of air concentrations of SO 2 , NO 2 and NH 3 at rural and remote sites in Asia[END_REF][START_REF] Carmichael | Measurements of sulfur dioxide, ozone and ammonia concentrations in Asia, Africa, and South America using passive samplers[END_REF][START_REF] Martins | Longterm measurements of sulphur dioxide, nitrogen dioxide, ammonia, nitric acid and ozone in southern Africa using passive samplers[END_REF]). [START_REF] Adon | Long term measurements of sulfur dioxide, nitrogen dioxide, ammonia, nitric acid and ozone in Africa using passive samplers[END_REF] presents ten year of gases monitoring on 7 IDAF sites. Mean annual concentrations from the whole database have been performed. Figure 10.3 presents the mean annual concentrations of NO 2 , HNO 3 and NH 3 for the different IDAF sites measured over the period 1997-2007. Concentrations are ranged from 1 to 2 ppb for NO 2 , from 0.2 to 0.5 ppb for HNO 3 and from 3.5 to 6 ppb for NH 3 , respectively.

The major uncertainty in the estimation of trace gases dry deposition is due to the computation of the dry deposition velocity. Monthly dry deposition velocities have been calculated for 4 years (2002, 2003, 2004 and 2006) to follow the monthly seasonal cycle of measured gases [START_REF] Delon | Atmospheric nitrogen budget in Sahelian dry savannas[END_REF]). In the present work, we have calculated annual means of deposition velocities for NO 2 , HNO 3 and NH 3 for each ecosystem type. Annual deposition velocities (2002)(2003)(2004)(2005)(2006) range from 2 to 5 mm s -1 for NO 2 , from 6 to 18 mm s -1 for HNO 3 and from 2 to 8 mm s -1 for NH 3 . We estimate the total uncertainty applied to the dry N fluxes. The uncertainties are mainly linked to the concentration measurements and the estimation of the deposition velocities. For dry deposition of gases, the total rate of uncertainty applied for deposition fluxes is 70 % for NO 2 , 31 % for NH 3 and 38 % for HNO 3 .

NO 2 dry deposition fluxes presents a small variability according to the type of ecosystem, with fluxes varying from 0.5 kg N ha -1 year -1 in the wet savanna, 0.82 kg N ha -1 year -1 in the dry savanna, to 1 kg N ha -1 year -1 in forested ecosystem.

NH 3 dry deposition fluxes present higher values with 2 and 2.3 kg N ha -1 year -1 in dry savanna and wet savanna, respectively and 6 kg N ha -1 year -1 over the forest. Major sources of NH 3 include bacterial decomposition from urea in animal excreta and emissions by natural or fertilized soils [START_REF] Schlesinger | A global budget for atmospheric ammonia[END_REF]. In Africa, another significant source of ammonia is produced by savanna fires and domestic fuelwood burning [START_REF] Delmas | Nitrogen compound emission from biomass burning in tropical African savanna, FOS/DECAFE 91 Experiment (Lamto, Ivory Coast)[END_REF]. Dry deposition fluxes of HNO 3 are very low compared with NO 2 and NH 3 , with values ranged between 0.4 and 0.8 kg N ha -1 year -1 for all the ecosystems. This result is correlated to very low HNO 3 concentrations measured on all the stations.

Nitrogen Deposition Budget

Dry deposition fluxes, estimated for the three African ecosystems were combined with those associated with wet deposition to provide a first estimate in western-central Africa for the annual nitrogen atmospheric deposition. The total N deposition is estimated to be around 6 kg N ha -1 year -1 , 6.5 kg N ha -1 year -1 and 13 kg N ha -1 year -1 respectively over dry savanna, humid savanna and over the forest. These values should be taken with caution and we estimated the uncertainties on the budget to be around 30 %. If the estimations of wet deposition fluxes are known within a 10 % margin, dry deposition fluxes present larger uncertainties mainly due to dry [START_REF] Adon | Long term measurements of sulfur dioxide, nitrogen dioxide, ammonia, nitric acid and ozone in Africa using passive samplers[END_REF] deposition calculation. It is also important to note that our budget does not take into account all nitrogenous species, especially organic N species.

An important result highlighted by this budget is the importance of dry deposition processes in West Central Africa, especially for nitrogenous gaseous compounds. In dry savanna and forest, the relative contribution of dry deposition is about 60 %. In the wet savanna the contribution is around 50 %.

  The IDAF program is associated with the African Monsoon Multidisciplinary Analyses/Long Observation Program (AMMA/LOP, Lebel et al. 2010) over West/Central Africa and with the South African Climate Change Air Pollution-PICS (SACCLAP) Program. The main objectives of IDAF were to measure wet-and dry-deposition fluxes and identify the relative contribution of natural and anthropogenic source. In this way, the IDAF activity is based on high quality measurements of atmospheric chemical data (gaseous, precipitation and aerosols chemical composition) on the basis of a multi-year monitoring. The IDAF project implemented 8 monitoring sites covering the major African ecosystems over West and Central Africa: dry savanna (Niger, Mali, South Africa), wet savanna (Côte D'Ivoire and Benin) and equatorial forest (Cameroon, Congo) (Fig. 10.1).

  Fig. 10.1 Vegetation and location map of the 10 measurement stations of the IDAF network. Reused from Adon et al. 2010 (© Authors 2010. CC Attribution 3.0 License)

  Figure 10.2 presents annual volume weighed mean concentrations of nitrate and ammonium for selected sites and integrated periods. Mean nitrate and ammonium concentrations in the dry savannas sites measured from 1994 to 2005 in Niger and from 1997 to 2006 in Mali are around 11 µeq L -1 of NO 3 -and 19 µeq L -1 of NH 4 + (Galy-Lacaux et al. 2009). In the wet savanna, measurements performed from 1995 to 2002 give mean values of 8 µeq L -1 of NO 3 -and 18 µeq L -1 of NH 4 + (Yoboué et al. 2005). In a forested ecosystem, Sigha-Nkamdjou et al. (2003) measured mean values of 7 µeq L -1 of NO 3 -and 10 µeq L -1 of NH 4 + from 2000 to 2007.

  Fig. 10.2 Volume Weighed Mean (VWM) concentrations of nitrate and ammonium (µeq l -1 )

  Fig. 10.3 Annual mean concentrations of nitric acid, ammonia and nitrogen dioxide over the period 1997-2007 in part per billion by volume (ppb).[START_REF] Adon | Long term measurements of sulfur dioxide, nitrogen dioxide, ammonia, nitric acid and ozone in Africa using passive samplers[END_REF]

Acknowledgments This work is part of the IDAF (IGAC/DEBITS/AFRICA) project; and was funded by INSU/CNRS "Institut National des Sciences de l'Univers/Centre National de Recherche Scientifique" and the API "African Monsoon Multidisciplinary Analysis" (AMMA in 2005). The authors are grateful to all the field technicians at the African station and IRD (Institut de Recherche pour le Developpement) for logistical support.