L T Duarte

Interpreting the contribution of sensors in blind source extraction by means of Shapley values

Guilherme D. Pelegrina, Leonardo T. Duarte, Senior Member, IEEE, and Michel Grabisch.

Abstract-Several practical applications can be formulated as a problem of estimating a source of interest from a set of mixed data collected by different sensors. Although a lot of effort has been done to address the optimization task in signal extraction, there is a lack in the literature on how to evaluate the contribution of each sensor in the extraction process. In this letter, we propose a model-agnostic approach that can be used to interpret both the contribution of each sensor in the estimated source and the interaction effects between them. Our proposal is based on a solution concept from game theory, called Shapley value. Numerical experiments on synthetic and real data attest the use of our proposal in blind source extraction problems.

Index Terms-signal extraction, sensors design, Shapley value

I. INTRODUCTION

I N Blind Source Extraction (BSE) [1] problems, the aim is to retrieve a signal of interest based on a set of observed mixed signals collected by a set of sensors. Several practical situations can be modeled as BSE problems. For instance, in fetal electrocardiogram (fetal ECG) [START_REF] Shi | Semi-blind source extraction for fetal electrocardiogram extraction by combining non-gaussianity and time-correlation[END_REF], [START_REF] Li | Sequential blind extraction adopting secondorder statistics[END_REF], [START_REF] Jaros | Non-adaptive methods for fetal ECG signal processing: A review and appraisal[END_REF], one extracts a signal that better represents the fetal heart beats based on the set of signals captured by electrodes on the maternal abdomen. In a recent work [START_REF] Wang | Blind source extraction of acoustic emission signals for rail cracks based on ensemble empirical mode decomposition and constrained independent component analysis[END_REF], the authors adopted a BSE formulation to extract acoustic emission signals and detect rail cracks. Another example consists in detecting bearing faults [START_REF] Zhao | Underdetermined blind source extraction of early vehicle bearing faults based on emd and kernelized correlation maximization[END_REF] from vibration signals and, therefore, prevent machine breakdowns.

BSE is generally tackled by an optimization task whose goal is to retrieve a signal that optimizes a predetermined property of interest, such as time-correlation [START_REF] Barros | Extraction of specific signals with temporal structure[END_REF], [START_REF] Ferdowsi | Removing ballistocardiogram artifact from EEG using short-and longterm linear predictor[END_REF], [START_REF] Taha | Efficient blind source extraction of noisy mixture utilising a class of parallel linear predictor filters[END_REF], sparsity [START_REF] Nadalin | Blind extraction of the sparsest component[END_REF], smoothness [START_REF] Mitianoudis | Smooth signal extraction from instantaneous mixtures[END_REF], [START_REF] Duarte | Blind extraction of smooth signals based on a second-order frequency identification algorithm[END_REF], quasi-periodicity [START_REF] Tsalaile | Sequential blind source extraction for quasi-periodic signals with time-varying period[END_REF] and cyclostationarity [START_REF] Xiang | Secondorder cyclostationary statistics-based blind source extraction from convolutional mixtures[END_REF]. One may also have strategies that exploit more than one property simultaneously [START_REF] Shi | Semi-blind source extraction for fetal electrocardiogram extraction by combining non-gaussianity and time-correlation[END_REF], [START_REF] Pelegrina | A multi-objective approach for blind source extraction[END_REF]. As these properties vary for different applications (e.g., time-correlation for ECG data and cyclostationarity for communication signals), different algorithms have been proposed in the literature to deal with the optimization task.

Another research topic in BSE lies in evaluating the influence of sensors in sensor design. For example, in fetal ECG, there is an interest in strategically positioning the electrodes in order to improve the extraction of the fetal components [START_REF] Sameni | Electrode selection for noninvasive fetal electrocardiogram extraction using mutual information criteria[END_REF].

Clearly, due to physical and financial constraints (e.g., cost reduction), only a subset of them (from all possible positions in the maternal abdomen) is selected to compose the array of sensors. Indeed, the use of an excessive number of electrodes during labor brings an inconvenience for the clinicians and ergonomic concerns for the mother [START_REF] Dia | Fetal heart rate estimation by non-invasive single abdominal electrocardiography in real clinical conditions[END_REF]. Therefore, the use of a subset of sensors is of importance in real situations.

Although existing works deal with sensor selection [START_REF] Soleimani Gilakjani | Improved sensor selection method during movement for breathing rate estimation with unobtrusive pressure sensor arrays[END_REF] or sensor placement [START_REF] Ghayem | Robust sensor placement for signal extraction[END_REF] problems, to the best of our knowledge, no effort was done to investigate the contribution of sensors based on a game theory paradigm widely used in machine learning explainability [START_REF] Štrumbelj | Explaining prediction models and individual predictions with feature contributions[END_REF], [START_REF] Lundberg | A unified approach to interpreting model predictions[END_REF], [START_REF] Molnar | Interpretable machine learning[END_REF], [START_REF] Pelegrina | Shapley value-based approaches to explain the robustness of classifiers in machine learning[END_REF], called Shapley value [START_REF] Shapley | A value for n-person games[END_REF]. Therefore, in this paper, we propose an approach to interpret the importance of each sensor in BSE based on the Shapley value. When a set of players cooperate in a game, the Shapley value of a specific player indicates his/her marginal contribution on the achieved payoff. The cooperative game in this case refers to the BSE problem in the following way: the sensors (mixtures) can be seen as the players, and the retrieved signal can be seen as the game payoff when a subset of mixtures is available in the extraction procedure. Therefore, the Shapley value associated with each sensor will indicate its marginal contribution on the extracted signal. In the problem of designing an array of sensors (e.g., in fetal ECG), sensors with high marginal contributions should be considered to compose the array. Numerical experiments in both synthetic and real datasets attest our proposal.

The rest of this letter is organized as follows. In Sections II and III, we describe the theoretical aspects about the BSE problem and the Shapley value, respectively. Section IV describes the proposed Shapley value-based approach to evaluate and interpret sensors contribution in BSE. The numerical experiments are conducted in Section V. Finally, we present our conclusions and future perspectives in Section VI.

II. BLIND SOURCE EXTRACTION PROBLEM Assume a set of N signal sources s(k) = [s 1 (k), s 2 (k), . . . , s N (k)]
T , where k = 1, . . . , K are the time samples. Consider that these sources are mixed by a linear and instantaneous process A ∈ R M ×N such that

x(k) = As(k), (1)
is the set of M mixtures (i.e.,

x(k) = [x 1 (k), x 2 (k), . . . , x M (k)]
T ) collected by different sensors.

Among s(k), suppose that we are interested in retrieving a single source s n * (k), n * ∈ {1, . . . , N }. In BSE, one assumes that both the sources and the parameters of A are unknown.

Therefore, in order to estimate s n * (k), one needs to adjust an extraction vector w = [w 1 , w 2 , . . . , w M ] T such that

ŝ(k) = w T x(k) (2)
is as close as possible to s n * (k). For this purpose, one formulates an optimization problem whose cost function represents a property expected from the extracted source ŝ(k).

III. SHAPLEY VALUE IN COOPERATIVE GAME THEORY

The Shapley value is a classical solution concept in cooperative game theory [START_REF] Shapley | A value for n-person games[END_REF]. Assume that a set M = {1, 2, . . . , M } of M players cooperate in order to achieve a common goal (e.g., increasing the profit from supply chain [START_REF] Gao | Uncertain Shapley value of coalitional game with application to supply chain alliance[END_REF] or software development [START_REF] Fahimullah | Collaboration formation and profit sharing between software development firms: A Shapley value based cooperative game[END_REF] alliances). Consider also a function υ : P(M) → R, where P(M) is the power set of M, that maps subsets of players to real numbers. υ(D) is referred to as the payoff (e.g., the profit) achieved in the game when players in D cooperate. When there is no player in the coalition, one assumes1 υ(∅) = 0. The Shapley value of a player m indicates his/her marginal contribution on the payoff υ(M) achieved by the cooperation of all players. It is defined as follows:

ϕ m = D⊆M\{m} (M -|D| -1)! |D|! M ! ∆ m υ(D), (3)
where

∆ m υ(D) = υ(D ∪ m) -υ(D)
and |D| represents the cardinality of subset D. An important aspect of the Shapley value is that it satisfies several desired properties when sharing benefits from a cooperative game [START_REF] Young | Monotonic solutions of cooperative games[END_REF]. Among them, three are of interest for this work: symmetry, null player and efficiency. Symmetry means that if two players equally cooperate in the game, the associated Shapley values must also be equal.

According to the null player property, if there is no gain when player m joins any coalition, his/her Shapley value must be zero (there is no profit to receive). Finally, the efficiency property ensures that M m=1 ϕ m = υ(M) -υ(∅) = υ(M), i.e., one may decompose the payoff of the grand coalition among players according to the Shapley values.

Besides the marginal contributions of players, one may also be interested in the interaction degree between them2 . In other words, for two players m, m ′ , one may evaluate if there is a positive (the contributions of both players is greater than the sum of their individual contributions) or a negative (the contributions of both players is lower than the sum of their individual contributions) effect between them. The Shapley interaction index is calculated as follows [START_REF] Murofushi | Techniques for reading fuzzy measures (iii): interaction index[END_REF], [START_REF] Grabisch | Alternative representations of discrete fuzzy measures for decision making[END_REF]:

I m,m ′ = D⊆M\{m,m ′ } (M -|D| -2)! |D|! (M -1)! ∆ m,m ′ υ(D), (4)
where

∆ m,m ′ υ(D) = υ(D ∪ {m, m ′ }) -υ(D ∪ m) -υ(D ∪ m ′ ) + υ(D). In this case, γI m,m ′ > 0, γI m,m ′ < 0 or I m,m ′ = 0, where γ = sign (υ(M) -υ(∅))
is the sign function, indicate a positive (synergistic), a negative (redundant) or the absence of interaction effect between players m, m ′ .

IV. THE PROPOSED SHAPLEY VALUE-BASED APPROACH FOR BSE INTERPRETABILITY

The idea in this work is to use the Shapley value to interpret the importance of each sensor in BSE problems. We consider as players the set M of M sensors. Each sensor m provides a mixed signal which (possibly) contains information from the source of interest. Therefore, when we use more sensors, we have more available information in order to extract s n * (k).

Let us define ŝD (k) = w T D x D (k) as the estimated source when only the mixed signals x D (k) obtained by sensors in D ⊆ M are used to adjust the extraction vector w D . As the use of all sensors provides all possible information to retrieve s n * (k), we assume that ŝM (k) is the best estimation that we could obtain. By modeling the BSE problem as a cooperative game, we here define the payoff υ BSE (D) as a measure that quantifies the quality of ŝD (k) when estimating ŝM (k). By using the mean squared error (other measure could be adopted, such as the signal-to-interference ratio), the payoff can be defined by3 3) and (4), respectively. Note that, when D = M, υ BSE (M) = 0. Therefore, in our proposal and based on the efficiency property, the Shapley values indicate the marginal contributions of sensors that cooperates to improve the prediction (or reducing the error) from ŝ∅ (k) to ŝM (k). It remains to define ŝ∅ (k). When no mixture is available, we simply assume that the retrieved signal is a vector sampled from an i.i.d. Gaussian distribution N (0, 1). The idea of sampling from a Gaussian distribution is to define a signal that carries no information from the source of interest. It consists in the "worst choice" when no information is available, so the most uncertain guess (in the sense of differential entropy) is considered. Surely, other assumptions can be adopted to generate ŝ∅ (k). Algorithm 1 presents the steps of our proposal.

υ BSE (D) = 1 K min K k=1 (ŝ M (k) -ŝD (k)) 2 , K k=1 (ŝ M (k) + ŝD (k)) 2 , (5)
It is worth highlighting that our proposal is model-agnostic, i.e., it can be used regardless how the sources were mixed or which BSE algorithm was adopted. It only requires that the adopted model is able to estimate the source of interest for any D ⊆ M. Moreover, another important aspect is that, in order to calculate the Shapley values, one must calculate υ BSE (D) for all D ⊆ M. This means that one needs to address BSE problems for all coalitions of sensors. As this may pose a computational time constraint, existing techniques can be used to approximate the Shapley values with less effort [START_REF] Castro | Polynomial calculation of the Shapley value based on sampling[END_REF], [START_REF] Štrumbelj | Explaining prediction models and individual predictions with feature contributions[END_REF]. 4)), for all m, m ′ = 1, . . . , M

V. NUMERICAL EXPERIMENTS

In the sequel, we illustrate the application of our proposal in both synthetic and real mixed data.

A. Experiments on synthetic data

In the experiments with synthetic mixing data, we considered the signal sources presented in Figure 1, which were extracted from the ABio7 dataset available in ICALAB toolboxes [START_REF] Cichocki | ICALAB Toolboxes[END_REF]. We assumed as the source of interest the ECG (s 3 (k)). As this signal is sparse in time, we formulate an optimization problem whose goal is to retrieve the sparsest signal from the set of mixtures; in that respect, we consider an approach based on the ℓ 1 -norm. Therefore, for each coalition of sensors D, we deal with the following problem:

min w w T x D 1 / w T x D 2 , (6)
where ∥•∥ 1 and ∥•∥ 2 represent the ℓ 1 and ℓ 2 -norm, respectively.

Fig. 1: Signal sources (synthetic data).

We evaluate our proposal by assuming the mixing matrix

A =     α β β β β λ η β β β α β β β β α     , (7)
where the elements α, β, λ and η are defined according to the following scenarios (U(a, b) indicates a random value generated from a uniform distribution in the range [a, b]): 1) α, λ ∼ ±U(0.95, 1.05) and β, η ∼ ±U(-0.05, 0.05);

2) α, λ ∼ ±U(0.95, 1.05) and β, η ∼ ±U(0.45, 0.55);

3) α, η ∼ ±U(0.95, 1.05) and β, λ ∼ ±U(-0.05, 0.05); 4) α, β, η, λ ∼ ±U(0.45, 0.55). The obtained results (average contribution and standard deviation over 500 simulations) are illustrated in Figure 2. The green bar represents the mean squared error υ BSE (∅) and the red bars indicate the contribution of sensors in decreasing the error until υ BSE (M) = 0 (recall that, lower is the Shapley value, greater is the contribution when estimating ŝM (k)). In all cases, we could recover the ECG signal when all sensors were used. In scenario (1), sensor 3 captured much more information from the source of interest in comparison to the other ones. Therefore, its contribution is practically maximal while the contribution of the other sensors are minimal (see Figure 2a). This illustrates the null player property, as there is no contribution from sensors 1, 2 and 4 when joining coalitions. For scenario (2), sensor 3 is still capturing more information from s 3 (k). However, the others sensors also captured some information from it. Therefore, although the contribution of sensor 3 is the highest one, it is not maximal, as the others sensors also contribute to retrieve the ECG (see Figure 2b). In scenario (3), both sensors 2 and 3 captured the same amount of information from s 3 (k). As a consequence, their contributions were the same (see Figure 2c). This reflects the symmetry property, as sensors 2 and 3 provide the same gain when joining coalitions. Finally, in scenario (4), all sensors captured similar information from s 3 (k). As also expected by the symmetry property, all of them have the same contribution towards the estimated source (see Figure 2d).

B. Experiments on real data

Aiming at validating our proposal in a real BSE scenario, we considered the ECG data provided by [START_REF] De Moor | Daisy: Database for the identification of systems[END_REF]. There are eight ECG signals (mixed data, as presented in Figure 3) obtained from a pregnant women and the goal is to extract the fetal ECG. In order to estimate such a signal, we adopted the strategy proposed by [START_REF] Barros | Extraction of specific signals with temporal structure[END_REF], whose idea is to extract a signal that maximizes its autocorrelation given a predetermined delay τ . We assumed the same delay as identified by the authors from x 1 (k) (an ECG where the fetal heart beats can be visualized), i.e., τ = 112 (see [START_REF] Barros | Extraction of specific signals with temporal structure[END_REF] for more details on how to define τ as well as on the extraction algorithm). Figure 4 presents the estimated fetal ECG s f ECG (k) based on all mixtures.

The application of our proposal leads to the sensors contributions described in Figure 5. Sensors 3 and 1 have the highest contribution on the estimated signal. This is visually in accordance with Figure 3, where the fetal heart beats could be noted on both x 1 (k) and x 3 (k). The better performance of sensor 3 over sensor 1 can be explained by positive interactions (see Figure 6) between the former and sensors 6 (I BSE = -0.17). Therefore, the strong contribution of sensor 3 is due to both the presence of fetal heart beats in the captured signal and the potential gain when used in combination with other sensors that essentially captured maternal signals.

VI. CONCLUSIONS

This letter proposes an approach to evaluate the contribution of sensors in blind source extraction problems. Moreover, this approach also provides the interaction effects between sensors, i.e., if there are synergies or redundancies among them. An important aspect of our proposal is that it is model-agnostic, so it can be deployed in any signal extraction algorithm.

Numerical experiments illustrated the application of our proposal. Based on synthetic data, we demonstrated that the obtained sensors contributions are in accordance with the Shapley values properties. The greater the information from the source of interest captured by the sensor, the higher is its contribution when estimating such a signal. This conclusion is also attested by the results on real data. Moreover, we could note that synergies between sensors also increase their marginal contribution. It is worth mentioning that a mechanism to interpret sensors contributions can be useful, for instance, to identify the ones that have low impact on the extraction process and, therefore, could be removed from the set. As in the selection of array of electrodes for fetal ECG, this is useful to avoid unnecessary materials and save investments when (re)designing the set of sensors.

As we deal with recovering information from mixed signals, future works aim at extending our proposal to other signal processing tasks involving sensor arrays, such as the blind source separation problem. Moreover, another perspective consists in interpreting the importance of sensors in nonlinear mixture as well as in overdetermined models.

  where the minimum operator min {•} avoids miscalculations when ŝD (k) ≈ -ŝ M (k). The Shapley value of sensor m (ϕ BSE m ) and the Shapley interaction index between sensors m, m ′ (I BSE m,m ′ ) are defined by replacing υ(D) by υ BSE (D) in Equations (

Algorithm 1 ( 1 : 2 : 3 : 5 :

 11235 Shapley values for BSE interpretability) Input: x(k) Output: ϕ BSE m and I BSE m,m ′ , for all m, m ′ = 1, . . . , M Based on all mixtures, apply a BSE algorithm and conduct signal extraction: ŝM (k) ← w T M x M (k) Normalize ŝM (k): ŝM (k) ← (ŝ M (k) -µ ŝM ) /σ ŝM Calculate the payoffs for all coalitions of sensors: for D ⊆ M do if D = ∅ then Generate: ŝD (k) ∼ N k (0, 1) else if D = m then Define: ŝD (k) ← x m (k) else Apply a BSE algorithm and extract the signal: ŝD (k) ← w T D x D (k) end if Normalize ŝD (k): ŝD (k) ← (ŝ D (k) -µ ŝD ) /σ ŝD Calculate the payoff: υ BSE (D) (Equation (5)) end for Calculate the Shapley values and interaction indices: ϕ BSE m (Equation (3)) and I BSE m,m ′ (Equation (

  (c) Scenario (3). (d) Scenario (4).

Fig. 2 :

 2 Fig. 2: Shapley values for each sensor (synthetic data).

Fig. 3 :

 3 Fig. 3: ECGs (mixed signals).

Fig. 4 :

 4 Fig. 4: Estimated fetal ECG.

3 , 6 = 7 =

 367 -0.37), 7 (I BSE 3,-0.21) and 8 (I BSE 3,8

Fig. 5 :

 5 Fig. 5: Shapley values for each sensor (fetal ECG).

Fig. 6 :

 6 Fig. 6: Interaction effects between sensors (fetal ECG).

In cooperative game theory, one frequently adopts υ(∅) = 0. As will be discussed latter, in this work, we consider another value for υ(∅).

It is worth highlighting that the Shapley interaction index may be defined for any coalition of players. However, one only has clear interpretations for at most two players. See[START_REF] Grabisch | Alternative representations of discrete fuzzy measures for decision making[END_REF] for further details.

After extracting both signals, we normalize them in order to avoid scaling ambiguity: ŝD (k) ← ŝD (k) -µ ŝD /σ ŝD , where µ ŝD and σ ŝD represent the mean and the standard deviation of vector ŝD (k), respectively.

Work supported by São Paulo Research Foundation (FAPESP) under the grants #2020/09838-0 (BI0S -Brazilian Institute of Data Science), #2020/10572-5 and #2021/11086-0. L. T. Duarte would like to thank the National Council for Scientific and Technological Development (CNPq, Brazil) for the financial support. G. D.


Abdelhakim Zeghoudi
email: hakooumzeghoudi@gmail.com

Abdelber Bendaoud

Baghdadi Benazza

Seyf Eddine Bechekir

Laurent Canale

Influence of Shielding on the Magnetic Field Emitted by the LED Driver

Keywords: LED driver, Near Magnetic Field, Magnetic probe, Magnetic Shielding, Frequency domain (FD)

LED lamps are an extremely efficient, economical and durable lighting option that has revolutionized the lighting industry. They are increasingly used in many areas due to their significant advantages in energy efficiency and sustainability, To operate the LED lamps, you only need an AC/DC driver.

The aim of this study is to quantify the magnetic near field above LED driver, for the three components (Hx, Hy, Hz). in the frequency domain were performed by using spectrum analyzer. Then, the magnetic field with shielding was determined. The magnetic shielding techniques used are total shielding, partial shielding or double total shielding. All the cases are compared with each other in all directions x, y and z. The shielding material that we used is Aluminum foil of about 0.1 mm thickness, , the LED driver size which is equal to (x, y) = (2.8 cm, 1.6 cm) supplied by an AC 220 V source, 50 Hz of frequency.

I. INTRODUCTION

A LED driver is an essential component in LED lighting systems, ensuring that LEDs operate efficiently, reliably and long-lasting. It provides the right amount of current and voltage to optimize LED performance, while providing features such as protection and regulation [START_REF] Zeghoudi | Experimental Measurement of Near Magnetic Field by Different Probe for AC/DC LED Driver[END_REF][START_REF] Zeghoudi | Common Mode and Differential Mode noise of AC/DC LED Driver[END_REF][START_REF] Lehman | Professional Education Seminars: LED lighting: trends, standards, optics and power electronics[END_REF][START_REF] Simanjuntak | Power quality of energy saving lamps under wide voltage variations[END_REF]. However, LED drivers are sources of conducted and radiated electromagnetic disturbances due to high-frequency switching electronic switches such as MOSFET, IGBT, as well as rectifier diodes that generate low-frequency current harmonics, passive components (capacitors, inductors), and parasitic elements [START_REF] Zeghoudi | Determination of Power Factor and Harmonic Distortion of AC/DC LED Driver[END_REF][START_REF] Zeghoudi | Power Impact And Electromagnetic Disturbances Of Different Lighting Modes From Spot LED Lamp[END_REF][START_REF] Zeghoudi | Determination of electromagnetic disturbances in a buck chopper[END_REF][START_REF] Slimani | Experimental evaluation of conducted disturbances induced during high frequency switching of active components[END_REF][START_REF] Yao | Modeling, Analysis, and Reduction of Radiated EMI Due to the Voltage across Input and Output Cables in an Automotive Non-isolated Power Converter[END_REF][START_REF] Karol | Reduction of electromagnetic disturbances emission generated by LED lighting fixture[END_REF][START_REF] Ishida | EMC of Wireless Medical Telemeters and Noise Radiated from Light Emitting Diode Lamps[END_REF].

Electromagnetic radiation can be measured in the far field or in the near field (NF). However, near-field measurement has attractive advantages in terms of cost, accuracy, and scope. In addition, the NF scanning technique is less dependent on test conditions. Thus, it has been used to obtain EM fields, provide EMC studies and source diagnostics [START_REF] Zeghoudi | Study of the impact of the power dimming and lighting modes of LED lamps on the near magnetic field[END_REF][START_REF] Zeghoudi | Experimental Measurement of Near Magnetic Field by Different Probe for AC/DC LED Driver[END_REF][START_REF] Frikha | Modeling of the shielding effectiveness of enclosures in near field at low frequencies[END_REF][START_REF] Benyoubi | An Efficient Method for Modeling the Magnetic Field Emissions of Power Electronic Equipment from Magnetic Near Field Measurements[END_REF][START_REF] Song | A novel electromagnetic radiated emission source identification methodology[END_REF][START_REF] Essakhi | Characterization of radiated emissions from power electronic devices: Synthesis of an equivalent model from near-field measurement[END_REF][START_REF] Saidi | A near-field technique based on PZMI, GA, and ANN: Application to power electronics systems[END_REF].

For this case, the protection materials against radiated electromagnetic in sensitive electrical and electronic devices to avoid disturbances of the external and internal electromagnetic field between the system components itself is very interesting [START_REF] Frikha | Prediction of the shielding effectiveness at low frequency in near magnetic field[END_REF][START_REF] Frikha | A new methodology to predict the magnetic shielding effectiveness of enclosures at low frequency in the near field[END_REF][START_REF] Ott | Electromagnetic compatibility engineering[END_REF].

Electromagnetic shielding is one of the possible solutions, usually metal housing is used. Unfortunately, the housing is equipped with openings and slots that lead to a degradation of the shielding performance [START_REF] Ott | Electromagnetic compatibility engineering[END_REF][START_REF] Paul | Introduction to Electromagnetic Compatibility[END_REF][START_REF] Rachidi | Notes de cours, Compatibilités électromagnétique[END_REF][START_REF] Hedia | Electromagnetic Time Reversal in the Near Field: Characterization of Transient Disturbances in Power Electronics[END_REF][START_REF] Zeghoudi | Experimental Measurements of Shielding Influences on Near Magnetic Field for LED Lighting Sources[END_REF].

Magnetic shielding works on the principle of deflection of magnetic field lines. When the shielding material is placed between the source of the magnetic field and the object or device to be protected, it attracts the magnetic field lines, creating a path of least resistance. This has the effect of reducing the magnetic field within the protected area [START_REF] Zeghoudi | Effectiveness of Shielding on the Magnetic Emissions Emitted by a Toroidal Inductor[END_REF]. This work is structured as follows: Section II introduces the AC/DC LED drive converter under examination, providing a comprehensive overview of its characteristics. Section III elaborates on the near-field scanning method in the FD, including a description of the magnetic field probes utilized in the bench measurement. Finally, Section IV outlines and analyzes the experimental findings pertaining to the near magnetic field, both with and without shielding positioned above the driver. The conclusion is presented in section V.

II. AC/DC LED Driver Study

The focus of this paper is on the analysis of an AC/DC converter LED driver that powers a 12 W lamp. The driver operates with a supply voltage of 220 V at a frequency of 50 Hz. It comprises a MB10F bridge rectifier, along with passive elements such as capacitors, inductors, resistors, and an ES1J diode Figure 1 illustrates the placement of its components on the printed circuit board (PCB).

III. Magnetic Field Measurement Bench

In this study, the radiation measurement of the studied AC/DC converter of the LED driver is carried out in measurement for different magnetic probes of type Rohde Schwarz Probe (1), RSH400-1, Probe (2), RSH50-1 and Probe (3), RSH 2.5-2 [START_REF]R&S®HZ-15/R&S®HZ-17Probe Sets R&S®HZ-16 Preamplifier "E and H near-field emission measurements with test receivers, spectrum analyzers and oscilloscopes[END_REF]as shown in Figure 2. The frequency measurement technique employs a highprecision spectrum analyzer with a wide bandwidth. This equipment is utilized for visualizing and recording the frequency signals acquired through the magnetic field probe's measurement above the LED driver, as illustrated in Figure 3.

In Figure 4, we observe the probe being systematically scanned over the LED Driver using a tracing table. This manual process involves moving the probe across the circuit under test to gather magnetic field measurements at precise points on the surface designated for mapping, with a step size of 0.5 cm.

IV. Results and Discussions v current shape and Total Harmonic Distortion (THD)

Figure 5 shows the current and voltage at the input of the LED Driver. We notice a sinusoidal voltage signal against the periodic current signal period strictly identical to the initial sinusoid but very different form because of the presence of the non-linear load which is a rectifier with diode bridge (MB10F), so it's a source of pollution for the line current.

Figure 6 shows the odd harmonics of the line current at the base frequency. We notice a very important amplitude of harmonic current of order 3 compared to the fundamental current, also the odd harmonics of order 5, 7, 9, etc…, this phenomenon can explain by the presence of a source of harmonics, and therefore a poor power factor and a very high total harmonic distortion (THD).

v Magnetic Shielding

In order to measure the near magnetic field above the LED driver, it is necessary to look for some technique to reduce the radiated emissions. Among the techniques of the electromagnetic shielding, we used in our study the Aluminum foil as a shield against the magnetic field generated by the LED driver. We made a total circuit shielding and a partial shielding just on the inductor because this last one is generating much more radiated emissions compared to the other electronic components, as shown in the figure 8. The thickness of shielding is equal to 0.1 mm.

Figure 10 shows the magnetic field close to the Hx component for the three cases: without shielding, with total or partial shielding using the probe (2) RSH 50-1 above the driver in the center of the inductor at "z" = 1 .5 cm. We note that the magnetic field decreases with the shielding, so the intensity of the field is less in the case of the total shielding compared to the partial shielding although we measured the field above the inductance which can explain that the magnetic field above the LED is not only generated by the inductor, but also by the neighboring circuit.

Figure 11 shows the near magnetic field of the Hy component above the LED driver at the center of the inductor. We notice the radiated emissions with the reduced shielding, so, the magnetic field almost identical for total and partial shielding. For the magnetic field of the component Hz above the LED driver in the center of inductor, we find more intense magnetic field in the case of shielding, then a reduction towards the high frequencies as shown in the Figure 12. Therefore, with the partial shielding the near magnetic field is reduced, and for the total shielding the radiated emissions reduced again more than the partial shielding, beyond the frequency 200 MHz we observe an identical between the two curves of shielding. Figure 13 shows the magnetic field of the component Hz above the LED driver using the probe (1) which has a surface diameter of about 25 mm. We measured the magnetic field in the center of the circuit which is (x, y) = (1.4 cm, 0.8 cm) at the height z = 1.5 cm. For total or partial shielding or double shielding, we notice that the radiated emissions in the case of double shielding is very low because of the increase of aluminum thickness, with a minimization towards high frequencies, the case of partial shielding of the inductor, the reduced magnetic field and the margin between the case without shielding and partial shielding defines the magnetic field effect by the inductor, and the margin between the case without shielding and total shielding presents the magnetic field effect by the driver circuit.

V. Conclusion

This paper presented the more preferable near-field scanning method to quantify the radiated EMI noise of magnetic near-field for a printed circuit board (PCB) with the electronic components based on AC/DC LED driver in frequency domain in three directions. The near magnetic field generated by the inductor exceeds the limit of EMI standards CISPR 11 class A. In this case, the magnetic shielding is one of the most known solutions to reduce the radiated emissions emitted by the inductor that we used. The near magnetic field scan with partial shielding of the inductor or the total shielding of the whole driver circuit have been studied after a scan, where we observe the reduction of the radiated disturbances

Fig. 1 .

 1 Fig. 1. LED driver modules integrated onto the PCB.

Fig. 2 .

 2 Fig. 2. Rohde Schwarz type magnetic probes.

Fig. 3 .

 3 Fig. 3. Bench measuring the near magnetic field with and without shielding above the driver.

Fig. 4 .

 4 Fig. 4. Principe of Scanning over the LED Driver.

Fig. 5 .

 5 Fig. 5. Input current and voltage for LED Driver.

Fig. 6 .

 6 Fig. 6. Frequency spectrum of harmonics generated by the AC/DC LED Driver.

Figure 7 Fig. 7 .

 77 Figure7illustrates the dimensions of the examined circuit, with lengths denoted as x = 2.8 cm and y = 1.6 cm, and the maximum component height designated as z = 1.4 cm. Magnetic field measurements were conducted in three directions (x, y, z) at z = 1.5 cm, with and without shielding. The magnetic field probe employed was the RSH 50-1 probe[START_REF] Zeghoudi | Common Mode and Differential Mode noise of AC/DC LED Driver[END_REF]

Fig. 8 .

 8 Fig. 8. LED driver and type of Shielding a) without Shielding, b) Partial shielding, c) Total shielding, d) Double shielding.

Fig. 9 .

 9 Fig. 9. Magnetic field scan component Hz above the LED driver at z = 1.5 cm: (a) without shielding, (b) partial shielding (c) total shielding.

Fig. 10 .

 10 Fig. 10. Near magnetic field of component HX with and without shielding.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Near magnetic field of component HY with and without shielding

Fig. 13 .

 13 Fig. 13. Near magnetic field with and without Shielding at the center of LED Driver at z = 1.5 cm, measured with the probe RSH400-1.

above the LED driver. Then the effect of different shielding techniques above the center of the inductor have been presented and compared with each other, for the different magnetic probe.