Yoan Miche

Patrick Bas

Amaury Lendasse

Christian Jutten

Olli Simula

Extracting Relevant Features of Steganographic Schemes by Feature Selection Techniques

This paper presents a methodology for steganalysis based on a set of 193 features with two main goals. The first goal is to determine a sufficient number of images for effective training of a classifier in the obtained high-dimensional space. Second goal is to use feature selection to select most relevant features for the desired classification. Dimensionality reduction is performed using a forward selection and reduces the original 193 features set by a factor of 13, with overall same performance. Additionally, two new tools are proposed for feature selection in order to validate and possibly analyze further the current results.

Introduction

Steganalysis' main goal is to automatically detect, with the possibly highest accuracy, the presence of a secretly embedded content in another document. This can be seen as a typical classification problem since an optimal separation between stego images and genuine ones is seeked.

For this matter, various features are extracted from the considered images. This number of features is nowadays growing consequently (as seen on Fig 1) and might be too important for the classifiers typically employed, as well as for an interpretation of the results.

The very first examples of LSBbased steganalysis made use of less than ten features, with an adapted and specific methodology for each stego algorithm. The idea of "universal steganalyzers" then became popular. In 1999, Westfeld proposes a χ 2 -based method, on the LSB of DCT coefficients [START_REF] Westfeld | Attacks on steganographic systems[END_REF].

Five years after, Fridrich in [START_REF] Fridrich | Feature-based steganalysis for jpeg images and its implications for future design of steganographic schemes[END_REF] uses a set of 23 features obtained by normalizations of a much larger set, whilst Farid et al. already proposed in 2002 a set of 72 features [START_REF] Lyu | Detecting hidden messages using higher-order statistics and support vector machines[END_REF]. Since then, an increasing number of research works are using supervised learning based classifiers in very high dimensional spaces. The recent work of Y. Q. Shi et al. [START_REF] Shi | A markov process based approach to effective attacking jpeg steganography[END_REF] is an example of an efficient result but using 324 features based on JPEG blocks differences modeled by Markov processes.

All these results are achieving better and better performance in terms of good classification. Meanwhile, there are some clear side-effects to this growing feature spaces. First of, as long as performances seem to increase, people tend to keep on adding new features to their sets, regardless of the resulting dimensionality. It has been shown [START_REF] François | High-dimensional data analysis: optimal metrics and feature selection[END_REF][START_REF] Verleysen | The curse of dimensionality in data mining and time series prediction[END_REF] that the feature space dimensionality in which the considered classifier is trained in, can have a crucial impact on its performances. Second, the computational complexity of the classifier's training is another important aspect, since most of the widely used classifiers have at least a linear relationship to the dimensionality of the space. Third, a high dimensional space requires an important number of samples (images in this case) for a correct training of the classifier and accurate model parameters. The relationship between dimensionality and number of samples is not trivial and a sufficient number of samples has to be determined for the considered problem.

Another matter is about the interpretability of such results on high dimensional spaces. A correct analysis of the results and of the possible weaknesses of the stego algorithms seems to be hard and long to perform, with so many features.

We propose a general methodology to address some high dimensionality issues. The next section proposes to illustrate the main problems related to this dimensionality. Section 3 presents the main tools and classifiers used in this paper which are part of the methodology. Results using the stego algorithm Outguess [START_REF] Provos | Defending against statistical steganalysis[END_REF], and a set of 193 features from Fridrich [START_REF] Pevny | Merging markov and dct features for multi-class jpeg steganalysis[END_REF] follow, with an overall analysis in section 5. Some new tools to investigate the current results and possibly extend them are finally presented in section 6.

Side-effects of the growing number of features

The common term "curse of dimensionality" [START_REF] Bellman | Adaptive control processes: a guided tour[END_REF] refers to a wide range of problems related to a high number of features. Our concern is focused on four main aspects of this high-dimensionality, namely the lack of interpretability, the complexity increase, the possible sensitivity to irrelevant features and the "need" for more data points, related to the empty space phenomenon as presented in [START_REF] François | High-dimensional data analysis: optimal metrics and feature selection[END_REF][START_REF] Verleysen | On the effects of dimensionality on data analysis with neural networks[END_REF][START_REF] Verleysen | The curse of dimensionality in data mining and time series prediction[END_REF].

The need for data points

As an example, when considering as few as five points in a three dimensional space, the underlying structure is impossible to infer, and so is the creation of a model for it. On the contrary, with hundreds to thousands of points it becomes possible to see clusters, relationships between dimensions and such.

More generally, in order for any tool to be able to analyze and find a structure within the data, the number of needed points is growing exponentially with the dimension. Indeed, consider a d-dimensional unit side hypercube, the number of points needed to fill the Cartesian grid of step ǫ inside of it, is growing as O((1/ǫ) d ). Thus, using a common grid of step 1/10 and a dimension of 10, it requires 10 10 points to fill the grid.

In practice, most of data analysis use at least 10 to 20 dimensions, implying a "needed" number of points impossible to achieve. As a consequence, the feature space may be not correctly filled with data points, which can give wrong models when building classifiers, having to extrapolate for the missing points.

The situation is globally not limited to this lack of data points because of the dimensionality. The reason why so many points are needed when dimension increases is the empty space phenomenon, related to the positioning of randomly distributed points in high dimensions.

The empty space phenomenon and distance concentration

On a theoretical point of view: draw points from a normal distribution and consider the probability to have a point at distance r from the center of the distribution. It is given by the probability density function:

f (r, d) = r d-1 2 d/2-1 . e -r 2 /2 Γ (d/2) (1)
having its maximum at r = √ d -1. Thus, when dimension increases, points are getting further from the center of the distribution. The phenomenon is the same for the uniform distribution as In practical cases it is also observed [START_REF] François | High-dimensional data analysis: optimal metrics and feature selection[END_REF][START_REF] Verleysen | On the effects of dimensionality on data analysis with neural networks[END_REF][START_REF] Verleysen | The curse of dimensionality in data mining and time series prediction[END_REF] that high-dimensionality has this effect on the data distribution: average distances distribution between two points increases with dimension while the distances variance distribution tends to decrease, as in the theoretical cases above.

This distance concentration tends to decrease the relevance of distance-based classification since the common notions on which these classifiers are based are no longer fully reliable. This fact and the possible disturbance of the classifier by unuseful dimensions especially motivates the use of feature selection techniques for dimensionality reduction: the forward technique presented in section 3 realizes such reduction and makes possible analysis and work in resulting lower dimensionality spaces.

Problems regarding irrelevant features

Here is presented the problem of the irrelevant and "parasite" dimensions: these dimensions bringing no useful information to the considered classification problem. An overall view of the effects of such sort of data on a SVM can be seen in [START_REF] Hermes | Feature selection for support vector machines[END_REF].

The two points sets called Whirls and Chessboard are depicted in Table 1 then presents results of three widely used classifiers on these two special examples to which have been added Gaussian distributed dimensions, with no information relevant for the classification process.

As can be seen from Table 1, the classification rate drops significantly when adding these unuseful dimensions. The sensitivity of the classifier to these seems to be more noticeable for the SVM classifier than for the KNN. This robustness of the KNN to high dimensionality as well as to noisy or irrelevant data is well known.

Lack of interpretability

Eventhough the nearest neighbours classifiers keep good performances in high dimensions [START_REF] François | High-dimensional data analysis: optimal metrics and feature selection[END_REF][START_REF] Verleysen | On the effects of dimensionality on data analysis with neural networks[END_REF][START_REF] Verleysen | The curse of dimensionality in data mining and time series prediction[END_REF], other obvious problems of high dimensionality motivate the idea of feature selection. The interpretability is an important one: high performances can indeed be reached using the whole 193 features set proposed by Fridrich [START_REF] Pevny | Merging markov and dct features for multi-class jpeg steganalysis[END_REF] for classification. Meanwhile, if looking for the weaknesses and reasons why these features react vividly to a specific algorithm, it seems rather impossible on this important set.

Reducing the required number of features to a small amount through feature selection enables to understand better why a steganographic model is weak on these particular details, highlighted by the selected features.

Increase of complexity

Computational time is another main reason. Nearest neighbours methods are usually implemented with a O(d) dimension relationship, as for SVMs. Clearly, reducing the dimensionality by a significant order of magnitude gives much more achievable computational times. As a consequence, one can use more data points and lower this "missing points" effect.

The next section presents the proposed methodology for dimensionality reduction by feature selection. The classifiers used as well as the chosen selection techniques are detailed.

Methodology and techniques used

Classifiers and appropriate number of images

The "appropriate" number of images (or at least the minimum required for the dimensionality of our data) should be determined. For this matter, a KNN classifier is used with a Monte-Carlo technique [START_REF] Christian | Monte Carlo statistical methods[END_REF]. This enables to estimate the noise and give a confidence interval for our results. We randomly draw (without repetitions) a subset of the whole data set, and use the obtained classifier on it.

For our experiments, two different types of classifiers have mainly been used: the first one, KNN, for its overall good performance even in high dimensional spaces, but most of all, because it is computationally very fast. SVM was also chosen because it is among the classifiers giving the best results. Major drawback is of course the computational time. The main principles of these two major classifiers are briefly explained in the following. Cross-validation of the KNN model is achieved in this paper through Leave-One-Out [START_REF] Efron | An Introduction to the Bootstrap[END_REF].

Support Vector Machine (SVM) SVM have been created by Vapnik [START_REF] Vapnik | Statistical Learning Theory[END_REF] in 1963 and then improved more recently (1992,1995) by Boser, Guyon and Vapnik [START_REF] Boser B E, Guyon | A training algorithm for optimal margin classifiers[END_REF]. The original idea was to separate data using a hyperplane: this was a linear classifier. The extension of this method adds a non-linear part by the use of kernel functions.

Taking a set of d-dimensional data {x i } ∈ R d , i ∈ 1, N labeled with classes y i ∈ {-1, 1} (restricted to two classes), the problem is to solve the optimization problem min w,b,ξ

||w|| 2 + C N i=1 ξ i , (2)
under the constraint

(w T ψ(x i ) + b)y i ≥ 1 -ξ i , (3)
where w is the separating hyperplane normal vector and ξ i are slack variables enabling clean separation even if data is not linearly separable in the mapped feature space. The training data x i is indeed mapped into a higher dimensional space by the function ψ, space into which it is hoped to be more linearly separable.

This ψ function is related to the kernel function K(x i , x j ) by

K(x i , x j ) = ψ(x i ) T ψ(x j ). (4)
In the study, Radial Basis Function kernels have been used, defined as

K(x i , x j ) = exp -γ||x i -x j || 2 , (5)
since they are the most commonly used kernels because of their good efficiency [START_REF] Wand | Gaussian-based kernels[END_REF][START_REF] Park | Universal approximation using radial basis functions network[END_REF].

As said before, the main issue with SVM is computational time, since equation 2 is a NP hard optimization problem. Most SVM algorithms (such as SMO [START_REF] Platt | Fast Training of Support Vector Machines using Sequential Minimal Optimization[END_REF]) are running with a dependency of order O(N 2 d) up to O(N 3 d), depending if K(x i , x j ) (eq. 5) can be stored in memory. With the choice of this particular kernel, we have two hyper-parameters -C and γ -to determine, expanding the previous line search for the k of KNN, to a grid search. Note that the determination of a good model for classification is then even more time-consuming. The validation for SVM is finally performed with a 10-fold cross-validation [START_REF] Efron | An Introduction to the Bootstrap[END_REF].

This again motivates the proposed feature selection step; the following is about the technique used to perform it, called forward selection.

Feature selection technique: Forward selection

The forward selection algorithm is a greedy algorithm proposed in [START_REF] Whitney | A direct method of nonparametric measurement selection[END_REF]; the algorithm selects one by one the dimensions, trying to find the one that combines best with the already selected ones, as shown in the following algorithm 2 (with x i denoting the i-th dimension of the data set):

Algorithm 2 Forward R = {x i , i ∈ 1, d } S = ø while R = ø do for x j ∈ R do
Evaluate performance with S ∪ x j end for Set S = S ∪ {x k }, R = Rx k with x k the dimension giving the best result in the loop end while Even if its capacity to isolate efficient features is obvious, the forward technique has some drawbacks: in the case where two or more features would have a high dependency and be "inefficient" when alone but very good when put together, forward might not take these into account soon enough in the selection process. Nevertheless, the feature selection using forward has been showing very good results and seems to perform well on our feature set; this is presented in the next section.

Results in steganalysis

Monte-Carlo

Determination of number of points required Fig. 4. Schematic view of the proposed methodology: (1) An appropriate number of data points to work with is determined using a Monte-Carlo method for statistical stability; (2) The forward selection is performed using a KNN classifier; (3) A good feature set is selected and performance is improved using SVM. number of images to use for training with the prepared database. Using a Monte-Carlo method on low numbers of images with both SVM and KNN, averaged plots are obtained. From it, a correct idea of a sufficient number of images for the later study can be obtained, as shown in the following experiments.

Since KNN is the fastest classifier between the two presented, it is used for the next step with forward technique. This produces a ranking of the features showing how much each new feature contributes to the correct classification rate. The best features combination is selected. A SVM is finally used on this combination, to improve the performance and obtain the final best classification rate achieved.

Experimental setup

Our image base was constituted of 13 000 images of natural scenes, coming from 5 different digital cameras. Images are then all reduced to a size of 800 × 600 (multiples of 8) to avoid some possible block effects and artifacts due to JPEG re-compression on another grid. At the same time, they are changed from their original colorspace to grayscale colorspace (256 gray levels).

A cropping operation to 512 × 512 follows, since our implementation of the extractor of Fridrich's 193 features works on 512 × 512 image blocks (powers of 2). In the end, the whole set of images is separated into two equal parts: one is kept genuine while the other one is steganographied with the Outguess algorithm at an embedding rate of 25%. This choice of half steganographied and half genuine can be discussed as it does not reflects a real world situation. Meanwhile, the whole steganalysis process presented is designed to be used on one image at a time, determining whether it is genuine or not. Furthermore, this choice has been done to be able to compare performances with the steganalysis community current research.

For classification and test purposes, the training set has been made with at most 8000 images. Test set is composed of the remaining, that is 5000 images. The 193 features proposed by Fridrich are used as in [START_REF] Pevny | Merging markov and dct features for multi-class jpeg steganalysis[END_REF].

Determination of sufficient number of images

Presented first is the result of the evaluation of a sufficient number of images, as explained in the methodology. The Monte-Carlo is used on randomly taken subsets of 200 up to 2000 images with 10 iterations. Each model built -using KNN and SVM -is also evaluated on the test set of 5000 images.

A single point is evaluated with a randomly chosen set of 4000 images, since computational time becomes very important with such number of images. In practice, on Fig. 5 presenting the results of this study, the two cross-validation results (SVM and KNN) should not be strictly compared since they do not use the same number of images to validate the model: SVM uses a 10-fold crossvalidation, while a Leave-One-Out (LOO) method is used for KNN. Test results are, on the other hand, comparable. One can really see on these plots that an apparently sufficient number of images is over 2000 for both classifiers, since the classification rate seems to increase very slowly over this value. For the experiments, a bigger set of 4000 images has been chosen.

Forward selection and optimization by SVM

Here, the results of the forward selection are presented shortly. As can be seen from Fig. 6, the whole process of forward selection is not fully shown since we do not go over 21 features. Meanwhile, as will be more detailed in the analysis part of these results, good performance is already performed before this value. Since the goal is to have the smallest possible feature set, while keeping average same performances, the forward selection could be stopped at this point. The rest of the features selected by the forward algorithm only make the classification rate oscillate slightly around 95%.

Test and 10-fold cross-validation remain in a much thinner interval than for our only-KNN tryouts. Moreover, the performance gain with SVM is significative as expected and reaches up to 2% in the frame of these plots. Table 2 presents the main values obtained using 193 features set. Our feature selection gives interesting results on this set. Indeed, using as few as 14 features, we are less than 1.9% behind the value obtained with all features for 10-fold cross-validation. Test values are following the exact same pattern.

Analysis

Some details from the previous results can be interpreted in both fields as follows.

From machine learning point of view: through the previous results, a major achievement was obtained: reducing the dimensionality by more than 13 and keeping roughly the same performance, in the variance interval.

This result is interesting for different reasons:

-Computational time is drastically reduced, since the classifiers complexity relationships to dimensionality are linear (see Tab. 2). -By this result, the number of features is reduced by 13 and it allows future new analysis and experiments previously not possible. -This specific 14 features set has proven to be relevant for the classification problem, leading to the next point. -The comprehension and interpretability is kept through our method and even improves it in an important way, by highlighting Outguess' weaknesses.

From steganalysis point of view: for the steganalysis field, the obtained results are behind the actual best values, obtained for the Outguess algorithm by Fridrich. Nevertheless, the two advantages coming out of these results -namely the decrease of computational time and the gain in interpretability -can counterbalance this opinion.

The 11 features selected by forward technique from the previous 23 features set were already showing some details about Outguess' weaknesses [START_REF] Miche | A feature selection methodology for steganalysis[END_REF]. The major advance using the extended 193 features set is that it gives an even more detailed view on them. Since all vectors are no longer normed using the L 1 norm, informations about the precise weaknesses can be inferred.

The Table 3 lists the set of obtained 14 features out of the full 193 set. 3. Names of all 14 selected features from the 193 features set.

g -2 C-1,-1 C-2,-1 h 21 [4] h 12 [5] C-1,-2 h 21 [5] g 0 [6] C-2,-2 g -1 [9] C0,1 g -2 [4] C2,-2 C0,-2 Table
From its algorithm, it is known that Outguess does not embed the information into coefficients with values 0 and 1. For this reason, and from the 7 first selected features, it is likely that it changes the coefficients with values -2 and -1, having several sensitive features to such changes react: dual histograms of coefficients -1 and -2, co-occurrence matrix coefficients related to -1 and -2 values changes. . .

In the end, this set of features describes in a more precise way the functioning and problems of the Outguess algorithm. Taking these into account might help improve the scheme and become less detectable.

New algorithms and ideas for further investigations

Two algorithms thought to investigate the current selection results achieved are presented here. These algorithms are derivations of the forward algorithm. They have been thought to try to address the drawbacks of the forward algorithm, that is, the one-by-one selection process by incremental adding of features.

"Anti-forward"

The "anti-forward" algorithm is an inverse forward in this sense that it selects the worst features first, instead of the best ones. The result on the very same feature set as before can be seen on Fig 7. A line has been drawn at 65% of correct classification to illustrate how many variables are below this limit. Among the last selected features -that is, the most relevant ones, in a sense -, many belong to the set of 14 features that has been retained for the previous results and analysis. As can be seen, the "linear" relation for the beginning of the graph is broken around 174 features, to increase a lot more.

Eventhough there is a clear correlation between the forward selected set of features and the 193 -174 = 19 remaining features, some new features are present and their importance and influence can be investigated.

"Grouped forward"

The idea behind the "grouped" forward was to try to solve the forward problem when considering groups of relevant features when the standalone features are not as relevant. For this matter, a size S for the groups is set. The classical forward algorithm is then used on the whole data set R, until the number of selected features reaches S. These S features are removed and a forward is launched again on the Card(R) -S remaining features of the data set. Until none remains in the data set.

A plot of the correct classification rate is obtained for each group: 1 to 10 in our case since we used a group size of 10 on the 193 features set. This plot is shown on Fig 8. This plot clearly separates the whole set of features into two: the first four groups, achieving each at least 73% of correct classification with only one feature, and the other six ones which seem to be not relevant enough for classification.

A research on the appropriate size S of the groups depending on the goals, has to be done.

Conclusion

This paper has presented a new methodology for dimensionality reduction by feature selection in the framework of steganalysis.

The issues of dimensionality have been addressed and the first step of our methodology proves that the theoretically required number of images for correct training is far from being needed. By the use of a Monte-Carlo technique on up to 4000 images, it has been shown that such numbers of images are sufficient for stable results. A set of 193 features extracted from all images serves the classification process, preceded by the dimensionality reduction step. This part of our methodology is achieved using a forward selection with a KNN classifier. It enables to reduce the number of required features to 14, while keeping roughly the same classification results. Computational time is thus greatly improved, divided by about 11. Further analysis becomes again possible with this low number of features: conclusions and precisions about the steganographic scheme can be inferred from the obtained feature set. The last step using SVM for improvement over the previous KNN results achieves high classification results for so small a feature set, proving that many features among the full 193 set might not be relevant enough to be kept for classification purposes.

A comparison between the obtained reduced sets of features for various steganographic algorithms might reveal some common sensitive features. An analysis of these common points could help design a more generic steganalysis method using a "low" number of features.

Meanwhile, the new tools proposed might be of great help when analyzing the retained features and trying to improve the results. Crossing these results will likely lead to validation of the current set, with possible improvements when combining with the new discovered features. The "group forward" might also lead to developments on the selection of groups of features instead of one-byone processing, thus taking into account possible relationships and dependencies between features.

Fig. 1 .

 1 Fig. 1. Dimensionality of features sets

  Fig 2 illustrates.

  Dimension 100, uniform dist.

Fig. 2 .

 2 Fig. 2. Overview of pairwise distances for uniformally distributed points in 2 and 100 dimensions: in dimension 2, distances are varying among the full possible amplitude (0 to √ 2); in dimension 100, all distances get very far from zero and tend to be much more concentrated close to the highest possible values.

Fig 3 .Fig. 3 .

 33 Fig. 3. Two classes on two different examples: the double-whirl and the chessboard one, not trivial to classify. Red points stand for one class and blue ones for another.

  K-Nearest Neighbours (KNN) KNN classifier is a supervised distance-based classifier, proposed by Devijver and Kittler in[START_REF] Devijver | Pattern recognition : a statistical approach[END_REF], usually using the Euclidean metric. It uses a majority vote among the k nearest neighbours classes to assign the class of the new considered point. The basic algorithm follows these steps:Algorithm1 KNN for a fixed K value for each point do Compute pairwise distances with all others for k = 1 : 2 : K do Take majority class of k nearest neighbours end for end for Compare obtained classes with ground truth for each k Keep k value giving the best good classification rateIn practice, this algorithm is quite efficient, fast and only needs one parameter to be determined, which makes model structure selection easier. KNN algorithms usually run with a O(N 2 d) dependency (with N the number of data points and d the dimension).

4. 1

 1 Our methodologyThe three main points of the proposed methodology are detailed in the following. Fig.4illustrates the process. First is seeked a possibly good candidate for the

Fig. 5 .

 5 Fig. 5. Correct classification rate for SVM (circles, top curve) and KNN (squares, bottom curve) with associated variance.

Fig. 6 .

 6 Fig.6. Plot of the correct classification rate for SVM: 10-fold cross-validation (circles, top curve) and test (squares, bottom curve). Crosses are for performance for random sets of 14 features with a KNN classifier.

Fig. 7 .

 7 Fig. 7. Anti-forward

Fig. 8 .

 8 Fig. 8. Grouped Froward

Table 1 .

 1 Whirls ChessboardNoise Without 1 dim. 2 dim. 3 dim. Without 1 dim. 2 dim. 3 dim. Effects of adding irrelevant Gaussian distributed dimensions to the classification problem.

	SVM	90%	61% 53% 52%	89%	63% 53% 51%
	KNN	89%	74% 61% 60%	89%	75% 64% 62%
	LVQ	62%	55% 51% 51%	64%	65% 52% 51%

Table 2 .

 2 Results of the different classifiers for cross-validation and test.

		LOO / 10-fold Test Comp. time
	KNN 193	86.65%	85.89%	4.5min
	KNN 193→14	93.20%	89.02%	60s
	SVM 193	96.92%	96.76%	49h
	SVM 193→14	95.08%	94.86%	4.5h


Sajad Sadough
email: sajad.sadough@ensta.fr

Mahieddine M Ichir

Emmanuel Jaffrot

Pierre Duhamel

Wavelet Based Semi-blind Channel Estimation For Multiband OFDM

This paper introduces an expectation-maximization (EM) algorithm within a wavelet domain Bayesian framework for semi-blind channel estimation of multiband OFDM based UWB communications. A prior distribution is chosen for the wavelet coefficients of the unknown channel impulse response in order to model a sparseness property of the wavelet representation. This prior yields, in maximum a posteriori estimation, a thresholding rule within the EM algorithm. We particularly focus on reducing the number of estimated parameters by iteratively discarding "unsignificant" wavelet coefficients from the estimation process. Simulation results using UWB channels issued from both models and measurements show that under sparsity conditions, the proposed algorithm outperforms pilot based channel estimation in terms of mean square error and bit error rate and enhances the estimation accuracy with less computational complexity than traditional semi-blind methods.

I. INTRODUCTION

A UWB radio signal is defined as any signal whose bandwidth is larger than 20% of its center frequency or greater than 500 MHz [START_REF]First report and order, revision of part 15 of the commission's rules regarding ultra-wideband transmission systems[END_REF]. In recent years, UWB system design has experienced a shift from the traditional "single-band" radio that occupies the whole 7.5 GHz allocated spectrum to a "multiband" design approach [START_REF] Roy | Ultrawideband radio design: The promise of high-speed, short range wireless conectivity[END_REF]. That consists in dividing the available UWB spectrum into several subbands, each one occupying approximately 500 MHz.

Multiband Orthogonal Frequency Division Multiplexing (MB-OFDM) [START_REF] Batra | Design of multiband OFDM system for realistic UWB channel environments[END_REF] is a strong candidate for multiband UWB which enables high data rate UWB transmission to inherit all the strength of OFDM that has already been shown for wireless communications (ADSL, DVB, 802.11a, 802.16.a, etc.). This approach uses a conventional coded OFDM system [START_REF] Prasad | OFDM for Wireless Communications Systems[END_REF] together with bit interleaved coded modulation (BICM) and frequency hopping over different subbands to improve diversity and to enable multiple access.

Basic receivers proposed for MB-OFDM [START_REF] Batra | Design of multiband OFDM system for realistic UWB channel environments[END_REF], estimate the channel by using pilots (known training symbols) transmitted at the beginning of the information frame, implicitly assuming a time invariant channel within a single frame. Thus, for an accurate channel acquisition, one must send several pilot patterns resulting in a significant loss in spectral efficiency.

Recent works [START_REF] Ma | EM-based channel estimation algorithms for OFDM[END_REF], [START_REF] Al-Naffouri | Semi-blind channel identification and equalization in OFDM: An expectation-maximization approach[END_REF] have reported promising results on the combination of channel estimation and data decoding process by using the Expectation-Maximization (EM) algorithm [START_REF] Mclachlan | The EM Algorithm and Extensions[END_REF] . Though the latter scheme outperforms pilot based receivers, it has a higher complexity that may be of a critical concern for its practical implementations. This complexity is mainly dominated by the number of estimated parameters for channel updating and the decoding algorithm within each iteration.

In this work, we consider a semi-blind joint channel estimation and data detection scheme based on the EM algorithm, with the objective of minimizing the number of estimated parameters and enhancing the estimation accuracy. This is achieved by expressing the unknown channel impulse response (CIR) in terms of its discrete wavelet series, which has been shown to provide a parsimonious representation [START_REF] Johnstone | Neddles and straw in haystacks: Empirical bayes estimates of possibly sparse sequences[END_REF], [START_REF] Sadough | A wavelet packet based model for an ultrawideband propagation channel[END_REF]. Thus, we choose a particular prior distribution for the channel wavelet coefficients that renders the maximum a posteriori (MAP) channel estimation equivalent to a hard thresholding rule at each iteration of the EM algorithm. The latter is then exploited to reduce the estimator computational load by discarding "unsignificant" wavelet coefficients from the estimation process. Moreover, since the probability of encoded bits are involved in the EM computation, we naturally combine the iterative process of channel estimation with the decoding operation of encoded data.

This paper is organized as follows. Section II introduces MB-OFDM and its wavelet domain channel estimation observation model. In section III, we first describe a MAP version of the EM algorithm for channel estimation and then show how the number of estimated parameters can be reduced through the EM iterations. The combination of the channel estimation part with the decoding operation and implementation issues are also discussed. Section V illustrates, via simulations, the performance of the proposed receiver over a realistic UWB channel environment and section VI concludes the paper.

Notational conventions are as follows: D x is a diagonal matrix with diagonal elements x = [x 1 , . . . , x N ] T , E x [.] refers to expectation with respect to x, I N denotes an (N × N ) identity matrix; . , (.) * , (.) T and (.) H denote Frobenious norm, matrix or vector conjugation, transpose and Hermitian transpose, respectively.

II. SYSTEM MODEL AND WAVELET DOMAIN PROBLEM

FORMULATION MB-OFDM system divides the spectrum between 3.1 to 10.6 GHz into several non-overlapping subbands each one occupying 528 MHz of bandwidth [START_REF] Batra | Design of multiband OFDM system for realistic UWB channel environments[END_REF]. The transmitter architecture for the MB-OFDM system is very similar to that of a conventional wireless OFDM system. The main difference is that MB-OFDM system uses a time-frequency code (TFC) to select the center frequency of different subbands which is used not only to provide frequency diversity but also to distinguish between multiple users (see figures 1 and 2). Here, we consider MB-OFDM in its basic mode ie. employing the three first subbands. We consider the multiband OFDM transmission of figure 2 using N data subcarriers. At the receiver, assuming a cyclic prefix (CP) longer than the channel maximum delay spread and perfect synchronization, OFDM converts a frequency selective channel into N parallel flat fading subchannels [START_REF] Prasad | OFDM for Wireless Communications Systems[END_REF] for each subband as

lower subbands

                       Band #
y i,n = D si,n h i,n + z i,n i ∈ {1, 2, 3}, n = 1, . . . , N sym (1)
where (1 × N ) vectors y i,n , s i,n and h i,n denote received and transmitted symbols, and the channel frequency response respectively; the noise block z i,n is assumed to be a zero mean white complex Gaussian noise with distribution CN (0, σ 2 I N ) ; i is the subband index and n refers to the OFDM symbol index inside the frame. The observation model corresponding to all three subbands can be written in frequency domain as

Y m = D Sm H m + Z m m = 1, . . . , M sym (2)
where

Y m = [y 1,n , y 2,n , y 3,n ] T , S m = [s 1,n , s 2,n , s 3,n ] T , H m = [h 1,n , h 2,n , h 3,n ] T and Z m = [z 1,n , z 2,n , z 3,n ] T are (M × 1)
vectors, with M = 3N and M sym = N sym /3. In the remainder, unless otherwise mentioned, we will not write the time index m for notational convenience.

In order to take advantage of the wavelet based estimation, the channel impulse response is expressed in terms of its orthogonal discrete wavelet coefficients. Let F M,L be the truncated fast Fourier transform (FFT) matrix constructed from the (M ×M ) FFT matrix by keeping the first L columns where L is the length of the CIR over a group of three subbands. We define W as the (L×L) orthogonal discrete wavelet transform (ODWT) matrix. The unknown channel can be expressed as H = F M,L W H g, where g is the (L × 1) vector of the CIR wavelet coefficients. The Observation model 2 is rewritten as

Y = D S T g + Z (3)
where

T = F M,L W H .
Although at the transmitter, the channel is practically used by slices of 528 MHz bandwidth that corresponds to one of the subbands, at the receiver side we gather three received OFDM symbols for estimating the wavelet coefficients of the CIR, taken over all of the subbands (1.584 GHz bandwidth). This is motivated by the fact that estimating the channel over a wider bandwidth leads to a sparser wavelet representation. Besides, this approach simplifies the receiver architecture since there is no need to change the central frequency for down converting different subbands.

III. THE EM-MAP ALGORITHM FOR WAVELET DOMAIN CHANNEL ESTIMATION

The EM algorithm proposed in this section is able to integrate the advantages of wavelet based estimation via the prior choosen for channel wavelet coefficients. Next, we see how the MAP estimator leads to a thresholding procedure which is used for reducing the number of estimated coefficients at each iteration of the EM algorithm.

A. An equivalent model and the EM principle

Our first step consists in decomposing the AWGN in (3) into the sum of two different Gaussian noise terms as

Z = D S Z 1 + Z 2 (4)
where Z 1 and Z 2 are (M × 1) independent Gaussian noise vectors such that p(Z 1 ) = CN (0,

α 2 I M ) and p(Z 2 ) = CN (0, σ 2 I M -α 2 D S D H S )
. Since we are using normalized QPSK symbols, D S D H S = I M and the covariance matrix of Z 2 reduces to Σ 2 = (σ 2α 2 )I M . We define the positive design parameter ρ α 2 /σ 2 , (0 < ρ ≤ 1) and notice that setting ρ = 1 leads to Z 2 = 0 which is equivalent to working with the initial model (3). However, for 0 < ρ < 1, the above noise decomposition allows the introduction of a hidden channel vector H defined as

H = T g + Z 1 Y = D S H + Z 2 .
(5)

The hidden vector H provides a direct relation between true and estimated wavelet coefficients corrupted by an additive white Gaussian noise, allowing the two-stage observation model [START_REF] Ma | EM-based channel estimation algorithms for OFDM[END_REF] which is equivalent to (3). However, the difference with a standard denoising problem is that S and H are unknown. Hence, the observation model has missing datas and hidden variables and the MAP solution of g has no closed form. In such situations, the EM algorithm [START_REF] Mclachlan | The EM Algorithm and Extensions[END_REF] is often used to maximize the expectation of the posterior distribution over all possible missing and hidden variables. Let X = {Y, S, H} be the complete data set in the EM algorithm terminology. Note that the observation set Y determines only a subset of the space X of which X is an outcome. We search g that maximizes log p(g|X). After initialization by a short pilot sequence at the beginning of the frame, the EM algorithm alternates between the following two steps (until some stopping criterion) to produce a sequence of estimates {g (t) , t = 0, 1, . . . , t max }.

• Expectation Step (E-step): The conditional expectation of the complete log-likelihood given the observed vector and the current estimate g (t) is calculated. This quantity is called the

auxiliary or Q-function Q g, g (t) = E S, H log p(Y, S, H|g) y, g (t) (6)
• Maximization Step (M-step): The estimated parameter is updated according to

g (t+1) = arg max g Q g, g (t) + log π(g) (7)
where π(g) is a prior distribution for the wavelet coefficients. Next, we derive the specific formulas of each step, according to [START_REF] Ma | EM-based channel estimation algorithms for OFDM[END_REF].

B. E-step: Computation of the Q-function

The complete likelihood is p(Y, S, H|g) = p(Y|S, H, g) p(S| H, g) p( H|g).

According to [START_REF] Ma | EM-based channel estimation algorithms for OFDM[END_REF], conditioned on H, Y is independent of g. Furthermore, S which results from coding and interleaving of bit sequence is independent of H and g. Since Z 1 is a complex white Gaussian noise, the complete log-likelihood can be simplified to log p(Y, S, H|g) = log p(Y|S, H) p(S) p( H|g)

= log p( H|g) + cst. = - g H T H Tg -2g H T H H α 2 + cst. ( 8
)
where cst. are different constant terms that do not depend on g. According to [START_REF] Al-Naffouri | Semi-blind channel identification and equalization in OFDM: An expectation-maximization approach[END_REF] we have

Q g, g (t) = E S, H - g H T H Tg -2g H T H H α 2 + cst. Y, g (t)
= -

H (t) -Tg 2 α 2 + cst. ( 9
)
where H (t) E S, H [ H|Y, g (t) ]. From [START_REF] Sadough | A wavelet packet based model for an ultrawideband propagation channel[END_REF], it is obvious that the E-step involves only the computation of H (t) , we have

H (t) = S∈C H∈H H p( H|Y, g (t) ) d H p(S|Y, g (t) ) (10)
where the last equation results from the independence between S and H belonging respectively to the sets C and H which contain all of their possible values.

In order to evaluate H (t) , we first have to evaluate the conditional mean µ

(t) H of H as µ (t) H = H∈H H p( H|Y, g (t) ) d H (11)
Since both p(Y| H) and p( H|g (t) ) are Gaussian densities, p( H|Y, g (t) ) ∝ p(Y| H) p( H|g (t) ) is also Gaussian. By standard manipulation of Gaussian densities, we obtain

µ (t) h = Tg (t) + ρ D H S Y -D S Tg (t) . ( 12
)
By using ( 12) in [START_REF] Bernardo | Bayesian Theory[END_REF] and after some simplifications we get

H (t) = (1 -ρ) Tg (t) + ρ D H S Y (13)
where

D S = s∈C D S p(S|Y, g (t)
).

The E-step is then completed by inserting H (t) into Q(g, g (t) ), equation [START_REF] Sadough | A wavelet packet based model for an ultrawideband propagation channel[END_REF].

C. M-step: Wavelet Based MAP Estimation

In this step the estimate of the parameter g is updated as given in [START_REF] Mclachlan | The EM Algorithm and Extensions[END_REF] where Q(θ, θ (t) ) is given by ( 9)

g (t+1) = arg max g - H (t) -Tg 2 α 2 + log π(g) . (14)
Due to the orthonormality of both Fourier and wavelet transforms, T H T = I L and we can replace H (t) -Tg 2 by g (t)g 2 , where

g (t) = T H H (t) = (1 -ρ) g (t) + ρ (D S T) H Y (15)
The M-step can be written as

g (t+1) = arg max g - g (t) -g 2 α 2 + log π(g) . (16)
Actually g (t+1) in ( 14) is no more than the MAP estimate of g from the observation model

g (t) = g + Z ′ 1 (17) where Z ′ 1 = T H Z 1 ∼ CN (0, α 2 I L ).
From the Bayes theorem, the posterior distribution of g is given by

p g| g (t) ∝ p g (t) |g π (g) (18)
where p( g (t) |g) is the Gaussian likelihood, g ∼ CN (g, α 2 I L ).

In this approach, we adopt the Bernoulli-Gaussian prior distribution π (g) for the wavelet coefficients g of the unknown CIR described by

π (g j ) = λ δ(g j ) + (1 -λ) CN gj 0, τ 2 (19)
for j = 1, . . . , L, which allows us to model a sparseness property of UWB channels in wavelet domain. This amounts considering that the wavelet coefficients have a probability λ to be zero and a probability 1λ to be distributed as CN (0, τ 2 ).

In order to deal with that particular model, we introduce an additional state variable (or indicator) β j ∈ {0, 1} such that we can express this prior conditionally as    (g j |β j = 0) ∼ δ(g j ) with probability λ, (g j |β j = 1) ∼ CN gj 0, τ 2 with probability 1λ.

(20) This prior model, conditionally on that state variable, leads to a Gaussian posterior for g j which makes the estimation explicit; from the direct observation model g (t) j = g j + Z ′ 1,j , we can express these posterior probabilities of β j as

p β j = 0| g (t) j = λ N 0, α 2 /c p β j = 1| g (t) j = (1 -λ) N 0, α 2 + τ 2 /c (21)
where the constant c = λ N 0,

α 2 + (1 -λ) N 0, α 2 + τ 2 .
From this set of equations, we easily notice that the indicator variable β j allows us to discriminate between the noise coefficients (for β j = 0) and the effective channel wavelet coefficients (for β j = 1), eventually corrupted by noise. The indicator variables β j are estimated, in the MAP sense, by

β (t+1) j =      0, if p β j = 0| g (t) j ≥ 0.5 1, elsewhere. (22)
Therefore, the MAP estimates of the channel wavelet coefficients are obtained by a simple denoising/thresholding rule as

g (t+1) j =    0, if β (t+1) j = 0 τ 2 α 2 + τ 2 g (t+1) j , if β (t+1) j = 1 (23)
1) τ and λ updating: The prior parameters τ and λ stand respectively for the (significant)-wavelet coefficients energy and unsignificant coefficient probability. The update rules for these two parameters are MAP based rules derived from assigning conjugate priors to these parameters [START_REF] Bernardo | Bayesian Theory[END_REF]:

λ = ( L -1/2)/(L -1), τ 2 = η/(L -L) ( 24
)
where L = Card{j β j = 0} and η = βj =1 g (t+1) j 2 ; Card{.} denoting the set cardinality.

2) Reduction of the number of estimated parameters:

The thresholding procedure derived in this section, provides an easy framework for reducing the number of estimated coefficients. This can be done by discarding at each iteration, the elements of g (t+1) that are replaced by zero in (23). The underlying assumption is as follows: whenever the estimator attributes an unknown wavelet coefficient to noise (replace it by zero), this coefficient will always be considered as noise and so will not be estimated in future iterations.

This operation is shown in figure 3 and can be modeled as:

g (t+1) tr = Θ g (t+1) , T tr = Ξ T (25)
where the truncation operator Θ(.) gathers in g (t+1) tr the components of g (t+1) that must be kept and the operator Ξ(.) constructs T tr from T by keeping the rows corresponding to kept indexes. During the first iteration (t = 0), the algorithm does not perform any truncation and the EM algorithm estimates all coefficients. However, after each M-step, the number of unknown parameters to be estimated in the next iteration is reduced according (25) by using g (t+1) tr and T tr in the update formula of the E-step (13).

IV. DECODING METHOD AND IMPLEMENTATION ISSUES

According to equation [START_REF] Bernardo | Bayesian Theory[END_REF], we make use of the information on transmitted symbols, obtained from the decoder, to update the channel estimate at each iteration. Besides, the decoder requires an estimate of the channel in order to provide the probability of encoded bits. Hence, the semi-blind channel estimation algorithm is naturally combined with the process of data decoding. The a posteriori probability of the unknown symbol S k , p(S k |Y k , H (t) k ), is calculated using the a posteriori probabilities provided by the decoder at the end of the t-th iteration as

p(S k |Y k , H) = B i=1 P dec (c k,i ) (26)
where P dec (c k,i ) is the a posteriori probability corresponding to the i-th bit of S k , c k,i . At the first iteration, where no a priori information is available on bits c k,i , P dec (c k,i ) are set to 1/2.

Θ

P (0) dec (c k,i ) = .5
Observed Frame Decoded Bits Fig. 3. EM-MAP channel estimation combined with the decoding process.

Among several possible ways to practically implement a joint channel estimation and decoding receiver, we adopt the following global procedure (see figure 3).

• Initialization (t = 0) -Set all probabilities of coded bits P dec (c k,i ) to 1/2 and derive p(S|Y, g (0) ) according to (26). -Initialize the unknown vector g by g (0) obtained from pilot symbols.

• for t = 1, . . . , t max -Use the current estimate g (t) to calculate g (t+1) according to (23). -Discard the wavelet coefficients that are replaced by zero for the next iteration by evaluating g

(t)
tr and T tr from (25).

-if t = t max : Use the current estimate g

(t)
tr to update the probability of encoded bits P dec (c k,i ) and derive p(S|Y, g (t) ) from (26). else: Decode the information data by thresholding the uncoded bit probabilities with 1/2.

V. SIMULATION RESULTS

In this section we present a comparative performance study of the proposed EM-MAP algorithm. The binary information data are encoded by a non-recursive non-systematic convolutional encoder with rate R = 1/2 and constraint length 3. Each frame has a payload of 1 KB along with 3 pilot symbols at the beginning for initializing the channel of each subband. The interleaver is random and operates over the entire frame. Among different wavelet families, "symmetric" wavelet basis functions [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] providing the sparser representation [START_REF] Sadough | A wavelet packet based model for an ultrawideband propagation channel[END_REF] have been considered.Unless otherwise mentioned, the curves are obtained after t max = 4 iterations.

First, a sparse channel model where only 20 wavelet coefficients out of total 96 have non zero values, is considered. The second channel, referred to as Corridor, is a line of sight (LOS) scenario issued from realistic UWB indoor channel measurements [START_REF] Bories | UWB indoor channel measurement[END_REF] where the receive and transmit antennas are located in a corridor separated by 9 meters.

Performance comparison is made with two pilot-only based approach using ML and minimum mean square error (MMSE) channel estimation, referred to as pilot-ML and pilot-MMSE. We also compare the proposed algorithm with two semi-blind channel estimation based on the EM algorithm, called respectively EM-Freq and EM-Wav. The first approach, consists of estimating the channel over all of the three subbands, using the model (3), similar to [START_REF] Ma | EM-based channel estimation algorithms for OFDM[END_REF] while the second scheme is a wavelet domain EM based estimation of the channel where the prior model is set to have a uniform distribution.

Figure 4 depicts the mean square error (MSE) between true and estimated channel as a function of E b /N 0 . It can be noticed that, although the pilot-MMSE approach improves the estimation accuracy for low SNR values, the performance of pilot based channel estimation methods are very far from the family of semi-blind methods. Comparing the wavelet domain semi-blind approach (EM-Wav) and the frequency domain approach (EM-freq), shows that significant gain is achieved by the former method. As shown, the best performance is achieved by the EM-MAP method. We see that by using EM-MAP, a gain of almost 4 dB in SNR is achieved at MSE=2 × 10 -3 , as compared to the EM-Wav method. This clearly shows the adequacy of the EM-MAP method for the case where the unknown channel has few non zero wavelet coefficients, which is in perfect agreement with the prior model.

Figure 5 shows the BER results along with the BER for the case of perfect channel state information (CSI). It can be seen that at a BER of 10 -3 , the pilot-ML and the EM-Freq approaches are respectively 3.9 and 2 dB of SNR far from the BER obtained with the perfect channel. Furthermore, the performance of the Pilot-MMSE approach is not shown since it was very close to that of Pilot-ML. Also, we observe that wavelet based semi-blind methods perform closely to the perfect CSI case. For example, at BER=10 -4 , the EM-MAP and EM-Wav method have respectively about 0.2 dB and 0.5 dB of SNR degradation from the performance obtained with perfect CSI. We now evaluate the performance of EM-MAP by considering the Corridor channel. Figure 6, shows that wavelet based methods again outperforms pilot based and EM-Freq methods in terms of MSE and BER. However, the EM-MAP performance is now comparable to that of EM-Wav method. This can be explained by noting that when the channel is not sparse, small values are attributed to λ by the algorithm (see ( 24)). This leads to a gaussian prior model with a large variance compared to the noise variance, which can be approximated with a uniform prior. As a results, the prior becomes less informative and the EM-MAP performs close to EM-Wav, as shown in figures 6. Thus, the proposed EM-MAP algorithm is able to adapt its prior model parameters for each propagation environment. Figure 7 shows the average number of estimated parameters versus the iteration number different channel scenarios. As observed, the EM-MAP approach tends to reduce significantly the number of estimated parameters. This can be seen for the sparse channel where the number of estimated parameters is reduced up to 20 parameters at the fifth iteration. Furthermore, under non-sparse Corridor channel, the figure shows that EM-MAP method is preferred to EM-Wav, due to its lower computational load.

VI. CONCLUSION

This paper proposed a semi-blind MAP channel estimation algorithm that integrates the advantages of wavelet based estimation. The investigated method naturally combines the EM iterations with the decoding process. We derived an equivalent data model for the multiband OFDM system involving the channel over all 3 subbands expressed in the wavelet domain. By choosing a Bernoulli-Gaussian prior distribution for the channel wavelet coefficients, the MAP estimator yields a thresholding procedure at the M-step of the EM algorithm which we used to reduce the number of estimated coefficients. With only few iterations, the EM-MAP method provides significant reduction in the number of estimated parameters and outperforms all considered pilot based and semi-blind methods.

Fig. 2 .

 2 Fig. 2. TX architecture of the multiband OFDM system.

Fig. 4 .

 4 Fig. 4. Mean square error between the true and estimated coefficients for the sparse channel model.

Fig. 5 .

 5 Fig. 5. BER performance of the EM-MAP method over the sparse channel model.

Fig. 6 .

 6 Fig. 6. Mean square error between the true and estimated coefficients over the Corridor channel.

Fig. 7 .

 7 Fig. 7. Reduction of the number of estimated parameters through iterations, Eb/N 0 = 8 dB.


M Brison

M Bensetti

R De Lacerda

F Vinci

Dos Santos

C Taurand

Commercial Power Line Communication Adaptation for Avionic Applications

Aircraft designers believe that Power Line Communication (PLC) technology is a solution to reduce the number of cables aboard. One strategy to implement PLC networks in aircrafts is adapting commercial residential PLC solutions to the avionic environment. We propose in this paper a coupling architecture that improves PLC performance in terms of signal to noise ratio (SNR) when deployed on monofilar avionic networks. We also show that our proposal complies with the aeronautical electromagnetic standard (DO-160) through experimental validation. The proposed coupling architecture is investigated and validated through simulations and measurements. We analyze the performance of the PLC in terms of 3 parameters: S-parameters (transfer function), SNR and currents circulating on the aircraft circuit that must comply with the aeronautical electromagnetic regulations. The study also includes a real case scenario with the presence of a switch-mode power supply, which introduces distortion on the network and deteriorates the PLC performance. A comparison with a standard coupling circuit is performed and the results are promising.

I. INTRODUCTION

Aircraft designers are faced with the challenge of increasing the number of cables/connexions without increasing the weight of the aircraft. This increase is linked to the electrification of previously hydraulic or mechanical functions as well as the advent of new on-board features. Power Line Communication (PLC) appears as a solution to limit the proliferation of cables, given that the aircraft is an environment with strict electromagnetic interference (EMI) regulations that hamper the use of wireless communication technology. PLC is now a well-established communication technology for the Home Area Network (HAN) market. This type of technology uses the residential power distribution network (PDN) to ensure communication between devices. On the residential case, the network is built through the superposition of high-frequency signals on the 50/60 Hz network.

From the semiconductor industry perspective, the aircraft PLC market is too small to justify the conception of a new PLC standard. As a consequence, our efforts are geared to the implementation of an aeronautical PLC solution based on a HAN PLC chipset available on the market. Such a solution must comply with the EMI regulations and simultaneously support enough data throughput to ensure the communication among on-board equipments. This adaptation shall involve custom programming and hardware modifications, which are required due to physical differences between the aircraft and the residential power network.

In this paper, we focus on the adaptation of the coupling circuit (CC) for a residential broadband PLC modem with high data rate (>10MBps). The broadband PLC unit is an off-the-shelf product marketed for residential power networks. We propose a coupling architecture adapted to the aeronautical monofilar power network. We compare our solution with the provided HAN coupling circuit in terms of signal-to-noise ratio (SNR), with and without a Switched-Mode Power Supply (SMPS) unit connected. Emitted current measurements are performed on the cable to check the compliance of the solution with the avionic standard DO-160 [1].

II. COUPLING A HAN PLC MODEM TO AN AVIONIC

POWER NETWORK Typically, two types of electrical PDNs are deployed in an aircraft, a three-phase alternating current (AC) network 115V/400Hz and a 28V direct current (DC)network. We focus herein on the 28V DC power line. The 28 DC network is deployed mostly for medium or low power loads and it is spread throughout the whole aircraft. This particular PDN is rarely cabled as a bifilar network. The power distribution unit is connected to the load by a single wire, so the return current goes through the fuselage of the aircraft. Hence, the fuselage is the electrical ground and/or the return current path for all electrical equipment aboard.

Studies on PLC to avionic power networks have already been published. One of the main endeavors that tackled this area is the "Transmissions in Aircraft on Unique Path Wires" (TAUPE) project, whose focus was to explore the general use of PLC technology in aircrafts. Within the TAUPE collaboration, commercial broadband PLC modems were used to perform EMC and communications tests [START_REF] Dominiak | A bifilar approach to power and data transmission over common wires in aircraft[END_REF], [START_REF] Dominiak | Achieving emc emissions compliance for an aeronautics power line communications system[END_REF] and yielded encouraging results. However, these results were obtained over a bifilar network, which is not representative of the majority of 978-1-5386-0365-9/17/$31.00 ©2017 IEEE Frequency in Hz avionics PDNs. More recently, PLC solutions have been employed for specific aircraft subsystems like actuators command [START_REF] Larhzaoui | Ofdm plc transmission for aircraft flight control system[END_REF] or motor control [START_REF] Kilani | Feasibility of powerline communication on a three-phase cable between pwm inverter and motor for avionics applications[END_REF]. This work is complementary to these earlier efforts and aims to provide further information about the deployment of PLCs over monofilar networks, which is the most common PDN cabling.

From an electrical perspective, residential and aircraft PDNs can be equally modeled as transmission lines (TLs). The physical differences between both networks results on different TL characteristic impedances (Zc). Commercial PLC modems are designed to work on power-lines with a Zc of around 100Ω. Avionics power-lines, configured as a single wire 5cm above a ground plane representative of the 28VDC PDN, exhibit a Zc around 300 Ω [START_REF] Grassi | Channel characterization and emc assessment of a plc system for spacecraft dc differential power buses[END_REF]. This impedance mismatch leads to additional loss and, as a consequence, communication degradation. A solution to improve the efficiency of the PLC link is to modify the coupling circuit of the PLC modem to be adapted to the avionic characteristic impedance.

A. The Coupling Circuit (CC)

The conventional coupling circuit is optimised to typical HAN PDN conditions. Depending on the cabling, it is necessary to adapt the coupling circuit to optimise the transfer function, which affects the communication quality between devices [5]- [START_REF] Van Rensburg | Step-by-step design of a coupling circuit with bi-directional transmission capabilities[END_REF]. One of the solutions could be the use of an adaptive coupling circuit, but it is still a research challenge.

For the narrowband PLC case, detailed design considerations of coupling is provided in [START_REF] Van Rensburg | Step-by-step design of a coupling circuit with bi-directional transmission capabilities[END_REF]. The comparison between the differential and common mode coupling for the aerospace context is presented in [START_REF] Grassi | Channel characterization and emc assessment of a plc system for spacecraft dc differential power buses[END_REF], and on the three-phase avionic power network at [START_REF] Kilani | Feasibility of powerline communication on a three-phase cable between pwm inverter and motor for avionics applications[END_REF].

In this work, we introduce the implementation and test of a modified coupling circuit based on the adaptation of the Sparameter using the G.hn standard. The G.hn standard uses the [1.8-86] MHz frequency band. Due to the high noise intensity generated by aircraft equipments on the frequency band below 20MHz, we rely on the [20-86] MHz frequency band to establish communication. Our design objective is to set the Analog Front End (AFE) input impedance close to the characteristic impedance of the line over the upper frequency band to limit the impedance mismatch. To design the circuit, we employ a custom BB transformer, to reach the correct impedance matching, connected to the Tx ports and to the power lines with decoupling capacitors. A high frequency filter designed around the input impedance of the AFE chipset is connected between the BB transformer and the Rx ports.

A comparison of the transfer function of the conventional G.hn CC and our CC is presented in Fig. 1. The conventional CC is the one used for the off-the-shelf G.hn PLC solution. We verify that the conventional CC is adapted to the [1.8-30] MHz frequency band but not for the whole [1.8-86] MHz band. As a consequence, it is not optimal for our avionic environment. We can observe that our CC is matched within the [20-86] MHz frequency band, as expected, and it provides an insertion loss of at most -3dB. The schematic of our CC and a photo of the board with our CC are shown in Fig. 2. We removed the original coupling circuit from the board to connect ours directly at the signal input/output of the AFE.

B. Comparison between the two coupling circuits

To verify the efficiency of the proposed coupling circuit, we compare the SNR perceived by the PLC modems with both We show in Fig. 3 the SNR of the link with the conventional CC in blue and with the improved CC in dotted red. The modified CC clearly improves the SNR , up to 20 dB at some frequency bands.

III. EMI AND SNR MEASUREMENTS WITH A SMPS

Our concept on the deployment of a avionic PLC is to include the PLC modems within the equipments to avoid designing a power supply for each modem. The equipment chosen in this case study is a 20W SMPS designed by Thales Avionics with a switching frequency of 150kHz. This SMPS is set in two modes : a standard behaviour and a deteriorate behaviour. The deteriorate behaviour introduces higher perturbation levels, particularly present beyond 10 MHz. The EMI difference between the two modes reach up to 40 dBμA. The electromagnetic emissions of this equipment in these two configurations are shown in Fig. 5. The complete system, composed off the SMPS and the BB PLC modem, must comply with the standard DO-160 [1].

A. Measurements Protocol

Based on the DO-160, our measurement setup is described in Fig. 4 and shown in Fig. 6, These tests are done with both CCs in the two configurations of the SMPS (standard and deteriotate). The measurement procedure is the following:

• Switch on the power supply and wait until the modems finished the synchronization phase; • Establish a real data transfer between the modems; • Measure the average SNR provided by the modems; • Measure cable currents with a spectrum analyser on MaxHold mode.

B. Measurement Results

In this section we present the measurements performed with both CCs on different configurations. The purpose of these results is to verify the performance of our CC and compare it with the conventional PLC HAN CC. The performance of our CC is presented in red dotted line and the conventional CC with the blue line. The PLC system uses an OFDM modulation where equally spaced subcarriers in frequency are used to carry the information. These modems use subcarriers that are 24.414kHz spaced to each others.

The first setup is set with the presence of the classic SMPS. Figs. 7 and8 show the results respectively of the measured SNR and measured currents with both CCs setup. In terms of the SNR, we verify that our circuit outperforms the conventional CC in almost all frequencies in the [20-80] MHz band, providing gains above 20dB in certain frequencies. In terms of currents, we observed that both setups are in conformity with the DO-160. However, as expected, the improvement of the communication link increased the currents between modems and our CC comes very close to the authorised levels within the [20-30] MHz band.

The second setup is set with the presence of the deteriorate SMPS, which is a worst case scenario in terms of noise distortion. Figs. 9 and 10 present the SNR and the EMI measurements, respectively. In this scenario, the modem with the conventional CC is not able to establish communication. SNR and EMI measurements are thus unavailable for this configuration. With our CC, the PLC modems establish communication achieving in some frequencies SNRs above 10dB. However, the large noise generated by the deteriorate SMPS has an impact on the PLC performance and the data-rate is lower than the first setup. In terms of current emissions, the result shows that it does not comply with the DO-160 Category B but it complies with the others.

IV. CONCLUSIONS

In this paper we demonstrate that coupling circuits used in commercial BB PLC systems are not well adapted to avionics 28V DC PDNs and we introduce a modified CC. Our modified CC was able to improve SNR and satisfy the Aeronautical EMI conditions to most of the cases. We have demonstrated that the concept of adding broadband PLC modems to SMPS equipment are feasible, even for worst case of noise emissions. Moreover, SNR and EMC measurements proved that communication is possible while complying to current emission limits specified in the DO-160, when our proposed CC is used.

Fig. 2 .

 2 Fig. 1. S-parameter comparison between standard CC and our modified CC

Fig. 3 .

 3 Fig. 3. SNR comparison between an old coupling circuit and the new one

Fig. 4 .Fig. 5 .

 45 Fig. 4. Schematic of the test setup

Fig. 6 .

 6 Fig. 6. Picture of the test setup

Fig. 7 .Fig. 8 .

 78 Fig. 7. SNR in the case SMPS + modem

REFERENCES [ 1 ]

 1 Environmental conditions and Test procedures for airborne equipment, DO-160, RTCA DO-160, 2010.

Fig. 9 .

 9 Fig. 9. SNR of the PLC link with the degraded SMPS