Achraf Skander

Lionel Roucoules
email: lionel.roucoules@utt.fr

Jean Sébastien

Klein Meyer

J S Klein

Meyer Charles Delaunay

Keywords: Concurrent engineering, Design and manufacture interface, Feature-based product modelling, DFM, Knowledge synthesis, Skins and skeleton concepts, Manufacturing knowledge management

The research results presented in this paper are related to the specification of a method and models that tackle the problem of manufacturing processes selection and the integration, as soon as possible, of their constraints in the product modelling (i.e. information synthesis). This method is based on a skin and skeleton design/manufacturing interface model that ensures connection between design and manufacturing information. The use of these features is justified by their capacity to make a product representation which allows integration of both design and manufacture data and therefore assists the product breakdown definition (including the 3D forms) by least commitment. This method first analyses the product data issued from functional analysis and component selection (form, roughness, tolerance interval, etc.). Then, it deals with manufacturing information (manufacturing processes constraints). The approach is formalised with IDEF and UML models and has been consolidated with software developments based on C++ and open CASCADE technologies.

Design and manufacturing interface modelling for manufacturing processes selection and knowledge synthesis in design [START_REF] Sohlenius | Concurrent engineering[END_REF]

Introduction

In a context of deep change of industrial market related to globalisation and delocalisation, being competitive becomes a challenge for all industries. In order to increase product quality and reduce production cost, the right information must be provided at the right time to the right person. DFX (design for X) methods provide approaches that take into account a maximum of product life cycle information during the design process. A design solution is then the result of multiple expert product assessments and a collaborative multidisciplinary decision making process.

The authors' research activity is based on a framework to support manufacturing information synthesis in design. That research work is included within a general context of concurrent engineering (CE) and integrated design. Figure 1 presents the global concept of the framework in which each design expert involved in the design project can access a central information kernel where information is shared. Product information is therefore first managed individually in asynchronous mode by each member of the design group and then shared in asynchronous or synchronous mode for collaborative activities. A manufacturing processes selection activity is specifically studied by the authors here.

The most important result of the paper is a methodology of process selection and manufacturing constraints integration in design. Specific product information modelling called design/ manufacturing interface model is defined as the interface of product modelling and expert's (X) assessment. This interface is dedicated to support the manufacturing information synthesis. The design is then a knowledge intensive process and no longer a 3D modelling-based process. The design solution is then progressively defined by knowledge synthesis by least commitment.

Principal objectives of the proposed synthesis approach are:

-Progressive selection of potential manufacturing processes in the early design stage. This is led by a manufacturing data management strategy. -Definition of a list of eligible (i.e. alternative) process plans and associated manufacturing constraints (roughness, tolerances, prices, technologies, etc.). -Integration (synthesis) of manufacturing constraints in the product definition using the interface model. -Proposition of alternative design solutions with respect to alternative manufacturing process plans.

The second section of this article presents a literature review concerning concurrent engineering, DFM and feature-based product modelling and depicts how alreadypublished results are used as fundamental concepts and are improved to support the manufacturing information synthesis process. The third section outlines, on the one hand, the knowledge intensive synthesis method for manufacturing processes selection, while on the other hand, the information model is detailed as an interface between expert manufacturing information and shared collaborative design information. Specifications and development of a software demonstrator that supports this information modelling is finally detailed. Section four presents an example that illustrates the approach. The last section gives a conclusion and recommendations for further work.

2 Literature review

Concurrent engineering

One of concurrent engineering's (CE) goals is to integrate product life cycle knowledge earlier during the design process. CE must reduce iterations between product design and manufacturing. One definition of concurrent engineering postulates that "Concurrent engineering is a way of work where the various engineering activities in the product and production development process are integrated and performed as much as possible in parallel rather than in sequence" [START_REF] Sohlenius | Concurrent engineering[END_REF]. Approaches of integrated design are also presented which specify that knowledge integration aims at taking into account the whole events which must appear early or late in the product life cycle [START_REF] Tichkiewitch | Methodology and product model for integrated design using a multiview system[END_REF][START_REF] Alting | Life cycle engineering and design[END_REF]. DFX [START_REF] Tsai | Design for manufacture and design for 'X': concepts, applications and perspectives[END_REF], seen as assessing and integrating "X-field" information, is then linked to CE and therefore can no longer be treated independently.

Beyond such very large concepts the main limit remains in modelling the information related to product design. Indeed, current software solutions (CAD, project planning, etc.) have not really followed such industrial (organisational) evolution and current industrial needs. Today, the information must link enterprise organisation data, design process data and product data [START_REF] Nowak | Towards a design process model enabling the integration of product, process and organization[END_REF][START_REF] Krause | Simulation approaches to optimise the management of product development processes[END_REF].

In that paper, the authors specifically tackle product data modelling for manufacturing knowledge intensive design.

Design for manufacture

Design for manufacture (DFM) is a product design approach that takes into account design goals and manufacturing information as soon as possible in product definition. Many studies deal with presentation and implementation of DFM concepts:

-Presentation of DFM concepts and tools [START_REF] Tsai | Design for manufacture and design for 'X': concepts, applications and perspectives[END_REF][START_REF] Ong | A fuzzy set AHP-based DFM tool for rotational parts[END_REF] -Analysis of manufacturability and providing solutions to improve design [8, 9] -Manufacturing processes selection based on processes classification and attributes comparison. Specification of appropriate method and techniques of materials and processes selection [10-13] -Specification of techniques that will reduce cost and ease handling of components [14] -Providing some fundamental rules of design that give the "best" solutions for manufacturing point of view [START_REF] Edwards | Designing of engineering components for optimal materials and manufacturing process utilisation[END_REF] In summary, the scientific community proposes many studies of product manufacturability analysis based on different parameters (tasks, attributes, characteristics, cost, etc.) but it does not propose a synthesis (integration) method. Current approaches require pre-definition of product geometry and take place late in the detailed design stage.

That paper then focuses on a methodology and a product model that simultaneously manage manufacturability analysis (processes selection) and synthesis (constraints integration) to let the product definition (including 3D Based on the original definition given by Shah, a feature is described "as a semantically endowed object that accompanies product development from the customer request through to product release" [START_REF] Shah | Assessment of feature technology[END_REF]. A huge number of product meta-modelling proposals have been based on that concept of feature to define design solution breakdown [START_REF] Krause | Product modelling[END_REF][START_REF] Kjelberg | Product modelling and information-integrated engineering systems[END_REF][START_REF] Martino | Design and engineering process integration through a multiple view intermediate modeller in a distributed object-oriented system environment[END_REF]. Current work should therefore not be focussed on developing new meta-models but to define concepts that support the interoperability among those which exist. With respect to general concepts given in Fig. 1, two groups of information modelling have to be tackled:

-Information managed in "X-field' expert activities for specific product behaviour assessments with respect to the entire product life cycle (manufacturability, assemblability, recycling, etc.).

-Information shared among the design group to set relationships among product data. That model must propose a multiple point of views breakdown operator [START_REF] Roucoules | Ippop: an Opensource collaborative design platform to link product[END_REF]. The relations are used to propagate design experts' constraints and to progressively converge from a set of alternatives to a design solution (knowledge intensive design versus 3D-based design).

Design and manufacturing interface

Concerning design and manufacturing interface, various scientific papers are focussed on CAD-CAM integration. They provide solutions for exchanging data based on standards (e.g. STEP-AP203); nevertheless, that standard is not currently defining experts' concepts but only from features. Moreover, STEP standards cannot be changed easily and dynamically to add some new concepts concerning product modelling. Some results are based on feature recognition [START_REF] Miao | CAD-CAM integration using machining features[END_REF]. Other papers propose the integration of manufacturing system constraints [START_REF] Zifang | Proactive support for product conceptual design incorporating manufacturing system considerations[END_REF] but it is not really the same as manufacturing process constraints as considered in the paper.

Twigg [START_REF] Twigg | Managing the design/manufacturing interface across firms[END_REF] gives a more generic overview of what could be mechanisms for design and manufacturing interface and classify the different problems that should be faced (cf. Fig. 2).

The research work presented here focuses on the design phase and could be used to tackle A5, A6, A7 and D3. As shown on the Fig. 1, the authors describe a specific structure to achieve the manufacturing information synthesis in design. That model is called an interface model, which is necessary because it represents only a few parts of the whole expert knowledge, but then has to be translated.

Skin and skeletons concepts are then used to define the interface model. Those concepts have already been presented in the literature to ensure a description of product functional surfaces. In [START_REF] Muh-Cerng | A skeleton approach for modelling assembly products[END_REF] and [START_REF] Tollenaere | Presented at feature modeling & recognition in advanced CAD/CAM systems[END_REF] the authors define skeleton features as skins links. Skeletons corre-Fig. 2 Classification of design/ manufacturing interface [START_REF] Muh-Cerng | A skeleton approach for modelling assembly products[END_REF] spond to neutral material fibre and define part material topology. In [START_REF] Song | FORMS: A flexible object recognition and modeling system[END_REF] and [START_REF] Ogniewicz | Skeleton-space: a multiscale shape description combining region and boundary information[END_REF] the authors describe a method for representing objects, they also use the notion of skeletons defined by the central axis of objects and shapes that cover the skeleton. The aim of the aforementioned research is a representation of object motion for robotic studies.

With regard to these works, the next section specifies skin and skeleton concepts with a specific set of attributes that ensure design/manufacturing interface modelling and let the product emerge by least commitment instead of being centred on CAD modelling.

Knowledge synthesis method and design/ manufacturing interface modelling

According to fundaments issued from the previously given survey, this section presents the details of the method and the models developed. Figure 3 presents the objectives and benefits of our contribution.

In the current design process (Fig. 3 column 1) there are very few feedbacks from "processes selection" to "design" (i.e. CAD modelling). The presented approach defines new models to support that kind of back loop (Fig. 3 column 2). Further on, the "processes selection" activity moreover should lead to a list of eligible process plans that also clearly impact the design activity proposing several design alternatives (Fig. 3 column 3). Each manufacturing process plan then constrains a design solution.

After detailing those benefits in the next sections an illustration is given (see Fig. 10).

Method for knowledge synthesis

According to the general framework of integrated design (Fig. 1), the proposed synthesis approach has been based on a formal analysis and representation of design expert activity. This work is specifically based on the manufacturing process selection activity in order to give, as soon as possible, recommendations to CAD models (Fig. 3). For instance, from the same required functional surfaces, a final CAD model may not be similar depending on forging, casting or milling process.

With respect to IDEFØ modelling (Fig. 4), that activity (A-0) is split into two main tasks: 2). This model is therefore the input of A2, where only the two lateral plane surfaces are specified (i.e. usage) (see Fig. 10). -A2: to select manufacturing processes and to identify process plans and their constraints. With respect to the requirements (i.e. interface model), the expert realises the first manufacturing process selection. The output is a set of process plans and their respective information (tolerances, roughness, etc.). That information is described using the manufacturing interface model and then fed back to the shared kernel of information. That feedback is then used either to add extra information (and associated range of values) on the product that has to be shared for the collaborative decision making process or to constrain the alreadydefined information range of values. The shaped surface was not defined as requirements but described according to manufacturing information (see Fig. 10). The activities A1 and A2 manage data flows representing design and manufacture requirements, product and process data resources, etc. This management consists of:

-Design information analysis: study of technological components constraints and elaborate a product model [START_REF] Klein-Meyer | Assessment of a functional and DFX approach for mems design[END_REF]. -Manufacturing processes analysis: choose processes alternatives and analyse manufacturing processes to identify constraints. -Creation of method and model of manufacturing processes selection and constraints synthesis (integration).

This IDEFØ modelling makes decomposition of every task into several ones possible at different granular levels. Accurate details of the entire activity modelling can be found in [START_REF] Skander | Méthode et modèle DFM pour le choix des procédés et l'intégration des contraintes de fabrication vers l'émergence de la solution produit[END_REF]. For example, A1 activity (Fig. 5) is broken down into:

-A11 Identify usage interface model from already existing requirements in the shared kernel of information. -A12 Characterize manufacturing interface model based on manufacturing process selection (A2). -A13 Apply inclusion relation (design product is included in manufacturing product) to keep the coherency in the synthesis (i.e. integration) loop as previously enounced.

The usage interface model is not modifiable in A-0; it is then A11 activity control information. Manufacturing interface model is on the contrary refined in A-0, it is the result of manufacturing attributes range of values synthesis; this information is an input for A12 activity.

The inclusion relation (A13) is defined as: "designed (i.e. required) product has to be included in manufacturing solution (i.e. manufactured product)". In fact, the manufacturing solution must include the required (design and functional) specifications and other specifications that result from other expertises. This relation is given by a rules and relations list between both "usage" and "manufacturing" interface model attributes. For example, the roughness value that can be realised by a selected process must be lower than the required roughness value.

With respect to the activity IDEFØ model, it is then easy to identify static concepts used while processing the knowledge synthesis method. The next step of the research work is therefore to propose a static design view of the information structure. To do so, the authors use UML -Name : char -Shape : Skin Shape -Ra : float -IT : float -Skin material direction : bool

Skin

-Name : char -Initial section form : Section Form -Final section form : Section Form -Section variation : Section Variation -Neutral fibre : Neutral Fibre -Skeleton material direction : int formalism (Unified Modelling Language) [START_REF] Booch | The unified modeling language user guide[END_REF] and, specifically, the class diagrams.

Skeleton

Neutral Fibre Section

That information modelling method is very similar to knowledge management approaches. They first give an overview of the activity and then extract the static concepts (i.e. knowledge) [START_REF] Ermine | La gestion des connaissances[END_REF]. That work has also been supported by specifying scenarios (sequence diagrams) of the design method (see Fig. 8).

Design and manufacturing interface model for manufacturing knowledge synthesis

Concepts overview

From IDEFØ formalism (input, control, mechanism and output information), information modelling concepts are extracted:

-Human resources: experts that run the "manufacturing process selection" activity (A-0). -Product solution alternatives: list of potential design solutions resulting from experts' constraints integration. -Manufacturing processes database.

-Process plan alternatives: list of eligible process plans that result from selected processes. -Design/manufacturing interface model: model that simultaneously supports product and manufacturing information in order to achieve the manufacturing information synthesis during the design process. -Information resources of product, processes and product/processes: all product and process information used in the product definition. That information concerns either physical parameters or technological parameters.

These concepts have then been structured and presented in a UML class diagram. Relations, attributes and operations (Fig. 6) are also detailed. The class diagram is made up of two parts:

-Interface model that represents the minimal information required to process the synthesis in design as presented in Fig. 1. The authors use skin and skeleton features and specific attributes to depict that model. -Manufacturing data model that represents manufacturing process selection information. That information, as expert information, has a priori not to be shared among the design group. It would be shared only if needed by another expert.

Relationships between the two models are represented by "processes selection" and "constraints integration" associations with respect to common skin and skeleton attributes.

The "project" class is made up (aggregation relation) of "product alternatives" and "process plan alternatives" classes. To define product model, skin and skeleton features are used and represented in the diagram by the skin and skeleton classes and their inheritance classes "usage skin", "usage skeleton", "manufacturing skin" and "manufacturing skeleton" (the diagram does not represent all the inheritance links of skin shape, section shape, section variation, and neutral fibre for clarity reasons).

Skin and skeleton feature definition

As introduced above, the interface model supports synthesis of both design and manufacturing specifications. This model allows emergence of product solutions that respect manufacturing constraints. To describe the interface model authors use specific skin and skeleton features.

According to the literature review and the definition of the interface model, generic concepts of skin and skeleton have first been defined. Skins must describe the product's functional surfaces and skeletons describe trajectory of flow. Secondly, two kinds of specific (i.e. multiple representations) of skin and skeleton features have been detailed: usage features (i.e. design requirements) and manufacturing features [START_REF] Roucoules | Manufacturing process selection and integration in product design. Analysis and synthesis approaches[END_REF]:

-Usage skin: surface through which energetic flows circulate [START_REF] Roucoules | CoDE: a cooperative design environment. A new generation of CAD systems[END_REF]. ). These skins support geometric attributes and tolerances. -Usage skeleton: trajectory of an energetic (mechanical, electrical, magnetic, etc.) flow circulating in the product (e.g. see, for example, Fig. 10 where the U trajectory represents that skeleton specified by designers according to specific required magnetic behaviour). -Manufacturing skeleton: material flow trajectory processed during manufacturing. Indeed authors make the hypothesis that every manufacturing process is based on a material flow (e.g. casting or injection moulding create material, machining removes material, forging deforms material).

-Manufacturing skin: surface generated by a manufacturing process. These features are partly issued from manufacturing skeletons by a sweeping operation (e.g. surfaces issued from a pocketing milling operation can be defined by sweeping the end mill tool section on its trajectory. The shaped surface is issued from both extrusion and drilling processes, cf. Fig. 10). In the case of deformation based processes, the manufacturing skin is not deducted from the sweeping operation but from deformation algorithms.

Owing to the manufacturing skeleton concept, material flow can be globally (i.e. part level) or accurately (i.e. tool level) described depending on the detail of information that has to be taken into account in the design process. Figure 7 presents an example of usage and manufacture skins and skeletons to model the product solution of what could be a "U" magnet. The required solution is not totally defined and would then be constrained by manufacturing interface modelling coming from the manufacturing process selection task. Several design alternatives are then available.

Specific sets of attributes are associated with skin and skeleton features:

-Skin attributes are shape (cylinder, plane, etc.), tolerance, roughness and material direction (depending on the form). -Skeleton attributes are initial section form, final section form (circular, rectangular, etc.), section variation (follow variation law: null, increasing, decreasing, etc.), neutral fibre (line, curve, plate, etc.).

For manufacturing skeletons, an extra attribute defines the material flow direction that could be set as: removal (e.g. machining), addition (e.g. casting) or deformation (e.g. forging, stamping).

Based on that simple concept it is very easy to analyse what could be the final part definition based on manufacturing process selection. Indeed, the final 3D model of a part (made of manufacturing skins) is constructed by sweeping (Fig. 7) or deforming the skeleton section on the skeleton trajectory. The initial model described with "usage" skin and skeleton can be compared to the "manufacturing" one.

The main benefit of these modelling concepts is that the manufacturing knowledge is taken into account very early in the product development process (and the CAD modelling) instead of waiting for an initial CAD model which would be modified afterwards.

Software demonstrator

In order to support the knowledge synthesis approach, a software demonstrator has been developed to manage the UML model presented above. It is specified using a UML Software developments have been based on the C++ language in the MS Visual 6.0 environment. Figure 9 presents an actual graphic user interface used to:

-Manage processes information (technological, physical, etc.) in order to select processes. -Model product (based on skin and skeleton) alternatives issued from manufacturing process alternatives. -Show product's form features (i.e. shapes) in a 3D viewer developed with Open CASCADE libraries. It is then possible to compare design requirements (requirements specifications) and manufacturing alternatives (final solution).

Illustration

Figure 10 presents two examples of a skin-skeleton model. First, an example of usage representation indicates the expected part (shape form, roughness, tolerance, etc.) issued from functional analysis. Second, the manufacturing process selection provides the manufacturing skeletons and manufacturing skin (generated by skeleton covering and represents manufactured part). Manufactured part solutions must conform to the inclusion relation. In fact, manufacturing solutions do not necessarily have the same attribute values of usage solutions (shape form, roughness, tolerance, etc.) but must include them, so these solutions can be different.

It should be noted that the proposed solutions represent only the manufacturing point of view. The skin and skeleton model could be enlarged and used to study other expert activities (Fig. 1).

Conclusion and recommendations for further work

This research work presents a new knowledge synthesis method that aims at integrating manufacturing constraints as early as possible during the product definition. The approach is based on two principal stages: processes selection and progressive integration of manufacturing constraints. To do so the authors present a method supported by a design/manufacturing interface model that uses skin and skeleton features specified in usage (i.e. design data issued from components selection) and manufacturing (i.e. data related to manufacturing processes). This model includes two parts: product interface model (product representation) and manufacturing data model (manufactur-ing process representation). Material selection and economic parameters are obviously linked to that examined but not formalized in the paper.

The software demonstrator ensures product design assistance. The result is the emergence of the product definition with manufacturing constraints and reduction of design/manufacturing iterations.

The first recommendation for further work concerns the creation of process knowledge structure with respect to the specified manufacturing data model and using existing databases like CES41 . Generalization of the method and model to other expertise fields (recycling, assembly, etc.) could also be investigated to enlarge the DFX framework.

-Fig. 4

 4 Fig. 3 Method benefits

Fig. 5

 5 Fig. 5 A1 activity decomposition. Identification of the interface model requirements

Fig. 6

 6 Fig. 6 Presentation of modelling concepts with UML class diagram

2 Fig. 7

 27 Fig.7Skin and skeleton illustration: example of alternative solutions of a "U" magnet manufacturing

Fig. 8

 8 Fig. 8 Example of UML sequence diagram

Fig. 9

 9 Fig. 9 Graphic user interface of the software demonstrator

In Collaborative activity Shared Information Model M a n u f a c t u r i n g p r o c e s s s e l e c t i o n

	E
	x p
	e
	r
	t
	a
	c
	t i v
	i
	t y
	i +
	1

M e c h a n i c a l O p t i m i s a t i o n a n d A n a l y s i s " X -f i e l d " i n f o r m a t i o n m o d e l Interface model

	modelling) progressively emerge. It takes place earlier
	during the product definition process (in the conceptual
	and embodiment design stages). The methodology is
	defined by a set of activities whereby input/output is the
	modelled product data.
	2.3 Feature-based product modelling
	Fig. 1 DFX in a CE framework where X = manufacture (DFM) or X =
	other expert activity

  Those surfaces' characteristics (IT, Ra) therefore depend on process selection. -A synthesis loop is defined between those two activities to model the fact that the integrated design process is no longer sequential. Requirements are then evolving each time an expert integrates new information. The interface model is then evolving too and the expert can therefore run a new manufacturing process selection task to either refine his first solution or to validate that new requirements are still coherent with the first selection. The final design solution takes into account the manufacturing information and is really based on knowledge synthesis and not only on CAD model assessment. This new CAD modelling approach based on experts' knowledge (e.g. manufacturing) respects experts' requirements "the first time" versus a "redo until right" process.

	Design requirements	Resources product
		Identify usage	Attributes values of designed product
		interface model		List of products alternatives solutions
			A11		(definition of : geometry, tolerance, etc) (if inclusion relation verified)
		Humans
		resources :
		DFM Actor
					Apply inclusion relation
					(Designed product is
					included in
					manufacturing product)	Interface model
	Manufacture requirements	Resources processes	A13	(evolve in the synthesis loop)
					Humans
					resources :
		Characterize	Attributes values of	DFM Actor
	manufacture interface	Manufactured product
		model
			A12
	Attributes values of Manufacturing interface model	Humans resources : DFM Actor

http://www.grantadesign.com

Acknowledgement This research is part of the regional French project IFP2R: "Manufacturing constraints integration in rapid prototyped part design" with IFTS (Higher Technical Formation Institute of Charleville Mézières-France).

Examples

Examples of usage interface modelling

Examples of manufacturing solutions: manufacturing interface modelling Example 1 : cube-shape creation

The expected part has cube-shape. Design requirements are described with two usage skins and one usage skeleton features.

Manufacturing Skeleton

An example of selected manufacturing process to make the hole is: Drilling. Initial and final section forms.

Neutral fibre.

Manufacturing skin. Initial and final section forms (rectangular).

Neutral fibre.

Initial and final section forms (circular).

Fig. 10 Examples of the knowledge synthesis approach using skin and skeleton modelling


Shuhua Zhang

Laurent Girin

Antoine Liutkus

INFORMED SOURCE SEPARATION FROM COMPRESSED MIXTURES USING SPATIAL WIENER FILTER AND QUANTIZATION NOISE ESTIMATION

Keywords: Informed Source Separation, Wiener Filter, Denoising, NTF, AAC

In a previous work, we proposed an Informed Source Separation system based on Wiener filtering for active listening of music from uncompressed (16-bit PCM) multichannel mix signals. In the present work, the system is improved to work with (MPEG-2 AAC) compressed mix signals: quantization noise is estimated from the AAC bitstream at the decoder and explicitly taken into account in the source separation process. Also a direct MDCT-to-STFT transform is used to optimize the computational efficiency of the process in the STFT domain from AAC-decoded MDCT coefficients.

INTRODUCTION

Active listening of music is a topic of both artistic and technological interest that consists in modifying the elements and structure of the music signal during the listening experience. It involves online advanced remixing processes such as generalized karaoke (the ability to mute any musical element, hence not limited to voice), respatialization, or application of instrument-specific audio effects (e.g., adding distortion to an acoustic guitar). To enable active listening from compliant 2-channel stereo mix signals, recent technologies have been proposed. In Spatial Audio Object Coding (SAOC) [START_REF] Engdegård | MPEG Spatial Audio Object Coding-the ISO/MPEG standard for efficient coding of interactive audio scenes[END_REF], standardized by MPEG, the stereo mix signal is enriched with spatial coding parameters at the coder stage, that enable to decode the multitrack source signals from the downmix signal at the decoder stage. Recently, this idea was revisited from the source separation [START_REF] Comon | Handbook of Blind Source Separation -Independent Component Analysis and Applications[END_REF] point of view (see [START_REF] Liutkus | Informed source separation: a comparative study[END_REF] and references therein). In particular, a Wiener-based Informed Source Separation (ISS) system was proposed in [START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF], where the side-information is composed of power spectral density (PSD) models of the source signals plus the mixing matrix, and separating Wiener filters are used at the decoder.

The new contributions of this paper with respect to previous works are threefold. First and more importantly, although ISS (and SAOC) purpose is to reduce the overall bitrate compared to multitrack coding (in addition to 2-channel stereo compliance with usual formats), the effects of compressing the mix signal on source signals decoding/separation is not being taken into account in those state-ofthe-art methods, leading to suboptimal performance. In the present paper, we present an improved version of the Wiener-based ISS system, where the mix signal is compressed, for instance using MPEG-2 Advanced Audio Coding (AAC [START_REF]11 MPEG[END_REF]), and the resulting quantization noise is explicitly accounted for in the separation process. This enables to significantly reduce the distortion in the separated source signals. Moreover, we propose to estimate this quantization noise at the decoder, so that no additional side-information needs to be transmitted. Second, we use here the spatial Wiener filtering scheme, as recently introduced in the ISS context in [START_REF] Liutkus | Spatial coding-based informed source separation[END_REF], which permits to fully benefit from the spatial dispersion of the sources within the mix to improve separation. And finally, the recently developed direct MDCT-to-STFT conversion technique of [START_REF] Zhang | Fast and accurate direct MDCT to DFT conversion with arbitrary window functions[END_REF] is used to enable fast and efficient STFT-domain processing directly from AAC-decoded MDCT coefficients.

This paper is organized as follows. First, the proposed separation technique and system are described in Section 2. Then explicit handling of the AAC quantization noise is presented in Section 3. The improved system is shown in Section 4 to provide significantly better quality compared to when the quantization noise is not taken into account, for identical side-information rate.

SYSTEM OVERVIEW

Spatial Wiener Filtering

The core source separation module of the proposed ISS system is based on the so-called spatial Wiener filter, or multichannel MMSE linear estimator [START_REF] Liutkus | Spatial coding-based informed source separation[END_REF][START_REF] Comon | Handbook of Blind Source Separation -Independent Component Analysis and Applications[END_REF]. Let us assume that we have a J-source signals vector

s[n] = s1[n] s2[n] • • • sJ [n] (n for time sample index, for transposition), which is downmixed (I < J) into a I-channel vector x[n] = x1[n] x2[n] • • • xI [n]
using linear time-invariant (LTI) filters, i.e. underdetermined convolutive mixing. It is classical to process the separation problem, i.e. recovering of s from x, in the frequency domain, where the mixture may be approximated as instantaneous at each frequency f [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF], using Short-Time Fourier Transform (STFT) (t denotes the frame index):

X(f, t) = A(f )S(f, t) + N (f, t), (1)
where

X(f, t) = STFT x[n] , S(f, t) = STFT s[n] , A(f )
is the I × J frequency-dependent downmix matrix, and N (f, t) is the noise. Here, the sources are assumed to be independent zero-mean complex Gaussian processes, i.e., S(f, t) ∼ N (0, R SS (f, t)), where R SS (f, t), the J × J covariance matrix of the sources, is diagonal and the j-th diagonal term is the PSD of source j at TF bin (f, t). The noise is also assumed to be Gaussian:

N (f, t) ∼ N (0, R N N (f, t)). Let us denote R SX (f, t) ≡ R SS (f, t)A(f ) .
Under the above parameters and given the mixture X(f, t), we have the MMSE estimator S(f, t) of the source signals vector:

S(f, t) = W(f, t)X(f, t), (2)
with

W(f, t) ≡ R SX (f, t)[A(f )R SX (f, t) + R N N (f, t)] -1 . ( 3
) STFT STFT STFT | • | 2 | • | 2 | • | 2 NTF Encoding Mixing AAC Encoding s1[n] s2[n] sJ [n] S1(f, t) S2(f, t) SJ (f, t) x1[n] xI [n] NTF sideinfo AAC Bitstream A(f ) Spatial Wiener Filtering NTF Decoding Noise Variance Estimation Modified AAC Decoding M→F M→F M→F M→F STFT -1 STFT -1 STFT -1 NTF sideinfo | S1(f, t)| 2 | S2(f, t)| 2 | SJ (f, t)| 2 A(f ) AAC bitstream Y1(f, t) YI (f, t) X1(f, t) XI (f, t) σ 2 Y 1 (f, t) σ 2 Y I (f, t) σ 2 1 (f, t) σ 2 I (f, t) S1(f, t) S2(f, t) SJ (f, t) ŝ1[n] ŝ2[n] ŝJ [n]
Fig. 1. Functional representation of the spatial Wiener filter based ISS system: encoder (left) and decoder (right). Here M → F is for MDCT-to-STFT conversion, see Section 3.3.

ISS encoder and decoder

The ISS encoder based on the spatial Wiener filter is schematized in Fig. 1

(f, t)| 2 [9], so that we rede- fine R SS (f, t) = diag {|S1(f, t)| 2 , |S2(f, t)| 2 , . . . , |SJ (f, t)| 2 }.
This covariance matrix also has to be transmitted to the decoder for Wiener filter construction. However, spectrograms generally occupy a very large amount of side-information, so they must be substantially compressed. This is done using Non-negative Tensor Factorization (NTF) modeling of the spectrograms, arranged as 3D tensors (frequency × time × source), as in [START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF][START_REF] Liutkus | Spatial coding-based informed source separation[END_REF], and subsequent NTF parameters quantization.

The ISS decoder side is shown in Fig. 1-right. The NTF models of spectrograms and mixing matrix are decoded from the side-info to provide R SS (f, t) and A(f ) from which the Wiener filter (3) is built. Filtered source coefficients S1(f, t), S2(f, t), . . . , SJ (f, t) are then obtained from mix coefficients X1(f, t), . . . , XI (f, t), and synthesized to time-domain signals ŝ1[n], ŝ2[n], . . . , ŝJ [n] using inverse STFT with overlap-add. Other processes are described in the next sections.

NOISE VARIANCE ESTIMATION FROM AAC BITSTREAM

In our previous works [START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF][START_REF] Liutkus | Spatial coding-based informed source separation[END_REF], 16-bit PCM mix signal x was processed at the coder and decoder. Hence it was considered as unquantized and the noise N (f, t) in ( 1) was assumed to be null 1 . In the present work, lossy MPEG-2 AAC compression [START_REF]11 MPEG[END_REF] is applied on the mix signal, introducing compression noise (mainly due to quantization) at each TF bin. Therefore, we propose to specifically model the degradation of the mixture due to compression as the additive noise N (f, t) in (1), independent from the sources. Doing so, we allow for the separation procedure (3) to take it into account to better recover the sources. This noise N (f, t) may be estimated simply at the coder as the difference between original and compressed mixture, and its PSD may be encoded and transmitted to the decoder using 1 Actually, N (f, t) in ( 1) can be used to model the mixing model error (resulting from STFT of convolution into product) when mixing filters are long, but this is very different and independent from compression noise. the NTF model. However, this would increase the side-information bitrate. Instead, we show below how the noise variance can be estimated directly from AAC bitstream.

Nonlinear Quantization Noise

In AAC, compression is based on the quantization of Modified Discrete Cosine Transform (MDCT) [START_REF] Malvar | Signal processing with lapped transforms[END_REF] coefficients. A coefficient Y (k) on the k-th frequency bin (frame index is omitted here for clarity) within subband b is quantized to an integer value by scaling,

Ȳ (k) = sgn (Y (k)) (|Y (k)|/2 1 4 s(b) ) 3 4 + C , (4)
where C ≡ 1-2 -3/4 and s(b) is the scalefactor for subband b. Now suppose Ȳ (k) = q is positive. Then by (4), any MDCT coefficient

Y (k) ∈ 2 1 4 s(b)
[Tq, Tq+1) will be quantized to q, where Tq ≡ (q -C) 4/3 is the lower boundary of the q-th quantization interval. A similar result can be derived for q ≤ 0:

Y (k) ∈ 2 1 4 s(b) ×      (-T |q|+1 , -T |q| ], q < 0, (-T1, T1), q = 0, [ Tq,
Tq+1), q > 0.

(5)

Due to the 3 4 -power compression, this quantization is nonlinear and the intervals have increasing lengths as |q| increases. Furthermore, assuming MDCT coefficients are uniformly distributed on each interval, the noise variance is

σ 2 (k) = 1 12 (T |q|+1 -T |q| ) 2 2 1 2 s(b) , q = 0, 1 12 (2T1) 2 2 1 2 s(b) , q = 0. (6)
Thus, after retrieving quantization index q and scalefactor s(b) from an AAC bitstream, (6) can be used at the decoder to estimate the variance of the quantization noise for a single channel2 .

Stereo Coding Noise

In AAC, each channel is either encoded independently or paired with another channel for stereo coding. In the latter case, both redundancy and irrelevancy between the paired channels (left and right) are exploited to save bitrate using either the Mid/Side (M/S) [START_REF] Johnston | Perceptual transform coding of wideband stereo signals[END_REF] or Intensity Stereo (IS) [START_REF] Herre | Intensity stereo coding[END_REF] coding tools. We shall consider these two tools separately in the following. M/S coding is noiseless but changes the noise distribution. Suppose that M/S is applied at bin k. The coefficients to be quantized are the middle and side of the MDCT coefficients YL(k) and YR(k) from left and right channels:

Y0(k) = 1 2 (YL(k) + YR(k)) and Y1(k) = 1 2 (YL(k) -YR(k)).
The middle Y0(k) and side Y1(k) are quantized by [START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF] and their quantization noises follow [START_REF] Liutkus | Spatial coding-based informed source separation[END_REF]. Assuming the noises are independent, then noise variances of the left and right channels are

σ 2 L (k) = σ 2 R (k) = σ 2 0 (k) + σ 2 1 (k), (7)
in which σ 2 0 (k) and σ 2 1 (k) are the noise variances determined by (6) for Y0(k) and Y1(k), respectively.

Conversely, IS coding is noisy, introducing noise while changing noise distribution. Let YS(k) = YL(k) + YR(k) be the sum channel and denote by I L/S (b) and I L/R (b) the ratios of intensities (square root of total energy) between the left and the sum channels, and between the left and the right channels, respectively, for subband b. In IS encoding, a pair of new left and right channels are built from the sum channel and the intensity ratios:

Y l (k) = YS(k)I L/S (b), Yr(k) = Y l (k)/I L/R (b). (8)
σ 2 L (k) ≈ σ 2 l (k), σ 2 R (k) ≈ 2 -p(b)/2 σ 2 (k) + B2 -p(b)/2 Y 2 l (k), (10)
in which B = 0.0037 is the variance of 2 δp(b) , and

Y l (k) is Y l (k)
after quantization and dequantization.

MDCT-to-STFT Conversion

Decoding of AAC bitstream is done in the MDCT domain. But the spatial Wiener filter building and filtering process is to be carried out in the STFT domain, where time-domain convolutive mixing can be approximated as instantaneous mixing [START_REF] Stein | Fourier analysis: an introduction[END_REF]. Therefore, we propose to use the direct MDCT-to-STFT conversion technique of [START_REF] Zhang | Fast and accurate direct MDCT to DFT conversion with arbitrary window functions[END_REF] to convert both the estimated quantization noise variances and the mix signal coefficients. Accordingly, the AAC decoder is modified such that it outputs the quantized MDCT coefficients ŶL/R (f, t) and other AAC parameters necessary for estimation and conversion (quantization indexes, scalefactors, stereo parameters, window shapes and sequences), and skip the MDCT-to-time signal conversion. This saves computational complexity by avoiding full AAC decoding and STFT analysis.

In [START_REF] Zhang | Fast and accurate direct MDCT to DFT conversion with arbitrary window functions[END_REF], STFT coefficient X(f, t) for frame t is obtained from MDCT coefficients of the previous, current and next frames using a linear FIR filtering form:

X(f, t) =φ(f ) M -1 k=0 (-1) k h0(f -k-1) + h0(f +k) Y (k, t) + h-1(f -k-1) + h-1(f +k) Y (k, t-1) + h+1(f -k-1) + h+1(f +k) Y (k, t+1) (11)
for f = 0, 1, . . . , M , where M is the MDCT coefficients vector size, equal to half the STFT framesize, φ(f ) = exp[-π 2M (1+M )k] is a pure phase function, and h0(f ), h-1(f ), and h+1(f ) are filter taps depending on the MDCT and STFT windows. Assuming that the quantization noises are independent between different frames, frequencies and channels, the noise variance in the STFT domain is

σ 2 (f, t) = M -1 k=0 h0(f -k-1) + h0(f +k) 2 σ 2 Y (k, t) + h-1(f -k-1) + h-1(f +k) 2 σ 2 Y (k, t-1) + h+1(f -k-1) + h+1(f +k) 2 σ 2 Y (k, t+1) . ( 12
)
It is also shown in [START_REF] Zhang | Fast and accurate direct MDCT to DFT conversion with arbitrary window functions[END_REF] that h0(f ), h-1(f ), and h+1(f ) all rapidly tend to 0 as |f | increases. For example, with M = 1024, the KBD window [START_REF] Bosi | Introduction to digital audio coding and standards[END_REF] for the MDCT and the Hanning window for the STFT, keeping a total of only 20 taps in ( 11) leads to a conversion SNR higher than 60 dB. Therefore, we use a similar setup in the experiments.

EXPERIMENTS

Implementation and Experiment Settings

The proposed ISS system has been implemented in Matlab. The AAC encoder, however, is a binary one from Nero3 known for its high audio coding quality. The AAC decoder is the open source4 FAAD2 in C, which is modified as mentioned above.

A set of 14 musical excerpts (10s long, sampled at 44.1 kHz, 5 to 10 source signals) from the Quaero database5 is used. For each excerpt, sources are downmixed to 2-channel stereo using Head Related Impulse Responses (HRIR) filters collected from the CIPIC database [START_REF] Algazi | The CIPIC HRTF database[END_REF] (with order 200). The corresponding mixing matrix A(f ) is transmitted to the decoder as single-precision floating-point numbers. Since A(f ) does not depend on the length of an excerpt, the corresponding bitrate is assumed to be negligible for a complete song, and is not accounted for here.

For each source and each frame, a STFT spectrum may come from one long window size of 2048 or eight short windows size of 256. In the latter case, STFT coefficients of the short windows are intertwined from low to high frequency. To save bitrate, a cutoff frequency is set to 16 kHz. The resulting tensor of spectrograms are compressed by NTF whose parameters are linearly quantized (usually 8 bits) in the logarithmic domain [START_REF] Ozerov | Informed source separation: source coding meets source separation[END_REF]. NTF compression ratio is mainly controlled by the number of NTF components [START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF], and values of 5, 10, 20, 30, 50, are tested here. The quantized NTF parameters are entropy coded before transmitting to the decoder, using here the data compression utility of the 'save' Matlab function with option

Results and Discussion

We used two metrics to measure separation performance: Perceptual Similarity Measure (PSM) of PEMO-Q [START_REF] Huber | PEMO-Q-a new method for objective audio quality assessment using a model of auditory perception[END_REF], and Signal to Distortion Ratio (SDR) computed using the BSS Eval toolbox [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]. PSM provides an assessment of perceived quality of separated sources against the original sources, ranging from 0 (worst) to 1 (best). Since the performance heavily depends on excerpts, and to compare performance across all the test excerpts, each excerpt is finally measured by relative scores: ∆PSM ≡ PSM(oracle method) -PSM(test method); ∆SDR ≡ SDR(oracle method) -SDR(test method). The so called oracle method gives the highest possible performance using the spatial Wiener filter: Original spectrograms (instead of NTF compressed ones) are used to construct the separation filter and original mixtures (instead of AAC compressed ones) are input to the filter.

To study the effectiveness of quantization noise variance estimation and noise reduction in the ISS process, we run four configurations (all with NTF-compressed spectrograms and AAC compressed mixtures): (a) no noise reduction; (b) NTF coding of noise spectrograms + noise reduction; (c) noise variance estimation from AAC bitstream + noise reduction; (d) exact noise variance + noise reduction. The resulting ∆SDR and ∆PSM scores for all the 14 excerpts are smoothed to a single rate-quality curve by the LOWESS method [START_REF] Cleveland | Locally weighted regression: An approach to regression analysis by local fitting[END_REF] and then plotted against side-info rate in Fig. 2, for the five tested AAC bitrates. Each curve is then understood as a smoothing of 5 × 14 points (#NTF compression settings × #excerpts). Across the 14 excerpts, the mean PSM score of the oracle method is 0.882 (min 0.837, max 0.932), the mean SDR of the oracle method is 11.12 dB (min 7.10 dB, max 17.94 dB).

It can be seen from Fig. 2 that under all the tested AAC and sideinfo bitrates, configuration (c) performs better than configuration (a): increase of SDR upto 2.1 dB and increase of PSM upto 0.08,which represents substantial sound quality improvement, also confirmed by informal listening test. Quality improvement due to noise reduction generally decreases as AAC bitrate increases. This is because at higher bitrates, compression noise becomes smaller. Nevertheless, upto 128 kbps, the improvement in term of PSM is still significant (0.07). We also observed with noise reduction (configuration (c)), using bitrates higher than 128 kbps brings very limited quality gain. Thus we consider 128 kbps a sweet spot bitrate for AAC coding of the mixture under configuration (c).

Configuration (b) also improves quality over configuration (a), however, at the cost higher sideinfo rate, up to 4 kbps per source for the same improvement of configuration (c). Configuration (d) gives slightly better PSM scores (maximally 0.008 higher) and SDRs (maximally 0.9 dB higher) than configuration (c). The differences are generally imperceptible and shrink to 0 as AAC bitrate increases. This indicates that the estimated noise is accurate enough for the purpose of noise reduction. Obviously, configuration (d) is not realistic since exact noise is not available at the decoder in a practical system. Therefore, configuration (c) is always preferred.

This ISS system is efficient. Implemented mainly in Matlab (excluding AAC encoding and decoding), the system runs 1 3 X to 1 2 X real-time at the encoder side and 6X to 10X real-time at the decoder side (CPU 3.0 GHz). If implemented in C or other compiled languages, the system can be faster.

CONCLUSION

In this paper, we have proposed an informed source separation system which permits to handle mixtures compressed with standard audio coders. This was done through spatial Wiener filtering with compression noise reduction. Moreover, we showed that for the noise reduction, all the needed parameters, i.e., noise variance at each TF bin, can be estimated directly from the compressed mixture bitstream, without transmission of additional side-info. A large evaluation demonstrated that by taking in account of compression noise, the proposed system significantly increases perceptual quality of separated audio sources, at no cost of side-info bitrate, and the estimated noise variances provided almost identical separation quality as the exact noise variance. In the further, we shall drop the assumption that the compression noise is independent from the sources and consider not just compression noise but modeling errors: for example, long convolutive mixing (up to seconds) and nonlinear/nonstationary mixing, using the recently proposed coding-based informed source separation technique.

3 4 -

 4 power compression, and rounding:

  Suppose within subband b, YL(k) and YR(k) are sufficiently correlated such that YL(k) ≈ Y l (k) and YR(k) ≈ Yr(k) (roughly the point of applying IS). Then, only the new left Y l (k) for each bin k and the intensity ratio I L/R (b) for the whole subband b are needed for the decoder, substantially saving bitrate. The new left Y l (k) will be quantized by (4) with noise variance σ 2 l (k) determined by (6). The intensity ratio I L/R (b) will also be quantized to the so called IS position p(b) in the logarithmic domain:p(b) = 4 log 2 (I L/R ) + 1 2 = 4 log 2 (I L/R ) + δp(b),(9)in which δp(b) is the rounding error assumed to be uniformly distributed on (-0.5, 0.5]. Thus the L/R noise variances can be estimated as

  Rate-SDR (top) and Rate-PSM (bottom) curves for the proposed Wiener-based ISS system. '-V7' . Resulting NTF side-info bitrate per source ranges from 0.3 to 10 kbps.The AAC bitrates that were tested for encoding the mix signal are 32, 96, 128, 192, 256 kbps, covering the most common bitrates in AAC encoding for stereo (low complexity profile).

	∆SDR/dB	-3 0		AAC bitrate = 32 kbps		∆SDR/dB	-3 0		AAC bitrate = 64 kbps		∆SDR/dB	-3 0		AAC bitrate = 128 kbps		∆SDR/dB	-3 0		AAC bitrate = 192 kbps		∆SDR/dB	-3 0	AAC bitrate = 256 kbps
		-6						-6						-6						-6						-6
		-9						-9						-9						-9						-9
	-15 -12	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-15 -12	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-15 -12	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-15 -12	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-15 -12	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12
				bitrate per source / kbps					bitrate per source / kbps					bitrate per source / kbps					bitrate per source / kbps			bitrate per source / kbps
	∆PSM	0		AAC bitrate = 32 kbps		∆PSM	0		AAC bitrate = 64 kbps		∆PSM	0		AAC bitrate = 128 kbps		∆PSM	0		AAC bitrate = 192 kbps		∆PSM	0	AAC bitrate = 256 kbps
	-0.05					-0.05					-0.05					-0.05					-0.05
	-0.10					-0.10					-0.10					-0.10					-0.10
	-0.15					-0.15					-0.15					-0.15					-0.15
	-0.25 -0.20	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-0.25 -0.20	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-0.25 -0.20	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-0.25 -0.20	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12	-0.25 -0.20	0	2	4 no noise reduction 6 8 10 NTF noise coding noise estimation exact noise	12
				bitrate per source / kbps					bitrate per source / kbps					bitrate per source / kbps					bitrate per source / kbps			bitrate per source / kbps
					Fig. 2.

Another tool, called Temporal Noise Shaping (TNS[START_REF] Herre | Enhancing the performance of perceptual audio coders by using temporal noise shaping (tns)[END_REF]), changes noise distribution. We ignore this tool here for simplicity. Experiments show this neglect does not noticeably impair sound quality of separated sources.

http://www.nero.com/enu/technologies-aac-codec.html

http://www.audiocoding.com/faad2.html

http://www.quaero.org

* This work is supported by the French National Research Agency (ANR) as part of the DReaM project -ANR CONTINT 2009-006.


G Perrin

C Soize

D Duhamel
email: nis.duhamel@enpc.fr

C Funfschilling
email: christine.funfschilling@sncf.fr

International Workshop on Uncertainty Quantification in fluids simulation 16-18 December, Bordeaux, France Karhunen-Loève based sensitivity analysis

The identification of the most dangerous combinations of excitations that a non-linear mechanical system can be confronted to is not an easy task. Indeed, in such cases, the link between the maximal values of the inputs and of the outputs is not direct, as the system can be more sensitive to a problematic succession of excitations of low amplitudes than to high amplitudes for each kind of excitations. This work presents therefore an innovative method to identify the combined shapes of excitations that are the most correlated to problematic responses of the studied mechanical system.

I. Introduction

For years, mechanical engineers have been working on the identification of the most dangerous excitations that a mechanical system can be confronted to during its lifecycle. In the case of a linear system excited by a scalar stationary process, these researches have been mostly based on correlation analysis between the spectral density function of the excitation and the eigenfrequencies of the studied system. When confronted to non-linear mechanical systems excited by vector-valued stochastic processes that are a priori non-stationary, such comparisons cannot be achieved. In this prospect, Section II proposes an original sensitivity method based on a scaled Karhunen-Loève expansion. This method is then illustrated on an application based on experimental data in Section III.

II. Karhunen-Loève based sensitivity analysis

Theoretical frame

For all P ≥ 1, let P (P) (Ω) be the space of all the second-order R P -valued stochastic processes, indexed by the compact interval Ω = [0, T], where T < +∞. Let H P = L 2 (Ω, R P ) be the space of square integrable functions on Ω, with values in R P , equipped with the inner product •, • P , such that for all u and v in H P ,

u, v P = Ω u T (t)v(t)dt.
(1)

For P and Q in N * , let X ∈ P (P) (Ω) and Y ∈ P (Q) (Ω) be two vector-valued stochastic processes. It is then supposed that X gathers the time evolution of the excitations, whereas Y refers to the time evolution of the corresponding quantities of interest of a dynamical mechanical system M (M is a differential operator that is linear or not, whose coefficients depend on t or not), such that:

Y (t) = M [t, {X(s), 0 ≤ s ≤ t}] , t ∈ Ω.
(2)

Let d be a particular element of H Q , which is supposed to correspond to the time evolution of a problematic response for system M. For instance, function d can refer to a dangerous combination of oscillations for the components of Y. In a sensitivity analysis prospect, we are thus interested in identifying the evolution for input X, which would be the most correlated to such a response d for Y. In other words, we therefore search the optimal function f in H P , such that:

f = arg max u∈H P E X, u P Y, d Q .
(3)

Optimality of the Karhunen-Loève expansion

Mathematically, the Karhunen-Loève (KL) expansion associated with any stochastic process Z in P (N) (Ω), N ∈ N, corresponds to the orthogonal projection theorem in separable Hilbert spaces, for which the Hilbertian basis, K = k (m) , m ≥ 1 , is constructed as the eigenfunctions of the covariance operator of Z, which is assumed, for instance, to be square integrable on Ω × Ω. This Hilbertian basis is thus optimal in the sense that, for all M ≥ 1, it can be extracted from K the M-dimensional family that minimizes the total mean square error associated with Z among all the M-dimensional families of H N . In particular, it can be shown that the element of K of highest eigenvalue, that is denoted by k * , verifies the following equality:

k * = arg max v∈H N E Z, v 2 N . (4)

Sensitivity analysis

For any κ in ]0, 1[, let Z(κ) ∈ P (P+1) (Ω) be the scaled stochastic process such that:

Z(κ) = κX (1 -κ) Y, d Q I , I = {I(t) = 1, t ∈ Ω} ∈ H 1 , (5)
where X and Y are the input and output stochastic processes previously defined. Based on the developments achieved in [START_REF] Perrin | Karhunen-Loève expansion revisited for vector-valued random fields: scaling, errors and optimal basis[END_REF], it can be shown that the KL expansion associated with such a scaled stochastic process, Z(κ), allows defining projection basis that can favor or put at a disadvantage on purpose the characterization of the last component of Z(κ). In particular, let k * (κ) be the eigenfunction of highest eigenvalue associated with the covariance operator of Z(κ), which can be decomposed as:

k * (κ) = h(κ) g(κ) , h(κ) ∈ H P , g(κ) ∈ H 1 . ( 6
)
In a sensitivity analysis prospect, such a method can therefore be used to identify, at a very low computational cost, very relevant approximations of optimal function f , defined by Eq. ( 3), as it can be deduced from Eqs. ( 4) and ( 6) that:

lim κ→0 h(κ) = f . (7)

III. Application based on experimental data

A railway simulation can be seen as the dynamic response of a complex and strongly non-linear system (the train) excited by a vector-valued stochastic process (the track irregularities). For the last decades, the identification of the most dangerous and most uncomfortable track irregularities for the train dynamics has been based on correlation analysis between dynamical quantities of the train, such as vertical and transverse accelerations of any mass bodies and loads between the rail and the wheel, and the amplitudes, the wavelengths or the maximal values of the corresponding measured track irregularities. However, due to the high non-linearities of the train suspensions and of the wheel-rail contact, even if the maximal amplitudes or the variances of the track irregularities seem to be representative quantities for the track quality, such analysis have shown that they are not at all sufficient. Indeed, it can easily be extracted track conditions with high track irregularities that would less excite the train than track conditions with low track irregularities. On the contrary, based on available measurements of the track geometry and of the corresponding train dynamical responses, it can be shown that the method developed in Section II is very relevant to identify the successions of track irregularities that are the most correlated to problematic responses of the train [Perrin, 2013].

IV. Conclusions

An innovative sensitivity analysis method has been presented in this work. Based on an original scaled KL expansion, it has therefore been shown to what extent this method can open new ways to identify problematic successions of excitations for the analysis of non-linear mechanical systems. At last, the possibilities of such a method, which can be applied to any scientific field, are illustrated in the railway fields.