Gabriel Hermosillo

Lionel Seinturier

Laurence Duchien

Complex Event Processing for Context-Adaptive Business Processes

Keywords: Complex Event Processing, BPEL, Context adaptation

As the amount of data created by today's pervasive environments increases exponentially, there is a stronger need to decipher the important information that is hidden among it. By using complex event processing, we can obtain the information that really matters to our organization and use it to improve our processes. However, even when this information is discovered, the processes remain static and cannot be changed to adapt to the actual scenario, diminishing the advantages that can be achieved. In this paper we present CEVICHE, a framework that combines the strengths of complex event processing and process adaption. CEVICHE deals with the different languages used to detect complex events while presenting a single interface to the user.

Introduction

In today's pervasive and ubiquitous environments, the amount of data sent by the different devices in the environment is constantly growing and there is a strong need to transform this data into relevant information that can help to make the right decisions at the right moment. To take advantage of this information, it is required to discover time, cause and belonging relationships between the different data to give them a more useful meaning according to the context in which they are obtained, to generate complex events from simple and isolated events. Complex Event Processing (CEP) is an emerging technology for finding relationships, in real-time, between series of simple and independent events from different sources, using previously defined rules [1]. For example, if pressure decreases, the air is humid and the weather is warm, a CEP system could infer that a hurricane is being formed.

The CEP technology can be used, among a lot of other things, to enrich the enterprise's existing processes, by introducing rules that will allow the capture of relevant information from the different steps of the process [START_REF] Ku | A Novel Complex Event Mining Network for Monitoring RFID-Enable Application[END_REF]. For example, in the scenario of a retail store that keeps a record of its inventory in an existing ERP system and wants to keep a live monitoring of its stocks in order to prevent shortage. To achieve this, the store installs a CEP engine that will monitor the products movements through their life cycle in the store's process by receiving and analyzing all the events generated by every change in the state. Since the objective is to monitor inventory, the CEP engine will only keep the events related to changes in the inventory and forget about the rest. By creating the necessary CEP rules, the configuration is set to specify the lowest acceptable stock of product that the store can have to avoid a shortage, e.g. a 10% for normal products and a 5% for some low-demand products. Whenever a product reaches a minimum, the CEP engine alerts the managers so they can make a supply order.

In addition to that, CEP can also be used to predict unexpected situations. To complement the previous example, we can say that because of a global pandemic alert, hand sanitizers are very popular and are selling a lot more than usual. Given this demand, the store will run out of hand sanitizer before they can resupply it, even with the minimum stock alert. By adding some specialized CEP rules to analyze the frequency of sells of each product during the last 4 or 5 hours, the engine could polarize these values to know in advance (if the sells rates are kept) that it will need to resupply before the expected time, which will allow them to react in time even before it reaches the minimum level.

By doing the analysis of these kind of complex events, we can also prevent the theft of merchandise from stores, by creating relationships between the amount, kind and movement of the products inside the store [START_REF] Huber | Minimizing Product Shrinkage across the Supply Chain using Radio Frequency Identification: a Case Study on a Major Australian Retailer[END_REF]. However, there are some occasions in which it is not enough to be able to obtain relevant information from simple raw data, but there is also a need to adapt the process according to the new context in order to continue in an optimal way, and this is why we developed CEVICHE (Complex EVent processIng for Context-adaptive processes in pervasive and Heterogeneous Environments).

The purpose of CEVICHE is to create a system capable of generating, from a process definition, a context adaptation according to a set of previously defined rules, using the information obtained from each step of the process, and also from additional data sources that can be incorporated to cover different needs. For example, if the retail store needs to send a load of perishable products and the sensors detect that the temperature outside is too hot, e.g. 40°C, CEVICHE could adapt the process so that all the deliveries of perishable products are held until dusk (when the temperature should be lower).

With CEVICHE, we will improve the adoption of CEP by providing a solution to one of the problems it has, which is that until today there is no standard way to define the rules that relate the simple events to generate complex events, so we bring forward the Standard Business Process Language (SBPL) and a translation framework that will be able to adapt a single process definition to different CEP engines. The syntax of the SBPL will be based on the ❊✈❡♥t Pr♦❝❡ss✐♥❣ ••♦ss❛r② [START_REF][END_REF] created by the Event Processing Technical Society and the ❈♦r❡ ❇✉s✐♥❡ss ❱♦❝❛❜✉•❛r② created by EPCglobal [START_REF]EPCglobal's Core Business Vocabulary. ❤tt♣✿✴✴✇✇✇✳❡♣❝❣•♦❜❛❞❛r❞s✴❝❜✈[END_REF] to make it as standard as possible. The translation is done by using a special plug-in for each CEP engine, so at the end the users only have to specify the process definition once even if they want to change to another engine, just by using the corresponding plug-in.

The rest of this paper is organized as follows. Section 2 explains the CEVICHE framework and its architecture. Section 3 presents some of the related work. Finally, section 4 concludes and discusses some future work.

The CEVICHE framework

CEVICHE is a framework that intends to facilitate the integration of CEP into existing business processes and to allow these processes to be dynamically adapted to different circumstances. With this framework we want to address mainly four challenges: adaptation, dynamicity, integration to business process, and non-dependency to a specific CEP engine. To face the first challenge, adaptation, we use an aspect-oriented approach, which allows the system to add or change functionality and facilitates the task of separating concerns. By analyzing the current events with CEP and using context information, we can decide when and how to adapt the system at run-time, thus giving a solution to the dynamicity challenge. For the third challenge, we use these solutions in a business process environment by integrating them with the Business Process Execution Language (BPEL) . Finally, CEVICHE aims to be able to work with any CEP engine available. For that, as part of this framework, we defined the SBPL, which will gather all the information about the processes, contextual environment, business rules, and adaptation conditions. Using the specifications in the SBPL file, CEVICHE will translate them to the chosen CEP engine using a translation plug-in, making the system independent and allowing the user to choose the engine she prefers.

CEVICHE architecture

CEVICHE is composed of three main parts: a user interface to create the SBPL file, a translation framework to manage the plug-ins for each CEP engine, and an aspect manager to deal with the process adaption. The aspectoriented approach allows us to separate cross-cutting concerns from the core business processes and to integrate new functionalities or change the existing ones at run-time. CEVICHE also relies on different technologies to achieve the process adaptation, as shown in Fig. 1. First of all, CEVICHE needs a CEP engine that will be in charge of capturing and filtering all the events surrounding the process to create complex events that are relevant to the business. In order to manage the business processes, we need a Business Process Management (BPM) engine or a CEP engine capable of doing this. For the process definition, CEVICHE uses BPEL, since it is a standard language used in the industry for this purpose. BPEL is an orchestration (focused on the execution of the process) XML-based language that allows the definition of a process and its interaction with different entities using Web Services.

To deal with context-awareness we use COSMOS [START_REF] Rouvoy | Software architecture patterns for a context-processing middleware framework[END_REF], a component-based development framework that manages the context information in ubiquitous environments. COSMOS allows the building of a hierarchical network of context nodes that will gather the information obtained by each component and send it to the upper nodes to get a general response. Finally, we use an aspect-oriented extension to BPEL4WS called AO4BPEL [START_REF] Charfi | Ao4bpel: An aspect-oriented extension to bpel[END_REF], to adapt the existing processes. The advantage of using AO4BPEL is that we can change the business process specifications at run-time without the need to redeploy them, avoiding to lose all the ongoing transactions by doing that.

To configure the system, the user is provided with an interface where she can capture the business rules, process definitions and all the context data sources needed to adapt the processes. This information is sent in an SBPL file to the translation framework, which uses a specialized translation plug-in to adapt the information in the file to the specific CEP engine's format. This way, whenever the user wants to use another CEP engine, the only thing that needs to be done is to change the plug-in, without rewriting all the specifications of the business processes. The translation framework will then use that information to provide the CEP engine with the business rules in the appropriate format and to send the corresponding part of the information to each component of the architecture, leaving all the parts ready to interact during the process execution. Once the initial setup is ready and all the components have been properly configured, the process starts and the information begins to flow from one component to the other, as seen in Fig. 2. In the first step, the CEP engine subscribes to the different sources of events, here called the ❡✈❡♥ts s♣❛❝❡ , which will provide the engine with the information it needs to take decisions and create complex events. At the same time, in step 2, COSMOS will use the context policies provided by CEVICHE during the setup process to generate the context hierarchies needed to monitor the context information. Whenever there is a change in this information, it will be reported as an event and sent to the CEP engine (step 3). The CEP engine will gather all the events, filter the interesting ones according to the business rules and find relations that can generate complex events, which are then sent to the BPM to be considered in the process (step 4). The CEP engine is provided with some rules, which specify the conditions under which the process would need to be adapted. Whenever these conditions are met, the engine sends the information about the process to the CEVICHE Aspect Manager (step 5). Depending on the information received, the Aspect Manager will select the steps of the process that need to be changed and the corresponding changes to be done. Then, it uses AO4BPEL (step 6) to modify the process at run-time to be considered by the BPM (step 7).

Related work

CEP and BPEL. As we mentioned earlier in this paper, CEP is an emerging technology and the use of it in the business processes is a recent topic of interest and research [START_REF] Alistair | Complex events in business processes[END_REF][START_REF] Hermosillo | A Traceability Service to Facilitate RFID Adoption in the Retail Supply Chain[END_REF]. An analysis of scenarios of composite event patterns comparing BPEL and BPMN is done in [START_REF] Alistair | Complex events in business processes[END_REF]. The authors analyze patterns of events that go from conjunction and cardinality to time relations and event consumption possibilities. The conclusion of their study is that neither BPEL nor BPMN are capable of supporting complex event scenarios in their specifications, so there is a need to integrate event pattern descriptions into the process definitions language, but they do not mention the need to adapt the process according to those complex events.

In [START_REF] Hermosillo | A Traceability Service to Facilitate RFID Adoption in the Retail Supply Chain[END_REF], we present a service to add traceability to the RFID tagged products by using Complex Event Processing. When the RFID events are captured, they are transformed into business events that correspond to the business rules defined in the process which allows the users to have a better understanding of the status of their products, i.e. the product's location and the environment it has been exposed to.

Business process adaptation. The need to adapt a process has been a topic of interest in the recent years and there have been different approaches that offer solutions to it [START_REF] Antônio | Adaptive web service composition[END_REF][START_REF] Sánchez | A flexible architecture to build workflows using aspect-oriented concepts[END_REF][START_REF] Saif Ur Rahman | An adaptive eca-centric architecture for agile service-based business processes with compliant aspectual .net environment[END_REF][START_REF] Koning | Vxbpel: Supporting variability for web services in bpel[END_REF]. In [START_REF] Antônio | Adaptive web service composition[END_REF], the authors propose to deal with process adaptation by adding a web service repository that will handle the web services to invoke in each case. Whenever an invocation of a web service is done, the call is intercepted and the repository is checked for changes in the process definition, before the invocation of a web service. If there have been some changes, then it examines the available web services in the repository and chooses the one that best suits the criteria, otherwise the invocation is executed as usual.

The authors in [START_REF] Sánchez | A flexible architecture to build workflows using aspect-oriented concepts[END_REF] use an aspect-oriented approach introducing ❡①❡❝✉t❛❜•❡ ♠♦❞❡•s, which are used to represent the cross-cutting concerns. They use ♦♣❡♥ ♦❜•❡❝ts, which are representations of the state of the elements in the model, to monitor the invocation of services and adapt the process by weaving the interaction with other models before (activation) and after (deactivation) the call to the service.

Another aspect oriented implementation, using the Spring .NET framework, is presented in [START_REF] Saif Ur Rahman | An adaptive eca-centric architecture for agile service-based business processes with compliant aspectual .net environment[END_REF]. They use a contract-based approach to assign a web service to each instance of an execution call. To achieve adaptation of the process they can change the contract at run-time to assign a new web service for the call. They are also capable of adapting an existing implementation of a web service by using aspects to weave the new behavior.

An adaptation of the BPEL language called VxBPEL is presented in [START_REF] Koning | Vxbpel: Supporting variability for web services in bpel[END_REF]. Here, the authors insist on the need of flexibility and variability in the service-based systems and the lack of them when deploying BPEL processes. They extend the BPEL language to add new elements like ❱❛r✐❛t✐♦♥ P♦✐♥ts, which are the places where the process can be adapted and ❱❛r✐❛♥ts, which define the alternative steps of the process that can be used. VxBPEL also accepts new ❱❛r✐❛♥ts to be added at run-time, allowing the systems to be adapted without redeploying the process.

BPEL context adaptation.

The work that is closer to our proposal is the one presented in [START_REF] Charfi | A plug-in architecture for self-adaptive web service compositions[END_REF]. Here the authors present a plug-in based architecture for self-adaptive processes that uses AO4BPEL. Their proposal is to have different plug-ins with a well-defined objective. Each plug-in will have two types of aspects: the ♠♦♥✐t♦r✐♥❣ ❛s♣❡❝ts that will check the system to observe when an adaptation is needed and the ❛❞❛♣t❛t✐♦♥ ❛s♣❡❝ts that will handle the situations detected by the monitoring aspects. Whenever the conditions of a ♠♦♥✐t♦r✐♥❣ ❛s♣❡❝t are met, it uses AO4BPEL to weave the ❛❞❛♣t❛t✐♦♥ ❛s♣❡❝ts into the process at run-time. In our approach we deal with the monitoring part using the rules deployed in the CEP engine, which will detect special situations (by relating simple events) and select the aspects to be used to adapt the process.

Advantages of CEVICHE. The advantage of CEVICHE's architecture is that it is not necessarily linked to the third party technologies that it uses. So, even though it is not its main purpose, just as it can change from one CEP engine to another, the way to adapt the processes could also be changed from AO4BPEL to some other approaches, by creating the proper plug-in for it.

For the work presented in [START_REF] Charfi | A plug-in architecture for self-adaptive web service compositions[END_REF], an advantage of their work is that the monitoring aspects can be hot-deployed to their BPEL engine while with our approach the changes in the rules might not be considered at run-time, depending on the CEP engine. On the other hand, this difference also shows an advantage for our proposal, since we are not tied to a single engine and we can use any CEP rules already defined for the monitoring process, while in their case we would need to create a new plug-in for each new situation we want to monitor.

Conclusions and future work

Process adaptation and Complex Event Processing are two topics that are creating a lot of interest in the research community, however there is still no integration of both domains. In this paper we presented CEVICHE, a framework that integrates CEP and process adaption according to the context and that deals with adaptation, dynamicity, integration to business process, and non-dependency to a specific CEP engine. As part of the CEVICHE framework, we proposed the SBPL to deal with the different languages of the CEP engines, allowing the users to write their process specifications only once and deploy it in the engine they want. As future work, we plan to work on the definition of a RESTful architecture to leverage on the deployment of CEVICHE components and facilitate the evolution of the architecture by adding or changing components.

Figure 1 :

 1 Figure 1: The architecture of CEVICHE

Figure 2 :

 2 Figure 2: Information flow for adaptation


P F Morlat

J C Nuñez Pérez

G Villemaud

J Verdier

J M Gorce

On the Compensation of RF Impairments with Multiple Antennas in SIMO OFDM systems

This paper provides an analysis of the impact of radio frequency (RF) front end impairments (I/Q gain and phase imbalance, phase noise, non-linear distortion and direct current offset) on the performance of a single input multiple output orthogonal frequency division multiplexing (SIMO OFDM) receiver. We developed a new estimation/compensation scheme to jointly compensate for the effect of multipath and RF nonidealities on baseband signals in the special case of Zero-IF receivers. Some first results illustrating this approach are presented for a 4 antenna 802.11g receiver with the SMI algorithm and with or without IQ imbalance, as well as BER curves for different phase noise models.

I. INTRODUCTION

In the field of emerging high rate wireless systems, OFDM is a very popular modulation technique because of its performances in severe multi-path environments. Moreover Wireless LAN systems generally deal with Rice or Rayleigh fading channels and so multiple antenna techniques appear to be an essential way of increasing throughput. At the same time, considerable effort is spent to develop reconfigurable receivers using more or less Software Defined Radio (SDR) principles [START_REF] Rykaczewski | Signal Path Optimization in Software-Defined Radio Systems[END_REF]. Thus multiple standards can coexist in the same simple architecture with good processing capabilities.

A global approach combining multiple antennas, OFDM and SDR principles appears to be a very good choice particularly for future broadband wireless systems. Unfortunately, this combination is not so evident: multiple antenna algorithms and SDR are resource-expensive and the OFDM technique is sensitive to RF front-end performances [START_REF] Woo | Combined Effects of RF Impairments in the Future IEEE 802.11n WLAN Systems[END_REF]. Moreover, the impact of RF impairments on a receiver depends on RF front end topologies [START_REF] Brandolini | Toward Multistandard Mobile Terminals -Fully Integrated Receivers Requirements and Architectures[END_REF]. The direct conversion receivers often referred to as zero intermediate-frequency (Zero-IF) receivers have become very popular in wireless mobiles. Nevertheless, the RF impairments are much more critical in this case, as compared with super-heterodyne receivers for instance.

In this paper, we discuss the relevance of multiple antenna usage for an OFDM receiver taking into account the most important limitations of the analog RF front-end such as I/Q gain and phase imbalance, non-linear distortion, phase noise and direct current (dc) offset. The aim of our work is to point out the potential of compensating RF impairments jointly with the fading effect by the usage of a classical SMI algorithm and under a Multi-* (multi-antenna, multi-mode and multi-channel) receivers approach related to a SDR point-of-view. Assuming that OFDM will be a popular technique for future developments, our study primarily focuses on an 802.11g system. Problems of zero-IF receiver topology and particular impacts on OFDM signals are investigated in detail. All simulations have been realized with the Advanced Design System (ADS) software (from Agilent technologies [4]).

II. MULTIPLE ANTENNAS 802.11G RECEIVER

It is well known that one of the most important causes of degradation of the performances of high data rate WLAN is multipath propagation. Using multiple antennas at the receiver in a Single Input Multiple Output (SIMO) configuration improves the performances of wireless systems significantly. Indeed, spatial diversity is a practical and widely applied technique for reducing the effect of this multipath fading.

In this paper, we will show via quantitative analysis that the SIMO configuration also helps in decreasing the impact of non-ideal characteristics of the RF front-end on OFDM receivers.

A. Studied Architecture

Our aim is to study the benefits of the traditional Sample Matrix Inversion (SMI) [START_REF] Gupta | SMI adaptive antenna arrays for weak interfering signals[END_REF] for an 802.11g receiver. Lots of different algorithms exist to take advantage of spatial diversity and to combine different signals arriving at the same receiver [START_REF] Godara | Application of antenna arrays to mobile communications, part II: beam forming and direction of arrival considerations[END_REF]. In our case, WLAN OFDM receivers are mostly used in indoor environments, with large angular spread, so it is very difficult to calculate the Direction of Arrival of Signal of Interest with algorithms such as ESPRIT or MUSIC [START_REF] Sarkar | Smart Antennas[END_REF] which also have an important computational complexity.

Therefore we prefer to take advantage of the training sequence used in OFDM 802.11g receiver [START_REF]Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: Further Higher Data Rate Extension in the 2.4 GHz Band[END_REF] and to compute the optimal combination (figure 1) of different signals with the help of an adaptive algorithm with a Minimum Mean Square Error (MMSE) criterion.

A simple adaptive algorithm is the Least Mean Square (LMS) which has a low complexity of O(N). But this algorithm has a very slow convergence.

Taking into account that the training sequence time of an 802.11g burst is small, it is preferable to use SMI which offers where T denotes the transpose operation. The covariance matrix R and the cross covariance vector r are estimated with K samples of the signal vector (K is the number of sample of the training sequence d):

= ⋅ = K n H i S i S K R 1 ) ( ) ( 1 (1)
where H denotes the transpose and conjugate operation. (3)

= ⋅ = K n H i d i S K r 1 ) ( ) ( 1

B. Simulated Performances

Our objective is to simulate a realistic transmission system and to have a better estimation of the advantage of the SMI. A complete 802.11g transmission system was modelled with ADS, with a particular attention on the different types of channel used (average white Gaussian noise or various multipath models).

At first we verify that the SMI improves significantly an ideal OFDM receiver's performances under different channel conditions.

Figure 2 presents the BER performances of the four arms SMI receiver compared to a single antenna receiver with a 36 Mbps transmission rate (16QAM). All further presented results are based on this configuration.

To simulate multipath fading, we use typical office environment properties based on ETSI channel models [START_REF]Channel Models for HIPERLAN/2 in different indoor Scenarios[END_REF]. In this simulation, we consider the different channels are totally uncorrelated.

Denoting h(t) the channel response and n(t) a white Gaussian noise, the input signal at each antenna element p is defined as:

) ( ) ( ) ( ) ( t n t x t h t s p p + * = (4)
where * is the convolution operation, and x(t) the transmitted signal.

We can observe that the SMI algorithm ensures an average 10 dB gain, if channels are totally uncorrelated. But in practice, correlation exists between the signals received at different antennas, by the way limiting this gain. Further studies have to be done to modify our channel models in order to apply spatial correlation and antenna coupling.

C. First Measurements

In this global system approach to evaluate the improvement of such techniques, a rapid solution is to use a complete connected solution with multiple antennas capabilities. The simulated system could be connected with measurement materials to replace one or several simulated blocks. To perform BER measurements, we use an Arbitrary Waveform Generator (Agilent ESG 4438C) as a 802.11g source and a Vector Spectrum Analyser (Agilent VSA 89641) with two 6 GHz RF channels used for downconversion. The measured baseband signal is then re-injected in the simulated receiver.

At this time, only first measurements on a single channel are presented in figure 3, more complete results will be presented later.

This figure presents the BER performances of a 36 Mbps transmission rate system. For each point of the measured BER, 10 000 frames of 100 bytes were simulated, that is to say a 520 msec long signal.

For a 4 antennas system, two sets of measurements are required. Nevertheless this connected solution seems relevant for a direct evaluation of simulated structures. About 1 dB deviation could be observed for a single antenna receiver between a simulated AWGN channel and a close line of sight measurements (assuming that those measurements were not realised in an anechoid chamber).

III. EVALUATION OF RF IMPAIREMENTS IMPACT

Zero-IF receivers are very appealing, because they avoid costly IF filters. However, Zero-IF front-ends also introduce significant additional front-end distortion, such as IQ imbalance. Moreover Zero-IF does not solve the phase noise problem or the dc offset. The intermodulation distortion and consequently the impact of the second-and third-order intercepts strongly appear in this receiver topology too.

In this section, we present and compare specially the effect of IQ imbalance with an AWGN channel and a multipath channel, and we also focus on the phase noise influence, detailing flicker noise and PLL spurious, all results being based on BER curves. Of course, other types of impairments have to be explored with the same approach.

A. IQ Imbalance in AWGN Channel

Multiple antenna techniques are known to be efficient for signal-to-noise ratio improvement or interference rejection. By the way, RF impairments such as IQ imbalance could be considered as additional interference [START_REF] Tubbax | Joint Compensation of IQ Imbalance, Frequency Offset and Phase Noise in OFDM Receivers[END_REF]. IQ imbalance is characterized by two parameters: gain imbalance (or amplitude mismatch) and phase orthogonality mismatch. Thus a first evaluation of the SMI gain is presented in figure 4.

This result is obtained in an AWGN channel for an IQ gain imbalance varying from 0 to 1 dB. The single antenna case was realised for a 10 dB signal-to-noise ratio and the four antennas case with a 6 dB one, in order to obtain a comparable range of BER (around 10 -3 for zero gain imbalance). For each point, 1000 frames of 100 bytes were simulated.

Results are plotted in relative BER (zero gain imbalance BER value taken for reference). It clearly appears that the SMI algorithm efficiently compensates RF impact even for a gain imbalance up to 1.2 dB.

The same approach was used to evaluate the improvement in the presence of phase imbalance from 0 to 10 degrees. Figure 5 shows that a great performance could be expected even up to 15°.

More results have to be obtained to clearly show the limitation of this technique, but our goal is to focus on more realistic evaluation.

Actually, these first results confirm the relevance of the study, but here the same imbalance is applied to each arm of the receiver, and remain constant during the simulation time.

Moreover, AWGN is not the most interesting case of study for WLAN OFDM receivers.

B. IQ Imbalance in Multipath Channel

The same characteristics were also studied in a multipath environment. As expected we obtain an important increase of the relative BER for a single antenna compared to the AWGN case. Furthermore we can see in figure 6 that the SMI gain is very interesting in such large angular spread conditions for IQ gain imbalance.

For phase imbalance a great improvement is also obtained as shown in figure 7. But it is important to note that these results would be degraded if we consider correlated channels. In addition we have studied the effect of non-uniform gain imbalance on the four arms of the receiver. For independent random values from -1 to 1 dB we obtain quite a stable compensation: starting from a zero gain imbalance BER of 1.9. 10 -3 , a 1.99.10 -3 mean BER is obtained with a standard deviation of 1.2.10 -4 .

C. Influence of Phase Noise

It is well known that the degradation a of multi-carrier OFDM signal because of the oscillator phase noise results from two kinds of effects. The first one is a close-in phase error common to all sub-carriers, which rotates the whole constellation. This error can be estimated and compensated [START_REF] Muschallik | Influence of RF oscillators on an OFDM signal[END_REF] [START_REF] Wu | Performance analysis on the effect of phase noise in OFDM systems[END_REF].

The second one, which has a higher frequency Gaussian-like noise behaviour, can not be corrected. It results in intersubcarrier interference.

The analysis developed here derives the error probability of SIMO OFDM systems in the general case of an arbitrary phase noise spectrum, pointing out the difference influence of the slopes of the phase noise spectrum on the BER performances but also the effects of PLL spurious (Phase locked Loop IC). The influence of receiver phase noise on the performance of SIMO receiver is only studied here. AWGN channel is the only one modelled in this work. In our set-up with ADS software, a phase noise characteristic is specified in the PhaseNoiseData list.

This list contains values of offset frequency (Hz) and single sideband relative power level (dBc/Hz) in pairs. Interpolation is applied between these frequency domain points as needed to give a full time domain simulation definition for this phase noise.

The standard model for the phase noise in free-running oscillators [START_REF] Wu | Performance analysis on the effect of phase noise in OFDM systems[END_REF] [START_REF] Parker | Characteristics and sources of phase nois in stable oscillators[END_REF] is given in terms of the slope of the phase noise spectrum in a Bode diagram.

Flicker frequency noise relating to -30 dB/dec and Wiener phase noise relating to -20 dB/dec is studied for both single and four antennas receiver structures. Phase noise levels have great impact on the BER performances and it clearly appears that the SMI algorithm efficiently compensate phase noise degradation except for high level of phase noise.

Moreover, we note that the shape of the phase noise power spectral density has a very little impact since no change has been observed when a Wiener phase noise model is used. Indeed, performances are mainly determined by the total integrated phase noise. Generally, synthesized oscillators combine a voltage controlled oscillator (VCO) with a Phase-locked Loop IC (PLL), frequency reference and a loop filter. The loop filter design must integrate all of the components to establish, among other things, a trade-off between noise and transient response. Inserted between the phase comparator and the VCO control voltage input terminal, it must eliminate the high frequency component of the phase correction pulse generated by the phase comparator so that only the DC component is provided to the VCO. In general, increasing loop filter cutoff frequency provides faster PLL response and shorter PLL lockup time. Consequently, while phase noise near the carrier frequency is "suppressed", the reference leak is not [START_REF] Razavi | Phase locking in high performance systems : From Devices to Architectures[END_REF]. It turns out that the PLL output signal is frequency-modulated and contains high level spurs. In figure 9, the BER performances versus SNR taking into account different spurs levels are reported for both receiver structures. The signal to spur ratio is : symbol --: noiseless local oscillator and no spurs symbol --: Flicker frequencynoise (-110 dBc/Hz at 1 kHz offset)

The spurs at ± 100kHz is -60 dBc symbol --: Flicker frequencynoise (-110 dBc/Hz at 1 kHz offset)

The spurs at ± 100kHz is -50 dBc symbol --: Flicker frequencynoise (-110 dBc/Hz at 1 kHz offset)

The spurs at ± 100kHz is -40 dBc For AWGN channel, the efficiency of the SMI algorithm does not clearly appear. The same study would be made in a multipath environment. We can hope for a substantial improvement with this algorithm.

IV. CONCLUSION

In order to develop new generation of wireless mobiles with high rate and multi-standards capabilities, SIMO or MIMO associated with OFDM and SDR appear to be competitive field. Key points are the global evaluation of systems including all these principles and particularly the impact of RF architectures on the overall performances.

In this paper we have presented a system approach to simulate and measure a complete transmission scheme taking into account all parts of the structure, allowing a realistic link budget. Thus, the influence of RF impairments can be evaluated, depending on architecture choices, SIMO algorithm and channel conditions.

First simulated and measured results based on a 1x4 antennas transmission with a 802.11g receiver are discussed depending on the environment, and then the impact of IQ imbalance is studied.

Therefore we have the opportunity for a complete study of all the effects of RF impairments, and we want to focus especially on the potential of large bandwidth acquisition approach for reconfigurable terminals, multiple antennas allowing interference cancellation and multi-channel capabilities.

=

  the complex weight to apply to different signals, an approximate solution to the MMSE problem is calculated as:

Figure 2 .

 2 Figure 2. BER vs. Eb/No in multipath channel

Figure 3 .

 3 Figure 3. BER vs. Eb/No in AWGN channel

Figure 4 .Figure 5 .

 45 Figure 4. Relative BER vs. IQ gain imbalance in AWGN channel

Figure 6 .

 6 Figure 6. Relative BER vs. IQ gain imbalance in multipath channel

Figure 7 .

 7 Figure 7. Relative BER vs. IQ phase imbalance in multipath channel

Figure 8

 8 presents the BER performances versus SNR relating to the flicker frequency noise. The three PhaseNoiseData lists are respectively: PN1 : noiseless local oscillator (symbol : --) PN2 : -60 dBc/Hz at 100 Hz offset (symbol : --) -90 dBc/Hz at 1 kHz offset -150 dBc/Hz at 100 kHz offset PN3 : -20 dBc/Hz at 100 Hz offset (symbol : --) -50 dBc/Hz at 1 kHz offset -110 dBc/Hz at 100 kHz offset

BERFigure 8 .Figure 9 :

 89 Figure 8. BER vs. Eb/No in AWGN channel (flicker frequency noise)


G Royer

R Rousseau

On the liquid drop model mass formulae and charge radii

Keywords: 21, 10, Dr Binding energies and masses -21, 60, Ev Collective models

An adjustment to 782 ground state nuclear charge radii for nuclei with N,Z ≥ 8 leads to R0 = 1.2257 A 1/3 fm and σ = 0.124 fm for the charge radius. Assuming such a Coulomb energy

, the coefficients of different possible mass formulae derived from the liquid drop model and including the shell and pairing energies have been determined from 2027 masses verifying N,Z ≥ 8 and a mass uncertainty ≤ 150 keV. These formulae take into account or not the diffuseness correction (Z 2 /A term), the charge exchange correction term (Z 4/3 /A 1/3 term), the curvature energy, the Wigner terms and different powers of I = (N -Z)/A. The Coulomb diffuseness correction or the charge exchange correction term plays the main role to improve the accuracy of the mass formulae. The different fits lead to a surface energy coefficient of around 17-18 MeV. A possible more precise formula for the Coulomb radius is R0 = 1.2332A 1/3 + 2.8961/A 2/3 -0.18688A 1/3 I fm with σ = 0.052 fm.

Introduction

The prediction of the binding energies of new exotic nuclei both in the regions close to the proton and neutron drip lines and in the superheavy element region requires very accurate mass formulae. Historically, within a modelling of the nucleus by a charged liquid drop, semimacroscopic models including a pairing energy have been firstly developed to reproduce the experimental nuclear masses [1,2]. Later on, macroscopic-microscopic approaches have been developed , mainly the finite-range liquid drop model and the finite-range droplet model [3]. They take into account the non-smooth values of the masses (due to the shell effects and other microscopic properties) and the deformation of the nuclei investigating the fission and alpha emission exit channels or the fusion entrance channels. Nuclear masses have also been obtained accurately within a recent version of the statistical Thomas-Fermi model [4,5]. More recently, microscopic Hartree-Fock self-consistent theories using mean-fields and Skyrme or Gogny forces and pairing correlations [6,7] as well as relativistic mean field theories [8] have also been used to reproduce these nuclear masses.

In the macroscopic-microscopic liquid drop model approaches the evolution of the nucleus is mainly governed by the balance between the effects of the repulsive Coulomb forces and attractive surface tension and proximity forces [9][10][11]. In these approaches the charge radius is simply given by R 0 = r 0 A 1/3 and r 0 varies essentially from 1.12 to 1.24 fm according to the models. In the second section a value of r 0 is extracted from a whole set of recent experimental data. Later on, different possible additional terms entering into the mass formulae derived from the liquid drop model are briefly detailed in section 3. The coefficients of the liquid drop model development have been determined in section 4 using the value of r 0 extracted from experiments to calculate the Coulomb energy. All the studied mass formulae contain the usual volume, surface, Coulomb energy terms and the shell and pairing energies and include or not the diffuseness correction to the Coulomb energy, the charge exchange correction term, the curvature energy, a constant term, different forms of the Wigner term and different powers of the relative neutron excess I = (N -Z)/A.

Nuclear charge radii

For nuclei with N,Z ≥ 8 the experimental set of 782 ground state nuclear charge radii presented in ref. [12] leads to a rms charge radius of 0.94944A 1/3 which gives for the charge radius defined as

R 0 = 5 3 < r 2 > 1/2 (1)
the value R 0 = 1.2257 A 1/3 fm with σ = 0.124 fm. It is worth-while to mention that the experimental data show that the ratio R 0 /A 1/3 is far from being constant. For example, R 0 /A 1/3 = 1.312 fm for 40 Ca and R 0 /A 1/3 = 1.234 for 48 Ca while R 0 /A 1/3 = 1.217 for 190 Pb and R 0 /A 1/3 = 1.201 fm for 214 Pb. In a preceding paper [13] several nuclear mass formulae have been determined leaving the parameter r 0 free to evolve. It is quite satisfying to observe that r 0 converges to 1.22-1.23 fm in good agreement with the experimental data for the charge radius. More precise formulae allow to better reproduce the experimental charge radius. For example, the expression proposed in Ref. [14] has been fitted again and the new formula

R 0 = 1.2332A 1/3 + 2.8961 A 2/3 -0.18688A 1/3 I fm (2)
gives a good accuracy (σ = 0.052 fm). The introduction of the ground state deformation via a promiscuity factor depending on the distance of Z an N to the proton and neutron magic numbers can still improve the precision [15]. Nevertheless, the introduction in the mass formulae of such expressions to calculate the Coulomb energy does not allow to improve the accuracy of the mass formulae since the liquid drop model mass formulae basically suppose the proportionality of the charge and mass radius to A 1/3 .

Nuclear binding energy

The nuclear mass is directly connected to the nuclear binding energy B and several subsets of the following expansion of the nuclear binding energy in powers of A -1/3 and |I| have been investigated :

B = a v 1 -k v1 |I| -k v2 I 2 A (3) -a s 1 -k s1 |I| -k s2 I 2 A 2 3 -a k 1 -k k1 |I| -k k2 I 2 A 1 3 -a 0 A 0 - 3 5
e 2 Z 2 1.2257 A 1 3 + f p Z 2 A +a c,exc Z 4 3 A 1 3 -E pairing -E shell -E W igner .
The first term is the volume energy in the infinite nuclear matter. I 2 A is the asymmetry energy term of the Bethe-Weizsäcker mass formula. The second term gives the surface energy and expresses the deficit of binding energy of the nucleons at the nuclear surface. The Bethe-Weizsäcker mass formula does not take into account the dependence of the surface energy on I. The following term is the curvature energy which appears to correct the surface energy when the local curvature properties are considered. This term is not taken into account in the FRLDM [3] but appears in the Lublin-Strasbourg Drop (LSD) model [16] and the TF model [5]. In the three first terms a dependence on |I| has been introduced since they have been proposed to better reproduce the fission barrier heights [17]. The A 0 term appears when the surface term is developed to include higher order terms in A -1/3 and I. The fifth term gives the decrease of binding energy due to the repulsion between the protons. The charge radius is fixed to the value derived from the recent experimental data. This Coulomb energy is proportional to Z(Z -1) in the Bethe-Weizsäcker mass formula. The Z 2 /A term is the diffuseness correction to the basic sharp radius Coulomb energy term (called also the proton form-factor correction to the Coulomb energy in Ref. [3]) and the term proportional to Z 4/3 /A 1/3 is the charge exchange correction term. The pairing energy has been calculated within the formulae proposed in the recent version of the Thomas-Fermi model [5]. The theoretical shell effects used in the TF model (7 th column of the table in [4] and [5]) have also been used since they have allowed to reproduce correctly the masses from fermium to Z = 112. They have been derived from the Strutinsky shell-correction method. As an example, they give a contribution of 12.84 MeV to the binding energy of 208 Pb. The fits on nuclear masses depend necessarily on the choice of the selected theoretical shell effects. The sign for the shell energy term comes from the adopted definition in [4].

The Wigner energy has been introduced to reproduce the kink in the nuclear mass surface. It is not a usual shell effect. It appears in the counting of identical pairs in a nucleus and depends on I. The original expression is simply W |I| [START_REF] Myers | Droplet Model of Atomic Nuclei[END_REF]. Its effect is to decrease the binding energy when N = Z.

The congruence energy term is expressed as :

E cong = -10exp(-4.2|I|) M eV. ( 4
)
It is an extra binding energy associated also with the presence of congruent pairs [5]. Within an Hartree-Fock approach [START_REF] Goriely | [END_REF] it has been argued that a gaussian expression

E = V W exp(-λI 2 ) ( 5 )
is as acceptable. Another term has also been proposed in Ref. [19]

E = β|N -Z| exp -(A/A 0 ) 2 . ( 6
)
Another expression has also been tested

E = β|N -Z| exp [-(A/A 0 )] . ( 7
)

Coefficients of the mass formulae

The coefficients of different expansions in powers of A -1/3 and |I| have been obtained by a least square fitting procedure to 2027 nuclear masses verifying N,Z ≥ 8 and a mass uncertainty ≤ 150 keV [20]. To compare the relative efficiency of the different selected sets of terms the rootmean-square deviation σ defined by

σ 2 = Σ [M T h -M Exp ] 2 n (8)
has been used. In the Table I, the improvement of the nuclear mass reproduction according to the introduction of different additional terms is displayed. The basic A, AI 2 , A 2/3 , A 2/3 I 2 and Z 2 /1.2257 A 1/3 terms lead already to a deviation of only 0.99 MeV.

The diffuseness correction term (in Z 2 /A) as well as the charge exchange correction term (in Z 4/3 /A 1/3 ) are very efficient to reduce the deviation in the same order of magnitude and towards σ = 0.67 MeV. When both the diffuseness correction to the Coulomb energy and the charge exchange correction term are taken into account the deviation remains the same and spurious coefficients appear. The introduction of a constant term is useless and can induce a discontinuity during the transition from one to twobody shapes as in fission, fusion or α emission. The first expression of the Wigner term W|I| allows to diminish the error of only 0.13 MeV. Its main drawback is that it is approximately independent of the nuclear shape [START_REF] Myers | Droplet Model of Atomic Nuclei[END_REF]. Then, in a division into 2 fragments, both with the same value of |I|, the Wigner energy jumps at scission to 2 times its original value leading, for an example, to a discontinuity of the potential energy of around 6.7 MeV at the contact point between spherical fragments of 258 Fm. The influence of the congruence term is very weak at least with the fixed coefficients adopted here. When the coefficients before the exponential and the exponent are free the congruence energy tends to the usual Wigner term since the coefficient before the exponential diminishes while the exponent increases. The congruence energy term leads also to an important discontinuity : 3.9 MeV for 258 Fm. The Coulomb diffuseness correction term has the main advantage to be almost continuous at the scission point in the entrance or exit channels. The introduction of the curvature energy in A 1/3 alone does not allow to significantly decrease the deviation from the experimental masses but when both the terms in A 1/3 and A 1/3 I 2 are taken into account σ is reduced to 0.64 MeV. These terms have the advantage to be almost continuous in a symmetric decay but their values and signs lack of stability. The values of A 0 which minimize the mass rms deviation when the formulae ( 6) and (7) are introduced are respectively 48 and 35 for the two expressions. These formulae for the Wigner energy are supposed to be approximately independent of the nuclear shape. Their discontinuity at the scission point of fission or fusion barriers is less important than that of the congruence and W|I| terms. For example, for the expressions |N -Z| exp -(A/48) 2 and 1.5|N -Z| exp [-(A/35)] and symmetric decay of the 258 Fm the discontinuities at the contact point of the nascent fragments are respectively 0 and 0.05 MeV. The introduction of an |I| dependence in the volume and surface energy terms improves the accuracy by 0.03-0.04 MeV. When both the Coulomb diffuseness correction term and the pure Wigner term are taken into account the deviation diminishes till σ = 0.61 MeV. The lowest value which can be attained when adding other terms seems to be 0.58 MeV. With only 6 adjustable parameters different combinations lead to σ = 0.61 MeV. Recently, the mutual influence of terms in semi-empirical formulae has been deeply investigated [21].

As an example, Fig. 1 indicates the difference between the theoretical and experimental masses using the formula -E shell -0.7614|N -Z|e -(A/35) (9) leading to σ = 0.61 MeV. For most of the nuclei with A ≥ 110 the difference is less than 1 MeV and it appears no strong structure in the set of errors. For A ≤ 110 the cloud of points is less uniform and more structured. This is probably due to the values of the pairing and shell energies.

B = 15.3543 1 -1.7445I 2 A -17.2293 1 -1.5765I 2 A 2/3 - 0.6e 2 Z 2 1.2257A 1/3 + 1.24413 Z 2 A -E pair

Conclusion

An adjustment to 782 recent ground state nuclear charge radii for nuclei with N,Z ≥ 8 leads to R 0 = 1.2257 A 1/3 fm. This value is close to the value R 0 = 1.2249 A 1/3 fm proposed in [22]. Assuming such a Coulomb energy E c = The Coulomb diffuseness correction Z 2 /A term or the charge exchange correction Z 4/3 /A 1/3 term plays the main role to improve the accuracy of the mass formulae. With a less efficiency, the Wigner term and the curvature energy can also be used separately. The introduction of an |I| dependence in the surface and volume energies improves slightly the efficiency of the expansion. The pure Wigner, the congruence and the constant terms lead to an important discontinuity of the deformation energy during the transition from one to two-body shapes as in fission, fusion or α emission. The congruence energy term is much less efficient than the Wigner term W|I|. Finally, the different best fits lead to a root-mean-square mass deviation of 0.58 MeV and a surface energy coefficient of around 17-18 MeV. For comparison the values adopted in [16], [22] and [23] are respectively 16.9707, 17.9439 and 21.7 MeV.

Fig. 1 .

 1 Fig. 1. Difference between the theoretical and experimental nuclear binding energies for the 2027 selected nuclei versus the mass number.

3 5 e 2 1 3

 321 Z 2 /1.2257 A , the coefficients of different possible mass formulae derived from the liquid drop model and including the shell and pairing energies have been revisited and determined from 2027 masses verifying N,Z ≥ 8 and a mass uncertainty ≤ 150 keV.

Table 1. Dependence of the energy coefficient values (in MeV) on the selected term set and root mean square deviation. The theoretical shell and pairing energies are taken into account.


Ronan Champagnat
email: ronan.champagnat@univ-lr.fr

Pascal Estraillier
email: pascal.estraillier@univ-lr.fr

Armelle Prigent
email: armelle.prigent@univ-lr.fr

Adaptative execution of game: unfolding a correct story

INTRODUCTION

Computer games have improved this last decade on technical aspects (graphical, sound design, etc.) as well as on the content. Actually games are getting more and more complex. Plot has been improved and is getting close to scenario movie or scenario novel.

Interactive narrative game corresponds to a game in which player unfolds a story. As a consequence, he is more or less free to choose actions to be executed according to a narrative framework a scenario writer has defined.

ADAPTATIVE EXECUTION OF GAME

An adaptative execution of game consists in keeping the unfolding within the bounds of the narrative frameworks. This implies that we have to be able to change the game unfolding by controlling the player interactivity. And also we have to be able to change the bounds of the narrative framework.

In [CPE05, PCE05] we have been proposed an architecture for adaptative execution. This architecture is made up of a simulation of a process and story director (or planner). It defines a set of agent that analyses the unfolding of a game and compute an adaptative execution of narrative during a game. The simulation process catches player's interactions and react to them and proposes a consistent unfolding. The story director can enable or disable parts of stories. As In the framework of complex and critical system, a fault tolerant conception allows designer to produce a system that will be able to follow correct narrative even if faced to inconsistent inputs.

The planner is made up of an observation agent which interprets player actions; a narrative agent which performs a supervisory control of the storytelling; analysis agent which makes "on-the-fly" verification; scripting agent that defines a new set of possible action and a director agent that chooses a set of relevant actions to be executed.

NARRATIVE FRAMEWORK

In [CPE05] a formal model for interactive narration has been introduced as a set of resource, character, action, and outcomes.

This modelling has been improved by adding the notion of narrative framework. The narrative framework does not describe the whole scenario in details but gives some part on different situations that hold in this one.

A narrative framework is divided into situations that player will encounter during game execution. These situations can be in a sequence or as a tree. Situations are being connected to each other as bridges between a situation input and another situation output.

A situation is a specific state of the game involving actors, objects and constraints by system capacity. Actions characterize dynamics of a situations.

ADAPTATIVE GAME EXECUTION

The dynamics of game execution has been characterized by a state vector made up by actors, objects, resources, player and context.

Let us consider S o has the observed state of the system and S e the expected state of the system. In order to make an optimization we only consider the significant variables and states to observe (given by the action to be executed)

The distance between the expected state and the observed one corresponds to the difference of their state vector

The execution loop for director agent is: select an action, configure the observer, perform action (at that point, we have a new state S e , compute S o then compute the distance between S o and S e and go back to select an action.


G Lemarquand

V Lemarquand

Calculation Method of Permanent Magnet Pickups for Electric Guitars

Keywords: Permanent magnets, pickups for electric guitar, analytical magnetic field calculation

This paper first presents the structures of permanent magnet pickups for electric guitar and the considered device: string, magnet, coil. It then describes a method to calculate the induced electromotive force (EMF) in the pickup coil when the string moves. The method of calculation links the EMF in the pickup coil with the flux cut by the string when it moves. The EMF is of course nonlinear. The harmonics of the EMF can be calculated. This is a first step towards the final aim that will consist in studying the frequency spectrum of the electrical signal given by the pickup, which is quite complicated, in order to link the magnetic structure and the string movement with the musical effect. Analytical calculations using the coulombian model of magnets are used to evaluate the magnetic field created by the magnet and the electromotive force in the pickup coil.

I. INTRODUCTION

W HO has never heard some music, played with electric guitars? These musical instruments are nowadays well-known and of current use in pop groups. Their story backs up in the 1930s, when Rickenbacker fitted out a guitar with a magnet and coils, thus designing the first magnetic circuit for electric guitar pickups. The way these pickups are designed and tuned has an empirical basis and is often related to the know-how and the feeling [START_REF] Wheeler | Pickups-A primer[END_REF]. Patents have been taken out by pickups designers, especially in the U.S.A. by well-known makers like Rickenbaker, Fender, Gibson, Seymour Duncan, Di Marzio [START_REF][END_REF] [3], but few papers are to be found in scientific journals [START_REF] Queen | From boom boxes to beatles, baez and boccherini. the electric guitar at the crossroads[END_REF]. We want to describe these pickups from a scientific point of view and to give quantitative guidelines for their design. The description can be done from two points of view that have to be joined: the magnetic one and the acoustic one. We have first to look at the types of magnetic circuit for the guitar pickups. We consider in this paper the most usual one, which is a cylindrical piece of permanent magnet axially magnetized. There is one magnet for each string. A whole pickup can be described as the juxtaposition of six magnets, one for each string, with the coil around them. We suppose that the magnet is rigid and that its magnetic relative permeability is equal to one, which is the case for hard ferrite as well as for rare earth magnets. This study doesn't include Alnico magnets, that have been used for a long time -and are still used for vintage pickups -but do not have these properties. We neglect the eddy currents, either in the magnet or in the string.

The string and its behavior have then to be considered. We consider that the guitar neck and the string define the x axis.

The movement of the string is constituted by two elementary transversal movements. Some authors [START_REF] Valette | Mecanique de la corde vibrante[END_REF] [6] consider that they are not coupled for small movements, within the linear approximation, and that they become coupled for higher amplitudes. The transversal modes have close but nevertheless separate frequencies. This comes from the limit conditions that are slightly different for each mode at the bridge and at the nut [START_REF] Lambourg | Measurements and modeling of the admittance matrix at the bridge in guitars[END_REF] . We study each transversal movement separately. We first consider that the string movement in front of the pickup is sinusoidal and occurs in a xy plane above and parallel to the pickup (Fig. 1). We then consider that the string movement in front of the pickup is sinusoidal and occurs in a xz plane above the pickup. In fact, a good player is able to exert a force on the string in the y direction only, while a less experienced player will also exert a force in the z direction. The string always has an elliptical movement, but the amplitude along the z axis is smaller than along the y axis and we may consider that the latter is between 10% and 20% of the former. We assume that the string moves freely, and we neglect the attraction force of the magnetic field on the string. The string movement possibly contains mechanical harmonics [START_REF] Watzky | Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string[END_REF], but we only consider the fundamental vibration of the string by now. The amplitude of the movement is of course of great importance, as the non linear behavior depends on it.

The string dimensions depend on the corresponding note: the smaller the diameter, the sharper the note. A guitar has six strings corresponding to the notes E (83Hz), A (110Hz), D (147Hz), G (196Hz), B (247Hz), E (330Hz) in the standard tuning. In fact the string diameter is chosen with regard to the note but also to the string tension, so that the tensions of the six strings are nearly identical, in order not to twist the guitar neck. The length of the string depends on the choice of the sounding length of the guitar string made by the guitar maker. The height of the string above the pickup is also an important parameter, and it is not always the same for all of the six strings.

The strings are metallic and the usually used materials are pure nickel, nickelplated steel or stainless steel, all ferromagnetic, but that give different types of sound, because of their different mechanical properties.

A coil of a highly thin copper wire (diameter 0.07mm) and very numerous windings (around 6500), is wound around the magnets. When the string moves in front of the pickup, the flux in the magnetic circuit changes and so does the electromotive force in the coil. This voltage, its amplitude and its frequency spectrum, constitutes the signal of the pickup. The magic of the sound depends on it and so, on the dimensioning of the pickup [START_REF] Skopp | Series of articles about pick-up tricks[END_REF].

We will consider only one magnet to explain the method and calculations. To dimension a pickup for an electrical guitar, one has to consider that it is a three dimensional device, including a cylindrical magnet and a string. Average dimensions are a diameter between 4 and 5mm and a height of 10mm for the magnet, between 0.2 and 1.3mm for the strings' diameter and a length of around 5mm for the distance between the magnet and the string. This means that the string is rather small in comparison with the magnet and also rather distant. As a consequence, it is not possible to do the usual approximate calculations as for a two dimensional device. Finite elements calculations still take too many computing resources to take correctly into account the differences of scale in the dimensions. The difficulty is to obtain a good mesh for devices in which one dimension is small compared with the others. Moreover, we want to calculate the effects of the variation of the flux in the magnetic circuit and we need a precision compatible with the spectral analysis of the signal [START_REF] Markovic | Determination of tooth cogging force in a hard-disk brushless dc motor[END_REF]. Such a precision on the flux value is not reachable with finite elements calculations programs.

We are confronted with the same problem as numerous authors which study permanent magnet electrical machines and try to find alternative methods to finite element analysis [START_REF] Ren | Improvement on windings flux models for a slottless self-bearing motor[END_REF][12] [START_REF] Lebec | Coil and magnet design for nuclear magnetic resonance in inhomogeneous field[END_REF]. The considered system has the same problematics as permanent magnet motors for which the EMF has to be calculated [START_REF] Zhu | Analysis of electromagnetic performance of flux-switching permanent-magnet machines by nonlinear adaptive lumped parameter magnetic circuit model[END_REF] [START_REF] Wang | Tubular modular permanent-magnet machines equiped with quasi-halbach magnetized magnets-part1:magnetic field distribution, emf, and thrust force[END_REF]. We describe a method to calculate the induced electromotive force (EMF) in the pickup coil when the string moves. The method of calculation links the EMF in the pickup coil with the flux cut by the string when it moves: we show that the EMF in the pickup is proportional to the magnetic flux cut by the string in its movement. Analytical calculations using the coulombian model of magnets are used to evaluate the magnetic field created by the magnet and the EMF in the pickup coil. The EMF is of course nonlinear. The harmonics of the EMF can be calculated. This is a first step towards the final aim that will consist in studying the frequency spectrum of the electrical signal given by the pickup, which is quite complicated, in order to link the magnetic structure and the string movement with the musical effect.

II. DESCRIPTION OF THE METHOD

We consider that the elementary pickup is constituted by a cylindrical modern permanent magnet and a coil around it. The diameter, d m , of the magnet is 4.5mm, its height, h m , is 10mm. The altitude, d, of the string above the pickup is 5mm at rest. The initial position of the string is centered above the upper face of the magnet. It has to be noticed that the magnet, the string and the coil constitute a magnetic circuit and we will work on the equivalent magnetic circuit [START_REF] Wang | Analytical prediction of the incremental inductance of the permanent magnet synchronous motors[END_REF]. In fact, the system works as a polarized reluctant system [START_REF] Jufer | Electromecanique, Traite d'electricite, d'electronique et d'electrotechnique[END_REF] [START_REF] Wildi | Electrotechnique. Les presses de l[END_REF], with a fixed source of magnetic field, the magnet, and a ferromagnetic moving part, the string.

All the hypotheses we make lead to the following expression of Ampere's theorem ( 1): where E is the magnetomotive force in the circuit, R, the reluctance of the circuit -string, magnet, air-, J, the magnetic polarization of the magnet, in Tesla, h m , the height of the magnet, N the number of windings of the coil and i the current in the coil. As the interesting signal is the electromotive force in the coil, the measurement electronics have a high input impedance and the current in the coil is zero. Equation [START_REF] Wheeler | Pickups-A primer[END_REF] shows then that the magnetomotive force in the circuit is a constant.

E = Rϕ = N i + Jh m µ 0 (1)
The magnetic circuit that has to be considered is constituted by a finite juxtaposition of flux tubes. All the tubes have the same section. Most of the tubes contain air and magnet. Some tubes go through the string, and contain iron, magnet and air. We index with j the tubes that contain the string at the time t and that the string is going to leave at the time t + ∆t. We index with k the tubes that don't contain the string at the time t but that are going to receive it at the time t + ∆t. We write ϕ j the magnetic flux in the j tubes where the string is at the time t. At the time t + ∆t, when the string has moved and has left them, we write ϕ′ j the new flux it these tubes. Identically, we write ϕ k the magnetic flux in the k tubes at the time t, and ϕ′ k the new flux it these tubes at the time t + ∆t, when the string has reached them. We want to calculate the flux variation in the coil, ∆ϕ , between the time t and the time t + ∆t. The tubes that contribute to the flux variation are the tubes j and k and we don't need to consider the other tubes. The part of the flux that contributes to the variation and that is seen by the coil at the time t is the sum ϕ j + ϕ k for all the tubes j and k. This flux becomes the sum ϕ′ j + ϕ′ k at the time t + ∆t. The total variation is then the following (2): ∆ϕ = all tubes j , k

(ϕ′ j + ϕ′ k ) -(ϕ j + ϕ k ) (2)
The reluctance of a j tube going through the string at the time t can be considered as the sum of the reluctance of the portion of tube going through the string R s , and the reluctance of the remaining length of the tube, R j . At the time t + ∆t, the reluctance R s becomes R a , because this portion of tube no longer contains iron, but contains air instead. The same kind of definitions are given for the tubes k. R s appears as the "passing reluctance" of the string. Table I sums up all these notations. There is an assumption on the reluctances: we consider that the string passing reluctance, R s , is approximately the same at the time t and at the time t + ∆t.

We assume that the flux density in a tube is uniform and corresponds to the value in the middle of the tube.

As the tubes are parallel connected, we also have the following relations (3) and ( 4):

E = (R j + R a ) ϕ j = (R j + R s ) ϕ′ j (3) = (R k + R s ) ϕ k = (R k + R a ) ϕ′ k (4)
From ( 2) and (3) we deduce (5):

j Tube k Tube Flux Reluctance Flux Reluctance t ϕj Rj + Rs ϕ k R k + Ra t + ∆t ϕ′j Rj + Ra ϕ′ k R k + Rs

TABLE I NOTATIONS FOR THE FLUXES AND RELUCTANCES

Where Rs is the string passing reluctance of a tube, Ra, is the reluctance of the preceeding portion of tube with air instead of iron, Rj and R k , the remaining reluctances of the considered tubes.

∆ϕ = all tubes j , k R s -R a R k + R a ϕ k - R s -R a R j + R a ϕ′ j (5)
And then we deduce ( 6) and [START_REF] Lambourg | Measurements and modeling of the admittance matrix at the bridge in guitars[END_REF].

∆ϕ = all tubes j , k R s -R a E (ϕ k ϕ′ k -ϕ j ϕ′ j ) (6)
≃ all tubes j , k R a -R s E (ϕ k -ϕ j )(ϕ k + ϕ j ) (7)
We make the additional hypothesis that the sum of the fluxes varies very slightly and can be considered as a constant.

Finally, the flux variation in the coil can be expressed as [START_REF] Watzky | Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string[END_REF]:

∆ϕ ≃ all tubes j , k K(ϕ k -ϕ j ) ( 8
)
where K is a constant corresponding to [START_REF] Skopp | Series of articles about pick-up tricks[END_REF]:

K = all tubes j , k 1 E (R a -R s )(ϕ′ k + ϕ′ j ) = all tubes j , k (R a -R s ) 1 R j + R s + 1 R k + R a ≃ all tubes j , k (R a -R s ) 1 R j + 1 R k (9)
Equation [START_REF] Watzky | Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string[END_REF] shows that the variation of the flux in the coil is proportional to the difference of the fluxes in the tubes left by the string and the tubes reached by the string, so the variation of the flux between two positions of the string, which can also be qualified as the flux, ϕ c , cut by the string when it moves.

Equation [START_REF] Skopp | Series of articles about pick-up tricks[END_REF] shows that the variation of the flux in the coil depends on the reluctance variation divided by the total reluctance of the magnetic circuit. Shortly, it depends on the relative reluctance variation, which is normal. Equation ( 9) also shows that ∆ϕ will be large if the difference (R a -R s ) is great, so, if the passing reluctance of the string is small. This reluctance can be expressed by [START_REF] Markovic | Determination of tooth cogging force in a hard-disk brushless dc motor[END_REF]:

R s = l s πµ 0 µ s δ s ( 10
)
where δ s is the diameter of the tube occupied by the string, l s the length of the tube occupied by the string and µ s the relative magnetic permeability of its material. We thus see the importance of having a ferromagnetic string, and the influence of the string diameter -thick strings will give a greater variation. The electromotive force (EMF), e, which is also the signal in the pickup, is given by the Faraday law and can be written as in [START_REF] Ren | Improvement on windings flux models for a slottless self-bearing motor[END_REF]:

e = relevant tubes -N ∆ϕ ∆t (11)
≃ relevant tubes -N K ϕ c ∆t
where N is the number of turns of the pickup coil. The link established by [START_REF] Watzky | Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string[END_REF] between the variation of the flux in the coil and the flux cut by the string is very from a calculation point of view, because the cut flux, ϕ c , can be evaluated with [START_REF] Mao | Design of magnet arrays for a permanent magnetic linear motor[END_REF]:

ϕ c = S B dS, ( 12
)
where B is the component of the magnetic flux density along the normal to the cut surface S.

We will now present how to calculate the magnetic field created by the magnet and then discuss how the movement of the string creates the signal.

A. Permanent Magnet Field Calculation 1) Principle:

We calculate the magnetic flux density, B, created by the permanent magnet in the space around it when the magnet is alone. We assume that the magnet is axially and uniformly magnetized, with a magnetic polarization, J, of 1T . We use a magnetic masses model for the magnet [START_REF] Yonnet | Magnetomechanical devices[END_REF]. The cylindrical magnet can be replaced by two discs of same diameter, d m , as the magnet, representing the top and bottom faces of the cylinder. They are separated by the magnet height, h m . We consider that the material between and around the discs is air. Both discs are charged with a uniform magnetic masses surface density, σ * :

σ * = J.n (13)
but with opposite signs. Let us call M 1 a point on the upper disc (surface S1, north pole of the magnet), and M 2 a point on the lower disc (surface S2, south pole of the magnet). The magnetic flux density, B, created by the magnet at a point M of the space is calculated with [START_REF] Zhu | Analysis of electromagnetic performance of flux-switching permanent-magnet machines by nonlinear adaptive lumped parameter magnetic circuit model[END_REF].

B(M ) = J 4π S1 MM 1 |MM 1 | 3 dS 1 - J 4π S2 MM 2 |MM 2 | 3 dS 2 (14)
We calculate the values of the magnetic field (modulus and direction) in the space where the string will be located. Fig. 2 shows values along the center of the string, as if the string had no thickness.

2) Circular section and square section: Previous calculations are achieved numerically, as we do not have an analytical expression for B when the surfaces of the magnet are circular. But when the magnet's surfaces are rectangular or square, the analytical expressions for the magnetic field B are available [START_REF] Yonnet | Magnetomechanical devices[END_REF]. Let us compare the values of B for two cases. In the first case, B c is calculated numerically for a magnet with a circular section. In the second case, B s is calculated analytically for a magnet with a square section. The dimensions are chosen so that the surfaces have the same value. We evaluate the relative difference, e r , for the three components of the magnetic density at a given distance, d, of the magnet. Of course, d can't be too small: the comparison is not valid on the surface of the magnet, but for the generally chosen string height above the magnet, the distance d is large enough to legitimate the comparison.

The relative difference for the component along the x axis is maximal when the string is located above the edge of the square magnet. The maximum difference is around 1.4%. For the component along the y and the z axis, the maximal difference occurs when the string is centered above the magnet. Its value is respectively around 1.5% and 0.6%. The magnet with a circular section will be replaced by a magnet with a square section in the further study, as it allows us to calculate the electromotive force analytically.

B. String motion

The real movement of the string is elliptical in the space, but can be described as the composition of two movements in two perpendicular planes. An approach is to study the movement of the string in two planes xy and xz [START_REF] Fletcher | The physics of musical instruments[END_REF] [21] [START_REF] Gough | The theory of strings resonances on musical instruments[END_REF].

When we consider that the string is moving in a plane parallel to the xy plane, which is parallel to the upper face of the magnet and at a distance d above the magnet, we only need to calculate the B z component to be able to evaluate the cut flux. The shape of the string is assumed to be a sine and is described by the following equation [START_REF] Wang | Tubular modular permanent-magnet machines equiped with quasi-halbach magnetized magnets-part1:magnetic field distribution, emf, and thrust force[END_REF]:

y(t, x) = Y I cos(2πf t) sin( πx L ) ( 15
)
where L is the length of the string, f is the frequency of the vibration and Y I is the initial amplitude of the excitation of the string . When we consider that the string is moving in a plane parallel to the xz plane, which is perpendicular to the upper face of the magnet, we only need to calculate the B y component to be able to evaluate the cut flux. The string is initially at a distance d above the magnet -along the z axis. The position along the y axis can't be y = 0, because the B y component would always be zero. The string has thus to be shifted along the y axis. The shape of the string is assumed to be a sine and is described by the same kind of equation as previously [START_REF] Wang | Analytical prediction of the incremental inductance of the permanent magnet synchronous motors[END_REF]:

z(t, x) = Z I cos(2πf t) sin( πx L ) + d (16)
where Z I is the initial amplitude of the excitation of the string. Of course, we consider that Z I remains smaller than d, to avoid the contact between the string and the magnet.

III. ILLUSTRATION

The numerical values we take for our calculations are the characteristics of a Fender Stratocaster guitar with Seymour Duncan single pickups and we consider the neck pickup. The length of the vibrating part of the string, L, is 65cm. The neck pickup is situated at a quarter of the string length from the bridge: this means that the magnet is at a distance of 16.25cm from the bridge extremity of the string. The magnet is circular of diameter 4.5mm, but as demonstrated we consider a square one of same section, so with a 4mm side.

A. Movement in the xy plane 1) Centred string:

We consider here that the initial position of the string is 5mm above the center of the upper magnet face, parallel to the x edges of the magnet. The cut surface considered to calculate the cut flux is the surface delimited in the xy plane by the shape of the string at rest -a straight line-and the sine shape it has when it moves. For small values of the mechanical excitation, the cut flux seems to vary sinusoidally with the time, this means, like the string amplitude. This is true, as long as the string remains in an area where the magnetic flux density is almost constant (Fig. 3). When the amplitude of the string movement increases, the cut flux should increase, but as the string comes into areas where the flux density amplitude decreases, the cut flux doesn't increase as rapidly as before and its shape is deformed, like for a saturation, as shown in Fig. 4. The corresponding phenomenon is observed on the EMF, Fig. 5, as it is the derivative of the former. This implies that the obtained electrical signal is no longer sinusoidal but that a distortion appears. A FFT decomposition of the EMF shows which harmonic frequencies appear when the excitation amplitude varies. We note E n the amplitude of the n th harmonic of the the EMF. We study their relative amplitudes, with regard to the fundamental, expressed in decibels (dB). This allows us to see how much a harmonic frequency is attenuated with regard to the fundamental and to conclude whether it is important or not. Table II shows the values for the EMF. There are only odd rank harmonics. This is because of the symetry of the movement in the symetrical flux density. When the excitation increases, the level of the harmonics increases and so, the number of significant harmonics increases. This corresponds to the increasing distortion.

2) Shifted string: When the initial position of the string is not above the center of the face of the magnet, the flux density is no longer symetrical for the string movement. As a consequence, the EMF shows another type of distortion, that is dissymetrical (Fig. 6). The FFT decomposition shows that even harmonics also appear.

It is interesting to notice that if the fundamental frequency is 110Hz, which means that the note A has been played, the second harmonic frequency is 220Hz and corresponds to the note A, but an octave higher. The third harmonic frequency is 330Hz and the corresponding note is the E, which is the fifth of a A. This shows that added notes appear, that are different from the played one. We must remark here that a pickup is not considered as a linear movement sensor: we don't look for a linear response, but we want to generate harmonics. The question is rather which harmonics and with what amplitude. And although this analysis considers only a sinusoidal vibration of the string, in reality any note played on the guitar will excite many higher string modes, which will have the same harmonic frequencies (neglecting string stiffness) as the EMF distortions treatd here. Hence the importance of being able to calculate each contribution to a frequency.

B. Movement in the xz plane

As said in the introduction, the real movement of the string is elliptical and we study both movements in the xy and in the xz planes. So, we want to study now the movement in a plane parallel to the xz plane. The cut flux is then calculated with the B y component of the flux density. If the string is centred above the magnet and its position along the y axis is y = 0 nothing will happen, as the B y component equals zero. So, we have to consider an initial position of the string shifted along the y axis, of 1mm, for example, which corresponds to the quarter of the magnet's side. We consider an excitation along the z axis, Z I = 6mm. This is largely exaggerated with regard with the real amplitude, but allows to emphazise the phenomenon for a better understanding. The cut surface is then the surface delimited in the xz plane by the straight line of the string at rest and the sine shape of the string when it moves. Fig. 7 shows how the EMF gets distorted in this case and we can see that the distortion is not the same as the ones shown in the preceeding section. The real EMF will contain all the distortions, but the contribution of the ones in the xy plane have a heavier weight in the total signal.

This study only gives a hint of the real EMF. Indeed, during the ellipsoidal movement, the cut flux is not the same for each alternance of the string. And as the frequencies of the modes are slightly different, the string never goes through exactly the same area. The composition of the movements has to be made on the vectors, moduli and phases, so, is quite complicated.

Nevertheless, the interesting thing is that we are able to calculate the harmonics generated for more or less complicated hypotheses.

IV. CONCLUSION

This paper presents a method to study PM pickups for electric guitars. Through the study of the magnetic circuit, we show that the electromotive force in the pickup can be considered as proportional to the magnetic flux cut by the string when it moves. This is interesting, because the cut flux can be calculated analytically, and the analytical approach allows the calculation of magnetic fluxes and of their variations with a precision good enough to study quantitatively the harmonics of the electromotive force, so, of the sound. The paper thus gives a model of the pickup, which is simplified but that can be used to design a pickup on a quantitative basis. Indeed, a pickup is not a linear movement sensor, but is designed to generate harmonics, which are very important from an acoustical point of view, as the richness of a sound is clearly related to them. So, this study is a first step towards the dimensioning of pickups to have a specifically desired sound.

Fig. 1 .

 1 Fig. 1. Pickup geometry: the six magnets and one string. The coil is wound around the six magnets and isn't represented here

Fig. 2 .

 2 Fig. 2. z component of the magnetic flux density for a circular section of 4.5mm diameter along the x axis at a distance d = 5mm along z above the magnet.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Bz component of the flux density above the square section magnet. Altitude:d = 5mm

Fig. 5 .

 5 Fig. 5. Electromotive force for an excitation of the string YI = 10mm along Oy. String altitude:d = 5mm. Magnet position L/4: neck pickup.

Fig. 6 .

 6 Fig. 6. Electromotive force for an excitation of the string YI = 6mm along y, string initially shifted of 3mm . String altitude:d = 5mm. Magnet position L/4: neck pickup.

Fig. 7 .

 7 Fig. 7. Electromotive force for an excitation of the string ZI = 6mm along z. String altitude: d = 5mm. Magnet position L/4: neck pickup.

TABLE II RELATIVE

 II AMPLITUDE IN dB OF THE HARMONICS FOR AN EXCITATION YI = 6mm ALONG y, STRING INITIALLY CENTRED.

	Harmonic rank	2	3	4	5	6
	E n E 1	(dB)	-7 -25 -25 -48 -50

TABLE III RELATIVE

 III AMPLITUDE IN dB OF THE HARMONICS FOR AN EXCITATION YI = 6mm ALONG y, STRING INITIALLY SHIFTED OF 3mm.