Jacky Montmain
email: jacky.montmain@mines-ales.fr

Lamia Berrah
email: lamia.berrah@univ-smb.fr

Vincent Clivillé
email: vincent.cliville@univ-smb.fr

Laurent Foulloy
email: laurent.foulloy@univ-smb.fr

Gilles Mauris
email: gilles.mauris@univ-smb.fr

Gilles Mauris How

How to handle the Decision-Maker's awareness of industrial context in his objectives declaration?

Keywords: industrial context, objectives declaration, Decision-Maker Awareness, Awareness Unit Cube, objectives re-declaration. I

This study investigates some new approaches that concern the handling of human considerations in the decisionaiding field. We particularly look for the industrial context change and the awareness of the Decision-Maker with regards to such a change, in his decisions. Beyond the Routine regular situation, three situations are thus considered, namely the Control, the Emergency Crisis and the Loss of Control. A proposition of definition and use of the Decision-Maker awareness is applied to the case of re-declaring initial objectives. Awareness is defined through three parameters which are the Belief, the Level and the Graduality, which make up to the socalled Awareness Unit Cube. This is assumed to be evolving with regards to the Decision-Maker perception evolution. According to the evolving values of these parameters on the one hand, and to the characterisation of the occurred situation on the other hand, mathematical adjustments of the considered objectives values are proposed. Some illustrations of the proposal are extracted from a case study submitted by a steel manufacturer.

INTRODUCTION

Management involves a set of operational, financial and human actions, with regards to the industrial objectives achievement leading thus to making decisions according to these dimensions [START_REF] Griffin | Management[END_REF][START_REF] Weihrich | Management: A Global Perspective[END_REF]. For instance, management decisions can be concerned with aspects such as the objectives declaration, the launch of improvement actions, the validation of the declared objectives, the workers' rewards… Incidentally, one parameter that widely impacts on such decisions is the situation which is encountered by the system that deals with the objective achievement. Namely, in the industrial context, external events as well as internal events may occur and may more or less favour the achievement of the declared objectives. One can imagine the financial crisis, on a world scale, or more particularly some kinds of breakdowns and other unforeseen events related to the customer demands variability, the competitive aspect of the market… dealing thus with situations other than the classical regular situation, which is associated with "normal" business conditions... By "normal" condition, we mean that the operational system is under a routine functioning [START_REF] Montmain | Operators aids: automation and supervision. 9th symposium on information control in manufacturing[END_REF]. Such a system is well-controlled, i.e. it supplies the expected production by the correct processing of the resources [START_REF] Ducq | A contribution of system theory to sustainable enterprise interoperability science base[END_REF].

To be more precise, the Routine situation is implicitly the reference for declaring the objectives, launching the actions, and, more generally, for all the structured decisions that are made. Hence, the risk becomes related to the focalisation on the Routine conditions, forgetting in this sense some kind of awareness of the real encountered conditions. In accordance with the topic of our paper, let us refer to the concept of Situation Awareness (SA), as it has been developed by the Human Factors community in the mid-nineties [START_REF] Endsley | Measurement of situation awareness in dynamic systems[END_REF][START_REF] Endsley | Toward a theory of situation awareness in dynamic systems[END_REF]. SA involves becoming aware (i.e. decision-makers and other responsible actors), of all that is happening in the vicinity in order to understand how information, events and one's own actions will alter the objectives achievement, both immediately and in the near future. With a lack, an inadequacy or a weakness, SA has been identified as one of the primary factors in accidents attributed to human error [START_REF] Hartel | Defining aircrew coordination: searching mishaps for meaning[END_REF]. According to us, the SA generally varies from not-at-all to total, with regards to the encountered situation on the one hand, the concerned Decision-Maker (DM), as well as the evolution of his perception and knowledge about the situation, on the other hand.

Hence, we choose to focus in this study on one particular aspect of management which is the objective declaration. We look for a type of handling of the soundness of such a declaration with regards to industrial encountered situations on the one hand and to the SA of the Decision-Maker (DMSA) on the other hand. To be more precise, the objective declaration is generally made by a DM, who has the authority and the control of the considered business unit or physical system that is concerned with the achievement of the objective. Moreover, knowing that an objective is declared by a target value and a temporal horizon [START_REF] Ducq | A contribution of system theory to sustainable enterprise interoperability science base[END_REF][START_REF] Foulloy | Fuzzy interpretation of performance scorecards for decision-making in the industrial context[END_REF][START_REF] Berrah | Towards a unified descriptive framework for industrial objective declaration and performance measurement[END_REF], both these parameters of the objective have to be, from a technical point of view, realistic with regards to the achievement. They also have to, from a strategic point of view, make the performance perennial. Hence, the DM becomes responsible for what can be called the soundness of the objective declaration. However, declaring accurate and relevant context-dependent objectives is thornier. A reliable knowledge and a strong SA thus allow the DM to first be able to identify the right situation and then deduce the right objectives values from this diagnosis.

Hence, the achievement of the initial declared objectives becomes more or less relevant/certain, depending on the current situation and leads thus to either the impossibility of performing, or bad performances. The purpose of this work is to adapt objectives declaration to the real situation that is encountered by the company or the physical considered system. Beyond the Routine situation, three situations are then considered: the Control situation, the Emergency Crisis situation and the Loss of Control situation. We suggest more particularly that such adaptation depends on the industrial situation, and that the characterisation of the industrial situation is related to the DMSA. That is why we look to handle the DMSA in the objective re-declaration mechanism, assuming in this sense that the DM is more or less aware of what is happening in the current situation. We imagine that the DM becomes more or less able to give relevant pieces of information about this situation, and, consequently, is more or less able to re-declare relevant values for the objectives. Moreover, we assume that the DMSA may evolve, and generally increase, by the occurrence of internal or external events that may impact on the DM's way of thinking. The developed idea here is then the modelling of such awareness by a triplet that involves the Belief of the DM on what he gives in terms of knowledge, his ability to characterise the Level of the situation as well as its Graduality. We assume moreover that the DMSA is something which needs to be captured, and types of decision-aiding systems can be useful in this sense.

Beyond the quantification of the DMSA but also depending on it, the situation is described through parameters such as, at minima, its name, and possibly the quantification of its level, the grade of membership or characterisation associated with it... The objective is then re-declared according to the expressed DMSA and the encountered situation characterisation. Mathematical translations and thresholds are thus proposed in order to obtain the re-declared values.

This paper is thus organised as follows. After a presentation of both the characteristics of the industrial context and its four typical situations, and the SA concept in Section 2, we introduce, in Section 3, a parametric way of quantifying the DMSA concept as well as a methodology for capturing such a SA. After a brief recall of the objective declaration principle, some examples of objectives re-declaration are then presented, on the basis of numerical initial declared objectives. We give after, in Section 4, illustrations that are extracted from a steel manufacturer. Concluding remarks will thus emphasise the prospects of this study.

II. SOME ELEMENTS CONCERNING THE INDUSTRIAL CONTEXT SITUATIONS AND THE DECISION-MAKER AWARENESS

A. The industrial context situations

According to the industrial management literature, four situations are considered as identifying the context of industrial systems (see Fig. 1) [START_REF] Montmain | Operators aids: automation and supervision. 9th symposium on information control in manufacturing[END_REF]. These situations are defined according to, on the one hand, the ability of the operational system to normally operate and, on the other hand, to a kind of control degree, by the DMs, of the considered systems.

By considering Fig. 1, one can see that the reciprocal function pairs (Control versus Emergency Crisis) and (Routine versus Loss of Control) are generalised, for example as respectively "incentives to action" and "incentives to inaction", with regards to the considered objectives. Control consists of acting to reach a desired state or event, namely a revisited Fig. 1. Industrial management context initial objective. Emergency Crisis involves opposing an undesired state or event. Routine and Loss of Control both consist of doing nothing in particular, because action is either unnecessary or impossible. The Emergency Crisis situation may be considered as the situation which is thornier to be correctly perceived because of its un-foreseeability (unlisted events, data transmission errors, events misunderstanding...) and that the reaction is inappropriate. Loss of Control reflects a situation in which it is impossible to implement coherent action with the objectives.

In Control situations that arise when unexpected events do appear, actions and decisions imply sophisticated strategies to face threatened incidents and unforeseeable consequences. Building a representation of such situations and planning actions are thorny tasks because of the uncertain environment and the unpredictability of phenomena that cannot be wholly modelled. In these situations, the organisation does the best it can to avoid threatened incidents and progressively improve the business unit functioning.

Emergency Crisis situations appear when threatened incidents occur or at least are pending. They can no longer be avoided. DMs have much less flexibility as in previous situations because the activity is already suffering damageable consequences. Decisions then aim at reducing consequences of threatened incidents on the activity, staff and environment.

Decision and action can still be inhibited in Loss of Control situations when DMs are faced with unthinkable situations. Such situations had never been envisaged in the framework of the company activity. Note that unpredictability can be caused by inadequate action plans with regard to the situation, by ineffective action plans, etc. In such cases, surprise appears as the crisis factor. Finally, Loss of Control situations can be the result of the deterioration of Emergency Crisis situations, when resources cannot prevent the disaster any longer or disturbances become recurrent. The system is destabilised and uncertainty is maximal.

Let us note that in this study only a few generalities are given, and that it is for introducing the four situations. For further information, one can consider [START_REF] Montmain | Operators aids: automation and supervision. 9th symposium on information control in manufacturing[END_REF][START_REF] Imoussaten | Modélisation et pilotage de la phase de délibération dans une décision collective : vers le management d'activités à risques[END_REF][START_REF] Sanchez | Spécification et Implémentation du Système d'Aide à la Décision Multicritère pour la Maintenance Préventive et la Gestion du Patrimoine de la Société d'Autoroute ESCOTA : le projet SINERGIE[END_REF].

B. The Situation Awareness SA

SA-related cognitive mechanisms (perception, comprehension, projection into the future), decision-making and performance assessment are all embedded into a cognitive feedback loop (see Fig. 2) [START_REF] Endsley | Measurement of situation awareness in dynamic systems[END_REF][START_REF] Endsley | Toward a theory of situation awareness in dynamic systems[END_REF].

The perceptual level 1 involves the detection, recognition and identification of elements that define a specific situation. Perceptual SA relies on available sensory information (e.g. from sensors in control situations, economic indicators in management situations) as well as the operator's/DM's prior knowledge (e.g. object patterns/diagrams activated by memory) in order to identify individual situation elements and object groups. Comprehension level 2 reflects an understanding of the current state of affairs and involves Fig. 2. Situation Awareness and Decision-Making [START_REF] Endsley | Toward a theory of situation awareness in dynamic systems[END_REF].

making inferences about activities in the current situation. As such, the comprehension level maps perception products into object functions. Finally, projection level 3 consists of interpretations concerning the trajectory of the situation based on the output of Comprehension SA and prior knowledge [START_REF] Curiel | Developing Situation Awareness Metrics in a Synthetic Battlespace Environment[END_REF].

Note that, for the sake of simplicity, we are looking here to handle each SA level (Perception, Comprehension, Projection) by one parameter, by looking for some kind of elicitation and quantification of it as it is described in the next section.

III. RE-DECLARING THE OBJECTIVES

A. Some indications with regards to the DMSA capture DMSA capture is one of the key points of our approach. As has been seen above, the DMSA concept handles the awareness of a DM with regards to an industrial encountered situation. Hence, we choose to consider the DMSA using three parameters. Even if these parameters are associated with the DM, the first one is related to his perception of the situation (SA Level 1) while the two others are related to his ability to characterise the occurred situation (Comprehension SA Level 2) and its possible evolutions (Projection SA Level 3) (see Fig. 2). Namely, we have:  the Belief, which is the ability of the DM to evaluate the truth of his knowledge about the situation;  the Level, which is the ability of the DM to quantify a level in a given situation;  the Graduality, which is the ability of the DM to qualify the situation with a membership grade.

Note that the Graduality parameter intervenes only when the system is changing and the DM is facing a transition between two situations. When only one situation is identified, the Graduality is naturally not taken into consideration.

Representing each criterion on a [0, 1] scale leads to the Awareness Unit Cube (AUC) represented in Fig. 3.

Graduality

Level Belief

Fig. 3. The Awareness Unit Cube

Let us represent the vertices as BLG coordinates, i.e as respectively Belief, Level, Graduality vectors. Let us note that vectors are expressed as BLG coordinates while the DM decided to represent the Belief on the conventional z-axis. By doing so they wanted to emphasize the importance of belief increase as an elevation.

Vertices such that the Belief is equal to 0, i.e. [0 0 0], [0 1 0], [0 0 1] and [0 1 1], correspond to DM whose situation analysis can hardly be trusted. Vertices such that the Belief is equal to 1 correspond to DM who is confident in the qualification and/or quantification of the situation. The Belief parameter becomes the most impacting one. A high level of Belief is thus a requirement to trust information on Level or Graduality.

We propose to differentiate some typical DMSA with several *. The greater number of *, the better in terms of situation characterisation.

 The vertex [1 0 0] is associated to 1*-DM. DM has a total Belief with regards to the situation he is facing, namely: Routine, Control, Emergency Crisis, Loss of Control ( §II.B).

 The vertex [1 1 0] is associated to 2*-DM. The DM is able to qualify not only the situation but also its Level. The quantification can either be numerical, e.g. Emergency Crisis with a Level 5 which means that the DM has a total ability to give the Emergency Crisis situation Level, and gives, for instance, the "5" value to it. The quantification can also be linguistic, e.g. a Severe Emerging Crisis, which means that the DM has a total ability to give the Emergency Crisis situation Level, and gives, for instance, the "Severe" value to it.

 The vertex [1 0 1] is associated to 3*-DM. The DM is, moreover, able to give the Graduality of the encountered situations. For instance, the DM is able to consider that the situation can be Routine with a certain grade of membership, e.g. 0.6/Routine -that is Routine with a grade of membership equal to 0.6 -but also Control or Emergency Crises to different grades of membership.

 The vertex [1 1 1] is associated to 4*-DM. In this case, the DM has a total SA, based on his Belief as, e.g., "I believe that the situation is 0.6/Emergency Crisis. In that case, I consider the Emergency Crisis as Very Severe. But the situation could also be 0.2/Loss of Control. And in that case, I consider that we are faced with a Medium Loss of Control situation."

Let us note that this typology corresponds to crisp cases that are linked with the vertices of the UAC (see Fig. 3). In the general case, one can imagine that the DMSA is a point in the UAC. This point in then defined by its BLG coordinates. Moreover, one can imagine also that the DMSA can evolve in time, according to the increasing perception level of the DM. In this sense, the DMSA becomes a non-static notion. Many parameters can make the SA of the DM change, such as the occurrence of some events, the analysis of what is happening, some oriented discussions with experts… Our aim is to make the DM increase his SA. Let us recall that with a better SA, the DM will better characterise the encountered situation. He will then be able to re-declare the objectives in a relevant manner. We propose here to proceed to a questioning of the DM, in such a way that answering our questions makes him become more and more aware of what the situation is. We assume, as a starting point for the DMSA capture, the worst case, namely the case where the DM has no SA. This starting point is the origin of the UCA, i.e. the BLG vector is [0 0 0]. In this logic, the ideal ending point is when the DM becomes a 4*-DM, namely with a BLG vector [1 1 1], having thus a total SA. The time-dependent trajectory in the UCA between these two points represents the DMSA evolution.

In our vision, this DMSA capture process relies on several structured steps.

 The first one consists of working on the Belief axis. Indeed, the DM has to be able to have a strong Belief in his characterisation of the occurred situation.

 When this step is achieved, the DM is close to a 1*-DM. Then, the Level axis can be considered such that the DM becomes aware of the Level of the encountered situation.

 When this step is achieved, the DM is close to a 2*-DM. Finally, the Graduality axis can be considered to drive the DM BLG vector towards the vertex [1 1 1] and makes him a 4*DM.

Note that we assume that the SA capture is made independently from the encountered situation. The questions asked to the DM are oriented towards some generic parameters that may allow him to be aware of what is happening.

B. The objective re-declaration mechanism

Let us recall that an objective is considered through a target value to achieve at a given time [START_REF] Foulloy | Fuzzy interpretation of performance scorecards for decision-making in the industrial context[END_REF][START_REF] Berrah | Towards a unified descriptive framework for industrial objective declaration and performance measurement[END_REF][START_REF] Ducq | Coherence analysis methods for production systems by performance aggregation[END_REF]. The value as well as the temporal horizon is declared with regards to a considered variable [START_REF] Berrah | Towards a unified descriptive framework for industrial objective declaration and performance measurement[END_REF]. Such a variable is generally associated with the actions to launch in order to achieve the objective. Previous works have dealt with the definition of a unified framework for handling several kinds of declarations; such as numerical, linguistic, crisp or imprecise… [START_REF] Berrah | Towards a unified descriptive framework for industrial objective declaration and performance measurement[END_REF][START_REF] Berrah | Global vision and performance indicators for an industrial improvement approach[END_REF]. Moreover, the objective declaration concerns the step where the DM gives this value. According to our classification of the situations, the objective declaration corresponds to the Routine situation and, to the best of our knowledge, very few works have been published concerning industrial objectives declaration in other situations that the Routine situation ( §II.A). Even if these situations are less frequent and could be separately addressed, our proposal is to consider them in the unified framework of the objective redeclaration, leading thus to analyse, in further works, the transitions between these situations.

Re-declaring the objective consists thus of changing the objective value according to the DMSA first and, secondly, the characterisation of the encountered situation. The basic ideas are to obtain fewer and fewer challenging objectives as the severity of the situation increases. Obviously, the more aware the DM is, the sounder the re-declaration of the objective. For the sake of simplicity, we consider an objective such as the greater its value, the greater the DM satisfaction with regards to its achievement. Moreover, we choose to focus here only on the value changes, leading thus to maintain as constant the corresponding temporal horizon. For instance, whatever a characterised DMSA, we can imagine the following rules.  In the Routine situation, the objective value is not modified.

 In the Control situation, the objective value is reduced.  In the Emergency Crisis situation, the objective value is reduced and some imprecision in its declaration is possible. Thus the aim is to stop the drift that causes the disturbed situation and recover as soon as possible a control situation (see figure 1). The emergency and damage characteristics of the crisis situation prevent managers to fit precise objective because they are faced with a management situation they are not used to run. It corresponds to knowledge and responsibilities outside of their traditional purview.  In the Loss of Control situation, the objective value is reduced with a high imprecision degree, leading thus to a translation of the value into a satisfaction (i.e. performance) expression function.

These ideas are emphasised in Fig. 4. We consider the case of a numerical objective value, where z y x w , , , are real values. Besides, m is the variable related to the measurement which is associated with the objective achievement [START_REF] Berrah | Global vision and performance indicators for an industrial improvement approach[END_REF]. Simple bounded values are given for the three first cases, while a function is proposed with regards to the fourth case.

Depending on the DMSA, the general previous principles can be refined. As an illustration, let us consider a refinement scenario in the case of the Control situation, assuming once again a numerical objective value. In the same previous assumptions with regards to the objective declaration and satisfaction, the general principle is to re-declare the objective value such that the new objective value is lower than the Routine objective value. Possible refinements are given in the following.

 1*-DM is able to provide re-declaration such as "reduction of the objective by 5%" which leads to y = 0.95z.

 2*-DM provides the same re-declaration as 1*-DM, but also defines a computation rule in order to take into account the Level in the Control situation, by considering, for instance, a linear function.

 3*-DM provides the same re-declaration as 1*-DM, but also defines a computation rule to take into account the Graduality of the situation, by using, for instance, a weighted mean.

 4*-DM provides the same re-declaration as 2*-DM and 3*-DM.

Fig. 4. The re-declaration values for a numerical objective

To summarise, one can say that in the general case, the re-declaration of the objective value is a more or less complex function which depends on:  the encountered situation;  the DMSA, i.e. the BLG vector;

 the characterisation of the encountered situation, by pieces of information that are provided by the DM.

Let us specify once again that, in this framework, the DM characterises the situation, according to his SA. The DM may hence give pieces of information such as the quantification of the Level, the grades of membership, the DM Beliefs. Now, before describing the steel manufacturer case study, let us introduce some necessary notations, according to our previous developed framework [START_REF] Berrah | Towards a unified descriptive framework for industrial objective declaration and performance measurement[END_REF]. When the DMSA is totally lacking, no information is assumed to be available concerning the encountered situation. In the same manner, other parameters like the grades of membership can be introduced for the re-declaration by a 3*-DM or a 4*-DM.

IV. CASE STUDY

A. Presentation

Let us illustrate our proposition by considering a steel manufacturer which produces black coils and sheets with a capacity of four million tons per year. The means of production of the plant are continuous processes, including blast furnaces, oxygen converters, continuous casters, one hot strip mill and six finishing lines. The main customers of the company are the automotive industry, the shipyards, the heavy-metal... Incidentally, the steel production is strongly sensitive to the demand, and the demand depends on the classical industry cycles of ten years [START_REF] Tan | Identification and analysis of industry cycles[END_REF]. Usually, the company is in work, in a continuous manner, for about 150 hours per week, according to the work load, as is planned by the MRP. Four types of products are manufactured, respectively black coil, pickled and oiled coil and sheet, cut to length sheet, slit coils. Each kind of product is proposed in several thicknesses, widths and finishings (surface aspects). The daily production is about 10000 tons. The production is defined with regards to the long term forecasts, and inventories allow the DMs to face the possible order variations. So the manufacture looks for economic lot sizing that involves both important work in progress and inventories. This way of managing production was satisfying in a Routine context, where the demand variations were not very fast and the competition conditions with the other steel producers were relatively well known and controlled. However, during these last ten years, many events have occurred, disturbing the factory's activity. Namely, a fusion with another company has been effected. Moreover, the recession of 2008 has impacted the performance of the company and, on a tactical level, internal reengineering has modified the production plant functioning. Adding to these points, the manufacturer has observed the competition of new steel actors as well as demand evolution (new finishings for instance), geographical location of the main customers (more and more of them are located in Asia or South America), and, on the operational level, incidents related to the use of some obsolete means of production.

The consequences of these diversified events were a chaotic evolution of the inventories. There were both numerous stock outs with regards to certain references, on the one hand, and very high levels of inventory on other references, on the other hand.

In this context, discussions have been undertaken with the plant head manager -the considered DM in this case -in order to help him to take into account the particularities of the encountered situation and the involved changes, for making his decision, especially for declaring and re-declaring his manufacturing objectives. To be more precise, the submitted problem was related to the adjustment of the lot sizing according to the demand and the finish line flexibility.

Taking into account such particularities and changes require the capture SA of the DM. That is what we have begun to do, according to pre-established awareness capture questionnaires.

B. The DMSA capture

The plant activity has known many situations changes, with regards to the long temporal horizon of ten years on the one hand and to important number of events that have been encountered, on the other hand. Thus, we have distinguished, during our study, several periods in the lead discussions and questionnaires. These periods are coherent with the evolution of DMSA and its capture. The starting point was corresponding to no SA for the DM. The ending point

Objective = [w, x] < y < x Routine Control Emergency Loss of Control Objective = z Objective = y < z z y [w, x]
Satisfaction function m correspond to a SA that allows the DM to characterise not only the situation but also its Level. The DM became then almost a 2*-DM, with a BLG vector close to [1 1 0].

1) Period 1 -the DM has no SA

Before the beginning of this work, the DM was lacking from SA. In that sense, his BLG vector was equal to [0 0 0].

2) Period 2 -the DM becomes a 1*-DM

Based on his long experience in the steel industry, the DM was able to easily answer the questionnaire, highlighting thus his SA. The DM was able to qualify and characterise the plant situation, using the four typical situations scale, namely the Routine, the Control, the Emergency Crisis and the Loss of Control ( §II.A). It was easily agreed that the Routine corresponds to the conventional regular situation that is associated with the first period. For instance, the Emergency Crisis qualifies the situation at the end of 2008 when the demand of the automotive sector fell by more than 50%, while the Control situation was related to the unusual problem of quality and maintenance which significantly worsened the line productivity. The DMSA was characterized by a BLG vector very close to [1 0 0] and the DM was rated as a 1*-DM.

3) Period 3 -The DM becomes a 2*-DM

After some weeks of discussions, the DMSA continued to increase. The DM was able to provide more explicit information concerning the situation and became aware of the situation Level. He was able to qualify this level by a relevant additional word among a set of possible words. For instance, the Control situation has been qualified as Partial, Intermediary or Total. Thus the DMSA was characterised by a vector close to [1 1 0] and the DM was tending towards a 2*-DM.

For the sake of simplicity, let us note that the last dimension of the SA, i.e. the Graduality between typical situations, has not been considered in this study.

For each period, the objective re-declaration was analysed with the DM, according to the approach which is summarised in Fig. 4.

C. Objective re-declaration by a DM with no SA

No re-declaration is required in this case. The lot sizing objective was 2 days of production with 10 000 tons/day for the finish line.

Formally, the declaration is 20000

) (  LotSize o T.

D. Objective re-declaration by a 1*-DM a) Routine

In the Routine situation no objective re-declaration is needed and the objective defined when the DM was with no SA is kept. Formally, the situation is introduced in the objective leading to

c) Emergency Crisis

In this situation, the DM proposed to reduce the Routine objective in order to permit a better reaction to the demand as well as a better correction of the line deviation due to major breakdowns or quality problems.

               b m if b a m if a b a m a m if m rol LossOfCont LotSize o 1 , 0 ) , , ( and )
, ( ] , [ risis EmergencyC LotSize o b a  1 .
The DM considered that the sustainability of the company could not be guaranteed if the main customer requirement could not be satisfied which gave the lower bound a. It was also considered that, in this situation, delivering the three main customers was, at least, what could be done to be satisfied, thus proving the upper bound b. Thus, the values a = 4000 T and b = 7000 T have been chosen. 1

Let

 

b a U ,  and   d c V
,  E. Objective re-declaration by a 2*-DM After the first formalisation a new step was undertaken while the DMSA was increasing with his ability to quantify the Level in a situation. Refinements in the re-declaration were proposed. They are described in the next sub-sections.

a) Control

The DM was able to define a Level described by one the following words: Partial, Intermediate, Total. The DM was also able to consider that the quantification of the Level has a strong relation with the quantification of the objective redeclaration. The Level was formally introduced in the objective re-declaration as follows:

        3 2 1 ) , , ( ) , , ( ) , ,
                      ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ) , , ( ] , [ ) , , ( ] , [ ) , , ( ] , [ ) , ,

c) Loss of Control

No Level was introduced by the DM in this situation, because of the perceived uncertainty with the eventual given information.

F. Synthesis

The importance of the SA was emphasised in the case study. For the sake of comprehension, a synthesis is provided in Fig. 5 This study has dealt with one major aspect in the decisionmaking field, which is related, in a general manner, to the behaviour of the DM when he has to make decisions. To be more precise, we have focused in this work on the DM "awareness" concept and made the assumption, according to the human factor literature, that the awareness notion is the key point for having a relevant and sound decision-making process.

Preliminary assumptions and limits have been expressed here, with regards to the kind of decision, on the one hand, and the context of decision-making, on the other hand. More particularly, the considered decision was concerned with objective declaration and adjustment. And the context of decision-making was handled by four classical situations that are the Routine, the Control, the Emergency Crisis and the Loss of Control. Objectives are generally declared with regards to normal Routine situation. Thus the aim of this study was then to look for the soundness of the objectives declaration whatever the occurred situation.

The developed idea is simple. With regards to the considered situation, the objective value is re-declared, the redeclared value being adjusted to the real encountered situation and conditions. Namely, the DM characterises the encountered situation and according to this characterisation, re-declares the objective. Mathematical operations can be used in this sense. However, if the idea is simple and almost classic, the novelty of the work is in the establishment of a link between the DM situation awareness -DMSA -and the way that he characterises such a situation. Some definitions and rules have been given in order to formalise this link. The AUC -Awareness Unit Cube -has been thus introduced, as a basis for the DMSA evolvement capture.

Fig. 5 .

 5 Fig. 5. Synthesis of the case study

  In this situation, the DM proposed to decrease the Routine objective value in order to avoid the saturation point where the production will get worse. As explained before ( §III.B), the

							b) Control
						corrected	objective	should	be	such	that
						( o	, LotSize	) Control		( o	, LotSize	) Routine	. In our considered
						case, it leads to a reduction of the production by 2000 T, i.e.
						( o	, LotSize	) Routine		18000	T.
	o (	LotSize ,	Routine )		T 20000	.

  The correction became:In this situation it is difficult for the DM to give pieces of information about the correction of the initial declared value. It is a very unusual situation which is, by nature, unpredictable. Finally, the DM agreed with our proposition to consider the objective as a satisfaction function. Let p be a value in [0, 1] reflecting the DM satisfaction.

	( o	, LotSize	) risis EmergencyC		  , x w	with
	w		x		( o	LotSize	,	Control	)	.
	After discussion, it was proposed to consider the interval     16000 8000, ,  x w .
				d) Loss of Control
											p		0	means a total
	dissatisfaction and		p		1	a total satisfaction. It was proposed
	to define two values, a and b, respectively corresponding to
	objectives where	p		1	and	p		0	. Between these values, the
	DM satisfaction is linearly varying. It led to:

  (The meanings of each term qualifying the Level have been discussed. The DMSA made it possible to distinguish between the objective re-declaration without the Level awareness, i.e.

		o	LotSize	Control	Partial	y
		o	LotSize	Control	te Intermedia	y
		o	LotSize	Control	Total	y
	( o	, LotSize	) Control	and the re-declaration when the Level is
	considered as Partial. Finally, it was easily agreed to satisfy
	the following order relation:
		3 y		y	2		( o	, LotSize	) Control		1 y		( o	) LotSize
		with	1  y	19000 T,		y	2 	17500 T and	3  y	16000 T.
		b) Emergency Crisis
		In this case, the DM was able to distinguish four intensity
	Levels: Light, Declared, Severe, Major. It was proposed to
	refine the re-declaration by a partition of the interval provided
	by the 1*-DM, i.e.	( o	LotSize	,	EmergencyC	risis	)		[	, w	x	]	[16].
	Formally it was proposed to let the re-declaration satisfy the
	following constraints:

  for the different DSMAs.

									DM with no SA
	( o	) LotSize		T 20000
											1*-DM
	( o	, LotSize	) Routine		T 20000
	( o	, LotSize	) Control		T 18000
	( o	, LotSize		) risis EmergencyC		T] 16000 , T 8000 [
											 0	if	m		T 4000
	o	(	, LotSize	, rol LossOfCont	) m		 1    w if   m z z m	 if	 T 7000  4000 T , 7000 T m	
											2*-DM
	( o	, LotSize	) Routine		T 20000
	( o	, LotSize	, Control		) Partial		T 19000
	( o	, LotSize	, Control	) te Intermedia		T 17500
	( o	, LotSize	, Control	) Total		T 16000
	( o	, LotSize		, risis EmergencyC	) Light		T] 16000 , T 14000 [
	( o	, LotSize		, risis EmergencyC	) Declared		T] 14000 , T 12000 [
	( o	, LotSize		, risis EmergencyC	) Severe		T] 12000 , T 10000 [
	( o	, LotSize		, risis EmergencyC	) Major		T] 10000 , T 8000 [

This article represents a first step in the investigation of the handling of the DM behaviour. Many axes can be envisaged in order to enrich these thoughts. Transitions between situations have to be considered as well as the handling of the sincerity and relevance of the DM when he estimates his SA. Qualitative and imprecise objectives can be imagined and redeclared using fuzzy mechanisms… Further developments are in progress in this sense.


Jianxiao Yang

Benoit Geller
email: geller@ensta.fr

Cedric Herzet
email: cedric.herzet@irisa.fr

Jean-Marc Brossier
email: jean-marc.brossier@gipsa-lab.inpg.fr

Smoothing PLLs for QAM Dynamical Phase Estimation

Keywords: Dynamical Phase Estimation, Phase-Locked Loop (PLL), QAM, Smoothing Algorithm. I. 0BINTRODUCTION

published or not. The documents may come   L'archive ouverte pluridisciplinaire

proposed a CA belief-propagation (BP) algorithm for the BPSK dynamical phase estimation but the computation complexity of the proposed BP algorithm is rather high. This paper is concerned with a very simple synchronizing scheme for any QAM modulated signal which is able to operate near the off-line time-varying phase bounds. To our knowledge, it was first proposed without any justification and without any performance evaluation in X [START_REF] Brossier | Procédé d'estimation de la phase dans un système de communication numérique et boucle à verrouillage de phase[END_REF]X-X [START_REF] Geller | Contribution à l'étude des systèmes de communications numériques[END_REF]X; contrarily to X [START_REF] Noels | Performance Analysis of ML-Based Feedback Carrier Phase Synchronizers for Coded Signals[END_REF]XX, [START_REF] Noels | Effectiveness Study of Code-Aided and Non-Code-Aided ML-Based Feedback Phase Synchronizers[END_REF]X, it takes advantage of averaging two phase trajectories provided by two PLLs, so that this S-PLL algorithm is able to have such a near off-line Cramér-Rao bound performance.

The rest of the paper is organized as follows. In section XIIX, we give the system model. In section XIIIX, we derive from the MAP estimation theory the proposed algorithm where the smoothing effect is achieved through two PLLs working in opposite time directions. Finally in section XIVX, we present the simulation results before giving some conclusions.

II. 1BSYSTEM MODEL

We consider the transmission of a complex-valued QAM modulated sequence ), the residual phase distortion and the zero mean complex-valued circular Gaussian noise with known variance 2 n  . We suppose that the system operates in a non-data aided (NDA) mode. Hence, the conditional probability based on the known phase k  is:

        M 2 2 2 1 | | , e x p , , M k k k k k k k k k k k k c c n n y p y p y c p c c y                        S , (2)
where

    2 2 2 R e , , e x p k j k k k k k k n c y c e cy             .
In practice, due to the rapid variant channel and the imperfections of the functional blocks before the phase Smoothing PLLs for QAM Dynamical Phase Estimation estimation, the residual phase distortion can be modeled efficiently by a Brownian motion. The corresponding phase model is:

1 , k k k w        (3)
where k  is the unknown phase offset at time k ,  is the unknown constant frequency offset (linear drift), k w is a realvalued white Gaussian noise with zero mean and variance 2 w  . This model is commonly used X [START_REF] Amblard | Phase tracking: what do we gain from optimality? Particle filtering versus phase-locked loops[END_REF]X-X [START_REF]Digital Video Broadcasting (DVB) User guidelines for the second generation system for Broadcasting, Interactive Services, News Gathering and other broadband satellite application (DVB-S2)[END_REF] X in order to describe the behavior of practical oscillators for which the frequency is randomly perturbed. Based on (3), the corresponding conditional probability can be expressed as:

    2 1 1 2 1 | , e x p . 2 2 kk kk w w p                  (4)

III. 2BRATIONALE FOR A FORWARD / BACKWARD APPROACH BY MAP ESTIMATION THEORY

In the MAP estimation approach, one classically chooses θ to maximize the posterior pdf X [START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF]:

      ˆa rg m a x | a rg m a x ln | ln p p p       θ θ θ θ yy θ θ (5)
Based on the model described by ( 2) and ( 4), the joint pdf of the observations and the parameters can be written as:

                      1M M 1 1 1 1 2 1 2 1 1 1 1 1 2 2 2 2 1 2 2 2 , , | | | | | , 1 1 e x p , , 2 exp , , 2 k k k k k k c n n w k k k k k k c k n w p p p p p y p y p y p c y M y cy                                                                                          S S y θ y θ θ      (
:           1M 1M 1 1 1 1 21 1 2 1 1 1 1 1 ,, ln , , ln ,, c w
c cy pp cy                                  S S y θ ,         M M 11 2 ,, ln , , 2 ,, k
k k k k c k k k k k k k k w c cy p cy                        S S y θ for 21 k     ,         M M 1 2 ,, ln , , , ,, c w c cy p cy                                    S S
y θ [START_REF] Yang | Near-optimum Low-Complexity Smoothing Loops for Dynamical Phase Estimation[END_REF] where

      2 ,, 2 , , Im k k k k j k k k k k kn cy c y y c e         
. Looking for the MAP estimator implies that one must set the first derivative

  ln , , k p        y θ
equal to 0, so that:

      1M 1M 1 1 1 1 22 1 12 1 1 1 1 ,, ln , ,, c ww c cy p cy                         S S         MM MM 22 11 , , , , 1 2 2 , , 2 ,
,

kk kk k k k k k k cc kk ww k k k k k k k k k cc c y c y c y c y                                                          SS SS for 21 k     ,     M M 2 1 ,, . ,, c w c cy cy                              S S (8) Rewrite     M M ,, ,, k k k k k c k k k k c cy cy           S S
as below in order to get the physical meaning of the term: Thus the term

                M M M M M M 2 2 2 2 2 2 2 R e 2 Im ,, e x p ,, 2 R e e x p |, 2 Im . |, k k k k k k k k k k j j k k k k k k k k c n n c k j k k k k k k c c n k k k j kk c n k k k c c y

c e y c e cy cy c y c e p y c y c e p y c

                                                S S
    M M ,, ,, k k k k k c k k k k c cy cy           S S
can be interpreted as the soft decision phase detector output. ( 9) can then be written as:

                1 1M M 2 1 2 1 2 1 1 1 1 1 2 1 2 1 2 ln 2 P r | , Im , 1 fo r 2 1, 2 2 P r | , Im , j w w c n F B k k k j w c n p c y y c e k c y y c e                                                              S S       (10)
where

            M M 2 1 2 2 1 2 P r | , Im , P r | , Im . k k k k F j w k k k k k k k c n B j w k k k k k k k c n c y y c e c y y c e                                S S   (11) Furthermore in (11),   F k  and   B k 
can be regarded as softdecision based first-order PLL outputs which are respectively updated in the increasing (Forward) and decreasing (Backward) time directions. The physical meaning of (9) can thus be summarized as following; assuming that we do not have any a priori information about the initial phase

  :                         1 M 1 M 2 11 2 2 11 2 ˆˆP r | , Im , ˆˆP r | , Im . F k k B k k F F F j w k k k k k k k c n B B B j w k k k k k k k c n c y y c e c y y c e                                  S S   (12)
From ( 8) the "Forward / Backward" (F/B) estimator can thus be written as: [START_REF] Zhang | Iterative Carrier Phase Recovery suited for Turbo-Coded systems[END_REF] Note that this structure is similar to that of the Kalman smoother valid for linear Gaussian problems. The name "Forward / Backward" stems from the fact that the off-line phase estimation is just the average of a classical (Forward) phase-locked loop and of a Backward phase-locked loop working in the reverse time direction and that can be initialized at the end of the forward PLL. This process can then be iterated, i.e. the estimation error at the end of the previous backward loop can be further used as the estimation error at the beginning of the next forward recursion, and several forward and backward recursions can sequentially be proceeded. We call this process in the sequel as "multiple forward / backward". Restricted by the paper size, we shall give further analysis of the proposed algorithm at the oral presentation.

                                2 1M 1 M 2 / 1 2 1 1 2 1 1 2 / 2 / 11 2 2 ˆˆˆP r | , Im , 1 ˆˆˆ for 2 1, 2 2 ˆˆˆP r | , Im . B F F B B B j w c n F B F B k k k F B F F j w c n c y y c e k c y y c e                                                            S S
IV. 3BS IMULATION AND DISCUSSION In a practical system, a frame header can be used and one could take advantage of it to get rid of the phase ambiguities. In our simulations, we thus assume that the phase ambiguity problem is solved. We evaluate the MSEs in the centre position of the block after 3 F/B iterations over 5 10 Monte-Carlo trials. The block length  for BPSK and QPSK is 60, and is 800 for the 16QAM constellation. We use the following notations in the figures of the present paragraph. "Forward (Sim)" means that the simulation MSE is measured after one (on-line) forward estimation without any backward estimation. The "Forward / Backward (Sim)" means that the MSE of the F/B estimation is measured after three (off-line) F/B iterations.

A.

5BPerformance with no linear drift Since all the parameters are random, we compare the estimation MSE of the "Forward (Sim)" (resp. "Forward / Backward" (Sim)) with the on-line BCRB (resp. the off-line BCRB) X [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF], [START_REF] Yang | Bayesian and Hybrid Cramer-Rao Bounds for QAM Dynamical Phase Estimation[END_REF], [START_REF] Yang | Approximate Expressions for Cramer-Rao Bounds of Code Aided QAM Dynamical Phase Estimation[END_REF], on Fig. 1 to Fig. 3 for different constellations. At high SNR, we notice that the forward MSE and the F/B MSE curves logically merge. In this case the observations are reliable enough to only take into account the present observation k y in order to estimate k  ; this is why the off-line BCRBs (corresponding to the F/B MSE) converge to the online BCRBs (corresponding to the forward MSE), and this is also why the NDA bounds converge to the DA bounds. As the a priori distribution of θ then has very little influence, the Bayesian problem tends to a deterministic phase estimation problem where we estimate independent observations. In more realistic mid-range SNRs, the F/B performance is definitely superior to the forward only recursion and the maximum difference is 3dB. In this range of SNRs, the a priori knowledge on θ plays a very important role in the phase estimation and this is why there is a larger difference between the F/B and forward recursions compared to higher and lower SNR range.

Finally, at low SNRs, because of the decision error, the MSE increases rapidly and the non-data-aided (NDA) BCRBs do not coincide anymore with the DA BCRBs. However, generally, the performance gain using a data-aided scenario is relatively low compared to the performance difference between the off-line and the on-line scenarios, and logically, when comparing with the forward recursion, there is still an appreciable gain in favor of the F/B recursion. Since the parameters contain both some random parameters k  and a deterministic linear drift  , we compare the MSEs to HCRBs of interest X[4]- [START_REF] Yang | Approximate Expressions for Cramer-Rao Bounds of Code Aided QAM Dynamical Phase Estimation[END_REF] for different constellations on Fig. 4 to Fig. 6.

At high SNR, the off-line HCRB coincides with the on-line HCRB, and so are the corresponding MSEs. Because in this range of SNR, the information provided by the observation is dominating over the a priori knowledge on θ , the observation k y is self-sufficient to estimate k  and the error on  does not disturb the estimation performance on k  . At mid-range SNRs, there is not enough information provided by k y to estimate the phase and one can take advantage of the a priori knowledge on θ (see the difference between the on-line and the off-line BCRBs). The F/B estimation is definitely superior (up to 5 dB on Fig. 4) to the forward MSE not only thanks to the a priori knowledge on θ ; this superiority also comes from the fact that the F/B scheme remains unbiased contrarily to the forward 1 st order loop which suffers from the high linear drift as the corresponding MSE does not coincide anymore with the on-line HCRB. At low SNRs, there is still an advantage for the F/B recursion; however the F/B performance of Fig. 4 deteriorates rapidly, because in practice the F/B recursion is made out of two unidirectional loops, and these loops are not able to operate anymore as wanted with the considered large linear drift. This phenomenon is attenuated with a smaller linear drift (see Fig. 6) or if we had replaced our simple first order PLL components by other component loops such as second order PLLs. The classical on-line PLL has a very low gradient-like complexity and has been employed in real systems for several decades. The complexity price for the off-line improvement is only two times that of the on-line algorithm as we combine two elementary PLLs. In addition, three forward-backward needs to be proceeded which both involves a very reasonable delay and the memorization of K symbols and of 2K phase values.

V.

4BCONC LUSION

In this paper, we presented a near-optimum smoothing phase locked loop (S-PLL) algorithm made out of two very simple first order PLLs. The performance of the S-PLL algorithm does not suffer from the poor transient behavior even with a small number of observations. The proposed scheme provides a gain of several dBs over a forward only on-line algorithm and its performance is near the Cramer-Rao bounds of interest. Finally it is very easy to implement and should be very useful in practice.

.

  ) over an AWGN channel affected by some carrier phase offsets stacked in a vector Assuming that the timing recovery is perfect without any inter-symbol interference (ISI) and k n are respectively the i.i.d. -t h k unknown transmitted constellation symbol (   1M k pc 

  classical hard decision phase detector output based on the decision k c , while the factor

1  1 

 11 estimator is just estimated from 2  with a backward recursion PLL; for the last position k  element, the estimator is estimated from  with a forward PLL; and in the general case ( 1 k    ), the phase estimator is estimated both from the previous and the following samples, i.e. as the average of a forward and of a backward PLL. Since in practice it is impossible to get the actual phase value k  before estimating it, we replace in (11) the true phase k  by the forward estimate   1 ˆF k   and the backward estimate   1 ˆB k

 w 2 =Fig. 1 BPSK

 21 Fig. 1 BPSK MSEs and BCRBs versus SNR

  Fig. 2 QPSK MSEs and BCRBs versus SNR

Fig. 4 BPSKFig. 5 QPSK

 45 Fig. 4 BPSK MSEs and HCRBs versus SNR

Fig. 6 16QAM

 6 Fig. 6 16QAM MSEs and HCRBs versus SNR

J. Yang,B. Geller, C. Herzet, and J.M. BrossierF[START_REF] Noels | Performance Analysis of ML-Based Feedback Carrier Phase Synchronizers for Coded Signals[END_REF]

 [START_REF] Noels | Performance Analysis of ML-Based Feedback Carrier Phase Synchronizers for Coded Signals[END_REF]This work was partially funded by the ANR LURGA program. J.


Jianxiao Yang

Benoit Geller
email: geller@ensta.fr

Anne Wei
email: hanne.wei@lattis.univ-toulouse.fr

Anne Wei Approximate

Approximate Expressions for Cramer-Rao Bounds of Code Aided QAM Dynamical Phase Estimation

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

instabilities X [START_REF] Amblard | Phase tracking: what do we gain from optimality? Particle filtering versus phase-locked loops[END_REF]X-X [START_REF]Digital Video Broadcasting (DVB) User guidelines for the second generation system for Broadcasting, Interactive Services, News Gathering and other broadband satellite application (DVB-S2)[END_REF]X. [START_REF] Barbieri | On the Cramer-Rao bound for carrier frequency estimation in the presence of phase noise[END_REF]X considered the DA CRB for the phase estimation with some noise variance. X [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF]X has derived a Bayesian CRB restricted to the case of NDA BPSK signals. When a deterministic parameter such as the frequency offset is taken into consideration, the hybrid CRB becomes relevant. HCRBs were derived in the case of NDA signals in [START_REF] Bay | On the general form of the Hybrid CRB and its application to the dynamical phase estimation[END_REF], [START_REF] Yang | Bayesian and Hybrid Cramer-Rao Bounds for QAM Dynamical Phase Estimation[END_REF]X and were applied as benchmarks in [START_REF] Yang | Smoothing PLLs for QAM Dynamical Phase Estimation[END_REF], [START_REF] Yang | Near-optimum Low-Complexity Smoothing Loops for Dynamical Phase Estimation[END_REF].

Differently from X [START_REF] Noels | The Cramer-Rao Bound for Phase Estimation from Coded Linearly Modulated Signals[END_REF] X-X [START_REF] Noels | The True Cramer-Rao Bound for Estimating the Carrier Phase of a Convolutionally Encoded PSK Signal[END_REF], [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF], [START_REF] Bay | On the general form of the Hybrid CRB and its application to the dynamical phase estimation[END_REF]X, the goal and the contribution of the paper is to give for the dynamic time-varying phase, both the BCRB and the HCRB in the case of coded and QAM modulated signals. In addition, we give analytical expressions to the various CRBs which greatly reduce the computation complexity. The rest of the paper is organized as follows. In section XIIX, we recall the various kinds of CRBs. After describing the system model in section XIIIX, we derive the BCRBs and the HCRBs for both on-line and off-line estimations for the CA scenario in section XIVX. The derivation for the CA case is different than the one used in X [START_REF] Noels | The Cramer-Rao Bound for Phase Estimation from Coded Linearly Modulated Signals[END_REF]X-X [START_REF] Noels | The True Cramer-Rao Bound for Estimating the Carrier Phase of a Convolutionally Encoded PSK Signal[END_REF]. Moreover, we also present analytical expressions for the various CRBs which do not require to compute the inversion of any information matrix. The various results are finally illustrated in section V. 2

II. 1BCRAMER-RAO BOUNDS (CRBS) REVIEW

In the following, we briefly review the links between the HCRB, the standard CRB, and the BCRB. In the most general case, the parameters to be estimated include both deterministic and random parameters. Denote this parameter vector as   ûy as an estimator of u where y is the observation vector. The HCRB satisfies the following inequality [START_REF] Bay | On the general form of the Hybrid CRB and its application to the dynamical phase estimation[END_REF] on the MSE:

          1 ,| | ˆr d d dd T d E            y u u u uu u y u u y u H u , (1)
2

The notational convention adopted is as follows: italic indicates a scalar quantity, as in a ; boldface indicates a vector quantity, as in a and capital boldface indicates a matrix quantity as in A . The ( , )

Hu

is the so-called hybrid information matrix (HIM)

and is defined as:

    | ,| lo g , | dd r d d d r d Ep           u u uu y u u u H u y u u . (2)
It is shown in X [START_REF] Bay | On the general form of the Hybrid CRB and its application to the dynamical phase estimation[END_REF]X that inequality (1) is still respected when the deterministic and the random parts of the parameter vector are dependent. The HIM can be rewritten as:

      || , lo g | dd r d d r d d d d r r d E E p              u u uu u u u u u u H u F u u u u (3)
where

    | |, , lo g | , dd r d d d r d r Ep           u u uu y u u u F u u y u u (4)
is the Fisher information matrix (FIM).

If in particular

      | | lo g | d d dd dd d d d Ep            u u uu y u u H u F u y u . ( 5
)
Then, the inverse of ( 5) is just the standard CRB [START_REF] Trees | Detection, Estimation and Modulation Theory[END_REF]X X.

On the contrary, if r  uu , then (3) becomes:

    lo g r r r r rr E E p         u u u u H F u u (6)
where

    | lo g | r rr rr Ep      u y u u F u y u . ( 7
)
In this case, the inverse of ( 6) is the Bayesian CRB X[8]X.

III. 2BSYSTEM MODEL

We consider the transmission of a modulated sequence  . We assume that the channel coding maps a message of  bits to a codeword of N bits denoted as

  12 , , , N c c c  c  with   0 ,1 n c  ( 1 nN  ) and   | 1, , 2 v    cc  
; moreover, the constellation vector corresponding to code  c  is denoted as

        1 ,, L ss     s c c c      
. Hence, the conditional probability based on the known vector phase is:

                    2 2 1 1 2 2 2 2 2 2 1 1 | | , | , Re 1 1 e x p 2 . 2 l v v v v v L j L l l v l v l v l n n n p p p p p y s e s y                                              y θ y c c θ c c y s s c θ s s c c c             (9)
In practice, clocks are never perfect, and oscillators suffer from jitters. This results in a Brownian phase model with a linear drift:

1 l l l w        , ( 10
)
where l  is the unknown phase offset at time l ,  is the unknown constant frequency offset (linear drift), l w is a white Gaussian noise with zero mean and variance 2 w  . This model is commonly used [START_REF] Amblard | Phase tracking: what do we gain from optimality? Particle filtering versus phase-locked loops[END_REF]X - [START_REF]Digital Video Broadcasting (DVB) User guidelines for the second generation system for Broadcasting, Interactive Services, News Gathering and other broadband satellite application (DVB-S2)[END_REF] XX in order to describe the behavior of practical oscillators for which the frequency is randomly perturbed. The corresponding conditional probability is:

    2 1 1 2 1 | , e x p 2 2 ll ll w w p                 . ( 11
)
IV. 3BCRB S FOR THE DYNAMICAL PHASE ESTIMATION

In practical receivers, phase estimation can actually be considered following two main scenarios:  Off-line synchronization: the receiver waits until the whole observation frame y has been received. Then afterwards, it processes all the observations to compute the estimates of the carrier phase offsets θ .  On-line synchronization: the receiver estimates l  upon the arrival of the -t h l observation, i.e. .

In this section, we derive some analytical expressions for both the on-line and the off-line CRBs. The parameters of the phase model in [START_REF] Bobrovsky | Some classes of global Cramer-Rao bounds[END_REF] include some random parameters

  1 ,, T L   θ 
(i.e. the dynamical phase) and a deterministic parameter  (i.e. the linear drift). So the parameters vector can be written as:

r d       u θ u u . ( 12
)
Equation ( 3) thus becomes:

              | | | | | | , lo g | lo g | lo g | lo g | T E p p E p p                                                   θ θ θ (13)
where  

  | |, , lo g | , Ep              θ θ y θ F θ y θ
.

We then decompose the HIM into several sub-matrices that will be useful in the sequel:

1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 , T              H H H H H H H H H (14)
where

            11 | | , | | 12 | | , | | 22 | | , | | lo g , | lo g | lo g , | lo g | lo g , | lo g | . E p E p E p E p E p E p                                                                                               H y θ ξ θ ξ H y θ ξ θ ξ (15) A. 6BComputation of   | , E       θ F θ
From ( 9), one obtains after some calculation the first derivative with respect to m  :

        2 1 ln | , , ln | , P r | , , m m m v m v v m m y s s p              c y θ c c y θ    (16)
where

  P r | , , v   c c y θ 
is the a posteriori probability of a code word. Discriminating in [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF] the code words according to the M possible values of the m th received constellation symbol, and noting

    P r | , , , m i v ss   cy θ  
the APP when all possible codewords are known, one further obtains:

        M M 2 1 ln | , 2 P r | , , , Im l i j m i v l i s i mn p s s y s e             S y θ cy θ    (17)
Taking the second derivative with respect to k  and m  , respectively for km  and for km  , one obtains:

              1 2 1 2 1 2 2 M M 2 2 1 1 ln | , ln | , ln | , 2 Im 2 Im P r , | , , , , k m k m k m j j k i m i k i m i v i i n n p p p
y s e y s e s s s s

                                y θ y θ y θ c y θ    (18)
and:

            1 1 1 1 2 2 2 2 M 2 2 1 ln | , ln | , 2 R e 2 Im + P r | , , , . k k k k j j k i k i k i v i n n p p y s e y s e s s                                                 y θ y θ c y θ  (19)
If the codeword has been sufficiently interleaved before the mapping from bits into symbols, then:

            1 2 1 2 P r , | , , , P r | , , , P r | , , , , k i m i v k i v m i v s s s s s s s s         c y θ c y θ c y θ        (20)
so that taking into account ( 17), ( 18) just becomes :

  2 ln | , 0. km p      y θ (21)
It is important to note that differently from [START_REF] Noels | The Cramer-Rao Bound for Phase Estimation from Coded Linearly Modulated Signals[END_REF]X - [START_REF] Noels | The True Cramer-Rao Bound for Estimating the Carrier Phase of a Convolutionally Encoded PSK Signal[END_REF], we take the second derivative to calculate the FIM so as to reduce the complexity of the computation with the help of equation [START_REF] Yang | Bayesian and Hybrid Cramer-Rao Bounds for QAM Dynamical Phase Estimation[END_REF]. Then the FIM can be approximately written as a diagonal matrix and one

 

,, DL EJ      θ F θI [START_REF] Yang | Smoothing PLLs for QAM Dynamical Phase Estimation[END_REF] where L I is the LL  identity matrix and D J is defined as:

  2 2 ln | , D k p J E            y,θ y θ  . ( 23
)

B. 7B Computation of the HIM

Like detailed in [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF], [START_REF] Bay | On the general form of the Hybrid CRB and its application to the dynamical phase estimation[END_REF], we now calculate the different sub-matrices of H (see [START_REF] Amblard | Phase tracking: what do we gain from optimality? Particle filtering versus phase-locked loops[END_REF]). From [START_REF] Mcneill | Jitter in ring oscillators[END_REF], due to the model of Section XIIIX and under the assumption that we have no priori knowledge of 1  , i.e.

0 0 0 1 1 0 0 1 1 A A b A A                    H      , (24)

H

thus become:

  2 2 12 1 2 1 , , 1
T w w L          H 0 (26)   2 22 1. w L   H (27)
We now inverse the HIM so as to obtain the analytical HCRB.

C. 8BAnalytical Expressions of HCRBs

Similarly to Appendix I of [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF]X , from ( 24)-(25) one obtains:

      1 1 1 1 1 1 1 1 1 2 2 2 1, 11 l L l L l l b r r b r r b               H H , ( 28
)         2 2 2 3 2 3 1 1 1 2 2 2 1 11 1 , 2 2 1 1 2 11 1 2 1 2 1 2 L L ll L L l L l L b r r r b r r r r A                                H H , (29)
where

    1 2 2 1 1 2 2 2 1 1 1 4 2 1 1 1 4 2 w D w D w D w D r J J r J J                                  , ( 30
)
and

            1 1 1 2 2 2 1 1 1 1 2 2 2 2 1 4 1 2 2 1 4 1 4 1 2 2 1 4 D w D w D w D w D w D w J J J J J J                                                        . ( 31
)
Thanks to the block-matrix inversion formula X[10]X, we have:

1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 L T                H V H H H HH , ( 32
)
where we define

1 1 2 1 1 1 2 2 1 T w L      H H H  and 1 1 1 1 1 1 2 1 2 1 1 T L     V H H H H  .
To obtain the analytical expression of  and of the diagonal terms of L V , we just exploit that 12 H in (26) has only two non-zero terms and that 11 H has a particular structure (see [START_REF]ETSI EN 300 744[END_REF]); substituting (26),( 28),(29) into the definition of  and L V , we directly have:

        2 2 2 2 1 1 1 1 2 2 2 1 1 1 2 , L L L w L b r r b r r b b               H (33)
and the diagonal elements   , L ll V written as:

    2 11 1 1 1 1 4 , 1 , 1 , 1 1 L ll l L l w             V H H . ( 34
)
We now derive an analytical expression of the diagonal elements of 1 11 L

 

HV corresponding to the minimum bound on θ (see (32)). Using (29) and (34), one can also get the analytical expression of the upper diagonal elements H (see [START_REF]ETSI EN 300 744[END_REF]), one also readily obtains the analytical expression of the on-line HCRB associated to the estimation of l  ( 3 l  ):

                    2 2 2 3 2 3 1 1 1 2 2 2 1 1 , 2 2 1 1 2 11 1 2 1 2 2 1 1 1 2 1 1 1 2 2 2 2 2 2 11 1 1 1 2 2 2 1 2 . L L ll L L l L l L l L l L l L l l l b r r b r r b r r r r A b r b r r b r b b r b r r b r                                                               H H H (35)
                      2 2 2 3 2 3 1 1 1 2 2 2 1 2 2 1 1 2 11 1 2 1 2 2 1 1 1 2 1 1 1 2 2 2 2 2 2 1 1 1 1 1 2 2 2 1 2 . l l l l l l l l b r r b r r C l b r r r r A b r b r r b r b l r b r r b r                                                       H H H (36)
One can notice that (35) and (36) do not depend on the value of the parameter  .

D. 9BThe Bayesian Cramer-Rao Bounds (BCRBs)

When there is no linear drift i.e. 0   , the parameter vector u contains only random parameters θ , i.e. r  uu θ . In this scenario, the BCRB is the lower bound of the MSE. Moreover, the Bayesian information matrix (BIM) L B is equal to the upper left sub-matrix of the hybrid information matrix (HIM): of the inverse of matrix L B is the off-line BCRB associated to the estimation of l  . Consequently, from the previous sections, the corresponding analytical expressions associated with the off-line and the on-line BCRB are respectively:

        2 2 3 3 1 1 1 2 2 2 1 1 , 2 2 1 1 2 1 2 1 2 1 , 2 L L L ll L L l L l L L b r r b r r b r r r r A                                B B ( 38
)         22 33 1 1 1 2 2 2 1 2 2 1 1 2 1 2 1 2 1 a n d , 2 l ll ll l b r r b r r C b r r r r A                    B B (39)
where 1 r , 2 r , 1  and 2  are given by ( 30) and (31). Note that (38) (resp. ( 39)) is the first term on the right side of (35) (resp. (36)). The second terms in (35) and (36) represent the additional positive uncertainty brought by  so that the HCRB is always lower bounded by the BCRB.

V. 4BSIMULATION AND DISCUSSION

We assume the transmission of Gray mapped symbols. We display results for the rate 12 and 64 states non-recursive convolution code adopted in DVB-T [START_REF]ETSI EN 300 744[END_REF]X with a pseudo random interleaver. For codes that are described by means of a trellis, the marginal symbol APPs can be computed from the trellis state APPs and state transition APPs, which in turn can be determined efficiently by the famous BCJR algorithm. Constrained by the paper size, only the HCRB will be discussed in the following. First like in [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF]X , one readily sees on XFig. 1 the superiority of off-line approach compared to the on-line approach in the different positions of the block. Also, there is little improvement for the CA scenario (compared to the NDA scenario) when using a BPSK modulation but the gain becomes obvious for a larger constellation. Moreover, as the observation number increases, both the on-line and off-line CRBs decrease and tend to reach the corresponding asymptote values. We now illustrate the behavior of the HCRB on l  as a function of the SNR (Fig. 2 to Fig. 4).  At high SNR (above 30dB), we notice that the various CRBs logically merge independently of the constellations, on-line/off-line and DA/CA/NDA schemes. The received symbols are reliable enough to make a correct decision and the additive noise can be neglected. Hence, the estimation problem tends to a deterministic phase estimation problem where we estimate L independent phases l  with L independent observations. ) for a 16QAM.

 In mid-range SNRs, the on-line HCRBs leave their corresponding off-line HCRBs, because one observation is not sufficient to estimate the phase offset and a block of observations can improve the estimation performance. This also explains why the NDA CRBs do not merge anymore with the CA and the DA CRBs. Moreover, we note on XFig. factor of 4, the thresholds where the CA bounds leave the DA bound are increased by 6dB.  At low SNR, the lack of symbol knowledge directly affects the estimation on l  . This illustrates why the NDA CRBs increase quicker at low SNR than at high SNR. Note that the CRBs of the BPSK do not merge with the CRBs of the other two-dimensions constellations. ) for BPSK and QPSK.

VI. 5BCONCLUSION

In this paper, we have applied the general form of the CA BCRB and of the CA HCRB in order to evaluate the ultimate performance of a dynamical phase estimator. This illustrates the respective possible advantage of decoding and of the off-line scenario on the synchronization performance. In particular, besides the off-line synchronization gain, there is some space for additional CA synchronization gain at low SNR.

  a priori probability density function (pdf)  

  the residual phase distortion and the zero mean circular Gaussian noise with known variance 2 n

  l y . The phase estimate is then computed based on the current and previous observations only, i.e.

11 H

 11 which has a particular mathematical structure just like in X[START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF]

  ) i.e. the off-line HCRB associated to the estimation of k  :

Fig. 1

 1 Fig. 1 HCRBs in the various block positions for two constellations (BPSK, QPSK) and two block lengths ( 30 L  , 120 L ).

Fig. 2

 2 Fig. 2 HCRBs in the center of the block ( 60 l 

Fig. 3

 3 Fig. 3 HCRBs in the center of the block ( 60 l 

Fig. 4

 4 Fig. 4 Off-Line CA HCRBs in the center of the block ( 60 l ).

 [START_REF] Herzet | Code-aided turbo synchronization[END_REF]This work was partially funded by the ANR LURGA program. J.


Jianxiao Yang

Benoit Geller

A Geller

Wei

Bayesian and Hybrid Cramer-Rao Bounds for QAM Dynamical Phase Estimation

Keywords: Bayesian Cramer-Rao Bound (BCRB), Hybrid Cramer-Rao Bound (HCRB), Synchronization Performance

Bayesian and hybrid Cramer-Rao bounds for QAM dynamical phase estimation

I. INTRODUCTION

here are three types of estimators widely used in communication systems [START_REF] Herzet | Code-aided turbo synchronization[END_REF]: data-aided (DA), code-aided (CA) and non-data-aided (NDA) estimators. DA estimation techniques obtain the better performance but may lead to unacceptable losses in power and spectral efficiency. CA synchronization allows an improved data efficiency but requires additional interactive impairments between decoding and synchronization units. Finally, NDA synchronization algorithms may sometimes lead to poor results but they exhibit the highest transmission efficiency and are still attractive.

To know whether an imposed algorithm is good or not, one often compares its performance with lower bounds. Although there exists many lower bounds, the Cramer-Rao bound (CRB) is the most commonly used and the easiest to determine [START_REF] Trees | Detection, Estimation and Modulation Theory[END_REF], [START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF].

Many works [START_REF] Meyr | Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, ser. Telecommunications and Signal Processing[END_REF]- [START_REF] Noels | The true Cramer-Rao bound for carrier frequency estimation from a PSK signal[END_REF] concern the CRBs for the carrier phase and frequency estimation in DA and NDA scenarios. But all these papers refer to an idealized situation in which the phase offset is constant. However, in modern burst-mode communications, a time-varying phase noise due to the oscillator instabilities has to be considered [START_REF] Mcneill | Jitter in ring oscillators[END_REF], [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF]. Taking into account such a phase noise variance, [START_REF] Barbieri | On the Cramer-Rao bound for carrier frequency estimation in the presence of phase noise[END_REF] considered the DA CRB for the phase estimation with a phase noise variance and [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF] derived a BCRB for NDA BPSK signals; [START_REF] Bay | A General Form of the Hybrid CRB. Application to the Dynamical Phase Estimation[END_REF] added the consideration of a deterministic parameter to obtain the HCRB for BPSK signals. However, the synchronization of BPSK signals is relatively simpler than that of QAM signals. The goal of this paper is then two-folded. First we detail the derivation of the BCRB and of the HCRB for QAM modulated signals. Second we want to provide benchmarks for the QAM dynamical phase estimation, since we present a generalization of the off-line synchronizing scheme [START_REF] Yang | Near-optimum Low-Complexity Smoothing Loops for Dynamical Phase Estimation[END_REF] to QAM modulated signals in [START_REF] Yang | Smoothing PLLs for QAM Dynamical Phase Estimation[END_REF] whose performance can exactly reach the bounds derived in this paper. For lack of space reasons, this study is extended to the code-aided case in [START_REF] Yang | Approximate Expressions for Cramer-Rao Bounds of Coded Aided QAM Dynamical Phase Estimation[END_REF]. This paper is organized as follows. In section II, we recall the various kinds of Cramer-Rao bounds. After describing the system model in section III, we derive the BCRBs and the HCRBs for both on-line and off-line estimations in section IV. Moreover, we also present the analytical expressions for the various CRBs; this avoids computing the inverse of any information matrix. A discussion about the various CRBs and a conclusion are respectively provided in section V and VI. 2

II. CRAMER-RAO BOUNDS (CRBS) REVIEW

It is known that the parameters to be estimated can be categorized as deterministic or random parameters. Denote this parameter vector as ( )

, T T T r d = u u u
, where is a

d u ( ) 1 n m -×
deterministic vector and is a random vector with an a priori probability density function (pdf) . The true value of will be denoted is the observation vector. The HCRB satisfies the following inequality [START_REF] Bay | A General Form of the Hybrid CRB. Application to the Dynamical Phase Estimation[END_REF] on the MSE:

y ( ) ( ) ( ) ( ) ( ) 1 , | | ˆr d d d d T d E Δ Δ - Δ = = ⎡ ⎤ - - ≥ ⎢ ⎥ ⎣ ⎦ y u u u u u u y u u y u H u , (1) where ( ) d Δ
H u is the so-called hybrid information matrix (HIM)

and is defined as:

( ) ( ) | , | log , | d d r d d d E p Δ Δ Δ = = r d ⎡ ⎤ = -Δ ⎣ ⎦ u u u u y u u u H u y u u . ( 2
)
It is shown in [START_REF] Yang | Near-optimum Low-Complexity Smoothing Loops for Dynamical Phase Estimation[END_REF]

( ) ( ) ( ) | | , l o g | d d r d d r d d d d r rd E E p Δ Δ Δ Δ Δ = = = ⎡ ⎤ ⎡ ⎤ = + -Δ ⎣ ⎦ ⎣ ⎦ u u u u u u u u u u H u F u u u u , (3)
where ( )

( ) | | , , l o g | , d d r d d d r d r E p Δ Δ Δ = = ⎡ ⎤ = -Δ ⎣ ⎦ u u u u y u u u F u u y u u (4)
is the Fisher information matrix (FIM).

In particular, if d = u u , (4) reduces to:

( ) ( ) ( ) | | log | d d d d d d d d d E p Δ Δ Δ Δ = = ⎡ ⎤ = = -Δ ⎣ ⎦ u u u u y u u H u F u y u . ( 5
)
The inverse of ( ) 5) is just the standard CRB [START_REF] Trees | Detection, Estimation and Modulation Theory[END_REF].

d Δ H u in (
On the contrary, if (

( ) log r r r r r E E p ⎡ = ⎡ ⎤+ -Δ ⎣ ⎦ ⎣ u u u u H F u u r ⎤ ⎦ , )
where ( ) ( )

| log | r r r r E p ⎡ ⎤ = -Δ ⎣ ⎦ u y u u F u y u r , (7)
and the inverse of H in ( 6) is the Bayesian CRB (BCRB) [START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF].

III. SYSTEM MODEL

We consider the transmission of a sequence

[ ] 1 , , T L s s = s of M-QAM signals from a set
, rotated by some random carrier phases , over an additive white Gaussian noise (AWGN) channel. Assuming that the timing recovery is perfect without inter-symbol interference (ISI), the sampled baseband signal can be written as:

M S [ 1 , , T L θ θ = θ [ 1 , , T L y y = y ] ] ( ) l j l l l l l y se n a jb e n θ = + = + + l j θ l , (8)
where l s , l θ and are respectively the transmitted complex symbol (

l n l l -th l l s a jb = +
), the residual phase distortion and the zero mean circular Gaussian noise with variance 2 n σ .

For the data aided (DA) scenario, the transmitted symbols are independent and identically distributed (i.i.d.) and the conditional probability based on the known phase is:

( ) ( ) { } 2 2 2 2 1 1 Re 1 | e x p e x p2 l L j L L l l l l k k l l n n y s e s y p py θ θ πσ σ σ - * = = ⎧ ⎫ ⎧ ⎫ + ⎛ ⎞ ⎪ ⎪ ⎪ = = - ⎜ ⎟ ⎨ ⎬ ⎨ ⎝ ⎠ ⎪ ⎪ ⎪ 2 n ⎪ ⎬ ⎩ ⎭ ⎩ ⎭ ∏ ∏ r | θ ⎪ (9)
For the non-data aided (NDA) case, the transmitted symbols are also i.i.d and thus ( )

p r | θ has a similar form: ( ) ( ) { } M 1 2 2 2 2 1 | 2Re 1 1 exp . M l l L l l l L j L l l ll s l n n p py s y yse θ θ πσ σ = - * ∈ = = ⎧ + + ⎛ ⎞ ⎪ ⎪ = - ⎜ ⎟ ⎨ ⎬ ⎝ ⎠ ⎪ ⎩ ∏ ∑ ∏ S r | θ ⎫ ⎪ ⎭ l (10)
In practice, the oscillators are never perfect and suffer from jitters. [START_REF] Mcneill | Jitter in ring oscillators[END_REF] and [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF] have provided a mathematical model which has been widely used to describe the oscillator behavior: σ . The corresponding conditional probability can be expressed as:

( ) ( ) 2 1 1 2 1 | , exp 2 2 l l l l w w p θ θ ξ θ θ ξ σ πσ - - ⎧ ⎫ - - ⎪ = - ⎨ ⎪ ⎪ ⎩ ⎭ ⎪ ⎬ ] . ( 12
)
IV. CRBS FOR THE DYNAMICAL PHASE ESTIMATION

In practice, phase estimation can be considered using the off-line scenario and the on-line scenario. The former uses the whole observations frame and only begins estimation when the whole frame has been received, while the latter directly uses the current and previous observations. In the following, we will give both the on-line and the off-line lower bounds.

The parameters of the phase model in ( 9), (10) include some random parameters (i.e. the dynamical phase) and a deterministic parameter

[ 1 , , T L θ θ = θ
ξ (i.e. the scalar linear drift). So the parameter vector can be written as:

r d ξ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ u θ u u (13)
Equation ( 3) thus becomes:

( ) ( ) ( ) ( ) ( ) ( ) | | | | | , log log | log | log | T E p p E p p ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ Δ Δ Δ Δ Δ Δ Δ = = = = ⎡ ⎤ = ⎣ ⎦ ⎛ ⎞ -Δ -Δ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ -Δ θ (14)
where ( )

( ) | | , o g |, E p ξ ξ ξ ξ ξ ξ Δ Δ Δ = = , l ⎡ ⎤ = -Δ ⎣ ⎦ y θ F θ y θ H θ θ

22

. We then decompose the HIM into smaller matrices that will be useful in the sequel:

T ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ H H H H H H H H H , (15)
where

( ) ( ) ( ) ( ) ( ) ( ) 11 | | , | | 12 | | , | | 22 | | , | | log , | log | log , | log | log , | log | . E p E p E p E p E p E p ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ = = = = = = = = = = = = ⎧ ξ ξ ξ ξ ξ ξ ⎡ ⎤ ⎡ = -Δ + -Δ ⎤ ⎣ ⎦ ⎣ ⎪ ⎪ ⎦ ⎡ ⎤ ⎡ = -Δ + -Δ ⎨ ⎤ ⎣ ⎦ ⎣ ⎪ ⎦ ⎡ ⎤ ⎡ ⎪ = -Δ + -Δ ⎤ ⎣ ( ) ( ) ( ) ln | , ln | , p p ξ ξ = y θ y θ s ∑ s p . (17)
Using the i.i.d condition among data, the FIM can be written as:

( ) [START_REF] Yang | Approximate Expressions for Cramer-Rao Bounds of Coded Aided QAM Dynamical Phase Estimation[END_REF] where L I is the L L × identity matrix and D J is defined as:

, D L J = θ F I E⎡ ⎤ ⎣ ⎦ θ
( )

2 2 log | , l l D l p y J E ⎡ ξ θ θ ⎤ ∂ - . ( 19
) ⎢ ⎥ ∂ ⎣ ⎦ y,θ
Starting with the DA scenario, from [START_REF] Steendam | Low-SNR limit of the Cramer-Rao bound for estimating the carrier phase and frequency of a PSK or QAM waveform[END_REF] we have that:

( )

2 2 2 2 ln | 2 2 SNR . s l n p E σ θ σ ⎧ ⎫ ∂ ⎪ ⎪ = - ⎨ ⎬ ∂ ⎪ ⎪ ⎩ ⎭ y θ = -⋅ (20)
We now turn to the NDA scenario, and from [START_REF] Noels | The true Cramer-Rao bound for carrier frequency estimation from a PSK signal[END_REF], one has that: p y s θ ξ has been shown in [START_REF] Noels | The true Cramer-Rao bound for carrier frequency estimation from a PSK signal[END_REF]. Unfortunately, the expectation of (21) has no simple analytical solution and one must resort to numerical methods.

( ) ( ) ( ) ( ) ( )
θ ξ θ ξ ξ θ θ ξ θ ξ ∈ ∈ ∈ ∂ ⎛ ∂ ⎞ ⎜ ⎟ ∂ ∂ ∂ ⎜ ⎟ = -⎜ ⎟ ∂ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ ∑ S S S y θ S , (21

B. Analytical Expressions of HCRBs

From ( 16), due to [START_REF] Demir | Phase noise in oscillators: a unifying theory and numerical methods for characterization[END_REF] and without any a priori knowledge on 1 θ (i.e.

), we obtain matrix :

( )

1 1 1 1 log 0 E p θ θ θ θ ⎡ Δ ⎣ ⎤ = ⎦ 0 1 11 H 11 1, 1 0 0 1 1 0 0 0 1 1 0 0 1 A A b A A + ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + ⎣ ⎦ H , ( 24
)
where and (25) For both the DA and NDA scenarios (see ( 9) and ( 10)), we can see that

2 12 1 2 1 , , 1
T w L σ × - ⎡ = - ⎣ H 0 2 w σ ⎤ ⎦ (26) ( ) 2 22 1 w L . σ = - H (27)
We now invert the HIM. Starting with and just similarly to Appendix I of [START_REF] Bay | A General Form of the Hybrid CRB. Application to the Dynamical Phase Estimation[END_REF], we find:

1 11 - H ( ) ( ) ( 1 1 1 1 11 1 1 1 2 2 2 1, 11 l L l L l l b r r b r r b ρ ρ - - -- -- ⎡ ⎤ = + + ⎣ ⎦ H H ) + , (28) ( ) ( ) ( ) (
)

2 2 2 3 2 3 1 1 1 2 2 2 1 11 1 , 2 2 1 1 2 11 1 2 1 2 1 2 L L l l L L l L l L b r r b r r b r r r r A ρ ρ - - - - - -- -- - ⎧ + + + ⎪ ⎡ ⎤ = ⎨ ⎣ ⎦ - + - ⎪ ⎩ H H ⎫ ⎪ ⎬ ⎪ ⎭ , (29) where ( ) ( ) 1 2 2 1 1 2 2 2 1 1 1 4 1 1 1 4 w
D w w D w r J r J σ σ σ σ - - ⎧ ⎛ ⎞ = + - + ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎨ ⎛ ⎞ ⎪ = + + + ⎜ ⎟ ⎪ ⎝ ⎠ ⎩ 2 2 D D J J , (30) and ( ) ( ) ( ) ( ) ( ) ( )
1 1 2 2 1 1 1 2 2 2 1 4 1 2 2 1 4 1 4 1 2 2 1 4 D w D w D w D w D w D w J J J J J J ρ σ σ σ ρ σ σ σ - - - - ⎧ ⎛ ⎞ ⎛ + -- + ⎜ ⎟ ⎜ ⎪ ⎪ ⎝ ⎠ ⎝ ⎨ ⎛ ⎞ ⎛ ⎪ + + + + ⎜ ⎟ ⎜ ⎪ ⎝ ⎠ ⎝ ⎩ 1 2 1 2 - - ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ . ( 31
)
Thanks to the block-matrix inversion formula [START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF], we have:

1 1 1 1 11 11 12 1 1 1 12 11 L T λ λ λ - - - - - - - ⎡ + - = ⎢ - ⎣ H V H H H H H , (32)
where we define Due to the particularly light structures of and , we find:

1 1 1 11 12 12 11 T L λ -- - V H H H H 11 H 12 H ( ) ( ) { 2 2 2 1 1 1 2 2 2 2 11 1 2 L L w L b r r b r r b b λ ρ ρ σ - - - = - + + + + H } 1 L - . ( 33
)
From the definition of L V , the diagonal elements [ ] ,

L l l V can be written as:

[ ] ( ) ( ) 2 1 1 11 11 , 1, 1, 1 4 L w l l l L l λσ - - + - ⎡ ⎤ ⎡ ⎤ = - ⎣ ⎦ ⎣ ⎦ V H H 1 2 l l l l l l b r r b r r C b r r r r A ρ ρ - - - -- - - ⎧ ⎫ + + + ⎪ ⎪ = ⎨ ⎬ - + - ⎪ ⎪ ⎩ ⎭ B B . ( 39
)
Note that (38) (resp. (39)) is the first term on the right side of (35) (resp. (36)). The second terms in (35) and (36) represent the additional positive uncertainty brought by ξ and the HCRB is always lower bounded by the BCRB.

) ⎫ ⎪ ⎬ ⎪ ⎭ ⎫ ⎪ ⎬ (35) V. DISCUSSION
Constrained by the paper size, only the HCRB will be discussed in the following. First, one readily sees on Fig. 1 the superiority of the off-line approach compared to the on-line approach in the different positions of the block. Also, there is little improvement for the DA scenario (compared to the NDA scenario) when using a BPSK modulation but the gain becomes obvious for a larger constellation. Moreover, as the observation number increases, both the on-line and off-line CRBs decreases and tends to reach the corresponding asymptote values.

We now illustrate the behavior of the M-QAM (M=4, 16, 64 and 256) HCRB of l θ as a function of the SNR in Fig. 2.

At high SNR (above 30dB), we notice that the various off-line CRBs logically merge and do not depend on the constellation. In this range of SNR, the received constellations are reliable enough to make correct decisions and it is sufficient to only take the present observation into account to estimate In mid-range SNRs, one observation is not sufficient to estimate the phase offset and a block of observations can improve the estimation performance. This explains why the NDA CRBs do not merge anymore with the DA CRBs. Moreover, we notice that every time the constellation size is multiplied by 4, the threshold where the NDA bounds leave the DA bound is increased by 6dB.

At low SNR, the AWGN has more influence than the phase noise and that results in many decision errors. That is why the NDA CRBs increase quicker at low SNR than at high SNR, and particularly for the 2-dimension QAM signals (compared to the real BPSK signals).

VI. CONCLUSION

In this paper, we have applied the general analytical form of the BCRBs and HCRBs to evaluate the performance of dynamical phase estimation. We have illustrated the phase estimation performance for arbitrary constellations. In particular, we can measure the advantage of using an off-line scenario. We also point out the difference between the QAM bounds and the traditionally studied BPSK bounds.

u

  and ( ) û y is the estimator of u where

2A

  The notational convention adopted is as follows: italic indicates a scalar quantity, as in ; boldface indicates a vector quantity, as in a and capital boldface indicates a matrix quantity as in . The entry of matrix A is denoted as . The transpose matrix of is indicated by a superscript , is the determinant of . represents the vector [, where and are positive integers ( and are respectively the real and imaginary parts of a . ] denotes the expectation over y ∇ u and Δ v u represent the first and second order derivative operators.

  l θ is the unknown phase offset at time l , ξ is the unknown constant frequency offset (linear drift), is a white Gaussian noise with zero mean and variance l w 2 w

  θ is independent of ξ , i.e.

θFig. 1

 1 Fig. 1 HCRBs in the various block positions ( and 30 L = 60 L = ).

Fig. 2

 2 Fig. 2 HCRBs in the center of the block () for various constellations. 30 l =

u u u u , the HIM can be rewritten as:

											that inequality (1) still holds when the
	deterministic and the random parts of the parameter vector are dependent. By expanding ( ) log , | r d p y u u as
	log	p	(	y	| , r	d	)	+	log	p	(	r	|	d	)

This work was partially funded by the ANR LURGA program. J. Yang is with SATIE, ENS Cachan (email: yang@satie.ens-cachan.fr); B. Geller is with LEI, ENSTA PARISTECH (email: geller@ensta.fr); A. Wei is with LATTIS, Université Toulouse II (email: anne.wei@lattis.Univtoulouse.fr).

.

(34)

Using ( 29) and (34), we can then get the analytical expression of the upper diagonal elements in (32) i.e.

the off-line HCRB associated to the estimation of

11 l H in (35), one also obtains the analytical expression of the on-line HCRB associated to the estimation of l θ ( ):

where ( )

is the left-upper submatrix of (see ( 24)).

We can notice that ( 35) and (36) do not depend on the value of the parameter

C. Analytical Expressions of BCRBs

When there is no linear drift i.e.

) ( )