Pietro Aricò

Gianluca Borghini

Gianluca Di Flumeri

Alfredo Colosimo

Ilenia Graziani

Jean-Paul Imbert

Géraud Granger

Raïlane Benhacene

Michela Terenzi

Simone Pozzi

Reliability over time of EEG-based mental workload evaluation during Air Traffic Management (ATM) tasks

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

I. INTRODUCTION

In the recent decades, research is focusing on the evaluation of user's mental states based on his/her neurophysiological activity in operating environments. In this regard, the mental workload (MW) monitoring is of particular interest especially in safety-critical applications where human performance is often the least controllable factor. In fact, as the MW increases, it became harder to maintain the user's task performance within an acceptable range, resulting then into an increasing of errors' occurrence [START_REF] Norman | On the analysis of performance operating characteristics[END_REF]. P. Aricò is with the Dept. Physiology and Pharmacology, University "Sapienza" of Rome, Italy and with the IRCCS Fondazione Santa Lucia, Rome, Italy (corresponding author to provide phone: +39 3292973269; email: p.arico@hsantalucia.it).

G. Borghini is with the Dept. Physiology and Pharmacology, University "Sapienza" of Rome, Italy and with the IRCCS Fondazione Santa Lucia, Rome, Italy I. Graziani is with the Dept. Physiology and Pharmacology, University "Sapienza" of Rome, Italy.

G. Di Flumeri and A. Colosimo are with the Dept. Anatomical, Histological, Forensic & Orthopedic Sciences, University "Sapienza" of Rome, Italy.

J.P. Imbert, G. Granger, R. Benhacene are with Ecole Nationale de l'Aviation Civile Toulouse, France M. Terenzi, S. Pozzi are with the Deep Blue Research and Consulting Rome, Italy.

F. Babiloni, is with the Dept. Molecular Medicine, University "Sapienza" of Rome, Italy and BrainSigns srl. Rome, Italy.

The theta (θ: over the frontal and the occipital sites) and alpha (α: over the parietal sites) rhythms of the electroencephalographic (EEG) signal have been taken into account in several studies because of their strong correlation with the MW variations [START_REF] Aricò | Towards a multimodal bioelectrical framework for the online mental workload evaluation[END_REF], [START_REF] Borghini | Quantitative Assessment of the Training Improvement in a Motor-Cognitive Task by Using EEG, ECG and EOG Signals[END_REF].

In this regard, the application of machine-learning techniques (MLTs) to the MW evaluation based on the measurement of brain activity is growing continuously. In general, the use of these techniques allows to assess subjects' MW in a short time (few seconds), reaching high binary discrimination accuracy (DA~90%) [START_REF] Aricò | Towards a multimodal bioelectrical framework for the online mental workload evaluation[END_REF], [START_REF] Kohlmorgen | Improving human performance in a real operating environment through real-time mental workload detection[END_REF]. These algorithms are able to extract from a big amount of physiological data within a training dataset the most significant features closely related to the user's mental state (i.e. MW). Then, based on those features it should be possible to assess the user's MW level and to keep the DA stable across different days. Actually, one of the big concerns of the MLTs is related to the capability of such algorithms to extract from the training dataset only those neurophysiological features by which the reliability of the measure could remain stable over time, providing a high DA across different days. Nowadays, the effects of day-to-day fluctuations in the operator's brain signals have not been thoroughly assessed while operators are engaged in complex tasks. Different studies showed that the performance of classifiers in evaluating the different MW levels of the user dramatically decrease over days [START_REF] Christensen | The effects of day-to-day variability of physiological data on operator functional state classification[END_REF], [START_REF] Mcevoy | Test-retest reliability of cognitive EEG[END_REF]. Despite the EEG is not a stationary process, let us assume that few neurophysiological features of the user related to the MW could remain enough stable over time. In this way, a classifier trained with those features should not degrade his DA over time. On the contrary, it will be indicative that it becomes too specific to the training dataset (overfitting). The reduction of features used by the classifier during its calibration phase could mitigate the overfitting and improve the reliability and stability of the DA over time [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF].

In this work, we hypothesized that if a simple classifier is calibrated by using a low number of brain spectral features between those ones strictly related to the MW (i.e. Frontal and Occipital θ, Parietal α rhythms), those features will make the classifier DA high across the days. In particular, the DA of a linear classifier across an entire week has been investigated by using EEG data related to MW collected in a professional school of Air Traffic Management (ATM) while Air Traffic Controllers (ATCos) trainees were performing several ATM scenarios characterized by increasing level of difficulty.

Reliability over time of EEG-based mental workload evaluation during Air Traffic Management (ATM) tasks

Pietro Aricò Member, IEEE, Gianluca Borghini Student Member, IEEE, Gianluca Di Flumeri, Alfredo Colosimo, Ilenia Graziani, Jean-Paul Imbert, Géraud Granger, Railene Benhacene, Michela Terenzi, Simone Pozzi, and Fabio Babiloni Member, IEEE

II. MATERIALS AND METHODS

A. Experimental protocol

Twelve ATCo trainees (age 25±3) from ENAC (Ecole Nationale de l'Aviation Civile, Toulouse, France) have been involved in this experimentation. In particular the experimental procedures involving human subjects described in this paper were approved by the Institutional Review Board. The experimental task used in this study was the Labyrinth (LABY, Figure 1), a functional simulated ATM environment, developed by ENAC [START_REF] Imbert | The LABY Microworld A Platform for Research, Training and System Engineering[END_REF]. The difficulty of the task can be altered according to how many aircraft the participant have to control, the number and type of clearances required over the time and the number/trajectory of other interfering flights. Subjects were asked to execute the LABY task under three difficulty conditions (Easy [E], Medium [M] and Hard [H]) chosen by ATM Experts at ENAC. Controllers have been trained to use LABY before starting with the experimentation. The experimental protocol has been composed of three recording sessions performed on three different days. The first two sessions have been performed in two consecutive days named hereafter as Day 1 and Day 2. The last session has been performed after one week from the last one (Day 9). Each session consisted in twelve runs, in which subjects performed the three LABY conditions (E, M, H) four times each in a randomized sequence. Each condition lasted 3 minutes. Also, in order to avoid habituation and expectation effects, some task parameters have been randomly changed across the experimental sessions. In summary, the whole dataset has been composed of twelve triplets of conditions (4 triplets of E, M, H conditions for each of the three experimental days).

B. EEG-based workload index

Neurophysiological signals have been recorded by the digital monitoring BEmicro system (EBNeuro, Italy). Thirteen EEG (FPz, F3, Fz, F4, AF3, AF4, P3, Pz, P4, POz, O1, Oz, O2) and one vertical EOG channels have been collected simultaneously with a sampling frequency of 256 (Hz). All the EEG electrodes have been referenced to both the earlobes, and the impedances of the electrodes were kept below 10 (kΩ). The acquired EEG signals have been digitally band-pass filtered by a 4 th order Butterworth filter [1÷30] (Hz) and the EOG signal has been used to remove eyes-blink artifacts from the EEG data. The EEG signal has been then segmented into epochs of 2 seconds, shifted of 0.125 seconds and the Power Spectral Density (PSD) has been calculated for each EEG epoch by using only the frequency bands directly correlated to the MW (frontal and occipital θ and parietal α bands, [START_REF] Borghini | Quantitative Assessment of the Training Improvement in a Motor-Cognitive Task by Using EEG, ECG and EOG Signals[END_REF][START_REF] Gevins | Monitoring working memory load during computerbased tasks with EEG pattern recognition methods[END_REF][START_REF] Kohlmorgen | Improving human performance in a real operating environment through real-time mental workload detection[END_REF][START_REF] Christensen | The effects of day-to-day variability of physiological data on operator functional state classification[END_REF][START_REF] Mcevoy | Test-retest reliability of cognitive EEG[END_REF][START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF][START_REF] Imbert | The LABY Microworld A Platform for Research, Training and System Engineering[END_REF][START_REF] Draper | Applied Regression Analysis[END_REF][START_REF] Bamber | The area above the ordinal dominance graph and the area below the receiver operating characteristic graph[END_REF][START_REF] Zhang | A Simple Statistical Parameter for Use in Evaluation and Validation of High Throughput Screening Assays[END_REF] (Hz)).

C. Classifier features selection

In this study we chose to use a StepWise Linear Discriminant Analysis (SWLDA) regression [START_REF] Aricò | Towards a multimodal bioelectrical framework for the online mental workload evaluation[END_REF] that is one of the best outperforming classifiers, in fact with respect to other methods it has the advantage of having automatic features extraction, so that insignificant terms are statistically removed from the model. In particular, a three-classes SWLDA has been used to select the most relevant EEG spectral features, within the training dataset, to discriminate the MW level among the three task conditions (E [Label = 0], M [Label = 0.5] and H [Label = 1]), and then the linear discriminant function has been evaluated to test the reliability of the feature selection criteria. For each PSD epoch, only the frequencies strictly related to the MW have been considered to train and test the classifier.

In a SWLDA regression, the input features are usually weighted by using ordinary least-squares regression to predict the target class label (i.e. 0, 0.5, 1). At each step, a new term can be added to or deleted from the model (if pvalue < α ENTER or if p-value > α REMOVE ). This process goes on until the predefined number of significant features is reached (Iteration MAX ), unless there are no more features satisfying the entry (α ENTER ) and the removal (α REMOVE ) conditions [START_REF] Draper | Applied Regression Analysis[END_REF]. Normally, it is possible to optimize a SWLDA regression by tuning all or few of the three parameters available in the algorithm (α ENTER , α REMOVE and Iteraction MAX ). For reducing the degrees of freedom of the problem, we chose to not impose constrains over the first two parameters α ENTER , α REMOVE , but to tune only the Iteraction MAX value. In other words, a "forward SWLDA" has been implemented. In this way, it was possible to impose the SWLDA to select a features' number equal to the number of iterations (Iteration MAX ). Despite no constrains on the α ENTER and the α REMOVE parameters, features are included in the model in order of significance (i.e. the first feature added into the model will be the most significant one, and so on). In this way, we can be sure that the first features included in the model are also the most significant ones. In particular, we performed simulations by using three different values of the Iteration MAX parameter, that is 5%, 50% and 100% of the available features (Frontal and Occipital θ, Parietal α). We used the 5% to test our hypothesis, so that to have a low number of features used for the calibration of the classifier. As we stated before, we expected that by using a low number of features (i.e. 5%) the DA of the classifier would be reliable over time. On the contrary, in the other cases (50% and 100%) the DA should decrease over time.

D. Mental workload index based on EEG activity

The linear discriminant function (y test (t)) for each window of 2 seconds has been computed by using the coefficients (weights: w itrain and bias: b train ) returned by the SWLDA function (equation 1, where f itest (t) represents the PSD matrix of the testing dataset at the time sample t, and of the i th feature). Finally, we applied a moving average of 8 seconds (8MA) to the y test (t) function in order to smooth it out by reducing the variance of the measure, and we defined it as EEG-based workload index (W EEG ). Here below the SWLDA discriminant function (1) and the W EEG index (2) equations are reported.

y test (t) = ∑ i w i train • f i test (t)+b train

(1)

W EEG = 8MA (y test (t)) (2)

E. Cross-validations between days

For each subject, different cross-

F. Performed analyses

Discriminant Accuracy (DA) analysis: We performed analysis by using different Iteration MAX values, the 5%, 50% and 100% of the available features. For each testing triplet, we calculated the W EEG indexes. At this point, Area Under Curve (AUC) values of the Receiver Operating Characteristic (ROC, [START_REF] Bamber | The area above the ordinal dominance graph and the area below the receiver operating characteristic graph[END_REF]) have been calculated by considering couples of W EEG distributions (E vs H, M vs H and E vs M) in order to test the DA of the classifier.

Workload distribution analysis (W EEG ):

For each subject we evaluated the W EEG distributions over the testing dataset, by considering the best Iteration MAX value resulting from the DA analysis. Two two-ways repeated measures ANOVA (CI = .95) analyses have been performed, one on the AUC values and the other one on the W EEG distributions. In the first one, we averaged the AUC values related to the three difficulty levels (E vs H, M vs H and E vs M) considering as within factors the "Iteration MAX values" (5%, 50%, 100%) and the "three cross-validation types" (Intra, Short term, Medium term). In the second one, we fixed the Iteration MAX value (based on the first ANOVA results) and we considered the W EEG by using as within factors the three "conditions" (E, M, H) and the three "cross-validation types" (Intra, Short term, Medium term). Post-hoc tests (Bonferroni correction for multiple comparisons) have been performed to assess significant differences among all pairs of levels of the considered factors. Before every statistical analysis, we used the z-score [START_REF] Zhang | A Simple Statistical Parameter for Use in Evaluation and Validation of High Throughput Screening Assays[END_REF] correction formula to normalize the different behaviors of the subjects. In particular, we calculated this score by using the mean and the standard deviations of the related values (i.e. AUC, scores) over the conditions (i.e. classifiers, cross-validations, difficulty levels).

III. RESULTS

Discriminant Accuracy (DA): Figure 2 represents the error bars (CI = .95) related to the mean AUC values of the classifier over the E vs H, M vs H and E vs M conditions by using the three Iteration MAX values (5%, 50%, 100%) and the three cross-validation types (Intra, Short term and Medium term). ANOVA results highlighted a significant main effect between the two factors (F(4,44)=3.81, p=.01). The post-hoc test highlighted a significant decrement of the AUC values within all the cross-validation types (Intra, Short term and Medium term) by considering both the 50% and the 100% of the available EEG features (all p<10 -3 ), but no significant differences across the different cross-validation types have been found by considering the 5% of features (all p=1). Also, despite there was no significant difference among the AUC values achieved by using the 5%, 50% and 100% of the features within the Intra cross-validations (all p>.3), the posthoc test highlighted a significant decrement of the AUC values between the 5% of features and the 50% and 100% of features within the Short term and the Medium term crossvalidation types (all p<10 -3 ).

Neurophysiological workload distribution (W EEG ):

ANOVA results showed a significant interaction between the two factors (F(4,44)=19.14, p=10 -6 ). The post-hoc test highlighted that the W EEG distributions related to the three difficulty levels (E, M, H) were significantly different for each cross-validation type (all p<10 -6 ). Furthermore, no significant differences have been shown between each W EEG distribution related to the three MATB difficulty levels among the cross-validation types (all p>0.6). Figure 3 represents the W EEG values measured for each difficulty level (E, M, H) and for each subject.

IV. DISCUSSION

The ATM environment imposes multiple concurrent demands on the operator, including air traffic monitoring, anticipating loss of separation between aircrafts, and intervening to resolve conflicts. The assessment of the MW associated with operating in such complex systems has long been recognized as an important issue [START_REF] Gopher | Workload: An examination of the concept[END_REF]. MLTs have been widely used to assess the MW of the operators by using their brain activity [START_REF] Aricò | Towards a multimodal bioelectrical framework for the online mental workload evaluation[END_REF], [START_REF] Kohlmorgen | Improving human performance in a real operating environment through real-time mental workload detection[END_REF]. One of the big problems in using these approaches is that the MW measure does not remain stable over time, unless a frequent recalibration procedure. The reliability of such methodologies is of great importance for their effective use in real work contexts. The necessity to recalibrate the system every day makes such kind of approach unsuitable in operative environments.

In addition, there are evidences in literature in which it has been demonstrated that, as long as a subject is trained to do a specific task, the cognitive processes required for performing such task would be always present over time [START_REF] Borghini | Quantitative Assessment of the Training Improvement in a Motor-Cognitive Task by Using EEG, ECG and EOG Signals[END_REF]. In other words, it would be possible to take into account certain cerebral features strictly related to the MW of the user (Frontal and Occipital θ, Parietal α), that remain enough stable over time. In this work, we observed that if a classifier is calibrated by using a low number of cerebral features (among the ones strictly related to the MW of the operator), the reliability of the system will not degrade across days. In this context, we could speculate that the classifier is able to identify those brain features always involved in the execution of the proposed task, since the DA remained stable over time. The described algorithm has been tested on twelve ATCos trainees while performing an ATM task under three different difficulty levels (E, M, H) resembling different possible ATM scenarios. Results demonstrated that if a SWLDA regression is calibrated by using a low number of features (5% out of the total available features) a stable MW measure across the days is obtained. The consequence of this finding is that within a week, it will not be necessary to recalibrate the algorithm and the classification system with new user's EEG data. In addition, the proposed algorithm has been able to differentiate significantly the MW over the three experimental conditions (E, M, H).

V. CONCLUSION

In this work, we have provided evidences that if a SWLDA classifier is used to select a low number of EEG spectral features related to the MW of the user, those features will make the classifier DA stable across different days. In other words, if the classifier selects certain cerebral features on Monday, those features can be used within one week, without degradation of the system's reliability. However, the study has several limitations. The first one is that the reliability of the measure has been tested only in a short period of one week. A second limitation is that different variables have been kept under control for the purpose of this experimentation. For example, each subject took part to the experiment at the same time over the different days; in addition the simulation scenario was not too much ecological in terms of time duration (the work shift for an ATCo is of ~40 minutes) and performed tasks (the ATCos use to talk with their colleagues and with the pilots). In addition, an open question that has to be investigated following the results of the actual study is if it exists the possibility to identify a subset of EEG features that are maintained stable even between different subjects. Taking into account all these limitations, there is the need to perform further experiments to test the proposed approach in a more real ATM work scenario, by considering a longer period of stability ("Long-term") than one week.

Figure 1 .

 1 Figure 1. The LABY task, developed by ENAC

  validations have been performed by training the classifier with one triplet of E, M, H conditions and by testing it over the other triplets. In particular, to investigate the stability of the measure across different days, we considered three types of crossvalidations. The Intra cross-validation type, where the training and testing triplets belonged to the same day; the Short term cross-validation type, where the training triplets belonged to Day 1 (Day 2) and the testing triplets to Day 2 (Day 1); and finally, the Medium term cross-validation type, where the training triplets belonged to Day 1 or Day 2 (Day 9) and the testing triplets to Day 9 (Day 1 or Day 2).

Figure 2 .

 2 Figure 2. Error bars (CI = .95) related to the normalized AUC values of the classifier over the E vs H, M vs H and E vs M conditions by using the three IterationMAX values (5%, 50%, 100%) and the three cross-validation types (Intra, Short term, Medium term). The absolute AUC values have been also reported (red writings).

Figure 3 .

 3 Figure 3. WEEG values distribution across the three difficulty levels across subjects.

ACKNOWLEDGMENT

This work is co-financed by EUROCONTROL on behalf of the SESAR Joint Undertaking in the context of SESAR Work Package E -NINA research project. The grant provided by the Italian Minister of University and Education under the PRIN 2012 scheme to F.B. is also gratefully acknowledged.


 Perso-Indica Conferences

The project also aims to stimulate a wide and critical reflection on this cultural movement, especially through the organization of a series of conferences and lectures, allowing us to look into several important features of Indo-Persian translation and knowledge transmission and to explore them from different perspectives, including religious studies, philological and historical approaches, as well as the history of sciences and arts.

 The 1st Perso-Indica Conference:

!!!

  Introduction: Aims and Methods......................................................1 Published Preliminary Entries..........................................................3 FICTION AND STORIES.......................................................................3 EPIC...................................................................................................3 HISTORICAL WORKS..........................................................................4 VEDĀNTA, YOGA AND DHARMA........................................................4 TREATISES ON INDIC RELIGIONS........................................................4 MATHEMATICS, ASTRONOMY AND ASTROLOGY................................4 MEDICINE...........................................................................................5 NATURAL SCIENCES...........................................................................5 OCCULT SCIENCES..............................................................................5 SEXOLOGY..........................................................................................6 MUSIC.................................................................................................6 TREATISES OF MIXED CONTENTS.......................................................6 LANGUAGE MANUALS AND GLOSSARIES...........................................6 Published Final Articles.....................................................................7 ! The Dynamic Indexes of the Survey..................................................7 ! Perso-Indica Conferences..................................................................9 ! Other Lectures and Seminars at University of Sorbonne Nouvelle, Paris.................................................................11 ! Franco-German Program in Social Sciences and Humanities ........12 ! Perso-Indica Visiting Fellowships...................................................12 ! Collaboration with McGill University Library................................13  Introduction: Aims and Methods The last decades have witnessed a growing number of studies on Indo-Persian literary culture and societies as well as a resurgent interest in cross-cultural issues, and translation movements between different cultures, and their role in world intellectual history. However, among the great cross-cultural enterprises, the movement of translation and writing of Persian texts on Indic traditions remains certainly among the least studied. Perso-Indica is a long-term project aimed to produce an analytical survey of Persian works on Indian traditions and culture, written in South Asia between the 13th and the 19th centuries. This project intends to offer a new perspective on the movements of translation from non-Muslim cultures in the intellectual history of Muslim societies. The production of Persian texts on the Indic traditions represents one of the greatest processes of knowledge transmission that have occurred between different Asian cultures. For seven centuries, a large number of Persian studies were written on Indic religions, history, sciences, arts, geography, flora, fauna, etc. These works were often translations of older sources, mainly from Sanskrit, although many of them were original treatises written in Persian. Perso-Indica stands within the tradition of bibliographical surveys of Persian sources, yet it is very different from traditional catalogues. By the use as an online resource with free access to its entries. Perso-Indica intends to become the first reference work for this field of study and provides a unique contribution to our understanding of the history of Persian and Indian literary traditions and theirs interactions. The Survey is arranged by subject, according to the different domains of the study, each constituting a chapter. Every article is devoted to a single work, its author or translator. Entries will include among other things: a biography of the author/translator, an analytical description of the work, the Indian sources mentioned in it, details of the manuscripts and editions. Preliminary entries with a short description of the text will precede the publication of the final entries. Preliminary entries can include all the metadata of the survey and constitute a very important element of the project since they will give a synoptic view of this translation movement well before the publication of the final articles. The system allows browsing the entries of the survey through the Table of Contents of the thematic chapters and through the Indexes. One of the main features of the project has been to develop a system that acquires metadata on authors and sources, which can be used for indexation as well as for quantitative and qualitative analysis of the information included in the database. Twenty-seven Indexes created by now are accessible online (see below). It is possible to search authors and translators according to their religion, with subsections for Muslim and Indian groups; we can see them subject wise, by geographical area, or according to their cultural background, for instance, those who were historians, Sufis or poets. Likewise, we can see dedicatees and commissioners of works according to their religion and ruling elites, and also browse the full list of the original works in chronological order, according to regions and places of composition and copying, etc. These indexes are dynamic which means that they are updated automatically every time a new text and/or data are added to the database and that the menus of the vocabularies for acquiring data are extensible and modifiable. This will allow us to adapt the system and the possible choices for a given category as the need arises. For instance, the system allows to extend and modify the menu describing the religious and cultural background of authors and translators, by adding a new element to the list or by merging together two existing elements. One of the important objectives of the project is to study the different genres of texts produced within this movement as well as their evolution through the centuries. The survey enables us to distinguish between direct translations of Indic sources, new original works, and works of encyclopaedic character -that means containing chapters on different Indian sciences -as well as the chapters and the important descriptions included in heterogeneous Persian works which are not monographs on Indian learning. Moreover, the survey and its indexes will allow us to discriminate between pseudepigraphic works, illustrated ones, and also the ones written in verse. It will list the commentaries and the translations of these Persian texts into other languages, as well as the Indic authors and sources translated and referred to; it may also indicate the cases in which these Indic sources should be considered as unknown or lost.

TABLE OF CONTENTS

 OF

  Perso-Indica. An Analytical Survey of Persian Works on Indian Learned Traditions, F. Speziale -C. W. Ernst, eds., available at http://www.perso- indica.net/work/sams_al-aswat. Index of Original Titles of Sources Translated into Persian [14 item]. Index of Commissioners [20 items].

	 Published Preliminary Entries FICTION AND STORIES 1. 'Imād ibn Muḥammad Ṯaġarī, Jawāhir al-asmār 2. 'Abd al-Quddūs Gangohī, Čandāyan 35. Aśvamedha Parvan 36. Debīdās Kāyasth, Rāmāyan HISTORICAL WORKS Żiyā' 57. Šihāb al-Dīn ibn 'Abd al-Karīm Nāgawrī, Šifā al-maraż 58. Sa'd Allāh Niẓāmī Zanjānī, Tajribāt al-mujarrabāt-i Ġiyāṯ-šāhī SEXOLOGY 87. Żiyā' al-Dīn Naḫšabī, Kokaśāstra (Laḏḏat al-nisā') 88. Faqīr Allāh ibn Muḥammad 'Azīz, Kokaśāstra (Laḏḏat al-nisā') 89. Kokaśāstra (Laḏḏat al-nisā')  Published Final Articles 101. Keshavmurthy, Prashant, 2013, 6. Searchable Subject wise 8. Index of Titles of unknown or nonexistent Sources (Translated and/or Quoted) [4 items]. Searchable Subject wise	ḥaqā'īq 18. Candarman Rām, Rāmāyan Kāyath ibn Shrī Rām Kāyath of TREATISES ON INDIC RELIGIONS ibn Candarman Shrī 46. Rām Mohan Roy, Tuḥfat al-muwaḥḥidīn 47. Gītā-Subodhinī NATURAL SCIENCES 71. Qurrat al-mulk 72. Zayn al-'Ābidīn ibn Abū al-Ḥasan Karbalā'ī Hāšimī, Faras-nāma 73. Tarjuma-yi Sālōtar-i asbān LANGUAGE MANUALS AND GLOSSARIES 97. Yusūf ibn Muḥammad, Qaṣīda dar luġāt-i hindī http://www.perso-indica.net/work/matnawi-i_madavanal-kamakandala_mausum_ba_mahz-i_ijaz Religion -By Dedicatee Ruling Elite -Alphabetical Order -Chronological Order 18. Searchable Subject wise -By Commissioner Religion -By Commissioner Ruling Elite -Alphabetical Order -Chronological 98. Ārzū, Nawādir al-Alfāẓ Order 74. Qāżī Ḥasan Dawlatābādī, Tuḥfa al-faras 75. Muḥammad Qāsim ibn Šarīf 99. Śukla Mathurānātha, Sanskrit-Persian 100. Śukla Mathurānātha, Saṃskṛtā 19. Index of Commissioners of Manuscripts [5 items]. Searchable By Alphabetical Order -Chronological
	3. Čaturbhūjdās bin Mihrčand, Šāhnāma or Singhāsanbattīsī 4. Muḥammad Riżā Nu'ī Ḫabūšānī, Sūz wa gudāz 5. 'Abd al-Qādir Badā'ūnī, Ḫirad-afzā 6. Muḥammad Qādirī, Ṭūṭī-nāma 7. Ḥamīd Kalānwarī, 'Iṣmat-nāma 8. Mullā 'Abd al-Šakūr Bazmī, Rat Padam 9. Mīr 'Asgarī 'Āqil Ḫān-i Rāzī, Šam' wa Parwāna 10. Anand Rām Muḫliṣ, Hingāma-yi 'išq EPIC 11. Gopal ibn Govind Satri, Tarjama-i 37. Abū al-Fażl 'Allāmī ibn Mubārak, Ā'īn-i Akbarī 38. Sayyid Fażl 'Alī Šāh Qādirī, Kulliyāt-i Gwāliyārī 39. Šayḫ Jalāl Ḥiṣārī, Gwāliyār-nāma 40. Hirāman b. Girdhardās Munšī, Gwāliyār-nāma 41. Ḫayr al-dīn Muḥammad Ilāhābādī, Gwāliyār-nāma VEDANTA, YOGA AND DHARMA 42. Tarjuma-i dharm šāstr MUSIC 9. Index of Titles of Illustrated Persian 59. 'Alī Muḥammad ibn Ismā'il Aṣawālī Aṣīlī, Ṭibb-i šifā-yi Maḥmūd-šāhī 60. Miyān Bhuwa ibn Ḫawāṣṣ Ḫān, Ma'dan al-šifā'-i Sikandar-šāhī 61. Yusūf ibn Muḥammad, Qaṣīda dar luġāt-i hindī 62. Muḥammad Qāsim Hindūšāh Firišta, Dastūr al-aṭibbā' 63. Bīnā ibn Ḥasan, Ḫulāṣa-yi Bīnā 64. Amān Allāh Ḫān 'Amānī', Dastūr al-hunūd 65. Darwīš Muḥammad, Ṭibb-i Awrang-90. Ġunyat al-munya Works [7 items]. Searchable Subject wise -By Alphabetical Order -Chronological Order 91. Yaḥyā al-Kābulī, Lahajāt-i Sikandar-šāhī TREATISES OF MIXED CONTENTS 92. 'Abd al-'Azīz Šams Bahā' Nūrī, Tarjuma-yi Bārāhī 93. Abū al-Fażl 'Allāmī ibn Mubārak, Ā'īn-i Akbarī 103. Speziale, Fabrizio, 2013, "Majmū'a-yi šamsī", Perso-Indica. An Analytical Survey of Persian Works on Indian Learned Traditions, 10. Index of Nonexistent Persian Works [2 items]. Searchable Subject wise -By Alphabetical Order -Chronological Order F. Speziale -C. W. Ernst, eds., available at http://www.perso-indica.net/work/majmua-yi_samsi. 104. D'Hubert, Thibaut, 2013, "Bayān-i 'ibādat-i mukh-hā ba-nām-i Takādībā", Perso-Indica. An Analytical Survey of Persian Works 11. Index of Titles of Pseudepigraphic Persian Works [1 item]. Searchable Subject wise -By Alphabetical Order -Chronological Order 12. Index of Works in Verses [13 items]. Searchable Subject wise -By Alphabetical Order -Chronological on Indian Learned Traditions, F. Speziale -C. W. Ernst, eds., available Order šāhī 66. Šāh Ahl Allāh, Takmila-yi hindī 94. Salāmat 'Alī Ṭabīb ibn Muḥammad 'Ajīb al-Benārsī, Maṭāli' al-hind at http://www.perso-indica.net/work/bayan-i_ibadat-i_muh-ha_ba-nam-i_takadi-ba. 13. Index of Manuscripts [108 items]. Searchable Subject wise -By Region of Library 43. Sītal Singh "Bī-khwud", Silsila-i jōgiyān 67. José da Silva, Mufradāt-i hindī 68. Šayḫ Ḥaydar Miṣrī, Mu'ālajāt-i hindī 95. James Skinner, Tašrīḥ al-aqwām 96. Ġulām Dāwud, Ġarīb-i hind 105. Keshavmurthy, Prashant, 2013, "Maṯnawī-i 'irfān", Perso-Indica. An Analytical Survey of Persian Works on Indian Learned Traditions, F. Order -Chronological Order 14. Index of Lithographs [14 items]. Searchable Subject wise -Alphabetical	19. Candarman "Bedil", Nargisistān 20. Amar Singh, Amar-Prakāsh (Rāmāyan) 21. Sumīr Chand, Rāmāyan of Sumīr Chand 22. Naqīb Ḫān, Rāmāyan of Akbar 23. Mal'ūn-nāma 24. Munšī Jagan Kashūr "Ḥusn", Nayrang-i Ḥusn 25. Haribansa Purāna 26. Munšī Bankī Lāl "Ẓār", Khulāṣa-yi Rāmāyan 27. Makkhan Lāl "Ẓafar", Jahān-i ẓafar MATHEMATICS, ASTRONOMY AND ASTROLOGY 48. 'Abd al-'Azīz Šams Bahā' Nūrī, Tarjuma-yi Bārāhī 49. Fayżī, Abū al-Fayż ibn Mubārak, Tarjuma-yi Līlāwatī 50. 'Aṭā' Allāh Rašīdī, Tarjuma-yi Bīj ganit 51. Mīdnī Mal ibn Dhārma Narāyāṇ, Badā'i'-i funūn 52. Abū al-Ḥasan Ajūdhanī, Miftāḥ al-Ḫān, Tuḥfa-yi kān-i 'ilāj-i asp Ratnākara Order 76. Anand Rām Muḫliṣ, Rāḥat al-faras  The Dynamic Indexes of the Survey 20. Index of Years of Composition of Works [40 items]. 77. Muḥammad Ṣādiq, Ḏaḫīrat al-fu'ād 78. Bāqir Ḥusayn Ḫān, Jāmi' al-ašyā' 79. Ġulām 'Alī, Timtāl-i ašyā' wa azhār al-adwiya 80. Ġulām 'Alī, Timtāl-i ašyā' wa azhār al-adwiya 81. Sa'd Akbar, Fīl-nāma wa šikār-nāma-yi Šāh-Jahāndat 82. Dastūr al-'amal be-qawl aṭibbā-yi hindī 1. Index of Authors of Original Persian 21. Index of Years of copying of Manuscripts [44 items] Works [37 items]. Searchable by Geographical Area of Activity -By Religion and Status -Subject wise -22. Index of Years of publication of Lithographs [13 items] Alphabetical Order -Chronological Order 23. Index of Regions of composition of Works [8 items] 2. Index of Translators into Persian [35 items]. Searchable by Geographical 24. Index of Places of composition of Area of Activity -By Religion and Status Works [14 items] -Subject wise -Alphabetical Order | Chronological Order 25. Index of Places of copying of Manuscripts [14 items] 3. Index of Authors of Sources nujūm 53. Uday Lāl, Dastūr al-siyāq Translated into Persian [12 items]. 26. Index of Places of publication of Lithographs [11 items] 28. Virāṭa Parvan 29. Udyoga Parvan 30. Rāmāyan 54. Muḥammad Irtaża 'Alī Ḫān, Nuqūd al-ḥisab OCCULT SCIENCES 83. Fażl Allāh, Qiyāfa-šināsī Searchable by Geographical Area of Activity -Subject wise -By city -Order Alphabetical Order -Chronological 27. Index of Publishers of Lithographs [10 items]
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14. Naqīb Ḫān, Rāmāyan of Akbar 15. Sa'd Allāh Masīḥ Pānīpatī, Dāstān-i Rām ū Sītā 16. Giridhar Dās, Rāmāyan of Giridhar Dās 17. 'Abd al-Raḥmān Čištī, Mir'āt al-33. Bhagavad-gītā attributed to Abū al-Fażl ibn Mubārak 34. Bhagavad-gītā attributed to Abū al-Fayż ibn Mubārak Fayżī (Fayyāżī) 44. 'Abd al-Raḥmān Čištī, Mir'āt alḥaqā'īq 45. Sītal Singh "Bī-khwud", Silsila-i jōgiyān MEDICINE 55. Muḥammad Šarīf Ḫān, Ta'līf-i Šarīfī 56. Żiyā' Muḥammad Mas'ūd Rašīd Zangī 'Umar Ġaznawī, Majmū'a-yi "Tuḥfat al-hind", Perso-Indica. An Analytical Survey of Persian Works on Indian Learned Traditions, F. Speziale -C. W. Ernst, eds., available at http://www.persoindica.net/work/tuhfat_al-hind-1. 102. Fallahzadeh, Mehrdad, 2013, "šams al-aṣwāt", Searchable Subject wise. 4. Index of Local Informants [8items].

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  February 7 th -8 th , 2014. Friedrich-Wilhelm University -DFG -Université Sorbonne Nouvelle, Paris 3 -CNRS UMR 'Mondes Iranien et Indien' -Iran Heritage Foundation. -Naveen Kanalu, "Variations in translation: Two Persian versions of the Bhagavadgītā in early modern South Asia" -Perwaiz Hayat, "The Theme of Siddha in Dārā Shukōh's Writings" -Owen Cornwall, "Astrology and Translation at Emperor Aurangzeb's Court" -Martin Gansten, "The Sanskrit and Arabic sources of Nīlakaṇṭha's Tale from Mirzā 'Abdul Qādir Khān Bidel's Masnavi-e 'irfān" -Chandhar Shekhar, "The story of Rani Chander Kiran and Raja Chaturmukh"

				-Prashant Keshavmurthy, "Reading
				Puranic Time: Mirza Bedil and his
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				-Sreeramula Rajeswara Sarma, "From
	Location: Friedrich-Wilhelm University,	Persian	to	Sanskrit	Texts,
	Bonn. Organized by: Eva Orthman -Fabrizio Speziale.	Translations/Adaptations Patrons/Authors"	and
	Translating and Writing Indic Learning in Persian Date : May 30 th -31 st , 2012 Location : Université Sorbonne Nouvelle -Paris 3. Organized by : Fabrizio Speziale. Funding Institutions : Institut Français de Recherche en Iran -Université Sorbonne Nouvelle, Paris 3 -CNRS UMR 'Mondes Funding institutions: -Anna Kollatz, "Integration of	-Christopher Minkowski, "Jyotiṣa Authors at the Mughal Court: Muhūrta and Tājika" -Susanne Kurz, "Role and relationship of Graeco-Islamic and Indian elements in Persian adaptations of the Koka Shastra" -Carl W. Ernst, "Indian Lovers in Arabic and Persian Guise: Azad Bilgrami's Depiction of nayikas" -Eloïse Brac de la Perrière, "Persian paintings on Indian learned traditions" -Supriya Gandhi, "Commanding Right and Forbidding Wrong in the Mazhab Praśnatantra" -Fabrizio Speziale, "The Persian translation of the tridoṣa: Lexical analogies and conceptual incon-gruities" -Eva Orthmann, "Sanskrit-Persian Glossaries" -Prashant Keshavmurthy, "Hindi Gnosticism: a -Anna Martin, "Translation and retranslation of Indic narrative texts (dastan or qissa literature in Persian)
	iranien et indien'. Thought? Persian and Indic concepts	of the Hindus" : Situating Persian in the Mughal era"
	For more information see: http://perso-reflected in Majālis-i Jahāngīrī" indica.net/events-news.faces?news=3	Translations of Dharmaśāstra texts" -Arthur Dudney, "The Translated
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 The Second Perso-Indica Conference:

The Persianisation of Indian Learning: Texts, Approaches and

Forms of Expression

Date: For more information see: http://persoindica.net/events-news.faces?news=12

List of presentations -Audrey Truschke, "A Padshah like Manu: Political Advice for Akbar in the Razmnāmah"

 The Third Perso-Indica Conference:

The Sultanate Period and the Early Mughal Empire

Date: September 3 rd -4 th , 2015

Location: University of Delhi, New Delhi.

Organized by: Chander Shekhar -Eva Orthman -Fabrizio Speziale


F Chevy

Unitary polarized Fermi gases

Keywords:

come

-Introduction

Pairing lies at the core of the standard Bardeen-Cooper-Schrieffer mechanism for metal superconductivity, and the very natural question to know whether it could survive population imbalances between the two spin states naturally arose very soon after its development [1,2]. It was pointed out that pairing was indeed robust to some amount of mismatch between the chemical potentials of the two species, but the fate of the system after the critical imbalance is reached has long been a mystery. The absence of clear answer to this problem was due in particular to the absence of an experimental system on which the various scenarios envisioned could be tested: existence of a spatially modulated order parameter (Fulde, Ferrel, Larkin and Ovshinikov, or FFLO, phases) [3,4,5], or the extension to trapped systems [6,7,8,9], deformed Fermi surfaces [10], interior gap superfluidity [11], phase separation between a normal and a superfluid state through a first order phase transition [12,13,14,15], BCS quasi-particle interactions [16] or onset of p-wave pairing [17]. When the strength of the interactions is varied, a complicated phase diagram mixing several of these scenarios is expected [18,19,20].

This issue was revived by the possibility of obtaining fermionic superfluids with ultra cold atoms [21,22,23,24,25,26], where spin imbalance could be controlled and maintained for a long time. This led to a series of experiments performed at MIT [27,28] and Rice [29,30] which clearly demonstrated a phase separation between regions characterized by different polarizations (i.e. spin population imbalances, by analogy with magnetism). The number of phases obtained by the two groups is however different. In Rice experiment, the cloud is constituted of a core where both spin populations are equal, surrounded by a shell of majority atoms only while at MIT a third phase mixing both species with unequal densities is intercalated between the previous ones, a discrepancy which is not yet fully explained [31,32,32,33,34,35,36,37,38].

In what follows we wish to explore the various consequences of these experiments. By contrast to most recent works on the subject, we would like to avoid the use of BCS mean field, which is known to give good qualitative insight to the problem under study, but fails when precise quantitative estimates are needed. Our scheme is based on a combination of exact variational analysis and Monte-carlo simulations. We will demonstrate that, in agreement with MIT experiments, three phases are expected in homogeneous systems. To compare with experimental results, we will make use of Local Density Approximation (LDA) which leads to quantitative agreement with MIT's data. Finally, following [31], we will show how Rice's apparently contradictory results can be interpreted as a breakdown of local density approximation in elongated traps.

-Universal phase diagram of a homogeneous system

Let us first consider an ensemble of spin 1/2 fermions of mass m trapped in a box of volume V . In the s-wave approximation, the hamiltonian H is given by

H = k,σ ǫ k a † k,σ a k,σ + g b V k,k ′ ,q a † k+q,↑ a † k ′ -q,↓ a k ′ ,↓ a k,↑ , (1)
where ǫ k = 2 k 2 /2m, a k,σ annihilates a particle of spin σ and momentum k and g b is the coupling constant characterizing s-wave interactions between atoms. This choice of interaction potential is singular and yields unphysical results and to get rid of the divergencies resulting by the zero range of the potential, we introduce an ultraviolet cutoff q c in momentum space (or equivalently, we work on a lattice of step 1/q c ). When q c goes to infinity, the Lippmann-Schwinger formula obtained by the resolution of the two-body problem yields the following relationship between the bare coupling constant and the scattering length a

1 g b = m 4π 2 a - 1 V k 1 ǫ k , (2)
where the sum over k is restricted to k < q c .

To anticipate the analysis of inhomogeneous systems, we work in the grand canonical ensemble, where the atom numbers fluctuate and only their expectation values are kept constant. Introducing the chemical potentials µ ↑,↓ as Lagrange multipliers associated with the constraints on atom numbers, we need to find the ground state of the grand potential Ω given by

Ω = H -µ ↑ N ↑ -µ ↓ N ↓ . (3)
In what follows, we replace the minimization condition on Ω = Ω by a maximization problem on the pressure P , using the thermodynamical relation Ω = -P V . Moreover, we assume µ ↑ > µ ↓ and we restrict ourselves to the unitary limit where a = ∞. This choice of scattering length leads to a deep simplification of the formalism, due to the universality characterizing this regime. Indeed, from dimensional analysis [39], we can show that for an arbitrary scattering length, the pressure P of a given phase is given by some relation

P (m, , a, µ ↑ , µ ↓ ) = P 0 (µ ↑ , , m)f (µ ↓ /µ ↑ , 1/k F↑ a),
where P 0 is the pressure of an ideal Fermi gas with chemical potential µ ↑ and k F ↑ is the Fermi wave vector associated with µ ↑ . At unitarity, 1/k F a = 0 and f is therefore function of η = µ ↓ /µ ↑ yielding the universal relation

P P 0 = g(η), (4)
where

g(µ ↓ /µ ↑ ) = f (µ ↓ /µ ↑ , 0).
Although the general minimization of the grand potential is an extremely challenging and still open problem, we first note that two exact eigenstates of the system can be found.

1. Fully polarized ideal gas. If we consider a fully polarized system containing no minority atom, the interaction term in H disappears, and we are left with a pure ideal gas of majority atoms. The pressure of this normal phase is simply the Fermi pressure, and we have in particular P/P 0 = 1.

2. Fully paired superfluid. Let |SF µ be the ground state of the balanced potential

Ω ′ = H -µ( N ↑ + N ↓ ).
Since Ω ′ commutes with the atom number operators, |SF µ can be chosen as an eigenstate of both N ↑,↓ , with

N ↑ |SF µ = N ↓ |SF µ .
Going back to the unbalanced problem, we write Ω as

Ω = H + µ ↑ + µ ↓ 2 ( N ↑ + N ↓ ) + µ ↑ -µ ↓ 2 ( N ↑ -N ↓ ). ( 5
)
We see readily that for µ = (µ 1 + µ 2 )/2 we have Ω|SF µ = Ω ′ |SF µ , which proves that |SF µ is also an eigenstate of the imbalanced grand potential. The pressure in this superfluid phase can be calculated using known results for the unitary balanced superfluid for which the universal relationship between chemical potential and density reads

µ ↑ = µ ↓ = ξ 2 2m (6π 2 n ↑ ) 2/3 , (6)
where ξ ∼ 0.42 is a universal number that was evaluated both experimentally [24,41,40,42,29] and theoretically [43,44,45,14]. Integrating Gibbs-Duhem identity (see appendix), one then obtains for the imbalanced system

P SF = 1 15π 2 m ξ 2 3/2 (µ ↑ + µ ↓ ) 5/2 , ( 7
) hence P SF /P 0 = (1 + η) 5/2 /(2ξ) 3/2 .
The variation of the pressure versus η is displayed in Fig. 1. We see that for small imbalances, i.e. η smaller than η c = (2ξ) 3/5 -1 ∼ -0.10, the fully paired superfluid is more stable than the fully paired normal phase, confirming the stability of pairing against a small mismatch of the Fermi surfaces. The experimental results presented in ref. [28] suggest that the two classes of states we have until now restricted ourselves are not sufficient to fully capture the physics of imbalanced systems. In particular, a mixed normal phase, containing atoms of both species in unequal proportions, must be taken into account. A sketch of g(η) for this intermediate phase is shown in Fig 1 . On this more general phase diagram, the parameters η α and η β are of special importance, since they characterize the phase transitions between the three different phases. A glance at Fig. 1 shows that they must satisfy the inequality

η α < η c < η β ,
and the next section is devoted to an improvement of these bounds.

-The N+1 body problem

Theoretically, the existence of the intermediate phase can be demonstrated by the study of the N+1 body problem, in other word the study of the ground state of the majority Fermi sea in the presence of a single minority atom. This particular system corresponds to an intermediate phase with η → η + β and we will prove that it yields the inequality η β < η c .

To address the N+1 body problem, we use a variational scheme, that we will compare to recent predictions based on Monte-Carlo simulations [46]. Let us consider the following trial state |ψ

|ψ = φ 0 |FS + k,q φ k,q |k, q ,
where |FS is a spin up Fermi sea plus a spin down impurity with 0 momentum, and |k, q is the perturbed Fermi sea with a spin up atom with momentum q (with q lower than k F ) excited to momentum k (with k > k F ). To satisfy momentum conservation, the impurity acquires a momentum qk.

F. Chevy

The energy of this state with respect to the non interacting ground state is

H = H 0 + V , with ψ| H 0 |ψ = k,q |φ k,q | 2 (ǫ k + ǫ q-k -ǫ q ),
and

ψ| V |ψ = g B V   q |φ 0 | 2 + k,k ′ ,q φ k ′ ,q φ * k,q + k,q,q ′ φ k,q φ * k,q ′ + q,k (φ * 0 φ k,q + φ 0 φ * k,q )   ,
where ǫ k = 2 k 2 /2, and the sums on q and k are implicitly limited to q < k F and k > k F . As we will check later, φ k,q ∼ 1/k 2 for large momenta (see below, eqn. ( 10)), in order to satisfy the short range behavior 1/r of the pair wave function in real space. This means that most of the sums on k diverge for k → ∞. This singular behavior is regularized by the renormalization of the coupling constant g B using the Lippman-Schwinger formula. It implies that g B vanishes for large cutoff, thus yielding a finite energy. However, it must be noted that the third sum in ψ| V |ψ is convergent and when multiplied by g B will give a zero contribution to the final energy and can therefore be omitted in the rest of the calculation.

The minimization of H with respect to φ 0 and φ k,q is straightforward and yields the following set of equations

g B V q φ 0 + g B V q,k φ k,q = Eφ 0 (8) (ǫ k + ǫ q-k -ǫ q )φ k,q + g B V k ′ φ k ′ ,q + g B V φ 0 = Eφ k,q , (9)
where E is the Lagrange multiplier associated to the normalization of |ψ , and can also be identified with the trial energy. Let us introduce χ(q) = φ 0 + k φ k,q . We see from eqn. 9 that

φ k,q = - g B V χ(q) ǫ k + ǫ q-k -ǫ q -E . ( 10
)
As expected, we note here the 1/ǫ k ∼ 1/k 2 dependence for large k. Inserting this expression in the definition of χ, we obtain

χ(q) = φ 0 - g B V k χ(q) ǫ k + ǫ q-k -ǫ q -E , that is χ(q) = φ 0 /g B 1 gB + 1 V k>kF 1 ǫ k +ǫ q-k -ǫq-E
Finally, eqn. ( 8) can be recast as Eφ 0 /g B = q<kF χ(q)/V , that is, using the explicit expression for χ(q):

E = 1 V q<kF 1 1 gB + 1 V k>kF 1 ǫ k +ǫ q-k -ǫq-E .
We get rid of the bare coupling constant g B by using the Lippman-Schwinger equation, which finally yields the following implicit equation for

E E = 1 V q<kF 1 m 4π 2 a -1 V k<kF 1 2ǫ k + 1 V k>kF 1 ǫ k +ǫ q-k -ǫq-E -1 2ǫ k . ( 11
)
Before addressing the unitary limit case, let us show that this formula allows us to recover the known exact results in the limit of small scattering lengths where the denominator is dominated by the 1/a term. The correction to the energy is therefore

E ∼ 1 V q<kF 4π 2 a m = 4π 2 a m N V
where N is the total number of majority atoms. We thus see that the trial state recovers the mean-field prediction for low interactions. For a → 0 + (BEC regime), a little more involved calculation allows one to recover the classical molecular binding energy E ∼ -2 /ma 2 . Finally in the case of the unitary regime relevant to experiments, eqn. ( 11) is solved numerically and yields E ∼ -0.3 2 k 2 F /m, that is η β < -0.60, a value remarkably close to that obtained in Monte-Carlo simulations [46].

-Trapped system and comparison with MIT experiment

The model presented in the previous section adresses only the situation of a homogeneous system and to compare with experiments, we need to extend the formalism developed in the previous section to the case of trapped systems. To this purpose we make use of the Local Density Approximation (LDA), in which we assume that the chemical potential of species σ varies as

µ σ (r) = µ 0 σ -V (r), ( 12
)
where V is the trapping potential. From this relation, we see that varying r is equivalent to varying the chemical potentials of the two species, and in particular their ratio η(r) The two phase transitions described in the previous section will happen for radii r = R α,β such that µ ↓ (R α,β )/µ ↑ (R α,β ) = η α,β . Moreover, since the outer rim is constituted by a normal ideal gas, the boundary R ↑ of the majority component is given by the condition µ ↑ (R ↑ ) = 0.

In an isotropic harmonic trap, we can combine these three relations to eliminate the parameters µ (0) σ , thus obtaining the general relation relating the three radii R α,β,↑ :

R α R ↑ 2 = (R β /R ↑ ) 2 -q 1 -q , ( 13
)
where q = (η α -η β )/(1 -η β ). One striking consequence of this equation is the prediction of a threshold at which R α vanishes, corresponding to the disappearance of the fully paired superfluid. This transition happens when the ratio (R β /R ↑ ) 2 reaches the critical value q. From the upper and lower bounds obtained for η α and η β , we see that q > 0.30.

This prediction of LDA is remarkably well verified in MIT's experiments [28] for which the three phases discussed above were indeed observed, and eq. ( 13) could be tested experimentally (Fig. 2). On this graph, we see that for large imbalance, the linear scaling predicted by eq. ( 13) is indeed satisfied, with q ∼ 0.32, in agreement with the lower bound obtained earlier. The deviation from theory observed for (R β /R ↑ ) 2 0.5 is not yet fully understood. However, it must be noted that the discrepancy takes place in a regime of low imbalance, where the phase transitions take place in the tail of the density distribution. In these regions of low density, we may observe a breakdown of the LDA, or of the hydrodynamical expansion that was used to infer the experimental radii.

The value q ∼ 0.32 obtained from the comparison with experimental data can help us improve the bounds for η α,β . Indeed, this relation fixes the relative values of η α and η β . When combined with the bounds found in the previous section, we obtain indeed -0.62 < η β < -0.60 (14) -0.10 < η α < -0.088 (15) From the previous analysis, we see that the combination of theoretical arguments and analysis of experimental data allows for a precise determination of the thresholds of the different phase transitions. Knowing the values of η α,β as well as the exact equation of state in the fully polarized and fully paired phases, we can even obtain some upper and lower bounds for the equation of state of the mixed phase, using the concavity of the grand potential [36]. 0,00 0,25 0,50 0,75

0,0 0,1 0,2 0,3 0,4 0,5 (R > /R ↑ ) 2 (R = /R ↑ ) 2 n r R α R β R ↑ Fig. 2.
-Comparison of equation ( 13) with experimental data from MIT. A fit to the data yields q ∼ 0.32. Inset: sketch of the density profile. The full (resp. dashed) line corresponds to the density of the majority (resp. minority) component. Rα marks the end of the superfluid region, R β that of the mixture and R ↑ is the frontier of the majority cloud.

-Elongated systems and Rice's experiment

Surprisingly, similar experiments performed at Rice University showed no evidence of an intermediate phase, but rather the coexistence of the fully paired and fully polarized phases only. Measurements of the axial radii of the two phases from ref. [29] are presented in Fig. 3 and can be compared with the model presented above when omitting the intermediate mixed phase [37]. In these conditions, the inner superfluid region is now defined by the condition µ ↓ (r)/µ ↑ (r) < η c and is bounded by the radius R ↓ defined by

R 2 ↓ = 2 mω 2 µ 0 ↓ -η c µ 0 ↑ 1 -η c . (16)
Atoms of the minority species are located in the paired superfluid phase only. We thus have where R2 = (µ 0 ↑ + µ 0 ↓ )/mω 2 and

N ↓ = r<R ↓ n ↓ (r) d 3 r = 2 3πξ 3/2 µ 0 ↑ + µ 0 ↓ ω 3 g(R ↓ / R), (17) 0,0 0,2 0,4 0,6 0,8 1,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 R/
g(x) = x √ 1 -x 2 -3 + 14 x 2 -8 x 4 + 3 arcsin(x) 48 . (18)
Excess atoms of the majority species are located between r = R ↓ and r = R ↑ such that mω 2 R 2 ↑ /2 = µ 0 ↑ . The number of excess atoms is therefore

N ↑ -N ↓ = R ↑ R ↓ n 1 (r) d 3 r, hence N ↑ -N ↓ = 2 3π 2µ 0 ↑ ω 3 (g(1) -g(R ↓ /R ↑ )). ( 19
)
Dividing by ( 19) by ( 17) yields the implicit equation for

η 0 = µ 0 ↓ /µ 0 ↑ as a function of N ↑ /N ↓ N ↑ N ↓ = 1 + ξ 3/2 8 (1 + η 0 ) 3 g(1) -g(R ↓ /R ↑ ) g(R ↓ / R) . (20)
Equation ( 20) is solved numerically and the value obtained for η 0 is then used to calculate the radii R ↑ and R ↓ . The predicted evolution of the R i versus the population imbalance P = (N ↑ -N ↓ )/(N ↑ + N ↓ ) is shown in Fig. 2. To follow Ref. [29], we have normalized each R i to the Thomas-Fermi radius R TF associated to an ideal gas containing N i atoms. The agreement with the experimental data is quite good as soon as P 0.1, a remarkable result, since the model presented here contains no adjustable parameter, as soon as the value of ξ is known.

Despite this remarkable agreement, this simple two phase+local density approximation model fails to captures all experimental features. In particular, a qualitative discrepancy occurs in the comparison between the theoretical and integrated density profiles. Indeed, as shown in [47], LDA at unitarity implies a constant density difference in the paired superfluid region, in contradiction with experimental data. One solution to this problem was presented in [32,31]. In these papers, it is noted that in the presence of phase transitions, the description of the sharp frontier separating to adjacent phases involves the introduction of density gradient terms in the energy. When the interface in thin enough, they can be encapsulated in a new surface tension energy term reading [31]

Ω ST = S γ(µ ↑↓ (r))d 2 S,
where S is the interface between the two phases, and γ is the surface tension constant, which should dimensionally vary as

γ = λ mµ 2 ↑ 2 .
Here, λ is a numerical factor that will be determined by comparison with experiments and we have used the fact that at the coincidence between the phases, the ratio µ ↓ /µ ↑ is fixed and equal to η c , meaning that the two chemical potentials are no longer independent. We can minimize the total grand potential Ω = Ω bulk + Ω ST , where Ω bulk = -(P N + P SF ) d 3 r is the bulk contribution to the energy. Following [31] we simplify the analysis by assuming that the interfaces are ellipsoidal, and for λ ∼ 10 -4 one obtains the results presented in Fig. 4, which coincides with experimental data. The absence of capillary effects at MIT can be explained by a smaller trap aspect ratio and a larger atom number of atoms compared with Rice's experimental situation, as shown by a simple scaling argument [31].

-Conclusion

The formalism presented here allows for a simple and quantitative description of macroscopic properties of polarized Fermi gases in the regime of strong interaction. This analysis is nevertheless far from being complete, since it does not give any information on the superfluid nature of the various phases. For instance, the mixed region of the phase

-Appendix: thermodynamical relations for the grand potential

Let us consider a homogeneous many-body system characterized by a hamiltonian H 0 and containing particles of p different species labelled by i = 1..p. In the grand canonical ensemble, one looks for the ground state of this system by letting the atom numbers fluctuate, but keeping the expectation values N i=1..p fixed. This therefore requires to find the ground state of the grand potential Ω(µ i ) = H -p i=1 µ i N i , where the µ i are Lagrange multiplier that we interpret as chemical potentials.

Let |ψ(µ i , V ) be the ground state of the grand potential, we set Ω(µ i , V ) = ψ(µ i , V )| Ω|ψ(µ i , V ) . Using Hellman-Feynman relation, we can write that

∂Ω ∂µ i = ψ(µ i , V )| ∂ Ω ∂µ i |ψ(µ i , V ) = -N i , (21)
from which we deduce that

dΩ = i -N i dµ i + ∂Ω ∂V dV. ( 22
)
By definition, and by analogie with classical thermodynamics, we identify ∂ V Ω with -P , the pressure in the system.

Let us now us the extensivity of the potential: when the volume is multiplied by some scaling factor λ, Ω is multiplied by the same factor. In other words, we have Ω(λV, µ i ) = λΩ(V, µ i ). Taking λ = 1/V , we get Ω(V, µ i ) = V Ω(1, µ i ). Differentiating this with respect to V , we note that Ω(1, µ i ) = -P , hence Ω = -P V (23) From this equation, we see that the minimum grand potential is state has also the highest pressure. P can moreover be calculated by differentiating Ω and using equations ( 23) and (22). We then obtain the Gibbs-Duhem relation

dP = i n i dµ i , (24)
where n i = N i /V is the density of species i. From equation (24), we see that the pressure (hence the grand potential) can be obtained simply from the knowledge of the equation of state n i (µ j ).

8

. 1. Concavity. -Since, by definition, |ψ(µ i ) is the ground state of Ω(µ i ), we have for any δµ i ψ(µ i + δµ i )| Ω(µ i )|ψ(µ i + δµ i ) ≥ ψ(µ i )| Ω(µ i )|ψ(µ i ) (25) Moreover, if one notes that Ω(µ i ) = Ω(µ i + δµ i ) + j δµ j N j , we see that for any δµ i Ω(µ i + δµ i ) + j δµ j N j (µ i + δµ i ) ≥ Ω(µ i ) (26) Finally, recalling that N i = ∂ µi Ω and after expansion of equation ( 26) to second order in δµ i , we obtain ∂ 2 Ω ∂µ i ∂µ j δµ i δµ j ≤ 0, (27) hence proving the concavity of the grand-potential (or conversely the convexity of the pressure).

Fig. 1 .

 1 Fig. 1. -Sketch of the grand potential Ω as a function of η = µ ↓ /µ ↑ . Ω is normalized to the grand potential Ω0 of the pure ideal gas of chemical potential µ ↑ . Full line: paired superfluid; dotted line: fully polarized normal phase; dashed line: intermediate mixed phase. ηα and η β designate the critical values for the two transitions between the superfluid/mixed phase and the mixed phase/fully polarized Fermi gas.

Fig. 3 .

 3 Fig.3. -Rice's radius measurement and comparison with a two phase model. The radius Ri is scaled in units of the Thomas Fermi radius of an ideal gas with a the same atom number Ni.

Fig. 4 .

 4 Fig. 4. -Integrated density difference in Rice experiments, and comparison with the surface tension model (data from [31]). Dashed line, LDA prediction: the density difference is flat in the superfluid region, in contradiction with experimental date. Full line: Two phase model incorporating surface tension effect. The same surface tension parameter λ =∼ 10 -4 is used for all three graphs.

  The author gratefully acknowledges support by the IFRAF institute and the ACI Nanosciences 2004 NR 2019. The author thanks the ENS ultracold atoms group, S. Stringari, C. Lobo, A. Recati, A. Bulgac, E.A. Mueller, X. Leyronas, C. Mora and R. Combescot for stimulating discussions. Laboratoire Kastler Brossel is a research unit No. 8552 of CNRS, ENS, and Université Paris 6.


Jordane Lorandel
email: jordane.lorandel@insa-rennes.fr

Jean-Christophe Prevotet
email: jean-christophe.prevotet@insa-rennes.fr

Maryline Helard
email: maryline.helard@insa-rennes.fr

Dynamic Power Evaluation of LTE Wireless Baseband Processing on FPGA

Mobile networks and user equipments continuously evolve to circumvent the data traffic growth and the increasing number of users. However, the complexity and heterogeneity of such systems (3G, LTE, LTE-A, etc.) makes power one of the most critical metric. In this context, power estimation has become an unavoidable task in the design process. In this paper, a dynamic power estimation methodology for FPGA-based systems is presented. It aims at providing accurate and fast power estimations of an entire system prior to its implementation. It also aims at making design space exploration easier. We introduce an innovative scenario-level in order to facilitate the comparison of domain-specific systems. We show the effectiveness of our approach on several LTE baseband configurations which leads to a low absolute error, compared to classic estimations. It also exhibits a high speed-up factor which is determinant during design space exploration.

I. INTRODUCTION

Today, the data traffic that is generated on mobile networks continues to grow rapidly. According to [START_REF]Cisco Visual Networking Index: Global Mobile Data Traffic Forecast Update[END_REF], global mobile data increases of 69% in 2014 and it will have a compound annual growth rate of 57% from 2014 to 2019.

To deal with these issues, mobile networks and user equipments tend to constantly adapt their processing capabilities. Among all possible solutions, a popular example is the LTE standard. The complexity of systems like LTE makes their design and development a challenging task, especially when they are implemented in embedded systems in which specific constraints have to be taken into account (power, size, performance, etc.). The number of parameters that can have an impact over power consumption makes the power estimation even more difficult. As the new technologies clearly enhance the performance in terms of throughput, QoS, it also implies a higher power consumption and more heat dissipation.

One of the most popular families of digital circuits in embedded systems are the Field Programmable Gate Arrays (FPGA). These devices represent an attractive technology and make it possible to implement complex systems due to their high density of gates and heterogeneous resources. As compare to ASIC that can achieve better performance [START_REF] Kuon | Measuring the gap between fpgas and asics[END_REF], FPGAs offer more flexibility. FPGA-based systems can be made of IP (Intellectual Property) which are hardware cores that facilitate design reuse and speed up development time. Their power consumption is generally divided into static and dynamic power. Static power comes from leakage currents whereas dynamic power is generated by the transistors switching activity as soon as the circuit is active.

In wireless communication systems, complexity of baseband processing is still increasing and estimating power and performance of such systems has become a hard task. Moreover, the power consumption depends on system parameters (e.g. the number of antennas, the modulation and coding scheme, the type of frame to process, etc.) as well as technological parameters (e.g clock frequency, data width, the FPGA type, its speed, etc.).

The purpose of this paper is to propose a power estimation methodology for FPGA-based wireless communication systems. Its objectives are multiple : 1) it aims at facilitating the design space exploration and providing fast power estimation at high level 2) it aims at considering dynamic IP time activities that depend on the application and the testbench which have a deep impact on the power estimation. In this work, we deliberately chose to focus on hardware design without taking any software considerations into account. Moreover, we decided to only focus on the baseband processing. The study of RF circuits will be out of scope of this study.

This paper is organized as follows. Section II deals with the related works on high level power estimation. Section III promotes our proposed methodology. Section IV provides results that have been obtained by applying our methodology on a LTE DL SISO-OFDM system. Finally, we conclude and discuss about prospects in section V.

II. RELATED WORKS

Power estimation in FPGAs has been a wide topic of research for several decades. The various related techniques are commonly classified according to the abstraction level of the system to implement i.e. the level of details to describe this system.

Generally, as designers want to implement a complete system into a FPGA, they follow a generic top-down design flow, described in Figure 1. This multi-levels flow starts from a behavioural description of the design and ends at the layout level in which the bit-stream is generated. Note that these description levels are common to all hardware design flows and are obviously adapted to wireless communication designs.

It has already been noticed that estimating power at lowlevel of abstraction or performing real measurements can obviously deliver accurate results, but is also very timeconsuming. This precise estimation relies on the fact that the system is completely defined at this last level. This also forces Fig. 1. Representation of the different abstraction levels from specification to FPGA bit-stream generation designers to take a decision very late in the design flow.For this reason, it is clear that low-level approaches are not suitable for fast power estimation of future systems composed of billions of gates.

At System level, early decisions can be taken and may have a more significant impact than lower level decisions on the overall system performance and power consumption [START_REF]International Technology Roadmap for Semiconductors (ITRS)[END_REF]. This is especially true when the primary objectives are the design of low-power systems. Although a lot of tools make it possible to work at this level, few of them actually enable both power and performance estimations because implementation details are not available at this level.

In the wireless communication domain, a system is usually described using modelling and simulation tools such as the Matlab/Simulink tool from Mathworks [4] which is widely used in the community. Other tools such as Ptolemy [START_REF]The ptolemy project[END_REF], Labview [START_REF]Labview system design software[END_REF], Cadence System development Suite [START_REF] Cadence | System development suite[END_REF] are some examples of behavioural-level tools and Electronic Design Automation (EDA) environment for modelling, simulating, prototyping that are used in the signal processing and wireless communication domains.

Although very powerful, some tools basically do not integrate power estimation in their design flow. These tools do not allow designers to get any information related to the energy spent by the algorithms. For example, Matlab basically does not enable power estimation without an additional package to complete the FPGA chip development such as System Generator from Xilinx.

To address the lack of detailed information at high level, power and performance details can be feedback from lower levels to Matlab/Simulink based on the synthesis of param-eterized designs for FPGA using the System Generator tool [START_REF] Ou | Pygen: a matlab/simulink based tool for synthesizing parameterized and energy efficient designs using fpgas[END_REF].

Another way of obtaining global estimation of circuits consumption consists in using general analytical power models. In [START_REF] Desset | Flexible power modeling of lte base stations[END_REF], [START_REF] Deruyck | Modelling the power consumption in femtocell networks[END_REF], power models or estimations are derived from datasheet values, real measurements or are possibly based on other works presented in the literature. The accuracy of the power estimation is generally not the priority since the main goals are to determine a global trend in terms of performance to achieve.

Other approaches are based on specific power models in which hardware is totally defined. In [START_REF] Abdelli | High-level power estimation of fpga[END_REF], the development of multi-level power models enable to estimate power of FPGA designs all along the design development in function of the available information at each level. Power models are built using the Functional Level Power Analysis (FLPA) methodology [START_REF] Elleouet | A high level soc power estimation based on ip modeling[END_REF], [START_REF] Rethinagiri | Hybrid system level power consumption estimation for fpga-based mpsoc[END_REF].Another power modelling approach is described in [START_REF] Chandrasekaran | A new behavioural power modelling approach for fpga based custom cores[END_REF]. The methodology called FLPAM, allows to model area and power of FPGA-based Custom IP cores. The authors focus on modelling the dynamic power (clock, logic, signal, I/O) of the different component of the FPGA and on performing non linear regressions in function of system variables such as area, frequency etc. The overall dynamic power is then estimated from these parameters, for each IP in the design.

Power models of IP cores for Multi-Core System-on-Chip targets, are built from low-level simulations and mode recognitions based on key signals in [START_REF] Hsu | Powerdepot: Integrating ip-based power modeling with esl power analysis for multi-core soc designs[END_REF]. Although very interesting in practise, this approach seems too power-oriented and does not take into account other metrics such as the global performance of the system.

FPGA vendors have also contributed to the development of early power estimation tools called spreadsheets as Xilinx Power Estimator (XPE) [START_REF]Xpower estimator user guide[END_REF] from Xilinx. The objective of these tools is to provide early power estimates based on the user design specifications, prior to any implementation. Average power consumption is estimated using analytical formulas based on parameters such as the number of resources, the clock frequency, the signal activity etc. Spreadsheet power estimations can be refined throughout the design implementation. Nevertheless, obtaining power estimations is generally a long and tedious task, especially when large and complex systems are considered.

An efficient way to deal with system complexity consists in using high level languages such as C/C++ or high level description languages e.g. System-Verilog or SystemC [START_REF]Systemc[END_REF]. As an example, SystemC consists of a C++ library and its associated simulation kernel which enables to run cycle-accurate simulations. SystemC seems to be convenient to model hardware designs. However, it does not basically enable power estimation and additional information has to be provided. To this purpose, power can be estimated using macro-models [START_REF] Damasevicius | Estimation of power consumption at behavioral modeling level using systemc[END_REF] or Power State Machines (PSM) [START_REF] Bergamaschi | State-based power analysis for systemson-chip[END_REF].

Finally, as related in the previous examples, power estimation in FPGAs can be performed using many methodologies, modelling techniques or languages. It can also be performed at different levels of abstraction. At system level, power estimation may be a hard task, due to the lack of information, and may lead to poor accuracy. Power can also be estimated from generic analytical models or from values coming from the literature. Although very interesting in practice, exploiting these results may not be well adapted to the design of large and complex systems since available studies are generally performed on specific targets and are not easily scalable. Several approaches require dedicated tools to realize the design implementation of an entire system in order to estimate the power consumption accurately. This prevents an efficient design space exploration. From our knowledge, there is also a lack of high-level tools that enable to perform an efficient comparison between several configurations of FPGA-based systems in terms of power consumption and performance, especially in the wireless communication domain. Power modelling of cores is also interesting but it usually do not consider the performance of the overall system.

III. PROPOSED METHODOLOGY

The proposed methodology is based on the assumption that any system can be represented by a group of hardware blocks such as IPs that are dedicated to a specific function e.g. Fast Fourier Transform (FFT). The main idea consists in providing power estimations of a global system, only by taking accurate information from each of its sub-elements. Each sub-element has been fully characterized and is available in a library. One of the main purpose of this approach is to prevent high development time of the entire system and related costs by encouraging models re-use.

The critical entry point of the methodology is the innovative definition of a scenario. This term has already been introduced in [START_REF] Gheorghita | System-scenario-based design of dynamic embedded systems[END_REF], [START_REF] Zompakis | Enabling efficient system configurations for dynamic wireless applications using system scenarios[END_REF] but refers to a different concept as in this work. In our case, a scenario refers to a set of parameters which are common to several applications in the same domain. It is composed of systems and hardware-oriented parameters which have an impact over power and/or performance.

An example of scenario in the wireless communication domain may refer to the modulation type, the coding scheme, the IFFT size, and also technological parameters such as the clock frequency, the FPGA target, the data quantization etc. It constitutes the meta-model of the system. As depicted in Figure 2, each application corresponding to a defined set of parameters is considered as an instance of a scenario.

The proposed methodology is performed in two stages: an IP characterization phase and a global system simulation using SystemC as modelling language. The first stage of the methodology consists in building a library of systemC IP that have been fully characterised in terms of power and behaviour at both low and high levels. To this purpose, we use a classical low level power estimation tool from Xilinx called XPower Analyzer (XPA) that enables to estimate the average power consumption and describe the behaviour of each IP at system level.

A classical characterisation flow has been applied to each IP in order to obtain accurate power estimation. A post-place and route VHDL simulation model, including timing properties, is generated under the Xilinx design software environment called ISE (v14.4). This model is then simulated using the Modelsim SE-64 10.1c [START_REF]Modelsim simulation tool[END_REF] tool in order to record the internal activities of all signals that constitute the IP. Based on specific test-benches, two simulations are performed. The first one is performed when the IP is active and the other one when the IP is idle. Finally, power is estimated using XPA in a classical way based on the activity file and additional implementation files i.e. constraint and design netlist files. Note that XPA delivers a complete report on the average power used by clock, logic, signals, memories, DSP blocks, etc.

The characterisation process using XPA and the timing simulation have to be re-run several times for each set of parameters. The characterisation process is effective when all power estimation results have been obtained for every IP that are used in the system. This first stage is quite tedious but has been relieved by the use of automated scripts that also spare time and reduce the number of errors.

After obtaining the power metrics for each IP configuration, this information is added to SystemC models that have been built. All IPs share the same implementation model that consists of a data path and a control path. These paths can be described at different levels of abstraction regarding the expected accuracy and the simulation speed. A major contribution of the work is the introduction of key signals that aim at identifying at high level if the IP is active or not and thus determine which power model has to be considered. Indeed, two patterns of test that are applied to a given circuit can lead to very different power consumption results. In a real system that is made of several IPs, the behaviour of one IP can have a large impact on the others. The importance of taking into account the timing activities of each IP based on the application parameters are exhibited in Figure 4. From this figure, it can also be noticed that different sets of IP parameters can lead to different timing behaviours and finally to different power consumption models. Although XPA delivers average power consumption results, dynamic effects of the application can be considered with our approach.

The second stage of the methodology is described in Figure 3. The system is built by connecting the SystemC models that have been developed during the first stage. Once the SystemC Fig. 3. High-level system modelling and simulation using SystemC Fig. 4. IP time-activities for different applications model of the entire system is realized, the users can define their applications i.e. instances of a scenario by setting up the system parameters. Simulations are then performed and time activity coefficients of all the IP models can be obtained. Finally, the dynamic power consumption of the entire system can be estimated by considering the power contribution of each IP that constitutes the system.

Thanks to the proposed methodology, several systems as LTE or WiMax baseband processing can be studied. Each block that composes such systems has to be characterized and then modelled using SystemC. A library of hardware and SystemC models is already being developed.

IV. USE CASES

Long-Term Evolution (LTE) is the fourth generation (4G) of radio technology that has been standardized by the 3GPP cooperation. The main objectives of LTE aim at reducing the latencies, improving throughputs while dealing with an increased number of users.

In order to meet these objectives, the LTE physical layer combines several technologies such as Orthogonal Frequency Division Multiple Access (OFDMA) for downlink (DL) and Single Carrier Frequency Division Multiple Access (SC-FDMA) for Uplink (UL).

According to the standard [START_REF]Evolved universal terrestrial radio access (e-utra) physical channels and modulation[END_REF] and to an overview presented in [START_REF] Zyren | Overview of the 3gpp long term evolution physical layer[END_REF], the main OFDM parameters in LTE are summarized in Table I.

The LTE frame, with a 10 ms duration, is built around different physical channels and signals that carry and exchange specific data between a base station and the user mobile equipment. Two frame structures are defined in LTE. We focus on the structure of type 1 that supports Frequency Division Duplex (FDD) mode whereas type 2 uses the Time Division Duplex (TDD) mode. As depicted in Figure 5, the channel concept enables to identify the specific type of data that are transmitted on the radio interface. For DL, there are 6 physical channels and 2 types of physical signals. We focus on Physical Downlink Shared CHannel (PDSCH) that is dedicated to user data. Different processes (modulation, coding, etc.) are performed according to the nature of the physical channels as shown in Figure 5. Physical Signals are generated by the physical layer and are known at the receiver. They distinguish Reference Signals (RS) that are allocated both in time and frequency and enable channel estimation, beamforming, channel demodulation and Synchronisation Signals (SS) that are mainly used for time and frequency synchronisation, CP length determination, cell search, etc.

As depicted in Figure 6, we have developed a simplified LTE DL transmitter that corresponds to the PDSCH processing (user data) and CS. The description has been made using the VHDL hardware description language.

The transmitter is composed of a Xilinx IP called DLChannelEncoder that adds checksum i.e. crc, enables transport block segmentation, channel coding, performs rate matching and code block concatenation. Then, binary data are sent to the scrambler that mixes the binary stream with a cellspecific pseudo-random sequence. A QAM modulation is then performed in order to map the binary stream into I/Q complex symbols. Cell-Specific Reference Signals (CRS, also named as pilots) are generated by a dedicated block. The resource mapper aims at generating the corresponding LTE frame as defined in the standard. Finally, the OFDM modulation and cyclic prefix insertion are performed.

For this system, some assumptions have been made:

• the simulation time corresponds to the generation of half of the LTE frame (70 OFDM symbols, 10 slots)

• some parameters for the CRS generation have been set such as the ID cell

• all the RB used for PDSCH and CRS are considered ; data from other channels are deliberately set to 0. A scenario has also been defined in order to efficiently compare different applications in terms of power and performance. This is summarized in Table II. It can be noticed that (I)FFT sizes are different for the 4 studied applications.

Each hardware IP that composes the hardware system has been characterized independently and SystemC models have been built according to the proposed methodology. Simulations have been performed at high-level for the 4 applications and corresponding IP time-activities have been obtained. Finally, the results based on the proposed methodology have been compared with classical XPA estimation that have been performed on the entire system. Results are provided in Table III for the different applications.

It can be noticed that power estimations resulting from our methodology are close to the XPA results. The maximum absolute error is around 10% (due to relatively small power estimation values). A comparison is also performed with the results obtained by a typical spreadsheet approach (sum of average powers) that are indicated by the red curve. Our methodology clearly outperforms the classical way of power estimation Moreover, as described in Table III, we estimate the power consumption of the different applications in few seconds. Note that cycle accurate simulations took hours using both Modelsim and the XPA tool, in a classical way. Moreover, we did not consider the time spent for the design implementation that is also time-consuming as compared to our approach that only requires the compiling of the SystemC models.

According to the Figure 7, if the power consumption related to the circuit is only considered, application 1 may be selected, at first glance, because it consumes less power than the others. In addition to power, domain-specific metrics can also be evaluated very fast in the proposed approach. For example, the Energy Efficiency (EE) of a wireless communication system can be used to evaluate the capacity of a system to transmit the maximum of data with a minimum of energy. EE is usually defined as follows in a SISO (Single Input Single Output (SISO)) channel:

EE(bit/J) = C(bit/s) P T otal (W = J/s) (1)
with C, the average channel capacity in SISO configuration:

C = W.E[log2(1 + |h| 2 P t N o )] (2)
with W the bandwidth, h the channel fading coefficient, N o the normalized unilateral spectral density of noise, P t the power allocated for data transmission. P T otal is the average total power (P T otal = P c + P t ) with P c , the average power consumption of the circuit. P c , is often neglected or omitted when large base stations (BS) are considered. However, this power consumption is no longer negligible for micro, pico or femto BS [START_REF] Auer | Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]. In our case, we take into account the power consumption of the baseband processing during the EE evaluation. As shown in Figure 8, the EE was computed for the 4 applications. It can be noticed that the EE of application 4 is higher than the others. This is due to the largest bandwidth and thus the largest capacity. However, more hardware resources are used for application 4 which leads to a higher power consumption. As previously mentioned, if only power is considered, application 1 should be selected. According to the EE metric, the application 4 has to be favoured. This clearly underlines the benefit to take into account both high and lowlevel details when such metrics are evaluated.

V. CONCLUSION AND FUTURE WORKS

In this paper, we have presented a system-level power estimation methodology for FPGA-based designs that are built around IP hardware blocks. The methodology has been evaluated on the LTE DL physical layer and provides fast and accurate power estimations. The introduction of the scenario concept enables to efficiently observe the impact of one parameter or another on the power consumption and the performance. Moreover, domain-specific metrics can be studied such as the energy efficiency based on realistic values. Finally, the proposed methodology allows to compare several systems in an efficient way in order to obtain the best trade-off between consumed power and performance.

As future works, the proposed approach will be used to compare several baseband processing schemes of various wireless communication systems such as LTE and WiMAX. Extended configurations such as multiple antennas systems will be investigated. Moreover, we aim at integrating additional power consumption models of RF stages in order to consider all the elements of a wireless communication system and provide an overall power estimation.

Fig. 2 .

 2 Fig. 2. Introduction of the scenario-level

Fig. 5 .

 5 Fig. 5. LTE TX Downlink processing for each physical channel without references and synchronisation signals

Fig. 6 .

 6 Fig. 6. Scheme of LTE TX DL processing for PDSCH channel in SISO configuration

Fig. 7 .

 7 Fig. 7. Average TX dynamic power for the 4 applications

Fig. 8 .

 8 Fig. 8. Energy Efficiency versus Total Power for the 4 applications

TABLE I MAIN

 I DOWNLINK OFDM PARAMETERS IN LTE

	Spectral Bandwidth
	(MHz)	1.4	3	5	10	15	20
	Sub-carrier spacing				15 kHz
	Used sub-carriers	72	180	300	600	900	1200
	Used Resource Blocks	6	12	25	50	75	100
	(I)FFT Size	128	256	512	1024	1536	2048
	OFDM length		66.67µs (without Cyclic Prefix)
	Cyclic Prefix length	normal: 5.21µs (1st symbol) then 4.67µs
				extended: 16.67µs