Juliette Lancel

THE DREAM AND THE SIN. EROTIC DREAM IN THE FRANCE OF THE SEVENTEENTH AND EIGHTEENTH CENTURIES

The Dream and the Sin. Erotic Dream in the France of the Seventeenth and Eighteenth Centuries.

Introduction

Is dreaming a sin? What does God think of our dreams? The sexual dream [i] , which is caracterised by erotic contents and may come along with nocturnal emissions, crystallises the numerous questions emerging from the relationship to the dream in the France of 17 th and 18 th centuries. At this time, the oneiromancy, which is the art of divination by the interpretation of dreams, spreads primarily through books of dreams. These are small treatises, giving indications on the essence of dreams, explaining how they emerge and how to acknowledge prophetics dreams and to make them happen. These books of dreams include also a series of interpretations, arranged by thematic or alphabetical order. This constitutes a distinct genre, present since the ancient times and self-perpetuating, where each author borrows from previous authors. There is in France, several types of dreambooks: scholarly works, mostly in latin, and some light works intended to a broader audience wich may include the uneducated masses. Books for the common man belong mostly to the Bibliothèque bleue (Blue library), a format born in the early XVIIth century after two families of printers, the Oudot and the Garnier who used to recycle knightly stories, tales and other popular works, including dreambooks.

In order to reduce the production costs, they were cheaply printed on small blue covered booklets, hence their name. Similar to the english chapbooks although with less illustrations, they use to account for a large part of the popular book culture. Yet not, it would be risky to consider these books as a typical representation of popular culture of the time, they were books intended for the popular classes.

Most of these books weren't even published for this reason in the first place and were recycled earlier works. Le Palais des curieux ('Palace of the Curious'), for instance, was a successful dreambook which first edition wasn't intended to be published at the Bibliothèque bleue. Its content is even quite scornful against the populace, described as unable to have premonitory dreams. Furthermore, we don't know how theses books were spread. There probably used to be group readings aloud which might have encouraged the oral dissemination of these dreambooks but it may as well have blent with pre-existing oneiromancy practices. But over the period of time considered, the stance on premonitory dream is evolving. When, at the beginning of the seventeenth century, it is commonly accepted to be worried after having dreamt of bad news, admitting to believe in one's dreams is getting less and less socially accepted during the century. Most of the treatises proposing interpretations to predict the future play the card of entertainment: interest in dreams is only for fun and should not be considered as a serious matter. Thus, in Le Palais du Prince du Sommeil ('The Palace of the Prince of Sleep'), which he publishes in 1667, Célestin de Mirbel offers sophisticated readers, that he calls 'courtiers of the Prince of Sleep', a recreational use, making it even a way to seduce 'the most strict women' [ii] . Being lawyer at the Parliament of Paris, and author of law and philosophy works written in latin, he almost apologises in his preface for publishing such a futile work.

It is in this context that fifteen years later, in july 1682, Louis XIV promulgates in Versailles an edict "for the punishment of poisoners, diviners and others". This edict accompanies the police repression following the scandal of the Poisons affair, which involved great names of the French Court. This struggle against superstitions by State and police goes along with a similar struggle from French authors publishing their works at this time and until the 18 th century.

But this does not prevent the authors of books of dreams, in the foreword as well as in the following text, from maintaining the ambiguity, from using subtle nuances and from quite often keeping the underlying assertion of a predictive capability. It would be difficult to assume that this ancient belief is not lingering undercover.

In the context of this widespread evolution towards a more endogenous explanation of dream, how is considered the erotic dream? Is it a sin, a symbol, a mere physical manifestation? Is the dreamer responsible for his or her dreams? The responsibility, that is the accountability for one's actions, seems a priori extraneous to the dream, which is in the time of the sleep, out of reach of the consciousness: that is why the actions of somnambulists are not punished by the French law in the seventeenth century [iii] . However, what is not governed by law may be governed, for some authors, by moral responsibility. They assert that one has the ability to influence on one's dreams through diurnal behaviour, which implies moral responsibility for these dreams. As one may act upon the contents of his/her dreams, there comes a series of duties distinguishing the good dreamer from the bad. In order to specifically deal with the discourse on the erotic dream, this study does not consider dream accounts but focuses on theoretical texts on dreams, from various origins.

I Sexual content in dream interpretation

According to oneiromancy, the erotic dream is a divine symbol to be deciphered. Thus, it has no moral significance by itself but is only conveying good or bad significances. For the historian of dream, treatises on oneiromancy (named in French 'clé des songes', literally 'dream key') may shed light on the perception of sexuality at a given time.

In 1984, Michel Foucault makes a founding contribution in starting the third volume of his History of Sexuality, The Care of the Self, by a first chapter devoted to sexual dreams in Artemidorus Daldianus, a Greek oneirocritic of the 2 nd century AD, author of Oneirokritika, the only treatise on dream interpretation which reached us in its whole. According to Artemidorus, sexual acts seen positively, i.e. in conformity with law and nature, convey positive predictions, while wrong sexual acts are considered as adverse predictions. Using this approach, Foucault succeeds in giving an overview of sexual practices accepted or condemned. Thus, in the dream of a relationship between two men, an older active partner means the dream is positive. If he is younger or poorer, the dream is negative, because it reverses the social hierarchy. The feasibility of applying this interpretation scheme to premodern history is an open question.

During the seventeenth century, the approach to sexual dreams is far less explicit, but may inform on a social system of representations. In Le Palais des Curieux, a small treatise published in 1646 and republished numerous times, about fifteen interpretations of sexual contents may be found -more than in other treatises -with a rather allusive description. The author, Marc de Vulson de la Colombière, is a French heraldist, native of Dauphiné and living in Paris. Being keen on history, he is also a prolific writer who publishes books on heraldry, genealogy, divination and oneiromancy.

In this treatise, as was already the case with Artemidorus, the man is the general case and the woman the specific: interpretations specific to women are given when needed. It should be noted that, for a woman, erotic dreams are always associated to bad omens, except if she dreams of her husband: 'To dream of one's husband naked means safety and luck in one's activities' [iv] . For men, associations are less accurate and sometimes contradictory. Dreaming of one's wife naked predicts disappointments, but to dream 'to go to bed with a ribald woman signals business safety' [v] ). Thus, the virtue of the man's partner does not presage favourable events, even quite the opposite. However, she must be pretty to bring good luck: 'To dream to see a beautiful woman means honour and happiness, provided that she is white and good looking; but if she is humpbacked, old, with a wrinkled skin, deformed or with a dark complexion, this will mean shame, repentance and bad luck' [vi] ).

These gendered predictions conform to the mental and moral landscape of the authors as well as of the readers, however with an analysis less clear than in Artemidorus. Indeed, the predictions may be contradictory, even from one page to the next, which is a result of the nature of these treatises, made of successive additions where plagiarizing played an important role.

II Can a dream offend God?

Are we responsible for our dreams? This question is debated because it raises questions about so-called guilty dreams and their nature. That a dream displaying an evil or impure act may come from God seems impossible. Yet the question is important as it goes to the foundations of oneiromancy and of the enigma dream, whose meaning is not literal.

For Célestin de Mirbel, the answer is categorical, one cannot be responsible for his/her dreams: 'I state firstly that in the dreams there is no sin and that one man cannot be guilty of what he does and says during his sleep' [vii] ). This sentence takes place at the beginning of a chapter dealing with the topic 'If it is sin to like and study dreams' [viii] . His purpose is to describe the practice of oneiromancy, but in order to do it he must start by demonstrating that dreams cannot be considered, even partly, as sins. This is a sensitive issue as christianism judges not only acts but also thoughts [ix] . However, Mirbel quotes Saint-Grégoire, who says that a mind whose judgement is not free cannot be guilty ('Tunc enim mens rea non fit, quae suo judicio libera non est' [x] ). The mind was unconscious during sleep, its movements cannot be held against it. Another argument put forward by Mirbel is that no law or sentence have ever been issued to punish dreams. As a matter of fact, the dream is not reprimanded by State, this is the critical issue in the debate on the witches Sabbath, since depending on whether it is dreamed or factual, the accused would be declared innocent or not according to law [xi] .

The author also stresses the usefulness of dreams, which allows to avoid unwanted events in real life ('That if the action produced by the vehemence of dreams is not criminal, much less are the reflections we make on them in order to avoid the accidents of which they threaten us' [xii] ). Classic biblical examples are then cited to show once again that God uses the media of dream to warn his believer against faults or dangers. The study of dreams is not criminal but imperative, as the inverse would mean to neglect divine warnings.

As one might suspect, this argument is used by Mirbel to defuse the attacks which could be launched against his book. If a dream cannot be guilty, it is then more difficult to condemn oneiromancy. Furthermore: the idea of a divine message conveyed by each dream, even when it is extremely disturbing, allows in some way to purify the dreamer, to reassure him/her, in a word to free him/her of the questions attached to an incomprehensible or unacceptable dream. To interpret one's dreams with a reading grid is a way to escape the anxiety which may raise from discovering the creations of one's imagination. the biblical examples to accept their existence, he confines himself to remind their extreme rarity, which corresponds, as a matter of fact, to a common view since the early Middle Ages [xiii] . He gives several arguments aiming to denounce the illogicality of these means of communication. One of them in particular concerns abominable dreams:

Of course

Why should we say, if Caesar dreams, being in Spain, that he commits an abominable incest with his mother, that the Heaven promises him by that the Empire of the Earth? I very well know that she is the common mother of all men. But it is a weak reason and there is no appearance that such a dirty way of explaining oneself come from Heaven ; as if God could not have told Caesar what he had ordered him without making his imagination criminal. [xiv] The theme of maternal incest, which marked so profoundly the Freudian psychoanalysis, stands in the seventeenth century as one of the worst conceivable crimes. But in the oneirocritic tradition, the mother is linked to earth, to mother country, so that, even in Artemidorus, the dream of maternal incest is considered as positive. La Mothe Le Vayer reminds that, according to Herodotos, Hippias as well as others 'have suffered of these disorders of the imagination' [xv] , and that, nevertheless, no one experienced the glory of Caesar, which proves according to him the futility of such a prediction. The dream of incest is thus, for La Mothe Le Vayer, a disruption of the imagination which brings it to being criminal. Although he is not aware of it, the dreamer is polluted by this dream and is to blame through his/her imagination. That the work of God may be the cause of a crime seems inconceivable. La Mothe Le Vayer, however, is very far from being a puritan. He has published, although in a discrete manner, books with an erotic content, not to say wholly obscene [xvi] . But, on the one hand, incest has not an ordinary status and, on the other hand, these arguments aim mostly to deconstruct the belief in the premonitory dream without putting him at risk. Scipion Dupleix, magistrate from Gascogne, became the protégé of the queen Marguerite de Valois and followed her when she came back from exile [xvii] .He is a historian but also an author of philosophy and physics books. Scipion Dupleix was, at the end of his career, Historiographer of France and State Councillor. In 1606, when he writes Les causes de la veille et du sommeil, des songes & de la vie & de la mort ('The causes of wakefulness and sleep, of dreams and of life and of death') , he is in Paris for just over a year. In his chapter 'If God may be offended by our dreams' [xviii] , he chooses a level purely moral and religious. If his treatise addresses oneiromancy theories -numerous passages have been reused in latter treatises on oneiromancy -for him, as he explains it, the true dreams remain an exception: That if we experience sometimes true dreams, this is only by chance and by dint of dreaming, as a poor archer hits sometimes the target right in the centre through shooting a great many arrows: so that it is much wiser to value them vains as a whole, in order to stray from superstition, that to work ourselves through a too strange search for the truth among this amount of vanity& provide clarity from the obscure confusion of darkness: & nevertheless praise and thank God if sometimes it pleases him to send us revelations during our sleep. [xix] But any dream, being premonitory or not, may have huge implications, insofar as it is part of the battle of good against evil, as well as the waking state.

Since God admonishes us to protect ourselves while sleeping, & that the evil spirits, immortal enemies of mankind do not trap us in vain to make us stumble & succumb to sin, we must believe that undoubtedly God may be offended by our dreams (…) [xx] .

The theory of Dupleix is simple. When a man has a dream that flatters one of his vices and derives pleasure from it, he is guilty, just as if he had really sinned. Conversely, dreaming of martyrdom is considered a good deed.

Yet we can offend Him during sleep by the images of the same objects & by the same actions & affections by which we offend him when awake. Thus the miser dreaming that he makes some illicit gain through usury, fraud or otherwise & likes in his imagination to receive this gain, sins against God.

The cruel & bloodthirsty man who dreams that he kills his enemy & who relishes in his revenge & bloodshed of his fellow man, seriously offends God. The lewd who is defiled by pollution in a dream imagining he enjoys his dirty loves, & gets from it some carnal pleasure sins similarly against God. [xxi] The case of nocturnal pollutions is dealt with particular care a step further, these being considered as 'one of the most damnable sins that are committed in a dream' [xxii] . Fasting and chaste thoughts are advised to overcome them, even the consumption of… water lilies and lettuce. 'But holy meditations, study, work & fasting tame the flesh more than anything else' [xxiii] , adds Dupleix. This evocation of guilty dreams, in particular sexual, and of the resulting physical effects, directly implicates the responsibility of the dreamer. While in La Mothe Le Vayer the guilty dream is indeed the result of a disorder, it is in Dupleix the consequence of a demonic temptation and of a reprehensible carelessness on the part of the dreamer. This literal conception of the dream, nuanced in Dupleix by the notion of pleasure, places the sleeping mind on a slippery ground. But in both cases, if the mind is responsible for his dreams that can make it criminal, he thereby becomes an active subject, and not the mere receptacle of ideas coming from above.

III Provoking sexual dreams: a catharsis?

In contrast to thinkers who hold the erotic dream as intrinsically sinful, two anonymous works of the eighteenth century present impure dreams as a mean avoiding to sin in the real life, through directed dream. The directed dream -the term 'lucid dream' only appears in the nineteenth century -is a way to control in one's sleep the dream content so as to direct it according to one's will and to remember it upon awakening. These two works, which greatly differ one from another in nature and in approach, appear to be the only French books advocating the directed dream in this period of time.

L'art de se rendre heureux par les songes ('The art of making oneself happy through dreams'), published anonymously in 1747 in Frankfurt, possibly to avoid censorship, is nothing less than a handbook of erotic magic recipes. In a 'Warning' serving as preamble, the publisher gives himself over to a presentation of the author which is supposed to be none other than his uncle, a French chemist died at the venerable age of eighty-six years and who would have requested the publication of this text as a last wish, 'as a gift to mankind'. This addition is not surprising since such a context -most likely -fictional is not uncommon in the eighteenth century and seems to aim in this case at increasing the legitimacy of the work. Although poor and elderly, this scholar would also have retained until his last day a look perpetually cheerful, evidence indeed that its recipes were a success. By making mixtures and applying them appropriately, the reader is supposed to manage to make the dreams that he is promised to, so as to fulfil all his desires. Some allow you to find in dreams fabulous treasures, experience thrilling adventures, have genius or eat delicious meals. But the majority of the book recipes gives priority to 'lascivious dreams', available in multiple scenarios.

For instance, in order to 'dream about sleeping with a woman and receive her favors', the postulant dreamer has to rub the feet, hands, temples and neck every night with this préparation : Take two ounces of calcined scammony root & Roman chamomile, three ounces of cod bones & turtle scales calcined as well. Mix the lot in five ounces of the fat of a male beaver then add two ounces of blue scammony flower oil picked during the morning the first days of spring. Boil the brew with an ounce of honey and six grams of morning dew harvested on a poppy. You may add to the ointment a sixth part of opium & once poured and sealed in a glass bottle, let it rest under the sun of two long summer months, then store the bottle in a fresh cellar until winter, when you will keep it buried in sand to unearth only once the spring is back. Then you will break the bottle and recover the ointment and put it into a stone jar, ready for your own use. [xxiv] Some other ointments make it possible 'to dream that you see naked women, & especially a particular woman which you know or whose you saw the portraits', 'to dream that you are at a table with naked women' or even 'to dream that you enjoy a woman on a flowery grass or in a grove or on the edge of a fountain' [xxv] . None of these recipes, however, offers the opportunity to engage in these activities with a man. The target audience, as one might expect, is as much masculine as heterosexual, joining thus the majority of erotic or pornographic works of the time.

However, throughout the first part of the book, the author sets up a moral justification. Although man aspires to happiness and that his dissatisfaction generates anxiety, it may nevertheless attempt to make himself happy on this earth, without waiting for Paradise, and without however violating the divine law.

On the contrary, the directed dream would be even for the author a way to purge passions from man ('My goal on the contrary is not to flatter them, but to destroy them' [xxvi] ) and to preserve his innocence in the manner of a catharsis that would prevent the carnal sin in real life. To prevent abuse of misspent youth, the author even claims to have purposely used complex recipes, long and difficult to implement. The ideal according to him would be that these recipes be made by an apothecary who would sell them bottled. He even raises the possibility of creating a dispensary of dreams, where we would choose the dream for our next night.

More realistically, we can presume that the book's interest seems to be some kind of simple reading of recipes as an erotic entertainment, more than their extremely difficult and tedious preparation.

The second book, L'art de faire des songes ('The art of making dreams'), is also an anonymous publication, which specifically raises the question of the sexuality of girls. This small text of twelve pages printed in format in-12, probably in France judging by the examination of typographic material, is dated 1750 by the French National Library (BNF) according to the study of the paper. The author addresses, in the form of a letter, perhaps imaginary, a girl to whom he wants to be nice. The lady having told him of her frightening nightmares, he explains what are the dreams, in the manner of an educational presentation. This is also in support of his argument that takes place the initial presentation, the girl being presented as wishing 'always proofs in all that we advance, very far in that from the custom of the sex, which often adopts paradoxes without examination & without proof, of which most men lull their credulity' [xxvii] . Better, he proposes her to turn her dreams to her advantage by controlling them through the concentration on an object that is kept in mind until sleep. This practice focused on concentration is consistent with those developed in the nineteenth century in particular by Hervey de Saint-Denys.

Despite its didacticism, the tone of the letter is gallant and the words are explicit. It's a matter of obtaining happiness through dreams, but above all pleasure. Indeed, directing his dreams is presented as an ideal way to enjoy the pleasures without affecting one's purity or reputation: 'Free from the constraints of the day, at night you would give to all the transports of love, without fear of evil speech from the Public, and even without costing anything to your virtue' [xxviii] . In addition, the dream seems even a way to mock her lover: 'What pleasure of constantly tormenting a lover by an inflexible rigor, and fulfilling your wishes in the sweet illusion of sleep, without him being for half in your pleasure, without him being able to boast of his triumph, and even without knowing anything' [xxix] .

Beyond the lightness of tone, the dream appears here as a real escape from a society where girls do not have the right to dispose of their bodies. Their reputation of virtue being a guarantee of their value (a French proverb present since the Middle Ages says 'Good repute is better than golden belt' [xxx] , thus implying that a young girl repute is her most precious asset, even more precious than a belt of considerable value), to see the loved one brag about his triumph is a risk absolutely real for the social future of the young woman. It should be noted that daydreaming, dream awakened, is considered in the eighteenth century as a serious threat to adolescent girls, which can lead to nymphomania [xxxi] .

Should we see in this call to licentious dream an evocation of masturbation? In L'harmonie des plaisirs [xxxii] , Alain Corbin studies the strategies developed by mothers and doctors to prevent girls to yield to onanism. The novels which may lead to daydreaming or even some music are forbidden in favour of readings together with healthy and moral activities, such as botany. The girls suspected must be subjected to a constant surveillance, must not be left alone under any circumstances. If need be, their mother may go to bed with them to ensure that they do nothing but sleep. In this context, the lucid dream appears as a solitary pleasure all the more valuable in that it is unnoticeable. Prisoner of her condition in the real world, the girl can find the way through the dream to a form of freedom, if not emancipation.

Conclusion

If the interpretation of premonitory dreams perceives sexual dreams as symbols and nothing more, they are not less significant for the historian who can read there valuable clues for the perception of sexual practices at a given time.

During the seventeenth and especially the eighteenth century, the dream is subject to suspicion.

Premonitory dreams are associated with superstitions and daydreams with guilty inclinations, even if this rational behaviour coexists with the persistence of a diffuse belief in premonitory dream, still alive in the depths of the French Enlightenment. This theoretical evolution of dream, from a divine and exogenous manifestation towards a purely natural and endogenous phenomenon, entails a restructuring of the relationship of one individual with his/her dreams. For if the impure dream is no longer the symbol to be deciphered of a message from God, how to serenely confront the bizarreness of its fantasy? For the dream has this capacity, as Foucault points out, 'to tell the subject about its inner truth, the truth of its nature (...)' [xxxiii] . This new uncertain status of the dream in the eighteenth century is a source of disquiet, which a purely recreational use of erotic dreams fails to fully dispel.

Sources

  Célestin de Mirbel is not the first to take sides on the possible guilt of the dreamer. Two other examples may be quoted, which conflict with the vision of the Palace of the Prince of Sleep: La Mothe Le Vayer and Scipion Dupleix. In his treatise Du sommeil et des songes ('About Sleep and Dreams'), published in 1643, the sceptical philosopher La Mothe Le Vayer opposes to an everyday conception of prophetic dreams. Forced by

  Anonymous. ca 1750. L'Art de faire des songes. Anonymous. 1746. L'art de se rendre heureux par les songes c'est à dire en se procurant telle espèce de songe que l'on Le Moyne de Bienville, Jean-Baptiste. 1771. La Nymphomanie, ou Traité de la fureur utérine: dans lequel on explique, avec autant de clarté que de méthode, les commencemens & les progrès de cette cruelle maladie, dont on développe les différentes causes. Amsterdam. Vulson de La Colombière, Marc de. 1655 (first ed. 1646). Le Palais des curieux, où l'algèbre et le sort donnent la décision des questions les plus douteuses, et où les songes et les visions nocturnes sont expliquées selon la doctrine des anciens. Troyes. Dupleix, Scipion. 1619 (fist ed.1606). Les causes de la veille, et du sommeil, des songes, & de la vie & de la mort.

	puisse	désirer	conformément	à	ses	inclinations.	Francfort.
	Paris.

Mirbel, Célestin de. 1667. Le Palais du Prince du Sommeil, où est enseignée l'oniromancie autrement l'art de deviner les songes. Bourges. La Mothe Le Vayer, François de. 1761 (first ed. 1654). Du sommeil et des songes. OEuvres. Dresde.


Simon Bicaïs
email: simon.bicais@cea.fr

Jean-Baptiste Doré

José Luis Gonzalez Jimenez

Adaptive PSK Modulation Scheme in the Presence of Phase Noise

Keywords: Millimeter wave, Phase noise, Phase shift keying, Adaptive Systems, Performance analysis, Channel estimation

Phase noise is one of the major impairments affecting severely performance of millimeter-wave systems. This paper addresses the problem of link adaption for coherent and non-coherent phase modulated signals subject to Gaussian and Wiener phase noise. We first derive closed-form approximations of the bit error rate. Then, in contrast to usual link adaptation techniques, we propose a simple scheme exploiting estimations of not only the signal-to-noise ratio but also of the phase noise variance, which is essential to achieve reliable communications.

I. INTRODUCTION

Future 5G millimeter-wave (mmWave) bands should lead to user throughputs above 1 Gbit/s. However, additional breakthrough technologies are necessary to reach this requirement. Therefore, for the first time in the telecommunication history, bands above 90 GHz are studied for future communications [START_REF] Fettweis | Entering the path towards terabit/s wireless links[END_REF]. In particular, mmWave systems are critically impacted by phase noise (PN) due to the poor performance of high frequency oscillators [START_REF] Voicu | Performance and trends in millimetrewave cmos oscillators for emerging wireless applications[END_REF]. This has motivated extensive work on PN estimation and compensation [START_REF] Yi | Phase estimation for coherent optical ofdm transmission[END_REF], [START_REF] Munier | Estimation of phase noise for qpsk modulation over awgn channels[END_REF]. Further, achieving high-rate communications over wireless links in the mmWave domain demands tremendous amount of power, and so, highefficiency and wide-bandwidth power amplifiers [START_REF] Song | Present and future of terahertz communications[END_REF]. Therefore, coherent Phase Shift Keying (PSK) modulations are highly valuable since they demonstrate a constant envelope property, and thus, offer an efficient use of power amplifiers. Yet, PSK modulations are highly sensitive to phase related impairments: Carrier Frequency Offset (CFO), PN. Differential PSK (DPSK) has been introduced as a more robust scheme. At the expense of a noise enhancement, DPSK enables noncoherent communications [START_REF] Proakis | Digital Communications, ser. McGraw-Hill series in electrical and computer engineering : communications and signal processing[END_REF]. Prior work in [START_REF] Burr | Comparison of coherent and noncoherent modulation in the presence of phase noise[END_REF] has confronted the robustness of PSK and DPSK when affected by a Tikohonov PN. Setting the modulation scheme to maximize spectral efficiency while maintaining robustness is correspondingly the motivation behind link adaption. Such adaptive systems are now implemented in most of practical systems [START_REF] Technologies | LTE and the Evolution to 4G Wireless: Design and Measurement Challenges[END_REF] and rely on a channel quality estimation to set the modulation.

Contributions: In this paper, we seek to improve the design of inherently robust communication systems impacted by PN. First, we assume a high Signal-to-Noise Ratio (SNR) and a Gaussian PN to derive a closed-form approximation of the Bit Error Rate (BER) for any M -ary PSK. Considering that the superposition of a Gaussian and a Wiener process is a more realistic PN model [START_REF] Demir | Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and1/f noise[END_REF], [START_REF] Khanzadi | Calculation of the performance of communication systems from measured oscillator phase noise[END_REF], we derive and confront the BER performance of PSK and DPSK for such PN. Furthermore, in contrast to usual link adaptation techniques, we propose a simple scheme exploiting estimations of not only the SNR but also of the PN variance, which proves to be truly essential to achieve reliable communications. We hence derive the appropriate Maximum Likelihood (ML) estimators of the thermal and phase noise variances. Eventually, we propose a statistical test to determine whether the coherent or noncoherent modulation is the most robust one and should be employed.

Organization: The remainder of this paper is structured as follows. Section II introduces a brief description of the channel and PN models. Section III is devoted to the performance analysis of PSK and DPSK impacted by Gaussian and Wiener PN. Finally, Section IV derives the appropriate channel estimation to implement the link adaptation scheme.

II. SYSTEM MODEL A. Channel model

Considering a single carrier communication system with a perfectly mitigated channel, the received symbol at time instant k is defined by

r k = s k • e jφ k + n k , (1)
where s is the modulated symbol from constellation C with average symbol energy E s , φ is the oscillator PN and n represents independent identically distributed samples of a zero-mean complex Additive White Gaussian Noise (AWGN) with variance 2σ 2 n . A PSK constellation with modulation order M is defined by

C = { √ E s exp j 2π M •i |i = 1, ..., M }.
In the case of a DPSK, information is encoded in the phase difference from one symbol to another denoted δθ k = s ks k-1 . We denote by s and |s| the phase and amplitude of a symbol s.

B. Phase noise model

Oscillators PN is generated from the transformation of amplitude fluctuations into phase fluctuations [START_REF] Chorti | A Spectral Model for RF Oscillators With Power-Law Phase Noise[END_REF], such that it describes a cumulative random process [START_REF] Demir | Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and1/f noise[END_REF], [START_REF] Hajimiri | A general theory of phase noise in electrical oscillators[END_REF]. If only white noise sources are considered 1 , the oscillator PN φ may thus be modeled by the superposition of a Wiener (Gaussian random-walk) process φ w and a Gaussian one φ g [START_REF] Demir | Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and1/f noise[END_REF]. This may be expressed by

φ k = φ w,k + φ g,k , φ g,k ∼ N (0, σ 2 g ), φ w,k = φ w,k-1 + δφ w,k , δφ w,k ∼ N (0, σ 2 w ), (2)
where σ 2 w and σ 2 g denote respectively the variances of the Wiener PN increment δφ w and the Gaussian PN. Besides mathematical convenience [START_REF] Khanzadi | On models, bounds, and estimation algorithms for time-varying phase noise[END_REF], [START_REF] Krishnan | Constellation optimization in the presence of strong phase noise[END_REF], the Gaussian distribution is also a relevant PN model. When considering wide bandwidth systems 2 , the oscillator noise floor represents the greatest contribution to the overall PN [START_REF] Khanzadi | Calculation of the performance of communication systems from measured oscillator phase noise[END_REF], such that the Wiener PN becomes negligible compared to the Gaussian one, i.e. σ 2 g >> σ 2 w . Thereby, Section III-A and IV-A use the Gaussian PN model to pursue a simple analytical analysis while Section III-B and IV-B extend it to the sum of Wiener and Gaussian PN.

III. PERFORMANCE ANALYSIS A. PSK over Gaussian phase noise

To begin with, let us assume that the channel is dominated by Gaussian PN. Regarding symbol-by-symbol detection, it is known that the ML criterion minimizes the Symbol Error Probability (SEP) [START_REF] Proakis | Digital Communications, ser. McGraw-Hill series in electrical and computer engineering : communications and signal processing[END_REF], [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. In order to design such optimum receiver, let us derive the likelihood function, which may be rewritten as

p(r k |s k ) = p |r k | , r k |s k | , s k . ( 3
)
By definition, the AWGN is invariant under rotation such that, for simplicity of notation, n may stand for n • e jθ for any θ fixed. The received symbol should be studied through its amplitude and phase as

|r k | = (|s k | + n k ) • e j( s k +φ k ) = (|s k | + (n k )) 2 + (n k ) 2 E s + (n k ) (4)
r k = arg[(|s k | + n k ) • e j( s k +φ k ) ] = s k + φ k + arctan (n k ) |s k | + (n k ) s k + φ k + (n k ) √ E s . (5)
These first-order approximations are tight for a high SNR scenario [START_REF] Fu | Exact phase noise model and its application to linear minimum variance estimation of frequency and phase of a noisy sinusoid[END_REF], [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF]. By Eq. ( 2), the likelihood function follows a bivariate Gaussian distribution and is thus expressed by

p(r k |s k ) = exp - 1 2 (|r k | - √ E s ) 2 σ 2 n + ( r k -s k ) 2 σ 2 g + σ 2 n /E s 2π σ 2 n (σ 2 g + σ 2 n /E s ) . (6)
This expression, originally derived in [START_REF] Krishnan | Constellation optimization in the presence of strong phase noise[END_REF], can be exploited in our study to approximate the BER of a PSK. With respect to the ML decision rule for a PSK, the detected symbol is

ŝk = argmax s ∈ C p(r k |s k ) = argmin s ∈ C ( r k -s k ) 2 , (7)
which enables to derive the SEP as follows

P se = 1 -Pr( ŝk = s k |s k ) = 1 -p - π M < r k -s k < π M = 2Q π M σ 2 g + σ 2 n /E s . ( 8
)
To directly relate the BER to the SEP, we shall make two commonly used assumptions. First, the bit labeling of the constellation satisfies a Gray coding. Furthermore, misdetections

10 0 32PSK -σ 2 g = 1x10 -3 16PSK -σ 2 g = 2x10 -3 QPSK -σ 2 g = 2.5x10 -2
Eb/N 0 (dB) only occur on the nearest neighbors of the sent symbol. Finally, the BER of a PSK affected by Gaussian PN is approximated by

P g be,PSK 2 log 2 (M ) Q π M σ 2 g + 1 2•log 2 (M ) 1 E b /N0 , (9)
where the average bit energy E b satisfies E s = E b log 2 (M ) and the noise power spectral density is defined by N 0 = 2σ 2 n . The BER performance of PSK for arbitrary orders of modulation and PN variances are illustrated in Fig. 1. As exhibited, the closed-form expression holds accurate approximations of the BER at high SNR.

B. DPSK over Wiener phase noise

DPSK enables a non-coherent demodulation at the receiver. As a consequence, estimation of the carrier phase is not required. Moreover, a differential modulation is not subject to the cumulative nature of the PN, and only slightly to the CFO. This robustness is achieved at the expense of a BER performance degradation. Yet, the stronger these impairments, the more valuable it seems to opt for differential encoding. The pending question is when DPSK is more robust than PSK. Therefore, let us derive the performance of PSK and DPSK when the PN is described by the sum of a Wiener and a Gaussian PN. The optimum decision for a DPSK receiver [START_REF] Proakis | Digital Communications, ser. McGraw-Hill series in electrical and computer engineering : communications and signal processing[END_REF] is given by

δθ = argmin δθ ∈ C ( r k -r k-1 ) -δθ 2 . ( 10
)
With regard to the PN model in Eq. ( 2) and to the high SNR approximation in Eq. ( 5), we have which is directly equivalent to

r k -r k-1 δθ k + δφ w,k + φ g,k -φ g,k-1 + (n k -n k-1 ) √ Es , (11)
r k -r k-1 δθ k + φ eq,k ,
where φ eq ∼ N (0,

σ 2 w + 2σ 2 g + 2σ 2 n /E s ). (12)
Thereupon, the BER for a DPSK subject to Gaussian and Wiener PN may be approximated by

P w be,DPSK 2 log 2 (M ) Q π M σ 2 w + 2σ 2 g + 1 E b /N0 1 log 2 (M ) . ( 13
)
To establish a comparison with the PSK modulation, we consider a genie estimation at the PSK receiver. With one pilot symbol, the receiver is able to estimate and compensate perfectly the Wiener process. The pilot period is defined by T , such that if k = n•T then φ w,k+1 = δφ w,k . The estimation problem in presence of Wiener PN exceeds the scope of this paper, but is addressed in [START_REF] Munier | Estimation of phase noise for qpsk modulation over awgn channels[END_REF]. Though the derivation is not developed here, the BER for a PSK in this case is given by

P w be,PSK 1 T -1 T -1 k=1 2 log 2 (M ) • Q π M k • σ 2 w + σ 2 g + T /(T -1) E b /N0 1 2•log 2 (M ) . (14)
Figure 2 confronts the performance of a 16PSK to the ones of a 16DPSK for Gaussian and Wiener PN. The PSK receiver presents a pilot density of 10% with a genie estimation. Conversely, the DPSK receiver does not implement any channel estimation. Figure 2 shows that analytical expressions are tight to the simulation results. Moreover, Fig. 2 confirms an aforementioned intuition. The DPSK presents a loss in BER performance, still, when considering stronger Wiener PN then it is more robust than the PSK.

IV. LINK ADAPTION A. Estimation of thermal and phase noise variances

To improve spectral efficiency, practical communication systems implement adaptive modulation scheme [START_REF] Technologies | LTE and the Evolution to 4G Wireless: Design and Measurement Challenges[END_REF]. Namely, the modulation order is inferred from a SNR estimation. In the case of a channel dominated by Gaussian PN, the analytical approximation of the BER in Eq. ( 9) enables to determine the greatest value of M while maintaining the error rate below a fixed target. Figure 3 depicts the modulation scheme regions for M -ary PSK, i.e. the highest modulation order M achieving BER < 10 -4 given E b /N 0 and σ 2 g . It is shown that the performance is highly related to the PN variance, which should necessarily be considered in the link adaptation to maintain robustness.

Such link adaptation requires an estimation of σ 2 n and σ 2 g . This may be realized by inserting N pilot samples possibly distributed and known from the receiver denoted s = (s 1 , s 2 , ..., s N ) and by deriving the appropriate estimators. From Eq. ( 6), the joint likelihood function of the N received samples r = (r 1 , ..., r N ) is given by

p N (r|s, σ 2 n , σ 2 g ) = N k=1 p(r k |s k , σ 2 n , σ 2 g ). (15)
We derive the ML estimate of the covariance matrix for the multivariate normal distribution [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. It yields here the ML estimators of σ 2 n and σ 2 g expressed by

σ2 n = 1 N N k=1 (|r k | -E s ) 2 , σ2 g = 1 N N k=1 ( r k -s k ) 2 - σ2 n E s . (16)
Writing these estimators as χ 2 (N ) distributions enables us to quantify their performance in terms of bias and dispersion respectively with the means and variances given by

E[σ 2 n ] = σ 2 n V[σ 2 n ] = 2σ 4 n N , E[σ 2 g ] = σ 2 g V[σ 2 g ] = 2(σ 2 g +σ 2 n /Es) 2 N + 2σ 4 n N Es . (17)
Since these estimators are unbiased and the regularity conditions clearly satisfied in Eq. ( 6), we may compare the estimators to their Cramer-Rao Lower Bounds (CRLB) [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF] written as

V σ2 n ≥ 2σ 4 n N , V σ2 g ≥ 2(σ 2 g + σ 2 n /E s ) 2 N . (18)
It follows easily that the thermal noise variance estimator is efficient in the sense of achieving the CRLB cf. [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. As for the PN variance, the estimator is tight to the CRLB but does not reach it. However, both estimators demonstrate the smallest Mean Square Error (MSE) among any unbiased estimators. From Eq. ( 15), it is straightforward that the joint density p N belongs to the exponential family such that S = (

N k=1 (|r k | - √ E s ) 2 , N k=1 ( r k -s k ) 2
) is a complete sufficient statistic for parameter (σ 2 n , σ 2 g ). By the Lehman-Scheffé theorem, the unbiased estimators (σ 2 n , σ2 g ), defined upon S, are respectively the unique Minimum-Variance Unbiased Estimators (MVUE) of σ 2 n and σ 2 g .

B. Is the phase noise a cumulative process?

As illustrated in Fig. 2, the DPSK should be favored at the detriment of PSK only if the cumulative nature of the PN is non-negligible. Hence, the link adaptation problem can be expressed as a simple detection problem: is the cumulative process of the phase noise strong enough to advantage a differential modulation. Let us formulate the corresponding binary hypothesis test as

H 0 : Gaussian PN, if σ 2 w /σ 2 g < λ fixed, H 1 : Wiener PN, otherwise. (19)
The link adaptation decision δ is then expressed by

δ(r) = PSK, if S(r) < ϕ(λ) DPSK, otherwise , (20)
where r = (r 1 , ..., r N ) is a received sequence of N consecutive pilots, S denotes a sufficient statistic of the received pilots sample and ϕ is a test threshold function of λ. Fortuitously, this problem has been extensively studied in the financial field under a different name: the random walk hypothesis for stock market prices [START_REF] Charles | Variance ratio tests of random walk: An overview[END_REF]. Numerous statistical tests have been designed, yet, one in particular has retained most of the attention: the variance ratio test [START_REF] Charles | Variance ratio tests of random walk: An overview[END_REF], [START_REF] Lo | Stock market prices do not follow random walks: Evidence from a simple specification test[END_REF]. This test relies on the fact that the variance of the random walk increments increases linearly with p the sampling interval. Applied to our context, this is expressed by

V [φ k+p -φ k ] = p • σ 2 w + 2σ 2 g , (21)
which is denoted further by σ 2 ∆p . Letting σ 2 w = λ • σ 2 g and summing over p, the statistical test becomes where σ2 ∆p is the estimator of σ 2 ∆p and evaluated by

S(r) = M p=1 σ2 ∆p σ2 ∆1 < ϕ(λ) = λ • M (M + 1) + 4M 4λ + 2 , (22)
σ2 ∆p = 1 N -p N -p k=1 ( r k+p -s k+p ) -( r k -s k ) 2 . (23)
The upper bound of summation M satisfies 1 < M < N , and is optimized heuristically 3 . Figure 4 depicts the mean probability of taking decision δ(r) = H 1 and thus use DPSK. Several pilot lengths are considered and λ is set to 1/2 as in Fig. 2 . As exhibited, the longer the pilot sequence, the more accurate the test. The performance degradation caused by an incorrect decision should be evaluated to properly specify the length of the training sequence. Due to space limitation, it is not carried out here but using the proposed framework, it is straightforward to derive.

V. CONCLUSION In this paper, we have addressed the problematic of link adaption for phase modulated signals affected by PN. We have first considered PSK modulations impacted by a Gaussian PN and provided the closed-form approximation of the BER. Optimum ML estimators of thermal and phase noise variances have been derived to propose a link adaptation scheme. Exploiting estimations of both the SNR and the PN variance proved to be essential to maintain robustness. Then, the PN has been described with a more realistic model: a sum of a Gaussian and a Wiener process. Performance of PSK and DPSK were compared in order to determine when the use of a differential modulation is beneficial. Accordingly, we proposed a simple detection test, inspired from the financial field, to perform the link adaptation and exploit the most robust modulation between PSK and DPSK. Nevertheless, these results remain an analytical analysis and it is of particular interest to investigate this adaptation problem under practical implementation considerations.

Fig. 1 :

 1 Fig. 1: Analytical and simulated BER performance of M -ary PSK by Gaussian PN with variance σ 2 g

Fig. 2 :

 2 Fig. 2: Analytical and simulated BER performance of 16PSK and 16DPSK for Gaussian PN (σ 2 g = 10 -3 , σ 2 w = 0) and Wiener PN (σ 2 g = 10 -3 , σ 2 w = σ 2 g /2)

Fig. 3 :

 3 Fig. 3: Modulation Scheme Regions for M -ary PSK: highest modulation order M achieving BER < 10 -4 as a function of E b /N 0 and the inverse of PN variance 1/σ 2 g

Fig. 4 :

 4 Fig. 4: Mean probability of taking decision δ(r) = H 1 when λ = 1/2 for different lengths of pilot sequence N

Flicker noise is diregarded in this study.

The upper bound M in Eq. 22 must be large enough to yield a robust test. However, as the sampling interval p gets bigger, the dispersion of σ2 ∆p in Eq. 23 increases. Thus, if M is too large, the test is defined upon unreliable estimations.

ACKNOWLEDGMENT

The research leading to these results received funding from the French National Research Agency (ANR-17-CE25-0013) within the frame of the project BRAVE.


Simon Bicaïs
email: simon.bicais@cea.fr

Jean-Baptiste Doré

José Luis Gonzalez Jimenez

On the Optimum Demodulation in the Presence of Gaussian Phase Noise

Keywords: Millimeter wave, Phase noise, Maximum likelihood detection, Complete metric space, Channel estimation

Millimeter-wave systems appears to be one of the foremost solutions to face the exponential growth of the data traffic. Yet, their performance are severely degraded by oscillators phase noise. This paper addresses the design of optimum receivers affected by phase noise. We first derive the optimum decision metric for symbol detection under the assumption of a Gaussian phase noise and a high signal-to-noise ratio. In contrast to state-of-the-art approaches, we propose to represent within a complete metric space the demodulation over the phase noise channel. The provided framework enables the use of efficient algorithms to perform the demodulation. Computable probabilistic demapper values are derived upon the optimum metric to implement soft channel decoding. This work shows that knowledge of the channel statistics can be capitalized to enhance performance of both coded and uncoded systems. Eventually, we propose the necessary channel estimation scheme and evaluate the performance degradation due to estimation errors.

I. INTRODUCTION

With regard to the spectrum shortage in the cellular bands, the millimeter-wave (mmWave) spectrum has aroused lots of interest in the research domain. With unprecedentedly large bandwidth, mmWave systems are considered to be one of the foremost solutions to face the exponential growth of the data traffic. Within the next couple of years, a requirement of 10+ Gbit/s user throughput is foreseen. Nonetheless, the poor performance of high frequency oscillators [START_REF] Voicu | Performance and Trends in Millimetre-Wave CMOS Oscillators for Emerging Wireless Applications[END_REF] plagues mmWave systems with strong Phase Noise (PN). This has motivated extensive work on the design of optimum receivers for systems impacted by PN. Numerous of these optimum receivers are listed in [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF] for the common PN models. Still, most of the literature's detectors present complex structures hindering any analytical study. The development of telecommunications has proved that simple analytical frameworks are fundamental for the design of communications systems as they provide means to pursue comprehensive analyses and thus develop efficient algorithms.

Contributions: Through this work, we aim at improving the design of high-rate mmWave communications. This paper mainly pursues the study of Krishnan [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF]. Similarly, we assume a high SNR and a Gaussian PN to derive the already known Maximum-Likelihood (ML) decision premetric for symbol detection. In contrast to [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF], we exploit further the high SNR approximation to properly define the polar metric. Our metric achieves the same optimum performance of the ML premetric but enables us to come within the scope of complete metric spaces. Representing signals upon an appropriate metric space allows the use of efficient algorithms to perform the demodulation. Further, to enhance performance of coded systems, we propose computable probabilistic demapper values using the polar metric to implement efficient soft decision decoding. Though knowledge of the channel statistics improves the performance, it requires a channel estimation. Accordingly, we propose ML estimators of the thermal and phase noise variances. The performance degradation due to estimation errors of the polar metric are eventually evaluated and compared to the ML premetric one.

Organization: The remainder of this paper is structured as follows. Section II introduces a brief description of the channel and PN models. Section III derives the polar metric decision rule for optimal symbol detection impacted by Gaussian PN. Further, Section IV is devoted to the evaluation of probabilistic demapper values. Finally, an appropriate channel estimation is proposed in Section V to implement the optimal demodulation.

II. SYSTEM MODEL

A. Channel

We consider the complex Additive White Gaussian Noise (AWGN) channel impacted by PN. The received signal is then written as

r = s • e jφ + n, (1)
where s is the modulated symbol, φ is the oscillator PN and n represents a zero-mean circular complex Gaussian noise with variance 2σ 2 n . The transmitted symbol s belongs to the constellation C with average energy E s and modulation order M . We will use the polar representation such that s ρ and s θ stand respectively for the amplitude and phase of symbol s.

B. Phase noise

The Gaussian distribution is mathematically convenient and thus often exploited to pursue simple analytical analyses [START_REF] Krishnan | Constellation Optimization in the Presence of Strong Phase Noise[END_REF]. In fact, it appears to be an appropriate PN model for mmWave systems. When considering wide bandwidth systems, the oscillator noise floor represents the greatest contribution to the overall PN [START_REF] Khanzadi | Calculation of the Performance of Communication Systems From Measured Oscillator Phase Noise[END_REF]. Then the oscillator PN φ is to be described by a zero-mean Gaussian distribution with variance σ 2 p as follows φ ∼ N (0, σ 2 p ).

III. MAXIMUM LIKELIHOOD SYMBOL DETECTION A. Optimum decision metric at high SNR Regarding symbol-by-symbol detection, the Symbol Error Probability (SEP) is minimized by the Maximum A Posteriori (MAP) decision criterion [START_REF] Proakis | Digital Communications 5th Edition, ser. McGraw-Hill series in electrical and computer engineering : communications and signal processing[END_REF]. When transmitted symbols are equiprobable and mutually independent, this criterion simplifies to the ML decision rule [START_REF] Proakis | Digital Communications 5th Edition, ser. McGraw-Hill series in electrical and computer engineering : communications and signal processing[END_REF]. The optimum receiver decision is hence defined upon the likelihood function p(r|s). With regard to the channel model in Eq. ( 1), it is relevant to rewrite p(r|s) = p(r ρ , r θ |s ρ , s θ ).

(

) 3
Let us denote by n the shifted noise n • e -j(s θ +φ) which is identically distributed to n as the complex AWGN is circular.

We then study the received symbol through its amplitude and phase as

r ρ = (s ρ + n ) • e j(s θ +φ) = s ρ + (n ) 2 + (n ) 2 s ρ + (n ) (4)
and

r θ = arg[(s ρ + n ) • e j(s θ +φ) ] = s θ + φ + arctan (n ) s ρ + (n ) s θ + φ + (n ) s ρ . (5)
These first-order approximations clearly hold under a high SNR assumption. Further, exploiting the high SNR assumption enables to derive a simple closed-form expression of the channel likelihood function [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF]. We conclude from the thermal and phase noise model in Eq. ( 1) and (2) that

r ρ ∼ N (s ρ , σ 2 n ), r θ ∼ N (s θ , σ 2 p + σ 2 n /s 2 ρ ). (6)
It is straightforward that the likelihood function follows a bivariate Gaussian distribution defined by

p(r|s) = exp - 1 2 (r ρ -s ρ ) 2 σ 2 n + (r θ -s θ ) 2 σ 2 p + σ 2 n /s 2 ρ 2π σ 2 n (σ 2 p + σ 2 n /s 2 ρ ) . (7)
This expression has been originally derived in [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF] and leads to the ML detection decision at the receiver given by

ŝ = arg max s ∈ C p(r, s) = arg min s ∈ C d 2 (r, s), ( 8
)
where d is the ML premetric defined by

d 2 (r, s) = (r ρ -s ρ ) 2 σ 2 n + (r θ -s θ ) 2 σ 2 p + σ 2 n /s 2 ρ + log 1 + 1 σ 2 p • σ 2 n s 2 ρ . (9)
The ML premetric has proved to yield significant gain in performance compared to the Euclidean distance detector [START_REF] Krishnan | Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise[END_REF].

This decision rule is the minimization of the Mahalanobis distance [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. However, in this case the covariance matrix is defined upon the amplitude of the sent symbol s ρ . Therefore, considering d is not convenient since it does not define a metric. Indeed, the map d : C × C → R + is neither symmetric nor subadditive -even if the log term is considered negligble in the decision.

Without loss of generality, we can assume that σ 2 p +σ 2 n /s 2 ρ σ 2 p +σ 2 n /E s , which is a tight approximation at high SNR. This leads to the polar metric decision rule

ŝ = arg min s ∈ C d 2 γ (r, s), d 2 γ (r, s) = (r ρ -s ρ ) 2 + (r θ -s θ ) 2 γ 2 , ( 10
)
where

γ 2 = σ 2 p /σ 2 n + 1/E s .
One of the advantages of exploiting the polar metric rather than the ML premetric is that distances are simpler to evaluate on practical systems. Besides, in doing so, the polar metric d γ is well-defined and enables us to pursue the analysis within the scope of complete metric space.

B. Framework

Lemma 1. Let the complex number s be represented by its polar coordinates (s ρ , s θ )

∈ P = [0, ∞) × [-π, π].
Then the space P N -of sequences s = {s ρ,k , s θ,k } 1≤k≤N of N numbers in P -is a complete metric space when equipped with the metric d γ :

P N × P N → R + defined by d 2 γ (x, y) = N k=1 (x ρ,k -y ρ,k ) 2 + (x θ,k -y θ,k ) 2 γ 2 , γ 2 ∈ R + * , (11
) for every x, y vectors of P N . Proof: It is sufficient to demonstrate that P N is a closed subset of the complete metric space (R 2N , d γ ). First, it is trivial that the map d γ is a weighted Euclidean distance on R 2N . It follows immediately that (R 2N , d γ ) is a complete metric space. Ultimately, the metric space (P N , d γ ) is complete as

P N is clearly a closed subset of R 2N .
From now on, we will abbreviate by p 2 the metric space (P N , d γ ) providing the framework to study the Gaussian PN channel. Metric spaces, by generalizing the notion of distances, provide a fundamental framework for telecommunications. Lemma 1 ensure that efficient algorithms may be designed for the demodulation of signals over the Gaussian PN channel. By way of example, the Nearest Neighbor (NN) search to realize the symbol detection in Eq. [START_REF]IEEE Standard for Information technology-Local and metropolitan area networks-Specific requirements-Part 11: Wireless LAN Medium Access Control (MAC)and Physical Layer (PHY) Specifications Amendment 5: Enhancements for Higher Throughput[END_REF] or to compute the probabilistic demapper valueslater defined in Eq. (17) -may be performed very efficiently within p 2 using projections or sphere decoding. However, design of such algorithm exceeds the scope of this paper and is to be addressed in a future dedicated one.

To conclude, it is important to highlight the strong similarity of the Gaussian PN channel and the complex AWGN one. This is well illustrated by the fact that the system model in Eq. ( 1) and ( 2) can be approximated within P by

(r ρ , r θ ) (s ρ + n ρ , s θ + n θ ), (12)
where n ρ and n θ follows Gaussian distributions with respect to Eq. ( 6). In fact, we entitled the space p 2 by analogy to the space l 21 , usually exploited to study discrete signals over linear channels.

C. Application: demodulation of uncoded M -QAM

We depict in Fig. 1 the BER performance of an uncoded 16-QAM with detection based either on the Euclidean distance, or on the ML premetric, or on the polar metric. First, it should be highlighted that the error floor, due to PN, is significantly reduced by using an appropriate decision rule. These simulations also demonstrate that performance achieved by the polar metric, with a simpler expression, are identical to the ML premetric one at high SNR. Fig. 1 is an illustration of the performance maximization through Channel-State-Information (CSI) capitalization. It is worth mentioning that using channel statistics to properly represent the signal and hence optimize the communication performance is a well-known scheme, implemented for instance in the Orthogonal Time Frequency Space (OTFS) waveform [START_REF] Hadani | Orthogonal Time Frequency Space Modulation[END_REF].

IV. PROBABILISTIC DEMAPPER

Channel coding is indivisible of modern communication systems, for it provides a means to achieve robust communications over noisy channels. Furthermore, it is usually complemented with soft decision decoding, which is known to improve significantly the performance of channel decoders.

A. Bit LLR values

Let us here consider a Bit-Interlaved-Coded-Modulation (BICM) architecture such that the symbol s maps the binary word b = (b 1 , ..., b log 2 (M ) ). The optimum hard decision on the simple detection problem regarding bit b i is achieved by the ML criterion [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF] bi

(r) = 1, if L i (r) > 0 0, otherwise , (13)
where L i , the Log-Likelihood-Ratio (LLR) of decision bi , is defined by

L i (r) = log p(b i = 1|r) p(b i = 0|r) . ( 14
)
While the sign of the LLR supplies the decision to be taken, its absolute value quantifies the reliability of this decision. By applying Bayes' rule and assuming equiprobable symbols, the LLR may be written as

L i (r) = log s1∈C i 1 p(r|s 1 ) s0∈C i 0 p(r|s 0 ) , (15)
where C i β denote the subset of the constellation symbols satisfying b i = β. This expression are tedious to evaluate directly. However, simplified but suboptimal LLR can be obtained through the commonly used max-log sum approximation [START_REF] Tosato | Simplified soft-output demapper for binary interleaved COFDM with application to HIPERLAN/2[END_REF], viz. log( k x k ) max k log(x k ). This approximation is tight at high SNR and leads to

L i (r) max s1∈C i 1 log p(r|s 1 ) -max s0∈C i 0 log p(r|s 0 ) . ( 16
)
In addition, most of implemented decoder algorithms are not sensitive to normalization, e.g. the min-sum decoder for LDPC. Such that, we can express the LLR upon the polar metric defined in Eq. ( 10) by

L i (r) min s0∈C i 0 d 2 γ (r, s 0 ) -min s1∈C i 1 d 2 γ (r, s 1 ). ( 17
)
Several comments should be pointed out. First, this expression is identical to the LLR of an AWGN channel if d denoted the Euclidean distance [START_REF] Tosato | Simplified soft-output demapper for binary interleaved COFDM with application to HIPERLAN/2[END_REF]. Furthermore, the bit LLR values in Eq. ( 17) can be computed simply -which is not the case for the ML premetric -since the polar metric d γ is no more than a weighted Euclidean distance within p 2 . Although the proposed demodulation is applied to QAM constellation in the following, it is valid for any modulation scheme. To the best of our knowledge, performance gains provided by an appropriate decision metric for demodulation of coded systems over the Gaussian PN channel have never been evaluated. Besides, in the high-rate communications context, Low-Density-Parity-Check (LDPC) codes are recognized as a foremost solution for error-correcting codes. Due to their excellent performance and highly parallelizable decoders, they have been adopted in several recent communication standards, e.g. 5G-NR [START_REF]3rd Generation Partnership Project (3GPP)[END_REF]. Therefore, a LDPC code appears as the most sensible choice to evaluate the proposed LLR expression. To be specific, the implemented code follows the specifications of the IEEE 802.11 standard [START_REF]IEEE Standard for Information technology-Local and metropolitan area networks-Specific requirements-Part 11: Wireless LAN Medium Access Control (MAC)and Physical Layer (PHY) Specifications Amendment 5: Enhancements for Higher Throughput[END_REF] with a min-sum decoder, a block length of 1944 bits and a coding rate of 2/3. Fig. 2 presents the BLock Error Rate (BLER) performance of a 16-QAM for arbitrary values of PN variance. In particular, it is shown that the polar metric yields significant gain in performance compared to the Euclidean distance. If the SNR is low and the PN variance is minor, then the Euclidean detector slightly outperforms (< 0.1 dB) the polar metric demodulation -as expected since the polar metric relies on the high SNR assumption. Yet, as PN gets stronger, the performance gain becomes more and more significant. Eventually for strong PN scenarios, the polar metric is the only one exhibiting the waterfall feature of LDPC, while the iterative decoder never converges with the Euclidean distance. Performance of the ML premetric are not shown as they are strictly identical to the ones of the polar metric, even though they are more complex to evaluate.

σ 2 p = 1x10 -2 σ 2 p = 5x10 -2 σ 2 p = 1x10 -1

V. CHANNEL ESTIMATION FOR OPTIMAL DEMODULATION

In previous sections, it has been emphasized that knowledge of the CSI is highly valuable to maximize the performance of communication systems affected by PN. Conversely to the Euclidean distance, the ML premetric and the polar metric are directly defined upon the thermal and phase noise variances. Greatly improving the performance, both decisions rules hence require an estimation of the values σ2 n and σ 2 p .

A. Thermal and phase noise variances estimation

Estimation of thermal and phase noise variances σ 2 n , σ 2 p can be performed by transmitting N pilot symbols. These pilot samples are known from the receiver and denoted s = (s 1 , s 2 , ..., s N ). The joint likelihood function may be derived from Eq. ( 7) and is expressed by

p N (r|s, σ 2 n , σ 2 p ) = N k=1 p(r k |s k , σ 2 n , σ 2 p ), (18)
where r = (r 1 , ..., r N ) stands for the received samples. We consider in the following that pilots have a constant amplitude 2 of √ E s . The ML estimate [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF] of the covariance matrix yields the ML estimators expressed by

σ2 n = 1 N N k=1 (r ρ,k -E s ) 2 , σ2 p = 1 N N k=1 (r θ,k -s θ,k ) 2 - σ2 n E s . (19)
We can write such estimators as χ 2 (N ) distributions to quantify their bias and dispersion from the means and variances:

E[σ 2 n ] = σ 2 n V[σ 2 n ] = 2σ 4 n N , E[σ 2 p ] = σ 2 p V[σ 2 p ] = 2(σ 2 p +σ 2 n /Es) 2 N + 2σ 4 n N Es
.

(20) Both of these estimators are unbiased. Therefore, it is relevant to compare their dispersions to the Cramer-Rao Lower Bounds (CRLB) [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. Since Eq. (18) satisfies the regularity conditions, the CRLBs are given by

V σ2 n ≥ 2σ 4 n N , V σ2 p ≥ 2(σ 2 p + σ 2 n /E s ) 2 N . (21)
Since the variance of σ2 n equals the CRLB, the estimator is efficient cf. [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. Regarding the PN variance, the estimator σ2

p is close to the CRLB but does not perfectly achieves it. Still, both estimators demonstrate the smallest Mean Square Error (MSE) among any unbiased estimators. From Eq. (18), it is straightforward that the joint density p N belongs to the exponential family such that B. Performance degradation due to estimation errors Though we have been able to evaluate the performance of the estimators. Yet, definitions of the ML premetric and the polar metric are subject to estimation errors. So that, the performance degradation due to such estimation errors has to be quantified to specify the required number of pilots. Therefore, Fig. 3 and4 present respectively the performance of an uncoded and a LDPC coded 16-QAM for different number of pilots symbols. We may remark that the polar metric is significantly more resilient than the ML premetric to estimation errors for hard symbol detection, and only slightly in the case of soft decision decoding. A rather low number of pilots is required to achieve near optimal performance. VI. CONCLUSION In this paper, we have addressed the problem of demodulation in the presence of PN. We have assumed a high SNR and a Gaussian PN to derive the optimum symbol detection decision rule. This ML decision premetric significantly reduces the BER error floor due to PN. Besides, it is possible to define a simpler metric, entitled polar metric, upon this decision rule without any loss in performance. Thereupon, we have proposed a framework to properly represent signals upon the Gaussian PN channel. Further, computable probabilistic demapper values have been defined upon the polar metric, which is not direct from the ML premetric. It has been shown that the polar metric, in comparison to the Euclidean distance, greatly enhances performance of coded systems. Since optimum demodulation requires the knowledge of CSI, we have proposed ML estimates of the thermal and phase noise variances. By evaluating the performance degradation due to estimation errors, we are able to state that a low number of pilots is sufficient to achieve near optimal performance. It is also worth mentioning that these results could be extented to non-mmWave applications and Rayleigh fading channels.

T = ( N k=1 (r ρ,k - √ E s ) 2 , N k=1 (r θ,k -s θ,k ) 2 ) is a com- plete sufficient statistic for parameter (σ 2 n , σ 2

Fig. 1 :

 1 Fig. 1: Performance of the different decision rules for an uncoded 16-QAM with PN variance σ 2 p = 10 -2

Fig. 2 :

 2 Fig. 2: Comparison of a LDPC coded 16-QAM performance exploiting LLR based either upon the Euclidean distance or on the polar metric for different values of PN variance σ 2 p

Fig. 3 :

 3 Fig. 3: Performance degradation due to estimation errors for an uncoded 16-QAM and different number of pilot samples with PN variance σ 2 p = 10 -2

Fig. 4 :

 4 Fig. 4: Performa.nce degradation due to estimation errors for a LDPC coded 16-QAM and different number of pilot samples with PN variance σ 2 p = 10 -1

The set of complex-valued sequences equipped with the Euclidean inner product Namely, soft decision decoding exploits the reliability information of the encoded bits. Nevertheless, it may be too complex for practical systems to evaluate directly the exact values of a probabilistic demapper. Still, they can be closely approximated in order to be efficiently computed.

The average symbol energy Es is considered perfectly known as it can be estimated blindly through all sent symbols.

ACKNOWLEDGMENT

The research leading to these results received funding from the French National Research Agency (ANR-17-CE25-0013) within the frame of the project BRAVE.