Thi-Ngoc-Diep Do

Michaud Alexis

Castelli Eric

TOWARDS THE AUTOMATIC PROCESSING OF YONGNING NA (SINO-TIBETAN): DEVELOPING A 'LIGHT' ACOUSTIC MODEL OF THE TARGET LANGUAGE AND TESTING 'HEAVYWEIGHT' MODELS FROM FIVE NATIONAL LANGUAGES

Keywords: Acoustic models, automatic speech recognition (ASR), multilingual modelling, under-resourced languages, endangered languages, Yongning Na, Naish languages, language portability, statistical language modeling, crosslingual acoustic modelling and adaptation

Automatic speech processing technologies hold great potential to facilitate the urgent task of documenting the world's languages. The present research aims to explore the application of speech recognition tools to a littledocumented language, with a view to facilitating processes of annotation, transcription and linguistic analysis. The target language is Yongning Na (a.k.a. Mosuo), an unwritten Sino-Tibetan language with less than 50,000 speakers. An acoustic model of Na was built using CMU Sphinx. In addition to this 'light' model, trained on a small data set (only 4 hours of speech from 1 speaker), 'heavyweight' models from five national languages (English, French, Chinese, Vietnamese and Khmer) were also applied to the same data. Preliminary results are reported, and perspectives for the long road ahead are outlined.

INTRODUCTION

The present exploratory research is part of a long-term endeavour to tap the potential of automatic speech processing technologies to facilitate the urgent task of documenting the world's languages. This paper is aimed at the interdisciplinary readership of the SLTU (Spoken Language Technologies for Under-resourced Languages) workshop series. Linguists who would like to skim the paper for information about what linguistic findings can be made in the early stages of speech-recognition software development are advised to read §1.2-1-3, then jump straight to section 4 ("Linguistic findings"). Speech processing specialists looking for the key facts about what has been done so far with the Yongning Na data can start from §2.3 (presentation of the online data set), and continue into §3 ("Method").

As an introduction, some reflections will be set out about current bottlenecks in language documentation and the contribution that speech recognition tools could make.

Bottlenecks in language documentation: transcription and alignment

The necessity to document the world's languages is now well known to linguists and the general public. In recent years, a number of archives have been created to address this major need of the linguistic community -including the Academia Sinica Collections, AILLA (Univ. of Texas), ELAR (SOAS, London), the Language Archive at the Max Planck Institute for Psycholinguistics, and PARADISEC (ANU/Univ. Sydney). The Open Language Archives Community (OLAC) and the Language Archive at the Max Planck Institute for Psycholinguistics list many other repositories. Each of these archives makes a contribution to the world-scale effort of documenting the diversity of spoken languages. However, while it becomes increasingly easy to collect large amounts of data, data annotation remains highly time-consuming. After a trained linguist has worked out the phonological system of a given language or dialect, and trained their ear to be able to transcribe with accuracy, they still need about one hour of work -usually in collaboration with a native language consultant -to transcribe every minute of recording. This "pernicious transcription bottleneck" [START_REF] Bird | Androids in Amazonia: recording an endangered language (Blog entry)[END_REF] puts severe limitations on the size of the data sets available for languages without a writing system: languages for which the only available resources are those created by professional linguists. In documentation projects, the proportion of transcribed materials is often significantly less than one fourth of the total recorded audio data. Further enrichment of the annotation, such as the addition of phoneme-level alignment (which would be extremely useful for phonetic studies), is seldom carried out, again because it would require more time than the busy schedule of fieldwork and linguistic analysis generally allows.

Automatic speech processing technologies could facilitate the realization of these tasks. The following subsections deal with phonemic alignment and document transcription, respectively.

A perspective for the short to mid term: forced alignment of manual transcriptions

Phoneme-level alignment of transcriptions can now be conducted in automatic mode for a range of English dialects. This allows for applications in research such as the FAVE (Forced Alignment and Vowel Extraction) Program Suite, which automatically aligns and extracts large quantities of vowel formant measurements from orthographically transcribed data [START_REF] Rosenfelder | FAVE (Forced Alignment and Vowel Extraction) Program Suite[END_REF]. Such automated measurements hold great promise for shedding light on disputed issues of phonology, such as vowel deletion in French [START_REF] Martinet | Nature phonologique d'e caduc[END_REF]: a recent study relies on 4,000 tokens of words produced as variants with and without schwa in a corpus of radio-broadcasted speech [START_REF] Bürki | What affects the presence versus absence of schwa and its duration : A corpus analysis of French connected speech[END_REF]; the scale of the corpus allows for new empirical insights into this classical issue. The extension of such fine-grained investigation to all the languages for which transcribed recordings are available would lead to progress (i) in the study of language-specific issues of phonological analysis and (ii) in the typology and modelling of sound systems. Forced-alignement tools, taking as their input the manual transcriptions created by linguists, would be a major step in this direction. Given the current state of the technology, this can confidently be planned in the short to mid term.

A perspective for the long term: automatic speech recognition as a tool to facilitate transcription

Another desirable tool for the linguist would be a fullfledged speech recognition system for the target language, which would provide a draft transcription -and, ultimately, translations into other languages, using automatic translation tools [START_REF] Do | Mining Parallel Data from Comparable Corpora via Triangulation[END_REF] -whose manual correction would be less time-consuming than fully manual transcription. This perspective currently appears distant, however, given the number of obstacles to overcome. Speech recognition technology requires large amounts of transcribed data in the first place; available data sets are typically too small to allow for the training of software for speech recognition and synthesis. For instance, in the course of two decades, the Pangloss Collection, an open archive of endangeredlanguage data [START_REF] Michailovsky | Documenting and researching endangered languages: the Pangloss Collection[END_REF], has reached the size of about 190 hours of recordings in 70 languages; about 1/3 rd of the total (60 hours) have a full transcription and annotation. This makes the Pangloss Collection one of the world's largest archives of "rare" or "endangered" languages. But this size -on average: one hour per language -remains extremely small in comparison with the amount of data currently required to train systems of recognition and synthesis. As a result, Human Language Technologies remain restricted to a small fraction of the world's languages.

In view of the situation summarized above, a suggestion that has been made in the literature is to improve "the portability of speech and language technologies for multilingual applications, especially for under-resourced languages" [START_REF] Besacier | Automatic speech recognition for under-resourced languages: A survey[END_REF], see also [START_REF] Schultz | Language-independent and language-adaptive acoustic modeling for speech recognition[END_REF]. The present paper reports on preliminary work intended as a contribution to this research strand, focusing on automatic speech recognition (ASR) for an unwritten Sino-Tibetan language of Southwest China that has less than 50,000 speakers.

PRESENTATION OF THE TRAINING CORPUS

Before presenting the data set used as a training corpus for the present study, it may be useful to present some general observations about data collection in fieldwork.

How the data set develops: the choice of recorded materials is guided by the research topics

In fieldwork, word lists are elicited first, to work out the phonemic system. But narratives (such as folk tales, life stories, and explanations about traditional techniques) and dialogues are the backbone of linguistic documentation.

They offer examples of the use of words in context, and constitute a reliable basis for an open range of research purposes.

In addition to these basic types of documents, recordings are guided by the issues encountered in research, and reflect the diversity of linguists' interests. In the case of the documents presented here, the balance is tilted towards phonetic/phonological topics, which the researcher had the interest to pursue in greatest detail. Specifically, the tone system of the target language of the present study -Yongning Na -is an area which calls for in-depth description: the language has a complex system of morphologically conditioned tone change [START_REF] Michaud | The tone patterns of numeral-plus-classifier phrases in Yongning Na: a synchronic description and analysis[END_REF]. Numerous elicitation sessions have therefore been devoted to an investigation of the tone system, each focussing on a specific issue, such as tone changes that take place when an object is associated with a verb.

There is therefore no sharp distinction between a documentation agenda and a research agenda: all documents recorded in the course of research are relevant additions to the online collection, gradually enriching the record.

Advantages of fieldwork conditions for collecting abundant and reliable data

Data collection is an underestimated challenge, and perhaps a weak spot of some current linguistic studies. It is obvious that the empirical basis of one's research is of paramount importance for all later stages. On the other hand, the importance of good communication with language consultants is not always recognized. The consultants' perception of the investigator's intentions exerts considerable influence on their behaviour [START_REF] Bowern | Linguistic fieldwork: a practical guide[END_REF]- [START_REF] Mosel | Field work and community language work[END_REF]. In this respect, the fieldworker's experience may be useful to the "laboratory worker". Documents collected in fieldwork compare favourably in many respects with those collected in the lab. In fieldwork, the investigator can gain familiarity with her or his consultants. (The second author of this paper stayed in the village of Yongning about two months a year from 2006 to 2009; from August 2011 to October 2012, he worked with his main consultant on a day-to-day basis.) This allows for the thoughtful design of materials to be recorded.

The transcription and annotation of the Yongning Na data was entirely manual, as is common for fieldwork materials. For some languages, adding complete glosses at word level can be done semi-automatically; however, semiautomatic treatment is painstaking in languages that contain numerous homophonous words, making it sometimes more appropriate to do all the glossing by hand rather than wade through lists of homophonous lexical entries.

Creating reliable, fine-grained transcriptions and annotations for documents in less-documented languages is a labour of love, into which investigators and their consultants put great amounts of time and effort. The quality of these hand-made annotations is usually excellent. This is a good start in life for these resources, which can later be further enriched and used for a variety of purposes.

Less positively, the way language documentation is conducted under fieldwork conditions entails some drawbacks for speech processing, as the data set is not tailored to match the needs of the speech recognition algorithms.

Current state of the online collection, used as training corpus for an acoustic model: a singlespeaker data set

This section presents the audio data set available for the target language, Yongning Na, also known as Mosuo [START_REF] Michaud | Phonemic and tonal analysis of Yongning Na[END_REF]- [START_REF] Michaud | A description of endangered phonemic oppositions in Mosuo (Yongning Na)[END_REF]. It updates an earlier presentation of these data [START_REF] Michaud | Combining documentation and research: Ongoing work on an endangered language[END_REF].

In the Glottolog inventory of languages, the language code of Yongning Na is yong1270; in the Ethnologue inventory maintained by the Summer Institute of Linguistics, its code is NRU. Na is a tonal language whose syllable structure is essentially CV [START_REF] Michaud | Phonemic and tonal analysis of Yongning Na[END_REF]. As most of the Yongning Na data were collected from one speaker, it was decided to create a speaker-specific language processing tool, not a speaker-independent tool.

One hundred narratives, making up a total of eleven hours, have been recorded by this speaker since 2006. Eight texts with complete transcription and Chinese and French translations are now available online (two of them with English translation as well), corresponding to 1.3 hour. Sixteen more (also available online) are transcribed and translated but without word-level glosses. Phonetic/phonological elicitation sessions with full Chinese, English and French annotations amount to more than two hours (over 40 documents). For narratives, the annotations are synchronized with the recordings at a level loosely referred to as the Sentence, with an average duration on the order of three seconds. Phonological materials are made up of phrases or short sentences, and the time-alignment is based on these units, which are less than three seconds in length.

The resources (recordings and annotations) are freely available for online browsing and download, under a CreativeCommons licence. The Pangloss Collection is hosted in a broader repository named CoCoON, via which the resources are referenced by various search engines including OLAC and OAIster. The metadata follow Dublin Core/Open Language Archives Community standards. The Yongning Na data are presented on the following page: http://lacito.vjf.cnrs.fr/archivage/languages/Na_en.htm A technical description of archiving and web hosting falls outside the scope of this paper; see [START_REF] Michailovsky | Documenting and researching endangered languages: the Pangloss Collection[END_REF]. Let us simply mention that long-term conservation is guaranteed through a partnership with a perennial archiving institution: CINES [START_REF]CINES[END_REF].

METHOD

"Porting Human Language Technologies (e.g. a speech recognition system) to an under-resourced language requires techniques that go far beyond the basic re-training of the models" [START_REF] Besacier | Automatic speech recognition for under-resourced languages: A survey[END_REF], as exemplified by previous work [START_REF] Caelen-Haumont | Automatic labeling and phonetic assessment for an unknown Asian language[END_REF]. The presentation of the method followed in the present preliminary work aims to emphasize the adjustments required by the processing of the target language.

Issues encountered when using spontaneous speech as a training corpus

As explained in section 2, one half of the training corpus is made up of narratives, and the other half of elicited phrases recorded to document the morpho-phonology of Yongning Na. This results in a degree of heterogeneity in the data, as elicited materials are pronounced with a slower speaking rate. In terms of the continuum between hypoarticulated and hyperarticulated speech [START_REF] Lindblom | Explaining phonetic variation: a sketch of the H&H theory[END_REF], elicited materials tend to be hyperarticulated, whereas spontaneous speech contains rapid alternations between hypoarticulation -typically, for backgrounded words -and hyperarticulation -for words that carry a greater informational and/or affective load. Spontaneous speech also contains dysfluencies and loanwords.

Dysfluencies

The narratives that constitute one half of the training corpus contain, like any set of continuous, non-read speech, some dysfluencies. Two types of filled pauses are distinguished in the narratives, those with the mouth open, transcribed as /əəə.../ (200 occurrences), and those with the mouth closed, transcribed as /mmm.../ (170 occurrences). One of the shortcomings of the preliminary version of the recognition system developed for Na is that it ignores filled pauses. In future versions, attempts will be made to identify various types of pauses, and indicate them in the automatic transcription.

Passages that were said by mistake (speech repairs) are transcribed in the annotation, aiming to approach as closely as possible (in consultation with the speaker) the actual flow of speech on the recording: for instance, the angle brackets in /ɖɯ˧-sɑ˥ | <le˧-ʂ...> ɖɯ˧-mə˧-ʁo˧-tsɯ˩ | -mv̩ ˩/ indicate that <le˧-ʂ...> was said by mistake. In the first test reported here, the angle brackets were simply removed, and the dysfluent data were fed into the acoustic model along with canonical realizations. In later work, they will need to be treated separately, to avoid the presence of noncanonical realizations in the training data.

Loanwords

Minority languages in today's world are subject to increasing pressure from national languages and other socially dominating language varieties. This is reflected in the presence of loanwords. While older loanwords tend to be integrated into the language's phonological system, in cases of bilingualism loanwords tend to introduce new sounds and new combinations of sounds. This increases the inventory of sounds, and the number of choices among which the speech recognition algorithms have to make a choice. In the case of Na, recent borrowings from Mandarin Chinese introduce new sounds, such as /ẽ/ in /ɕi˧ɕjẽ˧/~/ɕi˧ɕɥẽ˧/ 'fresh' (Chinese 新鲜). A more frequent case is that Chinese borrowings introduce new combinations among phonemes: for instance, in /sjae˧pʰjɛ˧/ 'photograph' (Chinese 相片), the first syllable combines an initial /s/ with a rhyme /jae/: this constitutes an unattested combination in Yongning Na. The second syllable contains a rhyme /jɛ/ which is altogether unattested in Na. In the initial version of the acoustic model, the sound /ɛ/ was simply merged with /e/; as the model is segment-based and not syllable-based, it was not necessary to face phonotactic issues, but these will need to be addressed at later stages.

Technical operations for building the 'light' acoustic model

Conversion of the recordings and transcriptions

The original recordings are WAV files, 24-bit, 44,100 Hz. One fifth consists of mono audio files, recorded with an AKG C-535EB or AKG C-900 microphone. Another fifth are stereo files with an audio channel and an electroglottographic channel. (Electroglottography is the ultimate reference for measuring fundamental frequency; it also allows for the evaluation of other glottal parameters [START_REF] Fabre | Un procédé électrique percutané d'inscription de l'accolement glottique au cours de la phonation: glottographie de haute fréquence[END_REF]- [START_REF] Baken | Electroglottography[END_REF].) The rest (60%) are stereo audio files comprising, in addition to the signal from a table microphone, a signal from a Sennheiser HSP 2 omnidirectional headworn microphone.

The electroglottographic recordings were not used so far; they will be useful for the analysis of tone, not undertaken here (see §5.1). The audio recordings were downgraded to 16-bit, 16,000 Hz to comply with the requirement on the input to CMU Sphinx. Stereo files were demultiplexed and fed into the training tool as if they constituted different data sets.

The annotation is logically structured text, in XML format. Narratives have the following structure: a TEXT is divided into S (sentences in a loose sense), which are divided into W (words) [START_REF] Jacobson | Linguistic documents synchronizing sound and text[END_REF]. These data were converted to plain text, retaining only the sentence-level transcription and time codes.

Construction of an acoustic model, and speech recognition tests

An acoustic model was created by feeding the training data into CMU Sphinx. This 'light' model was trained on a small data set: only 4 hours of speech from 1 speaker. The resulting tool was then tested on a recording by the same speaker with the same linguistic and technical properties: a narrative recorded in a quiet room with the same equipment as the training corpus.

Application of 'heavyweight' recognition systems from other languages

In addition to the 'light' acoustic model developed for Yongning Na, 'heavyweight' recognition systems from five other languages (English, Mandarin Chinese, French, Vietnamese [START_REF] Lê | Reconnaissance automatique de la parole pour des langues peu dotées[END_REF], [START_REF] Nguyen | Reconnaissance automatique de la parole continue à grand vocabulaire en vietnamien[END_REF], and Khmer [START_REF] Sam | Vers une adaptation autonome des modèles acoustiques multilingues pour le traitement automatique de la parole[END_REF]) were applied to the Na data, to determine to what extent Na sounds similar to sounds found in these five languages could be accurately recognized. A multilingual acoustic model for these five languages was built on the basis of [START_REF] Sam | Vers une adaptation autonome des modèles acoustiques multilingues pour le traitement automatique de la parole[END_REF]. This multilingual acoustic model combines five monolingual acoustic models. The total number of phones is 194: 40 English phones, 43 French phones, 34 Mandarin phones, 41 Vietnamese phones, and 36 Khmer phones.

The idea was to use the output from this 'heavyweight' multilingual model by applying rule-of-thumb acoustic correspondences between the target language's sounds and those of the five national languages selected. However, the error rate at first pass was extremely high. This is due in part to speaker characteristics: it was observed when doing spectrographic analysis of data from the Yongning Na speaker that her relatively high-pitched voiced, her speaking rate (often rapid) and the F-pattern of her small-sized vocal tract were a challenge to experimented spectrogram readers.

In future work using speaker-independent recognition tools from other languages, it appears advisable to devise and apply speaker adaptation modules, following a strand of current research in automatic speech recognition [START_REF] Wang | Comparison of acoustic model adaptation techniques on non-native speech[END_REF], [START_REF] Sam | Online unsupervised multilingual acoustic model adaptation for nonnative ASR[END_REF]. The passage shown in Figure 1 is /hĩ˧-ɳɯ˩ | pɤ˧~pɤ˧/, 'The people carried...'. It was recognized as /h ĩ ɳ ɯ p ɤ t o q v/ by the 'light' recognition tool developed for Na. Thus the first two syllables were correctly identified, whereas the two syllables /pɤ˧~pɤ˧/ were identified as /p ɤ/ and /t o/ respectively, i.e. with a mistake concerning the place of articulation of the initial stop. The initial of the second token of /pɤ˧/ in Figure 1 can hardly be confused visually with a /t/, since it does not present the F2 transition from about 1,800 Hz down to the target F2 value of the syllable's rhyme which would be a telltale sign of /t/. By comparison, the 'heavyweight' reconstruction models devised for other languages (see §3.3) yielded grossly inaccurate results for the unfamiliar sequence /hĩ.ɳɯ/ (identified as /k ɥ n/), and for the vowel /ɤ/ (identified as /o/ in the first case, and as /w/ in the second), but they identified the two labial stops accurately, suggesting that the mistaken identification proposed for /p/ by the Na acoustic model should be avoidable if a larger training corpus is used. (The passage shown in Figure 1 was identified as /k ɥ n p o p w/.) Finally, the sound on the right hand-side of the spectrogram in Figure 1 is a nonlinguistic sound: the unvoiced release of a glottal stop, which could be approximated as [ʔə̥ ]. The Na acoustic model identified it as a syllable made up of an unvoiced uvular sound and a /v̩ / rhyme; in order to avoid such mistaken identifications, it appears useful to provide more fine-grained notations for non-speech sounds than was done so far in the training corpus.

An example of the results

INITIAL LINGUISTIC FINDINGS

This section presents a small sample of linguistic findings based on the application of the very first version of the Yongning Na phoneme recognition tool. It is likely to be the most rewarding section of this paper for linguists; it is hoped that it will also be of interest to engineers working on speech processing, as an illustration of scientific side benefits of language technology.

Side benefits of data preparation

Data preparation offered a chance to check that all the data conformed to the phonological description. At data conversion from XML to plain text ( §3.2.2), a list of segments was produced, and compared with the list of sounds provided for the language (by the second author). This comparison brought out a handful of inconsistencies in the notation, such as the use of /ẽ/ for an interjection appearing in some of the texts. This prompted a return to the data, which revealed that these were in fact instances of 'yes' (canonical transcription: /ĩ/) that had been transcribed before the second author identified the nature of this morpheme. Systematic examination of the passages at issue revealed that this /ẽ/ was in most cases a response to a comment or yes/no question on the part of someone in the audience, confirming the interpretation of this morpheme as a sign of approval.

Allomorphic variation of grammatical morphemes

Cross-linguistically, grammatical morphemes tend to be hypoarticulated, by comparison with lexical morphemes. This is brought out clearly by identification mistakes of the speech recognition tool.

Fricatives and affricates

The topic marker /-ʈʂʰɯ˥/ tends to be identified as /ʂ ɯ/, with a fricative instead of an affricate, and without aspiration (e.g. in sentence 1 of the narrative Housebuilding2). This may be revealing of its customary hypoarticulation (although the algorithm's mistake could also be due to other causes, such as problems in the learning process). Such phenomena argue for a different encoding of grammatical words vs. content words.

Lack of identification of nasals in grammatical morphemes

Weaker phonetic realization may lead to some syllables being entirely overlooked by the recognition system. Nasalinitial grammatical morphemes are a case in point: they tend to be overlooked by the recognition algorithm when they are followed by an item with an initial nasal. For instance, the negation /mə˧-/ was not identified in the passage /ə˧tso˧-mə˧-ɲi˩/ (in sentence 1 of the narrative Housebuilding2), for which the algorithm's output was /ə ts o ɲ i/.

This stimulates further reflections on the linguist's part concerning the hypoarticulation of grammatical morphemes. The fact that the negation in Yongning Na undergoes a relatively high degree of vowel harmony with the verb that follows had led to its analysis as /mə˧-/, with a neutral vowel (schwa) which was defined phonologically as the neutralization of all vowel oppositions [29, p. 70]: a vowel that does not stand in opposition to the others. But the output of the recognition algorithm suggests that weakening also affects the initial. This leads to a questioning of the phonological analysis, in which the initial was considered identical to the /m/ found in other environments whereas a special treatment was set up for the rhyme. In this light, the second author's current thinking is that weakening should be considered a characteristic of the entire syllable, applying to a segmental structure that remains identical with that of lexical morphemes (content words). In turn, this raises the issue of the negation's phonemic vowel; phonetic evidence and historical-comparative evidence both point to an analysis as /mɤ˧-/. This reanalysis constitutes a step forward: arriving at a higher degree of generalization, while dispensing with a cumbersome theoretical device.

CONCLUSION: CHALLENGES AND PERSPECTIVES FOR FURTHER WORK

It is too early at present to launch into a performance evaluation for the system; instead, this section recapitulates challenges and perspectives for future work, emphasizing fundamental linguistic issues that will need to be integrated into the speech recognition software.

Handling linguistic tone

Tone was left out of the scope of this preliminary work, for the simple technical reason that it is not handled in CMU Sphinx. This is a major shortcoming, because tone has a high functional yield in Yongning Na. The common ancestor of Yongning Na and all other Sino-Tibetan languages was non-tonal, and replete with consonant clusters [START_REF] Ferlus | What were the four divisions of Middle Chinese?[END_REF]; dramatic phonological erosion took place, giving birth to tone [START_REF] Haudricourt | Comment reconstruire le chinois archaïque[END_REF]- [START_REF] Matisoff | Tibeto-Burman Tonology in an Areal Context[END_REF]; this erosion is especially advanced in Yongning Na [START_REF] Jacques | Approaching the historical phonology of three highly eroded Sino-Tibetan languages: Naxi, Na and Laze[END_REF]. For the Yongning Na speech recognition system to progress towards lexical identification, it will be necessary to handle tone recognition.

The need for tools that fit the structure of the target language: syllable-based recognition tools

Some contextual information is taken into account by the CMU-Sphinx software package: the preceding and following sound. In the case of Yongning Na, however, it appears necessary to go beyond these default settings and experiment with syllable-based recognition tools in future: a salient phonological characteristic of Yongning Na -as also of the closely related language Naxi [START_REF] Michaud | Three extreme cases of neutralisation: nasality, retroflexion and lip-rounding in Naxi[END_REF] -is the tight association of initial and rhyme within syllables, which are essentially of CV structure. Coarticulation within Na syllables is so strong that it is sometimes an issue how to analyze a given syllable into an initial and rhyme. Given this state of affairs, it definitely seems interesting to attempt to identify entire syllables, rather than successive sounds.

Identifying intonational phenomena

Intonational phenomena were entirely overlooked in the first-pass acoustic model. This is clearly a major shortcoming, and a source of errors in the automatic identification of vowels and consonants. Vowel lengthening is a case in point: there is no phonemic opposition of vowel length in Na; the last lexical item in a prosodic group tends to be greatly lengthened. In the first version of the software, lengthened syllables are often misinterpreted as a sequences of two syllables, the second with a continuant initial such as /ɭ/. This is due to the absence of a parameter for syllable length in the first-pass version of the system. There is so much intonationally conditioned variation in syllable length in Yongning Na (and in the closely related languages Naxi [START_REF] Michaud | Reassociated tones and coalescent syllables in Naxi (Tibeto-Burman)[END_REF] and Laze [START_REF] Michaud | The phonology of Laze: phonemic analysis, syllabic inventory, and a short word list[END_REF]) that this issue will definitely need to be carefully considered in future work.

An important task for our future work thus consists in finding ways to integrate intonational phenomena into the system. It is well-known that details in the articulation of lexically distinctive units (consonants, vowels, and stress or tone) carry intonational information, reflecting phrasing and prominence patterns [START_REF] Erickson | Effects of contrastive emphasis on jaw opening[END_REF]- [START_REF] Keating | Domaininitial articulatory strengthening in four languages[END_REF] as well as speaker attitudes. In the long run, it would be desirable for the recognition system to identify intonation-conditioned phenomena: locating each recorded token within the acoustic space of variation of the corresponding phoneme, and deducing properties such as the degree of strength with which the token is articulated (about the "strength coefficient" as a relevant parameter for speech processing, see [START_REF] Kochanski | Prosody Modelling with Soft Templates[END_REF]).

There already exists some information about intonation in the documents used as training corpus, which contain:

• systematic indications of prosodic-group boundaries (by the symbol [ | ]), reflecting the division of the utterance into tone groups [START_REF] Michaud | Phrasing, prominence, and morphotonology: How utterances are divided into tone groups in Yongning Na[END_REF] • indications about information structure: focusmarking, indicated by addition of the letter F after the word at issue; and emphatic stress, transcribed as an upward arrow ↑ before the syllable at issue [START_REF] Michaud | Information structure in Asia: Yongning Na (Sino-Tibetan) and Vietnamese (Austroasiatic)[END_REF] • punctuation marks: commas, colons, semi-colons, full stops, exclamation marks, and interrogation marks. These pieces of information were not used in the first-pass training of the acoustic model. They are all ultimately relevant for modelling, however, so that sources of variation across realizations of the same sound can be identified and analyzed, instead of treating variability as random.

Figure 1 Fig. 1 .

 11 Figure1shows a sample of the acoustic data to which the recognition tool was applied.

ACKNOWLEDGMENTS

Many thanks to the Yongning Na language consultants and friends, and to Martine Toda and two anonymous reviewers for useful comments. We gratefully acknowledge financial support from CNRS through the PEPS (Projets Exploratoires Premier Soutien) grant APRIL, and from Agence Nationale de la Recherche through the projects HimalCo, ANR-12-CORP-0006, and LabEx EFL, ANR-10-LABX-0083 -Investissements d'Avenir.


Moustapha Mbaye
email: moustapha.mbaye.sn@gmail.com

Moussa Diallo
email: moussa.diallo@ucad.edu.sn

Mamadou Mboup
email: mamadou.mboup@univ-reims.fr

Unimodular-Upper polynomial matrix decomposition for MIMO Spatial Multiplexing

We present a simple algorithm to compute the factors of a Unimodular-Upper (UU) polynomial matrix decomposition. The algorithm relies on the classical LU factorization and the inverse of the unimodular factor is also provided. Such decomposition is useful for spatial multiplexing in MIMO channel transmission system since it enables to reduce the MIMO channel matrix into independent SISO channels by a pre-and post-filtering. Unlike the classical QR-based polynomial matrix Singular Values Decomposition (QR-PMSVD), the proposed UU method allows to completely cancel the co-channel interference (CCI). Moreover, most of the resulting independent SISO channels are likely to be reduced to simple additive noise channels, i.e. with no InterSymbol Interference. However, the noise is coloured and possibly enhanced due to the non unitary property of the corresponding post filter. The complexity and sum rate capacity performance of the proposed method are studied and compared with QR-PMSVD.

I. INTRODUCTION

Let H(z) ∈ C p×q a polynomial matrix, i.e.

H(z) = L k=0 H k z k
where each coefficient H k is a p × q-complex matrix and L is the formal degree of the polynomial.

We consider the problem of the decomposition of H(z) in the form

H(z) = U (z)D(z)V (z),
where the factors V (z), D(z) and U (z) are simple and structured polynomial matrices. The difficulty lies in the constraint that the factors be also polynomial. Such decomposition is useful in Multiple-Input Multiple-Output (MIMO) channel transmission system. The polynomial matrix H(z) then represents the FIR channel transfer function.

The inverse of the factors U (z) and V (z) are used as postand pre-coders respectively in order to reduce the MIMO channel to a simpler form D(z). These inverses are not only assumed to exist but also, it is required that they are Laurent polynomials and easy, if not immediate, to compute.

Ideally, the form of D(z) is diagonal [START_REF] Brandt | Polynomial matrix decompositions: Evaluation of algorithms with an application to wideband MIMO communications[END_REF], [START_REF] Razavi | Subspace beamforming via block SVD for MIMO system[END_REF], [START_REF] Zamiri-Jafarian | A polynomial matrix SVD approach for time domain broadband beamforming in MIMO-OFDM systems[END_REF]. Diagonalization of the system in space means the transformation of the MIMO channel in N = min(p, q) separate and independant single input single output (SISO) channels, so that there is no co-channel interference (CCI). This is achieved for example when a singular value decomposition (SVD) is considered. This decomposition assumes that V (z) and U (z) are (FIR) paraunitary polynomial matrices, i.e. they satisfy

U (z)U (z) = I and V (z) V (z) = I, for all z ∈ C,
where the tilde stands for the paraconjugation, defined by: X(z) = X * (1/z). For instance, the paraconjugate of

X(z) = k X k z k is given by X(z) = k X * k z -k .
The sign * denotes the transpose-conjugate.

Note however that a polynomial matrix SVD with polynomial factors does not exist in general [START_REF] Icart | Some properties of Laurent polynomial matrices[END_REF]. Two main approaches are found in the literature:

Forster, McWhirter et al. [3]:

Apply successive elementary Laurent polynomial Givens rotations to iteratively zero the largest coefficient in magnitude, beneath the diagonal. This produces FIR paraunitary factors U (z) and V (z). The factor D(z) is also a Laurent polynomial but only approximately diagonal: the largest coefficients in magnitude, beneath the diagonal are less than a prescribed tolerance parameter ǫ. The effect of ǫ in the CCI as well as in the system performance is studied in [START_REF] Zamiri-Jafarian | A polynomial matrix SVD approach for time domain broadband beamforming in MIMO-OFDM systems[END_REF]. In addition, the degrees of the involved polynomials grow fast in general, this is the reason why a truncation step is introduced. However, the paraunitary property is lost with the truncation, which degree is measured by some parameter µ.

D. Cescato and H. Bolcskei [2]:

Apply classical SVD decomposition to samples of the original matrix

H(z i ) = U (z i )D(z i )V (z i ), i = 1,
2, . . .. The factors of the polynomial decomposition are deduced by polynomial interpolation of the samples of the factors respectively. However the nice properties of the sample matrices (easy invertibility, paraunitary,...) are not guaranteed to hold after interpolation.

In this paper, we investigate a Unimodular-Upper (UU) decomposition for polynomial matrices, from the classical LU decomposition. More precisely, given a C p×q polynomial matrix H(z), the problem is to write it in the form

H(z) = U (z)R(z),
where R(z) is an upper triangular polynomial matrix and U (z) is a unimodular polynomial matrix. Recall that a unimodular matrix is a square polynomial matrix with constant nonzero determinant. It therefore admits a polynomial inverse. Applying the same decomposition to the transpose of the resulting factor R(z) above provides a factorization of the form:

H(z) = U (z)D(z)V (z),
where V (z) and U (z) are two unimodular matrices and D(z) is a diagonal polynomial matrix. The rest of this paper is organized as follow. The proposed UU decomposition is detailed in section II. It is well known that such decomposition always exists and can be obtained from solutions of Bezout equations. However, we present a simple and explicit algorithm to compute the factor D(z) and the unimodular matrices U (z) and V (z) together with their respective inverses. Then section III is devoted to the calculation of the sum rate. In the section IV, some numerical results regarding the sum rates attained by the different transmission schemes and their cost in complexity is presented.

II. UU POLYNOMIAL MATRIX DECOMPOSITION

The proposed UU decomposition follows the same steps as the classical LU factorization (or Gauss elimination). In each step, however, a preprocessing is required in order to reduce the pivot element to a constant, subsequently set to 1. To begin, let us take a simple example of a 4 × 4 polynomial matrix.

H(z) =      h1,1(z) h1,2(z) h1,3(z) h1,4(z) h2,1(z) h2,2(z) h2,3(z) h2,4(z) h3,1(z) h3,2(z) h3,3(z) h3,4(z) h4,1(z) h4,2(z) h4,3(z) h4,4(z)      (1)
Assume that the polynomials h 1,1 (z) and h 2,1 (z) are coprime.

Then, let us denote by [h ♯ 1,1 (z), h ♯ 2,1 (z)] any pair of polynomials solving the Bezout equation.

h ♯ 1,1 (z)h 1,1 (z) + h ♯ 2,1 (z)h 2,1 (z) = 1. (2)
If necessary, we may consider a permutation of the rows of H(z) in order to get [START_REF] Cescato | QR decomposition of Laurent polynomial matrices sampled on the unit circle[END_REF]. This may however be insufficient, e.g. when the first column of H(z), noted h 1 (z), is not an irreducible polynomial vector. In such case we may write:

h 1 (z) = d(z) ĥ1,1 (z) ĥ2,1 (z) ĥ3,1 (z) ĥ4,1 (z) t △ = d(z) h 1 (z)
where d(z) is the greatest commom divisor (gcd) of the components of h 1 (z). The remaining polynomial vector h 1 (z) is thus irreducible and we assume that the Bezout equation ( 2) holds by replacing the polynomials h i,1 (z) by their reduced counterparts ĥi,1 (z). In this latter cased we set

d 1 (z) = d(z). Otherwise, d 1 (z) = 1 and h 1 (z) = h 1 (z). If, nonetheless (2)
still fails for all pairs while h 1 (z) is irreducible, then the right hand side of ( 2) is replaced by the gcd of the considered pair and the algorithm proceeds, following the procedure above. Next, we introduce the block diagonal matrix

A1(z) =      h ♯ 1,1 (z) h ♯ 2,1 (z) -h2,1(z) h1,1(z) 1 1      = B1(z) 0 0 I2 (3)
where I k is the k × k identity matrix. A direct verification shows that

H 1 (z) = A 1 (z)H(z) =       d 1 (z) h 1,2 (z) h 1,3 (z) h 1,4 (z) 0 h 2,2 (z) h 2,3 (z) h 2,4 (z) h 3,1 (z) h 3,2 (z) h 3,3 (z) h 3,4 (z) h 4,1 (z) h 4,2 (z) h 4,3 (z) h 4,4 (z)       (4)
Now, one can apply a first step of a classical Gaussian elimination to zero the off-diagonal elements of the first column. This amounts to multiplying the above matrix by the lower triangular polynomial matrix:

L1(z) =      1 0 1 -ĥ3,1(z) 1 -ĥ4,1(z) 1      (5)
The result is denoted by

H 1 (z) = L 1 (z)A 1 (z)H(z)
and it reads as:

H 1 (z) =        d 1 (z) h 1,2 (z) h 1,3 (z) h 1,4 (z) 0 h 2,2 (z) h 2,3 (z) h 2,4 (z) 0 h 3,2 (z) h 3,3 (z) h 3,4 (z) 0 h 4,2 (z) h 4,3 (z) h 4,4 (z)        (6)
The same steps are now repeated for the lower-right block of H 1 (z). In the second column we obtain:

H 2 (z) = L 2 (z)A 2 (z)H 1 (z)
Finally, we obtain the triangular polynomial matrix

R(z) = A3(z)H2(z) =       d1(z) h1,2(z) h1,3(z) h1,4(z) 0 d2(z) ȟ2,3(z) ȟ2,4(z) 0 0 d3(z) ĥ3,4(z) 0 0 0 ĥ4,4(z)       (7)
To summarize, note that this last matrix reads as

R(z) = A 3 (z)L 2 (z)A 2 (z)L 1 (z)A 1 (z)H(z) (8)
Each matrix L i (z) is lower triangular unimodular and its inverse is obtained by simply changing the sign of the offdiagonal entries. Each B i (z) as in ( 3) is also unimodular and:

-JB t i (z)J = B -1 i (z), where J = 0 1 -1 0 , i = 1, 2, 3.

Now on, we set:

A i (z) = A i (z) -1 , L i (z) = L i (z) -1 .
Then, the preceding steps provide a UU decompostion of the polynomial matrix H(z):

H(z) = U (z)R(z) (9)
where This decomposition can be generalized for any given matrix H ∈ C p×q . In the general case, the decomposition ends up after a number of iteration given by min(p, q). The factors V -1 (z) and U -1 (z) can be used as post-and pre-coders respectively in order to reduce the MIMO channel to its equivalent D(z). Since D(z) is exactly diagonal, there is no co-channel interference (CCI) unlike with other methods [START_REF] Foster | An algorithm for calculating the QR and Singular Value Decompositions of polynomial matrices[END_REF], [START_REF] Cescato | QR decomposition of Laurent polynomial matrices sampled on the unit circle[END_REF]. Note also that we may verify by direct inspection that, for exemple, the first entry of D(z) will correspond to the gcd of all the polynomials in the first row and first column of H(z). This shows that the probability that the diagonal elements of D(z) be equal to one is high, except for the last one.

U (z) = A 1 (z)L 1 (z)A 2 (z)L 2 (z)A 3 (z)

III. SUM RATE CAPACITY

Let H(z) ∈ C p×q a wideband MIMO channel and U po (z) and V pr (z) the post-and pre-coders respectively. The communication process in spatial multiplexing context is:

y = U po (z)H(z)V pr (z)x(z) + U po (z)n(z) (10)
where y ∈ C p×1 is the received signal, x ∈ C q×1 the signal vector with covariance E(xx * ) = σ 2 x I q and n a white Gaussian noise vector with covariance E(nn * ) = σ 2 n I p By introducing the decomposition of H(z), (10) becomes:

y = U po (z)U (z)D(z)V (z)V pr (z)x(z) + U po (z)n(z) (11)
The reduced MIMO channel depends on the decomposition and we consider the following two possibilities:

• UU decompostion with U po = U -1 and V pr = V -1 : In this case, y = D(z)x(z) + U -1 (z)n(z). There is no CCI because D(z) is diagonal. However U (z) is not paraunitary and this will affect the filtered noise properties. By considering an equal gain transmission scheme [START_REF] Tsai | An equal gain transmission in MIMO wireless communications[END_REF], the sum rate can be simply calculated by:

min(p,q) k=1 2π 0 log 2 1 + σ 2 x D k,k (e ω )D * k,k (e ω ) σ 2 n U -1 k,: (e ω )(U -1 k,: ) * (e ω ) dω (
y = U (z)U (z) IN r +Ue D(z) ∆+De V (z) V (z) IN t +Ve x(z) + U (z)n(z) w ( 13
)
where U e and V e are the errors caused by the truncation and D e is the off diagonal part of D(z) caused by the fact that the convergence criterion ǫ = 0. Thereby (13) can be rewritten as:

y = ∆(z)x(z) + C(z)x(z) + w(z) (14)
where C(z) represents the CCI filter and w(z) the noise component. Since ∆(z) is diagonal, the sum rate can be calculated by:

min(p,q) k=1 2π 0 log 2 1 + σ 2 x ∆ k,k (e ω )∆ * k,k (e ω ) σ 2
x Φ(e ω ) + σ 2 n U * k,: (e ω )U k,: (e ω ) dω.

(

) 15
where we have set Φ(e ω ) = C * k,: (e ω )C k,: (e ω ).

IV. SIMULATIONS RESULTS

For the numerical simulations, we consider a spatial multiplexing scheme with N t transmit antennas and N r receive antennas. The MIMO channel H(z) is modeled as:

H(z) = L-1 l=0 H l e -αl z l (16)
where the elements of the H l 's are obtained from a zeromean circularly symmetric normalized Gaussian distribution and α ∈ R + is added in order to obtain an exponentially decaying power-delay-profile.

The figures 1 and 2 show the impulse responses of the diagonalized matrix D(z) obtained respectively by QR-PMSVD and UU decomposition. The matrix D(z) is not diagonal with QR-PMSVD algorithm. However, the coefficients beneath the diagonal are in magnitude less than ǫ which is set to 0.01 in this simulation. Clearly, the decomposition error decreases for decreasing ǫ. By contrast, the complexity for post-and pre-coding may increase significantly, as shown in Table I which illustrates the maximum length of the post-and pre-coders. Unlike QR-PMSVD, D(z) is diagonal with UU and the post-and precoder are much less complex (see Table I).

The comparison in terms of sum rate versus SNR obtained through a MIMO 2×2 and MIMO 3×3 schemes, with several values of QR-PMSVD algorithm parameters, is represented respectively in Fig. 3 and Fig. 4. Firstly, the sum rate obtained with UU algorithm can not achieve the channel capacity. This is due to the fact that the post-coder (U -1 ) is not para-unitary and then the noise component is affected (see (12)).

Secondly, in the QR-PMSVD case, the sum rate is almost equal to the channel's capacity when ǫ is selected close to zero (here ǫ = 10 -3 ). However, if this parameter increases for obtaining more realistic FIR post-and pre-coder, as close to that obtained with UU (see Table .I), the sum rate with QR-PMSVD becomes worse due to the important CCI.

To finish, we analyze the elements in D(z) obtained here by UU decomposition. All diagonal elements of D(z) are equal to 1, except the last one. Thus, the first N G = min(N t , N r ) -1 separate SISO channels are equivalent to Gaussian channel. Hence, there is no inter-symbol interference (ISI) on these channels and the equalization becomes very simple. Figure 5, which presents the received signal constellation with 4-QAM modulation in MIMO 3 × 3 context, shows the effect of ISI. As we can see in figure boxes UU-antenna 1 and UU-antenna 2, which represent the first N G = 2 SISO channels, there is no ISI. We just have the noise component effect. However in the other figure boxes we note the presence of ISI. As we have previously noted, this situation of no ISI happens in general with high probability.

V. CONCLUSION

The proposed UU polynomial decomposition algorithm completely eliminates the CCI in MIMO spatial multiplexing scheme. In addition, except in the last SISO channel, the ISI in all channels is likely to be absent. The sum rate obtained with the decomposition can not, however, achieve the channel capacity because of the non unitary effect of the post-coder. Nevertheless, with equal complexity, the obtained sum rate is better than that achieved by the classical QR-based polynomial matrix approximate SVD decomposition.

  is a unimodular polynomial matrix and R(z) is the upper triangular polynomial matrix obtained in the last step of the algorithm described above. Applying the same decomposition to the right factor in (9), namely R(z) t = V (z) t D(z) provides a factorization of the form H(z) = U (z)D(z)V (z), where V (z) and U (z) are two unimodular matrices and D(z) is a diagonal polynomial matrix.

  12) where D k,k (•) are the diagonal elements of D(•), U k,: stands for the k-th line of U . • QR-PMSVD of Forster, McWhirter et al. [3]: In this case, D(z) is only approximately diagonal. Equation (11) becomes:

Fig. 1 .Fig. 2 .

 12 Fig. 1. Impulse response of diagonalized Matrix D(z) obtained by QR-PMSVD decomposition with ǫ = 0.01 and µ = 10 -6 .

Fig. 3 .

 3 Fig. 3. Sum Rate averaged over 100 MIMO 2 × 2 Channel realization for UU and QR-PMSVD The length was L = 7, the truncation parameter µ = -6 for various ǫ.

Fig. 4 .

 4 Fig. 4. Sum Rate averaged over 100 MIMO 3 × 3 Channel realization for UU and QR-PMSVD decomposition. The Channel length was L = 7, the truncation parameter µ = 10 -6 for various ǫ.

Fig. 5 .

 5 Fig. 5. Received signal constellation for 4-QAM modulation in MIMO 3 × 3 context. The SNR is set to 30dB.