Katia Chirkova

REVITALIZATION OF DUOXU: A FIRST-HAND ACCOUNT

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BACKGROUND ON DUOXU

Duoxu (/do³³ɕu³³/) is a Tibeto-Burman language traditionally spoken by a small eponymous ethnic group living in the Anning River valley-the largest mountain plain and the main agricultural strip of Southwest China. 1 The Anning River valley lies in the historically multi-ethnic borderlands between China and Tibet. Unsurprisingly, given this context, the history of the Duoxu people is marked by contact with these Chinese and Tibetan groups as well as its local ethnic neighbors (most importantly, Nuosu, also known as Yi). The arrival of ethnic Tibetans in the area dates back to the 7th century, when the traditional Duoxu settlement areas became the south-eastern border of the expanding Tibetan empire. Long-standing cultural and religious ties with Tibet had a lasting impact on the development of the cultural identity of the Duoxu people, who to this day mostly identify themselves as Tibetans. Starting with the Mongol conquest of China in the 13th century, Duoxu areas came under ever-increasing administrative control of the expanding Chinese empire. From the 14th through the 18th century, the Duoxu group was part of the Native Chieftain System that the Chinese established to exercise some form of control over their southwestern frontier areas. 2 During that period, as a group that provided one of the chieftains, Duoxu were at the peak of their power and influence, enjoying considerable autonomy from the Chinese state and belonging to the local frontier elite. From the 17th century onwards, China intensified its presence in the southwestern regions and consolidated its control over semiautonomous indigenous frontier societies. As a result, the Native Chieftain System was disbanded in order to incorporate those areas more solidly into the Qing. The system's disintegration in the 1 The name Duoxu is based on the Chinese spelling of the autonym of the group. The language is also known under the name Tosu [START_REF] Tatsuo | Tosu yakugo no kenkyū: shin gengo tosugo no kōzō to keitō / A study of the Tosu-Chinese vocabulary, Tosu i-yu: the structure and lineage of Tosu, a new language[END_REF]. According to a traditional belief, the Duoxu group has its origins in Tibet, but migrated and settled in the Anning River valley as early as the beginning of the Common Era [START_REF] Han | Duoxuyu qiangjiu gongzuo ji jingyan jieshao [Fieldwork on a Highly Endangered Language: A Case Study of Duoxu[END_REF]). 2 The Native Chieftain System is an institution created during the early Ming dynasty (1368-1644) to extend Chinese state control over the non-Han peoples beyond China's administrative borders (Herman 1997: 50-51).

second half of the Qing dynasty coincided with a major demographic change in the area resulting from a considerable influx of ethnic Nuosu (who were historically the southern neighbors of the Duoxu). Nuosu settled in many traditional Duoxu areas in the Anning River valley, thereby cutting those areas off from the Tibetan-speaking cultural region to its northwest. In the course of the 20th century, Nuosu and Chinese gradually outnumbered Duoxu in the areas where Duoxu traditionally settled.

After the establishment of the People's Republic of China (PRC) in 1949, Duoxu were surveyed as part of the PRC's State Ethnic Classification Project, which laid the foundation for the currently recognized framework of nationalities (see Harrell 1995;[START_REF] Bradley | Introduction: Language Policy and Language Endangerment in China[END_REF]Sun et al. 2007: 16-29). The Duoxu group was not granted the official status of "national minority," as it was grouped into the Tibetan nationality. This is because final decision making in the process of ethnic identifications often relied on traditional distinctions that were already part of folk beliefs (Harrell 1995: 66, see also Sun et al. 2007: 30-32), including the purported historical relatedness of groups, as in the case of Duoxu and Tibetans.

Mirroring the history of the group, the history of the Duoxu language has all along been marked by intense contact with its neighboring languages. As an unwritten language, Duoxu has always co-existed with written languages of administration, including, in this order: (Sde dge) Tibetan; Chinese; for a brief period in the 1960-70s, Yi; and at present, again, Chinese. During the Duoxu chieftainship period, the Duoxu language must have enjoyed high prestige as the language of the local frontier elite, as it was selected as one of the local languages recorded in the 18th century as part of the vocabularies Xifan Yiyu [Tibetan-Chinese Bilingual Glossary] in Chinese and Tibetan transcriptions [START_REF] Tatsuo | Tosu yakugo no kenkyū: shin gengo tosugo no kōzō to keitō / A study of the Tosu-Chinese vocabulary, Tosu i-yu: the structure and lineage of Tosu, a new language[END_REF]. Later in its history, however, the association with the Duoxu chieftainship had a negative effect on the Duoxu language, as it was stigmatized and virtually forbidden during the Cultural Revolution (1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975)(1976) for representing a "feudal" or "capitalist" culture of traditional China. By the end of the 1970s, Duoxu practically went out of use, being solely restricted to communication in Duoxu households.

As ethnic Duoxu came to be classified as members of the Tibetan nationality, their language was formally recognized as a variety of Tibetan, which is in stark contrast to the distant genetic relationship between Duoxu and Tibetic languages. In the mid-1970s, the PRC government reopened the ethnicity and language issue and accepted new applications from ethnic groups not designated as national minorities in the 1950s. These included applications from ethnic groups in Southwest China which (similar to Duoxu) shared religion and culture with Tibetans, but spoke their own languages. These groups also included two closely related sister-languages of Duoxu: Ersu and Lizu. 3 Although none of the applications succeeded, and none of the groups obtained the official status of "national minority," their languages were surveyed and received a de facto recognition in linguistics as independent Tibeto-Burman languages. These languages were thereupon grouped together as members of the Qiangic subgroup of the Tibeto-Burman language family [START_REF] Sun | Liujiang liuyu de minzu yuyan ji qi xishu fenlei [Minority languages of the Six River Valley and their genetic classification[END_REF]Bradley 1997: 36-37;[START_REF] Chirkova | The Qiangic subgroup from an areal perspective: A case study of languages of Muli[END_REF]. That generated enduring interest and research in these languages both in China and abroad. By a sad twist of fate, however, this new situation brought little improvement to the case of Duoxu. The already weakened and outnumbered Duoxu was grouped together with Ersu and Lizu as three dialects of one Ersu language [START_REF] Hongkai | Ersu (Duoxu) hua jianjie [An outline of Ersu (Duoxu)][END_REF][START_REF] Da | Three tongues and two identities: A case study of Ersu ethnic identities in Sichuan, China[END_REF]. According to the general linguistic policy for developing national minority languages, when a language has multiple dialects, only one dialect is generally selected as a "standard" variety to subsequently receive attention in documentation and research. In the case of Ersu, the choice fell on the Ersu "dialect," presumably owing to the large community of Ersu speakers, and leaving Lizu and Duoxu understudied.

At present, Duoxu are a small minority of circa 2,000 people who live in Mianning County, which is situated in a Yi nationality autonomous region (Liangshan Yi Autonomous Prefecture in Sichuan Province), and which is home to majority populations of Chinese and Nuosu. Hardly any of the 2,000 Duoxu people speak the traditional language of the Duoxu community. In fact, the Duoxu language is currently spoken by no more than 9 elderly speakers, and hence is critically endangered. There are practically no published data on Duoxu from before the language went out of use: the entire documentation record consists of two short lexical lists of 14 words (Sun 1982: 242) or 30 words (Nishida and Sun 1990:17) and one grammatical sketch (Huang and Yin 2012). Given that background, and unless urgent measures are taken, the Duoxu language is likely to disappear without a trace once its last speakers fall silent.

ASSESSING PROSPECTS FOR REVITALIZATION OF DUOXU

Previous studies in language revitalization identify a number of key variables in assessing the possibilities for the revitalization of a particular language (e.g. Grenoble and Whaley 2006: 21-49). These include variables on the national level (national language policy, education policies, regional autonomy, federal support) and variables on the local level (language attitudes, human resources, religion, literacy). None of these appear favorable in the case of Duoxu.

China has a clear and supportive legal framework for minority languages (see [START_REF] Bradley | Introduction: Language Policy and Language Endangerment in China[END_REF]. In particular, in areas where minority nationalities are concentrated, these nationalities enjoy considerable autonomy and may have local administrative rights and responsibility for education and culture. The system is, however, uniquely designed for the framework of officially recognized nationalities. Consequently, when a group lacks the officially recognized status of national minority, has a small population, and shares its area of residence with an officially recognized national minority, that ethnic group has neither right nor access to any form of official support (see [START_REF] Roche | Tibet's Minority Languages: Diversity and Endangerment[END_REF]. This is the case with Duoxu. Being formally a language of the Tibetan nationality, spoken in the autonomous prefecture of the Yi national minority, the Duoxu language is beyond any institutionalized framework of support.

Variables on the local level are not favorable either [START_REF] Han | Duoxuyu qiangjiu gongzuo ji jingyan jieshao [Fieldwork on a Highly Endangered Language: A Case Study of Duoxu[END_REF]. With regard to language attitudes, parts of the Duoxu community accept the loss of their traditional language as a fact and wish to invest more in their Tibetan identity. They consider Tibetan a better choice for their community language. This decision, in their eyes, is in agreement with their Tibetan nationality status, their strong cultural ties with Tibet, and with the history of Tibetan as a language of administration in Duoxu areas. As an informal implementation of this view, it is common and popular among Duoxu to hire private Tibetan teachers to teach their children written Tibetan during school breaks. Some other parts of the community have a soberer outlook on the issue of language revitalization. They find it unrealistic to revitalize the moribund Duoxu language or to acquire, as a community, proficiency in the Tibetan language. They call for recognition of the fact that the community has already largely assimilated into the Han Chinese nationality through intermarriage and cultural amalgamation. In their opinion, the Chinese language has already become the community language. Finally, a small part of the community would like to see their traditional Duoxu language revitalized for they worry that a loss of Duoxu could lead to a loss of their cultural traditions and deplete their sense of identity.

Local Duoxu officials are sympathetic to the cause (see below), but they do not have the autonomy to make decisions that deviate from approved national and regional policies.

In sum, the entire Duoxu community is not interested and invested in the cause of revitalization of their traditional language. Taken together with the non-existent support on either federal or regional level and a lack of resources, Duoxu's prospects for revitalization appear rather bleak.

REVITALIZATION EFFORTS IN THE DUOXU COMMUNITY

Language is neither the major constructive element of ethnic boundaries nor an exclusive marker of ethnicity (Haarmann 1999: 63),4 but language does always play a role in ethnic relations. Hence, when threatened with the degradation and possible loss of their distinct linguistic traditions, communities tend to mobilise themselves, if not as a whole, then at least in part. The Duoxu community illustrates this general tendency. Numerous individual efforts within the Duoxu community can be noted on both official and informal levels. These efforts include studying the Duoxu language, collecting linguistic and ethnographic data, and disseminating the results.

Local government officials (whether of Duoxu or Han Chinese ethnicity) have all along carried out important work in collecting and publishing traditional Duoxu songs and stories, albeit in Chinese translation (e.g. Yu 1988). The Gazetteer of Mianning County (Sichuan Sheng Mianning xian Difangzhi Bianzuan Weiyuanhui 2009) is another valuable source of information about the history, culture, and language of the Duoxu group, containing even one traditional Duoxu song in a Romanized pinyin transcription (as used for Standard Chinese) (p. 659).

Remarkable work by individual community members is also carried out on the informal level. One example is the initiative to collect Duoxu vocabulary lists recording the speech of the last fluent Duoxu speakers. Although recorded in Chinese transcription, hence being phonetically imprecise and at times difficult to interpret [START_REF] Chirkova | Shiyong Duoxuyu yufa[END_REF], Duoxu words on these lists are still sufficiently recognizable. Two unpublished manuscripts, both compiled in the 1990s, are particularly notable: These manuscripts contain extensive vocabulary lists, thematically organized into numerous subject areas (such as body parts, clothes, house, people). In addition, Wu Wancai's manuscript contains an outline of Duoxu history, language, and culture as well as records of folk stories, proverbs, and traditional song lyrics. The manuscripts circulate in a limited number of copies and are available to all interested community members. The manuscript by Jin Wanxiang deserves a special note as the author is a second language learner of Duoxu, who invested considerable time and effort into his work.

In addition to these local initiatives, efforts of one more community member, Yuan Xiaowen, were most decisive for the currently ongoing revitalization work on Duoxu. Yuan is a Duoxu historian and ethnographer working at the Research Institute for Nationalities of Sichuan Province. Yuan's strategy to safeguard the language and promote revitalization is to actively seek assistance of linguists. As a well-established scholar, Yuan is in a good position to promote the cause, contact scholars, and seek funding. My own involvement in the documentation of Duoxu owes to my meeting with Yuan Xiaowen at a linguistic conference, where Yuan made an eloquent appeal to all participants to lend their assistance in documenting the moribund Duoxu language (Yuan 2010). As I was working on Lizu (the closely related sister-language of Duoxu) at the time, I was interested in joining efforts with Yuan and extending my work to Duoxu. Together with Yuan, I applied for a documentation project which included Duoxu as one of its pillars. Right from the application phase, the Duoxu research component included a clear revitalization agenda, reflecting the views by Yuan and other involved Duoxu community members on the most urgent steps to take. The project received funding from the Endangered Languages Documentation Programme. 5Our work on Duoxu proceeded in the following commonsense steps. As a first step, we began with language assessment: a survey of the last speakers of Duoxu [START_REF] Chirkova | The Duoxu language and the Duoxu-Ersu-Lizu relationship[END_REF][START_REF] Han | Duoxuyu qiangjiu gongzuo ji jingyan jieshao [Fieldwork on a Highly Endangered Language: A Case Study of Duoxu[END_REF]. Needless to say, undertaking a survey of that kind required a high degree of organization and community involvement, and would have been impossible were I to work on the language alone. The survey revealed a sad statistic: only nine elderly members of the entire Duoxu community still had some knowledge of the language. All of the last speakers spoke the local variety of Southwest Mandarin as their dominant daily language and had no chance to speak Duoxu regularly. As a result, their proficiency in Duoxu varied considerably, from highly insecure rememberers and semi-speakers to reasonably fluent language users. Quite unsurprisingly, the most proficient speaker, who also happened to be the oldest member of the group (83 years of age at the time of the survey), Mr. Wu Rongfu, had a classical NORM profile, that is non-mobile, older, rural male (Chambers and Trudgill 1980: 33), which usually represents the most traditional section of a speech community.

As a second step, we concentrated on long-term documentation and description work with the most proficient speakers. We attempted to create communicative contexts and relied on previous documentation records (both linguistic and community-based, as the aforementioned handwritten manuscripts and collections of traditional Duoxu stories and songs) used to prompt and guide elicitations and conversations.

As a third step, on the basis of the collected data we prepared a range of linguistic and pedagogical materials [START_REF] Chirkova | The Duoxu language and the Duoxu-Ersu-Lizu relationship[END_REF][START_REF] Chirkova | A phonological sketch of Duoxu[END_REF][START_REF] Chirkova | Shiyong Duoxuyu yufa[END_REF]. Given the number and age of the last proficient speakers, the most appropriate way to assure the intergenerational continuity of the Duoxu language would ideally be an immersion program, similar to California's Master-Apprentice Language Learning Program [START_REF] Hinton | How to Keep your Language Alive[END_REF], whereby community members can informally learn the language from the last native speakers. However, the idea of such a program has not yet found much support in the Duoxu community. We opted instead for basic revitalization materials, including a pedagogical grammar with exercises, a Romanization system on the basis of the pinyin transcription used for Standard Chinese (which is familiar to all Duoxu speakers and hence easy to use), reading materials, and a dictionary. The grammar is currently in press [START_REF] Chirkova | Shiyong Duoxuyu yufa[END_REF], while a collection of traditional Duoxu stories in IPA transcription and Duoxu Romanization system, and a bilingual Duoxu-Chinese dictionary are in preparation. As a rule, all output of the project team is accompanied by sound files, which are in open access with the Endangered Languages Archive (https://elar.soas.ac.uk/Collection/MPI655546).

The joint documentation project is presently coming to an end, but neither Duoxu community members nor myself are planning to stop our work on the Duoxu language. A new documentation project is underway on the Duoxu community side, sponsored by a Chinese funding agency, while I continue to analyse and publish collected Duoxu data.

In sum, despite the unfavourable revitalization context, the Duoxu community showed itself resourceful and successful in actively working on the preservation of their language and implementing a beginning revitalization program.

DISCUSSION

What lessons can be learned from the Duoxu case? The most important lesson seems to be that of the significance of individual efforts at the community level directed at the preservation of the community language. As persuasively argued by Leanne Hinton (2001a: 6), "[a]ll that is really needed for language revitalization to begin is a minimum of one person who is dedicated to the cause. That one person can do a great deal with no support from the community … Community support may come later, after the prime mover has accomplished something the community can trust. Even if it does not, that one interested person can produce something of value that future generations may appreciate more than that person's peers." The Duoxu case nicely illustrates this point. Naturally, not all communities can necessarily be as resourceful as Duoxu or have community members who are in a strategic position to search for research collaborations and funding. However, in my experience working with small Tibeto-Burman communities of Southwest China, all communities have concerned members who work in one way or another on preserving their languages, collecting vocabulary lists, writing down stories and proverbs, and -with the advent of mobile phones and internet -also conveniently recording and disseminating linguistic data on blogs and other social media. Hence, the potential for revitalization is always there, waiting for an opportunity to realize itself.

The second lesson is that with these individual efforts in place, revitalization always remains possible (Hinton 2001a: 6). Even when everything seems to be conspiring against the cause, much important work can still be accomplished. The sobering reality of the current endangered language crisis is that policy-making and economic development are not commensurable with the speed of language endangerment and loss. That means that in the majority of cases of language endangerment, conditions for successful revitalization simply cannot be met. That being the case, many communities naturally may have more pressing priorities and concerns than their language preservation. In such situations, it is unrealistic to opt for the top levels of [START_REF] Fishman | Reversing Language Shift: Theoretical and Empirical Foundations of Assistance to Threatened Languages[END_REF] famous Graded Intergenerational Disruption Scale with the goal of creating new speakers and re-establishing the language as the main language of communication. At the same time, it is possible to strive for and attain more modest goals: collect word lists, prepare teaching materials, produce language-learning programs or simply assure that a language continues through a few speakers per generation (e.g. Hinton 2001a, Grenoble 2013: 50-51). Accomplishing these smaller goals naturally lays foundation for reaching higher levels on Fishman's revitalization scale, once circumstances become more favorable and/or the community at large recognizes the importance of individual revitalization efforts.

The third lesson is that a close collaboration between community members and linguists is a promising way to carry out revitalization work even under most unfavorable conditions. This asserts the need for a closer integration of linguistic work with the community needs, uniting documentation and revitalization (e.g. [START_REF] Grenoble | Endangered Languages: Current issues and future prospects[END_REF]: 220-221, Hinton 2001b[START_REF] Grenoble | Responses to Language Endangerment: New Directions in Language Documentation and Language Revitalization[END_REF]. This further entails that in her work a linguist needs (a) to focus on collection of a wide range of information, including pragmatic and paralinguistic, and (b) to measure descriptive and documentary adequacy of collected data by the possibility for a learner to extract sufficient information from those data to become a fluent speaker (Grenoble 2013: 53). This also means that a fieldwork linguist needs to accept work as part of a language revitalization team (writing teaching materials, helping with teacher training, and teaching the language) [START_REF] Gerdts | Beyond expertise: The role of the linguist in language revitalization programs[END_REF]. In addition to furthering grassroots revitalization, such an approach yields an enriched and more representative documentation corpus, which is essential for adequate analysis of lesser-known languages of smaller communities, such as Duoxu.

The story of Duoxu has an open ending, as the revitalization of Duoxu is in its infancy. Will the community succeed in keeping their traditional language alive? How much of the language can still be documented and explored before the last speakers fall silent? Which step on the revitalization scale will the Duoxu community be able to reach? The first results of the revitalization efforts are encouraging, but a lot of work remains to be done.

( a )

 a Liangshan zhou Zangzu jianshi: Mianning xian Zangzu Duoxu zhi pu [A Brief History of the Tibetans of Liangshan Prefecture: The Duoxu Tibetans of Mianning county] by Wu Wancai (b) Mianning Zangzu renmin de lishi he yuyan [History and Language of Mianning Tibetans] by Jin Wanxiang

Lizu is spoken to the northwest of Duoxu in the counties of Mianning, Muli, and Jiulong. Ersu is spoken to the northeast of Duoxu in the counties of Ganluo, Yuexi, Shijian, Hanyuan, all in Sichuan Province.

See also[START_REF] Chirkova | Between Tibetan and Chinese: Identity and language in Chinese Southwest[END_REF] in relation to Tibeto-Burman communities in Chinese Southwest.

Project: "Ersu and Xumi: Comparative and Cross-Varietal Documentation of Highly Endangered Languages of South-West China" (ELDP MDP0257), April 2013 -January 2017.


Bernard Picinbono
email: bernard.picinbono@lss.supelec.fr

Fellow

On Instantaneous Amplitude and Phase of Signals

In many questions of signal processing, it is important to use the concepts of instantaneous amplitude or phase of signals. This is especially the case in communication systems with amplitude or frequency modulation. These concepts are often introduced empirically. However, it is well known that the correct approach for this purpose is to use the concept of analytic signal. Starting from this point, we show some examples of contradictions appearing when using other definitions of instantaneous amplitude or frequency that are commonly admitted. This introduces the problem of characterizing pure amplitudemodulated or pure phase-modulated signals. It is especially shown that whereas amplitude modulated signals can be characterized by spectral considerations, this is no longer the case for phase modulated signals.

I. INTRODUCTION

INSTANTANEOUS amplitude and phase are basic concepts in all the questions dealing with modulation of signals appearing especially in communications or information processing. Let us remember that a purely monochromatic signal such as a cos(ωt + φ) cannot transmit any information. For this purpose, a modulation is required, and one of the simplest possible to introduce is amplitude modulation. Let m(t) be a positive function corresponding to the information o be transmitted. By multiplying the carrier frequency signal cos(ω 0 t) by m(t), we obtain the signal

x(t) = m(t) cos(ω 0 t) (1)
and it is commonly admitted that is m(t) the instantaneous amplitude of the signal x(t). This appears in many textbooks, especially in [1, p. 237]. On the other hand, the need for phase or frequency modulation requires the definition of the instantaneous phase. By a reasoning similar to the previous one concerning the amplitude, it is commonly admitted that the signal

x(t) = a cos(φt) (2)
has a constant amplitude a and an instantaneous phase φ(t). Furthermore, the instantaneous angular frequency is given by the derivative of φ(t). This is a generalization of the procedure applied in the case where the phase is linear in time or in the form φ(t) = ω 0 t + φ, giving the frequency . These definitions also appear in many textbooks or papers. For example, the following is written on [2, p. 645]: The argument θ(t) in any signal having the form cos[(θ(t)] is called the instantaneous phase, and

(1/2π)d[θ(t]/dt is called the instantaneous frequency. The same definition appears, for example, in [3, p. 480], [4, p. 260], and [5, p. 144].

Even if the previous definitions appear quite natural and are widely used in practical applications dealing with signal modulation, we immediately note that they cannot be satisfactory. In order to make this point clear, let us discuss a very simple example. Suppose that the function m(t) appearing in [START_REF] Panter | Modulation, Noise and Spectral Analysis[END_REF] is bounded or satisfies 0 ≤ m(t) ≤ a. As a result, we have |1/a| ≤ 1. It is then possible to introduce a unique function φ(t) satisfying

0 ≤ φ(t) ≤ π ; cos[φ(t)] = (1/a)x(t). (3)
By using this well-defined function, we obtain

x(t) = m(t) cos[ω 0 t] = a cos[φ(t)]. (4)
This shows that the signal x((t) can be considered frequency as well as amplitude modulated. In other words, its instantaneous amplitude is m(t) as well as a, and moreover, its instantaneous phase is either

ω 0 t or φ(t).
More generally, starting from a given signal x(t), it is possible to introduce an infinite number of pairs [a(t), φ(t)] such that

x(t) = a(t) cos[φ(t)]. (5)
This leads to the conclusion that the definitions given previously, even if they are widely used, are incoherent because they do not associate with a given real signal a well-defined pair of functions that are the instantaneous amplitude and phase of x(t). Therefore, these definitions must be reformulated in such a way that any given signal x(t) corresponds to one well-defined pair [a(t), φ(t)], allowing us to write x(t), as in [START_REF] Taub | Principles of Communication Systems[END_REF].

In reality, the solution of this problem is well known and explicitly introduced in [6, p. 50]. A recent review paper on this question [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF] also gives a good introduction on the discussion leading to the standard definition of the quantities studied hereafter. Moreover, this paper contains a list of references corresponding to the history of this problem.

The classical definition of the instantaneous amplitude and phase of a real signal is recalled in the next section. This definition introduces the concept of a canonical pair [a(t), φ(t)], and it is therefore interesting to find conditions ensuring that a given pair of functions a(t) and φ(t) is canonical. When the instantaneous phase is linear, or when the signal looks like (1), it is possible to characterize a canonical pair only from the spectral properties of the signal. This becomes much more difficult when the instantaneous phase is no longer linear, and this especially appears for signals with constant amplitude, which are called phase signals, which are at the foundation of phase-or frequency-modulation systems. The properties of such signals are analyzed in the following section, and from them, we can deduce various properties of canonical pairs. The practical consequences of these results and their extension for random signals are finally investigated.

II. DEFINITIONS AND CANONICAL PAIRS

A. Definitions

Let us recall the classical way to define without ambiguity the instantaneous amplitude and phase of a real signal x(t). The problem is to write x(t) as in (5) but by using a pair of functions [a(t),

φ(t)]
that is in a one-to-one correspondence with x(t). For this purpose, we associate with x(t) its analytic signal (AS) z(t) (see [6, p. 48]). It is obtained from x(t) by filtering it using a filter with the frequency response H(ν) equal to 2 for ν > 0 and to 0 for ν < 0. Conversely, it is obvious that

x(t) = Re[z(t)]
, where Re means the real part. Therefore, if z(t) is real, there is indeed a one-to-one correspondence between x(t) and z(t). On the other hand, a complex function is an AS if its Fourier transform is zero for negative frequencies. It is clear that this function is the AS of its real part.

January 5, 2018 DRAFT

As z(t) cannot be a real function because its Fourier transform Z(ν) is zero for ν < 0, it can be written as

z(t) = a(t) exp[φ(t)], (6)
where the phase φ(t) is defined modulo 2π, and a(t) is nonnegative. As a conclusion, using the AS makes it possible to associate with any real signal a unique pair [a(t), φ(t)] called in the following the canonical pair associated with x(t).

Definition: Let be a real signal and the canonical pair [a(t), φ(t)] associated with it. The function a((i)

appearing in this pair is the instantaneous amplitude of x((t), and φ((t) is its instantaneous phase. The instantaneous frequency is the derivative with respect to time of φ((t).

The introduction of the AS is not at all recent, and among the principal papers in this field, we can note [8] [START_REF] Bedrosian | The analytic signal representation of modulated waveforms[END_REF]. There are some questions concerning the physical meaning of the AS, and some of them are mentioned and discussed in [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF]. Furthermore, it is shown in [START_REF] Vakman | On the definition of concepts of amplitude, phase and instantaneous frequency of a signal[END_REF] and [START_REF] Vakman | On the analytic signal, the TeagerKaiser energy algorithm, and other methods for defining amplitude and frequency[END_REF] that starting from some a priori physical assumptions, the only possible definition of the instantaneous amplitude and phase is the one given just above. However, it is worth pointing out that other physical conditions lead to another definitions that are not discussed below [START_REF] Loughlin | On the amplitude and frequency-modulation decomposition of signals[END_REF].

Once the definition is given, the question that immediately follows is to characterize a canonical pair or to show what the conditions are on a(t) and φ(t) in order to ensure that ( 6) is an AS, which means that its Fourier transform is zero for negative frequencies. This is essential when verifying whether or not the classical definitions given above are correct. In fact, there is no a priori reason for [m(t)], ω 0 t] appearing in [START_REF] Panter | Modulation, Noise and Spectral Analysis[END_REF] or [a(t), φ(t)] appearing in (2) to be canonical.

B. Spectral Characterization of a Canonical Pair

Consider first a signal of the form [START_REF] Panter | Modulation, Noise and Spectral Analysis[END_REF]. It corresponds to an amplitude modulation of a pure monochromatic signal with the carrier frequency ω 0 . The nonnegative function m(t) is the instantaneous amplitude is equal to a(t) sin[φ(t)] (see [6, p. 49]). It is therefore appropriate to make use of the so-called Bedrosian theorem [START_REF] Bedrosian | A product theorem for Hilbert transforms[END_REF] dealing with the Hilbert transform of a product of two real functions x 1 (t) and x 2 (t). A very simple derivation of this theorem and some extensions can be found in [START_REF] Picinbono | Representation des signaux par amplitude et phase instantanees[END_REF]. The main result is as follows: Let X 1 (ν) and X 2 (ν) be the FTs of x 1 (t) and x 1 (t), respectively. If X 1 (ν) = 0 for ν > B and

of x(t) if and only if m(t) exp(jω 0 t) is an AS. Let M (ν) be the Fourier transform (FT) of m(t). As m(t) is real, M (ν) = M * (-ν). The FT of m(t) exp(jω 0 t) is, of course, M (ν -ν 0 ) , with ν 0 = ω/2π
X 2 (ν) = 0 for ν < B, then H[x 1 (t)x 2 (t)] = x 1 (t)H[x 2 (t)] (7)
where

However, this does not at all imply that

H{cos[φ(t)]} = sin[φ(t)] (9)
as stated by many authors and even recently in [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal[END_REF]. If [START_REF] Ville | Theorie et applications de la notion de signal analytique[END_REF] were true, it would also be possible to characterize a canonical pair only by spectral considerations, as for amplitude modulation. Furthermore, as the constant signal has an FT limited to the frequency zero, (9) would be true for any signal cos[φ(t)] without a lowfrequency component. This result would be especially attractive, suppressing all the questions discussed in the introduction, when presenting some comments on (2). Let us now show that [START_REF] Ville | Theorie et applications de la notion de signal analytique[END_REF] has no reason to be true when only spectral properties of cos[φ(t)] are introduced. For this purpose, we shall propose an elementary counterexample of this property. It is obvious that (9) implies that

cos 2 [φ(t)] + sin 2 [φ(t)] = 1. (10)
Let x(t) be defined by

x(t) = sinc(2Bt) cos(ω 0 t), (11)
where sinc(x) = sin(πx)/πx , and ω 0 > 2πB . As |x(t)| < 1, it is possible to introduce a phase φ(t) This shows that contrary to a common well-established idea, it is not possible to justify (9) by introducing only spectral considerations. This point will become much clearer in the next section. In fact, [START_REF] Ville | Theorie et applications de la notion de signal analytique[END_REF] implies that

uniquely defined if 0 ≤ φ(t) ≤ π
z(t) = exp[jφ(t)] (13)
is an AS or that the pair[1, φ(t)] is canonical. We shall now see that this requires very specific properties of the structure of the phase φ(t).

III. PHASE SIGNALS

A. General Structure

Phase signals are real signals with constant instantaneous amplitude. They can be expressed as (2) but with the condition that exp[jφ(t)] is an AS. As a consequence, ( 9) is satisfied. For such signals, all the information is contained in the instantaneous phase (or frequency), and phase signals are then the basic elements of phase or frequency modulation.

The condition that exp[jφ(t)] is an AS requires very specific properties on the phase φ(t). These properties have been analyzed in the framework of coherence problems in optics [START_REF] Edwards | The form of the general unimodular analytic signal[END_REF] but, more precisely, in the framework of the study of analytic functions, and especially in [19, ch. 7]. We present here the results that are the most important for our arguments without the proofs, which are out of the scope of this discussion and can be found in [START_REF] Nevanlinna | Analytic Functions[END_REF].

The most general structure of the AS of a nonsingular phase signal is

z(t) = b(t) exp[j(ω 0 t + θ)], (14)
where θ is arbitrary, ω 0 is nonnegative, and b(t) is a Blaschke function defined by

b(t) = N k=1 t -z k t -z * k , z k ∈ P + , (15)
where P + is the half plane of the complex plane defined by Im(z) > 0 . The quantity ω 0 is the carrier frequency and can be equal to zero. The expression nonsingular means that the instantaneous phase of z(t) remains finite for finite values of t. When the number N of factors in the product is not finite, there are other constraints on the complex numbers z k due to convergence problems. Here, we avoid these questions by assuming that N is finite. In this case, the interpretation of ( 15) is very simple. In order to be an AS, the function b(t), for complex values of t, must have all its poles in the half plane Im . In order to have a modulus equal to one, each pole must be associated with a corresponding zero symmetric of this pole with respect to the real axis. This procedure is well known in filter theory: Stable phase DRAFT January 5, 2018

filters have the same number of poles and zeros, and these zeros are symmetric to the poles with respect to the imaginary axis. The stability and causality conditions imply that all the poles are in the left half plane of the complex plane.

It

(t) = 1 + N k=1 c k t -z * k , (16)
where

c k = lim t→z * k (t -z * k )b(t) (17)
As a consequence, we have

B(ν) = δ(ν) + N k=1 C k (ν), (18)
where

C k (ν) is the FT of c k (t -z * k ) -1 .
Because of the localization of z * k in the complex plane, we deduce that C k (ν) = 0 for ν < 0, which implies that B(ν) = 0 for ν < 0 and ensures that b(t) is an AS. Finally, as ω 0 > 0, z(t) also is an AS. The reasoning can be extended without difficulty when some poles z k are no longer distinct.

The phase of b(t) is, of course

φ k (t) = Arg[b(t)], mod2π, (19)
and, as a result, we can say that any phase signal can be written as [START_REF] Wozencraft | Principles of Communication Engineering[END_REF], where φ(t) must have the form

φ(t) = θ + ω 0 t + φ b (t), mod2π. (20)
In practice, the continuity of the phase leads to suppress the term mod(2π), and this convention is adopted in all that follows.

This most general phase is defined by N complex parameters z k and two real parameters ω 0 and θ.

Furthermore, it is obvious that the phases φ k (t) are the sum of N phases of the factors appearing in the product [START_REF] Loughlin | On the amplitude and frequency-modulation decomposition of signals[END_REF]. Let b k ((t) be equal to

(t -z k )(t -z * k ) -1
, and let φk(t) be its phase. This gives

φ(t) = θ + ω 0 t + N k=1 φ k (t). (21)
By introducing the real and imaginary parts of z k or z k = a k + jb k , one obtains

φ k (t) = 2Arctg b k a k -t , -π/2 ≤ φ k (t) ≤ π/2. (22)
At this step, we come back to the problem discussed in the introduction. The signal ( 2) is phase modulated only if its phase takes the form [START_REF] Voelcker | Toward a unified theory of modulation. Part 1, Phase envelopes relationships[END_REF], and as this is not in general the case, its amplitude is not constant, and it must be expressed as in [START_REF] Taub | Principles of Communication Systems[END_REF].

B. Properties of Phase Signals

Having the most general structure of phase signals, we will now present some of their properties, which allows a better understanding of their structure.

Property 3.1:A phase signal contains only two spectral lines corresponding to its carrier frequency.

This is a direct consequence of the structure of the Blaschke function appearing in [START_REF] Loughlin | On the amplitude and frequency-modulation decomposition of signals[END_REF]. Its FTB(ν) is

given by [START_REF] Edwards | The form of the general unimodular analytic signal[END_REF], which can be written as

B(ν) = δ(ν) + B c (ν). (23)
The function B c (ν) describes the continuous part of the FT of B(ν). It is a sum of N components C k (ν) that are bounded and equal to zero for ν < 0. This implies that B c (ν) is also bounded and equal to zero for ν < 0, and thus B(ν) exhibits only one Dirac component, or a spectral line, at the frequency zero. Because of the exponential term in ( 14), the FT of z(t) is Z(ν) exp(jθ)B(ν -ν 0 ), and this means that there is only one spectral line at the carrier frequency ν 0 = ω 0 /2π. By using the Hermitian symmetry, we deduce that z(t) has only two spectral lines at the frequencies ±ν 0 .

This property can be used in a reciprocal way, indicating that all the signals containing more that one spectral line in the range of positive frequencies cannot be phase signals and, therefore, exhibit a nonconstant instantaneous amplitude.

Property 3.2: A phase signal with a nonzero carrier frequency) is a high-frequency(ν 0 signal. This is a direct consequence of the form of the FT Z(ν) analyzed just above. As B(ν) = 0 for ν < 0, Z(ν) = 0 for ν < ν 0 and X(ν) = 0 for |ν| < ν 0 .

The converse is, of course, not true. There is no reason for a high-frequency signal to be a phase signal because this frequency condition does not imply the structure [START_REF] Vakman | On the analytic signal, the TeagerKaiser energy algorithm, and other methods for defining amplitude and frequency[END_REF]. A simple counterexample appears with a high-frequency signal containing more than two spectral lines and, thus, does not satisfy Property 3.1.

Property 3.3: The FT of the AS of a phase signal is zero for all the frequencies smaller than the carrier frequency ν 0 where the spectral line is located. This is a direct consequence of (23) and of the fact that the FT Z(ν) ofz(t) is proportional to B(ν -ν 0 ).

Property 3.4: A phase signal cannot be a low-frequency(B) signal except when it is monochromatic.

The monochromatic case appears when b(1), and z(t) is therefore cos(ω t + θ), that is, of course, a low-frequency signal. Except in this case, the property means that it is impossible to find a frequency B such that Z(ν) = 0 for ν > B. This property can be shown by two procedures. In the first, we simply note that the functions C k (ν) of ( 18) are exponential functions for ν > 0 when the poles are distinct.

DRAFT January 5, 2018

However, it is well known that a sum of a finite number of exponential functions cannot be zero for all the frequencies satisfying ν > B.

It is also possible to show this property by contradiction. Suppose then that there exists a frequency B such that Z(ν) = 0 for ν > B. It results from this assumption and from Property 3.3 that the FT of z(t) is zero outside the frequency interval ν 0 < ν < B and has a spectral line at the carrier frequency ν 0 . Consider now the signal z * (t). Its FT is equal to Z * (-ν), and this FT is zero outside the frequency interval -B < -ν < -ν 0 and has a spectral line at the frequency -ν 0 . Let us now introduce the signal

w(t) = z * (t) exp(2πjf t), (24)
with f > B. This signal obviously satisfies |w(t)| = 1, and the frequency condition ensures that it is an AS. It then has the general form ( 14) and ( 15) and must satisfy Property 3.3. However, its spectral line is at the frequency f -ν 0 , and W (ν) is not zero for ν < f -ν 0 . This is in contradiction with Property 3.3, which shows the result.

Property 3.5: Frequency Shift. If in [START_REF] Vakman | On the analytic signal, the TeagerKaiser energy algorithm, and other methods for defining amplitude and frequency[END_REF] we replace ω 0 with ω 1 , with ω 1 > ω 0 , we obtain a complex signal that is still an AS. As a consequence, if x(t) cos[φ(t)] is a phase signal, which means that its phase has the structure (20), x (t = a cos |∆ωt + φ(t)]), where ∆ω = ω 1 -ω 0 , is still a phase signal with the carrier frequency ω 1 . This especially implies that x (t) is a high-frequency(ω 1 ) signal. (25)

A similar equation has been obtained in [START_REF] Voelcker | Toward a unified theory of modulation. Part 1, Phase envelopes relationships[END_REF] and [START_REF] Voelcker | Toward a unified theory of modulation. Part 2, Zero manipulation[END_REF] by using a rather different procedure.

As the coefficients b k are positive, because of the localization of the zeros z k , we deduce that the instantaneous frequency ω(t) is always greater than ω 0 . This is another illustration of the fact that the FT of z((t) is zero for ν < (1/2π)ω 0 . If all the z k s are zero, then b(t) defined by ( 15) is equal to one, and the instantaneous frequency is tsimply ω 0 .

Some other comments can be presented on the structure of the instantaneous frequency of a phase signal, and this is especially relevant in all those questions dealing with frequency or phase modulation where the index m stands for the modulation term. We note that this function tends to zero when |t| → ∞.

This especially means that ω m (t) cannot be a periodic function, and this is related to the fact that a phase signal cannot have spectral lines, except those coming from the carrier angular frequency ω 0 .

Furthermore, we note that ω m (t) is a rational function in t. The polynomials appearing in the numerator and the denominator have the degrees 2N -2 and 2N , respectively. As N isarbitrary, we deduce that by using the 2N parameters a k and b k , it is possible to approximate a large class of functions. The most limiting constraint on these functions comes from the necessary behavior for |t| → ∞. In fact, ω m (t)

decreases at infinity in |t| -2 , which is a strong restriction on the instantaneous frequency.

Property 3 . 6 : 2 N k=1 b k b 2 k

 3622 Instantaneous Frequency of a Phase Signal: It is obtained by differentiating the instantaneous phase. The most general form of this phase is given by (21) and (22), and differentiating this equation yieldsω(t) = ω 0 + + (a k -t) 2 .

  of signals. The information carried by the instantaneous frequency of a phase signal is entirely in the termω m (t) = 2 N k=1 b k b 2 k + (a k -t) 2 ,

  H[.] means the Hilbert transform. A direct application of this result shows that if a(t) is a low-

frequency (B) signal , and cos[φ(t)] a high-frequency (B) signal, or if their spectra do not overlap, then H{a(t) cos[φ(t)]} = a(t)H{cos[φ(t)]}.

  and such that x(t) = cos[φ(t)] . As the FT of sinc2Bt is zero for |ν| > B , the FT of x(t) does not contain low-frequency components because of the assumption on ω 0 .

Applying

[START_REF] Ville | Theorie et applications de la notion de signal analytique[END_REF]

then gives H[x(t)] sin[φ(t)], and as a consequence, x 2 ((t) + H 2 [x(t)] = 1. However, it is obvious that this equality is not correct. In fact, by applying the Bedrosian theorem, we obtain

H[x(t)] = sinc(2Bt) sin(ω 0 t)

(12)

January 5, 2018 DRAFT and x 2 (t) + H 2 [x(t)] == sinc 2 (2Bt).

  is obvious that |b(t)| = 1, which implies that |z(t)| = 1. Let us now explain why z(t) is an AS. For this, we must analyze the structure of the FT B(ν) of b(t). As N is finite, b(t) is a rational function in t. If all the z k s are distinct, we can write b

January 5, 2018 DRAFT


MD. Abraham B Bohadana
email: abohadana@szmc.org.il

MD Pascal Wild
email: pascal.wild@inrs.fr.

PhD Gabriel Izbicki

Symptom evaluation during the methacholine test: Does it add to the interpretation of the test results based on the PC20FEV1?

Keywords: methacholine challenge test, symptom perception, "asthma-like" symptoms, asthma, airway hyperresponsiveness

    Symptom evaluation during the methacholine test: Does it add to the interpretation of the test results based on the PC20FEV1? Abraham B. Bohadana, Pascal Wild, Gabriel

Introduction

Asthma is defined by the history of recurrent episodes of dyspnea, chest tightness, coughing and/or wheezing, combined with expiratory airflow limitation that is reversible in response to a bronchodilator [START_REF]From the Global Strategy for Asthma Management and Prevention, Global Initiative for Asthma (GINA)[END_REF] . In subjects with typical asthma symptoms and normal spirometry, one option is to perform a methacholine challenge test (MCT) in order to assess airway hyperresponsiveness (AHR) [START_REF]From the Global Strategy for Asthma Management and Prevention, Global Initiative for Asthma (GINA)[END_REF] . Conventionally, a positive test is defined as the provocative concentration of methacholine causing a ≥20 % fall in forced expiratory volume in one second (PC20FEV1) <8 or <16mg/mL, and a negative test as a non-response to the highest concentration [START_REF] Crapo | American Thoracic Society. Guideline for Methacholine and Exercise Challenge Testing[END_REF] . In contrast with the definition of asthma, the interpretation of the MCT results relies solely on the percent drop in FEV1, without consideration of the symptoms justifying referral or those that may occur during the test. Although very popular, this approach ignores prior studies emphasizing the relevance of symptom evaluation during provocation tests [START_REF] Boulet | Perception of scoring of induced bronchoconstriction as an index of awareness of asthma symptoms[END_REF][START_REF] Levesque | Does the methacholine test reproduce symptoms?[END_REF][START_REF] Stenton | Asthmatic symptoms, airway responsiveness and recognition of bronchoconstriction[END_REF][START_REF] Janssens | Predicting asthma outcome at diagnosis: The role of symptom perception during a histamine challenge test[END_REF] . Furthermore, since it can only be assessed in subjects with AHR, a threshold index like the PC20FEV1 is of limited value to examine the relationship between symptoms and responsiveness at sub-threshold (i.e. <20%) levels of fall in FEV1. This has practical importance because studies using non-censored measures of responsiveness such as the dose-response slope (DRS) found a significant association between asthma symptoms and airway responsiveness both above and below the threshold of AHR [START_REF] De Meer | Airway responsiveness to hypertonic saline: dose-response slope or PD15?[END_REF][START_REF] Peat | Relation of dose-response slope to respiratory symptoms and lung function in a population study of adults living in Busselton, Western Australia[END_REF] .

In this study, we investigated whether the evaluation of methacholine-induced symptoms (MIS) in a population of adults with suspected asthma added to the interpretation of the test results based on the PC20FEV1. To do so we determined the prevalence of MIS in responders and non-responders, checked their similarity with symptoms justifying referral, and examined their relationship with airways responsiveness assessed for all individuals, including those who did not reach the threshold of AHR.

Methods

Patients

The study was made up of 82 consecutive patients undergoing MCT at the Pulmonary Institute of the Shaare Zedek Medical Center, in Jerusalem. General practitioners, allergists and pulmonary specialists referred the subjects as part of the investigation of symptoms compatible with asthma. All subjects stopped short-acting bronchodilators for at least 8 hours, long-acting beta-agonists for at least 12 hours, and inhaled corticosteroids for at least 1 week before the MCT. No patient reported acute respiratory infections in the preceding 6 weeks. The hospital's Ethical Committee approved the study, but a consent form was not deemed necessary because the data to be collected were already included in the clinical MCT protocol.

Methacholine Challenge Testing

The MCT was done using the two-minute tidal breathing protocol 2 . At baseline, at least three reproducible measures of forced vital capacity (FVC) and FEV1 were obtained [START_REF] Miller | ATS/ERS TASK FORCE: STANDARDISATION OF LUNG FUNCTION TESTING'' Standardisation of Spirometry[END_REF] . Then the subject inhaled 0.9% saline followed by doubling concentrations of methacholine (0.03 mg/mL to 8 mg/mL) until a fall in baseline FEV1 by at least 20% occurred, or the last concentration of methacholine was inhaled. Aerosols (particle size=3.6 microns) were generated from a Hudson RCI micro mist nebulizer (Teleflex-Hudson RCI, USA) with an output of 0.13 mL/min and were inhaled through the mouth through a mouthpiece for 2 minutes by means of tidal breathing. The PC20FEV1 was obtained from the log-dose response curve and AHR was defined as a PC20FEV1<8mg/mL 2 . At end-test, responders and subjects reporting MIS inhaled four puffs of salbutamol (400 g) from a metered dose inhaler. The methacholine concentration-response slope was computed by applying a mixed model through all data points obtained by plotting the percent fall in FEV1 from prechallenge FEV1 divided by the amount of methacholine inhaled.

Questionnaire

Smoking history, respiratory health, the reason for the test, pre-challenge symptoms and symptoms occurring during the test were evaluated using the ATS questionnaire for MCT 2 (Appendix). At baseline and immediately after the administration of each methacholine concentration, before spirometry, subjects were asked whether they were experiencing symptoms of either dyspnoea, chest tightness, cough or wheezing. In case of positive response, they were asked whether the symptoms were similar to those for which they consulted with a physician and that motivated the test. In addition, an experienced observer (AB) assessed wheeze sounds occurring during the MCT objectively by placing an electronic stethoscope (Littman Model 4000) over the patient's trachea, 2 cm above the suprasternal notch [START_REF] Bohadana | Potential for lung sound monitoring during bronchial provocation testing[END_REF][START_REF] Bohadana | Fundamentals of lung auscultation[END_REF] .

Clinical responsiveness

Based on MIS and airway responsiveness, subjects were classified into four groups of clinical responsiveness: 1."Asymptomatic non-responders" (ANRs): no symptom and/or wheeze by auscultation during MCT; PC20FEV1≥8mg/mL; 2."Asymptomatic responders" (ARs): no symptom and/or wheeze by auscultation during MCT; PC20FEV1<8mg/mL; 3."Symptomatic non-responders" (SNRs): ≥1symptom and/or wheeze by auscultation during MCT; PC20FEV1≥8mg/mL; and 4."Symptomatic responders" (SRs): ≥1symptom and/or wheeze by auscultation during MCT; PC20FEV1<8mg/mL.

Statistical analysis

Statistical analysis was carried out using the Stata Package (Stata 13, College Station, TX, USA). Demographic characteristics were summarised using descriptive statistics, including means and standard deviations. To explore whether the pre-test characteristics were predictive of the outcome of the MCT, we modelled separately being a reactor, experiencing symptoms during the test and the combination of the two outcomes with simple logistic regressions including each pre-challenge characteristic at a time. In addition, three of the post-hoc groups namely SRs, SNRs and ANRs were compared using Fisher's exact test or one-way analysis of variance (ANOVA) as relevant. To assess the relationship between MIS and airway responsiveness below the level of AHR (i.e. at <20% fall in FEV1) we used a mixed model to compare the methacholine concentration-response curve slope (MCRS) of SNRs with that of ANRs over the whole range of the MCRS curve. The end-test FEV1 of SNRs and ARs was compared using t-tests. Finally, multiple logistic regression analysis was used to determine the factors associated with the occurrence of asthma-like symptoms during the MCT. The discrimination power of the fitted logistic model was expressed by the area under the curve (AUC) of receiver operating characteristic (ROC) curves. A p-value of less than 0.05 was considered significant.

Results

Of the 82 patients studied, 26 (31.7%) were classified as ANRs; 6 (7.3%) as ARs; 16 (19.5%) as SNRs and 34 (41.5%) as SRs. Table 1 shows the baseline characteristics of the four groups. Being a female was significantly associated with both being a responder and being a symptomatic responder. A lower baseline FEV1 was significantly associated with symptoms during MCT and with being a SR, but not with being a responder. Finally, dyspnoea as a reason for the test was strongly and significantly associated with symptoms during the MCT, being a responder, and being a SR. No other associations were statistically significant.

Pre-test characteristics

Table 2 shows the pre-challenge characteristics of the four groups. Reporting previous, physician-diagnosed asthma and all respiratory symptoms in the two weeks preceding the test -except chest tightness -were related to symptoms during MCT and being a SR. These associations were less strong and not always significant with being a responder. Use of respiratory medicine was significantly related to the three outcomes, but this association was strongest for symptoms during MCT.

Compared with ANRs, SRs presented significantly lower baseline FEV1 (p=0.002), more dyspnoea as a reason for performing the MCT (0.039), more physician-diagnosed asthma in the past (p=0.001) and more recent symptoms of dyspnoea (p<0.001), wheeze (p=0.03), or any symptom (p=0.02), as well as more use of pulmonary medication (p<0.001). In turn, SNRs looked very similar to SRs, no differences between the two groups emerging whatever the parameter considered (p values between 0.14 for pre-test dyspnoea, and 0.98 for baseline FEV1). Conversely, SNRs differed from ANRs by showing significantly lower baseline FEV1 (p=0.005), greater proportion of subjects reporting physician-diagnosed asthma (p<0.001), and a greater proportion of subjects using respiratory medication (p=0.03); for wheeze as a reason for the test the difference was of borderline significance (p=0.06).

Symptoms occurring during the MCT

Dyspnoea and wheeze detected by auscultation (Table 3) were, by far, the most common findings in each group, followed by cough, subjective wheeze, and chest tightness. SRs and SNRs were similar in the proportion of subjects reporting symptoms during the test and in the proportion of subjects who recognized these symptoms as similar to those justifying referral (SNRs=81.2% vs. SRs=79.4%; p=1.000). Figure 1 shows that the lower the dose of methacholine necessary to trigger symptoms the more likely the subjects were to be responders. Figure 2 shows that no linear relationship existed between the magnitude of percent fall in FEV1 and the onset of symptoms: some subjects presented symptoms at negligible falls in FEV1, while others remained asymptomatic until the FEV1 had fallen by values as great as 40%.

Relationship between MIS and airway responsiveness below the threshold of AHR

This was evaluated by assessing the differences between the MCRS of the two groups of non-responders. As shown in figure 3, SNRs had significantly steeper MCRS (p<0.001) values than ANRs a finding indicating relatively greater airway responsiveness in SNRs.

Accordingly, SNRs had significantly greater mean percent fall in FEV1 at end-test (14.0±4.7) than ANRs (8.6±5.9) (p=0.003).

Predictors of "asthma-like" symptoms during MCT

Multiple logistic regression analysis showed that three factors were significantly associated with the occurrence of MIS: (a) physician-diagnosed asthma in the past; (b) baseline FEV1 % predicted; and (c) percent fall in FEV1 at end-test (Table 4). The discrimination ability of the logistic model, assessed by means of ROC curve, showed a curve close to the optimal left upper corner. The resulting AUC was 0.9494, indicating the excellent discrimination ability for the model (Figure 4).

Discussion

In an outpatient population of adult patients referred for MCT because of suspected asthma, we found that 85% of the responders and 36% of the non-responders experienced methacholine-induced symptoms during the test, and that in ~80% of the subjects in each group these symptoms corresponded to the symptoms justifying referral. Furthermore, among non-responders, SNRs were found to exhibit a steeper methacholine concentrationresponse slope than ANRs indicating a relative increase in airway responsiveness in the former group. These results suggest a possible role for symptom evaluation during the methacholine test.

The observed high proportion of SRs is in keeping with studies showing high prevalence rates of histamine [START_REF] Boulet | Perception of scoring of induced bronchoconstriction as an index of awareness of asthma symptoms[END_REF][START_REF] Brand | Perception of airway obstruction in a random population sample[END_REF][START_REF] Salome | Perception of airway narrowing in a general population sample[END_REF] or methacholine [START_REF] Levesque | Does the methacholine test reproduce symptoms?[END_REF][START_REF] Devereux | Perception of respiratory symptoms after methacholine-induced bronchoconstriction in a general population[END_REF][START_REF] James | Increased perception of airway narrowing in patients with mild asthma[END_REF] induced symptoms in responders. It is also in agreement with studies showing that AHR is not only associated with paroxysms of bronchospasm and bronchial irritability [START_REF] Holgate | The pathogenesis and significance of bronchial hyperresponsiveness in airway disease[END_REF] but also a major determinant of the clinical manifestations of asthma [START_REF] O'byrne | Airway hyperresponsiveness[END_REF][START_REF] Cockroft | Mechanisms of airway hyperresponsiveness[END_REF] . Finally, our data confirm the common clinical observation that patients with AHR often complain of dyspnoea, cough, chest tightness and/or wheezing upon exposure to stimuli such as irritants, cold air or exercise. Different factors may influence the perception of pharmacologically induced bronchoconstriction, including the speed and severity of the airway narrowing, dynamic hyperinflation, obesity, airway inflammation and emotional status, among others [START_REF] Cockroft | Mechanisms of airway hyperresponsiveness[END_REF] . Our results build on previous studies by showing that the onset of MIS may occur at very low levels of fall in FEV1, within the range of the within-day variation of the FEV1 in healthy subjects (Figure 2). This suggests that, in the initial steps of the bronchial challenge, the level of airway obstruction as assessed by the FEV1 did not fully explain the appearance of symptoms. This notwithstanding, the association between MIS and percent fall in FEV1 at end-test (Table 4) supports the view that the role of airway narrowing increased throughout the methacholine test.

A small group of methacholine responders (n=6) denied MIS despite falls in FEV1 as large as 25%. One possible explanation is that these subjects underreported their symptoms. However, the short recall time resulting from the fact that the questionnaire was administered immediately after the inhalation of methacholine makes this possibility very unlikely and suggests these subjects were actually poor perceivers of acute airway narrowing. Poor perceptiveness of bronchoconstriction in hyperreactive subjects has been reported previously in subjects undergoing investigation for asthma [START_REF] Boulet | Perception of scoring of induced bronchoconstriction as an index of awareness of asthma symptoms[END_REF][START_REF] James | Increased perception of airway narrowing in patients with mild asthma[END_REF] or recruited from the general population [START_REF] Salome | Perception of airway narrowing in a general population sample[END_REF] . Factors such as older age, airway inflammation and lower baseline inflammation could explain our findings [START_REF] Rosi | Perception of airway obstruction and airway inflammation in asthma: A Review[END_REF] . However, while we did not measure airway inflammation, no differences existed between ARs and SRs in age and baseline FEV1 (Table 1). Whatever the explanation, in practice, the important point is the identification of poor-perceivers themselves, as they may incur an increased risk of severe asthma attacks [START_REF] Boulet | Perception of scoring of induced bronchoconstriction as an index of awareness of asthma symptoms[END_REF][START_REF] James | Increased perception of airway narrowing in patients with mild asthma[END_REF] .

Explaining methacholine-induced symptoms in non-responders is not straightforward. Surprisingly, we found SNRs to be strikingly similar to SRs in many clinical parameters including the proportion of subjects (~80%) in whom the methacholine-induced symptoms corresponded to the symptoms justifying referral; incidentally, this finding has been found to be associated with a positive test result [START_REF] Levesque | Does the methacholine test reproduce symptoms?[END_REF][START_REF] Stenton | Asthmatic symptoms, airway responsiveness and recognition of bronchoconstriction[END_REF][START_REF] Devereux | Perception of respiratory symptoms after methacholine-induced bronchoconstriction in a general population[END_REF] . In contrast, SNRs differed significantly from ANRs by having worse clinical parameters and milder AHR as detected by the methacholine concentration-response slope. The obvious question that arises from the above two comparisons is whether using the FEV1 as sole outcome measure in the methacholine test could have resulted in false negative tests in some SNRs. Indeed, other parameters such as, for instance, measures of airway resistance, have been reported to be more sensitive detectors of changes in airway calibre than the FEV1 [START_REF] Nensa | Assessment of airway hyperreactivity: comparison of forced spirometry and body plethysmography for methacholine challenge tests[END_REF][START_REF] Bohadana | Dose-response slope of forced oscillation and forced expiratory parameters in bronchial challenge testing[END_REF][START_REF] Khalid | Specific conductance criteria for a positive methacholine challenge test: Are the American Thoracic Society guidelines rather generous?[END_REF] . To quote but one study, Khalid and colleagues [START_REF] Khalid | Specific conductance criteria for a positive methacholine challenge test: Are the American Thoracic Society guidelines rather generous?[END_REF] found that 27 of 138 subjects undergoing a methacholine test because of suspected asthma had a significant fall in sGaw (>52%) but did not drop their FEV1 by 20%.

Secondly, for the methacholine test, a 12% drop in FEV1 has been found to be more accurate than the 20% threshold in separating healthy controls from subjects with exercise-induced bronchospasm [START_REF] Eliasson | Sensitivity and specificity of bronchial provocation testing. An evaluation of four techniques in exercise-induced bronchospasm[END_REF] . Had we used the 12% value no fewer than 12 of our 16 SNRs would have turned into SRs. Finally, in the quality of global test, the FEV1 is not ideally suited to detect regional lung changes often associated with AHR [START_REF] Chapman | Increased airway closure is a determinant of airway hyperresponsiveness[END_REF][START_REF] Zeidler | Montelukast improves regional airtrapping due to small airways obstruction in asthma[END_REF][START_REF] Farrow | Airway closure on imaging relates to airway hyperresponsiveness and peripheral airway disease in asthma[END_REF] . Had they been present in our subjects, such changes could have explained both the clinical picture of SNRs and the poor correlation between wheezesa sign produced locally in the central airways [START_REF] Bohadana | Fundamentals of lung auscultation[END_REF] -and spirometry observed in SNRs and SRs (Table 2). However, since we did not measure regional lung function we can only speculate on this matter.

Our study provides a good illustration of the limited application of PC20FEV1 as an index of severity in patients with MIS who failed to achieve a 20% fall in FEV1 during the methacholine test. Using the MCRS, we were able to demonstrate that the SNRs had milder AHR compared with the ANRs. This finding is compatible with results from prior studies demonstrating a significant relationship between airway responsiveness -assessed by the dose-response slope (DRS) -and asthma symptoms at sub-threshold levels of AHR in both children [START_REF] De Meer | Airway responsiveness to hypertonic saline: dose-response slope or PD15?[END_REF] and adults [START_REF] Peat | Relation of dose-response slope to respiratory symptoms and lung function in a population study of adults living in Busselton, Western Australia[END_REF] . With the MCRS, we were able to assign a measure of severity of bronchial responsiveness to the subjects with falls in FEV1 of between 5 and 19% who otherwise would be merely classified as having normal responsiveness. Furthermore, in quality of continuous measure, the MCRS is better suited than a dichotomous index to monitor changes in airway responsiveness over time. This aspect has practical importance because we cannot rule out the possibility that the milder AHR in SNRs might be an early manifestation of airway inflammation carrying an increased risk of future asthma. Although only prospective, follow-up studies of SNRs can prove or disprove this hypothesis, there is evidence to suggest this might be the case. In one study, Khalid and colleagues [START_REF] Khalid | Predictive value of sGaw, FEF25-75, and FEV1 for development of asthma after a negative methacholine challenge test[END_REF] found that 15 of 77 methacholine non-responders presenting asthma-like symptoms developed overt asthma over a 3-yr period; noticeably, 43% of patients with 10 to 20% decline in FEV1 during the MCT eventually developed asthma. More recently, Peiman and colleagues [START_REF] Peiman | Fluticasone propionate in clinically suspected asthma patients with negative methacholine challenge tests[END_REF] reported a marked improvement in asthma symptoms and the disappearance of wheeze in methacholine non-responders with clinically suspected asthma receiving treatment with high doses of inhaled fluticasone. However, as acknowledged by the respective authors, the above studies were somehow limited either by their retrospective design [START_REF] Khalid | Predictive value of sGaw, FEF25-75, and FEV1 for development of asthma after a negative methacholine challenge test[END_REF] or by the lack of a control group [START_REF] Peiman | Fluticasone propionate in clinically suspected asthma patients with negative methacholine challenge tests[END_REF] . Furthermore, none of the studies assessed airway responsiveness in nonresponders experiencing symptoms during the test (i.e. SNRs), so the significance of milder AHR in this population remains unclear.

We think methodological errors did not influence our results. First, we used a wellestablished method for the provocation challenge protocol along with the standard ATS questionnaire. Second, although subject blinding to the incremental nature of the MCT was not feasible, all patients lacked blinding so this factor cannot explain the observed betweengroup differences. Finally, one may argue that because methacholine is a direct stimulus, our results are not generalizable. However, a previous investigation in mild asthmatics showed that symptom perception did not differ appreciably during the early response to a direct stimulushistamine -and two indirect stimuli namely antigen exposure and exercise [START_REF] Turcotte | Perception of breathlessness during bronchoconstriction induced by antigen, exercise and histamine challenges[END_REF] .

In conclusion, our study shows that symptom evaluation during the MCT helped identifying patients with milder AHR that would be overlooked by the PC20FEV1. Further prospective, longitudinal studies of SNRs should be conducted to determine the prognostic validity of this approach. To gather information on this matter, based on our results we planned a longitudinal, prospective study aiming to examine the effect of inhaled corticosteroids on SNRs outcomes [START_REF] Bohadana | Effect of inhaled corticosteroid on methacholineinduced symptoms, airway responsiveness and FE NO levels in methacholine nonresponders: study protocol for a randomized controlled trial[END_REF] .

Legend to figures

Figure 1: Proportion of responders according to the inhaled methacholine concentration at first symptom among subjects experiencing MIS. The numbers at each step correspond to the number of responders in whom symptoms first manifested either at that step or before. For instance, of the 10 subjects whose symptoms appeared at a methacholine concentration of 0.0625 mg/mL or earlier, 100% were responders. In turn, of the 23 subjects whose symptoms appeared at a concentration of 0.250 mg/mL or lower, 20 (87%) were responders. The curves show that the fall in FEV1 was significantly greater in the SNRs across the challenge. Since responsiveness is defined by the ratio of fall in FEV1 to the amount of methacholine inhaled, for a given methacholine concentration the greater fall in FEV1 in SNRs indicates a relatively increased responsiveness in these subjects compared with the ANRs 4 including, as dependent variable, the occurrence of asthma-like symptoms during the challenge and, as independent variables, physician-diagnosed asthma, baseline FEV1 in percentage of the predicted, and percent fall in FEV1 at the end of the challenge. The curve is displaced toward the left upper corner, giving an area under the curve (AUC) of 0.9511, indicating the excellent discriminant power of the model.

Figure 2 :

 2 Figure 2:Scatter plot of percent fall in FEV1 vs. inhaled methacholine (mg/mL) at first symptom in responders and non-responders reporting MIS. A wide variation in the fall in FEV1 is seen in both responders (0-40%) and non-responders (~0-19%). In 10 responders and 8 non-responders, asthma-like symptoms manifested at decreases in FEV1 ~ 6% or lower.

Figure 3 :

 3 Figure 3: Mixed model assessing airway responsiveness below the threshold of AHR by computing the slope of the methacholine concentration-response curve in SNRs and ANRs.The curves show that the fall in FEV1 was significantly greater in the SNRs across the challenge. Since responsiveness is defined by the ratio of fall in FEV1 to the amount of methacholine inhaled, for a given methacholine concentration the greater fall in FEV1 in SNRs indicates a relatively increased responsiveness in these subjects compared with the ANRs

Figure 4 :

 4 Figure 4: Receiver operating characteristic curve (ROC) expressing the discrimination power of the logistic model presented in table4including, as dependent variable, the occurrence of asthma-like symptoms during the challenge and, as independent variables, physician-diagnosed asthma, baseline FEV1 in percentage of the predicted, and percent fall in FEV1 at the end of the challenge. The curve is displaced toward the left upper corner, giving an area under the curve (AUC) of 0.9511, indicating the excellent discriminant power of the model.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. Percent of reactors according to the methacholine concentration at onset of symptoms in subjects experiencing MIS 243x157mm (300 x 300 DPI)

Table 1 : Baseline characteristics of subjects groups classified by the occurrence of "asthma- like" symptoms during the challenge test and airway responsiveness a OR by decade of increasing age b OR by kg/m2 c OR by decreasing baseline FEV1 in % predicted * p<0.05, ** p<0.01 Non-responders Responders Odds Ratios Symptomatic during MCT No/Yes No/Yes

 1

	Association with	Association with	Association with
	being symptomatic	being reactor	being symptomatic
	during MCT		reactor

Table 2 : Pre-challenge clinical characteristics [n, (%)] of subjects in the four groups * p<0.05 ** p<0.01 § p=0.06 ¥ p=0.075

 2

		Non-responders	Responders		p values
	Symptomatic during MCT	No	Yes	No	Yes	Association	Association	Association
						with being	with being	with being
						symptomatic	responder	symptomatic
						during MCT		responder
	n	26	16	6	34
	Clinical history
	Physician-diagnosed asthma	2 (7.7)	10 (62.5) 0 (0.0) 18 (52.9)	34.1**	2.18	3.60*
	Hospitalization for asthma	1 (3.8)	1 (6.2)	0 (0.0)	2 (5.9)	4.09	1.36	2.25
	Respiratory dis. childhood	2 (7.7)	5 (31.2) 1 (16.7) 7 (20.6)	3.05	1.25	1.30
	Recent symptoms (2 weeks)
	Dyspnoea	5 (19.2)	6 (37.5) 2 (33.0) 21 (61.8)	4.19**	3.81**	4.35**
	Cough	11 (42.3)	8 (50)	1 (16.7) 23 (67.6)	2.72*	1.82	2.98*
	Chest tightness	3 (11.5)	3 (18.8)	0 (0.0) 10 (29.4)	3.40 ¥	2.00	2.92 §
	Wheeze	1 (3.8)	4 (250	1 (16.7) 15 (44.1)	9.19**	4.93**	5.53**
	Any symptom	14 (53.8) 12 (75) 3 (50.0) 28 (82.3)	3.53*	2.12	3.06*
	Current/past medication	4 (15.4)	8 (50)	2 (33.0) 21(61.8)	5.98**	3.38**	3.92**

Table 3 : Symptoms [n, (%)] experienced during the methacholine challenge test by responders and non-responders p value =0.300 for dyspnoea and 1.000 for all other comparisons.

 3

	Symptom

Table 4 : Logistic model for the prediction of appearance of symptoms during the methacholine challenge test Dependent variable Response variable Odds ratio

 4

	Standard	Z
	Error	p value

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Acknowledgements

Guarantor AB is guarantor for the study and takes responsibility for the integrity of all data and the accuracy of the data analysis

This study was funded by an unrestricted grant from Novartis, Israel

Funding

This study was supported by an unrestricted grant from Novartis Israel Financial/Nonfinancial disclosures  An unrestricted grant from GSK, Israel partially supports the respiratory research unit (RUPI).

Author contributions

 AB contributed to the study design, data collection, data interpretation, drafting the manuscript, critical review of the manuscript and approval of the final version to be published.

 PW contributed to the data analysis and interpretation, drafting the manuscript, critical review of the manuscript and approval of the final version to be published.

 GI contributed to the study design, data interpretation, drafting the manuscript, critical review of the manuscript and approval of the version to be published.

Role of the sponsors

The sponsors had no role in the design of the study, the collection and the analysis of the data, or the preparation of the manuscript.

Other contributions

The authors thank Dr. L-P Boulet for critically reviewing an earlier version of the manuscript, Mrs. Ahouva Tasama for performing the methacholine challenge tests, and the patients for their participation