SEDYL, Joseph D Thach

klah IN CONTEMPORARY KHMER Quantitative and qualitative plurality

Particles

1SG.

1 st person of singular personal pronoun 2SG.

2 nd person of singular personal pronoun 3SG

3 rd person of singular personal pronoun S 0 / S1 Speaker/Addressee

The present paper deals with klah in contemporary Khmer. In the existing grammars and manuals (such as Gorgonief (1966), [START_REF] Huffman | Modern spoken Cambodian[END_REF], Khin Sok (2000)), klah is described as an indefinite pronoun and often considered (at least in translations) as the equivalent of some in English. However, as we will demonstrate below -through a semantic analysis of its uses and values -, klah has its own syntactic and semantic values, distinct from those of some, and is a marker of plurality (it should be recalled that in Khmer the number category is not morphologically marked). For this reason, our study of klah will come within the framework of a wider discussion about the notion of plurality in Khmer (also linked to nominal and adjectival reduplication1 ).

Our paper endeavors to describe all the uses and values of klah without any exclusion.

Firstly, it deals with the cases where klah is combined with a (count or mass) N. For these two categories of N, we will show that klah comes under two interpretations: a partition of all the instances of the N; b. the construction of a set of singular instances of the N. Secondly, we will study the cases where klah relates to a V and constructs various instances of this V.

As regards reduplication, we will distinguish between two cases: a. klah in case of reduplication of the N; b. and klah reduplicated and introducing the notion of vagueness; compare:

(1) cam klah remember klah "I clearly remember some of the things"

(2) cam klah klah remember klah klah "I faintly remember some of the things/a few things now and then"

We will close our study with the analysis of cases where klah is combined with the indefinites/interrogatives ʔey and naa. In this respect, we will point out a difference of behaviour. With ʔey, klah can only stand in postposition and ʔey is an interrogative2 :

(3) prɑɑcum prɨk mɨɲ kee niʔyiey pii rɨəŋ ʔey Klah meeting morning DEICT people talk about story ʔey Klah "Which were the topics broached during this morning meeting?" With naa, klah can stand either in ante-position (naa is an indefinite: example (4)) or in postposition (naa is an interrogative: example (5)):

( "In this village, some inhabitants (whom I cannot or don't want to point out) don't even have rice to eat" (5) prɑɑcum prɨk mɨɲ kee niʔyiey pii rɨəŋ naa klah Meeting morning DEICT people talk about story naa klah "Among the topics on the agenda, which ones have actually been discussed this morning?"

The analysis of these various uses will allow us to verify our hypothesis on klah as a marker of quantitative and qualitative plurality.

The examples used in this paper were constructed by us (native speakers) or heard in daily conversations. All these examples were checked by ten native speakers of different social origins (students, teachers, actors, moto-taxi drivers).

In order to discuss the various uses of klah, we will use the notion of plurality, but with a meaning different from that usually given to this notion, especially when referring to the number category in languages where the opposition singular vs plural is morphologically marked. In their article entitled The semantics and pragmatics of plurals (2008), D. Farkas and H. de Swart give the following definition of the opposition between singular and plural: « singular nominals take values from the domain of atoms; plural nominals take values from the domain of sum ». In this view, the singular / plural opposition comes down to « atomic vs sum reference », in other words to a purely quantitative matter. Following M. Jarrega works on the plural in French, a different characterization of plural can be given [START_REF] Paillard | (éds) Indéfinis et prédication[END_REF] associating both quantitative and qualitative information, as shown by the following representation: X ( x i …. x k ….. x n ...) -quantitative component (Qnt). X corresponds to a given but not explicited quantity of instances of the category associated to the N under plural. In order to characterize this quantity, we will resort to the expression inclusive plural, adopted by several authors. Inclusive plural is neutral as regards the « atomic vs. sum reference » distinction (see -Do you have children? -Yes, one daughter);

-qualitative component (Qlt). The instances x i …. x k ….. x n of the category N are differentiated from each other, the criteria of differentiation being left unspecified [START_REF] Lasersohn | Plurality, conjunctions and events[END_REF].

This definition for plurality agrees with that given by R. [START_REF] Jackendoff | Parts and boundaries[END_REF] as "an aggregate of distinguishable individuals" (aggregate: X, distinguishable individuals:

x i …. x k ….. x n ).
Our hypothesis on klah is the following: klah is a marker of inclusive plurality actualizing both components: the quantitative (Qnt) and the qualitative (Qlt). In (6a) N + klah introduces some instances of the N as they fit a differential property which is not made explicit: the points to be discussed are identified for the speaker. In (6b) N + muəy-cɑmnuən refers to a sum of x (quantificational reference), out of all qualification. In (6c) N + Ø refers to one or several instances of the category N (leaving it to the context to make the difference).

In some cases, klah is impossible, and only Ø and muəy-cɑmnuən can be used: "In this group, there are some good students, in others there aren't any" (9a) with Ø plainly states the existence of good students, whereas (9b) with klah states that there is a given number of students different from the others as they fit the property "be good" in their studies.

(
The series (10a-c) confirms that N Ø, contrary to N + klah, cannot refer to a partition on a whole set contextually introduced. Example (10c) is possible owing to the fact that a subset is built through an independent source of determination, making it possible to identify a group of NGOs. only mədɑɑŋ once "In city schools, it is possible to find some good students, but in rural schools, there are none at all" In (12) contrary to (9b), klah comes after the property « be good »: this property does not work here as the basis for differentiating a subset. In this position, klah just means that there are good students, each student fitting specifically the property « be good ». The same is to be found in the series of examples (13a -c) under the form of questions. The context is the following: an organism must make an assessment in schools in order to grant scholarships to the students. mədɑɑŋ once "Are there some students who are good (students)?"

In (13a), the speaker has no prejudice: the question is whether there are (or not) good students. In (13b) with klah, the question is not about the existence (or not) of good students but is about the relevance of the property « be good » in order to qualify some of the students, which implies that for the speaker, it is not obvious that there are good students at all. As in (12), klah comes after the property 'be good'. In (13c) we have a different order (N + klah + property): the speaker wants to know if there are students who are different from other students by "being good in school".

In short, when the scope of klah is a count noun, it has two interpretations according to the context, corresponding to a 'weighting' either on the component X, or on the component (x i …. x k ….. x n ...): in one case it stands for the partition of a set based on a differential property; in the other, it introduces a series of instances of the category N taken in a qualitative variation.

Mass noun as the scope for klah

Combined with a mass noun klah has two interpretations: (14) tɨk klah ɲam tɨw cʰɨɨ puɜh water klah drink go suffer stomach "Some brands of water cause stomach ache when you drink it" In ( 14), klah can switch with muəy-cɑmnuən. klah expresses a partition in the whole of the water brands. Owing to mass nouns properties, the series coming under the partition operated by klah is formed by the whole of the brands of bottled water. In ( 15) klah can switch with bɑntec 'a little'. bɑntec stands only for a small quantity, whereas klah refers to a given quantity, i.e. a quantity which is qualitatively defined: in the case of (15), this corresponds to the quantity the interlocutor will deem suitable as regards the number of people attending the party. This quantity is distinguished in the set of the possible quantities of alcohol. As R. Jackendoff writes about mass nouns: "With a mass noun like water, one can divide its referent and still get something describable as water. For this second interpretation of klah, we put forward the hypothesis that klah marks a fragmentation of the mass noun, which means that in X (x i …. x k ….. x n ...) x i …. x k ….. x n ... each one of the x corresponds to a specific quantity (or portions) of alcohol. As for X, it refers to a set of a priori possible quantities

We now give one more example of klah marking a quantitative fragmentation of the mass noun (in ( 16) bɑntec is possible).

( 16) baə ʔɑt məhoop ɲam yɔɔk treyŋiet klah tɨw if NEG.

side-dish eat take driedfish klah go ɲam tɨw eat go "If you have no side dishes, take some fair quantity of dried fish to eat."

In ( 16) the quantity of dried fish is not just any quantity: take whatever you need, but leave some for me as well (sharing the quantity)

In short, with the mass nouns the two interpretations already identified for the count nouns are to be met again:

-a partition which, owing to the mass nouns properties, distinguishes a subset X (x i ….

x k ….. x n ...) of varieties of N; this subset is based on a differential property. The focus is on X, and the inner subset differences are not taken into account -the fragmentation of the N in a series of distinct quantities X (x i …. x k ….. x n ...); comparing (17a) with Ø / (17b) with klah shows that the fragmentation of the N results in given specific quantities: the focus is on the series (x i …. x k ….. x n ...).

klah as a pronoun

We will speak of klah as a pronoun when its scope is not an N, whether it is contextually given (ex. (17) or klah corresponds to the predicate internal object (ex. ( 18 klah "I remember some of the things but not others" (18b) cam klah remember klah "There are things I remember quite well I can tell you."

With the pronoun, we find again the two interpretations previously brought out: partition in (18a), differentiated instances of the category N in (17b) and (18b). Comparing (19a) with (19b) shows that when klah refers to a N standing in the left context, klah is compulsory to mark a partition. In (19b) the only possible interpretation is that the request involves the mango tree proper and not part of its fruit.

Directly related to this second interpretation, it must be noted that there are cases when it's the process itself which stands as klah scope (in such a case, bɑntec is possible as well):

(20) -In a meeting, the same person (S 1 ) keeps speaking all the time, which irritates S 0 who therefore stops him: "Just shut up a little [...] !"

In this series (20), sŋat moat means 'shut up'. moat which can often be translated by 'mouth' is part of the predicate. In (20a), with the form ø, the injonction takes a sharply categorical tone I want you to shut up right now. In (20b), klah stands for the construction of one or several instances of the process: all I'm asking you is shut up a little / from time to time. In (20c) with bɑntec the request is limited to one time for a limited period quantitatively defined. The principle here is quite similar to that working with the mass nouns: klah can be interpreted as referring to a series of occurrences of the V. In accordance with the notion of inclusive plurality, the actualization may concern one or several occurrences of the V, each occurrence being differentiated. In short, with klah as a pronoun, the two interpretations appearing with the count nouns and the mass nouns (partition and series of occurrences qualitatively differentiated) are available.

On the other hand, when klah has the process for its scope, the second interpretation is the only possible one (with a V there is no available set of Vs)

klah and the reduplication

As regards reduplication, we will study two different points:

klah in case of reduplication of the N; klah reduplicated.

In [START_REF] Paillard | Réduplication du nom et de l'adjectif en khmer[END_REF], we have shown that the reduplication of the N puts N in a qualitative variation out of any quantitative variation: each one of a series of instances (x i …. x k ….. x n ...) of the category N specifically fits the predicate of which it is an argument (or the qualifying property): NN p means….x i -p i …. x k -p k ….. x n -p n .... . It appears that the reduplication on the one hand and klah on the other have a quite similar semantic value: both involve a series of qualitatively differentiated instances. We hereafter take a series where for a given N, we consider all the possibilities for its determinations: N-, NN, N-klah, NN-klah 3.1. klah in case of reduplication of the N (22a) -A popular TV program presenter (S1) goes to the provinces in order to recruit girls for his program. On his arrival in a village, he asks a local inhabitant (S 0 ):

S

beauti. klah

In (a) N Ø property p ('beautiful') N Ø property p' ('not beautiful') corresponds to the construction of two subsets of girls in relation with the property p : The two subsets do not exhaust the whole set, but nothing is said about their number (the "beautiful-girls" subset can count for one single unit).

In (b) N N p ('beautiful') N N p' ('not beautiful') according to our hypothesis on the reduplication of the N corresponds to the construction of two series of N in relation with p and p', each occurrence (individual) of the two series specifically fitting p (or p'). Contrary to (a) the two series are not in a contrastive relation and those two series do not exhaust the whole set of the village girls (some of them are not taken into account as regards the property p).

In (c) N klah 1 -p ('beautiful') N klah 2 -p' ('not beautiful') klah 1 and klah 2 correspond to the construction of two subsets of N (in relation with the property p or p'), each instance of the two series are considered as individuals. As in (b) those two subsets don't exhaust the whole of the village girls.

In (d) N N klah 1 -p ('beautiful') N N klah 2 -p' ('not beautiful'), two series of differentiated N are at first introduced. In those series, klah 1 et klah 2 distinguish each one for its part a group of girls (previously differentiated), fitting or not the property p. The copresence of the reduplication of the N and of klah reinforces the differentiation between the individuals in each series. As in (b) and (c) those two subsets don't exhaust the whole of the village girls. In (c) and (d) klah is placed before p (and p'). In (e) and (f) klah is placed after p (and p'). As was noted before through the examples ( 12) and (13b), in the second case, the partition is not based on the property p (or p').

In (e) N p ('beautiful') klah 1 N p ('not beautiful') klah 2 , klah 1 and klah 2 stand for the construction of two subsets (partition of the set N) -this partition is not based on the property p (or p').

In (f), N N p ('beautiful') klah 1 N N neg p ('beautiful') klah 2 two series of differentiated N are at first introduced and in those series, klah 1 et klah 2 distinguish, each one for its part a group of girls previously differentiated as fitting respectively the properties p and p'. The distinction between those two groups is not based on the property p (or p').

The combinations illustrated by (22d-f) show the modes of interaction between the qualitative plurality (reduplication) and the <quantitative and qualitative> plurality, where both quantitative and qualitative components of the plurality get actualized.

klah reduplicated

As a noun determiner and as an indefinite pronoun as well, klah can be reduplicated (but it must be noted that it's impossible to have at the same time reduplication of the N and reduplication of klah: * NN klah klah). As regards the above examples (22 d, f) where the N klah has for its scope is reduplicated, we have put forward our hypothesis on the N reduplication: each instance of the category N specifically fits the predicate. An extension of this hypothesis to the reduplication of klah can be validated. In relation to the process, klah 1 and klah 2 construct, each for its part, a qualitatively distinct subset, entailing an effect of vagueness (no way to centre/focus on a given subset "There is just a very little left, it's just enough for two people"

In ( 24b) and ( 25) the reduplication of klah gives rise to a feeling of vagueness: there is no construction of qualitatively distinct subset. It is therefore impossible to identify the rehearsed sketches (ex. 24b) or the actualized recollections (ex. ( 25)). In (26b) klah 1 and klah 2 construct two distinct quantities of rice left, which stops from assessing the exact quantity of rice left, therefore interpreted as (very) insufficient.

The working of klah when reduplicated is similar to that of klah not reduplicated with the construction of two differentiated series (qualitative partition). The criterion liable to make an occurrence belong to such or such series not being specified, this co-presence of two series results in a kind of interference, with a depreciating effect on the occurrences involved, owing to the fact that although differentiated, they are not taken as full individuals.

klah combining with the interrogatives / indefinites ʔey and naa.

ʔey is a free choice type of indefinite and an interrogative. As an indefinite-interrogative naa means that the items previously made out are related to an undifferentiated set through the introduction of a new property: [START_REF] Thach | L'indéfinition en khmer : du groupe nominal au discours[END_REF]).

x i …. x k ….. x n  ( x (i) …. x (k) ….. x (n) ...) set (see
Whereas with ʔey, klah can only stand in postposition, with naa, klah can stand before naa (naa is an indefinite) or after naa (naa is an interrogative) (note that muəy 'one' shows the same distribution as klah. On this point, see [START_REF] Thach | L'indéfinition en khmer : du groupe nominal au discours[END_REF] "Not all of them, some are bad, but I can't tell which ones". Note that naa alone is not possible in this example.

In (27a) klah marks a partition on the set of the French films and constructs the subset of films fitting the property « be bad». It refers to instances of films which can be identified, but that the speaker cannot list.

In (27b) the presence of naa after klah means that the films of the subset of distinctive instances constructed by klah reduces to an undifferentiated subset: the speaker is unable to identify them; naa makes uncertain the previous distinction in the instances of bad films: with ʔey alone, is an open question: ʔey points the whole of the possible and conceivable dishes. In (29b) klah following ʔey means that for S 0 all the dishes of the set are not on the same level for S 1 and that the question aims at identifying a subset of dishes fitting the property 'be preferred by S1'. The difference between (29c) with naa and (29a) with ʔey is due to the fact that at first with naa the set is not that of all the possible and conceivable dishes, but a set of dishes on a menu. The question means that S 0 doesn't know which dishes on the list S 1 does prefer. We shift from a set of identified dishes to a set of unidentified dishes. In (29d) as well as in (29b), S 0 asks S 1 to identify a subset of dishes as those S 1 wants to eat on the evening.

Another example: (30) -S 0 who didn't attend the meeting asks S 1 : (30a) prɑcʊm prɨk mɨɲ kee niʔyiey pii rɨəŋ ʔey meeting morning DEICT. people speak about story ʔey "Which were the topics broached in this morning meeting?" (30b) prɑcʊm prɨk mɨɲ kee niʔyiey pii rɨəŋ ʔey klah meeting morning DEICT. people speak about story ʔey klah "What were the topics broached in this morning meeting? (I know nothing of the topics liable to be discussed, but since a meeting took place, I take it for granted that some subject(s) were broached)" (30c) prɑcʊm prɨk mɨɲ kee niʔyiey pii rɨəŋ naa meeting morning DEICT. people speak about story naa "What were the topics on the agenda discussed this morning?" (30d) prɑcʊm prɨk mɨɲ kee niʔyiey pii rɨəŋ naa klah meeting morning DEICT. people speak about story naa klah "Among the topics on the agenda, which ones were actually discussed this morning?"

Placed after ʔey and naa, klah introduces a qualitative partition on the set established by ʔey and naa. This qualitative partition leads to an heterogeneity of the set, which is virtual.

Since it comes through a question, ʔey and naa as question markers are in the locutor's area and klah in the interlocutor's.

Conclusion

We claim that /klah/ associates both a quantitative and a qualitative information, as shown by the following representation: X (x i …. x k ….. x n ...) This characterization is at work in all its various uses and values. If /klah/ can be considered as the main marker for plurality in Khmer, it must be pointed out that the qualitative component is nonetheless crucial, as regards the notion of partition as well as the construction of differentiated instances of the N

  t buy only fruit juice, buy a certain amount of alcohol (leaving the quantity up to you)!"

  tree bears actually a lot of fruits, may I ask you for some?

7a) siewpʰɨw muey- cɑmnuən nih trəw yɔɔk tɨw ʔaoy rooŋ- pʊm wɨɲ book muey- cɑmnuən

	DEICT. must	take go give printing-	PART.
		house
	"This set of books must be taken back to the printing house !"
	(7b)

siewpʰɨw Ø nih trəw yɔɔk tɨw ʔaoy rooŋ- pʊm wɨɲ

	(8bis) -At the market place, a customer (S 1 ) and a seller (S 0 ).
	S1 : ʔaoy	Sway kɲom muey kiiloo mɔɔk
	give	Mango 1SG.	one	kilo	come
	"Give me one kilo of mangoes!"
	S 0: yɔɔk	Sway	prɑɑpʰɛɛt naa
	take	Mango	sort	naa
	book	Ø		DEICT. must	take go give printing-	PART.
								house
	"This / these book(s) books must be taken back to the printing house !"
	(7c) *siewpʰɨw klah	nih	trəw	yɔɔk tɨw ʔaoy rooŋ-	wɨɲ
								pʊm
	book	klah	DEICT. must	house take go give printing-	PART.
	In (7) the impossibility of klah means that the books in question are copies of one and the
	same book, which precludes any qualitative differenciation of those books.
	Contrary to this case, examples can be met where klah only possible :
	(8) Context: a police inspector (S 1 ) tries to get the manager (S 0 ) of a restaurant tell him if the
	wanted person comes to eat daily in the restaurant:
	S1 : koat	mɔɔk ɲam	baay nɨŋ	roal	tŋay	rɨɨ	yaaŋmec
	3SG	come	eat	rice	DEICT.	every	day	or	how
	« Does he come to eat here every day? »
	S 0: mɨn	Tieŋ	tee	tŋay	klah	mɔɔk tŋay	klah	ʔɑt
	NEG.	Regular PART.	day	klah	come	day	klah	NEG.
	"It is not regular, some days he comes, and others he doesn't"
	In (8) muey-cɑmnuən and Ø cannot be used. Considering the whole set of the days, klah1
	and klah2 work out two subsets of days, the first one fitting « come », and the second one
	« not come ».

pruəh Sway klah cuu swaay klah pʔaem haey swaay

	Because Mango	klah	acid mango klah sweet	and	mango
	klah	Tiet	cuu-
			ʔaem
	klah	furthermore sweet-and-sour
	"What kind of mangoes do you want? For there are acid mangoes, sweet mangoes and
	the rest, sweet-and-sour mangoes".
	In (8bis), the mangoes as a whole are divided into three subsets, each subset fitting a
	differential property: acid, sweet and sweet-and-sour.
	(9) -To the question "in which group are there good students?", S 0 anwers:
	(9a) knoŋ Krom nih	mien sɨh	Ø	rien	puukae krom
	in	Group DEICT. have	student Ø	learn	gifted	group
	pseeɲ Tiet	ʔɑt	mien tee
	other	More	NEG.	have	PART.
	"In this group, there are good students, in others there aren't any"
	(9b) knoŋ Krom nih	mien sɨh	klah	rien	puukae krom
	in	Group DEICT. have	student klah	learn	gifted	group
	pseeɲ Tiet	ʔɑt	mien tee
	other	More	NEG.	have	PART.

cuəp kɲom muəy pʰlɛɛt Baan tee kɲom mien

	klah kɨt (11b) knoŋ krom	tae nih	pii kɲom dəŋ rɔɔk	lʊy tʰaa	tee mien neak Ø
	only DEICT. of coolcət kɲom tee group 1SG klah Think in mɨn	look for know that money PART. have	people Ø	only
	to make money" NEG. Like		1SG		PART.
	"In this group, I know that there are people who don't like me!"
	(10b) *nɨw srok In Country khmer have organisation much be-kmae mien ʔɑŋkaa craən mɛɛn tae ʔɑŋkaa true Example (11b) with N + Ø states the existence of one or several unfriendly people; in (11a) only organisation the speaker knows who is /are the one (s) in question but do not want to name him / them
	Ø (let's point out the inclusive interpretation of klah: one or several people can be concerned). kɨt tae pii rɔɔk lʊy tee
	Ø (12) saalaa	Think knoŋ	only kroŋ kroan mien sɨh of look for money PART. rien puukae klah
	(10c) nɨw school		srok In	kmae mien ʔɑŋkaa craən mɛɛn tae ʔɑŋkaa city enough have student learn be gifted klah
	In baə ʔoostraa-if	country Khmer have organisa-tion saalaa taam pʰuum weɲ ʔɑt much be-true mien sɑh only organisa-tae tion kɨt tae pii rɔɔk lʊy tee school follow village PART. NEG. avoir PART.
	lii
	Australia think	only	of	look for money PART.
	come meet	1SG.	one	moment possible PART. 1SG.	have
	rɨəŋ	klah trəw	niʔyiey
	story	klah must	talk
	Other examples can be given:
	(11a) knoŋ krom		nih		kɲom dəŋ	tʰaa	mien neak klah
	In	group		DEICT.	1SG	know that	have	people klah
	mɨn	coolcət kɲom tee
	NEG.	Like		1SG		PART.

(10a) nɨw Srok kmae mien ʔɑŋkaa craən mɛɛn tae ʔɑŋkaa in Country khmer have organisation much betrue only organisation "It's true that in Cambodia, there are many NGOs, but some (of them) are there "It's true that in Cambodia, there are many NGOs, but the Australian ones are there only to make money" In all the above examples, klah marks a partition: a differential property is used to identify a group of instances of the N. But the N count + klah pattern does not necessarily refer to a partition: N+klah can refer to a series of instances differentiated from each other. Whereas in the case of the "partition" interpretation, the property is used to distinguish a subset of instances of the N, in the second case what is emphasized is the fact that each one of the instances in the set has an identity. (6a) is a first example coming under this second interpretation: the different points to be mentioned by the speaker are not presented as a whole and in his view, each one of the points is important (6a) mɔɔk "Could you come and see me for a short while? I have things to talk (to you)"

"In this group, I know that some people don't like me!"

  , the presence of tiet 'extra, more' is due to the fact that a first quantity of honey has already been given. In (17a), the question is just about « wanting some more honey », without the extra quantity being determined. In (17b), the question is prejudiced: S 0 is ready to give an extra quantity, but not just any quantity. It depends on what S 0 can / or wants to give on a second time (the new quantity being less than the first one).

	S 1 :	ʔɑh		haəy
		finish		already
		"I have already finished it"
	(17a) S 0 :	cɑŋ	baan	Ø	tiet	tee
			want	obtain	Ø	more	PART.
		"Do you want some more?"
	(17b) S 0 : cɑŋ	baan	klah tiet	tee
			want	obtain	klah more	PART.
		"Do you want some extra?"
	In (17)(18) The topic is about S 0 having old recollections that S 1 would like to know:
	(18a)	cam	klah		ʔɑt	klah
		remember klah		NEG.
							) :
	(17) -context: S 0 gave some honey to S 1 , and one week later, S 0 asks S 1 :
	S 0 :	tɨk-		kɲom ʔaoy tɨw	ʔɑh	haəy	nɨw
		kmʊm
		honey	1SG	give	go	finish already yet
	"The honey I gave you, did you finish it or is there some left?"

  The combination between klah with ʔey and naa can be summed up as follows:

	ʔey: (29a)
		( x i …. x k ….. x n ...) set
		S 0	(S 1 ?)
	ʔey klah:	(29b)
		( x i …. x k ….. x n ...) set  X ( x i … x n ...) subset
		ʔey (S 0 )	klah (S 1 ?)
	naa:	(29c)
		x set
		list	S 0	(S 1 ?)
	naa klah: (29d)
	x

i …. x k ….. x n  ( x (i) …. x (k) ….. x (n) ...) i …. x k ….. x n  ( x (i) …. x (k) ….. x (n) ...) set  X ( x i … x n ...) subset list

naa (S 0 ) klah (S 1 ?)

On reduplication in keeping with plurality, see[START_REF] Paillard | Réduplication du nom et de l'adjectif en khmer[END_REF]

About naa and ʔey see[START_REF] Thach | L'indéfinition en khmer : du groupe nominal au discours[END_REF].

In all examples of this 29's series, reduplication of "girl" in second sequence of the sentence is required.


IMJ Benjamin Girard
email: bgirard@math.jussieu.fr

ON A COMBINATORIAL PROBLEM OF ERD ŐS, KLEITMAN AND LEMKE by

Keywords: Mathematics Subject Classification (2010): 05E15, 11B75, 11A25, 20D60, 20K01

In this paper, we study a combinatorial problem originating in the following conjecture of Erdős and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.

Introduction

Let G be a finite Abelian group, written additively. If G is cyclic of order n, it will be denoted by C n . In the general case, we can decompose G as a direct product of cyclic groups C n 1 ⊕ • • • ⊕ C nr such that 1 < n 1 | • • • | n r ∈ N, where r and n r are respectively called the rank and exponent of G. Usually, the exponent of G is simply denoted by exp(G). The order of an element g of G will be written ord(g) and for every divisor d of exp(G), we denote by G d the subgroup of G consisting of all elements of order dividing d:

G d = {x ∈ G | dx = 0} .
In this paper, any finite sequence S = (g 1 , . . . , g ℓ ) of ℓ elements of G will be called a sequence over G of length |S| = ℓ. We will also denote by σ(S) the sum of all elements contained in S, which will be referred to as a zero-sum sequence whenever σ(S) = 0. Given a sequence S over G, we denote by S d the subsequence of S consisting of all elements of order d contained in S, and by k(S) the cross number of S = (g 1 , . . . , g ℓ ), which is defined as follows:

k(S) = ℓ i=1 1 ord(g i )
.

By t(G) we denote the smallest integer t ∈ N * such that every sequence S over G of length |S| ≥ t contains a non-empty zero-sum subsequence S ′ | S with k(S ′ ) ≤ 1. Such a subsequence will be called a tiny zero-sum subsequence.

The investigations on t(G) originate in the following conjecture, addressed by Erdős and Lemke in the late eighties (see [START_REF] Lemke | An addition theorem on the integers modulo n[END_REF]Introduction]). Is it true that out of n divisors of n, repetitions being allowed, one can always find a certain number of them that sum up to n? Motivated by this conjecture, Kleitman and Lemke [START_REF] Lemke | An addition theorem on the integers modulo n[END_REF]Theorem 1] proved the following stronger result.

Theorem 1.1 (Kleitman and Lemke [START_REF] Lemke | An addition theorem on the integers modulo n[END_REF]). -For any given integers a 1 , . . . , a n there is a non-empty subset I ⊆ 1, n such that n | i∈I a i and i∈I gcd(a i , n) ≤ n.

Meanwhile, Lagarias and Saks framed a graph-theoretic approach, called graph pebbling, thanks to which the problem of Erdős and Lemke could be reduced to the study of a combinatorial game, played with pebbles on the vertices of a simple graph. In this context, Chung [START_REF] Chung | Pebbling in hypercubes[END_REF]Theorem 6] found a new elegant proof of Theorem 1.1, and, under one extra assumption made on the prime factors of n, Denley [3, Theorem 1] could sharpen this theorem. Graph pebbling led to a rapidly growing literature as well as many open problems. The interested reader is referred to the surveys [START_REF] Hurlbert | A survey of graph pebbling[END_REF][START_REF] Hurlbert | Recent progress in graph pebbling[END_REF] which contain many references on the subject.

In addition, let us underline that, with our notation, Theorem 1.1 simply asserts that t(C n ) ≤ n, and this upper bound is easily seen to be optimal. In the general case, Kleitman and Lemke [15, Section 3] conjectured that t(G) ≤ |G| holds for every finite Abelian group G. This conjecture was proved, using tools from zero-sum theory, by Geroldinger [START_REF] Geroldinger | On a conjecture of Kleitman and Lemke[END_REF]. An alternative proof was then found by Elledge and Hurlbert [5, Theorem 2] using graph pebbling.

In this paper, we prove that in the general case of finite Abelian groups, the currently known upper bound on t(G) can be improved significantly. More precisely, our main theorem shows that, for finite Abelian groups of fixed rank, t(G) grows linearly in the exponent of G, which gives the correct order of magnitude. To do so, we prove that t(G) can be bounded above using a result of Alon and Dubiner [START_REF] Alon | Dubiner A lattice point problem and additive number theory[END_REF] on the following classical invariant in zero-sum theory.

By η(G) we denote the smallest integer t ∈ N * such that every sequence S over G of length |S| ≥ t contains a non-empty zero-sum subsequence

S ′ | S with |S ′ | ≤ exp(G). Such a subsequence is called a short zero-sum subsequence. Since k(T ) ≤ 1 implies |T | ≤ exp(G), one can notice that η(G) ≤ t(G) always holds.
Let us also mention that η(G) is one out of many invariants studied because of their arithmetical applications. In this respect, the interested reader is referred to [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups: a survey[END_REF][START_REF]Geroldinger Additive group theory and non-unique factorizations[END_REF][START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF] for comprehensive surveys on non-unique factorization theory. For instance, it is known [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF]Theorem 5.8.3] that for all integers m, n ∈ N * such that m | n, one has

(1) η(C m ⊕ C n ) = 2m + n -2.
In particular, the equality t(C n ) = η(C n ) = n holds. Moreover, and even if little is known on the exact value of η(G) for finite Abelian groups of rank r ≥ 3, its behavior is better understood than it is for t(G), and the following key result, due to Alon and Dubiner [1, Theorem 1.1], will be extensively used throughout this article.

Theorem 1.2 (Alon and Dubiner [START_REF] Alon | Dubiner A lattice point problem and additive number theory[END_REF]). -For every r ∈ N * there exists a constant c r > 0 such that for every n ∈ N * , one has η (C r n ) ≤ c r (n -1) + 1. From now on, we will identify c r with its smallest possible value in Theorem 1.2. On the one hand, a natural construction shows that c r ≥ (2 r -1). Indeed, if (e 1 , . . . , e r ) is a basis of C r n with ord(e i ) = n for all i ∈ 1, r , it is easily checked that the sequence S consisting of n -1 copies of i∈I e i for each non-empty subset I ⊆ 1, n contains no short zero-sum subsequence, so that

(2) (2 r -1)(n -1) + 1 ≤ η(C r n ) ≤ t(C r n ). In particular, one has t(C r 2 ) = η(C r 2 ) = 2 r .
For the time being, the exact value of t(G) is known for cyclic groups and elementary 2-groups only.

On the other hand, the method used in [START_REF] Alon | Dubiner A lattice point problem and additive number theory[END_REF] yields c r ≤ (cr ln r) r , where c > 0 is an absolute constant. Also, it readily follows from (1) that it is possible to choose c 1 = 1 and c 2 = 3, with equality in Theorem 1.2, and it is conjectured in [START_REF] Alon | Dubiner A lattice point problem and additive number theory[END_REF] that there actually is an absolute constant d > 0 such that c r ≤ d r for all r ≥ 1. For a complete account on η(G), see [START_REF] Edel | Zero-sum problems in finite abelian groups and affine caps[END_REF][START_REF] Gao | Thangadurai On short zero-sum subsequences II[END_REF] and the references contained therein.

New results and plan of the paper

Let P = {p 1 = 2 < p 2 = 3 < • • • }
be the set of prime numbers. Given a positive integer n, let D n be the set of its positive divisors. By P -(n) and P + (n), we denote the smallest and greatest prime elements of D n respectively, with the convention P -(1) = P + (1) = 1. Finally, the p-adic valuation of n will be denoted by ν p (n).

In this paper, we prove that in the general case of finite Abelian groups, the currently known upper bound on t(G) can be improved significantly.

Our starting point will be to prove it first in the case of finite Abelian p-groups. For this purpose, a classical variant of t(G), introduced by Geroldinger and Schneider in [START_REF] Geroldinger | The cross number of finite abelian groups II[END_REF], will be studied in Section 3. Even though a single by-product of this investigation (see Corollary 3.6) will effectively be used in subsequent sections, we include this study in full, since it may be of interest in view of arithmetical applications.

Then, in Section 4, we prove the main theorem of this paper. This theorem provides a new upper bound for t(G), which depends on the rank and exponent of G only. It is proved thanks to an appropriate partition of the divisor lattice of exp(G). In addition, an interesting special case is the one of finite Abelian groups of rank two, where this theorem can be applied specifying c 2 = 3.

Theorem 2.1. -Let G be a finite Abelian group of rank r and exponent n. Then

t(G) ≤ c r d|n d P + (d) ν P + (d) (d) -1 + c r (n -1) + 1.
In Section 5, we then obtain, as a corollary of Theorem 2.1, the following Alon and Dubiner type upper bound for t(G). In particular, it is easily deduced from (2) that this upper bound has the right order of magnitude.

Theorem 2.2. -For every r ∈ N * there exists a constant d r > 0 such that, for every finite Abelian group G of rank r and exponent n, one has

t(G) ≤ d r (n -1) + 1.
The qualitative upper bound of Theorem 2.2 is voluntarily given in a form allowing to stress the connection between d r and c r . In this regard, a key argument in our proof of Theorem 2.2 actually comes from a simple, albeit somewhat surprising, property of the following arithmetic function

f (n) = d|n d P + (d)
.

Indeed, one can show that f (n) ≤ n always holds (see Proposition 5.2). Consequently, and even if no particular effort has been made to optimize the constant in Theorem 2.2, it turns out that d r can be chosen to satisfy 1 ≤ d r /c r ≤ 2, thus being at most twice as large as the best possible constant we could hope for.

Finally, in Section 6, we propose and discuss two open problems on t(G).

On a variant of t(G)

Let G be a finite Abelian group, and

d ′ , d ∈ N * be two integers such that d ′ | d | exp(G)
. This section is devoted to the following variant of t(G), which was first introduced by Geroldinger and Schneider [11, Section 3] (see also [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF]Section 5.7]):

ρ(G) = max{k(S) | S contains no tiny zero-sum subsequence}.
Using a compression argument, we will obtain a new upper bound for ρ(G), which has the correct order of magnitude and applies to any finite Abelian group G. For this purpose, we consider the following invariant, which was introduced in [START_REF] Girard | A new upper bound for the cross number of finite Abelian groups[END_REF].

G υ(d ′ ,d) ⊆ G such that η (d ′ ,d) (G) = η(G υ(d ′ ,d) ).
In order to define this particular subgroup G υ(d ′ ,d) properly, we introduce the following notation. Given the decomposition of G as a product of cyclic groups

G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | • • • | n r ∈ N, we set, for all i ∈ 1, r , A i = gcd(d ′ , n i ), B i = lcm(d, n i ) lcm(d ′ , n i ) , υ i (d ′ , d) = A i gcd(A i , B i ) .
Therefore, whenever d divides n i , we have

υ i (d ′ , d) = gcd(d ′ , n i ) = d ′ , and in particular υ r (d ′ , d) = d ′ . We can now state our result on η (d ′ ,d) (G). Proposition 3.1. -Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | • • • | n r ∈ N, be a finite Abelian group and d ′ , d ∈ N * be such that d ′ | d | exp(G). Then η (d ′ ,d) (G) = η C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d) .
Using Theorem 1.2, our Proposition 3.1 yields the following useful estimate on the numbers η (d ′ ,d) (G). Proposition 3.2. -For every r ∈ N * there exists a constant c r > 0 such that for every finite Abelian group G of rank r, and every

d ′ | d | exp(G), one has η (d ′ ,d) (G) ≤ c r (d ′ -1) + 1.
Proof. -Let us consider the group

G υ(d ′ ,d) = C υ 1 (d ′ ,d) ⊕ • • • ⊕ C υr(d ′ ,d) .
On the one hand, Proposition 3.

1 states that η (d ′ ,d) (G) = η(G υ(d ′ ,d) ). On the other hand, it is easily seen that υ i (d ′ , d) | d ′ for all i ∈ 1, r and υ r (d ′ , d) = d ′ , which implies that G υ(d ′ ,d) is a subgroup of C r d ′ of exponent d ′ . Now, since η(H) ≤ η(G) holds for all groups H ⊆ G such that exp(H) = exp(G), Theorem 1.2 gives η (d ′ ,d) (G) = η(G υ(d ′ ,d) ) ≤ η(C r d ′ ) ≤ c r (d ′ -1) + 1.
Now, we can prove the main theorem of this section.

P G = {x ∈ N m | f d (x) ≥ 0, d ∈ D n }, where f d (x) = min d ′ ∈D d \{1} η (d ′ ,d) (G) -1 -x d .
Proof. -Let S be a sequence over G containing no tiny zero-sum subsequence, satisfying k(S) = ρ(G) and being of minimal length regarding this property. Suppose also that the m-tuple x = (x d 1 , . . . , x dm ), where x d = |S d | for all d, is not an element of the polytope P G .

Then, there exists d 0 ∈ D n such that f d 0 (x) < 0, which means there is a

d ′ 0 ∈ D d 0 \{1} satisfying x d 0 ≥ η (d ′ 0 ,d 0 ) (G)
. So, the sequence S contains X elements g 1 , . . . , g X of order d 0 , with 1 < X ≤ d ′ 0 , the sum σ of which is an element of order d0 dividing d 0 /d ′ 0 . Let T be the sequence obtained from S by replacing these X elements by their sum. Reciprocally, for every subsequence T ′ of T containing σ, let us denote by ϕ(T ′ ) the subsequence of S obtained from T ′ by replacing σ by g 1 , . . . , g X . In particular, note that

k(ϕ(T ′ )) = k(T ′ ) - 1 ord(σ) + X i=1 1 ord(g i ) = k(T ′ ) + X d 0 - 1 d0 ≤ k(T ′ ) + X d 0 - d ′ 0 d 0 ≤ k(T ′ ).
First, T contains no tiny zero-sum subsequence. Indeed, if T ′ is a non-empty zerosum subsequence of T , then either T ′ contains σ so that ϕ(T ′ ) is a non-empty zero-sum subsequence of S, which yields

k(T ′ ) ≥ k(ϕ(T ′ )) > 1,
or T ′ does not contain σ, which implies that T ′ is a subsequence of S and k(T ′ ) > 1.

Second, it follows from the equality ϕ(T ) = S that k(T ) ≥ k(S) = ρ(G). Therefore, T is a sequence over G containing no tiny zero-sum subsequence such that k(T ) = ρ(G) and

|T | = |S| -X + 1 < |S|, a contradiction.
Keeping in mind the notation used in Theorem 3.3, we now obtain the following immediate corollary, giving a general upper bound for ρ(G), expressed as the solution of an integer linear program.

ρ(G) ≤ max x∈P G d|n x d d .
It is now possible, using Proposition 3.2, to deduce the following quantitative result from Theorem 3.3. Proof. -Let G ≃ C r p α , where α, r ∈ N * and p ∈ P. Using Theorem 3.5, one has

ρ(G) ≤ c r α i=1 p -1 p i = c r p α -1 p α .
Then, for every sequence S over G containing no tiny zero-sum subsequence, one has

|S| p α ≤ k(S) ≤ ρ(G) ≤ c r p α -1 p α ,
which gives |S| ≤ c r (p α -1) and completes the proof.

As already mentioned, an interesting special case is the one of finite Abelian groups of rank two. In this case, specifying c 2 = 3 in Corollary 3.6 implies that, for all primes p and α ∈ N * , the equality

(3) t (C p α ⊕ C p α ) = 3p α -2 = η (C p α ⊕ C p α )
holds. Building on the case where G has prime power exponent, we can now turn to the general case of finite Abelian groups.

Proof of the main theorem

Let G be a finite Abelian group of exponent n = q α 1 1 • • • q α ℓ ℓ , with q 1 < • • • < q ℓ , and let S be a sequence over G. We consider the following partition:

D n \{1} = ℓ i=1 A i , where A i = {d ∈ D n \{1} : P + (d) = q i }.
In particular, for all d ∈ A i , one has d ≤ ∆ i = q α 1 1 • • • q α i i . Now, for every d ∈ A i , we denote by k d the smallest integer such that

|S d | < k d d q νq i (d) i + η d q νq i (d) i ,d (G).
The number k d has the following combinatorial interpretation.

S ′ 1 , . . . , S ′ k d such that, for all j ∈ 1, k d , σ(S ′ j ) ∈ G q νq i (d) i and k(S ′ j ) ≤ 1
q νq i (d) i
.

Proof. -First, we shall prove by induction on k ∈ 0, k d that S d contains at least k disjoint non-empty subsequences S ′ 1 , . . . , S ′ k such that, for all j ∈ 1, k ,

σ(S ′ j ) ∈ G q νq i (d) i and |S ′ j | ≤ d q νq i (d) i
.

If k = 0 then this statement is clearly true. Now, assume that the statement holds for some k ∈ 0, k d -1 , and let us prove that it holds for k +1 as well. By the induction hypothesis, we already know that S d contains k disjoint non-empty subsequences S ′ 1 , . . . , S ′ k such that, for all j ∈ 1, k ,

σ(S ′ j ) ∈ G q νq i (d) i and |S ′ j | ≤ d q νq i (d) i
.

Moreover, the sequence T d obtained from S d by deleting all elements of S ′ 1 , . . . , S ′ k satisfies

|T d | = |S d | - k j=1 |S ′ j | ≥ (k d -1) d q νq i (d) i + η d q νq i (d) i ,d (G) -k d q νq i (d) i ≥ η d q νq i (d) i ,d (G), so that S d contains a non-empty subsequence S ′ k+1 disjoint from S ′ 1 , . . . , S ′ k such that σ(S ′ k+1 ) ∈ G q νq i (d) i and |S ′ k+1 | ≤ d q νq i (d) i , which completes the induction. Therefore, S d contains k d disjoint non-empty subsequences S ′ 1 , . . . , S ′ k d such that, for all j ∈ 1, k d , σ(S ′ j ) ∈ G q νq i (d) i and |S ′ j | ≤ d q νq i (d) i .
In addition, for all j ∈ 1, k d , one clearly has

k(S ′ j ) = |S ′ j | d ≤ 1
q νq i (d) i
,

and
σ(S ′ j ) ∈ G q νq i (n) i and k(S ′ j ) ≤ 1 ord σ S ′ j . Proof. -By Lemma 4.1, S d contains at least k d disjoint non-empty subsequences S ′ 1 , . . . , S ′ k d such that, for all j ∈ 1, k d , σ(S ′ j ) ∈ G q νq i (d) i and k(S ′ j ) ≤ 1 q νq i (d) i
.

Thus, the desired result directly follows from the fact that, for all j ∈ 1, k d , one has

σ(S ′ j ) ∈ G q νq i (d) i ⊆ G q νq i (n) i . Lemma 4.3.
-Let G be a finite Abelian group of exponent n, and let S be a sequence over G containing no tiny zero-sum subsequence. Then, for all i ∈ 1, ℓ ,

d∈A i k d ≤ t G q νq i (n) i -1. Proof. -Let us set t = t G q νq i (n) i
, and assume that

d∈A i k d ≥ t.
Then, it follows from Corollary 4.2 that S contains t disjoint non-empty subsequences S ′ 1 , . . . , S ′ t such that, for all j ∈ 1, t ,

σ(S ′ j ) ∈ G q νq i (n) i and k(S ′ j ) ≤ 1 ord σ S ′ j . Now, let us consider the sequence T over G q νq i (n) i defined by T = t j=1 σ(S ′ j ).
Since T is a sequence of length |T | = t, then it contains a tiny zero-sum subsequence T ′ | T . In other words, there exists a non-empty subset J ⊆ 1, t such that

T ′ = j∈J σ(S ′ j ).

Now, let us set

S ′ = j∈J S ′ j .
Since σ(S ′ ) = σ(T ′ ) = 0, then S ′ is a non-empty zero-sum subsequence of S. In addition, we have the following chain of inequalities:

k(S ′ ) = j∈J k(S ′ j ) ≤ j∈J 1 ord σ S ′ j = k(T ′ ) ≤ 1.
Therefore, S contains a tiny zero-sum subsequence, and the proof is complete.

We can now prove the main theorem of this paper.

Proof of Theorem 2.1. -Let S be a sequence over G containing no tiny zero-sum subsequence. For every i ∈ 1, ℓ , Corollary 3.6 and Lemma 4.3 yield

d∈A i k d ≤ t G q νq i (n) i -1 ≤ c r q νq i (n) i -1 , so that, setting ∆ 0 = 1, one obtains d∈A i   |S d | -   η d q νq i (d) i ,d (G) -1     ≤ d∈A i k d d q νq i (d) i ≤ ∆ i-1 d∈A i k d ≤ c r ∆ i-1 q νq i (n) i -1 = c r (∆ i -∆ i-1
) . Now, using Proposition 3.2, we have

|S| = ℓ i=1 d∈A i |S d | = ℓ i=1 d∈A i   η d q νq i (d) i ,d (G) -1 
 + ℓ i=1 d∈A i   |S d | -   η d q νq i (d) i ,d (G) -1     ≤ c r ℓ i=1 d∈A i d q νq i (d) i -1 + c r ℓ i=1 (∆ i -∆ i-1 ) = c r d|n d P + (d) ν P + (d) (d) -1 + c r (n -1) ,
which completes the proof of the theorem.

A special sum of divisors

In this section, we derive Theorem 2.2 from the study of a particular arithmetic function. We recall P = {p 1 = 2 < p 2 = 3 < • • • } denotes the set of prime numbers, and start by proving the following easy lemma.

Lemma 5.1. -For every integer ℓ ≥ 1, one has

ℓ i=1 1 + 1 p i -1 ≤ p ℓ+1 -1.
Proof. -To start with, let us prove the following statement by induction on ℓ ≥ 1.

ℓ i=2 1 + 1 2(i -1) ≤ ℓ.
One can readily notice that this statement is true for ℓ = 1 and ℓ = 2. Assume now that the statement holds for some ℓ ≥ 2. Then, let us show that it holds for ℓ + 1 also. Indeed,

ℓ+1 i=2 1 + 1 2(i -1) = ℓ i=2 1 + 1 2(i -1) 1 + 1 2ℓ ≤ ℓ 1 + 1 2ℓ ≤ ℓ + 1,
and we are done. The desired result now follows from the following chain of inequalities, using the trivial bound p ℓ ≥ 2ℓ -1, for all ℓ ≥ 2, in the following fashion.

ℓ i=1 1 + 1 p i -1 = 2 ℓ i=2 1 + 1 p i -1 ≤ 2 ℓ i=2 1 + 1 2(i -1) ≤ 2ℓ = (2(ℓ + 1) -1) -1 ≤ p ℓ+1 -1.
We can now prove the main result of this section. . For every integer n ≥ 1, one has f (n) ≤ n.

Proof. -Writing n = q α 1 1 • • • q α ℓ ℓ , where q 1 < • • • < q ℓ , we prove the desired result by induction on ℓ = ω(n) ≥ 1.

If ℓ = 1, then one has

f (q α 1 1 ) = d|q α 1 1 d P + (d) = 1 + α 1 i=1 q i-1 1 = 1 + q α 1 1 -1 q 1 -1 ,
so that we now have f (q α 1 1 )

q α 1 1 = 1 q α 1 1 + 1 q 1 -1 q α 1 1 -1 q α 1 1 ≤ 1 q α 1 1 + q α 1 1 -1 q α 1 1 = 1,
and we are done.

Assume now that the statement holds true for some ℓ ≥ 1. Then, setting σ(n) = d|n d, we obtain the following equalities.

f (q α 1 1 • • • q α ℓ+1 ℓ+1 ) = d|q α 1 1 •••q α ℓ+1 ℓ+1 d P + (d) = d|q α 1 1 •••q α ℓ ℓ d P + (d) + α ℓ+1 i=1 d|q α 1 1 •••q α ℓ ℓ dq i ℓ+1 q ℓ+1 = f (q α 1 1 • • • q α ℓ ℓ ) + σ(q α 1 1 • • • q α ℓ ℓ ) α ℓ+1 i=1 q i-1 ℓ+1 = f (q α 1 1 • • • q α ℓ ℓ ) + σ(q α 1 1 • • • q α ℓ ℓ ) q α ℓ+1 ℓ+1 -1 q ℓ+1 -1 so that f (q α 1 1 • • • q α ℓ+1 ℓ+1 ) q α 1 1 • • • q α ℓ+1 ℓ+1 = f (q α 1 1 • • • q α ℓ ℓ ) q α 1 1 • • • q α ℓ ℓ 1 q α ℓ+1 ℓ+1 + σ(q α 1 1 • • • q α ℓ ℓ ) q α 1 1 • • • q α ℓ ℓ 1 (q ℓ+1 -1) q α ℓ+1 ℓ+1 -1 q α ℓ+1 ℓ+1 .
First, by the induction hypothesis, we have

f (q α 1 1 • • • q α ℓ ℓ ) q α 1 1 • • • q α ℓ ℓ ≤ 1.
Second, since σ(n) is multiplicative, Lemma 5.1 yields

σ(q α 1 1 • • • q α ℓ ℓ ) q α 1 1 • • • q α ℓ ℓ = σ(q α 1 1 ) q α 1 1 • • • σ(q α ℓ ℓ ) q α ℓ ℓ ≤ ℓ i=1 1 + 1 q i -1 ≤ ℓ i=1 1 + 1 p i -1 ≤ p ℓ+1 -1 ≤ q ℓ+1 -1.
Thus, we obtain

f (q α 1 1 • • • q α ℓ+1 ℓ+1 ) q α 1 1 • • • q α ℓ+1 ℓ+1 ≤ 1 q α ℓ+1 ℓ+1 + q α ℓ+1 ℓ+1 -1 q α ℓ+1 ℓ+1 = 1,
which completes the proof.

As an immediate corollary of Proposition 5.

d P + (d) ν P + (d) (d) -1 + c r (n -1) + 1 ≤ c r d|n d P + (d) -1 + c r (n -1) + 1 ≤ 2c r (n -1) + 1.

A few concluding remarks

As previously stated, the exact value of t(G) is currently known for cyclic groups and elementary 2-groups only. In this context, the special case of finite Abelian groups of rank two is of particular interest, and the following conjecture appears to be inviting.

Conjecture 1. -For all integers m, n ∈ N * such that m | n, one has t(C m ⊕ C n ) = 2m + n -2.
If true, the statement of Conjecture 1 would nicely extend the theorem of Kleitman and Lemke. In view of equality (3), this conjecture readily holds true for all groups G ≃ C p α ⊕ C p α , where p ∈ P and α ∈ N * . Moreover, note that t(C m ⊕ C n ) ≥ 2m + n -2 easily follows from (1).

Even though far less is known on the exact value of η(G) for finite Abelian groups of higher rank, it would be worth knowing how close it actually is to t(G) in the general case. A first step in this direction is the following lemma, which gives a simple property shared by all finite Abelian groups for which t(G) = η(G) holds. Lemma 6.1. -Let G be a finite Abelian group such that t(G) = η(G). Then for every subgroup H of G, one has η(H) ≤ η(G).

Proof. -Let G be as in the statement of the lemma, and let H be a subgroup of G. By definition, the inequality t(H) ≤ t(G) holds. Therefore, η(H) ≤ t(H) ≤ t(G) = η(G), and the required result is proved. However, it turns out that the above property is restrictive enough to guarantee that for every r ≥ 4, there exists a finite Abelian group G of rank r for which t(G) > η(G). The proof of this fact actually relies on the following key invariant in zero-sum combinatorics. Given a finite Abelian group G, let D(G) denote the smallest integer t ∈ N * such that every sequence S over G of length |S| ≥ t contains a non-empty zero-sum subsequence. The number D(G) is called the Davenport constant of G, and we refer to [START_REF]Geroldinger Additive group theory and non-unique factorizations[END_REF][START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF] for background and connections with algebraic number theory.

In what follows, we will need a classical theorem, independently proved in the late sixties by Kruyswijk [START_REF] Van Emde | A combinatorial problem on finite abelian groups II[END_REF] and Olson [START_REF] Olson | A combinatorial problem on finite abelian groups I[END_REF], stating that Therefore, H is a subgroup of G such that η(H) > η(G), and the desired result follows from Lemma 6.1.

It would certainly be interesting to know whether the equality t(G) = η(G) holds for all finite Abelian groups of rank three. In another direction, we would also like to address the following conjecture. It can readily be seen that Conjecture 2 holds whenever G is an elementary p-group, since all non-zero elements of G have same order in this case. In addition, our Corollary 3.6 already gives a "nearly optimal" answer when G is of the form C r n , with n a prime power.

  By η (d ′ ,d) (G) we denote the smallest integer t ∈ N * such that every sequence S over G d of length |S| ≥ t contains a non-empty subsequence S ′ | S of length |S ′ | ≤ d ′ and with sum in G d/d ′ . The numbers η (d ′ ,d) (G) and η(G) are closely related to each other. First, note that the two definitions coincide when d ′ = d = exp(G). In addition, and as shown in [12, Proposition 3.1], there exists a subgroup

Theorem 3 . 3 .

 33 -Let G be a finite Abelian group of exponent n and D n = {d 1 , . . . , d m }. Let also S be a sequence over G containing no tiny zero-sum subsequence, reaching the maximum k(S) = ρ(G), and being of minimal length regarding this property. Then, the m-tuple x = (x d 1 , . . . , x dm ), where x d = |S d | for all d, is an element of the polytope

Corollary 3 . 4 .

 34 -For every finite Abelian group G of exponent n, one has

Theorem 3 . 5 .

 35 -Let G be a finite Abelian group of rank r and exponent n. Then ρ(G) ≤ c r d|n P -(d) -1 d . Proof. -Using Theorem 3.3 and Proposition 3.2, we indeed obtain ρ(G) ≤ d|n η (P -(d),d) (G) -1 d ≤ c r d|n P -(d) -1 d . Theorem 3.5 actually improves on the best known upper bound ρ(G) ≤ |G|/P -(n) proved by Geroldinger and Schneider [11, Lemma 2.1]. In addition, a simple study of the arithmetic function involved in our result (see [12, Lemma 5.1]) shows there exists a constant δ r > 0 such that, for every finite Abelian group G of rank r and exponent n, one has ρ(G) ≤ δ r ω(n), where ω(n) denotes the number of distinct prime divisors of n. On the other hand, Theorem 3.5 provides us with the following useful result on t(G) in the case of finite Abelian p-groups. Corollary 3.6. -Let p ∈ P. Then, for all α, r ∈ N * , one has t(C r p α ) ≤ c r (p α -1) + 1.

Lemma 4 . 1 .

 41 -Let G be a finite Abelian group of exponent n, and d ∈ A i . Let also S be a sequence over G. Then S d contains at least k d disjoint non-empty subsequences

Proposition 5 . 2 .

 52 -Let f (n) = d|n d P + (d)

( 4 ) 3 ⊕

 43 D(C p α 1 ⊕ • • • ⊕ C p αr ) = r i=1(p α i -1) + 1 for all primes p and positive integers α 1 , . . . , α r . Our result now is the following. Proposition 6.2. -For every integer r ≥ 4, there exists a finite Abelian group G of rank r for which t(G) > η(G).Proof. -Let r ≥ 4 be an integer. It is an easy exercise to prove there is an integer α ≥ 2 such that ln(2r -1) ln 3 ≤ α ≤ ln(2 r -r) ln 3 .Now, let us considerG = C r-1 C 3 α and H = C r 3 . Since G is a finite Abelian 3-group, it follows from (4) that D(G) = 2(r -1) + 3 α ≤ 2 exp(G) -1, so that [17, Theorem 1.2] yields η(G) ≤ D(G) + exp(G) -1 = 2r + 2.3 α -3.On the other hand, we deduce from (2) that η(H) ≥ (2 r -1)(3 -1) + 1 = 2 r+1 -1.

Conjecture 2 .

 2 -For all integers r, n ∈ N * , one has t(C r n ) = η(C r n ).

  2, we now prove Theorem 2.2. Proof of Theorem 2.2. -Let G be a finite Abelian group of rank r and exponent n. Then, by Theorem 2.1 and Proposition 5.2, one has

	t(G) ≤ c r
	d|n

Acknowledgments

This work was started at IPAM in Los Angeles. Thus, I would like to warmly thank its staff, as well as the organizers of the program Combinatorics: Methods and Applications in Mathematics and Computer Science, held in Fall 2009, for all their hospitality and for providing an excellent atmosphere for research. I would also like to thank W. Schmid and the referees for helpful comments on an earlier version of this paper.


Damien Roque
email: damien.roque@gipsa-lab.grenoble-inp.fr

Cyrille Siclet
email: cyrille.siclet@gipsa-lab.grenoble-inp.fr

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

A Low-Complexity Multicarrier Scheme with LDPC Coding for Mobile-to-Mobile Environment

In this paper, we consider a mobile ad-hoc radio network in an urban area. The propagation environment between two endpoints can be modeled by a double-Rayleigh fading multipath channel. Such a mobile scenario justifies the use of filter bank based multicarrier (FBMC) transmission systems. This technique generalizes traditional cyclic prefix orthogonal frequency-division multiplexing (CP-OFDM), allowing the design of non-rectangular pulse shape filters. We show that this approach leads to a better interference mitigation in time-variant channels. We restrict our study to short filters and singletap per sub-channel equalization in order to preserve a lowcomplexity transmultiplexer. In this study, we compare FBMC with short filters to CP-OFDM in terms of coded bit-error-rate performances, using a realistic mobile-to-mobile channel model.

I. INTRODUCTION

The propagation constraints experienced by land tactical communication systems are usually characterized by the decentralized structure of the network and the uncertainty of the hostile environment. In practice, it results in a multipath mobile-to-mobile transmission scenario, with low rise antennas and limited a priori knowledge of the channel [START_REF] Bullington | Radio propagation for vehicular communications[END_REF]. These assumptions reveal the need of accurate channel models that override traditional cellular models [START_REF] Molisch | A survey on vehicle-to-vehicle propagation channels[END_REF].

Cyclic-prefix orthogonal frequency-division multiplexing (CP-OFDM) is an attractive modulation scheme for multipath time-invariant channels since it removes inter-block interference (IBI). It relies on a low-complexity implementation using fast Fourier transform (FFT) and it allows perfect reconstruction of the transmitted signal thanks to a single-tap per subchannel equalizer [START_REF] Wang | Wireless multicarrier communications[END_REF]. However, the Doppler spread introduced by time-variant channels breaks the orthogonality between sub-channels [4, p. 753], causing inter-carrier interference (ICI). Despite this performance degradation, CP-OFDM has been adopted for several wireless mobile applications (e.g. IEEE 802.16e, 3GPP LTE Advanced).

Filter bank based multicarrier modulations (FBMC) represent a more general transmission scheme, also known as filtered multitone (FMT) or oversampled OFDM [START_REF] Kozek | Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels[END_REF], [START_REF] Cherubini | Filtered multitone modulation for very high-speed digital subscriber lines[END_REF]. This technique allows the design of various pulse shape filters, as compliant as possible with the multipath time-variant channel [START_REF] Strohmer | Optimal OFDM design for time-frequency dispersive channels[END_REF]. For example, a frequency-localized waveform may mitigate the ICI term described above. Nevertheless, this shaping is performed at the expense of a greater complexity, increasing with the filters length [START_REF] Moret | Design of orthogonal filtered multitone modulation systems and comparison among efficient realizations[END_REF].

In this work, we focus on short prototype filters, introduced in [START_REF] Pinchon | Closed-form expressions of optimal short PR FMT prototype filters[END_REF]. This particular class of pulses preserves a low-complexity transmultiplexer, comparable with CP-OFDM. Furthermore, a single-tap per sub-channel equalizer may be sufficient to yield superior uncoded bit-error-rate (BER) performances in several mobile scenarios [START_REF] Roque | A performance comparison of FBMC modulation schemes with short perfect reconstruction filters[END_REF].

Our contributions include the development of a mobileto-mobile channel model suitable for multipath propagation in urban areas. We also discuss BER performances, through simulation results, using a low-density parity check (LDPC) code.

This paper is organized as follows. Section 2 introduces the assumptions of the mobile-to-mobile propagation in an urban environment and derives a discrete-time equivalent channel model. Section 3 presents the general transmuliplexer structure and specifies pulse shaping filters, resulting in CP-OFDM and non-rectangular orthogonal transmission schemes. Section 4 compares the BER performances of the different transmission systems, including low-density parity check (LDPC) channel coding. Finally, conclusions are presented in section 5.

II. MOBILE-TO-MOBILE CHANNEL MODEL

Reflections, diffraction and scattering are three physical phenomena that justify a multipath propagation beyond the line of sight (LOS) in very/ultra high frequency (V/UHF) bands. In an urban environment this non-line of sight communication (NLOS) scenario is likely to happen between two mobile transceivers.

Let s(t) be the baseband equivalent transmitted signal that occupies a band B. We denote H the multipath timevarying channel with I uncorrelated resolvable paths (they are mostly due to reflections in the propagation environment). The noise process z(t) is a complex bandlimited Gaussian process characterized by its power spectral density σ 2 z /B for |f | ≤ B/2 and 0 otherwise. The received signal is given by

r(t) = (Hs)(t) + z(t) (1)
= I i=1 α i (t)s(t -τ i ) + z(t) (2)
where α i (t) is the time-varying complex gain of the ith path with delay τ i . Note that we choose static delays for the reproducibility of simulations. They are usually standardized through experimental results (e.g. COST 207, 3GPP...).

A. Double-Rayleigh model

In a NLOS urban environment, the transmitter and the receiver are surrounded by N 1 and N 2 scatterers that can be bounded by two rings (fig. 1a). The diameter of the rings is expected to be sufficiently small to neglect the excess delay occurring in the vicinity of the transmitter and the receiver. Assuming omni-directional antennas and planar propagation, the complex coefficients associated to the ith path expands as

α i (t) = N1 n1=1 ρ (1) i,n1 (t)e j ω (1) i,n 1 (t)+φ (1) i,n 1 (t) • N2 n2=1 ρ (2) i,n2 (t)e j ω (2) i,n 2 (t)+φ (2) i,n 2 (t) (3)
where

• ρ (1)
i,n1 (t) and ρ

(2) i,n2 (t) are the real valued gains of the n 1 th and n 2 th scatterer close to the transmitter and the receiver, respectively.

• ω

(1) i,n1 (t) and ω

(2) i,n2 (t) are the Doppler phase terms introduced by the motion of the transmitter and the receiver. One may develop ω

n1 (t) = 2πf 1 t cos β (1)
n1 where f 1 is the maximum Doppler frequency introduced by the transmitter, β

i,n1 is the angle of departure of the n 1 th ray with respect to the velocity vector v 1 . A similar expansion holds for ω

(2) i,n2 (t). • φ (1)
i,n1 (t) and φ

(2) i,n2 (t) are phase terms depending on the carrier frequency f 0 , intra-ring delays associated with each path, interactions between the waves and matter. Without a perfect knowledge of the terrain, the signals stated above are random processes. We model φ [START_REF] Bullington | Radio propagation for vehicular communications[END_REF] i,n1 (t) and φ

(2) i,n2 (t) as independent processes, uniformly distributed over [0, 2π). This hypothesis remains realistic in a dense scattering environment (N 1 , N 2 → ∞) and if the phase shifts occuring inside the rings are much larger than one [START_REF] Clarke | A statistical theory of mobile-radio reception[END_REF]. Since amplitudes ρ (1) i,n1 (t) and ρ

(2) i,n2 (t) are independent, the central limit theorem allows us to write

α i (t) = α (1) i (t)α (2) i (t) (4)
where α

(1)

i (t) and α (2)
i (t) are two independent centered Gaussian processes, characterized by their respective standard deviation σ i (t)|, has been justified in [START_REF] Kovacs | Investigations of outdoor-to-indoor mobile-to-mobile radio communication channels[END_REF]. It can be shown from [START_REF] Proakis | Digital communications[END_REF], that the probability density function of

|α i (t)| is f (|α i (t)|) = |α i (t)| σ (1) αi 2 σ (2) αi 2 K 0 |α i (t)| σ (1) αi σ (2) αi (5)
where K 0 (•) is the zeroth order modified Bessel function of the second kind. The phase arg{α i (t)} remains a uniform process over the interval [0; 2π), independent of the modulus [START_REF] Patel | Simulation of rayleigh-faded mobile-to-mobile communication channels[END_REF].

A particular case has been documented earlier in [START_REF] Akki | A statistical model of mobile-to-mobile land communication channel[END_REF] with the hypothesis ρ

(1) i,n1 (t) = ρ (2)
i,n2 (t) = 1 and n 1 = n 2 . Such a scenario relies on the approximation of a single incoming ray per Rx scatterer, leading to single Rayleigh fading model (fig. 1b).

• Tx v 1 • Rx v 2 × × × × × Scatterers (a) Full scattering configuration. • Tx v 1 • Rx v 2 × × × × × × Scatterers (b) Limited scattering configuration.
Fig. 1: Double scattering in a mobile-to-mobile transmission scenario (a small subset of scatterers is displayed).

B. Double-Jakes power spectrum

We first consider a displacement of the receiver over a (small) distance ξ along the axis x (fig. 2). Let φ

i,n2 (t) be the initial phase at point x 1 of the path induced by the n 2 th scatterer. Assuming a constant speed v 2 and denoting ∆t = ξ/v 2 , the receiver reaches x 2 at time t + ∆t, and the phase term can be written

φ (2) i,n2 (t + ∆t) = φ (2) i,n2 (t) + 2πf 2 ∆t cos β (2) i,n2 . (6)
At the transmitter side, the same displacement scenario yields a similar result in terms of phase offset with respect to the maximum Doppler frequency f 1 and the angle of departure β

i,n1 .

x

× n 2 th scatterer • x 0 • x 1 v 2 ξ β (2) i,n 2 (t) β (2) ′ i,n 2 (t)
Fig. 2: Motion of the receiver over a distance ξ.

Wide sense stationarity is usually expected in order to establish a suitable statistical model for randomly time-variant

This assumption allows a time-independent formulation of α i (t) correlation function, so that

Γ i (∆t) = E α (1) i (t)α (1) i (t + ∆t) • E α (2) i (t)α (2) i (t + ∆t) (7)
where E{•} denotes the expectation operator and • is the complex conjugate operator. Confined scattering areas and small displacement ξ justify invariant amplitudes over a time ∆t, namely

ρ (1) i,n1 (t + ∆t) ≈ ρ (1) i,n1 (t) (8) ρ (2) i,n2 (t + ∆t) ≈ ρ (2) i,n2 (t). (9)
If we recall the phase terms independence, we obtain

Γ i (∆t) = 4σ (1) αi σ (2) αi 2 N 1 N 2 N1 n1=1 E e j2πf1∆t cos β (1) i,n 1 • N2 n2=1 E e j2πf2∆t cos β (2) i,n 2 . ( 10
)
As explained in [START_REF] Patel | Statistical properties of amplify and forward relay fading channels[END_REF], the large number of scatterers and the use of isotropic antennas allow us to write

Γ i (∆t) = 4σ (1) αi σ (2) αi 2 4π 2 π -π e j2πf1∆t cos β (1)
i,n 1 dβ

(1) i,n1 • π -π e j2πf2∆t cos β (2) i,n 2 dβ (2) i,n2 (11)
= 4σ (1) αi σ (2) αi

2 J 0 (2πf 1 ∆t)J 0 (2πf 2 ∆t) (12)
where J 0 (•) is the zeroth order Bessel function of the first kind. The Fourier transform of Γ i (∆t) with respect to ∆t is referred to as the Doppler spectrum and describes the statistical distribution of motion-induced frequency shift. As explained in [START_REF] Akki | A statistical model of mobile-to-mobile land communication channel[END_REF], this function is written as

S(f ) = 4σ (1) αi σ (2) αi 2 π 2 √ f 1 f 2 K (f 1 + f 2 ) 2 -f 2 4f 1 f 2 (13)
where K(•) is the complete elliptic integral of the first kind. The resulting power spectrum presents a "U" shape with infinite values at |f | = f 1f 2 (fig. 3).

To summarize, the mobile-to-mobile non-LOS model is more stringent than the traditional base-to-mobile model in terms of fading characteristics because α i (t) is the product of two centered Gaussian processes. As for the Doppler shift, the mobile-to-mobile scenario introduces more degrees of freedom than base-to-mobile scenario. For example, if two mobile stations move at the same speed, a null relative velocity vector may exist (i.e. no Doppler shift).

C. Discrete-time equivalent model

For simulation purpose, it is necessary to develop a discretetime version of the channel model denoted H. Since s(t) and z(t) are bandlimited signals, the Whittaker-Shannon interpolation formula gives and the discrete-time equivalent input-output relation becomes

s(t) = k∈Z s[k]sinc(Bt -k) (14)
z(t) = k∈Z z[k]sinc(Bt -k) (15)
r[k] = Hs [k] + z[k] (16)
= l∈Z αl [k]s[k -l] + z[k] (17)
where

αl [k] = I i=1 α i k B sinc(Bτ i -l). (18)
If we consider a number I of uncorrelated α i (t) coefficients, the overlying discrete-equivalent model provides an infinite number of αl [k]. Furthermore, the 1/B-spaced channel taps are no longer uncorrelated and we have

E{α l [k]α l [k ]} = Ω l,l [k -k ] (19)
where, from [START_REF] Strohmer | Optimal OFDM design for time-frequency dispersive channels[END_REF], the correlation coefficient expands as

Ω l,l [k -k ] = I i=1 sinc(Bτ i -l)sinc(Bτ i -l )Γ i k -k B .
(20) In practice, the sequence

{α l [k]} l∈Z is truncated to L taps whenever {|α l [k]| 2 } l>L ≈ 0 [16].

III. FBMC TRANSMISSION SYSTEM WITH SHORT FILTERS

Faced with a time-frequency selective channel, several studies have proven the interest of FBMC transmission systems in order to mitigate ISI and ICI [START_REF] Kozek | Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels[END_REF], [START_REF] Strohmer | Optimal OFDM design for time-frequency dispersive channels[END_REF], [START_REF] Moret | Design of orthogonal filtered multitone modulation systems and comparison among efficient realizations[END_REF]. The design of the transmultiplexer is usually performed at the expense of an increased complexity brought by non-rectangular pulse shape filters (often overlapping between symbols) and the overlying matched equalizer.

As a trade-off between interference reduction and complexity, our approach focuses on short prototype filters (i.e. non-overlapping pulses) with a single-tap per sub-channel equalizer. A previous study showed attractive results in terms

IFFT c 0,n c M -1,n CP γ[0] γ[N -1] P/S H s[k] z[k] S/P r[k] γ[0] γ[N -1] CP -1 FFT ĉ0,n ĉM-1,n
Fig. 4: Efficient implementation of a FBMC transmultiplexer with short filters. of uncoded BER performances, in the case of base-to-mobile scenario [START_REF] Roque | A performance comparison of FBMC modulation schemes with short perfect reconstruction filters[END_REF].

A. General transmultiplexer structure

Information is carried by a sequence of complex symbols denoted {c m,n } (m,n)∈Λ where Λ = {0, . . . , M -1} × Z. The symbols are assumed to be independent and identically distributed. Their mean power is denoted σ 2 c = E{|c m,n | 2 }. Each c m,n is placed in the time-frequency plane at coordinates (m/M, nN ) where M is the number of sub-channels and N represents the number of sample per sub-channel symbol period. The oversampling ratio is given by N/M . The transmitted signal can be written

s[k] = (m,n)∈Λ c m,n γ m,n [k], k ∈ Z (21)
where γ m,n [k] is a time-frequency shifted version of the prototype pulse γ[k] defined as

γ m,n [k] = 1 √ M γ[k -nN ]e j2π m M k , γ[k] ∈ R. ( 22
)
A necessary condition for perfect reconstruction of the transmitted symbols is the linear independence of the vectors {γ m,n } (m,n)∈Λ . This requires N/M ≥ 1, but orthogonality and completeness are not mandatory [START_REF] Kozek | Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels[END_REF]. Let x and y be two sequences of 2 (Z), we define the inner product x, y = k∈Z x[k]y[k], its induced norm x = x, x and the infinity norm x ∞ = max{|x[k]|} k∈Z . At the receiver side, the dual family of {γ m,n } (m,n)∈Λ is written {γ m,n } (m,n)∈Λ and we have

γm,n [k] = 1 √ M γ[k -nN ]e j2π m M k , γ[k] ∈ R. ( 23
)
The biorthogonality condition ensures perfect reconstruction of the transmitted symbols, namely γp,q , γ m,n = δ m,p δ n,q for all (m, n) and (p, q) in Λ; δ m,p represents the Kronecker delta. We let Λ p,q = Λ\{(p, q)}, then the qth received symbol on the pth sub-channel can be written

ĉp,q = γp,q , r (24)
= γp,q , Hs + γp,q , z (25)

= c p,q A p,q Sp,q + (m,n)∈Λp,q c m,n A m,n,p,q Ip,q +Z p,q (26)
where

A m,n,p,q = γp,q , Hγ m,n (27)
= 1 √ M k∈Z γ[k -qN ]e j2π (m-p) M k • l∈Z αl [k]γ[k -l -nN ]e -j2π m M l (28)
A p,q = γp,q , Hγ p,q = A p,q,p,q (29) and

Z p,q = γp,q , z = 1 √ M k∈Z z[k]γ[k -qN ]e -j2π p M k . ( 30
)
The expression of the received symbol splits into three terms: S p,q represents the useful part of the signal, I p,q denotes the interference term, including inter-carrier and inter-symbol interference, Z p,q refers to the noise filtered by the receiver. The coefficient A m,n,p,q is the complex gain introduced by the system, assuming an estimated symbol (p, q) and an interfering symbol (m, n).

Assuming perfect knowledge of the channel coefficients, interference mitigation is performed thanks to a low-complexity equalizer, with a single-tap per sub-channel. This type of equalizer may amplify noise and interference terms when the channel experiences severe fading.

B. Filters specification : short length prototypes

This study focuses on short length prototype filters, that is to say γ[k] = γ[k] = 0, if k > N -1 and k < 0. This particular FBMC setup allows a N -size block processing. It leads to an efficient implementation depicted by figure 4 anddocumented in [START_REF] Moret | Design of orthogonal filtered multitone modulation systems and comparison among efficient realizations[END_REF]. The transmission process consists in a projection of M complex symbols over an exponential basis (IFFT block). This step is followed by a cyclic extension of the data block (CP block). The resulting N samples are weighted by γ[k], 0 ≤ k ≤ N -1. That is why this scheme is referred to as weighted cyclic prefix (WCP)-OFDM in the general case. At the receiver side, the dual operation is performed in order to ensure the perfect reconstruction (PR) of the transmitted symbols, that is to say ĉm,n = c m,n for any

(m, n) ∈ Λ assuming r[k] = s[k].
In the short filters case, if we set ∆ = N -M , PR conditions reduce to

• γ[k]γ[k] + γ[k + M ]γ[k + M ] = 1 for 0 ≤ k ≤ ∆ -1, • γ[k]γ[k] = 1 for ∆ ≤ k ≤ N -1.
First of all, let us define the rectangular filters leading to the traditional CP-OFDM scheme:

γ CP [k] = 1, if 0 ≤ k ≤ N -1, 0, otherwise, (31)
γCP [k] = 1, if ∆ ≤ k ≤ N -1, 0, otherwise. (32)
The first ∆ coefficients of γ CP form the cyclic prefix in order to circumvent channel-induced inter-block interference (IBI). At the receiver side, the cyclic prefix is removed thanks to the ∆ first null coefficients of γCP . Since γ CP m,n , γCP p,q = δ m,p δ n,q and γ CP = γCP , this modulation scheme constitutes a biorthogonal system. We notice that, for

∆ ≥ 1, γ CP m,n 2 > 1 whereas γCP m,n 2 = 1.
In presence of noise, such an unbalanced amplification scenario results in a sub-optimal transmission setup.

Apart from the CP-OFDM approach, there exists nonrectangular filters that ensure perfect reconstruction. We restrict our analysis to the linear phase orthogonal case, such that γ

[k] = γ[k] = γ[N -1 -k]. Thus, PR conditions become • γ 2 [k] + γ 2 [k + M ] = 1 for 0 ≤ k ≤ (∆ -1)/2, • γ 2 [k] = 1 for ∆ ≤ k ≤ (N -1)/2.
Short orthogonal prototype filters have been proposed in [START_REF] Pinchon | Closed-form expressions of optimal short PR FMT prototype filters[END_REF], assuming N/∆ = M/∆ + 1 and odd values of M/∆. One of them is optimized with respect to time-frequency localization (TFL) criterion (fig. 5). A closed-form expression of this prototype is given as a good approximation of the optimized impulse response for 2 ≤ M/∆ ≤ 20:

γ TFL [k] =            sin (2k+1)π 4∆ , if 0 ≤ k ≤ ∆ -1, 1, if ∆ ≤ k ≤ M -1, sin (2(∆-k)+1)π 4∆ , if M ≤ k ≤ N -1, 0, otherwise. (33)
Compared to rectangular pulses used in CP-OFDM, a previous study confirms the interest of WCP-OFDM with TFL filters for mobile channels [START_REF] Roque | A performance comparison of FBMC modulation schemes with short perfect reconstruction filters[END_REF]. The orthogonal setup also offers an optimal approach over additive white Gaussian noise channels [4, p. 160].

IV. BER PERFORMANCE ANALYSIS

In this simulation framework, we consider a land mobileto-mobile transmission system, operating in an urban area. A band B = 5 MHz is used, centered around a frequency f c = 1 GHz. We use a 10-path WSSUS model where the last path occurs at 12 µs (JTCOutUrbLRLAC [START_REF] Pahlavan | Wireless information networks[END_REF]) and whose discrete-equivalent impulse response is truncated to L = 45 taps. This empirical model fits an outdoor propagation with low-rise buildings and low-antennas. We develop here two mobility scenarios:

• low speed, with v 1 = v 2 = 3 km/h ; • high speed, with v 1 = 140 km/h and v 2 = 110 km/h. We consider M = 2048 sub-channels that carry quaternary symbols and we set N/M ∈ {17/16, 9/8, 5/4}. The LDPC encoder uses 32400 bit codewords and operates at a coding rate R c = 3/4. Each codeword is randomly interleaved and split into 8 blocks. Ideally, the transmission of each block should be spaced by a period greater than the coherence duration of the channel. In a similar way, sub-channels should by spaced by a band greater than the coherence bandwidth. Due to the time-frequency duality, it remains impossible to guarantee both conditions. In addition, applications constraints may restrict even more the channel coding scheme (e.g. interactivity, medium access control...). In this work, we choose the best diversity scenario where blocks are sent over independent channel realizations.

At the receiver side, a single-tap per sub-channel equalizer is used. After the deinterleaver, a LDPC decoder performs 10 iterations over each codeword without parity check 1 . We compare the performances of CP-OFDM with TFL-OFDM (i.e. WCP-OFDM with TFL pulses). Bit-error-rate (BER) is plotted as a function of E b /N 0 , where E b = N σ 2 c /2M B and N 0 = σ 2 z /B (fig. 6). First of all, we notice that specified oversampling ratios lead to ∆ > L. In the particular case of CP-OFDM, this condition is sufficient to cancel channel-induced IBI. However, for a general WCP-OFDM transmultiplexer, IBI may remain present.

In the low mobility scenario, we show that TFL-OFDM obtains better performances results than CP-OFDM regardless of the oversampling ratio. This emphasizes the need of orthogonal signaling in mainly noisy channels. Indeed, since the TFL impulse response shows smooth transitions, it justifies a low IBI term in presence of multipath propagation. We also see that CP-OFDM performances decrease as N/M increases. It summarizes the fact that the energy used to form the cyclic prefix is not recovered by the receiver.

In the high mobility scenario, the same behavior is observed for CP-OFDM. However, for high E b /N 0 values, the performances of TFL-OFDM increase with N/M . One may show that the frequency localization of the TFL pulse increases with N/M . As a consequence, such a good frequency containment leads to ICI mitigation and provides a lower BER.

V. CONCLUSION

We have defined a multipath mobile-to-mobile channel model that fits a non-LOS propagation in UHF band. We then proposed a low-complexity multicarrier scheme using short length pulse shape filters and a single-tap per sub-channel equalizer. We have shown, through simulation results, that an orthogonal transmission setup, provided with optimized TFL filters, outperforms CP-OFDM in a realistic transmission scenario. These results extend the work presented in [START_REF] Roque | A performance comparison of FBMC modulation schemes with short perfect reconstruction filters[END_REF], introducing a double-Rayleigh fading model and a LDPC channel coding scheme.

  αi . This model, referred to as double-Rayleigh fading according to the distribution of |α

Fig. 3 :

 3 Fig. 3: Mobile-to-mobile Doppler power spectrum comparison for f 1 /f 2 ∈ {1, 2}.

2 Fig. 5 :

 25 Fig. 5: Time and frequency responses of the prototype filters with Γ(ν) = 1/ γ ∞ N -1 k=0 γ[k] exp(-j2πνk/N ), using M = 1024 and N = 1280.

  BER performance in low mobility scenario.

  BER performance in high mobility scenario.

Fig. 6 :

 6 Fig. 6: BER performances for two mobility scenarios in a mobile-to-mobile channel. Comparison between CP-OFDM and TFL-OFDM for M = 2048 and several values of N/M .

Such a channel coding configuration (Rc = 3/4, 10 iterations) is chosen in order to achieve a statistically representative simulation with a reasonable computing time.

ACKNOWLEDGMENT

This work has been supported by the French defense procurement agency (DGA) and Orange-Labs. The authors would like to thank D. Pinchon and P. Siohan who designed the filters discussed in this article.