Mélodie Andrieu

Léo Vivion

Minimal Complexities for Infinite Words Written with d Letters

published or not. The documents may come   L'archive ouverte pluridisciplinaire

General motivation

Sturmian words (1940) form a class of infinite words over the binary alphabet, which sheds light on the remarkable interactions between combinatorics, dynamical systems, and number theory. These interactions are reflected in the various ways to define them (see the historic paper [START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF], or refer to the book [START_REF] Lothaire | Combinatorics on Words[END_REF] for a modern introduction). For instance, Sturmian words are equivalently

• words with subword complexity n + 1, i.e., admitting exactly n + 1 factors of length n for all n (a factor of w of length n is a subword of w consisting of n consecutive letters);

• binary aperiodic words with imbalance equal to 1: all factors of a given length contain, up to one, the same number of occurrences of 1s (and thus, up to one as well, the same number of 2s);

• the symbolic trajectories of a ball in a square billiard, launched with a momentum with rationally independent components.

They give rise to several generalizations over the d-letter alphabet for d ≥ 3, depending on the considered definition: Arnoux-Rauzy ([AR91]) and episturmian words ([JP02], [START_REF] Glen | Episturmian words: a survey[END_REF]), other words associated with d-dimensional continued fraction algorithms (see [START_REF] Schweiger | Multidimensional Continued Fractions[END_REF], [START_REF] Berthé | Multidimensional Euclidean algorithms, numeration and substitutions[END_REF], [START_REF] Berthé | Beyond substitutive dynamical systems: S-adic expansions[END_REF]), interval exchange transformations ( [START_REF] Viana | Ergodic theory of interval exchange maps[END_REF], [START_REF] Yoccoz | Interval exchange maps and translation surfaces[END_REF]), polygonal ( [START_REF] Tabachnikov | Geometry and Billiards[END_REF], [START_REF] Cassaigne | Complexity and growth for polygonal billiards[END_REF] and references therein) or cubic billiard words ( [START_REF] Arnoux | Complexity of sequences defined by billiard in the cube[END_REF]). A large program, initiated by Rauzy in the 80s ([Rau83, Rau84, Rau85]), is to determine which properties are still equivalent in higher dimensions, and which are not.

The present manuscript is part of this program. It mostly focuses on one of the dynamical representations of Sturmian words: as words generated by a billiard on a square table, which generalizes itself to a billiard in the cube, and in the cube of dimension d; and on two combinatorial quantities which characterize Sturmian words: the subword complexity and the abelian complexity. The manuscript is directed by the question: are the subword and abelian complexities of cubic billiard words in dimension d minimal among the complexities of d-ary words, as is the case when d = 2?

The manuscript is organized as follows. Hypercubic billiard words are defined in Section 2. Section 3 is devoted to the subword complexity. Section 4 is devoted to the abelian complexity.

Hypercubic billiard words

We start by defining cubic billiard words in dimension d, which we generically call hypercubic billiard words.

Let d ≥ 1. The set R d is equipped with the usual inner product and canonical basis (e 1 , ..., e d ). We are interested in the sequence of the faces successively hit by a billiard ball, initially located in x ∈ [0, 1] d , which is given an initial momentum θ = (θ 1 , ..., θ d ) ∈ (R\{0}) d . We will use the letter k to code a hit against one of the two faces orthogonal to the vector e k . The sequence that we thus obtain, denoted by w(x, θ), is an infinite word over the alphabet {1, ..., d}, that we call a cubic billiard word in dimension d. Remark. This definition omits to specify what happens when the trajectory of the ball intersects several faces simultaneously (for instance, for d = 2, when the ball hits a corner of the square). It has no importance: first, once the momentum θ is set up, the set of all starting positions x ∈ [0, 1] d for which such an event occurs has Lebesgue measure 0; secondly, for these rare "bad" starting positions, there still exists a reasonable coding strategy (not detailed here) that endows the words obtained with the same properties as regular hypercubic billiard words. In particular, all the results stated in this manuscript (noticeably Theorems 2, 3, 8, 9, and Corollary 1) are true for all hypercubic billiard words, including those generated by "bad" starting positions.

In the sequel, we focus on the billiard words generated by momenta θ with rationally independent components. The reason for this choice is that square billiard words generated by momenta with rationally independent components are exactly Sturmian words. Note that the rational independence of the momentum's components is equivalent to the minimality of one/all trajectories: balls launched with such momenta pass arbitrarily close to every point in the hypercube.

Subword complexity of hypercubic billiard words

Most of the literature defines Sturmian words as right-infinite words with subword complexity p : n → n + 1. We recall that the subword complexity of w (often simply called complexity) is the function that counts the number of factors of length n in w. This function is intricately linked with the notion of entropy in dynamics, which measures how well a trajectory can be predicted from the knowledge of a small portion of it. A fundamental fact is that n → n + 1 is the smallest complexity function for a "non-trivial" word.

Theorem 1 (Morse and Hedlund, 1938 [MH38]). Let w be an infinite word. If there exists an integer m such that p(m) ≤ m, then the word w is ultimately periodic, and its subword complexity p is ultimately constant.

The particular equality p(1) = 2 tells us that Sturmian words are written with exactly 2 letters. A natural question, when it comes to generalizing Sturmian words, is the following one.

Question 1. Let d ≥ 3. What is the subword complexity of cubic billiard words in dimension d? Is this function the smallest subword complexity for non-ultimately periodic words written with exactly d letters?

Subword complexity of hypercubic billiard words

The first part of the question has been fully answered by Arnoux, Mauduit, Shiokawa, Tamura, Baryshnikov, and Bédaride between 1994 and 2009. First, when d = 3, we have the following expression, which was originally conjectured by Rauzy in 1983 ([Rau83]).

Theorem 2 (Arnoux, Mauduit, Shiokawa, and Tamura, 1994 [START_REF] Arnoux | Complexity of sequences defined by billiard in the cube[END_REF] [AMST94b]; corrected by Bédaride [START_REF] Bédaride | Billiard complexity in rational polyhedra[END_REF]). Let x ∈ [0, 1] 3 , and θ ∈ R 3 with rationally independent components. If the inverses θ -1 1 , θ -1 2 and θ -1 3 are also rationally independent, then the number of distinct factors of length n in the cubic billiard word w(x, θ) is

n 2 + n + 1.
This formula extends itself to arbitrary dimensions in the following way.

Theorem 3 (Bédaride [START_REF] Bédaride | Directional complexity of the hypercubic billiard[END_REF]). Let d ≥ 1, x ∈ [0, 1] d , and θ ∈ R d with rationally independent components. If d ≥ 3, assume, moreover, that for any three indices i, j, k, the numbers θ -1 i , θ -1 j and θ -1 k are rationally independent (♣). Then the number of distinct factors of length n in the billiard word w(x, θ) is

min(n,d-1) k=0 k! n k d -1 k .
Borel [START_REF] Borel | Complexity of degenerated three dimensional billiard words[END_REF] and Bédaride [START_REF] Bédaride | Classification of rotations on the torus T 2[END_REF] showed that the condition (♣) is necessary.

This general formula was first conjectured by Arnoux, Mauduit, Shiokawa, and Tamura in [START_REF] Arnoux | Complexity of sequences defined by billiard in the cube[END_REF], and proved one year later, in 1995, under a strictly stronger -and thereby non optimal -condition by Baryshnikov [Bar95].

Minimal subword complexity for d-ary words

Regarding the second part of Question 1 ("is this function the minimal subword complexity for non-ultimately periodic words written with exactly d letters?"), one quickly convinces oneself that the question is poorly stated. Indeed, the uninteresting ternary word w = 3•w , where w is any Sturmian word on the alphabet {1, 2}, is non-ultimately periodic, and has subword complexity n → n + 2. It is not difficult to prove that this complexity is the lowest for a non-ultimately periodic ternary word. Note that there exist other families of words with subword complexity n → n + 2. They are classified in [START_REF] Alessandri | Classification et représentation des mots de complexité n + 2. Rapport technique[END_REF] (see also the reference [START_REF] Kaboré | Combinatoire de mots récurrents de complexité n + 2[END_REF], which is easier to find.)

In our opinion, an essential property of Sturmian words is that their letter frequencies are rationally independent (this is a manifestation of the action of the continued fraction). We recall that the frequency of a letter a in a right-infinite word w is the limit, if it exists, of the proportion of as in growing prefixes of w. By contrast to Sturmian words, the frequencies of letters of ultimately periodic words are rational, and thereby, as soon as there are at least two letters, rationally dependent. Thus, Sturmian words are exactly words with minimal complexity function among binary words whose letter frequencieswhich exist by a classical argument of [START_REF] Boshernitzan | A unique ergodicity of minimal symbolic flows with linear block growth[END_REF] -are rationally independent. From this perspective, our question can be better formulated.

Question 2. Let d ≥ 3. What is the smallest subword complexity for a d-ary word with rationally independent frequencies of letters?

This question was answered by Tijdeman in a mostly forgotten article from 1999.

Theorem 4 (Tijdeman, 1999, [START_REF] Tijdeman | On the minimal complexity of infinite words[END_REF]). Let d ≥ 1.

(1) Let w be a d-ary word that admits frequencies of letters. If there exists an integer m such that p(m) ≤ (d -1)m, where p denotes the subword complexity of w, then the frequencies of letters in w are rationally dependent.

(2) There exist d-ary words, with rationally independent letter frequencies, whose subword complexity is p : n → (d -1)n + 1.

In other words, the smallest subword complexity of d-ary words, with rationally independent letter frequencies, is p : n → (d -1)n + 1.

To our knowledge, there is no classification of the words, with rationally independent letter frequencies, which realize this minimal complexity. However, some families of such words are known, as illustrated below.

On the first hand, one finds the words associated with the interval exchange transformations defined by a partition of [0, 1) into d sub-intervals, with rationally independent lengths, and by the cyclic permutation (2, 3, ..., d, 1) (this is the example exhibited in [START_REF] Tijdeman | On the minimal complexity of infinite words[END_REF]). To learn more about interval exchange transformations, we refer to the lectures [START_REF] Viana | Ergodic theory of interval exchange maps[END_REF], [START_REF] Yoccoz | Interval exchange maps and translation surfaces[END_REF].

On the other hand, there are the words associated with certain multidimensional continued fraction algorithms. Noticeably, for d = 3, one finds the primitive C-adic words ([CLL17]), and the Arnoux-Rauzy words, for which the rational independence of the letter frequencies was only proven two years ago (see [START_REF] Andrieu | A Rauzy fractal unbounded in all directions of the plane[END_REF] or [START_REF] Dynnikov | Dynamical systems around the Rauzy gasket and their ergodic properties[END_REF]). More generally, for d ≥ 2, one finds the strict episturmian words over the d-letter alphabet.

Theorem 5 (Andrieu, in preparation, but a proof can be found in [START_REF] Andrieu | Exceptional trajectories in the symbolic dynamics of multidimensional continued fraction algorithms[END_REF]). Let d ≥ 2. The frequencies of letters of any strict episturmian word over the d-letter alphabet are rationally independent.

Note that the fact that strict episturmian words have subword complexity n → (d -1)n + 1 stems from their definition. To learn more about episturmian words (starting with their definition), we refer to the survey [START_REF] Glen | Episturmian words: a survey[END_REF].

Coming back to billiard words and to Question 1, one observes that: -the frequencies of letters of a hypercubic billiard word are rationally independent if, and only if, the components of its momentum θ are rationally independent. Indeed, these two vectors are equal up to a dilatation, and a change in the signs of some components.

-BUT its subword complexity function is far from being minimal when d ≥ 3. Indeed, one checks that the complexity of a cubic billiard word in dimension d, under the assumption (♣), is polynomial with degree d 1.

A long-standing conjecture

We conclude this section with a long-standing conjecture, which aims at generalizing Theorem 1 to bi-dimensional words (instead of d-ary words, as we did). Conjecture 1 (Nivat, 1997, [START_REF] Nivat | Invited talk at ICALP[END_REF]). Let A be an alphabet, and w an infinite bi-dimensional word over A, i.e., an element of A Z 2 . If there exist n, m ∈ N such that the number of distinct n × m patterns is at most nm, then the word w is periodic in some direction.

Here, "periodic in some direction" means that there exists t ∈ Z 2 such that for every z ∈ Z 2 , the letters at positions z and z +t coincide. A lot of efforts have been, and continue to be, given towards Conjecture 1, which remains unsolved at the time of writing (

[ST02], [EKM03], [QZ04], [CK15], [CK16], [KS20], [KM19]).

Abelian complexity of hypercubic billiard words

In this section, we consider a variation in the definition of complexity.

Two finite words u and v are abelian equivalent if u is the anagram of v (e.g., twelveplusone and elevenplustwo). The abelian complexity of an infinite word w is the function that counts the number of non-abelian equivalent factors of length n, for all n. For instance, for n ≥ 1, the ultimately constant word w = 21222222... admits exactly two non abelianequivalent factors of length n: the one which contains 1, and the one written with the letter 2 only. For an introduction to abelian complexity, or, more generally, to abelian combinatorics on words, we refer to the recent survey [START_REF] Fici | Abelian combinatorics on words: a survey[END_REF].

It turns out that Sturmian words are also characterized by a remarkable abelian complexity.

Theorem 6 (derived from Coven and Hedlund, 1973, [CH73]). A word is Sturmian if and only if its abelian complexity is constant, equal to 2, for all lengths n ≥ 1, and if its frequencies of letters (which exist) are rationally independent.

Interestingly, this is the smallest abelian complexity for non-periodic words. Indeed, it is not difficult to see that if, in an infinite word w, all factors of length n are abelian equivalent, then w is periodic with period n.

Thus, a natural question, first asked by Rauzy in 1983, is the following: does there exist a generalization of Sturmian words (among which are Arnoux-Rauzy and episturmian words, words associated with other multidimensional continued fractions, hypercubic billiard words, etc.), which somewhat preserves such a nice abelian complexity.

Question 3 (Rauzy, 1983, [Rau83] (Section 6), free translation from French). Does there exist a class of infinite words over the alphabet {1, 2, 3} whose abelian complexity is constant, equal to 3? It seems that no, except perhaps for very particular frequencies of letters. What is, then, the "good" generalization of the Coven and Hedlund's theorem?

Proving the conjecture of Rauzy

Question 3 splits into a conjecture, and a question. We start by examining the conjecture.

Conjecture 2. Under a condition A (to be determined), there exists no ternary word w with abelian complexity equal to 3 for all n ≥ 1.

With no condition A, Conjecture 2 is false. Indeed, one readily checks that the ternary words:

w 1 = 12333333333... and

w 2 = 1 • w 2 ,
where w 2 is any Sturmian word over the alphabet {2, 3}, admit exactly 3 non-abelian equivalent factors of length n, for all n ≥ 1. One may ask the word w to be uniformly recurrent (a word w is uniformly recurrent if each factor appears infinitely often, with bounded gaps). The motivation for choosing this condition is that Sturmian words are uniformly recurrent. Furthermore, with this additional condition, the trivial counterexamples w 1 and w 2 are eliminated. However, even with the condition A: "w must be uniformly recurrent", Conjecture 2 remains false [START_REF] Richomme | Abelian complexity of minimal subshifts[END_REF], and the counterexamples exhibited by Richomme, Saari, and Zamboni are no good generalizations of Sturmian words.

For the reason stated in Section 3, on the strength of Tijdeman's theorem (Theorem 4), and following our reformulation of Coven and Hedlund's theorem (Theorem 6), we think that the condition A should be: "the frequencies of letters are rationally independent". Under this condition A, Conjecture 2 turns out to be true.

Theorem 7 (Andrieu, Vivion, in preparation). Let w ∈ {1, 2, 3} N be a word with rationally independent letter frequencies. Then there exists an integer n ≥ 1 such that w admits at least 4 non-abelian equivalent factors of length n.

It remains to answer the second part of Question 3: what is, then, the good generalization of the Morse and Hedlund's theorem? We could, for instance, look for a class of ternary words with rationally independent letter frequencies and abelian complexity constant, equal to 4. Of course, this would only be possible for n ≥ 2.

Abelian complexity of hypercubic billiard words

Abelian complexity has been extensively studied since 2007 ([KT07]), in particular after the progress of Richomme, Saari, and Zamboni towards Question 3 ([RSZ11]), see for instance [START_REF] Saarela | Ultimately constant abelian complexity of infinite words[END_REF], [START_REF] Richomme | Balance and abelian complexity of the Tribonacci word[END_REF], [START_REF] Currie | Recurrent words with constant abelian complexity[END_REF], [START_REF] Balkovà | Abelian complexity of infinite words associated with quadratic Parry numbers[END_REF], [START_REF] Madill | The abelian complexity of the paperfolding word[END_REF], [START_REF] Turek | Abelian complexity and abelian co-decomposition[END_REF], [START_REF] Blanchet-Sadri | On the asymptotic abelian complexity of morphic words[END_REF], [START_REF] Blanchet-Sadri | Abelian complexity of fixed point of morphism 0 → 012,1 → 02,2 → 1[END_REF], [START_REF] Turek | Abelian complexity function of the Tribonacci word[END_REF], [START_REF] Blanchet-Sadri | Computing abelian complexity of binary uniform morphic words[END_REF], [START_REF] Lü | On the abelian complexity of the Rudin-Shapiro sequence[END_REF], [START_REF] Chen | On the abelian complexity of generalized Thue-Morse sequences[END_REF], [START_REF] Kaye | The abelian complexity of infinite words and the Frobenius problem[END_REF], [START_REF] Whiteland | Asymptotic abelian complexities of certain morphic binary words[END_REF], [START_REF] Shallit | Abelian complexity and synchronization[END_REF], etc. Experience showed that, even for friendly words (for instance, the Tribonacci word), the abelian complexity can be a surprisingly tricky function, and one generally obtains bounds instead of exact values.

Question 4. What can be said of the abelian complexity of hypercubic billiard words? Can we explicitly compute this function for some easy instances?

The answer is positive, beyond hope: not only is it possible to compute the abelian complexity of all hypercubic billiard words generated by momenta with rationally independent components, but this function, remarkably simple, turns out to be the same for all of them (which is, remember the condition (♣), noticeably not the case for the subword complexity, whenever d ≥ 3). In other words, the abelian complexity of cubic billiard words in dimension d is ultimately constant, equal to 2 d-1 . For d = 3, cubic billiard words have an abelian complexity constant, equal to 4, for all n ≥ 2. This result answers Question 3: cubic billiard words generated by momenta with rationally independent components (or, equivalently, and very interestingly, those with rationally independent letter frequencies) are a "good" generalization of Sturmian words, from the point of view of abelian complexity.

We conclude this subsection with a stronger result: not only do we know the number of non-abelian equivalent factors of length n of a hypercubic billiard word, but we are in fact able to compute them easily.

Theorem 9 (Andrieu, Vivion, in preparation). Let d ≥ 1, x ∈ [0, 1] d , and θ ∈ R d with rationally independent components. Let n ∈ N. The set of equivalence classes of length n factors of w(x, θ), for the relation "being abelian equivalent", can be computed from the sole knowledge of the first n -1 letters of the billiard word w(0, θ). This contrasts sharply with the difficulty of listing the factors of a given length in a hypercubic billiard word. For instance, in dimension 3, and for the parameters x = (0, 0, 0) and θ = ( √ 2, √ 3, √ 5), we know that the billiard word w(x, θ) admits 43 factors of length 6 (it is a quick computation from Theorem 3); nonetheless, they do not all appear in the prefix of length 10000000 of w(x, θ) ([AL22]).

Another old question

We conclude this extended abstract with a long-standing question, first asked by Arnoux, Mauduit, Shiokawa, and Tamura in 1994: why is the formula of the subword complexity (given in Theorem 3) symmetric in n (length of factors) and d-1 (cardinal of the alphabet minus one)?

Question 5 (Arnoux, Mauduit, Shiokawa, Tamura, 1994, [AMST94a]). Can we give a meaningful bijective map between the factors of length n of a hypercubic billiard word over the alphabet {1, . . . , d}, and the factors of length d -1 of a particular/any hypercubic billiard word over the alphabet {1, . . . , n + 1}?

To our knowledge, this question is still open at the time of writing.

Figure 1 :

 1 Figure 1: The billiard word w(x, θ) starts with 11211121...

Figure 2 :

 2 Figure 2: Example of a bi-dimensional word over the alphabet { , }.

Theorem 8 (

 8 Andrieu, Vivion, in preparation). Let d ≥ 1, x ∈ [0, 1] d , and θ ∈ R d with rationally independent components. Let n ∈ N. The number of non-abelian equivalent factors of length n in the billiard word w(x, θ) is min(n,d-1) k=0 d -1 k . Corollary 1. Let d ≥ 1, x ∈ [0, 1] d , and θ ∈ R d with rationally independent components. Let n ≥ d -1. There are exactly 2 d-1 non-abelian equivalent factors of length n in w(x, θ).

Acknowledgements We would like to express our gratitude to Boris Solomyak and to Julien Cassaigne for numerous exciting discussions on this topic; and to Shalom Eliahou for his kind help in improving the quality of the manuscript.


Mélodie Andrieu

Léo Vivion

Vivion Léo

Imbalances

Imbalances in Hypercubic Billiard Words

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

Motivation

Sturmian words form a class of infinite words over the binary alphabet which sheds light on the remarkable interactions between combinatorics, dynamical systems, and number theory. These interactions are reflected in the various ways to define them (see the historic paper [START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF], or refer to [START_REF] Lothaire | Combinatorics on Words[END_REF] for a recent general presentation). For instance, Sturmian words are equivalently

• words with complexity n + 1, i.e., admitting exactly n + 1 factors of length n for all n (a factor of w of length n is a subword of w written with n consecutive letters);

• binary aperiodic words with an imbalance equal to 1: all factors of a given length contain, up to one, the same numbers of 1s (and thus, as well, the same numbers of 2s);

• the symbolic trajectories of a ball in a square billiard, launched with a momentum with rationally independent entries (see Remark 2 for a rigorous statement). In this work, we focus on hypercubic billiard words, which we will call cubic billiard words, regardless of the dimension. Cubic billiard words are defined in Section 2. Roughly speaking, a cubic billiard word in dimension d is a word on the d-letter alphabet denoting the sequence of the faces successively hit by a billiard ball moving in the unit cube of R d , in which two parallel faces are encoded by the same letter. This class of words (and, first of all, their complexity) was studied in [START_REF] Arnoux | Complexity of sequences defined by billiard in the cube[END_REF], [START_REF] Baryshnikov | Complexity of trajectories in rectangular billiards[END_REF], [START_REF] Bedaride | Billiard complexity in the hypercube[END_REF] and [START_REF] Bedaride | Directional complexity of the hypercubic billiard[END_REF].

Hereafter, we focus on their imbalance, which is the maximal difference of occurrences of a letter in two factors of the same length: Imb(w) = sup{|u| a -|v| a a letter and u, v factors of w with same length},

where |u| a counts the numbers of a in u. The finiteness of this combinatorial quantity is associated with a fast convergence of the observed frequencies of letters in growing factors of w towards their limit value [START_REF] Adamczewski | Balances for fixed points of primitive substitutions[END_REF].

In [START_REF] Vuillon | Balanced words[END_REF], Vuillon provides a -somewhat optimal-upper bound for the imbalance of hypercubic billiard words.

Theorem 1 (Vuillon [START_REF] Vuillon | Balanced words[END_REF]). Let d ∈ N >0 . In dimension d: -the imbalance of any cubic billiard word, whose momentum has rationally independent entries, is bounded above by d -1; -there exists a cubic billiard word whose momentum has rationally independent entries and whose imbalance is exactly d -1. This work answers the natural questions that follow: is the imbalance of such words always equal to d -1, as it is the case for d = 2? If not, which values can be taken by the imbalances of cubic billiard words, and which sets of parameters (i.e., initial position and momentum of the billiard ball) lead to these values?

Remark 1. In our study, we focus on the case of initial momenta θ = (θ 1 , ..., θ d ) with rationally independent entries, i.e., dim(span Q (θ 1 , . . . , θ d )) = d. Indeed: -the set of such momenta has full Lebesgue measure; -the trajectory of a billiard ball with initial position x and initial momentum θ is dense in the unit cube of R d if and only if θ has rationally independent entries (this follows from the minimality of the associated linear flow on R d /Z d , see for instance [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF]); -it is clear that for any d ≥ 2, there exist cubic billiard words with rational momenta whose imbalance collapses to 1 (Example 1).

Cubic billiard words in dimension d: definition and properties

We consider the trajectory of a billiard ball (formally, a point mass) in the unit cube in dimension d ≥ 1, which moves with constant speed along straight lines until it hits a face, then bounces elastically, moving backwards along the direction given by the law of reflection. Given x ∈ [0, 1] d , θ ∈ R d \ {0} and t ∈ R >0 ∪ {+∞}, we denote by w(x, θ, t) ∈ {1, . . . , d} * ∪ {1, . . . , d} N the [finite or infinite] sequence of the faces successively hit by the ball, with initial position and momentum (x, θ), during the time interval (0, t]: the k-th letter of the word w is i if and only if the k-th face hit by the ball is x i = 0 or x i = 1. In the sequel, we shorten the notation w(x, θ, +∞) to w(x, θ).

Example 1. Let d ≥ 2. Consider θ = 1, 1 d -1 , ..., 1 d -1 and x = 1 2 , d -2 d -1 , d -3 d -1 , ..., 1 d -1 , 0 .
One checks that w(x, θ) is the periodic word with period 1213...1d. An easy computation shows that the imbalance of this word is 1. For the initial position y = (1 -ε 1 , ε 2 , ..., ε d ), where ε 1 and ε 2 > ε 3 > ... > ε d are chosen positive, small enough, and rationally independent, w(y, θ) is the periodic word with period 1 d-1 234...d, which has imbalance d -1.

Observe that the trajectory of the ball is well defined for any set of parameters (x, θ, t), while its symbolic coding w(x, θ, t) may encounter some problems: 1. when the ball has its entire trajectory on a face; 2. more generally, when the ball hits an edge, i.e., the intersection of several faces. For θ ∈ R d \{0}, we define the set Init(θ) ⊂ [0, 1] d of all initial positions x such that w(x, θ) is unambiguously defined. For example, one checks that if θ has rationally independent entries, then 0 ∈ Init(θ).

Proposition 1. Let d ≥ 1 and θ ∈ R d \{0}. The set Init(θ) is measurable and has Lebesgue measure 1.

Yet, if θ has rationally independent entries, the set Init(θ) has an empty interiorindeed, its complementary set contains the past trajectory of 0, which is dense in the cube.

Denote by L(w) the language of an infinite word w, which is the set of all its factors. Proposition 2. Let d ≥ 1 and θ ∈ R d with rationally independent entries. For any x, y ∈ Init(θ), we have L(w(x, θ)) = L(w(y, θ)).

Proposition 2 does not always hold when the entries of θ have rational dependencies, as illustrated by Example 1.

Proof (ideas). Proposition 2 comes from the continuity of the coding function

Init(θ) -→ {1, ..., d} N x -→ w(x, θ)
together with the minimality of the linear flow of direction (θ 1 , ..., θ d ) on the torus R d /(2Z) d .

Since the imbalance of a word only depends on its language, it will be sufficient to consider cubic billiard words [whose momenta have rationally independent entries] with initial position x = 0. These words will be called standard cubic billiard words. For d = 2, standard cubic (i.e., square) billiard words are standard Sturmian words (as already defined in [START_REF] Morse | Symbolic dynamics II. Sturmian trajectories[END_REF])-hence the name.

Remark 2. Link between square billiard words and Sturmian words. In dimension 2, the complementary set of Init(θ) is exactly the trajectories, back to the past, of the four corners of the unit square. For a momentum θ with rationally independent entries and an initial position x ∈ [0, 1] 2 \Init(θ), exactly one corner is hit, and it is hit exactly once. We can consider two "twin" square billiard words, in which this event is encoded by the finite words 12 and 21. In this more general setting, square billiard words whose momentum has rationally independent entries are exactly Sturmian words.

Main result and consequences

Below, L(θ) denotes the language of all/any cubic billiard words with momentum θ (following Proposition 2), L n (θ) the subset of all of these factors with length n, and pref n (w) the first n letters of a word w.

Theorem 2. Let d ≥ 3 and θ = (θ 1 , . . . , θ d ) ∈ R d with rationally independent entries. With no loss of generality, assume that

|θ 1 | > |θ i | for all i = 1. Then Imb(L(θ)) = 1 + max f ∈L d-2 (θ) |f | 1 = 2 + pref d-3 (w(0, θ)) 1 .
Proof (ideas). For i ∈ {1, ..., d}, we compute how many times the faces i are hit at least/at most during a time interval of length t, from which we obtain restrictions on the form of the vectors

(|u| i -|v| i ) 1≤i≤d for u, v ∈ L(θ).
This theorem immediately covers the results of Vuillon (Theorem 1): we trivially have |pref d-3 (w(0, θ))| 1 ≤ d -3, and this inequality becomes an equality when θ 1 is chosen large enough. It also answers the questions of Section 1: Corollary 1. For d ∈ {1, ..., 4}, the imbalance of any cubic billiard word generated by a momentum with rationally independent entries is exactly d -1.

Corollary 2. Let d ≥ 5.

-The imbalance of any cubic billiard word generated by a momentum with rationally independent entries belongs to {3, ..., d -1}.

-For every k ∈ {3, ..., d -1}, there exists a cubic billiard word generated by a momentum with rationally independent entries whose imbalance is equal to k.

We conclude this work with geometric and algorithmic remarks. Denote ∆ := {(θ 1 , ..., θ d ) ∈ (R ≥0 ) d | i=1..d θ i = 1}. Due to the symmetries of our billiard table, and because the trajectory of the ball does not depend on the norm of the momentum, we assume without loss of generality that the momentum of a cubic billiard word belongs to ∆. Observe that, for all k, Theorem 2 gives an explicit system of inequalities on the entries of θ which generates cubic billiard words with an imbalance equal to k. In other words, the simplex ∆ is partitioned into polytopes, on each of which the imbalance is constant for all momenta with rationally independent entries. For instance for d = 5, the imbalance takes the value 4 in the region Z 4 := ∪ i ∩ j =i θ i > 2θ j , and the value 3 in the rest of the simplex.

Finally, observe that it is in general hard to determine the imbalance of a language: the definition/formula (1) only enables us to compute larger and larger lower bounds. In the case of cubic billiard words with rationally independent entries, Theorem 2 gives an algorithm to compute it: -from the knowledge of θ (compute the d -3 first faces hit by a ball initially in 0); -from the knowledge of a large enough portion of the word (or, equivalently, of its language). Indeed, we are able to detect when all factors of length d -2 have been visited, since we know the cardinality of this set ([Bar95]):

card L d-2 (θ) = d-2 i=0 (d -1)!(d -2)! (d -1 -i)!(d -2 -i)!i! . (2)
However, the formula holds under a little additional condition: the inverses of any three coordinates of θ have to be rationally independent [START_REF] Bedaride | Directional complexity of the hypercubic billiard[END_REF]. Furthermore, no upper bound on the computation time can be guaranteed: the portion of the word/language required to compute the imbalance can be arbitrarily long, as illustrated by Example 2.

Example 2. For d = 5, it is possible to choose a momentum θ, with rationally independent entries, such that |θ 1 | ≈ 2|θ 2 | |θ i | for all i ∈ {3, 4, 5}, that would give: -w 1 := w(0, θ) = 112112...1123... if |θ 1 | > 2|θ 2 |; -w 2 := w(0, θ) = 12112112...1123... otherwise.

Observe that w 1 and w 2 share a factor 112112...112 that can be made arbitrarily long. Yet, by Theorem 2, we have Imb(w 1 ) = 4 = 3 = Imb(w 2 ).

  They give rise to several generalizations on the d-letter alphabet for d ≥ 3, depending on the considered definition: strict episturmian words ([JP02], [GJ09]), words associated with d-dimensional continued fraction algorithms ([Sch00], [Ber11], [BD14]), polygonal ([Tab05], [CHT02] and references therein) or cubic billiard words ([AMST94]). It has been, and it is, a large program to determine which properties are still equivalent in higher dimension, and which are not. 1 September 5, 2022


M Ait Assou

G Humbert

Member, IEEE A Crunteanu

C Decroze

Synthetic Photoconductive Antenna Array for Terahertz 3D Imaging and Spectroscopy

Keywords: Terahertz imaging, terahertz spectroscopy, material characterization, terahertz time-domain system, synthetic aperture, back-projection, 3D image reconstruction. I

We propose a 3D imaging and spectroscopy approach in the terahertz (THz) domain based on the synthetic aperture (SA) imaging technique achieved using a time domain spectroscopy system (TDS). The proposed system enables the imaging and spectroscopy of an object in a stationary position in transmission configuration. We demonstrate the reconstruction of 3D amplitude images with a sub-millimeter resolution of a multimaterial object from 0.6 to 1 THz using the back-projection algorithm. The reconstructed data is further used to characterize the optical properties of the substrates forming the object by calculating the refractive index and the absorption coefficient of its constituents in the reconstruction frequency band. We demonstrate the use of the SA not only for imaging, but also for material characterization. Consequently, we are able to reconstruct 2D high-resolution spectral maps of the real refractive index at different THz frequencies of the imaged object.

INTRODUCTION

T the junction between microwaves and infrared, THz imaging techniques offer numerous opportunities in non-destructive testing in several industrial sectors like building and construction industry and electrical cables industry [START_REF] Tao | Non-Contact, Non-Destructive Testing in Various Industrial Sectors with Terahertz Technology[END_REF], [START_REF] Nüßler | Terahertz based non-destructive testing (NDT): Making the invisible visible[END_REF], quality control of pharmaceuticals [START_REF] Patil | Terahertz Spectroscopy: Encoding the Discovery, Instrumentation, and Applications toward Pharmaceutical Prospectives[END_REF], archeological studies [START_REF] Mikerov | Analysis of ancient ceramics using terahertz imaging and photogrammetry[END_REF], medical and biomedical imaging [START_REF] Supriya | Terahertz technology: A Review on dental perspective[END_REF]- [START_REF] D'arco | THz Pulsed Imaging in Biomedical Applications[END_REF], etc.

In a typical THz imaging setup, the object under test is mechanically translated in the focus plane of a THz beam either in reflection or transmission configurations [START_REF] Hu | Imaging with terahertz waves[END_REF], [START_REF] Malhotra | Terahertz Antenna Technology for Imaging and Sensing Applications[END_REF]. The THz image is assembled pixel by pixel based on the complex spectrum measured at every scanned point by the focused beam, enabling spectroscopic images of the object. This approach is nevertheless limited to portable and mechanically translatable objects [START_REF] Zhang | Introduction to THz Wave Photonics[END_REF]. In specific imaging scenarios, it is more favorable to image an object without translating it, such as large, heavy, or fragile objects. In addition to that, the lateral resolution achieved using a typical imaging setup is defined by the spot size of the focused THz beam which depends on the optical alignment of focusing lenses or mirrors.

To overcome the challenges of the classical imaging setup, the SA technique is considered in the THz domain. SA is a wellestablished imaging technique initially invented for radar systems in the 1950s [START_REF] Soumekh | Synthetic Aperture Radar Signal Processing[END_REF] and it was later on adopted at RF and microwave frequency bands for a variety of applications in the commercial and defense sectors [START_REF] Obata | The Latest Status of Our Commercial Small Synthetic Aperture Radar Satellite Constellation[END_REF], [START_REF] Sun | Spaceborne Synthetic Aperture Radar Imaging Algorithms: An overview[END_REF], [START_REF] Ding | Target recognition in synthetic aperture radar images using binary morphological operations[END_REF] like topographic imaging [START_REF] Bürgmann | Synthetic Aperture Radar Interferometry to Measure Earth's Surface Topography and Its Deformation[END_REF], air traffic control [START_REF] Reigber | Very-High-Resolution Airborne Synthetic Aperture Radar Imaging: Signal Processing and Applications[END_REF], meteorology [START_REF] Puysségur | Tropospheric phase delay in interferometric synthetic aperture radar estimated from meteorological model and multispectral imagery[END_REF] and underground resource exploration [START_REF] Dang | Land subsidence induced by underground coal mining at Quang Ninh, Vietnam: persistent scatterer interferometric synthetic aperture radar observation using Sentinel-1 data[END_REF]. Imaging using SA consists of reconstructing an image by creating a numerical focusing lens by back-projecting acquired signals measured at different spatial positions from an illuminated area under test [START_REF] Ager | An Introduction to Synthetic Aperture Radar Imaging[END_REF]. As a result, the spatial resolution of the image is no longer defined by the aperture of each antenna, but it depends on the space/time localization of the mobile antenna, which enables a high spatial resolution with small radiating elements.

The implementation of the SA technique for imaging in the THz range was demonstrated in previous reports using either electronic or optoelectronic-based systems in reflection configuration. A Vector Network Analyzer (VNA) coupled with frequency extenders was used to demonstrate 3D imaging at four different frequency bands (220-330 GHz, 325-500 GHz, 850-1100 GHz, and 1100-1500 GHz). Four different antennas were needed to generate THz wave at the four frequency bands [START_REF] Batra | Short-Range SAR Imaging From GHz to THz Waves[END_REF], [START_REF] Batra | Sub-mm Resolution 3D SAR Imaging at 1.5 THz[END_REF]. To tackle the inherently limited frequency of the VNA, the addition of frequency extenders was necessary to upconvert the low frequency signal from the VNA into the desired frequency range, adding to the bulkiness of the VNA based imaging system [START_REF] Batra | Sub-mm Resolution 3D SAR Imaging at 1.5 THz[END_REF]. The object was mounted on a mobile platform in order to create a SA while its image was reconstructed from back-projected reflected signals. SA implementation with a THz-TDS system was firstly demonstrated using a photoconductive antenna (PCA) as both emitter and detector, in a monostatic configuration, scanned over the object to create a synthetic planar array [START_REF] Henry | 3-D broadband terahertz synthetic aperture imaging[END_REF]. Reflected electric fields acquired at different positions were numerically processed to reconstruct a 3D image of the object over a large frequency band from 0.3 to 1.2 THz. This imaging system was further exploited to differentiate several types of materials of an object from the intensity variations on the image at specific resonance frequencies [START_REF] Henry | 3-D Terahertz Synthetic-Aperture Imaging and Spectroscopy[END_REF]. The interest of SA-TDS was also demonstrated with 2D imaging of metallic objects within a very large bandwidth of 2.5 THz by measuring the reflected signal from the objects in rotation. The PCA emitter and detector were positioned in a reflection configuration with an angle between the two antennas (without collimating or focusing lenses) and the objects were rotated to mimic a synthetic array [START_REF] Damyanov | High Resolution Lensless Terahertz Imaging and Ranging[END_REF], [START_REF] Mcclatchey | Time resolved synthetic aperture terahertz impulse imaging[END_REF].

It is worth stating that SA-TDS offers a large bandwidth, coherent detection and does not require the optical focalization system associated with the classical imaging approach to have high range and cross-range resolutions. However, none of the SA systems reported so far, have yet paired SA imaging with spectroscopy analysis to extract the optical parameters of the materials forming the object. In this work, we experimentally demonstrate 3D THz imaging and spectroscopy analysis based on the SA imaging approach implemented on a TDS system, which is performed in a transmission configuration for the first time. In our system, the A synthetic array is realized by moving a PCA along a scanning plane. We present numerically reconstructed 3D images of a complex planar object (made of air, metal, magnesium oxide and quartz) using the back-projection algorithm. The reconstructed complex data is further used to compute the refractive index and the absorption coefficient of the object. Our proposed approach goes beyond the utilization of the SA imaging technique in previous studies by leveraging the phase and amplitude of the reconstructed data to calculate the real refractive index and absorption coefficient and pushing SA from being just an imaging tool to a valuable approach for material characterization. This approach is particularly useful for spectroscopy in fields where steady objects are more common to be characterized. The results show the potential of the proposed approach as an experimental technique for submillimeter 3D imaging and spectroscopy in the THz range of multi-material objects.

II. SA-BASED 3D IMAGING AND SPECTROSCOPY

In this section, we briefly recall the numerical process allowing the reconstruction of a 3D image from measured data using a SA. To synthesize a radiating aperture, a mobile antenna is periodically translated on a distance D, with a step d in a twodimensional plane perpendicular to the direction of propagation (figure 1). For each transmit antenna position (𝑥 𝑛 , 𝑦 𝑛 = 0, 𝑧 𝑛 ), with 𝑛 ∈ [1, N] and N the total number of spatial scanned points in the XOZ plan, a temporal signal 𝑠 𝑚𝑒𝑠 (𝑡, 𝑥 𝑛 , 𝑧 𝑛 ) is measured on the stationary receiving antenna. The 3D image is reconstructed in the frequency domain using an image reconstruction method known as the back projection (BP) algorithm [START_REF] Doerry | Basics of Backprojection Algorithm for Processing Synthetic Aperture Radar Images[END_REF]. The BP algorithm acts as a digital lens by compensating the phase of the signals propagated from the antennas to the scene to be imaged. The frequency-dependent reconstructed complex transmittivity of each voxel is expressed as:

𝑇 ̃𝑟𝑒 (𝑓 𝑚 , 𝑥 𝑝 , 𝑦 𝑝 , 𝑧 𝑝 ) ≈ ∑ 𝑆 𝑚𝑒𝑠 (𝑓 𝑚 , 𝑥 𝑛 , 𝑧 𝑛 ) . 𝑒 -𝑗 2𝜋 𝑐 𝑓 𝑚 𝑟 𝑛 𝑝 𝑁 𝑛=1 (1)
With c is the speed of the THz wave in vacuum, 𝑚 ∈ [1, 𝑀] with M the total number of frequency points, and 𝑝 ∈[1, P] with P the total number of voxels of the reconstructed image. 𝑆 𝑚𝑒𝑠 (𝑓 𝑚 , 𝑥 𝑛 , 𝑧 𝑛 ) is the Fourier Transform of the measured time domain signal with 𝑓 𝑚 the frequency. The parameter

𝑟 𝑛 𝑝 = √ (𝑥 𝑝 -𝑥 𝑛 ) 2 + (𝑦 𝑝 ) 2 +(𝑧 𝑝 -𝑧 𝑛 )
2 is the distance between an antenna n of the planar synthetic array, to a voxel p of the reconstructed 3D matrix. The intensity of each voxel of the reconstructed 3D image can be expressed as:

𝐼(𝑥 𝑝 , 𝑦 𝑝 , 𝑧 𝑝 ) = |∑ 𝑇 ̃𝑟𝑒 (𝑓 𝑚 , 𝑥 𝑝 , 𝑦 𝑝 , 𝑧 𝑝 ) 𝑀 𝑚=1 | (2)
The spatial theoretical resolutions of the resulting 3D image, using a planar synthetic array of isotropic antennas, are expressed as [START_REF] Soumekh | Synthetic Aperture Radar Signal Processing[END_REF]:

∆ 𝑥 = 𝜆.𝑅 𝐷 𝑥 , ∆ 𝑦 = 𝑐 𝐵𝑊 , ∆ 𝑧 = 𝜆.𝑅 𝐷 𝑧 (3)
Where ∆𝑥, ∆𝑦 and ∆𝑧 are the cross-range, range and elevation resolutions respectively, 𝐷 𝑥 is the length of the SA along (OX) axis, 𝐷 𝑧 is the length of the SA along (OZ) axis, BW is the available frequency bandwidth and 𝜆 is the wavelength of the propagating beam. However, in reality, the resolution is limited by the divergence of the antenna's radiation pattern [START_REF] Soumekh | Synthetic Aperture Radar Signal Processing[END_REF].

The SA allows a lateral field of view (FOV) of:

𝐹𝑂𝑉 𝑥 = 𝜆.𝑅 𝑑 𝑥 , 𝐹𝑂𝑉 𝑧 = 𝜆.𝑅 𝑑 𝑧 (4)
Where 𝑑 𝑥 and 𝑑 𝑧 are periodical antenna increments in the synthetic array along X-axis and Z-axis respectively. The complex reconstructed data can be further used to extract the frequency-dependent refractive index 𝑛(𝑓) and the absorption coefficient 𝛼(𝑓). Similar to classical THz-TDS measurement, our approach requires the measurement of both reference and sample signals using the same synthetic antenna array. The reference measurement is measured without the object under test. Following the acquisition of the reference data, we reconstructed a reference complex transmittivity 𝑇 ̃𝑟𝑒 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) independently from the sample data by applying Equation 1. This allowed obtaining a frequencydependent reconstructed spectrum at each pixel, which served as a reference spectrum for each corresponding pixel in the sample complex transmittivity 𝑇 ̃𝑟𝑒 𝑜𝑏𝑗𝑒𝑐𝑡 (𝑓, 𝑥 𝑝 , 𝑧 𝑝 ).

Based on the reference and sample reconstructed transmittivities, a complex transmission coefficient of the object is calculated at each frequency 𝑓 following this equation:

𝑇 ̃𝑟𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 (𝑓,𝑥 𝑝 ,𝑧 𝑝 ) 𝑇 ̃𝑟𝑒 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑓,𝑥 𝑝 ,𝑧 𝑝 )

= 𝜌 𝑟𝑒 (𝑓, 𝑥 𝑝 , 𝑧 𝑝 )𝑒 -𝑗𝜑 𝑟𝑒 (𝑓,𝑥 𝑝 ,𝑧 𝑝 ) [START_REF] Supriya | Terahertz technology: A Review on dental perspective[END_REF] This ratio leads to 𝜌 𝑟𝑒 (𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) and 𝜑 𝑟𝑒 (𝑓, 𝑥 𝑝 , 𝑧 𝑝 ), the amplitude ratio and phase difference, respectively, between the spectral components of the sample and the reference reconstructed transmittivities.

Consequently, the values of the real part of the refractive index 𝑛(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ), its imaginary part 𝜅(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) and the absorption coefficient 𝛼(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) of the object under test can be derived for every pixel of the reconstructed object image using the following equations [START_REF] Naftaly | Terahertz time-domain spectroscopy of silicate glasses and the relationship to material properties[END_REF]:

𝑛(𝑓,
With e the thickness of the sample measured with a digital micrometer gauge (having a precision of +/-1 µm) before performing the THz imaging and spectroscopy experiments.

The measured thickness is then used as an input parameter to calculate the parameters above from the reconstructed data.

III. EXPERIMENTAL SETUP: SA IMPLEMENTATION IN A THZ-TDS

SA-TDS approach was implemented in broadband THz-TDS system (TERA K15 all fiber-coupled Terahertz Spectrometer from Menlo Systems GmbH). The system illustrated in figure 2 is using a femtosecond laser operating at 𝜆 = 1550 𝑛𝑚. The laser beam is split into a probe and pump beam. The pump beam irradiates a biased photoconductive antenna (PCA) which will emit a THz pulse, while the probe beam is irradiating a second PCA which will generate an electrical signal proportional to the incident THz wave amplitude. The temporal THz pulse allows to acquire a complex spectrum from 0.1 THz to 6 THz. The emitter and receiver PCAs are commercial fiber-coupled TERA15-FC antenna modules in strip line and dipole configurations respectively, paired with a high refractive index silicon (HR-Si) hyperhemispherical lens at the output side, resulting in a divergence angle of THz radiation of about 14.5°. SA-TDS system was achieved by attaching the emitter antenna to a mechanical 2D translation arm to record the transmitted THz pulse at different spatial positions (figure 3). Both receiver and emitter antennas can be translated to create a synthetic antenna array as they are both lightweight, compact and fiber optic pigtailed PCAs. For simplicity, the emitter antenna was chosen because it is not attached to a current amplifier as in the case of the receiver, which otherwise restrains the range of movement and calls for a more complicated support system to securely move the amplifier together with the receiver.

Fig. 2. Classical focused configuration of THz TDS. The imaged object is mechanically translated in the XOZ plane using a two-axis translation stage. The focal plane of the focused THz beam is achieved using four TPX lenses.

Fig. 3. Transmission based SA configuration using a THz-TDS for 3D imaging and spectroscopy. A PTFE lens is used to focus the beam onto the fixed detector.

The merging of the SA approach into the available THz-TDS system allows to dispose of the optical focusing system illustrated in figure 2. However, the elimination of all TPX focusing lenses results in a sensitivity decrease in the system therefore a decline in the dynamic range (DR) of the measured THz pulse. In a four lenses configuration, the DR is 87 dB at 1 THz (figure 4, blue curve) and in lens-free configuration, the DR falls to 36 dB at 1 THz (figure 4, yellow curve) that is associated with a reduction of the available BW from 6 THz to 3 THz.

In order to increase the DR without focusing the THz beam, a PTFE plano-convex lens was added in front of the fixed detector (figure 3). The addition of the lens does not affect the topology of the synthetic antenna array or the image reconstruction algorithm but improves the DR of the signal to 80 dB at 1 THz (figure 4, orange curve).

Fig. 4. Amplitude of spectral power recorded with the full optical system as in a classical focused TDS system (blue curve), with one lens (orange curve) and without any focusing lenses (yellow curve).

IV. MEASUREMENT RESULTS AND DISCUSSIONS

A thin metallic plate (2 x 2 cm²) with four open circular holes with different diameters from 3 mm to 6 mm (figure 5) was chosen to demonstrate SA-TDS imaging in transmission configuration with the TDS system (figure 6). A sample of MgO (e = 530 µm) is placed behind the hole of 6 mm diameter and another sample of quartz (e = 530 µm) is placed behind the hole of 5 mm diameter. These samples with different refractive indexes will be evaluated for their dielectric THz properties along with the 3D reconstruction of the overall object.

Fig. 5. Object under test: a metallic plate of 2x2 cm² with four holes, two holes with diameters 6 mm and 5 mm containing MgO (false-colored in orange) and quartz (false-colored in green) respectively. The other smaller holes of 3 and 4 mm contain air. The thickness of the two substrates is 530 µm.

Fig. 6. Top view picture of the 3D SA-THz imaging system taken from top view, including a sketch of the divergent THz beam and propagation direction.

The image reconstruction was performed from 0.6 THz to 1 THz since this frequency band provides a good tradeoff between DR, atmospheric absorption of the THz beam, and a smaller wavelength for a higher image resolution. To further improve the DR, each signal is generated from 78 averaged THz time-domain traces. All measurements are performed under typical indoor atmospheric conditions at room temperature of 23°C. The rest of the parameters of the synthetic array and the acquisition process are resumed in table 1.

TABLE I PARAMETERS OF THE SYNTHETIC ARRAY *Taking into account the divergence of the antenna

The cross-range resolution is 671 µm at FWHM at 1 THz using the antenna with a divergent beam of 14.5° and the array parameters indicated in table 1 and was estimated through a calculation of the point spread function (PSF) of the imaging system. This estimated experimental resolution is very close to the theoretical resolution of 600 µm. The intensity of the reconstructed 3D image using the BP algorithm of the object under test is shown in figure 7. The shapes of the four holes are correctly reproduced as well as their spatial positions on the metallic plate. The reconstructed image is located at a depth position of y=21 cm, which corresponds to distance between the object and the synthetic array.

Symbol

Parameter Value This result underlines the strength of the imaging technique that does not require any input information about the shape or the size of the sample. Because the numerical processing we perform for image reconstruction considers that the propagation is always through air and does not take into account the permittivities of the materials during the numerical processing, the amplitude variations between the four holes are apparent when projecting the reconstructed amplitude image on the XOZ plane as each projection displays a different level of intensity proportional to the refractive index of mediums in the holes. By projecting the 3D reconstructed image on the XOY and YOZ planes (figure 7), the distance between the different reconstructed zones can be used to retrieve the value of the group refractive index of the medium using the equation:

𝑩𝑾

𝑛 𝑔 = 1 + ∆𝑦 𝑒 ( 10
)
where ∆𝑦 is the between the projection of the reconstructed image of a dielectric material (behind an air hole) and the reconstructed image of an air hole (reference), and e is the thickness of the sample. Thus, for a distance between the reconstructed quartz area and air of ∆𝑦 𝑄 = 500 µ𝑚, the group refractive index is 𝑛 𝑔 𝑞𝑢𝑎𝑟𝑡𝑧 = 1.94 while for MgO (∆𝑦 𝑀 = 1000 µ𝑚), 𝑛 𝑔 𝑀𝑔𝑂 = 2.88.

The projections on XOY and YOZ planes coming from the areas of MgO, and the air are clearly separated on figure 7 since the spatial depth between the two reconstructed zones is higher than the depth resolution of 750 µm of our synthetic array while the separation corresponding to the projections of quartz and air is less visible (∆𝑦 𝑄 < 750 µ𝑚).

We should note that the SA trajectories deviations, caused by the error of the positioning of the antenna during data acquisition, are limited to ±5 µm in both directions according to the supplier of the translation motor used in our experimental setup. The effect of this positioning error on the SA reconstructed image can be neglected.

To further demonstrate the capabilities of the proposed approach for material characterization, the reconstructed data is used to extract the refractive index and absorption coefficient as indicated in the previous section. The reference time domain signal through air and through the quartz and MgO samples after the reconstruction of the image are presented in figure 8. The selected materials are non-dispersive and each of them have a thickness of 500 µm, which is thick enough for the propagated short pulse to create an isolated echo pulse from the main signal. The transmitted temporal signal through the quartz does not display any visible multiples (figure 8.b.), while MgO exhibits one reflection peak after a trip in the substrate since it has a higher refractive index than quartz (figure 8.a.). In this particular case, the echo on the temporal signal can be removed using simple windowing techniques in the same way echoes could be removed when using a classical THz-TDS set up for material characterization.

Based on the reference and sample reconstructed transmittivities, a complex transmission coefficient of the object is calculated at each frequency following the equation 5. After computing 𝑛(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) using equation 6, the frequency dependent refractive index, taken at the center pixel of the holes containing MgO and quartz, are presented in figure 9.a from 0.6 THz to 1 THz. The average refractive index on the investigated frequency band of MgO is 𝑛 𝑀 𝑎𝑣𝑔 = 3.01 and the average refractive index of quartz is 𝑛 𝑄 𝑎𝑣𝑔 = 2.07, values which are consistent with previous reports [START_REF] Han | Terahertz Dielectric Properties of MgO Nanocrystals[END_REF], [START_REF] Naftaly | Terahertz Time-Domain Spectroscopy for Material Characterization[END_REF].

The absorption coefficient of each pixel of the reconstructed image of the object is calculated using equation [START_REF] D'arco | THz Pulsed Imaging in Biomedical Applications[END_REF]. In figure 9.b, the absorption coefficient of the center pixel of the hole containing MgO (blue curve) and the center pixel of the hole containing quartz (orange curve) are presented from 0.6 THz to 1 THz. The absorption coefficient of MgO is extremely less than 3 𝑐𝑚 -1 in magnitude, the material being nearly transparent between 0.6 THz and 1 THz [START_REF] Han | Terahertz Dielectric Properties of MgO Nanocrystals[END_REF]. The quartz sample also presents a very low absorption coefficient of less than 1.5 𝑐𝑚 -1 between 0.6 THz and 1 THz [START_REF] Naftaly | Terahertz Time-Domain Spectroscopy for Material Characterization[END_REF]. From the computed 𝑛(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ), 2D maps of the refractive index of the object could be reconstructed at each frequency of the reconstruction frequency band. As shown in figure 10, the index of the areas of the air holes, MgO sample and quartz sample are correctly reconstructed at 603 GHz and 990 GHz, emphasizing the interest of SA-TDS for imaging and material characterization in the THz rang. A higher resolution could be attained by placing the system in a dry air chamber to access higher frequencies of the THz spectrum generated by the TDS. The area of the metallic plate is set with a null refractive index for the visual aspect.

V. CONCLUSION

We presented an approach to broadband 3D imaging and spectroscopy in the THz domain. The measurement setup combines the non-focused radiation from a THz-TDS system and the SA imaging technique implemented in the transmission configuration. The experimental results show that the technique can generate 3D images with sub-millimeter resolution in a frequency band up to 1 THz of a stationary planar object integrating different materials. The 3D amplitude image can be used to compute the group refractive index of different constituents of the sample, providing a first information on the different integrated materials. We further demonstrate the pairing of SA technique and spectroscopy analysis of the object under test, by using the reconstructed amplitude and phase to obtain the refractive index and the absorption coefficient, which This firstly reported approach, combining imaging and spectroscopy for material characterization based on SA TDS THz in transmission configuration, can be highly relevant for non-destructive testing and spectral characterization of fixed objects with different permittivity. This method, which is extremely appropriate for the characterization of low absorption materials, could be easily implemented in reflection configuration, which is more adapted for the characterization of absorptive or opaque materials in the THz range.

The particular implementation we proposed opens robust perspectives for real-time 3D imaging and spectroscopy, by replacing the mechanical scanning of the antenna used to create a SA with a real antenna array illuminating the scene instantly.

Fig. 1 .

 1 Fig. 1. Schematic of SA imaging in a transmission configuration. The moving divergent antenna is mechanically scanning the plane XOZ to create a planar synthetic antenna array. At each position of the mobile antenna, the emitted beam passing through the stationary object is focused on the fixed, detecting antenna.

6 ) 2 ( 2 ( 8 )

 6228 𝑥 𝑝 , 𝑧 𝑝 ) = 1 + 𝑐 𝜑 𝑟𝑒 (𝑓,𝑥 𝑝 ,𝑧 𝑝 ) 2𝜋𝑓𝑒 (𝜅(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) = -𝑐 4𝜋𝑓𝑒 ln (𝜏(𝑓, 𝑥 𝑝 , 𝑧 𝑝 )𝜌 𝑟𝑒 (𝑓, 𝑥 𝑝 , 𝑧 𝑝 )) (7) With 𝜏(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) = 1 -(𝑛(𝑓,𝑥 𝑝 ,𝑧 𝑝 )-1) 𝑛(𝑓,𝑥 𝑝 ,𝑧 𝑝 )+1) 𝛼(𝑓, 𝑥 𝑝 , 𝑧 𝑝 ) = 4𝜋𝑓 𝑐 𝜅(𝑓, 𝑥 𝑝 , 𝑧 𝑝 )

Bandwidth 0. 6 Fig. 7 .

 67 Fig. 7. 3D image and associate projections on XOY, XOZ, and YOZ planes, reconstructed between 0.6 THz and 1 THz with sub-millimeter range and cross-range resolutions.

Fig. 8 .

 8 Fig. 8. The normalized amplitude of the complex envelope of the reconstructed temporal signals through the MgO and quartz samples: (a) MgO at x=6mm and z=5mm (orange curve) and the reference temporal signal corresponding to the same reconstructed coordinates (blue curve). (b) Quartz at x=-3mm and z=-3mm (orange curve) and the corresponding reference reconstructed temporal signal (blue curve).

Fig. 9 .

 9 Fig. 9. (a) Frequency dependent refractive index computed from reconstructed data using equation (6) at two different coordinates: MgO at x=6mm and z=5mm (blue curve) and quartz at x=-3mm and z=-3mm (orange curve). (b) Frequency dependent absorption coefficient computed from reconstructed data using equation (9) at two different coordinates: MgO at x=6mm and z=5mm (blue curve) and quartz at x=-3mm and z=-3mm (orange curve). The reference measurement for the absorption coefficient is obtained using the metallic object without the dielectric substrates.

Fig. 10 .

 10 Fig. 10. 2D maps of the reconstructed refractive index of the dielectric samples within the imaged object at two different frequencies (a) at 603 GHz and (b) at 990 GHz. The 3D imaging performed beforehand is used to define the areas of interest, in which the refractive index reconstruction could be performed. The area of the metallic plate is set with a null refractive index for the visual aspect.

  mapping for each frequency of the investigated reconstruction band.

FUNDING

This research was partially funded by the Nouvelle Aquitaine region under the project MIMO, AAP2020 -R529.

MANAL AIT ASSOU received the M.S. degree in telecommunication, high frequencies and optics from the University of Limoges, Limoges, France, in 2020. She is currently pursuing the Ph.D. degree with the XLIM research institute, University of Limoges, France on the subject of synthetic aperture imaging and spectroscopy in the terahertz range using a time domain spectroscopy system. Her main research interests concern terahertz imaging and material characterization.


Baptiste Bouvier

Emmanuel Ponsot

Patrick Susini

Reorganization of temporal local/global processing with auditory salience

Keywords: attention, auditory perception, local/global

The so-called "forest before the trees" phenomenon refers to the perceptual processing of the global information before the fine-grained analysis in vision. This advantage can be reversed by the presence of local salience. The same advantage for processing global temporal information is found in audition, but not for musical experts. Here, we ask whether and how salience modifies the local/global temporal auditory processing. To do so, participants (13 non-musicians, 7 expert-musicians) had to search, within melodic sequences of 9 notes organized in 3 triplets, for local or global modification applied to one triplet. In some conditions, these sequences were further manipulated in timbre (brightness) to make one of the triplets salient. Results show that the advantage for processing global information was substantially decreased with salience, regardless of its congruence with the modification to detect. Furthermore, the performance to find a local modification was improved for non-musicians when salience was congruent with that modification. We conclude that salience draws attention to local information, and that once at the local level, attention is directed by salience to the locally modified information. This re-organization of local/global temporal auditory processing with salience appears to be more efficient for non-musicians than for expert musicians.

INTRODUCTION

Perception is a complex process involving the integration of sensory information at different scales to form a coherent perceptual experience. In the visual modality, studies using Navons's paradigm [START_REF] Navon | Forest before trees: the precedence of global features in visual perception[END_REF] revealed that the processing of the global information took priority over the local processing [START_REF] Mevorach | Opposite biases in salience-based selection for the left and right posterior parietal cortex[END_REF]. Similarly, in hearing, an advantage of the global processing of information has been revealed through studies using an adaptation of the visual paradigm [START_REF] Bouvet | Global precedence effect in audition and vision: evidence for similar cognitive styles across modalities[END_REF][START_REF] Ouimet | Auditory local-global processing: effects of attention and musical experience[END_REF]. The ability to process local vs. global information has since been shown to be influenced by top-down processes shaped by musical expertise [START_REF] Black | The role of musical experience in hemispheric lateralization of global and local auditory processing[END_REF][START_REF] Susini | Auditory local-global temporal processing: evidence for perceptual reorganization with musical expertise[END_REF]. More precisely, the advantage for processing the global information was found in nonmusicians, and, on the contrary, a local advantage was revealed for expert-musicians [START_REF] Susini | Auditory local-global temporal processing: evidence for perceptual reorganization with musical expertise[END_REF]. An important bottom-up process in perception is the salience of physical features. It has been shown to modulate attention and perception in both visual and auditory modalities [START_REF] Mevorach | Opposite biases in salience-based selection for the left and right posterior parietal cortex[END_REF][START_REF] Elhilali | Interaction between attention and bottom-up saliency mediates the representation of foreground and background in an auditory scene[END_REF]. In the auditory domain, salience can be modulated by features such as loudness, pitch or timbre [START_REF] Kim | Automatic detection of auditory salience with optimized linear filters derived from human annotation[END_REF][START_REF] Huang | Auditory salience using natural soundscapes[END_REF][START_REF] Kaya | Pitch, timbre and intensity interdependently modulate neural responses to salient sounds[END_REF][START_REF] Bouvier | Revealing the stimulus-driven component of attention through modulations of auditory salience by timbre attributes[END_REF]. Thus, in an orchestra, for example, a trumpet's clarion may be salient because of its level and brightness, but a musician could still be able -more than a non-musician -to pay attention to other instruments if he wishes. In this study, we aim to investigate the effect of bottom-up salience on auditory local-global processing. As this processing depends on musical expertise, we selected nonmusicians and expert-musicians to observe how salience affects the global and local advantage respectively. We hypothesize that salience will affect these processing hierarchies differently for musicians and non-musicians due to top-down processes modulated by musical expertise. Specifically, we will use a modified version of the auditory local-global paradigm [START_REF] Justus | Auditory attention to frequency and time: an analogy to visual local-global stimuli[END_REF] to investigate how the salience of auditory features influences the local/global processing, in musicians and non-musicians.

MATERIALS AND METHODS

Participants and Setup

Twenty volunteer participants took part in the experiment: 13 non-musicians (4 women, mean age: 31.4 ± 11.0 year) and 7 expert-musicians (1 woman, mean age: 41.4 ± 15.9 year). Sounds were presented to listeners through a Beyerdynamic DT-770 PRO headset (Heilbronn, Germany) and a Focusrite Scarlett 2i2 soundcard at a level of 70 dB SPL. The sound level was measured with the sound level meter type 2250-S of Bruel & Kjaer. Participants were tested in a double-walled Industrial Acoustics Company (IAC) sound-insulated booth. The test interface was coded with Max (v8) (https://cycling74.com/products/max) on a Mac Mini.

Stimuli

Stimuli construction were based on Susini et al., 2020 [START_REF] Black | The role of musical experience in hemispheric lateralization of global and local auditory processing[END_REF]: they were pairs of melodies made of 9 notes segmented into 3 triplets. The local level was defined as the pitch structure within the triplets, and the global level was defined as the pitch structure formed by the mean pitch of the three triplets. In the present study, the rising and falling pitch profiles were kept. Stimuli were presented as pairs of melodies: a target and a comparison, separated by a 500 ms silence. Four types of modifications could distinguish the pitch profile of the target and the comparison melodies: no modification, a local modification, a global modification, or both local and global modifications (hereafter called Condition No, Condition L, Condition G and Condition L + G, respectively). Local variations were changes in the pitch of one note within one triplet, while global variations were changes in the mean pitch of an entire triplet.

All notes followed the harmonic structure of Bouvier et al., 2023 [START_REF] Bouvier | Revealing the stimulus-driven component of attention through modulations of auditory salience by timbre attributes[END_REF]: each note with a fundamental frequency f0 had n harmonics, the n th harmonic fn having a frequency n*f0 and a weight 1/n 𝛼 . Thus, varying 𝛼 changed the sound spectral centroid (SC), and therefore its perceived brightness. For the ordinary notes, 𝛼 = 5. For the so-called bright notes, 𝛼 = 1.5. The stimulus pair could be affected or not by a salience manipulation (2/3 of all the trials containing a salience manipulation). If yes, in the congruent salience condition (1/3 of all trials), the modified triplet consisted of bright notes, both in the target and the comparison stimulus. In the non-congruent salience condition (1/3 of all trials), one of the other two triplets was made salient with equal probability.

Procedure

Participants took part in two separate attention-directed tasks in separate sessions: a "local" and a "global" session. They were asked to perform a "similar/different" discrimination task by focusing on the local or global scale depending on the session. In the local task, they had to determine whether the pitch profiles of the three triplets were similar or different in the target and the comparison stimuli, irrespective of the global profile. In the global task, they had to determine whether the global profile was identical or not, irrespective of the local profiles of each triplet. Each session was divided into 5 blocks of 96 trials and lasted approximately 1h30.

RESULTS

Analyses

Within the framework of SDT (Signal Detection Theory), we computed confusion matrices to derive sensitivity (d') and decision criterion (c) values for each participant, in each task, as a function of the modalities of the different factors.

Results

DISCUSSION

A global advantage was found for non-musicians when they have to detect modifications in melodies at a local or global scale. Indeed, in the No-salience condition, they performed better in the global task than in the local one. This effect does not stand for expert-musicians, which is in line with Susini et al. [START_REF] Susini | Auditory local-global temporal processing: evidence for perceptual reorganization with musical expertise[END_REF].

More importantly, the results with salience manipulations show that the global advantage is reversed in non-musicians when salience is congruent with a local change. In this condition, they become better at detecting local changes. Furthermore, the global information processing is always disrupted when information is locally salient, regardless of the salience congruence.

In sum, salience disrupts the global advantage by systematically degrading performance in the global task, and by enhancing performance in the local task when it is congruent. This result is in line with the results found in vision: Mevorach et al. [START_REF] Mevorach | Opposite biases in salience-based selection for the left and right posterior parietal cortex[END_REF] showed that the presence of local salience degraded performance in the global identification task.

The same pattern could be true for expert-musicians: the local advantage is increased when salience is congruent with the modification to detect. But their performance in the local task reveals a ceiling effect and does not allow a perfectly reliable observation. However, their performance in the global task is also degraded by the presence of local salience. This re-organization of the local/global auditory processing appears to be less efficient than for nonmusicians.

CONCLUSION

In summary, local/global information processing is reorganized because of salience. The re-organization seems to be more pronounced in non-musicians.

-*

  Corresponding author: baptiste.bouvier@ircam.fr. Copyright: ©2023 First author et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

  Analyses followed those proposed by Susini et al. (2020). This experiment is based on a 2 × [2 × 2 × 4 × 2 × 3] factorial design: one between-factor "Group" (musicians|non-musicians) and five within-factors "Task" (Local|Global) × "Profile" (R|F) × "Condition" (No|L|G|L + G

Figure 1 .

 1 Figure 1. Sensitivity (d') in the local and the global task for non-musicians (left) and expert-musicians (right) depending on the salience condition. Error bars: standard error of the distribution of participants' paired scores between each condition and the No-Salience condition.Results reveal a global advantage for non-musicians in the No Salience condition: these participants are better in the global task (d' = 2.45) than the local task (d' = 1.45). A ttest revealed that this advantage in the global task compared to the local one was significant (T(10) = 3.38, p = .007, cohen-d = 1.02, power = 0.86). The expert-musicians, on the contrary, are better in the local (d' = 5.48) than in the global task (d' = 4.37). A t-test revealed that this advantage in the local task compared to the global one was significant (T(5) = 3.24, p = .02, cohen-d = 1.32, power = 0.74).When salience is congruent with the modification to find, the pattern is changed: non-musicians no longer show a global advantage but, on the contrary, a better performance in the local task (d' = 1.77 for the local task, d' = 1.26 for the global one). This inversion is due to a significant


Louise-Anne Charles

Retour d'expériences : publication de données dans l'entrepôt de données SHS Nakala labedoc.hypotheses.org/10424 L'année scolaire 2021-2022 du centre de documentation du CRESSON a été dédiée à la gestion des données, en s'appuyant sur deux projets ANR. Pour ce type de projet, la gestion des données est devenue obligatoire, notamment à travers la rédaction du plan de gestion de données attendue par l'ANR. Mais que faire pour les projets dont cet aspect n'est pas obligatoire ou ceux qui sont terminés ? C'est grâce aux ateliers du réseau Scripto du Réseau National des Maisons des sciences de l'homme, et à une présentation de leur travail sur les plans de gestion de données rétrospectifs, que nous avons commencé à nous intéresser aux projets qui sont terminés. Cela permet d'avoir accès à des données stabilisées, documentées, qui peuvent faire l'objet d'un partage. En octobre 2021, lors d'une journée d'études (que vous pouvez découvrir ici), nous avons été marquées par la qualité esthétique et scientifique des Itinéraires et des cartes produites dans le cadre du projet de recherche-création Les Ondes de l'Eau : mémoires des lieux et du travail dans la vallée de la Romanche (AAU-CRESSON, Regards des Lieux). Par exemple, cette carte des mémoires sensibles de la Romanche longue de 6 mètres ! Ou le travail d'enquête sous forme d'itinéraires formalisés en livrets qui retracent un parcours avec un habitant ou un professionnel: 2/13 Image de la journée d'études du projet les Ondes de l'Eau, Sylvie Laroche, ENSAG, 1/10/2021 Ces données ont résonné avec une formation suivie en septembre 2021 sur l'entrepôt de données Nakala développé par Huma-Num et dédié aux SHS. Les atouts de cet entrepôt par rapport à ces données sont :

Une visionneuse qui permet de zoomer en conservant une grande qualité de l'image ; L'attribution de DOI aux données que l'on peut lier à un dépôt d'article sur HAL ; Le partage des données dans des publications web comme les billets de blog.

Un espace numérique qui semblait pouvoir faire la part belle aux aspects visuels des données tout en gardant leur qualité. Nous avons alors contacté l'équipe du projet représentée par Laure Brayer, la coordinatrice et Ryma Hadbi, doctorante et membre active du groupe et nous leur avons proposé de les accompagner, en nous appuyant sur l'expérience acquise avec les projets ANR. Cela s'est fait selon plusieurs axes : Regrouper les données du projet sur un serveur sécurisé du laboratoire et les trier. En effet, plusieurs espaces de stockage avaient été utilisés et des doublons de fichiers existaient ; Valoriser les itinéraires et les cartes produites sur Nakala ; Contacter les archives départementales dans l'idée d'archiver des données de recherche.

Ce billet est dédié à la publication des données sur Nakala (sans dépôt par lot) : documentation, préparation, publication et valorisation. Il s'agissait de prendre en main la plateforme avec un nombre assez réduit de données (5 itinéraires et 3 cartes) pour explorer ce qui était possible de faire. L'idée est maintenant de capitaliser ce qui a été appris, sous la forme d'un guide et de ce présent billet, pour que d'autres projets de recherche du laboratoire puissent investir la plateforme et que les réponses que nous avons eues puissent profiter à tous et toutes.

I. Processus

Le choix des données et leur organisation en collection ont été déterminés lors de plusieurs réunions entre documentalistes et chercheurs. La documentation des données s'est faite sur un tableur en suivant l'ordre des métadonnées de Nakala et a servi de base de discussion.

II. Les choix éditoriaux 1) La construction des collections

Deux possibilités s'offraient à l'équipe de recherche, comme il ne s'agissait que d'un nombre réduit de données : Soit regrouper dans une seule et unique collection toutes les données du projet ; Soit distinguer les itinéraires des cartes en les mettant dans deux collections différentes, en plus de faire la collection dédiée à toutes les données du projet.

Cette dernière solution a finalement été privilégiée par l'équipe car, même si une collection ne bénéficie pas d'un DOI, il est toujours possible d'envoyer son URL. Cela leur permet de renvoyer vers un ensemble de données. De plus, nous avons pu ainsi décrire avec plus de finesse chacune des collections.

2) Le choix de la licence des données

Ces données ont déjà été montrées plusieurs fois (lors de manifestations) et l'objectif était de faciliter leur visualisation et leur communication. Cependant, elles n'ont pas encore fait l'objet de publication. L'équipe a alors décidé d'opter pour la publication des données, sans embargo, mais sous une licence « tous droits réservés ».

L'idée étant de faire évoluer ce choix vers une licence plus ouverte au réusage lorsque leur travail d'analyse sera finalisé dans des publications.

III. Le travail sur la documentation 1) De l'exposition aux métadonnées

Le projet Les Ondes de l'Eau a fait l'objet d'une exposition « Cartographier les paysages sensibles » à l'École Nationale supérieure d'architecture de Grenoble du 1 au 15 octobre 2021 accompagné de travaux d'étudiants. Dans ce cadre-là, des panneaux ont été réalisés pour décrire les supports affichés. Certains de ces textes ont ainsi pu alimenter les descriptions des données sur Nakala. Les livrets contenaient également des informations réutilisables.

2) La description d'une donnée en plusieurs étapes

4) Les mots-clés

Pour les mots-clés, nous nous sommes appuyés sur deux référentiels, Archirès et Rameau grâce à l'interface IDRef, pour compléter ce champ.

Astuce : pour pouvoir rentrer une série de mots-clés en une fois, il suffit de séparer les mots-clés par des ; !

5) Préciser le rôle des auteurs et ajouter le nom des personnes enquêtées

Lors de l'ajout des auteurs sur Nakala, il n'est pas possible de préciser leur fonction dans la production de la donnée.

Nous avons alors utilisé le type Dcterms:creator sous la forme : nom (fonction).

Le type Dcterms :contributor a lui servi pour mentionner les personnes enquêtées qui acceptaient de ne pas être anonymes.

6) Géolocalisation : utiliser le référentiel TGN

6/13

La dimension de l'espace est importante dans ce projet puisqu'il s'agit d'un travail sur un territoire, la basse vallée de La Romanche (38). Nous avons, pour l'exprimer sous forme de métadonnées, utilisé le référentiel TGN (Getty Thesaurus of Geographic Names).

Le référentiel TGN permet à la fois d'avoir : un identifiant, un lieu exprimé sous forme hiérarchique et une URI. Nous avons conservé ces trois informations en dupliquant une propriété dcterms :spatial, où le type devait être précisé pour l'ID et l'URI.

Voici un exemple pour le lieu Livet-et-Gavet :

Captures d'écran des métadonnées supplémentaires "dcterms:spatial" sur Nakala

7) Autres métadonnées supplémentaires

Nous avons également précisé le format des données (jpeg, pdf) dans des métadonnées supplémentaires de propriété dcterms:format.

8) Relations entre les données

VII. Conclusion

Avec ce billet dense, nous le concevons bien !, nous espérions pouvoir donner quelques clés sur la prise en main de Nakala, sur

  Ce travail de publication des données et de documentation s'est fait grâce à l'aide de Sara Tandar qui fait partie du pôle « DONNÉES et coordination de l'accompagnement utilisateur » d'Huma-Num et qui a été très disponible ces derniers mois pour nous répondre. Un grand merci à elle ! Sommaire I. Processus II. Les choix éditoriaux III. Le travail sur la documentation IV. Le dépôt et la publication V. Nakala et son lien avec d'autres plateformes VI. Les problèmes rencontrés et les solutions VII. Conclusion

  Trois types d'information ont été mis en description de chaque donnée : Contexte du projet ; Résumé de la donnée ; Méthode de création de la donnée.Nous avons choisi de dupliquer le champ description pour mettre en valeur les méthodes spécifiques utilisées.Capture d'écran des champs "description" dupliqués sur Nakala er3) Le problème du type de donnéePour rappel, l'un des types de données que nous souhaitions publier sur Nakala est un livret qui donne à voir un itinéraire avec un habitant ou un professionnel. Il y a à la fois des photos, une carte du trajet, des verbatims de l'enquêté. Nakala propose de choisir dans une liste contrôlée le type de la donnée. Or aucun ne semblait correspondre, étant soit réducteur soit inapproprié. La solution proposée par Sara Tandar a été d'opter pour le type de document « autre » et d'utiliser les métadonnées supplémentaires pour préciser le type qui nous semblait correspondre, et ne perdre ainsi aucune information. Nous avons utilisé la propriété Dcterms:subject et dupliqué autant de fois que nécessaire. Capture d'écran des métadonnées supplémentaires "dcterms:subject" sur Nakala

L

  'option de créer des relations entre les données s'est révélée intéressante : nous avons pu créer une relation entre deux cartes et une carte qui cite le travail fait pour un livret. Ces informations apparaissent sur la page de la donnée, de manière plus transparente. Une fois qu'une relation était créée sur une donnée, il n'y avait pas besoin de modifier l'autre donnée concernée. Capture d'écran sur la relation à une autre donnée sur Nakala 10/13 Capture d'écran d'un data set sur ORCID Ou il est également possible de faire un ajout par DOI, le type de document est alors « autre ». ORCID récupère le titre, le résumé, la date (de mise en ligne), le lien et crée la citation. La citation bibtex peut être importée dans Zotero.

intérêt de la gestion des données pour les projets terminés et

  la documentation qu'il est possible de faire et sur ce que nous avions pu noter tout au long du parcours. souligné comment celle-ci pouvait répondre à des besoins concrets.De plus, une réflexion nécessaire semble se dessiner sur l'articulation à faire entre publications et données : comment le stade de publication influence le partage mais également comment les outils qui se développent communiqueront à l'avenir, à l'instar d'HAL etde Nakala. L

	confirmé l'
	ère
	ème
	13/13

Il s'agissait d'un projet d'un genre nouveau pour le centre de documentation qui, même s'il a acquis de l'expérience dans le partage et l'archivage des données sonores, s'est confronté cette année à une multitude de typologie de données et à des nouveaux outils ! Ce travail a

'enjeu est d'également communiquer et d'accroitre la visibilité de ces données pour

  les chercheurs et les usagers.Source de l'image mise en avant : gallica.bnf.fr / INSEP, "Foss bat le record du saut à la perche", Le miroir des sports, 1920 (mis en ligne en 2013). Disponible à l'adresse : https://gallica.bnf.fr/ark:/12148/bpt6k65253135/f10.item# Cite this article as: Louise-Anne Charles et Françoise Acquier, "Retour d'expériences de publication de données : l'entrepôt de données SHS Nakala," in Lab & doc, 22/07/2022, https://labedoc.hypotheses.org/?p=10424.

Le conseil qui nous a été donné est de créer une nouvelle fois cet auteur avec son ORCID. Deux formes apparaissent alors lorsque nous tapons son nom. La deuxième forme est celle plus récente avec l'ORCID. Nous avons repris les données créées par cet auteur pour lui attribuer la bonne forme. Nakala a pour axe d'amélioration le développement de la base auteurs.

2) Impossible d'ajouter la donnée à une collection

Nous avons déposé les données avant de les publier afin de permettre une vérification par les membres de l'équipe projet. Or lorsque nous avons repris les données pour les publier, il n'était plus possible de les ajouter aux collections. Ces dernières n'apparaissaient plus.

Comme nous les avions préalablement déposées, il a fallu les publier avant de pouvoir les ajouter à des collections publiques.


F Lacressonniere
email: fabien.lacressonniere@fsa.univ-artois.fr

B Cassoret

Converter used as a battery charger and a motor speed controller in an industrial truck

Keywords: Asynchronous motor», «Battery charger», «DC power supply», «Power factor correction»

This paper present a bi-directional power converter used on one hand as a battery charger and on the other hand as a motor speed controller. Different possibilities are presented and a comparison of these structures is carried out. In battery charger mode, the power converter has to absorb a sinusoidal current with a unit power factor. According to the power range, two possibilities are presented : the isolated CUK converter with a 1,5kW DC motor and a pulse width modulation (P.W.M.) inverter/rectifier to supply a 6kW asynchronous motor.

Introduction

Nowadays, in an electrical industrial truck, there is two distinct power converters : the speed controller for the traction motor and the battery charger. According to the nature of the truck (pallet truck or fork lift truck), the power and the type of the traction motor are different. Generally, in a pallet truck, the traction motor is a DC machine with a power range of 1-2kW, this one is supplied by a chopper. On the other hand, in the fork lift truck, the power is higher (6-10kw) and the asynchronous machine is preferred and it is associated with a PWM inverter. A second converter is used to recharge the battery. This one is connected at the single phase network for low power(<6kW) whereas for higher powers, it is connected at the three phase network. Between the network and the converter, there is a rectifier. Usually, this one generates some current harmonics in the network and a bad power factor. To reduce these harmonics, we use the Power Factor Correction (P.F.C) technique. In order to decrease the weight and the cost of the battery charger, we use the same power converter for the traction and charger modes. The aim of this paper is to present different possibilities of converters for the two operations modes. According to some criteria, the isolated CUK converter and the P.W.M. inverter/rectifier are chosen for the pallet truck and the fork lift truck applications. The experimental results are shown for the CUK converter.

Power Converter for the electric pallet truck

Choice of the topology

A pallet truck is equipped with a lead acid battery which has a nominal voltage of 24V. The traction motor is a DC machine. Nowadays, in this truck, the nominal voltage of the DC machine and the battery are the same. Hence, a chopper (Buck converter) allows to control the speed of the truck. The problem of the DC machine is the damage of the brushes, especially when the voltage of the motor is low. To reduce this damage, the voltage of the DC machine has to be increased. Hence, a nominal voltage of 220V is chosen. The nominal voltage of the battery is 24V and we use a 220V DC motor. In the speed controller mode, the power converter has to convert the DC voltage of the battery into a variable voltage for the DC motor (0 -220V). The system has to work in Buck or Boost mode. The power converters which have at the same time the Buck and Boost mode are indirect choppers (with intermediate storage of energy). In the battery charger mode, there is a rectifier between the network (220V/50Hz) and the converter. This one generate some current harmonics and reduce the power factor. To improve it, the converter has to be able to operate in PFC mode [START_REF] Rossetto | Control techniques for power factor correction converters[END_REF]. The galvanic isolation between low and high voltage is realised with a high frequency transformer, which operates at 20kHz. There are two possibilities to include those two operations modes in the same converter :

• At first, an unidirectional converter (Fig. 1) can be used. In this case, the battery has to disconnect according to the operation mode : speed controller (1) or battery charger [START_REF] Spiazzi | Design criteria for power factor preregulators based on Sepic and Cuk converters in continuous conduction mode[END_REF].

Fig. 1 : The configuration for an unidirectional converter

• The other possibility is to use a bi-directional converter. Then, the battery is always connected at the low voltage side (Fig. 2).

Fig. 2 : The topology for a bi-directional converter

The drawback of the unidirectional converter is the stress on the semiconductors components. That is to say, for the same power in both operations modes, the components have to support high currents and high voltages because the input and output voltages are different according to the operation mode. Then, the bi-directional topology are chosen.

In the speed controller mode, the converter has to convert the battery voltage into a variable voltage for the DC motor. In battery charger, the converter has to charge the battery with a unity power factor.

To achieve those two operations modes, we have two possibilities :

• The Buck-Boost converter. The storage element is an inductor. The Flyback converter has the same topology with a galvanic isolation (Fig. 3). • The Cuk converter. The storage element is a capacitor. Figure 4 presents this converter with a galvanic isolation (Fig. 4).

Network 220V DC Motor 220V 24V 24 V (1) (1)
(2) (2)

Network 220V DC Motor 220V 24 V (2)
(1)

Fig. 3 : The Flyback converter Fig. 4 : The Cuk converter

The main difference between these converters is the nature of the current. For the Flyback converter, the input and output currents are discontinuous whereas in the Cuk topology, they are continuous. Hence, at the nominal power, the maximal current in the semiconductors is twice higher in the Flyback converter. Moreover, the Cuk allows to work in PFC continuous mode [START_REF] Spiazzi | Design criteria for power factor preregulators based on Sepic and Cuk converters in continuous conduction mode[END_REF][3] and so to reduce the input filter and conduced electromagnetic disturbances [START_REF] Reis | Matching conducted EMI to international standards[END_REF]. Then, the proposed DC/DC converter (Fig. 5) is based on the Cuk principle and only contains two transistors and two diodes for both energy flow directions. The final structure is given by the Fig. 6. The switch K allows to realize the connection according to the operation mode.

Experimental results

The experiments was made with a 1,5kW prototype. At the low voltage side, the transistor T1 is a MOSFET 130V-130A. At the high voltage side, T2 is a IGBT 600V-50A. The operating frequency is equal to 20kHz. The UC3854 integrated circuit of Unitrode has been used for the command of the battery charger in PFC mode. Fig. 7 shows the currents in the DC machine and in the battery during the start of the truck. A regulation was done in order to limit it. The maximal current in the battery at the nominal power is equal to 100A. 9 shows the waveform of the battery current during this operation mode. This one is a typical waveform when a single converter are used which operates as a battery charger with an unity power factor and a sinusoidal input current. Then, its mean value is at the origin of the recharge of the battery [START_REF] Lacressonniere | Conception de structures de conversion d'énergie à double fonctionnalité pour chariots électriques de manutention[END_REF]. The efficiency of converter is around 90%. This result has been obtained by the utilization of a soft switching snubber [START_REF] Lacressonniere | Characterization and dimensioning of the Cuk converter in power factor correction[END_REF].

Power Converter for the fork lift truck

In this application, the nominal voltage of the battery is 48V and we use a three phased 48V asynchronous motor. To improve the efficiency of the traction system, the speed reducer between the shaft of the motor and the traction wheel has been eliminated. In the speed controller mode, a PWM inverter to supply the traction motor can be used and the space vector modulation strategy can be employed. In the battery charger mode, the inverter will be used as a PWM rectifier. The advantages of this converter are : bi-directional power flow, low harmonic distortion of the line current and the regulation of the input power factor to unity [6][7]. A transformer has to be placed in the structure to provide a galvanic isolation. To reduce its size and weight, the asynchronous motor are used like a transformer in the battery mode operation. The nominal power of the system is 6kW. The diagram of the system is given by the Fig. 10.

Characteristics of the asynchronous motor

The characteristics of the 6kW asynchronous motor described in detail in [START_REF] Lacressonniere | Conception de structures de conversion d'énergie à double fonctionnalité pour chariots électriques de manutention[END_REF] are reported in the table I. The outer diameter and the active length of the machine are equal to 45,8cm and 13cm. The weight of the machine is approximately equal to 125kg and its efficiency equal to 85% for the nominal power.

Table I : Technical data of the asynchronous motor

Combination of the asynchronous machine and PWM rectifier

During the battery charger mode, the rotor windings are connected to the 3 phase network. Thee stator windings are always connected to the PWM converter. The system of the battery charger is presented to the Fig. 11.

Fig. 11 : The battery charger topology

During the charging mode, the traction motor creates a torque. In order to the truck does not move, the rotor has to be blocked. The stator and rotor leakage inductances serve as filter inductance and the inverter operates as a PWM rectifier. This one allows to control the RMS value of U S and the phase shift between the control voltage U S and the source voltage U R . The phase shift between U S and U R is noted '. The RMS value of U S is controlled by the parameter r :

2 2 3 V r U bat S =
The Fig. 12 presents the evolution of the parameters of the PWM rectifier according to the power of charge (Pbat) in order to obtain a unity power factor. Pbat(W)

Conclusion

For a pallet truck and a fork lift truck, the speed controller and the battery charger can be realised with only one power converter. The proposed solutions allows to control the battery current during the charge, with a good input power factor. The cost, the size and the weight of the battery charger is reduced. The advantages and drawbacks of each possibilities have been shown.

Fig. 5 :

 5 Fig. 5 : Bi-directional DC/DC Cuk converter

Fig. 6 :

 6 Fig. 6 : Structure for the pallet truck

Fig. 7 :

 7 Fig. 7 : Battery and motor currents during the speed controller mode

Fig. 8 :

 8 Fig. 8 : Line voltage and current in power factor correction mode

Fig. 9 :

 9 Fig. 9 : Battery current in battery charger mode

Fig. 10 :

 10 Fig. 10 : Structure for the fork lift truck

Fig. 12 :

 12 Fig. 12 : Evolution of r and ' in battery charge mode