Gourab Ghatak
email: gourab.ghatak@cea.fr

Antonio De Domenico
email: antonio.de-domenico@cea.fr

Marceau Coupechoux
email: marceau.coupechoux@telecom-paristech.fr

Accurate Characterization of Dynamic Cell Load in Noise-Limited Random Cellular Networks

The analyses of cellular network performance based on stochastic geometry generally ignore the traffic dynamics in the network. This restricts the proper evaluation and dimensioning of the network from the perspective of a mobile operator. To address the effect of dynamic traffic, recently, the mean cell approach has been introduced, which approximates the average network load by the zero cell load. However, this is not a realistic characterization of the network load, since a zero cell is statistically larger than a random cell drawn from the population of cells, i.e., a typical cell. In this paper, we analyze the load of a noise-limited network characterized by high signal to noise ratio (SNR). The noise-limited assumption can be applied to a variety of scenarios, e.g., millimeter wave networks with efficient interference management mechanisms. First, we provide an analytical framework to obtain the cumulative density function of the load of the typical cell. Then, we obtain two approximations of the average load of the typical cell. We show that our study provides a more realistic characterization of the average load of the network as compared to the mean cell approach. Moreover, the prescribed closed form approximation is more tractable than the mean cell approach.

I. INTRODUCTION

Stochastic geometry has emerged as an important tool for modeling and analyzing large scale wireless cellular networks [START_REF] Elsawy | Stochastic Geometry for Modeling, Analysis, and Design of Multi-Tier and Cognitive Cellular Wireless Networks: A Survey[END_REF], wherein the performance is typically characterized by studying metrics such as signal to interference plus noise ratio (SINR) coverage probability and user throughput. To effectively model the user throughput and to efficiently dimension a cellular network from the operators' perspective, the characterization of the cell load is necessary. The existing literature in stochastic geometry models the cell load by considering the average number of associated full buffer users, uniformly distributed over the cell area, see e.g., [START_REF] Elsawy | Stochastic Geometry for Modeling, Analysis, and Design of Multi-Tier and Cognitive Cellular Wireless Networks: A Survey[END_REF], [START_REF] Elshaer | Downlink and Uplink Cell Association With Traditional Macrocells and Millimeter Wave Small Cells[END_REF]. This is not realistic since it ignores the effect of dynamic traffic on the user distribution: users with low data rate tend to stay longer in the system, and as a result, the user distribution becomes inhomogeneous in space.

However, studying dynamic traffic using elements of queuing theory in stochastic geometry based analyses is still an open problem [START_REF] Elsawy | Stochastic Geometry for Modeling, Analysis, and Design of Multi-Tier and Cognitive Cellular Wireless Networks: A Survey[END_REF]. In this regard, Blaszczyszyn et al. [START_REF] Błaszczyszyn | How user throughput depends on the traffic demand in large cellular networks[END_REF], have introduced the mean cell approach which avoids extensive simulations by approximating the spatial SINR distribution of a cell with the SINR distribution of the typical user. Thus, in essence, the mean cell approach characterizes the load of the cell containing the typical user, i.e., the zero cell [START_REF] Chiu | Stochastic geometry and its applications[END_REF]. Although this approach enables to model the cell load, it may lead to intractable expressions consisting of multiple integrals for evaluation of the SINR coverage probability.

Moreover, a characterization of the load of the zero cell is not a reliable metric for evaluating the network wide load distribution since the zero cell is statistically larger than a random cell drawn from the population of cells, i.e., a typical cell. To understand this intuitively, one can assume a random sample point and select the cell containing the point. By stationarity, the distribution of this cell coincides with that of the zero cell. Since the sample point tends to fall with greater probability into larger cells, the zero cell tends to be larger than the typical cell. In this paper, for the case of noise-limited networks, we provide approximations for the network load by characterizing the load of the typical cell. This provides a more realistic characterization of the network load. This noiselimited assumption can be applied to a variety of contexts. For example, in millimeter wave (mm-wave) networks that utilize directional antennas and advanced interference management mechanisms, the performance tends to be noise-limited. Singh et.al [START_REF] Singh | Interference analysis for highly directional 60-ghz mesh networks: The case for rethinking medium access control[END_REF] have shown the validity of the noise-limited network assumption in mm-wave mesh networks. Furthermore, this noise-limited scenario enables us to visualize our results in light of the seminal work of Bonald et al. [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF] who derived the cell load expressions for a single cell with dynamic traffic.

The contribution of this paper is as follows. We obtain a closed form expression for the cumulative density function (CDF) of the load of the typical cell in a noise-limited network by considering dynamic traffic. We use it to obtain the fraction of stable cells for a given deployment density of small cells. Then, we obtain a single integral based approximation, and a closed form expression, for the average load of the typical cell. We show that the first approximation models the cell load from a network perspective more accurately than the mean cell approach. Whereas, the closed form expression provides a faster and more tractable alternative to calculate the network load, since it does not require evaluation of integrals.

The rest of the paper is organized as follows. In Section II we introduce the single tier network and the associated parameters. In Section III, we present our main results on the CDF and the average of the load of the typical cell of the network. In Section IV, we present the results on the stable fraction of the network and we show the accuracy of our derived approximations with respect to Monte-Carlo simulations. Finally, the paper concludes in Section V.

II. SYSTEM MODEL

We consider a single-tier cellular network equipped with advanced interference management algorithms, so that the user performance is noise-limited.

The positions of the base stations (BS) are modeled as points of a Poisson point process (PPP) φ with intensity λ [m -2 ]. The BSs operate with a transmit power P t , and the product of the gains of the antennas at the transmitter and the receiver is G 0 . We consider a fast fading that is Rayleigh distributed with variance equal to one. Furthermore, we assume a path loss model where the power at the origin received from a BS located at a distance r is given by P

r = K • P t • h • G 0 • r -α ,
where K is the path loss coefficient, h is the exponentially distributed fading power, and α is the path loss exponent. Thus, the average SNR can be written as

K•Pt•G0•r -α N0•B = ξr -α , where ξ = K•Pt•G0 N0•B
is the average SNR at 1 m. N 0 and B are the noise power density and the operating bandwidth, respectively.

In this network, we assume that the users arrive in the system, download a file, and leave the system. Any new download by the same user is considered as a new user. The arrival process of the new users is Poisson distributed with an intensity λ U [users •s -1 •m -2 ] and these new users are uniformly distributed over the network area A. The average file size is σ [bits/user]. When there are n users simultaneously served by a BS, the available resources are equally shared between them in a Round Robin fashion. Accordingly, we define the traffic density w in the network as

w = λ U • σ [bits•s -1 •m -2 ]
. Note that, while the user arrivals are uniform in space, as the space-time process evolves, users farther from the serving BSs, i.e., characterized by lower data rates, stay longer in the system, resulting in an inhomogeneous distribution of active users in the network.

III. CHARACTERIZATION OF THE NETWORK LOAD A. Static vs Dynamic Load

Before proceeding to our main results, it is necessary to discuss the distinction of dynamic cell-load characterization as compared to the approaches that model the cell load as simply the average number of associated users to a BS [START_REF] Singh | Joint Resource Partitioning and Offloading in Heterogeneous Cellular Networks[END_REF]. For this we compare the downlink user throughput using the two approaches to study the difference. First, in our system model, we calculate the dynamic cell load using the study of Bonald et.al. [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF], say ρ. Then use it to calculate the downlink user throughput R dyn , which is given by [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF]:

R dyn = w 1 - ρ ρ • A, (1)
where, the area of the typical cell A is approximated as A = 1 λ , and the average number of active users in the dynamic traffic model [START_REF] Błaszczyszyn | How user throughput depends on the traffic demand in large cellular networks[END_REF] is: N = ρ 1-ρ To compare R dyn with that obtained using the analyses in the existing literature [START_REF] Elshaer | Downlink and Uplink Cell Association With Traditional Macrocells and Millimeter Wave Small Cells[END_REF], [START_REF] Singh | Joint Resource Partitioning and Offloading in Heterogeneous Cellular Networks[END_REF], we assume that the users are located homogeneously in each cell of the network following a PPP with an average of N users per cell. Using this assumption, we carry out simulations to obtain the downlink user throughput T P P P as:

R P P P = E n B n log 2 (1 + SIN R) , (2)
where the expectation is taken with respect to the number of users in each cell (n), which is Poisson distributed with mean N , as well as the SINR of each user in each cell. The difference between R dyn and R P P P is highlighted in Fig. 1.

Even though the average number of users in both cases are same, as the space-time process with dynamic traffic evolves, the user distribution is no longer homogeneous in space which is not taken into account in existing studies.

B. Preliminaries

In case of single-tier random cellular networks, the cell of a BS is given by the Poisson-Voronoi (PV) partition of the space [START_REF] Chiu | Stochastic geometry and its applications[END_REF]. In the R 2 plane, the PV region of a BS located at

x 0 ∈ φ is: A = {y : ||y -x 0 || < ||y -x i ||; ∀x i ∈ φ\{x 0 }}.
The mosaic of the cells formed for all such x 0 from a PPP is called a PV network. To investigate the geometry-dependent characteristics of the cells (e.g., the cell load), in a stationary random PV network, it is imperative to define the notion of the 'average' cell. Thus, we recall the following definitions that provide a characterization of the average cell. Definition 1. The zero cell or the Crofton cell of the PV network is defined as the cell containing a given fixed point in its interior [START_REF] Chiu | Stochastic geometry and its applications[END_REF].

Definition 2. The typical cell of a PV network is defined as a cell selected at random within a large region of the network with equal chances for each cell to be picked [START_REF] Chiu | Stochastic geometry and its applications[END_REF]. Thereafter, the network is translated so that the center of the typical cell becomes the origin.

The zero cell versus typical-cell approaches of modeling the network performance corresponds to the evaluation from the perspectives of the user and the network operator, respectively. Hence, for the case of analyzing the network load, the zero cell perspective is not an accurate way of characterization. In what follows, we first define the load of the typical-cell and then, discuss how the load of the zero cell is generally obtained. Then we present our analysis to characterize the load of the typical cell and hence the average network load.

1) Average Load of the Typical Cell: The load of the typical cell in the network can be calculated as:

ρ = A w C(s) ds, (3)
where C(s) is the rate that a user located at s receives in the typical cell A, calculated using the Shannon formula. The random variable ρ characterizes the load of the cell centered at x 0 , and depends on the shape and size of A.

The average load of the typical cell is then calculated by taking the expected values of loads for different realizations of the PPP itself:

ρ = E[ρ],
2) Mean Cell Approximation: In case of PV cells, the average load of the typical cell is generally difficult to evaluate because the shape and size of the typical cells is random. However, by using the ergodicity of the PPP, the area of the typical cell can be approximated as 1 λ [START_REF] Singh | Joint Resource Partitioning and Offloading in Heterogeneous Cellular Networks[END_REF]. Then, by assuming the network to be noise-limited, the average load can be approximated using the mean cell approach [START_REF] Błaszczyszyn | How user throughput depends on the traffic demand in large cellular networks[END_REF], as:

ρMC = T w Bλ log 2 (1 + T ) p(T )dT, (4)
where the expectation is taken with respect to the signal to noise ratio (SNR) (T ) variations averaged over the fast fading, and

p(T ) = -dP C (T )
dT is the probability density function (pdf) of the SNR of the typical user obtained by differentiating the SNR coverage probability, P C (T ). Thus, the cell load can be calculated numerically, given the SNR distribution.

However in the mean cell approach, as the expectation is taken with respect to the SNR variations of the typical user, it calculates the expected load of the zero cell which is statistically larger than the typical cell [START_REF] Chiu | Stochastic geometry and its applications[END_REF]. Thus, the mean cell approach, always overestimates the load of the typical cell. In the next sections, we propose a new approximation, which is both more accurate and more tractable. First, we derive the CDF of the cell load of the typical cell using the distribution of its area. Then, we obtain a single-integral based and a closed form approximation for the average load of the typical cell.

C. Distribution of the Area of the Typical Cell

The reduced area of a PV cell is defined as [START_REF] Chiu | Stochastic geometry and its applications[END_REF] :

s = A/E[A], (5)
where A is the area of the typical cell, and E [•] is the expectation operator. The pdf of the reduced area of the typical PV cell in two dimensions is given by [START_REF] Tanemura | Statistical distributions of Poisson Voronoi cells in two and three dimensions[END_REF]:

f s (x) = 343 15 7 2π x 5/2 exp - 7 2 x . (6)
Using this, we can obtain the CDF of the area as given below.

Lemma 1. The CDF of the area of the typical PV cell for a PPP with intensity λ is given by:

F A (x) = 343 15 7 2π 2 7 7/2 γ inc 7λx 2 , 7 2 , (7)
where γ inc (•) is the lower incomplete gamma function given by: γ inc (x, a) = x 0 t a-1 exp(-t)dt. Proof. The CDF can be easily derived using the relation [START_REF] Singh | Interference analysis for highly directional 60-ghz mesh networks: The case for rethinking medium access control[END_REF] and integrating [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF].

D. Distribution of the Load of the Typical Cell

For obtaining the CDF of the load of the typical cell, we assume that the shape of PV cells is circular. Although in a real PV network, almost surely no circular cells occur, our results show that the circular assumption does not greatly deteriorate the derived approximation. Accordingly, the load of a typical cell (3) with area A is approximated as [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF]:

ρ T C (A) ≈ ρ AP (A) = 2π 0 √ A π 0 wr Bλ log 2 (1 + ξr -2 )
drdθ.

Theorem 1. The distribution of the load of the typical cell, ρ is given by:

F ρ (l) = F A π 1 ξ exp - α 2 Ei -1 - l K -2 α ( 8
)
where K = 4wπ ln(2)ξ α 2 Bλ ξ 2 α and the symbol Ei -1 (x) is the inverse of the exponential integral. For the special case of α = 2, it is approximated as:

F ρ (l) ≈ F A   ξπ 1 + exp -l K1 exp -l K1   , (9)
where

K 1 = wπ ln(2)ξ Bλ .
Proof. According to our assumption of high SNR for dense networks, we can approximate 1 + ξr -α as ξr -α . Substituting ln (ξr -α ) = y, we have:

ρ AP (A) = K ∞ ln ξ(π/A) α 2
exp(-y) y dy

= K E 1 2 α ln ξ π A α 2 , (10)
where, K = 4wπ ln(2)ξ α 2 Bλ ξ 2 α and E 1 (•) is the exponential integral function [START_REF] Barry | Approximation for the exponential integral (theis well function)[END_REF]. The CDF is then simply obtained by some algebraic manipulations of the expression P(ρ AP (A) ≤ l). For the special case of α = 2, the CDF of the approximated load of the typical cell ρ AP (A) is derived as:

P(ρ AP (A) ≤ l) = P -ln ξπA -1 ≤ Ei -1 - l K 1 ,
where, the symbol Ei -1 (x) is given by Ei(x) = -E 1 (-x). Although an explicit expression for Ei -1 (.) does not exist, Pecina [START_REF] Pecina | On the function inverse to the exponential integral function[END_REF] provided piece-wise functions to approximate Ei -1 (x) for different ranges of x. The asymptotic approximation for -1

K1 → 0 is [10]: Ei -1 - l K 1 ≈ exp -l K1 1 + exp -l K1 .
In our analysis, we assume P t = 30 dBm, a noise density of -174 dBm/Hz, and B = 1 GHz. The path loss coefficient K is derived from the Umi model for data transmission [START_REF] Gpp | TR 38.900[END_REF]. As the load l varies from 0 ≤ l ≤ 1, for G 0 = 20 dB, and λ = 1e -5 m -2 , we have 0 ≤ l K1 ≤ 1e -8. Thus, our asymptotic approximation is valid. Using this approximation completes the proof.

E. Proposed Approximations for the Average Load of the Typical Cell

Using the distribution of the area of the typical cell ( 7) and [START_REF] Pecina | On the function inverse to the exponential integral function[END_REF] we obtain the following approximation of the average load:

ρAP = ∞ 0 K E 1 2 α ln ξ π A α 2 f A (A)dA, ( 11
)
ρ AP = ∞ 0 λ 2π ln(2)ξ B F1(A) 343 15 7 2π (Aλ) 5/2 exp - 7 2
Aλ dA [START_REF]TR 36.814, E-UTRA; Further advancements for E-UTRA physical layer aspects[END_REF] where,

F1(A) =            exp(-ln( ξπ A )) ln G 0 ln( ξπ A ) +G 0 +(1-G 0 )β(ln( ξπ A )) G 0 +(1-G 0 ) exp -ln( ξπ A ) 1-G 0 ; A ≤ πξ exp(1) -γ -ln -ln ξπ A + -ln ξπ A - (-ln( ξπ A )) 2 8 ; A > πξ exp(1)
where f A (x) = dF A (x) dx . Since solving [START_REF] Gpp | TR 38.900[END_REF] is tedious, we provide two results to approximate the average load of the typical cell for the special case of α = 2 1 . In Section IV, we will highlight the advantage of each approximation.

1) Exponential Integral (EI) based Approximation:

Theorem 2. The average load of the typical cell can be approximated as [START_REF]TR 36.814, E-UTRA; Further advancements for E-UTRA physical layer aspects[END_REF].

Proof. We rely on an approximation of the exponential integral provided by Barry et. al. [START_REF] Barry | Approximation for the exponential integral (theis well function)[END_REF]:

E 1 (x) = exp(-x) ln K2 x + K 2 + (1 -K 2 )β(x) K 2 + (1 -K 2 ) exp -x 1-K2 , (13)
where,

K 2 = exp(-γ) = 0.56, β(x) = 1 - 1 (h(x)+bx) 2 , h(x) = 1 1 + x √ x + 0.46x √ 31 26 1 + 0.43x √ 31 26 , b ≈ 1.04207,
and γ is the Euler's constant. This interpolated version of the exponential integral provides a good approximation for 1 ≤ x ≤ 50. This corresponds to the range πξ exp(50) ≤ A ≤ πξ exp [START_REF] Elsawy | Stochastic Geometry for Modeling, Analysis, and Design of Multi-Tier and Cognitive Cellular Wireless Networks: A Survey[END_REF] .

For the region of A greater than this range, we use the asymptotic expansion of E 1 (x) as:

E 1 (x) = -γ -ln(x) + x - x 2 8 + ... (15)
For practical ranges of cell sizes, the lower bound in (14) always holds (e.g., with G 0 = 0 dB, the lower bound on the area is A ≥ 2e -8). Now, substituting [START_REF] Pecina | On the function inverse to the exponential integral function[END_REF] in [START_REF] Gpp | TR 38.900[END_REF] and using ( 13) and (15) to evaluate the integral, completes the proof.

2) Closed Form (CF) Approximation: Substituting ln ξπ A = t, the approximated average load of the typical cell [START_REF] Gpp | TR 38.900[END_REF] becomes:

ρAP = χ 2 ∞ -∞ E 1 (t) exp - 7 2 t F 2 (t)dt, (16)
where χ 2 = wπ ln(2)ξ λB (ξπλ) 3.5 and F 2 (t) = exp -7λξπ 2 exp(-t) . Now, a closed form solution to this integral does not exist. However, in what follows, we derive an approximate closed form solution for the average load of the typical cell, which we show to be very accurate in Section IV.

Theorem 3. The average load of the typical cell is approximated by the closed form expression:

ρ AP = χ 2 (I 1 (t 2 ) -I 1 (t 1 ) + (y 1 -1)I 2 (t 2 ) -y 1 I 2 (t 1 ))
where,

I 1 (x) = 2 7 2 E 1 (4.5x) -(1 + 3.5x)e -3.5x E 1 (x)+ 7 9 exp(-4.5x) I 2 (x) = 2 7 E 1 (4.5x) -e -3
.5x E 1 (x) , and, t 1 =ln -2 7λξπ ln (0.1) , t 2 =ln -2 7λξπ ln (0.9) and, y 1 = 0.9t1-0.1t2 t1-t2 .

Proof. We can approximate F 2 (t), with a piece-wise defined ramp and step function as follows:

F2 (t) =      0; t ≤ t 1 , 0.8t t2-t1 + y 1 ; t 1 < t ≤ t 2 , 1; t > t 2 (17)
where t 1 and t 2 are the points corresponding to 10 and 90 percentile values of F 2 (t), and y 1 is the intercept. In the next section, we will show that this approximation provides accurate results for the average cell load. With the approximation of (17) in (16), we have:

ρAP = χ 2 t2 t1 E 1 (x)e 3.5x 0.8 t 2 -t 1 x + y 1 dx+ ∞ t2 E 1 (x)e -3.5x dx
Geller and Ng [START_REF] Geller | A table of integrals of the exponential integral[END_REF] provided closed form expressions for both of the above integral types, which we employ to obtain the closed form for the average cell load.

IV. SIMULATION RESULTS

A. CDF of the Load and Stable Fraction of the Network

To validate the approximation of the CDF of the cell load, we compute the stable fraction of the network, which is defined as the fraction of non-overloaded cells. Mathematically, this is the probability that the load of the typical cell is less than 1. In Fig. 2 we compare the stable fraction of the network for a file size of σ = 100 Mb, and a user arrival rate of λ U = 100 km -2 , obtained with the approximation of the CDF derived in Theorem 1 and the one computed from Monte-Carlo simulations of the PPP. This provides dimensioning rules for the operator in terms of the minimum deployment density of BSs required to achieve a given stable fraction. For example, with a directive antenna gain of G 0 = 20 dB and for a load target of 0.5, the operator must deploy at least 50 BSs km -2 . We also observe that the closed-form CDF provides a good approximation of the numerical values, specially for a larger antenna gain (G 0 = 20 dB). Accordingly, the circular assumption of the cell shape is not detrimental for evaluating the performance of the network.

B. Accuracy of the EI Approximation of the Network Load

In Fig. 3, we compare the average load of the typical cell, computed with the EI approximation (Theorem 2), the CF expression (Theorem 3), and that obtained using the mean cell approach with the network load calculated using Monte-Carlo simulations. For the Monte-Carlo simulations, we find the average cell load in one realization of the PPP φ, for a given λ, λ U , and σ and then we perform the same calculations and average over 1000 PPP realizations. As seen in the figure, the EI approximation provides a more accurate characterization of the network load than the mean cell approach. The mean cell approach always overestimates the actual load, because, the zero cell is, on average, larger than the typical cell, resulting in higher load. Therefore, from the perspective of an operator, we provide a more realistic, and hence reliable method to characterize the network load and to dimension the network. As an example, for σ = 100 Mb, and λ U = 0.01 users per second, the EI approximation accurately estimates that the operator must deploy 10 BS less (120 as compared to 130) than that prescribed by the mean cell approach.

C. Advantages of the CF Approximation of the Network Load

As we see in Fig. 3, the CF approximation provides the loosest approximation to the network load; however, it does not require numerical evaluation of integrals. Moreover, we see that for higher file sizes (σ = 100 Mb) and denser deployment of small cells (λ ≥ 1e -4 m -2 ), even the CF approximation provides an excellent approximation of the network load.

From a practical perspective, it provides a fast method of accurately estimating the network load without the need of running extensive simulations, which can become infeasible. Particularly, recall that BS locations are Poisson distributed. For every realization of BS locations, SNR distribution should be computed by drawing all required random variables. Moreover, a dynamic traffic of users arriving in the system, downloading a file, and leaving should be simulated for a sufficient duration to reach the mixing time of the Markov process. This procedure should be repeated for every set of possible parameters if we want to characterize the network load. At last cell overloading may be undetectable as any simulation has a finite duration. For all these reasons, the analytical model presented in this analysis is necessary to provide very quick results and interesting insights to the system.

V. CONCLUSION

The realistic assessment of the mobile network performance need to take dynamic traffic into account in order to characterize the network load, which is still an open problem. Towards this end, we have derived a simple approximation for the CDF of the cell load of the typical cell in a noiselimited network, which is characterized by high SINR and low inter-cell interference. Furthermore, we have obtained a single approximation-based expression and a closed-form expression for the average load of the network by using the distribution of the area of the typical cell. Our derivations present a more realistic characterization of the cell load, as compared to the recently introduced mean cell approach, since we consider the typical cell of the network rather than the zero cell. The analysis provides a tractable and accurate characterization for the cell load that can be utilized, e.g., for evaluating the user throughput and dimensioning 5G networks. However, accurate characterization of the dynamic network load in case of an interference prone network is not straightforward. This will be addressed in a future work.

Figure 1 :

 1 Figure 1: Throughput comparison of mean cell approach with PPP distributed users.

Figure 2 :

 2 Figure 2: Stable fraction of the network.

Figure 3 :

 3 Figure 3: Analytical approximation accuracy, G 0 = 36 dB.

For dense deployments, the serving BS is generally in line of sight (LOS) which has a path-loss exponent close to

for sub-6GHz[START_REF]TR 36.814, E-UTRA; Further advancements for E-UTRA physical layer aspects[END_REF] and mmwave[START_REF] Gpp | TR 38.900[END_REF] transmissions.


Aghilas Sini

Damien Lolive

Gaëlle Vidal
email: gaelle.vidal@irisa.fr

Marie Tahon
email: marie.tahon@univ-lemans.fraghilas.sini

Élisabeth Delais-Roussarie
email: elisabeth.delais-roussarie@univ-nantes.fr

SynPaFlex-Corpus: An Expressive French Audiobooks Corpus Dedicated to Expressive Speech Synthesis

Keywords: Speech corpus, audiobooks, emotion, expressiveness

This paper presents an expressive French audiobooks corpus containing eighty seven hours of good audio quality speech, recorded by a single amateur speaker reading audiobooks of different literary genres. This corpus departs from existing corpora collected from audiobooks since they usually provide a few hours of mono-genre and multi-speaker speech. The motivation for setting up such a corpus is to explore expressiveness from different perspectives, such as discourse styles, prosody, and pronunciation, and using different levels of analysis (syllable, prosodic and lexical words, prosodic and syntactic phrases, utterance or paragraph). This will allow developing models to better control expressiveness in speech synthesis, and to adapt pronunciation and prosody to specific discourse settings (changes in discourse perspectives, indirect vs. direct styles, etc.). To this end, the corpus has been annotated automatically and provides information as phone labels, phone boundaries, syllables, words or morpho-syntactic tagging. Moreover, a significant part of the corpus has also been annotated manually to encode direct/indirect speech information and emotional content. The corpus is already usable for studies on prosody and TTS purposes and is available to the community.

Introduction

To build an expressive Text-To-Speech (TTS) system able to read books of different literary genres, using various discourse modes and speaking styles, a corpus that covers all these specificities is required. Usually, corpora built for TTS purposes are less than ten hours long and monospeaker. In addition, the content is carefully controlled to maximize the homogeneity of the synthetic speech. Long and coherent speech data is very interesting as it gives the possibility of studying voice expressiveness under different situations. Audiobooks are a good example of such data and are valuable for prosody modeling, especially in the field of storytelling. For instance, [START_REF] Montaño | Prosodic analysis of storytelling discourse modes and narrative situations oriented to text-to-speech synthesis[END_REF][START_REF] Aparicio | Prosodic and Voice Quality Cross-Language Analysis of Storytelling Expressive Categories Oriented to Text-To-Speech Synthesis[END_REF] show that expressive categories might exist in the storytelling speaking style. Some works have also been done to detect the speaking style in audiobooks [START_REF] Székely | Detecting a targeted voice style in an audiobook using voice quality features[END_REF] and to evaluate the usability of audiobooks data for the generation of conversational speech (Székely et al., 2012a). In the last decade, many corpora were built from audiobooks. For instance, [START_REF] Panayotov | Librispeech: an ASR corpus based on public domain audio books[END_REF] presents a multi-speaker English corpus built for speech-text alignment purposes containing thousands of hours of speech. In [START_REF] Stan | TUNDRA: a multilingual corpus of found data for TTS research created with light supervision[END_REF], a multilingual corpus that contains approximately sixty hours of speech data from audiobooks in 14 languages is introduced. Since this corpus contains an average of four hours per language and only one book per language, it prevents studies on speaking styles particularly for TTS synthesis. The same analysis is true for the GV-Lex corpus [START_REF] Doukhan | The GV-LEx corpus of tales in French: Text and speech corpora enriched with lexical, discourse, structural, phonemic and prosodic annotations[END_REF] which focuses on tales analysis and contains twelve tales but only one hour of speech. In other works such as [START_REF] Zhao | Constructing stylistic synthesis databases from audio books[END_REF][START_REF] Wang | Exploring expressive speech space in an audio-book[END_REF], authors use an audiobook recorded by a pro-fessional speaker, and explore the speech recording in different spaces (expressive, acoustic and perceptual). Nevertheless, none of these works proposes a large monospeaker audiobook corpus in French language including various literary genres. This work introduces a new corpus of read French speech, suitable to build an expressive speech synthesis or to build robust storytelling prosodic models. It is built in the context of the SynPaFlex1 research project aiming at improving text-to-speech synthesis. The main motivation to build a corpus of audiobooks is to study a large coherent speech recording, made by a single amateur voice and containing different linguistic, acoustic, phonetic and phonological phenomena. To this end, this new corpus contains an eighty seven hours collection of good quality audiobooks, extracted from Librivox, uttered by one speaker and covering various types of literary genres (novels, short stories, tales, fables, and poems). When reading the books, the speaker has an expressive speaking style and uses personification to make the characters distinguishable. Moreover, whole books or chapters are used, thus enabling to study long term discourse strategies used by a speaker. To build such large corpora from audiobooks, many techniques have been proposed such as in [START_REF] Braunschweiler | Lightly supervised recognition for automatic alignment of large coherent speech recordings[END_REF][START_REF] Boeffard | Towards Fully Automatic Annotation of Audio Books for TTS[END_REF][START_REF] Mamiya | Lightly supervised GMM VAD to use audiobook for speech synthesiser[END_REF]Charfuelan and Steiner, 2013). In this paper, we use an automatic large-scale speech-to-text aligner for the French language [START_REF] Cerisara | JTrans, an open-source software for semi-automatic text-to-speech alignment[END_REF] to perform the segmentation into phones. The remainder of the paper is organized as follows. Section 2 explains how data used to build the corpus were chosen. In sections 3 and 4 , the different manual and automatic an-

Corpus Construction

Corpus Constraints

Designing a speech corpus requires the definition of some criteria for data selection. As this corpus is built in the framework of the SynPaFlex project, it will be used to study prosody, pronunciation, and also to build expressive speech synthesis models for French. Considering this, we have made the following requirements:

• Availability of a large quantity of data uttered by a single speaker;

• Availability of the corresponding texts;

• Good audio signal quality and homogeneous voice;

• Various discourse styles and literary genres;

• Conveying emotions in speech.

Data Selection

Investigations were conducted using two main types of sources: audio CD and on-line servers. Because of the lesser accessibility to CD audiobooks, online servers were found to be the most appropriate search areas, even if they sometimes provides data patchy in quality.

After several trials, we concentrated our efforts on the recordings made by a female speaker and available on Librivox, one of the public domain projects that provide audio recordings of book readings on a Web server. In total, eighty seven hours of audio files and the corresponding texts have been collected. As shown in Table 1, the novel genre represents about 90% of the corpus. Among the list of selected books for this literary genre, there are "Les Misérables" (Victor Hugo), "Madame Bovary" (Gustave Flaubert) and "Les Mystères de Paris" (Eugène Sue).

As the selected audiobooks are read by a non-professional speaker, the acoustic conditions might be different between chapters and books. Some listening evaluations have been done on audio signal in order to identify those of lesser quality, allowing to potentially exclude them from further processes.

Automatic Annotations

Data Preparation

The whole annotation process has been conducted relying on the ROOTS toolkit [START_REF] Chevelu | ROOTS: a toolkit for easy, fast and consistent processing of large sequential annotated data collections[END_REF], that allows storing various types of data in a coherent way using sequences and relations. This toolkit allowed us to incrementally add new information to the corpus. Once audio data have been selected and the corresponding texts have been collected, a few manual operations have been applied to simplify further processing. Notably, as recordings were performed in different technical and environmental conditions, loudness has been harmonized using the FreeLCS tool2 . Despite of that, audio data acoustic features remain more or less heterogeneous.

As texts were coming from diverse sources, their formats were unified. Then the exact orthographic transcriptions of the readings were achieved by inserting introductions and conclusions the speaker added in the recording, and by placing footnotes and end-of-book notes where they appear in the reading stream. The next step has been to normalize the texts using rulebased techniques appropriate for the French language, and split them into paragraphs. For the rest of the process, we keep each chapter in a separate file so as to keep long term information accessible.

Speech Segmentation

The broad phonetic transcription, based on the French subset of Sampa, has been extracted and aligned with the speech signal using JTrans [START_REF] Cerisara | JTrans, an open-source software for semi-automatic text-to-speech alignment[END_REF].

To evaluate the accuracy of the phone segmentation, an expert annotator performed a manual validation using Praat [START_REF] Boersma | Praat: Doing phonetics by computer[END_REF]. Since there is only one speaker, half an hour of the SynPaFlex Corpus was taken into account to evaluate the quality of the phone labels and boundaries. The set of data used for the evaluation task has been selected respecting the proportion of the different literary genres in the corpus.

Results related to the validation are presented in Table 2. We can observe that the Phoneme Error Rate (PER) is low for every literary genres, and the average PER is 6.1%.

Concerning the average alignment error, results are reported in the fourth column of Table 2. Globally, on average, the error is 11ms.

As far as errors on label assignment are concerned, they mostly occur on vocalic segments. Most of the deletion observed involve /@/ (83.31%), this phoneme being generally optional in French. Table 3: Amounts of linguistic units in the corpus is of 6 syllables/s on average. Secondly, the acoustic models used to achieve the automatic segmentation [START_REF] Cerisara | JTrans, an open-source software for semi-automatic text-to-speech alignment[END_REF] have been trained on the ESTER2 corpus [START_REF] Galliano | The ester 2 evaluation campaign for the rich transcription of french radio broadcasts[END_REF] which is a French radio broadcasts corpus. The resulting models could thus be slightly unadapted for poem reading data.

Linguistic Information

Additional linguistic information has been added to the corpus, such as syllables and Part-Of-Speech tags using the Stanford parser [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF]. Table 3 sums up the content of the corpus in terms of linguistic units. We also plan to include syntactic information in a near future.

Acoustic and Prosodic Information

The speech signal is stored using a sampling frequency of 44.1kHz. From the signal, we have extracted (i) the energy and 12 mel-frequency cepstral coefficients (MFCC 1-12) which we have added delta and delta-delta coefficients using [START_REF] Gravier | Spro: "speech signal processing toolkit[END_REF], (ii) the instantaneous fundamental frequency (F 0 ) using the ESPS get_f0 method implementing the algorithm presented in [START_REF] Talkin | A robust algorithm for pitch tracking (rapt)[END_REF], and (iii) pitchmarks using our own software. Additionally, we have added some prosody related features as the articulation rate (in syllables/s), the speech rate (in syllables/s), and F0 mean/min/max/range (in Hz) at the syllable and word levels.

Manual Annotations

Audio tracks corresponding to chapters of different books have also been annotated manually according to prosodic units, characters, emotions, and other events that were heard. The annotation method had first been defined on a small subset of readings, and then tested on audiobook recordings completed by other readers. It was found to be generic enough to render a global perceptive description of the speech. As Table 1 shows, 38% of the whole corpus have been processed manually to provide characters annotation, and 15% -included in those 38% -to describe emotional and prosodic patterns contents.

Prosodic Patterns

After considering the whole speech data, eight prosodic descriptors were defined, then encoded and assigned by an expert annotator to a large number of audio tracks corresponding to chapters of 18 different books, and defining a 13h25m sub-corpus. As far as possible, labels were assigned according to the perceived prosody, without taking into account the linguistic content. They characterize units which could range in length from a word to several sentences. Seven of them correspond to speech showing the following types of prosodic patterns: QUESTION (interrogative), NOTE, NUANCE, SUSPENSE, RESOLUTION (authority, or imperative), SINGING, and IDLE (no particular prosodic pattern, or declarative). The eighth label, EMO-TION, was used to report -but without describing it -the presence of any perceived emotional content. Let's notice as of now that the tag EXCLAMATION is not listed above. This is because this information can be simply deduced from another level of description: in this corpus, the Exclamation pattern was found strictly correlated with the emotional content of surprise, which is reported in the emotion labeling level (presented in Section 4.3.). Manual annotation is costly in time and redundancy is not desirable in its process. In the following analysis of the prosodic manual labeling, emotion labels surprise will therefore be assimilated to hidden prosodic labels for EXCLAMATION. Another important point is that, when needed for a more precise description, labels were combined (e.g. Emo-tion+Question+Nuance). Hence, simply summing the labels duration for each type of prosodic pattern gives a value which exceeds the duration of the sub-corpus. Among the prosodic parameters, the perceived pitch-curve during voice production takes an important role in assigning the labels. For instance, the NUANCE pattern, which is one of the reading strategy of the speaker, maintains listener's attention. This pattern is characterized melodically by a high pitch at the beginning, then a decrease with modulations, and finally a slight increase when it doesn't end the sentence (see Figure 1). Table 4 shows total duration for each manual prosodic labels in the 13h25 sub-corpus.

A non-IDLE prosodic tag has been assigned to 68% of the speech. As shown in More than a half of the speech showing particular prosodic figures is described with combined labels, pointing out where prosody may be more complex.

Most of all, it was found that the EXCLAMATION pattern happens very frequently, especially in narration. In a way, it is an inherent part of the speaker's style.

The generic EMOTION prosodic indicator is assigned to 39% of the whole sub-corpus (5h18m), showing a large amount of emotional data. Its manual description is presented in Section 4.3.

Characters

The speaker, who is the same for the whole corpus, can personify the different characters of the book by changing her voice or her way of speaking. The character's tags were identified from the text and any turn of speech has been labeled according to the following annotation scheme:

• CHARACTER ID: indicates which character is talking according to the text, and refers to Meta-data where each character is summarily described (name, age, gender, prosody and timbre features). For instance, to personify a gloomy man, the speaker uses a low pitch, low energy and devoiced voice.

• VOCAL PERSONALITY ID: indicates which character is talking according to the vocal personality. Indeed, even if the speaker is very talented and coherent along the books, she can for example forget to change her voice when comes a new character. Therefore, for such speech intervals, the timbre remains the own speaker's timbre or corresponds to another character.

The characters labeling was annotated on more than one third of the whole corpus (33h39m) mined from 18 different books. Dialogue tags were reported as parts of the narrator's speech.

First estimates indicate that one third of the speech content is personified. The average duration for speech turns being of 7s, against 29s for the narrator. In some chapters, direct speech segments can also be very long, typically when a character becomes a narrator who tells his own story. 370 characters were identified, and the full data of their vocal personality labeling indicates a not negligible amount of prosody and vocal tone personification. Covering a wide range and typology, the speaker's voice is thus more or less radically far from her natural style (males, children and elderly people embodiments, psychological singularization, imaginary figures). These vocal personality changes often happen: around 20% of the speech is concerned and, for the half, in stark contrast with the natural speaker's voice.

Emotions

Different theoretical backgrounds are classically used to identify emotional states, principally based on either distinct emotion categories or affective dimensions [START_REF] Cowen | Self-report captures 27 distinct categories of emotion bridged by continuous gradients[END_REF]. Usually, choosing the emotion categories and their number, or the emotion dimensions is an issue.

In the present study, the basic scheme used to manually encode emotions has three items:

• Emotion category: Six categories are available, those selected by the Basic Emotions theory [START_REF] Ekman | Basic emotions[END_REF]: SADNESS, ANGER, FEAR, HAPPINESS, SURPRISE, DISGUST. Two other categories were added to better represent the content of the different books: IRONY and THREAT.

• Intensity level: This item, on a scale from 1 to 3, is meant to give a measurement of the experienced emotion intensity according to the speech. For instance, one can interpret its values as follows: SLIGHTLY AN-GRY (1), ANGRY (2) , and STRONGLY ANGRY (3). The second and third items may have strong correlations with some of the widely used affective dimensions, as activation and arousal. Furthermore, an important feature of the manual emotion annotation used for the corpus is that the three items labels can be mixed together to provide a more precise description of the perceived emotion. For instance, speech can continuously convey strong and very expressive SADNESS as well as FEAR through some words, which could be tagged as [sadness-3-E + fear-1-E].

Manual emotion labeling was done on the same sub-corpus as for prosody (13h25m). A large amount of emotional content was reported (39% of the speech, including 13% with combined tags). Duration of tagged speech for each category of emotion is given in Table 6, and the number and average duration of labels are indicated in Table 7.

Significant observations have emerged during the annotation. A challenging one is that two radically different types of JOY can be conveyed by the speech, whereas none of the three items could take over their differentiation: on the one hand suave joy, and on other hand elation or gladness. Also, it is suggested that labels should be interpreted in context, notably in conjunction with the discourse mode. In particular, the expressive strategy implemented in the corpus narration is very specific, conveying almost continuously positive valence but in a subtle way, through pitch modulation and with focus words. The SURPRISE label was widely assigned to those recurrent patterns showing (i) a sudden pitch shifting upwards (ii) at least one accentuation onto the first syllable of a focus word (iii) a phonetic elongation or a short silence before this first syllable. Thus, as introduced in section (see 4 1 )SURPRISE describes a recurrent emotional attitude of the reader, attracting the listener attention by regularly emphasizing the text. Other types of variation occur when the speech conveys emotion, some examples are related Table 5 .

Other Events

Besides acoustic indications of loud noises or music, different unexpected speech events were also reported:

• Linguistic events: for example, the use of foreign languages;

• Phonetic events which are not written in the text: phoneme substitutions, elisions and insertions, high elongations, breaks and pauses, specific voice quality (e.g. whispered voice).

All these features can be of high interest for rendering a more human synthetic voice [START_REF] Campbell | Conversational speech synthesis and the need for some laughter[END_REF].

The manual data-sets could provide valuable guidance for further analysis, especially by linking with linguistic information, acoustic measurements, and other descriptions. Examining how manual labels are distributed among literary genres could also be of great interest.

Emotion classification

This section presents binary emotion classification experiments conducted on emotional labels of the SynPaFlex corpus. The use of a state of the art methodology aims at positioning our mono-speaker read expressive speech corpus among existing multi-speaker acted or spontaneous emotional speech corpora.

Data analysis

The manual segmentation and labeling of emotion -which concerns 15% of the whole corpus -results in a total number of 8 751 segments as shown in Table 7. Among them, 5 387 convey an emotional content, while 3 364 do not. To get around the issue of a "neutral" emotion, we decided to label these segments as Idle. As mentioned previously, label combinations were used during the annotation phase to better characterize some expressive content. Consequently, these annotations are considered as new emotional labels which can not be merged with single labels easily. A deeper investigation of these label combinations is needed in order to manage them in a speech synthesis system. Interestingly, the SURPRISE label is highly represented among other single emotional labels. Actually, as described in Section 4 , SURPRISE better corresponds to an emotional attitude of the reader to keep the listener's attention, than an emotion conveyed from the text. Emotional segments are defined as segments consisting of an homogeneous emotion, be it characterized by single or combined labels. Therefore, there is not constraint on segments' duration. As a consequence, some segments can be very long. For example, one IDLE segment lasts more than 43s. On average (cf Table 7), IDLE segments have the highest durations (8.76s), then comes SURPRISE segments (3.83s.) and COMBINATION labels (3.45s.).

Methodology

The following experiments aim at classifying the manual annotations with binary emotional models. We know that for multi-speaker acted emotions, classification rates usually reach high performance (for example with corpora such as EMO-DB). However, with multi-speaker spontaneous speech, the classification rates are much lower, thus reflecting the difficulty to discriminate emotions in such a context (Schuller et al., 2009b). The present corpus gives the opportunity to bring a new benchmark of performances on mono-speaker read speech.

To do so, our experimental set up follows a standard classification methodology (Schuller et al., 2009a;[START_REF] Schuller | Paralinguistics in speech and language-state-of-the-art and the challenge[END_REF]. By this way, our results are comparable with those obtained on other existing emotion corpora. In other words, emotional models are trained in cross-validation conditions (here 5 folds to keep enough data) on acoustic features. 384 acoustic features -16 Low-Level Descriptors (LLD) × 12 functionals + ∆ -are extracted on emotional segments with OpenSmile toolkit and Interspeech 2009 configuration (Schuller et al., 2009a). To avoid over fitting the data, different subsets of features are tested:

• OS192: 16 LLD × 12 functionals without ∆

• ∆ OS192: 16 LLD × 12 functionals with ∆ only

• OS24: 2 LLD (range + amean) × 12 functionals without ∆

In order to have homogeneous segment durations, we decided to chunk manual segments every 1 s keeping the remaining part. This operation helps in increasing the amount of data available for the experiment, as reported in Table 7.

As aforementioned, COMBINATION labels are not taken into account because merging them with single labels is clearly not obvious. Also, IRONY and THREAT segments are discarded regarding to the small number of labels. To better identify the pairs of labels that can be easily discriminated from those which can not, only binary models are trained thus resulting in an emotion confusion matrix.

The number of segments is equally balanced among the two classes. .580 .633 Dis.

Results

Models

.584

Table 8: Unweighted Average Recall (UAR) results for binary emotion classification using the three feature subsets.

In bold, UAR > 60%

Regarding the results obtained with the small OS24 feature subset, classification between non emotional (IDLE) and emotional segments is over 60% (bold font in Table 8) for ANGER, SADNESS, FEAR and DISGUST. Two emotion groups emerge from the results:

• IDLE/JOY (58.0%), IDLE/SURPRISE (56.3%) and JOY/SURPRISE (56.7%)

• SADNESS/FEAR (53.0%), SADNESS/DISGUST (54.8%), SADNESS/ANGER (56.7%), FEAR/DISGUST (58.0%), FEAR/ANGER (57.8%) and ANGER/DISGUST (58.0%)

The second group clearly contains negative emotions with different arousal levels. Further experiments are needed to deeper investigate these groups such as unsupervised clustering, feature selection, etc. For example, ∆ features are clearly relevant for ANGER/DISGUST classification. Moreover, emotions are likely to be strongly correlated with direct/indirect speech and also with characters. Additional analyses are required to confirm this observation. The addition of phonological and linguistic information could also help in understanding the emotional distribution of the SynPaFlex corpus.

Conclusion and Perspectives

This paper describes a new large speech corpus composed of eighty seven hours of audiobooks from several literary genres read by a single speaker. By being mono-speaker, this corpus can be used to study the strategy of a speaker over entire books. Annotations and speech segmentation into phones are also provided. Among them, we can mention that a manual annotation of emotional contents and characters has been done for respectively 15% and 38% of the whole corpus.

Preliminary emotion classification experiments show that the expressive read speech contained in the SynPaFlex corpus is much closer from spontaneous speech than acted speech. From binary classification results, two emotional groups emerge, one clearly containing negative content. Deeper investigations and analyses of the corpus are planned in a future work: correlation between direct and indirect phrases, emotional speech and characters. Further experiments on feature selection and clustering could also help in investigating the emotional content of this corpus. Moreover, the full corpus has already been used to build a speech synthesis voice, and informal evaluations show that the output of the unit-selection speech synthesis system is relatively good. The corpus is available on our website3 . This corpus could also be used to study prosodic and phonological aspects of expressive read speech (character personification, speaking styles) and to develop expressive synthesized speech of good audio quality. The amount of data available with the SynPaFlex corpus is consequent enough to allow carrying a deeper analysis of the linguistic, acoustic and prosodic features associated with some of these aspects. Machine Learning techniques such as deep learning can also be used to build prosodic models.

Table 2 :

 2 Validation results for the segmentation step per literary genre : lengths of the validation subsets, Phoneme Error Rate (PER), and average alignment error.

		Validation PER	average
		subset	(%) alignment
				error (ms)
	Novels	25m36s	5.8	11.5
	Short stories	3m49s	7.1	9.4
	Tales	2m47s	0.8	14.3
	Fables	1m47s	6.5	12.1
	Poems	1m07s	6.3	28.3
	Total	35m52s	6.1	11.4
	Unit type			Number
	Paragraphs			23 671
	Sentences			54 393
	Words			799 773
	Orthographically distinct words		38 651
	Phonemically distinct words		28 734
	Non Stop Words			411 210
	Syllables			1 154 714
	Distinct syllables			8 910
	Open			808 503
	Closed			346 211
	Phonemes			2 613 496
	Distinct phonemes			33

The majority of substitutions concern mid vowels (37.04% for the substitution of /E/ by /e/, and 31.04% for /o/ by /O/), these realizations being the result of a specific pronunciation or simply phonetization errors. As for boundary alignment, in 77.17% of cases, boundaries are misplaced from less than 20ms. In poems, however, errors in alignment are more important: in 35% of the vowels, boundaries have been shifted by more than 20ms. It could be explained by two distinct factors. First, the speech rate is relatively slow in poems (with an average of 5 syllables/s) in comparison to other literary genres where the speech rate

Table 4

 4

	Prosodic label EXCLAMATION IDLE	NUANCE	RESOLUTION SUSPENSE QUESTION NOTE
		(hidden label)
	Duration	4h42m	4h21m	3h58m	45m	41m	38m	39m
	Sub-corpus %	34.8%	32.2%	29.5%	5.6%	5.1%	4.7%	4.8%

, the hidden EXCLAMATION tag is very largely represented (more than 4h42), before the IDLE one (4h21m). The first particular prosodic pattern that comes after is NUANCE (3h58m), then come all the other prosodic patterns that are relatively well represented Figure 1: Manual annotations -NUANCE prosodic pattern example

Table 4 :

 4 Manual annotations -Total duration of prosodic patterns (including combinations) in the 13h25 sub-corpus and evenly distributed (around 40m): RESOLUTION, SUS-PENSE, QUESTION and NOTE. SINGING was found to be exceptional and is not reported here.

Table 5 :

 5 Examples of perceived impacts of emotion on the speech the way the emotion is rendered through the speech (discreetly, prudently / obtrusively, ostentatiously)

	Emotion	SURPRISE	SADNESS	JOY	ANGER	DISGUST	FEAR
	Effects on
	the first syllable accentuation disappearance		accentuation accentuation
	of focus word(s)
	Pitch median	high	low	according to	low	low	low
				joy type
	Pitch curve		flat	flat (suave joy)	flat or	flat or	flat
					top-down	top-down
	Rate		slow	according to	fast	fast on	varying with
				joy type		focus words fear intensity
	Loudness		low	loud		low
				(intense joy)
	Timbre changes		breath during	breath during		yes	yes
			the speech	the speech

• Introversion/Extroversion: This binary item reflects

Table 6 :

 6 Manual annotations -Total durations of emotion categories labels (including combinations) in the 13h25 subcorpus

	Emotion	IDLE	SURPRISE SADNESS JOY	ANGER DISGUST FEAR	IRONY THREAT
	Duration	8h11m	4h42m		44m	32m		31m	15m	11m	10m	3m
	Sub-corpus %	61.0%	35.0%		5.4 %	3.9 %		3.9%	1.9%	1.3%	1.2%	0.4%
	Emotion	Idle Anger	Joy Sadness	Fear Surprise Disgust Other Comb.	Total
	# Seg. manual	3 364	147	115	295		76	2 895	47	23	1 699	8751
	Avg. dur (s)	8.76	2.62 2.99	2.67	2.20	3.83	2.26	2.30	3.45	5.55
	# Seg. 1 s. max 30 989	447	397	929		199	12 794	125	0	0 45 880

Table 7 :

 7 Number of manual annotated emotional segments and segments resulting from a 1 s. max chunking. The latest are used in the classification experiments. Other includes IRONY and THREAT labels.

https://synpaflex.irisa.fr/

http://freelcs.sourceforge.net/

https://synpaflex.irisa.fr/corpus/

Acknowledgements

This study has been realized under the ANR (French National Research Agency) project SynPaFlex ANR-15-CE23-0015 and the LABEX EFL (Empirical Foundations in Linguistics) ANR-10-LABEX-0083.


Anne Baillot
email: anne.baillot@univ-lemans.fr

François Vignale
email: francois.vignale@univ-lemans.fr

EXPLORING THE DIGITAL HERITAGE OF READING: THE READ-IT PROJECT

in the years to come.

The goal of the READ-IT project is to build a unique large-scale, user-friendly, open access, semantically-enriched investigation tool to identify and share groundbreaking evidence about 18th-21st century Cultural Heritage of reading in Europe. READ-IT will investigate innovative ways of gathering new resources through crowd-sourcing and web-crawling as well as linking and reusing preexisting datasets, and try to defne a common denominator of European reading experiences across times and cultures. READ-IT will thus ensure the sustainable and reusable aggregation of qualitative data allowing an in-depth analysis of the Cultural Heritage of reading.

In terms of its structure, READ-IT consists in a consortium of 5 academic partners from 4 different countries (Institute of Czech Literature, Czech Academy of Sciences; Open University, London; Utrecht University, Netherlands; CNRS-IRISA, Rennes, France and Le Mans University for France). The consortium is led by Le Mans University (Professor Brigitte Ouvry-Vial). The Institute of Czech Literature coordinates the use cases and Open University the dissemination actions, thus taking a signifcant part in the defnition of the ontology which is at the core of the project. Utrecht University is in charge of the web interface, CNRS-IRISA is taking care of the core computer science research and Le Mans Université is managing the project, cataloging requirements and taking part in the defnition of the ontology, the data model and the vocabularies associated. The complete research team consists in a pool of 16 scholars 2 . Regarding history of reading practices, knowledge has signifcantly increased over the last decades about what, where and when people read. Still, two major questions remain unanswered: why and how do people read?

These two main research questions can also be decomposed in a series of subordinated questions:

 What kind of transaction does there exist between a reader and a text?  What role does the environment play in this transaction?  Is it possible to list and model the emotions caused by reading?  Have these emotions changed throughout time and space in Europe?  Is it possible to sketch out the portrait of something like the « European reader »?

In order to provide an answer to these questions, one must consider the act of reading as an individual experience with modalities specifc of each iteration of the experience, and not following one single pattern. This approach also implies that one must take into account not only the reading testimony itself, but also the circumstances or the context that accompany its production.

However, in doing so, one faces a major methodological issue, namely that reading is a brain activity that leaves no direct or tangible traces. Within the READ-IT project, the research team thus has to deal with direct or indirect testimonies, concomitant of the reading experience or memories which, in addition to their great heterogeneity in terms of reliability and completeness, will take the form of representations, narratives or reports. The major challenge here is to fnd a structure that allows to grasp different narratives, different modalities, different temporalities, and still be able to aggregate them in a fruitful way in order to address the research questions listed above.

Defining reading exiperiences and reading sitaations

Within READ-IT, we make a difference between reading experience and reading situation. This distinction is based on the media through which the experience or situation is transcribed or depicted. We speak of an experience when the testimony consists in a narrative or a report. This occurs mainly in textual sources and audio or video recordings. Conversely, we speak of a situation when the trace consists in a representation such as an image, painting, photograph, engraving … Broadly speaking, at this preliminary stage, we consider that a reading experience (or situation) is composed of 4 large groups of items. An experience (or situation) engages 1) a reader, 2) a text read in a certain context that can give rise to 3) certain modalities and 4) certain consequences or results. Each of these items can then be subdivided into lower levels according to the degree of granularity required to avoid ambiguities and ensure data quality. Information sought about the reader are mainly biographical (name, date of birth, place of birth, social status, religion, …) while information sought about the text read are mostly bibliographical (title, author, form, language, provenance, …). Circumstantial or contextual elements will inform about the date and place of the experience, whether it was public or solitary, about the material conditions (lighting, position), whether it was repeated or not. developed within the French ANR P-RECIHC project (2014-2017; http://eured.univ-lemans.fr).

Elements related to the effects or consequences of the reading will inform on its silent (or not silent) character, on its intensity and especially on the emotions caused by the reading.

Reader and text read answer the "what" question, while circumstances and effects address the "how. One of the major elements of the characterization of any reading experience or situation is that we always have in the one hand a reader and a text (which corresponds to the "what) and on the other hand circumstances (the "how).

This combination leads to a theoretical -and minimal -defnition of the reading experience/situation as a testimony (or trace) that provides at least one piece of information regarding the what and one piece of information regarding the how. In practical terms, the combination or association of at least two of the items listed above -as long as they belong to the what and the how-is potentially a reading experience/situation and, therefore can be indexed in the database through a data model and an ontology.

Workplan and soarces

The workplan in READ-IT follows two purposes: 1) identifying and describing reading experiences/situations; 2) connecting reading experiences/situations with their context.

The sources used or reused within READ-IT are multimodal and consist of textual sources (digitized but not necessarily ocr-ized), images (paintings, photographs, drawings, engravings …), audio recordings (transcribed), digital-born content (blogs, social medias). These sources have to be in the public domain or CC-like, already digitized, FAIR data principles compliant, available in English, French, German, Dutch, Italian or Czech and they must be dated from the 18th to the 21st century.

The two examples below show the kind of information READ-IT wants to retrieve, extract and index.

The extract presented here is part of a transcribed interview that can be found in an oral history project called "Memories of fction" where people are asked about their reading practices. A corpus like this one is of particular interest to the READ-IT team in order to address the challenge of automatically identifying scattered pieces of information that, once they will be gathered together, could lead to the characterization of a reading experience. In this example, we have a text read (what?): Voyage of the Dawn treader; a location where the experience took place (where?): Italy; circumstances (how?): on holiday with family; a date (when?): not before 1957 and some feelings expressed by the reader (effects?): "go to another world, which I have always loved".

After this identifcation phase, the next major technological challenge of READ-IT is to ensure that these dispersed elements, when confronted to the ontology and the data model, can be gathered and indexed in a unique consistent record.

An example illustrating this work step can be elaborated on based on the painting by Mary Cassatt below.

When it comes to images, the method is slightly different. In this case, the algorithms developed within the project will be required to recognize some pieces of information that could ultimately lead to the characterization of a reading situation, but due to the flter of the artist's perception, only a certain degree of certainty can be achieved. Here, the major challenge consists in making the computer recognize a potential interaction between one or several human beings and a written support, and only then to propose a maximum of description elements. In this painting, we can see two individuals, seating outside in the daylight, possibly holding a book -and for that reason, maybe reading.

As a consequence, computers are only asked to identify candidate images. After that, human beings brought together via crowd-sourcing campaigns will validate or not the machine's interpretation and then contribute to its training.

Technological foandations

Technically speaking, the IT architecture and infrastructure follows the schema above. Two different kinds of sources are envisioned: community generated content and, more important here, curated content emanating from cultural institutions or other research projects (Memories of Fiction, UK-RED …).

The goal is to gather and connect state-of-the-art technology in semantic web, linked data and multimedia analytics in order to provide structured descriptions of reading experience resources and to facilitate reuse and interoperability with related web data across languages. The choice of technologies and standards, such as RDF and SPARQL, combined with ontology formalisms and widespread vocabularies, allow READ-IT to publish its structured data based on a highly expressive and universally understood data model. This will facilitate the conception and implementation of fexible yet powerful interfaces for search and exploration purposes, allowing groups of users with different research agendas to engage with READ-IT resources. Text mining and multimedia information retrieval are intended to be extensively exploited: natural language processing (NLP) will play a pivotal role in identifying new sources of information on the web; multimodal content analysis and image retrieval will enable the curation of rich multimodal resources; entity detection and linking in key languages (English, French, German, Italian) will facilitate the description and curation of data.

READ-IT and caltaral heritage institations

The realization of this ambitious endeavor depends on the availability of curated digital collections. The three major reasons for this desideratum are the following. First, curated collections are organized, described and often indexed. In addition to that, Intellectual Property Rights issues are most often dealt with. Second, these collections are more than likely to contain the traces or the testimonies needed in READ-IT in order to construct part of the answer to the research questions we raised about the how and the what that we want to answer. Third, READ-IT needs a considerable amount of multimodal data to train the machine and improve the performance of the algorithms that will be implemented.

In exchange to the access to this digital heritage of reading, this project aims at helping cultural institutions such as libraries or museums to highlight their own collections by giving its users the possibility to annotate and enrich their collections (making use of the possibility to create special digital collections about reading within the READ-IT portal, or building one's own). APIs developed within READ-IT will be made open source-available. They should be made replicable and transposable to other esthetical experiences such as music or artworks.

1

  http://www.jpi-culturalheritage.eu/ 2 READ-IT has been inspired by the 2006 pioneering manually entered UK-Reading Experience Database (http://www.open.ac.uk/Arts/reading/UK) and by the EuRED proof-of-concept database READ-IT and history of reading practices.

Illustration 2: Mary Cassatt. Nurse reading to a little girl, 1895, Metropolitan Museum of Art, New-York

Acknowledgments.

This work has been funded by the Agence Nationale de la Recherche (ANR-17-JPCH-0001-01).


Alice Mizrahi

Nicolas Locatelli

Rie Matsumoto

Akio Fukushima

Hitoshi Kubota

Shinji Yuasa

Vincent Cros

Joo- Kim

Julie Grollier

Damien S Querlioz

Magnetic Stochastic Oscillators: Noise-Induced Synchronization to Underthreshold Excitation and Comprehensive Compact Model

Keywords: Spintronics, bioinspired computing, stochastic resonance, magnetic tunnel junctions

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

I. INTRODUCTION

M AGNETIC tunnel junctions are considered to be a breakthrough technology for non volatile memories as the cell of magnetic random access memories (MRAM). However, they can also be engineered to be stochastic oscillators powered by noise, in which case they are called superparamagnetic tunnel junction (SMTJs) [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF]- [START_REF] Cheng | Nonadiabatic Stochastic Resonance of a Nanomagnet Excited by Spin Torque[END_REF]. It has been shown that these junctions can exhibit stochastic resonance [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF]- [START_REF] Cheng | Nonadiabatic Stochastic Resonance of a Nanomagnet Excited by Spin Torque[END_REF], a phenomenon where noise enables a non-linear system to detect or synchronize on weak signals [START_REF] Gammaitoni | Stochastic resonance[END_REF], [START_REF] Bahar | Phase synchronization and stochastic resonance effects in the crayfish caudal photoreceptor[END_REF]. Neuroscience research suggests that our brain takes advantage of stochastic resonance to perform more uncertainty tolerant and less power consuming computations [START_REF] Samardak | Noise-Controlled Signal Transmission in a Multithread Semiconductor Neuron[END_REF], [START_REF] Mori | Noise-Induced Entrainment and Stochastic Resonance in Human Brain Waves[END_REF]. In addition to their ability to harness noise, the small size and CMOS compatibility of magnetic tunnel junctions makes them promising candidates to implement novel computing schemes [START_REF] Locatelli | Spin-torque building blocks[END_REF]. By harnessing thermal energy, bioinspired computing applications based on SMTJs would tackle the issue of increasing power consumption and poor resilience to noise in processors. However, using noise-induced phenomena in SMTJs for bioinspired application requires a better understanding of the behavior of SMTJs and in particular of how they can be integrated in hybrid SMTJ/CMOS circuits.

Because of the utility of MTJs for memory applications [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF], [START_REF] Kawahara | Spin-transfer torque RAM technology: Review and prospect[END_REF], there is a lot of interest in developping so-called "compact models" of MTJs that can be used along with CMOS within standard integrated circuit design tools such as the Cadence platform [START_REF] Vincent | Analytical Macrospin Modeling of the Stochastic Switching Time of Spin-Transfer Torque Devices[END_REF]- [START_REF] Panagopoulos | Physics-Based SPICE-Compatible Compact Model for Simulating Hybrid MTJ/CMOS Circuits[END_REF]. Nevertheless, these models consider traditional MTJs in their switching regime and not SMTJs. Therefore, in this work, we propose a compact model of superparamagnetic tunnel junction and demonstrate its validity by comparing simulations to experimental results.

Corresponding author: A. Mizrahi (email: alice.mizrahi@u-psud.fr).

This paper is organized as follows: we introduce SMTJs and their behavior as noise-powered stochastic oscillators, we present our compact model and compare it with a simple experimental characterization of a SMTJ, and finally we confront our model with experimental results in the case of a complex phenomenon: noise-induced synchronization.

II. SUPERPARAMAGNETIC MTJS A NOISE-POWERED

OSCILLATORS

MTJs are composed of two magnetic layers separated by a tunnel barrier. The magnetization of the top magnetic layer (free layer) has two stable states: parallel or anti-parallel to the magnetization of the lower magnetic layer (reference layer), which is pinned (Fig. 1). Parallel magnetizations (P state) lead to a low electric resistance (R P ) whereas anti-parallel magnetizations (AP state) lead to a high electric resistance (R AP ). MTJs are mainly used for data storage [START_REF] Chappert | The emergence of spin electronics in data storage[END_REF], [START_REF] Kawahara | Spin-transfer torque RAM technology: Review and prospect[END_REF] and are therefore usually designed to have a high energy barrier separating the P and AP state. On the contrary, SMTJs are engineered to have a low energy barrier, so that the thermal energy at room temperature is high enough to induce switching of the magnetization between the P and AP states (Fig. 1(b)). SMTJs thus behave as stochastic oscillators (Fig. 1(c)) which operate with thermal noise and do not require any external supply of energy [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF]. The SMTJ is characterized by its dwell times, which are the time intervals spent in the AP and P states (Fig. 1(c)). We work with the assumption that the free layer can be seen as a single domain magnetization element, therefore its magnetization reversal can be considered within the Kramer's transition rates theory [START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF]. The magnetization has a probability to switch from the state AP (P) to the state P (AP) between time 0 and time t: as described by the Néel-Brown model [START_REF] Brown | Thermal Fluctuations of a Single-Domain Particle[END_REF], [START_REF] Li | Thermally assisted magnetization reversal in the presence of a spin-transfer torque[END_REF]. Thus the dwell times follow a Poisson process of characteristic time < τ AP/P > which corresponds to the mean dwell time in the AP (P) state. Therefore, even though the SMTJ does not oscillate with a constant period we can define its mean number of oscillations by second -or natural frequency:

P AP/P (t) = 1 -exp(- t < τ AP/P > ), (1)
F = 1 < τ AP > + < τ P > . (2)
The mean dwell times are given by the Arrhenius equations: < τ AP/P >= τ 0 exp( ∆E k B T ) where τ 0 is the effective attempt time, ∆E is the energy barrier between the AP and P states, k B is the Boltzmann constant and T is the temperature. The stability of both states can be influenced magnetically and electrically. The application of an external magnetic field parallel (anti-parallel) to the magnetization of the reference layer (i.e. the easy axis of the junction) stabilizes the parallel (antiparallel) state by modifying the energy landscape. Through spin transfer torque (STT) effect, a positive (negative) current injected in the SMTJ destabilizes the parallel (anti-parallel) state [START_REF] Slonczewski | Current-driven excitation of magnetic multilayers[END_REF]- [START_REF] Slonczewski | Currents, torques, and polarization factors in magnetic tunnel junctions[END_REF] -as illustrated on Fig. 1(a-b) -which can be described as a modification of the effective temperature [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF], [START_REF] Li | Thermally assisted magnetization reversal in the presence of a spin-transfer torque[END_REF]:

< τ AP/P >= τ 0 exp( ∆E kT (1 ± V V c )(1 ∓ H -H 0 H k ) n ), ( 3
)
where V is the voltage across the junction, V c is the threshold switching voltage, H is the external magnetic field applied along the easy axis of the junction, H k is the anisotropy and n is a real number exponent. H 0 is the residual stray field from the reference layer. Optimal fabrication of the junction with a perfectly balanced synthetic antiferromagnet (SAF) would lead to a situation where H 0 = 0, but in practice H 0 can reach several Oersteds. The SMTJ used for the experimental results in this work are elliptical pillars of 60 × 180 nm 2 composed of a SAF trilayer of CoFe (2.5 nm)/Ru (0.85 nm)/CoFeB (3 nm) as reference layer, an MgO tunnel barrier (1.05 nm), and a CoFeTiB (2 nm) free layer (Fig. 1(a)). Twelve samples were studied, here we present we results for two of them.

III. COMPREHENSIVE COMPACT MODEL FOR FAST SIMULATIONS AND SAMPLE CHARACTERIZATION

It is important to have models of SMTJs that are compatible with the tools used by both academic and industrial researchers to design integrated circuits including CMOS technology. In consequence, we propose a compact model written in the VerilogA language, that can be used within standard design tools, such as the Cadence Spectre simulator. In order to perform fast simulations, this model is not based on the full magnetic Landau-Lifshitz-Gilbert equation but on the Néel-Brown model presented in section II. At each step the program computes the mean dwell-times < τ AP > and < τ P > and the corresponding probability to switch from the current state to the other, according to equations ( 1) and (3). A random number is generated to take the decision to switch or not. The model does not take into account the dynamics of the magnetization inside each energy well (intra-well dynamics) and hence does not take switching duration into account. Therefore the model is valid if the dwell-times are large compared to the time scale of the intra-well dynamics, which will be the case in this paper because < τ AP/P > is larger than 10 -5 s whereas intrawell dynamics have time constant in the order of nanoseconds [START_REF] Vincent | Analytical Macrospin Modeling of the Stochastic Switching Time of Spin-Transfer Torque Devices[END_REF]. In order to have time efficient simulations and not lose the advantage of the compact model, it is important that the time step is not too small. On the other hand, to accurately reproduce the Néel-Brown model, the time step must be small compared to the dwell times. We use the VerilogA command boundstep which allows to set a variable upper limit on the time step. A satisfying compromise is to compute boundstep as a hundredth of the current mean dwell time at each step. The model allows the user to set all the parameters present in equation ( 3), as well as the resistance of the parallel and anti-parallel states. The influence of the field like torque can also be taken into account by modifying the effective magnetic field in equation ( 3) [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF] but is neglected in this paper.

We test our model by comparing it with the experimental characterization of a SMTJ. In sample n1, R P 130 Ω and R AP 165 Ω. The measurements are conducted at room temperature. Fig. 2(a) represents the mean dwell times for the parallel and anti-parallel states as functions of the injected DC current, for various external magnetic fields, at room temperature. Fig. 2(b) illustrates the magnetic field dependency of the SMTJ. The injected current for which the probability to be in each sate AP or P is 50% (< τ AP >=< τ P >) is plotted as a function the applied magnetic field. For both cases, we observe that the simulations results (solid line) match the experimental results (symbols). Three parameters are extracted from the experimental data then finely tuned to obtain this match: ∆E/kT = 11.3, V c = 0.18 V and H k = 57 Oe. We set τ 0 = 1 ns [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF] and n = 2 which are the appropriate values for a single domain particle with uniaxial anisotropy [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF], [START_REF] Victora | Predicted time dependence of the switching field for magnetic materials[END_REF]. These results suggest the validity of the Néel-Brown model with spin transfer torque to describe SMTJs as well as its implementation in our compact model. Fig. 2(b) enables us to extract the value H 0 = 7.8 Oe for the sample presented here. As this is usually not optimal for applications, we experimentally apply an external magnetic field along the easy axis of the junction in order to compensate H 0 .

IV. CONFRONTATION OF THE COMPACT MODEL WITH A COMPLEX PHENOMENON: NOISE-INDUCED SYNCHRONIZATION

Here, we want to prove the validity of our compact model for the understanding of more complex phenomena, with view of exploring bioinspired computing applications. Therefore, we confront our model to experimental results that exhibit highly non linear behavior: stochastic resonance and noiseinduced synchronization of a SMTJ [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF], [START_REF] Locatelli | Spintronics devices as key elements for energy-efficient neuroinspired architectures[END_REF]. Stochastic resonance has been observed in spin valves [START_REF] Finocchio | Micromagnetic understanding of stochastic resonance driven by spintransfer-torque[END_REF], [START_REF] Aquino | Stochastic resonance in noise-induced transitions between selfoscillations and equilibria in spin-valve nanomagnets[END_REF] and more recently in MTJs [START_REF] Cheng | Nonadiabatic Stochastic Resonance of a Nanomagnet Excited by Spin Torque[END_REF] as a mean to detect a weak AC signal. In Ref. [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF] we demonstrated that stochastic resonance can be used to synchronize a SMTJ to a weak periodic current signal. Here, we reproduce these experimental results with Cadence Spectre simulations based on our compact model. Experiments are conducted with the sample n2, a SMTJ for which R P 150 Ω and R AP 200 Ω. At room temperature, a square periodic current is applied to the SMTJ, with a subthreshold amplitude. This means that the STT effect itself is not sufficient to trigger the switching, but instead modulates the probability for the magnetization to switch. The switches are thus only allowed by the presence of thermal noise and are thus of stochastic nature. We monitor the resistance of the SMTJ for different current amplitudes and frequencies in order to perform a time resolved analysis of its response.

We use two criteria to quantify synchronization, both represented on Fig. 3. The first criterion is the synchronization rate (Fig. 3(a)), which corresponds to the proportion of time spent in the same state (up/AP or down/P) by both the excitation and the oscillator. The second criterion is the excitation influence on the frequency of the stochastic oscillator (Fig. 3(b)). For both criteria, we observe that simulations results (lines) match experimental results (symbols). The following values are used the simulations: n = 2, τ 0 = 1 ns, ∆E/kT = 16.2 and V c = 0.25 V. The values for ∆E/kT and V c were extracted from prior experimental data then finely tuned.

At high input frequencies the excitation is too fast to be followed by the oscillator. Therefore the synchronization rate is low (Fig. 3(a)) and the oscillator frequency does not vary with the excitation frequency and only depends on the current amplitude (Fig. 3(b)). As the input frequency decreases, the system is increasingly able to follow the excitation. In consequence, the matching time increases (Fig. 3(a)) and the oscillator frequency is pulled toward the excitation frequency (Fig. 3(b)). When the excitation frequency is very small compared to the natural frequency of the oscillator at the considered amplitude, supplementary switches (glitches) appear [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF]. These are very short oscillations of the SMTJ, leading the oscillator frequency to rise above the excitation frequency (Fig. 3(b)). Because glitches are very short, they do not significantly affect the synchronization rate, which keeps increasing (Fig. 3(a)). These behaviors are the signature of stochastic resonance [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF], [START_REF] Neiman | Stochastic resonance: Noise-enhanced phase coherence[END_REF]- [START_REF] Freund | Frequency and phase synchronization in stochastic systems[END_REF].

Higher current amplitudes allow wider frequency ranges of synchronization (as the natural frequency of the oscillator is increased) and higher synchronization rate. We observe here that noise-induced synchronization is achieved even at amplitudes as low as I ac = 100 µA, less than a tenth of the required current to obtain deterministic switching

I c = V c * R 1.3 mA.
Noise-induced synchronization of superparamagnetic tunnel junctions is hence promising for low power computing applications involving synchronization.

V. CONCLUSION

We have proposed a compact model for superparamagnetic tunnel junctions and shown its validity by reproducing experimental results in the case of a complex phenomenon. Moreover, the model allows predicting the behavior of SMTJs with different parameters and therefore will be useful for designing optimal SMTJs for future applications. For instance smaller energy barriers -which can be achieved at smaller junction dimensions -lead to higher synchronization frequencies, while lower V c allows achieving synchronization at even lower power consumption. This model allows for fast simulations: the totality of the results presented in this paper can be computed in one hour on a standard Intel Xeon CPU. Therefore, the model will enable exploring bioinspired applications involving CMOS and several SMTJs, such as associative memories based on arrays of synchronized SMTJs [START_REF] Csaba | Spin torque oscillator models for applications in associative memories[END_REF].

Fig. 1 .

 1 Fig. 1. (a) Schematics of a MTJ. (b) Spin transfer torque and thermal switching in a SMTJ: the energy of the system is plotted versus the magnetization state of the free layer. (c) Experiments: Resistance of a SMTJ as a function of time, featuring the dwell-times in the Anti-Parallel (red) and Parallel (green) states. The asymmetry between the P and AP states is due to the spin transfer torque effect generated by a small DC current into the junction.

Fig. 2 .

 2 Fig. 2. Experiments on sample n1. (a) Mean dwell times for the antiparallel (up triangle) and parallel (down triangle) states as functions of the applied current for various applied magnetic fields. (b) Current for which < τ AP >=< τ P > as a function of the applied magnetic field. H 0 is the necessary field to compensate the residual field of the SAF. For both graphs experimental measurements (symbol) are compared with simulations based on the compact model (lines).

Fig. 3 .

 3 Fig. 3. Experiments on sample n2. (a) Measurements of the synchronization rate of the SMTJ with subthreshold inputs, as a function of input frequency. (b) Mean response frequency of the SMTJ as a function of input frequency. For both graphs, experimental measurements (symbols) are compared with simulations based on the compact model (lines). On the dashed line, the frequency F of the stochastic oscillator is equal to the frequency Fac of the excitation.

ACKNOWLEDGMENT

The authors acknowledge financial support from the FET-OPEN Bambi project No. 618024. A. M. acknowledges financial support from the Ile-de-France regional government through the DIM nano-K program.