Blandine Feneuil
email: bffeneui@math.uio.no

Patrick Aimedieu

Mario Scheel

Jonathan Perrin

Nicolas Roussel

Olivier Pitois

Stability criterion for fresh cement foams 1

Keywords: foam, rheology (A), cement paste (D), micro-tomography 19

We prepare and study cement foam samples with well-controlled structure, 9 i.e. containing monodisperse bubbles. We observe that the foam structure often 10 changes before cement setting and identify ripening as the major destabiliza-11 tion mechanism at stake. Drainage plays only a minor role in cement foam 12 destabilization except when bubble size is large. Then we show that a single 13 stability criterion can be defined, for a large range of cement foams with differ-14 ent formulations. This criterion involves the bubble radius and the yield stress 15 of the cement paste such as confined by and between the bubbles, at a given 16 characteristic time after sample preparation.

Introduction

When it is unconstrained, a bubble has a spherical shape because of the air-liquid surface tension. Though, in a foam, bubbles are deformed by their neighbors. The structure of a foam was studied in 1873 by Joseph Plateau, who stated three laws known as Plateau's laws [START_REF] Cantat | Foams -Structure and dynamics[END_REF]: [START_REF] Cantat | Foams -Structure and dynamics[END_REF] two bubbles are separated by a soap film of constant average curvature, (2) three films join in a channel, called Plateau border, forming 120°angles, (3) four Plateau borders join into a node at angle 109.5°. The resulting morphology of foam tends to evolve with time due to downward flow of the interstitial fluid due to gravity (drainage), air exchange between bubbles (ripening) and film breakage (coalescence).

In cement foams, these destabilization mechanisms are expected to occur until cement paste hardening. Thus, to control the final bubble size and air distribution in the foam, one has to stop or slow down the three mechanisms.

These mechanisms are affected by initial bubble size. For instance, the increase of bubble size for a given gas volume fraction results in the increase of the size of the film areas between the bubbles, which enhances coalescence. It also increases the size of the Plateau borders and nodes, which favors drainage [START_REF] Cantat | Foams -Structure and dynamics[END_REF].

Ripening, on the contrary, is reduced when bubble size increases. Indeed, it is caused by the capillary pressure inside the bubbles P c ≈ 2γ/R 0 , where γ is the air-liquid surface tension and R 0 the bubble radius.

To avoid coalescence, liquid film must be stabilized by molecules or partially hydrophobic particles, which adsorb at air-water interfaces. The molecules, called surfactants, must be compatible with the highly alkaline cement solution and be present in sufficiently high amount [START_REF] Cantat | Foams -Structure and dynamics[END_REF].

Consistency of cement paste is also expected to play a major role in foam stability. High yield stress can stop drainage and ripening [START_REF] Guignot | Liquid and particles retention in foamed suspensions[END_REF][START_REF] Lesov | Factors controlling the formation and stability of foams used as precursors of porous materials[END_REF]. However, we have shown in a previous paper [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF] that, in a cement foam, the effective yield stress of the cement paste confined between the bubbles, noted τ y,int , can differ significantly from the reference yield stress of the cement paste τ y,0 , measured in the bubble-free paste. On the one hand, when τ y,0 is low, i.e. a few Pascals, cement grains remain stuck in the channels and nodes between the bubbles, whereas gravity makes the liquid flow to the bottom of the foam. This drainage of the liquid leads to a decrease of the water-to-cement ratio of the interstitial cement paste, and therefore, to an increase of the interstitial yield stress τ y,int up to about 100 Pa. On the other hand, when the yield stress of the reference cement paste τ y,0 is high, i.e. a few tens of Pascals, no densification of the cement paste through drainage occurs, so that τ y,int ≈ τ y,0 during the first 10 min after sample preparation. For the cement foam formulations studied in [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF], cement paste densification through liquid drainage was found to be essential to ensure the foam stability.

Therefore, the stability of fresh cement foams is expected to be observed for pastes with sufficiently high yield stress values, but stability can be observed also for low yield stress values. The aim of this paper is to reconcile those contradictory results and to propose a single stability criterion for cement foams.

In the materials and methods, we describe how we prepare cement foams with controlled morphology and formulation, which allows for the factors controlling the stability of these cement foams to be investigated. First, the leading destabilization mechanism is identified. Then, the effects of bubble size and of cement paste yield stress are investigated. Finally, a global criterion for cement foam stability is defined.

Materials and methods

Materials

Cement

We use two cements. The first will be referred to as C1, it is manufactured by Lafarge, in Saint-Vigor factory and C2 is a CEM I cement from Lafarge, Lagerdorf. Their compositions and physical properties are specified in Table 1.

Surfactants

Two surfactants are used to produce the precursor foam. Tetradecyltrymethyl ammonium bromide (TTAB) is a cationic surfactant at purity above 99% pro- anionic surfactant provided by Stepan. Its molar mass indicated by the manufacturer is 382 g/mol and active content 68-72%. Surfactant chemical formulas can be found in [START_REF] Feneuil | Effect of surfactants on the yield stress of cement paste[END_REF].

Previous study showed that both surfactants are able to form stable foams in the highly alkaline cement paste interstitial solution [START_REF] Feneuil | Effect of surfactants on the yield stress of cement paste[END_REF].

Methods

Precursor foam and mixing

Cement foams are prepared by mixing precursor aqueous foam and cement paste. Precursor foams are generated using the method described in [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF]. Here we recall the main outlines of the method: bubbles are generated with a T-junction (inner diameter ≈ 100 µm). All the bubbles have approximately the same radius R 0 (the relative dispersion was measured to be ∆R 0 /R 0 ≈ 3%) which is set by the entrance rates of the gas (nitrogen) and the foaming liquid (water and surfactant). Bubbles are collected in a column, where liquid fraction is controlled thank to addition of foaming liquid from the top of the foam. Studied bubble radii are comprised between 200 µm and 900 µm. In TTAB precursor foams, TTAB concentration is 10 g/L and liquid fraction is between 0.5 and 2%. In Steol precursor foams, Steol concentration is 1 g/L, and liquid fraction is 1.6±0.1% when bubble radius is below 350 µm and 1.4±0.1% otherwise.

Mixing of cement paste and precursor foam is carried out with a flow focusing device as described in [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF]. The main advantage of this method is that bubbles are not broken during the mixing process. Gas volume fraction in the cement foams depends mainly on the flow rates of both the precursor foam and the cement paste. It is however also affected by gas compressibility. By weighting our samples after preparation, we check that air fraction for all of them is between 81% and 84% (average value Φ = 83%). Note that the flow focusing method involves the flow of the cement paste in small channels (2 mm diameter)

and that therefore, the yield stress of the paste must be limited to a few tens of Pascals to avoid jamming in these channels.

Each sample is obtained by filling a mold (diameter 2.6 cm, height 6 cm), through a layer-by-layer deposition process. Therefore, cement foam does not undergo strong shearing after its preparation, and bubbles are not broken during sample production.

Protocol

For each surfactant, all cement foam samples are prepared using the same mixing procedure, from water and cement mixing to casting. In the case of Steol samples, large amount of surfactant is added to cement paste 20 minutes after cement paste preparation to make the consistency of the paste decrease [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF]. For both surfactants, precursor foam and cement paste are mixed 30 min after cement paste preparation. Our protocol is schematized in Fig. 1.

Observation of stability

Samples are demolded one week after casting and the stability is visually assessed from the final morphology of the cement foam. Sample stability is evaluated according to the scale described below. Illustrations of the scale can be found in [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF]. • 3: fully stable sample. All bubbles have kept their original size

• 2: large stable area. In some areas (oftern the top of the bottom of the sample) the bubble size has changed before cement hardening, but most of the sample is stable

• 1: small stable area(s). The morphology of most bubbles has evolved, yet, one or several area(s) containing at least several tens of bubbles can be observed.

• 0: fully unstable sample, no stable area.

Moreover the evolution of the bubble morphology of fresh cement foams is followed across the transparent molds. Note however that parietal bubbles are not fully representative of the bulk bubbles. In particular, Plateau borders are larger. For samples with large bubble size or low cement paste yield stress, this leads to the full coverage of the mold walls by cement paste. Obtained images are analyzed with freeware ImageJ program [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF] to calculate the apparent radius of the parietal bubbles.

Besides, several samples are prepared by following a different procedure in order to stop one or two destabilization mechanisms. To prevent ripening, we use perfluorohexane saturated nitrogen instead of pure nitrogen [START_REF] Cantat | Foams -Structure and dynamics[END_REF][START_REF] Bey | Stabilization of foams by the combined effects of an insoluble gas species and gelation[END_REF]. Chemical formula of perfluorohexane is C 6 F 14 and it has a very low solubility in water, which reduces the global gas transfer rate between the bubbles. In addition, the effects of drainage were significantly reduced by making the samples rotate at 10 rpm around a horizontal axis for several hours after preparation.

Stability of aqueous foam

To check the ability of surfactant to stabilize the foam films during several hours, we prepare an initial height h 0 =11 cm of aqueous foam in a glass column with diameter 2.6 cm. The generation method is the same as described in part 2. Note that the presence of perfluorohexane tends to make the foam swell.

Indeed, there is C 6 F 14 in the bubbles but not in the air in the column above the foam. Difference in C 6 F 14 chemical potentials leads to transfer of nitrogen and oxygen from the air above the foam to the top bubbles. This swelling effect, which intrinsically results in a slight increase of foam height h, has been disregarded in the reported values of h.

X-ray tomography

Several cement foams have been studied by X-ray tomography one or two months after the preparation. Two types of experiments have been performed.

Images of one whole 2.6-cm-diameter and 6-cm-high sample and an addi- Pores appeared black on the reconstructed images and interstitial cement pate was light grey. This allowed us to analyze the images with the freeware Im-ageJ program [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF] to compute the gaz volume fraction of the sample and the pore size distribution using the following procedure: first, a closing filter from

MorphoLibJ plugin [START_REF] Legland | MorphoLibJ: integrated library and plugins for mathematical morphology with ImageJ[END_REF] with a 5 voxel-radius ball element was applied to reduce noise from the images. Then, image threshold was calculated using the Otsu method [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF]. On the one hand, gas volume fraction (see Fig. 9) was deduced from the number of black pixels in each binarized horizontal slice. The binary images were also used to obtain the 3D visualization of the sample shown in Fig. 2, top, with the 3D Viewer plugin. On the other hand, further processing was required to obtain the pore size distribution. 3D Watershed from MorphoLibJ was applied, then 1-voxel dilatation filter. Finally, 3D Object Counter plugin [START_REF] Bolte | A guided tour into subcellular colocalization analysis in light microscopy[END_REF] returned the volume V p of the pores. The obtained equivalent pore radii shown in Fig. 2, bottom, were deduced from V p under the assumption of spherical pore shape. "Exclude on Edge" option prevented the incomplete bubbles on the image edge to be taken into account. Histograms in Fig. 2 confirms that that the pore size distributions obtained for two samples are narrow around the initial bubble radius, which validates our preparation protocol. The microstructures of the elaborated materials have been characterized by means of synchrotron X-Ray microtomography on the recently constructed Anatomix beamline [START_REF] Weitkamp | The tomography beamline ANATOMIX at Synchrotron SOLEIL[END_REF] of the Soleil synchrotron facility, located in Saint Aubin,

France, where we could add our sample during the measurement time of the proposal 20171213. Anatomix is a 200 m-long beamline based on a cryogenic in-vacuum undulateur (U18). The tomography setup was therefore still in a temporary state and was composed of a small air-bearing rotation table from LAB (RT150v3) equipped with a small manual goniometer head from X-Huber, which maintained a 20 mm long small PMMA rod at the end of which the sample, approximately 5 mm in diameter, was glued. X-Ray radiographs of the latter were obtained with a 20 µm-thick LuAG scintillator mounted on a right angle kinematic cage from Thorlabs, containing a mirror reflecting the optical image towards the CMOS sensor of a ORCA flash 4.0 v2 camera from Hamamatsu, through a 10x microscopic objective from Mitutoyo, with a numerical aperture of 0.28 and a working distance of 33.5 mm. The optical definition of the sensor is 2048×2048 pixels and the pixel size is 6.5 µm, inducing a voxel size of the tomographic reconstructions, assuming a perfect parallel projection, of 650 nm. The undulator was set with a 7 mm gap and the beam was filtered by both a 10 µm Au filter and a 200 µm Cu filter, resulting in a large band pink beam centered on about 30 keV. Images were recorded synchronously with the continuous rotation of the table, at a frequency of 1 Hz, so that 2000 radiographs were recorded over a 180°total angle, in about 35 minutes. In addition, 50 so-called "dark" images were recorded in the same conditions without the beam and 50 so-called reference images were recorded in presence of the beam but without the sample. The averages of these darks and references allowed us to compute the attenuation of the sample, assuming classically an affine dependance of grey levels with attenuation. Finally, 3D tomographic reconstruction were performed at laboratoire Navier with the X-Act commercial software developed by RX-Solutions. Obtained volumes were images of 2048 3 voxels, coded on 16-bits, providing 3D attenuations maps over a 1.33×1.33×1.33 mm 3 subvolume of the samples. Even if their voxel size is of 650 nm, their actual spatial resolution is expected to be closer to 1 µm in their central part. Because of some residual imperfections of the manual alignment of the scintillator and the optics, images were slightly unfocused on their lower and upper parts, but the central zone of about 2048×2048×1200 voxels were of excellent quality.

Properties of the reference cement pastes 2.3.1. Yield stress

We estimate the yield stress of the reference cement pastes, i.e. prepared following the protocol described in 2.2.2 with addition of foaming solution (without bubbles) instead of foam. Yield stress is measured using simple spread tests:

the paste is poured on a flat horizontal surface and the yield stress is obtained by the following formula [START_REF] Roussel | Fifty-cent rheometer for yield stress measurements: From slump to spreading flow[END_REF]:

τ y,0 = 225ρgΩ 2 128π 2 R 5 spread (1)
where ρ, Ω and R spread are respectively the density, the volume and the average radius of the spread cement paste. Note that this formula requires that 1 P a τ y,0 100 P a [START_REF] Roussel | Fifty-cent rheometer for yield stress measurements: From slump to spreading flow[END_REF][START_REF] Feneuil | Effect of surfactants on the yield stress of cement paste[END_REF].

TTAB cement foams are prepared with C1 cement. We have seen in a previous study [START_REF] Feneuil | Effect of surfactants on the yield stress of cement paste[END_REF] that TTAB partially adsorbs on cement grains and has only a small effect on the yield stress. In such a case, the cement paste yield stress depends mainly on the final water-to-cement ratio W/C f . Water-to-cement ratio of the cement paste before mixing with the precursor foam is W/C i =0.37. Within our experimental conditions, both W/C f and final TTAB concentration in cement foam depend only on the liquid content in the precursor foam. Measured yield stress is given as a function of W/C f in Fig. 3.

To get a relation between W/C f and τ y,0 ,which will allow us to calculate τ y,0 for all cement foams samples containing TTAB, we fit these data point using the so-called Yodel [START_REF] Flatt | Yodel: a yield stress model for suspensions[END_REF]. This model has been proposed to describe the yield stress of solid suspensions such as cement pastes. According to the Yodel, the yield stress can be obtained by the simple combination of a parameter m 1 which accounts for the interparticle forces, and a function of the solid volume fraction Φ p :

τ y,0 = m 1 Φ 2 p (Φ p -Φ perc ) Φ max (Φ max -Φ p ) (2)
where Φ perc is the percolation threshold and Φ max is the maximal solid frac-

Surface tension of cement paste

The global TTAB concentration after mixing of the precursor foam with the cement paste is between 0.7 and 2.4 g/L. Adsorption isotherms measured on pastes at W/C f =0.5 showed that partial adsorption on cement grains leaves a residual concentration in solution between 0.2 and 1 g/L and corresponding surface tensions (in synthetic cement pore solution) are comprised between 37 and 42 mN/m [START_REF] Feneuil | Effect of surfactants on the yield stress of cement paste[END_REF]. For the sake of simplicity we will assume that γ T T AB 40 mN/m.

In the case of Steol surfactant, for both investigated concentrations, yield stress values are smaller than the yield stress of the same cement paste without surfactant (see paragraph 2.3.1). Therefore, all cement foams made with Steol are in the high Steol concentration regime, for which surface tension is γ Steol = 27 mN/m.

Results

Stability of aqueous foams

During the six experiments presented in Fig. 4, we observe that the foam becomes more and more dry and that the size of the air bubbles increases.

However, the height of the foam did not decrease for 10 hours in all cases.

Foams made with synthetic cement pore solution are less stable than those made from distilled water, and the presence of C 6 F 14 decreases the collapse velocity; however, both these effects can be seen only after 10 hours. as illustrated in Fig. 5. Bubble size after cement hardening in these samples is kept unchanged, except at the bottom of the sample. Note that for the smaller bubbles, mixing of cement paste and foam is sometimes not fully homogeneous at the bubble scale: small volumes of cement paste appear to be surrounded by foam of higher air content than the average value.

For instance, structure of foams in pictures (c) and (d) in Fig. 6 should be the same, but mixing was more inhomogeneous in sample (c). However, the quality of the mixing does not change the fact that samples are stable only if ripening is prevented: the final pore distribution measured from the tomography images after cement hardening is narrow around the average value.

An example for the evolution of the morphology of a TTAB cement foam with R 0 = 365 µm is shown in Fig. 7. The average radius of the bubbles and the radius of the bigger bubbles (i.e. average of the bigger three radii) are given in Fig. 8 for two TTAB samples with different initial bubble sizes but approximately the same W/C ratio (i.e. 0.41). For both samples, the maximal radius and the average radius of the bubbles start to increase just after sample preparation. Bubble size increases faster when the initial bubble size is smaller. . Lines correspond to average value of all the bubbles and dots show the size of the defaults (average of the bigger three radius). Note that fast variations for the default size curves are due to the image analysis process: some of the bubbles cannot be identified on all the pictures because of unclean sample walls. In addition, the apparent radius are below the effective radius of the bulk bubbles due to the width of the Plateau borders on the pictures.

Bigger bubbles (R 0 500 µm)

In samples containing larger bubbles, gas fraction often increases from the bottom to the top of the sample when drainage is not prevented through rotation. Examples of slices obtained by X-ray tomography of a 11 cm high sample are shown in Fig. 9. Gas fraction Φ has been calculated by image analysis of the tomography slices, and is also plotted in Fig. 9 as a function of height: it increases from 80% at the bottom to 85% at the top. This curve additionally shows oscillations of Φ over a length scale close to 1300 µm, i.e. the diameter of the bubbles (see inset), which could be a signature of a bubble crystalline structure. Some local minima and maxima are still observed when the curve is averaged over a height bigger than the bubble size (black curve). These local variations may be created during the mixing process or appear later during the drainage of the sample.

In addition, drainage sometimes leads to a strong destabilization of bubbles at the bottom of the sample (see Fig. 10, left). Preventing drainage by sample rotation makes the sample more homogeneous (see Fig. 10, center) but often leads to weak samples, that break when they are demolded (see Fig. 10,right).

This occurs when the cement foam is flowing in the mold during the rotation stage, which is often the case when τ y,0 is too low.

h=3 cm h= 10 cm

Influence of initial yield stress and bubbles size

In the following we do not consider the drainage-induced destabilization of the cement foam samples. A sample is considered as unstable when a major change of bubble size has occurred in the whole sample. Sample stability observations are gathered in Figs. 11 and12. For Steol, each point corresponds to at least two experiments. In some cases, identical foams have different stability behavior.

We observe for both surfactants a significant effect of bubble size: all samples are stable when R 0 is high and unstable at very low R 0 . Minimum radius for stable foams depends however on the composition of the foams: 600 µm for TTAB and C1, and respectively 400 µm and 200 µm for Steol and C2 when τ y,0 =18 Pa and 3 Pa. On the contrary, the increase of the yield stress does not improve stability in the studied range, i.e. below 50 Pa (we recall that higher yield stresses can not be achieved with our mixing device). is the liquid density. Our results show that TTAB and Steol are able to prevent coalescence, even in the highly alkaline conditions in cement paste, which means that Π d,crit > 1000 P a. Note however that in foams, coalescence can take place even if Π d < Π d,crit , because bubble rearrangements can lead to film breakage due to dynamics, if liquid volume content is very low [START_REF] Biance | How topological rearrangements and liquid fraction control liquid foam stability[END_REF]. Rearrangements can for instance be a consequence of ripening. This could explain why coalescence occurs after a few hours in perfluorohexane-free Steol foams whereas foams made with perfluorohexane were stable for 30 hours. These observations on aqueous foams show that coalescence is not expected to be the leading destabilisation mechanism in our cement foam samples. Let us now investigate the other two mechanisms.

Drainage should stop if the yield stress of the cement paste exceeds a critical value τ c,d whose order of magnitude is given by [START_REF] Guignot | Liquid and particles retention in foamed suspensions[END_REF]:

τ c,d ∼ ρgr (3)
where r refers to the radius of curvature of Plateau borders, as measured within their transversal cross-section. At high air content Φ > 99%, r ≈ R 0 / 3(1 -Φ) [START_REF] Cantat | Foams -Structure and dynamics[END_REF] whereas for gas volume fractions close to the packing volume fraction of spherical bubbles (i.e. Φ → 64%), r → R 0 . As Φ = 83 % in the present study, we take r ∼ R 0 , so τ c,d ∼ ρgR 0 .

Ripening is expected to be slowed down or stopped when the yield stress of the interstitial material reaches the order of magnitude of the bubble capillary pressure [START_REF] Lesov | Factors controlling the formation and stability of foams used as precursors of porous materials[END_REF]. Therefore, we define the critical yield stress for the ripening process as:

τ c,r ∼ γ/R 0 (4)
Both critical stresses for drainage and ripening depend on the bubble radius R 0 . Eqs 3 and 4 are plotted in Fig 13 (we assume that ρ 2000 kg/m 3 and γ 30 mN/m). We notice that τ c,d < τ c,r within the range of studied bubbles sizes, i.e. 100 µm < R < 1000 µm. τ c,r is two orders of magnitude above τ c,d when R is a few 100 µm, which explains that ripening is the dominant destabilization mechanism at small bubble size. When R → 1 mm, critical stresses for drainage and ripening are of the same order of magnitude. In this case, to understand the major destabilization mechanism at stake, we have to compare the kinetics of destabilization for drainage and ripening.

Characteristic destabilization times

Let us first calculate a rough estimation of drainage characteristic time t d as given by the ratio of the sample height H and the drainage velocity v:

t d = H/v [1]
. The velocity is given by Darcy's law for pressure gradient induced by

gravity ρg: v = k µ app ρg ( 5
)
where k is the Darcy permeability and µ app the apparent shear viscosity of the cement paste. Permeability of a foam depends of the gas volume fraction and the bubble size [START_REF] Rouyer | Permeability of a bubble assembly: from the very dry to the wet limit[END_REF]:

k = 4(1 -Φ) 3/2 1700(1 -2.7(1 -Φ) + 2.2(1 -Φ) 2 ) 2 R 2 (6)
For Φ = 0.83, this gives k = 5.10 -4 R 2 .

To assess the apparent viscosity of the continuous phase, a rheological model must be chosen. The Bingham model describes well the rheological behavior of cement paste [START_REF] Roussel | Steady state flow of cement suspensions: A micromechanical state of the art[END_REF], the stress τ and the shear rate γ in the paste are related by:

τ = τ y + µ p γ (7)
where the yield stress τ y and the plastic viscosity µ p depend on the paste formulation. We have measured the flow curves of three cement pastes with no surfactant, at water-to-cement ratio from 0.37 to 0.5 (results not shown here).

For all three of them, stress at γ100 = 100 s -1 is close to 2τ y , i.e. µ p ∼ τ y / γ100 .

At the scale of the Plateau borders and nodes, the gravity-induced stress is τ ∼ ρgr where r is the characteristic size of the channels and is close to the bubble radius R 0 as already explained in the previous paragraph. Following the above assumptions, the apparent viscosity is therefore:

µ app = τ / γ ∼ ρgR 0 τ y (ρgR 0 -τ y ) γ100 (8)
From equations 5, 6 and 8, we find that the characteristic drainage time is

t d ∼ Hτ y 5.10 -4 R 0 (ρgR 0 -τ y ) γ100 (9)
This gives, for τ y = 1 P a, t d ∼ 30 min when R 0 = 200 µm and t d ∼ 1 min when R 0 = 700 µm.

The characteristic time for foam ripening is given by [START_REF] Pitois | Foam ripening[END_REF]:

t r = 2R 2 av K 2 (10)
where K 2 is a diffusion coefficient. For nitrogen and low molecular weight surfactants K 2 ∼ 50 µm 2 /s. For polydisperse foams, the characteristic time given by equation 10 is the time during which the average bubble size grows from its initial value < R > to 2 < R >. In the case of monodisperse foams, ripening is delayed: first, there is an induction period during which defaults appear and grow in the foam. The duration of this induction period is difficult to estimate. Here, we chose to use equation 10 with < R >= R 0 to assess the time evolution of the defaults due to ripening. We obtain t r ∼ 50 min when R 0 = 270 µm and t r ∼ 90 min when R 0 = 365 µm. These calculated times are of the same order of magnitude than the observed ripening times (see Fig. 8).

In table 3

Effect of Bingham capillary number

We have identified foam ripening as the major destabilization mechanism.

To go further, we introduce the Bingham capillary number, which compares the yield stress (stabilizing effect) with the capillary pressure inside the bubbles, which drives the ripening process:

Ca y = τ y γ/R (11)
We expect that a suitable criterion for foam stability would be in the form of a critical capillary number Ca * y . As a first approach we use the yield stress of the reference cement paste τ y,0 (obtained from Fig. 3) to calculate the Bingham capillary number. The calculated values are plotted in Fig. 11 and 12 and compared to the stability of TTAB and Steol samples, showing that a critical value Ca * y cannot be defined to describe the observed transitions between stable and unstable fresh cement foams. This can be better understood in Fig. 14 where we have reported the observed stability for all samples as a function of the calculated Ca y . One can see that stability is not controlled by the parameter Ca y : in particular, the range of Ca y values for optimal stability depends on the surfactant used to make the cement foam. τ y,0 is therefore not the adequate yield stress to be used to estimate the Bingham capillary number. Two effects should be considered:

• The yield stress of the interstitial cement paste, i.e. confined between the bubbles, can be higher than the yield stress of the reference cement paste, i.e. the unfoamed cement paste [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF]. The interstitial yield stress τ y,int is the effective stress that the bubbles must overcome to deform. It can be obtained from the macroscopic cement foam yield stress τ y,f oam (Φ), as measured for this purpose, using the following equation [START_REF] Gorlier | Yielding of complex liquid foams[END_REF]:

τ y,int = γ/R 0 c 3/2 (1 -Φ) 2 1 - τ y,f oam (Φ) τ y,aq (Φ) 3/2 (12)
where τ y,aq (Φ) = 0.6 γ R (Φ -0.64) 2 is the yield stress of aqueous foam with the same Φ value and c=110 is a constant.

• Yield stress of cementitious materials at rest increases with time due to flocculation and creation of hydrate bonds between the particles [START_REF] Roussel | Steady state flow of cement suspensions: A micromechanical state of the art[END_REF]. In addition, it has been shown in a previous study [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF] that for some samples, densification of the interstitial yield stress increases during several minutes after the foam preparation due to liquid drainage and the associated densification of the cement paste.

These two issues suggest that the interstitial yield stress τ y,int at a relevant time t * after foam preparation should be considered instead of the reference yield stress τ y,0 . The time t * is expected to be set by the interplay of several complex mechanisms, such as hydration, liquid drainage and foam ripening, and the detailed analysis for such effect is by far beyond the scope of this paper. We have noticed that all the unstable samples (including both samples presented in Figs. 7 and8) have started to destabilize before 15 min. Therefore, in the following we consider that t * = 15 min, keeping in mind that this value could be different for cement foams prepared with a completely different protocol than ours. In the next part we measured the interstitial Bingham capillary number at t * = 15 min.

15 min interstitial Bingham capillary number

We have measured the yield stress τ y,f oam of several cement foams at 15 min with the rheometry protocol described in [START_REF] Feneuil | Optimal cement paste yield stress for the production of stable cement foam[END_REF]. The interstitial capillary number is then deduced from τ y,f oam using equation 12. Results obtained for TTAB foams with different bubble sizes are presented in Fig. 15. The reported interstitial yield stress decreases for the largest bubble sizes. On the other hand, a possible effect of bubble size is revealed by the high resolution tomography pictures in Fig. 16. When the bubbles are small, the bigger cement grains, whose diameter before the start of hydration is about 100 µm, have similar size as the Plateau borders. Note that the synchrotron pictures have been taken two months after the sample preparation, when most of the cement has reacted with water and formed hydrates, resulting in a smaller apparent cement grain size than their initial size. The diameter of the larger particles that can enter the Plateau borders is given by equation [START_REF] Louvet | Capture of particles in soft porous media[END_REF]: Stability of samples is now plotted as a function the interstitial capillary number at t * = 15 min in Fig. 17. We can see for all the curves that a transition from unstable samples at Ca y,int (t * ) 1 to stable samples when Ca y,int (t * ) is higher. We can therefore define a critical value for the Bingham capillary number: Ca * y,int = 1. The stability criterion can therefore be expressed as follows: Ca y,int (t * ) > 1 [START_REF] Flatt | Yodel: a yield stress model for suspensions[END_REF] It is important to recall here the hypothesis leading to this criterion:

d P B = 2R 0.27 √ 1 -Φ + 3.17(1 -Φ) 2.75 1 + 0.57(1 -Φ) 0.27 (13
• Destabilization is due to ripening. We have shown in paragraphs 4.1 and 4.2 that this hypothesis is true for the samples studied here, but it may fail when bigger bubbles are studied (R 0 1 mm)

• Interstitial yield stress has been calculated using equation 12. This empirical relation has been obtained for interstitial fluids that are continuous compared to the size of the Plateau borders, and for values of the Bingham capillary number below 0.5. Therefore, care should be taken when Plateau border size is small (due to small bubble size or high gas volume content) and when the interstitial yield stress is high.

• The value of t * required to define this criterion is expected to depend on the preparation method of the cement foam samples and can be affected, for instance, by the polydispersity of the foam and the setting time of the cement. We are able to get a single value of t * for all the samples studied here because all are initially monodisperse, and although two different cements are used, hydration time for both is roughly similar. The obtained value, t * = 15 min is of the order of magnitude of the drainage characteristic time for the smallest studied bubbles (see paragraph 4.2).

Conclusion

We have investigated the mechanisms at stake in the destabilization of cement foam samples. We first note that a proper choice of surfactant can prevent coalescence of the bubbles up to cement hardening.

For most of the cement foam samples, prepared either with anionic or cationic surfactant, ripening is the leading destabilization mechanism. Drainage is also sometimes observed when the yield stress of cement paste is very low or bubble size is large. In these cases, we observed that drainage takes place faster than ripening, which was found to be consistent with our calculation of the drainage and ripening characteristic times.

Thanks to stability measurements performed on foams made from different formulations of cement pastes, i.e. surfactant and W/C ratio, with controlled bubble diameters, we have shown that the cement foam stability is governed by a single parameter, namely the Bingham capillary number calculated using the effective yield stress of the interstitial cement paste at time t * after the foam preparation, Ca y,int (t * ). Whatever the studied cement foam, the transition between stable and unstable foam is observed for the simple criterion Ca * y,int (t * ) 1.

In this work, the gas volume fraction of the samples has been kept constant and both cement used have similar setting times. In addition, we have studied foams with initially very narrow bubble size distribution. As setting time and bubble size distribution are expected to set the characteristic time t * , it would

Figure 1 :

 1 Figure 1: Preparation protocol of cement foams made with Steol anionic surfactant and TTAB cationic surfactant.

tional 11 -

 11 cm-high sample were obtained with a Ultratom scanner from RX solu-tions at Laboratoire Navier. Measurement involved a Hamamatsu L10801 X-ray source (160 kV) and a Paxscan Varian 2520V flat-panel imager. All scans were performed at 80 kV and 70 µA. To analyze the whole sample height, we have used stack type scans, i.e. horizontal sections of the sample were scanned independently and combined later by the reconstruction software. Frame rate was 3 images per second and 12 images were averaged to produce one projection.Resulting effective exposure time was therefore 4s. 3D tomographic reconstruction were performed at laboratoire Navier with the X-Act commercial software developed by RX-Solutions. Voxel size for the obtained images was 16.3 µm.

Figure 2 :

 2 Figure 2: Top: 3D reconstruction from X-ray tomography experiment on a Steol sample (τ y,0 = 3 Pa, R 0 = 685 µm). Cube size is 3 mm. Bottom: Final pore size distribution by volume (arbitrary unit on vertical axis) of the hardened cement foams, statistics on a 1.5 cm cube in the middle of the sample (about 2000 bubbles); black histogram illustrates a TTAB sample where W/C f =0.42 and R 0 =370 µm, and grey histrogram shows a Steol sample where τ y,0 = 3 Pa and R 0 = 685 µm.

  tion. In cement pastes, Φ p is related to the water-to-cement ratio: Φ p = (ρ w /ρ c )/(ρ w /ρ c + W/C f ). Equation 2 can be conveniently used for cement pastes containing TTAB because this surfactant hardly affects cement particle interactions, i.e. m 1 parameter is constant. The obtained fitting parameters are m 1 = 15 P a, Φ perc = 0.32 and Φ max = 0.46.

Figure 3 :

 3 Figure 3: Yield stress of cement pastes containing TTAB cationic surfactant and C1 cement (Fit with Yodel, equation 2 with m 1 = 15 P a, Φperc = 0.32 and Φmax = 0.46)

Figure 4 :

 4 Figure 4: Time evolution of the height of aqueous foams prepared from TTAB solutions at 5 g/L and Steol solutions at 1 g/L. Black dots or triangles refers to surfactant solutions in distilled water and grey signs to synthetic cement paste solutions. Dots refers to foams containing C 6 F 14 to slow down ripening and triangles to foams made of nitrogen only.

Figure 5 :Figure 6 :

 56 Figure 5: Bottom of cement foam sample 15 minutes after production (TTAB, R = 300 µm, W/C f =0.69 with perfluorohexane). Sample diameter is 2.6 cm.

Figure 7 :

 7 Figure 7: Evolution of morphology of an unstable fresh cement foam. Example of sample with initial bubble radius 365 µm, W/C=0.41 and 83% air content containing TTAB surfactant. Picture width is 1.5 cm.

Figure 8 :

 8 Figure8: Evolution of bubble radius with time for the TTAB samples with initial bubble radius 365 µm, W/C f =0.41 (grey curves) and 270 µm, W/C f =0.42 (black curves). Lines correspond to average value of all the bubbles and dots show the size of the defaults (average of the bigger three radius). Note that fast variations for the default size curves are due to the image analysis process: some of the bubbles cannot be identified on all the pictures because of unclean sample walls. In addition, the apparent radius are below the effective radius of the bulk bubbles due to the width of the Plateau borders on the pictures.

Figure 9 :

 9 Figure 9: Steol sample (τ y,0 =3 Pa, R 0 =685 µm). Top: Slices of cement foam sample obtained by X-ray tomography at different height, bubbles appear in black and cement paste is grey. Bottom : evolution of gas fraction with sample height. Grey curve is raw data, black curve is the data average over 2000 µm.

Figure 10 :

 10 Figure 10: Comparison of samples with big bubble size (TTAB, Φ=0.87%, R 0 =1000 µm) after free drainage (W/C f =0.44, left) and when drainage has been prevented by rotation (center and right). Rotation sometimes leads to weak samples that break when they are demolded (W/C f =0.49, right). Sample height is 6 cm.

Figure 11 :

 11 Figure 11: Stability of TTAB cement foam samples. Black lines show constant Bingham capillary number (as given by eq. 11 with γ T T AB = 40mN/m).

Figure 12 :

 12 Figure 12: Stability of Steol cement foam samples. Black lines show constant Bingham capillary number (as given by eq. 11 with γ Steol = 27 mN/m).

  Let us first consider coalescence. Experiments performed on aqueous foams made with both surfactants have shown that the foam volume was constant for more than 10 hours with C 6 F 14 and without. Stability of a thin liquid film depends on the ability of the surfactant layers on both interfaces to repel each other. Film breakage occurs when the disjoining pressure Π d , i.e. the pressure in the liquid film due the the repulsion on the air-liquid interfaces, reaches a critical value Π d,crit . In an aqueous foam in mechanical equilibrium, the disjoining pressure is maximal at the top of the foam and Π d = ρ liq gh where ρ liq

Figure 13 :

 13 Figure 13: Comparison of driving stresses for drainage and ripening. The curves define three possible behavior of cement foams when no experimental artifice is used to slow down ripening or drainage: stable foam if τ y,0 > τc,r, both drainage and ripening if τ y,0 < τ c,d and only ripening at intermediate yield stress values.

Figure 14 :

 14 Figure 14: Cement foam stability as a function of Bingham capillary number calculated from the yield stress of the reference cement paste. Black dots correspond to Steol samples with τ y,0 = 3 Pa, grey dots to Steol samples with τ y,0 = 18 Pa and empty triangles to TTAB samples.

Figure 15 :

 15 Figure 15: Interstitial yield stress calculated from foam yield stress measured at 15 min for samples containing TTAB, W/C f =0.42

  ) with Φ = 83%, we obtain d P B = 60 µm if R = 300 µm and d P B = 160 µm if R = 800 µm. As a consequence, when the bubbles are small, larger cement grains cannot enter the Plateau borders and are stuck in the foam nodes. This is expected to lead to a segregation of cement grains according to their size, which could result in measurable effect on the yield stress of the cement paste.

Figure 16 :

 16 Figure 16: Synchrotron tomography slices of TTAB cement foams stabilized by C 6 F 14 , for R=265 µm (left) and R=550 µm (right), two months after preparation. Image width is 1300 µm. Red arrows indicate some of the bigger cement grains.

Figure 17 :

 17 Figure 17: Cement foam stability as a function of Bingham capillary number calculated from interstitial yield stress at 15 min. Black dots correspond to Steol samples with τ y,0 = 3 Pa, grey dots to Steol samples with τ y,0 = 18 Pa and empty triangles to TTAB samples.

Table 1 :

 1 vided by Sigma-Aldrich. Its molar mass is 336 g/mol. Steol ® 270 CIT is an Chemical and physical properties of cements. C1 refers to CEM I cement from Lafarge, Saint-Vigor and C2 to CEM I cement from Lafarge, Lagerdorf.

		C1	C2
	CaO/SiO 2	3	3
	MgO	1.1%	0.8%
	Na 2 O + 0.658 K 2 O 0.34%	0.5%
	SO 3	2.58% 2.5 %
	Cl -	0.03% 0.04 %
	Gypsum	2.4%	4%
	Density (g/cm 3 )	3.21	3.15
	SSB (cm 2 /g)	3586	4330

Table 2 .

 2 Preventing drainage by a rotation of the samples does not stop the foam destabilization process. On the other hand, counteracting artificially the ripening process using perfluorohexane allows for stabilizing robustly the fresh Free drainage No drainage No ripening No ripening (µm) and ripening Free ripening Free drainage No drainage TTAB -C1, W/C f from 0.39 to 0.5

	cement foams.
	When W/C f is further increased above 0.5 for TTAB samples (i.e. a yield
	stress deduced from equation 2 below 1 Pa) containing perfluorohexane, we
	sometimes notice a segregation of cement grains at the bottom of the samples,

3.2. Drainage and ripening 3.2.1. Smaller bubbles (R 0 500 µm) Let us first consider the unstable samples, whose properties are summarized in

Table 2 :

 2 Effect of slowing down ripening and drainage on sample stability.

  we have reported both drainage and ripening characteristic times for several bubble sizes and several paste yield stresses. One can see that when cement paste yield stress is below the driving pressures for both drainage and ripening, drainage occurs faster than ripening as soon as R 0

				200 µm. This
	result is compatible with the experimental observation that drainage sometimes
	occurs in samples containing the larger bubbles.
		R = 200 µm	700 µm	1 mm
	τ y = 1 Pa Drainage + ripening ∼ 30 min Drainage ∼ 1 min Drainage ∼ 1 min
	10 Pa	Ripening ∼ 30 min	Drainage ∼ 1 h	STABLE
	100 Pa	Ripening ∼ 30 min	STABLE	STABLE

Table 3 :

 3 Characteristic destabilization times for cement foams with different bubble size and interstitial yield stress

Acknowledgments

The authors wish to thank David Hautemayou and Cédric Mézières for technical support for the foam generation and mixing devices, Sabine Carré for support on synchrotron measurements, Nicolas Ducoulombier for help for the 3D reconstructions of tomography measurements, and Michel Bornert for giving us time to study our samples on the synchrotron line and for useful comments on this paper. This work has benefited from two French government Grants managed by the National Research Agency (ANR) [Grants no. ANR-11-LABX-022-01 and ANR-13-RMNP-0003-01]. In addition, the construction and operation of ANATOMIX is largely funded by the French ANR through the EQUIPEX investment program, project NanoimagesX, grant no. ANR-11-EQPX-0031.

be interesting to investigate how robust is our stability criterion when these parameters are allowed to vary within a large range.


Graduate Student Member, IEEE Maryam Saeed
email: maryam.saeed@barry.cardiff

Graduate Student Member, IEEE, Olev Märtens, Senior Member, IEEE Qingyuan Wang
email: qingyuan.wang@ucdconnect.ie@barry.cardiff

Olev Martens
email: olev.martens@taltech.ee

Benoit Larras
email: benoit.larras@junia.com

Antoine Frappé
email: antoine.frappe@junia.com

Barry Cardiff

John M Deepu
email: deepu.john@ucd.ie

Q Saeed

B Wang

D Cardiff

Evaluation of Level-Crossing ADCs for Event-Driven ECG Classification

Keywords: level-crossing ADC, cardiac arrhythmia classification, convolutional neural networks, wearable sensors, eventdriven data

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

I. INTRODUCTION

I N recent years, the number of portable wireless biomed- ical systems have grown exponentially facilitating better health care monitoring and earlier diagnosis [START_REF] Wong | An integrated wearable wireless vital signs biosensor for continuous inpatient monitoring[END_REF]. As a result, high volumes of data are produced with increasing number of biomedical sensors and long-term monitoring. In these biomedical signal monitoring systems, the energy consumption of wireless communication modules generally exceeds all other modules combined, for example in [START_REF] Chen | Energyaware design of compressed sensing systems for wireless sensors under performance and reliability constraints[END_REF], [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF], it is reported that the wireless module consumes up-to two-thirds of the total power on-chip. To reduce energy consumption, on-chip feature extraction and compression techniques are applied after sampling from an analog-to-digital converter (ADC) [START_REF] Yin | A 2.63 w ecg processor with adaptive arrhythmia detection and data compression for implantable cardiac monitoring device[END_REF]- [START_REF] Deepu | A 3-lead ecg-on-chip with qrs detection and lossless compression for wireless sensors[END_REF]. Recently, a new class of ADCs, called levelcrossing ADCs (LC-ADCs), has been developed which embed compression into the data acquisition stage [START_REF] Van Assche | Power Efficiency Comparison of Event-Driven and Fixed-Rate Signal Conversion and Compression for Biomedical Applications[END_REF]- [START_REF] Zhao | A 13.34 µW Event-Driven Patient-Specific ANN Cardiac Arrhythmia Classifier for Wearable ECG Sensors[END_REF]. LC-ADCs utilize the time-domain sparsity of signals, such as, electrocardiogram (ECG) signals, which remain inactive for large time intervals. LC-ADCs produce asynchronous samples of data triggered by the threshold crossings of the analog signal. This reduces the sampling in low-activity regions of the signal, thus reducing the data rate and energy consumption on-chip. LC-ADCs have the advantage of built-in compression and a corresponding potential reduction in processing costs in comparison with Nyquist sampling ADCs.

The energy efficiency of a LC-ADC and traditional successive-approximation (SAR) ADC using various biosignals was studied in [START_REF] Van Assche | Power Efficiency Comparison of Event-Driven and Fixed-Rate Signal Conversion and Compression for Biomedical Applications[END_REF]. They concluded that LC-ADCs are more energy-efficient than SAR ADCs for low-to-medium resolution applications, while also producing less data for transmission.Many low-power LC-ADC architectures have been proposed in recent years for various biomedical applications, such as biomedical signal acquisition [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF], [START_REF] Li | A sub-microwatt asynchronous level-crossing adc for biomedical applications[END_REF]- [START_REF] Weltin-Wu | An event-driven clockless level-crossing ADC with signal-dependent adaptive resolution[END_REF], ultrasound measurement systems [START_REF] Kozmin | Level-crossing adc performance evaluation toward ultrasound application[END_REF], ECG signal acquisition [START_REF] Marisa | Pseudo asynchronous level crossing adc for ecg signal acquisition[END_REF], low-power QRS detection in ECG signals [START_REF] Zhang | A 300-mV 220-nW Event-Driven ADC With Real-Time QRS Detection for Wearable ECG Sensors[END_REF], [START_REF] Ravanshad | A Level-Crossing Based QRS-Detection Algorithm for Wearable ECG Sensors[END_REF], low data rate image sensors [START_REF] Darwish | 1-level crossing sampling scheme for low data rate image sensors[END_REF], and a reconfigurable event-driven wake-up system [START_REF] Wang | A 148-nw reconfigurable event-driven intelligent wake-up system for aiot nodes using an asynchronous pulse-based feature extractor and a convolutional neural network[END_REF]. Moreover, classification algorithms have been widely designed for various biomedical applications using traditional SAR ADCs. A detailed review of classification algorithms, their hardware designs and power consumption is presented in [START_REF] Wei | A review of algorithm & hardware design for ai-based biomedical applications[END_REF]. Recently, ECG signals have also been used for low-power authentication of devices which also utilizes neural networks and compression techniques [START_REF] Cherupally | Ecg authentication hardware design with low-power signal processing and neural network optimization with low precision and structured compression[END_REF]. Using event-driven data from LC-ADCs, an artificial neural network based arrhythmia classification has been presented in [START_REF] Zhao | A 13.34 µW Event-Driven Patient-Specific ANN Cardiac Arrhythmia Classifier for Wearable ECG Sensors[END_REF].

This article builds on the work presented in [START_REF] Saeed | Event-driven ecg classification using an open-source, lcadc based non-uniformly sampled dataset[END_REF], which evaluated several level-crossing ADC models for cardiac arrhythmia classification and compression performance. To the best of our knowledge, no other performance analysis of LC-ADC models for event-driven ECG classification is found in the literature. In this article, we propose a new design strategy for modeling LC-ADC parameters inspired by the work presented in [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF] and apply it to low-power event-driven cardiac arrhythmia classification. In this context, the major contributions of this paper are as follows: a) a new design technique for modeling level-crossing ADC parameters is presented that improves sampling accuracy and reduces data stream rate from the LC-ADC, b) several LC-ADC models are simulated and their performance is analyzed in terms of signal quality and compression using the MIT-BIH Arrhythmia dataset [START_REF] Goldberger | Physiobank PhysioToolkit and PhysioNet components of a new research resource for complex physiologic signals[END_REF], c) a deep learning model is proposed for eventdriven cardiac arrhythmia classification and, d) an open-source event-driven ECG database derived from MIT-BIH database using the proposed LC-ADC models with arrhythmia annotations is presented.

The rest of the paper is organized as follows. Section II presents the principles of event-driven sampling for ECG applications including, LC-ADC architecture, and performance metrics for evaluation of LC-ADC models. Section III proposes a new design methodology for modeling LC-ADC parameters and evaluates several models using performance metrics. Section IV proposes a deep learning model for classification of cardiac arrhythmias using event-driven ECG Fig. 2: (a) The proposed new methodology for selecting LC-ADC design parameters, inspired by the work presented in [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF]. The three design parameters (M, N, Fc) must meet the respective design conditions before the LC-ADC can be evaluated for acceptable SDR and CR. (b) The LC-ADC evaluation methodology. ECG input is converted to event-driven signal using selected design parameters, reconstructed using linear interpolation and assessed for SDR and CR performance. data. A complexity comparison with uniformly sampled data is presented. An event-driven ECG dataset is also presented in Section IV. Finally, in Section V, conclusions are drawn.

II. EVENT-DRIVEN ECG SAMPLING

This section presents the principles of level-crossing sampling with an LC-ADC architecture and the performance metrics for compression and signal-quality evaluation of LC-ADC Models.

A. LC-ADC Architecture

The LC-ADC architecture is presented in Fig. 1a. The ECG signal range is divided into a fixed number of quantization levels. The LC-ADC sampling processing is implemented by continuously comparing against an upper and lower threshold voltage, U QL and L QL respectively, each of which is quantized into LSBs each having size q Volts:

q ≜ A F S 2 M (1)
where A F S is the full-scale voltage range of the ECG signal, M is the resolution of the LC-ADC and 2 M is the total number of quantization levels. The dynamic range of 10mVpp is used as the full-scale voltage throughout this study, which ensures industry standard of ambulatory equipment [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF] and matches the dynamic range of the MIT-BIH Arrhythmia dataset [START_REF] Ravanshad | A Level-Crossing Based QRS-Detection Algorithm for Wearable ECG Sensors[END_REF] used to evaluate models in this study. The difference between the upper and lower threshold voltages, ∆v, is usually a fixed integer number of LSBs:

∆v = k * q (2)
where k ≥ 1 ∈ Z. Typically k = 1, but for higher resolution LC-ADCs k is often larger so as to maintain a ∆v that is greater than the noise thereby avoiding excessive threshold crossings.

Whenever the ADC's input signal crosses U QL a sample is taken, i.e. we set ECG out = U QL and both threshold are increased by q Volts (i.e. +1 LSB). Similarly when L QL is crossed we set ECG out = L QL and decrease the two thresholds by q Volts (i.e. -1 LSB). The control logic block in Fig. 1 manages this process. This way, an LC-ADC creates continuous-in-time discrete-in-amplitude ECG signals.

Compared to a uniform sampling ADC where the timestamp associated with each sample is implicit, a LC-ADC however must output both the amplitude level ECG out and the associated time, or equivalently the Time Interval (TI) between each sample, as a tuple (ECG out , T I) n , where n is an index meaning the n th tuple in time. In practice the Time Interval TI is quantized w.r.t. a clock having frequency F c and is represented using an N bit word. Whenever the Time Interval counter rolls-over, i.e. after (2 N -1) Fc sec during which time there were no level crossing events, an extra sample is output where the previous amplitude is simply repeated.

Using a 7-bit LC-ADC, the analog ECG signal (ECG in ) with event-driven samples (ECG out ) are presented in Fig. 1b. The horizontal grid lines represent the quantization levels spaced at q volts. The gray highlighted region on the third QRS peak is further illustrated in Fig. 1c with an LSB size of ∆v = 3q volts. While the distance between U QL and L QL is always maintained at 3q volts, whenever a threshold crossing occurs, the two thresholds are increased or decreased by 1q volts only. Furthermore, it is worth noting the difference in time intervals between the six event-driven samples illustrated in Fig. 1c.

B. Performance metrics

To measure the signal distortion and compression in the LC-ADC output, its output samples are interpolated using linear interpolation to reconstruct the ECG signal, as in [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF]. The two performance parameters are as follows:

1) Compression Ratio (CR): The compression ratio is the ratio of the average number of bits/sec in the signal obtained from a uniform ADC to the average number of bits/sec in the signal obtained from a LC-ADC. 2) Signal-to-Distortion Ratio (SDR): The temporal signalto-distortion ratio evaluates the distortion between the original signal, x, and the reconstructed signal, x is defined as: having units of decibels (dB). In [START_REF] Zigel | The weighted diagnostic distortion (WDD) measure for ECG signal compression[END_REF], the clinically acceptable values for SDR in ECG signal reconstruction are also reported. "Good" quality signals must follow the following criteria:

SDR =10 log 10 (x -x) 2 -10 log 10 (x -x) 2 (3)
SDR ≥ 21dB (4)

III. LEVEL-CROSSING ADC MODELING AND EVALUATION

This section presents a new LC-ADC design and evaluation methodology. In order to accurately model an LC-ADC, the following design parameters must be considered: the resolution of the LC-ADC, M , the counter clock frequency, F c, and the counter clock resolution N . These parameters simultaneously impact the sampling accuracy of the level-crossing ADC as well as accurate time tracking between two consecutive eventdriven samples. First, the resolution, M , maps the number of quantization levels (2 M ) in the LC-ADC. If M is too high, too many level-crossings are triggered which might produce more data than an SAR ADC [START_REF] Van Assche | Power Efficiency Comparison of Event-Driven and Fixed-Rate Signal Conversion and Compression for Biomedical Applications[END_REF]. If M is too low, important level-crossings will be missed and the signal-to-distortion ratio will be significantly decreased. For a fair evaluation, in this section, we design and evaluate ten LC-ADC models using resolutions M = 2, 3, . . . , 11, respectively. For each of these resolutions, the ideal counter frequency and counter resolution are modeled as presented in the following subsections.

A. Selecting the Counter Clock Frequency

The fastest signal variations in ECG signals can be seen around the QRS complex as shown in Fig. 1b. In fact, these amplitude variations are larger in ECG signals with pathological abnormalities than normal conditions. The counter clock frequency should be able to keep up with these fast amplitude variations between two consecutive LC-ADC samples. Violation of this condition will lead to clipped shapes in the QRS complex [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF]. Furthermore, F c should also be fast enough to minimize time-quantization errors in mapping T I as described in Section IIA. In [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF], a maximal slope condition is used to determine F c. In general the maximal obtained from any random sample can be a statistically unreliable measure and could lead to significant over or under estimates of the typical maximum which is what, we argue, one should design according to. Thus the approach we adapt here is to estimate the CDF (Cumulative Distribution Function) of the ECG slopes and take the 99.9% level as being a reliable estimate of the typical maximum slope.

Consider the vector ⃗ x containing the magnitude of slopes in the entire MIT-BIH arrhythmia dataset. Through a cumulative distributive function, find the value, x o , in ⃗ x, for which the magnitude of slopes in the dataset are less than or equal to x o , 99.9% of the time. Therefore, x o and corresponding F c can be calculated as:

P dECG in dt ≤ x o ≃ 0.999 (5)
⇒ F c ≥ x o q (6)
Fig. 3 shows the cumulative distributive function of magnitude of slopes in the MIT-BIH dataset with x o shown at 99.9%. Fig. 4 shows the value of F c obtained in this manner as a function of the LC-ADC resolution (M ) obtained by simulation over the entire MIT-BIH Arrhythmia dataset. These result are also summarized in Section IIIC, Table I. As M is increased, the LSB size, q decreases, thereby increasing the counter clock frequency required to capture fast slopes in the ECG signal. At M = 8 bits, the minimum required Fc is 4744.24Hz, which is less than half of that used in [START_REF] Tlili | Level-crossing ADC design and evaluation methodology for normal and pathological electrocardiogram signals measurement[END_REF] for the same LC-ADC resolution.

B. Determining the Counter Clock Resolution

Lastly, the final design parameter, counter clock resolution (N), directly impacts the number of bits per second generated by the LC-ADC, defined as the stream-bit-rate (sbr). Mostly, sbr is dominated by the level-crossing rate (lcrate), which is the average sampling rate (i.e. the average level crossing rate) of the LC-ADC. However, as described in Section IIA, the LC-ADC design presented in this study can also output a tuple at every clock rollover i.e, every (2 N -1) Fc seconds if there has been no level crossing during this time 1 . If N is chosen to be very small, too many tuples are generated. Similarly, if N is too high, the size of T I (N-bit word) will be large. In other words, the number of bits per tuple increases linearly as N is increased. It is worth noting that this approach does not add any additional overhead in terms of the tuple transfer rate as it occurs quite infrequently. We propose to find the optimal value of N for which the sbr is minimum. The sbr can be defined as,

tuple rate = max F c 2 N , lcrate (7)
sbr =tuple rate * (M + N )

We simulated sbr over N = 3, 4, . . . , 16 using the resolutions, M and counter clock frequency, F c, pairs estimated in the previous subsections. Fig. 5 shows the sbr for varying counter clock resolution (N) over different LC-ADC resolutions (M).

In each sub-figure, the optimal choice of N for which sbr is minimum is marked as N o . It can be observed that as N grows large, sbr grows linearly. When M and N are both small, sbr is large due the tuple rate being dominated by the clock rollover samples. When M is large enough, however, tuple rate is dependent on the LSB size. Fig. 6 summarizes the optimal N choice for each resolution of the LC-ADC. It can be observed that when M is big enough and the sbr is largely dependent on the LSB size (q), the counter clock resolution (N) does not need to be too large. However, when M is small, N must be chosen higher so as not to generate too many tuples from counter clock roll-over.

In the next subsection, the three chosen design parameters, i.e. counter clock frequency (F c) and counter clock resolution (N ) for each LC-ADC resolution (M ) are evaluated for compression performance and signal quality.

C. Evaluation of LC-ADC Models

The LC-ADC design parameters and the respective design conditions they must meet are summarized in Fig. 2a. It also presents the LC-ADC evaluation criteria using the two performance metrics, compression ratio (CR) and signal-todistortion ratio (SDR), introduced in Section IIB. These can help designers choose the right LC-ADC model for their application. Fig 2b shows the LC-ADC evaluation methodology used in this subsection. We converted the 48 recordings in the MIT-BIH Arrhythmia dataset to event-driven data using the ten (M, Fc, N) value sets summarized in Table I, then reconstructed the signals to Fc using linear interpolation and evaluated them using CR and SDR. The aim is to select LC-ADC models that maximize the SDR subject to a CR greater than 1.5. Fig. 7 shows the compression ratio at different resolutions of M. A CR of 1 indicates no compression, while for CR ≤ 1, the LC-ADC model generates more samples than a Nyquist Sampling ADC. A compression ratio greater than 1.5 would indicate acceptable compression. It can be observed that for M > 7 bits, the CR is less than 1, which means a Nyquist sampling ADC will be better for high-resolution applications. For models with M ≤ 7, good compression can be observed.

Ideally, an LC-ADC must maximize the SDR for a given CR. We extend this condition to also include the signal quality criteria presented in eq. ( 4). However, it must be noted that for some ECG applications, an SDR less than 21dB might also be acceptable. Fig. 8 shows the SDR at different resolutions of M. The models with M ≥ 7 have SDR close to the 21dB criteria. However, the most promising trade-off between CR and SDR can be observed at M = 7. To study this LC-ADC resolution further, we looked into varying the counter clock resolution (N). The MIT-BIH Arrhythmia dataset is originally sampled at 360Hz using an 11-bit ADC. In order, to sample close to the Fc of 2372.12 Hz for M= 7 bits, we first upsample the entire database to 38160Hz. From this up-sampled database, we then sample at Fc = 2385Hz, which is 0.54% greater than the minimum required Fc at this resolution. Then using M= 7 bits and Fc = 2385Hz, we varied N from 4 to 8-bits to search over the curve created with these values (see M=7 bits in Fig. 5). Beyond 8-bit counter clock resolution, the sbr starts increasing linearly and therefore, excluded from analysis here. The results are summarized in Table II.

In all N we simulated the case where the step-size between the two LC-ADC thresholds, ∆v, was set to q (i.e. 1 LSB) but additionally for the N = 6 to 8 bits models were also evaluated at ∆v = 2q as they have SDR > 21dB. From the eight models summarized in this table, MDL6, MDL7, and MDL8 follow the criteria of eq. ( 4). However, MDL8 gives the most ideal CR and SDR trade-off at a slightly higher sbr of 1320.92 bps. Moreover, MDL4, MDL5, and MDL6b, MDL7b, and MDL8b with ∆v = 2q might still be acceptable for certain applications with SDR > 20dB. All of these models are evaluated for eventdriven arrhythmia classification in the following section.

IV. CARDIAC ARRHYTHMIA CLASSIFICATION USING

EVENT-DRIVEN ECG In this section, a new deep learning based classifier is presented and the eight LC-ADC models summarized in Ta-

A. Deep Learning Based Classifier

We designed a one-dimensional convolutional neural network as illustrated in Fig. 9. This 1D-CNN model has three convolutional layers followed by two fully-connected layers. Each convolutional layer has 64 filters of size 3, each. A stride of 1 is maintained throughout the network. Batch normalization and max pooling is applied after each convolutional layer to accelerate training and to reduce over-fitting. Max pooling also helps reduce complexity of the network in subsequent layers. To further ensure that over-fitting is minimized a dropout of 50% probability is applied between the two fully connected layers. Sparse categorical cross-entropy is used as the loss function with an ADAM optimizer and a learning rate of 0.001. The exponential decay rate for the 1st and 2nd moment estimates is set to 0.9 and 0.99, respectively.

The size of the input layer varies for each model based on the number of features. The output layer uses a softmax layer and classifies each beat into one of the following four categories: normal beat (N), supraventricular ectopic beats (S), ventricular ectopic beats (V), and fusion beats (F). The prominent morphological features of an ECG beat is contained in approximately an 800ms window [START_REF] Zhao | A 13.34 µW Event-Driven Patient-Specific ANN Cardiac Arrhythmia Classifier for Wearable ECG Sensors[END_REF] centered around the QRS peak as shown in Fig. 1b. For every model, we take Fig. 9: Proposed one-dimensional CNN for event-driven ECG classification 300ms before the R peak and 400ms after the R peak as a feature set, the size of which is indicated by f eatures variable of the input vector in Fig. 9. For event-driven models, a second channel containing time at each sample relative to the R peak is added. The number of channels are indicated as depth of the input vector in Fig. 9. Finally, to handle the extreme classimbalance inherent in the dataset, we use Synthetic Minority Oversampling TEchnique (SMOTE) to statistically balance the training set.

B. Classification Performance

A global training and verification approach is applied for classification of arrhythmias in the MIT-BIH Arrhythmia dataset. The four paced records 102, 104, 107 and 217 are excluded from the analysis as recommended by the Association for the Advancement of Medical Instrumentation (AAMI) [START_REF] Ec ; | Testing and reporting performance results of cardiac rhythm and ST segment measurement algorithms[END_REF]. All the beats from the remaining records are split into 65% training set, 10% validation set, and 25% test set. For a fair comparison, we have also tested the classifier on uniformly-sampled data using the MIT-BIH database as is, named MDL0 from here on. The MIT-BIH database is sampled at 360Hz using an 11-bit ADC. Since time information is implicit in uniformly sampled data, we only use the amplitudes at each sample as input to the 1D-CNN. The input feature vector for the uniform model is of the size 238x1 (features, depth). While, all event-driven models have an input vector size of 120x2 (features, depth). Accuracy (ACC), sensitivity (SEN), positive predictivity (+PV), and false positive rate (FPR) are used to assess the classifier performance and are defined as follows:

ACC = (T P + T N ) (T P + F P + F N + T N ) (9)
SEN = T P (T P + F N ) (10)
+P V = T P (T P + F P )

F P R = F P (F P + T N ) (11)
where TP, TN, FP, and FN are the number of true positives, true negatives, false positive and false negatives per class.

The classification results for each using the eight event-driven models and the uniformly sampled, MDL0, are summarized in Table III. The classification performance of MDL0 is the benchmark for comparison with ACC, SEN and +PPV, all above 92%. In terms of overall performance considering all four classification parameters, MDL5 has the closest performance to MDL0. This is closely followed by MDL7b, MDL4 and MDL6b. If overall sensitivity is considered, MDL6b and MDL5 at 89.99% and 89.98% SEN, respectively, are closest in performance to MDL0. The overall sensitivity performance of event-driven models can be further improved by using a better class-balancing technique. Next, the overall +PV is greater than 90% for all event-driven models except MDL8b, while the FPR is lowest for MDL5 from all event-driven models. Classification performance by itself is not a good indicator for evaluation of event-driven models. We must also consider the complexity, therefore, we now present complexity analysis in the next subsection.

C. Accuracy-Complexity Trade-off

Table IV summarizes the complexity analysis of eventdriven and uniformly sampled 1D-CNN models. Since all event-driven models summarized in Table II use an LC-ADC resolution of 7 bits, their feature sets can be roughly captured in 120 features with 2 channels (depth). Therefore, the number of trainable parameters for the 1D-CNN classifier are consistently 132,676 for all event-driven models. The uniformly sampled model takes 238 features with 1 channel (depth), yet the number of trainable parameters are doubled at 255,748 compared to the event-driven models. We also calculated the number of floating point operations per second (FLOPS) required by the event-driven and uniformly sampled models. The uniformly sampled model requires 4883224 FLOPS, while the event-driven models only require 49% of this at 2397144 FLOPS. As expected, the uniformly sampled model has superior classification performance compared to all event-driven models. In order to better understand the accuracy-complexity tradeoff in event-driven models, we have analyzed the four most important parameters for a comparative analysis, namely the classification performance and complexity of the classifier as well as the compression and the signal quality of the ECG data. Fig. 10 shows the comparison with sensitivity of all models with their respective compression ratios. The models that follow the signal quality criteria of eq. ( 4) are colored blue, while those that do not are colored orange. Fig. 11 and Fig. 12 present a similar comparison using accuracies and positive predictive values of all models. In terms of accuracy, in Fig. 11, MDL6 offers the best accuracy and a In terms of positive predictive value, in Fig. 12, MDL6 still offers the best compression to signal-quality trade-off with excellent +PV and SDR greater than 21dB. This is closely followed by MDL4 and MDL7b, both of which have less than 21dB SDR. However, at CR=1.59, MDL4 offers the lowest compression. If higher signal compression is desired, MDL7b offers a better trade off with SDR of 20.75dB, which would still be acceptable for many applications. This higher compression can be attributed to the larger step-size between the upper and lower thresholds of the LC-ADC. It must be noted that higher compression rates imply lesser sampling by the LC-ADC and thereby, lesser power consumption in the circuit.

For biomedical applications, sensitivity is often considered the most important parameter as the cost of a misclassified true positives is much higher than the cost of misclassified true negatives. Therefore, in terms of sensitivity of different sampling schemes presented in Fig. 10, MDL5, MDL6 and MDL8b offer the best classification-compression trade off. All Fig. 11: Accuracy vs CR vs FLOPS for LC-ADC models Fig. 12: Positive predictivity vs CR vs FLOPS for LC-ADC models three of these models offer greater than 20dB SDR. With sensitivity at approximately 90%, MDL5 and MDL6 offer a compression of 2.34 and 2.93, respectively. In comparison with the uniformly sampled MDL0, MDL8b has 2.89% less SEN but offers a CR of 4.94. Overall, considering all three parameters, MDL6, offers the best performance with CR of 2.92 and SDR of 21.19. Therefore, a 7-bit LC-ADC with a 2385Hz counter clock frequency and a 6-bit counter clock resolution will offer comparative classification performance to a Nyquist ADC at 11-bits and 360Hz. The LC-ADC will save power by reducing the average sampling rate while maintaining good signal quality. Classification using the LC-ADC will require only 49% of the FLOPS required by data generated from a Nyquist sampling ADC.

D. Open-Source Event-Driven ECG Dataset

To the best of our knowledge, there is no publicly available non-uniformly sampled ECG dataset with arrhythmia labels for researching event-driven data processing and classification. Therefore, in this study, we used the popular MIT-BIH arrhythmia database and different LC-ADC models to derive a nonuniformly sampled event-driven ECG dataset. The dataset is generated using a MATLAB script and can generate LC-ADC data at different LC-ADC resolutions, clock frequencies and counter clock resolutions. The event-driven ECG dataset with arrhythmia annotations and corresponding MATLAB scripts is open-sourced in authors' website 2 to enable further research on the topic.

V. CONCLUSIONS

A new LC-ADC design and evaluation methodology was presented. For a given LC-ADC resolution, the minimum required counter clock frequency was estimated using the standard deviation means of 99.9 percentile magnitude of slopes. Furthermore, the counter clock resolution was chosen based on the number of bits that produce the minimum streambit-rate per second. The stream-bit-rate was estimated using the clock roll-over frequency, the average sampling rate of the LC-ADC and the size of the tuples generated by the LC-ADC. Using these design methodologies, ten different event-driven models were analyzed in terms of compression and signal-todistortion ratio. Since the 7-bit LC-ADC model offered the best compression and signal-quality, this model was further analyzed at different counter clock resolutions and different gaps between the LC-ADC thresholds. Then, a new 1D-CNN model was for cardiac arrhythmia classification. The event-driven models and a uniformly sampled model analyzed classification performance using the 1D-CNN. Then a thorough accuracy-complexity analysis was presented. It was concluded that a 7-bit LC-ADC at 2385Hz counter clock frequency and a 6-bit counter clock resolution offers the best accuracy-compression-complexity trade off. With classification performance comparable to a Nyquist sampled ADC, this LC-ADC model offers 3x data compression while maintaining the signal-to-distortion ratio of more than 21dB. Furthermore, this model requires only 49% FLOPS for classification in comparison with a uniformly sampled model. Therefore, the new LC-ADC evaluation and design methodology presented in this paper is a good tool for designers to choose an appropriate LC-ADC model for their event-driven applications, which may not be limited to ECG data. Finally, we developed an opensource event-driven ECG database with arrhythmia annotations for enabling further research on non-uniformly sampled data processing.

Fig. 1 :

 1 Fig. 1: (a) a level-crossing ADC architecture (b) event-driven ECG samples (ECG out ), the analog ECG signal (ECG in ), and horizontal lines indicating the quantization levels using a 7-bit LC-ADC. (c) illustrated threshold crossings with LSB size, ∆v = 3q.

Fig. 3 :

 3 Fig. 3: The CDF of magnitude of slopes in the MIT-BIH dataset with x o shown at 99.9%.

Fig. 4 :

 4 Fig. 4: Minimum Fc required to accurately model event-driven data at different LC-ADC resolutions (M).

Fig. 5 :

 5 Fig. 5: The stream-bit-rate for varying counter clock resolution (N) over different LC-ADC resolutions (M).

Fig. 6 :

 6 Fig. 6: Counter clock resolution (N) that minimizes the streambit-rate over different LC-ADC resolutions (M).

Fig. 7 :

 7 Fig. 7: The compression ratio (CR) over the different LC-ADC resolutions (M). A CR of 1 represents no compression.

Fig. 8 :

 8 Fig. 8: SDR vs LC-ADC resolutions (M). SDR = 21dB represents the cross over line for 'good' quality signals.

Fig. 10 :

 10 Fig. 10: Sensitivity vs CR vs FLOPS for LC-ADC models

Maryam

  Saeed received her M.S. degree in Electrical Engineering from National University of Sciences and Technology, Islamabad and her B.S. in Telecommunication Engineering from the National University of Computer and Emerging Sciences, Lahore. She is a PhD scholar at the University College Dublin and a Schlumberger Faculty for the Future Fellow. Her current research includes designing arrhythmia classifiers for low power circuits using event-driven ADCs. Qingyuan Wang received the B.E degree with the first honor from both University College Dublin (UCD) and Beijing University of Technology (BJUT) in 2019. He is working towards a Ph.D. degree with SFI Centre for Research Training in Machine Learning at UCD. His interests include deep learning and its implementation on hardware. Olev Martens (Senior member, IEEE) has born in Tallinn, Estonia, 1960. Dipl. engineer of electronics (CUM LAUDE) from 1983, PhD from 2000, both degrees from Tallinn University of Technology (TalTech). He has experience in the industrial R&D, and from 2000 in academy, being senior and lead researcher and now associate professor of measurement electronics at Thomas Johann Seebeck Department of Electronics, TalTech.

TABLE I :

 I Compression and Signal Quality Analysis of LC-ADC Models

	M	Fc (Hz)	N	CR	SDR (dB)
	2	74.13	8	83.96	-2.93
	3	148.26	8	60.04	-1.10
	4	296.51	7	30.34	2.72
	5	593.03	7	13.11	9.06
	6	1186.06	7	6.46	15.61
	7	2372.12	4	1.59	19.91
	8	4744.24	3	0.46	24.94
	9	9488.49	3	0.23	29.92
	10	18976.97	3	0.12	34.54
	11	37953.94	3	0.06	38.68

TABLE II :

 II Compression and Signal Quality Analysis of LC-ADC Models at M = 7 bits and Fc = 2385 Hz

	Model	N	LSB	CR	SDR (dB)	sbr (bps)
	MDL4	4	q	1.59	19.91	1036.82
	MDL5	5	q	2.34	20.46	1074.44
	MDL6	6	q	2.92	21.19	1147.75
	MDL7	7	q	3.25	21.99	1233.14
	MDL8	8	q	3.38	22.57	1320.92
	MDL6b	6	2q	3.66	20.66	-
	MDL7b	7	2q	4.45	20.75	-
	MDL8b	8	2q	4.94	20.25	-

ble II are analyzed for event-driven ECG classification. Then, a complexity analysis in terms of floating point operations per second for each model is presented and a comparison is made with uniformly sampled data. Lastly, an open-source event-driven ECG dataset is summarized based on the best performing LC-ADC models from this study.

TABLE III :

 III Arrhythmia Classification Summary

	Model	Class	ACC	SEN	+PV	FPR
		N	99.09	99.59	99.40	5.10
		S	99.45	86.97	92.46	0.20
	MDL0	V	99.65	97.73	97.29	0.20
		F	99.80	85.71	90.00	0.08
		Average	99.49	92.40	94.78	1.39
	MDL4	N	98.39	99.48	98.72	10.87
		S	98.94	72.65	86.55	0.32
		V	99.49	94.85	97.84	0.16
		F	99.75	82.86	86.57	0.11
		Average	99.14	87.46	92.42	2.86
	MDL5	N	98.53	99.20	99.14	7.19
		S	99.03	81.48	83.04	0.47
		V	99.51	96.88	96.12	0.29
		F	99.76	82.38	88.26	0.09
		Average	99.20	89.98	91.64	2.01
	MDL6	N	98.57	99.55	98.86	9.71
		S	99.13	76.98	90.01	0.37
		V	99.49	95.19	97.45	0.19
		F	99.74	80.48	87.56	0.10
		Average	99.23	88.05	93.47	2.59
	MDL7	N	98.34	99.32	98.82	9.97
		S	98.96	75.68	84.90	0.37
		V	99.42	95.07	96.06	0.16
		F	99.69	78.51	83.33	0.13
		Average	99.10	87.14	90.77	2.66
	MDL8	N	98.31	99.36	98.75	10.57
		S	98.93	76.55	83.31	0.43
		V	99.43	94.22	97.53	0.18
		F	99.71	77.61	86.70	0.10
		Average	99.09	86.92	90.42	2.82
	MDL6b	N	98.50	99.20	99.12	7.42
		S	99.07	81.91	83.98	0.44
		V	99.44	95.98	96.09	0.29
		F	99.71	82.86	82.86	0.14
		Average	99.18	89.99	90.51	2.07
	MDL7b	N	98.53	99.53	98.83	9.94
		S	99.13	75.40	91.24	0.20
		V	99.43	95.02	96.83	0.23
		F	99.72	81.43	84.24	0.13
		Average	99.20	87.84	92.78	2.62
	MDL8b	N	98.23	99.01	99.01	8.36
		S	98.92	81.04	80.00	0.57
		V	99.37	95.13	95.89	0.31
		F	99.69	82.86	80.93	0.16
		Average	99.05	89.51	88.95	2.35

TABLE IV :

 IV Complexity Analysis of Event-Driven and Uniformly Sampled 1D-CNN Models

	Type	Input Vector	Parameters	FLOPS
	Event-Driven	120x2	132,676	2397144
	Uniform	238x1	255,748	4883224

Other design options include increasing N such that the smallest slope in the ECG signal can be captured or to add an overflow flag in the hardware design. However, these would add further complexity in the downstream processing. Repeating a tuple is a simpler solution which adds very little complexity.

The open-source event-driven ECG dataset is available at https://github.com/jedaiproject/Open-Source-Event-Driven-ECG-Dataset

This work is supported by 1) JEDAI project under the Chist-Era Program; 2) Schlumberger Foundation's Faculty for the Future Program 3) Microelectronic Circuits Centre Ireland and 4) Science Foundation Ireland through the SFI Centre for Research Training in Machine Learning (18/CRT/6183).

Benoit Larras was born in Nancy, France in 1988. He received both his engineering degree and his Master in Telecommunications from IMT Atlantique in 2012. He received his Ph.D. in Electrical Engineering in 2015 from IMT Atlantique, Brest, France. He is now an associate professor at Junia, Lille, France, in the Electronics Team. His research topic is about analog/mixed-signal IC design and circuit implementation of neural networks and associative memories, in the context of "Near-sensor computing" and "Edge computing". He is the co-recipient of a Best Paper at the IEEE AICAS2020 conference.

Antoine Frappe (M'08-SM '17)