Kaijun Yi
email: kaijun.yi@femto-st.fr

Morvan Ouisse

Emeline Sadoulet

Gaël Matten

Emeline Sadoulet-Reboul

Active metamaterials with broadband controllable stiffness for tunable band gaps and non-reciprocal wave propagation

Keywords: metamaterials, piezoelectric materials, feedback control, vibration, non-reciprocal wave propagation

Introduction

Metamaterials are artificially engineered structures with unconventional effective properties. They are composed of unit cells (also called meta-atoms [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF]), whose sizes are smaller than the wavelengths at interested frequency ranges. Typically through a periodic arrangement (but not necessary [START_REF] Rupin | Experimental demonstration of ordered and disordered multiresonant metamaterials for lamb waves[END_REF]) of these unit cells, band gaps at low frequencies can be created for sound and vibration mitigation [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Zhu | A chiral elastic metamaterial beam for broadband vibration suppression[END_REF]. Since the effective properties of metamaterials can be delicately customized and tailored through the design of unit cells, metamaterials have also been widely used to control acoustic and elastic waves. A plenty of unconventional wave propagation effects have been realized using metamaterials, such as negative refraction [START_REF] Zhu | Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial[END_REF], cloaking [START_REF] Li | Acoustic cloak constructed with thinplate metamaterials[END_REF], topologically protected wave propagation [START_REF] S H Mousavi | Topologically protected elastic waves in phononic metamaterials[END_REF], etc. Although metamaterials have helped to broaden the frontiers of acoustic and mechanical techniques in the past decades, efforts have mostly been dedicated to passive and static unit cells, which are difficult to be altered after being manufactured. This limitation conflicts with the demands of more intelligent and adaptive structures.

Mainly motivated by the aforementioned reason, there has been a growing effort to study active metamaterials [START_REF] Collet | Adaptive metacomposites for vibroacoustic control applications[END_REF][START_REF] Wang | Multi-resonant piezoelectric shunting induced by digital controllers for subwavelength elastic wave attenuation in smart metamaterial[END_REF][START_REF] Sirota | Tunable and reconfigurable mechanical transmission-line metamaterials via direct active feedback control[END_REF]. The key distinguished features of these advanced metamaterials are their controllable properties. The control is mostly realized by using smart materials in the unit cell. For example, including piezoelectric patches shunted with resonant circuits into a unit cell can obtain tunable equivalent dynamic stiffness [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF][START_REF] Bergamini | Phononic crystal with adaptive connectivity[END_REF]. Negative capacitances have the ability to change the equivalent static Young's modulus of piezoelectric materials in a large frequency band. Therefore, they are more widely used to design controllable unit cell. The negative capacitance presents an unstable zone which must be avoided in practical applications [START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Zhu | Experimental study of an adaptive elastic metamaterial controlled by electric circuits[END_REF]. Active control is an alternative way to tune the structural properties. Parameters of some smart materials can be tuned through an external field, for example, magnetoelastic materials show a varying Young's modulus when they are placed in a changing magnetic field. Therefore, these materials have been used to actively modulate system properties [START_REF] Chen | A design of active elastic metamaterials for control of flexural waves using the transformation method[END_REF][START_REF] Mh Ansari | Application of magnetoelastic materials in spatiotemporally modulated phononic crystals for nonreciprocal wave propagation[END_REF]. Actively tuning the effective parameters using feedback control loops has also been proposed. A part of the efforts has been dedicated to design digital circuits [START_REF] Matten | Synthetic impedance for adaptive piezoelectric metacomposite[END_REF][START_REF] Sugino | Design and analysis of piezoelectric metamaterial beams with synthetic impedance shunt circuits[END_REF]. The digital circuit measures the voltage on a piezoelectric transducer and feedbacks current into the same transducer according to a designed control law therefore to mimic the behaviors of analog electrical elements or any behavior of interest [START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF]. For example, the digital circuit could be programmed to mimic a negative capacitance to control the effective Young's modulus. Direct feedback control is another active way to tune the structural properties, which has already been used to realize effective negative mass [START_REF] Wang | Active feedback control of elastic wave metamaterials[END_REF], or add a positive active stiffness into the system [START_REF] Li | Active tuning of vibration and wave propagation in elastic beams with periodically placed piezoelectric actuator/sensor pairs[END_REF].

Active metamaterials have been proposed for many applications. For example, periodic arrays of piezoelectric patches shunted with resonant circuits or negative capacitances are bonded on the surfaces of beams or plates to obtain tunable band gaps [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF][START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Yy Chen | Band gap control in an active elastic metamaterial with negative capacitance piezoelectric shunting[END_REF][START_REF] Fan | A wave-based design of semi-active piezoelectric composites for broadband vibration control[END_REF]. A self-adaptive metamaterial beam with digital circuit controlled mechanical resonators for broadband wave attenuation at sub-wavelength scales is proposed in [START_REF] Li | A self-adaptive metamaterial beam with digitally controlled for subwavelength broadband flexural wave attenuation[END_REF]. Active metamaterials are also explored to manipulate wave propagation. Piezoelectric patches with shunts are used to steer waves for effects like wave focusing [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF], wave redirecting [START_REF] Celli | Wave control through soft microstructural curling: bandgap shifting, reconfigurable anisotropy and switchable chirality[END_REF]. A programmable metasurface with sensing and actuating units is proposed to manipulate the amplitude and phase of transmitted and reflected waves in real-time [START_REF] Chen | A programmable metasurface for real time control of broadband elastic rays[END_REF]. The metasurface particularly shows potential applications in one-way blocking of waves and cloaking. An active metamaterial consisting of symmetrical double Helmholtz resonators is proposed in [START_REF] Ning | Active control of elastic metamaterials consisting of symmetric double helmholtz resonator cavities[END_REF] to realize cloak effect in fluid.

The progress of active metamaterials encourages studies on time-dependent structures. These types of structures possess parameters being modulated in time or in time and space simultaneously. It is shown that modulation of parameters of periodic waveguides in time domain significantly alters the transmission properties at frequencies near and within the band gap [START_REF] Wright | Acoustic wave transmission in time-varying phononic crystals[END_REF]. Piezoelectric patches shunted with time-varying resistance-inductance circuits have shown to provide broadband vibration control effect [START_REF] Casagrande | Smart panel with time-varying shunted piezoelectric patch absorbers for broadband vibration control[END_REF]. Recently, media with parameters modulated in both time and space in a traveling wave form have drawn lots of attention since the wave propagation in them is non-reciprocal. Dispersion curves of waves in these spatiotemporal periodic structures are no longer symmetrical [START_REF] Trainiti | Non-reciprocal elastic wave propagation in spatiotemporal periodic structures[END_REF]. Band gaps for waves propagating in opposite directions are at different frequency ranges. Within these band gaps, several unusual wave propagation behaviors have been observed, such as one-way wave transmission [START_REF] Trainiti | Non-reciprocal elastic wave propagation in spatiotemporal periodic structures[END_REF][START_REF] Nassar | Modulated phononic crystals: Nonreciprocal wave propagation and willis materials[END_REF][START_REF] Yi | One-way energy insulation using time-space modulated structures[END_REF], frequency conversion [START_REF] Yi | Frequency conversion induced by time-space modulated media[END_REF] and frequency splitting [START_REF] Croënne | Brillouin scattering-like effect and non-reciprocal propagation of elastic waves due to spatio-temporal modulation of electrical boundary conditions in piezoelectric media[END_REF].

Although several strategies have been proposed to design active metamaterials as introduced above, efficiently changing the structural parameters still remains an open challenge. The equivalent stiffness obtained using piezoelectric patches shunted with resonant circuits strongly depends on frequency and is only available in a narrow band near the resonant frequency of the circuit [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF][START_REF] Bergamini | Phononic crystal with adaptive connectivity[END_REF]. Negative capacitances are able to tune structural properties in a wide frequency band. Nevertheless, the controlled equivalent Young's modulus only varies dramatically at the vicinity of the unstable zone [START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Zhu | Experimental study of an adaptive elastic metamaterial controlled by electric circuits[END_REF], which means that to obtain significantly modulated system parameters, the system has to work very close to the unstable zone, a small variation of the applied negative capacitance value may make the system unstable or deviate the controlled parameters from the desired values. Direct active feedback control is an emerging technique to design controllable metamaterials. It has been proposed to control wave propagation and vibration properties of 1D periodic waveguides [START_REF] Li | Active tuning of vibration and wave propagation in elastic beams with periodically placed piezoelectric actuator/sensor pairs[END_REF][START_REF] Li | Active localization of wave propagation in elastic beams using periodic placement of piezoelectric actuator/sensor pairs[END_REF]. However, its ability to control structural parameters has not been satisfactorily explored yet.

This paper proposes new kinds of metamaterials with broadband controllable stiffness based on direct active feedback control. The designed basic active unit is composed of a host beam and piezoelectric patches bonded on the beam surfaces. Some of the patches serve as sensors to measure the input signal for the controller; the rest are used as actuators, a feedback voltage generated by the controller is applied on them. The bending stiffness of the active unit is controlled, the geometry and control law are introduced in section 2. Performance and stability issues of the designed active unit are discussed in section 3. The active units are used to form a 1D spatial periodic waveguide and a 1D spatiotemporal periodic waveguide, properties of the two waveguides are studied in section 4.1 and section 4.2, respectively. Finally, conclusions are drawn in section 5.

The designed unit and control law

Figure 1 shows the designed active unit. There are four sensors and two actuators in the cell all made of piezoelectric materials. The polarization of these patches is along the z axis. Electrodes of the patches are on the surfaces perpendicular to the z axis. The target is to control the bending stiffness associated with the flexural waves traveling along the x axis in beam-like structures. Therefore, the four sensors are connected in a way shown in the figure to filter the voltages generated by longitudinal and zero-order torsion waves. The measured voltage by the sensors is V s . The two actuators are also connected together, the applied voltage on them is V a . In practice, the two sensors and one actuator on the upper surface, also those on the lower surface, may be realized using a single complete patch by dividing the electrode into three segments [START_REF] Li | Enhanced electromechanical coupling of piezoelectric system for multimodal vibration[END_REF].

Under bending movement, according to the Euler-Bernoulli beam theory, the normal strain on the cross section (on YZ plane) of the active unit is Hereafter, b

x = -z ∂ 2 w ∂x 2 , (1)
x is used to represent the normal strain on the host beam cross section and p

x is used to represent that on the patch cross section.

The constitutive equations of the piezoelectric patches are

p x = σ p x Y p + d 31 E 3 , D 3 = d 31 σ p x + ε σ 3 E 3 , E 3 = - V h p , (2)
in which, Y p is the in-plane Young's modulus of the patch, d 31 is the piezoelectric constant under constant stress, and ε σ 3 represents the dielectric permittivity. σ p x is the normal stress on the cross section of the patch, D 3 and E 3 are the electric displacement and electric field, respectively. V denotes the voltage on the electrode, for the actuator it is V a , and for the sensor it is V s . Directions 1, 2 and 3 correspond to x, y and z axis, respectively.

The normal stresses on the cross section of the actuators are obtained according to equations (2):

σ p x = Y p ( p x + d 31 V a h p ). (3)
The normal stress on the host beam cross section is

σ b x = Y b b x , (4)
in which, Y b is the Young's modulus of the beam.

According to the Euler-Bernoulli beam theory and using the expressions in equations ( 1), ( 3) and ( 4), the bending moment of the active unit is obtained as

M = h b 2 +hp - h b 2 -hp b 2 -b 2 zσ x dydz = - Y b bh 3 b 12 ∂ 2 w ∂x 2 -{ Y p b a [(h b + 2h p ) 3 -h 3 b ] 12 ∂ 2 w ∂x 2 -Y p d 31 b a (h b + h p )V a }. (5)
On the right side of equation ( 5), the first term corresponds to the contribution of the host beam, and the rest terms are contributions from the actuators. Note that, the contributions of the sensors are ignored since the width of them is much smaller than that of the actuators.

It is more meaningful to consider the behavior of the whole unit rather than a single section of it since the patches act on the whole unit. Therefore, equation ( 5) is integrated from the left end (x L ) of the active unit to the right end (x R ):

x R x L M dx = - Y b bh 3 b 12 θ -{ Y p b a [(h b + 2h p ) 3 -h 3 b ] 12 θ -Y p d 31 b a l(h b + h p )V a }, (6) in which, θ = x R x L ∂ 2 w ∂x 2 dx = ∂w ∂x | x R - ∂w ∂x | x L , it
is the difference of the rotation angles at the left and right ends.

According to equation ( 6), it can be seen that, the bending moment can be changed by controlling the voltage V a . Therefore, if the voltage is controlled according to the following law

V a = αY b bh 3 b -Y p b a [(h b + 2h p ) 3 -h 3 b ] 12Y p d 31 b a l(h b + h p ) θ , (7)
in which, α is an input parameter to determine the bending stiffness of the unit after control as will be seen below, the integration of the bending moment accordingly turns into

x R x L M dx = -(1 + α) Y b bh 3 b 12 θ , (8)
which means that the bending stiffness of the active unit after control is

D b (α) = (1 + α) Y b bh 3 b 12 . (9)
From equation ( 9), it can be seen that the bending stiffness of the controlled unit is (1 + α) times of that of the bare host beam.

The sensors are open-circuited, the difference between the rotation angles θ in the control law in equation ( 7) is measured according to

θ = - 2l(ε σ 3 -Y p d 2 31 ) h p (h b + h p )d 31 Y p V s . (10)
More details related to equation ( 10) can be found in Appendix B. Using equations ( 7) and ( 10), the final control law is obtained as

V a = {αY b bh 3 b -Y p b a [(h b + 2h p ) 3 -h 3 b ]}(ε σ 3 -Y p d 2 31 ) 6Y 2 p b a h p (h b + h p ) 2 d 2 31 V s . (11)
In summary, applying the law in equation [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF], it is possible to change the bending stiffness of the active unit in a manner expressed in equation [START_REF] Wang | Multi-resonant piezoelectric shunting induced by digital controllers for subwavelength elastic wave attenuation in smart metamaterial[END_REF]. One can increase the bending stiffness by using a positive α or decrease it by using a negative α compared with that of the host beam.

Control effects and stability issues

In this section, the control effects and stability issues of the active unit are discussed based on theoretical and numerical studies on a clamped-free unit (namely, a cantilever). In the simulations, the host beam is made of aluminum and the piezoelectric patches are made of PIC 151. Parameters of these materials are listed in Appendix A. The geometrical parameters of the active unit are given in table 1.

Table 1: Geometrical parameters of the clamped-free active unit. The first bending mode of the clamped-free unit is studied to reveal the control performances. Assume that the damping is low therefore can be ignored, then the resonant frequency of the first bending mode of the clamped-free unit is analytically obtained

Length

f n1 = 3.516 l 2 D b (α) ρ b bh b + 2ρ p bh p , (12)
in which, ρ b and ρ p are the density of the host beam and that of the piezoelectric patches, respectively. From equation ( 12) it can be seen that the bending stiffness D b is positively correlated to the resonant frequency, therefore the latter can be used as an indicator of the former.

The first bending mode of the clamped-free unit is also numerically studied using the finite element method (FEM). The simulations are done in the COMSOL Multiphysics software. 3D quadratic Lagrange elements are used in the FEM model, as shown in figure 2. The control law in equation ( 11) is applied on the actuators as electric boundary condition. Figure 3 shows the variation of the resonant frequency of the first bending mode when α changes. Both the theoretical and numerical results are illustrated in the figure. Figure 3 also shows the ratios between controlled and measured voltages (V a /V s ) for different α obtained using equation [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF]. From the numerical results, it can be seen that the applied control strategy is able to reduce the bending stiffness to be close to zero or increase it to some extent compared with that of the host beam. The theoretical and numerical results match well with each other when α satisfies -0.7 ≤ α ≤ 2.6, the relative difference between them is less than 10%. Obvious discrepancies between the theoretical and numerical results are observed when α < -0.7 or α > 2.6. This difference is mainly caused by the actuators. The actuators are designed to bend the unit along the length direction (x axis). However, an unwanted bending along the width direction (y axis) is also caused since the patches are transversely isotropic. The importance of this unwanted bending increases as the absolute value of the ratio V a /V s increases. Consequently, when α < -0.7 or α > 2.6, the unwanted bending becomes non-negligible (for instance, see the mode shape at α = 3 in figure 3), the beam theory no longer holds very well in the unit's behaviors. With respect to the stability, according to equation ( 9), it can be seen that a positive α will cause no stability problem since the bending stiffness is increased. On the contrary, a negative α decreases the bending stiffness, more larger the absolute value of the negative α is, more the stiffness is reduced. Therefore, after a certain critical value the static bending stiffness becomes negative, the system is unstable. Theoretically, the active unit becomes unstable when α ≤ -1. However, as revealed in figure 3, the active unit behaves differently from the theoretical prediction when α is close to -1, which means that the critical point of the stable zone could differ from the theoretical one. Therefore, for systems composed of the designed active units, if negative α values are used in the control law, the stability issue must be checked first.

The stability can be checked by only studying the pole related to the first resonant mode (the term "resonant" indicates that all rigid body modes are excluded since they are not controlled). According to the control theory, a linear system is stable when no pole of it is located in the right half part of the complex plane in the Laplace domain [START_REF] G F Franklin | Feedback control of dynamic systems[END_REF]. There are usually thousands of poles for a system. However, for the active systems presented here, there is no need to study all the poles, because as the bending stiffness decreases, the first resonant mode becomes unstable before others since it has the minimum resonant frequency. Therefore, only the pole related to the first resonant mode needs to be considered. For example, figure 4 shows the variation of the pole related to the first mode of the clamped-free active unit when α decreases from 4 to -1. These results are obtained using the FEM. It can be seen that the clamped-free unit becomes unstable when α < -0.8726, before the theoretical critical point.

Applications

Spatial periodic waveguide with tunable band gaps

First, the proposed active units are used to form a 1D spatial periodic waveguide. Band gaps in periodic structures are useful for vibration and noise control. Realizing band gaps nowadays is not a big challenge, however wider and even tunable band gaps in real time are still not easy to be obtained. It will be demonstrated that periodic waveguides composed of the proposed active units can have broad and controllable gaps.

Figure 5 shows the designed spatial periodic waveguide. It is obtained by alternating active units with passive beams. The waveguide can be divided into 20 identical cells. A zoom in on one of these cells is shown in figure 5. It should be clarified that, the term "active unit(s)" always only denotes the part composed of the patches and the host beam covered by the patches, as illustrated in figure 1, and the term "cell(s)" refers to the repetitive basic part composing a periodic waveguide. For instance, in figure 5, the cell contains an active unit and passive beams. The applied α values for all the active parts are the same. Geometrical parameters of the cells are listed in table 2.

The dispersion curves of flexural waves (A0 mode) corresponding to different α values are studied and the results are illustrated in figure 6. In the figure, k represents the wavenumber and l 1 is the length of the cell. The simulations are done in COMSOL. Only a single cell illustrated in figure 5 is used in the simulation, Floquet periodic conditions are applied on the left and right boundaries of the cell to obtain the dispersion curves.

In figure 6(a) to (d), α equal to or smaller than 0 are used to soften the waveguide. Note that, the system is stable in all these studied cases. When α = 0 is used, the waveguide is very close to a homogeneous beam, therefore the band gaps in this case are quite narrow. As the absolute value of the negative α increases, the lower boundaries of the first and second band gaps decrease, the upper boundaries remain nearly unchanged. Consequently, the widths of the gaps are significantly broadened. In figure 6(e) to (h), α with values larger than 0 are used to stiffen the waveguide. In these cases, as the α increases, the upper boundaries of the gaps move to higher frequencies and the lower boundaries remain almost at the original location, leading to wider gaps. Figure 7 more clearly illustrates how the applied α value will influence the widths of the first and second band gaps. In the figure are also shown the ratios of the wavelengths at the upper frequencies of the second band gaps to the length of the patch. It can be seen that the control law works well until the wavelength is close to 1.7 times of the patch's length. Below this ratio, increasing α leads to less enlargement of the second band gap.

The vibrational properties of the finite waveguide are studied using the FEM in frequency domain. A transverse harmonic force is applied on the left end of the whole waveguide, and the displacement of a corner on the right end is studied (see figure 5). The structural loss factor for the aluminum and patches is set as 1 × 10 -4 . Figure 8 shows the frequency response curves of the displacement at the studied location when different α values are applied. The frequency bands where the vibration level is low correspond to the band gaps. A solo peak inside a band gap observed in some cases are caused by localized modes [START_REF] Fan | A wave-based design of semi-active piezoelectric composites for broadband vibration control[END_REF]. One can see that results in figure 8 further verify the tunable band gaps of the proposed active waveguide. One can use a negative α to broaden the gaps and make them cover low frequency ranges. This feature could be very useful for low frequency vibration and noise control, which is still a challenge in many situations. One can also use a positive α to make the gaps cover wider and higher frequency ranges for the purpose of vibration and noise reduction at interested frequencies.

Spatiotemporal periodic waveguide for non-reciprocal wave propagation

In this section the active units are used to form a spatiotemporal periodic waveguide to realize non-reciprocal wave propagation. Spatiotemporal periodic waveguides possess properties being modulated in both space and time. The non-reciprocal wave propagation effects inside them have been studied by many authors, as introduced in section 1. However, very few propositions can be found on realization of such structures especially for guided elastic waves.

The designed waveguide has 20 cells with identical geometrical parameters, as shown in figure 9. Each cell contains 5 of these active units proposed in figure 1. Therefore, in total there are 100 active units. The reason to design such a compact cell is motivated by the thought that in practice such cell may be realized by using only one complete patch with divided electrode segments on each surface of the host beam. The designed geometrical parameters of the cells are illustrated in table 3. The local bending stiffnesses of the waveguide are modulated by changing the α values applied on the active units according to

α(x i , t) = α 1 + (α 2 -α 1 )H[cos(2πf m t -k m x i )], i = 1, 2, ..., 100, (13)
in which, x i is the central coordinate of the ith active unit, H(•) represents the Heaviside function, f m and k m are the frequency and wavenumber of the modulation wave, respectively. The wavenumber k m is determined by the wavelength λ m according to k m = 2π/λ m . In the simulations, 10 active units per wavelength are used to realize the modulation wave in a piecewise form. Therefore, the wavelength λ m is equal to two times of the cell's length, as illustrated in figure 9. The function in equation ( 13) approximates a rectangular wave as illustrated in figure 10, the α alternates between α 1 and α 2 . Consequently, the bending stiffness of the ith active unit alternates between

K b (α 1 ) = (1 + α 1 )Y b bh 3 b /12 and K b (α 2 ) = (1 + α 2 )Y b bh 3 b /12.
The transfer functions from left to right and from right to left of the waveguide are studied using the FEM. One pair of piezoelectric patches is placed on each side of the waveguide, as shown in figure 9. To obtain the transfer function from left to right, the patches "L" are excited by a tone-burst voltage signal v L (t), the transient voltage responses v R (t) of the patches "R" are measured. The left to right transfer function is therefore obtained using F F T (v R (t))/F F T (v L (t)), F F T (•) means Fast Fourier Transform. Similarly, the right to left transfer function is obtained by exciting the patches "R" and measure the response of patches "L". In the time domain simulations, the time step is 1 × 10 -5 s, which is sufficient since as will be shown the interested frequencies are below 3000 Hz. The damping is included by using the Rayleigh damping model, the coefficient for the mass matrix is 0.005, and the one for the stiffness matrix is 3.18 × 10 -8 . As examples, α 1 = 0 and α 2 = 2 are used in the simulations for demonstration. First, the modulation frequency is set to be zero, namely f m = 0 in equation [START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF]. In this case, the designed waveguide only has periodicity in space, the band gaps of it for left-and right-going waves are the same. To estimate the location of the first band gap, the one cycle tone-burst voltage with the central frequency equal to 1500 Hz shown in figure 11(a) is applied on the patches "L". The voltage responses of the patches "R" are measured, as shown in figure 11(b). Note that, in the simulations the left and right passive beams are chosen to be long enough therefore the measured signals do not include the reflected waves from the two free ends of the whole waveguide. Using the measured voltage and the excitation voltage, the transfer function curve is obtained, as shown in figure 11(c). It can be observed that there is a band gap from around 1250 to 2000 Hz.

When the modulation frequency is no longer zero, for example, if it is a positive value, the band gaps for left-going waves are moved to lower frequency bands, on the contrary, the gaps for right-going waves are moved to higher frequency bands. The shifted frequency value for the first gap of the flexural wave can be estimated using equation ( 14) when a harmonic modulation wave is used [START_REF] Trainiti | Non-reciprocal elastic wave propagation in spatiotemporal periodic structures[END_REF]. In our cases, a rectangular modulation wave is used. The major component of the Fourier series of a rectangular wave is the fundamental harmonic. Therefore, equation ( 14) could also be used in our cases to approximately estimate the shifted frequency bands.

f shif t = f m 4 ( πf m √ 3ρ b k 2 m Y b h b + 2) ≈ f m 2 . ( 14
)
According to equation ( 14), to totally separate the left-going waves' band gap and the right-going waves' band gap, namely to obtain complete unidirectional gaps, the modulation frequency needs to be equal to or larger than the width of the gap. The width of the band gap of the waveguide with α 1 = 0 and α 2 = 2 is revealed in figure 11(c), which is 750 Hz. Therefore, to have completely separated unidirectional band gaps, the modulation frequency must satisfy f m ≥ 750 Hz.

In figure 12, the left panel illustrates the corresponding left to right and right to left transfer functions when the modulation frequency is not zero, the right panel shows the measured voltage on patches "R" and "L" due to the excitation on patches "L" and "R". The transmission coefficients (namely the transfer functions here) are recommended tools to study the reciprocity [START_REF] Jalas | What is and what is not an optical isolator[END_REF][START_REF] Aa Maznev | Reciprocity in reflection and transmission: What is a phonon diode?[END_REF]. In figure 12(a), the modulation frequency is 400 Hz.

From the left panel it can be observed that for the left-going waves, a gap from 1444 to 2221 Hz is created; on the other hand, for the right-going waves, the gap is from 1055 to 1832 Hz. Comparing these two gaps with the one obtained when f m = 0, it can be seen that the frequency shift caused by the moving modulation is close to the estimated value by using equation ( 14), which to some extend backs the accuracy of the simulations. The two gaps in figure 12(a) are not totally separated since the applied modulation frequency is smaller than the critical value. Therefore, in other simulations, the modulation frequency is chosen as 800 Hz, larger than the critical value. In these cases, two totally separated unidirectional gaps are obtained, as shown in figure 12(b). Regarding the recorded voltages, from the right panel of figure 12 it can be seen that due to the non-reciprocity the recoded voltages on opposite sides are different.

Figure 13 shows the control signals of the 1st, 20th, 60th and 100th active units for f m = 400 Hz and f m = 800 Hz. The excitation is applied on the patches "L". The 1st active unit is the left-most one and the 100th is the right-most one in the waveguide. The control is local for each active unit, therefore the control acts only when the waves reach the corresponding active unit. The excitation voltage has a maximum amplitude equal to 0.65 V, as shown in figure 11(a). Under this excitation, the control signal of the 1st active unit has the maximum amplitude among all the active units, around 0.06 V for both simulations. These results demonstrate that the required control voltages are totally within the reasonable range.

The duration of the incident wave is short in the simulations. Therefore, the phase difference between the incident wave and the modulation wave has some influences on the transmission properties of the waveguide. To study these influences, a initial phase ϕ 0 is introduced into the modulation wave, as shown in equation [START_REF] Chen | A design of active elastic metamaterials for control of flexural waves using the transformation method[END_REF], while the initial phase of the incident wave keeps zero. 8 different ϕ 0 values evenly chosen between 0 and 2π have been studied, the corresponding transfer functions are illustrated in figure 14. The excitation is applied on the left in each simulation. It can be observed that the phase changes the depth of the gap. Increasing the cycle number of the incident wave can reduce the influences of phase, since it has been demonstrated that the phase has no influence on the transmission properties in frequency domain [START_REF] Yi | One-way energy insulation using time-space modulated structures[END_REF].

α(x i , t) = α 1 + (α 2 -α 1 )H[cos(2πf m t -k m x i + ϕ 0 )], i = 1, 2, ..., 100. (15)

Conclusions

An active unit with controllable bending stiffness is proposed. The active unit is composed of a host beam and piezoelectric patches bonded on the beam surfaces. Some patches are used as sensors to measure the difference between the rotation angles at the two ends of the active unit. The other patches are used as actuators. A feedback control loop is used between the sensors and actuators. An appropriate control law is applied to control the bending stiffness. Due to the control, the bending stiffness is (1 + α) times of that of the bare host beam. By choosing different α values, it is possible to obtain different stiffnesses for the active unit. A positive α stiffens the unit and a negative one softens it. Systems containing the designed active units are stable if α is larger than a certain critical value, which is negative and depends on the studied system. A simple method to check the stability is studying the pole related to the first resonant mode of the system. The pole must not be located in the right half part of the complex domain to guarantee a stable system.

The active units are included in a 1D spatial period waveguide to obtain tunable band gaps. Numerical results show that, by softening or stiffening the waveguide, the band gaps are broadened. Particularly, when the waveguide is softened, the first gap extends to low frequency ranges, which is very desired since controlling low frequency vibration and noise is not an easy task in many situations.

The active units are also used to realize a 1D spatiotemporal periodic waveguide for non-reciprocal wave propagation. The moving modulation of the local bending stiffnesses is realized by alternating the applied α for each active unit between two designed values according to a rectangular wave function. The non-reciprocal transmission through the waveguide is numerically verified. By choosing an appropriate modulation frequency, complete unidirectional band gaps are demonstrated. It is also demonstrated that the required control voltages are totally within a reasonable range.

Figure 1 :

 1 Figure 1: The designed active unit.

  Width Height Host beam l = 0.05 m b = 0.05 m h b = 0.005 m Actuator and sensor l = 0.05 m b a = 0.044 m, b s = 0.0025 m h p = 0.0005 m

Figure 2 :

 2 Figure 2: Mesh of the FE model.

Figure 3 :

 3 Figure 3: Theoretical and numerical resonant frequencies of the first bending mode as well as the ratios V a /V s obtained using equation (11) for different α values. The mode shape at α = 3 obtained using the FEM is also shown in the figure in Y Z view. The horizontal dashed line indicates the resonant frequency of the host beam.

Figure 4 :

 4 Figure 4: Variation of the pole related to the first mode of the clamped-free active unit when α decreases from 4 to -1 obtained using the FEM. The crosses represent the poles for different α values.

Figure 5 :

 5 Figure 5: A spatial periodic waveguides containing 20 active units.

Figure 6 :

 6 Figure 6: The dispersion curves of flexural waves (A0 mode) in the active waveguide corresponding to different α values. Color blocks indicate the band gaps. k represents the wavenumber and l 1 is the length of the cell. In (a) to (d) α values smaller than or equal to 0 are used to soften the waveguide; in (e) to (h) α values larger than 0 are used to stiffen the waveguide.

Figure 7 :

 7 Figure 7: Left Y axis: widths of band gaps versus α. Right Y axis: ratios of the wavelengths at the upper frequencies of the second band gaps to the length of the patch.

l 1

 1 = 0.042 m, l 2 = 0.0005 m b = 0.01 m h b = 0.005 m Actuator and sensor l p = 0.008 m b a = 0.008 m, b s = 0.0005 m h p = 0.0005 m

Figure 8 :

 8 Figure 8: Frequency response properties of the displacement at the studied location when different α values are used to (a) soften the structure or (b) stiffen the structure.

Figure 9 :

 9 Figure9: A spatiotemporal periodic waveguide containing 100 active units. A passive beam is connected to the active waveguide on each side, a pair of piezoelectric patches denoted by "L" or "R" is placed on each side of the waveguide.

Figure 10 :Figure 11 :

 1011 Figure 10: The applied modulation wave. v m = 2πf m /k m is the velocity and λ m represents the wavelength.

Figure 12 :Figure 13 :

 1213 Figure 12: Left panel: left to right and right to left transfer functions of the waveguide. Right panel: measured voltage on patches "R" and "L" due to the excitation on patches "L" and "R". The modulation parameters are α 1 = 0, α 2 = 2, and (a) f m = 400 Hz, (b) f m = 800 Hz.

Figure 14 :

 14 Figure 14: Influences of the initial phase ϕ 0 of the modulation wave on the transfer function. The excitation is applied on the left in each simulation.

Using the expression p x = -z ∂ 2 w ∂x 2 ∂ 2 w ∂x 2 = ε σ 3 -Y p d 2 31 d 31 Y p E 3 . (B. 3 ) 2 w ∂x 2 = - 2 (ε σ 3 -

 23313133223 and equation (B.2), one can obtain z Integrating the above equation along the thickness of the sensors leading to∂ Y p d 2 31 ) h p (h b + h p )d 31 Y p V s . (B.4)in which, V s = -E 3 dz. Note that there are sensors on lower and upper surfaces, the integration must be performed from (-h b /2 -h p ) to -h b /2 and from h b /2 to (h b /2 + h p ).Further integrating equation (B.4) from x L to x R , one can have the final relationship between the sensed voltage and the difference of the rotation anglesθ = x R x L ∂ 2 w ∂x 2 dx = -2l(ε σ 3 -Y p d231 ) h p (h b + h p )d 31 Y p V s . (B.5)

Table 3 :

 3 Geometrical parameters for cells in the spatiotemporal periodic waveguide.

Acknowledgement

This work was supported by the EUR EIPHI project (contract ANR-17-EURE-0002) and Bourgogne Franche-Comté region.

Appendix A. Materials parameters

The Young's modulus and density of aluminum are 70 GP a and 2700 kg/m


Eva Cerio
email: eva.cerio@u-pem.fr

Alain Debenedetti
email: alain.debenedetti@u-pem.fr

FROM DISPOSSESSION TO OBJECTS' REUSE: TRAJECTORIES AND PRACTICES IN THE CONTEXT OF CLOTHING

Introduction

Sustainability is a key societal challenge, and many institutions point the damages done by overconsumption and the need to reduce and better manage waste. In France, ADEME (the public agency for environment) calls for new ways of consuming and has defined the concept of "second life of objects" (SVO in French). This term refers to the practices that allow an object to be used again by keeping or not its shape and use: objects, once produced, can be successively used and possessed by many consumers in their lives. SVO includes practices of reuse (the shape and the use do not change), repair and redistribution -consumers transform the use of the object-but it differs from recycling -the object is destroyed and only its raw materials are used. If giving an object a second life can be done in every step of consumptionacquisition, use and disposal-this research focuses on the third step, when consumers dispossess an object, by selling it, giving it or bartering it. This research aims at understanding consumers' representations of dispossession and their associated practices in the context of objects' reuse.

The dynamics of dispossession in the context of objects' reuse

If practices of gift-giving, sales or barter are not new -flea markets have existed since the 16 th century-several phenomena have influenced their redevelopment: an ecological awareness of hyper consumption's negative effects, various economic crises, which have impacted consumers' purchasing power, trades opening allowed by Internet and the progressive detachment concerning possession (Botsman & Roger, 2011). The renewed interest for this set of practices calls for a deeper understanding of this phenomenon, in line with research on second hand markets (e.g. [START_REF] Belk | A Naturalistic Inquiry into Buyer and Seller Behavior at a Swap Meet[END_REF][START_REF] Denegri-Knott | I'll sell this and I'll buy them that': eBay and the management of possessions as stock[END_REF], practices of bartering [START_REF] Dalli | From ownership to sharing. Motivations, behaviors and Value at zerorelativo.com[END_REF] or gift-giving [START_REF] Bergadaà | Le don d'objets : dimensions centrales et profils de donneurs aux oeuvres de bienfaisance[END_REF].

Research shows that giving an object a second life is explained by ecological, economic, social and hedonic elements: consumers reuse their objects to preserve the environment or to meet financial needs [START_REF] Ertz | Life after death? Study of second life practices[END_REF]. By doing this, consumers search for practicality, self-expression or socialization and their practices are influenced by objects' value (a more valuable object is less reused) or by sociodemographic characteristics such as age or financial resources (Van de Walle, Hébel & Siounandan, 2012).

If reuse behaviors have been predicted [START_REF] Guiot | A Second-hand Shoppers' Motivation Scale: Antecedents, Consequences, and Implications for Retailers[END_REF] and symbolic meanings of these practices emphasized [START_REF] Belk | A Naturalistic Inquiry into Buyer and Seller Behavior at a Swap Meet[END_REF], research mainly focuses on specific practices and on consumers who retrieve and appropriate previous owned goods -showing how these objects raise paradoxical feelings (e.g. [START_REF] Roux | Identity and Self-Territory in Second Hand Clothing Transfers[END_REF]. The counterpart of appropriation -the ways consumers dispossess objects to allow them to be reused-has been overlooked, which is problematic given the importance of this issue from an ecological, social, or market perspective. This research is intended to fill this gap, by exploring the dynamics of dispossession and go beyond the role of possession in the self-construction (e.g. [START_REF] Belk | Possessions and the Extended Self[END_REF][START_REF] Wallendorf | My Favorite Things": A Cross-Cultural Inquiry into Object Attachment, Possessiveness, and Social Linkage[END_REF].

When consumers choose to dispose an object, they can commit themselves to disinvestment rituals to help them breaking away from their objects and purifying them. This is related to dispossession that is defined as a psychological and emotional process in which a consumer gives up his material and symbolic links with an object [START_REF] Roster | Letting Go: the Process and Meaning of Dispossession in the Lives of Consumers[END_REF][START_REF] Young | Ashes to Ashes, Dust to Dust »: Conceptualizing Consumer Disposition of Possessions[END_REF]. These rituals are essential to allow the object to pass from a domestic and personal space to another and depends on the object's, individual and communal characteristics [START_REF] Albinsson | From trash to treasure and beyond: the meaning of voluntary disposition[END_REF]. Prior works on dispossession in the context of transmission [START_REF] Price | Older Consumers' Disposition of Special Possessions[END_REF] or sales [START_REF] Lastovicka | Three Paths to Disposition: The Movement of Meaningful Possessions to Strangers[END_REF] show that dispossession is a means to defining oneself and constructing one's identity, which can help "rejecting a past-self (what is not anymore or has never been "me"), claiming a present self (a "me" I think to be or I want to be) and constructing a future self (a "me" I want to become)" (Lemaitre & De Barnier, 2015, p. 14).

Using Practice Theory to study the process of dispossession

Prior research on dispossession and reuse have mainly focused on discursive analysis. As dispossession in the context of reuse is an observable and ordinary performance of consumption and spatiotemporal contexts, using a practice-based approach seem relevant [START_REF] Schatzki | Social Practices: A Wittgensteinian Approach to Human Activity and the Social[END_REF]. Practice Theory, initially defined by [START_REF] Reckwitz | Toward a Theory of Social Practices: A Development in Culturalist Theorizing[END_REF] and [START_REF] Schatzki | Social Practices: A Wittgensteinian Approach to Human Activity and the Social[END_REF] is inspired by [START_REF] Bourdieu | La Distinction : Critique sociale du jugement[END_REF] and [START_REF] Giddens | The Constitution of Society: Outline of the Theory of Structuration[END_REF] but differs from their approach: [START_REF] Bourdieu | La Distinction : Critique sociale du jugement[END_REF] introduces practices in his social action theory and habitus concept with the objective of understanding how practices evolve, are reproduced and contribute to social order. For [START_REF] Giddens | The Constitution of Society: Outline of the Theory of Structuration[END_REF], practices are continually recreated by social actors and are stabilized in social structures.

In consumer studies, Practice Theory gets out from consumption behavior's analysis that sees in consumption a means to symbolically communicate with society [START_REF] Warde | Consumption and Theories of Practice[END_REF]. Practices are seen as a space of social representations and an organized manifestation of human actions [START_REF] Schatzki | The Site of the Social: A Philosophical Account of the Constitution of Social Life and Change[END_REF]. A practice is made of doings and sayings, routinized behavior resulting from individual performances and made of corporal and mental activities, rules, knowledge, things, technologies, temporal and social structures [START_REF] Reckwitz | Toward a Theory of Social Practices: A Development in Culturalist Theorizing[END_REF]. Practices are "forms of social realizations linked to material arrangements" (Roques & Roux, 2018, p.39).

From an empirical point of view, Practice Theory considers the different dimensions that characterizes practice, beyond the purely individual aspect and the discourses. These dimensions (cf. Table 1) have raised debates. [START_REF] Schatzki | The Site of the Social: A Philosophical Account of the Constitution of Social Life and Change[END_REF] identifies practical understandings (know-how, routines), rules (prescriptions, instructions on ways of doings) and teleo-affective structures (goals, emotions, meanings, which can also be considered as engagement for [START_REF] Warde | Consumption and Theories of Practice[END_REF] but does not consider the material culture -even though, objects and technologies are essential to understand practices [START_REF] Reckwitz | Toward a Theory of Social Practices: A Development in Culturalist Theorizing[END_REF][START_REF] Warde | Consumption and Theories of Practice[END_REF]. For analytic purposes, different frames have been used to examine consumption's practices. [START_REF] Magaudda | When materiality 'bites back': Digital music consumption practices in the age of dematerialization[END_REF], [START_REF] Shove | Consumers, Producers and Practices: Understanding the invention and reinvention of Nordic walking[END_REF] or [START_REF] Arsel | Taste Regimes and Market-Mediated Practice[END_REF] opt for an object-doingsmeanings approach respectively in the contexts of recorded music listening, Nordic walking or home design. [START_REF] Woermann | Timeflow: How Consumption Practices Shape Consumers' Temporal Experiences[END_REF] use a concurrent frame to examine the concept of timeflow in the context of freeskiing and paintball -material set-up, bodily routines and skills, teleoaffective structures, rules and cultural understandings.

In these works, temporality and space are central: practices are spatially and temporally situated. For example, [START_REF] Evans | Beyond the Throwaway Society: Ordinary Domestic Practice and a Sociological Approach to Household Food Waste[END_REF], in his study of household's food waste practices, identifies that professional and personal schedule constraints influence waste production. [START_REF] Gregson | Moving things along: the conduits and practices of divestment in consumption[END_REF], by exploring conduits of ordinary objects' divestment, suggest that divestment is a spatialized practice and that objects have trajectories: how or where placing objects constitute narratives of consumers and their social relationships.

Suggested field and methodology

We think that working on clothes as a fieldwork would be appropriate, as clothes are everyday life mundane objects that are likely (1) to have different status for the consumer and (2) to cover different practices of dispossession. In addition, depending on the context, the object is likely to have different trajectories within the owner's home. It can be associated to different knowledge and expertise (for example, how to sell it, how to manage the relationship within the family/couple following the decision to dispossess from it etc.), goals (making money, generously helping someone etc.), emotions (relief, anxiety etc.), rules (for example, rituals of divestment) or meanings etc.

In terms of methodology, given the dynamic aspect of the research, a longitudinal multi-sited ethnography of consumers' ways to get separated from objects to be reused seems relevant as it would enable to cover a wide range of contexts. Data collection could therefore combine introspection, logbooks filled by consumers, interviews, observations and photos/videos to grasp the complexity of this practice.

In sum, this paper opens a new landscape of research on the ways people dispossess their objects to give them a second life. It also allows us to discuss people/objects' relationships in the context of durability and to introduce practice theory in the understanding of objects' reuse.

Table 1 . Dimensions of practice theory

 1

	Schatzki, 2002	Reckwitz, 2002	Warde, 2005	Shove and	Woermann &
				Pantzar, 2005 ;	Rokka, 2015
				Magaudda,
				2011 ; Arsel and
				Bean, 2013
	Practical	Body	Understandings	Doings	Bodily routines
	understanding	Mind			and skills
		Agent
		Process
	Rules	Knowledge	Procedures		Rules
	Teleoaffective	Discourse/	Engagement	Meanings	Teleoaffective
	structures				structures
		Language			Cultural
					understandings
		Things	Items of	Objects	Material set-up
			consumption


Diego Di

Antoine Deleforge

Nancy Bertin

MIRAGE: 2D SOURCE LOCALIZATION USING MICROPHONE PAIR AUGMENTATION WITH ECHOES

Keywords: Sound Source Localization, Image Microphones, TDOA Estimation, Supervised Learning

It is commonly observed that acoustic echoes hurt performance of sound source localization (SSL) methods. We introduce the concept of microphone array augmentation with echoes (MIRAGE) and show how estimation of early-echo characteristics can in fact benefit SSL. We propose a learningbased scheme for echo estimation combined with a physicsbased scheme for echo aggregation. In a simple scenario involving 2 microphones close to a reflective surface and one source, we show using simulated data that the proposed approach performs similarly to a correlation-based method in azimuth estimation while retrieving elevation as well from 2 microphones only, an impossible task in anechoic settings.

INTRODUCTION

Sound source localization (SSL) consists in determining the position of a sound source from microphone signals in 3D space. In polar coordinates, most existing methods focus on estimating the directional of arrival, namely, azimuth and elevation angles. Though this task is performed routinely by humans, it still challenges today's computational methods, in particular in the presence of reverberation or interfering sources (see [START_REF] Rascon | Localization of sound sources in robotics: A review[END_REF] and [START_REF] Argentieri | A survey on sound source localization in robotics: From binaural to array processing methods[END_REF] for a review). Computational approaches consist in two components. First, extracting features from audio data that are as independent as possible from the source's content while preserving spatial information. Second, mapping these features to the source position. Two lines of research have been investigated to obtain such mappings: physics-based and learning-based approaches.

Physics-based approaches rely on a simplified sound propagation model [START_REF] Rascon | Localization of sound sources in robotics: A review[END_REF][START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF][START_REF] Dibiase | Robust Localization in Reverberant Rooms[END_REF][START_REF] Lebarbenchon | Evaluation of an open-source implementation of the SRP-PHAT algorithm within the 2018 LOCATA challenge[END_REF]. The free-field model is by far the most widely used one and assumes a single direct sound path from the source to each microphone. When the source is placed far enough, this yields a closed-form mapping from the sound's time-difference-of-arrival (TDOA) in a microphone pair and the source's azimuth angle in this pair. If multiple The research presented in this paper is reproducible. Code and data are available at https://github.com/Chutlhu/MIRAGE s 1 m 1 m 2 Fig. 1: Typical setup with one source source recorded by two microphones. The illustration shows direct sound path (blue lines) and resulting first-order echoes (orange lines). microphone pairs are available and form a non-linear array, their TDOAs can be aggregated to obtain 2D directions of arrival [START_REF] Dibiase | Robust Localization in Reverberant Rooms[END_REF]. These methods strongly suffer in environments where the free-field assumption is violated, e.g., in the presence of strong acoustic echoes and reverberation [START_REF] Scheuing | Disambiguation of tdoa estimates in multi-path multi-source environments (datemm)[END_REF].

Learning-based approaches use an annotated training dataset to implicitly learn a mapping from audio features to source positions [START_REF] Deleforge | Acoustic space learning for sound-source separation and localization on binaural manifolds[END_REF][START_REF] Vesperini | A neural network based algorithm for speaker localization in a multiroom environment[END_REF][START_REF] Sharath Adavanne | Direction of arrival estimation for multiple sound sources using convolutional recurrent neural network[END_REF][START_REF] Perotin | CRNN-based multiple DoA estimation using Ambisonics acoustic intensity features[END_REF][START_REF] Gaultier | VAST : The Virtual Acoustic Space Traveler Dataset[END_REF]. Such data can be obtained from real recordings [START_REF] Deleforge | Acoustic space learning for sound-source separation and localization on binaural manifolds[END_REF] or using physics-based simulators [START_REF] Vesperini | A neural network based algorithm for speaker localization in a multiroom environment[END_REF][START_REF] Sharath Adavanne | Direction of arrival estimation for multiple sound sources using convolutional recurrent neural network[END_REF][START_REF] Perotin | CRNN-based multiple DoA estimation using Ambisonics acoustic intensity features[END_REF][START_REF] Gaultier | VAST : The Virtual Acoustic Space Traveler Dataset[END_REF]. These methods were showed to overcome some limitations of the free-field model, but are usually trained for specific microphone arrays and fail whenever test conditions strongly mismatch training conditions.

Most sound source localization methods, including the above listed, regard reverberation and in particular acoustic echoes as a nuisance. In contrast, some recent work that we refer to as echo-aware methods have showed that the knowledge of early acoustic echoes could be used to reconstruct the geometry of an audio scene [START_REF] Nakashima | A localization method for multiple sound sources by using coherence function[END_REF][START_REF] Dokmanić | Acoustic echoes reveal room shape[END_REF][START_REF] Inkyu An | Reflection-Aware Sound Source Localization[END_REF] or to improve performance of signal enhancement methods [START_REF] James L Flanagan | Spatially selective sound capture for speech and audio processing[END_REF][START_REF] Dokmanić | Raking the cocktail party[END_REF][START_REF] Scheibler | Separake: Source separation with a little help from echoes[END_REF]. In [START_REF] Nakashima | A localization method for multiple sound sources by using coherence function[END_REF], some ad-hoc reflectors are used as artificial pinnae to estimate elevation based on a simple reflection model. In [START_REF] Inkyu An | Reflection-Aware Sound Source Localization[END_REF], cameras, depth sensors and laser sensors are used to identify reflectors and build a corresponding acoustic model that helps SSL.

In this work, we combine ideas from physics-based, learning-based and echo-aware approaches to introduce the framework of microphone array augmentation with echoes (MIRAGE) for SSL. We consider a simple yet common scenario to illustrate this idea: two microphones, one source and a nearby reflective surface, as illustrated in Fig. 1. This may occur, for instance, when the sensors are placed on a table such as in voice-based assistant devices or next to a wall. The reflective surface is assumed to be the most reflective and closest one to the microphones in the environment, hence generating the strongest and earliest echo in each microphone. Under this close-surface model, we ask the following questions:

1. Can early echoes be estimated from two-microphone recordings of an unknown source? 2. Can they be used to estimate both the azimuth and elevation angles of the source, an impossible task in free field conditions?

We propose to use a deep neural network (DNN) trained on a simulated close-surface dataset to estimate early echoes properties from audio features. The MIRAGE framework then exploits these estimated properties by expressing them as TDOAs in the virtual 4-microphone array formed by the true microphone pair and its image with respect to the reflective surface. We show that the framework approximately estimates echo properties, perform similarly to a correlationbased method in azimuth estimation for the considered scenario and estimates impossible elevation angles with good accuracy in noiseless settings using two microphones only.

BACKGROUND IN MICROPHONE ARRAY SSL

In this section, we briefly review some necessary background in microphone array SSL. Let us assume a microphone array of I sensors is placed inside a room and records the sound emitted by one static point sound source. In all generality, the relationship between the signal m i (t) recorded by the sensor placed at fixed position m i and the signal s(t) emitted by the source at fixed position s is defined by:

m i (t) = (h i * s)(t) + n i (t), (1)
where the convolution with room impulse response (RIR) h i (t) embodies the fact that sensor i receives a so-called spatial image of the source and n i denotes possible measurement noise. The RIR depends on the spatial parameters of the scene: microphone positions, source position w.r.t the room, as well as the room acoustic properties (size, absorption and diffuseness of the wall materials.) RIRs can be typically modelled as the sum of the direct path and multiple reflections of the sound. This can boil down to modelling h i as a Dirac impulse at time τ i accounting for the time delay from the source to microphone i, plus an error term. In the frequency domain, this leads to:

H i (f ) = α i (f ) e -2πf τi + ε i (f ), (2)
where the error term ε i (t) collects echoes, the reverberation tail, diffusion, and noise. The term α i (f ) captures the air attenuation phenomenon. A time-domain example of RIR is shown in Fig. 2 (left). Fig. 2: Left, a simulated RIR with annotated components. Right, superposition of two RIRs and visualization of time difference of arrival between direct paths (TDOA), first echoes (iTDOA) and direct path and first echo (TDOE).

2-channel 1D-SSL

Let us first consider the stereo case (I = 2). Under the farfield assumption, traditional SSL methods use the time difference of arrival (TDOA), τ τ 2 -τ 1 , as a proxy for the estimation of the angle of arrival (AOA), since:

AOA = arccos (c τ / d) , (3)
where c is the speed of sound and d the inter-microphone distance. SSL then reduces to estimating the TDOA, which can be done by cross-correlation-based methods such as the widely used and well performing generalized crosscorrelation with phase transform (GCC-PHAT) method [START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF][START_REF] Blandin | Multi-source TDOA estimation in reverberant audio using angular spectra and clustering[END_REF]. Given short-time Fourier transforms M 1 and M 2 of the two microphones signals, the GCC-PHAT angular spectrum is defined as:

Ψ GCC (τ ) = f,n M 1 (f, n)M * 2 (f, n) | M 1 (f, n)M * 2 (f, n) | e -2πf τ . (4)
Then, the TDOA estimate is given by τ = arg max τ Ψ GCC (τ ).

Note that Ψ GCC can also be expressed directly as a function of the AOA using (3), hence the term angular spectrum.

Multichannel 2D-SSL

When more microphones are available and the array is not linear, 2D-SSL can be envisioned. A possible approach is to use 1D-SSL on all pairs and combine their results, a principle which was successfully applied in the steered response power with phase transform (SRP-PHAT) method [START_REF] Dibiase | Robust Localization in Reverberant Rooms[END_REF]. SRP-PHAT exploits the geometry of the microphone array and the estimated TDOAs from microphone pairs to return the DOA. In a nutshell, this algorithm aims to estimate a global angular spectrum Ψ SRP (θ, φ) which will exhibit a local maximum in the direction of the active source. First, a global grid of possible DOAs is defined according to a desired resolution and

MIRAGE: MICROPHONE ARRAY AUGMENTATION WITH ECHOES

We now introduce the proposed concept of microphone array augmentation with echoes (MIRAGE). Let us first expand formula (2) to account for more echoes:

H i (f ) = K k=0 α k i (f ) e -2πf τ k i + ε i (f ) (5)
where the sum now comprises the direct path (k = 0) and the K earliest reflections (K = 1 in this paper) and ε i collects the remaining RIR components. Here, α k i (f ) accounts for both air attenuation and wall absorption phenomena. In the remainder of this paper, we make the approximation of frequency-independent α k i . Eq. 5 then corresponds to the well known image-source (IS) model, where reflections are treated as mirror images of the true source with respect to reflective surfaces, emitting the same signal. We will employ here a less common but equivalent interpretation of IS, namely, the image-microphone (IM) model. As illustrated in Fig. 3, virtual microphones are mirror images of the true microphones with respect to reflective surfaces. In this view, the echoic signal received at a true microphone is the sum of the anechoic signals received at this microphone and its images. If we consider the virtual array consisting of both true and image microphones, multiple microphone pairs are now available. For each of them, it is then possible to define a corresponding time difference of arrival. Among them, we will refer to the one between the two real microphones as TDOA, the one between the two image microphones as image TDOA (iTDOA) and the one between the first microphone and its image as time difference of echoes (TDOE). We have:

TDOA = ( m 2 -s -m 1 -s )/c = τ 0 2 -τ 0 1 , (6) iTDOA
= ( m2 -s -m1 -s )/c = τ 1 2 -τ 1 1 , (7)
TDOE = ( m1 -s -m 1 -s )/c = τ 1 1 -τ 0 1 , (8)
where mi denotes the image of position m i . These three quantities are directly connected to RIRs, as illustrated in Fig. 2(right). Let V = {TDOA, iTDOA, TDOE} ∈ R 3 . Following the 2D-SSL scheme described in Sec. 2.2 and given the virtual microphone-array geometry (which depends on the relative position of microphones to the surface), V could in principle be used to estimate the 2D directional of arrival of the source. In the next section, we present a learning-based method to estimate V using audio features obtained from only two microphones.

LEARNING-BASED ECHO ESTIMATION

Our approach is to train a deep neural network (DNN) on a dataset simulating the considered close-surface scenario. We model the problem as multi-target regression, with interaural level difference (ILD) and interaural phase difference (IPD) as input features, and V ∈ R 3 as output parameters. ILD and IPD features are defined in the frequency domain as follows:

ILD(f ) = 1 T T t=1 log | M2(f,t) M1(f,t) | IP D(f ) = 1 T T t=1 M2(f,t)/|M2(f,t)| M2(f,t)/|M1(f,t)| (9)
More precisely, the input of the network is x = [ILD, Re(IP D)], Im(IP D)], where Re and Im denote real and imaginary part operators, respectively. Note that for the IPD, the frequency f = 0 is discarded because it is constant for every observation. In general, the mapping between V and the proposed feature is not unique. In particular, this happen when τ1 2 = τ 1 1 . In order to avoid this, we preventively pruned all the entries with |τ 1 2 -τ 1 1 | < 10 -6 from the dataset. We use a simple fully-connected DNN architecture consisting of a D-dimensional input layer, a 3-dimensional output layer, and 3 fully connected hidden layers with respective input sizes 500, 300 and 50. Rectified linear unit (ReLU) activation functions are used except at the output layer, and each hidden layer has a dropout probability p do = 0.3. We use the mean square error loss function for training and the Adam optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF]. The normalized root mean square error (nRMSE) is taken as validation metric 1 . The network is manually tuned on a validation set to find the best combination of number of hidden layers, their sizes and p do . Once time delay estimates V are returned by the DNN, they are converted to synthetic local angular spectra and passed to Ψ SRP (See Sec. 2.2) together with the relative positions of true and image microphones which are assumed known. We call this algorithm MIRAGE. The synthetic local angular spectra consist of Gaussians centered at V and with variances equal to the prediction errors made by the DNN on the validation set.

IMPLEMENTATION AND RESULTS

To the best of the authors' knowledge, no reference implementation of algorithms for 2D-SSL using only 2 microphones is available to date. To check the validity of TDOA estimation, it is compared to GCC-PHAT using the true microphones (see Sec. 2.1). For training and validation of the DNN we generate many random shoe-box room configurations using the software presented in [START_REF] Steven | A fast and accurate shoebox room acoustics simulator[END_REF]. This software implements both the image-method for simulating reflections and a ray-tracing algorithm for diffusion. Room widths are uniformly drawn at random in [START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF][START_REF] Sharath Adavanne | Direction of arrival estimation for multiple sound sources using convolutional recurrent neural network[END_REF] m, heights in [START_REF] Argentieri | A survey on sound source localization in robotics: From binaural to array processing methods[END_REF][START_REF] Dibiase | Robust Localization in Reverberant Rooms[END_REF] m. Random source/microphones positions and absorption coefficients for the 6 surfaces are used, respecting the close-surface scenario. In particular, the microphones are at most 30 cm from the close-surface, placed 10 cm from each other, the absorption coefficients of the other walls are uniformly sampled in (0.5, 1) and the one of the close-surface is in (0, 0.5). The same realistic diffusion profile [START_REF] Gaultier | VAST : The Virtual Acoustic Space Traveler Dataset[END_REF] is used for all surfaces. Around 90, 000 audio scenes are generated this way, yielding reverberation times (RT 60 ) between 20 ms and 250 ms. For training and validation, the RIRs are convolved with 1 sec of white-noise (wn) with no additional noise. All signals and RIRs are sampled at 16 kHz. The STFT is performed on 1024 point with 50% overlap. Finally the features are computed as in (9) yielding a vector of size D = 1534 for each observation. While we validate the DNN on a portion of the dataset in a holdout fashion, the test is conducted on 200 new RIRs convolved with both wn and speech (sp) utterances. This set is generated similarly to the training and validation sets. Moreover the recordings are perturbed by external white noise at 10 dB SNR (wn+n, sp+n). The speech signals are normalized speech utterances of various lengths (from 1 s to 6 s), randomly selected from the TIMIT corpus. A free and opensource Matlab implementation of SRP-PHAT 2 is used to aggregate local angular spectra obtained from the DNN's output. A sphere sampling with 0.5 • resolution and coordinates θ ∈ [-179, 180] and φ ∈ [0, 90] is used for the DOA search.

TDOA estimation errors using the proposed approach and GCC-PHAT are presented in Table 1. Training a DNN to estimate TDOAs brings similar performances as GCC-PHAT in terms of nRMSE. Estimation of iTDOA and TDOE seems to be a harder task for the simple DNN we used. Nevertheless, our results confirm the possibility of retrieving early echoes from only two-microphone recordings. When some external noise is added, performance of both methods severely degrades. This is a well-know and expected behaviour for GCC-PHAT. It suggests that noise should be considered in the training phase of MIRAGE. When we compare the performance in terms of AOA, the two methods yield the same accuracy within a 20 • threshold, as can be see in Table 1.

When a smaller tolerance is considered, GCC-PHAT outdoes the proposed approach in accuracy, with comparable errors. Again, when adding noise, performance decreases. In Table 2 the performance of the full 2D-SSL pipeline is showed. Within a tolerance of 20 • , the MIRAGE model allows estimation of both azimuth and elevation of the target source. However since in our data the 2 microphones were free to move, the inclinations of the true and image pairs are rarely flat. While this helps elevation estimation, it reduces the accuracy of predicting the right azimuth. While external noise is again decreasing the accuracy dramatically, it is interesting to notice that our DNN model trained and validated with white noise sources somewhat generalizes to speech sources.

CONCLUSION

In this paper we demonstrated how a simple echo model could allow 2D SSL with only two microphones, using simulated data. Future research will focus on extending this proof-ofconcept to real data. The problem of echo-delay estimation proved to be very challenging, and extensions of the proposed learning scheme will be developed to obtain more reliable estimations of angular spectra. Extensions of the method to better handle various types of noise and emitted signals will also be sought. Finally, applications of the MIRAGE framework to larger microphone arrays, higher order echoes and a variety of tasks beyond SSL will be explored.

Fig. 3 :

 3 Fig.3: Illustration of the images m1 and m2 of microphones m 1 and m 2 in the presence of a reflective surface and a source. Blue lines correspond to direct paths, orange lines correspond to echo paths. computational load. Second, for each pair of microphones, a local set of AOAs is defined and a TDOA-based algorithm (e.g. GCC-PHAT) is used to compute the associated local angular spectrum. Finally all the local contributions (a collection of local Ψ GCC (τ )) are geometrical aggregated and interpolated back to the global DOA grid to form Ψ SRP (θ, φ), and the DOA maximizing Ψ is used as estimate.

Table 1 :

 1 2 http://bass-db.gforge.inria.fr/bss_locate/ nRMSE ACCURACY Input TDOA iTDOA TDOE θ < 10 • θ < 20 • Normalize root mean squared error for TDOA estimation and mean angular error in • (with accuracies (%)) for AOA estimation with 10 • and 20 • angular tolerance.

	MIRAGE	wn	0.18	0.28	0.25 4.10 (77) 5.97 (97)
	MIRAGE	wn+n 0.68	0.69	0.89 5.00 (26) 9.89 (54)
	MIRAGE	sp	0.31	0.34	0.56 4.83 (63) 7.26 (82)
	MIRAGE	sp+n	0.99	0.98	1.48 4.60 (16) 9.88 (35)
	GCC-PHAT wn	0.21	-	-	4.22 (81) 6.19 (97)
	GCC-PHAT wn+n 0.68	-	-	4.03 (65) 5.34 (83)
	GCC-PHAT sp	0.32	-	-	4.08 (82) 5.34 (97)
	GCC-PHAT sp+n	1.38	-	-	4.70 (19) 8.38 (32)
	DoA		ACCURACY		ACCURACY
				< 10 •		< 20 •
		Input	θ	φ		θ	φ
	MIRAGE wn	4.5 (59) 3.9 (71)	6.8 (79)	5.9 (88)
	MIRAGE wn+n	4.4 (18) 5.5 (26)	9.4 (35)	11.1 (66)
	MIRAGE sp	4.6 (45) 4.8 (59)	8.1 (71)	7.2 (83)
	MIRAGE sp+n	5.2 (17) 5.9 (12)	10.7 (38) 12.3 (43)

Table 2 :

 2 Mean angular error in • (with accuracies (%)) for 2D SSL (azimuth and elevation) with 10 • and 20 • tolerance.

The nRMSE takes values between 0 (perfect fit) and ∞ (bad fit). If it is equal to 1, then the prediction is no better than a constant.