Léo Girier
email: leo.girier@ec-lyon.fr

Michel Roger

Hadrien Bériot

Anthony Lafitte

Hélène Posson

A Two-Dimensional Model of Sound Transmission Through Curved and Staggered OGV: Effect of Inter-Vane Channel Mode Transitions

The present work extends a recent two-dimensional analytical model for sound generation and transmission in an axial-flow outlet guide vanes row, by taking into account vane camber and stagger, as well as cut-on/cut-off or cut-off/cut-on transitions of inter-vane channel modes. It is aimed at demonstrating that such transitions may have a significant role in turbomachinery aeroacoustics, typically by changing the balance of upstream and downstream scattered waves. The model is based on a mode-matching procedure and generates a uniformly valid description of the sound field. Previous implementations neglecting the vane curvature have shown significant limitations at high frequencies compared to numerical simulations performed with a finite element method code. The extension presented in this paper introduces curvature by a slowly-varying duct formalism. Comparison with numerical simulations show substantial improvements at the cost of a reasonably higher computation time.

I. Introduction

With the exponential growth of air traffic since the 1980s, International Civil Aviation Organization (ICAO) noise standards have become more and more stringent in order to limit noise annoyance for people living in the vicinity of airports. Consequently, aircraft and engine manufacturers need to maintain a constant research effort to keep reducing overall aircraft noise. In particular, with the constant increase in bypass ratio of turbofan engines, fan related noise is today one of the main lever to reduce Effective Perceived Noise Level (EPNL). In this context, the mode-matching technique described in Bouley et al. [START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF] showed promising capabilities as a tool for fan stage pre-design, especially to further understand acoustic transmission and reflection phenomena. The present work is aimed at i) developing an analytical tool based on the mode-matching technique to model acoustic transmission phenomena in a rotor-stator stage and ii) bringing a deeper understanding of sound propagation in blades row. The objective is to include as many realistic design parameters as possible in the model while preserving analytical tractability, so that fast computations are ensured especially for broadband noise. The model should also perform efficiently in an optimization strategy.

The paper focuses on the study of sound transmission through an Outlet Guide Vanes (OGV) row using a two dimensional formulation of the analytical model presented by Roger et al. [START_REF] Roger | Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise[END_REF]. An improvement of the model, taking into account the effect of vane camber on sound propagation, is presented in this paper.

The implementation of camber also reproduces the diffuser effect of the OGV row by the equivalent stream-wise variation of the inter-vane channel cross-section. This has a significant effect on the amplitudes and wavenumbers of the acoustic modes. In particular the mechanism known as cut-off/cut-on or cut-on/cut-off transition for inter-vane channel modes is also expected. The results shown in this paper, as a preliminary implementation, are obtained without mean flow, though including flow is not a difficulty as shown in Bouley et al. [START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF].

The outline of the paper is as follows. State of the art on the mode-matching technique and sound propagation in ducts of varying cross-section are presented in section II alongside the geometrical approximations used to describe the inter-vane channels. The newly added implementation of vane camber in the modified mode-matching equations is also described. Pressure fields computed with the developed algorithm are then compared with numerical simulations in section III. Finally, the influence of inter-vane channel mode transitions on the acoustic power balance is discussed in section IV.

II. Mode-Matching Technique

A. Background

The mode-matching technique is used to solve boundary value problems with linear frequency-domain differential equations. This technique is well suited when the geometry of the problem can be seen as the junction of multiple sub-regions. Such problems can arise for wave propagation in ducts with liners on specific portions of the wall or for wave scattering through bifurcated channels. In particular, sound propagation through an OGV row seen in a 2-D unwrapped cut at a constant radius can be viewed as an example of the latter case. An explanatory scheme is given in Fig. 1. An incident mode, described by its velocity potential φ i , is scattered at the OGV Leading-Edge (LE) interface, generating reflected modes φ r and transmitted modes φ d in the channels that scatter again at the Trailing-Edge (TE) interface, giving rise to reflected channel modes φ u and transmitted modes φ t . Hard-walled boundary conditions at the walls of inter-vane channels and periodic boundary conditions in the y-direction are imposed. In the mode-matching technique, the Helmholtz equation is solved for all scattered potentials. The method can be described in three steps: partitioning, solving and matching. The partitioning consists in dividing the domain into different sub-domains in which the boundary conditions are uniform, allowing a solution of the Helmholtz equation on a local modal basis. In Fig. 1, the sub-domains are the upstream medium, the inter-vane channels and the downstream medium. The solving step is self-explanatory and provides the solution in each sub-domains in a modal form. Finally, the matching procedure connects the different modal solutions at both leading-edge and trailing-edge interfaces (Fig. 1). To do so, it uses continuity equations specific to the problem in order to build a set of equations on the modal coefficients that can be solved by matrix inversion.

The mode-matching technique was first used for electromagnetic fields by Whitehead [START_REF] Whitehead | The theory of parallel-plate media for microwave lenses[END_REF] and the method was later described in details by Mittra and Lee [START_REF] Mittra | Analytical techniques in the theory of guided waves[END_REF]. In the context of cascade aeroacoustics, Bouley et al. [START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF] recently developed a mode-matching technique to study sound transmission and generation through OGV in axial-flow turbofan engines (an exhaustive list of references can also be found in the paper). Bouley et al. [START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF] used an unwrapped 2-D model at a given duct radius and modeled the vanes by infinitely thin flat plates with zero stagger, which corresponds exactly to the problem in Fig. 1 with the addition of a mean flow. The model was subsequently improved to account for stagger angle at the leading edge using Green's second identity, as described by Roger & François [START_REF] Roger | Combined analytical models for sound generation and transmission in cambered axial-flow outlet guide vanes[END_REF]. Roger et al. [START_REF] Roger | Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise[END_REF] later improved the model with a natural extension of mode-matching to make the solution of the potentials φ d and φ u available in the entire inter-vane channel for any stagger. They also did a preliminary investigation on the diffuser effect of the OGV row due to an increasing inter-vane channel cross-section. This effect is introduced when vane mean camber is taken into account. To do so, vane profiles were modeled using circle arcs (see Fig. 2a). With this approximation, the sound field was only available in the domain BA'C'C. To address this problem, they slightly modified the geometry to allow the definition of acoustic modes in the triangle part of the inter-vane channels defined by the points A, A' and B in Fig. 2b. Good agreements were obtained by Roger & Moreau [START_REF] Roger | Coupled mode-matching and slowly-varying duct formulations for predicting noise generation and transmission in axial-flow rotor-stator stages[END_REF] when applying the mode-matching technique with cambered vanes compared to Hixon's [START_REF] Hixon | Computational Aeroacoustics Prediction of Acoustic Transmission Through a Realistic 2D Stator[END_REF] results with the NASA Glenn Research Center BASS code which solves the fully nonlinear Euler equations. Roger et al. [START_REF] Roger | Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise[END_REF] and Roger & François [START_REF] Roger | Combined analytical models for sound generation and transmission in cambered axial-flow outlet guide vanes[END_REF] then used Ovenden's [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF] solution for the velocity potential to highlight the effect of cut-on/cut-off transition on sound propagation but did not implement this mechanism in the mode-matching technique. To derive the solution of the velocity potentials inside the cambered part of the inter-vane channels (from the BA' interface to the channel outlet CC' in Fig. 2b), two effects need to be addressed : the curvature of the channel centerline and the stream-wise variation of the cross-section (diffuser effect). In a first approach, the centerline curvature was not included in the mode-matching technique [START_REF] Roger | Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise[END_REF][START_REF] Roger | Combined analytical models for sound generation and transmission in cambered axial-flow outlet guide vanes[END_REF][START_REF] Roger | Coupled mode-matching and slowly-varying duct formulations for predicting noise generation and transmission in axial-flow rotor-stator stages[END_REF] because i) its impact on sound transmission phenomena has been considered of secondary importance compared to the diffuser effect and ii) this hypothesis ensures faster computations. In that case, the velocity potentials were given by Rienstra's [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] linearized slowly-varying solution, based on a multiple-scale analysis. This formulation provides closed-form expressions for sound transmission through the OGV row. The model proved to perform well in some cases [START_REF] Roger | Coupled mode-matching and slowly-varying duct formulations for predicting noise generation and transmission in axial-flow rotor-stator stages[END_REF] but showed limitations when increasing frequency. At higher frequencies, the effect of curvature can no longer be neglected. Assuming a slowly-varying curvature, Brambley & Peake [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] developed an extension of Rienstra's work [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] including the effects of curvature. This solution can be implemented to compute the velocity potentials in the inter-vane channels. Both Rienstra [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] and Brambley & Peake's [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] formulations are tested in this paper. The following section describes how these solutions are implemented in the mode-matching technique of Roger et al. [START_REF] Roger | Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise[END_REF] and gives all needed velocity potential formulations and matching equations.

B. Matching Equations

Conservation Laws for Axial-Flow Rotor-Stator Stages

The basic equations that need to be satisfied through the OGV row are the conservation of mass and the conservation of total enthalpy. Assuming an inviscid perfect isentropic irrotational gas flow and harmonic fluctuations, Bouley et al. [START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF] showed that the aforementioned conservation laws reduce to the continuity of fluctuating pressure p and fluctuating axial velocity u x written as

p = i ρ 0 kc 0 φ + ρ 0 U 0 • ∇φ, u x = ∂φ ∂ x ,
in which k is the acoustic wavenumber and ρ 0 , c 0 and U 0 are the mean fluid density, sound speed and velocity. Finally, without mean flow, the continuity equations simply concern the velocity potential φ and its gradient ∂φ/∂ x. Two sets of equations are then written at the leading-edge (x = 0) and the trailing-edge (x = c x ) interfaces. The velocity potential and its gradient are gathered into a vector Γ γ . The index γ stands either for the incident (i), reflected (r) or transmitted (t) waves, or for the downstream (d) and upstream (u) acoustic waves inside the channels

Γ γ (x, y) = φ γ ∂φ γ /∂ x , γ = i, r, t, d, u.
The matching equations for the leading-edge (LE) and trailing-edge (TE) interfaces then read

Γ i + Γ r = Γ d + Γ u at LE, Γ d + Γ u = Γ t at TE. (1)
These equations can be directly rewritten on the acoustic modal coefficients. The related developments are described in the following sections.

Leading-Edge Interface

Matching equations at the leading-edge interface involve the incident wave φ i , the scattered reflected φ r and transmitted φ d waves. The velocity potential φ u generated from trailing-edge scattering will be added in section II.B.4. Since the inter-vane channel is staggered by an angle Ψ, the transmitted φ d velocity potential is written in the channel system of coordinates (ξ,η) shown in Fig. 2b. With the modal function for a straight duct of constant height being ψ q (η) = cos α q η , the potentials read

φ i (x, y) = e iα j y e ik + j x , φ r (x, y) = +∞ p=-∞ R p e iα p y e ik - p x , φ d (ξ, η) = +∞ q=0 D q ψ q (η)e ik + q ξ ,
where α j , α p , α q , k j , k p and k q are the transverse and axial wavenumbers defined by

α j = j R 0 , α p = α j + p 2π h M , α q = qπ h 0 and k ± ζ = ± k 2 -α 2 ζ , ζ = j, p, q.
Indices ( j, p) ∈ Z 2 are the azimuthal modal orders of the incident and reflected waves respectively and q ∈ N * is the order of the inter-vane channel mode. R 0 is the radius of the cut through the OGV row. From Roger et al. [START_REF] Roger | Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise[END_REF], the matching equations at the leading-edge interface (1) can be recast to yield a first equation (2) on the modal coefficients D q of the transmitted channel modes and a second one on the reflected modes coefficients R p (3), as

+∞ q=0 D q ϕ + q,p k - p -K + q,p = h M k - p -k + j δ p,0 , (2)
R p = 1 h M +∞ q=0 D q ϕ + q,p -δ p,0 , (3)
with

ϕ ± q,p (Ψ) =              -iu ± q,p (Ψ) (qπ/h M ) 2 -u ± 2 q,p (Ψ) 1 -(-1) q e ih M u ± q , p (Ψ) , h M 2 (1 + δ q,0 ) if u ± q,p (Ψ) = qπ h M , δ q,0 = 1 if q = 0, 0 otherwise, K ± q,p = k ± q cos Ψ - qπ h M 2 tan Ψ u ± q,p (Ψ) , u ± q,p (Ψ) = k ± q sin Ψ -α p .

Trailing-Edge Interface

At the trailing-edge interface the downstream wave φ d is scattered into a reflected upstream wave φ u in the channel and a transmitted wave φ t downstream of the OGV row. The channel velocity potentials φ d and φ u are now expressed in terms of a curvilinear abscissa s, varying from 0 at the inlet (BA') to L c at the outlet (CC'), and its associated curvilinear ordinate n(s) represented in Fig. 2b. The cross-height h(s) varies from h 0 at the inlet (BA') to h M at the outlet (CC'). The velocity potentials φ d and φ u can be expressed using either Rienstra's solution [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] if the curvature is neglected or Brambley & Peake's solution [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] otherwise. The transmitted potential φ t is expressed in its local system of coordinates ( x, ỹ) linked to (x,y) by x = xc x and ỹ = y -R c (1 -cos Ψ). This gives (details are provided in Appendix A)

φ d (s, n) = +∞ q=0 D q e ik + q h M sin Ψ +∞ ν=0 B q ν (0) Υ + ν (s) Υ + ν (0) +∞ µ=0 A ν µ (s)ψ * µ (s, n), φ u (s, n) = +∞ q=0 U q +∞ ν=0 B q ν (L c ) Υ - ν (s) Υ - ν (L c ) +∞ µ=0 A ν µ (s)ψ * µ (s, n) and φ t ( x, ỹ) = +∞ p=-∞ R p e iα p ỹ e ik + p x,
where the star symbol * denotes modal functions and wavenumbers of the slowly-varying part of the channel. The coefficients A q µ , B q ν , the modal function ψ * q and the modal amplitude variation Υ ± q are computed differently in the cases of Rienstra [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] and Brambley & Peake [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF].

When using Rienstra's solution, A q µ = δ µ,q , B q ν = δ ν,q , and

ψ * q (s, n) = cos α * q (s) n + h(s) 2 , Υ ± q (s) = exp i ∫ s s 0 k ± * q ds ρ 0 k ± * q 1/2 with s 0 = 0 for Υ + q , L c for Υ - q ,
while the wavenumbers are given by

α * q (s) = qπ/h(s) and k ± * q (s) = ± k 2 -α * 2 q (s).
If Brambley & Peake's solution is used instead, the modal shape ψq and the axial wavenumber are computed numerically using a Chebyshev collocation method [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF]. Since the modal basis varies from the leading-edge interface to the trailing-edge interface, a projection is needed and coefficients A q µ and B q ν are computed as

ψq (s, n) = +∞ µ=0 A q µ (s)ψ * µ (s, n) and ψ * q (s, n) = +∞ ν=0 B q ν (s) ψν (s, n).
The modal amplitude variation Υ ± q is also changed to

Υ ± q (s) = exp i ∫ s s 0 k ± * q ds ρ 0 k ± * q ∫ h(s)/2 -h(s)/2 ψ2 q /(1 -κn) dn 1/2 ,
in which κ is the channel centerline curvature. In Brambley & Peake's formulation [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF], ψq is normalized so that the integral of ψ2 q over the cross-height is equal to one. Hence, this solution generalizes Rienstra's solution [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] given that the modal shapes and wavenumbers computed numerically converge toward the analytical solution as κ tends to zero.

Using the continuity equations at the trailing-edge interface (1) gives the equations on both sets of reflected U q and transmitted T p modal coefficients as

+∞ q=0 U q (k + p -k - * q (L c ))ϕ 0 q,p = +∞ q=0 D q e ik + q h M sin Ψ +∞ ν=0 B q ν (0) Υ + ν (L c ) Υ + ν (0) (k + * ν (L c ) -k + p ) +∞ µ=0 A ν µ (L c )ϕ 0 µ,p , (4)
T p = 1 h M +∞ q=0 D q e ik + q h M sin Ψ +∞ ν=0 B q ν (0) Υ + ν (L c ) Υ + ν (0) +∞ µ=0 A ν µ (L c )ϕ 0 µ,p + U q ϕ 0 q,p , (5)
The projection function ϕ 0 q,p = ϕ ± q,p (Ψ = 0) stands for the particular case of a zero-stagger interface, such as the trailing-edge interface for an OGV row.

Iterative Procedure

The mode-matching technique runs an iterative procedure of leading-edge/trailing-edge matchings until a converged solution on modal coefficients is reached. At the beginning only the incident wave φ i is known. At the first iteration, the leading-edge matching equations ( 2) and ( 3) are solved to get D q and R p . The modal coefficients D q are then used to compute the trailing-edge matching equations ( 4) and ( 5), giving U q and T p . At the following iterations, the modified leading-edge matching equations ( 6) and ( 7), taking into account the reflected channel wave φ u , are solved. Then the same trailing-edge matching equations ( 4) and ( 5) as for the first iteration are used. This iterative procedure ends when modal coefficients are considered converged, i.e when the difference in acoustic power between two consecutive iterations is sufficiently low.

The modified equations at the leading-edge interface are

+∞ q=0 D q k - p -K + q,p ϕ + q,p = h M k - p -k + j δ p,0 + +∞ q=0 U q +∞ ν=0 B q ν (L c ) Υ - ν (0) Υ - ν (L c ) +∞ µ=0 A ν µ (0) K - µ,p -k - p ϕ - µ,p e -ik - µ h M sin Ψ , (6) and
R p = 1 h M +∞ q=0 D q ϕ + q,p + U q +∞ ν=0 B q ν (L c ) Υ - ν (0) Υ - ν (L c ) +∞ µ=0 A ν µ (0)ϕ - µ,p e -ik - µ h M sin Ψ -δ p,0 , (7)

III. Comparison With Numerical Simulations

A. Methodology A couple of test cases have been defined to test the present analytical solution against numerical simulations computed with the commercial software Simcenter 3D Acoustics [START_REF]Simcenter 3D Acoustics[END_REF]. The two-dimensional Helmholtz problem is addressed using a high-order adaptive Finite Element Method (FEM) [START_REF] Bériot | Efficient implementation of high-order finite elements for Helmholtz problems[END_REF][START_REF] Bériot | Anisotropic adaptivity of the p-FEM for time-harmonic acoustic wave propagation[END_REF]. Periodic boundary conditions are enforced and the far-field reflections are avoided using Perfectly Matched Layers (PML) [START_REF] Bécache | Perfectly Matched Layers for the Convected Helmholtz Equation[END_REF] as shown in Fig. 3. Qualitative and quantitative comparisons in terms of real values of fluctuating pressure between FEM results and the mode-matching technique are presented below.

B. Results

Vane Camber Effect

Four vanes (V = 4) are taken at a duct radius of R 0 = 38 mm, with a leading-edge camber angle Ψ = 30 • and an axial chord length c x = 60 mm. No mean flow is considered (M = 0) and the mean density and sound speed are assumed constant and set to ρ 0 = 1.225 kg/m 3 and c 0 = 340 m/s. Incident acoustic waves of unit amplitude are scattered by the OGV. Computations have been performed for two distinct incident mode orders, j = 1 at kc x = 2.44 ( f 2200 Hz) and j = 5 at kc x = 12.2 ( f 11000 Hz), with corresponding angles of incidence from the x-direction given by θ j = arcsin( j/k R 0 ) to ensure periodicity in the y-direction.

Figure 4 shows the instantaneous pressure maps computed with the FEM code and the mode-matching technique at j = 1 and kc x = 2.44. The pressure patterns are in good agreement upstream and downstream the OGV row, as well as in the inter-vane channels, for both analytical models with and without camber effects. Figure 5 shows the instantaneous pressure profiles extracted upstream, downstream and through the OGV row as shown in dashed black lines in Fig. 4. These quantitative results comfort the conclusion that the model based on Rienstra's solution [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] performs well at low frequencies even without accounting for vane curvature. Yet accounting explicitly for the curvature produces a better match with the numerical solution.

When increasing the frequency to kc x = 12.2 and changing the incident mode order to j = 5, the same fairly good agreement is still observed upstream (Fig. 7a). However the simplified model displays significant discrepancies beyond x = 2c x /3 compared to the numerical simulation (Fig. 7b). These differences are also observable on the downstream profile (Fig. 7c). Including the vane camber using Brambley & Peake's solution [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] then gives much better results (Fig. 6 and Fig. 7). The new model proves to be efficient to predict the pressure pattern evolution inside the inter-vane channels but there are still some differences with FEM results when looking at the downstream pressure profile in Fig. 7c. Work is still in progress to understand the residual errors at high frequencies. The new algorithm already shows promising capabilities at the cost of a reasonably higher computation time. The mode-matching code without camber effects runs in less than 2 seconds for a given frequency while it takes around 17 seconds accounting for vane camber, with a not yet optimized MATLAB code.

Vane Profile Approximation Effect

In the previous analysis, numerical simulations were performed using the same geometry as in the mode-matching model. In an industrial case of vane row pre-design the vane profile could have thickness and camber varying along the chord. It is therefore important to quantify the effect of the approximated vane geometry to see if a more refined description will be needed for future acoustics predictions. To test this effect, the NASA Source Diagnostic Test (SDT) OGV geometry studied by Hixon [START_REF] Hixon | Computational Aeroacoustics Prediction of Acoustic Transmission Through a Realistic 2D Stator[END_REF] has been chosen. The cascade is made of fifty-four vanes (V = 54) of chord length c = 40 mm taken at a duct radius of R 0 = 22.35 cm. Numerical simulations using the real geometry have been carried out to compare the results with those of the mode-matching technique including camber effects and using two different approximated vane geometries (Fig. 8). The camber-angle model uses the same leading-edge camber angle of Ψ = 33.66 • to fit the real deviation from the leading edge to the trailing edge. It has therefore a greater stagger angle (16.83 • ) than the real stagger angle of 11 • and a slightly longer chord length c = 41 mm. The stagger-angle model has instead the same stagger angle of 11 • but a substantially lower leading-edge camber angle (Ψ = 22 • ). No mean flow is considered (M = 0) and the mean density and sound speed are constant and set to ρ 0 = 1.225 kg/m 3 and c 0 = 340 m/s. Computations have been performed for two different incident acoustic waves: mode order j = 6 at the frequency f = 5726 Hz and mode order j = -18 at the frequency f = 11452 Hz.

Figures 9 and10 show the instantaneous pressure fields obtained at f = 5726 Hz with the incident mode order j = 6. The scattered sound field is well reproduced by the modified circle arc geometry used in the mode-matching technique (Fig. 9) but the approximated vane geometry with the real stagger angle gives slightly better results upstream the OGV row and inside the inter-vane channels than the geometry using the real leading-edge camber angle (Fig. 10a and Fig. 10b). Note that the higher stagger angle in the camber-angle vane model does not have any significant effect on the transmitted sound field at this frequency (Fig. 10c). At the frequency f = 11452 Hz, with an incident mode order j = -18, discrepancies between the analytical solution and the numerical solution are more significant (Fig. 11 and Fig. 12). These deficiencies are expected since high frequency waves are more sensitive to small details of the vane geometry. Note that the pressure field computed with the camber-angle vane model is in better agreement with the numerical solution in this case, especially upstream the OGV row where the stagger-angle model seems to miss a cut-on reflected mode.

Up to the frequency f = 11452 Hz, the modified circle arc geometry is able to generate a pressure field in fairly good agreement with the FEM results. The model using the same leading-edge camber angle should be preferred as it reproduces more precisely the reflected scattered waves.

IV. Inter-Vane Channel Mode Transitions

A. Mode Transition in Ducts of Varying Cross-Section

An important fact when dealing with mode transmission through channels of varying cross-section is the possibility for a cut-on mode to turn cut-off, or vice versa. This phenomenon was highlighted by Rienstra [START_REF] Rienstra | Sound propagation in slowly varying lined flow ducts of arbitrary cross-section[END_REF]. In the case of a converging duct, an incident cut-on mode undergoing transition would be totally reflected back, leaving only an evanescent mode beyond. Those phenomenons can occur in OGV inter-vane channels showing cross-section variation.

In case of such a transition, the wave behavior in the inter-vane channels of an OGV row is changed and there could be a significant influence on the acoustic power balance between the reflected and transmitted powers. Both cut-on/cut-off and cut-off/cut-on transitions and their possible effects on modal energy distribution are highlighted below using a solution derived from Ovenden's work [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF] explained in Appendix B. One important underlying hypothesis is the energy conservation of a mode when undergoing transition, avoiding transfer of energy to neighboring modes. Finally, even if the curvature is not accounted for in this preliminary discussion, the transition mechanisms a priori remain the same. The implementation of the transitions in Brambley & Peake's [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] solution is presently in progress.

B. Energy Pumping by Acoustic Tunnel Effect

A cut-off mode generated at the leading-edge interface which undergoes transition expectedly has a higher amplitude at the trailing-edge interface than if was simply cut-off, especially if the transition occurs sufficiently close to the leading edge as shown in Fig. 13. Hence, a possibly significant quantity of energy could propagate into the inter-vane channel via this mode, generating what is called acoustic energy pumping. This mechanism allows energy transmission from upstream to downstream that would not occur without cut-off/cut-on transition. It is then expected that acoustic tunnel effect increases the relative part of transmitted power through the OGV. In Fig. 13b, the amplitude variation for a given curvilinear abscissa plotted in dashed red line in Fig. 13a shows the difference between a cut-off mode which undergoes transition and the equivalent fully cut-off mode computed with a constant cross-height taken at the inlet. The difference before the transition is partially due to the reflection of the cut-off mode where the transition occurs and the variation of the cross-section. Note that the amplitude of the cut-off/cut-on solution is still decaying after the transition but only because the channel cross-section is increasing.

C. Standing Wave Pattern

An upstream propagating mode, generated at the trailing-edge interface, which undergoes transition in the channel will be reflected at the transition location. The upstream propagating mode added to its reflection forms a standing wave pattern. A typical pressure amplitude map can be seen in Fig. 14. The pressure amplitude at the trailing edge location is therefore highly affected by transition. On the other side of the transition, only an evanescent wave is transmitted, which substantially reduces the energy transferred upstream of the OGV, especially if the transition occurs far enough from the leading edge.

V. Conclusion

An improved mode-matching model including vane camber for OGV aeroacoustics studies has been developed in this paper. In particular, comparisons with numerical simulations have shown the importance of accounting explicitly for vane camber when modeling sound transmission through vanes row at high frequencies. The improved model still shows residual discrepancies with numerical simulations in the transmitted field. This is under investigation to further improve the current model, looking at possible curvature-gradient effects. Only a reasonably higher computation time is needed to account for vane camber with a non optimized MATLAB code, which makes the model attractive for broadband-noise predictions. However, the simplified model without camber should be preferred for predictions at low frequencies since it has proved accuracy in this range.

Encouraging results have been obtained with the mode-matching model compared to numerical simulations using a more realistic vane geometry. Even if more discrepancies can be expected at higher frequencies than presented in this paper, the model proves to give accurate results hence a more refined vane geometry description is not necessary in this range. Results with a mean potential flow have also been performed in Moreau et al. [START_REF] Moreau | Two-Dimensional Sound Transmission in Realistic Turbomachinery Cascade[END_REF], using the simplified mode-matching model, and give a reasonably good match with numerical simulations carried out using the Simcenter 3D Acoustics software.

The formalism used in the mode-matching technique is also well suited to implement cut-off/cut-on and cut-on/cut-off transitions in the inter-vane channels. These transitions have already been implemented in the mode-matching technique without camber, with a uniformly valid solution derived from Rienstra [START_REF] Rienstra | Sound propagation in slowly varying lined flow ducts of arbitrary cross-section[END_REF] and Ovenden's work [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF]. Ongoing derivations will allow the addition of transitions when also accounting for curvature [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF]. It could be argued that the effect of those transitions on broadband noise could be negligible at low and medium frequencies since only few and relatively small frequency ranges would be significantly affected. At high frequencies though, these frequency ranges become wider and several modes can undergo transition at the same frequency. A greater effect may be expected then but this needs to be confirmed quantitatively. Furthermore, the selective modal content of tonal noise in turbomachines could be more sensitive to the occurrence of transitions. One limitation in the present solutions [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF][START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF][START_REF] Rienstra | Sound propagation in slowly varying lined flow ducts of arbitrary cross-section[END_REF] could be that the energy conservation of a cut-on mode prevents it from transferring energy to neighboring modes at a transition. Ovenden et al. [START_REF] Ovenden | Cut-on Cut-off Transition in Flow Ducts: Comparing Multiplescales and Finite-element Solutions[END_REF] have presented FEM results showing examples of modes transferring a significant amount of energy to neighboring modes when undergoing transition. They found that such modal scattering may be expected when both the radial mode order and the frequency are high enough in slowly-varying cylindrical ducts. Smith et al. [START_REF] Smith | Flow and geometry induced scattering of high frequency acoustic duct modes[END_REF] then studied flow and geometry induced scattering when a transition occurs at such high frequencies. Their work could be implemented in the current mode-matching technique if deemed necessary. Otherwise, the multimodal approach of Felix & Pagneux [START_REF] Felix | Sound propagation in rigid bends: A multimodal approach[END_REF] could also be used but it would substantially increase the computation time.

Other developments presently in progress include the implementation of a Kutta condition in the mode-matching formulation with vane camber. Then the next objective is to model sound generation from impinging wakes at the leading edge.

A. Matching Constant and Slowly Varying Velocity Potential Formulations

A. Potentials without Curvature

The potential φ d (s, n) of the wave propagating downstream from the leading edge is expressed from Rienstra [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF][START_REF] Rienstra | Sound propagation in slowly varying lined flow ducts of arbitrary cross-section[END_REF] as

φ d (s, n) = +∞ q=0 Q q,d ψ * q (s, n)Υ + q (s).
In the constant cross-section part of the channel, from the leading-edge interface AA' to the interface BA', the same mode is given by

φ d (ξ, η) = +∞ q=0 D q ψ q (η)e ik + q ξ .
At the BA' interface, the systems of coordinates are

s = 0, n = η -h 0 /2 and ξ = h M sin Ψ, η ∈ [0, h 0 ].
Hence ψ * q (s = 0, n) = ψ q (η) and matching the two solutions gives

Q q,d = D q e ik + q h M sin Ψ Υ + q (0)
, so that the velocity potential in the slowly-varying part is recast to

φ d (s, n) = +∞ q=0 D q ψ * q (s, n) Υ + q (s) Υ + q (0) e ik + q h M sin Ψ . (8)
Scattering at the trailing-edge interface generates reflected modes φ u (s, n) in the slowly varying part of the inter-vane channels. The solution is expressed from Rienstra [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF][START_REF] Rienstra | Sound propagation in slowly varying lined flow ducts of arbitrary cross-section[END_REF] as

φ u (s, n) = +∞ q=0 Qq,u ψ * q (s, n)Υ - q (s).
For easier integration in the mode-matching technique, a modification is needed to match cosine modes generated at the trailing-edge interface

φ u ( x, ỹ) = +∞ q=0
U q ψ q ( ỹ)e ik - q x .

At the trailing-edge interface, the systems of coordinates are s = L c , n = ỹh M /2 and x = 0, ỹ ∈ [0, h M ].

Hence ψ * q (s = L c , n) = ψ q ( ỹ) and matching the two solutions gives Qq,u = U q 1 Υ - q (L c )

Then, in the constant part of the channel, the reflected modes are given by φ u (ξ, η) = +∞ q=0 Q q,u ψ q (η)e ik - q (ξ-h M sin Ψ) , and matching at the BA' interface yields

Q q,u = U q Υ - q (0) Υ - q (L c )
.

Finally, the reflected modes in the constant part of the inter-vane channels are written as φ u (ξ, η) = +∞ q=0 U q ψ q (η)

Υ - q (0) Υ - q (L c ) e ik - q (ξ-h M sin Ψ) . (9)

B. Uniformly Valid Solution with Transition

As shown by Rienstra [START_REF] Rienstra | Sound propagation in slowly varying lined flow ducts of arbitrary cross-section[END_REF], a transition is a singularity in the linearized slowly varying solution. Since transitions can occur close to a matching interface, a regularized solution as proposed by Ovenden [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF][START_REF] Ovenden | Cut-on Cut-off Transition in Flow Ducts: Comparing Multiplescales and Finite-element Solutions[END_REF] is needed to avoid wrong values of potentials at matching interfaces that would lead to erroneous modal coefficients. For sound propagation inside inter-vane channels, both cut-on/cut-off and cut-off/cut-on transitions are possible. A solution for the velocity potential of a given mode of order q is then proposed that can be used for both transition phenomena. In the inter-vane channel reference frame (S, n), where S = s is the slow variable, the expression reads φ =Q 2 √ πe iπ/4 ρ 0 k ψ * q (n; S)

where Q = constant to be set with a known value of the potential at some point, σ = propagated part of the axial wavenumber, M = Mach number, β = √ 1 -M 2 , Prandtl-Glauert factor, S t = transition location.

A i and B i are the Airy functions of the first and second kinds. I d and I u represent the relative initial amplitudes of the downstream cut-off mode and of the upstream cut-on mode respectively. To use this solution in the leading-edge problem of section IV.B, so for a cut-off mode coming from the left-side of the transition, I d is set to 1 and I u to 0 since there is no reflected upstream mode coming from the right-side. This yields φof f /on = Q √ πe iπ/4 ρ 0 k ψ * q (n; S) .

Note that the particular case of a cut-on/cut-off solution is exactly Ovenden's result [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF] with a change of coordinate S → -S.

Fig. 1

 1 Fig. 1 Scattering of an acoustic wave through an OGV row seen in 2-D at a given radius.

Fig. 2

 2 Fig. 2 Radial cut showing vane mean camber (a) and the inter-vane channel with the modified vanes profile used in the mode-matching problem statement (b). (Fig. (a) taken from Roger et al. [2])

Fig. 3

 3 Fig. 3 Explanatory scheme of the numerical setup showing how boundary conditions are treated.

Fig. 4 Fig. 5

 45 Fig. 4 Instantaneous pressure maps computed with FEM (a), Mode-Matching without camber (b) and Mode-Matching with camber (c) for j = 1 and k c x = 2.44. Dashed black lines are the locations for quantitative comparisons and solid black lines in Fig. 4a show the limit of PML.

Fig. 6 Fig. 7

 67 Fig. 6 Instantaneous pressure maps computed with FEM (a), Mode-Matching without camber (b) and Mode-Matching with camber (c) for j = 5 and k c x = 12.2. Dashed black lines are the locations for quantitative comparisons and solid black lines in Fig. 6a show the limit of PML.

Fig. 8 Fig. 9 Fig. 10

 8910 Fig. 8 Approximated vane profiles used in the mode-matching technique compared to the real geometry used in the numerical simulation.

Fig. 11 Fig. 12

 1112 Fig. 11 Instantaneous pressure maps computed with FEM (a), Mode-Matching with the same leading-edge camber angle (b) and Mode-Matching with the same stagger angle (c) for j = -18 and f = 11452 Hz.

Fig. 13 (

 13 Fig. 13 (a) Pressure amplitude map of an evanescent mode generated at the leading-edge interface and undergoing cut-off/cut-on transition at 14.6 % of the channel length. (b) Amplitude decay along the red dashed line in Fig. 13a.

Fig. 14 (

 14 Fig. 14 (a) Pressure amplitude map of an upstream propagating mode generated at the trailing-edge interface and undergoing cut-on/cut-off transition at 8.8 % of the channel length. (b) Amplitude decay along the red dashed line in Fig. 14a.

2 √

 2 F B A { χ(S)} ,where F B A ≡ B ii A i . In the case of section IV.C, for a cut-on mode traveling upward from the trailing-edge interface, I u is set to 1 and I d to 0. The solution then reads φon/of f = Q πe

Acknowledgements

This work was performed within the framework of the Labex CeLyA of the Université de Lyon, within the programme 'Investissements d'Avenir' (ANR-10-LABX-0060/ANR-16-IDEX-0005) operated by the French National Research Agency (ANR), and is also partially supported by the industrial Chair ADOPSYS co-financed by Safran Aircraft Engines and the ANR (ANR-13-CHIN-0001-01).

B. Potentials with Curvature

In the case of Brambley & Peake's solution [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] including curvature, a projection has to be made at the BA' interface to change from the straight modal basis ψ * q q ∈N * to the curved modal basis ψq q ∈N * . The projection coefficients are defined as

where ψ * q is a generalized form of ψ q accounting for slow variations of the cross-section. Hence a cosine mode φ d,q of order q propagating downstream from the leading-edge interface is expressed in the slowly varying curved passage as a sum of curved modes

.

In order to simplify the implementation in the mode-matching technique, a solution in the form of a sum of cosine modal functions is sought. This, and summing on all modes q, yields

It is possible to describe the reflected waves following the same methodology. A given cosine mode of order q propagating upstream is then expressed in the slowly varying curved passage as a sum of curved modes

.

The expression in the constant part of the inter-vane channels is then, summing on all modes q

As mentioned in the core text, the formalism for curved ducts of Brambley & Peake [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] is an extension of the straight-duct formalism of Rienstra [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF] given that the modal shape functions and axial wavenumbers computed numerically converge to the straight-duct analytical solution when the curvature tends to zero in the Chebyshev collocation solver. In this case the projection coefficients become A q µ (s) = δ µ,q and B q ν (s) = δ ν,q , ∀s, and the formulations [START_REF] Brambley | Sound transmission in strongly curved slowly varying cylindrical ducts with flow[END_REF] and [START_REF]Simcenter 3D Acoustics[END_REF] become exactly equivalent to [START_REF] Ovenden | A uniformly valid multiple scales solution for cut-on cut-off transition of sound in flow ducts[END_REF] and [START_REF] Rienstra | Sound transmission in slowly varying circular and annular lined ducts with flow[END_REF].


Ali Mestrah
email: ali.mestrah@unicaen.fr

Hicham Oualla

Mathieu Pouliquen

Eric Pigeon

An Identification Algorithm for FIR Systems from Binary Output Measurements

Keywords: System Identifiation, Binary Measurements, FIR system

The present study deals with a new identification algorithm from binary output measurements. The study focuses on the class of Finite Impulse Response (FIR) systems. The proposed algorithm is based on the estimation of correlation functions. A geometric interpretation is proposed and leads to a formulation of the algorithm using a Singular Value Decomposition (SVD). A convergence analysis is proposed showing the mean-square convergence with a rate of OpN ´1q, Monte Carlo simulations are proposed to confirm performance.

I. INTRODUCTION

The system parameters identification is a process that requires usually some information on input, output and noise. The input is often known by the user, but the output not always as one might want. In this study, our interest is the identification with binary output, in which the real output of the system is unknown, we only have access to binary measurements. This type of identification is widely used in the domain of micro-electronics and telecommunications for instance. The choice of binary measurements is due to the facility of measuring, e.g knowing the sign of the output in binary measurements is much easier than knowing the real output, binary measurements are also easy to store and occupy less memory. An other important aspect is the fact that binary sensors have lower cost and are more robust against noise than ordinary sensors.

Several researches focused on the identification of systems from binary measurements, we cite for instance [START_REF] Wang | System identification using binary sensors[END_REF], [START_REF] Jafari | A recursive system identification method based on binary measurements[END_REF], [START_REF] Colinet | A weighted least-squares approach to parameter estimation problems based on binary measurements[END_REF], [START_REF] Depraetere | Identification of linear systems with binary outputs using short independent experiments[END_REF], [START_REF] Chen | Impulse response estimation with binary measurements: A regularized FIR model approach[END_REF], [START_REF] Jafari | Convergence analysis of an online approach to parameter estimation problems based on binary observations[END_REF], [START_REF] Guo | Recursive projection algorithm on FIR system identification with binary-valued observations[END_REF], [START_REF] Marelli | Identification of ARMA models using intermittent and quantized output observations[END_REF], [START_REF] Goudjil | Identification of systems using binary sensors via support vector machines[END_REF], [START_REF] Mathieu Pouliquen | Recursive system identification algorithm using binary measurements[END_REF], [START_REF] Mathieu Pouliquen | Continuous-time system identification using binary measurements[END_REF], [START_REF] Son | Recursive identification of systems with binary-valued outputs and with ARMA noises[END_REF], [START_REF] Pouliquen | Identification using binary measurements for IIR systems[END_REF], [START_REF] Mestrah | Adaptation of recursive least squares identification for binary output systems[END_REF] and [START_REF] Mestrah | Subspace identification from binary output measurements[END_REF]. What differ these studies are the process of implementation (online or offline), the impulse response of the system (finite or infinite), the representation of the system (as a state representation or a transfer function representation), the condition on the input and noise (statistical characteristics, periodicity, distribution function, etc..). In this study, our contribution focuses on Finite Impulse Response (FIR) systems. In most of studies developed for this type of systems, the input and the noise are subject to strict conditions: in [START_REF] Wang | System identification using binary sensors[END_REF], [START_REF] Wang | Identification input design for consistent parameter estimation of linear systems with binary-valued output observations[END_REF] and [START_REF] Wang | Adaptive tracking control of FIR systems under binary-valued observations and recursive projection identification[END_REF] the input must be periodic, in [START_REF] Mathieu Pouliquen | Impulse response identification from input/output binary measurements[END_REF] the input must be binary, in [START_REF] Colinet | A weighted least-squares approach to parameter estimation problems based on binary measurements[END_REF] a known dithering signal is added on the input of the binary sensor, in [START_REF] Jafari | Convergence analysis of an online approach to parameter estimation problems based on binary observations[END_REF] the algorithm is developed in a noise free framework, in [START_REF] Guo | Recursive projection algorithm on FIR system identification with binary-valued observations[END_REF] the noise distribution function must be known, in [START_REF] Son | Recursive identification of systems with binary-valued outputs and with ARMA noises[END_REF] the input must be an Independent and Identically Distributed (IID) random variable sequence, in [START_REF] Wang | Adaptive tracking control of FIR systems under binary-valued observations and recursive projection identification[END_REF] the noise distribution function must be estimated before the identification process or it must be known. Sometimes some user defined parameters are difficult to adjust as in [START_REF] Mathieu Pouliquen | Recursive system identification algorithm using binary measurements[END_REF] where there are two parameters that must be well adjusted to have consistent estimates, sometimes the threshold must be variable as in [START_REF] Wang | a unified identification algorithm of fir systems based on binary observations with time-varying thresholds[END_REF] and sometimes there is no theoretically convergence proof as in [START_REF] Jafari | A recursive system identification method based on binary measurements[END_REF] and [START_REF] Mestrah | Adaptation of recursive least squares identification for binary output systems[END_REF].

Our first contribution in this paper is to propose a simplest identification algorithm with respect to other algorithms, simplest in the sense that the practical framework is less restrictive: simplest in terms of condition on the input signal, simplest in terms of required knowledge on the noise, simplest in terms of user-defined parameter. Roughly speaking, our algorithm is based on the estimation of correlation functions. These correlation functions can be estimated from knowledge of the input and the binary output data, and this leads to the estimation of the system parameters. Our second contribution is to propose an algebraic formulation, allowing an implementation more computationally robust.

The article is organized as follows: the identification problem is described and some notations are introduced in section II. The proposed identification algorithm is introduced, interpreted and analyzed in section III. Numerical simulations are proposed in section IV. Section V concludes the paper.

II. PROBLEM STATEMENT

The discrete time system considered in this paper is depicted in Fig. 1. The system is described by Hpz ´1q defined by:

Hpz ´1q " n ÿ i"0 h i z ´i ( 1
)
where n is the order and th i u iPr0;ns is the impulse response. y t and u t are respectively the system output and input at the time instant tT , T is the sampling period.

From (1) and Fig. 1 the output is expressed as follows:

y t " n ÿ i"0 h i u t´i `vt (2)
where v t is the noise on the output. As shown in Fig. 1 the output of the system is unknown but is measured by a 1-bit ADC (Analog to Digital Converter). We thus define the known binary output s t as follows:

" s t " 1, if y t ě C s t " ´1, if y t ă C . ( 3
)
Remark 1: It was possible to choose the values 1 and 0 for the binary output, but we choose the values 1 and ´1, in order to simplify the algebraic interpretation in section III-B. In the following, let us denote the parameter vector θ as

θ T " `h0 h 1 ¨¨¨h n ˘(4)
The objective of the paper is to estimate the parameter vector θ from N available data tu t , s t u N t"1 .

Some assumptions are taken throughout the document in order to solve the identification problem:

' A.1
The input is a stationary zero mean sequence with gaussian distribution. ' A. [START_REF] Jafari | A recursive system identification method based on binary measurements[END_REF] The noise is zero mean sequence of stationary random variables uncorrelated with the input. ' A. [START_REF] Colinet | A weighted least-squares approach to parameter estimation problems based on binary measurements[END_REF] The order n of the system is known. ' A. [START_REF] Depraetere | Identification of linear systems with binary outputs using short independent experiments[END_REF] The L 2 norm of the parameter vector θ is known and different from 0. Note that assumption A.4 is a normalization assumption. Such an assumption is commonly used in system identification from binary measurements ( [START_REF] Jafari | Convergence analysis of an online approach to parameter estimation problems based on binary observations[END_REF] and [START_REF] Mathieu Pouliquen | Recursive system identification algorithm using binary measurements[END_REF] for instance). An other similar assumption can be an assumption on the static gain of the process which is the sum of the values of θ .

III. IDENTIFICATION ALGORITHM

The identification algorithm is presented in this section. First, the principle is described in subsection III-A, then an algebraic formulation is proposed in subsection III-B, finally a convergence analysis is provided in subsection III-C.

A. Principle

Let us define Er.s as the expected value. From (2) it can easily be shown that

R uy piq " n ÿ j"0 h j R uu pi ´jq `Ruv piq (5)
where R uy piq " E ru t y t`i s and R uv piq " E ru t v t`i s are respectively the cross-correlations between the variables u t , y t and u t , v t with lag equal to i and R uu piq " E ru t u t`i s is the auto-correlation of the variable u t with lag equal to i. From Assumptions A.1 and A.2 the noise is uncorrelated with the input and zero mean, it follows that R uv piq " 0. Using (5) for i from 0 to n we get ¨Ruy p0q R uy p1q . . .

R uy pnq ‹ ‹ ‹ ' " ¨Ruu p0q R uu p1q . . . R uu pnq R uu p1q R uu p0q . . . R uu pn ´1q . . . . . . . . . . . . R uu pnq R uu pn ´1q . . . R uu p0q ‹ ‹ ‹ ' θ . (6)
In order to use [START_REF] Jafari | Convergence analysis of an online approach to parameter estimation problems based on binary observations[END_REF] to estimate θ note that @i P r1; ns R yy p0q y t`i ¯ı " 0. Then from [START_REF] Marelli | Identification of ARMA models using intermittent and quantized output observations[END_REF] we have

E " u t ˇˇyt`i ı " R uy piq R yy p0q E " y t`i ˇˇyt`i ı " R uy piq R yy p0q y t`i . (9)
The cross-correlation between u t and s t is defined by R us piq " E ru t s t`i s, it can then be expressed by

R us piq " E ru t s t`i s " E " E " u t s t`i ˇˇyt`i ıı " E " s t`i E " u t ˇˇyt`i ıı " R uy piq R yy p0q E rs t`i y t`i s " αR uy piq (10)
where α "

R sy p0q
R yy p0q is a constant. It follows from ( 5) and ( 10) that θ satisfies

¨Rus p0q R us p1q . . . R us pnq ‹ ‹ ‹ ' " α ¨Ruu p0q R uu p1q . . . R uu pnq R uu p1q R uu p0q . . . R
‹ ‹ ‹ ' ´1 ¨Rus p0q R us p1q . . . R us pnq ‹ ‹ ‹ ' , ( 12
)
then θ is such that

θ " 1 α θ ( 13
)
where α is given above. 1 α corresponds to a normalization term, if R sy p0q and R yy p0q are not known (which is probably the case in practice), then it is possible to expressed α using Assumption A.4 as follows α " }θ } 2 }θ } 2 .

The algorithm proposed in this paper is then based on (13): we propose the estimation of θ as follows

θ " 1 α θ (14) with θ " ¨ R uu p0q R uu p1q . . . R uu pnq R uu p1q R uu p0q . . . R uu pn ´1q . . . . . . . . . . . . R uu pnq R uu pn ´1q . . . R uu p0q ‹ ‹ ‹ ' ´1 ¨ R us p0q R us p1q . . .
R us piq " 1 N N ÿ t"1 u t s t`i , R uu piq " 1 N N ÿ t"1 u t u t`i (16)
and

α " } θ } 2 }θ } 2 . ( 17
)
Note that the implementation of ( 14) and ( 15) requires two conditions. First the Toeplitz matrix of the estimated autocorrelation R uu piq must be full rank in order to compute θ . From Assumption A.1 this should be the case. However, for a finite number of available data this condition can be not satisfied. In the following we consider the implementation of the above algorithm under the following assumption:

' A. [START_REF] Chen | Impulse response estimation with binary measurements: A regularized FIR model approach[END_REF] The smallest eigen-value of the Toeplitz matrix of the estimated auto-correlation R uu piq is greater than or equal to a strictly positive user-defined value (10 ´9 for instance).

Second, } θ } 2 must be different from 0 in order to realize the normalization. This condition leads to the question of the value of the threshold C. Indeed, for a finite number of available data, if C is too low (respectively too high) with respect to y t , then s t " 1 (respectively s t " ´1) for all t and then θ " 0 from the fact that the input is zero mean. In the following, in order to avoid such a pathological case, we consider the implementation of the above algorithm only if } θ } 2 is greater than or equal to an user-defined value (10 ´9 for instance). In practice, it is sometimes suggested (as in [START_REF] Wang | System identification using binary sensors[END_REF] and [START_REF] Colinet | A weighted least-squares approach to parameter estimation problems based on binary measurements[END_REF]) to add a dithering signal on the input of the binary sensor. Such a "complementary" noise allows to avoid the pathological case described above.

B. Geometric interpretation and algebraic formulation

In this subsection we exhibit a geometric interpretation of the proposed method which leads to an algebraic formulation of [START_REF] Mestrah | Subspace identification from binary output measurements[END_REF]. This geometric interpretation is divided into several steps as detailed below. Some illustrations are depicted in Fig. 2 where, for simplicity, we assume that the number of parameters to be estimated is n `1 " 2 and C " 0. ' Fig. 2. a) shows the scatter plot of the dataset x t " ˆut u t´1 ˙for t from 1 to N, the yellow and blue points represent respectively the dataset for s t equal 1 and ´1.

The red line in Fig. 2. a) corresponds to the equation h 0 u t `h1 u t´1 " 0. Visual inspection shows that the scatter plot has two main directions: ⃗ i is the primary vector and ⃗ j is the secondary vector.

' Let X be the regression matrix defined by

X " `x1 x 2 . . . x N ˘. (18)
Applying the Singular Values Decomposition (SVD) on X we get

X " U 1 `Σ1 0 ˘ˆV T 1 V T 2 ˙" U 1 Σ 1 V T 1 ( 19
)
where

Σ 1 P R 2ˆ2 is a diagonal square matrix, U 1 P R 2ˆ2 and V T 1 P R 2ˆN such that U T 1 U 1 " I 2 and V T 1 V 1 " I 2 .
Multiplying the data by Σ ´1 1 U T 1 we get the scatter plot of Fig. 2. b) (U 1 provides the rotation and Σ ´1 1 realizes the decompression of data).

' Multiplying each vector in Fig. 2. b) by its corresponding binary output, we get Fig. 2.c).

' Computing the mean of the scatter plot of Fig. 2.c), we get the black star in Fig. 2.d). The vector corresponding to this black star is given by 1

N ř N t"1 pΣ ´1 1 U T 1 x t qs t " Σ ´1 1 U T 1 1 N ř N
t"1 x t s t . Note that this vector is orthogonal to the separating line in Fig. 2

.b).

' Multiplying the previous vector

Σ ´1 1 U T 1 1 N ř N t"1 x t s t by U 1 Σ ´1 1 , we obtain the vector U 1 Σ ´2 1 U T 1 1 N ř N
t"1 x t s t . This vector is the vector with the black star depicted on Fig. 2.e) and it is orthogonal to the red line h 0 u t h1 u t´1 " 0. This vector is consequently proportional to ˆh0 h 1

˙.

Note that 1 N U 1 Σ 2 1 U T 1 " 1 N XX T corresponds to the matrix of the estimated auto-correlation R uu piq used in [START_REF] Mestrah | Subspace identification from binary output measurements[END_REF]. It follows that θ in [START_REF] Mestrah | Subspace identification from binary output measurements[END_REF]

corresponds to θ " `1 N U 1 Σ 2 1 U T 1 ˘´1 1 N ř N t"1 x t s t " NU 1 Σ ´2 1 U T 1 1 N ř N t"1 x t s t . (20)
This shows that θ is proportional to the vector with the black star depicted on Fig. 2.e).

From the above steps we obtain the following formulation on θ :

θ " NU 1 Σ ´2 1 U T 1 meanpU 1 Σ 1 V T 1 d Sq ( 21
)
where meanpU 1 Σ 1 V T 1 d Sq denotes the vector built with the mean of each line of U 1 Σ 1 V T 1 d S, d denotes the Hadamard product and S the matrix defined by

S " ˆs1 s 2 . . . s N s 1 s 2 . . . s N ˙. (22)
Finally ( 21) provides the following formulation for the computation of θ :

θ " NU 1 Σ ´1 1 meanpV T 1 d Sq. ( 23
)
This algebraic formulation is equivalent to [START_REF] Mestrah | Subspace identification from binary output measurements[END_REF], its interest is the use of a robust linear algebra tool (i.e. the SVD) for the estimation of θ ( [START_REF] Lloyd N Trefethen | Numerical linear algebra[END_REF]).

C. Analysis

An analysis of the algorithm is provided in this subsection. It requires a complementary assumption. Let us first define F k t the σ -algebra of events generated by the random variables tu l ,t ď l ď ku and let L 2 pF b a q denotes the collection of all second-order random variables which are F b a -measurable. Define ρpkq the maximal correlation coefficient as

ρpkq " sup APL 2 pF 0 ´8q,BPL 2 pF 8 k q |covtA, Bu| σ A σ B ( 24
)
where σ A denotes the standard deviation and covtA, Bu the covariance. The stationary process tu t u is called ρ-mixing ( [START_REF] Kolmogorov | On strong mixing conditions for stationary gaussian processes[END_REF]) if it satisfies lim kÑ8 ρpkq " 0 (25)

Our complementary assumption is thus the following:

' A.6 tu t u is a ρ-mixing process. Roughly speaking, Assumption A.6 means that samples of tu t u sufficiently well separated in time are approximately independent. By now we can establish the following convergence theorem.

Theorem 1: Assume A.1, A.2, A.3, A.4, A.5 and A.6 hold, then θ defined by ( 14) is asymptotically unbiased, mean-square convergent and there exists Σ such that lim

NÑ8 NE " } θ ´θ } 2 2 ı ď Σ. (26)
■ Proof 1: The proof is divided in two parts: first we focus on θ ´θ , then we focus on θ ´θ .

' To begin let us consider θ ´θ . θ is defined by [START_REF] Mestrah | Subspace identification from binary output measurements[END_REF]. Denote

M " ¨Ruu p0q R uu p1q . . . R uu pnq R uu p1q R uu p0q . . . R uu pn ´1q . . . . . . . . . . . . R uu pnq R uu pn ´1q . . . R uu p0q ‹ ‹ ‹ ' (27)
and

N " ¨Rus p0q R us p1q . . . R us pnq ‹ ‹ ‹ ' , (28)
we have θ " M ´1 N and then

θ ´θ " M ´1 N ´M´1 N (29)
This gives

θ ´θ " M ´1p N ´Nq `p M ´1 ´M´1 qN " M ´1p N ´Nq ` M ´1pM ´ MqM ´1N " M ´1p N ´Nq ` M ´1pM ´ Mq θ . (30)
Using the triangular inequality and the Frobenius norm this gives

} θ ´θ } 2 ď } M ´1} F ´} N ´N} 2 `} M ´M} F } θ } 2 ¯. (31)
It is considered in Assumption A.5 that the smallest eigenvalue of M is greater than or equal to a strictly positive user defined value. Denote a this user defined value, this gives

} M ´1} F ď ? n `1 a (32)
and then

} θ ´θ } 2 ď ? n `1 a ´} N ´N} 2 `} M ´M} F } θ } 2 ¯(33)
' By now let us consider θ ´θ . θ is defined by ( 14), then we have

θ ´θ " 1 α θ ´1 α θ " 1 α p θ ´θ q ``1 α ´1 α ˘θ " }θ } 2 } θ } 2 p θ ´θ q `}θ} 2 }θ } 2 } θ } 2 p}θ } 2 ´} θ } 2 qθ " }θ } 2 } θ } 2 p θ ´θ q `1 } θ } 2 p}θ } 2 ´} θ } 2 qθ . (34)
Using the triangular inequality and the fact that

|}θ } 2 } θ } 2 | ď }θ ´ θ } 2 we get } θ ´θ } 2 ď 2 }θ } 2 } θ } 2 } θ ´θ } 2 . (35)
It has been considered that } θ } 2 is greater than or equal to an a priori user defined value. Denote b this user defined value, this gives

} θ ´θ } 2 ď 2 }θ } 2 b } θ ´θ } 2 . ( 36
)
Together with (33) we obtain

} θ ´θ } 2 ď 2 }θ } 2 b ? n `1 a ´} N ´N} 2 `} M ´M} F } θ } 2 ( 37)
which gives

} θ ´θ } 2 2 ď 8 }θ } 2 2 b 2 n `1 a 2 ´} N ´N} 2 2 `} M ´M} 2 F } θ } 2 2 ¯(38) and then NE " } θ ´θ } 2 2 ı ď 8 }θ } 2 2 b 2 n `1 a 2 ´NE " } N ´N} 2 2 ı `NE " } M ´M} 2 F ı } θ } 2 2 ¯. (39)
Note that from Assumption tu t u is ρ-mixing process, it follows from ( [START_REF] Masry | The estimation of the correlation coefficient of bivariate data under dependence: Convergence analysis[END_REF]) that for all j, R uu p jq is mean-square convergent and there exists Σ uu j such that lim

NÑ8 NE " p R uu p jq ´Ruu p jqq 2 ı " Σ uu j . (40)
It follows that there exists Σ M such that lim

NÑ8 NE " } M ´M} 2 F ı " Σ M . (41)
Note also that ty t u is the output of a FIR filter with tu t u as input, then tu t ; y t u is also a ρ-mixing process. ts t u being the output of a 1-bit ADC with ty t u as input, then it can be shown that tu t ; s t u is a ρ-mixing process too. It follows from ( [START_REF] Masry | The estimation of the correlation coefficient of bivariate data under dependence: Convergence analysis[END_REF]) that for all j R us p jq is mean-square convergent and there exists Σ IV. NUMERICAL EXAMPLE In this section some numerical simulations are depicted so as to show the efficiency of the proposed algorithm and confirm the analysis presented in subsection III-C. We choose the same example as in [START_REF] Wigren | Adaptive filtering using quantized output measurements[END_REF]. The system is then described by h 0 " 1, h 1 " ´0.7, h 2 " 4, h 3 " ´2.8 and n " 3. In this example, we assume }θ } 2 to be known (Assumption A.4).

The input tu t u is a zero mean colored gaussian noise. It follows that tu t u satisfies Assumption A.1 and tu t u is a ρmixing process satisfying Assumption A.5. The noise tv t u is a zero mean white gaussian noise uncorrelated with tu t u (Assumption A.2). The threshold is C " 1.

A. First experiment: influence of N

In a first experiment, we investigate the behavior of the algorithm as a function of N. In this experiment the variance of the noise is adjusted so as to have a Signal to Noise Ratio (SNR) equal to 0dB pr 20db. Estimates of all h i are depicted in Fig. 3 as function of N. These results show that the estimates of system parameters are asymptotically unbiased and are well estimated even if N is not very large.

B. second experiment: influence of the noise level

In a second experiment, we investigate the noise effect on the estimates. A Monte Carlo simulation is carried out with 100 runs for several SNR (from 0dB to 20dB with a step of 5dB) and with N " 5000 available data.

The Mean Square Error E " } θ ´θ } 2 2 ı is depicted in Fig. 4 as function of SNR. We notice that the higher the SNR is, the more accurate the estimate is. We conclude from this experiment that the algorithm performs well even if the noise range is wide.

C. Third experiment: confirmation of the analysis

In a third experiment we confirm the analysis of subsection III-C. A Monte Carlo simulation is carried out with 100 runs for several N (from 10 3 to 10 5 with a step of 5.10 3 ) and with SNR" 0dB and SNR" 20dB. ı in logarithmic scale as function of N. We notice that these results confirm the previous conclusion with respect the noise impact.

V. CONCLUSION

An identification algorithm from binary output measurements is proposed in this paper. The algorithm is adapted to the identification of FIR systems. It is based on the estimation of correlation functions. From an implementation point of view, with respect other algorithms, the algorithm requires few on the input signal and on the noise, a main interest of the proposed algorithm is then the fact that the practical framework is less restrictive. An algebraic formulation using a SVD is also proposed in order to facilitate the implementation through a robust algebra tool (robust from a numerical implementation point of view). A convergence analysis is provided, it is demonstrated that the algorithm is asymptotically unbiased and mean-square convergent. Numerical simulations confirm the analysis. The extension of the approach to Infinite Impulse Response systems or to an online formulation are subjects for future works.

Fig. 1 .

 1 Fig. 1. Block diagram of the system model.

  where R us piq and R uu piq are computed as follows

Fig. 2 .

 2 Fig. 2. 2D Geometric interpretation: scatter plot of ˆut u t´1 ˙in differentbasis and estimation of a vector orthogonal to h 0 u t `h1 u t´1 " 0.

Fig. 3 .

 3 Fig. 3. First experiment: Plots of estimated components of θ as a function of N for SNR " 0dB and 20db.

Fig. 5

 5 depicts the variation of E " } θ ´θ } 2 2

Fig. 4 .

 4 Fig. 4. Second experiment: The Mean Square Error as function of SNR for N " 5000.

Fig. 5 ." } θ ´θ } 2 2 ı

 52 Fig. 5. Third experiment: The Mean Square Error as a function of N for SNR" 0dB and SNR" 20dB.