Stéphane Monteil
email: stephane.monteil@in2p3.fr

CKM Fits: the Standard Model and the New Physics

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In the Standard Model (SM) of electroweak interactions, the CP violation arises from a single non-vanishing complex phase in the (3 × 3) unitary Cabibbo-Kobayashi-Maskawa [1] matrix describing the quark flavour-mixing in charged currents interactions. Four fundamental constants are enough to parametrize this matrix. An exact, unitary to all orders and phaseconvention independent parametrization [2] is used throughout this document, inspired from the one proposed by Wolfenstein et al. [START_REF] Wolfenstein | [END_REF]:

λ = |V us |/ |V ud | 2 + |V us | 2 , Aλ 2 = |V cb | |V ud | 2 + |V us | 2 and ρ + iη = -V ud V * ub /V cd V * cb .
While λ and A are accurately determined (λ is measured from superallowed nuclear transitions (|V ud |) and semileptonic kaon decays (|V us |), A comes from the inclusive and exclusive semileptonic B decays), the parameters ρ and η, being the real and imaginary coordinates of the unitarity triangle (UT) apex, are much less constrained. The CP violation experiments are aiming at measuring both the sides and angles of the triangle, overconstraining the apex. The metrology of the apex paramaters within the SM allows to measure the size of the CP violation through the Jarlskog invariant [START_REF] Jarlskog | CP Violation[END_REF]. Yet, the main purpose of a global analysis of the sides and angles measurements is certainly to find inconsistencies, which would suggest new contributions above the Standard Model. Alternatively, it is worth to examine, in a generic way, which room is left to New Physics (NP) by the current B data. A relevant natural laboratory to study NP effects is the mixing of B mesons. Assuming NP contributions only in the short-distance part of the mixing process, it can be parametrized by two additional parameters: r 2 q e 2iθq = B0 q M SM+NP 12 B 0 q / B0 q M SM 12 B 0 q [5], where q stands for d or s quarks. It might be interesting to distinguish the NP amplitude, which leads to the following alternate parametrization: h q e 2iσq = B0 q M NP 12 B 0 q / B0 q M SM 12 B 0 q [6,7].

The Inputs

The inputs of the global fit we consider in the analyses reported in this document, performed within a frequentist statistical treatment, are the observables where the theoretical uncertainties are quantitatively under control :

- A complete review of the inputs used in this analysis can be found in [8].

|V
In the presence of NP contributions, the observables transform as follows: sin2β

-→ sin (2β + 2θ d ), α -→ π -γ -β -θ d and ∆m q -→ r 2 q • ∆m SM q .
Only few information on the phase 2θ d can be obtained from the previous set of inputs (α together with sin2β is actually a tree-level γ determination). As advocated for instance in [9], there are other observables, basically unuseful for the CKM metrology in the Standard Model, but emphasizing a double dependence to r q and θ q : this is the case of A q SL or ∆Γ s for which the complete expressions can be found in [10] with NLO calculations. A d SL is of major importance to constrain the (r d ,θ d ).

The Gobal Fit

Figure 1 displays the result of the global fit together with the 95 % CL contours of the individual constraints. α (together with β), γ and |V ub | determines ρ and η from tree-level processes in satisfactory agreement with the mixing loop-induced observables, as underlined in Figures 2 and3. Table 1 gives a selection of the numerical values obtained from the global fit. The CKM paradigm is thus the dominant source of CP violation in B processes. The allowed region of coordinates at 95 % CL is shown in yellow with a red contour. The only differences in the inputs for this conference with respect to [8] are the α and |V ub | values, very slightly modified (see the web update in [2]). A nice overall agreement is observed between the individual constraints.

The New Physics in B -B Mixing

The global CKM fit provides so far a successful test of the SM hypothesis. It is not necessary to allow for supplementary CP-violation phases to describe the current data. Yet, it is useful to establish their constraints to the NP parameters in ∆F = 2 transitions. The inputs of the fit are those discussed in [8]. Figures 4 and5 display the contraints on the (h q ,σ q ) planes, for the B d and B s systems, respectively. While the B d system is tightly constrained (New CPviolating phases could only enter as a typical 30 % correction to the Standard Model), there are still large opportunities for NP contributions in the B s mixing. Let us mention the remarkable interplay between the B-factories and the Tevatron measurements. Clearly, a measurement of the weak phase of the B s mixing, denoted β s and accurately predicted in the Standard Model, will constitute the real step forward on the subject. This is a first-year measurement of the LHCb experiment [11,12,13] , which can rapidly exclude (or observe !) phase values receiving a large NP contribution. The near future is appealing : significant improvements can be expected in the lattice QCD predictions and the LHC start-up will bring precise measurements of the so far missing or badly measured observables as illustrated Figure 6. In flavour physics also, exciting times are in front of us.

  ud |, |V us |, |V cb | determine the λ and A parameters and fix accordingly the length scale of the UT. -|V ub | (including B(B → τ ν)), ∆m d and ∆m s are CP-conserving observables (sensitive to ρ and η) measuring the sides of the UT. -α, γ, sin2β, cos2β are CP-violating observables measuring the UT angles and ε K is the measure of CP-violation in the kaon mixing.

Figure 1 .

 1 Figure 1. Individual constraints and the global fit result on the (ρ,η) plane. Shaded areas have 95 % CL.The allowed region of coordinates at 95 % CL is shown in yellow with a red contour. The only differences in the inputs for this conference with respect to[8] are the α and |V ub | values, very slightly modified (see the web update in[2]). A nice overall agreement is observed between the individual constraints.

Figure 2 .Figure 3 .

 23 Figure 2. . (ρ,η) constraints from tree-level observables only.

Figure 4 .Figure 5 .

 45 Figure 4. . (h d ,σ d ) constraints.

Figure 6 .

 6 Figure 6. Individual constraints and the global fit result on the (ρ,η) plane as we could expect them from one nominal year of data taking from the LHCb experiment. The uncertainty on ∆m s , γ, α et sin2β are 0.01 ps, 10 • , 5 • and 0.02. Most of the improvement is due to the γ measurement. As in Figure1, shaded areas have 95 % CL. The allowed region of coordinates at 95 % CL is shown in yellow and incribed by the red contour.

  The CKM mechanism is successful in describing the flavour dynamics of the present observables in B and K physics. It is now established that the CKM phase is the dominant source of CP violation in B systems and this is a major achievement of the B factories. The current limitations of the CKM consistency tests are of three origins: LQCD, γ and the B s measurements.

Table 1 .

 1 Fit results: the CKM parameters within the framework of the Standard .

	Parameter Value and Uncertainties (95 % CL)
	λ A ρ	0.2258 +0.0016 -0.0017 0.817 +0.030 -0.028 [0.108, 0.243]
	η	[0.288, 0.375]
	J	(2.74 +0.63 -0.22 ) × 10 -5

Acknowledgements

I warmly thank my collaborators of the CKMfitter group for their support in preparing this contribution.


Julien Babault
email: julien.babault@uab.es

Stéphane Bonnet

Jean Van Den Driessche

Alain Crave

High elevation of low-relief surfaces in mountain belts: does it equate to post-orogenic surface uplift?

Keywords: relief dynamics, uplift, erosion, sedimentation, denudation, experimental modeling, low-relief, high-level, geomorphic surface, applanation

We present experiments of upraising and relaxing topographies showing that peneplanation can occur above the ultimate base level (sea level). After active uplift, the erosion of a topography bounded by a piedmont generates a final smooth and highly elevated topography. Smoothing at high elevation is even possible during active uplift if the evolution of topography is disrupted by the deposition of the products of erosion on its piedmont which is the case at the transition from underfilled to overfilled conditions in foreland basins.

INTRODUCTION

 [START_REF] Leonard | Geomorphic and tectonic forcing of late Cenozoic warping of the Colorado piedmont[END_REF]The origin of low relief erosion surfaces lying at high elevation in mountain belts throughout the world has been long debated (e.g. [START_REF] Davis | The Colorado Front Range, a study on physiographic presentation[END_REF][START_REF] De Sitter | Pliocene uplift of Tertiary mountain chains[END_REF], England & Molnar 1990, Molnar & England 1990[START_REF] Gregory | Tectonic and climatic significance of a late Eocene lorelief, high-level geomorphic surface, Colorado[END_REF][START_REF] Leonard | Geomorphic and tectonic forcing of late Cenozoic warping of the Colorado piedmont[END_REF][START_REF] Mcmillan | Postdepositional tilt of the Miocene-Pliocene Ogallala Group on the western Great Plains: Evidence of late Cenozoic uplift of the Rocky Mountains[END_REF][START_REF] Molnar | Late cenozoic increase in accumulation rates of terrestrial sediment: How might climate change have affected erosion rates[END_REF]. Most previous interpretations have considered that those were remnants of peneplain worn near to sea level, which has been later uplifted by post-orogenic tectonics, then dissected by recent streams (e.g., [START_REF] Davis | The Colorado Front Range, a study on physiographic presentation[END_REF][START_REF] Tweto | Laramide (Late Cretaceous-Early Tertiary) Orogeny in the Southern Rocky Mountains[END_REF][START_REF] Epis | Summary of Cenozoic geomorphic, volcanic and tectonic features of central Colorado and adjoining areas[END_REF]).

An implicit concept to those interpretations is that relief subduing is concomitant with surface elevation lowering near to sea level, and thus remnants of peneplain at high elevation require tectonic uplift to occur. On the other hand, [START_REF] Molnar | Late Cenozoic uplift of mountain ranges and global climate change: chicken or egg ?[END_REF] have proposed that uplift of the peneplain remnants was the isostatic consequence of Pliocene-Quaternary climate-driven dissection, not the cause. But erosion-driven isostatic uplift appears to have a limited effect and cannot account for overall assumed uplift of peneplain remnants from near sea-level to their present elevation (e.g., [START_REF] Gilchrist | Landscape dissection, isostatic uplift and the morphologic development of orogens[END_REF][START_REF] Montgomery | Valley incision and the uplift of mountain peaks[END_REF][START_REF] Small | Pleistocene relief production in Laramide mountain ranges, western United States[END_REF][START_REF] Leonard | Geomorphic and tectonic forcing of late Cenozoic warping of the Colorado piedmont[END_REF]. Alternatively, other works have appealed for deep mantle Four figures in Supplementary Material Babault et al., High elevation of low-relief surfaces in mountain belts submitted to Terra Nova, June 2006 revised version 3 processes to account for large scale uplift of such low-relief surfaces (e.g., [START_REF] Leonard | Geomorphic and tectonic forcing of late Cenozoic warping of the Colorado piedmont[END_REF][START_REF] Mcmillan | Postdepositional tilt of the Miocene-Pliocene Ogallala Group on the western Great Plains: Evidence of late Cenozoic uplift of the Rocky Mountains[END_REF].

[3] Several works have recently suggested that piedmont sedimentation in foreland basins corresponds to the rise of the base level for mountain belts erosion, resulting in reducing erosion rate [START_REF] Baldwin | Implications of the shear stress river incision model for the timescale of postorogenic decay of topography[END_REF], Babault 2004[START_REF] Pelletier | Influence of piedmont deposition on the time scale of mountain-belt denudation[END_REF], Babault et al. 2005a[START_REF] Carretier | How does alluvial sedimentation at range fronts modify the erosional dynamics of mountain catchments?[END_REF]. In particular, from the relation between piedmont sedimentation and erosion pattern in the Pyrenees during the Cenozoic, Babault et al. (2005b) reach the conclusion that relief subduing does not necessarily equate to surface elevation lowering, so that relief of mountain chains can be smoothed at high elevation. We use a similar experimental approach as Babault et al. (2005a) who have investigated the influence of piedmont sedimentation on the relation between denudation rate and uplift rate of upraising mountain belts. Here, we investigate the effect of piedmont sedimentation on local relief and we describe the mechanisms by which a relief is subdued at high elevation.

EXPERIMENTAL MODELS

[4] The experimental design we used is the same as described in Babault et al. (2005a). It is made of an erosion box (600 × 400 mm and 500 mm deep) filled by a material to erode, a silica paste which consists of silica powder (D50 = 10 μm) mixed with water (20%). The silica paste is uplifted within the erosion box by moving the base of the erosion box, the movements being driven by a screw and a computer controlled stepping motor. In some experiments, a plateau is added at the top of the erosion box, allowing deposition of the eroded products from upstream (herein referred to as "piedmont sedimentation"). The erosion box is located in a rainfall simulator, where four sprinklers deliver a high-pressure water-air mixture * . All experiments run under the same rainfall rate of 120 ± 5 mm/h, as in experiments in Babault et al. (2005a). To construct 0.5 mm square-grid digital elevation models, we stop [5] At first we use the experiments previously performed by Babault et al. (2005a) to analyze the evolution of local relief within a rising zone, without piedmont sedimentation. We also refer to experiment where piedmont sedimentation is allowed after a first phase of erosion of an experimental landscape submitted to constant uplift but without deposition (Babault et al., 2005a). Subsequently, we present new experiments that investigate the effect of piedmont sedimentation on the relationship between the local relief and the mean elevation during a post-uplift decay of the topography. The dynamics of the local relief is studied by analyzing the variation of the mean local slope of the experimental landscapes (within the erosion box), the local slopes being computed over two adjacent pixels of the DEMs in the steepest direction.

Effect of an emergent piedmont on the local relief of an uplifting topography

[6] The first set of experiments (Babault et al. 2005a) runs under constant uplift and rainfall rates, without piedmont deposition. Different uplift rate values have been applied [START_REF] Birot | Recherches sur la morphologie des Pyrénées orientales franco-espagnoles[END_REF][START_REF] Baldwin | Implications of the shear stress river incision model for the timescale of postorogenic decay of topography[END_REF]15 and 20 mm/h). In each experiment, rills first develop from the borders toward the centre of the erosion box when the mean elevation of the experimental landscapes increases, defining a growth stage of the topography. After 300 min, the mean elevation reaches a constant value defining a macroscale steady state stage for the relief dynamics (Hack, 1960) where the output eroded flux balances the input uplifted flux. The mean elevation (<h>) of the steady state topography increases linearly with the uplift rate (U) [START_REF] Lague | Laboratory experiments simulating the geomorphic response to tectonic uplift[END_REF], Babault et al. 2005a). As for the mean elevation vs. uplift rate relationship, the mean local slope <s> increases with U (Fig. 1).

Four figures in Supplementary Material

[7]

In a second step, we add a plateau around the steady-state topography (for U = 15 mm/h) at t = 600 min and we carry on the experiment at the same uplift and rainfall rates (Fig. 2B). The piedmont sedimentation that starts to surround the uplifted topography induces the onset of the mean elevation growth and the disruption of the previous macroscale equilibrium.

Although the uplift and rainfall rates remain continuous, the end of the steady-state at t = 600 min and the revival of surface uplift mean that the mean denudation does not longer balance the input flux (Fig. 2B) (Babault et al. 2005a). At the local scale, the inhibition of erosion goes with a modification of the local relief. The transversal topographic profiles evolution (Fig. 2A, 2C and 2D) show an increase of the elevations with time, concomitant with a decrease of the local relief. The smoothing of the upraising topography starts downstream and propagates upwards. At t = 615 min, the topography is only smoothed in the downstream parts (Fig. 2C) whereas the local relief of the upstream parts is not modified until t = 630 min (Fig. 2D).

[8] The piedmont deposition raises the limit between the erosion zone and the deposition zone, i.e. the base level for erosion. As a result, the uplifted zone rises at a relative uplift rate (U r ) that depends on the aggradation rate (U f ) (Babault et al. 2005a). Since the onset of the piedmont sedimentation, the relative uplift rate decreases from the initial value of U = 15mm/h down to U = 10 mm/h (Fig. 2B). As the local slopes depend on the uplift rate (Fig. 1), the decrease of the relative uplift rate then induces the smoothing of topography † . It is also responsible for the decrease of the local denudation rates (Figure 3) ‡ , before and after the onset of the piedmont sedimentation. Four figures in Supplementary Material only for the channelised downstream parts). It is to note that the ongoing uplift leads eventually to a dynamic equilibrium of the topography, characterized by a progressive reincrease of the local relief as the piedmont aggradation rate decreases (see Babault et al. 2005a).

Effect of piedmont deposition on mean elevation dynamics following the cessation of active uplift

[9]

In this new set of experiments, we first look at the evolution of the topographic decay of a previous steady state topography with free borders (Figure 4). At t = 0 the denudation is at equilibrium with U = 15 mm/h and afterwards the experiment runs without uplift. Figure 4 shows the decay of both the mean elevation <h> and the mean local slope <s> from the initial steady state topography to the final relaxed one. The mean elevation evolves following an exponential decay of the form:

(1)

τ / 0 x Ce y y - + =
with y 0free = 6.4 mm, C free = 15.4 and τ free = 90 y -1 (r² = 99.5 %).

We perform another relaxation experiment but we add a plateau just after uplift is stopped, allowing piedmont sedimentation to develop. The pattern of the decay of topography is similar to the previous experiment with free borders, the respective value of the terms of equation (1) being:

y 0piedmont = 11.8 mm, C piedmont = 8.2 and τ piedmont = 100 y -1 (r² = 98 %)
The value of the mean local slope at infinite time is similar in both experiments (Fig. 4, <s> final ∼ 3.5°), but a main difference exists that is the much higher final mean elevation reached in the experiment with piedmont (y 0piedmont = 11.8 mm) compare to the experiment with free borders (y 0free = 6.4 mm). This difference in elevation (5.4 mm) almost corresponds to the elevation of the fan apex at t = 425 min (<h> f = 6.5 mm).

Four figures in Supplementary Material DISCUSSION [START_REF] Baldwin | Implications of the shear stress river incision model for the timescale of postorogenic decay of topography[END_REF] Numerical modeling and laboratory experiments have already stressed the effect of piedmont deposition, on time scale of mountain denudation and erosion dynamics in mountain catchments, as corresponding to the rise of the efficient base level of mountain belts [START_REF] Pelletier | Influence of piedmont deposition on the time scale of mountain-belt denudation[END_REF], Babault et al. 2005a[START_REF] Carretier | How does alluvial sedimentation at range fronts modify the erosional dynamics of mountain catchments?[END_REF]. In the present work we focus on this effect with regard to the local relief in mountain belts. [START_REF] Epis | Summary of Cenozoic geomorphic, volcanic and tectonic features of central Colorado and adjoining areas[END_REF] The experiments show that relief subduing, which corresponds to local slope decrease, may develop during the new growth stage of an uplifting topography initially at steady state in response to the development of an aggrading piedmont. They also shows that the development of a piedmont during relief decay induces relief subduing at high elevation, the value of which is a function of the final elevation reached by the piedmont apex. In both types of experiments, relief subduing results from the rise of the efficient base level of the uplifting topography. [START_REF] Montgomery | Valley incision and the uplift of mountain peaks[END_REF] The dominating process by which the peneplanation develops is most probably a hybrid process between detachment-limited and transported-limited models. Indeed, in this type of experiments, local slopes and mean elevation decrease with increasing precipitation rates. A detachment-limited process predicts an inverse correlation suggesting that the rate limiting process governing the evolution of the experimental channels is the transport of sediments rather than the rate of bed incision [START_REF] Turowski | Experimental channel response to tectonic uplift[END_REF]). However the upward migration of the topography smoothing argues for a rather advective process, not a diffusive one as predicted by a transported-limited model. Note that hybrid-type numerical model encompasses these two-end members of landscape evolution model by introducing a varying transfer distance. This suggests that the transfer distance may vary during experimental peneplanation but this remains to investigate. More importantly, in both transport-limited and detachment-limited model slope depends on the uplift rate. Hence the decrease of the relative Four figures in Supplementary Material [13] The direct application of the experimental results to natural systems is not reliable, as perfect scaling of laboratory-scale reliefs is nearly impossible [START_REF] Crave | Analogue Modelling of Relief Dynamics[END_REF][START_REF] Hasbargen | Landscape instability in an experimental drainage basin[END_REF][START_REF] Lague | Laboratory experiments simulating the geomorphic response to tectonic uplift[END_REF][START_REF] Bonnet | Macroscale dynamics of experimental landscapes[END_REF]. However, results from modeling can be used to emphasize general concepts of landscape behavior [START_REF] Bonnet | Macroscale dynamics of experimental landscapes[END_REF]. Here, the experimental results provide an alternative explanation in the long-standing debate on the origin of high-elevation, low-relief surface remnants observed in several mountain belts throughout the world, including the Southern Rocky Mountains, the Pyrenees, the Atlas (e.g. [START_REF] Davis | The Colorado Front Range, a study on physiographic presentation[END_REF][START_REF] Birot | Recherches sur la morphologie des Pyrénées orientales franco-espagnoles[END_REF][START_REF] Choubert | Note Préliminaire sur le Pontien au Maroc (Essai de synthèse orogénique du Maroc Atlasique)[END_REF][START_REF] De Sitter | Pliocene uplift of Tertiary mountain chains[END_REF][START_REF] Scott | Cenozoic Surfaces and Deposits in the Southern Rocky Mountains[END_REF], Molnar & England 1990).

Most interpretations consider that uplift, whatever its origin is, is required to explain the high elevation of low-relief surface remnants that are supposed to have developed near sea-level.

Isostatic rebound resulting from heterogeneous erosion or thermal mantle anomaly may also account for post-orogenic uplift (e.g. Molnar & England 1990[START_REF] Leonard | Geomorphic and tectonic forcing of late Cenozoic warping of the Colorado piedmont[END_REF][START_REF] Mcmillan | Postdepositional tilt of the Miocene-Pliocene Ogallala Group on the western Great Plains: Evidence of late Cenozoic uplift of the Rocky Mountains[END_REF][START_REF] Molnar | Late cenozoic increase in accumulation rates of terrestrial sediment: How might climate change have affected erosion rates[END_REF][START_REF] Teixell | Effects of mantle upwelling in a compressional setting: the Atlas Mountains of Morocco[END_REF], but not for the whole elevation in most cases. For example, paleobotanical data show that the present-day elevation of the low-relief surfaces in the Southern Rocky Mountains was nearly the same as during the Late Eocene, suggesting that these surfaces formed at high elevation [START_REF] Gregory | Tectonic and climatic significance of a late Eocene lorelief, high-level geomorphic surface, Colorado[END_REF][START_REF] Wolfe | Paleobotanical evidence of Eocene and Oligocene paleoaltitudes in midlatitude western North America[END_REF]). On the other hand, according to Babault et al (2005b), the example of the Pyrenees strongly argues for relief subduing at high elevation during post-orogenic decay as shown by the second set of experiments. Indeed, neither post-Pyrenean tectonics nor mantle thermal anomaly exist that can explain the uplift of low-relief surfaces in this example (Babault et al. 2006). As well, Pliocene-Quaternary, erosionally driven isostatic rebound does not account either for most of the present-day elevation of these surfaces (Babault 2004, Babault et al. 2005b).

Four figures in Supplementary Material

9

[14] As mountain belt building is generally accompanied by piedmont aggradation, this suggests that relief subduing at high elevation is most likely an intrinsic erosional process to post-orogenic decay of mountain belts.

ACKOWLEDGMENTS

We thank Jean-Jacques Kermarrec and Jean-Pierre Caudal-Robin for making the experimental apparatus in working order; Philippe Davy for providing the GridVisual soft for experimental topographies display. The manuscript benefited from thorough comments by Sierd Cloetingh and an anonymous reviewer. This research was funded by the CNRS-INSU research program PNSE and ATI. The Ministère de l'Éducation Nationale, de la Recherche et de la Technologie funded Julien Babault's Ph.D. thesis.

REFERENCES CITED

Babault, J., 2004. Dynamique de l'érosion dans une chaîne de montagnes : Influence de la sédimentation de piedmont. L'exemple des Pyrénées. PhD, Université de Rennes1, Rennes (France), 218 pp. Babault, J., Bonnet, S., Crave, A. & Van Den Driessche, J., 2005a. Influence of piedmont sedimentation on erosion dynamics of an uplifting landscape: An experimental approach. Geology, 33, 301-304, doi: 10.1130/G21095.1. Babault, J., Van Den Driessche, J., Bonnet, S., Castelltort, S. & Crave, A., 2005b Local denudation rates are superimposed on the DEM of the topographies of experiments A, B and C at t = 600 min, at t = 615 min and at t = 630 min, respectively (points "A", "B" and "C" in Figure 2B). Colored topography, from blue to red, corresponds to increasing denudation rate, green corresponding to sedimentation. It shows the decrease of both the local erosion and local relief that is induced by the piedmont deposition although the uplift and rainfall rates (15 mm/h and 120 mm/h, respectively) are kept constant during the experiment. The plateau is 250 mm large (measured from the border of the uplifting zone).

Note that except for the topography B, the smoothed topography is an erosional surface, i.e.

smoothing is not achieved by the infilling of depressions with the products of erosion. Four figures in Supplementary Material Four figures in Supplementary Material

SUPPLEMENTARY MATERIAL FIGURE CAPTIONS

  Figure 3 also shows that the subdued topographies are erosional surfaces, i.e. smoothing does not result from the infill of the depressions by erosion products but from lowering of valley slopes by erosion (except between 600 and 615 min and † Figure 3 in supplementary material. ‡ Figure 4 in supplementary material.

  Babault et al., High elevation of low-relief surfaces in mountain belts submitted to Terra Nova, June 2006 revised version 8 uplift rate triggered by piedmont aggradation should allow the development of a peneplain as well in both end-members.

  Babault et al., High elevation of low-relief surfaces in mountain belts submitted to Terra Nova, June 2006 revised version

FIGURE CAPTIONS Figure 1 :

 CAPTIONS1 FIGURE CAPTIONS

  Babault et al., High elevation of low-relief surfaces in mountain belts submitted to Terra Nova, June 2006 revised version 13 110 mm from the border of the uplifted zone. The mean local slope and the local relief increase with the uplift rate. U: uplift rate; <s>: mean local slope.

Figure 2 :

 2 Figure 2: Effect of an emergent piedmont on the local relief of an uplifting topography.

Figure 3 :

 3 Figure 3: Effect of an emergent piedmont on the local denudation of an uplifting topography.

Figure 4 :

 4 Figure 4: Effect of piedmont deposition on the mean elevation in relaxation experiments.

Figure 1 :

 1 Figure 1: Drawing of the experimental apparatus that shows the erosion box filled by silica

Figure 2 :

 2 Figure 2: Detailed picture of the erosion box at the end of an experiment. The uplifted

Figure 3 :

 3 Figure 3: Perturbation of a Steady State topography by piedmont deposition. Photos of the

Figure 4 :

 4 Figure 4: Perturbation of a Steady State topography by piedmont deposition. DEMs movie of

  Babault et al., High elevation of low-relief surfaces in mountain belts submitted to Terra Nova, June 2006 revised version 14 Open squares and black circles: mean elevations (<h>) of relaxation experiments with and without piedmont sedimentation, respectively. Grey open squares: mean fan apex elevation (<h> f ). Relaxation experiments start from steady state topography. <s> s.s. is the mean local slope of the steady state topographies for similar uplift and rainfall rates, and <s> final is the mean local slope of the final topographies. The greater elevation of the smoothed topography surrounded by piedmont almost correlates with the final elevation of the fan apex.


Aurélien Chazottes-Leconte

Emmanuel Sonde

Loup Plantevin

Charles Joubert

Thibaut Chaise

Laurent Morel

Daniel Nelias

Hubert Razik

The Effect of an Electromagnetic Peening Process on Mumetal Properties

Keywords: Electromagnetic Peening processes, Peening process, Pulse current generator, Needle sensor, H-Coi, Magnetic properties, Magntic materials, mumetal properties

Peening processes are widely used in the industries to enhance the performance of the most solicited parts of mechanical pieces. Actual processes, like shot peening and low plasticity burnishing, are quite limited to a small depth of treatment circa 1 mm at maximum. A new process is developed based on pulsed electromagnetic fields. This method allows a better depth of treatment up to 5 mm and it can be easily monitored. A prototype was developed using a pulse current generator and specifically designed inductor. This prototype was used on mumetal for the preliminary tests. Indeed, its maximal permeability and coercivity are highly impacted by the presence of stresses. Using specifics sensors, some changes occur after treatment on mumetal samples and can be observed.

I. INTRODUCTION

Peening processes are widely used in industries to introduce compressive residual stresses in the most solicited parts of mechanical pieces. Properly applied, the process significantly increases the lifespan of these parts, especially under fatigue solicitations or corrosive environment, and will reduce or prevent the propagation of micro-cracks inside the material.

There are several different processes used to apply compressive residual stresses that can be thermal treatment, thermo-chemical, and mechanical. Nowadays, some of the main processes used in the industry are shot peening, ultrasonic shot peening, and low plasticity burnishing [1][START_REF] Chaise | On the effect of isotropic hardening on the coefficient of restitution for single or repeated impacts using a semi analytical method[END_REF]. These processes are well known however they share one common disadvantage of a depth of treatment circa 1 mm at maximum [START_REF] Jean | Precontraintes et traitements superficiels[END_REF].

Recently news processes breakthrough such as laser shock peening, cavitation peening and the new technology we described in the present paper. These technologies are very interesting for two aspects. First, there is no particles left on the piece which is not possible using the current classical process and second, it possible to reach several millimeters of treatment depth.

With the electromagnetic peening technology, electromagnetic pulses are used to generate eddy currents inside metallic materials to introduce plastic stresses with Lorentz forces. If the electromagnetic pulse is high and fast enough, it will generate a plastic flow and therefore residual stress. According to the simulation of the process, which is not presented here, it is possible to expect around 5 mm or even more of treated depth [START_REF] Sonde | Numerical simulation of electromagnetic surface treatment[END_REF]. The level of residual stress and the treatment depth can be controlled with the energy and the frequency of the electromagnetic pulse.

To generate electromagnetic pulses suitable to apply compressive residual stresses, a pulsed current generator with a custom inductor is used [START_REF] Lehr | Foundations of Pulsed Power Technology[END_REF]. This inductor was specially developed for this application as the current pulse generator. It was designed to prove the feasibility of this technology. It can reach high currents peaks of 180 kA nominal current with a pseudo frequency of 35 kHz.

To investigate the generation of residual stresses inside materials, a material with a low yield strength is chosen to begin with. It will require less energy to introduce compressive residual stress than an industrial material. With a highpermeability and a relative low yield strength when annealed, the mumetal is a good candidate for this experiment. Especially its maximal permeability and coercivity are affected by the presence of stresses. By recording the hysteresis cycle of mumetal's workpiece before and after an electromagnetic peening, a significant variation of the magnetic properties is expected to be observed.

II. PULSE CURRENT GENERATOR

A. Generator Characteristics

As it is a proof of concept for electromagnetic peening, this current generator must be versatile. Indeed, as we intended to work with a broad range of material samples we may have to change electrical parameters. The nominal characteristics of the pulse generator were defined as follows.

B. Circuit Schematic

A RLC circuit was chosen for this application. It is described schematically in fig. 1 [START_REF] Hartmann | Design of a High Current Pulse Generator for Magnetoforming[END_REF][START_REF] Nashilevskiy | High-Current Pulse Generator for Plasma Focus[END_REF]. A high voltage power supply provides the necessary current to charge a bank of capacitors. This capacitor bank includes 24 high voltage polypropylene capacitors of 2.6 μF in parallel to provide a maximum energy of 12.5 kJ for the system. Each capacitor can be easily removed or added from the bank to achieve a modularity in the overall impedance of the system. The main switch, SW1, which is a triggered spark gap, can discharge the energy stored at a specific voltage. With this flexibility on the rated voltage, it is possible to change the maximum current for each pulse. One side of the capacitor bank is grounded and the other side is linked to the high voltage power supply and the sparkgap. The capacitor bank is charged through a resistor R to limit the current and protect the power supply. Capacitors are placed in a circular shape around the central main power connection. The goal of this array is to balance impedances between all capacitors. In that way, all capacitors provide the same current.

The output of the sparkgap is directly connected to the branch with the inductor L_ind and the parasitic elements (R_line and L_line). This branch is exposed to high voltage only a short amount of time during pulses which limits the dielectric stress on the insulating coating of the inductor.

C. Inductor Design

A specific inductor was designed for this application based on the analytical model developed previously, it consist in a 3 turn coil with a spiral shape, an inner radius of 7 mm and outer radius of 13 mm. Each coil has a cross section of 4 by 1 mm with 1 mm of space between each turn. Base on analytic formulas [START_REF] Rosa | Inductance Calculations[END_REF], the inductance of the inductor is 220 nH. The inductor is formed with Cu-ETP copper alloy with a very low electrical resistivity (1.724 × 10 -8 Ω/m). During a test, the inductor is supposed to be at 1 mm from the sample.

To insure a good electrical insulation of the inductor, an acrylic socket is used with a clear epoxy to drown in the inductor into. The usage of these compounds allows the possibility of a visual inspection of the inductor [START_REF] Vanbenthysen | Effect of Specimen Planar Area on Electromagnetic Flanging[END_REF][START_REF] Thibaudeau | Analytical Design and Experimental Validation of Uniform Pressure Actuator for Electromagnetic Forming and Welding[END_REF]. An aluminum shield of 5 mm thick is used to contain the inductor and its insulating coating. Figure 2 shows a representation of the actual inductor with and without its protectives layers.

D. Experimental setup

The electromagnetic peening prototype is described in the fig. 3. As shown, the capacitor bank is fully equipped with capacitors and the maximum capacitance is used. The inductor and the sparkgap are connected together with the bank using busbars. A Spellman SLM20P1200 high voltage power supply is used to charge the capacitor with a maximum current of 60 mA. The inductor current is measured with a Powertek CWT 1500B * coil. The voltage is supposed to be read out with the power supply and an extra North Star high voltage probe PVM-1 is also used. To record all the data, a LeCroy WaveJet 314A oscilloscope, with 100 MHz bandwidth and 1 GSa/s is used and is triggered by the igniting system of the sparkgap. The workpiece in mumetal is placed between the inductor and acrylic plate which is on an XY table. It allows the workpiece to be in a precise position or repetitive shots on different places within a tenth of millimeter of precision. For the preliminary experiments on the mumetal, a single shot is made.

To hold down the inductor and the workpiece a press system is used to avoid recoil from the electromagnetic reaction. This pulsed current generator is placed in a faraday chamber to prevent electromagnetic interferences. Another aspect of this chamber, is that it can prevent certain mechanical failures of the inductor or the sample under the stress.

III. CHARACTERIZATION TOOLS

The electromagnetic properties like the permeability and the coercive field of ferromagnetic materials are highly sensitives to micro-structural variations, especially with high permeability materials like mumetal. When applying a peening process to a workpiece, it changes the micro-structural state for the first millimeters depth. So, a change of these parameters can be observed. To quantify these variations, the local flux density B inside the material and surface H field can be monitored when the material is properly excited.

A. Needles Sensors

For measuring the local induction B in a workpiece, the needle-probe technique is the only one nondestructive method available. The principle is described in fig. 4. This method is very interesting from a mechanical point of view because it does not change the mechanical stress state of the material during measures [START_REF] Wulf | Needle-Probe Techniques for Local Magnetic Flux Measurements[END_REF][START_REF] Tumanski | Handbook of Magnetic Measurements[END_REF]. Based on the Maxwell's second equation, it allows for the evaluation of the local magnetic flux in the material with the voltage drop measured between the two needles. It is possible to consider only a half thickness of the material if the induction B=B z x,t •l z has only a space dependency along x and if the needles are relatively far away from the edge. Under these conditions, the induced voltage e(t) is given by the flux variation inside the section S+S a . In our case, we minimized the section of air S a , to only consider the section S of the material.

e(t) ≈ 1 2 •d•L• dB dt 1
The previous relation (1) gives resulting voltage with the geometric parameter defined in the fig. 4. This signal must be amplified with a proper system because the measured voltage is less than 100 μV.

B. H-Coil Sensor

An H-coil or an induction coil is used to track the magnetic flux φ on the surface of the treated workpiece. The voltage given by this sensor results from the Faraday's law of induction [START_REF] Tumanski | Handbook of Magnetic Measurements[END_REF].

V= -n• dφ dt = -μ 0 •n•A• dH dt 2
Where n is the number of turns in the coil and A the effective area corresponding to the inner surface of the coil.

C. Sensors Prototype

A specific probe prototype was created to monitor B-H field. Figure 5 presents the existing sensors with its 3d printed support. The two brass needles are spaced by 1 mm and the H-coil is placed in between. The acquisition board connected to the sensors needs to be calibrated before each measurement. It means that coil parameters and needles don't have to be known to use it as long as it is properly calibrated. With this sensor it is possible to do a local measure of the B(t) and H(t) field simultaneously. So the static and dynamic shapes of the hysteresis curves can be plotted.

IV. RESULTS

As it mentioned earlier, we applied some residual stresses on mumetal. We used annealed mumetal with a low yield strength around 160 Mpa and high maximal permeability of around 7×10 4 . For these experiments, we used workpieces of 125 mm in length, 30 mm large and 0.5 mm thick. We chose a longer workpiece than the diameter of the inductor for observing the parameters variation.

A. Current Discharge

For first experiments, the pulse current generator is not used at full power and a single shot has been done in the middle of the workpiece. According to simulations, an electromagnetic pulsed of 3.1 kJ should be enough to observe some changes in mumetal properties. The measured experimental current through the inductor is shown in fig. 6. During this discharge, a peak current of 90 kA with a pseudo frequency slightly lower than 30 kHz was recorded.

B. Sample Shape Measurement

For a sufficiently thin workpiece with enough residual stresses in it, the accommodation of strains in the material may induce a macroscopic deflection. Thus, for a thin workpiece, containing compressive residual stresses at its surface, a concave shape is expected in the treated area [START_REF] Jean | Precontraintes et traitements superficiels[END_REF].

A stereo-digital image correlation is used to evaluate the deformation at the surface of the workpiece generated by oneshot at the middle of the workpiece. This measurement method is simple to implement compare to others methods and the precision of the system is circa one micrometer [START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF]. Only the treated area was measured and is represented as follow. The middle line along the length (x-axis) of the workpiece is taken as the reference line for the measurement. The results from the stereo-digital image correlation shows that a positive deflection does appear after treatment underneath the inductor area. The top height of the deflection is around 0.3 mm. This deflection can be directly related to the presence of residual stresses. Since this phenomenon is directly linked to the amount of stresses, it is possible to evaluate them by measuring the height of the deflection. However, this result is not accurate and it is more quantitative than quantitative.

C. Coercivity and Permeability Measurement

After treatment, the B-H sensors are used on the workpiece to record the local hysteresis cycle. These measurements are done along the same reference line established for stereo-digital image correlation. The interest is to relate the visible deformation due to residual stresses with the magnetic parameters.

A u-shape magnetization core is used to apply a variable magnetic field to the mumetal workpiece at a frequency of 10 Hz. Based on the local hysteresis cycles recorded, it is possible to extract the local coercivity and maximal permeability of the mumetal using a matlab routine and the sensors data. As we can see with these results, the maximal permeability and the coercivity are strongly impacted by the electromagnetic peening shot. The peening process induces a lot of dislocations inside the workpiece, residual stresses and maybe change the grain size. As it is shown in the literature [START_REF] Preece | The 77 Effects of Various Stresses on Mumetal: Percent Ni-14 Percent Fe-5 Percent Cu-4 Percent Mo[END_REF], the coercivity and maximal permeability are affected by these parameters.

The affected area seems to be larger than the corresponding deformed area (cf. Fig. 7). The observed deflection is related to sufficient amount compressive residual stresses in the inner area of the inductor (between -12 mm and 12mm). However, some residual stresses are present in the outer area but the accommodation of the strains might be not enough to induce a deformation. Some dislocations are present too and may affect the grain size. So, these effects and the stresses have an effect on the magnetic parameters as shown in fig. 8.

V. CONCLUSION

The results on the mumetal shows that the electromagnetic peening process has an influence on mumetal properties. Some investigations have to be done to evaluate the amount of residual stresses. By decorrelating the different measures, it maybe possible to determine the local residual stress. On another side with this sensor it might be possible to create a two-dimensional cartography of the stresses in a ferromagnetic workpiece and to check the uniformity of the process.

Fig. 1 .

 1 Fig. 1. Equivalent circuit of the pulse current generator based on a highvoltage capacitor bench with a closing switch SW1 (sparkgap) and the inductor. R-L_line represents the parasitic elements of the line.

Fig. 2 .

 2 Fig. 2. Design of the inductor: (a) spiral coil made of Cu-ETP, (b) Coil with its protective socket of polycarbonate and epoxy and its steel shield.

Fig. 3 .

 3 Fig.3. View of the prototype of pulse current generator with the capacitor bank, the sparkgap and the inductor. A xy table is used to move the workpiece for multiple shots on different locations with a good precision.

Fig. 4 .

 4 Fig. 4. Principle of the needle sensor for B measurement. The two needles are spaced by a length L.

Fig. 5 .

 5 Fig. 5. B-H sensors with its support. The needle sensor is made with two brass needles. H-coil consist in two layers of 100 turns each.

Fig. 6 .

 6 Fig. 6. Inductor current measured during a test on mumetal at 10 kV with a Rogowski coil.

Fig. 7 .

 7 Fig. 7. Results of stereo-digital image correlation along the x axis. the reference is taken in the middle of the workpiece.

Fig. 8 .

 8 Fig. 8. Relative coercivity and the maximal permeability of the mumetal along the reference line.

TABLE I .

 I MAIN CHARACTERISTICS FOR PULSE CURRENT GENERATOR

	Peak current	50 to 180 kA
	Pseudo-frequency	30 kHz Nominal
	Load inductance	250 nH Maximum
	Lifetime	> 1000 pulses

ACKNOWLEGMENT

This work is supported by the Carnot Institute of Lyon ("ElectroMagnetic Peening" (EMP) Project).


Fellow, IEEE Bernard Picinbono
email: bernard.picinbono@lss.supelec.fr

Messaoud Benidir

Some Properties of Lattice Autoregressive Filters

An autoregressive filter is defined either by the components of the regression vector or by the reflection coefficients appearing in its lattice representation. The mathematical expression of the regression vector in terms of the reflection coefficients is very complex but many structural properties can be obtained without this exact expression. In this paper, we present some examples of such structural properties, and we apply these results to prove some extrema1 properties of stable filters such as the maximum value of the components of the regression vector or the maximum value of its norm. Moreover, some properties of the boundary of the stability domain are discussed.

I. INTRODUCTION

An autoregressive (AR), or recursive, discrete time filter is a filter deduced from the difference equation

y k - n i=1 a i y k-i = u k , (1)
where u k , y k , and a i are the input, output, and the components of a vector a, respectively. This vector is called the regression vector because when u k is a white noise the linear prediction of y k in terms of all its past is

ŷk = n i=1 a i y k-i , (2)
which is a linear regression. The transfer function of the causal filter defined by ( 1) is, of course,

H(z) = z n z n -n i=1 a i z n-i , (3)
which clearly introduces the equivalent terminology of an all poles filter.

It is well known that the regression vector a can be expressed in terms of the so-called reflection coefficients k by using the Levinson recursion [ l , p . 271, [START_REF] Kay | Spectrum analysis. A modern perspective[END_REF] , which allows us to calculate a regression vector of order m, a m in terms of a m-1 by

a m m-1 = a m-1 -k m a (-) m-1 (4)
a m m = k m . (5)
In these equations, a m m is the last component of a m and a m m-1 is a vector of order (m -1) deduced from a m by suppressing its last component. Moreover, a (-) m-1 is deduced from a m-1 by inversion of the order of the components Then using recursively (4) and ( 5) from 1 to n, we deduce the vector a n in terms of the set of reflexion coefficients k i , 1 ≤ i ≤ n.

Unfortunately, the explicit expression of the components of a n in terms of k i is not simple and, as an example, we give the first three vectors

a 1 = k 1 (6)
a 2 = [k 1 -k 1 k 2 , k 2 ] T (7)
a 3 = [k 1 -k 1 k 2 -k 2 k 3 , k 2 -k 1 k 3 + k 1 k 2 k 3 , k 3 ] T . (8)
Finally, another way to represent the transfer function is to use a sequence of polynomials defined by

P m (z) = 1 -a T m Z m , (9)
where Z m is the vector

Z m = [z, z 2 , z 3 , . . . , z m ] T . ( 10
)
Of course, the transfer function ( 3) is given by

H(z) = [P n (z -1 )] -1 , (11)
and it is equivalent to study the polynomial P n (z) or the transfer function H(z).

As a consequence, it appears that if the roots of P n (z) are z k , 1 ≤ k ≤ n, then the poles of H n (z) are z -1 k and, in particular, the stability condition for the AR filter is that the roots z k are outside the unit circle.

The purpose of this paper is the following one. In Section I1 we present some general properties of the regression vector a which are direct consequences of the Levinson recursion and then obtained without calculating the components of this vector. In Section III we introduce the stability condition and we deduce some properties of the regression vector of stable filters. In particular, we give the expression of the vector a of maximum norm for a stable filter. This question was motivated by an interpolation problem appearing in estimation theory [START_REF] Papoulis | Constrained Wiener filtering[END_REF]. In Section IV we present some simple sufficient conditions for stability of an AR filter. The conditions are particularly important in an adaptive filtering context where the coefficients a i 's are time varying. In these cases, it is important to secure the stability with conditions involving almost no calculations. In Section V, the stability problem is considered more carefully and we specify the limits of the stability domain in the space of the regression vector a. This problem was partially considered in [START_REF] Shlien | A geometric description of stable LPC digital filters[END_REF]. More precisely we give some geometrical properties of the boundary of the stability domain in the a domain. For the points belonging to this boundary, we give the structure of the transfer function which allows us to indicate which poles are on the unit circle. Finally, Section VI is devoted to some general expressions of the regression vector in terms of the reflection coefficients.

II. SOME STRUCTURAL PROPERTIES

In this section we present a sequence of properties directly deduced from the Levinson recursion and interpreted in the lattice representation.

Property 2.1 : Each component of a n is a polynomial in k i with a degree smaller or equal to n.

Proof: The property is true until the order 3, from ( 6) to [START_REF] Bellanger | Truitement Numérique du Signal[END_REF]. If it is true at the order m, we deduce immediately from (4) that it is also true at the order m + 1.

Property 2.2 :

The powers of k i in the polynomials introduced by 2.1 are either 0 or 1.

Proof: The property is true for a i , i ≤ 3. If it is true for a n , it is also true for a n+1 because in (4) a n and a (-) n do not depend on k n+1 .

Property 2.3 : The polynomials introduced in 2.1 are sums of products of k i with the signs + if the number of k i is odd andif it is even.

Proof: It is also by induction, and a direct consequence of the multiplication by -k n+1 in (4). Property 2.4 : If the reflection coefficients are negative, the components of a are also negative.

Proof: It is a direct consequence of 2.3. May 25, 2018 DRAFT Property 2.5 : If Sg(k i ) = (-1) i+1 , 1 ≤ i ≤ n, then Sg(a i ) = (-1) i+1 , 1 ≤ i ≤ n, where Sg(f ) is the sign of f .
Proof: The property is true for a 1 , a 2 , and a 3 , from ( 6) to [START_REF] Bellanger | Truitement Numérique du Signal[END_REF]. Let us suppose that the signs of the componentsof a m-1 are

a m-1 : [+, -, +, -, . . . , (-1) m ]. (12)
For a

(-)

m-1 we obtain similarly

a (-) m-1 : (-1) m [+, -, +, -, . . . , (-1) m ]. (13)
Then as

k m = (-1) m+1 |k m |, we deduce that the signs of -k m a (-) m-1 appearing in (4) are -k m a (-) m-1 : [+, -, +, -, . . . , (-1) m ]. (14)
and by using ( 4), the property is true for a m m-1 . As it is true for the last component given by ( 5), it is true for a i , 1 ≤ i ≤ n.

Property 2.6 : The number of terms appearing in the polynomials introduced in 2.1 is for each

component [C 1 n , C 2 n , . . . , C n n ]
. Proof: These coefficients, often denoted ( n r ), are the binomial coefficients. The property is true for a 1 , a 2 , and a 3 . If it is true for n, the number of terms appearing in a n+1 is from (1-4)

[C 1 n +, C n n , C 2 n +, C n-1 n , . . . , C i n +, C n-i+1 n , . . . , C n n , C n n ]
Indeed a n and k n+1 a

+ C p n == C p n+1 , (15)
we deduce immediately

C i n + C n-i+1 n = C i n+1 . ( 16
)
As the last component of a n+1 is k n+1 , if the property is true for a n , it is also true for a n+1 .

Property As the last component of a 2n+1 is of degree 1, we deduce that the degree of a 2n+1 is given by (18).

By the same procedure, if a 2n+1 has the property (18), a 2n+2 has the property (17).

III. SOME CONSEQUENCES OF STABILITY

All the properties indicated in Section II are direct consequences of the Levinson recursion ( 4)-( 5), without any consideration of stability problems. In this section, we will discuss some consequences on the regression vector deduced from the stability condition

|k i | < 1, 1 ≤ i ≤ n.
Consequence 3.1: If an AR filter is stable, then the components of the regression vector satisfy

|a i | < C i n .
Proof: It is a direct consequence of 2.6. Indeed, a i is a polynomial with C i n terms which are product of reflection coefficients k i . As |k j | < 1, the absolute values of these terms are also smaller than 1 which gives immediately |a i | < C i n . Comment: It is clear that this condition does not ensure that the AR filter is stable. Indeed, it is well known that the stability condition cannot be expressed as a condition on each component, but is a condition on all the a i together. On the other hand, if one a i is greater than C i n , the filter is certainly unstable.

Consequence 3.2: If an AR filter is stable, then we have

|a i | < 2 n -1.
Proof: It is a direct consequence of 3.1. Indeed, we deduce from it that

|a i | < C i n = 2 n -1. (19)
Comments: As for 3.1, this condition is a necessary condition for stability of an AR filter, but does not secure this stability.

Consequence 3.3: If an AR filter is stable, then the magnitude of the regression vector is smaller than

[C n 2n -1] 1/2 .
Proof: We deduce that the square of the square of the magnitude of a n satisfies

||a i || 2 < n i=1 (C i n ) 2 = n i=0 (C i n ) 2 -1. (20)
In order to calculate this term, we write two expansions of (a + b) 2n

(a + b) 2n = 2n k=0 C k 2n a 2n-k b k , (21)
May 25, 2018 DRAFT

[(a + b) n ] 2 = i j C i n C j n a 2n-i-j b i+j , (22)
By identification we deduce immediately

C n 2n = n i=0 C i n C n-i n = n i=0 [C i n ] 2 , (23)
and with (20) we deduce

||a i || 2 < C n 2n -1. (24)
Consequence 3.4: If an AR filter is stable, then the maximum of the magnitude of the regression vector is reached for two vectors corresponding to k i = -1 and to k i = (-1) i-1 , 1 ≤ i ≤ n, respectively, and its value is [C n 2n -1] 1/2 . Proof: For k i = -1 we deduce from 2.3, 2.4, and 2.6 that the components of a n , are

a T n = -[C 1 n , C 2 n , , . . . , C n n ], (25)
which means that for this vector the upper bound given by (20) is reached.

For k i = (-1) i-1 we will prove that the corresponding vector is

a T n = [C 1 n , -C 2 n , , C 3 n . . . , (-1) n-1 C n n ]. (26)
Let us suppose that this property is true for n, and let us calculate a n+1 with k n+1 = (-1) n . We deduce that

-k n+1 a (-)T n = [C n n , -C n-1 n , C n-2 n , . . . , (-1) n-1 C 1 n ], (27)
and applying (4) and ( 5), we deduce that

a nT n+1 = [C 1 n+1 , -C 2 n+1 , C 3 n+1 , . . . , (-1) n-1 C n n+1 ], (28)
As the last component of a n+1 is k n+1 = (-1) n = (-1) n C n+1 n+1 , we see that a n+1 has the structure of a n given by (26). Of course, the upper bound of (20) is reached for this vector.

Comment: This property is important in some interpolation problems. Indeed it has been shown in another paper [START_REF] Papoulis | Constrained Wiener filtering[END_REF] that if we consider the signal y k defined by [START_REF] Blanc-Lapierre | Fonctions Aléatoires[END_REF] where u k is a white noise of variance 2 , the interpolation of y n in terms of all its past and future can be obtained with an error given by

η 2 = 2 [1 + a 2 i ] -1 = 2 [1 + ||a n || 2 ] -1 . ( 29
)
For a given value of 2 this error is minimum |a n | is maximum, provided that the AR filter is stable.

Using (24) we see that this minimum error is Proof: The denominator of H(z) defined by ( 3) is

η 2 min = 2 (n!) 2 /(2n)!. (30)
D(z) = z n - a i z n-i . ( 31
)
If the regression vectors are a n given by ( 25) or (26), we obtain, respectively,

D + (z) = (z + 1) n (32) D -(z) = (z -1) n . (33)
IV. SIMPLE SUFFICIENT CONDITIONS FOR STABILITY All the properties indicated in Section I11 are consequences of the stability and are then necessary conditions for stability. But conversely they can be satisfied for unstable filters and are then not sufficient conditions for stability.

Of course such necessary and sufficient conditions do exist and are well known [START_REF] Jury | Theory and Application of the z-Transform Method[END_REF], [START_REF] Rabiner | Digital Processing of Speech Signals[END_REF]. But, in general, they are very complicated to check and require a lot of calculation. In practice, they are equivalent to the calculation of all the reflections coefficients k i from a given regression vector in order to check that

|k i | < 1.
In many problems, and particularly in an adaptive context, it is necessary to have simple conditions securing the stability of the filter. By simple we mean conditions which can be verified almost without calculation.

Starting from the idea that if |a| is small, the k i s also are small and the filter is stable, we will introduce some very simple sufficient conditions for stability.

For this purpose, let us associate to any vector a the functions

S n (a) = n i=1 |a i | (34) T n (a) = n i=1 a 2 i = ||a|| 2 (35)
A n (a) = max 1≤i≤n (|a i |).

(36)

These functions have been considered in Section III, and, in particular, their upper bounds for stable filters have been obtained.

In this section we are interested in some sufficient conditions on these functions which secure the stability of the corresponding AR filter. Of course it is obvious that these functions are very simple, .in terms of calculation from the components of the regression vector.

May 25, 2018 DRAFT Result 4. 6: If A n > l/n, then the AR filter may be unstable.

Proof: It is the same as for 4.4. By taking all the a i 's equal to a, the condition means that a may be greater than l/n and, in this case, the filter is unstable.

Finally, it is worth noticing that .the condition A n < 1/n implies T n < 1/n which also implies S n < 1.

In other words, the simplest condition (A n < 1/n), which needs absolutely no calculation, is also the most restrictive.

These sufficient stability conditions are generalizations to all orders of a well-known conditions for second-order filters [8, p. 211].

May 25, 2018 DRAFT

  terms because k n+1 is not present in the expression of the components of a n . But as C p n = C n-p n and C p-1 n

DRAFT May 25, 2018 Consequence 3 . 5 :

 201835 The two regression vectors of 3.4 correspond to two AR filters with poles of order n located in +1 and -1.

  2.7 :The total degrees of the polynomials appearing in the components of a n , are Proof: f we consider a 3 given by (8), we see that the three components are polynomials in k i , respectively, with degrees 2, 3, 1. Let us suppose that (17) is true. The degrees of the components of k 2n+1 a 2n(-) are, becauseof the multiplication by k 2n+1

	k 2n+1 a	(-) 2n : [2, 4, . . . , 2n -2, 2n, 2n + 1, 2n -1, . . . , 5, 3].
		[2, 4, 6, . . . , 2n, 2n -1, 2n -3, . . . 3, 1]	(17)

for a 2n , and [2, 4, 6, . . . , 2n, 2n + 1, 2n -1, 2n -3, , . . . 3, 1] (18) for a 2n+1 . DRAFT May 25, 2018

At first let us recall a known result.

Result 4. I: If S n < 1, then the corresponding AR filter is stable.

Proof: We have to prove that if S n < 1, the polynomial P n (z) defined by (9) has all its roots outside the unit circle. Let us suppose the contrary and consider a root z 0 such that |z 0 | < 1. As z 0 is a root we have, from (9),

which gives

because |z 0 | < 1. Thus, S n ≥ 1 which is in contradiction with the initial condition. Then all the roots are outside the unit circle.

Comment: We deduce from 3.2. and 4.1. that if S n < 1, the AR filter is stable, and if

it can be either stable or unstable, as we will see now. Proof: From.the Schwarz inequality we deduce that

and if T n < 1/n, then S n < 1, which secures the stability.

Result 4. 4: For every t > 1/n it is possible to find an unstable AR filter with regression vector a such that T n (a) > t.

Proof: Let us consider the filter such that a i = a > 0, ∀i. For this filter T n = na 2 , and as t > 1/n, then a > 1/n, or can be written 1/n + , > 0. It is possible to calculate the reflection coefficients of this filter, using (4) and [START_REF] Shlien | A geometric description of stable LPC digital filters[END_REF]. For example, we obtain easily

if < [n(n -1)] -1 , which proves that this filter is unstable.

Result 4. 5: If A n < l/n, then the corresponding AR filter is stable.

Proof: This condition means that |a i | < 1/n, 1 ≤ i ≤ n, and it results that S n < 1, which gives the stability.

DRAFT May 25, 2018