Melvin Raulin

Bruno Drouillat

Jérome Marrot

François Couty

Karen Wright

Diastereoselective ring cleavage of azetidines with cyanogen bromide

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Dedicated to the memory of Pr. François Couty (1964-2019) Diastereoselective ring cleavage of azetidines with cyanogen bromide Melvin Raulin, Bruno Drouillat, Jérome Marrot, François Couty and Karen Wright* Université Paris-Saclay, UVSQ, CNRS, UMR 8180, Institut Lavoisier de Versailles, 78000, Versailles, France Abstract:N-alkyl azetidines bearing a chiral substituent at the nitrogen atom, and a variable substituent at C-3 as a prochiral centre, react readily with cyanogen bromide even at low temperatures to produce the corresponding 3-bromo-Ncyanamides as diastereoisomeric mixtures via ring opening. This reaction was found to be diastereoselective (up to 80:20 dr). In one case, the major diastereomer could be isolated by recrystallization, and the configuration of the newly created stereocentre was determined by X-ray crystallography Strain in azetidines allows for many useful ring-opening and ring-expansion processes. [START_REF] Couty | Synthesis of Azetidines[END_REF] When the nitrogen atom within this heterocycle is not acylated or tosylated, the nucleophilic character, as well as the basicity, of the intracyclic amine is close to a linear amine, in contrast to aziridines where the basicity is considerably lower. Therefore, N-alkyl azetidines react with many electrophiles (eg chloroformates, 2 alkyl halides, 3 alkyl sulfonates, 4 triphosgene, 5 cyanogen bromide 6 ) to give a stable or transient azetidinium ion that can react further either by ring cleavage or by ring-expansion. 7 In this area, it was recently reported 8 (Scheme 1A) that C-3 substituted azetidiniums can be opened in an enantioselective manner in the presence of a catalytic amount of a SPINOL-derived phosphonate acting as a chiral counteranion for the azetidinium cation.

Scheme 1. Previous reports of enantioselective opening of achiral C-3 substituted azetidines (A), and overview of the present work (B).

This study follows the pioneering work by the same group, involving enantioselective opening of N-acyl azetidines with chiral phosphonic acids acting as catalysts 9 (Scheme 1). These recent reports led us to disclose our results in this field based on a different strategy. We studied the diastereoselective opening of azetidines of the type shown in Scheme 1B, fitted with a nonracemic substituent at the nitrogen atom (R = Ph, Cy, 1 or 2naphthyl) and a functional group at C-3. These 3-substituted heterocycles constitute prochiral entities, which can be converted into a chiral synthon upon diastereoselective reaction at one of the symmetrical methylene groups at C-2 or C-4. We thus chose to react these azetidines with cyanogen bromide through the von Braun reaction, a process that we reported recently 10 which produces 3-bromo-N-cyanamides through ring cleavage in a regioselective manner. The 3-bromo-N-cyanamides produced are versatile bis-electrophilic building blocks that were used for the preparation of cyclic guanidines. 11 The azetidine substrates used herein were prepared in a straightforward manner by using Gaertner's seminal reported method for the preparation of 3-hydroxy azetidines 12 adapted to different (S)-ethylamines. Treatment of epichlorohydrin with ethylamines 1-4 in water, followed by basic treatment in acetonitrile gave average-to-good yields of azetidinols 5-8 as crystalline compounds, without the need for chromatographic purification (Scheme 2).

Scheme 2. Preparation of 3-hydroxy azetidines.

The tosylate 9 derived from 5 was used to prepare S, N and C substituted azetidines by nucleophilic substitution (Scheme 3). The 3-thio azetidines 10 and 11 were obtained by reaction with potassium thioacetate or sodium methoxybenzyl thiolate in 71 and 75% yields respectively. Similarly, reaction with potassium phthalimide, sodium azide or potassium cyanide gave the 3substituted azetidines 12 (53%), 13 (75%) and 17 (82%). The 3azido azetidine 13 was further transformed either by a CuAAc reaction to give the triazole 14 (79%), or by Staudinger reduction with triphenylphosphine followed by Boc or Fmoc protection to give 15 (23%) and 16 (74%). The 3-cyano azetidine 17 gave ethyl ester 18 through acidic ethanolysis (82%), which was reduced to the primary alcohol 19 (94%) with LiAlH4. NaOH, EtOH, 80°C; d) K phthalimide, EtOH, 80°C; e) NaN3, EtOH, 40°C; f) phenylacetylene, CuSO4, Na ascorbate, THF/H2O, rt; g, h) PPh3, THF/H2O, 70°C, then g) Boc2O, aq NaHCO3, rt, or h) FmocCl, MeCN, rt; i) KCN, EtOH, 80°C; j) H2SO4, EtOH, 80°C; k) LiAlH4, THF, rt.

The alcohols 5-8 were also alkylated in the presence of sodium hydride and an alkyl halide to give the ethers 20-29 (Figure 1). During these reactions some ring-opening of the azetidine by the alkyl halide occurred, leading to low yields of the ether product in some cases. This reactivity has already been observed by ourselves and others, 13 and we intend to study further the diastereoselectivity of these ring openings. The alcohols 5 and 6 were also acylated to give the esters 30-33, including both antipodes of a mandelate ester, introducing an additional stereogenic center around the azetidine ring (Figure 1).

The von Braun reaction of 3-S, 3-N and 3-C substituted N-((S)-1-phenylethyl) azetidines was investigated first under standard conditions (0.2 M of the azetidine in chloroform at -30°C, 3 equiv. of 3M BrCN in dichloromethane). Reactions were rapid and led to a diastereomeric mixture of ring-opened products with a diasteromeric ratio ranging from 55:45 to 70:30 (Table 1). No specific correlation could be deduced from the size or the nature of the functional groups at the 3-position on the selectivity of the reaction.

Substrate

The reaction of 3-O substituted N-((S)-1-phenylethyl) azetidines

with cyanogen bromide under the same conditions was studied next (Table 2). The reaction proceeded well with all the azetidines, but again diastereoselectivity was not correlated with the nature or the steric crowding of the O-substituents. Nevertheless, diastereoselectivity could be improved by carrying out the reaction at lower temperatures (eg. R = Bzh 50% de and R = Bn 60% de at -78°C) ( The major diastereoisomer of 44 could be purified by crystallization from a chromatographically inseparable mixture, allowing the determination of its (1S, 2S) configuration by X-ray crystallography (Figure 2). The influence of the N-substituent on the azetidine ring in the reaction of 3-O substituted azetidines was explored. Substrate 28 with an N-(S)-1-cyclohexylethyl group gave a lower de than the corresponding N-(S)-1-phenylethyl substrate 20, while increasing the bulk of the N-substituent with a 1-or 2-naphthyl group gave better diastereomeric ratios at -30°C than for 20. No further advantage was found in using a 3-O-2-CH2-naphthyl at C-3 instead of the 3-O-Bn group. An interesting match/mismatch effect was found with the substrates 32 and 33, bearing acetoxymandelate esters at C-3, with the (R)-mandelate ester giving a de of 50%, while for the (S)-mandelate no diastereoselectivity was observed (Table 3). The reaction of the azetidin-3-ols 5-8 bearing different chiral substituents at the nitrogen was finally investigated. In this case, the 2-hydroxy cyanamides formed cyclized on evaporation of the reaction solvent to give the corresponding 2-imino oxazolidines, isolated as their bromide salts. The corresponding O-acetyl cyanamides could be isolated if the reaction mixture was treated by addition of acetic anhydride before evaporation. The diastereomeric excess ranged from 20% to 40% at -30°C, and was again dependent upon the nature of the N-substituent of the azetidine (Table 4). Improved diastereoselectivity was found for reactions carried out at -78°C (60% de for 5). We previously carried out a detailed in silico study of the von Braun reaction applied to azetidine substrates to better understand the reaction pathways involved. [START_REF] Couty | [END_REF]

Figure 3. Reaction paths in the von Braun reaction of azetidines

This revealed that the azetidine reacts initially with cyanogen bromide yielding a quaternary ammonium salt, which can then undergo bromide substitution to give a mixture of two diastereomeric open chain products, and/or an N-cyanoazetidine and an alkyl bromide. Only the open chain products were obtained with the substrates used in this study. Using a simplified model, these theoretical studies showed little difference (0.2 kcal.mol -1 ) in the total free energies of the transition states leading to each diastereomer, leaving us with no valid tools to predict diastereoselectivity. Experimentally, no systematic relationship between the size or the nature of the functional groups at the 3-position was found to explain the outcome of the von Braun reaction.

In conclusion, we have prepared a large set of new, diversely 3-substituted azetidines bearing a chiral centre at the ring nitrogen atom and studied their reactivity towards cyanogen bromide. The diastereoselectivity of the von Braun reaction was low to moderate with no evident correlation between the size and the nature of the C3-substituent. However, changes in the nature of the chiral group on the nitrogen atom of these strained heterocycles appeared to influence diastereoselectivity. Unsurprisingly, by performing the reaction at a lower temperature (-78°C), it was possible to increase diastereoselectivity. Desymmetrization of azetidines with other electrophiles is currently under investigation in our laboratory.

Scheme 3 .

 3 Scheme 3. Preparation of 3-substituted azetidines. a) TsCl, DMAP, TEA, CH2Cl2, 5°C; b) KSAc, acetone, 60°C; c) MeOBnSH, aq. NaOH, EtOH, 80°C; d) K phthalimide, EtOH, 80°C; e) NaN3, EtOH, 40°C; f) phenylacetylene, CuSO4, Na ascorbate, THF/H2O, rt; g, h) PPh3, THF/H2O, 70°C, then g) Boc2O, aq NaHCO3, rt, or h) FmocCl, MeCN, rt; i) KCN, EtOH, 80°C; j) H2SO4, EtOH, 80°C; k) LiAlH4, THF, rt.

Figure 1 .

 1 Figure 1. 3-O-alkyl and 3-O-acyl azetidines.

14

 14

Figure 2 .

 2 Figure 2. X-ray structure of (1S, 2S)-44

Table 1 .

 1 Reaction

		R =	Product	dia. ratio a
	10	SAc	34	55:45
	11	SBnOMe	35	60:40
	12	NPhth	36	60:40
	13	N3	37	65:35
	14	Ph-triazole	38	65:35
	15	NHBoc	39	55:45
	16	NHFmoc	40	55:45
	17	CN	41	65:35
	18	COOEt	42	65:35
	19	CH2OH	43	70:30

of 3-S, 3-N and 3-C substituted N-((S)-1phenylethyl) azetidines in the von Braun reaction. a From 1 H NMR spectra, rounded down to nearest 5%.

Table 2

 2 ).

	Substrate	R =	Product dia. ratio (%) a
	20	Bn	44	70:30 (-30°C) 80:20 (-78°C)
	21	Bzh	45	70:30 (-30°C) 75:25 (-78°C)
	22	Tr	46	65:35
	23	All	47	70:30
	24	2-CH2Naph	48	70:30
	25	9-CH2Ant	49	60:40
	9	Ts	50	70:30
	30	Ac	51	55:45
	31	Piv	52	60:40

Table 2 .

 2 Reaction

	of 3-O substituted N-((S)-1-phenylethyl)
	azetidines in the von Braun reaction. a From 1 H NMR spectra,
	rounded down to nearest 5%.

Table 3 .

 3 Reaction of 3-O substituted azetidines in the von Braun reaction. a From 1 H NMR spectra, rounded down to nearest 5%.

Table 4 .

 4 Reaction of azetidin-3-ols. i) BrCN, CHCl3 -30°C; ii) solvent concentration; iii) Ac2O. a From 1 H NMR spectra, rounded down to nearest 5%.

Acknowledgments

The University of Versailles St-Quentin-en-Yvelines, University Paris-Saclay and the CNRS are acknowledged for financial support. MR thanks the Ministère de l'Enseignement Supérieur de la Recherche et de l'Innovation for a PhD fellowship.

Supplementary data

Supplementary data to this article can be found online at


Julien Sohier
email: sohier@math.jussieu.fr

Finite size

Keywords: Mathematics Subject Classification. 60K35, 82B41 Renewal Theory, Pinning Models, Criticality, Subordinators, Regenerative Sets. 1

come

Introduction

The model of directed polymers interacting with a one dimensional defect turns out to be quite satisfactory to describe various biological or physical phenomena, such as (1 + 1) interface wetting [START_REF] Derrida | Effect of disorder on two-dimensional wetting[END_REF], the problem of depinning of flux lines from columnar defects in type II superconductors [START_REF] Nelson | Effect of disorder on two-dimensional wetting[END_REF] and the denaturation transition of DNA in the Poland Sheraga approximation [START_REF] Kafri | Why is the dna denaturation transition first order?[END_REF]. The first results obtained by the physical community have been summarized in [START_REF] Fisher | Walks, walls, wetting, and melting[END_REF], a paper which received a lot of attention from mathematicians. Since then, this model has been under close scrutinity, including in the last few years (the recent mathematical monograph [START_REF] Giacomin | Random Polymer Models[END_REF], but also [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1 + 1)-dimension[END_REF], [START_REF] Isozaki | Weakly pinned random walk on the wall: pathwise descriptions of the phase transition[END_REF] or [START_REF] Deuschel | Scaling limits of equilibrium wetting models in (1 + 1)-dimension[END_REF]).

Here, we are going to deal with the homogeneous version of the model; this version is mathematically remarkable in the sense that it is essentially completely solvable. For example, the model can have a transition of any given order [START_REF] Giacomin | Random Polymer Models[END_REF] depending on the value of α, a parameter in the definition of the system (1 + α is sometimes called loop exponent in the literature).

A central role in statistical mechanics is played by the correlation length, a quantity that we will denote by ξ. Properly defining this quantity in an (infinite volume) model requires some work and in most of the cases, such a concept does not correspond to a unique mathematical object. All the same, it is essential that any "reasonably defined" correlation length behaves in the same way close to criticality. And precisely close to criticality, for a large class of models, ξ becomes the only "relevant" scale in the system (see below for a more precise explanation). This idea is one of the basic concepts of the so called "finite size scaling" theory (see [START_REF] Cardy | Finite size scaling[END_REF]). Here we illustrate in very concrete terms this concept via the scaling limit of pinning systems, obtaining a limiting behavior if the parameter is in a small, size dependent window near criticality as the size of the system goes to ∞.

In this first part, we will try to see in which way the above, rather unprecise statements, can be made quantitative. The second part should be seen as a warm up for the two last parts; actually, in the one dimensionnal wetting, the existing literature makes the statement quite easy. In the third part, we introduce our general return model, and make the precise statements of our result in terms of scaling limits of the zero level set of our system near criticality.

In this note, α will always be a real number in (0, 1). For positive sequences (a n ) n≥0 and (b n ) n≥0 , we will write a n ∼ b n if lim n→∞ a n /b n = 1, and for random variables (X n ) n≥0 and X with values in a Polish space (E, d), we will write X n ⇒ X if the sequence (X n ) n≥0 converges in law towards X.

1.1 A first model. Let τ be a recurrent renewal process with law P, that is τ = {τ n } n≥0 where τ 0 = 0 and (τ i+1τ i ) i≥0 are iid with common law P, where P is N valued and verifying, as k → ∞,

P(τ i+1 -τ i = k) ∼ C k 1+α .
(1.1)

We will actually consider a slightly more general model in section 3, allowing in particular τ to be transient, but for simplicity, we will restrict ourselves temporarly to this setup. The set τ ∩ [0, N ] can be considered as a random subset of {0, 1, . . . , N }. We define the law P N,β on the subsets of {0, 1, . . . , N } by

dP N,β dP (τ ) = 1 Z N,β exp(βN N (τ )), (1.2)
where N N (τ ) is the cardinality of the set τ ∩ [0, N ]. In (1.2), Z N,β is the partition function of the model and of course

Z N,β = E [exp(βN N (τ ))] . (1.3)
It is not difficult to show that the limit of the quantity F N (β) := 1 N log Z N,β exists and is non negative. We denote it by F (β). F (•) is a convex, non decreasing and non negative function, and there exists a critical value of β such that β ≤ β c implies F (β) = 0 and β > β c implies the positivity of F (β). If F (β) = 0, the system is said to be delocalized, and it is localized otherwise. As the underlying renewal is recurrent, we actually have β c = 0 (see [START_REF] Giacomin | Random Polymer Models[END_REF], chapter 2 and section 3 below).

1.2 Finite size scaling. We define the correlation length of our system by

ξ(β) := 1 F (β) , (1.4)
if β > 0, and ξ(β) = ∞ otherwise. This definition has first been introduced in the physical litterature in [START_REF] Fisher | Walks, walls, wetting, and melting[END_REF], and its mathematical relevance with respect to other quantities (in particular with the correlation function) is discussed in [START_REF] Giacomin | Renewal convergence rates and correlation decay for homogeneous pinning models[END_REF]. Also note that the concept of correlation length has been considered in depth in the inhomogeneous case (that is when the reward-penalty become site-dependent and random), see e.g. [START_REF] Toninelli | Correlation lengths for random polymer models and for some renewal sequences[END_REF]. It has been shown in [START_REF] Giacomin | Random Polymer Models[END_REF] that

ξ(β) βցβc ∼ c α (β -β c ) -1/α , (1.5)
for some (explicit) constant c α > 0.

We expect finite size effects to appear only if the system is of the scale of the correlation length, that is only if ξ(β)/N stays fixed (say equal to q). It is clear (considering (1.5)) that this is equivalent to keeping (β-β c ) 1/α N fixed (at least close to criticality). In this case, the observables of our system have a different behavior near criticality than their behavior in each of the two phases in the infinite volume limit. In an alternative way, we could say that close to criticality, the only relevant length scale is the correlation length. In order to make this precise, we need two basic objects. The first one is the stable subordinator with index α. Recall that a subordinator is an increasing Lévy process; a subordinator (σ (α) s ) s≥0 is said to be α stable (for α ∈ (0, 1)) if, for all M ∈ R + , the process (σ

(α)
Ms ) s≥0 has the same law than the process (M 1/α σ (α) s ) s≥0 . The second one is its local time (L (α) t ) t≥0 , which is simply the generalized inverse of (σ

(α) s ) s≥0 , that is L (α) t := inf s ≥ 0, σ (α) s ≥ t .
(1.6)

It will be shown in the last part (Lemma 1) that there exists C K > 0 such that

C K N N N α N ≥0 ⇒ L (α) 1 , (1.7)
and moreover the sequence C K NN N α N ≥0 is uniformly integrable (Lemma 2). Using these two facts and (1.5), one gets that the finite size correlation length ξ N (β) := 1/F N (β) verifies

ξ N (β) N N →∞ ∼ log E exp q α c α α N N N α -1 ∼ f (q), (1.8)
where f (q) = log E exp

q α c α α CK L (α) 1 -1
.

Let us now consider two very relevant observables (in the usual window): the (finite volume) contact fraction ρ N (β) (that is the expectation of N N /N with respect to the finite polymer measure) and the (once again finite volume) specific heat χ N (•) (that is the second derivative of the finite volume free energy). In our setup, these quantities can be written in an analogous way as above. Specifically, we get

ρ N (β) ∼ N α-1 1 C K E L (α) 1 exp q α c α α CK L (α) 1 E exp q α c α α CK L (α) 1 := N α-1 ρ(q).
(1.9)

In an equivalent way, introducing ρ(x) := x 1-α ρ(q), we get ρ N (β) ∼ ξ(β) α-1 ρ(q), that is we can express ρ N (•) (as N → ∞) as a function of ξ(•). Analogously, we have

∂ 2 ∂ 2 β F N (β) = χ N (β) ∼ N 2α-1 1 C 2 K E (L (α) 1 -E[L (α) 1 ]) 2 exp q α c α α CK L (α) 1 E exp q α c α α CK L (α) 1
, (1.10) so that, once again, we get the following equivalence:

χ N (β) ∼ ξ(β) 2α-1 χ(q).
(1.11)

We show here that these two particular (and physically relevant) examples are actually special cases for a much more general phenomenon, namely the convergence in law of the whole rescaled system as ξ(β)/N stays fixed. We are able to compute the scaling limit of the system in terms of a Radon Nykodym derivative with respect to the α regenerative set.

A look at the wetting model

As a warm-up, we will deal with the case in which the underlying law of our polymer model is the simple symmetric random walk conditioned to stay non negative, that is a sequence (S n ) n≥0 where S 0 = 0, the variables (S i -S i-1 ) i≥1 are iid and such that P

[S i -S i-1 = 1] = P [S i -S i-1 = -1] = 1
2 . We introduce the following probability law on Z N :

dP β,N dP (x) = 1 Z β,N exp(H β,N (x))1I x1≥0,...,xN ≥0 (2.1)
where the Hamiltonian of our system is given by

H β,N (x) = β N i=1 1I xi=0 = βN N . (2.2)
Here also, it is not hard to prove the existence of lim N →∞ [START_REF] Giacomin | Random Polymer Models[END_REF]). In particular, for N even, we have the inequality

1 N log(Z β,N ) = F (β) (see
Z β,N = E [exp(H β,N (S))1I S1≥0,...,SN ≥0 ] ≥ E [exp(H β,N (S))1I S1>0,...,SN >0 ] = P [S 1 > 0, . . . , S N > 0] ∼ 1 √ 2πN , (2.3)
where the last equivalence is well known (cf Feller (1971)), which entails that for every β ∈ R , F (β) ≥ 0.

In this model, it is possible to show that β c is actually equal to log(2) (see [START_REF] Giacomin | Random Polymer Models[END_REF] chapter 1, or [START_REF] Isozaki | Weakly pinned random walk on the wall: pathwise descriptions of the phase transition[END_REF]). We are interested in the scaling limit of the system near criticality. To be more explicit, we define the application:

X N : R N → C([0, 1]) by: (X N t (x)) t∈[0,1] = x [N t] N 1/2 + (N t -[N t]) x [N t]+1 -x [N t] N 1/2 . (2.4)
and we introduce the sequence of measures

Q β,N := P β,N • (X N ) -1 , N ≥ 0. Of course, in (2.4), [x]
denotes the integer part of the real number x.

Let (B t ) t≥0 be a standard Brownian motion defined on our probability space (Ω, F , P) in the sequel, and (L t ) t≥0 denote its local time at zero. It has been proven in [START_REF] Isozaki | Weakly pinned random walk on the wall: pathwise descriptions of the phase transition[END_REF] that the sequence (Q βc,N ) N ≥0 converges weakly to the law of the process (|B t |) t≥0 . This result is actually very natural since it is quite easy to see that the law of the process (S n ) n≥0 under P βc,N has the same law as (|S n |) n≥0 under P.

The following result gives some intuition about the more general theorem we are going to prove in the next section in terms of zero level sets.

Z40000(x) Z40000 x ε = 0 ε = -1 ε = 1 β = .1 β = -.2 Figure 2
.arabic@ ¸figure. The distribution of the last point of the system for N = 40000, where

Z N (x) := E [exp(βN N (τ ))1 SN =x ] for β ∈ {.1, -.2, ε/ √ 40000} and ε ∈ {-1; 0; 1}.
Proof . The following convergence in law is classical:

N N √ N ⇒ L 1 , (2.5)
but a much stronger statement can be proved: given a Brownian motion on (Ω, F , P), one can construct a simple random walk (S n ) n≥0 on the same probability space such that, for every η > 0, one has:

lim N →∞ N -1/4-η |L N -N N | = 0, (2.6)
almost surely as N → ∞ (see [START_REF] Révész | Random walk in random and non-random environments[END_REF], Theorem 10.1). In particular for η = 1/2, and by the scaling property of (L t ) t≥0 , this implies immediately the convergence of the joint law (S

[N t] / √ N , N N / √ N ) ⇒ (B t , L 1 ). A proof of the tightness of the sequence (Q βc+ ε √ N
,N ) N ≥0 can be found in [START_REF] Caravenna | Tightness conditions for polymer measures[END_REF]. We are therefore left with showing the convergence in law of the finite dimensional marginals of the process, that is we have to show that, for every n ∈ N and for every continuous bounded function F (•) from [0, 1] n to R, the following holds:

Q βc+ ε √ N ,N [F (ω t1 , . . . , ω tn )] → N →∞ E e εL1 F (|B t1 |, . . . , |B tn |) E[e εL1 ] .
(2.7)

We already know by [START_REF] Isozaki | Weakly pinned random walk on the wall: pathwise descriptions of the phase transition[END_REF] that (2.7) holds for ε = 0. The term in the left hand side of (2.7) can be written as is uniformly integrable. This is obvious if ε ≤ 0, and will be proved in the last section in a more general setup (see Lemma 2) for ε > 0, although it could be also proven directly using the exact distribution of N N , which is actually well known in the simple symmetric random walk setup (see for example [START_REF] Révész | Random walk in random and non-random environments[END_REF]). This completes the proof.

E βc,N F S [N t1] √ N , . . . , S [N tn] √ N e ε N N √ N /E βc,N e ε N N √ N . ( 2

The renewal setup

3.1 The model. As we did in the introduction, we denote by τ the points of a renewal process, that is the image set of a sequence of iid positive random variables (l i ) i≥0 whose law is P. More precisely, τ := {τ 0 , τ 1 , . . . , } where τ k = k i=1 l i . In what follows, P is an integer valued law verifying

K(n) := P(l 1 = n) = L(n) n 1+α , (3.1)
where L(.) is a slowly varying function. Furthermore, we allow transience for our renewal process, that is we may have n≥1 K(n) < 1 (or equivalently P(l 1 = ∞) > 0). For N ∈ N, we introduce K(N ) := n>N K(n). We recall that a positive measurable function L(.) is called slowly varying if for all c > 0, we have lim x→∞ L(cx)/L(x) = 1. Such functions are well known, see [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]. One basic example of slowly varying function (apart from the trivial case of constants) at infinity is x → log(1 + x), x ∈ (0, ∞); but logarithmic functions are not the only examples, x → exp(a(log(1 + x)) c ) is also slowly varying for every a ∈ R and c < 1.

We then define the homogeneous pinning polymer law exactly as in (1.2). In all what follows, we will denote by τ (N ) the set (τ ∩ [0, N ])/N ⊂ [0, 1]. When we use this notation, it will be implicit in the sequel that the set τ has the law P N,β , where β will be obvious from the context. This model has been essentially completely solved, see [START_REF] Giacomin | Random Polymer Models[END_REF]; basic facts about it are the existence of a localization delocalization transition for a critical β, whose value is given by β c =log(Σ K ), where Σ K := n≥1 K(n). In particular, we have the equivalence β c = 0 iff the underlying renewal process is recurrent. This transition is actually defined in terms of the free energy, which is defined exactly as in the previous sections. It is actually possible to give a very intuitive description of both phases in terms of the scaling limit of the zero level set. Intuitively, if the system is localized, we expect the set τ (N ) to converge in a suitable way to the full interval [0, 1]. Similarly, if β < β c , we expect it to converge to {0}.

To make such statements quantitative, we introduce F the set of closed subsets of [0, 1]. We endow it with the topology of Matheron, which can be described as follows. For F ∈ F and t ∈ R + , we set d t (F ) := inf(F ∩ (t, ∞)). Notice that t → d t (F ) is a right-continuous function and that F can actually be identified with

d (•) (F ) because F = {t ∈ R + : d t -(F ) = t}.
Then in terms of d (•) (F ), the Matheron topology is the standart Skorohod topology on càdlàg functions taking values in R + ∪ {+∞}. We point out that with this topology, the space F is metrizable, separable and compact, hence in particular Polish. We denote by ρ(•) the Hausdorff metric given by ρ(F 1 , F 2 ) := max i∈{(1,2),(2,1)} sup t∈Fi 1 inf s∈Fi 2 |t -s| where F 1 , F 2 ∈ F. It is then a well known result that this metric is equivalent to the metric engendered by the Skorohod distance via the identification through d (•) on F . Using this topology, both of the above statements have actually been proven in [START_REF] Giacomin | Random Polymer Models[END_REF]. The scaling limit at criticality (that is when β = β c ) is much richer than that. To describe it, we need the notion of α stable sets.

3.2 α regenerative sets and subordinators. Recall that a subordinator is a non decreasing Lévy process. A well-known fact about a subordinator (σ t ) t≥0 is the existence of its so-called Lévy Khintchine exponent Φ(.), that is a measurable function verifying ∀λ > 0, ∀t ≥ 0, E e -λσt = e -tΦ(λ) . When Φ(t) = t α for α ∈ (0, 1), the associated subordinator is refered to as α stable, and we will denote it by (σ (α) s ) s≥0 in what follows. We already introduced its local time in (1.6). These objects are very well-known and a classic reference is [START_REF] Bertoin | Subordinators: examples and applications[END_REF]. We then define the set A α = {σ (α) s , s ≥ 0} (∈ F) the regenerative set of index α, where for a subset A ⊂ R, A is the closure of A. An important property that will turn out to be useful is the fact that P [1 ∈ A α ] = 0 (see [START_REF] Kesten | Hitting probabilities of single points for processes with stationary independent increments[END_REF]).

3.3 Scaling limits at and near criticality. In [START_REF] Giacomin | Random Polymer Models[END_REF], it was shown that, if β = β c , we have the convergence:

τ (N ) ⇒ A α (3.2)
in the recurrent case (that is K(∞) = 0), and in the transient case

τ (N ) ⇒ Ãα , (3.3)
where Ãα is a random subset of [0, 1] whose law is absolutely continuous with respect to the law of A α with Radon-Nykodym density equal to (απ/ sin(απ)

) (1 - max(A α ∩ [0, 1])) α .
A first step towards this convergence has been made in [START_REF] Isozaki | Weakly pinned random walk on the wall: pathwise descriptions of the phase transition[END_REF] in a discrete random walk set-up; they actually proved the convergence of the entire process at criticality towards the brownian motion. This work has been strongly generalized in [START_REF] Deuschel | Scaling limits of equilibrium wetting models in (1 + 1)-dimension[END_REF], actually being the most important part of the proof of the convergence of more general pinning models towards the reflected brownian motion at criticality. This was in turn generalized in [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1 + 1)-dimension[END_REF] using powerful renewal techniques, on which the present work is based.

Let us focus for the moment on the recurrent case, that is we assume for the moment K(∞) = 0. First of all, we recall the following result (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF],XIII.6 Theorem 2): it is possible to choose a sequence (a n ) n≥0 such that

nΓ(1 -α)L(a n ) αa α n → 1, (3.4)
and furthermore, as soon as a sequence (a n ) n≥0 verifies (3.4), we have the convergence :

n i=1 l i a n ⇒ σ (α) 1 (3.5)
where the (l i ) i≥0 are iid with common law K(.) satisfying (3.1). We then define the sequence (b n ) n≥0 to be the inverse sequence of the (a n ) n≥0 , that is the unique sequence, up to asymptotic equivalence, which verifies

a [bn] ∼ b [an] ∼ n (3.6)
as n → ∞. Its existence is ensured by Theorem 1.5.12, page 28 of [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]. It is rather easy to see that b n ∼ (Γ(1α)K(n)) -1 as n → ∞. Let us quickly prove this fact. Using the definition of (a n ) n≥0 , we get

b n Γ(1 -α)L(a [bn] ) αa α [bn] ∼ b n Γ(1 -α)L(n) αn α ∼ 1, (3.7)
and then we note that K(N ) ∼ L(N )(αN α ) -1 (see [START_REF] Giacomin | Random Polymer Models[END_REF], P.201). We also used the fact that

x n ∼ y n yields L(x n ) ∼ L(y n ) as n → ∞ for (x n ) n≥0 and (y n ) n≥0 positive sequences. For n large enough, x n /y n ∈ [1 -η; 1 + η]
for a given η > 0, and the fact that the convergence L(cx)/L(x) → 1 is uniform in c on every compact subset of R + (a basic result on slowly varying functions, see [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]) yields the assumption.

To deal with the transient case, we introduce the recurrent law P given by

P(l 1 = n) := K(n) = L(n)e βc n 1+α = L(n) n 1+α Σ K . (3.8)
We have then exactly the same statements as in Eq.(3.4) and (3.5) making the obvious changes, and we introduce the sequence ( bn ) n≥0 , the analogous of (b n ) n≥0 for the new sequence (ã n ) n≥0 . We are now ready to state our main result: Theorem 3.1. Let ε ∈ R. The following statements hold:

(1) if K(∞) = 0, if τ is distributed according to P βc+ε/bN ,N , then τ (N ) ⇒ B α,ε , (3.9)
where B α,ε (⊂ [0, 1]) is the random set whose law is absolutely continuous with respect to the law of A α whith Radon Nykodym density equal to exp(εL

(α) 1 )/E[exp(εL (α) 1 )]. (2) if K(∞) > 0, if τ is distributed according to P βc+ε/ bN ,N , then τ (N ) ⇒ Bα,ε , (3.10)
where Bα,ε (⊂ [0, 1]) has a Radon-Nykodym density given by exp(εL

(α) 1 )(1 -max(A α ∩ [0, 1])) α / E[exp(εL (α) 1 )(1 -max(A α ∩ [0, 1])) α ] .

Proofs

Lemma 4.1. If K(∞) = 0, the following convergence holds:

|τ (N ) | b N ⇒ L (α) 1 . (4.1)
Proof . For M > 0, we have the equality:

P |τ (N ) | b N ≤ M = P   1 a [MbN ] [MbN ] i=1 l i ≥ N a [MbN ]   . (4.2)
The right hand side of (4.2) is easily seen to converge towards P M

1 α σ (α) 1 ≥ 1 . Actually, we have N/a [MbN ] → M -1 α as N → ∞ (since M b N ∼ b [M 1 α N ]
), and it is easy to see that, if X n ⇒ X, if a deterministic sequence u n → u and the distribution of X is continuous, then P(X n ≥ u n ) → P(X ≥ u). So we just have to recall that the distribution of σ (α) 1 is continuous, which is actually a well known fact about stable subordinators (see [START_REF] Bertoin | Lévy processes[END_REF], P.271).

Because of the scaling property of the stable subordinator, we get

P M 1 α σ (α) 1 ≥ 1 = P(σ (α) M ≥ 1) = P(L (α) 1 ≤ M ), (4.3)
which proves our claim.

Lemma 4.2. For K(∞) = 0 and for all ε > 0, the family exp(

ε|τ (N ) | bN ) N ≥0 is uniformly integrable.
Proof . It is enough to show that, for all a > 0, we have sup

N ∈N E e a|τ (N ) |/bN < ∞.
(4.4)

As

E e a|τ (N ) |/bN = ∞ 1 P |τ (N ) |/b N > 1/a log(x) dx, (4.5)
to show 4.4, we just have to see that for all C 1 > 0, there exists C 2 > 0 such that for y ≥ 1,

P |τ (N ) |/b N > y ≤ C 2 exp(-C 1 y), (4.6)
as soon as N is large enough. Using the Markov inequality, we see that for every λ > 0 and y ≥ 1 :

P |τ (N ) )| b N > y = P   [ybN ] i=1 l i < N   ≤ E e -λl1 [ybN ] e λN . (4.7)
Noting that τ is recurrent, we have :

log(E[e -λl1 ]) = log   1 -L(1/λ)λ 1+α n≥1 1 -e -λn (λn) 1+α L( nλ λ ) L(1/λ)   .
(4.8)

For any η > 0 and using the uniform convergence property on compact sets for slowly varying functions, one has the equivalence (for λ ց 0)

λ 1/η nλ≥η 1 -e -λn (λn) 1+α L( nλ λ ) L(1/λ) ∼ 1/η η 1 -e x x 1+α dx, (4.9)
and it is easily seen that this implies the equivalence (still for λ ց 0):

-log(E[e -λl1 ]) ∼ Γ(1 -α)/αλ α L(1/λ). (4.10)
This entails that there exists c(α) > 0 and λ 0 (α) > 0 such that for all λ ∈ (0, λ 0 ), we have

-log(E[e -λl1 ]) ≥ c(α)λ α L(1/λ). Note that (b N L(N )N -α ) N ≥0
is a positive bounded sequence. Then, using 4.7, we get, for every N :

P |τ (N ) )| b N > y ≤ inf 0≤λ≤λ0 exp -c(α)λ α L(1/λ)y N α L(N ) + λN , (4.11)
possibly modifying c(α) in doing so. Then we consider N 0 such that for N ≥ N 0 , C 1 /N < λ 0 , and we give us

C 2 ∈ (C 1 ; N 0 λ 0 ). For u in the compact set [C 1 , C 2 ]
, the convergence of L(N/u)/L(N ) towards 1 is uniform, so that, for η > 0, for N large enough, L(N/λN )/L(N ) ≥ 1η as long as C 1 /N ≤ λ ≤ C 2 /N . Thus, for N large:

P |τ (N ) )| b N > y ≤ inf C1/N ≤λ≤C2/N exp (-c(α)(1 -η)(λN ) α y + λN ) . (4.12)
And this last inequality clearly entails 4.6, and thus the lemma.

Lemma 4.3. For K(∞) = 0, we have the convergence:

(|τ (N ) |/b N , τ (N ) ) ⇒ (L (α) 1 , A α ). (4.13)
Proof . We introduce the process (N γN (t)) t≥0 , a Poisson process of rate γ N where we define

γ N := n≥1 (1 -e -n/N )K(n) -1
. We take (N γN (t)) t≥0 independent from τ . It is easy to see that the process τ Nγ N (t) /N t≥0 is a subordinator. In [START_REF] Giacomin | Random Polymer Models[END_REF], it was shown that the Lévy exponent of τ Nγ N (t) /N t≥0 converges to the one of σ where Im(f ) is the image set of f where we endow the space R × F by the topology σ (B(R) ⊗ F), and using the distance max(|.|, ρ) where ρ is the Hausdorff metric. We will show that, for all continuous bounded G : R × F → R, we have:

E G |τ (N ) | b N , τ (N ) = E G F, G ( τ Nγ N (s) N ) s≥0 + (o N (1)), (4.15)
that is that we can consider the joint law in the left hand side of (4.13) as a function of (τ Nγ N (s) /N ) s≥0 up to negligible corrections. Assume for the moment that (4.15) is true. Taking into account the convergence

τ Nγ N (t) /N t≥0 ⇒ σ (α) s s≥0
and the fact that (F, G) σ

(α) s s≥0
is exactly the limiting law we want, it is enough to show that (F, G) (•) is almost surely continuous at σ (α) s s≥0

to apply the continuous mapping theorem (see [START_REF] Billingsley | Convergence of probability measures[END_REF]) to prove our lemma. So let us show this almost sure continuity first, then we will turn to the proof of (4.15). For η ∈ (0, 1), we denote by B η the subset of D defined by

B η := {f ∈ D|[1 -η; 1 + η] ∩ Im(f) = ∅}. (4.16)
We already pointed out that P(1 ∈ A α ) = 0, so that with probability one, (σ

(α) s ) s≥0 ∈ ∪ η>0 B η . We show that (F, G) restricted to ∪ η>0 B η is continuous.
Let f ∈ B η with η > 0, let us show that F is continuous at f . We introduce s f := sup{s ≥ 0|f (s) ≤ 1}. Let g ∈ D such that d(f, g) < µ/2 where µ ∈ (0, η), and in the same way we define s g := sup{s ≥ 0|g(s) ≤ 1}. We consider λ ∈ Λ such that ||fg • λ|| < µ and ||λ -id|| < µ. Furthermore, for all ξ > 0, and as f ∈ B η , we have f (s fξ) < 1η and f (s f ) > 1 + η. Then we get

g(λ(s f -ξ)) ≤ f (s f -ξ) + µ ≤ 1 -η + µ < 1, (4.17)
and similarly

g(λ(s f )) ≥ f (s f ) -µ > 1 + η -µ > 1, (4.18) which gives λ(s f -ξ) ≤ s g ≤ λ(s f ), (4.19)
and this entails the desired continuity since ξ is arbitrarily small, λ is continuous and ||λ -id|| < µ. Note that, by the virtue of the continuous mapping's theorem, this implies the convergence

sup s ≥ 0, τ Nγ N (s) N ≤ 1 ⇒ L (α) 1 . (4.20)
For the second component, it is very easy also; for g as above, we have: We now go to the proof of (4.15). Let δ > 0, there exists a compact set K δ ⊂ R + such that for all N ∈ N, we have: For N ∈ N, we introduce the event

ρ (Im(f), Im(g)) = max sup
max P |τ (N ) | b N ∈ K c δ , P sup s ≥ 0 τ Nγ N (s) N ≤ 1 ∈ K c δ < δ. ( 4
H N,δ := |τ (N ) | b N ∈ K δ , sup s ≥ 0 τ Nγ N (s) N ≤ 1 ∈ K δ , τ (N ) ∈ L δ . (4.24)
Eqs. (4.22) and (4.23) say that P H c N,δ < δ. Let M > 0 be an upper bound for G . For every N ∈ N and taking into account the fact that γ N ∼ b N (see [START_REF] Giacomin | Random Polymer Models[END_REF], (A.48)), we have:

E G sup s ≥ 0 τ Nγ N (s) N ≤ 1 , τ (N ) -G |τ (N ) | γ N , τ (N ) ≤ E G sup s ≥ 0 τ Nγ N (s) N ≤ 1 , τ (N ) -G |τ (N ) | γ N , τ (N ) 1I H N,δ + 2M δ ≤ E G sup s ≥ 0 τ Nγ N (s) N ≤ 1 , τ (N ) -G sup N γN (s) γ N -s + s τ Nγ N (s) N ≤ 1 , τ (N ) 1I H N,δ + 2M δ.
(4.25)

For a given κ > 0, we introduce the event J κ,δ defined by:

J κ,δ := sup s∈K δ N γN (s) γ N -s ≥ κ . (4.26)
Note that the process

Nγ N (s) γN -s s≥0
is a martingale, so that, applying Doob's inequality, we get:

P [J κ,δ ] = P   sup s∈K δ N γN (s) γ N -s 2 ≥ κ 2   ≤ 1 κ 2 sup s∈K δ E   N γN (s) γ N -s 2   = max K δ γ N κ 2 . (4.27)
Cutting once again the expectation appearing in Eq.(4.25) with respect to the fact that J κ,δ occurs or not, we get:

E G sup s ≥ 0 τ Nγ N (s) N ≤ 1 , τ (N ) -G |τ (N ) | γ N , τ (N ) ≤ E sup |u|≤κ G sup s ≥ 0 τ Nγ N (s) N ≤ 1 , τ (N ) -G u + sup s ≥ 0 τ Nγ N (s) N ≤ 1 , τ (N ) 1I H N,δ 1I J c κ,δ + 2M δ + max K δ γ N κ 2 .
(4.28)

In the last inequality, we consider first κ very small, which deals with the expectation term due to the uniform continuity of G on the compact set K δ × L δ . Then, we consider δ small enough, and this achieves the proof of (4.15), since γ N N →∞ → ∞. Thus the lemma is proved.

We are now ready to show the main part of our theorem.

Proof of Theorem 2. We have just shown the following convergence: under the law K(.), assuming K(∞) = 0, for all F continuous bounded function on F and for all ε ∈ R, we have: we have actually shown the first part of the claim. We also used the fact that Z N,βc = 1 for all N ∈ N see [START_REF] Giacomin | Random Polymer Models[END_REF], (2.17)).

We now deal finally with the case K(∞) > 0; we can write:

E N,βc+ε/ bN F (τ (N ) ) = Ẽ F (τ (N ) )e ε|τ (N ) |/ bN K(N (1max(τ (N ) ))) + K(∞) Z N,βc+ε/ bN j>N (1-max(τ (N ) )) K(j) .

(4.31) The justification for (4.31) can be found in Giacomin (2007), equality (2.48). Then, on the event max τ (N ) < 1η for a given η ∈ (0, 1), it is known that ((2.50) of Giacomin ( 2007

  Theorem 2.1. Let ε ∈ R. The sequence of measures (Q βc+εN -1/2 ,N ) N ≥0 converges weakly as N → ∞; the law of the limiting process is absolutely continuous with respect to (|B t |) t≥0 with Radon-Nykodym density given by e εL1 /E e εL1 .

  {τ Nγ N (s) /N, s ≥ 0} ⇒ A α as well as the convergence in law of the entire process τ Nγ N (t) . (1985)) in the Polish space D. Here, D denotes the space of càdlàg functions on [0, 1] endowed with the standart Skorokhod topology (that isd(f, g) := inf λ∈Λ {max(||λ -id||, ||f • λ -g||)} where Λ is the set of non-decreasing homeomorphisms from [0, 1] to [0, 1]) and of course ||f || = sup t∈[0,1] |f (t)| for f ∈ D.We define the function (F, G) :D → R × F by (F, G)(f ) : f → (sup {s ≥ 0|f (s) ≤ 1} , Im(f )),(4.14)

  λ as above and s ∈ R, we have inft∈R |f (s)g(t)| ≤ |f (s)g(λ(s))| ≤ µ,which achieves the proof of the continuity of the couple (F, G) on ∪ η>0 B η .

  .22) This is due to the fact that the sequences sup s ≥ 0 τ Nγ N (s) /N ≤ 1 they converge in law (see (4.20) and Lemma 1). Similarly, the convergence of the sequence (τ (N ) ) N ≥0 implies the existence of a compact set L δ ⊂ F such that for every N ∈ N, P τ (N ) ∈ L c δ < δ.(4.23)

E

  F (τ (N ) )e ε|τ (N ) |/bN → N →∞ E F (A α )e εL (αcase where ε > 0 actually uses the fact (see Lemma 2) that the sequence (e ε|τ (N ) |/bN ) N ≥0 is uniformly integrable.We then haveE F (τ (N ) ) = Z N,βc Z N,βc+ε/bN × 1 Z N,βc E F (τ (N ) )e ε|τ (N ) |/bN . (4.30)As in the case K(∞) = 0, P N,βc is actually P (becauseβ c =log(Σ K ) = 0)and thanks to the easy remark that Z N,βc+ε/bN = E e ε|τ (N ) |/bN → N →∞ E e εL (α) 1 ,

  )))K(N (1max(τ (N ) ))) + K(∞) Z N,βc j>N (1-max(τ (N ) )) K(j) ∼ N →∞ απ sin(απ) (1max(τ (N ) )) α ,(4.32) uniformly in the trajectories of τ . Using once again the convergence of the joint law (τ (N ) , |τ (N ) |/ bN ), where this time τ is distributed according to P, we get :E N,βc+ε bN F (τ (N ) )1I max τ (N ) <1-η -→ N →∞ E F (A α )e εL (α) 1 1I max Aα<1-η απ sin(απ) (1max(A α )) α . (4.33)The fact that max(A α ) < 1 almost surely allows us to take the above limit for η ց 0. Taking F ≡ 1 in (4.33), we getZ N,βc+ε/ bN Z max(A α )) α . (4.34)This achieves the proof because we can writeE N,βc+ε/ bN F (τ (N ) ) = Z N,βc Z N,βc+ε/ bN Ẽ F (τ (N ) )e ε|τ (N ) |/ bN K(N (1max(τ (N ) ))) + K(∞) Z N,βc j>N (1-max(τ (N ) )) K(j) , (4.35)and the term in the right hand side converges towards the desired quantity.

Acknowledgement I am very grateful to my Ph.D. supervisor Giambattista Giacomin for his help and suggestions. I would like to thank Francesco Caravenna also, in particular for the proof of Lemma 2.

I am also grateful to an anonymous referee for his remarks and comments.


Amine Bouabdallah
email: amine.bouabdallah@ensica.fr

Member, IEEE Michel Kieffer

Jérôme Lacan
email: jerome.lacan@ensica.fr

Galina Sabeva

Gaël Scot
email: gael.scot@cnes.fr

Caroline Bazile
email: caroline.bazile@cnes.fr

Fellow, IEEE . Pierre Duhamel
email: pierre.duhamel@lss.supelec.fr

Evaluation of Cross-Layer Reliability Mechanisms for Satellite Digital Multimedia Broadcast

Keywords: Broadcasting, Channel coding, Multimedia communication, Video coding

This paper presents a study of some reliability mechanisms which may be put at work in the context of Satellite Digital Multimedia Broadcasting (SDMB) to mobile devices such as handheld phones.

These mechanisms include error correcting codes, interleaving at the physical layer, erasure codes at intermediate layers and error concealment on the video decoder. The evaluation is made on a realistic satellite channel and takes into account practical constraints such as the maximum zapping time and the user mobility at several speeds. The evaluation is done by simulating different scenarii with complete protocol stacks. The simulations indicate that, under the assumptions taken here, the scenario using highly compressed video protected by erasure codes at intermediate layers seems to be the best solution on this kind of channel.

I. INTRODUCTION

There is currently a growing demand for good-quality and location-independent access to multimedia contents. Applications such as video on demand or TV on mobile devices such as handheld phones, PDA, or notebooks should develop widely in the near future. Solutions based on radio access networks (RAN) such as GPRS, UMTS, or CDMA-2000 are well-adapted for unicast communications but not viable for the long term, since such networks are not likely to support, e.g., massive consultations of several TV channels in parallel.

The third-generation partnership project (3GPP) is currently investigating Multimedia Broadcast/Multicast Services (MBMS) extensions to these 2G and 3G RAN [START_REF] Etsi | Multimedia Broadcast/Multicast service. (MBMS)[END_REF]. Nevertheless, MBMS seems currently still not suited for the broadcasting of many TV channels over a wide area.

Satellite Digital Multimedia Broadcasting (SDMB) represents an interesting proposal to respond to this demand, as well as some other alternatives such as Terrestrial DMB [START_REF]Digital audio broadcasting (DAB); DMB video service; user application specification[END_REF] and Digital Video Broadcasting-Handheld (DVB-H) [START_REF]Digital video broadcasting (DVB); transmission system for handheld terminals (DVB-h)[END_REF]. SDMB is especially interesting to ensure coverage of rural and small towns environments, for which it provides moderate deployment costs, when compared to T-DMB or DVB-H.

In this context, the SDMB project conducted by CNES, connected with the Unlimited Mobile TV project of Alcatel, aims at delivering live TV channels directly to mobile handheld phones through an hybrid infrastructure composed of a satellite offering a broad coverage of a large area and terrestrial repeaters for urban complementary coverage (see Figure 1). The objective is to broadcast QVGA or CIF encoded video with a typical throughput of about 250 kbps per channel.

Typical issues that have to be addressed in such SDMB application are in part similar to those encountered in any wireless multimedia transmission context, see [START_REF] Stockhammer | H.264/AVC in wireless environment[END_REF]. The characteristics of the wireless channel are location-dependent and thus time-varying when the receiver moves. The receiver has to comply with signals varying in a large dynamical range, from good conditions, when the satellite or a repeater is in Line Of Sight (LOS), to quasi-loss of the received signal, when deep satellite shadowing occurs out of repeater coverage [START_REF] Pérez Fontán | Statistical modelling of the lms channel[END_REF]. Automatic Repeat reQuest (ARQ) mechanisms are very difficult to implement in SDMB, mainly due to the potentially high number of receivers. The transmitter has thus to provide a signal which may be well decoded by the largest proportion of receivers. Transmission delays are less critical than in conversational October 27, 2006 DRAFT Fig. 1. Representation of the SDMB system applications. More important is the zapping time, i.e., the time necessary to switch between two channels, which has to remain as small as possible and is a specific constraint of DMB.

Redundancy introduction and error concealment are the tools of choice for addressing these issues. In [START_REF] Stockhammer | H.264/AVC in wireless environment[END_REF], several application-layer mechanisms, parts of the H.264/AVC standard [START_REF]Advanced video coding for generic audiovisual services[END_REF], are presented to provide efficient video transmission over wireless error-prone channels. In the context of SDMB, the satellite propagation channel is such that redundancy has also to be introduced at lower protocol layers in order to cope with fading and possible deep shadowing.

The optimization of the allocation of redundancy between physical, intermediate and application layers is then a natural question which arises in this context [START_REF] Van Der Schaar | Cross-layer wireless multimedia transmission: Challenges, principles, and new paradigms[END_REF]. Nevertheless, the absence of feedback channel in the context of SDMB imposes a worst-case optimization in the place of continuous adaptations suggested by [START_REF] Van Der Schaar | Cross-layer wireless multimedia transmission: Challenges, principles, and new paradigms[END_REF]. These questions have also been considered in [START_REF] Jenkac | Cross-layer assisted reliability design for wireless multimedia broadcast[END_REF] in the context of MBMS.

The aim of this paper is to present a solution for SDMB combining redundancy introduction at October 27, 2006 DRAFT the physical, network, and application layers. Redundancy introduction and error concealment at the application layer concur to mitigate the effects of transmission channel variations, allowing good video quality for a given bit-budget with a reasonable zapping time. Performance evaluations are done on a realistic satellite channel model. Some interactions between the different protocol layers will be illustrated, evidencing the need for a cross-layer optimisation of the robustness introduction. More sophisticated tools such as robust decoders taking into account the redundancy left within the encoded video stream [START_REF] Nguyen | Compressed image and video redundancy for joint source-channel decoding[END_REF], [START_REF] Sabeva | Robust decoding of h.264 encoded video transmitted over wireless channels[END_REF] will not be considered here.

Possible places where these techniques may be helpful are pointed out.

The channel model presented in Section II evidences the previously mentioned difficulties which have to be faced in the context of SDMB. The proposed protocol stack and system architecture are then detailed in Section III. The various techniques involved in the protocol stack to increase the robustness of the compressed video to variations of the channel characteristics are presented in Section IV. Section V provides some simulation results for the whole system, before drawing some conclusions.

II. CHANNEL MODEL

The satellite transmission channel model largely determines the design and dimensioning of the different layers of the protocol stack employed for SDMB. In this paper we focus on the case of satellite-only reception. This situation represents a worst-case study, as repeaters may reduce the occurrence of loss of signals.

The satellite link budget is the most important constraint in the design of a SDMB system.

It determines the received signal-to-noise ratio in LOS. Usually, shadowing and blocking phenomena on the satellite link have a low coherence distance. An accurate determination of this distance is useful for dimensioning interleavers at the physical layer or at intermediate layers.

Contrary to propagation channel of repeaters, the satellite channel is usually considered as flat in the signal bandwidth of 5 MHz. To cope with frequency selectivity of the transmission channel in repeaters areas, selection of adapted waveforms such as OFDM is necessary. On satellite link, different physical layer options are possible depending on the frequency band used. For a system privileging frequency efficiency by the use of the same frequency band for both satellite and repeater links, the same type of OFDM signal may be used. If the two links use separate bands, a better optimisation of satellite link is possible and the use of a more classical TDM signal can October 27, 2006 DRAFT be preferred for its better resistance to non-linear satellite amplification.

The propagation part of the channel model considered in the paper is a three-states model (LOS/Shadowing/Blockage) based on Markov chains integrating three scales of propagation effects [START_REF] Itu-R | Propagation data and prediction methods required for the design of Earth-space telecommunication systems[END_REF], [START_REF] Pérez Fontán | Statistical modelling of the lms channel[END_REF]. The large scale effects correspond to the changes between the three states. The mid scale effect or "Slow" fading corresponds to shadowing. The small scale effect or "Fast" fading is due to multipath. Based on extensive measurement campaigns, [START_REF] Pérez Fontán | Statistical modelling of the lms channel[END_REF] defines transition matrices between the three states for five environments (Open, Intermediate tree shadowed, Heavy tree shadowed, Suburban, and Urban), for elevation angles above 40°. The environment selected in this paper is the Intermediate tree shadowed, which is rather severe in terms of shadowing attenuations compared with the open and even with the Suburban cases. For each state, slow fading and fast fading variations are defined according to a Loo distribution [START_REF] Loo | A statistical model for land mobile satellite link[END_REF]. The Loo distribution considers that the received signal is made up of the sum of two components, the direct signal and multipaths. Within one state, the dynamics of the shadowing affecting the direct path is given by its correlation distance, equal in Intermediate tree shadowed environment to 1.5 m.

Moreover, a perfect demodulation is assumed. Simulations are done with a TDM signal, which is expected to be representative for both TDM and OFDM cases. Indeed, for OFDM signals, the symbol duration is equal to a few hundreds of micro-seconds, which corresponds to a rather stable propagation attenuation even for the highest speed considered. Noise level adjustment compared with LOS signal level is based on the assumption of a margin of 10 dB, which corresponds to a typical link budget, as detailed in [START_REF] Bazile | Status on satellite mobile tv air interface issues[END_REF].

The receiver carrier-to-noise ratio (C/N ) is simulated over a distance of 1 km, in order to limit the duration of the simulations performed in the upper layers. The simulator accounts for the modulation, the propagation part, and the demodulation. The considered simulation distance is split into 220000 segments of 0.004545 m each. The simulated data, represented in Figure 2, cannot be considered as perfectly representative for the complete environment but give a good idea of the channel behavior. The distance may then be covered at various speeds.

As can be seen on Figure 2, the received carrier-to-noise ratio ranges from approximatively

III. PROTOCOL STACK FOR SDMB

The considered protocol stack, represented on Figure 3, has been largely inspired from DVB-H [START_REF]Digital video broadcasting (DVB); transmission system for handheld terminals (DVB-h)[END_REF], in order to improve interoperability with this broadcast technique and minimize terminal chipset modifications. This choice is well-suited for a signal which may be received either directly from the satellite or from a repeater. However, modifications of the DVB-H stack are proposed in order to improve the robustness to fading and shadowing of SDMB.

The next sections detail the protocol layers of Figure 3. The reliability mechanisms which may be integrated on these layers are considered in Section IV.

A. Application layer

The considered video coder, H.264 [START_REF]Advanced video coding for generic audiovisual services[END_REF], consists of two parts. The Video Coding Layer (VCL) is responsible for the video compression. The Network Abstraction Layer (NAL) performs the interface between the VCL and the lower protocol layers. The NAL produces NAL Units (NALUs) which may be concatenated in order to form a bitstream suitable for storage or encapsulation into Real-time Transport Protocol (RTP) [START_REF] Schulzrinne | RTP: A Transport Protocol for Real-Time Applications[END_REF] packets according to the RTP payload specification for H.264 [START_REF] Wenger | RTP Payload Format for H.264 Video[END_REF].

The VCL divides each frame n of the video sequence into macroblocks of 16 × 16 pixels.

A group of macroblocks forms a slice of the frame. Each slice is encoded independently. For which would require a feedback channel not realistic for satellite broadcast.

Since SDMB considers unidirectional broadcast, the transport layer is not in charge of the reliability nor the congestion control and UDP [START_REF] Postel | User Datagram Protocol[END_REF] is used. The length of the UDP header is 8 bytes, including the sender and receiver port and a 16-bit checksum detecting errors on the whole packet.

The network protocol is IP, two versions of which, IPv4 [18] and IPv6 [START_REF] Deering | Internet Protocol[END_REF] may be used.

Their header are respectively 20 and 40 bytes long.

The RTP, UDP, and IP encapsulations add thus headers of 44 bytes (with IPv4) and 64 bytes (with IPv6) per NALU. As will be seen in Section IV-A.1, the desirable length of the NALU in the context of SDMB is between 100 and 400 bytes. The headers thus represent an extremely large proportion of the IP packets. For this reason, the use of header compression mechanisms is recommended. For the protocols used here, ROHC [START_REF] Bormann | RObust Header Compression (ROHC): Framework and Four Profiles: RTP, UDP, ESP, and Uncompressed[END_REF] is the best candidate since it is able to consider together the RTP, UDP and IP headers in the compression process. Initially designed for bi-directional links, ROHC integrates an unidirectional mode that may also be used in the present broadcasting context, as detailed in [START_REF] Couvreur | Performance analysis of a header compression protocol: The rohc unidirectional mode[END_REF].

In the simulations, we chose to implement ROHC instead of RTP, UDP and IP. Since it is very difficult to estimate the average length of the ROHC header in unidirectional links, we fixed the header length to 20 bytes (estimated from the results of [START_REF] Couvreur | Performance analysis of a header compression protocol: The rohc unidirectional mode[END_REF]). In practice, this header is directly added to the NALUs provided by the H.264 video codec. At the decoder side, this header is October 27, 2006 DRAFT simply erased.

2) Link layer: Since the MPEG2 Transport Stream (MPEG2 TS) packets [START_REF]Generic coding of moving pictures and associated audio information: Systems[END_REF] are the data units handled by most of the DVB physical layers, the adaptation layer must be used to map variable-length IP packets onto fixed-length MPEG2 packets. Even if the ULE protocol [START_REF] Fairhurst | Unidirectional Lightweight Encapsulation (ULE) for Transmission of IP Datagrams over an MPEG-2 Transport Stream (TS)[END_REF] may be used, DVB recommends the use of Multi-Protocol Encapsulation (MPE) [START_REF]DVB specification for data broadcasting[END_REF]. The header added by MPE to each IP packet is 12 bytes long, a 4 bytes long CRC is also appended. For transmissions toward handheld devices, two optional mechanisms are proposed: time-slicing and MPE-FEC [START_REF]DVB specification for data broadcasting[END_REF].

The concept of time slicing is to send data in bursts. Between each burst, data of the elementary stream are not transmitted, allowing other elementary streams to use the bandwidth otherwise allocated. This enables a receiver to stay active only for a fragment of the time, while receiving bursts of the service it requests. Time slicing thus limits the receiver power consumption. The structure of the MPE-FEC erasure code is described in Section IV-B.

The aim of MPEG2 is to multiplex several sources of compressed data. As already explained, the packets provided by MPE are organized into MPEG2 packets of 188 bytes. Among the 188 bytes, 4 are dedicated to the packet header.

In the simulations, MPE uses the section-packing algorithm to build MPEG2 packets, i.e., it allows MPEG2 packets to carry parts of several MPE packets. The optional time-slicing was not implemented since the power consumption is not one of the main objectives of this work.

The second optional mechanism, MPE-FEC, was not used either, because an erasure code at transport level was preferred (see the justification in Section IV-B).

C. Physical layer

The modulation and channel model used by SDMB has been described in Section II. Here, we focus on the construction of the transport stream and on the channel code of the physical layer.

In DVB-H, which uses the same channel coding as DVB-T [START_REF] Etsi | Digital video broadcasting (DVB); framing structure, channel coding and modulation for digital terrestrial television[END_REF], convolutional codes are concatenated with Reed-Solomon codes. Several solutions may be considered to improve the decoding efficiency and thus increase the throughput of the physical layer for a fixed link budget.

One may consider turbo codes, such as those proposed by the 3GPP for W-CDMA [START_REF]Universal mobile telecommunications system (UMTS); multiplexing and channel coding (FDD) (3GPP TS 25.212 version 6.7.0 release 6)[END_REF] or by the 3GPP2 for CDMA2000 [START_REF]Physical layer standard for cdma2000 spread spectrum systems[END_REF]. LDPC codes [START_REF] Gallager | Low Density Parity Check Codes[END_REF] such as those adopted in DVB S2 [START_REF] Etsi | Digital video broadcasting (DVB); second generation framing structure, channel coding and modulation systems October 27, 2006 DRAFT for broadcasting, interactive services, news gathering and other broadband satellite applications[END_REF] are October 27, 2006 DRAFT also a potential alternative. Moreover, the use of a long interleaving at physical layer may be considered to enhance the air interface performance and to mitigate satellite channel impairments.

The increase of this physical layer interleaving has to be kept compatible with zapping time constraints.

In this paper, the rate 1/3 turbo code specified by [START_REF]Universal mobile telecommunications system (UMTS); multiplexing and channel coding (FDD) (3GPP TS 25.212 version 6.7.0 release 6)[END_REF] is considered. It processes each 188-byte MPEG2 packet independently to generate a codeword of 564 bytes. The rate of the channel code is one of the constraints of the "SDMB" and "Unlimited Mobile TV" projects respectively conducted by CNES and Alcatel Alenia Space, and we did not consider other options. Nevertheless, the opportunity to use long interleavers at the physical layer will be discussed in Section V.

IV. INCREASING ROBUSTNESS

At receiver side, a noisy version of the transmitted bitstream is fed to the turbo decoder. The output of the turbo decoder is put into MPEG2 packets which are delivered to the upper protocol layers. In the proposed scheme, as Reed-Solomon codes concatenated with convolutional codes have been replaced by turbo codes, it is not easy to determine whether an MPEG2 packet is erroneous, see [START_REF] Matache | Stopping rules for turbo decoders[END_REF].

The MPE layer implements a CRC protecting the whole MPE sections to detect residual errors.

Any MPE section detected as erroneous is discarded. The error channel is then transformed into a packet erasure channel. The aim of this section is to describe several robustness tools to recover or conceal lost packets.

A. Application layer

At application layer, the redundancy introduction and error-concealment mechanisms put at work are the same as those used in wireless conversational applications. They are briefly recalled here. For more details, see [START_REF] Stockhammer | H.264/AVC in wireless environment[END_REF].

1) Optimizing the size of the NALU: H.264 does not allow to get NALUs, and thus RTP packets, with a constant size. Nevertheless, using slice-structured coding [START_REF]Advanced video coding for generic audiovisual services[END_REF], it is possible to limit the size of each NALU. This may be helpful for controlling the proportion of lost RTP packets due to erroneously received MPEG2 packets.

October 27, 2006 DRAFT

A lost MPEG2 packet may result in the loss of one to several NALUs, depending of the size of the NALUs. When large NALUs are considered, the loss of a small MPEG2 packet may result in the loss of a large amount of data. From a NALU-loss point of view, using small NALUs is thus recommended. However, as NALUs are encoded independently, the probability counts of the adaptive entropy coders of H.264 are reinitialized at the beginning of each NALU, which is detrimental for the coding efficiency. Moreover, small NALUs result in RTP packets with poor ratios payload over total packet size. The size limit of the NALUs has thus to be carefully optimized. Note that the problem of optimization of the application-layer packet size as a function of the channel condition has been previously considered in [START_REF] Van Der Schaar | Adaptive cross-layer protection strategies for robust scalable video transmission over 802.11 WLANs[END_REF] in the context of video transmission over 802.11 WLANs. Here, it will be studied as a function of the MPEG2 packet loss rate.

A sequence of NALUs is randomly generated according to a gaussian probability distribution with mean µ and variance σ 2 = 100, truncated at µ + 10 to account for the size limit imposed by H.264. As seen in Section III-B, the average length of the ROHC headers is estimated to 20 bytes. Additionally, the header and CRC of the MPE layer have a total length of 16 bytes.

Thus headers of 36 bytes are appended to the generated NALUs. The resulting packets are then mapped to MPEG2 packets. Packet loss rates of 5%, 10%, and 15% are considered. Figure 4 gives the goodput, i.e., the ratio between the number of correctly received information bits (headers and CRC have been removed) and the number of bits sent into MPEG2 packets as a function of the packet-size limit (µ + 10). Each point is simulated with bitstreams of 2.10 4 NALUs.

The optimal NALU size limit depends on the MPEG2 packet-loss rate. For 15% of lost packets, a NALU size limit of about 190 bytes provides the best goodput. For a loss rate of 10%, NALU size limits between 150 and 350 bytes perform almost similarly, whereas for 5% of lost packets, NALUs of more than 200 bytes may be used.

The previous study does not consider the loss in coding efficiency when the NALU gets smaller. To study the effect of size limits, the JM 10. I. Even for very small NALU (120 bytes), the redundancy is relatively small (10.6%). The conclusions obtained from Figure 4 are thus unchanged. Nevertheless, an optimisation of the NALU size limit taking into account the effect of concealment performed by the video decoder would provide more accurate results. 2) Error concealment: When a NALU has been lost, error concealment techniques may be used to limit the visual impact of the lost slices on the reconstructed video stream. Most concealment techniques rely on the hypothesis that the video sequence is spatially and temporally correlated. Good reviews of such techniques may be found in [START_REF] Wang | Error control and concealment for video communication: A review[END_REF], [START_REF] Wang | Review of error resilient coding techniques for real-time video communications[END_REF].

Two concealment techniques have been implemented in the JM 10.2 of H.264. First, all received NALUs corresponding to a given frame are decoded. All macroblocks that were not decoded due to losses of NALU are marked and error concealment is performed. A first concealment technique consists in estimating the pixels of a lost macroblock in such a way that they are as close as possible to the pixels of their neighboring macroblocs, as suggested by [START_REF] Zhu | Coding and cell loss recovery in DCT based packet video[END_REF]. An alternative approach, suited for inter-encoded frames consists in estimating the motion vector of the macroblock to conceal with the motion vectors of the surrounding macroblocks and to use the previously decoded frames to perform motion reconstruction, as suggested by [START_REF] Lam | Recovery of lost or erroneously received motion vectors[END_REF], [START_REF] Hong | Error concealment algorithms for compressed video[END_REF].

Both concealment techniques are satisfying when there are only few consecutive lost slices and more generally when the lost macroblocks are spread on the frame. Picking the macroblocks randomly to fill a slice, as allowed by the Flexible Macroblock Ordering option of H.264, may help to improve the concealment efficiency.

B. Intermediate layers: Erasure codes

On the packet erasure channel seen by the intermediate layers, erasure codes can be used at several levels. First, since MPE has defined his own MPE-FEC erasure code [START_REF]DVB specification for data broadcasting[END_REF], the link Longer codes, such as Raptor codes [START_REF] Luby | Raptor forward error correction scheme for object delivery[END_REF], LDGM codes [START_REF] Roca | Design and evaluation of a low density generator matrix (LDGM) large block FEC codec[END_REF], or simply longer Reed-Solomon erasure codes [START_REF] Lacan | Reed-Solomon Forward Error Correction (FEC)[END_REF], which are candidates for standardization by DVB and/or IETF organisms, can be used. They are used between application and transport layers and directly work on the October 27, 2006 DRAFT data units produced by the application layer.

Since our goal is to evaluate the potential of the different mechanisms, punctured Reed-Solomon erasure codes have been implemented between application and transport layers, which allows much more freedom on the length of the code. The parameter T block represents the time needed to transmit all the packets belonging to a given codeword and thus determines the length of the code. For example, H.264 encoding a video at 150 kbps with NALUs limited to 120 bytes, produces roughly 180 NALUs per second. For T block = 1, the erasure code builds then 120 redundant packets and the 180+120 = 300 resulting packets constitute a codeword, see Figure 5.

The length of each redundant packet is the size limit of the NALUs. The performance of the erasure code strongly depends on the parameter T block because a code with long codewords is less sensitive to bursts of lost packets. However, the choice of a high value of T block has several negative consequences. The main drawback is that the lost packets can be recovered only when the whole block is received. Therefore, the real-time constraints of the application must be taken into account to the choice of the value of T block . A strong constraint is the zapping time, which is between T block and 2T block . Long codes also need more CPU and memory resource to encode and decode the messages. However, it has been shown in [START_REF] Nokia | Simulation results for the performance and complexity of RS Codes for MBMS FEC, 3GPP TSG SA WG[END_REF] that October 27, 2006 DRAFT the CPU resource needed to decode the erasure code can be neglected compared to that used for video decoding, especially for erasure codes based on sparse matrices.

C. Physical layer: Interleaving

One of the main characteristics of the channel described in Section II is a strong spatial correlation (characterized by a correlation distance of 1.5 m). A direct consequence is that a moving receiver observes bursts of erroneous bits or of packet losses. A classical solution to cope with these bursts is to use interleaving or spreading techniques in order to distribute in time (and thus in space for a moving receiver) the erased packets or erroneous bits of the codewords.

The spreading consist in organizing the emission of a codeword in order to send its different parts in different time slots. In practice, spreading is often implemented through interleaving techniques which allow to send simultaneously several codewords in order to keep an equivalent throughput. Figure 6 illustrates the use of an interleaving of depth 2 (the spreading factor is also equal to 2). As shown in Figures 6 (b) and (c), the use of a spreading factor of 2 for a receiver spreading). Actually, it can be observed that two receivers observe bursts with similar parameters if they have the same product speed × spreading factor. This observation is confirmed by the simulation results presented below.

The spreading with possible interleaving may be implemented at several layers. First, the DVB-H physical layer can mimic UMTS by distributing the data of a codeword into different time slots. Related to the bursty nature of the channel, this solution may be a good candidate to improve the global reliability, provided that the interleaving level complies with the limit on the zapping time.

Interleaving may also be used at MPE layer with MPE-FEC which encodes the rows of a matrix whose columns are filled by consecutive IP packets. If the columns are chosen sufficiently large, the FEC codes encodes data belonging to non consecutive IP packets.

However, it is well known that long codes are more efficient in terms of correction capability than interleaved short codes [START_REF] Byers | A digital fountain approach to reliable distribution of bulk data[END_REF]. Thus, we choose to use long erasure codes between application and transport level instead of a short code and interleaving at MPE layer. Actually, the use of long codes, which combines a large number of packets (and thus packets spaced in the time), can be considered as a form of spreading.

V. SIMULATIONS

The protocol stack presented in Section III, serves as a reference for the simulations. The influence of the reliability mechanisms described in Section IV is evaluated with four scenarii.

The section starts with a description of the common parts for all scenarii: the video coder and decoder and the description of simulations on the physical and intermediate layers.

A. Conditions 1) Video coder and decoder:

For the simulations, the same video coder as in Section IV-A. is enabled. NALUs are limited to 120 bytes in most of the cases, which allow an efficient error concealment, for a satisfying goodput at the cost of a reasonable redundancy introduction, see Section IV-A.1. Error concealment is activated at the decoder for all scenarii.

At decoder output, to evaluate the image quality, the mean squared error σ 2 n between the luminance of the n-th original frame and that of the reconstructed frame is evaluated. The PSNR for the n-th frame is then given by PSNR n = 10 log 10 255 2 σ 2 n (in dB).

The average mean squared error σ 2 is evaluated for the N frames of the video sequence and the average PSNR is PSNR = 10 log 10 255 2 σ 2 (in dB).

2) Intermediate layers:

The CRC of the MPE packets is used to detect erroneous MPE packets which are consequences of noisy MPEG2 packets provided by the physical layer.

Table II shows the average bit error rates (BER), packet loss rates (PLR), and bit error rates in the lost packets (BERLP) as a function of the speed. Though the average output BER is lower for high speeds than for low speeds, the number of erroneous blocks after processing by MPE is larger for high speeds than for low speeds. For low speeds, each erroneous packet contains more errors than for high speeds. This can be explained by remembering that an increase of the speed plays a role similar to an increase of the interleaving depth, see Section IV-C. Increasing the speed improves the performance of the turbo decoder in terms of bit-error probability. Nevertheless, as long as the spreading is not efficient enough to obtain a vanishing error probability at the output of the turbo decoder, bit errors will remain present and will be distributed in more packets than October 27, 2006 DRAFT without spreading, see [START_REF] Zorzi | On the statistics of block errors in bursty channels[END_REF] for more details. A further analysis of the influence of speed and interleaving is provided in Section V-C.

3) Common physical layer: Simulations for the modulation, the propagation part, and the demodulation are described in Section II. The bits forming the MPEG2 packets are encoded using the turbo code described in Section III-C. Each pair of encoded bits is then mapped to a QPSK symbol. The variance of the noise corrupting this symbol is adjusted using the C/N of the location this symbol has been received. Remember that the C/N has been evaluated over a distance of 1 km with a spatial resolution of 0.004545 m. Assuming a bitrate of 250 kbps at the output of the application layer and a header overhead of 20%, the number of bits received in each 0.004545 m-long space interval depends of the constant speed v of the receiver (1.5 m/s, The bit error rate (BER) at the input and output of the turbo decoder have been represented on Figure 7. For all speeds, results have been averaged over 1 s. There is a high variation of the BER with the location of the receiver. The C/N varies by several dB between two points distant of few centimeters. The consequence is that the input BER for each turbo codeword is highly variable. It follows that, though the average input BER (about 6.3 10 -2 ) is acceptable for the considered turbo code, the average output BER is relatively high (see Table II). As a result, many MPEG2 packets will be received without error and some will contain many errors. This illustrates that when the channel has a C/N above 5 dB, the received video is of good quality (PSNR larger than 32 dB).

15
C N / (dB) PSNR (dB)
Average PSNR = 27.28 dB In order to reduce the proportion of low quality frames, several solutions are investigated in the next sections. In the second scenario, presented in Section V-C, some spreading at the physical layer is introduced, in order to improve the efficiency of the turbo code. Next, one may think allowing some erroneously decoded packets to reach the upper layers, as suggested e.g. by [START_REF] Masala | MAC-level partial checksum for H.264 video transmission over 802.11 ad hoc wireless networks[END_REF] or UDP-lite, recently standardized [START_REF] Larzon | The Lightweight User Datagram Protocol (UDP-Lite)[END_REF], which modify the areas protected by the checksum to allow erroneous data to reach the application layer. Nevertheless, this option will not be presented October 27, 2006 DRAFT here: experimentations have shown that the reconstructed video quality is not acceptable, the current implementations of the H.264 decoder being particularly prone to transmission errors.

Robust video decoders taking into account source constraints have to be employed in this context [START_REF] Sabeva | Robust decoding of h.264 encoded video transmitted over wireless channels[END_REF]. The second scenario, described in Section V-D, thus considers a packet-erasure code, such as that detailed in Section IV-B, to recover lost RTP packets. Section V-E illustrates a facet of the various cross-layer optimisation solutions for the redundancy introduction in the protocol stack and focuses on the distribution of redundancy between the intermediate and the application layers.

C. Scenario 2: Spreading at physical layer

The high potential of the spreading at physical layer is evident when considering the errorcorrection capabilities of the turbo-code. Indeed, with an average input BER of about 6.3×10 -2 , a rate 1/3 turbo code should perform well. If the bursts of errors evidenced in Scenario 1 can be transformed into quasi-independent errors, the turbo code will produce quasi error-free data.

It has been shown in Section IV-C that the diversity of the data in the space, when the receiver moves, is function of the product spreading × speed. Clearly, the speed is not a parameter that can be modified by the system. Additionally, the results provided in Section V-B show that the maximum speed (31 m/s) is not sufficient to ensure a satisfying level of diversity.

It follows that the only way to improve the diversity is to spread data at the physical layer. This is done by transmitting the data of a turbo codeword in different time slots, and possibly by interleaving several codewords. This solution is used, e.g., by the UMTS physical layer.

This technique has been implemented in the simulator by spreading the data belonging to a codeword over a duration larger than its natural duration. For example, for the video rate R = 250 kbps, each turbo codeword (of 3 × 188 bytes) is sent in roughly 4.7 ms. Thus, a spreading of the codewords over a duration of T s = 100 ms is done by interleaving 21 codewords.

The influence of the spreading duration T s on the PSNR is provided in Figure 10. In order to compare the received video quality at various speeds, the PSNR has been represented as a function of the product T s v for v = 1.5 m/s and v = 15 m/s.

These results are compatible with the analysis of the interleaving performance provided in [START_REF] Zorzi | On the statistics of block errors in bursty channels[END_REF],

which shows that after a given threshold for the interleaving time, the maximum performance of the coding system is reached. From Figure 10, one may observe that for T s v 9 m, it is The average PSNR and corresponding PLR for all combinations of T block and v are given in Table IV. The characteristics of the code employed are also indicated. The reference case (without erasure code) is the first column. The erasure code clearly improves the performance of the system. Indeed, for all speeds, the performance with a video rate of 250 kbps is lower than that for a video rate of 150 kbps and an erasure code of rate 3/5. Additionally, except for the speed equal to 1.5 m/s, the quality of the video is significantly improved (about 2.5 dB for v = 15 m/s) each time the value of T block is increased. This is confirmed by Figure 12 showing the cumulative distribution of the frame PSNR for various T block . For T block = 1 s, about 30% of the frames have a PSNR lower than 30 dB; for T block = 2 s and T block = 3 s, this proportion reduces to 17% and 8% respectively. The better performance of the erasure code in this scenario may be explained by the bursty erasure patterns of the channel. Indeed, concealment techniques are efficient in the presence of randomly spaced erasures and much less in the presence of bursts of erasures. On the contrary, an erasure code is able to correct bursts of erasures provided that the burst length is lower than its correction capability.

The mean burst length explains the relatively poor performance of the erasure code for v = 1.5 m/s. Indeed, as explained in Section II, the channel considered in our simulations shows areas with different levels of reception. A mobile crossing an area with a bad reception level at First, it was shown that the spreading/interleaving of the data on the physical layer obtained definitely the best performance in terms of packet loss rate. Provided that the turbo codewords are spread over a distance of more than 9 m, the turbo code obtains its best performance by generating a packet loss rate lower than 1%. Nevertheless, for a pedestrian moving at 1.5 m/s, this corresponds to the spreading of a codeword over a duration of 6 s, which results in an unacceptable average zapping time of 9 s. Another limitation of this solution is the increase of the hardware complexity. Indeed, the receiver has to store and process an amount of data corresponding to the spreading duration.

Second, it has been mentioned that allowing erroneous data to reach the application layer is not a viable solution. Indeed, even if the recent video standards such H.264 are able to cope with packets corrupted with few errors, the satellite channel is too bursty and the channel code for each speed and for each length of codes. Their main limitation is the zapping constraint which indirectly limits their length and thus their performance in terms of correction capability.

However, this mechanism offers several advantages. Indeed, their software implementation on higher layers allows a global flexibility (easy updates, reception of the data for terminals which do not implement the mechanism, easy activation/deactivation...) which can be of great interest in the SDMB system.

Extensions of this work include more cross-layer interactions, in order to provide richer information from the physical layer to the upper layer. This may allow to use error-correcting codes at intermediate layers in place of erasure codes. Soft decoding of the video taking into account the redundancy left by the video coder, as illustrated in [START_REF] Sabeva | Robust decoding of h.264 encoded video transmitted over wireless channels[END_REF], may also significantly increase the performance. Finally, the use of the scalable extension of H.264 [START_REF] Schwartz | Scalable extension of h.264/AVC[END_REF], in order to be able to broadcast video to receivers with various display and computing capabilities, using a single video stream may also be interesting.

This paper has studied a set of solutions to improve the reliability in satellite-mobile transmissions. These techniques were evaluated and their strengths, the drawbacks and their relations with the constraints of the system were presented. It seems difficult to advocate a technique related to the others since their use strongly depends on the choices made on the system at the different layers. However, the results presented in this paper should be useful for the designers of such systems.

Fig. 2 .

 2 Fig.2. Evolution over 1 km of the receiver signal-to-noise ratio for the Intermediate tree shadowed scenario

Fig. 4 .

 4 Fig. 4. Proportion of transmitted information bits for various NALU size limit

  layer could be the right place to implement it. MPE-FEC is a Reed-Solomon erasure code of dimension 191 and length 255. Its input is a set of 191 columns of fixed-length obtained by segmenting the stream of the IP packets. It produces 255 -191 = 64 redundant columns that are considered as MPE packets. The number of rows of the matrix is a parameter determining the level of interleaving which is adapted to the context. It should be noted that the parameters of the erasure code can be modified by shortening and puncturing operations. The main drawback of this code is the limited maximum length. Indeed, the more the erasure code combines packets, the more efficient it is.

Fig. 5 .

 5 Fig. 5. Construction of the redundant NALUs with a punctured Reed-Solomon code

Fig. 6 .

 6 Fig. 6. Relation between (a) sequential transmission with mobile at speed v, (b) interleaved transmission with mobile at speed v and (c) sequential transmission with mobile with speed 2 * v

Figure 7 Fig. 7 .

 77 Figure 7 also shows that for high speeds, a smoother channel is seen by the receiver. This effect is similar to that obtained with the use of interleavers at the physical layer.

Figure 8 Fig. 8 .

 88 Figure 8 presents the PSNR evaluated for the luminance of each received video frame. There is always a degradation of the PSNR when long shadowing occurs. Moreover, since there are

Figure 9

 9 Figure 9 represents the PSNR of the luminance of each decoded video frame as a function of the average C/N observed during the transmission of its corresponding compressed bistream.

Fig. 9 .

 9 Fig. 9. PSNR of the reconstructed image as a function of the average C/N observed during its transmission for v = 1.5 m/s

Fig. 10 .

 10 Fig. 10. Influence of the product Tsv on the average PSNR of the reconstructed video sequence (R = 250 kbps)

Figure 11

 11 Figure 11 shows the influence of the erasure code by comparing the evolution of the PSNR for T block = 1, 2 and 3 s for v = 15 m/s to the case without erasure code but with a video with R = 250 kbps.

Fig. 11 .

 11 Fig. 11. PSNR for v = 15 m/s and for (a) Tblock = 0 s (no erasure code, but with a video rate at 250 kbps), (b) Tblock = 1 s, (c) Tblock = 2 s and (d) Tblock = 3 s,

Fig. 12 .

 12 Fig. 12. Cumulative distribution of the frame PSNR for various Tblock, v = 15 m/s

  generates a level of bit errors at the output of the physical layer that are not supported by the video decoder. Whether robust decoding of video can play a useful role in this situation is still an open question. This conclusion holds temporarily with the actual JM 10.2 decoder.Third, Scenarii 1, 3, and 4 have shown that reducing the nominal video quality to allow some redundancy introduction at lower protocol layers is necessary to achieve satisfying performance.The error concealment mechanisms of H.264 have a performance limit, and may only account for a moderate proportion of lost packets. Erasure codes, evaluated in Scenarii 3 and 4, obtained good performance in all the simulated cases. Even if they have some drawbacks, such as the increase of the CPU and storage resources, they improved the final PSNR of the videoOctober 27, 2006 DRAFT

  Even if it is possible to encapsulate several NALUs in a single RTP packet, in the context of this work where relatively high packet loss rate may be encountered, the

	B. Intermediate Layers
	The convergence of the different network technologies in the next few years is now established.
	For this reason, the main next generation networks are designed to transport IP data. In this
	context, the DVB consortium has introduced the IP Datacast architecture [16] defining the
	protocol stacks to transmit IP data over DVB-H. The proposed Transport and Network layers
	comply with this architecture.
	1) Transport and Network layers: As seen in Section III-A, RTP packets are provided by
	the NAL of H.264.
	Fig. 3. Considered protocol stack
	original macroblock M n,m . The result is referred to as a difference macroblock D n,m and is
	passed through a video processing module, performing discrete cosine transform, quantization
	and finally entropy coding. There are two types of prediction: intra and inter. For intra prediction,
	the encoder uses only information from the current slice. For inter prediction, data from the other
	encoded frames may also be used. Inter-encoded frames may be of types P and B. For P frames,
	only the previously encoded frames are used for prediction, whereas for B frames, bidirectional
	prediction is performed. Usually, B frames do not serve as reference. Within P or B frames,
	macroblocks may be encoded in intra mode.
	The most common NALU organisation is to place one slice per NALU. Loosing a NALU,
	except when it corresponds to a slice of a B frame results in errors in the corresponding frame,
	but also in the frames which use it as reference for prediction. The probability of loosing a
	NALU mainly depends of the size of the NALU and of the packets provided by the lower
	protocol layers. The optimization of the size of the NALU will be discussed in Section IV-A.
	October 27, 2006 October 27, 2006	DRAFT DRAFT

the m-th macroblock M n,m of a slice, a prediction P n,m is evaluated and subtracted from the encapsulation of a single NALU per RTP packet is more efficient. The main function of RTP is to add a timestamp, a sequence number and to identify the payload type. The length of the RTP header (with only fixed fields) is 16 bytes. Note that SDMB does not consider RTCP

[START_REF] Schulzrinne | RTP: A Transport Protocol for Real-Time Applications[END_REF]

,

TABLE I REDUNDANCY

 I INTRODUCED WHEN LIMITING THE SIZE OF NALUS (FIRST 197 FRAMES OF FOREMAN.CIF)

  m/s, and 31 m/s) and is given in TableIII.

	Speed (m/s)	1.5	15	31
	Number of received bits 2727 273 132

TABLE III NUMBER

 III OF RECEIVED BITS IN AN SPACE INTERVAL OF 0.004545

M

At low speed, several MPEG2 packets are received in the same space interval. When the corresponding C/N is bad, several MPEG2 packets may thus be lost.

TABLE IV AVERAGE

 IV PSNR (PLR) FOR v = 1.5, v = 15 AND v = 31 M/S AND FOR TBLOCK = 0 (NO FEC), 1, 2 AND 3 SECONDS.

  October 27, 2006 DRAFT crossing this area at 15 m/s. The simulations shows that bursts of erasures experimented by a receiver moving at 1.5 m/s are too long for the implemented erasure codes.Even if the spreading at physical layer outperforms the solution proposed in this section for v = 15 m/s and v = 31 m/s, introducing redundancy at the intermediate layers turns out to provide more robustness at v = 1.5 m/s, while keeping an acceptable quality at higher speeds.Nevertheless, the required zapping times, 1.5T block in average, between 1.5 s and 4.5 s are still large. Here again, robust source decoders may allow the use of a shorter T block at low speeds.Section V-D shows that using erasure codes at intermediate layer significantly improves the received video quality. Further improvements may be obtained by optimizing the redundancy introduced at intermediate layer for a given T block . Increasing the redundancy improves robustness,

	E. Scenario 4: Cross-layer redundancy optimization
				Tblock (s)
	R	0	1	2	3	Noise-free
	250 kbps 24.5				36.8
	200 kbps 24.6	26.2	26.5	27.5	35.8
			RS(261, 157) RS(521, 313) RS(782, 469)
	150 kbps 23.9	28.2	30.1	30.8	34.4
			RS(196, 118) RS(391, 235) RS(586, 252)
	100 kbps 23.1	28.4	30.1	30.1	32.3
			RS(131, 79)	RS(261, 157) RS(391, 233)

1.5 m/s observes a burst erasure length equal to 10 times the burst erasure length of a mobile but requires a reduction of the rate R at the output of the video coder, leading to a reduction of the maximum expectable video quality.

New simulations have been performed with erasure codes with rates 2/5, 3/5, and 4/5. The output rate R of the video coder has been adjusted accordingly, in order to keep the rate at the output of the packet erasure coder at 250 kbps. In this scenario, the size of the NALUs has been limited to 160 bytes.

TABLE V AVERAGE

 V PSNR (IN DB) FOR v = 15 M/S AND FOR VARIOUS COMBINATIONS OF THE VIDEO CODING RATE R AND OF TBLOCK.

	October 27, 2006	DRAFT

Table V

 V summarizes the results obtained for v = 15 m/s. For small values of T block , increasing the redundancy improves the reconstructed video quality. For larger values of T block , it is better to use less redundancy to get an increased video quality. A trade-off between redundancy introduction and compression has thus to be attained and this trade-off depends on the value T block . At fixed redundancy, only an increase of T block allows an increase of the nominal video quality.VI. CONCLUSIONThis paper compares several reliability mechanisms built around a protocol stack for a reliable satellite digital mobile video broadcast application. Several conclusions may be drawn from this study.

October 27, 2006 DRAFT

This work was partly supported by the CNES SDMB project.