V Mansuy-Schlick

R Delage-Mourroux

M Jouvenot

W Boireau
email: wboireau@lpmo.edu

Strategy of macromolecular grafting onto a gold substrate dedicated to proteinprotein interaction measurements

Keywords: Self-assembled-monolayer (SAM), DPDPB, Protein immobilization, Surface Plasmon Resonance (SPR), GEC1, tubulin

Many biotechnology applications use proteins immobilized on surface. For biosensor, the sensing layer is a key component interfacing the transducer and the sample. Strategies employed to activate the bidimensional surface act directly on the performance of the biosensor. In this paper we propose a novel strategy for engineered proteins self-assembly.

Our original supramolecular structure allows a direct and fast covalent attachment of proteins onto bare gold substrate through a homobifunctional crosslinker, 1,4-Di-[2'pyridyldithio)propionamido]butane (DPDPB). In this work, engineered proteins and linkerprotein complexes were synthesized and characterized by gel electrophoresis, chromatography and spectroscopy experiments. Macromolecular construction "DPDPB-GST tag-GEC1 protein" was conceived in order to guarantee a 2D architecture enhancing the capabilities of the target (tubulin) to recognize its partner (GEC1). Surface plasmon resonance measurements clearly showed potential of this particular self-assembled protein layer compared to a commercial immunosensor interface. At the concentrations tested, the recognition process occurs between tubulin and the immobilized GEC1; moreover enhanced binding was obtained with the home-made 2D sensing layer more than with 3D carboxymethyl dextran matrix.

1) Introduction

In the biosensor field, protein immobilization is a crucial point conditioning the properties of specificity, stability and usability. From the point of view of protein-protein interactions and kinetic measurements, immobilization induces heterogeneous binding characteristics compared to bio-recognitions in solution. Thus, strategies employed require high level of control of the macromolecular architecture in order to prevent steric hindrance and constraints of mass transfer. Most of strategies of functionalization deal with polymer deposition onto the biosensor surface as carboxymethyl dextran. Chemical modifications of this 3D matrix are well known and have been used in many cases in order to immobilize receptors, enzymes, and antibodies through their amino or sulfhydryl residues [START_REF] Sonezaki | Analysis of the interaction between monoclonal antibodies and human hemoglobin (native and cross-linked) using a surface plasmon resonance (SPR) biosensor[END_REF]. However, these techniques have some limitations in exposing ligands far from the surface, since they introduce diffusion restrictions inside the coating and deviations of the kinetic models [START_REF] Sadana | A kinetic study of analyte-receptor binding and dissociation, and dissociation alone, for biosensor applications: a fractal analysis[END_REF][START_REF] Yarmush | An analysis of transport resistances in the operation of BIAcore; implications for kinetic studies of biospecific interactions[END_REF]. At present, self assembled monolayers (SAMs) of macromolecular species on inorganic surfaces are used in extending domains especially in protein chips [START_REF] Park | Operational characteristics of an antibody-immobilized QCM system detecting Salmonella spp[END_REF]. Bi-dimensional grafting is a promising way to get a specific and sensitive sensor at the condition that highly controls of supramolecular constructions are performed (Boireau et al., 2005a;[START_REF] Nieba | BIACORE analysis of histidinetagged proteins using a chelating NTA sensor chip[END_REF][START_REF] Shriver-Lake | Antibody immobilization using heterobifunctional crosslinkers[END_REF]; SAM approach can bring enhanced performance in selectivity compared to commercial hydrogel biochip [START_REF] Frederix | Enhanced Performance of an Affinity Biosensor Interface Based on Mixed Self-Assembled Monolayers of Thiols on Gold[END_REF].

Many commercial bi-functional linkers are available and are normally used for linking two different macromolecules. For example, 1,4-Di-[2'-pyridyldithio)propionamido]butane (DPDPB) is a homo-bifunctional crosslinker that reacts specifically with sulfhydryl groups. It is commonly employed to produce protein-protein cross bridges or to confer disulfide links with modified nucleic acids (Boireau et al., 2005b). As previously demonstrated with other cross linkers, the intrinsic disulfide links allows chemisorption of the sulfhydryl terminus onto a metallic template leading to a highly controlled monolayer of macromolecular species [START_REF] Park | Thiolated Salmonella antibody immobilization onto the gold surface of piezoelectric quartz crystal[END_REF]. In our laboratory, we have recently identified a novel gene called GEC1 [START_REF] Vernier-Magnin | A novel early estrogen-regulated gene gec1 encodes a protein related to GABARAP[END_REF]. GEC1 has been shown to associate with tubulin and microtubules and may be involved in GABA A receptor trafficking in synapses along the cytoskeleton [START_REF] Mansuy | GEC1, a protein related to GABARAP, interacts with tubulin and GABA A receptor[END_REF][START_REF] Wang | Binding of the GABA A receptor-associated protein (GABARAP) to microtubules and microfilaments suggests involvement of the cytoskeleton in GABARAPGABA A receptor interaction[END_REF][START_REF] Wang | GABA A -receptor-associated protein links GABA(A) receptors and the cytoskeleton[END_REF]. The GABA A receptor is the main receptor in the central nervous system and is a target for drugs like Valium and Xanax, frequently used for anxiety, mood alteration, sleep, or epilepsy [START_REF] Kennedy | Sticking together[END_REF]. Thus, it is very important to establish whether this protein play a role in the transport processes. To demonstrate the interaction between GEC1 and tubulin, GST pull down and traditional BIAcore assays have been performed using recombinant purified GST-GEC1 and tubulin purified from microtubule-associated protein [START_REF] Mansuy | GEC1, a protein related to GABARAP, interacts with tubulin and GABA A receptor[END_REF].

In this study, we present a new method of protein immobilization leading to selfassembled protein monolayer reconstitution onto a gold surface in order to characterize of GEC1-tubulin interactions by surface plasmon resonance.

2) Material and methods

Construction of the GST fusion expression vector: GST-GEC1

Plasmids encoding glutathione S-transferase (GST)-GEC1 was generated by PCR using the human full-length pGEMT-GEC1 as a template. Reactions were performed using Taq polymerase from Life Technologies Inc. (Cergy Pontoise, France) according to the manufacturer's recommendations.

The following PCR primers were used (f = forward primer, r = reverse primer): GEC1 (1)f: 5'-CGGAATTCCGATGAAGTTCCAGTACAAGGAG-3'; GEC1(117)r: 5'-CCGCTCGAGCGGTCATTTCCCATAGACACTCTC-3'.

PCR fragments were purified, digested with EcoRI and XhoI, and cloned into the EcoRI/XhoI sites of pGEX-4T-2 (Amersham Biosciences, Orsay, France) in order to generate the pGEX-4T-2-GEC1 plasmid. The nucleotide sequence of this plasmid was confirmed by sequencing.

Protein expression and purification

The full-length gec1 was cloned in pGEX-4T-2 (Pharmacia) and expressed as a fusion protein with the glutathione S-transferase (GST) moiety on the N-terminus. Escherichia coli strain DH5α or BL21 containing the expression plasmid were grown to OD 600 nm of 0.8, induced by 0.8 mM isopropyl D-thiogalactopyranoside for 2h30', homogenized in phosphatebuffered saline (137 mM NaCl, 2.7 mM KCl, 10 mM NaHPO 4 and 1.7 mM KH 2 PO 4 , pH 7.4) with 1% Triton X-100 and protease inhibitors, and loaded onto glutathione sepharose (Amersham). The GST fusion proteins were eluted by 10 mM glutathione/ 50 mM Tris, pH 8.0. Protein concentration was determined by the Lowry method [START_REF] Lowry | Protein measurement with the Folin phenol reagent[END_REF]. The purified proteins were analyzed by sodium dodecyl sulfate (SDS)-polyacrylamide gel electrophoresis (PAGE).

GST fusion protein modification procedure

The protein GST-GEC1 and GST in phosphate buffer saline were treated for 15 minutes with dithiothreitol (DTT) 10-fold molar excess at 0.1 M to eliminate GST dimerized forms. DTT was separated using a gel filtration column filled with Sephadex G25 (Sigma Aldrich). Then, the fusion proteins were treated with a 100-fold molar excess of DPDPB (1,4-Di-[2'-pyridyldithio]propionamido)butane which is used as a homobifunctional reagent for 3 hours at 25°C. Unreacted DPDPB was separated using a gel filtration column filled with Sephadex G25. To confirm that unreacted DPDPB was well separated from the GST-fusion protein -DPDPB complex, excess of DTT (up to 20:1 mole) was incubated 5 minutes with the sample. DTT breaks the disulfide bond within the fusion protein-DPDPB complex and releases thiopyridine that is absorbed at 344 nm. Protein separation and percentage of modified proteins were monitored with a λ900 spectrophotometer (Perkin Elmer) by absorbance change at 344 nm. Indeed, bound DPDPB species were determined with the following formula:

SPR experiments

BIAcore experiments were performed with Biacore ® 3000 apparatus. All of the experiments were carried out at 25°C in Hepes Buffer Saline or HBS (10 mM HEPES, 0,15M NaCl, 50µM EDTA, 0,005% P20, pH 7.4) with a flow rate between 5 and 30µl/min. All the samples were prepared and diluted with HBS buffer. Two different kits were used: CM5 (carboxymethyl dextran matrix) and SIA-kit (bare gold substrates). The immobilization degree of proteins and the level of interactions in Biacore SPR technology are reported in a sensorgram (response unit (RU) versus time (in seconds)). 1000 RU correspond to a shift in resonance angle of 0.1 degree. Calibration of the apparatus gives a correlation between the shift in angle and the mass deposition on the surface of the biochip of 0.1° ~ 1 ng/mm 2 [START_REF] Stenberg | Quantitative determination of surface concentration of protein with surface plasmon resonance using radiolabeled proteins[END_REF].

Protein immobilization on carboxymethyl dextran surface

The immunosensor on the CM5 sensor chip was prepared as previously described [START_REF] Mansuy | GEC1, a protein related to GABARAP, interacts with tubulin and GABA A receptor[END_REF]. Briefly, anti-GST antibody was immobilized using an amine coupling kit, as described by the manufacturer. The purified protein GST or GST-GEC1 at 1 µM was then immobilized using antibody-antigen interactions. Surfaces with approximately 2500 RU of immobilized proteins were used for the interaction analysis.

Immobilization of fusion protein -DPDPB complex on bare gold surface

The gold surface was cleaned with octyl glucopyranoside (OG), 20 mM at 20 µl/min for 1 minute. Fusion Protein-DPDPB complexes were injected and immobilized by DPDPB-Au autoassembly. The injection was performed at 1 µM with a constant flow of buffer at 20 µl/minute for 17 minutes. The immobilized complex was washed with OG for an additional 1 minute.

Interaction assays on the BIACORE and data analysis

Each binding cycle was performed with a constant flow of buffer at 20 µl/min. Purified tubulin (Cytoskeleton ® ) was injected across the surface at 800 nM for 4 minutes.

Once the injection plug had passed the surface, the formed complex DPDPB-fusion GST GEC1 -tubulin was washed with running buffer for an additional 1000 seconds. Data were collected and analyzed using the BIA-Evaluation program 3.0 (BIAcore ® ).

Results and discussion

Expression and purification of GEC1

To perform BIAcore experiments, GST-GEC1 fusion protein and GST were expressed in E. coli and were purified by standard glutathion sepharose affinity chromatography.

Coomassie blue staining of SDS-PAGE showed that the purified GST-GEC1 and GST proteins migrated as a single band at approximately 40 kDa and 28 kDa respectively with no detectable contaminating bands (Figure 1).

Self-assembly of modified GEC1 onto gold sensor chip

GST-DPDPB and GST-GEC1-DPDPB couplings

We developed an original way of placing the protein assembly directly onto the gold layer of the sensor chip. This approach has been compared with a classical way of biosensor building based on antigen-antibody recognition.

The GST portion of the fusion protein provides four cysteine residues not present in the primary sequence of GEC1 (Figure 2). Sulfhydryl compounds are used as anchor sites in various supramolecular assemblies whether directly through disulfide bridges or through different cross linkers. 1, 4-Di-[2'-pyridyldithio) propionamido] butane (DPDPB) is a homobifunctional crosslinker that reacts specifically with sulfhydryl groups.

The coupling experiment is described below and illustrated in figure 3. In this experiment, 1.5 nmoles of GST fusion protein in phosphate buffer saline were treated for 15 minutes with a 10-fold molar excess of dithiothreitol (DTT) at 0.1 M to eliminate dimerized form of GST. DTT was separated using a gel filtration column filled with Sephadex G25.

Extemporary, the protein was treated with a 100-fold molar excess of DPDPB 3h at 25°C.

Unreacted DPDPB was eliminated by gel filtration chromatography using a column filled with Sephadex G25. Absorbance data indicate that 1.2 nmoles of GST-DPDPB were obtained after the second gel filtration (data not shown). To determine the yield of GST-DPDPB coupling complex, an excess of DTT (20/1, 60/1, 80/1 by mole) was introduced sequentially to 550 µl of the 750 µl total sample. Protein separation and percentage of modified proteins were monitored by absorbance change at 344 nm. Values were obtained by subtracting spectrum curves with reference (without DTT) leading to a shift in the optical density at 344nm (ΔOD 344nm ). After incubation 5 min with a 20-fold molar excess of DTT, a ΔOD 344nm of 0.074 was obtained (data not shown). The ΔOD 344nm reached 0.012 and stabilized after adding 60-fold or 80-fold molar excess of DTT (Figure 3). These data indicate that 1.1 nmol of the 1.2 nmol of GST was modified. Therefore, the increase of absorbance at 344 nm indicated that 92% of GST is coupled to DPDPB. In four experiments with GST or GST-GEC1, the rate of coupling was more than 80 % and macroscopically closer to an equimolecular coupling.

Supramolecular self-assemblies onto gold chip

Most of the modified proteins presented a 2-pyridyl-disulfide group that could be used directly as chemical anchor onto the gold substrate. It has been previously demonstrated that it is not necessary to disrupt the disulfide bonds with a reducing agent to get self-assembly processes onto gold substrate [START_REF] Lee | Nanoscale fabrication of protein A on selfassembled monolayer and its application to surface plasmon resonance immunosensor[END_REF]. In a novel process, we proposed to use the strong reactivity of disulfide bonds, bearing the dithiopyridyl group, to form cross bridges between macromolecular complexes and the gold substrate of the sensor chip. Figure 4 shows the procedure performed in the Biacore apparatus to build the biosensor by way of the macromolecular complexes self-assembly. First, the surface was cleaned with a detergent (OG). This process was followed by an injection of DPDPB-GST-GEC1 complexes (step 2).

Finally, macromolecular complexes that had not been covalently bound to the gold substrate were removed by OG (step 3). An ideal schematic description of GST-GEC1 immobilization by self-assembly of DPDPB linker onto gold substrate is presented below the sensorgram in figure 3 and transposes analogous to the DPDPB-GST immobilisation process. A strong selfassembly was observed leading to a saturation of the chip bidimensional surface at a level of 2850 RU. As in Stenberg's calibration, the signal was converted to 2.85 ng per square mm [START_REF] Stenberg | Quantitative determination of surface concentration of protein with surface plasmon resonance using radiolabeled proteins[END_REF]. This value corresponded to a surface coverage of 7 picomoles of protein per square centimeter, i.e close to highly packed monolayer of proteins. We observed for each experiment that up to 80% of the signal was reached before 120 seconds after injection, indicating the efficiency of this strategy to build a protein monolayer. Moreover, the self-assembled macromolecular complexes tolerated smooth washing used to remove non-specific interactions. In our study, we observed a negligible non-specific coupling (less than 5% response units) easily removed by two pulses of OG.

Homo or heterobifunctional linkers have been previously used to immobilize macromolecular species onto a metallic template. Usually, building of supramolecular layers are performed sequentially with a first step consisting of covering the surface with the linker and a second step consisting of in situ chemical reactivity with the protein or nucleic acid [START_REF] Hedges | Electrochemical control of protein monolayers at indium tin oxide surfaces for the reagentless optical biosensing of nitric oxide[END_REF][START_REF] Lee | Nanoscale fabrication of protein A on selfassembled monolayer and its application to surface plasmon resonance immunosensor[END_REF]. Few studies propose in vitro synthesis of complexes following by their direct self-assembly onto gold surfaces but, in these cases, the disulphide bridge is initially reduced in order to expose free sulfhydryl residues [START_REF] Lee | Fabrication of selfassembled protein A monolayer and its application as an immunosensor[END_REF][START_REF] Park | Thiolated Salmonella antibody immobilization onto the gold surface of piezoelectric quartz crystal[END_REF]. We demonstrated that our procedure freed from this additional step and the present study pointed out the simplest way to build bidimensional protein self-assembly onto bare gold substrate.

The kinetics of the self-assembly was compared with a conventional and commercial process. Thus, we performed GST captures through a CM5 sensor chip preliminarily coupled with anti-GST antibodies using an amine coupling kit. In this way, a large excess of antibodies was covalently coupled with the carboxymethyl dextran matrix (data not shown) and monitored during the antibody/antigen recognition. Superimposition of the two association kinetics during the first 120 seconds after injection is represented in figure 5.

Results indicate that self-assembly is at least equivalent to the immuno capture in terms of kinetics and level of association.

Studies of GEC1-tubulin interactions

In order to demonstrate the biorecognition capabilities of the self-assembled monolayer of GEC1, we proceeded to injection of tubulin (Figure 6). As expected, a strong interaction between these two proteins was observed (curve A), whereas the specificity of the sensor was magnified through the negligible level of tubulin interactions with a monolayer of GST (curve B). Thus, at submicromolar concentration of tubulin, we observed a significant and specific signal of interaction with less than 11% of non-specific coupling without any cleaning procedure.

Control of the orientation of the macromolecular probes is essential to obtain a highly specific bidimensional response [START_REF] Vijayendran | A quantitative assessment of heterogeneity for surfaceimmobilized proteins[END_REF]. Concerning the supramolecular complex presented herein, the lack of cystein in the primary sequence of GEC1 leads to a directed coupling between DPDPB and the GST tag. Thus, as represented in figure 3, GST tag acts as a macromolecular spacer keeping away GEC1 from the metallic surface. Recently, we have demonstrated the capability of tubulin to associate with GST-GEC1 through a dextran matrix functionalized with anti-GST antibodies [START_REF] Mansuy | GEC1, a protein related to GABARAP, interacts with tubulin and GABA A receptor[END_REF]. The biorecognition property of the highly ordered 2D protein layer was compared with this classical 3D carboxy methyl dextran matrix by SPR experiments (Figure 7). As illustrated, injections of tubulin at a concentration of 800 nM have been performed for these two biosensing layers covered with the same level of immobilized GST-GEC1 species (2600-2800 RU). We observed a significant difference of interaction of these two biological partners. Thus, for the CM5 curve, the plateau was quickly obtained at a rate of coupling that did not exceed 3%, whereas in the case of the bidimensional architecture, signal of association reached a significant higher level corresponding to 12% of the saturation. This difference in performance is mainly due to the limitations of the dextran matrix to present the probe in a homogeneous conformation with its target. CM5 matrix has intrinsic limitations in (i) steric hindrance, (ii) heterogeneities in the density and conformation of probes inside the gel, and (iii) kinetic resistance due to improper molecular orientations and mass transfers causing an apparent decrease in the reaction-rate constant [START_REF] Schuck | Reliable determination of binding affinity and kinetics using surface plasmon resonance biosensors[END_REF][START_REF] Yarmush | An analysis of transport resistances in the operation of BIAcore; implications for kinetic studies of biospecific interactions[END_REF][START_REF] Nygren | Kinetics of antibody binding to solid-phaseimmobilised antigen. Effect of diffusion rate limitation and steric interaction[END_REF].

Conclusion

Most of the macromolecular coupling strategies onto biosensor surface layers are based on the deposition, functionalization and activation of polymer cushions with an expected high density of probes. However, strategies of direct coupling onto the bidimensional surface of the sensor may be an interesting alternative if bio-functionalization processes are well controlled.

In this study, we produce a fusion protein "GST tag-protein of interest" containing accessible cystein suitable for the efficient coupling with DPDPB linker. The procedure could be obviously used with proteins already possessing internal accessible cysteins.

We demonstrated that efficient and specific self-assembly can be achieved directly onto bare gold substrate leading to a rapid and simple procedure of the "linker-GST tagprotein of interest" monolayer reconstitution. Moreover, the protein layer formed yields higher coupling rates with biological partner than a classical immunosensor. This original procedure gives a new strategy of macromolecular assembly for numerous applications based on gold sensing layer. (B) Sensorgram of immunocapture of GST-GEC1 (1µM in HBS) on carboxy methyl dextran substrate pre-coated with anti-GST antibodies (CM5 sensor chip).

Legends for figures:

Experiments were performed during the first 120 sec after the injection for both substrates. Results of specific GEC1-Tubulin interactions on CM5 immuno sensorchip (curve A) and SIA-gold sensorchip (curve B) were superimposed for the same level of GST-GEC1 immobilization.

Injections of tubulin (800 nM in HBS) were performed during the first 240 seconds after injection for both substrates.

Figure 1 :

 1 Figure 1: Expression and purification of recombinant proteins.

Figure 2 :

 2 Figure 2: GEC1 and GST amino acid sequences.

Figure 3 :

 3 Figure 3: Spectrophotometer characterization of GST-DPDPB coupling.

Figure 4 :

 4 Figure 4: Self-Assembly of "DPDPB-GST-GEC1" complexes onto gold sensor chip.

Figure 5 :

 5 Figure 5: Superimposition of two GST-GEC1 immobilization strategies onto sensor

Figure 6 :

 6 Figure 6: Specific interactions between tubulin and self-assembled DPDPB-GST-GEC1

Figure 7 :

 7 Figure 7: Comparisons of interaction levels of tubulin with two different GST-GEC1

Acknowledgements

This research work was supported by a fellowship from Ministère de l'Enseignement Supérieur et de la Recherche (MESR) and by grants from MESR (EA 3922) and Ligue Nationale Contre le Cancer (Comité du Doubs and Comité du Jura). The authors thank Didier DUPONT from INRA (Poligny, France), Christophe Quétard and Denis To Van from Biacore ® International SA for their help for SPR instrumentation. We thank Dr. Paolo Martini for reviewing the English.


Î T Beno

Claudon

NON ALGEBRAICITY OF UNIVERSAL COVERS OF K ÄHLER SURFACES

In this short note, we prove the following fact : if the universal cover of a Kähler compact surface is biholomorphic to an affine variety, then, up to some finite étale cover, the surface is a torus.

Introduction

In his book [START_REF] Shafarevich | Basic algebraic geometry[END_REF], I. Shafarevich splits the classification of universal covers of projective manifolds into two types. The type I corresponds to manifolds having a finite fundamental group and their universal covers still belong to the class of projective manifolds. The type II is (according to him) somehow more delicate to define but in this case the fundamental group is infinite and the universal cover is very far from being a projective manifold 1 .

The following question seems then to be natural : how far from being projective this universal cover can be ? Since there is the obvious example of complex tori, we can wonder if the universal cover of a projective manifold (or more generally of a compact Kähler manifold) can appear as a quasi-projective variety (or a quasi-Kähler manifold in the Kähler setting). The case of curves is easily settled : the elliptic curves are the only one having in the same time an infinite fundamental group and a universal cover which can be compactified.

In this article, we study the surface case and show the following result.

Theorem 1.1. Let X be a compact Kähler surface whith infinite fundamental group and whose universal cover is quasi-Kähler ( i.e. can be realized as a Zariski open subset of a compact Kähler surface). Then, X is one of the following :

(1) X is geometrically ruled on an elliptic curve and X ≃ C × P 1 , (2) X admits then a finite étale cover which is a torus and the universal cover is C 2 .

We can specialize it to the affine case.

Corollary 1.1. Let X be a compact Kähler surface whose universal cover X is (biholomorphic to) an affine variety. Then, X ≃ C 2 and, up to some finite étale cover, X is a two-dimensional torus.

It is quite natural to formulate the following hope :

Conjecture 1.1. Let X be a compact Kähler manifold whose universal cover is (biholomorphic to) an affine variety. Up to some finite étale cover, X is a torus.

This conjecture should be related to the Iitaka conjecture (and can be seen as a strengthening of the latter).

1 the exact quotation is the following : Type II is hard to define at present, beyond the fact the fundamental group is infinite. In this case, the universal cover is a "very big" complex manifold, very far from being a projective or complete variety.

Conjecture 1.2. A compact Kähler manifold covered by C n has a finite étale cover which is a torus.

Note that the conjecture 1.2 holds up to dimension 3 ; see [START_REF] Campana | Compact Kähler threefolds of π 1 -general type, Recent progress in arithmetic and algebraic geometry[END_REF].

To conclude this introduction, let us point out the fact that the Kähler hypothesis is essential to ensure the validity of theorem 1.1. There exist several examples of non Kähler compact surfaces whose universal covers appear as quasi-projective varieties. Among them, we can quote the following examples :

(1) Hopf surfaces: the universal cover is then C 2 \(0).

(2) Kodaira surfaces: these surfaces are elliptic submersion on an elliptic curve.

Their universal covers are isomorphic to C 2 and their fundamental group are solvable groups which sit in an exact sequence

1 -→ Z 2 -→ π 1 (X) -→ Z 2 -→ 1.
Les us note that these groups are not almost abelian (not even almost nilpotent). A detailed description of these examples can be found in [START_REF] Barth | Compact complex surfaces[END_REF].

The non-general type case

In this section, we study the case where the surface X satisfies κ(X) ≤ 1. For sake of brievity, let us denote (in the rest of this article) by (H) the following hypothesis :

(H) : π 1 (X) is infinite and X is quasi-Kähler. We begin this section by the following observation : the hypothesis (H) implies minimality.

Lemma 2.1. If a surface X satisfies the hypothesis (H), it is minimal : X does not contain a (-1)-curve.

Proof :

Since X is assumed to be quasi-Kähler, there exists a compact surface X such that X ֒→ X is an open subset. If C is a (-1)-curve contained in X, π -1 (C) can then be written as a disjoint union of rational curves in X and hence in X :

π -1 (C) = i∈π1(S) C i .
Since being a (-1)-curve is a local property of the embedding of a curve, each curve C i has self-intersection -1 in X. Here we have a contradiction : a compact surface cannot contain infinitely many disjoint (-1)-curves.

Remark 2.1. A compact surface cannot contain infinitely many disjoint (-1)curves because the second Betti number strictly decreases at each blow-down. It should be noted that a rational surface can contain infinitely many (-1)-curve.

For instance, the blow-up of P 2 along the base locus of a pencil of cubics has this property.

The main point is now to understand which elliptic surfaces can in the same time fullfill the hypothesis (H).

Proposition 2.1. Let X -→ C be an elliptic surface over a curve C. If X satisfies the hypothesis (H), then the universal cover of X is either C 2 (and X is covered by a torus) or P 1 × C. In particular, in both cases we have κ(X) ≤ 0.

Proof :

We can assume (replacing S by a finite étale cover) that the fibration induces an exact sequence (see for instance [Cam98, appendix C]):

(1)

1 -→ π 1 (E) X = Im (π 1 (E) -→ π 1 (X)) -→ π 1 (X) -→ π 1 (C) -→ 1.
The universal cover of X being quasi-Kähler do not admit any non constant bounded holomorphic function and this implies that g(C) ≤ 1. If the fibration has only multiple smooth fibres (of type mI 0 ), some finite étale cover of X will become a smooth elliptic fibration. We can then apply the proposition 2.2 below to conclude that

X ≃ C × C.
Let us assume now that the fibration has a genuine singular fiber (over y ∈ C). We can then infer that the quotient π 1 (E) X is finite (as in [START_REF] Gurjar | Covering spaces of an elliptic surface[END_REF]) and that g(C) = 1 ( X is assumed to be non compact). This eaxctly means that the induced fibration X -→ C is a proper map. The family of fibres of this fibration consists in an analytic family of compact cycles of X and then it induces a family of cycles of X. The cycles space C( X) (constructed in [START_REF] Barlet | Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, Fonctions de plusieurs variables complexes[END_REF]) has only compact irreducible components and the fibration can then be extended to a (proper) fibration X -→ Ĉ = P 1 . But this fibration has one singular fibre for each point ỹ ∈ Ĉ lying over y ∈ C ; since it consists of infinitely many points, we get a fibration of a compact surface with infinitely many singular fibers. This is clearly impossible and the proposition is proved.

In the course of the preceeding proof, we used the following fact.

Lemma 2.2. Let f : X -→ Y a submersion between compact complex manifolds whose fibers are genus 1 curves. If moreover X is supposed to be Kähler, then f is isotrivial and X is biholomorphic to C × Y .

Proof :

The j function j : Y -→ C y → j f -1 (y) is well-defined: use local (on Y ) analytic sections of f to have genuine elliptic curves and compute j. Since f is a submersion, j is holomorphic on Y hence constant. This exactly means that all fibers are isomorphic to a fixed elliptic curve E; in particular, up to a finite étale cover, f : X -→ Y is a principal bundle of fiber E. From the fact that X is Kähler, we infer that q(X) = q(Y ) + 1 or, equivalently, that the morphism i * : π 1 (E) -→ π 1 (X) induced by i : E ֒→ X is injective. We can then consider the Albanese tori of X and Y ; the map α(f ) : Alb(X) -→ Alb(Y ) is an elliptic bundle and induced clearly an isomorphism Alb(X) ≃ Alb(Y ) × C.

Pulling back this splitting to X yields the proposition.

Remark 2.2. The Kähler hypothesis is certainly not superfluous. If S -→ P 1 is an elliptic Hopf surface, the morphism

Z 2 = π 1 (E) i * -→ π 1 (S) = Z
is absolutely not injective and the universal cover of S is C 2 \ {0}, i.e. a C * -bundle over P 1 .

We can now prove the main theorem under the assumption that the surface X is not of general type.

Proof of the theorem 1.1, case κ(X) < 2 :

Being minimal (lemma 2.1) and not of general type, the surface X belongs to one of the following class : κ(X) = -∞: X is geometrically ruled on a curve C (X = P(E) for some rank 2 vector bundle E on C), κ(X) = 0: up to a finite étale cover, X is a torus or a K3 surface, κ(S) = 1: X is elliptic over a curve C.

In the first case, we can easily see that g(C) = 1 and, the bundle E becoming trivial on C = C, that X ≃ P 1 × C (which is actually a quasi-projective variety). In the κ(X) = 0 case, the fact that π 1 (X) is infinite implies that X is a torus. The last case is ruled out by proposition 2.1.

L 2 canonical forms and the general type case

In this final section, we have to prove that a (projective) surface cannot in the same time be of general type and satisfy the hypothesis H. To achieve this goal, we resort to L 2 theory on X.

3.1. Review of L 2 index theory. In this paragraph, we state a particular case of the L 2 index theorem of Atiyah [START_REF] Atiyah | Elliptic operators, discrete groups and von Neumann algebras, Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan[END_REF]; for more on this subject, see [START_REF] Atiyah | Elliptic operators, discrete groups and von Neumann algebras, Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan[END_REF] or the books [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF] and [START_REF] Roe | Elliptic operators, topology and asymptotic methods[END_REF].

On X, consider the spaces of (p, q) harmonic forms which are L 2 with respect to ω :

H p,q ( X) = α (p, q) -form| ∆α = 0, X |α| ω dV ω<∞ .
The case (p, 0) has the following interpretation; the Kähler identities show that (p, 0) harmonic forms are indeed holomorphic and theses spaces can be described as follows:

H p,0 ( X) = H 0 (2) (Ω p X ) = α ∈ H 0 (Ω p X )| X α ∧ α ∧ ωn-p < ∞ .
In particular, for p = n, we have the Bergman space and the integrability condition is independant of the metric ω.

These spaces of L 2 harmonic forms are Hilbert spaces which are moreover endowed with an isometric action of Γ = π 1 (X). Using this action (and the compactness of X), it is possible to define a renormalized dimension function (which takes a priori non-negative real values); we will use the following notations (we omit the additional subscript Γ):

h p,q (2) ( X) := dim Γ H p,q ( X) .
This notion of dimension enjoy some usual properties; we will need one of them in the sequel.

Proposition 3.1. If h p,q (2) ( X) > 0 and if Γ is infinite, the space H p,q ( X) is infinite dimensional (in the usual sense).

Proof :

If H p,q ( X) were finite dimensional (but non zero), we could consider the function:

k(x) = N j=1 |α j (x)| 2 ω (x ∈ X)
where (α j ) j=1..N is an orthonormal basis of H p,q ( X). Since this expression is independant of the chosen basis, it is Γ-invariant and the computation of its integral on X gives:

X k(x)dV ω = j=1..N X |α j | 2 ω dV ω = N.
Since we assume Γ to be infinite, the first integral is in fact infinite (k is Γ-invariant and not identically zero) and we get a contradiction.

In this setting, the L 2 index theorem can be stated in the following form.

Theorem 3.1 (M. F. Atiyah, [START_REF] Atiyah | Elliptic operators, discrete groups and von Neumann algebras, Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan[END_REF]). The L 2 Euler characteristic of X is the same as the Euler characteristic of X:

χ (2) (O X ) := n p=0 (-1) p h p,0 (2) ( X) = χ(O X ).
3.2. Existence of L 2 canonical forms and conclusion. We apply the L 2 index theorem to get L 2 canonical forms on universal covers of certain projective manifolds of general type.

Lemma 3.1. Let X be a projective manifold of general type. Then there exists an L 2 canonical form on X in the following situations:

(i) dim(X) = 2 and π 1 (X) is infinite.

(ii) dim(X) = 3, X has a smooth minimal model and X is not covered by compact surfaces.

Proof :

Let us consider first the surface case. Since everything is invariant under birational morphism, we can assume that the surface X is minimal: K X is then big and nef. Applying L 2 -index theorem and Miyaoka-Yau inequality, this yields:

h 0 (2) (K X ) -h 0 (2) (Ω 1 X ) = χ (2) (O X ) = χ(O X ) = c 2 1 + c 2 12 ≥ 1 9 c 2 1 > 0.
We use here the vanishing of h 0 (2) (O X ): being L 2 , an holomorphic function tends to zero at infinity; it has to be constant (maximum principle) and then zero since X has infinite volume. The positivity of c 2 1 (K X being nef and big) implies then the positivity of h 0 (2) (K X ). The three-dimensional case is based on the same arguments. Since X is not covered by compact surfaces, Gromov Theorem 3.2 below implies the vanishing of h 0

(2) (Ω 1 ) and the same computation as above gives:

h 0 (2) (Ω 2 X ) -h 0 (2) (K X ) = c 1 c 2 24 ≤ 1 64 c 3 1 < 0.
Here we use the fact that we can assume X smooth and minimal so as to apply the Miyaoka-Yau inequality. This proves the second statement.

Let us recall the statement of the theorem quoted in the previous proof.

Theorem 3.2 (M. Gromov, [START_REF] Gromov | Kähler hyperbolicity and L 2 -Hodge theory[END_REF]). Let X be a compact Kähler manifold with non vanishing first L 2 Betti number h 0 (2) (Ω 1 X ) and with infinite fundamental group. Then there exists a proper (surjective) map

f : X -→ D such that H 0 (2) (Ω 1 X
) is induced by f ; in particular, X is covered by compact hypersurfaces.

Remark 3.1. The smoothness of the minimal model cannot be ommited in lemma 3.1. To see this consider the example described in [START_REF] Ein | Singularities of theta divisors and the birational geometry of irregular varieties[END_REF]. Let π : C -→ E be a double cover of an elliptic curve with g(C) ≥ 2 and denote by τ the involution of π.

If A = E × E × E and X is a desingularization of the quotient Y = C × C × C/ < τ × τ × τ >,
X has a generically finite map X -→ A (its Albanese variety). It is obvious that X is of general type; nevertheless its universal cover has no non-zero L 2 canonical form. Actually, it is proven in [EL97] that χ(O X ) = 0; being generically finite on an abelian variety, X satisfies the following L 2 vanishings conditions : H 0

(2) ( X, Ω j X ) = 0, j = 0, 1, 2. The L 2 -index theorem is reduced to the equality we looked for h 0

(2) (K X ) = -χ (2) (O X ) = -χ(O X ) = 0. It should be pointed out that the minimal model of X is nothing but Y which is (fortunately) not smooth.

We can now finish the Proof of the theorem 1.1 : Indeed, if X is surface of general type, its universal cover carries some L 2 canonical forms (lemma 3.1). On the other hand, if X satisfies the hypothesis (H), we can consider some compactification X ֒→ X of X. It is now a well known fact the L 2 canonical forms extend through the divisor D = X\ X. Indeed, choosing coordinates adapted to D near a smooth point of D and using Taylor expansion to compute the L 2 norm, we see easily that the L 2 condition prevent the canonical forms to have poles on D (here we use the important fact that the L 2 condition does not involve the metric ω; this is true only for canonical forms). We get thus an injection:

H 0 (2) ( X, K X ) ֒→ H 0 ( X, K X ).
At this point, we have a contradiction: H 0 ( X, K X ) is finite dimensional (compactness of X) whereas H 0 (2) ( X, K X ) is infinite dimensional (proposition 3.1). This concludes the proof of the main theorem.


Salma Ben Jamaa

Michel Kieffer

Pierre Duhamel

Efficient MAP estimation of CABAC encoded data with objective adjustment of the complexity/efficiency tradeoff

Keywords: Arithmetic codes, Communication system performance, Error correction coding, Source coding I

This paper presents an efficient MAP estimator for the joint source-channel decoding of data encoded with a context adaptive binary arithmetic coder (CABAC). The decoding process is compatible with realistic implementations of CABAC in standards like H.264, i.e., handling adaptive probabilities, context modeling and integer arithmetic coding. Soft decoding is obtained using an improved sequential decoding technique, which allows to obtain various tradeoffs between complexity and efficiency. The algorithms are simulated in a context reminiscent of H264. Error detection is realized by exploiting on one side the properties of the binarization scheme and on the other side the redundancy left in the code string. As a result, the CABAC compression efficiency is preserved and no additional redundancy is introduced in the bit stream. Simulation results outline the efficiency of the proposed techniques for encoded data sent over AWGN and UMTS-OFDM channels.

losses, a popular approach is to use joint source-channel (JSC) decoding schemes, which maintain the efficiency in terms of data compression (since they decode the same bitstream as standard decoders) while improving robustness against errors. A significant part of the work in JSC decoding is related to the reliable decoding of variable length codes (VLC), see, e.g., [START_REF] Park | Joint source-channel decoding for variable length encoded data by exact and approximate MAP sequence estimation[END_REF]- [START_REF] Wen | Soft-input soft-output decoding of variable length codes[END_REF].

Arithmetic Coding (AC) [START_REF] Howard | Practical implementations of arithmetic coding[END_REF] is currently the object of a growing interest as it yields higher compression efficiency when compared to other compression methods. Nevertheless, the high compression rate makes AC particularly vulnerable to transmission errors. This point has motivated the recent development of JSC decoding techniques for AC encoded data [START_REF] Boyd | Integrating error detection into arithmetic coding[END_REF]- [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF].

In most existing works, error detection and correction is usually performed by introducing redundancy in the compressed bitstream, thus reducing the compression efficiency. In [START_REF] Boyd | Integrating error detection into arithmetic coding[END_REF], a forbidden symbol (FS) is introduced in the coding alphabet and used as an error detection device. This technique is coupled with ARQ in [START_REF] Chou | Arithmetic coding-based continuous error detection for efficient ARQ-based image transmission[END_REF]. In [START_REF] Pettijohn | Joint source/channel coding using arithmetic codes[END_REF], both depth first and breadth first decoding algorithms have been considered. Error detection is again achieved by testing the presence of a FS during the decoding process and error correction is based on a maximumlikelihood (ML) criterion. Sequential decoding with maximum a posteriori MAP estimation of the encoded sequence is combined with the use of a FS in [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF]. In [START_REF] Demiroglu | Joint source channel coding using arithmetic codes and trellis coded modulation[END_REF], a Trellis Coded Modulation is used jointly with AC; the FS is exploited to discard erroneous paths during a List Viterbi decoding process. The quasi-arithmetic coder used in [START_REF] Guionnet | Soft decoding and synchronization of arithmetic codes: Application to image transmission over noisy channels[END_REF] is represented by a finite state machine, where transitions between states are modeled by a Markov process.

Redundancy is introduced by considering a reduced precision AC and adding synchronization markers before the encoding process. Recently, in [START_REF] Dongsheng | State machine interpretation of arithmetic codes for joint source and channel coding[END_REF], all sequences an integer AC may generate are represented by a 3D trellis taking into account the presence of a FS in the source alphabet. This trellis representation allows the use of soft output decoders such as the BCJR algorithm [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate[END_REF].

Context-based Adaptive Binary Arithmetic Coding (CABAC) [START_REF] Marpe | Context based adaptative binary arithmetic coding in the H.264/AVC video compression standard[END_REF] is the encoder used in the H.264/AVC standard [START_REF] Wiegand | Overview of the H.264/AVC video coding standard[END_REF]. By combining an adaptive binary encoding technique with context modeling, an improved compression efficiency is achieved especially in presence of non-stationary sources. For instance, when CABAC is used, the H.264 compression results are up to 20% better than those obtained by the baseline entropy coder of H.264/AVC, based on variable length codes [START_REF] Saponara | The JVT advanced video coding standard complexity and performances analysis on a tool-by-tool basis[END_REF]. However, since CABAC uses binary AC and probabilities stored in look-up tables, techniques similar to those cited above cannot be applied unless important changes are performed in the CABAC encoder implementation. This work deals with the MAP estimation of CABAC encoded data. Its purpose is to provide an efficient MAP metric which is suited to an actual CABAC implementation, i.e., a metric taking into account the delays between the code reception and the decoded stream emission due to the buffer-based integer implementation. For that purpose, an estimator (observer) of the CABAC encoder state is required at decoder side. No extra redundancy is compulsory: only redundancy introduced during the binarization step of the CABAC and left in the code string is exploited to detect errors. Our decoder has the characteristic of using an exact metric, in contrast with that proposed in [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF], which would only be approximate on a realistic CABAC. Moreover, this exact computation is compatible with existing ways of adding redundancy, and a combination of both strategies would result in further improvements, but are not considered in this paper. Decoding is achieved using sequential decoders [START_REF] Anderson | Source and channel coding: an algorithmic approach[END_REF],

where the improvement in terms of error resilience usually goes with an increase in decoding complexity. This problem is addressed here by the use of objective tests based on the MAP metric evaluation which allow an explicit adjustment of the tradeoff between decoding complexity and error correction efficiency.

Section II introduces the context of the work and recalls the main features of the CABAC implementation. Then, the way the exact MAP estimator is derived is explained in Section III.

The decoding process is put at work by means of sequential decoding algorithms introduced in Section IV, where improvements provided to conventional sequential decoding schemes by allowing to adjust the complexity-efficiency tradeoff are presented. Finally, simulation results are shown in Section V.

II. CONTEXT OF THE WORK

This section first describes the considered transmission scheme and related notations. Then, the principles of arithmetic coding, derived from Elias Coding are recalled in Section II-B.

The practical implementation of this basic version of AC is explained in Section II-C.

A. Transmission scheme

Data are assumed to be processed by a CABAC encoder, then packetized and transmitted over a radio mobile channel. Packets undergo some alteration during the transmission. This paper is concerned with the means of detecting and correcting transmission errors at the source bitstream level.

Since CABAC handles only binary data, a binarization step consisting in converting nonbinary information into binary source symbols according to a binarization scheme [START_REF] Marpe | Context based adaptative binary arithmetic coding in the H.264/AVC video compression standard[END_REF] is needed. The encoder is then run, and sends packets of the resulting bitstream in the channel.

Soft estimates are obtained from the output of the channel and fed to the CABAC decoder.

The input of the CABAC encoder is a sequence of K bins denoted by S K 1 = {S 1 , ..., S K }. It consists of a succession of binarized source symbols belonging to a set of binary words denoted by C. Bins stand for bits obtained by the binarization process. The last binarized source symbol of S K 1 , denoted by EOS (End Of Sequence), indicates the end of the binarized stream. Let X N 1 = {X 1 , ..., X N } be the succession of CABAC output bits. They are mapped using BPSK signaling into Only N is assumed to be known at the decoder side. Capital letters are for random variables, and small letters for their values. Integers n and k denote respectively the current length of the received code stream (decoder input), and the current number of decoded bins (decoder output).

R N 1 = {R 1 , ..., R N }, with R i = ± √ E b , i = 1 . . . N,

B. Basic principles of binary arithmetic coding (BAC)

Basic principles of BAC are recalled here, for more details, see, e.g., [START_REF] Elias | Universal codeword sets and representations of the integers[END_REF].

At each iteration of the encoding process, a subinterval of [0, 1) is recursively constructed.

The current interval [low , high) is divided into two subintervals of lengths L 0 and L 1 , respectively proportional to the probabilities P 0 and P 1 of the source bins 0 and 1. One of these two subintervals is selected, depending on the value of the current bin. Once the last bin of EOS is encoded, the algorithm computes the real value V , belonging to the obtained interval, which can be represented by the minimum number of bits. The binary representation of V forms the code string. Based on V , the decoder is able to reconstruct intermediate intervals and to recover the source bin stream.

For sources with very dissymetric bin probabilities, and for long source sequences, L 0 or L 1 may get too small to be accurately handled by a finite-precision processor. This problem is solved with finite precision implementations of BAC.

C. Practical implementations of BAC

To overcome the precision problem, most AC encoders are implemented using integers [START_REF] Witten | Arithmetic coding for data compression[END_REF]. Since this process is of interest in the derivation of the MAP decoder, see Section III, the corresponding algorithm is recalled below.

The initial interval [0, 1) of reals is replaced by the interval [0, 2 p ) of integers, where p 2 is the chosen finite precision, i.e., p bits are used to represent low and high. Partition and selection are carried out every time a source symbol is encoded. Renormalization by doubling the size of the source interval is performed as long as one of the following conditions holds 1) If high 2 p-1 , low and high are doubled.

2) If 2 p-1 high, low and high are doubled after subtracting 2 p-1 .

3) If 2 p-2 low and high < 3 × 2 p-2 , low and high are doubled after subtracting 2 p-2 .

If the current interval (before renormalization) overlaps the midpoint of [0, 2 p ), no bit is output. The number of consecutive times this occurs is stored in a variable called follow . If the current interval (before renormalization) lies entirely in the upper or lower half of [0, 2 p ), the encoder emits the leading bit of low (0 or 1) and follow opposite bits (1 or 0). This is called the follow-on procedure [START_REF] Witten | Arithmetic coding for data compression[END_REF].

At decoder side, a p-bit buffer value, formed by the p first received bits, is used to determine the divisions and selections of [0, 2 p ) which have been performed at encoder side.

Decoding starts after an initialization step, consisting in loading p bits into value without outputting any estimate of the source bins. The partition and selection operations performed at encoder side are determined using value, which helps thus estimating the source bins.

Every renormalization results in multiplying by 2 the integer associated to value, maybe after subtracting 2 p-1 or 2 p-2 , and thus shifting the buffer value to the left. Therefore, the next bit in the code stream is loaded to occupy the vacant position on the right of the buffer. This is repeated until all the code stream has passed into the buffer.

In the context-based BAC, interval divisions are performed according to bins probabilities deduced from contexts. Context modeling consists in assigning probability distribution models to the binary source symbols according to statistical dependencies between them. Each context is characterized by its index, state and the value of the most probable bin (MPS: Most Probable Symbol) [START_REF] Marpe | Context based adaptative binary arithmetic coding in the H.264/AVC video compression standard[END_REF]. The index may be specified by the current bin position in the binary source symbol and by the type of the syntax element to be encoded. The state indicates the probability estimate of the least probable bin (LPS: Least Probable Symbol), and is updated iteratively. Selection depends on whether the current bin is the MPS or the LPS.

III. MAP ESTIMATION OF CABAC ENCODED DATA

In [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF], a MAP estimator is proposed to evaluate the length K and content of the CABAC encoder input bin sequence s K 1 using y N 1 . This estimator may be expressed as

K, s K 1 = arg max k,s k 1 P S k 1 = s k 1 |Y N 1 = y N 1 = arg max k,s k 1 P S k 1 = s k 1 P Y N 1 = y N 1 |S k 1 = s k 1 P (Y N 1 = y N 1 )
.

The main difficulty in using (1) lies in the evaluation of

P Y N 1 = y N 1 |S k 1 = s k 1 .
The problem is circumvented in [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF] by using a modified version of the previous estimator

K, s K 1 , x N 1 = arg max k,s k 1 ,x N 1 e(s k 1 )=x N 1 P S k 1 = s k 1 P Y N 1 = y N 1 |X N 1 = x N 1 P (Y N 1 = y N 1 ) , (2)
where e s k 1 represents the output of the CABAC encoder fed with the bin sequence s k 1 . The maximisation is thus performed among all sequences which are valid outputs of the CABAC encoder. Nevertheless, the set on which the maximisation is performed is quite complex, and may be described by a trellis of reasonable size only for reduced-complexity AC, such as those presented in [START_REF] Guionnet | Soft decoding and synchronization of arithmetic codes: Application to image transmission over noisy channels[END_REF]. For realistic implementations, sequential decoders (see [START_REF] Anderson | Source and channel coding: an algorithmic approach[END_REF] and Section IV) have to be put at work. These decoders perform an iterative decoding using only a part of the observations. For example, in [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF], at step n of the decoding process, (2) is replaced by

x n 1 = arg max x n 1 ∈Bn P S k(x n 1 ) 1 = d (x n 1 ) P (Y n 1 = y n 1 |X n 1 = x n 1 ) P (Y n 1 = y n 1 ) , (3)
where d (x n 1 ) represent the output (formed by k (x n 1 ) bins) of the CABAC decoder fed with x n 1 at its input. In (3), B n is the set containing the n bits long prefixes of all sequences that could be generated by the CABAC encoder described in Section II-C.

As will become clear at the end of Section III-B, the estimate provided by (3) is suboptimal when compared to the a posteriori estimate of x n 1 which could be obtained using y n 1 and the fact that x n 1 is the output of a CABAC encoder. In [START_REF] Sayir | On Coding by Probability Transformation[END_REF], Sayir proposes a MAP estimation of noisy arithmetic encoded data, which has inspired that work, though the decoding delay occurring at the CABAC decoder is not taken into account in [START_REF] Sayir | On Coding by Probability Transformation[END_REF]. In our work, a different definition of the MAP estimate is adopted. As [START_REF] Jeanne | Application of a joint source-channel decoding technique to UMTS channel codes and OFDM modulation[END_REF] DRAFT will be shown, this requires the estimation of some features of the encoder state (follow and probability estimates) at decoder side.

A. Definition and assumptions

The aim of this work is to determine at any n N, the sequence x n 1 maximizing the a posteriori probability (APP) given by

x n 1 = arg max x n 1 ∈Bn P (X n 1 = x n 1 |Y n 1 = y n 1 ), (4)
As in (3), the evaluation of (4) takes into account the fact that x n 1 is the output of a CABAC encoder fed with binarized source symbols. Once N is reached, x N 1 , maximizing (4) for n = N, may be decoded as s K 1 satisfying all constraints imposed by the binarization. As there is a one-to-one mapping between x N 1 and s K 1 , s K 1 also maximizes (1). Consider the decoder input sequence x n 1 , and the sequence of bins s

k(x n 1 ) 1 = d (x n 1 )
obtained at the output of a CABAC decoder fed with x n 1 (to make notations shorter, k (x n 1 ) will be denoted by k). The decoder input sequence x n 1 may then be decomposed into three parts: 1)

x n (s k 1 ) 1 = e s k 1
is the code string that would be emitted by an encoder fed with

s k 1 = d (x n 1 ), n (s k 1 ) being its length, 2) x n (s k 1 )+F (s k 1 ) n (s k 1 )+1
are the postponed bits [START_REF] Sayir | On Coding by Probability Transformation[END_REF], i.e., the F (s k 1 ) first bits that would be emitted during the next follow-on procedure by an encoder fed with s k 1 . The number F (s k 1 ) of postponed bits is equal to f ollow + 1, keeping in mind that f ollow depends on s k 1 . Postponed bits may only take the values {1, 0, . . . , 0} and {0, 1, . . . , 1}, depending on the coder internal state (low, range, f ollow, contexts) after being fed with

s k 1 , 3) x n n (s k 1 )+F (s k 1 )+1 , are bits assumed independent of s k 1 . As n (s k 1 ), F (s k 1 )
, and the current probability estimates are not directly available at the decoder (due to delays introduced by the encoding and decoding buffers), a possible way to estimate them is to reencode s k 1 . The additional encoder used for that purpose is called observer to avoid confusion with the encoder at emitter side, see Figure 1. Moreover, as X N 1 is the output of an arithmetic encoder, the X i , i = 1 . . . N, are assumed equally likely,

P (X i = 1) = P (X i = 0) = 1 2 , ∀i ∈ {1 . . . N}. (5)
In the remainder of the paper, the notations n , k, and F are adopted for n (s k 1 ), k (x n 1 ), and F (s k 1 ) when there is no risk of confusion. Moreover, the random variables X, Y , and S are omitted in the expressions of the probabilities, provided that there is no ambiguity. As x n n +F +1 does not depend on

s k 1 = d (x n 1 ) (see Section III-A), P x n n +F +1 |S k 1 = d (x n 1 ) = P x n n +F +1 . (12)
Moreover, assuming that X n n +F +1 does not provide more information about X n +F n +1 than S k 1 does, one gets

P x n +F n +1 |x n n +F +1 , S k 1 = d (x n 1 ) = P x n +F n +1 |S k 1 = d (x n 1 ) . (13)
Finally, combining (10), ( 11), [START_REF] Guionnet | Soft decoding and synchronization of arithmetic codes: Application to image transmission over noisy channels[END_REF], and (13), the APP (6) may be expressed as

P (x n 1 |y n 1 ) = P S k 1 = d (x n 1 ) P x n n +F +1 P x n +F n +1 |S k 1 = d (x n 1 ) P (y n 1 |x n 1 ) P (y n 1 ) . ( 14
)
When compared to the metric adopted in [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF], two additional terms are involved in [START_REF] Demiroglu | Joint source channel coding using arithmetic codes and trellis coded modulation[END_REF],

namely P x n n +F +1 and P x n +F n +1 |S k 1 = d (x n 1 )
. The evaluation of the probabilities making up ( 14) is described in Section III-C .

C. Practical implementation of the MAP estimator

In this section, we show the way the MAP estimate (4) is computed and updated using [START_REF] Demiroglu | Joint source channel coding using arithmetic codes and trellis coded modulation[END_REF] for a CABAC decoder.

1) Evaluation of the bin stream probability

P S k 1 = d (x n 1
) : Here, a priori probabilities of the form P (S k 1 = s k 1 ) have to be evaluated. The source bin stream is a sequence consisting of a succession of binarized symbols belonging to C. The probability of each source symbols may be estimated by algorithms such as the one described in [START_REF] Wen | Utilizing soft information in decoding of variable length codes[END_REF], or assumed to be known and deduced from the encoding tree as in [START_REF] Guivarch | Joint source-channel soft decoding of huffman codes with turbo-codes[END_REF], [START_REF] Jeanne | Etude de systmes robuste de dcodage conjoint source-canal pour la transmission sans fil de vido[END_REF].

As this paper deals with an adaptive context based AC, one can rely on the adaptive probabilities of the MPS and the LPS deduced from the contexts at the decoder side. Indeed,

P (S k 1 = s k 1 ) = P (S 1 = s 1 ) k i=2 P S i = s i |S i-1 1 = s i-1 1 ,
where

P S i = s i |S i-1 1 = s i-1 1 is estimated by P LPS if s i = LPS and 1 -P LPS if s i = MPS.
When incomplete binarization schemes are considered (i.e., when the Kraft inequality is strict), for some s k 1 , one may get

P (S k = s k |S k-1 1 = s k-1 1 ) = 0, (15)
which corresponds to values of s k not consistent with the binarization tree.

November 14, 2006 DRAFT

2) Evaluating probabilities related to the transmission channel: The likelihood P (y n 1 |x n 1 ) and the channel output probability P (y n 1 ) are deduced from the channel model. In Section V, the AWGN channel model and an UMTS-OFDM channel model are considered. Both channels being assumed memoryless, one may write

P (y n 1 |x n 1 ) = n i=1 P (y i |x i ). (16)
For an AWGN channel, P (y n 1 ) can be approximated by 2 -n , as in [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF], and the last term of [START_REF] Demiroglu | Joint source channel coding using arithmetic codes and trellis coded modulation[END_REF] becomes

P (y n 1 |x n 1 ) P (y n 1 ) = 2 n (2πσ 2 ) n/2 n i=1 exp -(r i -y i ) 2 2σ 2 ,
where r i = ± √ E b , depending of x i . When for each bit at channel output a Log Likelihood Ratio (LLR) is available, for the i-th received bit one has

LLR i = log P (X i = 1|y i ) P (X i = 0|y i ) .
Using (5), one gets

P (X i = 1|y i ) = 1 exp(-LLR i ) + 1 (17)
P (X i = 0|y i ) = 1 exp(LLR i ) + 1 . (18)
Consequently, for such channels, one obtains

P (y n 1 |x n 1 ) P (y n 1 ) = n i=1 2 exp -r i √ E b LLR i + 1
.

3) Evaluating P x n n +F +1 : All sequences x n n +F +1 are assumed to have equal a priori probability, i.e., P (x n n +F +1 ) = 2 -n+(n +F +1) . If more information is available (provided in the context of iterative decoding, for example), it may obviously be used.

4) Evaluating the postponed bits probability

P x n +F n +1 |S k 1 = d (x n 1
) : In this case, the k (x n 1 ) first encoder input bins are assumed to be equal to

s k 1 = d (x n 1
). Under that hypothesis, the aim is to determine the probability that X n +F n +1 = x n +F n +1 . F represents the number of bits to be output as soon as a follow-on procedure is performed. More bins s k k+1 are necessary to perfectly determine x n +F n +1 at the observer output. A possible way to estimate

P x n +F n +1 |S k 1 = s k 1 = P x n +F n +1 |s k 1
is to write this probability as

P x n +F n +1 |s k 1 = s k k+1 P s k k+1 |s k 1 P (x n +F n +1 |s k k+1 , s k 1 ),
which requires to feed the observer with all possible s k k+1 until a follow-on procedure is performed.

This procedure has a combinatorial complexity. Here, the procedure for a single additional bin s is described, its generalization to more bins is straightforward.

For a given value of k (x n 1 ), the tail of s k 1 corresponds to the beginning of a binarized source symbol. According to the binarization scheme, the following bin may take two values (MPS or LPS). If incomplete binarization schemes are involved, only one value of S k+1 may be possible. Considering both cases S k+1 = MPS and S k+1 = LPS, P x n +F n +1 |s k 1 becomes,

P x n +F n +1 |s k 1 = P MPS P x n +F n +1 |S k+1 = MPS, s k 1 + P LPS P x n +F n +1 |S k+1 = LPS, s k 1 ,
where P MPS = P (S k+1 = MPS|s k 1 ) and P LPS = P (S k+1 = LPS|s k 1 ) are deduced from the contexts estimated at the observer. If postponed bits are emitted, their values may be either {0, 1, . . . , 1} or {1, 0, . . . , 0}. If x n +F n +1 is not equal to one of these two sequences, one has

P (x n +F n +1 |s k 1 ) = 0. (19)
Now, if x n +F n +1 is equal to {0, 1, . . . , 1} or {1, 0, . . . , 0}, and

• only a MPS produces x n +F n +1 then P (x n +F n +1 |s k 1 ) = P MPS , • only a LPS produces x n +F n +1 then P (x n +F n +1 |s k 1 ) = P LPS , • both MPS and LPS produce x n +F n +1 then P (x n +F n +1 |s k 1 ) = 1. In all other cases, it is assumed that P (x n +F n +1 |s k 1 ) = 1 2 . Practical implementations could fed the observer with more than a single bit to improve the evaluation of P (x n +F n +1 |s k 1 ). In practical situations, satisfactory results are obtained with less than 3 additional bits. The additional complexity remains thus limited.

IV. SEQUENTIAL DECODING

In order to compute the MAP estimator [START_REF] Demiroglu | Joint source channel coding using arithmetic codes and trellis coded modulation[END_REF], the APP has to be evaluated for all paths of the decoding tree. To an observation Y n 1 , one may assign up to 2 n paths representing possible estimates of the code string X n 1 . For relatively large n, examining the whole decoding tree is infeasible. The purpose of sequential decoders is to find the best path, according to a chosen metric, without examining too many branches. The most popular sequential decoding algorithms are the stack algorithm (SA) [START_REF] Zigangirov | Some sequential decoding procedures[END_REF], [START_REF] Jelinek | Fast sequential decoding algorithm using a stack[END_REF], the M-algorithm (MA) [START_REF] Anderson | Source and channel coding: an algorithmic approach[END_REF] and their variants, such as the generalized stack algorithm [START_REF] Haccoun | Generalized stack algorithms for decoding convolutional codes[END_REF].

A. Stack Algorithm

The stack algorithm (SA) [START_REF] Zigangirov | Some sequential decoding procedures[END_REF], [START_REF] Jelinek | Fast sequential decoding algorithm using a stack[END_REF] is a metric first search performed iteratively: an ordered stack containing previously examined paths (of different lengths) is maintained. The best path (located in the top of the stack) is expanded by the exploration of the two branches following the current node. The top path is then removed and the two new paths are merged in the stack.

Algorithm 1: Basic Stack Algorithm

Step 1. Initialize the stack with a single line containing the root of the decoding tree.

Step 2. Extend the top path by creating two new paths, one for each possible value of the following emitted bit.

Step 3. Store the two created paths in the stack with their corresponding metric. Sort the stack according to the metric. Drop paths lying beyond the stack size limit.

Step 4. If the top path reaches the maximum depth on the decoding tree, stop. Otherwise, go to 2.

Many variations of this basic version have been proposed in the literature, such as the generalized stack algorithm [START_REF] Haccoun | Generalized stack algorithms for decoding convolutional codes[END_REF], which extends the first ≥ 2 paths in the stack instead of only the first one. This allows packet loss reduction especially when the beginning of the received sequence is strongly disturbed. The multiple stack algorithm [START_REF] Chevillat | A multiple stack algorithm for erasurefree decoding of convolutional codes[END_REF] is another approach claiming theoretically packet loss free decoding.

B. M-Algorithm

The M-algorithm (MA) is a breadth first search, the breadth being M. The decoding tree exploration is performed by iteratively incrementing the depth of the paths, keeping only the best M paths according to the metric. [START_REF] Jeanne | Application of a joint source-channel decoding technique to UMTS channel codes and OFDM modulation[END_REF] DRAFT

Algorithm 2: Basic M-Algorithm

Step 1. Initialize the list with a single line containing the root of the decoding tree to which a null metric is assigned.

Step 2. Extend all paths to the following branches, creating 2 new paths from each stored path.

Step 3. Keep only the M best paths in terms of metric.

Step 4. Stop if one of the M maintained paths reaches the maximum depth. Otherwise, go to 2.

C. Adapting sequential decoding to CABAC

In this section, the proposed sequential decoder is detailed. First, a metric based on the a posteriori probability derived in Section III-B is adopted. Then, drop conditions are put at work in order to lighten the decoding tree and avoid false tracks. These conditions can be considered as tools to detect errors in some transmitted bitstreams, without any additional redundancy. Finally, the way the decoding process stops is explained.

1) Decoding metric: Let I(it) be the set of paths maintained at the it-th iteration of a sequential decoding algorithm. For a breadth-first search, all stored paths in I(it) have the same length n = it. For a metric first search, paths with unequal lengths may be considered.

Thus, both length n and values of bits composing x n 1 vary from one path belonging to I(it) to an other. The metric assigned to every path x n 1 is derived from [START_REF] Demiroglu | Joint source channel coding using arithmetic codes and trellis coded modulation[END_REF] and given by

M MAP (x n 1 ) = log P (x n 1 |y n 1 ). ( 20
)
The best estimator of the beginning of X N 1 at the it-th iteration is the path x n 1 maximizing the APP among all paths belonging to I(it), and given by

( x n 1 ) it = arg max x n 1 ∈I(it) M(x n 1 ) .
2) Drop conditions: Drop conditions allow the decoder to identify some paths not deserving to be stored. Such paths are removed from the decoding tree. The most used drop conditions are those related to the complexity restriction as a maximum number of simultaneously stored paths is fixed. When this number is reached, any path having a metric smaller than the metric of the last path is dropped. If the correct path is dropped in this way, the decoding fails and decoder may not output any solution. This event is called erasure. Note, however, that this is a somewhat different definition compared to the classical one.

A constraint on s K 1 results from the assumption that the length of the code string (N bits) is known and that the last binarized symbol is the EOS symbol. This constraint leads to two dropping situations. First, if a path x N 1 of N bits is examined over the decoding tree, and if the associated s k 1 is not ended by EOS, the sequence s k 1 is then deemed erroneous and x N 1 is dropped. Second, if s k 1 contains EOS while less than N bits have been decoded, an error is detected and the path is dropped. In these cases, paths are dropped without any risk of packet loss as the decoded stream cannot correspond to s K 1 . Other drop conditions are directly deduced from the metric evaluation. Indeed, when the APP probability of a given path is equal to zero, it is very likely to be automatically discarded when explored paths are sorted and that the constraint on the maximum number of stored paths is applied. The APP can be equal to zero in the following cases: First, as CABAC mostly relies on incomplete binarization schemes (single infinite extended binarized symbol set consisting of zero order Exp-Golomb codes [START_REF] Marpe | Context based adaptative binary arithmetic coding in the H.264/AVC video compression standard[END_REF]), the situation described in (15) is likely to occur leading to a null APP. Second, when the case described by [START_REF] Saponara | The JVT advanced video coding standard complexity and performances analysis on a tool-by-tool basis[END_REF] occurs, a null APP is assigned to the corresponding path on the decoding tree. Notice that this drop condition exploits the little amount of redundancy left by the CABAC encoder in the semantic of the code string.

3) Stop conditions:

The decoding tree exploration stops when a path x N 1 of N bits yields a sequence of binarized symbols s k 1 ending with EOS. Then, x N 1 is the path maximizing M M AP (x N 1 ) among all the stored paths at the current iteration. The sequence s k 1 is the solution output by the decoder. In some cases, decoding stops when all paths have been dropped, or when the maximum allowed computational effort has been reached. Both of these situations may lead to the occurrence of an erasure. The lost packet may be re-emitted if ARQ is allowed.

D. Objective adjustment of the efficiency-complexity tradeoff

The decoding performance in terms of error resilience efficiency is improved when more paths are explored in the decoding tree. This is especially true when the beginning of the code string is strongly corrupted. Nevertheless, this improvement usually goes with increasing decoding complexity. Therefore, in order to judiciously adjust a tradeoff between decoding complexity and efficiency, an objective test providing a new drop condition is proposed.

Assume that the path to be extended is x n-1

1

. Two extensions can be considered, x n = 1 and x n = 0. The hard decoder discards systematically one extension and thus, explores only one path on the decoding tree. The idea is to derive a test allowing to decide if both of the two choices deserve being considered or if a hard decision on the current bit is sufficient. In [START_REF] Pettijohn | Joint source/channel coding using arithmetic codes[END_REF], a 2∆ width null zone is used to determine whether a hard decision on y n is reliable, error detection is then performed by means of a FS. Here, we consider that if a path has a relatively low metric, it is very likely to be dropped in the next iterations. Thus, omitting exploring and storing such paths saves computational effort. The idea is to derive an objective criterion in order to characterize a low metric for a controlled amount of decision errors.

At the node corresponding to n-1 1 on the decoding tree, three actions may be taken:

• A n 0 : only the branch corresponding to x n = 0 is explored. • A n 1 : only the branch corresponding to x n = 1 is explored. • A n 0|1 : both extensions are explored. Let

Λ n = M MAP (x n-1 1 , 1) -M MAP (x n-1 1 , 0)
be the logarithm of the a posteriori probability ratio. The purpose is to derive a threshold T such that

Λ n A n 0|1 ≶ A n 1 T (21)
-Λ n A n 0 ≶ A n 0|1 T (22)
According to [START_REF] Anderson | Source and channel coding: an algorithmic approach[END_REF], and using the assumption that the channel is memoryless, one may write

Λ n = log P (x n-1 1 , X n = 1|y n 1 ) -log P (x n-1 1 , X n = 0|y n 1 ) = log P (y n |X n = 1) -log P (y n |X n = 0). (23)
The probability of error P e , corresponding to the probability of loosing the correct path, may be evaluated as

P e = P (A n 0 , X n = 1) + P (A n 1 , X n = 0) = P (Λ n < -T, X n = 1) + P (Λ n > T, X n = 0) = P (X n = 1)P (Λ n < -T |X n = 1) + P (X n = 0)P (Λ n > T |X n = 0) = 1 2 P (Λ n < -T |X n = 1) + 1 2 P (Λ n > T |X n = 0).
Similarly, the probability P f of useless extension of a the path with two branches is

P f = P (A 0|1 , X n = 1) + P (A 0|1 , X n = 0) = 1 2 P (-T < Λ n < T |X n = 1) + 1 2 P (-T < Λ n < T |X n = 0).
To get T , one minimizes P f for a given tolerated value of P e .

Let us derive the threshold T assuming that the channel is AWGN, of variance σ 2 . For other kinds of symmetric channels, such as the UMTS channel, an equivalent AWGN channel may be estimated. Code bits are assumed to be mapped into a BPSK symmetric signaling such that r n = ± √ E b . For such channel, when x n is considered at the decoder output and y n is received, [START_REF] Wen | Utilizing soft information in decoding of variable length codes[END_REF] gets

Λ n = 2 √ E b σ 2 y n .
Thus, [START_REF] Elias | Universal codeword sets and representations of the integers[END_REF] and [START_REF] Witten | Arithmetic coding for data compression[END_REF] become

y n A n 0|1 ≶ A n 1 σ 2 2 √ E b T and y n A n 0 ≶ A n 0|1 - σ 2 2 √ E b T
Then P e and P f are expressed as

P e = 1 2 1 -erf E b 2σ 2 -T σ 2 2E b + 1 . P f = erf E b 2σ 2 T σ 2 2E b + 1 + erf E b 2σ 2 T σ 2 2E b -1 .
where erf(z) = 2 √ π z 0 e -t 2 dt. One can finally express the threshold T as

T (σ, P e ) = 2 E b σ 2 √ 2erf -1 (1 -P e ) - E b σ 2 .
Several choices may be considered for P e . We constrain the probability of making a wrong decision to be a fraction of the probability P Hard (σ) that the hard decoder locally fails, expressed in the AWGN case by

P Hard (σ) = +∞ 0 P (y|X n = 0)dy + 0 -∞ P (y|X n = 1)dy. = 1 -erf E b 2σ 2 . ( 24
)
Then, P e can be expressed as

P e (σ, n) = α • P Hard (σ, n), with α < 1. (25)
The parameter α allows one to adjust the decoding complexity-efficiency tradeoff, as shown in Section V.

V. SIMULATIONS Simulation are performed using the CABAC defined in the H.264/AVC standard. Binarized source symbols belong to the first 9 binary codewords of the zero-order Exp-Golomb scheme (EG0) [START_REF] Marpe | Context based adaptative binary arithmetic coding in the H.264/AVC video compression standard[END_REF]. A simplified context modeling with three contexts is considered.

Simulations using an AWGN channel and Pedestrian-B UMTS soft error patterns, see, e.g., [START_REF] Jeanne | Application of a joint source-channel decoding technique to UMTS channel codes and OFDM modulation[END_REF] are considered. For the AWGN channel, no channel coding is used and error correction relies only on the redundancy due to the EG0 binarization scheme (see Figure 1).

For the Pedestrian-B channel, the CABAC is followed by a rate is not decoded from this bitstream (erasure). When an erasure occurs, the decoder does not output any solution and all symbols emitted by the source are considered as erroneous, and are included in the SER evaluation.

A. Performances of the MAP estimator

Figure 2 compares the results obtained using the proposed exact MAP decoder and the one based on the metric proposed by [START_REF] Grangetto | Joint source/channel coding and MAP decoding of arithmetic codes[END_REF] on specific source sequences. These source sequences contain 100 binarized source symbols and every sequence is transmitted within packets of 640 bits. They were chosen in such a way that the number of follow-on procedures performed during encoding and the mean value of follow is much higher than for totally random

sequences. An M-algorithm with M = 10 is used for decoding. For a SER of 10 -3 , an improvement of 0.6 dB is achieved. The purpose, here is to show that the proposed MAP estimator derivation is, indeed more efficient than the previously proposed ones.

It is clearly seen on Figure 2 that the exact computation may lead to improved decoding in some situations. Note that the approximation used in the previous works is often quite accurate, and that, on many sequences, both computations provided almost the same performance. However, as illustrated, on some sequences the phenomenon shown on Figure 2 was observed. This clearly illustrates the usefulness of using an exact computation rather than an approximate one.

Figure 3 illustrates the results obtained using source sequences of 100 binarized source symbols and the M-algorithm with M = 20. The performance of a MAP decoder using (4) is compared to that obtained using a Maximum-Likelihood (ML) decoder. For a SER of 10 -3 , the improvement (in dB) obtained when using (4), compared to the ML metric is about 0.7 dB for both channels. The gain achieved by the soft decoding when compared to the hard one is up to 4 dB for the AWGN channel and 2 dB for the UMTS channel.

B. Adjustment of the complexity-efficiency tradeoff

MAP decoders based on both SA and MA are considered, and the following abbreviations are adopted. BSA and BMA(M) stand for the basic SA and MA, M being the number of paths kept at each MA iteration. GSA( ) is the generalized SA, extending paths at each iteration.

Finally FGSA( , α) and FMA(M, α) denote the Fast SA and the Fast MA embedding the test presented in Section IV-D.

Figure 5 illustrates the error resilience, in terms of SER, and complexity performance, in terms of average number of visited branches during erasure-free decoding, of four versions of the decoder using the SA. Figure 6 presents the same performance for four versions of the decoder using the MA.

When compared to a standard decoder carrying out hard decisions on noisy bits, the sequential decoders present an important gain (up to 3 dB for the SA based decoders and 4 dB for the MA) in error correction. The performance is improved as the number of simultaneously explored paths ( and M) increases, and as α decreases.

On the other hand, one may notice that the more efficient is the decoding in recovering errors, the higher is the complexity. Compared to the BSA, the FGSA(3, 10 -4 ) reaches a gain of 1.8 dB, at SER = 10 -3 for a doubled complexity at 12 dB. For the same SNR, FMA(20, 10 -4 ) presents a 20 times lower complexity when compared to MA [START_REF] Chou | Arithmetic coding-based continuous error detection for efficient ARQ-based image transmission[END_REF], with a gain of 2.5 dB at SER = 10 -3 . This shows that the effects of α and or M on the tradeoff between the complexity and the robustness may be combined in order to design JSCD schemes according to application needs. supposing that the channel rate is equal to 64 kb/s, the effective rate available to the source, at SNR = 9 dB and taking the ARQ into account, is 57.6 kb/s using a BSA, 63.7 kb/s using a FGSA(3, 10 -4 ), 62.08 kb/s using a MA [START_REF] Chou | Arithmetic coding-based continuous error detection for efficient ARQ-based image transmission[END_REF], and 63.93 kb/s using a FMA (20, 10 -4 ).

VI. CONCLUSIONS

The main drawback of the high compression rate CABAC is its vulnerability to transmission errors. In this paper, we have presented a soft decoding technique based on a MAP estimator, exploiting binarization scheme, information given by context modeling, and the semantic of the code string. This estimator is associated to sequential decoding algorithms to carry out a reliable soft CABAC decoder. An objective test allowing to adjust the decoding complexity according to the desired performances is elaborated, and shown to provide better performance in terms of symbol error rates for a given complexity.

In all variants of the decoder, the small redundancy present after the binarization step is exploited and no extra redundancy is added. The proposed sequential decoders can handle adaptive probabilities and context modeling, and can be directly embedded into a standard implementation of CABAC without impairing its compression efficiency. Current work is dedicated to embedding the soft MAP decoder and the objective test within the full H.264 decoder.

Figure 4

 4 Figure 4 depicts the packet erasure rate related to decoding failures for the same versions of the sequential algorithms. For the FGSA and FMA decoders, packet losses are mainly caused by the dropping strategy introduced in Section IV-D. Indeed, for low values of the SNR, the number of stored paths often reaches zero before any solution is obtained. Nevertheless, one can note, for example, that for a SNR of 10 dB, FGSA(3, 10 -4 ) presents a packet loss

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Transmission block diagram

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Packet erasure rate using SA and MA based decoders

  1/2 convolutionnal code with constraint length 9 and generators (561, 753) o . A standard UMTS interleaver of length 640 is also considered. At the channel output, a SOVA decoder is implemented, providing log-likelihood ratios (LLR n ). The Symbol Error Sate (SER) is evaluated for different values of the SNR. Hard decoding provides the bit value x n using the sign of the channel output y n in the AWGN case, and of the LLR n in the UMTS case. Hard decoding fails if debinarization fails or if the EOS

[START_REF] Jeanne | Application of a joint source-channel decoding technique to UMTS channel codes and OFDM modulation[END_REF] DRAFT

for providing the soft error patterns in the software platform of the NEWCOM NoE.

B. APP metric derivation

The APP in (4) may be written as

.

To evaluate [START_REF] Wen | Soft-input soft-output decoding of variable length codes[END_REF], one has to express the a priori probability P (x n 1 ) in terms of a priori probabilities of the source bins.

For s k 1 = d (x n 1 ), one has

As

, one gets

) = 1 and one gets

Therefore, using the fact that P (x n 1 ) = P (x n 1 , x n n +1 ), (8) leads to

Using ( 9), ( 6) may be rewritten as

.

Now, the decomposition of x N 1 into three parts may be further exploited to obtain a computable APP. The second term of the right part of (10) may be written as