Dr Andrej Jancarik
email: andrej.jancarik@cemes.fr

Dr Khanh Nguyen

Hung

Dr Dmitry Skidin

Dr Francesca Moresco

Dr Andre Gourdon
email: andre.gourdon@cemes.fr

] A Jancarik

N K Hung

Preparation of Tetrabenzo[4.4.2]undecastarphene by On-Surface Synthesis

A large dissymmetric starphene molecule, the tetrabenzo[a,c,u,w]naphtho[2,3-l]nonaphene, can be obtained by first preparing a soluble precursor which is then sublimated on a Au(111) surface in ultra-high vacuum. In a second step, controlled annealings from 200°C to 275°C initiate two successive cyclodehydrogenation steps with the formation of 3 new carbon-carbon bonds. A second conformer is also stable enough during the annealing step to give another compound in similar yield, the benzodibenzo[7,8,9,10]naphthaceno[2,1-h]phenanthro[9,10p]hexaphene. The formation of this more hindered species stresses the importance of strong molecule-surface interactions during the cyclodehydrogenations steps of these large polyaromatic hydrocarbons.

Introduction

Starphene [START_REF] Pozo | [END_REF] , molecules comprising three acene branches, have been recently proposed as single molecular gates in the Quantum Hamiltonian Computing approach. It has in particular been shown that [2,2,2]-heptastarphene could be implemented as a NOR logic gate [2] whereas the asymmetric [3,3,2]-nonastarphene was a NAND gate. [3] To go beyond and be able testing more powerful logic gates such as a half-adder, [4][5][6][7] it is necessary to increase the complexity and the asymmetry of these conjugated three branches molecules. However, the synthesis of large unsubstituted starphenes in solution is a difficult task due to their insolubility, and the largest starphene to-date, the [3,3,3]-decastarphene, [8,9] has been reported by Clar and Müllen more than fifty years ago. [10] Indeed, it is possible to increase the solubility of large polyaromatic hydrocarbon molecules by preventing complete planarity, and therefore the intermolecular -stacking forces. This can be done by designing sterically congested structures like cloverphenes [11] or by adding bulky substituents [12] , but this strategy also modifies electronic properties of the molecular gate. Alternatively, we have recently shown that it was possible to yield unsubstituted and planar [5,5,5]-hexadecastarphene by decarbonylation in the solid state or on a surface of a soluble precursor. [13] Another potential route is on-surface synthesis which allows the preparation of insoluble molecules by coupling smaller functionalized precursors on surfaces in ultra-high vacuum. [14,15] It allowed Fasel et al. [16] to synthesize the symmetric [4,4,4]-tridecastarphenefrom ortho-dibromotetracene on a Ag(111) surface and Grill, Hecht et al. [17] to get decastarphene by coupling of ortho-dibromoanthracene on a Au(111) surface. However, the preparation of an asymmetric starphene by on-surface Ullmann reaction would require a statistical asymmetric coupling of two different 2,3-dihaoloacenes, which is not a satisfactory approach indeed.

In this work, our objective was to test the feasibility of preparation of a large asymmetric starphene by first synthesizing soluble precursors (ie which can be purified, characterized and sublimated) by multistep in-solution synthesis, and in a second step complete the reaction scheme by several surface activated cyclodehydrogenations. To understand the potential and the limits of this strategy, we have designed a Y shape molecular gate comprising one naphthalene branch, and two symmetric dibenzo[a,c]naphthacene as a potential molecular gate. This extension corresponds to the addition of a phenanthrene group to the ends of the anthracene moieties of the [3,3,2]-nonastarphene NAND gate (Fig. 1), [3] ie a total of six supplementary benzene rings.

Scheme 1. Retrosynthetic strategy for the on-surface preparation of the molecular gate 1 from the molecule 2 in its conformation A.

Indeed, this large planar polyaromatic hydrocarbon 1, tetrabenzo[a,c,u,w]naphtho [2,3-l]nonaphene, is expected to be fully insoluble, thanks to strong - intermolecular interactions, so that its synthesis and purification in solution would not be possible. Furthermore, the sublimation of this type of large polyaromatics with strong intermolecular -stacking in the solid state, a necessary step to transfer the molecular gate on ultra clean surfaces in ultra-high vacuum (UHV), is known to be problematic. Increasing the sublimation temperature often leads to fragmentation of the molecules. In these conditions, on-surface synthesis appears as a potential answer to this problem. We therefore decided to synthesize the precursor 2, in which six single bonds together with internal steric crowding should ensure some solubility and after standard purification and characterization steps to transfer this non-planar molecule by sublimation on Au(111) surfaces. One potential expected difficulty was that the on-surface cyclodehydrogenation could yield by-products depending on the conformation of the molecules on the surface (Scheme 2). Indeed, rotations around the bonds linking the naphthalene moieties to the anthracene branches from the conformer A give also the conformers B and C on the surface, so that it should be possible to obtain also two other compounds. Indeed, one of the objectives of this work was to understand how the balance between intramolecular molecular hindering and molecule-surface adsorption energies controls the distribution between these different isomers.

Scheme 2 The three possible products by on-surface-cyclodehydrogenation of 2 in the conformation A (gives 1), B (gives 3), and C (not observed).

Results and Discussion

Synthesis. The synthesis of 2 is described in scheme 3. Our disconnection approach involves breaking down of 2 into a naphthalene moiety and two 2,3-diphenylanthracene units. The preparation starts from 1,2-bis(bromomethyl)-4-methoxybenzene 4 which can be obtained from dimethyl 4-methoxy-1,2-benzenedicarboxylate by reduction of the two ester functions followed by bromination of the two alcohol groups by PBr3 in dichloromethane. [18], [19] Scheme 3. Synthesis of the precursor 2: (a) LiAlH4, THF, 0°C (b) PBr3, DCM, 0°C (c) TMSA (4 equiv.), EtMgBr (4 Metallation of 4 by ethylmagnesium bromide followed by copper coupling with trilmethylsilyl acetylene gave 5. Deprotection by tetrabutylammonium fluoride in THF yielded 6. This latter was involved in [2+2+2] cyclotrimerization with diphenyl ethylene catalysed by nickel [20,21] to get the 6-methoxy-2,3-diphenyl-9,10-dihydroanthracene 7 which was then aromatized by DDQ leading to 8. Cleavage of the methoxy groups by boron tribromide in DCM was followed by triflatation of the so-obtained alcohol groups by triflic anhydride to give 10. Then, 2-(6,7-diphenylanthracen-2-yl)-4,4,5,5-tetramethyl-1,3,2-dioxaborolane compound 11 was obtained from 10 by palladium catalytic coupling with bis(pinacolato)diboron. The final step is a double Suzuki coupling with 2,3-dibromonaphthalene yielding the target molecule 2. This compound is a well-soluble crystalline compound which can be sublimated at 340°C under ultrahigh vacuum (UHV) conditions.

On-surface cyclodehydrogenation. After thermal sublimation of precursor 2 on the Au(111) surface kept at room temperature, the adsorbed molecules have been imaged by scanning tunnelling microscopy (STM) at low temperature (T = 5 K) (Figure 2a). In the STM images we observe only non-planarized precursor molecules 2, thus excluding that intramolecular reactions already happened in the crucible. The precursors form small self-assembled structures due to intramolecular interactions and are characterized by three bright lobes (see also Figure 3a). Such assemblies can be easily separated by STM lateral manipulation. To induce cyclodehydrogenation, we annealed the sample to 200°C and then further to 275 °C. After the final annealing (Figure 2b), all molecules reacted and are flat, adopting either the symmetric Y shape of molecular gate 1 or the asymmetric shape of compound 3, benzodibenzo [7,8,9,10]naphthaceno[2,1-h]phenanthro[9,10-p]hexaphene. The compound which could be obtained by cyclodehydrogenation of the conformer C is not observed on the surface. The molecules are preferentially aligned along the fcc sites of the Au surface. The analysis of 15 images containing about 300 molecules each reveals that the molecules are equally divided between product 1 and 3 (see SI). Therefore, about 50% of the products are molecular gates 1, in contrast to the [3,3,2]-nonastarphene (the NAND gate), for which their percentage was more than 90 %. [3] This difference shows that the extension of the polyaromatic system by four phenyl rings increases significantly the adsorption forces so that even more sterically crowded conformation are stable on the surface. The assignment of the structures is confirmed by the similar experiment performed to obtain the NAND gate. [3] We have performed dI/dV spectroscopy to investigate the positions and the geometry of the electronic resonances for the compound 1. We have identified (Figure 4a) electronic resonances at -1.63 V, -0.93 V, +1.4 V, and +2.38 V. The energy gap is therefore 2.33 eV.

The spatial distribution of the frontier electronic resonances probed by dI/dV mapping at -0.93 V and 1.4 V is shown in Figure 4b and 4c, respectively. Compared to the NAND gate [3,3,2]-nonastarphene, [3] shown in Figure 1, the energy gap has decreased from 2.9 eV to 2.33 eV. This reduction can be explained by the elongation of the two branches which pass from anthracene to dibenzotetracene.

Conclusion

To sum up, we have shown that it is possible to prepare large non-fully symmetric polyaromatic hydrocarbons by on-surface multiple cyclodehydrogenations of large, sterically crowded and soluble precursors. Indeed, the design of these precursors has to consider the possibility to observe several on-surface precursor conformers yielding several products, especially when the physisorption energy increases with the size of the molecule.

Experimental Section

General procedure

Solvents were dried over molecular sieves prior to use. [22] Dry DMF was from Acros Organics or Sigma Aldrich. THF was distilled on sodium/benzophenone. Triethylamine was distilled on CaH2 under argon atmosphere. Flash column chromatography was performed by using silica gel (60 Å pore size, 40-63 µm Merck). The reactions were monitored by thin layer chromatography (TLC) on silica gel-coated plates (Merck 60 F254). The yields refer to chromatographically and spectroscopically (1H-NMR and 13C-NMR) homogeneous materials, unless otherwise stated. The NMR spectroscopic data were recorded with Bruker Avance 300/400/500 MHz instruments and were calibrated by using the residual undeuterated solvent as an internal reference. Chemical shifts are reported in parts per million (ppm) on the δ scale and coupling constants (J) are in Hertz (Hz). The abbreviations used to describe the multiplicities are s= singlet, bs= broad singlet d= doublet, t= triplet= dd= doublet of doublets, q= quintuplet, m= multiplet. Mass spectra were recorded at the Service Commun de Spectrometrie de Masse of University Paul Sabatier (Toulouse 3), Toulouse (France). Elemental analyses were done by the Service d'Analyse de l'ICSN (Paris, France). All reagents of high quality were purchased from commercial suppliers, and used without further purification. Solvents were dried on 4°A activated molecular sieves (3,3'-(4-methoxy-1,2-phenylene)bis(prop-1-yne-3,1-diyl))bis(trimethylsilane) 5. In a three necked round bottom flask equipped with reflux condenser was prepared solution of ethynyltrimethyl silane (1 mL, 6.8 mmol, 4 equiv.) in absolute tetrahydrofurane (20 mL) and cooled to 0°C. A solution of ethyl magnesium bromide (3.0 M in diethyl ether, 2.27 mL, 6.8 mmol, 4 equiv.) was added dropwise and then the reaction mixture was heated to 40 o C for 1 hour. Copper chloride (85 mg, 0.85 mmol, 0.5 equiv.) and the dibromide 4 (500 mg, 1.7 mmol, 1 equiv.) were added and the reaction mixture was heated to reflux for 3 h. The reaction mixture was cooled to room temperature and stirred overnight. The reaction was quenched with aq. solution of NH4Cl and extracted with diethyl ether. The combined organic layers were washed with brine (40 ml) and dried over anhydrous Na2SO4. The solvents were removed in vacuo and the residue was chromatographed on a silica gel (petroleum ether -ethyl acetate 97 : 3) to get the desired product as a viscous yellow oil.

(418 mg, 75%). Rf = 0.61 (petroleum ether/ ethyl acetate = 95/5). [START_REF] Pozo | [END_REF]

4-methoxy-1,2-di(prop-2-ynyl)benzene 6.

The protected diyne 5 (400 mg, 1.22 mmol) was dissolved in tetrahydrofuran (20 ml) under argon and cooled to 0°C. A solution of n-tetrabutylammonium fluoride trihydrate (1 M in tetrahydrofuran, 2.7 ml, 2.7 mmol, 2.2 equiv.) was added dropwise and the reaction mixture was stirred at room temperature for 6h. Then the reaction mixture was quenched by water (20 ml) and extracted with diethyl ether. The combined organic portions were dried over anhydrous Na2SO4. The evaporation of the solvents afforded an orange oil, which was chromatographed on silica gel (petroleum etherethyl acetate 95:5) to get the desired product (157 mg, 70%) as a yellow viscous oil. 6-methoxy-2,3-diphenyl-9,10-dihydroanthracene 7. A well dried flask equipped with a reflux condenser was charged with the diyne 6 (1.24 g, 6.74 mmol), diphenylacetylene (12 g, 67.4 mmol, 10 equiv.) and (Ph3P)2Ni(CO)2 (0.43 g, 0.674 mmol, 0.1 mol%) under argon. The absolute toluene (125 mL) was added and the reaction mixture was placed into a preheated oil bath (135°C) for 1 h. After cooling to room temperature, the solvent was removed in vacuo and the residue was chromatographed on silica gel (cyclohexaneethyl acetate 95 : 5) to afford the desired product (1.28 g, 22%) as a yellow amorphous solid.

6-methoxy-2,3-diphenylanthracene 8.

A well dried Schlenk flask was charged with dihydroanthracene 7 (1.2 g, 3.3 mmol, 1 equiv.), DDQ (1.13 g, 4.97 mmol, 1.5 equiv.) under argon. The absolute toluene (100 mL) was added and the reaction mixture was stirred at 120°C for 3h. After cooling to room temperature, the solvent was removed in vacuo and the residue was chromatographed on silica gel (hexanechloroform 4:1) to afford the desired product (833 mg, 70%) as a yellow crystalline solid. 2,3-bis(6,7-diphenylanthracen-2-yl)naphthalene 2. A Schlenk flask was charged with 11 (480 mg, 1.05 mmol, 3 equiv.), potassium phosphate tribasic (297 mg, 1.40 mmol, 4 equiv.), XPhos Pd G2 (27 mg, 35 µmol, 10 mol%) and 2,3-dibromonaphthalene (100 mg, 0.35 mmol, 1 equiv.) under argon. The freshly distilled THF (10 mL) and water (2.8 mL) was added and the reaction mixture was degassed and stirred at room temperature for 24 h. The reaction mixture was evaporated and chromatographed on silica gel (hexane -chloroform 4:1) to get the desired product (200 mg, 73%) as a yellow crystalline compound. [START_REF] Pozo | [END_REF]

Experimental details.

The STM experiments were performed with a low-temperature (T = 5 K) STM under ultrahigh vacuum conditions (base pressure <10 -10 mbar). The Au(111) substrate was cleaned by repeated cycles of sputtering with Ar ions and annealing at 723 K. The precursors 2 were sublimated from a Knudsen cell at a temperature of 340°C onto the Au(111) surface kept at room temperature. The on-surface reaction was induced by annealing the sample to 275 °C. After sublimation and annealing, the sample was cooled to cryogenic temperature and transferred to the STM without breaking the vacuum. Scanning tunneling spectroscopy measurements were performed using a lock-in technique with a bias modulation amplitude of 40 mV at a frequency of 707 Hz. Voltages are applied to the surface respect to the tip.

Figure 1 .

 1 Figure 1. Polyaromatic logic gates for single molecule computing.

Figure 2 .

 2 Figure 2. Overview STM images (a) After deposition all molecules are non-flat: each precursor shows three bright lobes. (b) Annealing to 275 °C leads to a complete reaction and planarization of all molecules. All images: U = 0.5 V, I = 100 pA, 30 nm x 30 nm.

Figure 3 .

 3 Figure 3. STM images of single molecules at the different reaction steps. (a) and (b): molecule 2 after deposition, in the conformation A, and (c) in the conformation B. After annealing at T = 200°C, in some cases, a partial cyclodehydrogenation can be observed between the anthracene sub-units (d) for the isomer 1; (f) for the isomer 2. (e, molecule 1) and (g, molecule 3): after annealing at 275° C, the cyclodehydrogenation is complete, and the images show only planar molecules 1, and 3, respectively for the isomer A and B. Imaging parameters: U = 0.5 V, I = 100 pA, image size 3.3 nm x 3.3 nm.

Figure 4 .

 4 Figure 4. Scanning tunnelling spectroscopy of the product molecule 1. (a) dI/dV spectrum. In the inset, STM image of a single molecule with stars marking the positions where the spectra were taken. The colour of each spectrum corresponds to the color of the corresponding star. (b-c) dI/dV maps of the frontier resonances recorded at (b) U = -0.93 V and (c) U =1.4 V.

  Then temperature was allowed to increase to room temperature and stirring continued for another 8 h. The reaction mixture was quenched with cold water and then extracted with DCM. The combined organic portions were dried over anhydrous MgSO4. The solvents were removed in vacuo and the residue was chromatographed on a short plug (4cm) of silica gel (hexane-chloroform 1:4) to get the desired product (730 mg, 99%) as a brown In a dry Schlenk flask was dissolved previous hydroxyl compound 9 (730 mg, 2.11 mmol) in dichloromethane (30 mL) under argon and cooled to 0°C. Triethyl amine (879 μL, 6.32 mmol, 3 equiv.) was added dropwise and the yellow solution was stirred for 5 min. A solution of triflic anhydride (1M solution in dichloromethane, 2.74 mL, 2.74 mmol, 1.3 equiv.) was added dropwise. The reaction was stirred 1 hour at 0°C and then 1 hour at room temperature. The colour of the reaction mixture turn slowely to brown. The reaction mixture was adsorbed on silica gel and the volatiles were removed in vacuo and the residue was chromatographed on silica gel (hexanedichloromethane 5:1) to get the desired product (950 mg, 94%) as a yellow amorphous solid.[START_REF] Pozo | [END_REF] H NMR (300 MHz, CD2Cl2): 7.26-7.30 (10H, m), 7.37 (1H, dd, J = 9.3, 2.5), 7.94 (1H, d, J = 2.4), 8.06 (1H, s), 8.07 (1H, s), 8.11 (1H, d, J = 10.0), 8.49 (1H, s), 8.52 (s, 1H).13 C NMR (75 MHz, CD2Cl2): 119.62, 119.65 (J = 321 Hz, CF3), 119.90, 127.13, 127.18, 127.49 (2C), 128.58 (4C), 129.85, 130.09, 130.50 (4C), 130.75, 131.36, 131.82, 132.02, 132.12, 140.38, 140.55, 141.71 (2C), 147.32. DCI MS: 479 ([M+H] + ). HR DCI MS: calcd for C27H18F3O3S 479.0923, found 479.0942.

	6,7-diphenylanthracen-2-yl trifluoromethanesulfonate 10. 2-(6,7-diphenylanthracen-2-yl)-4,4,5,5-tetramethyl-1,3,2-dioxaborolane 11. A well dried Schlenk flask was charged with previous triflate 10 (940 mg,
	1.97 mmol), PdCl2(dppf) (58 mg, 78.6 μmol, 4 mol%), dppf (44 mg, 78.6 μmol, 4 mol%), bis(pinocolato)diboron (748 mg, 2.95 mmol, 1.5 equiv.) and
	potassium acetate (578 mg, 5.89 mmol, 3 equiv.) under argon. The absolute dioxane (50 ml) was added and the reaction mixture was stirred at 80°C for
	24 h. The reaction mixture was evaporated and chromatographed on silica gel (gradient hexane -dichloromethane 4:1 to dichloromethane) to get the
	desired product (820 mg, 91%) as a foam yellowish compound. H NMR (300 MHz, CD2Cl2): 1.44 (12H, s), 7.30 (10H, s), 7.81 (1H, d, J = 8.5), 8.03 (1H,
	d, J = 8.6), 8.07 (1H, s), 8.10 (1H, s), 8.47 (1H, s), 8.56 (1H, s), 8.60 (1H, s). 13 C NMR (75 MHz, CD2Cl2): 25.36 (4C), 84.57 (2C), 126.38, 127.19, 127.22,
	127.85, 128.41 (4C), 129.88, 130.03, 130.35, 130.49 (4C), 131.59, 132.07, 132.32, 133.67, 137.79, 139.55, 140.04, 142.04 (2C). DCI MS: 457
	([M+H] ). HR DCI MS: calcd for C32H30BO2 457.2333, found 457.2350.
	1 H NMR (300 MHz, CD2Cl2): 3.98 (3H, s), 7.18 (1H, dd, J = 9.2, 2.5), 7.25-7.30 (11H, m), 7.93 (1H,
	d, J = 9.3), 8.01 (1H, s), 8.02 (1H, s), 8.34 (1H, s), 8.40 (1H, s). 13 C NMR (75 MHz, CD2Cl2): 55.85, 104.16, 121.22, 124.50, 126.63, 127.03, 127.10,
	128.36 (4C), 129.23, 129.52, 130.15, 130.21, 130.34, 130.49 (4C), 132.07, 133.77, 138.57, 139.66, 142.19, 142.23, 157.97. DCI MS: 361 ([M+H] + ). HR
	DCI MS: calcd for C27H21O 361.1587, found 361.1509.
	6,7-diphenylanthracen-2-ol 9. A methoxy derivative 8 (770 mg, 2.14 mmol) was dissolved in dry dichloromethane (70 mL) under argon and cooled to
	0°C. A solution of BBr3 (1M solution in dichloromethane, 6.41 mL, 6.41 mmol, 3 equiv.) was added dropwise and the reaction mixture was stirred at 0°C
	for 30 min. amorphous solid. 1 H NMR (300 MHz, CD2Cl2): 5.67 (1H, s), 7.18 (1H, dd, J = 9.1, 2.5), 7.26 -7.28 (11H, m), 7.96 (1H, d, J = 9.2), 7.98 (1H, s), 8.01 (1H,
	s), 8.26 (1H, s), 8.41 (1H, s). 13 C NMR (75 MHz, CD2Cl2): 108.07, 120.07, 124.09, 126.82, 127.04, 127.12, 128.36 (4C), 129.05, 129.51, 130.19 (2C),
	130.48 (4C), 130.92, 132.13, 133.68, 138.60, 139.74, 142.16, 142.18, 153.85. DCI MS: 347 ([M+H] + ). HR DCI MS: calcd for C26H19O 347.1430; found
	347.1423.

  H NMR (300 MHz, CD2Cl2): 7.21 -7.26 (20 H, m), 7.31 (2H, dd, J = 6.2, 3.3), 7.60 (2H, dt, J = 6.2, 3.3), 7.75 (2H, d, J = 9.0), 7.97 (2H, s), 7.98 (2H, s), 8.01 (2H, dt, J = 6.2, 3.3), 8.13 (2H, dd, J = 1.7, 0.8), 8.16 (2H, s), 8.35 (2H, s), 8.37 (2H, s). 13 C NMR (126 MHz, CD2Cl2): 126.32, 126.72, 127.10, 127.13 (2C), 127.91, 128.35 (4C), 128.37, 128.93, 129.13, 129.98, 130.03, 130.45 (4C), 130.50, 131.42, 131.66, 131.75, 132.68, 133.57, 139.43, 139.49, 139.51, 139.59, 142.04 (2C). HR MALDI: calcd for C62H40: 784.3130, found 784.3136.

Acknowledgements

This research has been funded in parts by the Foundation EXPERIENTIA and from ERDF/ESF "UOCHB MSCA Mobility" (No. CZ.02.2.69/0.0/0.0/17_050/0008490), by the ICT-FET European Union Integrated Project PAMS (Agreement No. 610446) and has received funding from the EraNET Cofund Initiatives QuantERA under the European Union's Horizon 2020 research and innovation programme grant agreement ORQUID. Support by the German Research Foundation (DFG) Project number 432345501 is gratefully acknowledged.


Simple Asymptotic BER Expressions for LoRa

System over Rice and Rayleigh Channels Vincent Savaux

Advanced Connectivity Lab b<>com, Rennes, France Email: vincent.savaux@b-com.com Guillaume Ferré Univ. Bordeaux, CNRS, Bordeaux INP, IMS, UMR 5218, F-33400, Talence, France Email: guillaume.ferre@enseirb-matmeca.fr

Reference

Channel Approach exact [START_REF] Proakis | Digital Communications[END_REF] [3]- [START_REF] Proakis | Digital Communications[END_REF] AWGN approximation [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF] Monte-Carlo [3] [8] AWGN+channel coding approximation [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF] AWGN+interference approximation [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] Rayleigh, Rice, Nakagami exact have been suggested, but they are approximations based on error functions. In [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF], the exact expression of the BER for LoRa signal over different channel models has been derived. However, despite their exactness, the BER expressions in [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] are not tractable, and lead to problem of precision. This is mainly due to the fact that the solutions involve sums of binomial coefficients whose full precision computation requires hundreds of bits. We provide in Table I a summary of the literature dealing with BER derivation in LoRa system according to the channel model (AWGN, Rayleigh, etc.) and the approach (exact or approximated BER expression). Note that we have added [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF] and [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF] to enlarge the scope of this overview. In fact, these paper does not deal with BER of LoRa signal over specific channel model, but their BER analyses consider the effect of the channel coding and the impact of interference on the BER performance, respectively.

In this paper we derive a very simple and tractable asymptotic expression of the BER of LoRa signal over Rice and Rayleigh channels from that developed in [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF], which tends to the exact BER for low noise variance value. It is shown through simulations that this approximation matches well the exact BER behaviors [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] in wide BER ranges. Furthermore, the suggested solution only involves straightforward functions. Thus, the proposed BER expressions is a good trade-off between accuracy and simplicity, allowing for a fast assessment of the achievable performance of a LoRa signal over Rice and Rayleigh channels. Moreover, we provide a summary of the literature dealing with more sophisticated BER expressions in different channel models, as well as including the effect of interference [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF] and channel coding [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF].

The rest of the paper is organized as follows: Section II Abstract-This paper deals with the bit error rate (BER) of LoRa signal over Rayleigh and Rice channels. Exact analytical expressions of the BER in LoRa systems have been proposed in the literature, but they are not tractable and raise problems of computation precision. This is mainly due to the fact that these expressions involve sums of binomial coefficients, leading to extremely large numbers and leading to problem of precision. We then hereby suggest asymptotic expressions of the BER, which are at the same time tractable and accurate over a wide BER range. These expressions do not involve sums of binomial coefficients anymore, but only closed-form functions, and can then be easily obtained through computer simulations.

I. INTRODUCTION

Among the digital communication technologies used in the low power wide area network (LPWAN) world, LoRa is the most studied by the scientific c ommunity t o date. This is mainly due to the open deployment strategy chosen by Semtech. The digital communication technique used in LoRa modulation is the chirp spread spectrum (CSS) initially described by Winkler in [START_REF] Winkler | Chirp signals for communications[END_REF]. Despite a high sensitivity to poor time/frequency synchronization, this technique offers many advantages such as i) a constant envelope which allows to operate with power amplifiers ( PAs) i n s aturation m ode and thus to have excellent amplification efficiency, ii) a sensitivity that can be optimized at the cost of a loss of spectral efficiency, iii) a low complexity demodulation based on fast Fourier transform (FFT).

LoRa or more generally CSS has given rise to numerous studies and analyses in recent years. At the physical layer level, one will retain works proposing to optimize the receivers in order to improve the synchronization steps or to find solutions to destructive collisions [START_REF] Laporte-Fauret | An enhanced lora-like receiver for the simultaneous reception of two interfering signals[END_REF]. However, in order to be able to judge the effectiveness of the proposed processing, it is necessary to situate the performance in terms of bit error rate (BER) of the developed algorithms based on the theoretical results. Thus [START_REF] Ferré | LoRa Physical Layer Principle and Performance Analysis[END_REF] and [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF], [START_REF] Proakis | Digital Communications[END_REF] have proposed approximates and closed expressions of the binary error probabilities of LoRa system over additive white Gaussian noise (AWGN) channels. However, LoRa propagation channels are generally non frequency-selective for which only few literature proposes a closed form expression of binary error probability. In [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF], expressions of the BER of LoRa signal over Rayleigh channel is dedicated to the description of the LoRa modulation and demodulation principles and to the introduction of the exact BER expression. The derivation of a very simple approximation of theses closed form expressions is developed in Section III. The simulations results are presented in Section IV, and we draw conclusions in Section V.

II. BACKGROUND

LoRa uses several spreading factors (SF ) to control the bit rate, improve the range, and decrease the energy consumption. Initially, the binary information flow generated from the medium access control (MAC) layer is divided into subsequences, each of length SF ∈ [7 . . . 12]. The set of SF consecutive bits constitutes a symbol. The number of possible symbols is hence equal to M = 2 SF . In the LoRa context, SF also indicates the spreading factor and the relation between the bit rate D b and the symbol rate D s that can be written as: D s = D b /SF . Spread spectrum is obtained through a signal known as chirp that varies continuously and linearly in frequency. When the derivative of the frequency variation is positive, then we deal with an up chirp, conversely it is a down chirp.

When the chirp is up or down over the entire symbol period T s , which is also called the signaling interval, it is identified as raw chirp. Its complex envelope is mathematically expressed as follows, for t ∈ -Ts 2 , Ts 2 :

c(t) = e jθc(t) , (1)
with θ c (t) = ±π B Ts t 2 where B is the frequency excursion of the transmitted chirp. The '+' and '-' signs stand for up and down raw chirp, respectively. If we note f c (t) the chirp frequency variation in time, we have:

f c (t) = 1 2π dθ c (t) dt = ± B T s t. (2)
For digital communication systems with no spreading spectrum, the bandwidth used by the transmitted signal is proportional to the symbol rate. The coefficient of proportionality depends on the shaping filter, which is in general a half Nyquist one. In the CSS modulation used in LoRa, the signal bandwidth is fixed by B which has the following relationship with T s :

M = B × T s . (3)
To distinguish between the M different symbols of the constellation, M orthogonal chirps have to be defined so that each symbol exhibits a specific instantaneous phase trajectory. In the sequel, we denote m i the transmitted symbol at time iT s , with m i ∈ {0, ..., M -1} and i ∈ N. The chirp associated to the i-th transmitted symbol is then obtained from the raw chirp by applying a delay τ i = mi B and the raw chirp outside the interval

[-Ts 2 , Ts 2 ) is cyclically brought back into [-Ts 2 , -Ts 2 + τ i ].
As a consequence, the the instantaneous phase modulated chirp (e.g. up chirp) related to the transmission of the symbol m i breaks down into two parts:

θ i c (t) =π B T s (t -2τ i )t + 2Bt for t ∈ - T s 2 , - T s 2 + m i B θ i c (t) = πB T s (t -2τ i )t for t ∈ - T s 2 + m i B , T s 2 . (4)
Accordingly, the complex envelope of the LoRa symbol transmitting the symbol m i is s i (t) = e jθ i c (t-iTs) . In the following, we consider perfect time and frequency synchronizations and a one tap block fading channel h ∼ N C (µ, σ 2 h ). Thus, |h| obeys a Rice distribution if µ = 0, and a Rayleigh distribution if µ = 0. Moreover, we introduce the Rice coefficient

κ = λ σ 2 h
, where λ = |µ| 2 . Under these assumptions, the received signal sampled at rate T e and denoted by y i (nT e ) can be written as

y i (nT e ) = h × s i (nT e ) + w(nT e ), (5)
where w(nT e ) ∼ N C (0, σ 2 ) represents the complex noise assumed to be white, Gaussian and circular. Furthermore, n is the sample index with n ∈ -N 2 , N 2 -1 , N being the number of samples of a LoRa symbol. At Nyquist rate, we have T e = 1 B , and then M = N . At the receiver, the transmitted symbols are detected by multiplying every received signal y i by the conjugate version of the raw chirp (1) used in the transmitter, i.e. r i (nT e ) = y i (nT e )e -jθc(nTe) . Then, when a non coherent demodulation strategy is used, the optimal estimation of m i can be performed by searching for the maximum of the periodogram of r i (nT e ), computed as

P i [k] = |R i [k]| 2 , where R i [k],
for k ∈ 0, N -1 , is the discrete Fourier Transform (DFT) of N samples from r i (nT e ). Thus, the estimation of m i can be achieved by computing:

mi = N -argmax k∈ 0,N -1 (P i [k]) . (6)
The symbol error probability, denoted by P s , is defined as the probability that mi is not equal to the actual transmitted symbol m i , i.e. P s = P[ mi = m i ]. The BER is then obtained through:

P b = 2 SF -1 2 SF -1 P s . (7)
It has been proved in [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] that, in the absence of prior information, all the symbols can be considered as equiprobable so that the symbol error probability over a Rice channel, denoted by P Rice s , is expressed as

P Rice s = - N -1 k=1 N -1 k (-1) k σ 2 kN σ 2 h + (k + 1)σ 2 × exp - λkN kN σ 2 h + (k + 1)σ 2 . (8)
It must be noticed that the error probability over Rayleigh channel (denoted by P Ray s in the following) is straightforwardly deduced from (8) by setting λ = 0. Alternatively, the symbol error probability in AWGN channel is obtained by setting (λ = 1, σ 2 h = 0). In any case, despite the exactness of the expression [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF], it requires the computation of the high-valued binomial coefficient N -1 k . This may suffer from precision problems since we remind that, in LoRa we have N ∈ {128, 256, .., 4096}. It results that ( 8) is hardly computable in practice nor tractable, therefore limiting its application to theoretical results. This is the reason why we hereby provide simple expressions of P Rice s and P Ray s in asymptotic cases where the noise variance value is low.

III. ASYMPTOTIC BER EXPRESSION

In this section we provide simple expressions of the asymptotic BER value for LoRa system over Rice and Rayleigh channels.

Proposition. Let the noise variance value σ 2 be low enough to consider that k ≥ 1 >>

1 N. σ 2 h σ 2 +1
, and kN σ 2 h >> (k + 1)σ 2 . Then, for any N = 2 SF , we can make the following approximation

P Rice s ≈ e -κ × γ + ln(N -1)
N.

σ 2 h σ 2 + 1 , ( 9
)
where γ is the Euler-Mascheroni constant γ ≈ 0.577.

Proof. According to the assumption of low σ 2 value, then, for any k = 0, 1, .., N -1, the exponential functions in (8) can be approximated as

exp - λkN kN σ 2 h + (k + 1)σ 2 ≈ exp(-κ). ( 10
)
The substitution of (10) into (8) then leads to

P Rice s ≈ -e -κ N -1 k=1 N -1 k (-1) k σ 2 kN σ 2 h + (k + 1)σ 2 . (11)
Then, in conditions where 1 >>

1 N. σ 2 h σ 2 +1
, P Rice s in (11) can be simplified as

P Rice s ≈ - e -κ
N.

σ 2 h σ 2 + 1 N -1 k=1 N -1 k (-1) k k . (12)
Moreover, we consider N large enough to assume that the approximation

N -1 k ∼ N →+∞ (N -1) k k!
holds, which yields

P Rice s ≈ - e -κ
N.

σ 2 h σ 2 + 1 N -1 k=1 (N -1) k k.k! . (13)
Then, let

E 1 (z) = +∞ z e -t t dt,
with z ∈ C\R -, be the exponential integral function, defined in [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. The series representation of E 1 is

E 1 (z) = -γ -ln(z) - +∞ k=1 (-1) k z k kk! . (14)
Since lim z→+∞ E 1 (z) = 0, and reminding that we assume N → +∞, then the substitution of z by (N -1) in the sum of ( 14) and then in (13) leads to [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], which concludes the proof.

The asymptotic error probability in Rayleigh channel is straightforwardly obtained when κ = 0, leading to

P Ray s ≈ γ + ln(N -1)
N.

σ 2 h σ 2 + 1 . (15)
It should be noted that ( 9) and ( 15) are more tractable expressions than (8) as they only involve usual functions. Furthermore, the suggested approximations do not require the computation of N -1 k , which leads to problems of computation precision, especially when N is large (e.g. N = 1024, 2048, 4096). In the following, we investigate through simulations the accuracy and the domain of validity of our approximations of the error probability.

IV. SIMULATIONS RESULTS

Simulations have been performed to verify the accuracy of the suggested approximations of the error probability in ( 9) and (15). Theoretical results have been obtained using Python 3.6 with scipy and gmpy2 packages for mathematical computing and precision improvement, respectively 1 . It must be emphasized that the theoretical results presented in [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] had been validated through simulations. As a consequence, we do not reproduce the Monte-Carlo simulations to avoid excessive plots.

Fig. 1 shows the BER performance of LoRa system over Rayleigh and Rice channel models versus 1/σ 2 (in dB) for SF7 (a) and SF12 (b). Other series of simulations have been undertaken for SF8 to SF11, but the results are not shown in this paper, as the BER performance is equivalent to that of SF7, except that the BER curves are shifted to the left (a shift of about -3 dB per SF). The performance of the exact expression (8) obtained from [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] (referred as "exact") is compared with that of suggested approximations ( 9) and (15) (referred as "approx.") for κ = 0 (Rayleigh), κ = 1, and κ = 4.

It can be observed in Figs. 1-(a) and (b) that the larger the Rice coefficient κ, the lower the BER. This is a very wellknown result since asymptotically, the case κ = +∞ corresponds to the AWGN channel, which is a lower performance bound. Otherwise, we notice that the domain of validity of the suggested approximations depends on the coefficient κ. In fact, we can see that in the Rayleigh case (κ = 0), the asymptotic expression well matches the exact performance for BER values lower than 10 -1 , which is relevant as a BER range larger than 10 -1 is usually not of interest in practice. For the 1 The code can be found at https://github.com/b-com/ber LoRa (b) SF12. Rice channel with κ = 1, we observe that the approximation well matches the exact expression for any BER value, and for κ = 4, the asymptotic expression of the BER well matches the exact one for BER values lower than 10 -3 . We deduce that the suggested approximations are more relevant for lower values of the Rice coefficient κ, which is consistent with theory since no asymptotic BER expression can be derived for the AWGN channel corresponding to κ = +∞ [START_REF] Proakis | Digital Communications[END_REF].

The accuracy of the suggested BER expressions compared with the exact one is very interesting in practice, especially since (8) is not tractable, and raises some computing precision problems. In fact, to give an idea of the required precision, the full precision computation of the binomial coefficient N -1 k in [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF] in SF7 requires at least 150 bits, and even up to 4000 bits in SF12 (e.g. 4096 200 returns infinity in Matlab), which can be achieved only with the use of the package gmpy2 in python. This practically shows the relevance of the suggested asymptotic BER expressions, in addition to their accuracy.

V. CONCLUSION

In this paper, we have derived simple asymptotic expressions of the BER of LoRa signal over Rice and Rayleigh channel, considering low noise variance value. We have shown through simulations the validity of the suggested approximated expressions compared with the exact one. The advantage of the proposed analysis is to obtain tractable and easily computable solutions, whereas the computation of the exact BER expression may raise problems of precision, or requires large variables sizes. In general, the suggested expressions well fit the exact BER value over a wide BER range. Moreover, an overview of the available literature dealing with BER analysis in LoRa system according to different models and approaches has been provided as well.

Fig. 1 .

 1 Fig.1. BER versus 1/σ 2 (dB) for SF7 (a) and SF12 (b), comparison of the exact BER expression[START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF], and the suggested approximations (9) and (15) for κ = 0 (Rayleigh), κ = 1, and κ = 4.

TABLE I OVERVIEW

 I OF BER ANALYSIS OF LORA SYSTEM ACCORDING TO DIFFERENT MODELS AND APPROACHES.


Vincent Savaux
email: vincent.savaux@b-com.com

Christophe Delacourt
email: christophe.delacourt@b-com.com

Patrick Savelli
email: patrick.savelli@b-com.com

Considering Sync Word and Header Error Rates for Performance Assessment in LoRa System

Keywords: LoRa, Error Rate, Channel

This paper deals with a general and complete analysis of the frame error rate in LoRa system, not only considering the payload error rate, but also the sync word (carrying network information) and the header (carrying control information) error rates. It is proved that, in the high signal-to-noise ratio (SNR) range, it is more likely to reject a LoRa frame due to an erroneous sync word estimation rather than an erroneous header or payload decoding, due to the fact that it is not coded. The theoretical results are then verified through simulations.

the quality and the rigor of their theoretical developments, we consider that the analysis in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF] leads to what we call the payload error rate (PLER) instead of the FER. In fact, such as depicted in Fig. 1, a LoRa frame is rejected, i.e. not transmitted to the network, not only if the payload is erroneously decoded, but also if the header (containing control information such as the payload size or the coding rate applied to the payload) is erroneously decoded, or if the sync word (two LoRa symbols within the preamble that carry the network information) is badly estimated. Thus, considering only the PLER limits the LoRa performance analysis to a theoretical use case, whereas in practice the FER analysis requires the sync word and the header error rates (SWER and HER respectively) besides the PLER.

In this paper, we extend the results in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF] to suggest a more general FER expression considering at the same time the contributions of the SWER, the HER and PLER. To this end, we also consider the more general Rice channel model [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF] than the AWGN as in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF]. Moreover, an asymptotic performance analysis in the high SNR values region is also performed. The obtained theoretical results prove that above a given SNR value, the FER is dominated by the contribution of the sync word estimation errors. This is a interesting and counter-intuitive result as it shows that a LoRa frame may be rejected whereas the SNR would be sufficient for a proper payload decoding. The reason is that the sync word is not coded whereas the header and the payload are. Nevertheless, it is worth mentioning that the sync word is mandatory for the network deployment and then cannot be omitted in practical use cases. Thus, the FER considers the performance of the overall LoRa system whereas the BER is limited to a performance analysis considering the integrity of the data binary stream only. The theoretical results are then verified

I. INTRODUCTION

The last few years have seen the large scale deployment of Internet of things (IoT) networks. Among the different emerging technologies, LoRa addresses the low power wide area network (LPWAN) segment. LPWAN relies on low cost, long battery life devices, able to achieve long range data communication with low data rates. These requirements typically target remote sensing for smart cities, smart agriculture or industrial applications to name a few. The LoRa technology is originally based on a proprietary physical layer (PHY) developed by Semtech corp., and the system has been promoted by the LoRa Alliance, that specifies t he L oRaWAN o pen protocol.

The performance of the LoRa PHY in terms of bit error rate (BER) is being widely studied in the literature. In [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF], [START_REF] Ferré | LoRa Physical Layer Principle and Performance Analysis[END_REF], the LoRa receiver BER is analyzed in additive white Gaussian noise (AWGN) channel model. Thus, based on the results in [START_REF] Proakis | Digital Communications[END_REF], the authors of [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF] suggest an analytical but complex BER expression as well as simpler approximations, while in [START_REF] Ferré | LoRa Physical Layer Principle and Performance Analysis[END_REF] the BER is partially derived through a Monte-Carlo method. The theoretical BER performance of a LoRa signal receiver over more general channel models such as Rayleigh, Rice, and Nakagami is dealt with in [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF], and approximations for large SNR values are provided in [START_REF] Savaux | Simple Asymptotic BER Expressions for LoRaSystem over Rice and Rayleigh Channels[END_REF]. Alternatively in [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF], the BER performance issue is tackled by considering interferers, and the effect of the channel coding is dealt with in [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF] using both theoretical and experimental approaches.

However, all these performance analyses are carried out at the symbol or at the bit level, which does not reflect t he fact that a whole LoRa payload within a LoRa frame is considered as erroneous as soon as one bit is erroneously decoded. This is the reason why the authors in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF] suggest a more thorough performance analysis of what they called frame error rate (FER), considering the whole payload where the binary stream is encoded (Hamming code) and interleaved. Despite through simulations of LoRa signals in both AWGN and Rayleigh channels.

The rest of the paper is organized as follows: Section II presents the LoRa frame format and modulation scheme, as well as the demodulation process. It also provides the background related to the LoRa performance analysis. We develop the FER expression in Section III including the asymptotic analysis, and simulations results are shown in Section IV. Finally, we draw our conclusions in Section V.

Notations: P(A) means the probability of the event A, and F(.) is the Fourier transform operation. The notations P b , P s , P sw , P h P pl , and P f indicate the binary, symbol, sync word, header, payload, and frame error probabilities, respectively.

II. BACKGROUND ON LORA SYSTEM AND PERFORMANCE A. LoRa Signal Model

The LoRa modulation is based on chirp spread spectrum (CSS) featuring different spreading factors (SF ), allowing to control the bit rate and improve the range of the signal. The chirp symbol, originally described in [START_REF] Winkler | Chirp signals for communications[END_REF], is a simple sine wave x(t) = e jfc(t)t , t ∈ [0, T s ] where T s is a symbol period, and the instantaneous frequency f c (t) varies linearly with time within the range

[-Bw 2 , Bw 2 ]
, where B w is called the frequency excursion of the chirp. The binary stream provided by the medium access control (MAC) layer is split by the PHY layer into sequences of SF bits each, SF ∈ {7, 8, .., 12}, which are in turn Gray-mapped in one of the M = 2 SF possible chirp-based symbols. The modulation consists in a cyclic frequency shift of the base chirp x(t). The way the LoRa symbols are built from the chirp x(t) is described in numerous papers dealing with LoRa modulation [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF], [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF], [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF], [START_REF] Winkler | Chirp signals for communications[END_REF], and not further detailed in this paper as it is not mandatory for a good understanding of the hereby analysis.

A LoRa frame consists in three main parts as depicted on Fig. 2 andhereby described: 1) The preamble is used for detection of the LoRa frame and the time-frequency synchronization. Furthermore, it includes the sync word composed of two symbols, which carries the network identifier. The sync word is not coded.

2) The header contains the control information, such as the payload length and the channel coding rate applied to the payload. As described in [START_REF] Semtech | Datasheet SX1272/73[END_REF], it is always composed of eight symbols, and the binary control information is coded with a Hamming code [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF][START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF], where the first and second integers denote the word length and the codeword length, respectively.

3) The payload carries data composed of N pl LoRa symbols, and the data is coded with Hamming code (4, n H ) where n H ∈ {5, 6, 7, 8}. In the following, we assume that the LoRa bandwidth is narrow enough to consider a one-tap block fading channel h. Then, if we denote by x i (t), i ∈ N, the i-th transmitted LoRa symbol within a frame, the received symbol y i (t), under the assumptions of perfect time-frequency synchronization, can be expressed as

y i (t) = hx i (t) + w i (t), (1)
where w i (t) ∼ N C (0, σ 2 ) is the complex AWGN. Furthermore, we suppose that |h| obeys a Rice distribution, i.e. h ∼ N C (µ, σ 2 h ), as it is a general model that includes other usual channel models, line-of-sight (LOS) (where µ = 0) as well as non-LOS ones (where µ = 0). Since the chirp has a normalized module, the signal-to-noise ratio can be expressed

as SN R = E{|h| 2 } σ 2 , which reduces to SN R = 1 σ 2 in AWGN channel.
The demodulation and detection of the transmitted symbol from the observation (1) is deeply detailed in [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF], [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF], [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF], [START_REF] Winkler | Chirp signals for communications[END_REF]. The basic principle consists in multiplying the observation y i (t) by the conjugate of the base chirp, i.e. x(t) * . Then, the frequency index corresponding to the maximum of the periodogram |F(y i (t))x(t) * | 2 yields the estimated transmitted symbol. If we denote by m ∈ {0, 1, .., M -1} the actual transmitted symbol and by m its estimate, then the probability that the symbol is erroneously estimated is defined as P s = P( m = m). The remaining of the paper deals with the analysis of the error performance in the LoRa system.

B. LoRa Performance

Numerous papers deal with the performance of the LoRa system according to both BER [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF], [START_REF] Ferré | LoRa Physical Layer Principle and Performance Analysis[END_REF], [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF], [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF], [START_REF] Faber | A Theoretical and Experimental Evaluation on the Performance of LoRa Technology[END_REF] and payload error rate (PLER) [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF]. The latter is especially relevant as it reflects that, in practice, the whole payload is considered as erroneous and therefore rejected (i.e. not forwarded to the network) if any constituting bit is badly decoded. It has been shown in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF] that a good approximation of the PLER, denoted by P pl , can be expressed as

P pl = 1 -(1 -P b ) n H + n H P b (1 -P b ) n H -1 N pl SF n H , (2)
where P b is the bit error rate. In [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF], the authors focus their analysis to the AWGN channel and use the approximated expression of P b such as suggested in [START_REF] Elshabrawy | Closed-Form Approximation of LoRa Modulation BER Performance[END_REF]. We hereby extend the analysis to the more general Rice channel model. In fact, it has been proved in [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] that, in the absence of prior information, all the symbols can be considered as equiprobable so that the BER of the LoRa signal over a Rice channel, denoted by P Rice b , is expressed as AWGN+channel coding approximation [START_REF] Afisiadis | On the Error Rate of the LoRa Modulation with Interference[END_REF] AWGN+interference approximation [START_REF] Dias | Bit Error Rate Closed-Form Expressions for LoRa Systems under Nakagami and Rice Fading Channels[END_REF] Rayleigh, Rice, Nakagami exact [START_REF] Savaux | Simple Asymptotic BER Expressions for LoRaSystem over Rice and Rayleigh Channels[END_REF] Rayleigh, Rice approximation

P Rice b = - 2 SF -1 2 SF -1 M -1 k=1 M -1 k (-1) k σ 2 kM σ 2 h + (k + 1)σ 2 × exp - λkM kN σ 2 h + (k + 1)σ 2 , (3)
where λ = |µ| 2 is called the non-central parameter. It must be noticed that the BER over Rayleigh and AWGN channels are straightforwardly deduced from (3) by setting λ = 0 (Rayleigh) and (λ = 1, σ 2 h = 0) (AWGN), respectively. However, it must be emphasized that (3) is not tractable, and that the computation of M -1 k may raise accuracy problems. This is the reason why approximations of the exact error probability (3) have been suggested for different channel models. Thus, Table I summarizes the approaches for LoRa BER analysis considering different channel models, which can be used for the following suggested FER expression.

However, although the PLER in ( 2) is a relevant performance indicator (note that a more sophisticated PLER approximation is also provided in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF]), it does account for all the possible LoRa frame rejection as illustrated in Fig. 1. In fact, a LoRa frame is indeed rejected if the payload is erroneously decoded (highlighted by [START_REF] Ferré | LoRa Physical Layer Principle and Performance Analysis[END_REF]), but also if the sync word is not properly estimated (which means the network is not recognized) or if the header is badly decoded (the control information required for the payload demodulation is erroneous). Thus, the complete FER expression including both SWER and HER in addition to PLER is hereby developed in Section III.

III. FRAME ERROR RATE EXPRESSION

In this section we derive the expressions of the sync word and header error rates, leading to the general FER.

A. Sync Word Error Rate

As aforementioned, the sync word is composed of two noncoded symbol. Thus, since the symbol error probability P s is obtained from the bit error probability P s through

P s = 2 SF -1 2 SF -1 P b , (4)
then the probability that the sync word is badly estimated corresponds to the probability that at least one among the two independent symbols (in terms of observations) is badly estimated, which is expressed as

P sw = 1 -(1 -P s ) 2 , (5)
where [START_REF] Proakis | Digital Communications[END_REF] or any of the BER expressions developed in the literature (see Table I) can be substituted in (4) and then in [START_REF] Savaux | Simple Asymptotic BER Expressions for LoRaSystem over Rice and Rayleigh Channels[END_REF]. Moreover, note that the term (1 -P s ) 2 in (5) corresponds to the probability that both symbols are correctly estimated.

B. Header Error Rate

The header is composed of N pl = 8 symbols coded with Hamming code (4, n H = 8) [START_REF] Semtech | Datasheet SX1272/73[END_REF], therefore the header error rate P h can be deduced from (2) as

P h = 1 -(1 -P b ) 8 + 8P b (1 -P b ) 7 SF . (6)

C. Frame Error Rate

A frame is rejected when the sync word is badly estimated, or when the header or the payload is badly decoded. Thus, the general frame error probability P f can be expressed from ( 2), ( 5), and (6) using the inclusion-exclusion principle as

P f = 1 -(1 -P sw )(1 -P h )(1 -P pl ). (7)
It must be noticed in ( 7) that the FER is subject to the contributions of two coded sequences (the header and the payload), and one non-coded sequence (the sync word). We hereby develop an asymptotic analysis of the error probabilities P sw , P h , and P pl when the noise variance σ 2 reaches low values (or equivalently when the SNR is high).

D. Asymptotic Performance

In the following, we assume that σ 2 << 1 such that P b << 1. In this asymptotic condition, we analyze which contribution of the error rate P sw , P h , and P pl is dominant to the general FER P f , i.e. which operation among the sync word estimation, the header decoding and the payload decoding is more likely to be erroneous. To this end, we rewrite P pl in (2) by using the Newton's general binomial theorem. For clarity purpose, we define β =

N pl SF n H , then β ∈ Q * + . Then, since P b << 1 we have (1 -P b ) n H > n H P b (1 -P b ) n H -1 > 0,
allowing us to develop (2) as

P pl =1 - +∞ k=0 β k (1 -P b ) n H (β-k) (n H P b ) k (1 -P b ) (n H -1)k =1 - +∞ k=0 β k (n H P b ) k (1 -P b ) n H β-k , (8)
where β k is the generalized binomial coefficient, and (1 -P b ) n H β-k can also be rewritten by using the Newton's general binomial theorem:

P pl =1 - +∞ k=0 β k (n H P b ) k +∞ m=0 n H β -k m (-1) m P m b , = - +∞ k=1 β k (n H P b ) k +∞ m=0 n H β -k m (-1) m P m b , - +∞ m=1 n H β m (-1) m P m b . (9)
By keeping the terms of smallest degrees in [START_REF] Winkler | Chirp signals for communications[END_REF], we obtain, after some developments:

P pl = βn H (n H -1) 2 P 2 b + o(P 2 b ) = N pl SF (n H -1) 2 P 2 b + o(P 2 b ). (10)
Similarly, we have

P h = SF n H (n H -1) 2 P 2 b + o(P 2 b ). (11)
Furthermore, the development of (4) yields

P sw = 2 2 SF -1 2 SF -1 P b - 2 SF -1 2 SF -1 2 P 2 b . (12)
Then, by assuming that N pl > n H (which is always true in practice), we obtain the asymptotic limits:

lim P b →0 P h P pl = n H N pl < 1, (13)
lim

P b →0 P pl P sw = N pl SF (n H -1)2 SF -1 4(2 SF -1) P b = 0. ( 14
)
We deduce from (13) and ( 14) that, asymptotically we have P h < P pl < P sw . Moreover, we deduce from (13) that P h and P pl have a similar behavior, and from (14) that P sw decreases slower than P pl (of order O(P b ) for P sw versus order O(P 2 b ) for P pl ). This means that it exists an SNR value from which we have P pl < P sw . In practice, it signifies that in high SNR conditions (e.g. a device is close to the gateway), it is more likely to erroneously estimate the sync word and reject the frame, even when the header and the payload could be properly decoded. This behavior is verified through simulations, as presented in next Section.

IV. SIMULATIONS RESULTS

The simulations have been carried out using python 3.6 with scipy and gmpy2 packages used for mathematical computing and precision improvement, respectively. The code can be found at https://github.com/b-com/fer swer LoRa. In fact, it has been stated in [START_REF] Savaux | Simple Asymptotic BER Expressions for LoRaSystem over Rice and Rayleigh Channels[END_REF] that the computation of (3) leads to precision issues, due to the binomial coefficient M -1 k where M is up to 4096. In all simulations, we use similar parameters as [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF], i.e. the Hamming code for the payload is (k = 4, n H = 7), and the payload length is N pl = 32. Note that it has been shown in [START_REF] Afisiadis | Coded LoRa Frame Error Rate Analysis[END_REF] match the the theoretical performance, therefore we can hereby focus on theoretical performance. Fig. 3 shows the performance in terms of SWER, HER, PLER, and FER versus SNR (dB) for LoRa signal featuring SF7 over an AWGN channel. The theoretical results obtained from the asymptotic analysis are verified in Fig. 3. In fact, the HER and PLER have a similar behavior, in the sense that they have the same decreasing rate. Furthermore, SWER decreases slower than the HER and PLER, such that the FER is asymptotically equivalent to the SWER in the higher SNR range, and we observe that SWER>PLER for SN R > -9 dB.

Other series of simulations have been undertaken for a LoRa signal featuring SF7 over Rayleigh channel. The corresponding performance versus SNR (dB) is shown in Fig. 4. Similar comments as previously can be drawn from Fig. 4, except that the error rates are higher than those in Fig. 3, due to the random nature of the considered channel. In that case, we find that SWER>PLER for SN R > 5 dB. Thus, we conclude from Figs. 3 and4 that considering the BER performance only does not reflect all the source of errors and then underestimates the overall frame error rate.

V. CONCLUSION

In the LoRa protocol, a frame can be rejected if the frame header or the payload is badly decoded, or if the network identifier (sync word) is erroneously estimated. In this paper, we have proposed an improved expression of the frame error rate taking into account the three possible causes of rejection. Our analysis, corroborated with simulations, tend to prove that the impact of the SWER is not negligible in the whole SNR range and even tends to weaken the LoRa performance in the higher SNR range. To alleviate this problem, it may be possible to ignore the sync word at the PHY level and to let the MAC layer be responsible of frame rejection.

Fig. 1 .

 1 Fig.1. Overview of LoRa decoding process: the sync word, the header, and the payload must be properly estimated/decoded otherwise the frame is rejected.

Fig. 2 .

 2 Fig. 2. A LoRa frame is composed of a preamble including the sync word (2 symbols), the header composed of eight symbols, and the payloads composed of N pl symbols.

Fig. 4 .

 4 Fig. 4. Error rate versus SNR (dB), in Rayleigh channel and SF7.

  that the Monte-Carlo simulations

		10 0
		10 -1
	Error rate	10 -2		SWER
				HER
				PLER
				FER
		10 -3
		10 -4	-18	-16	-14	-12	-10	-8	-6	-4	-2
							SNR (dB)
	Fig. 3. Error rate versus SNR (dB), in AWGN channel and SF7.


MD Pauline Potet
email: paulinepotet@hotmail.fr

MD Guillaune De Bonnecaze

MD Emilien Chabrillac

MD Agnès Dupret-Bories

MD Sébastien Vergez

MD Benoit Chaput

Closure of radial forearm free Oap donor site: A comparative study between keystone flap and skin graft

Keywords: esthetic and functional results, donor site rnorbidity, keystone flap, radial forearrn free flap, skin graft ORCID

Background:

The aim was to investigate the feasibility of radial foreann free flap (RFFF) donor site closure by keystone flap (KF) and compare its outcomes to those of skin graft (SG) closure.

Methods:

One hundred and one patients who underwent RFFF for head and neck reconstruction were included (35 KF closure and 65 SG closure). Duration of wound healing and donor site complications was collected. After a minimal follow-up of 1 year, patients were questioned about functional and esthetic impairment.

Results: Coverage of donor site by KF was successful in ail cases. The duration of wound healing was longer after SG than after KF (32 days vs 18 days, P < .001). Healing complications, esthetic and functional results were not statistically different.

Conclusion:

Forearm donor site closure by KF is a feasible alternative to the tradi tional SG. lts main advantages are the reduced wound healing time and the avoid ance of a second donor site.

The fasciocutaneous radial forearm free flap (RFFF), increasingly used since Yang's description in a large series in 1981, [START_REF] Yang | Foreann free skin flap transplanta tion[END_REF] is commonly used in reconstructive surgery for head and neck tumors. Since the survival rate of the flap approaches 95%, 2 the attention of the surgeon has gradually focused on postoperative quality of life, most particularly on the donor site morbidity, which is often underestimated. [START_REF] Harris | Minimizing free flap donor site morbidity[END_REF] We propose to adapt the use of keystone local flap to the closure of RFFF donor site. Keystone perforator island flap is a multiperforator advancement fl ap based on musculocutaneous or fasciocutaneous perforators, described by Behan in 2003. [START_REF] Behan | The keystone design perforator Island flap in recontructive surgery[END_REF] This study presents the Keystone local flap surgical tech nique and the outcomes of the RFFF donor site closure with both techniques.

Our objective was to prove the feasibility of closure by keystone flap (KF) and to compare its outcomes with skin graft (SG) in terms of wound healing, esthetic, and func tional impairment.

Many possibilities of donor site closure have been described, ranging from direct closure to local flaps, includ ing thin or full skin grafting, artificial dermis, expanders, or negative pressure therapy. Although skin grafting is the most frequently used technique, the question of an ideal way to close the donor site remains unanswered. [START_REF] Pabst | Is there an ideal way to close the donor site of radial forearm free flaps?[END_REF]

1 METHODS

We performed a monocentric retrospective study from January department of the University Institute of Cancer in Toulouse, France. We included 101 patients who received RFFF for reconstruction after head and neck tumor. Patient's agreement was collected in order to use their data for research and publica tion purposes.

We divided the population into two groups: a group of 36 patients for which a first surgical team systernatically and consecutively perforrned a KF (group A), and a group of 65 patients for which a second surgical team systematically and consecutively perforrned a SG (group B).

Patients' dernographics and characteristics were collected in the medical record: age , sex, co-morbidities (including tobacco use, diabetes, and cardiovascular disease), local complications, wound healing disorders and healing duration (in days).

The area of substance loss was calculated using the length and width of the antebrachial flap harvested (elliptical area for Keystone reconstructions and rect. angu lar area for skin graft coverage ).

After at least 12 months, patients were asked (on the phone or during consultation) about donor site current pain (visual analog scale for pain, out of 10), loss of grip strength (yes/no), local sensitivity (normal/decreased/abolished), esthetic self assess ment from 1 to 5 (1: ugly, 3: fair, 5: perfect). If neces sary, the healing duration data were cross-referenced with the patient's staternents. They were also asked a free description of their current global fi.mctional irnpairment. An esthetic hetero evaluation was performed for ail donor sites by an independent examiner (sarne scale from 1 to 5), during consultation or by photograph.

1 Surgical technique

The KF can be used for elliptical defects, with transfer of adjacent tissue for better color and contour match. The flap

(A) (B)
is designed as two opposite V-Y flaps that are oriented paral lel to the log axis of the defect. [START_REF] Chaput | The keystone flap alternative in posttraumatic lower extrernity reconstruc tion[END_REF] In the case of RFFF closure, the KF will be harvested from the ulnar side, vascularized by a rich network of ulnar artery perforators (Figures 1 and2). [START_REF] Abraham | Keystone and pedicle perforator flaps in reconstructive surgeiy[END_REF] Skin laxity is tested preoperatively by pinch test and must be sufficient to al low a direct closure by interrupted suture with nonabsorbable thread, generally at the cost of excessive tension.

An incision at 90 ° from the tip of the defect is performed to join the curvilinear line of the flap mark out. The flap width is equal to that of the defect. Blunt dissection of the subcutaneous tissue ensures mobilization of the flap and pre serves venous and nervous networks. The closure is obtained with both subcutaneous and cutaneous interrupted suture.

1 ST A TISTICAL ANAL YSIS

Categorical variables were expressed as proportions and com parecl between groups using Fisher's exact test. Continuous var iables were expressed as means with ranges and comparecl between groups using the Mann-Whitney U test. A two-sided P value <.05 was considered significant. AU statistical analyses were performed using BiostaTGV software (Jussieu, France).

1 RESULTS

Between January 2013 and October 2017, 101 patients underwent a head and neck reconstruction with forearm free flap. A KF closure was performed for 36 of them (group A), and a SG coverage for 64 patients (group B).

One patient received an upfront SG closure, and a KF 15 days later due to SG loss with tendinous exposure. The latter was evaluated in group B concerning the initial outcomes, and then in group A.

KF postoperative care consisted of Redon drainage for 1 or 2 days, with a dry bandage. The sutures were removed after an average of 10 days.

The wound healing protocol after skin grafting consisted of an applied fixed fatty bandage removed after an average period of 5 days, as well as a wrist splint. Tuen the fat dress ing was reproduced every day until complete healing.

Both groups were comparable in terms of age (P = .696), sex ratio (P = .109), and BMI (P = .912). The skin defect was significantly larger in group B comparecl to group A with 41.1 ± 13.4 and '22,.7 ± 5.1 cm 2 , respectively (P < .0001) (Table 1).

Coverage of the RFFF donor site was successful for all patients in group A. The median duration of local care was 14 days in group A and 28 days in group B (P < .001) (Table 2).

In group A, 11 patients had at least one wound healing com plication. There were six wounds reopening (16.7%) with one In group B, 21 patients presented at least one wound healing complication. There were 16 partial graft necroses (24.6%) with nine tendon exposures, three local infections, and four hematomas. One of the tendon exposures motivated a secondary coverage at 15 days by KF.

The morbidity was assessed after a minimum follow-up of 12 months (max: 5 years), either by telephone or during a consultation (l' able 3). In group A, the average visual analog scale for pain was 0.61/10. Seven patients complained about hyperesthesia to touch or cold (27%), limited to the scar for three of them. Two patients (7%) deplored a "ca rdboardy" appearance of the reconstruction, and one patient described neuropathie pain (3.8%). In group B, the average visual analog scale for pain was 1.4/10 (Figure 3). Ten patients (21 %) had hyperesthesia, six patients (12%) complained about the cardboard-like appear ance, and one felt neuropathie pain (2.1 %).

Regarding the sensitivity of the reconstmcted skin pad dle, a disorder (hypoesthesia or anesthesia) was reported by nine patients (34%) in group A and 28 patients (59%) in group B (P = .049). The patients' mean esthetic score was 3.46/5 and 3.51/5, respectively, in group A and B. The average score given by the independent examiner was 2.76/5 and 2.70/5, respec tively. The esthetic score differences were not statistically significant.

S I DISCUSSION

This case-control study compares SG with KF for donor site closure after forearm free flap harvesting.

In our study, the overall early wound healing complica tion rate after skin grafting (partial graft necrosis, hematoma, infection, and tendon exposure) was 32%. In a prospective study on 86 patients, Richardson et al. 8 found 16% of partial SG necrosis, and 13% of tendon exposure, comparable to our results: 13% of tendinous exposure and 24% of partial graft necrosis.

Wound healing complications after donor site closure are responsible for an increased length of postoperative care (Figure 4). In our study, the average healing time after skin grafting was 4 weeks, which is consistent with the literature: 4.68 weeks in a study by Karini et al. 9

The major late complaints described in the literature are lack of cutaneous thickness, loss of grip strength, decrease in joint amplitude, dysesthesia, hypoesthesia in the tenitory of the super ficial sensory branch of the radial nerve, paresthesia, pruritus, hypersensitivity to cold, neuropathie pain, and unsightly scar. 4 • [START_REF] De Witt | Donor site morbidity of the fasciocutaneous radial foreann flap: what does the patient really bother?[END_REF][START_REF] Orlik | Long term functional donor site morbidity of the free radial forearm flap in head and neck cancer survivors[END_REF][START_REF]Radial forearm free flap don or site morbidity ulnar based transp osition flap vs Split thickness skin graft[END_REF][START_REF] Emerick | Incidence of donor site skin graft loss requiring surgical intervention with the radial forearm free flap[END_REF][START_REF] Fang | Upper extrernity morbidity after radial forearm flap harvest : a prospec tive stndy[END_REF] In group B, 59.5% of patients had sensitivity disorders of the reconstructed skin paddle (hypoesthesia or anesthesia). 34.6% patients reported hyperesthesia to touch. Twenty-two patients reported a tolerable pain at rest (V AS 1.4). 23.8% of patients reported a subjective decrease in strength or amplitude of the wrist.

The sequelae as well as the healing disorders may be related to the RFFF harvesting itself and are therefore not modifiable: consequence of the vascular flow changes after radial artery flow interruption, causing, for example, hyper sensitivity to cold.

These sequelae and complications may also be due to the flap dissection procedure, the patient's comotbidities and, finally, the covering technique. Severa] studies compared various closure techniques and failed to demonstrate the superiority of one of them. 4 • 15• 16 Sorne procedures are known to reduce the donor site mor bidity. Supra-fascial dissection can provide a complication rate of 6%. [START_REF] Shonka | Suprafascial bar vest of the radial forearm free flap decreases the risk of postopera tive tendon exposure[END_REF] Oosing by a purse string suture can reduce by up to 44% the area to be grafted and thus the risk of healing compli cation. 9• 1 8 • 1 9 A careful tendinous coverage by peritendinous tis sue ensures both better support for engraftment and a lesser risk of postoperative tendon exposure. [START_REF] Shonka | Suprafascial bar vest of the radial forearm free flap decreases the risk of postopera tive tendon exposure[END_REF] The use of a wrist splint in order to limit wrist mobility for a short time tends to improve healing, especially after skin grafting. [START_REF] Emerick | Incidence of donor site skin graft loss requiring surgical intervention with the radial forearm free flap[END_REF] The use of artificial der mis or collagen matrix associated or not with skin grafting, may allow for a reduction of scarring complications?>

The split-thickness skin graft is widely used because it is simple and not limited by the size of the defect. However, it requires an additional donor site (frequently the anteromedial side of the thigh), involves more pain and wound care, and can cause skin depigmentation. This thin coverage provides a cuta neous depression, a risk of pruritus, and a lesser protection of the noble structures (particularly tendinous and nervous ones) favoring hypersensitivity to touch and cold.

Full-thickness skin graft is a comparable type of wound do sure that provides slightly better esthetic outcomes than split thickness skin graft. Its donor site is closed primarily and is thus less painful. 21 Sorne authors harvested a full-thickness skin graft at the level of the RFFF pedicle, avoiding an additional donor site. 9 ,22

A primary coverage by local flap, with consistency and color similar to those of defect, theoretically provides greater protec tion against tendon exposure and skin paddle hyperesthesia Foissac et al. suggest the use of a perforator flap that arises distally to the ulnar artery for tendon exposure cover age after primary closure. 23 Elliot et al. in 1988 [START_REF] Elliot | Direct closure of the radial forearm fl ap donor defect[END_REF] and then Jaquet et al. in 2012 1 2 described the use of ulnar transposi tion flap, based on an ulnar perforator artery, for primary closure of small and medium donor site defects.

Our study brings to light that KF provides a reduction in healing time, with a median duration of 14 days in group A and 28 days in group B (P < .001). Moreover, we observed less ten don exposure after KF closure, at the cost of an increased risk of compartment syndrome. Barly wound healing disorders were not greater in group A.

Concerning esthetic and functional outcomes, sensitivity disorders were less frequent after KF closure than after skin grafting (34.6% vs 59.5%, P = .049) This can be explained by the preservation of deep sensory pedicles. The skin pad dle hypoesthesia, frequently reported after skin grafting, is limited to the scar with KF closure, thus decreasing the risk of injury. The analysis of other parameters (esthetic self and hetero-evaluation, grip strength, and distal pain) did not show any significant difference between the two groups.

Postoperative pain was not assessed because this criterion is neither reliable nor reproducible in a retrospective study. Nev ertheless, it seems to us that the KF is more painful during the first few days (because of the significant tension of the closure, leading to local edema), while the skin graft preferentially cau ses burning pain on the secondary harvesting site.

The main complication of this closure is the compartment syndrome, requiring close monitoring of edema, tenderness, and distal motricity in the early postoperative days.

The KF is suitable for small to medium size defects. In our study, the calculated area of group A flaps (elliptical) is statistically smaller than the estimated area of group B grafts (rectangular) (28.69 vs 41.05 cm 2 ; P < .001). The average area of group A flaps is comparable to literature data for local perforator flaps: between that of the series of Jaquet et al. (19.8cm 2 ) 12 and Hsieh et al. (47cm 2 ). [START_REF] Hsieh | Primary clo sure of radial forearm fl ap donor defects with a bilobed flap based on the fasciocutaneous perforator of the ulnar artery[END_REF] In our study, the preoperative assessment of KF coverage possibility is not based on the defect size, but on the ratio of the flap width to the forearm perimeter as well as on skin laxity.

This flap is less favorable on colored skin. 26 It is indeed based on tension distribution on scars, which is a risk factor for keloid scars.

The limitations of our study were the bias of retrospec tively collected data and the absence of morbidity analysis of the SG secondary harvesting site. Our study does not establish clear-cut criteria for choosing this type of recon struction, but it demonstrates that this technique is an inter esting alternative to skin grafting.

Future studies should prospectively evaluate postopera tive pain, skin graft donor site morbidity, and objective mea sures of the ratio of the flap width to the forearm perimeter in order to propose precise indications for KF.

1 CONCLUSION

While the radial forearm is the ideal choice for closure of many defects , the donor site morbidity remains an issue, with no method of closure conclusively superior to another. This study offers a valuable alternative to the closure of small to medium sized radial forearm flap donor sites. Keystone-type local flap closure is an interesting option to skin graft closure that provides coverage of the donor site with reduced healing time and acceptable outcomes.

F l G URE 2

 2 Surgical technique of coverage by keystone flap. A, Design of keystone flap. B, Blunt di ssection. C, Oosure with interrupted suture (nonabsorbable thread) F l G UR E 1 Design of keystone flap. A, Design of flap on ulnar side of defect. B, Distribution of tensions on the scar after closure

F

  l G UR E 3 &thetic results of radial foreann free flap donor site. A, Fîve years after closure by keystone flap. Esthetic score by patient two of fi ve. B, One year after closure by keystone flap. Esthetic score by patient three of fi ve. C, One y� after closure by Keystone flap. Esthetic score by patient ail of five. D, Two years after closure by skin graft. Esthetic score by patient one os five. F, R>ur years after closure by skin graft. Esthetic score by patient three of fi ve. F, Two years after closure by skin graft. Esthetic score by patient ail of five

FIGURE 4

 4 FIGURE 4 Wound complication examples. A, Hypertrophie scar after keyston e flap. B, skin trough. C, Tendinous exposure. D, Pigmentation disorder tendon exposure, three local infections, and four inflammations with significant distal edema (11 %) also occurred, requiring in one case a stitch release to avoid compartment syndrome.In group B, 21 patients presented at least one wound healing complication. There were 16 partial graft necroses (24.6%) with nine tendon exposures, three local infections, and four hematomas. One of the tendon exposures motivated a secondary coverage at 15 days by KF.The morbidity was assessed after a minimum follow-up of 12 months (max: 5 years), either by telephone or during a consultation (l' able 3). In group A, the average visual analog scale for pain was 0.61/10. Seven patients complained about hyperesthesia to touch or cold (27%), limited to the scar for three of them. Two patients (7%) deplored a "ca rdboardy" appearance of the reconstruction, and one patient described neuropathie pain (3.8%).

TABLE 1 Patient characteristics GroupA keystone Group B skin flap

 1 (n = 36) graft (n = 65) p

	Age, median	60.5 (34 79 )	61 ( 29 87)
	(years)
	Sex
	• Male	23	51	.109
	• Female	13	14
	BMI, median	23.15	23.3	.9
		(16.3 31.4)	(14.2 31.6)
	Comorbidities
	• Smoking	21	47
	• Diabetes	2	6
	• Cardiovascular	5	7
	disease
	Tumor site
	• Oral cavity	17	29
	• Oropharynx	15	22
	• Hypopharynx	3	9
	• Lip	0	2
	Flap surface (cm 2 )	28.7	41.1	.001

TABLE 2 Healing complication and duration GroupA GroupB keystone skin graft (n = 36) (n = 65) p Patients with at least one

 2

	skin graft
	• Wound reopening	6 (16.7%)	0
	• Tendinous exposure	1 (2.8%)	9 (13.8%)	.091
	• Hematoma	1 (2.8% )	4 (6.2%)
	• Infection	3 (8.3%)	3 ( 4.6%)
	• Edema	4 (11.1 %)	0
	(± epidermolysis)
	Healing lime, median	14 (7 42)	28 (14 168)	.001
	(days)

11 (30.5%) 21 (32.3%) complication • Partial necrosis of 16(24.6% )

  TABLE3 Esthetic and functional evaluation after at least 1 year

		GroupA	GroupB
		keystone	skin graft
		(n = 26)	(n = 47)	p
	Pain, mean (V AS from	0.61	1.4
	0 to 10)
	Skin paddle sensitivity	9 (34.6%)	28 (59.5%)	.049
	disorder
	Decreased grip strength	5 (19.2%)	11 (23.4%)
	Esthetic self evaluation,	3.46	3.54
	mean (from 1 to 5)
	Esthetic hetero evaluation,	2.76	2.70
	mean (from 1 to 5)
	Discomfort
	• Hyperesthesia	7 (26.9%)	10 (21.3%)
	• Cardboard like	2 (7.7%)	6 (12.8%)
	appearance
	• Neuropathie pain	1 (3.8%)	1 (2.1 %)

CONFLICT OF INTEREST

There are no conflicts of interest.