Hadiseh Geramirad
email: hadisehgeramirad@supergrid-institute.com

Florent Morel

Bruno Lefebvre

Christian Vollaire

Arnaud Breard

Experimental study of the self-disturbance phenomena in a halfbridge configuration of Si IGBT and SiC MOSFET switches

This paper investigates the gate-driver design challenges encountered due to the fast switching of wide band-gap semiconductors (here, SiC MOSFETs) in the half-bridge configuration. It discusses precisely the common-mode current generated by fast switching passing through gate-driver, which leads to drivingvoltage perturbations and disturbs the correct triggering of switches. This phenomenon is described here as self-disturbance and the experimental study shows that it is exacerbated with SiC MOSFETs, when compared with Si IGBTs. The solution proposed in this article is to block the generated noise linked to self-disturbance phenomenon by placing a common-mode choke into the gate path. The proposed solution is compared with the classical active Miller-clamp protection solution. The experimental results from 1.7kV, 300A SiC MOSFET validate the solution. The solution proposed in this work allows to keep the MOSFET gate-voltage perturbation lower than the MOSFET threshold voltage while it does not increase the switching losses. Experimental results show that by implementing the proposed solution, it is possible to reduce the input common-mode current through the gate driver by up to 20%.

Introduction

An important objective when designing a power converter is high efficiency. That is why, due to their lower switching losses, SiC MOSFETs became an alternative to silicon IGBTs [START_REF] Friedrichs | SiC power devices for industrial applications[END_REF], [START_REF] Kaplar | Characterization and reliability of SiC-and GaN-based power transistors for renewable energy applications[END_REF]. In spite of the advantages of the high switching speed including reducing the dead time in a phase-leg configuration and increasing switching frequency [START_REF] Friedrichs | SiC power devices for industrial applications[END_REF], [START_REF] Biela | SiC versus Si-Evaluation of potentials for performance improvement of inverter and DC-DC converter systems by SiC power semiconductors[END_REF], it adds difficulties to gate driver design [START_REF] Costa | An innovative gate drive to control electromagnetic perturbations emitted by a MOSFET chopper[END_REF], [START_REF] Rondon | Impact of SiC components on the EMC behaviour of a power electronics converter[END_REF]. By increasing the switching speed to obtain minimum switching losses, converter parasitic elements are subject to large dv/dt and di/dt values which can generate disturbances. In a half-bridge configuration, during switching transient, each switch affects the operating of the complementary switch. The fast variation of the voltage across a switch in a half-bridge produces an unavoidable perturbation on the control voltage of the complementary switch due to parasitic elements. In addition, since gate driver acts as an interface between a switch and a control circuit, common-mode current generated by fast switching passes through the gate driver isolation [START_REF] Ghossein | State of the Art of Gate-Drive Power Supplies for Medium and High Voltage Applications[END_REF] and can disturb the controller of the converter. Therefore, the main objective of this work is to keep the switching speed as high as possible while dealing with the generated disturbances. The paper is organized as follow. Section 2 explains the experimental set-up used in this work to investigate the problem and validate the solution. Section 3 explains the mechanism of the voltage perturbation in the gate which is called in this work "self-disturbance" phenomenon. Section 4 compares experimentally the difference between IGBTs and SiC MOSFETs which highlights the causes of self-disturbance phenomenon. The typical solution to reduce the voltage perturbation is discussed in section 5. The proposed solution and conclusion are presented in sections 6 and 7 respectively.

Experimental platform

To investigate the self-disturbance phenomenon, the experimental study of switching characteristics has been carried out on a full-bridge double-pulse test bench (see Fig. 1). In this test bench, by turning on and off switch 1 or switch 2 (see Fig. 2), it is possible to control the direction of the current through the inductor. Double pulse test with both current directions enable to characterize the highside and low-side switch of the half-bridge under test (DUT) in soft switching and hard switching which is the natural switching behaviour of a halfbridge configuration (see figures 1 and 2).

Problem statement

During turn-on transient of the lower switch, the increasing voltage across the upper switch creates an increasing voltage across the Miller capacitance of the semiconductor (C dg ). This rising voltage induces a current going to the gate loop and introduces a positive voltage perturbation (see Fig. 3). If this parasitic voltage exceeds the threshold voltage (V th ) of the switch, it can partially or fully turn-on the device. The partially turn-on results in a shoot-through current leading to additional losses in both switches and potential circuit oscillation [START_REF] Murata | Self turn-on loss of MOSFET as synchronous rectifier in DC/DC buck converter-in case of a low driving impedance[END_REF].

In case of fully turn-on of the switch, a short-circuit happens due to the simultaneous conduction of both switches of the half-bridge which can destroy the semiconductors [START_REF] Zaman | Suppression of switching crosstalk and voltage oscillations in a SiC MOSFET based half-bridge converter[END_REF].

Similar to turn-on transition, during turn-off transition of the lower switch, the decreasing voltage across the upper switch coupled to the Miller capacitance and due to the parasitic elements of the gate, creates a negative parasitic voltage across the gate of the switch. If this negative voltage exceeds the maximum allowable negative gate bias of the switch, it can over stress the device and degrade the switch stability [START_REF] Jahdi | Temperature and switching rate dependence of crosstalk in Si-IGBT and SiC power modules[END_REF].

It is noteworthy that the low-side switch is subject to the same phenomenon. This means that, during the turn-off and turn-on of the upper switch, a parasitic voltage appears in the gate of the low-side switch.

The presented results here are dedicated to highside switch due to space limitation but the same analysis can be applied to the lower switch also.

This positive and negative parasitic voltage due to complementary switching transient in halfbridge configuration is called "self-disturbance phenomenon" in this work since it can disturb the normal operation of the switches and consequently can disturb the converter. This phenomenon has been called in many dissertations "cross-talk", "selfturn-on", "false turn-on", "false triggering", etc. The mechanism of this induced voltage on the gate voltage has been studied in [START_REF] Murata | A self turn-on mechanism of the synchronous rectifier in a DC-DC converter[END_REF]- [START_REF] Zhang | Analysis of the switching speed limitation of wide band-gap devices in a phase-leg configuration[END_REF]. Analytical investigation of the circuit during turn-on and turn-off showed that the maximum gate voltage perturbation given by (Eq. 1) [START_REF] Murata | Self turn-on loss of MOSFET as synchronous rectifier in DC/DC buck converter-in case of a low driving impedance[END_REF]. where (see Fig. 3) the dv/dt is the slew rate of the voltage across the switch, C iss is the input capacitance of the DUT, C dg is the Miller capacitance of the DUT and R gate H is the gate resistance.

V gs H (max) = dv dt R gate H C dg H 1 -e V DC dv dt C iss R gate H (1)
Based on the figure 3 and equation 1, it can be concluded that the voltage perturbations mainly depend on the voltage variation and the value of the junction capacitances. The impedance of the gate path is an unavoidable factor in the design of the gate driver which has to be minimized. As IGBTs and MOSFETs show up with very different characteristics (e.g. C dg and C iss ), for the development of the SiC MOSFET gate drivers, it is necessary to consider the difference between SiC MOSFETs and IGBTs and to focus on the main challenge: self-disturbance. package (3.3kV, 450A) have been compared experimentally. Table 1 outlines the main differences between the two devices in terms of junction capacitance, input resistance and threshold voltage. Considering Fig. 3, which shows the elements related to the gate perturbation, in order to keep the effect of the gate impedance constant between both tests, the same gate driver has been used for both devices. L gate H is unavoidable in the design of the gate driver, due to the connection of the gate driver and the parasitic impedance of the electronic board. Therefore, the same gate driver in the same converter layout has been used for both DUTs in order to have a fair comparison. The external gate resistance of gate driver has been fixed to 0.2Ω for both cases. The only difference is the driver power supply. The SiC MOSFET gate is regulated between -6 and 20 volts while the IGBT is biased between -15 to 15 volts.

The experimental results shown in figure 4 prove that almost 4 times faster switching speed of the SiC MOSFET compared to IGBT module causes a higher gate voltage perturbation even if the value of the junction capacitance is lower. The significant gate voltage perturbation prevents to test the MOSFET at voltages higher than 500V while this power module is expected to be used in application with DC voltages of up to 1200V. Therefore, in a half-bridge configuration, in order to fully benefit from the potential advantage of fast switching rate of SiC MOSFET and guarantee the proper operation of the power devices, selfdisturbance phenomena should be suppressed. For IGBTs, the perturbation is lower and the bigger negative bias reduces the risk of short-through in a half-bridge configuration. That is the reason why the investigation is continued for SiC MOSFETs only.

Miller-clamp protection

As it has been shown in section 4, switching speed is the key contributor to self-disturbance phenomenon. Therefore, slowing down the dv/dt can reduce the perturbation in the gate voltage. This is obtained by prolonging the charging of the input capacitances of the device [14]- [START_REF] Sato | Forced-air-cooled 10 kW three-phase SiC inverter with output power density of more than 20 kW/L[END_REF].

Decreasing the switching speed comes with the penalty of increasing losses which is in contrary with the goal of using SiC MOSFETs. There are several works that present the reduction of the self-disturbance phenomena by taking advantage of negative biasing ability of the SiC MOSFETs [START_REF] Zaman | Suppression of switching crosstalk and voltage oscillations in a SiC MOSFET based half-bridge converter[END_REF], [START_REF] Wang | A novel RCD level shifter for elimination of spurious turnon in the bridge-leg configuration[END_REF]. In order to reduce the self-disturbance with minimum effect on the switching speed, these methods try to control the gate signal thanks to passive components. In general, they are called "passive methods". However, the passive methods are able to suppress only the positive voltage perturbation [START_REF] Zaman | Suppression of switching crosstalk and voltage oscillations in a SiC MOSFET based half-bridge converter[END_REF], [START_REF] Wang | A novel RCD level shifter for elimination of spurious turnon in the bridge-leg configuration[END_REF].

Another category of solutions that has been presented to overcome self-disturbance, is called "active method". For these methods, an assistant circuit is added to the circuit of the gate driver. The assistant circuit is activated during off-state of the switch by an external control signal. The main idea of the active methods is to regulate the impedance of the gate during off-state by proposing a path of lower impedance to the induced current towards the negative bias of the gate [START_REF] Zaman | Suppression of switching crosstalk and voltage oscillations in a SiC MOSFET based half-bridge converter[END_REF], [START_REF] Zushi | A novel gate assist circuit for quick and stable driving of SiC-JFETs in a 3-phase inverter[END_REF], [START_REF] Zhang | Active gate driver for crosstalk suppression of SiC devices in a phase-leg configuration[END_REF]. Most of these methods has been presented either for negative either for positive perturbation. Moreover, these methods have not been evaluated in maximum switching speed of high-power SiC MOSFETs modules. In this work, an active Millerclamp protection as an assistant circuit is added to the gate driver circuit (see Fig. 5). the assistant circuit proposes another path for the induced current to the gate driver and clamps the gate voltage to the negative bias of the gate driver.

A concern is that, in order to effectively clamp the gate voltage to the negative bias, the Miller-clamp protection MOSFET should turn-on when the device under test is fully turned-off. Therefore, the choice of the MOSFET is an important point. To do so, a resistance and a capacitance have been added to the circuit in order to tune the on-transition of the Miller-clamp protection MOSFET. Unlike the active solution proposed in [START_REF] Zushi | A novel gate assist circuit for quick and stable driving of SiC-JFETs in a 3-phase inverter[END_REF] and [START_REF] Zhang | Active gate driver for crosstalk suppression of SiC devices in a phase-leg configuration[END_REF], the external capacitance (green capacitance in Fig. 5) has not been added to the assistant circuit in order to reduce as much as possible the gate impedance.

The added external capacitance can prolong the charging of the input capacitance of the device and consequently slow down the turn-on transition while in this work the goal is to keep the maximum switching speed. The effect of the active Millerclamp protection is shown in figures 6 and 7. It can be seen that the protection circuit is able to keep the perturbation lower than the threshold voltage for DC voltages up to 500V. During these tests, the voltage between the mid-point of the halfbridge and ground varied with a slope of about 12.5kV /µs. By increasing the DC voltage level up to 1200V (corresponding to a voltage slope of about 30kV /µs), it can be seen that, the protection circuit cannot completely keep the parasitic voltage lower than the device threshold-voltage.

Proposed solution and experimental results

Experimental results in section 5 reveal that, thanks to Miller-clamp protection, the gate voltage perturbation is attenuated enough for DC voltage up to 500V. The effectiveness of Miller-clamp protection in maximum switching rate is not validated in higher voltage operation of the SiC MOSFET (see Fig. 7). So as, Miller-clamp protection circuit cannot keep the gate voltage less than the SiC MOSFET threshold-voltage. Hence, for investigating the perturbations, the commonmode (CM) current generated due to the fast switching has been measured across each switch. The details of the generated CM currents and measurement points are shown in figure 8 To prove this assumption and to reduce selfdisturbances, the proposed solution is to filter the associated CM current of fast switching before it goes through the gate driver. Therefore, a commonmode inductor has been implemented in the gate driver circuit. The chosen filter should have a low impedance for differential mode (DM) in order to not affect the behaviour of the gate drive signal during turn-on and turn-off while it has a big impedance in CM to block the high frequency induced current to the gate and protect the gate driver against voltage perturbation [START_REF] Chen | Experimental parametric study of the parasitic inductance influence on MOSFET switching characteristics[END_REF].

There are two types of windings for CM chokes: sectional and bifilar. The sectional components have a higher leakage inductance (DM inductor in low frequency) while the performance feature of the two coils is similar in terms of the CM (see Fig. 9). In order to have a minimal impact on the switching speed and gate oscillation, a bifilar choke (one-way winding) has been implemented in the gate driver. Experimental results (see Fig. 10) confirm that, by choosing bifilar winding choke, gate-voltage slope and consequently switching-voltage and switching losses will not be affected by the added component. Experimental results shown in figure 11, illustrate that the CM currents have been reduced in highside and low-side and proves the effect of the choke. By implementing the filter, input CM current has been reduced by up to 20% (see Fig. 11). Since the gate driver provides a path for common-mode current to the controller, this method helps to reduce the risk of loading the controller and reduces the self-disturbance phenomenon.

Conclusion

This article compared the self-disturbance phenomenon in phase-leg configuration with IGBTs Here, in order to overcome the self-disturbance, the common-mode current generated by fast switching has been measured and it has been proposed to block this common-mode current before it goes to the gate. By choosing bifilar choke, it is possible to block the associated current and reduce the gatevoltage perturbation. The proposed solution has been validated in 1200V, 150A switching transition and reduces significantly the input common-mode current of gate driver. Then this solution allows to use SiC MOSFETs at maximum switching speed and does not increase switching losses.

Fig. 1 :

 1 Fig. 1: Experimental set-up.

Fig. 2 :

 2 Fig. 2: Experimental set-up schematic.

Fig. 3 :

 3 Fig. 3: Gate voltage perturbation mechanism (a) Positive perturbation during turn-on (b) Negative perturbation during turn-off.

Fig. 4 :

 4 Fig. 4: Experimental results of comparison between the IGBT and SiC MOSFET in 500V, 150A double pulse test (a) Drain-source voltage (b) Gatesource voltage.

Fig. 5 :

 5 Fig. 5: Miller-clamp protection (the used circuit shown in red, an alternative circuit shown in green).

Fig. 6 :

 6 Fig. 6: Experimental result of VgsH considering Miller clamp protection in 500V, 150A double pulse test.

Fig. 7 :Fig. 8 :

 78 Fig. 7: Experimental result of VgsH considering Millerclamp protection in 500V, 700V and 1200V and 150A double pulse test.

  as

Fig. 9 :

 9 Fig. 9: Common-mode choke winding (a) sectional (b) bifilar (c) Impedance comparison of the commonmode chokes.

Fig. 10 :

 10 Fig. 10: Experimental results considering proposed solution in 1200V, 150A double pulse test (a) gate-source voltage (b) drain-source voltage.

Fig. 11 :

 11 Fig. 11: Experimental results of measured common-mode currents in 1200V, 150A double pulse test (a) ICML (b) ICMH (b) Input-ICM.

Unit SI IGBT SiC MOSFET

	Vth	V	5.2	2.5
	Ciss	nF	84	20
	Coss	nF	3.5	2.5
	Crss (Cdg)	nF	2	0.08
	Qg	µC	12.5	0.273
	Qrr	µC	230	324
	Rint	Ω	1.3	3.7
	4 Experimental comparison of
	SiC MOSFETs and IGBTs power
	modules
	To show the difference between modules based
	on SiC MOSFETs and IGBTs, a Wolfspeed SiC
	MOSFET CAS300M17BM2 package (1.7kV, 300A)
	and an Infineon IGBT FF450R33T3E3, XHP3

Tab. 1: Electrical characteristic of the considered MOSFET and IGBT module Parameter

Acknowledgement

This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Program (ANE-ITE-002-01).


L Marin

Herve ´ Combeau

Julien Zollinger
email: julien.zollinger@univ-lorraine.fr

Moukrane Dehmas

Bernard Rouat

Aude Lamontagne

Natalia Loukachenko

Lucile Lhenry-Robert

¨l Raphae

Marin

Herve ´combeau

r-Phase Formation in Super Austenitic Stainless Steel During Directional Solidification and Subsequent Phase Transformations

The solidification path and the r-phase precipitation mechanism in the S31254 (UNS designation) steel are investigated thanks to Quenching during Directional Solidification (QDS) experiments accompanied by scanning electron microscopy observations and electron backscattered diffraction (EBSD) analysis. Considering experimental conditions, the c-austenite is found to be the primary solidifying phase (1430 °C), followed by d-ferrite (1400 °C, % 87 pct solid fraction). The r-phase appears in the solid-state through the eutectoid decomposition of the d-ferrite: d fi r + c 2 (1210 °C), whereas the r-phase is predicted to form from the austenite at 1096 °C in equilibrium conditions. The resulting temperatures of solidification path and phase transformation are compared with Gulliver-Scheil model and equilibrium calculations predicted using Thermo-Calc Ó software. It is shown that the thermodynamics calculations agree with experimental results of solidification path. The EBSD analysis show that the d-ferrite has dNW2 ORs with the r-phase.

I. INTRODUCTION

STAINLESS steels are an important class of materials. Their applications are countless, from basic applications such as sinks and hubcaps to very complex ones like aircraft fuselage and space vehicles. Stainless steels are rich in chromium (at least 11 wt pct) in order to form a chromium oxide layer on the surface, which guards against pitting and rusting. However, stainless steels are most often used in more hostile environments such as desalination of seawater and petrochemical plants, [START_REF] Lai | Stainless Steels: An Introduction and Their Recent Developments[END_REF][START_REF] Stein | Materials Science Forum[END_REF][START_REF] Olsson | [END_REF] which requires high corrosion resistance. Therefore, molybdenum is added to enhance resistance against pitting and rusting. During processing, due to the high content of alloying elements contained in these steels, it may be possible to form intermetallic precipitates enriched in chromium and molybdenum. The formation of these intermetallic precipitates removes Cr and Mo from the matrix, which lead to a significant decrease in the corrosion resistance of the steel. One of this intermetallic phase was found in 1907 by Treitschke and Tamman [4] in the Fe-Cr system. In 1927, a hard and brittle phase was found by Bain and Griffiths [5] in the Fe-Cr-Ni system. This phase was called ''r-phase'' in 1951 by Jett and Foote. [6] The crystallographic structure of the r-phase in the Fe-Cr system was obtained by Bergmann and Shoemaker in 1951. [7] In 1966, the r-phase was discovered in austenitic stainless steels by Hattersley and Hume-Rothery [8] and Hall and Algie. [9] Particularly, as-cast ingots exhibit r-phase (space group 136, tetragonal [10] ) in interdendritic regions, which must be eliminated by heat treatments after solidification. Numerous studies are focused on the r-phase formation during heat treatments of different class of stainless steels (ferritic, austenitic, and duplex). Two mechanisms of r-phase precipitation are well established: (i) r-phase nucleation in austenite [11][START_REF] Koutsoukis | Solid State Phenomena[END_REF][START_REF] Koutsoukis | [END_REF][14][15] and (ii) r-phase nucleation from the eutectoid decomposition of the d-ferrite. [11,[START_REF] Koutsoukis | [END_REF][16][17][18][19] However, few studies report on the r-phase precipitation mechanism during the solidification in the stainless steels. Two scenarios are reported: it can be formed (i) during the last stage of the solidification through a eutectic reaction [20] due to the segregation of the molybdenum in the liquid during the solidification stage, [21] or (ii) during the decomposition of the d-ferrite dendrites, through a solid-state phase transformation. [22] The purpose of this paper is to study the solidification path and the phase transformations of the super austenitic steel (UNS S31254). A Quenching during Directional Solidification (QDS) apparatus was used and coupled with Thermo-Calc Ó simulations to establish the solidification path and the r-phase transformation mechanism of the steel. The crystallographic orientation relationships were determined using EBSD technique and compared with the literature.

II. MATERIAL AND EXPERIMENTAL PROCEDURES

A. Material

The steel used in this investigation was hot rolled grade, namely S31254 (UNS designation) and was provided by INDUSTEEL company. The nominal chemical composition (wt pct) is given in Table I.

The equilibrium liquidus and solidus temperatures of the S31254 steel were determined using a Setaram multi-detector high-temperature calorimeter apparatus. Three heating rates were applied, ranging from 2 to 10 °C/min. The equilibrium liquidus and solidus temperatures were found by extrapolation of the measurements to a 0 °C/min heating rate. The equilibrium liquidus and solidus temperatures were found to be 1413 °C ± 5 °C and 1376 °C ± 5 °C, respectively.

B. Quenching During Directional Solidification

The QDS experiments were performed with a Bridgman-type solidification setup, consisting of an induction furnace system and a water spraying cooling box, which was previously described by Charpentier et al. [23] The melting and the directional solidification were performed under Helium gas flow. The samples were introduced in an alumina tube of 5-mm inner diameter. The thermal history of the sample was recorded by a Pt/ Rh B-type thermocouple located inside the sample and protected by an alumina tube. The top of the QDS sample was heated at a temperature higher than its melting point, while the rest of the QDS sample was heated at lower temperature and remaining thus solid. Once the interface between the liquid and the solid was stabilized, the directional solidification was achieved by pulling the alumina tube and the sample along the vertical axis at a constant solidification velocity (V = 5 9 10 À5 m/s). The solidification velocity was chosen in order to get a directional solidification and not a radial solidification. The thermal gradient (G) was controlled by the temperature difference between the induction furnace and the water cooling box. At a constant solidification velocity, G controls the size of the different area on the sample, i.e., the thickness of the mushy zone. After a given solidified length, 10 cm, the sample was quenched in order to freeze the microsegregation and the microstructure, by pulling it instantaneously through the cooling box, by a pneumatic piston fixed at the bottom of the alumina tube. The induced quenching rate was 100 °C/s. This experiment allows the solidification velocity (V) and the thermal gradient (G) to be monitored separately. After the QDS experiments, the sample was polished along the longitudinal section, which is parallel to the growth direction during the experiment. The temperature distribution along the sample was recorded by thermocouple in contact with the sample. In the present study, the average temperature gradient in the liquid phase and the mushy zone were G = 3 °C/mm and G = 10 °C/mm, respectively. These conditions are close to the solidification conditions at the center of the ingot.

C. Microstructure Analysis

The QDS samples were prepared using standard metallographic polishing. Final sample preparation was accomplished with colloidal silica. The microstructural observations were performed using a scanning electron microscope (SEM) Quanta FEG 650 in backscattered electron mode (BSE), operated at 20 kV. Electron back-scattered diffraction (EBSD) was used to precise the crystallography of the different phases and their crystallographic orientations. These investigations were performed with a HKL Nordlys II F+ EBSD camera. The step size was 0.2 lm and the analyzed surface was 65 lm 9 44 lm. Wavelength-dispersive X-ray spectroscopy (WDS) measurements were used to determine solid fraction in the mushy zone. The WDS analyses of the samples were realized with a JEOL JXA-8530 F which was equipped with 5 WDS spectrometers, 10 crystal analyzers. All analyses were made using a 20-kV accelerating voltage and a beam current of 100 nA.

D. Thermodynamics Calculations

The Thermo-Calc Ó software [24] was used to predict the solidification path and the microsegregation evolution for lever-rule and Gulliver-Scheil corresponding assumptions. The calculations were conducted with the TCFe7 database. All the solute elements were accounted and all the possible phases were considered for the calculations. In addition to the equilibrium calculation, the non-equilibrium solidification simulations were realized with the Gulliver-Scheil model, which assumes infinite diffusion in the liquid phase and no diffusion in the solid phases for the substitutional solute elements. In these calculations, carbon and nitrogen are assumed to be fast diffuser and to follow thermodynamic equilibrium. Three different slopes can be observed. The first slope corresponds to the temperature evolution in the liquid phase. The thermal gradient in the liquid phase was determined to be 2.9 °C/mm. The second slope is associated with the mushy zone. The thermal gradient determiner in this zone was equal to 10.6 °C/mm. In the fully solidified zone (third slope), the thermal gradient was 12.5 °C/mm. Particular temperatures such as the liquidus temperature, assuming the dendrite tip undercooling negligible, and the end of solidification were determined by fitting tangential lines to the temperature evolution (as can be seen in Figure 1(b)).

III. RESULTS AND DISCUSSION

A. Experimental Solidification Path and Phase Transformations

The first tangential line was drawn to fit the thermal signal in the liquid, and the second one to fit the thermal signal in the mushy zone. The intersection of the two tangential lines gives the dendrites tip solidification temperatures. An uncertainty of ± 10 °C on the temperature determination was considered, due to the thermocouple accuracy. The same procedure was used to determine the end of solidification temperature. Thus, the solidification starts at 1430 °C ± 10 °C and ends at 1345 °C ± 10 °C on the sample presented in Figure 1. Using the solidification start temperature, the thermal gradient in the different phases, and the position of the dendrite tips, it is possible to know the temperature at every position of the sample. There is a difference between the tip temperature measured with this method and the temperature obtained by DTA measurements (1413 °C). A variability of 7 °C was observed for the tip temperature in between the different samples prepared by the QDS method. This difference between DTA and QDS samples might be due to the uncertainty on the thermocouple, the one associated by the temperature determination from each methods, and also minor changes in the chemical composition of the steel.

Figure 2 shows the evolution of the microstructure during the solidification stage with the corresponding temperatures. The temperature decreases from the top to the bottom of the figure. On the right side of Figure 2, the microstructure of the different locations of the QDS sample was schematized. The phases at the different locations (i.e., mushy zone and solid-state) were identified by EBSD analysis. Two different types of r phase and c-austenite islands were observed. The first type of islands, which is present in the quenched liquid and the mushy zone, is formed from the liquid phase during quenching to room temperature. These islands have a very thin microstructure with an interlamellar spacing of 300 nm. The second one is much coarser (interlamellar spacing ‡ 1 lm) and corresponds to the r-phase formed during the directional solidification process. The solidification starts with primary c-austenite dendrites at 1430 °C. At 1400 °C, in the mushy zone, d-ferrite grows in the interdendritic spaces, stabilized by the molybdenum segregation in the liquid during solidification. [21] The molybdenum apparent partition coefficient was determined by EDX measurements and is approximately equal to 0.68, which is close to the value predicted by Thermo-Calc Ó (0.69). The complete solidification occurs at 1345 °C. Below this temperature, the microstructure consists of c-austenite dendrite and d-ferrite in the interdendritic areas. As previously mentioned, the formation of r-phase in the quenched liquid and the mushy zone is due to high cooling rate from the quench; thus, we can conclude that no r-phase was formed during the solidification period. At the solid-state, from 1345 °C to 1210 °C, the microstructure is characterized by c-austenite dendrites and d-ferrite. Below 1210 °C, a partial decomposition of the d-ferrite into r-phase and secondary austenite (c 2 ) occurs. This decomposition starts at 1210 °C and ends at 1040 °C. Below 1040 °C, the d-ferrite is fully transformed into r-phase and c 2 and the microstructure consists of austenite dendrites surrounded by r + c 2 mixture, as shown in Figure 2.

Based on these results, the solidification path and solid-state phase transformation of the S31254 steel can be expressed as follows:

L ! 1430 C L þ c ! 1400 C L þ c þ d ! 1345 C c þ d ! 1210 C c þ d þ r þ c 2 ð Þ ! 1040 C c þ r þ c 2 ð Þ

B. Solidification Path: Comparison Between Experimental and Numerical Results

Table II compares the solidification interval determined from the DTA sample, QDS sample and from non-equilibrium and equilibrium Thermo-Calc Ó calculations. The solidification interval calculated under non-equilibrium condition is larger (124 °C) than the solidification interval obtained experimentally with QDS experiment (85 °C); this difference can be explained by the assumptions of the Gulliver-Scheil model. [START_REF] Scheil | Bemerkungen zur schichtkristallbildung[END_REF] This model assumes no diffusion in the solid phase for the substitution elements and an infinite diffusion in the liquid phase. In real conditions, the diffusivity of solute elements is nonzero into the solid and finite into the liquid during the solidification process. In the case of the equilibrium conditions, the solidification interval (32 °C) is close to the DTA result (37 °C) and also larger than the solidification interval obtained experimentally with QDS experiment.

Figure 3 compares the solidification paths obtained experimentally and numerically by considering the equilibrium conditions (level rule) and the Gulliver-Scheil assumptions. The temperature difference from the liquidus temperature as a function of solid fraction was considered. The undercooling of the dendrite tips was estimated with LGK model [START_REF] Dantzig | Solidification: -Revised & Expanded[END_REF] and is equal to 2 °C; it will be neglected in the following analysis. In the case of the experimental results, the solid fractions at different temperatures were estimated by image analysis (IA) from cross sections of the QDS sample and WDS measurements. The solid fraction in the mushy zone of the QDS sample is plotted in orange squares (IA) and blue squares (WDS). In the mushy zone of the QDS sample, the solid fractions measured with the two different methods were found to be close, except for the first point which corresponds to the lowest solid fraction. This can be explained by the fact that during the quenching, the c-austenite dendrites will experience growth that induces a large error in measurement by image analysis as shown by Pompe and Rettenmayr. [START_REF] Pompe | [END_REF] For this reason, the solid fraction of this point is probably likely overestimated. The solid fractions determined experimentally were found to be close to the Gulliver-Scheil solidification path.

From these results, three different solidification paths can be described:

The equilibrium solidification path as computed from

Thermo-Calc Ó : L þ c ! 32 C c
The experimental solidification path:

L þ c ! 30 C L þ c þ d ! 85 C c þ d
The Gulliver-Scheil solidification path computed from Thermo-Calc Ó :

L þ c ! 24 C L þ c þ d ! 74 C c þ r ! 124 C c þ d þ r
Considering the equilibrium conditions, the microstructure is fully austenitic at the end of the solidification, which is not the case for the experimental result and the non-equilibrium calculation. The d-ferrite temperature formation for the QDS sample (30 °C below the liquidus temperature) is quite close to the Gulliver-Scheil calculation (24 °C below the liquidus temperature). However, the solid fractions are different: 85 pct solid for the QDS sample and 65 pct for the non-equilibrium calculation. Moreover, the equilibrium calculations showed the beginning of r-precipitation at 1096 °C in the solid state from austenite, while the formation of r-phase starts at 1210 °C from d-ferrite in the QDS sample. This finding highlights the importance of microsegregation during the solidification of the S31254 on the sequence of precipitation. It leads to an increase of the r-phase precipitation temperature despite the back-diffusion in the solid.

The importance of the back-diffusion in the solid during microsegregation was estimated with the model developed by Brody and Flemings. [28] Since chromium and nickel have partition coefficient close to unity, only the microsegregation of the molybdenum was considered (k Mo % 0.67). The Fourier number obtained with the experimental data is close to 0 (0.0117), which correspond to Gulliver-Scheil assumptions. The comparison between the experimental (85 °C) and modeled solidification interval (84 °C) shows that considering microsegregation leads to a better agreement between experimental and modeling for the investigated processing conditions.

It has been shown that the r-phase does not precipitate during the solidification for the investigated processing conditions. However, it might be possible to form r-phase during solidification directly from the liquid phase in the case of rapid cooling. As an example, Lee et al. [29] reported the presence of r-phase in a super austenitic weld sample, in the interdendritic space. Figure 4 represents a quaternary phase diagram (Fe-Cr-Mo-Ni) showing the difference of liquidus temperature between the phases as a function of the chemical composition. For each chemical composition, three different systems were considered with only two phases per system: L + c, L + d, L + r. The liquidus temperatures for each system were obtained and compared. These results were obtained with Thermo-Calc Ó calculations. The chemical composition was modified for each calculation. Chromium, nickel, and molybdenum were considered in the range of 15 to 35 wt pct, 10 to 30 wt. pct and 5 to 25wt. pct, respectively. The minor solute elements (C, N, Mn, Cu, and Si) were considered as constants with values of the nominal composition.

The experimental values, represented by green and black points correspond to the average chemical composition of the liquid from the QDS sample at 55 and 87 pct volume solid fraction, respectively. The red point corresponds to the nominal composition of the steel.

Figure 4(a) represents the temperature difference between austenite liquidus temperature and d-ferrite liquidus temperature for each chemical composition and figure 4b gives the temperature difference between d-ferrite liquidus temperature and r-phase liquidus temperature.

The transition from a phase to another happens at DT = 0 °C, for example, in Figure 4(a) for positive temperature difference the austenite will be the stable phase but below DT = 0 °C, d-ferrite will be the stable phase. Figure 4(a) shows that the formation of d-ferrite during the solidification stage is thermodynamically possible, and it corresponds to the experimental results. (K-S). [30] This funding suggests that the d-ferrite nucleates at the L/c interface during solidification. dendrites at 1129 °C. The crystalline phase structure is identified with a color code on Figure 5(a), the austenitic phase (FCC) is identified in red, while the d-ferrite (BCC) and the r-phase (tetragonal) are in blue and yellow, respectively. The gray color corresponds to pixels that were not indexed due to the small size of the eutectoid microstructure. . This orientation relationship was described by Kurdjumov and Sachs orientation relationship is called dNW2 and was described by Redjaı¨mia. [30] Small angle deviations are observed (< 5 deg) from the coplanar and collinear conditions. Figure 5(d) shows the projection of the r-phase dense plane 001 ð Þ and d-ferrite dense plane 1 10 À Á respecting common directions and lattice parameters of each phase. The r-phase lattice parameters were obtained by TEM measurements and d-ferrite lattice parameters were obtained for literature. The secondary austenite (c 2 ), resulting from the d-ferrite eutectoid decomposition, grew at the d/r interface and showed a K-S OR with the d-ferrite. Redjaı¨mia [31] reported that the transformation of d-ferrite into r-phase was due to small shifts, with a value less than the interatomic distance. On Figure 5(d) the atomic rows are parallel to direction ½110 d and ½1 10 r . The atomic distribution on these two rows is almost identical, the transformation from the d-ferrite to r-phase requires only small adjustments of the atomic positions when the dNW2 orientation relationship is verified. The existence of ORs between d-ferrite and c phases leads to reduce the interfacial energy required to nucleate r-phase on the interface. However, an investigation conducted by Sato et al. [32] in a duplex stainless steels showed that a higher crystallographic misorientation between the austenitic and ferritic phases favored r-phase precipitation. Red-jaı¨mia proposed a transformation mechanism [START_REF] Scheil | Bemerkungen zur schichtkristallbildung[END_REF] : the r-phase nucleation starts at the interface d/c and grows in the d-ferrite with a lamellar morphology, due to the presence of Cr and Mo in d-ferrite, which have higher diffusivity. The partitioning of the r favoring elements (Cr, Mo) leads to the growth of the lamellae while the Fe and Ni are rejected in the d-ferrite. This diffusion length of solute elements is short through the d/r interface and locally modifies the chemical composition of the d-ferrite from the d/r interface. The partitioning destabilizes the d-ferrite and causes its decomposition into austenite (c 2 ) and r-phase. A mixture of r + c 2 lamellar morphology is obtained and grows at the expense of the d-ferrite. As a part of a future work, the phase transformation should be described with a diffusion model to improve the comprehension of these results and composition profiles at the interface between the c-austenite dendrites and the r-phase and r-phase/c 2 .

IV. CONCLUSION

Directional solidification of the S31254 alloy was investigated through QDS experiments. In addition to the solidification path, it has been showed that information on the solid-state transformations can also be obtained from the quenching that is performed. Considering experimental conditions, the austenite was found to be the primary solidifying phase, followed by d-ferrite in the interdendritic area, due to the segregation of molybdenum. It was shown that the austenite dendrites have K-S ORs with the d-ferrite. Gulliver-Scheil microsegregation model was in agreement with experimental results, especially for the solidification path. During the solidification, the microsegregation intensity was slightly decreased by the back diffusion in the solid. At the solid state, the r-phase appears through an eutectoid decomposition of the d-ferrite: d fi r + c 2 . The EBSD analysis showed that the d-ferrite has dNW2 ORs with the r-phase. Finally, this study proposes the following solidification path and the r-phase precipitation mechanism for the S31254 steel:

L ! 1430 C L þ c ! 1400 C L þ c þ d ! 1345 C c þ d ! 1210 C c þ d þ r þ c 2 ! 1040 C c þ r þ c 2

Figure 1 (

 1 Figure 1(a) displays an optical micrograph of the longitudinal section from the top part of the QDS sample. Two different microstructures can be observed, the as-quenched remaining liquid on the top, and the mushy zone with columnar dendrites on the bottom. The yellow dotted line indicates the solidification front. The dendrite primary arms grow in a direction close to the thermal gradient, as a consequence of directional solidification. During the experiment, as the thermocouple position is known, it is thus possible to access to the thermal history of the sample as a function of time and distance. The QDS sample thermal history as a function of time is shown in Figure 1(b). The red curve represents the evolution of the temperature with time.Three different slopes can be observed. The first slope corresponds to the temperature evolution in the liquid phase. The thermal gradient in the liquid phase was determined to be 2.9 °C/mm. The second slope is associated with the mushy zone. The thermal gradient determiner in this zone was equal to 10.6 °C/mm. In the fully solidified zone (third slope), the thermal gradient was 12.5 °C/mm. Particular temperatures such as the liquidus temperature, assuming the dendrite tip

Fig. 1 -

 1 Fig. 1-QDS results, (a) optical micrograph showing quenched liquid and mushy zone after LB (Lichtenegger and Blo¨ ch) etching (60 g NH 4 HF 2 , 2 g KHSO 3 and 100 mL of hot distilled water at 80 °C), (b) temperature evolution of the sample measured by the Pt/Rh B-type thermocouple.

Fig. 2 -

 2 Fig. 2-SEM observations at the different locations of the QDS sample. Schematic representation of the microstructure on the right side. The vertical scale bar on the right side indicates the size of each location.

Figure 4 (

 4 Figures 5 displays the EBSD maps of the QDS sample in the solid-state location. It can be found former interdendritic regions undergoing the eutectoid transformation d fi r + c 2 surrounded by c-austenite

Fig. 3 -

 3 Fig. 3-Experimental and numerical solidification path. Experimental data are given with the orange squares (image analysis) and blue squares (WDS measurements), equilibrium conditions are plotted with a dashed line and Gulliver-Scheil conditions are represented with a black dotted line. Arrows and dots represent the phases involved during the solidification process (Color figure online).

Figure 5 (

 5 c) displays three stereographic projection showing the ORs between d-ferrite and the r-phase grains. The d-ferrite poles and directions are marked with blue closed symbols, and the three r-phase grains r 1 , r 2 , and r 3 with green, orange, and black open symbols, respectively. The analysis of the superposition of the stereographic projection extracted from the crystallographic orientation map showed that the 110 f gd pole are superimposed with the three r-phase variants (r 1 , r 2 , r 3 ), which share the same (001) pole and are a common interface with the same ferrite grain:f1 10g d ==f001g r . The two stereographic projection show that the d-ferrite and the three r-phase variants exhibit two common direction types:½110 d ==½1 10 r and ½00 1 d ==½110 r . The d-ferrite holds an orientation relationship with the r-phases grains (r 1 , r 2 , r 3 ). This

Fig. 4 -

 4 Fig. 4-Quaternary phase diagram (Fe-Cr-Mo-Ni) representing the difference of liquidus temperature between the different phases for each different chemical composition with temperature isosurface. (a) Between c/d, (b) between d/r. Red square corresponds to the nominal chemical composition of the steel, while green and black squares correspond to the average chemical composition of the liquid from the QDS sample at 55 and 87 pct volume solid fraction, respectively (Color figure online).

Figure 5 (

 5 b) gives a crystallographic orientation map of the different phases on the EBSD map. The EBSD orientation map shows that the interdendritic precipitate partially is decomposed into r + c 2 , arose from a d-ferrite single grain. The analysis of the stereographic projection resulting from the EBSD map showed that the d-ferrite grains have an orientation relationship with the austenite dendrites with a small deviation (5 deg) between the growth directions. :ð110Þ==ð111Þ c ; 111 ÂÃ d == 110 ÂÃ c

Fig. 5 -

 5 Fig. 5-EBSD maps, (a) phase identification, (b) grain orientation with IPF coloring, (c) poles figures: overlaying ofð110Þ d ==ð001Þ r , 001 d == Â Â 110 r and 110 d == Â Â 110 r , showing the existence of dNW2 orientation relationship between d-ferrite and r-phase. The black open circles show the matching pole, (d) schematic representation of the superposition of the (001) r-phase dense plan and the (110) d-ferrite dense plan following the dNW2 orientation relationship: ð1 10Þ d ==ð001Þ r ½110 d ==½1 10 r ½00 1 d ==½110 r .

Table I .

 I Nominal Chemical Composition of the S31254 Steel in Weight Percent

	Fe	Cr	Ni	Mo Cu Mn	Si	N

S31254 bal. 19.68 17.77 6.08 0.67 0.65 0.20 0.19

ACKNOWLEDGMENTS

The assistance of E. Etienne and S. Mathieu (Centre de Compe´tences Microscopie, Microsondes et Me´tallographie, CC3M, Institut Jean Lamour) for the EBSD preparation and analysis was greatly appreciated. Financial support from Industeel and ANRT under a CIFRE Ph.D. fellowship (Grant Number 2016-0780) is gratefully acknowledged.


Jorge Luis

Gulfo Monsalve

Laurent Ros

Jean-Marc Brossier

Denis Mestdagh

Jorge L Gulfo Monsalve

An Improved GreenOFDM Scheme for PAPR Reduction

Keywords: OFDM, PAPR, Selected Mapping, GreenOFDM

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Orthogonal Frequency Division Multiplexing (OFDM) is a widely used modulation technique for wireless communication systems thanks to its high spectral efficiency and robustness against frequency-selective channel. However, a well-known limitation of OFDM (as well as other multi-carrier schemes) is its high Peak-to-Average Power Ratio (PAPR) [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF]. In addition, the PAPR in OFDM is a random variable. These characteristics of the PAPR in OFDM are troublesome for many reasons:

(i) the PAPR of OFDM signals needs to be correctly estimated in order to take its dynamic range into consideration for the Digital-to-Analog Converters specification [START_REF] Singh | Effect of Peak-to-Average Power Ratio Reduction on the Multicarrier Communication System Performance Parameters[END_REF][START_REF] Gustavsson | CMOS Data Converters for Communications[END_REF];

(ii) in order to correctly estimate the PAPR of OFDM signals in the digital domain, the computational complexity of the digital modulator is increased due to oversampling [START_REF] Tellambura | Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers[END_REF];

(iii) the high PAPR of these signals generally increases the Power Amplifier's (PA's) power dissipation as its linear region has to be extended to be able to accommodate signals with wide amplitude excursions. Not doing so results in non-linear distortions and out-of-band radiations that degrade the system performances [START_REF] Mestdagh | Analysis of clipping effect in DMT-based ADSL systems[END_REF].

To counteract the above-mentioned limitations, several PAPR reduction schemes have been proposed in the literature (with well documented surveys like [START_REF] Jiang | An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals[END_REF][START_REF] Rahmatallah | Peak-To-Average Power Ratio Reduction in OFDM Systems: A Survey and Taxonomy[END_REF][START_REF] Sandoval | Hybrid Peak-to-Average Power Ratio Reduction Techniques: Review and Performance Comparison[END_REF] that outline the most important schemes that have been proposed during the last couple of decades). These schemes could be classified like proposed by [START_REF] Rahmatallah | Peak-To-Average Power Ratio Reduction in OFDM Systems: A Survey and Taxonomy[END_REF][START_REF] Sandoval | Hybrid Peak-to-Average Power Ratio Reduction Techniques: Review and Performance Comparison[END_REF] into different but not exhaustive categories as follows:

• Signal distortion techniques: Clipping and Filtering [START_REF] Li | Effects of clipping and filtering on the performance of OFDM[END_REF][START_REF] Armstrong | Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering[END_REF], Companding [START_REF] Huang | Companding transform for reduction in peak-to-average power ratio of OFDM signals[END_REF].

• Multiple signaling and probabilistic techniques: SeLected Mapping OFDM (SLM-OFDM) [START_REF] Mestdagh | A Method to Reduce the Probability of Clipping in DMT-Based Transceivers[END_REF][START_REF] Bauml | Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by Selected Mapping[END_REF], Tone Injection [START_REF] Han | Tone injection with hexagonal constellation for peak-to-average power ratio reduction in OFDM[END_REF], Tone Reservation [START_REF] Tellado | Efficient algorithms for reducing PAR in multicarrier systems[END_REF][START_REF] Wang | Analysis of Clipping Noise and Tone-Reservation Algorithms for Peak Reduction in OFDM Systems[END_REF], Active Constellation Extension [START_REF] Krongold | PAR reduction in OFDM via active constellation extension[END_REF], Partial Transmit Sequence [START_REF] Muller | OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences[END_REF]. • Pre-coding techniques: Single Carrier Frequency Division Multiple Access (SC-FDMA) [START_REF] Myung | Peak-To-Average Power Ratio of Single Carrier FDMA Signals with Pulse Shaping[END_REF], Carrier Interferometry [START_REF] Wiegandt | Overcoming peak-to-average power ratio issues in OFDM via carrier-interferometry codes[END_REF][START_REF] Anwar | Simplified Realization of Pseudo-Orthogonal Carrier Interferometry OFDM by FFT Algorithm[END_REF].

Among these schemes, the SLM-OFDM is a well known one and has very recently been improved into the so-called GreenOFDM technique [START_REF] Mestdagh | GreenOFDM: a new selected mapping method for OFDM PAPR reduction[END_REF][START_REF] Mestdagh | OFDM transmission systems with reduced Peak-to-Average Power Ratio[END_REF]. The underlying idea of SLM-OFDM is to generate a number of different OFDM symbol waveform candidates from the same data and to select the waveform with the lowest PAPR for transmission.

For SLM-OFDM, the PAPR reduction depends on the number of waveform candidates; the more the number of candidates, the lesser the PAPR of OFDM. Essentially, GreenOFDM is a SLM-OFDM variant that increases the number of candidates in a square-scale while using the same number of IFFTs as in conventional SLM-OFDM. Thus, for a given number of computed IFFTs, a better PAPR reduction for GreenOFDM is attained as compared to SLM-OFDM.

GreenOFDM receives attention in the literature [START_REF] Yu | A New Method for Reducing PAPR in OFDM System Based on GreenOFDM[END_REF][START_REF] Zhou | Improved SLM Scheme for Reducing the PAPR of QAM OFDM Signals[END_REF][START_REF] Tas | Dual symbol optimization-based partial transmit sequence technique for PAPR reduction in WOLA-OFDM waveform[END_REF][START_REF] Tas ¸pınar | Cumulative symbol optimization-based partial transmit sequence technique for PAPR reduction in low complexity GFDM system[END_REF]. Some works compare the original GreenOFDM scheme with PTS (Partial Transmit Sequence) variants but without using the same scheme [START_REF] Tas | Dual symbol optimization-based partial transmit sequence technique for PAPR reduction in WOLA-OFDM waveform[END_REF][START_REF] Tas ¸pınar | Cumulative symbol optimization-based partial transmit sequence technique for PAPR reduction in low complexity GFDM system[END_REF]. Other works aim to improve the PAPR reduction capabilities as in [START_REF] Yu | A New Method for Reducing PAPR in OFDM System Based on GreenOFDM[END_REF], but the results seem to come from a not so exhaustive simulation study. Finally, other works like [START_REF] Zhou | Improved SLM Scheme for Reducing the PAPR of QAM OFDM Signals[END_REF] reducing the complexity by relaxing the PAPR reduction capability as it has been also proposed in other works [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF].

In contrast to previous studies, we propose in this paper a simple yet efficient improvement of the GreenOFDM that allows to obtain a higher number of candidates without increasing the number of IFFTs. As shall become apparent, this extension of the original GreenOFDM technique relies on the linearity of the IFFT that allows simplifications in the initial GreenOFDM and permits a reduction by a factor of 2 of the number of IFFTs for a same PAPR performance, providing a relevant gain in terms of performance/complexity trade-off. The proposed idea results in PAPR reductions that are statistically similar to those obtained independently in [START_REF] Heo | A Modified SLM Scheme With Low Complexity for PAPR Reduction of OFDM Systems[END_REF]. However, in this work we provide analytical expressions to predict the PAPR in a given configuration, which aids the design of the analog front-end [START_REF] Singh | Effect of Peak-to-Average Power Ratio Reduction on the Multicarrier Communication System Performance Parameters[END_REF][START_REF] Gustavsson | CMOS Data Converters for Communications[END_REF][START_REF] Mestdagh | Analysis of clipping effect in DMT-based ADSL systems[END_REF], and suggest further complexity reductions. These further complexity reductions are based on a strategy (IFFTs-on-demand and and hierarchical sampling methods) proposed in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF] to reduce the implementation complexity of the original GreenOFDM, and are shown to be achievable with great efficiency for the proposed technique.

This document is organized as follows: First, a brief OFDM and PAPR description is given. Then, SLM-OFDM and GreenOFDM algorithms are succinctly presented before introducing the proposed method. In the last part, computer simulations of the PAPR distribution are presented to clearly demonstrate the improvement of the proposed technique and finally conclusions are drawn.

OFDM and PAPR

OFDM is a multi-carrier modulation technique in which every time-domain symbol carries the sum of N overlapping orthogonal subcarriers mapped by {A k } symbols (e.g. Quadrature Amplitude Modulation: QAM symbols) with

0 ≤ k ≤ N -1.
The OFDM base-band time-domain symbol is obtained by applying the Inverse Discrete Fourier Transform (IDFT) 1to the frequency-domain inputs {A k }. The digital time-domain symbols are often oversampled to emulate the peaks that might appear in the analog domain and hence obtain a well-approximated digital version. A L.N points IFFT over

{X k } = {A 0 A 1 • • • A N 2 -1 0 • • • 0 (L-1).N A N 2 A N 2 +1 • • • A N -1 } (1)
achieves the oversampling, where L is the oversampling factor. The OFDM base-band time-domain symbol is hence given by:

x[n] = 1 √ N LN -1 k=0 X k . exp i2π kn LN (2)
with 0 ≤ n ≤ N -1 and i = √ -1. Generally an oversampling factor L = 4 is applied since it is enough to well approximate the analog symbols. Higher values will only add a higher computational complexity without substantial gain in the approximation.

The PAPR of the oversampled OFDM symbol is given by:

PAPR = max 0≤n≤LN -1 |x[n]| 2 E {|x[n]| 2 } (3)
where E{.} is the expectation and

E |x[n]| 2 = E |X k | 2 for X k i.i.d. random variables ({X k } ∈ 2 n -QAM with n = 2m, m ∈ N ).
The PAPR of OFDM symbols is high because the numerator of Eq. ( 3) is high. This is because sometimes many modulated subcarriers X k . exp i2π kn LN add themselves in-phase generating high peaks. Furthermore, since the PAPR is a random variable it can be characterized by its Complementary Cumulative Distribution Function (CCDF), i.e. the probability that the PAPR of an OFDM symbol exceeds a predetermined value γ. A semi-empirical approximation is given by [START_REF] Ochiai | On the Distribution of the Peak-to-Average Power Ratio in OFDM Signals[END_REF]:

CCDF OFDM (γ) ≈ 1 -(1 -exp {-γ}) 2.8N (4)
3 From SLM-OFDM to GreenOFDM

In conventional SLM-OFDM, U different symbol waveforms are generated from the same data set {X k }. To do so, the input data set {X k } is repeated U times and each copy is multiplied element-wise by pseudo-random sequences where the coefficients have unitary norm

φ u,k , 0 ≤ u ≤ U -1; i.e. X u,k = X k .φ u.k with |φ u,k | = 1.
It is noteworthy to mention that in the following, U represents the total number of computed IFFTs as well as the total number of pseudo-random sequences. The time-domain symbol waveforms x u [n] are computed by LN -IFFTs with inputs {X u,k }. The waveform that exhibits the lowest PAPR among the C SLM = U candidates is selected for transmission. The SLM-OFDM algorithm is summarized in Algorithm 1 and its block diagram depicted in Figure 1.

The CCDF of PAPR of conventional SLM-OFDM is well approximated by [START_REF] Bauml | Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by Selected Mapping[END_REF]:

CCDF SLM (γ) ≈ 1 -(1 -exp {-γ}) 2.8N CSLM (5)
The function CCDF SLM (PAPR) decreases when the number of candidates C SLM = U increases. However, higher values of C SLM lead to larger values of the computational burden because of the required number of IFFTs.

In order to mitigate this limitation of SLM-OFDM, the key idea of GreenOFDM is to generate the candidates by adding (with a normalisation factor √ 2) two coupleable IFFT-based waveforms. This allows for the generation of more candidates C Green corresponding to all the possible combinations, whilst maintaining the same PAPR reduction behaviour of SLM-OFDM with respect to the number of candidates (i.e. equation ( 5) with C Green instead of C SLM ).

The GreenOFDM technique generates C Green = U 2 /4 OFDM symbols candidates by first splitting the U set of copied and randomized data {X u,k } into two groups: the first one represented by the index g 1 , with 0 ≤ g 1 ≤ U

2 -1 and the second one represented by the index g 2 , with

U 2 ≤ g 2 ≤ U -1. The first group waveforms {x g1 [n]} is obtained, like in SLM-OFDM, through IFFTs over {X g1,k = X k .ϕ g1,k } (with {ϕ g1,k } = {φ u,k }, u < U 2 and φ u,k ∈ ±1). Similarly, the second group waveforms {x g2 [n]} are obtained through IFFTs over {X g2,k = X k .ϕ g2,k } (with {ϕ g2,k } = {i × φ u,k }, u ≥ U 2 and φ u,k ∈ ±1).
As it can be seen, the difference between the two groups of waveforms is the fact that the sequences {ϕ g2,k } use pure imaginary unitary values: {±i} whereas the sequences {ϕ g1,k } use pure real unitary values: {±1}. In this way, individual elements of the two sequences are orthogonal in order not to cancel out some subcarriers after adding the two IFFT outputs. The last part of the GreenOFDM method, which depicts the way the U 2 4 waveform candidates are obtained, is described in Algorithm 2. The equivalent block diagram of the GreenOFDM is depicted in Figure 2.

An improved GreenOFDM version

The goal of the improved version is to increase the total number of candidates without increasing the number of IFFTs. As aforementioned, the key idea of the GreenOFDM technique is based on the use of two different

{X k } {φ 0,k } {φ 1,k } . . . {φ U -1,k } IFFT {X 0,k } IFFT {X 1,k } . . . . . . IFFT {X U -1,k } LOWEST PAPR SYMBOL SELECTION {x 0 [n]} {x 1 [n]} {x U -1 [n]} . . . {x ũ[n]} Figure 1: SLM-OFDM block diagram. {X k } {ϕ 0,k } ϕ U 2 -1,k IFFT {X 0,k } . . . IFFT X U 2 -1,k {x 0 [n]} xU 2 -1 [n] {ϕ g1,k } ∈ {±1} IFFT X U 2 ,k ϕ U 2 ,k x U 2 [n] x 0, U 2 [n] IFFT {X U -1,k } {ϕ U -1,k } {x U -1 [n]} x 0,U -1 [n] {X k } {ϕ g2,k } ∈ {±i} . . . . . . . . . . . . . . . U 2 4 OFDM waveform candidates x U 2 -1, U 2 [n] xU 2 -1,U -1 [n]
Figure 2: Original GreenOFDM block diagram [START_REF] Mestdagh | GreenOFDM: a new selected mapping method for OFDM PAPR reduction[END_REF][START_REF] Luis | GreenOFDM a new method for OFDM PAPR reduction[END_REF].

Algorithm 1 The SLM-OFDM algorithm.

Require:

{X k }, {φ u,k } with φ u,k ∈ {±1} minPAPR ← +∞; for (u ← 0 to U -1) do {X u,k } ← {X k .φ u,k }; {x u [n]} ← IFFT{X u,k }; if (PAPR{x u [n]} < minPAPR) then minPAPR ← PAPR{x u [n]}; {x ũ[n]} ← {x u [n]}; end if end for SEND{x ũ[n]};
Algorithm 2 The GreenOFDM algorithm.

Require: ). In the proposed improved version, referred from now on as GreenOFDMv2, the full set of IFFTs will be used both for the first group, and then for the second group after a slight modification exploiting the linearity of the IFFT and ensuring orthogonality between the two groups.

{x g1 [n]} = IFFT{X k .ϕ g1,k }, {x g2 [n]} = IFFT{X k .i.ϕ g2,k } minPAPR ← +∞; for (g 1 ← 0 to U 2 -1) do for (g 2 ← U 2 to U -1) do {x g1,g2 [n]} ← {xg 1 [n]+xg 2 [n]} √ 2 ; if (PAPR{x g1,g2 [n]} < minPAPR) then minPAPR ← PAPR{x g1,g2 [n]}; {x g1,g2 [n]} ← {x g1,
The IFFTs of the initial sequence {X k } weighted by pure real binary sequences {φ u,k } with values in {±1} compose directly the first group :

{x u [n]} = IFFT{X k . φ u,k }, with 0 ≤ u ≤ U -1
The complex waveforms of the second group correspond conceptually to the IFFTs of the initial sequence {X k } weighted by pure imaginary binary sequences i × {φ u,k } with values in {±i}, but are generated in a less complex way. Indeed, due to the linearity of the IFFT, the complex waveforms of the second group {x u [n]} are equal to the complex waveforms of the first group {x u [n]}, only modified by a factor of i :

{x u [n]} = IFFT{X k . i.φ u,k } with 0 ≤ u ≤ U -1 = i × IFFT{X k . φ u,k } = i × {x u [n]},
So, the second group can be obtained from the IFFT outputs of the first group {x u [n]} by only shifting the complex values ( + i. ) into (-+ i. ), eliminating the need to compute the IFFTs again. For a given number of IFFT, the size of each group is then doubled compared to the original GreenOFDM method.

The improved GreenOFDM method is described in Algorithm 3. It can be observed that the first and second groups are redefined. The first group contains the U IFFTs (the equivalent to the SLM-OFDM generated candidates), and the second group is trivially generated by exploiting the linearity of the IFFT.

Algorithm 3 The GreenOFDMv2 algorithm.

Require: In the particular case u 1 = u 2 , there is no need to perform the additions {x u1 [n]+i.x u2 [n]} because it does not change the PAPR of the initial waveform {x u1 [n]}. In the proposed method, these extra operations are avoided in that particular case by choosing the candidate directly from the IFFT output, i.e. we choose the candidate to be equal to {x u1 [n]}.

{x u [n]} = IFFT{X k .φ u,k } minPAPR← +∞; for (u 1 ← 0 to U -1) do for (u 2 ← 0 to U -1) do if u 1 == u 2 then {x u1,u2 [n]} ← {x u1 [n]}; else {x u1,u2 [n]} ← {xu 1 [n]+i×xu 2 [n]} √ 2 ; end if if (PAPR{x u1,u2 [n]} < minPAPR) then minPAPR ← PAPR{x u1,u2 [n]}; {x ũ1,ũ2 [n]} ← {x u1,u2 [n]}; end if end for end for SEND{x ũ1,ũ2 [n]}; {X k } {φ 0,k } {φ 1,k } . . . {φ U -1,k } IFFT {X 0,k } IFFT {X 1,k } . . . . . . IFFT {X U -1,k } WAVEFORMS COMBINER & LOWEST PAPR SYMBOL SELECTION {x 0 [n]} {x 1 [n]} {x U -1 [n]} . . . {x ũ1 ,ũ 2 [n]} . . . i × {x 0 [n]} i × {x 1 [n]} i × {x U -1 [n]} . . . . . . . . . . . . U 2 OFDM waveform candidates {x 0 [n]} {x 0,1 [n]} {x 0,U -1 [n]} {x 0 [n]} . . . {x 1,0 [n]} {x 1 [n]} {x 1,U -1 [n]} {x 1 [n]} . . . {x U -1,0 [n]} {x U -1,1 [n]} {x U -1 [n]} {x U -1 [n]} . . . . . .
The number of candidates that are generated in the proposed method, here after referred to as the GreenOFDMv2, is equal to the number of times the inner loop is run. It corresponds to C Greenv2 = U 2 . This means that we have 4 times more candidate waveforms as compared to the initial version of the GreenOFDM by computing the same number of IFFTs, hence a better PAPR reduction is expected. Figure 3 depicts the proposed method block diagram.

Simulation results and discussion

Computer simulations were carried out to evaluate the CCDF of PAPR of the GreenOFDMv2 method proposed in this work and to compare it with the previous methods (SLM-OFDM and the former GreenOFDM) and to the conventional OFDM. Also, the PAPR threshold, for a given CCDF, and its evolution were investigated to know and predict the required number of IFFTs to stay under a given PAPR threshold for the different techniques and different number of subcarriers.

PAPR Complementary Cumulative Distribution Function (CCDF)

Results for 10 5 symbols with N = 64 QP SK-modulated subcarriers, with the oversampling factor L = 4 and U = 16 IFFTs for the SLM-OFDM, GreenOFDM and the GreenOFDMv2, are depicted in Fig. 4 where the dashed lines correspond to the approximated expressions of the CCDF in Eq.( 4) and Eq.( 5) and the marks correspond to the simulation results.

The simulation results are well approximated by the CCDFs expressions for OFDM, SLM-OFDM (with C SLM = U ), GreenOFDM (by replacing C SLM by C Green = U 2 /4) and GreenOFDMv2 (by replacing C SLM by C Greenv2 = U 2 ).

Moreover, it can be seen that the best performance for a fixed probability (CCDF(γ)) is obtained for the GreenOFDMv2 method as compared to SLM-OFDM and the previously proposed GreenOFDM, with a gain of 1.2 dB and 0.5 dB, respectively, for a CCDF(γ) = 10 -3 (the gain versus conventional OFDM is about 5 dB). This is because of the increase in the number of candidates (U 2 for GreenOFDMv2 rather than U for SLM-OFDM or U 2 4 for GreenOFDM).

PAPR threshold : simulations and approximated formula

In this section we present a different way to analyse the PAPR reduction capabilities in SLM-based schemes. For a fixed probability CCDF(γ) = p, it is possible to calculate the value of γ as a function of the number of subcarriers N and the number of candidates that depends on U . To do so, Eq.( 5) is reformulated as follows:

γ ≈ -log 1 -1 -p 1 C 1 2.8N (6)
where C is a generic notation for the number of candidates as a function of the number of IFFTs U , i.e.

C = C SLM = U for SLM-OFDM, C = C Green = U 2
4 for GreenOFDM and C = C Greenv2 = U 2 for the proposed method. The same way, for conventional OFDM, Eq.( 4) is reformulated as:

γ ≈ -log 1 -(1 -p) 1 2.8N (7)
These reformulations allow us to predict the PAPR value for a given configuration (N and U ) and at a given probability. Simulations were carried out for different values of N and U and for a fixed probability CCDF(γ) = 10 -3 .

Results are summarized in Fig. 5 where the performance evolution with N are depicted at each subplot for a fixed value of U . For all the schemes the marks represent the simulation results and the dashed lines correspond to the approximated expressions in ( 7) and [START_REF] Jiang | An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals[END_REF].

In all the cases, the value of γ[dB] is always lower for the proposed method as compared to the SLM-OFDM and the GreenOFDM. However, the difference with the original version of GreenOFDM reduces as U increases.

It can also be noted that, for all the tested values of N and U , the CCDF threshold γ obtained by simulation is very close to the one obtained from Eq. [START_REF] Jiang | An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals[END_REF]. It means that the PAPR performance of the proposed technique is also well deterministically linked to the number of candidates and can be accurately predicted by the use of the Eqs.( 5) and [START_REF] Jiang | An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals[END_REF].

Another way to interpret the results is in terms of the reduced number of IFFTs needed to attain the same PAPR performance. To facilitate the observation of this fact, Fig. 6 regroups the different configurations with equivalent performance (PAPR threshold for CCDF of 10 -3 ) for a number of subcarriers going from N = 64 to 1024. As expected, for all N , the proposed method requires the smallest number of IFFTs, keeping always a reduction factor of 2 with respect to the initial GreenOFDM method. For example, to ensure that the PAPR stays under 8 dB (with a probability 1 -10 -3 ), the required number of IFFTs is only 8 for the GreenOFDMv2 method, against 8 × 2 = 16 for the original GreenOFDM and 8 2 = 64 for the conventional SLM technique. And even for a large number of 1024 subcarriers, the GreenOFDMv2 technique maintains a quite reasonable complexity of 32 IFFTs contrary to SLM 64 128 256 512 1,024 6) are also plotted in dashed lines for GreenOFDMv2.

[dB] GreenOFDM v2 U = 8 GreenOFDM U = 16 SLM U = 64 GreenOFDM v2 U = 16 GreenOFDM U = 32 GreenOFDM v2 U = 32 GreenOFDM U = 64 5.5 6 6.5 7 7.5 8 8.5 N γ
(32 2 = 1024 expected IFFTs).

So far we have limited the simulations to QP SK-modulated subcarriers. To improve this analysis, we have obtained the results depicted in Fig. 7 for 16 -QAM (left side) and 64 -QAM (right side) modulated subcarriers. As it can be seen, with the same number of available candidates, equivalent PAPR reductions are obtained for all the schemes but with the smallest required number of IFFTs for our proposed method. However, this time, the PAPR values for a CCDF of 10 -3 are slightly above the analytical expressions (represented by the dashed lines) and that for all the schemes. These discrepancies tends to vanish as N increases.

The drawback that remains, as for all the SLM-based methods, is the need of a Side Information (SI) in order to correctly de-randomize the received data symbols on the receiver-side.

Discussion on additional possible complexity reduction

As seen before, the proposed method requires only the square-root of the number of IFFTs of the original SLM method for a same PAPR reduction performance (i.e. a same number of candidates), making it a feasible technique from a complexity point-of-view. In addition to the reduction in the number of required IFFTs, the strategy proposed in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF][START_REF] Luis | GreenOFDM a new method for OFDM PAPR reduction[END_REF] to reduce the implementation complexity of GreenOFDM can also be exploited for GreenOFDMv2 to further reduce the computational complexity.

The probabilistic strategy exploits the knowledge of the CCDF of the PAPR. Indeed, to attain a certain performance in terms of PAPR (at a CCDF of PAPR p, with a value of γ p deterministically predicted through Eq.( 6)), there is no need to find the lowest PAPR symbol candidate among all the candidates but the first that satisfies a PAPR below the threshold γ (denoted as γ p0 , p 0 being typically chosen to 10 -3 ). In addition, the idea of finding the first candidate with acceptable PAPR is complemented with the fact that calculating the PAPR requires to compute the instant power for each sample (see Eq.( 3)). Indeed, if the Instantaneous-to-Average Power Ratio (IAPR) of a sample of a certain candidate exceeds the imposed threshold, there is no need to continue with that candidate and it can be discarded to continue with another one.

These facts are taken into account and implemented through the IFFTs-on-demand and hierarchical sampling methods (please refer to [START_REF] Luis | GreenOFDM a new method for OFDM PAPR reduction[END_REF] sections 3.1 and 3.2 for the details of each method) in order to drastically reduce the Furthermore, in [START_REF] Luis | GreenOFDM a new method for OFDM PAPR reduction[END_REF] it was shown that relaxing the PAPR reduction capability to a slightly higher threshold γ pi with

γ pi[dB] = γ p0[dB] + i × 0.1[dB] i = 0, 1, 2, • • • (8)
attains considerably higher computational reductions with a low impact in the PAPR reduction which provides a good performance-complexity trade-off.

The same methods are implementable in the GreenOFDMv2 as described in Algorithm 4. In this algorithm, the IFFTs are computed step-by-step as the candidates are progressively calculated in the 'Hierarchical-sample' method that returns a flag in addition of the candidate itself. This flag being at '1' means that the corresponding candidate PAPR is below the imposed threshold γ pi and can be chosen for transmission. Otherwise the candidate has at least an IAPR exceeding γ i and is hence discarded.

All these strategies have allowed a relevant computational reduction of the former GreenOFDM in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF], especially in terms of the number of IFFTs around fifty percent (with N = 64 subcarriers, U = 64 IFFTs, and a back-off of 0.3 dB for the PAPR threshold). The reduction of the computational complexity of GreenOFDMv2 is expected to be better than those in the former version of GreenOFDM. The reason lies on the specific way in which the IFFTs are progressively computed and the possible available candidates. Indeed, in the former GreenOFDM method, the available candidates up to the v-th computed IFFT (with v ≤ U ) are C v ∝ v 2 /4 (section 3.1 in [START_REF] Luis | GreenOFDM a new method for OFDM PAPR reduction[END_REF]); while for the proposed method, the available candidates up to the v-th computed IFFT are C v = v 2 . This means that every additional computed IFFT

N CCRF M [%] γ p2 [dB] γ p3 [dB] γ p4 [dB] γ p5 [dB]
Figure 8: The CCRF between the GreenOFDM and SLM-OFDM as presented in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF] (left-side) and the CCRF between the proposed method and the conventional SLM-OFDM (right-side) with complexity reduction as a function of γ pi [dB] and N .

Conclusion

This paper has presented an efficient method to reduce the PAPR in OFDM. This method is a simplification of the recent GreenOFDM technique, which is itself an extension of the well-known SLM PAPR reduction technique. In all these techniques, several waveform candidates are generated from a certain number of IFFT to retain the one with the lowest PAPR as actual OFDM symbol. The PAPR performance of these methods is shown to be a decreasing function of the number of candidates. To increase the number of candidates for a given number of IFFT, the key idea of GreenOFDM was to combine two coupleable IFFT-based waveforms rather than only one in the SLM method. Moreover, the new version proposed in this paper allows to use plainly the full set of IFFTs for each group of waveforms while maintaining their ability to be coupled, which was not the case for the original GreenOFDM version in [START_REF] Mestdagh | GreenOFDM: a new selected mapping method for OFDM PAPR reduction[END_REF] that split the IFFTs into two smaller separate subsets.

Consequently, compared to the first version of the GreenOFDM method, the proposed method adds a simple yet efficient improvement to reduce the PAPR by increasing the number of available candidates by a factor of 4 without increasing the number of IFFTs. Alternatively, a same PAPR performance is obtained by using only half the number of IFFTs. Compared to the initial well known SLM technique, the improved GreenOFDM technique provides then definitively a drastic complexity reduction while keeping the same idea, requiring only the square-root of the number of IFFTs for a same PAPR reduction performance.

Finally, approximate formula (mainly accurate for small constellation size such as QPSK or for high values of N ) are provided to predict the PAPR performance of the proposed method as a function of the number of sub-carriers and the number of candidates, or alternatively to determine the number of required IFFTs for a given PAPR performance. This prediction is very useful not only to anticipate the complexity of the technique, but also to allow implementation of further complexity reduction methods dedicated to GreenOFDM proposed previously [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF][START_REF] Luis | GreenOFDM a new method for OFDM PAPR reduction[END_REF] as has been discussed and shown in the paper.

Further studies are needed to propose efficient blind methods adapted to the presented technique to recover the data on the receiver side without the need of implicit side-information.

Figure 3 :

 3 Figure 3: The proposed GreenOFDMv2 method block diagram.

Figure 4 :

 4 Figure 4: The CCDF of PAPR for conventional OFDM, SLM-OFDM, GreenOFDM and GreenOFDMv2 with the same parameter U = 16 and for N = 64 subcarriers per symbol, obtained by simulation (marks) and approximated expressions (dashed lines).

Figure 5 :

 5 Figure 5: The value of γ[dB] to attain CCDF(γ) = 10 -3 for different values of U and N for OFDM, SLM-OFDM, GreenOFDM and GreenOFDMv2, obtained by simulation and approximated expressions (dashed lines).

Figure 6 :

 6 Figure 6: The value of γ[dB] to attain CCDF(γ) = 10 -3 for different values of U and N for SLM-OFDM, GreenOFDM and GreenOFDMv2 obtained by simulation (marks). Approximated expressions in Eq.(6) are also plotted in dashed lines for GreenOFDMv2.

Figure 7 :

 7 Figure 7: The value of γ[dB] to attain CCDF(γ) = 10 -3 for different values of U and N for SLM-OFDM, GreenOFDM and GreenOFDMv2 obtained by simulation (marks). Approximated expressions in Eq.(6) are also plotted in dashed lines. Left side corresponds to the results for 16-QAM mapping and right side corresponds to the results for 64-QAM.

  groups (respectively generated from {±1} or {±i} orthogonal sequences) of IFFTs. In the original version of the GreenOFDM technique, only the first half of the IFFTs were used for the first group ({x g1 [n]} = IFFT{X k .φ g1,k } with g 1 < U2 ), the second half being reserved for the second group ({x g2 [n]} = IFFT{X k .ϕ g2,k }, with g 2 ≥ U 2

	g2 [n]};
	end if
	end for
	end for
	SEND{x g1,g2 [n]};

Digitally implemented with an Inverse Fast Fourier Transform (IFFT).

Acknowledgements The authors would wish to acknowledge Ali Waqar Azim for the helpful comments and remarks on this work.

Abbreviations

The following abbreviations are used in this manuscript: CCDF Complementary Cumulative Distribution Function CCRF Computational Complexity Reduction Factor IAPR Instantaneous-to-Average Power Ratio IFFT Inverse Fast Fourier Transform OFDM Orthogonal Frequency Division Multiplexing PA Power Amplifier PAPR Peak-to-Average Power Ratio SLM SeLected Mapping

Algorithm 4 Computational complexity reduction in the GreenOFDMv2 scheme. Require: {X k }, {φ u,k } with φ u,k ∈ {±1}, γ pi {x 0:U -1 [n]} ← zeros(U, LN );

allows to generate many more candidates for the proposed method and hence more chances to find the symbol whose PAPR is below the imposed threshold, reducing the expected total number of operations (in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF], it was pointed-out that the IFFTs computations are the greatest contributors to complexity in terms of the total number of operations).

Computer simulations were carried out to validate the improved complexity reduction capabilities of the proposed method. To be consistent with the comparisons, we have imposed the same set of parameters as in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF] which are 10 5 symbols iterations with N = 64, 128, 256, 512 and 1024 QP SK-modulated subcarriers, oversampling factor L = 4 and U = 1024 IFFTs for the SLM-OFDM, U = 64 for the GreenOFDM and U = 32 for the proposed method.

The same complexity reduction strategy applied to all the schemes as a function of the imposed threshold γ pi calculated with Eq.( 6)& Eq.( 8) at p = 10 -3 was computed. The CCDF distribution of the total number of operations (multiplications, additions and comparisons), noted M, was computed for different imposed threshold γ pi . The value of M at CCDF(M)= 10 -3 was obtained for comparison purposes and the Computational Complexity Reduction Factor (CCRF) was calculated.

Figure 8 depicts, on the left side, the CCRF M between the original GreenOFDM and the SLM-OFDM as introduced in [START_REF] Gulfo Monsalve | Computational complexity reduction of GreenOFDM[END_REF] i.e.,

On the right side we can find the CCRF M between the GreenOFDMv2 and again the SLM-OFDM, i.e.,

As it can be seen, for the same imposed threshold, the CCRF for the GreenOFDMv2 is always lower than the GreenOFDM. Furthermore, almost a factor two of reduction is appreciated between both CCRF which is in line with the previously evoked factor 2 in terms of the reduced number of IFFTs.