M Parthasarathy

G Jasniewicz
email: gerard.jasniewicz@umontpellier.fr

F Thévenin

The puzzling high velocity G5 supergiant star HD 179821 : new insight from Gaia DR2 data

Keywords: AGB and post-AGB | Stars, individual, HD179821 | Stars, distances | Stars, evolution

HD 179821 is classified as G5 Ia star. From the IRAS colors and spectral energy distributions it was classified as a post-AGB star. But some studies classify it as a massive (30 to 19 M sun ) post-red supergiant evolving to become a Type II supernova. Its mass and evolutionary status remained a hotly debated question even after several detailed spectroscopic studies as the distance was not known. We use the parallax of HD 179821 from the second Gaia data release, and deduce its distance 2959 ± 409 pc and its absolute magnitude MV = -5.7 ± 0.3. Using the absolute magnitude determined in this paper, we show that HD 179821 fits very well with post-AGB tracks in the H-R diagram. Our results clearly confirm that HD 179821 is a post-AGB star of mass in the range of 0.8 M sun . It is not a 30 M sun red supergiant. The progenitor mass of this post-AGB star may be in the range of 4 M sun but may be a bit more.

Introduction

HD 179821 (IRAS 19114+0002) was classified as a post-AGB star by Pottasch & Parthasarathy (1988) based on IRAS colors and SED. Its IRAS colours and SED are similar to that of high galactic latitude post-AGB star HD 161796 (Parthasarathy & Pottasch 1986). It was also classified as a massive (30 M sun ) red supergiant, post-red supergiant evolving to become a Type II supernova (Zuckerman & Dyck 1986, Jura & Werner 1999, Jura et al. 2001). Zuckerman & Dyck (1986) from CO observations estimated the distance of HD 179821 to be 6kpc. Adopting this 6kpc distance Hawkins et al. (1995) and Kastner & Weintraub (1995) also considered to be a very massive star with massive circumstellar shell.

The spectral type of HD 179821 is G5 Ia (Keenan 1983;Keenan & McNeil 1989;[START_REF] Buscombe | MK spectral classifications. Sixth gener al catalogue Ekström[END_REF]Hrivnak et al. 1989) and galactic latitude b = -4.96 o . Zacs et al. (1996) derived the Radial Velocity (RV) of the star from high resolution spectra to be +88.4 km s -1 which shows that it is a high velocity star.

On the other hand, the work of Josselin & Lèbre (2001) presents high sensitivity radio observations of HD 179821 in the circumstellar 12 CO and 13 CO. They detected HCO+ and found 12 CO/ 13 CO < 5 ± 1. Such a low value is characteristic of (post-)AGB stars with low or intermediate-mass progenitors, and not of a 30 M sun red supergiant.

Ferguson & Ueta (2010) made differential proper motion study of the circumstellar shell based on HST observations. They concluded that it is a post-AGB star and estimated distance of 3.91 ± 3.23 kpc.

Recently Ikonnikova et al. (2018) presented the results of multicolor photometry obtained during 2009 to 2017. They found that HD 179821 displays low amplitude 0.2 magnitude semi-periodic brightness variations, and concluded that the star's behaviour is similar to that of yellow hypergiants located near the very unstable Yellow Void, and differs significantly from ordinary post-AGB objects.

Thus HD 179821 is either classified as a post-AGB star descended from a low-mass progenitor or as a very massive red supergiant. This research note follows a previous paper (Thévenin et al. 2000) in which we concluded that HD 179821 is a low mass post-AGB star with T eff = 5660 K and not a massive red supergiant. In Section 2 of this paper we report accurate distance and M V based on the analysis of the accurate parallax of HD 179821 from the second Gaia data release (hereafter GDR2; Gaia Collaboration 2018). In Section 3 we discuss the new data, relate recent studies and give our conclusions in Sect.4.

New observational data from the second Gaia data release

The GDR2 parallax of HD 179821 is found to be 0.310239 ± 0.051225 mas (Gaia Collaboration 2018), giving a parallax precision about 16%. The distance of the star is thus approximately 3000 pc, but according to Bailer-Jones et al. (2018) going from a Gaia parallax to a distance is a non-trivial issue and cannot be obtained by simply inverting the parallax. In the following we adopt the distance given by the inference procedure developed by Bailer-Jones et al. ( 2018) : 2959 ± 409pc. Using this distance and observed V magnitude (8.19) and B-V color (1.509) from Høg et al. (2000) and G5Ia spectral type and intrinsic (B-V=1.02) color of a G5Ia star and A V = 3.1 E(B-V) we find A V = 1.50 ± 0.06 and M V = -5.7 ± 0.3.

The observed (B-V) color of HD 179821 consists of interstellar and circumstellar reddening which is taken into account when we calculate A V using the E(B-V). Le Coroller et al. (2003) and Arkhipova et al. (2009) monitored the photometric variability for many years and found small amplitude light variations similar to that found in other post-AGB supergiants. They derived E(B-V) = 0.7, which is compatible with the E(B-V) value we have adopted here.

From the Gaia parallax we further deduce that HD 179821 is approximately z = -260 pc below the galactic plane. The large V LSR = 105 kms -1 observed by Zuckerman & Dyck (1986) give the central velocity of the CO or HCN profile with respect to the local standard of rest. The 6 kpc distance being no more reliable, the large VLSR is not due to differential galactic rotation. HD 179821 is simply a high velocity star, confirmed by the Gaia RVS value +81.8 ± 3.7kms -1 (Sartoretti et al. 2018).

Discussion

The derived M V = -5.7 clearly establishes that HD 179821 is not a 30 or 19 M sun post-red supergiant. Indeed, for such a mass Ekström et al. ( 2012) give a luminosity log(L/L sun ) = 5.5, which corresponds to M bol = -9.0. Hereafter we take BC = -0.33 for the bolometric correction for a G5 supergiant, and thus M bol = -6.0 ± 0.4 and log(L/L sun ) = 4.3 ± 0.2 for HD 179821.

Concerning the effective temperature of HD 179821 there is no consensus among researchers. However we would like to emphasize here that there is not large error or uncertainty in the MK spectral type assigned to this star by Keenan (1983), Keenan & McNeil (1989), [START_REF] Buscombe | MK spectral classifications. Sixth gener al catalogue Ekström[END_REF], and Hrivnak et al. (1989). These authors have classified the spectrum of HD 179821 as G5 I, and never as that of a F0Ia-F2Ia (7350K) star. Sahin et al. (2016) derive T eff = 7350K which is not consistent with its observed spectral type and B-V color. The spectroscopic studies by Zacs et al. (1996), Reddy & Hrivnak (1999), Thévenin et al. (2000), Kipper (2008) and Sahin et al. (2016) yielded very different T eff values ranging from 5500 K to 7350 K. In the following we adopt the effective temperature 5660 K (log T eff =3.75) according to Thévenin et al. (2000), in agreement with the G5 MK spectral classification, to show its position in the H-R diagram. We have not used the T eff values in our calculations. We used only the observed V magnitude, observed B-V colour and observed spectral type. In the HR diagram the post-AGB stars evolutionary track is towards the left from the tip of the AGB and with constant luminosity even though their T eff varies as they evolve towards hot post-AGB stars and early stages of young planetary nebulae.

In order to study the evolutionary status of HD 179821 in the H-R diagram, we use the post-AGB evolutionary models from Miller Bertolami (2016) which are based on outdated micro-and macrophysics. The models are computed for initial masses between 0.8 and 4 M sun and for a wide range of initial metallicities, the solar metallicity being Z = 0.02. They include all previous evolutionary stages from the Zero Age Main Sequence to the White Dwarf phase. The evolutionary sequences in Fig. 1 correspond to the sequence presented in Table 3 of the article of Miller Bertolami. These new models are 0.1 to 0.3 dex brighter than the old post-AGB stellar evolution models with similar remnant masses. The final masses are in the relevant range for the formation of planetary nebulae with central stars masses in the range 0.5 to 0.8 M sun .

The location of HD 179821 indicates that the progenitor of this post-AGB star may have an initial main sequence mass of 4 ± 1 M sun . We can call it as a super post-AGB star resulting from an intermediate mass AGB star or from a star which is in the lower limit of a super-AGB star. Sahin et al. (2016) used the very broad and very strong Na D absorption feature in the high resolution spectrum of HD 179821. This Na D feature is strongly blended with stellar, interstellar and circumstellar Na D lines and it is impossible to disentangle the interstellar and circumstellar Na D lines to estimate reliable E(B-V) value. Considering the uncertainties and errors in the E(B-V) value they estimated it is similar to the value that we adopted here. In deriving the 19 M sun mass of HD 179821 Sahin et al. (2016) used the unreliable M V value derived from the equivalent width of OI 7774 Å triplet feature and spectroscopic surface gravity log g. The post-AGB supergiants and normal massive supergiants have similar spectroscopic surface gravity values. The spectroscopic surface gravity is not a reliable parameter to derive mass of a supergiant. Therefore the 19 M sun for HD 179821 derived by them is unreliable and not correct.

Conclusions

Based on the accuracy of the parallax of HD 179821 taken from the GDR2, and our confidence in the evolutionary tracks of post-AGB (Miller Bertolami 2016) and supergiants (Ekström et al. 2012) we definitively conclude after 30 years of debate, that HD 179821 is a post-AGB star of mass in the range of 0.8 M sun . The bolometric magnitude clearly excludes the possibility, defended by many authors, that it is a very massive (30 to 19 M sun ) post-red supergiant star.

Fig. 1 .

 1 Fig. 1. HR diagram. HD 179821 is located with T eff = 5660K and log(L/L sun ) = 4.3 ± 0.2 (see Sect.3). The post-AGB models come from Miller Bertolami (2016) ; the initial masses and metallicities are indicated.

Acknowledgements. This study has made use of the SIMBAD database, the VizieR catalogue access tool and of NASA's Astrophysics Data System, operated at CDS, Strasbourg, France. The authors are grateful to Olivia Mancuso for her review of English phrasing and grammar in this article.


Alex Chaves

Gabriel Azevedo Limeira

Roberto Coury Pedrosa

Vicente Zarzoso

Jurandir Nadal

Alex C Alberto

ECG-Based Predictors of Sudden Cardiac Death in Chagas' Disease

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Chagas' disease is a parasitic disease caused by Trypanossoma cruzi, which is endemic in Latin America with nearly six million infected subjects, becoming an alarming public health problem. Due to immigration, it spreads to other regions and about 300,000 people are estimated to be infected only in the United States [START_REF] Chagas Disease | Available on line in CDC website accessed[END_REF]. The parasite infects the heart tissues and causes an immune response, which affects myocardial contractile cells and nerve endings of the autonomic nervous system (ANS). The Chagas' heart disease (CHD) is associated with arrhythmias and high risk of sudden cardiac death (SCD). The early stages of the disease are characterized by the presence of multifocal premature ventricular contractions (PVC). The occurrence of SCD in early and intermediate stages of Chagas disease doubles or triples that observed in the most advanced stage, when ventricular dysfunction occurs [START_REF] Rassi | Challenges and opportunities for primary, secondary, and tertiary prevention of Chagas' disease[END_REF].

Heart rate variability (HRV) is useful for assessing ANS action over the heart. The control of heart rate is due to balance of sympathetic and parasympathetic nervous system branches of the ANS. The degree of variability provides information about the autonomic control of the heart rate, and a significant relationship between ANS control and cardiovascular mortality including SCD is observed [START_REF]Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology[END_REF].

Developed by Schmidt et al. [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF], heart rate turbulence (HRT) focuses on heartbeat's response to a PVC. This technique describes the biological phenomenon of spontaneous short-term fluctuations in sinus cycle length that follow the occurrence of the PVC. The expected response is an immediate cardiac rhythm increase followed by a progressive decrease in subjects without abnormalities or disease. HRT mechanisms have not been fully identified, and have been attributed to the baroreflex response to hemodynamic changes caused by the PVC [START_REF] Watanabe | Heart rate turbulence: A 5-year review[END_REF]. The turbulence trigger is the lower systolic blood pressure pulse related to the PVC, which is followed by an increased pulse due to the compensatory pause. This abnormal pressure variation is expected to trigger a baroreflex response. If ANS is intact, the HRT corresponds to its immediate response in the sinus node. However, in impaired autonomic heart rate control the reaction is weak or does not occur.

The HRT technique consists in the coherent average of tachogram segments using as anchors the short RR intervals due to each isolated PVC detected along the 24 h Holter ECG record. From this averaged tachogram, two indices are obtained: the turbulence onset (TO), that measures the immediate increase in heart rate; and the turbulence slope (TS), which measures the compensatory and progressive decrease in heart rate [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF]. TO and TS are powerful indicators of SCD, particularly when combined to other established parameters [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF]. Some studies [START_REF] Watanabe | Heart rate turbulence: A 5-year review[END_REF] show TS as an independent mortality predictor for subjects after post-infarction. Particularly in subjects with left ventricular ejection fraction (LVEF) below 30%, HRT indices showed to be powerful to stratify the risk of death [START_REF] Dias | Brazilian Consensus on Chagas Disease[END_REF]. Although HRT was proposed and validated for the assessment of patients after acute myocardial infarction, some studies point to TO and TS as useful parameters for risk stratification also in other cardiac diseases [START_REF] Savelieva | Heart Hate Turbulence After Atrial and Ventricular Premature Beats -Relation to Left Ventricular Function and Coupling Interval[END_REF], including Chagas cardiomyopathy [START_REF] Watanabe | Heart rate turbulence: A 5-year review[END_REF]. A possible limitation of the HRT approach is the use of the average response from Holter recordings over 24 h, which may neglect information of circadian rhythm variability [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF].

Computing in

This study aims at retrospectively analyzing 24 h Holter recordings from Chagas disease patients by HRT and HRV parameters. The respective parameters were obtained from 12 h periods corresponding to day and night, and evaluated by multivariate analysis to assess the risk of SCD.

2.

Material and methods

Database

Seventy five Holter ECG records from patients with acute and chronic Chagas' heart disease were selected for the study, from a set of 410 recordings of the Hospital Universitário Clementino Fraga Filho, Rio de Janeiro, Brazil. 1), following the Brazilian Consensus on Chagas' Disease [START_REF] Dias | Brazilian Consensus on Chagas Disease[END_REF] by three different tests: serology, Holter ECG, and echocardiography. The diagnosis of Chagas disease was based on Machado-Guerrero serology, in accordance with the World Health Organization guidelines [START_REF] Chagas Disease | Available on line in CDC website accessed[END_REF].

For the present study, patients were grouped according to outcome as survivals (Alive) or SCD. To avoid bias in statistical analysis due to unpaired number of cases between the groups, the number of alive patients was reduced to 11 by random selection (Table 2).

Heart Rate Turbulence

HRT analysis was performed with the original software developed by Schmidt et al. [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF]. Each accepted tachogram (Figure 1) consisted of 22 RR intervals comprising a short one related to a PVC, preceded by five normal intervals, and the respective compensatory pause followed by 15 post-PVC intervals after the compensatory pause. The select tachograms from the same patient record were aligned around the PVC interval to produce coherent averaged segments used to measure TO and TS parameters. TO was measured as the ratio between the first two intervals after the PVC and the two ones just before the PVC, while TS is retrieved from the maximum slope of the regression line obtained over any five consecutive RR intervals within the 15 intervals after the PVC in the coherent average [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF].

To take into account circadian changes in autonomic control, these parameters were also computed from the first 12h (day) and the last 12h (night) of the 24h Holter recordings.

Heart Rate Variability

The HRV analysis was assessed over the longest segment of RR intervals without ectopic beats (NN intervals) in each of the 12 h period considered. The analysis was performed in the time-domain following standard guidelines [START_REF]Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology[END_REF]. The following parameters were computed: mean heart rate (HR); standard deviation of all NN intervals (SDNN); the root mean square of the successive differences in NN intervals (RMSSD); number of NN intervals higher than 50 ms (NN50) and the ratio between the count of NN intervals higher than 50 ms divided by the total number of NN intervals (pNN50).

Statistical Analysis

All signal processing and statistical analysis were performed using MATLAB (The Mathworks, USA). Seven different parameters (TO, TS, HR, SDNN, RMSSD, NN50 and pNN50) were obtained for two different conditions (day and night), yielding a total of 14 features.

The gaussianity of the predictors was verified using Kolmogorov-Smirnov test.

For feature selection, it was performed both univariate and bivariate analysis, which was performed on all combinations of predictor pairs. Multivariate analysis including more than two predictors was not considered to avoid overfitting due to limited sample size.

To find a linear combination of features between two chosen predictors logistic regression (LR) and Fisher linear discriminant (LDA) were applied. LR model was developed using a multinomial logistic regression from the Statistical Toolbox Package of MATLAB. For LDA model, Statistical Pattern Recognition Toolbox was used. Both models were evaluated in terms of sensitivity, specificity and overall accuracy.

Results

The HRT parameters showed different prognostic outcome for a same patient over 24 h and 12 h recordings when considering abnormal values (TO > 0 and TS < 2.5), following the original criteria. Not all the available signals were useful to the analysis due to the need of at least one isolated PVC occurrence in the analyzed segment. Just nine Alive and eight SCD patients presented HRT results after the signal fractionation in day and night. The univariate analysis showed that all predictors had normal distribution, and no simple feature could discriminate between Alive and SCD groups.

The bivariate analysis led to three independent parameters retained in the resulting bivariate prognostic models: TS, SDNN and pNN50, all at night period. These predictors appeared in the two best combinations: SDNN vs TS (Comb1) and pNN50 vs TS (Comb2).

The best result of the all six possible combinations of TO and TS including day and night periods didn't exceed 70.6% accuracy. The LDA model performed 20,000 iterations with 0.01 margin of error to separate the groups. Further, a logistic regression (LR) model was developed for the same parameters, considering the binomial distribution.

In the bivariate analysis, the two types of linear separation models were applied to find fitted outcomes in early stages of Chagas disease (Figures 2 and3). The LDA model showed better performance indices than LR, when using pNN50 as a predictor (Table 3).

Discussion

The results obtained for 12 h data from day or night periods show changes in both TO and TS values. This suggests that the conventional estimation of these parameters as an average result from a 24 h period could cover different turbulence responses for diverse conditions in the circadian cycle, as originally suggested by Schmidt et al. [START_REF] Schmidt | Heart-rate turbulence after ventricular premature beats as a predictor of mortality after acute myocardial infarction[END_REF]. On the other hand, shortening the period of analysis increases the chance of lack of results in CHD patients. Independent of the analyzed period, TO and TS parameters were not adequate for separating subjects according the clinical outcome after follow-up, since most of SCD subjects presented low risk HRT parameters.

The combination of HRT and HRV parameters yielded reasonable results, with both LR and LDA models using combinations of parameters SDNN, pNN50 and TS, in all cases measured in the night period. All these parameters are related to vagal activity. Particularly, pNN50 measures fast changes in HR and TS estimates the rate of HR deceleration after stimulus. Additionally, all parameters came from the night period of signals, when the vagal tonus is expected to increase. However, these parameters may not be considered as independent predictors of SCD.

The reduced sample size and the retrospective analysis could be considered as study limitations. The discriminant power of these models was tested in the same data used for model estimation. Further analysis over an increased sample is required to confirm the reproducibility of this study and the prediction capability of the selected features and models.

Conclusion

When computed separately, HRT and HRV indices cannot discriminate Alive from SCD patients with Chagas disease. Their discrimination ability increases when HRT is combined with standard HRV indices and they are computed in night recordings, when vagal tonus is increased. A novel approach that considers the segmentation of Holter signals in periods of day and night and combines HRT and HRV analysis in multivariate linear models allows risk stratification of sudden cardiac dead in Chagas' heart disease.

Figure 1 -

 1 Figure 1 -(Top) ECG with isolated PVC and the TRC method. (Bottom) The coherent average tachogram used to calculate HRT indices is in thick black line.

Figure 2 -

 2 Figure 2 -Classification thresholds between Alive and SCD using LDA (line) and LR (line dashed). Predictors are SDNN and TS, both from night data.

Figure 3 -

 3 Figure 3 -Classification thresholds between Alive and SCD using LDA (line) and LR (line dashed). Predictors are pNN50 and TS, both from night data.

Table 1 .

 1 Clinical stages of Chagas' heart disease

	Stage	ECG	Echocardiogram	Heart Failure
	A	Abnormal	Normal	Absent
	B1	Abnormal Abnormal, LVEF ≥45%	Absent
	B2	Abnormal Abnormal, LVEF <45%	Absent
	C	Abnormal	Abnormal	Compensable
	D	Abnormal	Abnormal	Refractory

Table 2 .

 2 Studied Sample

	Group	Alive	SCD
	Number of Patients	11	11
	Male sex	4	7
	Mean age	50.4 ± 14.3	60.5 ± 10.6
	LVEF	54.5 ± 16.1	40 ± 13.2
	Follow up (months)	141.3 ± 78.7
	The resulting database contains 24 h records of ECG
	signals from patients with CHD (two channels, 128 Hz
	sample rate), including 49 survival subjects and 26 deaths
	(all causes), being only 11 of them caused by SCD. Such

Table 3 -

 3 Performance indices of the two models used for separating Alive and SCD subjects

		LR	LDA
		SDNN vs TS (Comb 1)
	Sensitivity	87.5%	87.5%
	Specificity	77.8%	77.8%
	Accuracy	82.4%	82.4%
		pNN50 vs TS (Comb 2)
	Sensitivity	75.0%	87.5%
	Specificity	77.8%	77.8%
	Accuracy	76.5%	82.4%

Acknowledgements

This work received financial support from a Brazil/France cooperation CAPES/COFECUB, and the Brazilian agencies CAPES, CNPq and FINEP. V. Zarzoso is a member of the Institut Universitaire de France.


Hussein Chour
email: hussein.chour@supelec.fr

Oussama Bazzi
email: obazzi@ul.edu.lb

Faouzi Bader
email: faouzi.bader@supelec.fr

Youssef Nasser

Analytical Framework for Joint Mode Selection and Power Allocation for Full Duplex D2D Network

Keywords: duplex transmission, device-to-device (D2D) communication, optimal power allocation, duplex mode selection

In this paper, a simple and accurate analytical framework for the duplex mode selection and the power allocation scheme for full duplex (FD) device-to-device (D2D) communications underlaying wireless cellular network is proposed. We first aim at maximizing the D2D links rate by properly selecting the duplex mode of the D2D pairs and allocating the power of the users while fulfilling the quality-of-service (QoS) of the cellular users (CUEs). Since the resulting optimization problem is non-convex, we derive a first-order optimal solution by leveraging an interesting optimization tool named as sequential convex optimization theory. We also discuss how to obtain an efficient sub-optimal solution in term of complexity and accuracy by deriving the optimal power ratio between the D2D devices assuming an interference limited system. The simulations have shown more than 95% accuracy of the proposed approaches and provided great insights on the solution design parameters taking into account the CU location and the self-interference threshold.

I. INTRODUCTION

With the rapid growth in the users' data traffic demands and the scarcity of the radio spectrum [START_REF] Mobile | Cisco visual networking index: Global mobile data traffic forecast update[END_REF], the need of new technologies which use the spectrum efficiently and meet the users' data requirements has become an essential part in the future cellular network (5G). In this context, device-to-device (D2D) communication and Full Duplex (FD) technologies are proposed to improve the cellular spectrum utilization and the users' experience in a cellular network. D2D allows two nearby devices to initiate a direct link without base station (BS) participation or with limited participation [START_REF] Doppler | Deviceto-device communication as an underlay to lte-advanced networks[END_REF]. On the other hand, FD allows a pair of devices to simultaneously transmit and receive in the same frequency band at the same time slot [START_REF] Sabharwal | In-band full-duplex wireless: Challenges and opportunities[END_REF].

The exploitation of the FD scheme in D2D technology can further improve the cellular spectrum efficiency and the users throughput due to the short distance property of the D2D communication [START_REF] Tapio | System scenarios and technical requirements for full-duplex concept[END_REF]. However, the practical full duplex transceivers add new challenges for the D2D communication. For instance, the existing FD devices cannot perfectly eliminate the self interference (SI) imposed on the receiver by the node's own transmitter [START_REF] Sabharwal | In-band full-duplex wireless: Challenges and opportunities[END_REF]. Thus, the residual self interference (RSI) which is tightly related to the transmitter power value highly affects the performance of FD transmission. In addition, using the FD technique creates additional interference in the network which may degrade its performance when compared with the half duplex (HD) transmission. Accordingly, the power allocation strategy and the selection of the D2D transmission mode are very important problems to be tackled.

The power optimization problem of an isolated full duplex D2D (FD-D2D) pair underlay cellular network was tackled in [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF]. Built on this, the authors of [START_REF] Zuo | Power allocation optimization for full-duplex d2d communications underlaying cellular networks[END_REF] derived a convex optimization problem to maximize the rate of FD-D2D link while satisfying the minimum rate requirement of the cellular users. However, neither [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF] or [START_REF] Zuo | Power allocation optimization for full-duplex d2d communications underlaying cellular networks[END_REF] provided an analysis on the best D2D transmission mode. Moreover, the solution therein is rather algorithmic. Without addressing the power allocation problem, the authors of [START_REF] Hemachandra | Sum-rate analysis for full-duplex underlay device-to-device networks[END_REF] derived and analysed a closed form expression for the Ergadic sum rate of a FD-D2D network. The mode selection problem of a FD-D2D link was tackled in [START_REF] Liu | On the performance of mode selection for hybrid-duplex d2d communications[END_REF]- [START_REF] Han | Duplex mode selection for device-todevice communications underlaying the cellular uplink[END_REF]. Using stochastic geometry the authors of [START_REF] Liu | On the performance of mode selection for hybrid-duplex d2d communications[END_REF] proved that by properly switching between HD and FD, the system spectrum efficiency of a FD-D2D pair can be improved. In [START_REF] Kadhim | On selecting duplex-mode and resource allocation strategy in full duplex d2d communication[END_REF], the mode selection problem was investigated for an overlay and underlay FD-D2D enabled cellular network. However, the optimal power allocation was not discussed either in [START_REF] Liu | On the performance of mode selection for hybrid-duplex d2d communications[END_REF] nor in [START_REF] Kadhim | On selecting duplex-mode and resource allocation strategy in full duplex d2d communication[END_REF]. In [START_REF] Han | Duplex mode selection for device-todevice communications underlaying the cellular uplink[END_REF] the duplex mode selection is investigated considering a D2D underlaying scenario and aiming to maximize the rate of the cellular user with a minimum constraint on the D2D users. Since the main goal of the D2D technology is to alleviate the load from the cellular network, it is more interested to study the performance of FD-D2D communication when the goal is to maximize the D2D rate while fulfilling the QoS of the CUE. In an interference limited scenario, the latter has a non-convexity structure and thus it will lead to a much challenging tasks. In addition, no closed form expression of the optimal power allocation was provided.

In this paper, we analytically investigate both the duplex mode selection and the power allocation problem of a FD-D2D underlaying cellular network. In particular, we provide two approaches to solve the power allocation problem. The first is based on the sequential optimization theory (SCO), [START_REF] Venturino | Energy-efficient scheduling and power allocation in downlink OFDMA networks with base station coordination[END_REF]- [START_REF] Venturino | Coordinated scheduling and power allocation in downlink multicell OFDMA networks[END_REF], which is iterative based solution. In the second solution, we develop an analytical framework which yields to a suboptimal solution but with high accuracy and low complexity.

Moreover, it provides a closed form expression for the optimal power allocation and an upper bound condition for the usage of full duplex.

The rest of this paper is organized as follows. Section II describes the system model and formulates the optimization problem. Section III and IV derive the optimal solution and section V provides the simulation results. Finally, the conclusion is drawn in section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a D2D communication underlaying cellular network where M D2D pairs coexist with M cellular users. Each D2D pair D2D j consists of two nearby devices denoted as D j 1 and D j 2 and they have the capability to operate on both Full duplex and Half duplex modes, while the CU is assumed to be operated only in HD mode. Moreover, the D2D pair shares the uplink resources of the CU. In particular, UL resources sharing is considered since the UL resources are underutilized comparing to that of downlink. Furthermore, sharing the UL spectrum only affects the BS and incurred interference can be handled by BS coordination. We also assume that D2D j can reuse only the uplink channel of CU j and the CUs are allocated orthogonal channels. In other word, the focus in this work is on the mode selection and power allocation and not on the channel assignment step.

An instance of this network is depicted in Fig. 1 in which two D2D pairs coexist with two cellular users (M = 2). The first D2D pair D2D 1 depicts the case of FD D2D communication while D2D 2 depicts the HD-D2D mode. The channel power gain between the jth cellular user and the BS is denoted as g j c,bs while the channel power gains between D j 1 and D j 2 and between D j 2 and D j 1 are denoted as g j d . The interference channel power gains from D j 1 and D j 2 to BS are respectively denoted by h j d1,bs and h j d2,bs , while h j c,d1 and h j c,d2 denote the interference channel power gains from the jth cellular user to D j 1 and D j 2 respectively. All direct/interference channels are assumed to be zero-mean complex Gaussian random variables (i.e., channels are expressing Rayleigh fading) with variance l -α ij , where i ∈ {c; d1; d2}, j ∈ {bs; d1; d2}, i = j, l ij denotes the distance between the nodes i and j, and α denotes the path loss exponent.

The channels h j 11 and h j 22 are the self interference channels at D j 1 and D j 2 . They can be modeled as complex Gaussian random variables with zero-mean and variance ηP j d1 and ηP j d2 respectively [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF], [START_REF] Zuo | Power allocation optimization for full-duplex d2d communications underlaying cellular networks[END_REF], [START_REF] Liu | On the performance of mode selection for hybrid-duplex d2d communications[END_REF]- [START_REF] Han | Duplex mode selection for device-todevice communications underlaying the cellular uplink[END_REF], [START_REF] Riihonen | Mitigation of loopback selfinterference in full-duplex mimo relays[END_REF], where P j d1 and P j d1 are respectively the transmission power of D j 1 and D j 2 , η denotes the SI cancellation capability of each D2D device. Hence, the residual SI power at D j 1 and D j 2 are respectively ηP j d1 and ηP j d2 .

A. Full duplex D2D link

In the FD D2D communication mode, both D j 1 and D j 2 reuse the spectrum of CU j to communicate with each other. Denote by N 0 the power of additive white Gaussian noise and let P j c be the transmission power of the jth cellular user, the received signal to interference plus noise ration (SINR) at BS,

BS D 1 1 D 1 2 g d g d h 1 11 h 1 22 D2D1 D 2 1 D 2 2 g d D2D2 h 1 d2 ,bs h 1 d1, bs h 2 d1, bs CU 1 h 1 c,bs h 1 c ,d 2 h 1 c ,d 1 CU 2 h 2 c ,b s h 2 c,d 2

Direct link

Interference link Fig. 1. A full duplex and half duplex D2D pairs sharing the UL resources of the cellular users.

D j 1 and D j 2 when D2D j shares the spectrum of CU j can be respectively expressed as

Γ j bs =
P j c g j c,bs P j d1 h j d1,bs + P j d2 h j d2,bs + N 0

(1)

Γ j FD,d1 = P j d2 g j d P j c h j c,d1 + ηP j d1 + N 0 (2)
Γ j FD,d2 = P j d1 g j d P j c h j c,d2 + ηP j d2 + N 0 (3)
Let B j be the bandwidth of the jth CU spectrum. Applying Shannon theorem, the rate of the jth D2D when FD mode has been selected can be expressed as:

R j FD =B j log 2 (1 + Γ j FD,d1 ) + B j log 2 (1 + Γ j FD,d2 ) (4) B.

Half duplex D2D link

For fair comparison with the FD mode, we assume that D j 1 and D j 2 are operating on two equally orthogonal portions of B j during the transmission. Thus, the received SINRs at BS from each couple (D2D j , CU j ) remains the same as in the FD mode, i.e., eq [START_REF] Mobile | Cisco visual networking index: Global mobile data traffic forecast update[END_REF]. However, and since in HD mode the RSI does not longer exist, the received SINR at D j 1 and D j 2 when D2D j is operating in HD changed to the follows:

Γ j HD,d1 = P j d2 g j d P j c g j c,d1 + N 0 (5)
Γ j HD,d2 = P j d1 g j d P j c g j c,d2 + N 0 (6)
Again by using Shannon theorem, the HD-D2D link capacity of D2D j is given by

R j HD = 1 2 B j log 2 (1 + Γ j HD,d1 ) + 1 2 B j log 2 (1 + Γ j HD,d2 ) (7)

C. Problem formulation

The main goal of D2D technology is to offload a portion of the data traffic from the cellular network. Thus, in this paper, we aim to maximize the total D2D capacity while satisfying the QoS of the cellular users. Since each D2D pair may operate either in HD or FD mode, two maximization problem must be studied.

Let R j min be the minimum data rate for the jth cellular user, and denote by P = [p j ] M ×3 the matrix of the users' transmission power with p j = (P j d1 , P j d2 , P j c ) being the transmission power of the couple (D2D j , CU j ), the D2D capacity maximization problem in the HD and the FD modes can be respectively formulated as follows:

P1 : max P R HD d2d = M j=1 R j HD s.t. P ∈ Φ (8) P2 : max P R F D d2d = M j=1 R j F D s.t. P ∈ Φ (9)
Φ = {Γ j bs ≥ γ j min = 2 R j min -1, j ∈ {1, ..., M } (10) 0 ≤P j i ≤ P i max , i ∈ {d1, d2, c}, j ∈ {1, ..., M }} (11)
where P i max is the maximum transmission power of a transmitter i. Solving P1 is the target of the following section.

III. POWER ALLOCATION AND DUPLEX MODE SELECTION

In this section, we focus on finding the best duplex mode and power allocation for all the users in the network with the assumption that the BS has perfect knowledge about the channel state information for both D2D users and CUs.

A. Problem reformulation

Observing that the couples (D2D j , CU j ), j ∈ {1, . . . , M }, are independent from each others, the maximization problems P1 and P2 can be respectively rewritten as follows:

P1 : R HD d2d = M j=1 max p j R j HD s.t. p j ∈ Φ (12) P2 : R F D d2d = M j=1 max p j R j F D s.t. p j ∈ Φ (13)
Moreover, depending on the channel situation of each couple (D2D j , CU j ), the HD or the FD mode may be the optimal duplex mode. Therefore, the maximum D2D capacity can be achieved by optimizing the power of each transmission mode and then selecting the duplex mode with higher rate for each (D2D j , CU j ). Thus the total D2D capacity maximization problem can be reformulated as follows:

P3 : R d2d = M j=1 max(PA j F D , PA j HD ) (14)
PA j F D : max

p j R j F D s.t. p j ∈ Φ PA j HD : max p j R j HD s.t. p j ∈ Φ
with PA j F D / PA j HD being the power allocation problem for the couple (D2D j , CU j ) operating in FD/HD mode. The couples (D2D j , CU j ) are independent, and thus in the following, and for ease of notation we derive the solution of P3 for a sample pair (D2D, CU ) i.e., we omit the j superscript from all the notation. The solution for any pair will be the same.

It can be observed that P3 is a non-concave problem, since both R F D and R HD are not concave function with p. Thus the global optimal solution of P3 is very hard to achieved. Therefore, in the following we present two suboptimal solutions. The first one obtains a first order optimal solution by using the sequential convex optimization theory (SCO). The SCO is an iterative based solution, and thus it may require high number of iteration to converge in some situation. In addition, it does not give a closed form solution. Therefore, to further reduce the complexity and derive a closed form solution we propose a new sub-optimal solution that finds the optimal power ratio between the users in the SIR domain which in turn leads to a high efficient sub-optimal solution of P3 in term of complexity and accuracy. Note that, even if it not mathematically proved, in most of cases the SCO theory finds the global optimal solution of the problem as reported in [START_REF] Zappone | Globally optimal energy-efficient power control and receiver design in wireless networks[END_REF] [START_REF] Matthiesen | Global sum rate optimal resource allocation for non-regenerative 3-way relay channels[END_REF]. Thus, the SCO method will be used to benchmark the results of our proposed method.

B. Sequential convex optimization method

In this section we solve the optimization problem defined in P3 by using the sequential convex optimization method. The SCO technique is an iterative algorithm which finds local optima of a non-convex maximization problem with objective f , by solving a sequence of simpler problems with convex objectives {f i } i . Moreover, SCO is guaranteed to converge to a first-order optimal solution of P3 when at each iteration i the following three properties are satisfied [12, Section IV].

• f i (x) ≤ f (x), ∀x; • f i (x * i-1 ) = f (x * i-1 ); • f i (x * i-1 ) = f (x * i-1 )
; where x * i-1 is the maximizer of f i-1 . Hence,the key point in SCO is to find a simpler objective {f i } i which fulfill the above three properties. In the following we show how to apply SCO in P 3.

Since both R F D d2d and R HD d2d have the same function structure we will demonstrate how to derive the first-order optimal solution only for the FD mode. Now, using the logarithm properties R F D d2d is reshaped as follows:

R F D =B log 2 (P c g c,d1 +ηP d1 +N 0 +P d2 g d ) + log 2 (P c g c,d2 +ηP d2 +N 0 +P d1 g d ) -log 2 (P c g c,d1 +ηP d1 +N 0 ) -log 2 (P c g c,d2 +ηP d2 +N 0 ) =r + f d (p) -r - f d (p) (15)
observe that r + f d and r - f d are concave functions and recall that any concave function is upper bounded by its first-order Taylor expansion at any point. Therefore, for any given power vector p t we can approximate [START_REF] Zappone | Globally optimal energy-efficient power control and receiver design in wireless networks[END_REF] as follows:

R F D ≥ r + f d (p) -r - f d (p t ) + ∇ p r - f d | p=p t T (p -p t ) (16)
Hence, ( 16) is lower bound of the utility in [START_REF] Zappone | Globally optimal energy-efficient power control and receiver design in wireless networks[END_REF]. Moreover, since the lower bound in ( 15) is tight when evaluated in p t , ( 16) is equal to (15) for p = p t . Next, by using the linearity property of the gradient it is easily verified that ∇ (( 15)) = ∇ (( 16)) | p=p t . Thus, all the above requirements are satisfied and by using this relaxation the convergence of SCO to the first order optimal solution is guaranteed.

Similarly, the approximation of R HD can be derived as:

R HD ≥ r + hd (p) -r - hd (p t ) + ∇ p r - hd | p=p t T (p -p t ) (17)
wherein r + hd = 1 2 B(log 2 (P c g c,d1 + N 0 + P d2 g d2,d1 ) + log 2 (P c g c,d2 +N 0 +P d1 g d1,d2 ) and r - hd = 1 2 B(log 2 (P c g c,d1 + N 0 ) + log 2 (P c g c,d2 +N 0 )).

Note that both ( 16) and ( 17) are convex function (since they are summation of concave and linear functions), and thus the solutions of the power allocation problems PA F D and PA HD can be easily found by using any convex optimization tools such as CVX [START_REF] Grant | Cvx: Matlab software for disciplined convex programming[END_REF]. After that we select the duplex mode with higher rate. Therefore, the solution of P3 can be obtained by applying the previous procedure on each pair (D2D j , CU j ). Since SCO is an iterative based solution and it require solving an approximated convex problem at each iteration, in the next section we look to provide a closed-form suboptimal solution and also find the upper bound condition for using FD mode.

IV. THE PROPOSED POWER ALLOCATION SCHEME

Despite the interest of the SCO solution, the latter does not provide a closed-form analytical expression for the optimal power allocation and also may require high iteration numbers in some situation. Thus we use the SCO for benchmark purpose, and we present in this section a suitable analytical solution for P3. The proposed solution finds the optimal power ratio between the D2D users in the SIR domain and then calculates the corresponding power for each user in the SINR domain. The simulation results in section IV shows that the performance of the proposed technique is within the 95 % of the benchmark results achieved by the SCO method but with much lower complexity. Indeed, the proposed solution finds a solution for the PA problem by only searching three points. In addition, the proposed solution provides valuable information about the necessary conditions to operate in FD mode.

A. Optimal Power Ratio

The aim of this subsection is to find the optimal power ratio between the D2D users. To that end, first we relaxed the power allocation problems PA F D and PA HD to PA' F D and PA' HD respectively by considering an interference limited system, i.e., the noise power is assumed to be negligible compared to the interference power.

PA' F D : max

p j log 2 (Q f d (x, y)) s.t. p j ∈ Ω (18)
PA' HD : max

p j log 2 (Q hd (x, y)) s.t. p j ∈ Ω (19) Q f d (x, y) = (1+ g d y g c,d1 +ηx )(1+ g d x g c,d2 +ηy ) (20) Q hd (x, y) = (1+ g d y g c,d1 )(1+ g d x g c,d2 ) (21) Ω = (x, y) : g c,BS ≥ γ c min (xg d1,bs +yg d2,bs ) (22) 0 ≤ x ≤ M x = g c,BS γ c min g d1,BS ,0 ≤ y ≤ M y = g c,BS γ c min g d2,BS (23)
where x = P d1 /P c and y = P d2 /P c are the power ratio between the D2D users and the cellular user. M x and M y denote the maximum allowed power ratio which guarantees the CUE's QoS, i.e., γ c min . Ω stands for the feasible set of PA' F D and PA' HD and it can be illustrated as in Fig 2 . The vertices o, v 1 and v 2 respectively represents the power ratio pairs (0, 0), (M x , 0), and (0, M y ). The lines ov 1 , ov 2 , and v 1 v 2 in Fig. 2 are the border lines of Ω. In order to find the optimal power ratios of ( 18) and ( 19) we first provide Lemma 1, and then we introduce Theorem 1.

Lemma 1. The optimal power ratio of PA' F D or PA' HD must reside on the border line v 1 v 2 .

Proof: First we provide the proof for the FD mode. For any scaling parameter λ > 1 and a power ratio pair (x, y) ∈ Ω we have the following,

Q f d (λx, λy) = log 2 [(1+ g d y g c,d1 λ +ηx )(1+ g d x g c,d2 λ +ηy )] ≥ Q f d (24)
Equation ( 24) states that given any (x, y) in the interior of the feasible region Ω, one can always find another power ratio, (λx, λy), that yields to a larger rate. Accordingly, the optimal point must lie on the border lines of Ω. It is easily verified that the maximum of (18) on the lines ov 1 and ov 2 are respectively the vertices v 1 and v 2 . Thus, the optimal point of (18) must reside on the line v 1 v 2 . A similar proof can be derive for the HD mode. Denote by v * f d and v * hd the optimal power ratio of ( 18) and ( 19) respectively. In the following theorem we show that both v * f d and v * hd can be found by only searching three points on the line v 1 v 2 Theorem 1. The optimal power ratios v * f d and v * hd belongs to the sets {v 1 ; v 2 ; v f d 3 } and {v 1 ; v 2 ; v hd 3 } respectively, where the points v f d 3 and v hd 3 lie on the line v 1 v 2 . Proof: Due to the monotonicity property of the logarithm, maximizing (18) or ( 19) is equivalent to maximizing Q f d (x, y) or Q hd (x, y) respectively. From Lemma 1 we know that the optimal point must resides on the line v 1 v 2 . Now, applying the equality case of (22), i.e., the line

v 1 v 2 , in Q f d (x, y)
and Q hd (x, y) and differentiating the result w.r.t x yields the following

∂Q f d (x) ∂x = Ax 2 + 2Bx + C F , (25)
∂Q hd (x) ∂x = ∂Q f d (x) ∂x | η=0 , (26)
where

A=g d η(g c,d2 + ηM y )(η -βg d ), B=(g c,d2 +ηM y ) M y g d βη 2 +ηg c,d1 g d +g c,d1 βg d (ηβ -g d ) , C=(g c,d2 +ηM y ) g c,d1 2 g d + M y g c,d1 g d 2 -(g c,d2 + ηM y )(ηM y g d + g c,d1 βg d ) , F =(ηx + g c,d1 ) 2 (ηβx -g c,d2 -ηM y ) 2 , β=g d1,bs /g d2,bs
Since F is seen to be always positive, the possible optima of Q f d can be found as the solution to Ax 2 + 2Bx + C, which is given by,

v f d 3 = 1 A (-B ± B 2 -AC), M y -βx (27)
The obtained point, v f d 3 , does not always guarantee a feasible solution (since the solution of ( 27) may be an imaginary point or it may lie outside v 1 v 2 ) and does not always present a global maxima on the line v 1 v 2 (since there is no guarantee that the second derivative is always negative). Thus, using Lemma 1, the global maximum of ( 18) is either one of the corner points, i.e., v 1 or v 2 , or v f d 3 (if v f d 3 is feasible). For the HD mode, it is easily to see that the second derivative of Q hd is always negative. Thus, the optimal solution of Q hd is the solution of (26) which is equal to v hd 3 = -C 2B | η=0 . However, v hd 3 is not always a feasible solution because it may reside outside v 1 v 2 and thus the solution of Q hd is also either v 1 or v 2 or v hd 3 (if v hd 3 is feasible). This theorem shows that the optimal power ratios in the SIR domain v * f d and v * hd are reduced to a set of three points. The equivalent power of each user in the SINR domain is derived in the next subsection.

B. Optimal transmission power and duplex mode

Here, we are interested in finding a suitable sub-optimal solution of PA F D and PA HD by using the optimal power ratio derived in the previous subsection. To that end, first we change the form of the the QoS constraint defined in [START_REF] Han | Duplex mode selection for device-todevice communications underlaying the cellular uplink[END_REF] as follows:

P c ≥ γ min g c,bs (g d1,bs P d1 + g d2,bs P d2 + N 0 ) (28)
By inspecting the objective function of PA F D and PA HD , it is clear that the maximum rate occurs only if P c is at its minimum level. In other words, to obtain the maximum D2D rate the equality in (28) has to be achieved. Now by using this information, the corresponding transmission powers in the three dimensional space (P d1 , P d2 , P c ), of v 1 , v 2 , v f d 3 , and

v hd 3 denoted respectively by V 1 , V 2 , V f d 3
, and V hd 3 can be calculated as follows:

V 1 = min{P d1 max ,M x (P c max -µN 0 )}; 0; µ(g d1,bs P V1 d1 +N 0 ) (29) V 2 = 0;min{P d2 max ,M y (P c max -µN 0 )}; µ(g d2,bs P V2 d2 + N 0 ) (30) V i 3 = min{P d1 max , θi µ(g d2,bs +θig d1,bs ) (P c max -µN 0 ); (31)
min{P d2 max , 1
µ(g d2,bs +θig d1,bs ) (P c max -µN 0 ); µ(g d1,bs P

v i 3 d1 + g d2,bs P V i 3 d2 + N 0 ) , i ∈ {f d, hd}
where

θ f d = x v f d 3 y v f d 3 , θ hd = x v hd 3 y v hd 3 and µ = γ c
min gc,BS . Note that, x t and y t denotes the x and y abscissa of the power ratio point t. Similarly, P t k , k ∈ {d1, d2, c}, denotes the P k abscissa of the power point t.

Let p f d and p hd be the sub-optimal solution of ( 18) and ( 19) respectively. Then, similar to theorem 1,

p f d ∈ {V 1 , V 2 , V f d 3 } and p hd ∈ {V 1 , V 2 , V hd 3 }. Now, denote by p * = max(R f d | p=p f d , R hd | p=p hd )
the sub-optimal solution of P3, the following theorem summarizes all the results. Theorem 2. p * can be found by only searching a set of four points as follows:

p * = max R f d | p=V1 , R f d | p=V2 , R f d | p=V f d 3 , R hd | p=V hd 3 (32) Proof: It is easily verified that R f d is always greater to R hd when p is equal to V 1 or V 2 ,
and thus only four points has to be searched instead of six. This is because the points V 1 and V 2 represent the case where only one of the D2D users is transmitting. Hence, in such case, the total bandwidth B must be used by the transmitter instead of using half of the bandwidth.

As a result, a suitable solution of P3 can be found by applying (32) on each couple (D2D j , CU j ). Here we should strengthen that the solution derived in this section is highly tractable as it has much lower complexity than the SCO solution.

C. Upper bound condition for using FD mode

First, we observe that the D2D pair operates in FD mode only when p = V f d 3 or equivalently when the power ratio is equal to v f d 3 . Thus, analyzing the performance of the D2D rate at this point may lead to an important characteristics of FD.

Here we are interesting in finding the upper bound condition of using FD mode, and thus we study the case of perfect SI cancellation (η = 0) in the SIR domain. In such case, v f d 3 is reduced to the follows:

v f d 3 = M y g d + g c,d1 -βg c,d2 2βg d , M y g d + βg c,d2 -g c,d1 2g d (33)
Moreover, when no RSI exist the second derivative of Q f d (x, y) can be written as

∂∂Q f d ∂∂x = 2B F = -2g c,d1 g c,d2 g 2 d F < 0. (34)
From (34), it is clear that in the case of perfect SI cancellation (18) is always concave. Thus, the FD mode will be the optimal transmission mode only if v f d 3 given in ( 33) is feasible and lies on v 1 v 2 . Hence, by comparing (33) with v 1 and v 2 the optimal condition of FD mode can be written as

γ c min (g d1,BS g c,d2 -g d2,BS g c,d1 ) g c,BS g d < 1 (35)
Accordingly, the D2D pair can operate in FD only when (35) holds true.

V. NUMERICAL RESULTS

We consider a single cell network with radius R and M CUEs and D2D pairs. The CUEs are uniformly located in the cell while each D2D pairs is uniformly located within a randomly located cluster with radius r. The different simulation parameters are summarized in Table I. Fig. 3 compares the proposed power allocation scheme with the SCO method for different cluster radius and different SI cancellation factors. In this figure we generate M = 10 CUEs and D2D pairs 200 times. Each iteration we averaged the results of all the couples (D2D j , CU j ) and finally we averaged the results of all the iterations. As it can be seen, the curves are very close for almost all the cases. This is because our approach finds the global optima in the SIR domain, so at high SINR regime the solutions will be almost the same. At low SINR regime the D2D rate will be very low and thus the difference between the two solution will not be that important. The main difference between the two solutions are at the medium SINR region. Moreover, Fig. 3 clearly shows that as the cluster radius r increases the average rate decreases. In addition, we can see that the average rate increases as the SI factor decreases. For a SI coefficient η < -70dB, i.e. for a good SI cancellation technique, the average rate is saturated. This is because at low SI, the interference terms of other users will play the major role in setting the FD rates as stated in (35).

In Table II we report the average number of iteration required for convergence for the SCO method. As it can be seen, on average SCO requires three to five iterations to achieve the solution. In addition, at each iteration SCO requires using a convex optimization solver to solve the approximated convex problem. On contrary, the proposed method can find a good solution with only one iteration and without using any optimization solver. To show the effect of the CUE location on the optimal transmission powers we first set D1 and D2 at a distance equal to 100m and 140m from the BS. Then we moved the CUE toward the D2D pair and at each step we computed the optimal transmission powers as shown in Fig. 4. From Fig. 4 we can see that when the CUE is near the BS and far from the D2D pair (d c,BS = 20m) both D1 and D2 can transmit with their maximum powers. However, when the CUE becomes close to D1 the transmission power of D2 decreases and at the point d c,BS = 70m only D1 must transmit to achieve the optima. This is because starting from this point D1 will face high interference from the CUE and thus it is better to send messages rather than receiving messages. The same analogy can be applied to the case where the CUE is closer to D2. On the other hand, P c increases with the increasing of d c,BS . This is because the CUE has QoS constraint. As a result, the CUE location highly affects the optimal mode selection.

To highlight the effect of the CUE location on the optimal D2D transmission mode, we reset the D2D users at 100m and 140m from the BS. Then, we moved the CUE on a random trajectory which covers almost all the cell. At each position we computed the optimal D2D transmission mode and we colored the CUE's location accordingly. As expected, when the CUE is far from the D2D pair, the yellow area in Fig. 5, the optima is achieved by using the FD mode. However, when the CUE is close to the D2D pair, the blue sector in

VI. CONCLUSION

In this paper we propose two frameworks to solve the joint mode selection and power allocation problem of a FD-D2D network. The first framework yields to a first-order optimal solution by using the SCO theory. The second framework gives a sub-optimal solution by finding the optimal power ratio between the users in the SIR domain and then computing the corresponding power in the SINR domain. The derived approach has been validated in simulation results. Among the different results, it is important to highlight the importance of the CUE location on the D2D transmission mode (HD or FD) as well the capabilities of the D2D devices to reduce the self-interference in the FD mode. As a future work, the maximization problem of the total rate, i.e., the rate of both cellular users and D2D users, will be considered. In addition, it might be interested to study the channel assignment problem of a FD-D2D network.

Fig. 2 .

 2 Fig. 2. Graphically illustration of the feasible set of P2.

Fig. 3 .

 3 Fig. 3. Comparison of FD-D2D rate obtained from the exhaustive search and from the proposed power allocation scheme.

  Fig 5, the optima can be achieved by turning off one of the D2D users.

Fig. 4 .

 4 Fig. 4. The effect of the CUE location on the optimal power (η = -70dB).

Fig. 5 .

 5 Fig. 5. The effect of the CUE location on the optimal D2D transmission mode (η = -70dB).

ACKNOWLEDGEMENT

This work was partially funded through French National Research Agency (ANR) project WONG5 with grant agreement code: ANR-15-CE25-0005603. A part of this work was also carried out thanks to the STSM grant of COST Action CA15104 (IRACON) and CNRS (GdR-ISIS).


Hussein Chour
email: hussein.chour@supelec.fr

Faouzi Bader
email: faouzi.bader@supelec.fr

Youssef Nasser

Oussama Bazzi
email: obazzi@ul.edu.lb

GALEN: A Geometric Framework for Global Optimal Power Allocation in a Full Duplex D2D Network

Keywords: transmission, device-to-device (D2D) communication, optimal power allocation, duplex mode selection

This paper studies the rate maximization problem of a Full duplex(FD) D2D underlaying cellular network. In the considered scenario, multiple FD-D2D pairs coexist with multiple cellular users which generate mutual interference between the two communication types. The complicated interference environment makes the optimization problem a non-concave problem.

To solve this problem, we discuss how to leverage the monotonic optimization theory to obtain the global optimal solution at the cost of high complexity. We also derive a geometric based optimization framework, denoted as GALEN, that achieves the global optimality with much lower complexity, and also it provides a closed form expression for the solution. The simulation results show the importance of the proposed solution. Index Termsfull-duplex

I. INTRODUCTION

Given the scarcity property of the cellular radio spectrum, coupled with the rapidly growing of the users' data traffic demands [START_REF] Mobile | Cisco visual networking index: Global mobile data traffic forecast update[END_REF], the need of new technologies which efficiently use the radio resources and fulfill the customers' demands has become an essential part in the upcoming cellular network (5G). In this context, Full Duplex (FD) and device-to-device (D2D) communication are proposed to enhance the radio spectrum efficiency and the users' experience in a cellular network. D2D allows two nearby devices to bypass the base station (BS) and communicate directly peer-to-peer as a D2D communication pair [START_REF] Doppler | to-device communication as an underlay to lte-advanced[END_REF]. On the other hand, FD allows a device to simultaneously transmit and receive in the same frequency band at the same time slot [START_REF] Sabharwal | In-band full-duplex wireless: Challenges and opportunities[END_REF].

Due to the short distance feature of the D2D communication, integrating the FD technology with D2D communication can further enhance the radio spectrum efficiency and the users' throughput [START_REF] Tapio | System scenarios and technical requirements for full-duplex concept[END_REF]. However, the practical FD devices add new challenges on both the D2D communication and the traditional cellular communication. For instance, the existing FD transceivers can not perfectly remove the self interference (SI) imposed on the receiver by the node's own transmitter [START_REF] Sabharwal | In-band full-duplex wireless: Challenges and opportunities[END_REF]. Thus, the residual self interference (RSI) which is tightly related to the transmitter power value highly affects the performance of FD transmission. In addition, using the FD technique creates additional interference in the network which may overwhelm the conventional cellular link. Thus, a proper power allocation is needed to exploit the benefits of FD and guarantee the quality of service (QoS) of the users.

The authors of [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF] tackled the power allocation problem of an isolated full duplex D2D pair operating on the cellular resources. Motivated by the work of [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF], the authors of [START_REF] Zuo | Power allocation optimization for full-duplex d2d communications underlaying cellular networks[END_REF] aimed to maximize the throughput of a FD-D2D link by properly allocating the power of the D2D pair and the CU while satisfying the minimum rate requirement of the CU. The ergodic rate of a FD-D2D pair coexisting with a CU was derived and analysed in [START_REF] Hemachandra | Sum-rate analysis for full-duplex underlay device-to-device networks[END_REF] considering a fixed transmission power strategy. The authors of [START_REF] Chour | Full-duplex or half-duplex d2d mode? closed form expression of the optimal power allocation[END_REF] proposed to maximize the ergodic sum-rate of a FD-D2D pair underlaying cellular network while satisfying the data-rate requirement of the CUs. In [START_REF] Li | Resource allocation for weighted sum-rate maximization in multi-user full-duplex device-to-device communications: Approaches for perfect and statistical csis[END_REF], a power allocation solution was proposed by using the DC programming theory [START_REF] Tao | The dc (difference of convex functions) programming and dca revisited with dc models of real world nonconvex optimization problems[END_REF]. However, none of the above works provided a global optimal power allocation scheme. Indeed, due to the complicated interference environment generated by the FD transmission, the power allocation is a nonconcave problem. Thus finding the global optimal solution is a challenging task. Instead, a first-order optimal solution was given in literature (e.g. [START_REF] Li | Resource allocation for weighted sum-rate maximization in multi-user full-duplex device-to-device communications: Approaches for perfect and statistical csis[END_REF]), or a sub-optimal solution was provided as in [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF], [START_REF] Hemachandra | Sum-rate analysis for full-duplex underlay device-to-device networks[END_REF], [START_REF] Chour | Full-duplex or half-duplex d2d mode? closed form expression of the optimal power allocation[END_REF].

In this paper, we aim at providing the global optimal solution of the rate maximization problem of a FD-D2D network.

To that end, we develop two novel optimization frameworks for a FD-D2D underlaying cellular network. The former provides the global optimal solution by means of monotonic optimization theory (MO) [START_REF] Zhang | Monotonic optimization in communication and networking systems[END_REF], [START_REF] Tuy | Monotonic optimization: Problems and solution approaches[END_REF] at the expense of high complexity, whereas the latter achieved the global optimality by exploiting the geometric structure of the feasible set with much reduced complexity. For the sake of readability, we named the second approach GALEN (Geometrical framework Approach for gLobal optimal powEr allocatioN). The major contribution of this work are as follows:

• Shedding the light on the optimal rate gain of a FD-D2D network • Providing two approaches that lead to the global optimal solution. To the best of the authors knowledge, none of the existing works provides the global optimal solution of a FD-D2D network. Global optimality is achieved by exploiting the MO theory and developing a new geometric based optimization framework.

• Providing a closed form expression for the optimal power allocation • Highlighting the impact of the SI cancellation factor and the D2D proximity distance on the throughput gain. The rest of this paper is organized as follows. Section II describes the system model and formulates the optimization problem. Section III derives the optimal solution by using the MO theory while section IV derives the optimal solution by using the geometric structure of the feasible set. The simulation results are given in section V, and finally, the conclusion is drawn in section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a D2D communication underlaying cellular network where M FD-D2D pairs coexist with M half duplex cellular users. Without loss of generality, we will consider in this work that each D2D pair will share the same resources as an uplink CU resource. We will denote that resource as j. In other words, Each D2D pair D2D j consists of two nearby devices, denoted as D j 1 and D j 2 , that have been assigned the uplink (UL) resources of the jth cellular users (CU j ). UL sharing is particularly considered since the UL resources are underutilized compared to that of downlink. Furthermore, sharing the UL spectrum only affects the BS and incurred interference can be handled by BS coordination.

An instance of this network is drawn in Fig. 1 in which two D2D pairs coexist with two cellular users (M = 2). The channel gain between the jth cellular user and the BS is denoted as g j c,bs while the channel gain between D j 1 and D j 2 and between D j 2 and D j 1 are denoted as g d . h j d1,bs and h j d2,bs respectively denotes the interference channel gains from D j 1 and D j 2 to BS. h j c,d1 and h j c,d2 denotes the interference channel gains from the jth cellular user to D j 1 and D j 2 respectively. All direct/interference channels are assumed to be zero-mean complex Gaussian random variables (i.e., channels are expressing Rayleigh fading) with variance l ij , where i ∈ {c; d1; d2}, j ∈ {bs; d1; d2}, i = j, l ij denotes the distance between the nodes i and j, and α denotes the path loss exponent. h j 1,1 and h j 2,2 stands for the residual self interference at D j 1 and D j 2 . They can be modeled as complex Gaussian random variables with zero-mean and variance ηP j d1 and ηP j d2 respectively [START_REF] Cheng | Optimal power allocation for fullduplex d2d communications over wireless cellular networks[END_REF], [START_REF] Zuo | Power allocation optimization for full-duplex d2d communications underlaying cellular networks[END_REF], [START_REF] Riihonen | Mitigation of loopback selfinterference in full-duplex mimo relays[END_REF], where P j d1 and P j d2 are respectively the transmission power of D j 1 and D j 2 , η denotes the SI cancellation capability of each D2D device.

A. Full duplex D2D link's rate

In the FD D2D communication mode, both D j 1 and D j 2 reuse the spectrum of CU j to communicate with each other. By denoting N 0 the power of the additive white Gaussian noise, and P j c the transmission power of the jth cellular user

BS D 1 1 D 1 2 g d g d h 1 11 h 1 22 D2D1 D 2 1 D 2 2 g d g d h 2 11 h 2 22 D2D2 h 1 d2 ,bs h 1 d1, bs h 2 d1,b s h 2 d2 ,b s CU 1 g 1 c,bs h 1 c ,d 2 h 1 c ,d 1 CU 2 g 2 c ,b s h 2 c ,d 1 h 2 c, d 2

Direct link

Interference link Fig. 1. Full duplex D2D pairs sharing the UL resources of the cellular users.

then the received signal to interference plus noise ratio (SINR) at the BS, at D j 1 and at D j 2 when D2D j shares the spectrum of CU j can be respectively expressed as

Γ j bs = P j c g j c,bs P j d1 h j d1,bs + P j d2 h j d2,bs + N 0 (1)
Γ j d1 = P j d2 g j d P j c h j c,d1 + ηP j d1 + N 0 (2)
Γ j d2 = P j d1 g j d P j c h j c,d2 + ηP j d2 + N 0 (3)
Applying Shannon theorem, the rate of D2D j (R j f d ) can be expressed as the summation of D j 1 's rate (R j d1 ) and

D j 2 's rate (R j d2 ). R j f d (p j )=B j log 2 (1 + Γ j d1 ) R j d1 + B j log 2 (1 + Γ j d2 R j d2 ) (4)
with p j = (P j d1 , P j d2 , P j c ) being the transmission power of the couple (D2D j , CU j ). B j denotes the bandwidth of CU j which is reused by D2D j .

B. Problem formulation

The main rationale behind introducing D2D communications is to mitigate the heavy data traffic in the cellular network. In this paper, we aim at maximizing the total D2D capacity while satisfying the QoS of the cellular users 1 . Let r j min be the minimum data rate of CU j , and denote by P = [p j ] M ×3 the matrix of the users' transmission power, the D2D sum-rate maximization problem can be formulated as

P1 : max P M j=1 R j f d s.t. P ∈ Φ (5) Φ = {Γ j bs ≥ γ j min = 2 r j min -1, j ∈ {1, ..., M } (6) 0 ≤P j i ≤ P i max , i ∈ {d1, d2, c}, j ∈ {1, .
.., M }}, where P i max is the maximum transmission power of a transmitter i, and γ j min is the minimum required SINR level to achieve r j min . Solving P1 is the target of the next section. 1 The QoS of the CU sharing the resource might be highly constrained

C. Reformulation of problem P1

First we observe that the couples (D2D j , CU j ), j ∈ {1, . . . , M }, are independent from each others, and thus P1 can be rewritten as:

P2 : M j=1 max p j R j f d s.t. p j ∈ Φ. (7)
Next, by using the Logarithm's properties, we rewrite R j f d as:

R j f d (p j )=B j log 2 Q j (p j ) = B j log 2 (1+Γ j d1 )(1+Γ j d2 ) . ( 8
)
The Logarithm is a monotonic increasing function, and thus maximizing R j f d (p j ) is equivalent to maximize Q j (p j ). Therefore, P2 can be equivalently written as:

P3 : M j=1 R j f d (p j ) (9)
such that p j = argmax

p j ∈Φ Q j (p j ) (10)
with p j = (P j, d1 , P j, d1 , P j, c ) being the optimal power allocation vector for the couple (D2D j , CU j ). Now, observe that Q j (p j ) is a non-concave function (since it is the multiplication of two linear fractional functions). Hence, P3 is a non-concave optimization problem. Thus, solving P3 is a challenging task. In the following we provide two approaches to solve P3. The first approach finds the global optimal solution by using the monotonic optimization theory at the expense of high complexity. The second approach utilizes the geometrical structure of the feasible set to find the global optimal solution with much lower complexity, and it is denoted as GALEN. Moreover, GALEN provides a closed form expression for the optimal solution which make it very useful for resource allocation.

III. OPTIMAL POWER ALLOCATION BY MONOTONIC

OPTIMIZATION

In this section, we focus on finding the optimal solution of P3, the equivalent problem of P1, with the assumption that the BS has perfect knowledge about the channel state information for both the cellular and the D2D users.

In general, globally solving a non-convex optimization problem can involve examining every point in the feasible set. Contrarily, monotonic optimization theory exploits the monotonicity property of the utility function and the constraints in the optimization problem in a much more efficient way but still with exponential complexity in the number of optimization variables. The key idea behind monotonic optimization theory is to search for the global solution on the outer boundary of the feasible set instead of exploring the complete feasible set of the problem. Thus, the usage of monotonic optimization theory is limited to the problems which have certain properties. In particular, the monotonic optimization tool can solve the general problem max x∈G∩H f (x) providing that the following three properties hold true [14, Chapter I].

• f (x) is an increasing function; • G is a normal set; • H is a co-normal set. A function f : R n → R is called an increasing function if for any two vectors x ∈ R N and y ∈ R N we have f (x) f (y) when 0 x y 2 . A set G ⊂ R n is normal (co- normal)
set, if for any point x ∈ G, all other points x such that x ≤ x (x ≥ x) are also in the set G. Based on the above requirements MO appears to be non-applicable to P 3 (Since Q j is non-increasing with p j ). In the following we show how to transform P 3 into an MO problem.

It is first straightforward to claim that Γ j d1 (p j ), Γ j d2 (p j ), and Γ j bs (p j ) are always strictly positive. Now let Γ j = (Γ j d1 , Γ j d2 , Γ j bs ) be the achieved SINR vector for all links, it is clear that the function Q(Γ j ) is an increasing function with Γ j . Thus, the optimization problem defined in (10) has a hidden monotonocity structure which can be extracted by using the reformulation of [15, Section III]. The latter is given below:

P4 : z j = argmax zj Q j (z j ) = (1 + z j d1 )(1 + z j d2 ) s.t. z j ∈ S = S n ∩ S c S n = {z j |z j Γ j , p j p max } (11)
S c = {z j bs ≥ γ j min , z j d1 ≥ 0, z j d2 ≥ 0} (12)
wherein z j = (z j d1 , z j d2 , z j bs ) and z j = (z j, d1 , z j, d2 , z j, bs ). It is easily verified that the new objective function Q j (z j ) is an increasing function with z j , and that S n and S c are respectively normal and co-normal sets. Hence P4 is MO problem and it can be solved in exponential time using the outer polyblock approximation algorithm (OPA) described in [START_REF] Qian | Mapel: Achieving global optimality for a non-convex wireless power control problem[END_REF] and reported in [14, Algorithm 3]. The OPA algorithm solves the rate maximization problem by constructing a sequence of shrinking polyblocks that eventually reaches the global optimal point.

Since the objective function of P4 is increasing with z j , the optimal solution Z j must occur at a place where z j = Γ j . Accordingly, the optimal solution of (10) denoted by p j can be recovered by solving the following linear equations system:

P j d1 ηz j, d1 -P j d2 g j d + P j c h j c,d1 z j, d1 + z j, d1 N 0 = 0, (13)
       -P j d1 g j d + P j d2 ηz j, d2 + P j c h j c,d2 z j, d2 + z j, d2 N 0 = 0, P j d1 h j d1,bs z j, d3 + P j d2 h j d2,bs z j, d3 -P j c g j c,bs + z j, d3 N 0 = 0. As a result, solving P4 for all j ∈ {1, 2, ..., M } yields to the optimal solution of P3. Due to the exponential complexity of OPA, in the next section, we develop an analytical framework to compute the global optimal point with low complexity.

IV. GALEN'S FRAMEWORK Despite the interest of MO optimization theory, the latter converges to the optimal solution at the expense of exponential complexity. In this section, we provide a mathematical framework named GALEN which finds the optimal solution of P2 in an efficient way, by analyzing the geometric structure of the feasible set Φ.

A. Geometric representation of Φ

The feasible set Φ of each couple (D2D j , CU j ) can be rewritten as follows: Φ = {P j c g j c,bs -γ j min (P j d1 h j d1,bs + P j d2 h j d2,bs

+ N 0 ) ≥ 0 (14) 0 ≤P j i ≤ P i max , i ∈ {d1, d2, c}} (15)
Equations ( 14) and ( 15) respectively represent the QoS constraint of CU j and the power constraints of the couple (D2D j , CU j ). Denote by P c the equality case of ( 14), i.e., the QoS plane. Moreover, let P max i (P 0 i ) be the maximum (minimum) power planes, i.e., P j i = P i max (P j i = 0) where i ∈ {c, d1, d2}. Hence, in the three dimensional space (P j d1 , P j d2 , P j c ), Φ can be represented as the intersection of P c with P 0 i and P max i

, where i = {c, d1, d2}. It can be observed that P max i (P 0 i ) with i ∈ {d1, d2, c}, are fixed in the 3D-space and they form a cuboid shape as shown in Fig. 2a-2e. However, the plane P c varies according to the channel situation of (D2D j , CU j ) and the required rate of CU j . Thus, Φ may have different possible shapes as shown in Fig. 2a-2e. Fig. 2a-2c shows the cases where P c has intersection with two maximum power planes and two minimum power planes. Fig. 2b shows the case where P c intersects all the maximum and the minimum power planes, while Fig. 2e shows the case where P c intersects the plane P max c , P 0 d1 and P 0 d2 . The thick lines in the above figures represent the intersection lines of P c with the maximum power planes. The corner of these lines are denoted as v i with i = {1, 2, ..., 7}. Moreover, we define the follows v1 Pc ∩ P

B. Optimal power allocation

Denote by

Ω = {v 1 v 3 ; v 2 v 3 ; v 4 v 5 ; v 6 v 7 ; v 4 v 6 ; v 5 v 7
} the set of the intersection lines between P c and the maximum power planes. The following lemma shows that the optima can be searched only in Ω instead of Φ. Here it is worthy mentioning that Ω is an exclusive set, i.e., not all its members occurs at the same time as shown in Fig. 2a-2e.

Lemma 1. The optimal power vector p j has at least one power bounded by the maximum power constraint and it lies on the QoS plane P c .

Proof: For any scaling parameter µ > 1 and a power vector p j = (P j d1 , P j d2 , P j c ) ∈ Φ we have:

Q j (µp j )= (1+ P d2 g d P c g c,d1 + ηP d1 + σ 2 N /µ
)× (16)

(1+ P d1 g d P c g c,d2 + ηP d2 + σ 2 N /µ ) > Q j (p j )
The power constraints defined in [START_REF] Qian | Mapel: Achieving global optimality for a non-convex wireless power control problem[END_REF] imply that the maximum µP j i is P i max , i ∈ {d1, d2, c} (due to inequality in ( 16)). Hence, the optimal point p j can be achieved only if at least one user is transmitting with the maximum allowed power. On the other hand, since Q j monotonically decreases with P j c , the maximum R j f d occurs when CU j transmits with the minimum required power. Thus, p j must reside on the plane P c . As a result, the optimal points must reside in Ω Lemma 1 reduces the feasible set Φ to Ω, and thus the optimal point can be obtained by solving the power allocation problem defined in [START_REF] Tao | The dc (difference of convex functions) programming and dca revisited with dc models of real world nonconvex optimization problems[END_REF] over the different points of Ω.

Being on v 1 v 3 ( see Fig. 2a) or v 1 v 6 ( see Fig. 2b and Fig. 2c) implies that P j d1 = P d1 max and P j c = (P d1 max h j d1,bs + P j d2 h j d2,bs + N0) . In such case, the objective function of [START_REF] Tao | The dc (difference of convex functions) programming and dca revisited with dc models of real world nonconvex optimization problems[END_REF] reduces to

Q j (p j ) = 1 + P j d2 P j d2 a 0 +a 1 1 + P d1 max P j d2 a 2 +a 3 , (17)
where the constant coefficients in the above equation are defined as

a 0 = γ j min h j d2,bs h j c,d1
g j c,bs g j d ; a 2 = ηg j c,bs + γ j min h j d2,bs h j c,d2

g j c,bs g j d a 1 = P d1 max g j d η+ γ j min h j d1,bs h j c,d1 g j c,bs + N 0 g j d 1+ γ j min h j c,d1 g j c,bs a 3 = γ j min P d1 max h j d1,bs h j c,d2 g j c,bs g j d + N 0 g j d 1 + γ j min h j c,d2 g j c,bs
.

Taking the first derivative of (17) w.r.t P j d2 leads to the following:

∂(17) ∂P j d2 = A 1 P j d2 2 + 2B 1 P j d2 + C 1 D 1 (18)
where the constants A, B, C and D are given as follows:

A 1 = a 2 2 a 1 -P d1 max a 0 a 2 (a 0 +1) ; B 1 =-P d1 max a 0 -a 3 a 2 a 1 ; C 1 = P d1 max (a 1 a 3 -a 1 2 a 2 )+a 1 a 3 2 ; D 1 =(P j d2 a 0 +a 1 ) 2 (P j d2 a 2 +a 3 ) 2 .
Since D 1 is always positive, a possible optima of (17) would be the solution of A 1 P j d2 2 +2B 1 P j d2 +C 1 . Denote such optima point as e 1 , the coordinates of e 1 are given by:

e 1 =   P d1 max ; 1 A1 -B 2 ± B 1 2 -A 1 C 1 ;
γ j min P d1 max h j d1,bs +P j d2,e1 h j d2,bs +N 0 /g j c,bs

  , (19)
with P j d2,e1 being the P j d2 abscissa of e 1 . The above solution does not always provide a global maxima of (17), since their is no guarantee that the second derivative is always negative. Moreover, e 1 may not be always a feasible point( since it may lies outside v 1 v 3 or v 1 v 6 , or it might be an imaginary point). Accordingly, the global maxima of (17) would be either one of the corner points of the line v 1 v 3 (v 1 v 6 ), i.e., v 1 or v 3 (v 6 ), or the extreme point e 1 ( if e 1 is feasible).

In the same manner, the optimal point of (10) on the line v 2 v 3 as in Fig. 2a ( or v 2 v 4 as in Fig. 2b and Fig. 2d) is either one of the corner points v 2 or v 3 ( v 4 ) or an extreme point e 2 which lies on v 2 v 3 ( v 2 v 4 ) and has the following coordinate:

e2=   1 A 2 -B2 ± B2 2 -A2C2 ; P d2 max ; γ j min P j d1,e 2 h j d1,bs +P d2 max h j d2,bs +N0 /g j c,bs   , (20)
where the constants A 2 , B 2 , and C 2 are given by Similarly, the optimal point of (10) on the line v 6 v 4 as in Fig. 2d ( or v 5 v 7 as in Fig. 2e) is either one of the corner points v 4 or v 6 (v 5 or v 7 ) or an extreme point e 3 which lies on v 6 v 4 (v 5 v 7 ) and has the following coordinate where the constants A 3 , B 3 , and C 3 are given by .

A 2 = b 2 2 b 1 -P d2 max b 0 b 2 (b 0 +1) ; B 2 = b 3 -P d2 max b 0 b 2 b 1 ; C 2 = P d1 max (b 1 b 3 -b 1 2 b 2 )+b 1 b 3 2 ,
e3=    1 A 3 -B3 ± B3 2 -A3C3 ;
A 3 =ηc 3 (η -c 2 ) -c 4 (c 4 -1) (c 0 c 2 + c 1 η) ; B 3 =((c 2 c 4 -c 2 + η) c 0 + η c 1 c 4 ) c 3 ; C 3 =c 3 (c 0 (c 0 + c 1 ) -(c 0 c 2 + c 1 η) c 3 ) ;
Accordingly, the optimal point would be the point which gives higher rate. The following Theorem summarizes all the previous discussion.

Theorem 1. The optimal power vector p j can be searched only in the set P opt defined as

P opt = {v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , e 1 , e 2 , e 3 } (22)
Here we should emphasize that Theorem 1 reduces the feasible set from a three dimensional shape to a set of points with known coordinates. Hence, with GALEN the global optimal point can be obtained in one iteration on contrary to the MO algorithm which requires a high number of iterations. Moreover, GALEN provides a closed form expression of the optimal solution.

V. NUMERICAL RESULTS

In this section we aim to find the ultimate performance of a FD-D2D enabled cellular network. To that end, we consider a single cell network with radius R and M CUEs and D2D pairs. The CUs are equally sharing the uplink bandwidth and they are uniformly distributed in the cell. Moreover, each D2D j pair is uniformly distributed within a randomly located cluster with radius r, and is sharing the same resource of CU j . All the simulation parameters are summarized in Table I.

A. GALEN VS MO

In this subsection we aim to validate GALEN by comparing its results with the well known monotonic optimization framework. Fig. 3 shows the variation of the D2D rate w.r.t the SI cancellation factor and the D2D cluster radius. Moreover, it compares the proposed power allocation GALEN with the MO method for different cluster radius and different SI cancellation factors. In this figure we consider 500 realizations each with M = 15 CUEs and M = 15 D2D pairs. At Each iteration we averaged the results of all the couples (D2D j , CU j ) and finally we averaged the results over all realizations. As it can be seen, the curves perfectly match in all cases. Moreover, Fig. 3 clearly shows that as the cluster radius r increases, the average rate decreases. As expected, we can see that the average rate increases as the SI factor decreases.

Moreover, to highlight the importance of GALEN, we show in Fig. 4 the complexity of MO in terms of the required number of iterations to achieve the global optimality. As it can be seen, the MO theory requires 10 4 to achieve the optimal point. However, from Theorem 2, we know that GALEN obtains the optimal solution by searching a set of ten points. This low complexity feature makes GALEN a perfect choice for power allocation. Moreover, it makes the analysis of a FD-D2D network easier. Fig. 4 also shows that the number of iterations declines with the increasing of the cluster radius. This is because when r increases the average number of admitted D2D pairs decreases.

B. FD-D2D or HD-D2D

Here we aim to investigate the performance of the FD-D2D communication, and see if it always outperforms the HD-D2D mode. To that end, and for fair comparison with FD, we assume that in the HD mode the D2D users in each D2D pair are equally sharing the bandwith B j . Therefore, the received SINR at the BS will remain the same as in (1). However, the received SINR at D j 1 and D j 2 will no longer contain the SI component (i.e. η component). Accordingly, denote by Γ j hd,d1 and Γ j hd,d2 the received SINR at the D2D devices in the HD mode, we have Γ j hd,d1 = Γ j d1 | η=0 and Γ j hd,d2 = Γ j d2 | η=0 . Consequently, the HD D2D rate (R j hd ) becomes R j hd = 1 2 R j hd | η=0 . Based on that, by setting η = 0, both GALEN and MO can be used to derive the global optimal solution of the rate maximization problem when the D2D users are operating in HD mode. In Fig. 5 we compare the FD-D2D performance with the HD-D2D in terms of the average D2D rate. In particular, Fig. 5 shows the variation of the D2D rate w.r.t the SI cancellation factor and for different D2D cluster radius. As it can be seen, at low self interference cancellation in the FD mode, the HD mode outperforms FD. In addition when the separation between the D2D users increases the FD mode needs higher SI cancellation to outperform the HD mode. As a result, the FD gain mainly appears when the D2D users are close to each other and they have high SI cancellation capability. Hence, to achieve the maximum gain from the D2D technology, a mode selection step must always be taken into consideration when allocating the resources for the users. The latter is out of scope of the paper and will be elaborated in the future.

VI. CONCLUSION

This paper proposed two approaches to globally solve the power allocation problem of a FD-D2D network. The first one is based on the MO theory to the detriment of high complexity, while the latter, denoted GALEN, is a geometric-based approach with very low complexity. The simulation results have validated the simplified GALEN approach and have shown its low complexity feature. Moreover, simulation results have shown that the HD-D2D might outperform the FD-D2D in some situations. As a future work, the resource allocation which contains mode selection and channel assignment of a FD-D2D network will be considered. In addition, it might be interesting to analyze the maximization problem of the global cell rate, i.e., the rate of both D2D and the cellular users.

Fig. 2 .

 2 Fig. 2. The possible shapes of the feasible set Φ. (Pc )

and the constant coefficients b 0 , b 1 , b 2 , and b 3

 123

  and the constant coefficients c 0 , c 1 , c 2 , and c 3 are given by max g c,d1 +N 0 ) ; c 1 =

Fig. 3 .Fig. 4 .

 34 Fig. 3. Comparison of FD-D2D rate obtained from GALEN and from the MO theory.

Fig. 5 .

 5 Fig. 5. Comparison of the FD-D2D rate and the HD-D2D rate.

In this paper we use x y to denote that the vector y ∈ R N is greater than or equal to x ∈ R N in a component-wise manner.

ACKNOWLEDGEMENT

This work was partially funded through French National Research Agency (ANR) project WONG5 with grant agreement code: ANR-15-CE25-0005603. A part of this work was also carried out thanks to the STSM grant of COST Action CA15104 (IRACON) and CNRS (GdR-ISIS).