Joint Spectrum Partition and Performance Analysis of Full-Duplex D2D Communications in Multi-Tier Wireless Networks

Yueping Wang,Xuan Zhangand Yixuan Zhang

Abstract:Full-duplex(FD)has been recognized as a promising technology for future 5G networks to improve the spectrum efficiency.However,the biggest practical impediments of realizing full-duplex communications are the presence of self-interference,especially in complex cellular networks.With the current development of self-interference cancellation techniques,full-duplex has been considered to be more suitable for device-to-device(D2D)and small cell communications which have small transmission range and low transmit power.In this paper,we consider the full-duplex D2D communications in multi-tier wireless networks and present an analytical model which jointly considers mode selection,resource allocation,and power control.Specifically,we consider a distance based mode selection scheme.The performance analysis of different D2D communications modes are performed based on stochastic geometry,and tractable analytical solutions are obtained.Then we investigate the optimal resource partitions between dedicated D2D mode and cellular mode.Numerical results validate the theoretical anlaysis and indicate that with appropriate proportions of users operated in different transmission modes and optimal partitioning of spectrum,the performance gain of FD-D2D communication can be achieved.

Keywords:Full-duplex,device-to-device communications,HetNets,power control,spectrum partition.

1 Introduction

The next generation cellular network(i.e.,5G)aims to provide significant improvement on system capacity,data rate,and spectrum/energy efficiency.A critical solution for satisfying the challenging requirements in 5G is to bring the transmitter and receiver closer by dense deployment of small cells[Lee and Quek(2015)]and enabling device-to-device(D2D)communications[Lin,Andrews,Ghosh et al.(2014);Andrews,Buzzi,Choi et al.(2014)].Specifically,small cell provides high data rate in a smaller coverage with reduced transmit power[Wang,Ju,Gao et al.(2018)].Small cell and traditional macro cell constitute the multi-tier heterogeneous wireless networks(HetNets).Also,D2D communication as an underlay coexistence with cellular networks add another tier to the HetNets.Both small cell and D2D communications can help improving the spectrum utilization efficiency,energy efficiency,and offloading traffic from base station.

Furthermore,full-duplex(FD)has been introduced to perform bidirectional communication on the same temporal and spectral resources for enhancing the spectrum efficiency[Sabharwal,Schniter,Guo et al.(2014);Choi,Jain,Srinivasan et al.(2010)].Theoretically,enabling wireless devices transmit and receive simultaneously can double the spectral efficiency,which raised great interest for next-generation wireless networks.However,the presence of self-interference is still the key challenge in implementing FD communication,and becomes the main restriction for wide application[Sultan,Song and Han(2014)].Considering the present status of self-interference cancellation(SIC)techniques,FD is potentially suitable in heterogeneous networks due to the applications of small cell as well as the low transmit power[Lee and Quek(2015)].

Note that D2D communications can be operated in different modes,i.e.,cellular mode,reuse mode,and dedicated mode.In cellular mode,the device performs exactly the same as a cellular device which still requires relaying via the base station.In reuse mode,D2D communications in performed in an underlay manner which reuse the spectrum allocated for cellular communications.In dedicated mode,dedicated spectrums are allocated for D2D communications to avoid interference.Naturally,when devices are operated in D2D mode,the question of spectrum resource sharing arises.The spectrum sharing is similar to cognitive radio networks.In general,the D2D spectrum sharing can be clarified into two types:spectrum overlay and spectrum underlay[Lin,Andrews and Ghosh(2014);Zhu and Hossain(2015)].With the overlay spectrum sharing,cellular and D2D transmitters use orthogonal spectrum,the underlay spectrum sharing refers to the scenario that D2D devices use the cellular spectrum occupied by cellular users in a reuse mode.Although the underlay spectrum sharing improve the spectral efficiency,it also complicates the network interference management.Comparing with the underlay approach,dedicated D2D mode has better SINRs since resources allocated to D2D and cellular links are orthogonal.Most existing research on D2D mainly focus half-duplex communication[Lin,Andrews and Ghosh(2014);Zhu and Hossain(2015);ElSawy,Hossain and Alouini(2014);Lin and Andrews(2013)],and only few work consider the full-duplex D2D communications[Ali,ElSawy and Alouini(2016);Mach,Becvar and Vanek(2015);Ali,Rajatheva and Latvaaho(2014)].However,only single-tier cellular network and one D2D mode are considered.More recently,small cell use cases have been identified in Wang et al.[Wang,Tian,Svensson et al.(2015)],and Ye et al.[Ye,Al-Shalash,Caramanis et al.(2014);Wu,Cai,Hu et al.(2015);Cho,Koufos,Jäntti et al.(2015)]designs the spectrum sharing with cooperative communication.In our previous work[Xia,Zhu,Chen et al.(2017)],we provide performance analysis on the D2D communications in multi-tier networks.However,the different modes and spectrum sharing are not considered.

In this work,we consider a system of full-duplex D2D communications in multi-tier wireless networks and present an analytical model which jointly considers mode selection,power control,and spectrum sharing.Specifically,we propose a distance based mode selection scheme.The performance analysis of different D2D communication modes are performed based on stochastic geometry which has been widely used to analyze the performance of heterogeneous cellular network[François and Bartłomiej(2009)].The transmit power,coverage probability,and rate are analyzed and tractable analytical solutions are obtained.Then,we investigate the optimal spectrum partitions between dedicated D2D mode and cellular mode.Numerical results show the validity of the obtained analytical solutions.Also,the results show that with appropriate proportions of users operated in different transmission modes and optimal partitioning of spectrum,the performance gain of FD-D2D communication can be achieved.

The rest of this paper is organized as follows:In Section 2,we present the system model and the channel model.In Section 3,the transmit power is analyzed.In Section 4,the detailed coverage and rate analysis are provided.The simulation results and analysis are presented in Section 5.Section 6 concludes the paper.A list of major symbols and notations used in this paper is provided in Tab.1.

Table 1:List of symbols and notations

2 System model

2.1 Network model

We consider a large D2D-enabled two-tier uplink cellular network,and the MBSs are regularly placed according to a hexagonal grid with density λb.The SBSs are randomly distributed in the network which is modeled by an independently marked Poisson point process(mPPP)with intensity λs,and the transmitting UEs also distributed in the given geographical area randomly which is modeled as an mPPP.To simplify analysis,we focus on a single cell A and approximated as a circle with radius.As depicted in Fig.1,the cellular users connect to the base station(i.e.,MBS,SBS),the D2D pairs communicate with each other directly.If the D2D users transmit on the reuse mode,it will reuse the channel with existing cellular users,and causing co-channel interference between D2D and cellular users.If the D2D users transmit on the dedicated mode,the co-channel interference would not occur.Similar to the definition in Zhu et al.[Zhu and Hossain(2015)]the transmitting UEs can be represented as,where

·{yi}represents the location of the receiver of transmitting UE Xi.

·{Di}denotes the length of radio links where Di=‖Xi-yi‖,for notational simplicity Diis an independently and identically distributed(i.i.d.)random variable.

From the PPP assumption,we denote the cellular link distance distribution by

where rcis a random variable which denotes the distance between a cellular UE and it transmitting base station.Moreover,the D2D pair transmit distance is Rayleigh distributed with pdf frd=2πλrde-πλrd2,and the SBS-cellular link length rsis given by frs=2πλrse-πλrs2.Also,we consider a general power-law path-loss model for both cellular and D2D link,which the signal power decays at the rate r-ηwith the distancer,η is the pathloss exponent where η ≥ 2.In this paper,we assume that the cellular and D2D links experience same propagation conditions.Accordingly,a target receiver can receive power P ·ℎ ·r-ηat a typical position with a distancerfrom its transmitter,Pandhare the transmit power and power gain,respectively.In a Rayleigh fading network,we denoteℎ～exp(μ)is an exponentially distributed random variable,and all channel gains are assumed to be i.i.d.

2.2 Mode selection

Here we consider a flexible distance based mode selection scheme.The cellular mode is used ifwhere a UE is a D2D UE.If the distance to its receiver is less than threshold,otherwise,D2D mode is selected.Note that,denote the probability of a UE to be a D2D UE,andto be cellular UE.When it comes to the selections of cellular users,the proportions denoted by a flexible selection criterion which based on the bias factor β to impose a tunable selection for the SBSs.The probability of a cellular UE select the SBSs is:

For the sake of simple presentation,we.In this case,the intensities of cellular UEs in SBS mode can be denoted byand the MBS cellular intensities is.Moreover,we denote by xr,xdthe proportions of D2D users transmit in reuse and dedicated mode,so·represent the intensities of reuse D2D mode and dedicated D2D mode.

2.3 Spectrum partitioning

Here,we consider the spectrum partitioning which specifically refers to the cellular link resource.In this paper,we divide the channel into F frequency subchannels,and the spectrum is divided into two orthogonal portions.We aim to determine the optimal partitioning(θ,1-θ),where a fraction θ are allocate to dedicated D2D communication while the rest are assigned for cellular and reuse D2D users.Moreover,we defined the θ as the D2D spectrum partition factor.

3 Transmit power analysis

Due to PPP assumptions we modeled,the transmit power of the MBS-cellular,SBS-cellular,reuse D2D,and dedicated D2D modes are all random variables.Moreover,we employ channel inversion based power control scheme,and the purpose is that the transmitting devices can compensate for the large scale path-loss.Note that for ease of analysis,we employ the minimum power of ρ received signal at the terminals.In this section we analyze the transmit power distribution for ease of the follow references.

Proposition 1.The average transmit power of a typical MBS-UE,a SBS-UE and a D2D-mode UE are respectively given by:

where,κ∈{f,r}represent the different D2D link.

Proof.According to the approximate approach in the above section,we obtain the typical distribution of the active cellular network and we obtain the PDF of different transmit link.And the average MBS-cellular transmit power is:

Accordingly,the average SBS-cellular transmit power is:

And the average D2D transmit power is:

4 Coverage probability and rate analysis

In this section,we analyze the average rate of transmit users in different modes.

Accordingly,we denote the point processthe set of interfering MBS-cellular UEs and the set of interfering SBS-cellular UEs in the same subchannel,andare both PPP.

4.1 Average rate for MBS-cellular mode

We assume an orthogonal resource access scheme and only one active uplink transmitter exist.For a MBS-cellular UE,the interference comes from the SBS-cellular UEs transmit in the same subchannel and D2D UEs in reuse mode accessing the same channel.In a uplink MBS cellular mode transmission,the base station is the receiver and the aggregate interference experienced by the MBS in the MBS-cellular mode on channel α is:

where g0represents the fading gain from interferes to the tagged receiver,ℐcχis the interference of UEs operated in mode χ∈{c,f,r,s},and similarly,Pχis the interference power.The SINR of the receiver is denoted as:

where ℎ0denotes the fading gain to the tagged receiver from its transmitter,δ2denotes the noise power,ρcdenotes the received threshold of the tagged receiver.Then,the average spectrum efficiency of a UE in the MBS-cellular mode is:

Then,we can express the following proposition.

Proposition 2.The average spectrum efficiency of a MBS-cellular UE is given by:

where,

In the MBS-cellular mode,the transmit resources are allocated by the base station in a round-robin manner,and the probability of a MBS-cellular UE transmit on channel α is,andepresent the expect MBS-cellular UEs numbers[Zhu and Hossain(2015)].Then,the expected channels are:

Then,the average spectrum efficiency of a MBS-cellular UE can be denoted as Rc=Fcτc.

4.2 Analysis of SBS-cellular mode

For an SBS-cellular UE accessing channel α,the interference comes from the MBS-cellular mode and reused D2D mode users.In this case,the aggregate interference in a typical SBS UEs receiver,specifically,the SBS experienced interference in channel α is given by:

Accordingly,the SINR at the tagged SBS receiver is:

Then,we derive the average spectrum efficiency of a SBS-cellular UE as

and we can obtain the following proposition.

Proposition 3.The average spectrum efficiency of a SBS-cellular UE is given by

where

The SBS-cellular user is modeled to reuse the spectrum resource and the expect spectrum can be denoted by kp,where k denotes the amount of subchannels of a SBS-UE chosen among the(1-η)F subchannels,and for each subchannel the SBS users decided to transmit in this channel with probability p.Then,we can obtain the average rate is

4.3 Analysis of reused full-duplex D2D mode

When analyze the FD communication,the self-interference need to be raised.And in this paper we only consider the residual self-interference which the imperfect interference cancellation is modeled,hence we approximate ξ denote the SI factor.We characterize the reused FD-D2D in both f-D2D links and r-D2D links,for notational convenience we set κ∈(f,r)where f,r denote the f-D2D and the r-D2D links.Moreover,the FD-D2D pairs reuse the spectrum resource allocated to the MBS UEs,and the aggregate interference is

The SINR at the tagged D2D receiver is given by

Accordingly,the average spectrum efficiency of a FD-D2D pair is

Proposition 4.The average spectrum efficiency of a FD-D2D pair is given by

The FD-D2D users in reuse mode reuse spectrum resources with kp similar to SBS users,and we can obtain the average rate is

4.4 Analysis of dedicated mode

In this section,we analyze the average rate of FD-D2D links in the dedicated network.We treat the existing interference as noise,and we use Shannon's capacity formula to approximate the rate,i.e.,,where Fdis available allocated bandwidth,can simply denote the average spectrum efficiency.In this case,the spectrum allocation is implemented by the MBS in a round-robin manner,and Fdcan be denoted as:

5 Numerical results and spectrum partition

In this section,we firstly validate out model by simulations and present some numerical results.Then,we analyze the optimizing spectrum partition and obtain the optimal spectrum partition factor θ⋆.Note that,θ⋆F subschannels are allocated for dedicated D2D user while the remains are for reused UEs.

5.1 Analytical results validation

In this part,we validate the analytical results based on the parameters we set,and all the results are functions of SINR which we derived in Section IV.Moreover,we compare the derived distribution to its corresponding Monte Carlo simulations of 2000 runs under the same parameters.Fig.2 and Fig.3 show the SINR coverage,the former is conducted under perfect SIC while the latter with residual SIC factor ξ.From Fig.2,we can see that the simulation results match the analytical results well,which validated that the proposed framework capture the FD-D2D based HetNet coverage features well.Besides,the Fig.3 shows the proportions of dedicated D2D users influence the network performance and with the increases of dedicated D2D users the coverage probability increased.

Table 2:Parameter values

Figure 2:Coverage probability without SI in a two-tier HetNets

Figure 3:Coverage probability with imperfect SIC,SIC factor ξ=10-8

Moreover,we investigate the D2D mode selection thresholdwhich impacts the average rate of the network.As shown in Fig.4,the average network throughput firstly increases withdue to the D2D offloading gain,because with increasingmore D2D UEs can be scheduled.But then,increasedmotivates more cellular UEs to transmit in D2D mode,which decrease the transmit performance due to the increased D2D co-channel interference,so Fig.4 demonstrated that with appropriate choice of,D2D-based cellular network can achieve much higher rate than traditional cellular network.

Figure 4:Network throughput of FD-D2D,HD-D2D,and Network without D2D

5.2 Optimizing spectrum partition

In this part we analyze the optimal spectrum partition factor θ⋆

where Trand Tdrepresent the reused and the dedicated rate expression,and from the above section we denote the rate expression as following:

Td=Rdd;

Tr=Rc+Rs+Rd.

And in this paper,we use the weighted proportional fair function:

u(Tr,Td)=ωrTr+ωdTd,

where weight factors ωr,ωd＞ 0 and ωr+ωd=1.From Lin et al.[Lin,Andrews and Ghosh(2014)],we obtain the following optimal weighted proportional spectrum partition factor.Lemma 1.The optimal weighted proportional spectrum partition θ⋆is given by

From Fig.5,we plot the utility value versus θ under different values of xr,which is the proportion of reuse D2D UEs,and we obtained that the optimal spectrum partition θ⋆≈ 0.4 and it can be seen that the optimal θ⋆are equal under different xr.However,in this paper we only consider a fixed mode selection threshold,andwe will analyze it future.

Figure 5:The utility value versus D2D spectrum partition factor θ under different values of χr

6 Conclusion

In this work,we have considered the full-duplex D2D communications in multi-tier cellular networks and have presented a tractable framework for analyzing the performance different communication modes.We have obtained tractable expressions for important performance metrics such as coverage probability and average rate.Based on the theoretical analysis,we derive the optimal spectrum partitioning for dedicated communication mode.Then numerical simulations have been performed which demonstrate the effectiveness of the proposed analytical framework.Also,the results show that with appropriate partition of the spectrum the network provide better performance.For the future work,the game theory based spectrum partition can be studied.

Acknowledgement:This work was supported by National Natural Science Foundation of China.The grant number is 61672283.