Beamforming and Interference Cancellation in D 2 D Random Network

Device-to-Device (D2D) communication is an important proximity communication technology. We model the hybrid network of cellular and D2D communication with stochastic geometry theory. In the network, cellular base stations are deployed with multiantennas. Two transmission strategies including beamforming and interference cancellation are proposed to boost system achievable rate in this paper. We derive analytical success probability and rate expression in these strategies. In interference cancellation strategy, we propose the partical BS transmission degrees of freedom (dofs) that can be used to cancel its D2D users (DUEs) interferences around the BS or to boost the desired signal power of associated cellular (CUE). In order tomaximize the total area spectral efficiency (ASE), the BS transmission degrees of freedom are allocated according to proper interference cancellation radius around the BS. Monte Carlo simulations are performed to verify our analytical results, and two transmission strategies are compared.


Introduction
Device-to-Device (D2D) communication is an important proximity communication technology, which has been in standard process of LTE-advanced system and it is a key technology for the future hybrid networks.With the development of mobile internet, the cellular network is not able to meet the requirements for the future localizing applications and D2D technology comes to an important complement for it [1][2][3].The performance of wireless communication can be analyzed accurately by stochastic geometry theory.Traditional model has Wyner model or hexagonal grid [4].The Wyner model or the hexagonal grid can be evaluated by system-level simulations.However, both the scalability and the accuracy of grid model were questionable in the context of network heterogeneity [4][5][6].An alternative is to model the locations of sites as random and drawn from a spatial stochastic process, such as the Poisson point process (PPP), which has been confirmed as accurate as the grid model [5].This stochastic model has been used recently in [7] to analyze success probability and average rate of heterogeneous network.
Reference [1] has studied spectrum sharing and derived analytical rate expressions for D2D communication in cellular networks by stochastic geometry theory and compared with signal to noise plus interference ratios (SINR) distribution using the hexagonal model by Monte Carlo.In [8], the spatial distribution of transmit powers and SINR are studied, and cumulative distribution function (CDF) of the transmit power and SINR have analytically been derived for a D2D network employing power control.In [9], mode selection and power control have been presented for underlay D2D communication in cellular networks, in which the proposed mode selection scheme for a user accounted for both the D2D link distance and cellular link distance (i.e., distance between the CUE and the BS).In [10], the small-scale fading experienced in the D2D direct link is modeled as Rician distribution.
Most of the previous works, for example [7][8][9][10], study results were based on single antenna deployed at BS in cellular network by stochastic geometry theory.Multiple antenna techniques are already relatively mature, and many standardization activities clearly indicate that multiantenna techniques and hybrid network will coexist and complement each other in the future wireless networks and should not be studied in isolation, as has been typically done in the literature [11].Multiple antenna techniques have many significant features [12][13][14], such as using precoding design for interference cancellation or using beamforming design for boosting diversity gain.In random network, average achievable rate and reliability can be improved by the multiple antenna techniques [15][16][17].In this paper, we model the hybrid network of cellular and D2D communication with stochastic geometry theory.In the network, cellular base stations are deployed with multiantennas.We analyze two primary performance measures: success probability and average achievable rate expression.The rate performance of beamforming and interference cancellation strategies is compared.
The rest of the paper is organised as follows.In Section 2, we introduce the system model.In Section 3, the success probability and average rate performance of beamforming strategy are investigated.In Section 4, the success probability and average rate performance of interference cancellation strategy are investigated.In Section 5, numerical simulation and analysis are discussed to verify these results.A conclusion is drawn in Section 6.
Notation.Let a denote a vector.Transpose and conjugate transpose are denoted by  and .The expectation of function () with respect to  is denoted as [()].The Laplace transform of () is denoted by   ().A circularly symmetric complex Gaussian random variable  with zero mean and variance  2 is denoted as  ∼ CN(0,  2 ).A Chi-square distributed random variable  with  degree of freedom is denoted by  ∼  2 ().An exponential distributed random variable with mean 1 is denoted by  ∼ exp (1).Let  1 be a set and let  2 be a subset of  1 ; then  1 \  2 denotes the set of elements of  1 that do not belong to  2 .

System Model
We consider a downlink hybrid network of cellular and D2D communication [1], as shown in Figure 1.The locations of base stations (BSs) are deployed as a PPP [5]   = {Γ  ,  ∈ N} with intensity   and constant transmission power   .Similarly, the cellular users locations are modeled as a PPP   = {Γ  ,  ∈ N} with intensity   .The locations of the D2D users are assumed to follow a PPP   = {Γ  ,  ∈ N} with intensity   and constant transmission power   .We assume the whole bandwidth  is divided into  subchannels.All the subchannels are available for BSs and D2D.Each D2D transmitter may randomly and independently access the subchannel.Each BS is configured with  antennas.Each CUE and DUE is configured with single antenna.The downlink channel is composed of path large-scale attenuation and fading for both cellular networks and D2D communication.Large-scale attenuation is modeled as the standard pathloss propagation represented as , where  is path-loss exponent and   is distance between transmitter  and receiver . may be BS or DUE transmitter, and  may be DUE receiver or CUE receiver.
Each CUE is associated with the nearest BS.Therefore, the probability density function (PDF) of the distance   can be derived as    () = exp(−   2 )2   according to the null probability of a 2D Poisson process [10].Each DUE transmitter and its DUE receiver have a fixed distance of   .Meanwhile, Rayleigh fading is assumed for the BS-CUE, DUE-CUE, BS-DUE, and DUE-DUE links.We consider the interference limited regime; that is, noise power is negligible compared to the interference power [10].In the following, we will characterize the performance of beamforming and interference cancellation strategies.

Beamforming Strategy
With multiple antennas, the BS chooses to increase its own cellular user signal power by performing beamforming. 0 and  0 are the received signal at the typical CUE ( 0 ) and typical DUE receiver ( 0 ), respectively, where The signal to interference ratio (SIR) in  0 with beamforming strategy is where   BF denotes the total interference at  0 and   BF,1 denotes the interference from BSs.   BF,2 denotes the interference from DUEs and Ĩ BF,1 and Ĩ BF,2 are simple transform of   BF,1 and   BF,2 by normalizing the desired signal power, respectively, According to the BF criterion via receiving power maximization, the precoding vector u  should align with the same direction as the channel itself u  = q  /‖q  ‖ 2 ; then the signal power at the  0 is |q 00 u 0 | 2 ∼  2 (2) and the interference power at the  0 from other BS is A performance metric of interest in this study is the success transmission probability of BS  Suc. (  ) with respect to a predefined SIR threshold   , similarly, the success transmission probability of D2D network  Suc. (  ) with respect to a predefined SIR threshold   .The D2D and cellular network success probability are given in Theorem 1.

Theorem 1.
For the cellular network underlay with D2D communication, the cellular BSs are configured with  antennas for beamforming strategy.The cellular and D2D success probability are given by where Proof.See Appendix A.
Theorem 2. For the cellular network underlay with D2D communication, the cellular BSs are configured with  antennas for beamforming strategy.The cellular and D2D average achievable rate in a shared subchannel are given by where ) . ( Proof.See Appendix B. It is difficult to derive the closed form expressions by directly integrating in (14).Based on this expression, a practical case of  = 4 and known   is applied to further derive the closed form of   BF , given by where  , , (  1 ,...,  , +1 ,...,   1 ,...,  , +1 ,...,  | ) is Meijer- function [10].The proof of Expression ( 16) is given by

Interference Cancellation Strategy
When the Interference cancellation strategy is exploited at the BS, the strategy employs a partial zero forcing (PZF) beamforming vector [18,19].It was found to be amenable to analysis and to explicitly balance interference cancellation and boosting of the desired signal power.With PZF, the BS uses  degrees of freedom to cancel its  DUE interferences inside a circle of radius   centered around the BS and uses the remaining  −  degrees of freedom to transmit the desired signal to its associated CUE receiver.In the following, we will characterize the performance of system based on interference cancellation (i.e., PZF) strategy. 0 and  0 are the received signal at the typical CUE ( 0 ) and typical DUE receiver ( 0 ), respectively.One has Let ,  ≤  − 1. g  is the channel between the BS and the th DUE inside a circle of radius   centered around the BS.Consider  = round([]) = round(   2  ), where [] =    2  denotes the DUEs number in circle of   centered around the BS, and round(⋅) is round function.u  lies in the null space of G  to null the interference towards the  DUEs and choose such that it maximizes the signal power |q  u  | 2 .From [17], u  = q   SS  /|q   SS  |, where S ∈ C ×− is the orthonormal basis of the null space of G  .The signal to interference ratio (SIR) in  0 based on PZF strategy is PZF denotes the total interference.One has The signal to interference ratio (SIR) in  0 based on PZF strategy is where   PZF denotes the total interference and   PZF,1 denotes the interference from BSs.   PZF,2 denotes the interference from DUEs.Consider At the typical DUE receiver, the sum function subscript Proof.The proof is similar to that of Theorem 2.
Area spectral efficiency (ASE) is defined as the product of the unconditioned success probability and the maximum sum rate (in bps/Hz) that can be sent per unit area [13].In order to maximize the ASE, we set a proper   to get . is the number of the canceled DUE interferences, which International Journal of Antennas and Propagation is also the degrees of freedom to null DUE interferences.So the BS transmission degrees of freedom can be allocated effectively between CUE beamforming and DUE interference cancellation.Consider The total ASE is In order to maximize the total ASE, it is difficult to get the close-form expression of optimal   .In this paper, we simulate the relation between ASE and   and get the optimal   .

Simulation Results and Discussion
In this section, we present Monte Carlo simulations to evaluate the performance of BF and PZF strategies and discuss the relation between ASE and   in this large random D2D underlaid cellular network.The simulated BS and DUE lie in a two-dimensional plane with independent Poisson processes.The default parameters are listed in Table 1 [10] unless otherwise stated.The analysis results developed in previous sections are validated with Monte Carlo simulations.Moreover, the analysis is performed to investigate the success probability as well as average achievable rate and illustrate the impact of the optimal   to system ASE.The simulation is run for 10000 times and the average was taken.
Figure 2 shows the analytical and simulation success probability versus the SIR threshold in BF strategy.From the figures we can see the analytical model fits the simulation results fairly well and thus can conclude that our analysis is International Journal of Antennas and Propagation well validated.The success probability of CUE in deployed multiantennas network is higher than single antenna case.addition, the more antennas BS has the more dofs CUE get, but the change of BS antennas number makes no difference to success probability of DUE.This is because all the BS transmitter dofs are allocated to CUE in BF strategy.Figure 3 validates the analysis results of average rate for both the cellular network and D2D communication in BF strategy.In addition, it illustrates how the average achievable rate is impacted as the increase of   with a fixed   .From Figure 3, we observe the rate performances are severely degraded for all scenarios when the intensity ratio rises up to 10 3 .When the intensity ratio   /  is a lower value, such as 10 0 , the average rate of DUE is higher than the average rate of CUE, but as the intensity ratio reaches 10 2. 16 , we find that the average rate of DUE is lower than the average rate of CUE from the enlarged part figure.This is because the rate gain caused by the increase of DUE intensity is lesser than the rate degradation caused by the increase of many DUE interferences at the moment.
Figure 4 shows the analytical and simulation success probability versus the SIR threshold in PZF strategy.From Figure 4, we find that the more dofs user get the higher success probability is.
Figure 5 validates the analysis results of average rate for both the cellular network and D2D communication in PZF strategy and compares the rate performance of two transmission strategies.When the BS deployed six antennas, the BS allocates 2 or 4 dofs to cancel the DUE interference in PZF strategy; then the BS allocates 4 or 2 dofs to increase the diversity gain of CUE accordingly.In BF strategy, all BS transmission dofs are allocated to their association CUE, and no dof is allocated to DUE.From Figure 5, we observe that when the intensity ratio is   /  < 10 1.6 , the average achievable total rate with PZF strategy is higher than the total rate with BF strategy.In order to maximize the total rate, we should allocate more dofs to DUE, but intensity ratio  /  ≥ 10 1.6 ; the average achievable total rate with PZF strategy is lower than the total rate with BF strategy.This is because when the BS transmission dofs and CUE intensity are fixed, the rate gain due to the increase of DUE intensity is less than the rate degradation due to the increase of many DUE interferences.
Figure 6 shows the relation between the ASE and   .SIR threshold of the cellular network and D2D communications set to be −5 dB.From the figure, we see that when   increases, the ASE of CUE decreases and the ASE of DUE increases.This is due to that fact that as   increases, more dofs are allocated to DUE for interference cancelation, and less dofs are allocated to CUE for increasing the diversity gain.In addition, there exists an optimal   to maximize the total ASE.

Conclusion
This paper analyzes the performance of the hybrid network of cellular and D2D communication with stochastic geometry theory.The analytical expressions of success probability and average rate are derived in BF and PZF transmission strategies, and the relation between ASE and interference cancellation radius is gotten.Simulation results show that the expressions can provide sufficient precision to evaluate the systems performance.In future study, we can consider the analysis of a hybrid network that the BS and CUEs/DUEs are both configured with multiple antennas by different precoding designs.These studies lay a theoretical foundation for network planning and base station deployment in hybrid network of cellular and D2D communication.(A.9)

Appendices
In the following, we will calculate the double integral where  0 and    are denoted in ( 9) and (10).

2 InternationalFigure 1 :
Figure 1: Cellular and D2D communication model.  is interference cancellation radius in interference cancellation strategy.

Figure 2 :
Figure 2: Cellular and D2D communication success probability in BF strategy.

Figure 3 :
Figure 3: Average achievable rate for cellular and D2D communication in BF strategy.

Figure 5 :
Figure 5: Average rate in PZF and BF strategies.CUE dof = 2, 4 and DUE dof = 4, 2 denote total simulation average rates include the cellular network and D2D communication in PZF strategy.
2 with PZF strategy is different from the subscript ( : Γ  ∈   ) in   BF,2 with BF strategy.{Γ 0 } denotes the BS which cancels the interference toward the typical DUE by PZF strategy.The typical CUE signal power is |q 00 u 0 | 2 ∼  2 (2( − )) with PZF strategy.It is different from the typical CUE signal power |q 00 u 0 | 2 ∼  2 (2) with BF strategy. is the degrees of freedom to null  DUE interferences.The D2D and cellular network success probability with PZF strategy are given in the following theorem.
+1 .(24) Proof.See Appendix C. Theorem 4. For the cellular network underlay with D2D communication, the cellular BS is configured with  antennas for PZF strategy.The cellular and D2D average achievable rate in a shared subchannel are given by