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 to maximize 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.
1. 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 [1–3]. 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 [4–6]. 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 [7–10], 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 [12–14], 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 [15–17]. 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 T and H. The expectation of function f(x) with respect to x is denoted as E[f(x)]. The Laplace transform of f(x) is denoted by Lf(s). A circularly symmetric complex Gaussian random variable x with zero mean and variance σ2 is denoted as x~CN(0,σ2). A Chi-square distributed random variable x with N degree of freedom is denoted by x~χ2(N). An exponential distributed random variable with mean 1 is denoted by x~exp(1). Let I1 be a set and let I2 be a subset of I1; then I1∖I2 denotes the set of elements of I1 that do not belong to I2.
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] ϕB={ΓBn,n∈N} with intensity λB and constant transmission power pB. Similarly, the cellular users locations are modeled as a PPP ϕU={ΓUn,n∈N} with intensity λU. The locations of the D2D users are assumed to follow a PPP ϕD={ΓDn,n∈N} with intensity λD and constant transmission power pD. We assume the whole bandwidth W is divided into V 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 N 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 dTR-a/2, where a is path-loss exponent and dTR is distance between transmitter T and receiver R. T may be BS or DUE transmitter, and R may be DUE receiver or CUE receiver. Each CUE is associated with the nearest BS. Therefore, the probability density function (PDF) of the distance dBC can be derived as fdBC(r)=exp(-πλBr2)2πλBr according to the null probability of a 2D Poisson process [10]. Each DUE transmitter and its DUE receiver have a fixed distance of dDD. 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.
Cellular and D2D communication model. rd is interference cancellation radius in interference cancellation strategy.
3. Beamforming Strategy
With multiple antennas, the BS chooses to increase its own cellular user signal power by performing beamforming. yC0 and yD0 are the received signal at the typical CUE (RC0) and typical DUE receiver (RD0), respectively, (1)yC0=∑n:ΓBn∈ϕBpBdBC,n-a/2q0nunxBn+∑n:ΓDn∈ϕDpDdDC,n-a/2f0nxDn,yD0=∑n:ΓDn∈ϕDpDdDD,n-a/2h0nxDn+∑n:ΓBn∈ϕBpBdBD,n-a/2g0nunxBn,where q0n∈C1×N is the channel between BS in ϕB and RC0. f0n is the channel between RC0 and DUE transmitter in ϕD. h0n is the channel between RD0 and DUE transmitter in ϕD. g0n∈C1×N is the channel between RD0 and BS in ϕB.un∈CN×1 is the beamformer used by the nth transmitter. xBn and xDn are the data signals transmitted from BS and DUE, respectively. xBn and xDn are ~CN(0,1). The signal to interference ratio (SIR) in RD0 with beamforming strategy is(2)SIRDBF=PDdDD-ah002IDBF,where IDBF denotes the total interference, IDBF,1 denotes the interference from DUEs, and IDBF,2 denotes the interference from BSs. I~DBF,1 and I~DBF,2 are simple transform of IDBF,1 and IDBF,2 by normalizing the desired signal power, respectively,(3)IDBF=∑n:ΓDn∈ϕD∖{ΓD0}pDdDD,n-ah0n2︸IDBF,1+∑n:ΓBn∈ϕBpBdBD,n-ag0nun2︸IDBF,2,I~DBF,1=∑n:ΓDn∈ϕD∖{ΓD0}dDD,n-ah0n2dDD-a,I~DBF,2=∑n:ΓBn∈ϕBdBD,n-ag0nun2dDD-apD/pB.The signal to interference ratio (SIR) in RC0 with beamforming strategy is (4)SIRCBF=PBdBC-aq00u02ICBF,where ICBF denotes the total interference at RC0 and ICBF,1 denotes the interference from BSs. ICBF,2 denotes the interference from DUEs and I~CBF,1 and I~CBF,2 are simple transform of ICBF,1 and ICBF,2 by normalizing the desired signal power, respectively,(5)ICBF=∑n:ΓBn∈ϕB∖{ΓB0}pBdBC,n-aq0nun2︸ICBF,1+∑n:ΓDn∈ϕDpDdDC,n-af0n2︸ICBF,2,I~CBF,1=∑n:ΓBn∈ϕB∖{ΓB0}dBC,n-aq0nun2dBC-a,I~CBF,2=∑n:ΓDn∈ϕDdDC,n-af0n2dBC-apD/pB.
According to the BF criterion via receiving power maximization, the precoding vector un should align with the same direction as the channel itself un=qon/qon2; then the signal power at the RC0 is q00u02~χ2(2N) and the interference power at the RC0 from other BS is q0nun2~χ2(2) as [15]. Similarly, g0nu02~χ2(2). Because f0n~CN(0,1), we have f0n2~χ2(2).
A performance metric of interest in this study is the success transmission probability of BS PSuc.C(βC) with respect to a predefined SIR threshold βC, similarly, the success transmission probability of D2D network PSuc.D(βD) with respect to a predefined SIR threshold βD. 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 N antennas for beamforming strategy. The cellular and D2D success probability are given by (6)PSuc.DBFβD=exp-Cd1βD2,(7)PSuc.CBFβC=πλBA0+∑k=1N-1∑j=1k-1kk!πλBAjkΓj+1A0j+1,where (8)Cd1=πcadDDa/2λD+λBpBpD2/a,(9)A0=Cd2′s2/a+πλBs=βc=Cd2′βC2/a+πλB,(10)Ajk=βjkj!Cd2′s2/ajs=βc=βjkj!Cd2′βC2/aj,(11)βjk=∑i=1j-1jji2iak,j=1,…,k,(12)xk=xx-1⋯x-k+1.
Proof.
See Appendix A.
Theorem 2.
For the cellular network underlay with D2D communication, the cellular BSs are configured with N antennas for beamforming strategy. The cellular and D2D average achievable rate in a shared subchannel are given by (13)RDBF=ΛCd1,(14)RCBF=∫0∞∫z=0∞∑k=0N-11/k!-zkQz,k1+zmmmmmm·2πλBrexp-πλBr2dzdr,where (15)Λx=πsinx-2sinxSix-2cosxCix,Six=∫0xsinttdt,Cix=-∫x∞costtdt,Qs,k=dLICBFsdsk=exp(-Cd2s2/a),k=0-1skexp-Cd2s2/a∑j=1kβjkj!Cd2s2/aj,k≠0,Cd2=πcadBC2λB+λDpDpB2/a.
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 a=4 and known dBC is applied to further derive the closed form of RCBF, given by(16)RCBF∣dBC=Elog1+ξ∣dBC=Λ(Cd2)+∑k=1N-1∑j=1kβjk-1kj!G1,33,1-j2-j2,0,12∣Cd224,where Gp,qm,na1,…,an,an+1,…,apb1,…,bm,bm+1,…,bq∣z is Meijer-G function [10]. The proof of Expression (16) is given by (17)Elog1+ξ∣dBC=∫x=0∞log1+xfξxdx=∫x=0∞∫z=0∞11+zfξxdzdx=∫z=0∞1-Fξz1+zdz=∫z=0∞∑k=0N-1(1/k!)-zkQ(z,k)1+zdz=Λ(Cd2)+∑k=1N-1∑j=1kβjk-1kj!G1,33,1-j2-j2,0,12∣Cd224.
4. 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 L degrees of freedom to cancel its L DUE interferences inside a circle of radius rd centered around the BS and uses the remaining N-L 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. yC0 and yD0 are the received signal at the typical CUE (RC0) and typical DUE receiver (RD0), respectively. One has(18)yC0=∑n:ΓBn∈ϕBpBdBC,n-a/2q0nunxBn+∑n:ΓDn∈ϕDpDdDC,n-a/2f0nxDn,yD0=∑n:ΓDn∈ϕDpDdDD,n-a/2h0nxDn+∑n:ΓBn∈ϕB∖{ΓB0}pBdBD,n-a/2g0nunxBn.
Let Gn=g1nTg2nT⋯gLnT, L≤N-1. gLn is the channel between the BS and the Lth DUE inside a circle of radius rd centered around the BS. Consider L=round(E[l])=round(πλBrd2), where E[l]=πλBrd2 denotes the DUEs number in circle of rd centered around the BS, and round(·) is round function. un lies in the null space of Gn to null the interference towards the L DUEs and choose such that it maximizes the signal power qnnun2. From [17], un=qnnHSSH/qnnHSSH, where S∈CN×N-L is the orthonormal basis of the null space of Gn. The signal to interference ratio (SIR) in RC0 based on PZF strategy is (19)SIRCPZF=PBdBC-aq00u02ICPZFICPZF denotes the total interference. One has (20)ICPZF=∑n:ΓBn∈ϕB∖{ΓB0}pBdBC,n-aq0nun2+∑n:ΓDn∈ϕDpDdDC,n-af0n2.The signal to interference ratio (SIR) in RD0 based on PZF strategy is (21)SIRDPZF=pDdDD-ah002IDPZF,where IDPZF denotes the total interference and IDPZF,1 denotes the interference from BSs. IDPZF,2 denotes the interference from DUEs. Consider(22)IDPZF=∑n:ΓDn∈ϕD∖{ΓD0}pDdDD,n-ah0nxDn︸IDPZF,1+∑n:ΓBn∈ϕB∖ΓB0pBdBD,n-ag0nun2︸IDPZF,2.
At the typical DUE receiver, the sum function subscript (n:ΓBn∈ϕB∖{ΓB0}) in IDPZF,2 with PZF strategy is different from the subscript (n:ΓBn∈ϕB) in IDBF,2 with BF strategy. {ΓB0} denotes the BS which cancels the interference toward the typical DUE by PZF strategy. The typical CUE signal power is q00u02~χ2(2(N-L)) with PZF strategy. It is different from the typical CUE signal power q00u02~χ2(2N) with BF strategy. L is the degrees of freedom to null L DUE interferences. The D2D and cellular network success probability with PZF strategy are given in the following theorem.
Theorem 3.
For the cellular network underlay with D2D communication, the cellular BSs are configured with N antennas for PZF strategy. The cellular and D2D success probability are given by(23)PSuc,DPZFβD=exp-λDπsinc2/aβD2/adDD21,1-2a;2-2a;-βDdDDapBpDrD-ammmmmm-2a-2πλBrD2-apBpDdDDaβD2mmmmmm×F11,1-2a;2-2a;-βDdDDapBpDrD-a,(24)PSuc,CPZF=πλBA0+∑k=1N-L-1∑j=1k-1kk!πλBAjkΓ(j+1)A0j+1.
Proof.
See Appendix C.
Theorem 4.
For the cellular network underlay with D2D communication, the cellular BS is configured with N antennas for PZF strategy. The cellular and D2D average achievable rate in a shared subchannel are given by (25)RDPZF=Elog1+ξDPZF=∫z=0∞1-Fξz1+zdz=∫z=0∞exp-λDπsinc2/az2/adDD2·F211,1-2a;2-2a;-zdDDapBpDrD-ammmmmmm-2a-2πλBrD2-apBpDdDDazmmmmmmm·F211,1-2a;2-2a;-zdDDapBpDrD-a1,1-2a;2πsinc2/a-2a;-zdDDapBpDrD-a·1+z-1dz,RCPZF=Elog1+ξCPZF=∫0∞∫z=0∞∑k=0N-L-1(1/k!)-zkQ(z,k)1+zmmmmmm·2πλBrexp-πλBr2dzdr.
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 rd to get L. L is the number of the canceled DUE interferences, which 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(26)TDPZF=λDlog21+βDPSuc,DPZFβD=λDlog21+βD·exp-λDπsinc2/aβD2/adDD2-2a-2πλBrD2-apBpDdDDaβD·F211,1-2a;2-2a;-βDdDDapBpDrD-a,TCPZF=λBlog21+βCPSuc,CPZFβC=λBlog2(1+βC)·πλBA0+∑k=1N-L-1∑j=1k-1kk!πλBAjkΓj+1A0j+1.The total ASE is (27)TPZF=TDPZF+TCPZF=λDlog21+βD×exp-λDπsinc2/aβD2/adDD21,1-2a;2-2a;-βDdDDapBpDrD-ammmmm-2a-2πλBrD2-apBpDdDDaβDmmmmm·F211,1-2a;2-2a;-βDdDDapBpDrD-a+λBlog21+βC·πλBA0+∑k=1N-L-1∑j=1k-1kk!πλBAjkΓj+1A0j+1.
In order to maximize the total ASE, it is difficult to get the close-form expression of optimal rd. In this paper, we simulate the relation between ASE and rd and get the optimal rd.
5. 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 rd 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 rd to system ASE. The simulation is run for 10000 times and the average was taken.
Parameter assumptions.
Parameter
Meaning
Default value
N
Number of BS antennas
1, 2, 4, 6
V
Number of subchannels
1000
α
Pathloss exponent
4
λB
Intensity of BS
6×10-6/m2
λD
Intensity of DUE
2.4×10-5/m2
λU
Intensity of CUE
6×10-4/m2
PC
Transmission power of BSs
43 dBm
PD
Transmission power of DUE transmitter
23 dBm
βC
SIR threshold of the cellular network
0 dB
βD
SIR threshold of the cellular network
0 dB
dDD
Distance between a D2D pair
45 m
XC
Maximum transmit range of CUE
400 m
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 well validated. The success probability of CUE in deployed multiantennas network is higher than single antenna case. In 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.
Cellular and D2D communication success probability 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 λD with a fixed λB. From Figure 3, we observe the rate performances are severely degraded for all scenarios when the intensity ratio rises up to 103. When the intensity ratio λD/λB is a lower value, such as 100, the average rate of DUE is higher than the average rate of CUE, but as the intensity ratio reaches 102.16, 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.
Average achievable rate for cellular and D2D communication in BF strategy.
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.
Cellular and D2D communication success probability for difference N in PZF strategy. DUE dof = 0 1 2 4 denotes 0, 1, 2, 4 dofs are allocated to DUE for interference cancellation, respectively. CUE dof = 0 1 2 4 denotes 0, 1, 2, 4 dofs are allocated to CUE for increasing the diversity gain, respectively.
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.
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.
From Figure 5, we observe that when the intensity ratio is λD/λB<101.6, 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 λD/λB≥101.6; 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 rd. SIR threshold of the cellular network and D2D communications set to be −5 dB. From the figure, we see that when rd increases, the ASE of CUE decreases and the ASE of DUE increases. This is due to that fact that as rd 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 rd to maximize the total ASE.
ASE with PZF strategy.
6. 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.
AppendicesA. Proof of Theorem <xref ref-type="statement" rid="thm1">1</xref>
(1) Derivation of DUE success probability in BF strategy is as follows:(A.1)PSuc.DBFβD=PpDdDD-ah002IDBF>βD=Ph002>βD(I~DBF,1+I~DBF,2)=LI~DBF,1βDLI~DBF,2βD.LI~DPZF,1(βD) is the Laplace transform of the interference from other D2D transmitters and LI~DPZF,2(βD) is the Laplace transform of the interference from other BS transmitters. One has(A.2)LI~DBF,1s=Ee-s∑n:ΓDn∈ϕD∖{ΓD0}(dDD,n-ah0n2/dDD-a)=E!0e-s∑n:ΓDn∈ϕD(dDD,n-ah0n2/dDD-a)=e-2πλDdDD2∫0∞1-Eexp-sr-ah0n2rdr=exp-πλDcadDD2s2/a,(A.3)LI~DBF,2s=exp-πλBcadDD2pBpD2/as2/a,where (A.2) follows the Campbells theorem as [5, 10, 20]. One has c(a)=(2π/a)/sin(2π/a). Substituting (A.2) and (A.3) into (A.1) yields the desired result in (6).
(2) Derivation of CUE success probability in BF strategy is as follows.
We rewrite SIR in RC0 based on BF strategy: (A.4)SIRCBF=PBdBC-aq00u02ICBF,ICBF=∑n:ΓBn∈ϕB∖{ΓB0}pBdBC,n-aq0nun2︸ICBF,1+∑n:ΓDn∈ϕDpDdDC,n-af0n2︸ICBF,2,where I~CBF,1 and I~CBF,2 are simple transform of ICBF,1 and ICBF,2 by normalizing the desired signal power, respectively:(A.5)I~CBF,1=∑n:ΓBn∈ϕB∖{ΓB0}dBC,n-aq0nun2dBC-a,I~CBF,2=∑n:ΓDn∈ϕDdDC,n-af0n2dBC-apD/pB.When dBC is known, (A.6)LI~CBF,1s≈exp-πλBcadBC2s2/a,LI~CBF,2s=exp-πλDcadBC2pDpB2/as2/a,Cd2=πcadBC2λB+λDpDpB2/a,LI~CBFs=LI~CBF,1sLI~CBF,2s=expCd2s2/a.Let the signal power SC=q00u02 and the complementary cumulative distribution function (CCDF) of SC is given by (A.7)FSct=e-t∑k=0N-1tkk!.The probability density function (PDF) of the distance dBC is given by (A.8)fdBCr=2πλBrexp-πλBr2.CUE success probability in BF strategy can be derived as (A.9)PSuc.CBFβC=EdBCPSIRCBF>βC∣dBCs=βC=∫0∞P(SIRCBF>βC∣dBC)fdBC(r)dr=∫0∞PpBr-aq00u02ICBF>βCfdBC(r)dr=∫0∞∫0∞FScstfI~CBFtfdBCrdtdrs=βC.In the following, we will calculate the double integral (A.10)∫0∞∫0∞FScstfI~CBFtfdBCrdtdr=∫0∞∫0∞e-st∑k=0N-1stkk!fI~CBFtdtfdBCrdr=∫0∞∑k=0N-11k!sk∫0∞e-ttkfI~CBFtdtfdBCrdr=a∫0∞∑k=0N-11k!snLtkfI~CBFtsfdBCrdr=∫0∞∑k=0N-11k!-skdkLI~CBFsdskfdBCrdr,where step (a) is achieved by utilizing the property of Laplace transform(A.11)tnft⟷-1ndndsnLfts.Let Cd2′=-πc(a)(λB+λDpD/pB2/a) and Q(s,k)=dkLI~CBF(s)/dsk. Employing the nth derivation of the composite function and after some algebra, we can obtain Q(s,k) as [10](A.12)Qs,k=dLI~CBFsdsk=exp(-Cd2s2/a),k=0-1skexp-Cd2s2/a∑j=1kβjkj!Cd2s2/aj,k≠0.Substituting (A.12) into (A.10) and setting a=4 and s=βC yield the desired CUE success probability: (A.13)PSuc.CBFβC=EdBCPSIRCBF>βC∣dBC=∫0∞PSIRCBF>βC∣dBCfdBC(r)dr=∫0∞∑k=0N-11k!-skQ(s,k)fdBC(r)dr=∫0∞2πλBrexp-Cd2′s2/a+πλBr2dr+∫0∞-1kk!2πλBrexp-Cd2′s2/a+πλBr2·∑k=1N-1∑j=1kβjkj!Cd2′s2/ajr2jdr=∫0∞2πλBrexp-A0r2dr+∫0∞-1kk!2πλB∑k=1N-1∑j=1kAjkexp-A0r2r2j+1dr=πλBA0+∑k=1N-1∑j=1k-1kk!πλBAjkΓj+1A0j+1,where A0 and Ajk are denoted in (9) and (10).
B. Proof of Theorem <xref ref-type="statement" rid="thm2">2</xref>
(1) Derivation of average rate of CUE is as follows:(B.1)Elog1+ξ=∫0∞Elog1+ξ∣dBCfdBCrdr=∫0∞∫z=0∞∑k=0N-1(1/k!)-zkQ(z,k)1+z2πλBrmmmmmmm·exp-πλBr2dzdr.
(2) Derivation of average rate of DUE is as follows:(B.2)RDBF=Elog1+ξDBF=∫z=0∞1-Fξz1+zdz=∫z=0∞exp-Cd1z2/a1+zdz=ΛCd1.
C. Proof of Theorem <xref ref-type="statement" rid="thm3">3</xref>
(1) Derivation of DUE success probability PSuc.DPZF(βD) is as follows.
The signal to interference ratio SIRDPZF in RD0 is denoted in (21). IDPZF,1 denotes the interference from BSs. IDPZF,2 denotes the interference from DUEs; I~DPZF,1 and I~DPZF,2 are simple transform of IDPZF,1 and IDPZF,2 by dividing the desired signal power, respectively:(C.1)I~DPZF,1=∑n:ΓDn∈ϕD∖{ΓD0}dDD,n-ah0n2dDD-a,I~DPZF,2=∑n:ΓBn∈ϕB∖{ΓB0}dBD,n-ag0nun2dDD-apD/pB.The success transmission probability of DUE is (C.2)PSuc,DPZFβD=PSIRDPZF>βD=Ph002I~DPZF>βD=Eexp-βDI~DPZF,1+I~DPZF,2=LI~DPZF,1βDLI~DPZF,2βD.Equation (C.2) follows from h002~exp(1) as [1, 10]. I~DPZF=I~DPZF,1+I~DPZF,2, LI~DPZF,1(βD) is the Laplace transform of the interference from other D2D transmitters, and LI~DPZF,2(βD) is the Laplace transform of the interference from BS transmitters. Consider(C.3)LI~DPZF,1ss=βD=exp-πλDcadDD2s2/as=βD=exp-λDπsinc2/aβD2/adDD2,LI~DPZF,2s=Eexp-spBdDD-apDr-aW=exp-2πλB∫rD∞1-11+spBdDDapD-1r-ardr=exp-2a-2πλBrD2-apBpDdDDasmmmmm·F211,1-2a;2-2a;-spBpDdDDarD-a,where W=g0nun2, 2F1(·) is Gause hypergeometry function [1]. Substituting (C.3) into (C.2) yields the desired result in (23).
(2) Derivation of CUE success probability PSuc.CPZF(βC) is as follows.
The typical CUE signal power is q00u02~χ2(2(N-L)) with PZF strategy, and the typical CUE signal power q00u02~χ2(2N) with BF strategy. There is only difference in degree of freedom in two strategies. So we change the N in (7) into N-L; then we get (24).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge anonymous reviewers who read drafts and made many helpful suggestions. This study is supported by the Natural Science Foundation of China (no. 60872149 and no. 61471056), BUPT Youth Research and Innovation Plan (2014RC0103), and the National High Technology Research and Development Program of China (863 Program) (no. 2014AA01A701).
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