Exploiting Joint Base Station Equipped Multiple Antenna and Full-Duplex D2D Users in Power Domain Division Based Multiple Access Networks

Abstract

In this paper, we investigate power domain division-based multiple access (PDMA) to support the base stations (BS) equipped with multiple antennas to serve mobile users. Such a system deploys multiple input single output (MISO)-based wireless transmission and a full-duplex (FD) scheme. Furthermore, such MISO PDMA system consists of BS employing transmit antenna selection to reduce complexity in signal processing at the receivers. We distinguish two kinds of mobile users, device-to-device (D2D) users and traditional users. In such MISO PDMA, there exists a trade-off between outage performance of each PDMA user and power allocation factors. Since the implementation of the FD scheme at PDMA users, bandwidth efficiency will be enhanced despite the existence of self-interference related to such FD. In particular, exact expressions of outage probability are derived to exhibit system performance with respect to D2D users. Finally, valuable results from the simulated parameters together with the analytical results show that MISO PDMA can improve its performance by increasing the number of transmit antennas at the BS.

1. Introduction

To adapt fast development in wireless techniques, many protocols and topologies have been introduced. One of these schemes, relaying networks has recently attracted the research community due to extended coverage and improved reliability [1,2,3,4]. Full-duplex is considered a scheme that exhibits higher bandwidth efficiency and such schemes are proposed in wireless powered relaying networks as in [2,3]. However, multiple users need to be served to access the core network, and it requires the base station in a cellular network that can transmit a mixture of signals to them. To address this shortcoming, PDMA or non-orthogonal multiple access (NOMA) has been recently introduced to improve the spectral efficiency and to provide fairness in resource allocation. These advantages can be performed by multiplexing multiple users on the same time/frequency resource [5,6,7,8,9,10,11,12]. In particular, NOMA employs the power domain to serve multiple access and harnesses interference via superposition coding at the transmitter, while successive interference cancellation (SIC) is required at the receiver. In [6], the authors investigated maximal performance of single-input single output (SISO) single-carrier (SC) NOMA systems in terms of system throughput and they explored the optimal power allocation design. The higher spectral efficiency can be achieved in SISO NOMA compared to conventional SISO OMA [6]. The suboptimal precoding design is presented for minimization of the transmit power in multiple-input single-output (MISO) SC-NOMA systems [7]. In addition, zero-forcing downlink (DL) beamforming was analyzed in MISO SC-NOMA systems [8].

Furthermore, it is very expensive to deploy a MISO system since both space-time codes and transmit beamforming require multiple RF chains [13]. Fortunately, the antenna selection scheme is proposed to overcome such disadvantage, providing a good trade-off between cost, complexity, and performance [13]. Antenna selection can be implemented at both ends, and transmit antenna selection with maximal-ratio combining (TAS/MRC) is presented in [14]. This scheme can be mainly described as follows: by using CSI feedback, the best transmit antenna out of all transmit candidates to maximize the post processing signal to noise ratio (SNR) at the MRC output of receive antennas, is selected to transmit data for the corresponding user. These analytical discussions motivate us to explore improved performance of MISO PDMA.

With distant transmission, relaying schemes are required but in close distance transmission, and so new protocols need to be exhibited. Recently, in project 3GPP for long term evolution (LTE), device to device (D2D) communication has been introduced. As one of the effective technologies of the forthcoming 5th generation (5G) cellular standard, D2D is explored as in [15]. By using new paradigm, i.e., without or limited controlling and signaling information from the base station (BS), two users can communicate in instant and direct ways with each other (when in proximity) in context of D2D scenario [16,17,18,19,20]. Furthermore, potential application in disaster-affected areas needs fast connections and D2D can be adopted in such case. In particular, the local connectivity is provided to devices even in a case of damage to the network infrastructure. D2D can be employed in several other emerging applications. For example, vehicular-to-vehicular (V to V) communication, vehicular-to-infrastructure (V to I) communication are introduced with applications of D2D communication to exhibit proximity based add-on services and multi-party gaming or public safety applications are studied as well [21,22]. It can be exhibited commercial D2D communication to improve the throughput, spectrum utilization, and energy efficiency of the cellular network. Other challenges are raised such as interference management security. To meet the capacity requirements of the 5G cellular system, a project was deployed and it is known as METIS (mobile and wireless communications enablers for the twenty-twenty information society). The METIS has recently been funded by the European Union [23].

Regarding exploiting advantages of D2D into NOMA, the authors in [24] investigated a model of the integration of a downlink NOMA system with D2D communications. D2D reported in [25] with resource allocation scheme is promising approach. They further derived expressions of the outage probability that both users obtain higher rates in NOMA under a fixed power control strategy. In addition, the uplink multi-carrier is considered in NOMA with support of D2D underlaid cellular networks [26]. More specifically, an iterative algorithm applying Karush-Kuhn-Tucker conditions is proposed to solve the power allocation problem in D2D NOMA [26]. The authors in [27] studied the device-to-device (D2D) assisted and NOMA-based mobile edge computing (MEC) system by deploying D2D communication for enabling user collaboration and reducing the edge server’s load.

In this paper, we consider a D2D transmission existing in a downlink PDMA system. The selected antenna at one BS communicating has two D2D far receivers with the aid of D2D implementation. Different from existing works on D2D PDMA [26,27], where the end-user operating half-duplex, we assume that the D2D users operate in the FD mode and investigate outage performance taking into account both the downlink and D2D links. The key contributions of this study are summarized as follows:

1. In the presence of a downlink under support of multiple antenna based BS, two D2D users exhibit different outage performance. We individually investigated the performance of each end-user in such a MISO NOMA system. Compared to most existing cooperative PDMA schemes, FD scheme is enabled at the end-user. To look how good performance two far users have, two far D2D users with different QoS requirements can be paired with each other and get benefit from D2D transmission.

2. Different from [25], transmit selection and full-duplex are joint investigated in this study. Most important is that we provide simulation results of integration of a D2D connection to a downlink two-user PDMA system.

3. We provide simulation results showing that, under the fixed power allocation strategies, D2D users achieve outage behavior in the NOMA scheme. The results also indicate the probability that both D2D users obtain improved outage performance in MISO PDMA depends on the power level of the BS and the required target rates.

4. For system performance evaluation, the closed-form expressions for the outage probabilities are derived for both two D2D users. To highlight the impact of the system parameters on the outage performance, outage probabilities achieved at both two D2D users are simulated to verify derived expressions.

Notation: The cumulative distribution function of a real-valued random variable X is denoted by_FX(.),_fX(.)stands for probability density functions, whilePr(.)symbolizes outage probability.

2. System Model

We consider a downlink MISO aided PDMA network as shown in Figure 1, in which the base station (BS) is equipped with multiple antennas to serve two PDMA users. There are conventional cellular users (CUE) in this model, such CUE devices are able to receive signal under coverage of this BS, but this paper focuses on more complex operations of D2D users. It is assumed that interference from CUE to D2D users is smaller than self-interference at each D2D user. In this case, two PDMA users (_D1,_D2) operate in full-duplex (FD) mode and they can communicate directly without helping of the BS as underlay topology. Two PDMA users are able to communicate directly on channel_hi,i=1,2. It is noted that_gi,kdenotes the channel gain between the BS and user_Di,i=1,2, the BS hask,(k=1,2,…K)antennas. Further, in this scenario PDMA users are double-antenna devices and operate in a FD mode, except for the BS equipped multiple antenna. The direct links between the source node and the users are assumed available, which is common in the scenarios where two PDMA users acquire device to device transmission in reliable coverage of such BS. We assume that all users are clustered very close such that a homogeneous network topology is considered in our paper. The channels associated with each link exhibit the Rayleigh fading and additive white Gaussian noise (AWGN).

In first phase, the BS sends signalx=_{a1 PSx1}+_{a2 PSx2}to_D1and_D2according to direct transmissions. Here,_PSis the transmitted power of the BS,_x1x2is the signal of_D1D2, and_a1,_a2is the power allocation coefficient with_a1+_a2=1,_a1>_a2.

2.1. Calculation of Signal to Noise Ratio (SNR)

The received signal at_Diis given by

_yDiFD−PDMA=_{gi,ka1 PSx1}+_{a2 PSx2}+_fiϖ_PDixFDi+_wi,

whereϖ=1denotes user 1 working in FD,_PDiis transmit power of_Di,i=1,2and_wiis the additive white Gaussian noise with zero mean and variance_N0. We call_xFDia signal related to self-interference at user i, and_fiis the self-interference channel and follows_fi∼CN0,_λfi .

Then, the received signal-to-interference-plus-noise ratio (SINR) at user 1_D1becomes

_{ySD1,kFD−PDMA}=_a1ρ^_g1,k2_a2ρ^_g1,k2+ϖρ^_f12+1,

whereρ=_{PSPS N0N0}is the transmit signal-to-noise ratio (SNR) which was measured at the BS.

In this scenario,_D2is so-called as SIC user, i.e., SIC is required to eliminate interference from signal of_D1. Firstly, the received SINR at user 2 to detect user1’s message_x1is given by

_{γSD1←2,kFD−PDMA}=_a1ρ^_g2,k2_a2ρ^_g2,k2+ϖρ^_f22+1.

Then SIC is activated to eliminate interference from_D1, the received SINRs at the user 2_D2is calculated to decode its own signal as

_{γSD2,kFD−PDMA}=_a2ρ^_g2,k2ϖρ^_f22+1.

2.2. D2D Transmission

In this phase, the cooperation signal is transmitted from the user with a stronger channel gain to the user with a weaker gain. The cooperation signal can help user 1 to decode its data, or user 2 to perform SIC better. The cooperation signal received by user 1_D1is given by

_zDiPDMA=_hiPSs+_fiϖ_PDixFDi+_ni,

where_h1is a Rayleigh fading channel coefficient from user 1 to user 2 and vice versa. As mentioned in the channel information exchange phase, when^_g1,k2>^_g2,k2, only_zD2PDMAexists, and when^_g1,k2<^_g2,k2, only_zD1PDMAis transmitted from_D2.

Generally, the received SINR at user i is given by

_χDiPDMA=ρ^_hi2ϖρ^_fi2+1.

The SINR for decoding_x1is given by

χ=minmax_γSD1,kPDMA,_χD1PDMA,_{γSD1←2,kPDMA},if^_g1,k2<^_g2,k2min_γSD1,kPDMA,max_{γSD1←2,kPDMA},_χD2PDMA,otherwise.

The antenna index can be selected to strengthen the BS to serve user i link as follows:

^k*=argmax︸k=1,…,K^_gi,k2.

In this case, CDF and PDF related selected channel are given respectively by

_F^gi,k*2 x=1−∑k=1KKk^−1k−1exp−kx_λg,

and

_f^gi,k*2 x=∑k=1KKk^−1k−1k_λgexp−kx_λg.

Here,_λuis the channel gain of u.

3. Outage Probability Performance Analysis

When the targeted data rates,_R1and_R2are determined by the users’ QoS requirements for user_D1,_D2. In fact, the outage probability is an important performance criterion which needs to be investigated. If the outage event occurs at the non-SIC user, the SIC user does not use the D2D signal, and the outage of the SIC user does not allow D2D transmission from the SIC user to the non-SIC user.

3.1. Outage Probability of D2D User 1

Considering outage probability of_D1: According to PDMA protocol, the complementary events of outage at_D1can be explained as:_D1can detect_x2as well as its own message_x1. From the above description, the outage probability of_D1is expressed as

O_PD1−bi=Pr_γSD1,k*PDMA<_ε1,_{γSD1←2,k*PDMA}<_ε1︸_B1+Prmax_γSD1,k*PDMA,_χD1PDMA<_ε1,_{γSD1←2,k*PDMA}>_ε1︸_B2,

Proposition 1.

The closed-form expression of outage probability at D1 is given by

O_PD1−bi=1−∑k=1KKk^−1k−1_ϑ11−∑k=1KKk^−1k−1 _ϑ2+1−∑k=1KKk^−1k−1 _ϑ1×1−ρ_λh1 ρ_λh1 +_ε1ϖρ_λf1 exp−_ε1ρ_λh1 ∑k=1KKk^−1k−1 _ϑ2

where_ε1=^22_R1,_R1is target rate for signal_x1.

Proof.

See Appendix A. □

3.2. Outage Probability of D2D User 2

The outage events of_D2can be explained as below. The first is that_D1cannot detect_x2. The second is that_D2cannot detect its own message_x2on the conditions that_D1can detect_x2successfully. Based on these, the outage probability of_D2is expressed as

O_PD2−bi=Pr_γSD2,k*PDMA<_ε2∪_{γSD1←2,k*PDMA}<_ε1,_γSD1,k*PDMA<_ε2︸_Ψ1+Pr_γSD2,k*PDMA<_ε2∪max_{γSD1←2,k*PDMA},_χD2PDMA<_ε1,_γSD1,k*PDMA>_ε1︸_Ψ2,

where_ε2=^22_R2,_R2is denoted as target rate for signal_x2 , and with the help of (5) and (7), terms_Ψ1and_Ψ2can be calculated, the first being

_Ψ1=Pr_γSD2,k*PDMA<_ε2∪_{γSD1←2,k*PDMA}<_ε1,_γSD1,k*PDMA<_ε2=1−Pr_γSD2,k*PDMA≥_ε2,_{γSD1←2,k*PDMA}≥_ε1︸_D111−Pr_γSD1,k*PDMA≥_ε2︸_D12.

Therefore,_D11can be expressed as

_D11=Pr^_g2,k2≥_ε2ϖρ^_f22+_ε2a2ρ,^_g2,k2≥_ε1ϖρ^_f22+_ε1a1ρ−_{ε1 a2}ρ=Pr^_g2,k2≥_ε2ϖρ^_f22+1_a2ρ,^_g2,k2≥_ε1ϖρ^_f22+1_a1ρ−_{ε1 a2}ρ=Pr^_g2,k2≥ϖρ^_f22+1max_ε2a2ρ,_ε1a1ρ−_{ε1 a2}ρ,

whereθ=max_ε2a2ρ,_ε1a1ρ−_{ε1 a2}ρ.

It worth noting that, we can achieve important computations as below:

_D11=Pr^_g2,k2≥ϖρ^_f22+1θ=1_{λf2 ∫0∞}∑k=1KKk^−1k−1exp−ϖρy+1θk_λ2−y_λf2 dy=∑k=1KKk^−1k−1_λ2θkϖρ_λf2 +_λ2exp−θk_λ2.

Similarly,_D12can be expressed as

_D12=1−Pr_γSD1,k*PDMA≥_ε2=1−Pr^_g1,k2≥_ε2ϖρ^_f12+_ε2a1ρ−_{ε2 a2}ρ=1−1_{λf1 ∫0∞}∑k=1KKk^−1k−1exp−_ε2ϖρx+_ε2k_a1ρ−_{ε2 a2}ρ_λ1−x_λf1 dx=1−∑k=1KKk^−1k−1 _υ1,

where_υ1=_a1ρ−_{ε2 a2}ρ_λ1a1ρ−_{ε2 a2}ρ_λ1+_ε2ϖρk_λf1 exp−_ε2k_a1ρ−_{ε2 a2}ρ_λ1.

From (16) and (17), we find the expression_Ψ1

_Ψ1=1−∑k=1KKk^−1k−1_λ2θkϖρ_λf2 +_λ2exp−θk_λ2×1−∑k=1KKk^−1k−1 _υ1.

Next,_Ψ2can be calculated by

_Ψ2=Pr_γSD2,k*PDMA<_ε2∪max_{γSD1←2,k*PDMA},_χD2PDMA<_ε1,_γSD1,k*PDMA>_ε1=Pr_γSD2,k*PDMA<_ε2+Prmax_{γSD1←2,k*PDMA},_χD2PDMA<_ε1︸_D21−Pr_γSD2,k*PDMA<_ε2∪max_{γSD1←2,k*PDMA},_χD2PDMA<_ε1︸_D22×Pr_γSD1,k*PDMA>_ε1.

Interestingly, we have the following result:

Pr_γSD2,k*PDMA<_ε2=1−Pr_γSD2,k*PDMA≥_ε2=1−Pr^_g2,k2≥_ε2ϖρ^_f22+_ε2a2ρ=1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2,

and

Pr_γSD1,k*PDMA>_ε1=Pr^_g1,k2>_ε1ϖρ^_f12+_ε1a1ρ−_{ε1 a2}ρ=1_{λf1 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρx+_ε1k_a1ρ−_{ε1 a2}ρ_λ1−x_λf1 dx=∑k=1KKk^−1k−1 _ϑ1.

After this step, two lemmas as shown below need to be considered.

Lemma 1.

The closed-form expression of_D21is calculated as

_D21=1−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 1−∑k=1KKk^−1k−1_ϑ2.

Proof.

See in Appendix B. □

Lemma 2.

_D22is computed in closed-form by

_D22=1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2×1−∑k=1KKk^−1k−1 _ϑ21−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 .

Proof.

See in Appendix C. □

From (20)–(23) we find the expression of_Ψ2as below:

_Ψ2=1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2+1−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 1−∑k=1KKk^−1k−1_ϑ2−1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2×1−∑k=1KKk^−1k−1 _ϑ21−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 ×∑k=1KKk^−1k−1 _ϑ1.

From (18) and (24), the outage probability of D2D user_D2can be examined through the formulation

O_PD2−bi=1−∑k=1KKk^−1k−1_λ2θkϖρ_λf2 +_λ2exp−θk_λ2×1−∑k=1KKk^−1k−1 _υ1+1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2+1−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 1−∑k=1KKk^−1k−1_ϑ2−1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2×1−∑k=1KKk^−1k−1 _ϑ21−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 ×∑k=1KKk^−1k−1 _ϑ1.

4. Analysis On Asymptotic Outage Probability

Based on the previous results, an asymptotic analysis for both_D1and_D2will be carried out to evaluate the outage behavior, i.e.,O_PD1−biandO_PD2−bi, respectively. Particularly, the following expressions are provide insight observation for the proposed system in the high SNR regime.

4.1. Asymptotic Outage Probability at D2D User 1

Based on the above analytical results in (12), by using^e−x≈1−xthe asymptotic outage probability of D2D User 1 with is given by

O_PD1−asym=1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ1}k_ε1ϖ_λf1 +_a1−_{ε1 a2λ1}1−_ε1k_a1ρ−_{ε1 a2}ρ_λ1×1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 1−_ε1k_a1ρ−_{ε1 a2}ρ_λ2+1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ1a1}−_{ε1 a2λ1}+_ε1ϖk_λf1 1−_ε1k_a1ρ−_{ε1 a2}ρ_λ1×1−_{λh1 λh1} +_ε1ϖ_λf1 1−_ε1ρ_λh1 ∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 1−_ε1k_a1ρ−_{ε1 a2}ρ_λ2.

To look lower bound, whenρ→∞, the asymptotic outage probability of D2D User 1 with is determined by

O_PD1−floor=1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ1}k_ε1ϖ_λf1 +_a1−_{ε1 a2λ1}×1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 +1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ1a1}−_{ε1 a2λ1}+_ε1ϖk_λf1 ×1−_{λh1 λh1} +_ε1ϖ_λf1 ∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 .

4.2. Asymptotic Outage Probability at D2D User 2

Similar to the derivation ofO_PD1−asym, an asymptotic outage expression forO_PD2−bi . It is noted that the related exact expression is presented in (25), now it can be derived as

O_PD2−asym=1−∑k=1KKk^−1k−1_λ2θkϖρ_λf2 +_λ21−θk_λ2×1−∑k=1KKk^−1k−1_a1−_{ε2 a2λ1a1}−_{ε2 a2λ1}+_ε2ϖk_λf1 1−_ε2k_a1ρ−_{ε2 a2}ρ_λ1+1−∑k=1KKk^−1k−1_{a2 λ2a2 λ2}+_ε2ϖk_λf2 1−_ε2k_a2ρ_λ2+1−_{λh2 λh2} +_ε1ϖ_λf2 1−_ε1ρ_λh2 1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 1−_ε1k_a1ρ−_{ε1 a2}ρ_λ2−1−∑k=1KKk^−1k−1_{a2 λ2a2 λ2}+_ε2ϖk_λf2 1−_ε2k_a2ρ_λ2×1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 1−_ε1k_a1ρ−_{ε1 a2}ρ_λ21−_{λh2 λh2} +_ε1ϖ_λf2 1−_ε1ρ_λh2 ×∑k=1KKk^−1k−1_a1−_{ε1 a2λ1a1}−_{ε1 a2λ1}+_ε1ϖk_λf1 1−_ε1k_a1ρ−_{ε1 a2}ρ_λ1,

and with regard to lower bound, it can be obtained lower bound of user_D2as

O_PD2−floor=1−∑k=1KKk^−1k−1_λ2θkϖρ_λf2 +_λ21−∑k=1KKk^−1k−1_a1−_{ε2 a2λ1a1}−_{ε2 a2λ1}+_ε2ϖk_λf1 +1−∑k=1KKk^−1k−1_{a2 λ2a2 λ2}+_ε2ϖk_λf2 +1−_{λh2 λh2} +_ε1ϖ_λf2 1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 −1−∑k=1KKk^−1k−1_{a2 λ2a2 λ2}+_ε2ϖk_λf2 ×1−∑k=1KKk^−1k−1_a1−_{ε1 a2λ2a1}−_{ε1 a2λ2}+_ε1ϖk_λf2 1−_{λh2 λh2} +_ε1ϖ_λf2 ×∑k=1KKk^−1k−1_a1−_{ε1 a2λ1a1}−_{ε1 a2λ1}+_ε1ϖk_λf1 .

Remark 1.

These approximate performances provide easy way to evaluate system performance rather than complex manner of derived expressions in term of outage probability. It is expected that these approximate expressions exhibits corresponding curves in simulation and they will match with exact curves achieved by analytical expressions presented in Section 3.

5. Numerical Results

In this section, numerical examples are performed to verify the outage performance of the downlink MISO PDMA network under Rayleigh fading channels with FD scheme. We denote_d1,_d2as distances between the BS and the first D2D user and the second one, repetitively. Such distance is normalized as unit. Moreover, Monte Carlo simulation is run in¹⁰⁶times to compare with analytical results as proved in previous section.

In Figure 2, the outage probability versus transmit SNR at the BSρis presented in different power allocation parameters. We assume the distance between BS and_D1is_d1=0.4, path loss exponent isα=2, channel gain_λ2=_d2−α, while_d2=0.2,_λh1 =_λh2 =1,_λf1 =_λf2 =0.01, the number of antenna at BS isK=1. As a clear observation, the exact analytical results and simulation results are in excellent agreement, and the outage probability will be constant at high-SNR regimes. Moreover, as the transmit SNR increases, the outage probability decreases. Another important observation is that the outage probability for User 2_D2outperforms User 1_D1 . Figure 3 shows outage performance for user_D1. The parameters for this case are_a1=0.7,_λ1=_d1−α,_d1=0.4,α=2,_λ2=_d2−α,_d2=0.2,_λh1 =_λh2 =1,_λf1 =_λf2 =0.01,K=1. It can be seen that lower target rate_R1results in better outage performance. It is intuitively that floor outage values match with analytical curves at highρ. Such observation confirms our analysis on finding lower bound of outage probability. While asymptotic lines also match with exact lines at several points within the range of considered transmit SNR at the BS.

In Figure 4, the outage probabilities are shown as a function of the transmit SNR. Reported from the impact of target rate_R2, there is a decrease in outage probability for such user as change to lower level of_R2. This figure requires several parameters as_a1=0.7,_R1=0.5,_λ1=_d1−α,_d1=0.4,α=2,_λ2=_d2−α,_d2=0.2,_λh1 =_λh2 =1,_λf1 =_λf2 =0.01,K=1 . Similar trends with Figure 3 in terms of approximate and floor value of outage for_D2can be observed in this figure.

Figure 5 plots the outage probability versus SNR with the different numbers of transmit antennas at the BS (other parameters as declarations in Figure 5 as_a1=0.7,_R1=0.5,_λ1=_d1−α,_d1=0.4.α=2,_λ2=_d2−α,_d2=0.2,_λh1 =_λh2 =1,_λf1 =_λf2 =0.01). More antennas at the BS indicates better outage probability in such PDMA.K=3case provides the best performance and an important observation in this study.

Due to self-interference related to FD scheme at users, it need be considered performance of user_D1in four cases of_λf1 as observation in Figure 6. In this case, we set_a1=0.7,_R1=0.5,_λ1=_d1−α,_d1=0.4,α=2,_λ2=_d2−α,_d2=0.2,_λh1 =_λh2 =1,K=1 for both Figure 6 and Figure 7. It is noted that_λf2=0.01,_λf1 =0.01 for Figure 6, Figure 7, respectively. Obviously, strong self-interference makes outage performance worse. The main reason is that achievable SNR will be smaller as existence of self-interference and hence outage event easily happens. Similarly, performance of_D2 in Figure 7 is changed as varying_λf2 .

6. Conclusions

This paper analytically investigated the impact of number of transmit antennas at the BS on outage performance of each D2D user in the MISO PDMA. Closed-form analytical expressions for the outage probability were obtained. Our theoretical analysis indicated that the outage performance gap between two D2D users exists due to different power allocation factors given. The best performance can be raised at a higher number of transmit antennas at the BS. Furthermore, we observed that target rates have only a small impact on outage performance.

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Author Contributions

D.T.D. provided idea, wrote paper and verified expressions, M.S.V.N. and T.A.H. derived mathematical problems and performed experiments; B.M.L. contributed to prepare manuscript and delivered valuable comments.

Funding

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by Korea government (MSIT) (Grant No.: NRF-2017R1D1A1B03028350).

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A.

Proof of Proposition 1.

The terms_B1and_B2 can be calculated with the help of (2) and (3). The former becomes

_B1=Pr_{γSD1,k*PDMAε1},_{γSD1←2,k*PDMAε1}=1−Pr_γSD1,k*PDMA≥_ε1︸_B111−Pr_{γSD1←2,k*PDMA}≥_ε1︸_B12.

Therefore,_B11can be expressed as

_B11=1−Pr_γSD1,kPDMA≥_ε1=1−Pr^_g1,k2≥_ε1ϖρ^_f12+_ε1a1ρ−_{ε1 a2}ρ=1−1_{λf1 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρx+_ε1k_a1ρ−_{ε1 a2}ρ_λ1−x_λf1 dx=1−∑k=1KKk^−1k−1_ϑ1,

where_ϑ1=_a1ρ−_{ε1 a2}ρ_λ1k_ε1ϖρ_λf1 +_a1ρ−_{ε1 a2}ρ_λ1exp−_ε1k_a1ρ−_{ε1 a2}ρ_λ1.

It is noted that_λ1is average channel gain of_g1,k, then_λ2is average channel gain of_g2,k,_λf1,_λf2are average channel gain of_f1,_f2, respectively;_λh1,_λh2are average channel gains of_h1,_h2, respectively.

Similarly,_B12can be expressed by

_B12=1−Pr_{γSD1←2,kPDMA}≥_ε1=1−Pr^_g2,k2≥_ε1ϖρ^_f22+_ε1a1ρ−_{ε1 a2}ρ=1−1_{λf2 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρy+_ε1k_a1ρ−_{ε1 a2}ρ_λ2−y_λf2 dy=1−∑k=1KKk^−1k−1 _ϑ2,

where_ϑ2=_a1ρ−_{ε1 a2}ρ_λ2a1ρ−_{ε1 a2}ρ_λ2+_ε1ϖρk_λf2 exp−_ε1k_a1ρ−_{ε1 a2}ρ_λ2.

From (A2) and (A3) we find the expression_B1as

_B1=1−∑k=1KKk^−1k−1_ϑ11−∑k=1KKk^−1k−1 _ϑ2.

In similar way,_B2can be computed as

_B2=Prmax_γSD1,k*PDMA,_χD1PDMAε1,_{γSD1←2,k*PDMA}>_ε1=Pr_{γSD1,k*PDMAε1}Pr_χD1PDMAε1Pr_{γSD1←2,k*PDMA}>_ε1=1−Pr_γSD1,k*PDMA≥_ε1︸_B211−Pr_χD1PDMA≥_ε1︸_B22×Pr_{γSD1←2,k*PDMA}>_ε1︸_B23.

In this step,_B21is formulated as

_B21=Pr_γSD1,k*PDMA≥_ε1=Pr^_g1,k2≥_ε1ϖρ^_f12+_ε1a1ρ−_{ε1 a2}ρ=1_{λf1 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρx+_ε1k_a1ρ−_{ε1 a2}ρ_λ1−x_λf1 dx=∑k=1KKk^−1k−1 _ϑ1.

Then,_B22is calculated as

_B22=Pr_χD1PDMA≥_ε1=Pr^_h12≥_ε1ϖρ^_f12+_ε1ρ=_∫0∞exp−_ε1ϖρx+_ε1ρ_λh1 1_λf1 exp−x_λf1 dx=1_λf1 exp−_ε1ρ_{λh1 ∫0∞}exp−_ε1ϖρρ_λh1 +1_λf1 xdx=ρ_λh1 ρ_λh1 +_ε1ϖρ_λf1 exp−_ε1ρ_λh1 .

Similarly,_B23is calculated as

_B23=Pr_{γSD1←2,k*PDMA}>_ε1=Pr^_g2,k2>_ε1ϖρ^_f22+_ε1a1ρ−_{ε1 a2}ρ=1_{λf2 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρy+_ε1k_a1ρ−_{ε1 a2}ρ_λ2−y_λf2 dy=∑k=1KKk^−1k−1 _ϑ2.

From (A6), (A7) and (A8) we find the expression_B2to be

_B2=1−∑k=1KKk^−1k−1 _ϑ1×1−ρ_λh1 ρ_λh1 +_ε1ϖρ_λf1 exp−_ε1ρ_λh1 ∑k=1KKk^−1k−1 _ϑ2.

This completes the proof of Proposition 1. □

Appendix B.

Proof of Lemma 1.

_D21=1−Pr_χD2PDMA≥_ε1︸_κ11−Pr_{γSD1←2,kPDMA}≥_ε1︸_κ2.

Here,_κ1can be calculated as

_κ1=1−Pr_χD2PDMA≥_ε1=1−Pr^_h22≥_ε1ϖρ^_f22+_ε1ρ=1−1_{λf2 ∫0∞}exp−_ε1ϖρy+_ε1ρ_λh2 −y_λf2 dy=1−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 .

Similarly,_κ2can be calculated as

_κ2=1−Pr_{γSD1←2,kPDMA}≥_ε1=1−Pr^_g2,k2≥_ε1ϖρ^_f22+_ε1a1ρ−_{ε1 a2}ρ=1−1_{λf2 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρy+_ε1k_a1ρ−_{ε1 a2}ρ_λ2−y_λf2 dy=1−∑k=1KKk^−1k−1_ϑ2.

This completes the proof of Lemma 1. □

Appendix C.

Proof of Lemma 2.

By definition, we have following outage probability:

_D22=Pr_{γSD2,k*PDMAε2}∪max_{γSD1←2,k*PDMA},_χD2PDMAε1=1−Pr_γSD2,kPDMA≥_ε2︸_Ξ11−Pr_{γSD1←2,kPDMA}≥_ε1︸_Ξ2×1−Pr_χD2PDMA≥_ε1︸_Ξ3.

Firstly,_Ξ1can be calculated as

_Ξ1=1−Pr_γSD2,kPDMA≥_ε2=1−Pr^_g2,k2≥_ε2ϖρ^_f22+_ε2a2ρ=1−1_{λf2 ∫0∞}∑k=1KKk^−1k−1exp−_ε2ϖρy+_ε2k_a2ρ_λ2−y_λf2 dy=1−∑k=1KKk^−1k−1_a2ρ_λ2a2ρ_λ2+_ε2ϖρk_λf2 exp−_ε2k_a2ρ_λ2.

Similarly,_Ξ2can be calculated as

_Ξ2=1−Pr_{γSD1←2,kPDMA}≥_ε1=1−Pr^_g2,k2≥_ε1ϖρ^_f22+_ε1a1ρ−_{ε1 a2}ρ=1−1_{λf2 ∫0∞}∑k=1KKk^−1k−1exp−_ε1ϖρy+_ε1k_a1ρ−_{ε1 a2}ρ_λ2−y_λf2 dy=1−∑k=1KKk^−1k−1 _ϑ2.

It is noted that_Ξ3can be formulated as

_Ξ3=1−Pr_χD2PDMA≥_ε1=1−Pr^_h22≥_ε1ϖρ^_f22+_ε1ρ=1−1_{λf2 ∫0∞}exp−_ε1ϖρy+_ε1ρ_λh2 −y_λf2 dy=1−ρ_λh2 ρ_λh2 +_ε1ϖρ_λf2 exp−_ε1ρ_λh2 .

This completes the proof of Lemma 2. □

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