# Performance Analysis for Downlink MIMO-NOMA in Millimeter Wave Cellular Network with D2D Communications

Authors: Jianguo Li; Xiangming Li; Aihua Wang; Neng Ye
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

Enabling nonorthogonal multiple access (NOMA) in device-to-device (D2D) communications under the millimeter wave (mmWave) multiple-input multiple-output (MIMO) cellular network is of critical importance for 5G wireless systems to support low latency, high reliability, and high throughput radio access. In this paper, the closed-form expressions for the outage probability and the ergodic capacity in downlink MIMO-NOMA mmWave cellular network with D2D communications are considered, which indicates that NOMA outperforms TDMA. The influencing factors of performance, such as transmission power and antenna number, are also analyzed. It is found that higher transmission power and more antennas in the base station can decrease the outage probability and enhance the ergodic capacity of NOMA.


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## Body

## 1. Introduction

With the rapid growth of a variety of 5th-generation wireless systems commercial requirements, like 8k video, cloud VR, unmanned driving, smart city, etc., the demands on high data rates, low latency, and high reliability of wireless communication are increasing rapidly. As the core technologies of 5G, the wide bandwidth and large scale antenna arrays can be used by mmWave MIMO to provide high data rates. Meanwhile, NOMA can increase the spectral efficiency of transmission so that more data is transmitted over the same bandwidth spectrum. At the same time, due to the short communication distance, D2D communication can obtain lower transmission power and lower delay to ensure the battery availability and system reliability. Therefore, combined with MIMO-NOMA and mmWave in D2D communications, the throughput of the entire cell system can be improved.D2D communication was noticed as a viable candidate for some certain applications such as proximity services, content sharing, multiparty games, and coverage discovered with business purposes in the 3rd-Generation Partnership Project (3GPP) Long-term Evolution (LTE) 12th and 13th editions [1, 2]. By sharing the same resource blocks with downlink cellular users, D2D communication is a useful technology which can enhance the spectrum efficiency and system capacity successfully. Due to the short communication distance, D2D pairs can communicate with each other directly without being relayed by the base station which can reduce transmission power, increase transmission reliability, and extend system range [3, 4]. Since the cellular user and the D2D user coexist, if the D2D user and the cellular user are not well coordinated, it will not bring any benefits and may affect the communication of the normal cellular user. According to the literature [5], based on the overlapping coalition formation game theory, the authors have proposed a method to conduct joint interference management and resource allocation in the D2D communications. In addition, a new interference management strategy has been discussed to enhance the overall sum rate of cellular networks and D2D pairs which has combined the conventional mechanism and δD-interference limited area control scheme [6]. Furthermore, a graph theoretic approach for transmission-order optimization scheme in bidirectional D2D communications underlaying cellular TDD networks has been introduced in the [7]. Meanwhile, the authors in [8] have considered a continuous beamforming vector design for all cellular users and the D2D pairs association vector search algorithm to maximize the capacity of the cellular users and D2D pairs. In order to reduce the interference between the cellular users and D2D pairs, a combining call admission control and power control scheme under guaranteeing QoS of every users in the cellular network has been presented in [9]. Besides, it is significant that the NOMA and MU-MIMO can also improve the overall system throughput in the cellular network with underlaid D2D communications, which have been proposed in [10].As the core technology of 5G, NOMA can serve multiple users in the same resource block, such as a time slot, frequency channel, or spreading code. Compared to the orthogonal multiple access, NOMA can provide a set of visible benefits such as greater spectrum efficiency and higher system throughout [11–15]. There are two available NOMA technology categories, namely, power-domain and code-domain NOMA, widely used in the cellular network. This paper focuses on the power-domain NOMA which divides the users in power-domain. On the transmitter sides, signals from multiple users are transmitted at the same resource blocks. On the receiver sides, the multiuser detection algorithms, such as successive interference cancellation, have been utilized to detect the signal, regarding the weak signal as the interference, and decode the strong signal while cancelling the strong signal to detect the weak signal [16, 17]. Aiming at enhancing system capacity and user fairness, the authors have presented a novel resource and power allocation technique which can provide a flexible balancing between capacity and fairness maximization in [18]. Meanwhile, the optimum received power levels of uplink NOMA signal have been proposed in [19]. Furthermore, comparing with OMA in massive connectivity, NOMA can supply more users than OMA because of the limited number of supported users by the amount of the available resources [20–22]. In addition, the analytical outage probability expression for each user has been derived in the downlink cooperative NOMA network over Nakagami-m fading channels in [23]. Since the perfect channel state information at the transmitter side is nearly impractical for many communication scenarios, the literature [24] presented a practical downlink NOMA system over the Nakagami-m fading channels with statistical CSI associated with each other. At the same time, the authors attempted to combine the NOMA and MIMO to accomplish high spectral efficiency [25, 26]. Additionally, the outage probability of the massive MIMO-NOMA system has been derived with perfect user-ordering and limited feedback [27]. Besides, the scheme discussed in [28] required that the number of antennas at the receiver is larger than that at the transmitter instead of the CSI at the transmitter. In [29], the optimal and low complexity suboptimal power allocation schemes have been proposed to maximize the ergodic capacity of MIMO-NOMA system with statistical channel state information in the transmitter over Rayleigh fading channel. Also, the authors in [30] have proposed a cluster beamforming strategy which can optimize beamforming vectors and power allocation coefficients in MIMO-NOMA system to decrease the total power. In addition, NOMA can improve the achievable rate greatly in the D2D-aided cooperative relaying system in the literature [31].Nowadays, as spectrum resources become increasingly scare, the mmWave band with a wider spectrum becomes a natural choice for large-volume content services [32]. Meanwhile, benefitting from the small wavelength of mmWave, the large scale antenna arrays can be adopted easily. Combining the high gain directional antenna and beamforming technology enabled by massive antennas, the high data rates can be reached within a 200-meter mmWave cellular network which has high path loss in mmWave band [33]. In the literature [34], the authors have proposed that the base station which is equipped with a large antenna array can serve a set of users through the users precoding with massive MIMO. Furthermore, the literature [35] has proved that the orders of magnitude are increased in spectral efficiency with massive MIMO which can offer more multiuser gain. However, as for MIMO in conventional cellular frequency band, the beamforming precoding is totally achieved in the digital domain which can cancel the interference between different beams. In order to realize the precoder in the digital domain, a dedicated RF chain is needed by every antenna. But it is difficult for the mmWave base station with large scale antennas whose energy consumption is a large part of the total energy consumption at mmWave frequencies due to the wide bandwidth [36]. At the same time, the authors in [36] have discussed a successive interference cancellation-based hybrid precoding method with near-optimal performance and low complexity which can maximize the achievable subrate of each subantenna array and avoid the need for singular value decomposition and matrix inversion. Additionally, the literature [37] has proposed an interference-aware beam selection which can avoid serious multiuser interferences to reduce obvious performance loss. Besides, the authors in [38] have proposed a practical design of hybrid precoders and combiners with low-resolution phase shifters in mmWave MIMO systems adopting an iterative algorithm for hybrid precoders and combiners to optimize the spectral efficiency. Moreover, the MIMO capacity has been computed over the Nakagami-m fading channel in [39] and the closed-form expressions for outage probabilities achieved by NOMA users in a multicell downlink mmWave network have been obtained over a Poisson cluster point process. Moreover, a fine-grained performance analysis over Poisson bipolar model of the mmWave D2D communication networks was provided in [40]. The literature [41] has presented an in-depth capacity analysis for the integrated NOMA mmWave-massive-MIMO systems which can achieve significant capacity improvements.Because of the large link attenuation and weak coverage in the MIMO-NOMA cellular network, the D2D communication can be used to enhance service for cell edge users. In this paper, the outage probability and the ergodic capacity are proposed in the downlink MIMO-NOMA cellular network with D2D communications. The remaining part of this paper is organized as follows. In Section2, we introduce a system model of the downlink MIMO-NOMA cellular network with multiple direct D2D pairs underlaying communication. The closed-form expressions of the performance analysis including the outage probability and the ergodic capacity are given in Section 3. Finally, the numerical results in Section 4 validate the theoretical analysis and demonstrate that the system capacity can be improved by the integrated MIMO-NOMA in the mmWave network with D2D communications.


## 2. System Model

The paper considers a downlink NOMA, MU-MIMO mmWave cellular network with multiple direct D2D pairs underlaying communications where the cellular users are randomly distributed. The same mmWave resources are used by the cellular user and D2D pairs with NOMA. The mmWave MIMO-NOMA D2D communication system model is shown in Figure1. The base station is equipped with multiple antennas, which can generate high directional and high gain beams for cellular users. There are Q cellular users with signal antenna and P D2D users with signal antenna in the cellular network, which are denoted as CU1,CU2,…,CUQ and DU1,DU2,…,DUP. The D2D pairs are randomly distributed in the edge of the cellular network and there is no direct link between the base station and the D2D pairs [10].Figure 1
System model of D2D-aided mmWave MIMO-NOMA.


### 2.1. NOMA Signal

In beamn, we assume that u(n,1),u(n,2),...,u(n,K) are scheduled on the same radio resource with NOMA, K≥2, where CU of kth in the beam n is denoted as u(n,k). xn is the transmitted signal by the base station in the beam n, which is the sum of all K user signals.(1)xn=∑i=1Kλun,iPnSun,i,where λu(n,k) is the power ratio of kth user, λu(n,1)≤λu(n,2)≤λu(n,K), and ∑k=1Kλu(n,k)=1. Pn is the total power in the beam n, Su(n,k) is the normalized transmitted signal of kth user in the beam n, and E(Su(n,k)2)=1.


### 2.2. Channel Model

As for the large scale fading, the mmWave link is similar to that used in [33]; the large scale fading L(r), in dB, is modeled as(2)Lr=ρ+10αlog⁡r,where ρ=32.4+20log⁡(fc), fc is the carrier frequency, α is the path loss exponents, and r is the distance from transmitter to receiver.As for the small scale fading, the Nakagami-m fading is considered for each link.hu(n,k),hpu(n,k),hp′p are donated as the link of base station to cellular user, D2D user to cellular user, and D2D user to D2D user whose modular square is normalized Gamma random variable [39]. And when H is normalized Gamma random variable, which is denoted as H~Gamma(ω,ψ), the probability density function f(x) and Cumulative Distribution Function F(x) of H are(3)fx=1ψωΓωxω-1e-x/ψ,(4)Fx=1-∑J=0ω-11j!ψjxje-x/ψ,x≥00,otherwise.


### 2.3. Directional Beamforming

The base station with mmWave band is equipped with multiple antennas which can generate high directional and high gain beams. The actual antenna pattern is modeled as the sectorized antenna model approximately for the sake of mathematical tractability [42]. Generally the maximum power gain is adopted to replace the array gain within the half-power beam width (main lobe gain), and the first minor maximum gain is used to replace the gains of the other DoAs (side lobe gain). According to the literature [39, 40], when the antenna pattern is a planar square, the total array gain from base station to the user is G(n,k) where(5)Gn,k=T,mainlobeτ,sidelobe, where T=L, τ=1/sin2⁡(3π/2L) and L is the number of antennas.


### 2.4. Received Signal

For downlink MIMO-NOMA transmission, the cellular useru(n,K) will receive the sum of signal from base station and the signal from D2D transmitter at the same time. In addition to receiving the D2D transmitted signal, the D2D receiver DUP will also receive the interference signal from the other D2D users. Without loss of generality, we assume there are N beams and M D2D pairs in the cellular network; then the received signal of cellular user yu(n,k) and the received signal of D2D user yDp can be formulated as(6)yun,k=hun,k∑n=1NGn,kxn+∑p=1MPDhpun,kSp+nun,k,(7)yDp=∑p′=1MPDhp′pSp′+nDp,where hu(n,k) is the channel gain for downlink u(n,K), hu(n,1)≥hu(n,2)≥hu(n,K), hpu(n,k) is the channel gain between DUP and u(n,K), and hp′p is the channel gain from DUp′ transmitter to DUp receiver. G(n,k) is the total beamforming array gain from base station to u(n,K). xn is the superimposed signal by the total K u(n,k) in the beam n. PD is the transmitted power of DUP. Sp is the signal transmitted by DUP, and E(Sp2)=1. Meanwhile nu(n,k) and nDp are the i.i.d. white Gaussian noise with zero mean and one variance at cellular user u(n,k) and D2D user DUP, which is denoted as nu(n,k),nDp~CN(0,1).We denote the SINRs ofu(n,k) and DUP in the downlink NOMA-MIMO cellular network as γu(n,k), γDUp. Without loss of generality, the interbeam interference is ignored in this paper. In the meantime, the perfect SIC is used to prevent error propagation in the NOMA users in the paper. Using (1) in (6) and (7), γu(n,k) and γDUp can be formulated as(8)γun,k=λun,kPnhun,kGn,k2Iun,kN+Iun,kD+σn2,(9)γDp=PDhpp2IDpD+σn2,where(10)Iun,kN=∑k′=1k-1λun,k′Pnhun,kGn,k2,(11)Iun,kD=∑p′=1MPDhp′un,k2,(12)IDpD=∑p′=1,p′≠pMPDhp′p2.


## 2.1. NOMA Signal

In beamn, we assume that u(n,1),u(n,2),...,u(n,K) are scheduled on the same radio resource with NOMA, K≥2, where CU of kth in the beam n is denoted as u(n,k). xn is the transmitted signal by the base station in the beam n, which is the sum of all K user signals.(1)xn=∑i=1Kλun,iPnSun,i,where λu(n,k) is the power ratio of kth user, λu(n,1)≤λu(n,2)≤λu(n,K), and ∑k=1Kλu(n,k)=1. Pn is the total power in the beam n, Su(n,k) is the normalized transmitted signal of kth user in the beam n, and E(Su(n,k)2)=1.


## 2.2. Channel Model

As for the large scale fading, the mmWave link is similar to that used in [33]; the large scale fading L(r), in dB, is modeled as(2)Lr=ρ+10αlog⁡r,where ρ=32.4+20log⁡(fc), fc is the carrier frequency, α is the path loss exponents, and r is the distance from transmitter to receiver.As for the small scale fading, the Nakagami-m fading is considered for each link.hu(n,k),hpu(n,k),hp′p are donated as the link of base station to cellular user, D2D user to cellular user, and D2D user to D2D user whose modular square is normalized Gamma random variable [39]. And when H is normalized Gamma random variable, which is denoted as H~Gamma(ω,ψ), the probability density function f(x) and Cumulative Distribution Function F(x) of H are(3)fx=1ψωΓωxω-1e-x/ψ,(4)Fx=1-∑J=0ω-11j!ψjxje-x/ψ,x≥00,otherwise.


## 2.3. Directional Beamforming

The base station with mmWave band is equipped with multiple antennas which can generate high directional and high gain beams. The actual antenna pattern is modeled as the sectorized antenna model approximately for the sake of mathematical tractability [42]. Generally the maximum power gain is adopted to replace the array gain within the half-power beam width (main lobe gain), and the first minor maximum gain is used to replace the gains of the other DoAs (side lobe gain). According to the literature [39, 40], when the antenna pattern is a planar square, the total array gain from base station to the user is G(n,k) where(5)Gn,k=T,mainlobeτ,sidelobe, where T=L, τ=1/sin2⁡(3π/2L) and L is the number of antennas.


## 2.4. Received Signal

For downlink MIMO-NOMA transmission, the cellular useru(n,K) will receive the sum of signal from base station and the signal from D2D transmitter at the same time. In addition to receiving the D2D transmitted signal, the D2D receiver DUP will also receive the interference signal from the other D2D users. Without loss of generality, we assume there are N beams and M D2D pairs in the cellular network; then the received signal of cellular user yu(n,k) and the received signal of D2D user yDp can be formulated as(6)yun,k=hun,k∑n=1NGn,kxn+∑p=1MPDhpun,kSp+nun,k,(7)yDp=∑p′=1MPDhp′pSp′+nDp,where hu(n,k) is the channel gain for downlink u(n,K), hu(n,1)≥hu(n,2)≥hu(n,K), hpu(n,k) is the channel gain between DUP and u(n,K), and hp′p is the channel gain from DUp′ transmitter to DUp receiver. G(n,k) is the total beamforming array gain from base station to u(n,K). xn is the superimposed signal by the total K u(n,k) in the beam n. PD is the transmitted power of DUP. Sp is the signal transmitted by DUP, and E(Sp2)=1. Meanwhile nu(n,k) and nDp are the i.i.d. white Gaussian noise with zero mean and one variance at cellular user u(n,k) and D2D user DUP, which is denoted as nu(n,k),nDp~CN(0,1).We denote the SINRs ofu(n,k) and DUP in the downlink NOMA-MIMO cellular network as γu(n,k), γDUp. Without loss of generality, the interbeam interference is ignored in this paper. In the meantime, the perfect SIC is used to prevent error propagation in the NOMA users in the paper. Using (1) in (6) and (7), γu(n,k) and γDUp can be formulated as(8)γun,k=λun,kPnhun,kGn,k2Iun,kN+Iun,kD+σn2,(9)γDp=PDhpp2IDpD+σn2,where(10)Iun,kN=∑k′=1k-1λun,k′Pnhun,kGn,k2,(11)Iun,kD=∑p′=1MPDhp′un,k2,(12)IDpD=∑p′=1,p′≠pMPDhp′p2.


## 3. Performance Analysis

In this section, we present the performance analysis of D2D-aided mmWave MIMO-NOMA system. Specifically, the closed-form expressions for the performance metrics (i.e., the outage probability and the ergodic capacity) are presented in the following. Without loss of generality, in the beamn, k-th CUs are adopted with NOMA in one beam, k∈{1,2}hu(n,1)≥hu(n,2), and one DU is randomly distributed at the edge of the beam. Three events are considered in this system.Event 1. According to the NOMA successive interference cancellation (SIC) principle, user 1 obtains the information intended for user 2 with γ1→2 and removes it. When decoding the information intended for user 2, user 1 cancels it successfully with γ1.(13)γ1→2=λun,2Pnhun,1Gn,12λun,1Pnhun,1Gn,12+PDhpun,12+σn2,(14)γ1=λun,1Pnhun,1Gn,12PDhpun,12+σn2.Event 2. User 2 decodes the signal with γ2, treating user 1 as interference.(15)γ2=λun,2Pnhun,2Gn,22λun,1Pnhun,2Gn,22+PDhpun,22+σn2.Event 3. The DU receiver only receives signals from the DU transmitter, whose SINR is denoted as γDp:(16)γD=PDhpp2σn2.


### 3.1. Outage Probability

In this section, we study the outage probability ofCU and DU. The outage probability of the user 1, user 2, and DU is given by P1out,P2out,PDout:(17)P1out=Plog⁡1+γ1→2<R2orlog⁡1+γ1<R1=1-Plog⁡1+γ1→2≥R2Plog⁡1+γ1≥R1,(18)P2out=Plog⁡1+γ2<R2,(19)PDout=Plog⁡1+γD<RD, where R1,R2,RD are the target rates of user 1, user 2, and DU.Firstly, we consider the outage probability of user 1P1out; then P(log⁡(1+γ1→2)≤R2) can be rewritten as P(γ1→2≤2R2-1), so we set(20)Fm,a,b,c,d=PaH1bH1+dH2+c≤m.According to (13),(20), we can get(21)Pγ1→2≤2R2-1=Fm,a,b,c,d=Fγ1→2m, where a=λu(n,2)PnG(n,1)2, b=λu(n,1)PnG(n,1)2, d=PD, c=σn2,m=2R2-1, and H1=hu(n,1)2~Gamma(ω,ψ), H2=hpu(n,1)2~Gamma(η,θ); then(22)Fγ1→2m=PaH1bH1+dH2+c≤m=∫0∞Payby+dH2+c≤mfH1ydy=∫0∞Pay-bmy-cmdm≤H2fH1ydy.In order to determine ifay-bmy-cm/dm≤0, we set Φ=cm/a-bm. When Φ≤0, ay-bmy-cm/dm≤0, so P(ay-bmy-cm/dm≤H2)=1; therefore(23)Fγ1→2m=1.WhenΦ>0, which is m<a/b,(24)Fγ1→2m=∫0ΦfH1ydy+∫Φ∞1-FH2ay-bmy-cmdmfH1ydy=∫Φ∞∑j=0η-11j!θja-bmdmy-cdje-a-bm/dmθy-c/dθ1ψωΓωyω-1e-y/ψdy+∫0ΦfH1ydy.We setα=a-bm/dm,β=-c/d, and(25)Fγ1→2m=∫0ΦfH1ydy+1ψωΓω∑j=0η-11j!θj∫Φ∞αy+βje-α/θy+β/θyω-1e-y/ψdy. As for (x+a)k=∑j=0kkjxjak-j, then(26)Fγ1→2m=1ψωΓω∑j=0η-11j!θj∑i=0jjiαiβj-ie-β/θ∫Φ∞yω+i-1e-α/θ+1/ψydy+∫0ΦfH1ydy.We set(27)Ja,n,x=eax∑k=0n-1kk!nkak+1xn-k.Then by substituting (27) into (26), Fγ1→2(m) can be denoted as(28)Fγ1→2m=1ψωΓω∑j=0η-11j!θj∑i=0jjiαiβj-ie-β/θ-J-αθ+1ψ,ω+j-1,Φ+∫0ΦfH1ydy=1ψωΓω∑j=0η-11j!θj∑i=0jjiαiβj-ie-β/θ-J-αθ+1ψ,ω+j-1,Φ+1ψωΓωJ-1ψ,ω-1,Φ-J-1ψ,ω-1,0.Similarly, by adopting different parameters witha,b,c,d, which is shown in Table 1, we can obtain Fγ1→2(m),Fγ1(m),Fγ2(m).Table 1
Parameters of the outage probability.
a b c d Fγ1→2(m) λ u ( n , 1 ) P n G ( n , 1 ) 2 λ u ( n , 2 ) P n G ( n , 2 ) 2 σ n 2 P D Fγ1(m) λ u ( n , 1 ) P n G ( n , 1 ) 2 0 σ n 2 P D Fγ2(m) λ u ( n , 2 ) P n G ( n , 2 ) 2 λ u ( n , 1 ) P n G ( n , 1 ) 2 σ n 2 P DIn addition, whenH3=hpp2~Gamma(ω,ϕ), PDhpp2/σn2~Gamma(ω,PD/σn2ϕ). Hence FγD(m) is denoted as(29)FγDm=1-∑J=0ω-11j!PD/σn2ϕjxje-xσn2/PDϕ.Finally, the outage probability of user 1, user 2, andDU can be evaluated by Fγ1→2(m),Fγ1(m),Fγ2(m),FγD(m), which are formulated as(30)P1out=1-1-Fγ1→22R2-1∗1-Fγ12R1-1,(31)P2out=Fγ22R2-1,(32)PDout=FγD2RD-1.


### 3.2. Ergodic Capacity

The ergodic capacity is the average capacity of channel which can be defined as the instantaneous end-to-end mutual information expectations and denoted as(33)Cerg=Elog21+γ1+Elog21+γ2+Elog21+γD.The ergodic capacity of the system can be obtained by substituting (14), (15), and (16) into (33), which is formulated as(34)Cerg=Elog21+λun,1Pnhun,1Gn,12PDhpun,12+σn2+Elog21+λun,2Pnhun,2Gn,22λun,1Pnhun,2Gn,22+PDhpun,22+σn2+E[log21+PDhpp2σn2=Elog21+λun,1PnGn,12σn2hun,12+PDσn2hpun,12-Elog21+PDσn2hpun,12+Elog21+λun,2PnGn,22σn2+λun,1PnGn,22σn2hun,22+PDσn2hpun,22-Elog21+PDσn2hpun,22+λun,1PnGn,22σn2hun,22+Elog21+PDhpp2σn2.In order to compute (34), we first compute the first item of the formula. As we set before, a=λu(n,1)PnG(n,1)2, d=PD, c=σn2, H1=hu(n,1)2~Gamma(ω,ψ), H2=hpu(n,1)2~Gamma(η,θ), we can get(35)Elog21+λun,1PnGn,12σn2hun,12+PDσn2hpun,12=Elog2(1+acH1+dcH2.According to the literature [43, 44], we can get (36)Eln⁡1+x≈ln⁡1+Ex-Ex2-Ex221+Ex2.Based on (36), we start with E[x] and E[x2], which can be written as (37)EacH1+dcH2=∫0∞∫0∞acx+dcyfxfydxdy=∫0∞acxfxdx+∫0∞dcyfydy=acEx+dcEy=acωψ+dcηθ,(38)EacH1+dcH22=∫0∞∫0∞acx+dcy2fxfydxdy=∫0∞axc2fxdx+∫0∞dyc2fydy+∫0∞∫0∞2adxyc2fxfydxdy=a2c2ω+1ωψ2+d2c2η+1ηθ2+2adc2ωηψθ.By substituting (37) and (38) into (35), we can obtain the first item of formula (34), which can be denoted as(39)Elog21+acH1+dcH2=log2eln⁡1+acωψ+dcηθ-a2/c2ωψ2+d2/c2ηθ221+a/cωψ+d/cηθ2.Similarly, we can set a, b, c, d as different parameters to obtain the other items of the formula (34) and then pull everything together. The asymptotic result for the ergodic capacity of the consider system can be obtained.


## 3.1. Outage Probability

In this section, we study the outage probability ofCU and DU. The outage probability of the user 1, user 2, and DU is given by P1out,P2out,PDout:(17)P1out=Plog⁡1+γ1→2<R2orlog⁡1+γ1<R1=1-Plog⁡1+γ1→2≥R2Plog⁡1+γ1≥R1,(18)P2out=Plog⁡1+γ2<R2,(19)PDout=Plog⁡1+γD<RD, where R1,R2,RD are the target rates of user 1, user 2, and DU.Firstly, we consider the outage probability of user 1P1out; then P(log⁡(1+γ1→2)≤R2) can be rewritten as P(γ1→2≤2R2-1), so we set(20)Fm,a,b,c,d=PaH1bH1+dH2+c≤m.According to (13),(20), we can get(21)Pγ1→2≤2R2-1=Fm,a,b,c,d=Fγ1→2m, where a=λu(n,2)PnG(n,1)2, b=λu(n,1)PnG(n,1)2, d=PD, c=σn2,m=2R2-1, and H1=hu(n,1)2~Gamma(ω,ψ), H2=hpu(n,1)2~Gamma(η,θ); then(22)Fγ1→2m=PaH1bH1+dH2+c≤m=∫0∞Payby+dH2+c≤mfH1ydy=∫0∞Pay-bmy-cmdm≤H2fH1ydy.In order to determine ifay-bmy-cm/dm≤0, we set Φ=cm/a-bm. When Φ≤0, ay-bmy-cm/dm≤0, so P(ay-bmy-cm/dm≤H2)=1; therefore(23)Fγ1→2m=1.WhenΦ>0, which is m<a/b,(24)Fγ1→2m=∫0ΦfH1ydy+∫Φ∞1-FH2ay-bmy-cmdmfH1ydy=∫Φ∞∑j=0η-11j!θja-bmdmy-cdje-a-bm/dmθy-c/dθ1ψωΓωyω-1e-y/ψdy+∫0ΦfH1ydy.We setα=a-bm/dm,β=-c/d, and(25)Fγ1→2m=∫0ΦfH1ydy+1ψωΓω∑j=0η-11j!θj∫Φ∞αy+βje-α/θy+β/θyω-1e-y/ψdy. As for (x+a)k=∑j=0kkjxjak-j, then(26)Fγ1→2m=1ψωΓω∑j=0η-11j!θj∑i=0jjiαiβj-ie-β/θ∫Φ∞yω+i-1e-α/θ+1/ψydy+∫0ΦfH1ydy.We set(27)Ja,n,x=eax∑k=0n-1kk!nkak+1xn-k.Then by substituting (27) into (26), Fγ1→2(m) can be denoted as(28)Fγ1→2m=1ψωΓω∑j=0η-11j!θj∑i=0jjiαiβj-ie-β/θ-J-αθ+1ψ,ω+j-1,Φ+∫0ΦfH1ydy=1ψωΓω∑j=0η-11j!θj∑i=0jjiαiβj-ie-β/θ-J-αθ+1ψ,ω+j-1,Φ+1ψωΓωJ-1ψ,ω-1,Φ-J-1ψ,ω-1,0.Similarly, by adopting different parameters witha,b,c,d, which is shown in Table 1, we can obtain Fγ1→2(m),Fγ1(m),Fγ2(m).Table 1
Parameters of the outage probability.
a b c d Fγ1→2(m) λ u ( n , 1 ) P n G ( n , 1 ) 2 λ u ( n , 2 ) P n G ( n , 2 ) 2 σ n 2 P D Fγ1(m) λ u ( n , 1 ) P n G ( n , 1 ) 2 0 σ n 2 P D Fγ2(m) λ u ( n , 2 ) P n G ( n , 2 ) 2 λ u ( n , 1 ) P n G ( n , 1 ) 2 σ n 2 P DIn addition, whenH3=hpp2~Gamma(ω,ϕ), PDhpp2/σn2~Gamma(ω,PD/σn2ϕ). Hence FγD(m) is denoted as(29)FγDm=1-∑J=0ω-11j!PD/σn2ϕjxje-xσn2/PDϕ.Finally, the outage probability of user 1, user 2, andDU can be evaluated by Fγ1→2(m),Fγ1(m),Fγ2(m),FγD(m), which are formulated as(30)P1out=1-1-Fγ1→22R2-1∗1-Fγ12R1-1,(31)P2out=Fγ22R2-1,(32)PDout=FγD2RD-1.


## 3.2. Ergodic Capacity

The ergodic capacity is the average capacity of channel which can be defined as the instantaneous end-to-end mutual information expectations and denoted as(33)Cerg=Elog21+γ1+Elog21+γ2+Elog21+γD.The ergodic capacity of the system can be obtained by substituting (14), (15), and (16) into (33), which is formulated as(34)Cerg=Elog21+λun,1Pnhun,1Gn,12PDhpun,12+σn2+Elog21+λun,2Pnhun,2Gn,22λun,1Pnhun,2Gn,22+PDhpun,22+σn2+E[log21+PDhpp2σn2=Elog21+λun,1PnGn,12σn2hun,12+PDσn2hpun,12-Elog21+PDσn2hpun,12+Elog21+λun,2PnGn,22σn2+λun,1PnGn,22σn2hun,22+PDσn2hpun,22-Elog21+PDσn2hpun,22+λun,1PnGn,22σn2hun,22+Elog21+PDhpp2σn2.In order to compute (34), we first compute the first item of the formula. As we set before, a=λu(n,1)PnG(n,1)2, d=PD, c=σn2, H1=hu(n,1)2~Gamma(ω,ψ), H2=hpu(n,1)2~Gamma(η,θ), we can get(35)Elog21+λun,1PnGn,12σn2hun,12+PDσn2hpun,12=Elog2(1+acH1+dcH2.According to the literature [43, 44], we can get (36)Eln⁡1+x≈ln⁡1+Ex-Ex2-Ex221+Ex2.Based on (36), we start with E[x] and E[x2], which can be written as (37)EacH1+dcH2=∫0∞∫0∞acx+dcyfxfydxdy=∫0∞acxfxdx+∫0∞dcyfydy=acEx+dcEy=acωψ+dcηθ,(38)EacH1+dcH22=∫0∞∫0∞acx+dcy2fxfydxdy=∫0∞axc2fxdx+∫0∞dyc2fydy+∫0∞∫0∞2adxyc2fxfydxdy=a2c2ω+1ωψ2+d2c2η+1ηθ2+2adc2ωηψθ.By substituting (37) and (38) into (35), we can obtain the first item of formula (34), which can be denoted as(39)Elog21+acH1+dcH2=log2eln⁡1+acωψ+dcηθ-a2/c2ωψ2+d2/c2ηθ221+a/cωψ+d/cηθ2.Similarly, we can set a, b, c, d as different parameters to obtain the other items of the formula (34) and then pull everything together. The asymptotic result for the ergodic capacity of the consider system can be obtained.


## 4. Numerical Results

In this section, the outage probability and the ergodic capacity of MIMO-NOMA mmWave cellular network with D2D communications are investigated. The effects of different parameters on the probability of outage and ergodic capacity are analyzed such as the base station transmission power, the number of base station antennas, the power ratio of NOMA user, and the distance between D2D users. In order to verify the performance of the system, the traditional TDMA theme is adopted as the comparison between the two users of each beam. In particular, the time slot is equally divided by the two users. Hence the capacity of this theme isRTDMA, which is denoted as(40)RTDMA=12log⁡1+γ1+log⁡1+γ2.A simplified cellular network system is discussed for the performance analyzed here. The carrier frequency is 28GHz which is commonly used for wireless broadband service. There are 16 antennas in the base station whose coverage radius is 100m. There is single antenna with D2D user. And the distance between D2D users is 30m. Meanwhile, the transmission power of base station and D2D users is 5 dbm. In addition, there are 8 NOMA users and 4 D2D users in the cellular network. The path loss exponent is set as 3. Furthermore, the small scale fading is denoted asH~Gamma(2,1) which is simplified for the simulation.


### 4.1. Outage Probability

In this section, we consider the outage probability of the NOMA far user and near user.Figure2 depicts the outage probability in the different base station transmission power with R1 = 5 bit/s/Hz and R2 = 3.32 bit/s/Hz. As the base station transmission power increases, it can be seen that the outage probability of the NOMA users decreases with the exponential form. Furthermore, the performance of each user’s outage probability in the NOMA scheme is significantly better than the TDMA, and the closed-form solution obtained is consistent with the Monte Carlo simulation results.Figure 2
Impact of transmission power on outage probability.In Figure3, the impact of antenna number in the base station on the outage probability (R1 = 5.64 bit/s/Hz and R2 = 4 bit/s/Hz) is presented. The simulation results effectively verify that the number of antennas of the base station can decrease the users’ outage probability in the MIMO-NOMA mmWave cellular network, thereby improving the throughput of the system under the limited time-frequency resources. As can be seen from Figure 3, the number of antennas has a greater impact on user 2 than user 1. When the number of antennas is 36, the outage probability of the system is satisfied which can balance the number of RF chains and the system performance. Then the NOMA scheme performs better than the traditional TDMA in the mmWave MIMO cellular network with D2D communications.Figure 3
Impact of antenna number on outage probability.In Figure4, we discuss the influence of the power ratio coefficient in the cellular network between the NOMA users. It can be seen that the outage probability (R1 = 4 bit/s/Hz and R2 = 3 bit/s/Hz) of the two users in NOMA is balanced when the power ratio coefficient is approximately 0.2.Figure 4
Impact of power ratio coefficient on outage probability.In Figure5, the effect of the distance between D2D users is considered. The figure indicates that the outage probability (R1 = 5 bit/s/Hz and R2 = 3.32 bit/s/Hz) of the NOMA users is reduced in the form of an exponent when the distance of D2D user is linear growth. Since the distance between D2D users is increasing, the interference from the D2D transmitter to the NOMA user is weak. Hence, the throughput of the NOMA users is improved, while the outage probability is dropping.Figure 5
Impact of D2D distance on outage probability.


### 4.2. Ergodic Capacity

In this section, the total ergodic capacity is considered in the MIMO-NOMA mmWave cellular network with D2D communications. It can be seen that the numerical results are consistent with the closed-form solution which is better than the traditional TDMA. Meanwhile, the transmission power and the number of base station antennas have a greater impact, and the power ratio coefficient and the distance between D2D users have less effect.In Figure6, the impact of the transmission power of base station on the ergodic capacity is considered in the MIMO-NOMA mmWave cellular network with D2D communications. It is shown that the ergodic capacity is growing linearly with the increase of transmission power. In addition, the ergodic capacity of the system we proposed is higher than the traditional TDMA. Hence, in order to improve the ergodic capacity of the system, we can increase the base station transmission power as much as possible without affecting others.Figure 6
Impact of transmission power on ergodic capacity.As the number of the base station antennas is increasing, it is indicated that the ergodic capacity can be improved better than the traditional TDMA in Figure7. Benefiting from the length of mmWave, more and more antennas can be equipped for the base station. At the same time, we need to balance the improvement in ergodic capacity brought by the increasing of the number of antennas and the power consumption and hardware requirements of the increase in RF chains to determine the final number of antennas.Figure 7
Impact of antenna number on ergodic capacity.In Figure8, the ergodic capacity is affected by the change of the power ratio coefficient of the NOMA users in the MIMO-NOMA mmWave cellular network with D2D communications. It can be seen that the total ergodic capacity changes slowly with the increase of power ratio.Figure 8
Impact of power ratio coefficient on ergodic capacity.In Figure9, since the interference from the D2D users is decreasing, the total ergodic capacity is improved with the increase of the distance between the D2D users in the MIMO-NOMA mmWave cellular network. It can also be seen that the ergodic capacity in the MIMO-NOMA mmWave cellular network is always better than the traditional TDMA.Figure 9
Impact of D2D distance on ergodic capacity.


## 4.1. Outage Probability

In this section, we consider the outage probability of the NOMA far user and near user.Figure2 depicts the outage probability in the different base station transmission power with R1 = 5 bit/s/Hz and R2 = 3.32 bit/s/Hz. As the base station transmission power increases, it can be seen that the outage probability of the NOMA users decreases with the exponential form. Furthermore, the performance of each user’s outage probability in the NOMA scheme is significantly better than the TDMA, and the closed-form solution obtained is consistent with the Monte Carlo simulation results.Figure 2
Impact of transmission power on outage probability.In Figure3, the impact of antenna number in the base station on the outage probability (R1 = 5.64 bit/s/Hz and R2 = 4 bit/s/Hz) is presented. The simulation results effectively verify that the number of antennas of the base station can decrease the users’ outage probability in the MIMO-NOMA mmWave cellular network, thereby improving the throughput of the system under the limited time-frequency resources. As can be seen from Figure 3, the number of antennas has a greater impact on user 2 than user 1. When the number of antennas is 36, the outage probability of the system is satisfied which can balance the number of RF chains and the system performance. Then the NOMA scheme performs better than the traditional TDMA in the mmWave MIMO cellular network with D2D communications.Figure 3
Impact of antenna number on outage probability.In Figure4, we discuss the influence of the power ratio coefficient in the cellular network between the NOMA users. It can be seen that the outage probability (R1 = 4 bit/s/Hz and R2 = 3 bit/s/Hz) of the two users in NOMA is balanced when the power ratio coefficient is approximately 0.2.Figure 4
Impact of power ratio coefficient on outage probability.In Figure5, the effect of the distance between D2D users is considered. The figure indicates that the outage probability (R1 = 5 bit/s/Hz and R2 = 3.32 bit/s/Hz) of the NOMA users is reduced in the form of an exponent when the distance of D2D user is linear growth. Since the distance between D2D users is increasing, the interference from the D2D transmitter to the NOMA user is weak. Hence, the throughput of the NOMA users is improved, while the outage probability is dropping.Figure 5
Impact of D2D distance on outage probability.


## 4.2. Ergodic Capacity

In this section, the total ergodic capacity is considered in the MIMO-NOMA mmWave cellular network with D2D communications. It can be seen that the numerical results are consistent with the closed-form solution which is better than the traditional TDMA. Meanwhile, the transmission power and the number of base station antennas have a greater impact, and the power ratio coefficient and the distance between D2D users have less effect.In Figure6, the impact of the transmission power of base station on the ergodic capacity is considered in the MIMO-NOMA mmWave cellular network with D2D communications. It is shown that the ergodic capacity is growing linearly with the increase of transmission power. In addition, the ergodic capacity of the system we proposed is higher than the traditional TDMA. Hence, in order to improve the ergodic capacity of the system, we can increase the base station transmission power as much as possible without affecting others.Figure 6
Impact of transmission power on ergodic capacity.As the number of the base station antennas is increasing, it is indicated that the ergodic capacity can be improved better than the traditional TDMA in Figure7. Benefiting from the length of mmWave, more and more antennas can be equipped for the base station. At the same time, we need to balance the improvement in ergodic capacity brought by the increasing of the number of antennas and the power consumption and hardware requirements of the increase in RF chains to determine the final number of antennas.Figure 7
Impact of antenna number on ergodic capacity.In Figure8, the ergodic capacity is affected by the change of the power ratio coefficient of the NOMA users in the MIMO-NOMA mmWave cellular network with D2D communications. It can be seen that the total ergodic capacity changes slowly with the increase of power ratio.Figure 8
Impact of power ratio coefficient on ergodic capacity.In Figure9, since the interference from the D2D users is decreasing, the total ergodic capacity is improved with the increase of the distance between the D2D users in the MIMO-NOMA mmWave cellular network. It can also be seen that the ergodic capacity in the MIMO-NOMA mmWave cellular network is always better than the traditional TDMA.Figure 9
Impact of D2D distance on ergodic capacity.


## 5. Conclusion

In this paper, the outage probability and the ergodic capacity of the NOMA in the MIMO-NOMA mmWave cellular network with D2D communications are studied. The closed-form solutions of the outage probability and the ergodic capacity are obtained, which are consistent with the numerical results. Meanwhile, the performance of NOMA is shown to be better than traditional TDMA in the MIMO mmWave cellular network with D2D communications. Furthermore, the higher transmission power of base station and the larger antenna array can also improve system performance.



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*Source: 1914762-2019-06-19.xml*