Multiple antenna aided NOMA in UAV networks a stochastic geometr....pdf

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I Introduction

I-A Motivations and Related Works

I-B Contributions

I-C Organization and Notations

II MIMO-NOMA Assisted UAV Network Model

II-A System Description

II-B Channel Model

II-C Downlink transmission

III Performance Evaluations

III-A New Channel Statistics

III-B Outage Probabilities

III-C Ergodic Rates

IV Numerical Studies

IV-A Outage Probabilities

IV-B Ergodic rates

V Conclusions

References

1 Multiple Antenna Aided NOMA in UAV Networks: A Stochastic Geometry Approach Tianwei Hou, Student Member, IEEE, Yuanwei Liu, Member, IEEE, Zhengyu Song, Xin Sun, and Yue Chen, Senior Member, IEEE, Abstract This article investigates the multiple-input multiple-output (MIMO) non-orthogonal multiple access (NOMA) assisted unmanned aerial vehicles (UAVs) networks. By utilizing a stochastic geometry model, a new 3-Dimension UAV framework for providing wireless service to randomly roaming NOMA users has been proposed. In an effort to evaluate the performance of the proposed framework, we derive analytical expressions for the outage probability and the ergodic rate of MIMO-NOMA enhanced UAV networks. We examine tractable upper bounds for the whole proposed framework, with deriving asymptotic results for scenarios that transmit power of interference sources being proportional or being ﬁxed to the UAV. For obtaining more insights for the proposed framework, we investigate the diversity order and high signal-to-noise (SNR) slope of MIMO-NOMA assisted UAV networks. Our results conﬁrm that: i) The outage probability of NOMA enhanced UAV networks is affected to a large extent by the targeted transmission rates and power allocation factors of NOMA users; and ii) For the case that the interference power is proportional to the UAV power, there are error ﬂoors for the outage probabilities. Index Terms MIMO, non-orthogonal multiple access, signal alignment, stochastic geometry, unmanned aerial vehicles. This work is supported by the Fundamental Research Funds for the Central Universities under Grant 2016RC055. This paper was presented at the IEEE Global Communication Conference, Abu Dhabi, United Arab Emirates, Dec. 2018 [1]. T. Hou, Z. Song and X. Sun are with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China (email: 16111019@bjtu.edu.cn, songzy@bjtu.edu.cn, xsun@bjtu.edu.cn). Y. Liu and Yue Chen are with School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.K. (e-mail: yuanwei.liu@qmul.ac.uk, yue.chen@qmul.ac.uk). 8 1 0 2 y a M 4 1 ] T I . s c [ 1 v 5 8 9 4 0 . 5 0 8 1 : v i X r a

2 I. INTRODUCTION In the ﬁfth generation (5G) communication systems and internet of things (IoT) networks, massive connectivity is required to support large number of devices in various scenarios with limited spectrum resources [2], [3]. Unmanned aerial vehicles (UAVs) communication is an effective approach to provide connectivity during temporary events and after disasters. Some initial research contributions in the ﬁeld of UAVs communication have been made by researchers in [4]–[8]. UAVs can provide connectivity to multiple users as wireless relays for improving wireless coverage. On the other hand, ground users can have access to UAVs, which can act as aerial base stations, for reliable downlink and uplink communications [4]–[6]. Additionally, UAV networks are capable of enhancing spectrum efﬁciency by having line-of-sight (LoS) connections towards the users, which provide higher received power for the users [7], [8]. Non-orthogonal multiple access (NOMA) has been considered as a promising technique in 5G mobile networks because of its superior spectrum efﬁciency [9], [10]. The key idea is that multiple users are served within the same frequency, time and code block [11], [12], which relies on the employment of superposition coding (SC) and successive interference cancellation (SIC) at the transmitter and receiver, respectively [13]. Moreover, SIC technique at receivers allows that users who have better channel conditions to remove the intra-channel interference. Ding et al. [14] estimated the performance of NOMA with ﬁxed power allocation (F-NOMA) and cognitive radio inspired NOMA, which demonstrated that only the user with higher channel gain inﬂuences the system performance. Timotheou and Krikidis optimized the user-power allocation problem to improve user fairness of a NOMA system [15]. To further exploit NOMA networks, the outage performance and capacity in the downlink and uplink transmission scenario with dynamic power allocation factors was estimated by Yang et al. [16], which can guarantee the quality of service in dynamic power allocation NOMA (D-NOMA). A. Motivations and Related Works 1) Studies on MIMO-NOMA systems: Current multiple-input multiple-output (MIMO) NOMA applications can be primarily classiﬁed into a pair of categories, namely, beamformer based structure and cluster based structure. In beamformer based structure, one centralized beamformer serves only one user in a beam [17]. Choi proposed a coordinated multi-point transmission scheme in [18], where two base stations (BS) serve paired NOMA users simultaneously by

3 beamformer based structure. The multiple antenna scenario of beamformer based structure was proposed by Shin et al. in [19], where a joint centralized and coordinated beamforming de- sign was developed for suppressing the inter-cell interference. Some related works on cluster based MIMO-NOMA have been investigated in [20]–[22], [22], [23]. More speciﬁcally, Ding et al. [20] proposed a MIMO-NOMA model with transmit power control and detection scheme. By adopting this design, the MIMO-NOMA model can be separated to multiple independent SISO- NOMA arrangements. Then, outage probabilities and the sum rate gap of multiple SISO-NOMA arrangements were estimated, which the model assumes that global channel state information (CSI) is unknown at the BS. The sum rate and multi-user capacity were investigated by Zeng et al. [24], which show that multi-user MIMO-NOMA is not a preferable solution of NOMA due to high computational complexity. In order to establish a more general framework of MIMO- NOMA, and circumvent the restrictive assumption of receiver antennas, the signal alignment technique was proposed for both downlink and uplink transmission scenarios in [21]. 2) Studies on NOMA in stochastic geometry systems: Stochastic geometry is an effective mathematical tool for capturing the topological randomness of networks. As such, stochastic geometry tools were invoked to model the impact of the locations for NOMA users [25]. Some research contributions with utilizing stochastic geometry approaches have been studied in [22], [23]. More particularly, Liu et al. [22] proposed an innovative model of cooperative NOMA with simultaneous wireless information and power transfer (SWIPT). In this model, the wireless power transfer technique was employed at users, where near users acted as energy harvesting relays for supporting far users. Ding et al. [25] evaluated the performance of NOMA with randomly deployed users. The analytical results show that it is more preferable to group users whose channel gains are more distinctive to improve the diversity order in NOMA system. With the goal of enhancing the physical layer security of NOMA networks, Liu et al. [23] proposed a NOMA assisted physical layer security framework in large-scale networks. The secrecy performance of both single antenna and multiple antenna aided BS scenarios have been investigated. 3) Studies on UAV: In UAV aided wireless networks, the probability that each device has a LoS link is dependent on the environment, location of the device, carrier frequency and the elevation angle [4], [8], [26]. Sharma and Kim estimated the outage performance of single antenna assisted NOMA in UAV networks [27], where UAV-ground channels are characterized by LoS transmission. Recently, the general form of both LoS and NLoS transmission scenarios,

4 namely, Nakagami-m fading channels, have been proposed in the literature. Hou et al. [28] estimated the outage performance of F-NOMA downlink transmission scenario in both LoS and NLoS scenarios. While the aforementioned research contributions have laid a solid foundation with providing a good understanding of single input single output (SISO) NOMA terrestrial networks, how MIMO-NOMA technique is capable of assisting UAV networks is still unknown. To the best of our knowledge, there has been no existing work intelligently investigating the effect of the network performance of MIMO-NOMA assisted UAV networks, which motivates us to develop this treatise. B. Contributions The novel structure design in this work–by introducing the MIMO-NOMA assisted UAV networks–can be a new highly rewarding candidate, which will contribute the following key advantages: • We propose a general MIMO-NOMA aided UAV framework with interference, where stochastic geometry approaches are invoked to model the locations of users and interference sources. Utilizing this framework, LoS and NLoS links are considered to illustrate the general case of NOMA assisted UAV networks. • We derive closed-form expressions for outage probability of paired NOMA users in the proposed framework. We provide tractable analytical upper bounds for both LoS and NLoS scenarios. Diversity orders are obtained for the paired NOMA users based on the developed outage probability. The obtained results conﬁrm that for the case that interference power is proportional to the transmit power, diversity orders of paired NOMA users are zero. • We derive exact analytical expressions for ergodic rate in both LoS and NLoS scenarios. We provide tractable analytical lower bounds for the general case. We also provide closed-form expressions for the special case when the path loss is three. We obtain high SNR slopes for the paired NOMA users based on the developed ergodic rate. The obtained results conﬁrm that the high SNR slopes of far users are zero in both LoS and NLoS scenarios. • We demonstrate that 1) the outage performance and ergodic rate can be enhanced by the LoS propagation; 2) the ergodic rates of near users have the same rate ceiling in both LoS and NLoS scenarios; 3) diversity orders and high SNR slopes of paired NOMA users are

5 not affected by the LoS transmission of the proposed framework. C. Organization and Notations The rest of the paper is organized as follows. In Section II, a model of UAV-aided transmission scenario is investigated in wireless networks, where NOMA users are uniformly allocated on the ground. Then precoding and detection strategies have proposed for UAV networks. Analytical results are presented in Section III to show the performance of UAV-aided MIMO-NOMA networks. Our numerical results are demonstrated in Section IV for verifying our analysis, which is followed by the conclusion in Section V. Table I lists all notations used in this article. TABLE I: TABLE OF NOTATIONS k and α2 α2 k Pu PI tk and tk α Rm and Rd Rk and Rk m Power allocation factors Transmit power of the UAV Transmit power of interference sources Detection vectors Path loss exponent The radius of small disc and large disc The target rate of the k-th user and of the k-th user Small-scale fading parameter II. MIMO-NOMA ASSISTED UAV NETWORK MODEL Consider a MIMO-NOMA downlink communication scenario in which a UAV equipped with K antennas is communicating with multiple users equipped with N antennas each. Fig. 1 illustrates the wireless communication model with a single UAV, which is supported by MIMO beamforming, namely the cluster-based MIMO-NOMA. Multiple users are grouped into one cluster, and the UAV serves multiple clusters simultaneously in the cluster-based MIMO-NOMA, which can perfectly improve the system performance [17], [29]. A. System Description The UAV cell coverage is a disc area, denoted by D, which has a coverage radius of Rd. It is assumed that the users are uniformly distributed according to HPPPs distribution, which is denoted by Ψ and associated with the density λ, within large disc D and small disc with radius Rd and Rm, respectively. For simplicity, we only focus our attention on investigating a

6 Fig. 1: Illustration of a typical UAV cellular network supported by beamforming. typical user pairing in this treatise, where two users are grouped to deploy NOMA transmission protocol. The downlink users also detect signals sent by interferers which are distributed in R according to HPPPs distribution ΨI with density λI [30]. It is assumed that the interference sources are equipped with one antenna each and use identical transmission power, which is denoted by PI. B. Channel Model Consider the use of a composite channel model with two parts, large-scale fading and small- scale fading. gk,kn denotes the channel coefﬁcient between the k-th antenna of the UAV and the n-th antenna of the k-th user, and the channel gain gk,kn is modelled as gk,kn = βk,knhk,kn, (1) where βk,kn and hk,kn denotes the large-scale fading and small-scale fading, respectively. It is assumed that βk,kn and hk,kn with k = 1, ...K, n = 1, ..., N, are independent and identically distributed (i.i.d.). In this paper, large-scale fading represents the path loss and shadowing between the UAV and users. Generally speaking, the large-scale fading between the k-th user zxhOriginRmRdBeamformingdirections

and the UAV changes slightly for different antenna pairs, which can be expressed as βk = βk,kn,∀k = 1··· K, n = 1··· N. For simplicity, the small-scale fading matrix from the UAV to user k is deﬁned as  Hk =  , hk,11 ... ··· hk,K1 ... ... hk,1N ··· hk,KN where Hk is a N × K matrix whose elements represent Nakagami-m fading channel gains. The density function of the elements is f (x) = mmxm−1 Γ(m) e−mx, where m denotes the fading parameter. In this paper, the UAV can be projected to the coverage disc by projection theorem. Thus, the distance between the UAV to user k can be written as where h denotes the height of the UAV, and rk is the horizontal distance between the k-th user and projective point of the UAV. Therefore, the large-scale fading can be expressed as 7 (2) (3) (4) (5) dk = h2 + r2 k, βk = d−α k , where α denotes the path loss exponent. For simplicity, we still use βk to represent the large- scale fading between the UAV and the k-th user in Section II. Thus, the received power for the k-th user from the UAV is given by Pk = Puβk|Hk|2, (6) where Pu denotes the transmit power of the UAV. Besides, it is assumed that the CSI of users is perfectly known at the UAV. The proposed MIMO-NOMA assisted UAV network model for downlink transmission is described in the following subsection. C. Downlink transmission In this subsection, we estimate the downlink quality of MIMO-NOMA assisted single UAV networks, where two users, i.e., the k-th user and the k-th user, are grouped to perform NOMA.

We consider a more practicable method that the number of antenna equipped on each user is assumed to greater than 0.5K, i.e., N ≥ 0.5K. In the beginning, it is assumed that the maximum clusters is K. Therefore, the UAV sends the following K × 1 information-bearing vector s to users as follows: s = , (7) where sk is the signal intended for the k-th user, αk is the power allocation coefﬁcient, and α2 k + α2 k = 1. The co-channel interference Ik can be further expressed as follows: 8 (8) (9)  α1s1 + α1s1 ... αksk + αksk ... αKsK + αKsK  j∈ΨI ∆= Ik PId−α j,k 1N×1, where 1k×n denotes an k×n all one matrix, dj,k denotes the distance from the k-th user to the j- th interference source. ΨI and PI are the HPPPs distribution and the single antenna transmission power of interference sources, respectively. The interference model omits the small scale fading, since the effect of path loss is the dominant part for the long distance interferences. Similar to [21], [31], the detection vectors t can be designed as follows:  = Dkxk,  tk tk where xk denotes a (2N − K) × 1 vector, which is normalized to 2, i.e., |xk|2 = 2. Dk denotes a 2N × (2N − K) matrix containing the (2N − K) right singular vectors of corresponding to its zero singular values. k −HH HH k ∆= HH Deﬁne bk ··· bK k tk as the effective channel vector shared by users in the cluster, B ∆= , and (·)k,k denotes the k-th element on the main diagonal of the matrix. Thus, with the design of precoding matrix P and detection vector t in [21], [31], the signal-to- b1 H

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