Outage Performance of Non-Orthogonal Multiple Access Based Unmanned Aerial Vehicles Satellite Networks

Ting Qi, Wei Feng＊, Youzheng Wang

1 Tsinghua Space Center, Tsinghua University, Beijing 100084, China

2 Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University,Beijing 100084, China

I. INTRODUCTION

Unmanned aerial vehicles (UAVs) have recently gained rapid development and wide applications in photography, videography, cargo delivery, communication etc. It is predicted that the number of commercial UAVs will increase by three fold by 2020. To support reliable remote command and control and enable high-rate mission-related information transmission, it is imperative to integrate UAVs into cellular and satellite networks [1]. For UAVs working at areas out of the coverage of the cellular network, accessing to the satellite networks is a reasonable choice [2].

To alleviate the shortage of radio spectrum,satellite systems have been moving toward Ku-band and above, which suffers from rain attenuation. Multi-satellite cooperation or satellite diversity is an efficient way to enhance the service capability. Distributed satellite multiple-input multiple-output (MIMO) communication system has advantages in cost, scalability and fault tolerance compared with conventional single-input single-output system [3]. A MIMO channel model and corresponding transmission technique for distributed dual-satellite was studies in [4]. In [5], the authors investigated the spatial multiplexing gain and receive selection diversity gain of the dual-satellite MIMO channel. The potential gain of satellite diversity techniques and the effects of separation angles between satellites were discussed in [6].

When connecting UAVs to multi-satellite networks, non-orthogonal multiple access(NOMA) can be applied to improve the spectral efficiency [7]. By sharing spectrum resource among users, NOMA can simul-taneously serve more users than traditional orthogonal multiple access (OMA) using the same orthogonal resources. The basic problem of user pairing in NOMA was investigated in [8], which revealed that pairing users with distinctive channel conditions can improve the capacity. Power allocation is crucial to NOMA because it basically influences the performance of successive interference cancellation(SCI) multiuser detection and achievable rate of users due to the existence of mutual interference [9-11]. Besides used in the cellular network, NOMA technique can also be applied to UAV-enabled wireless system [12] and integrated terrestrial-satellite networks [13, 14].

Different from previous work, we consider the NOMA based UAV satellite networks,where UAVs access the multi-satellite networks by NOMA scheme using Ku-band and above.Taking into account the rain attenuation and its spatial correlation, we analyze the outage performance and obtain the analytic outage probability. On this basis, we address the issue of power allocation and formulate the min-max fairness problem to minimize the maximum outage probability of users. A simple iteration algorithm is proposed to obtain the optimal solution. Simulation results have verified the outage analysis and confirmed the convergence of the proposed algorithm. The impact of rain attenuation correlation and the performance superiority of NOMA are demonstrated.

Fig. 1. The diagram of integrating UAVs into the satellite networks.

II. SYSTEM MODEL

As illustrated in figure 1, consider the UAV satellite networks, where UAVs working on the remote area or on the sea integrate into satellite networks for service. The UAVs may be used for maritime patrol, emergency response, communication assist and so on. Depending on their applications and equipment, UAVs access to satellites in different orbits. In this paper, we focus on a typical scenario therein, where two geosynchronous satellites cooperatively serve UAVs and a maritime ship-load mobile terminal (MT) using the Ku-band and above. The MT is equipped with two co-located antennas with a separation angle θ, which are highly directive to the two satellites respectively. To improve the system capacity, NOMA scheme is deployed for each satellite to serve a UAV and the MT simultaneously without additional spectrum resources. The UAV(denoted as user 1) is elaborately selected in the coverage area of each satellite to pair with the antenna of MT (denoted as user 2) to implement NOMA. The UAVs usually fly several hundred meters high and carry an ordinary antenna which has much lower antenna gain than the MT. The two users of satellite i are supposed to have similar elevation angle, denoted by φi.

Since the satellites work on Ku-band and above, the channel is line-of-sight (LOS). Besides path loss, the channel is susceptible to various atmospheric fading and rain induced attenuation is the dominant one among them.The rain attenuation of the link between satellite i and its user k (link i-k for short), denoted by Ai,kin dB, is statistically modeled as a lognormal variable, i.e.,where the statistical parameters μ and

i,kare related to the rainfall rate, the operating frequency, the elevation angle and so on. The channel gain from satellite i to its user k, denoted by gi,k, is model as

where Gtiand Grkare the antenna gain of satellite i and user k, respectively, Li,kis the coefficient due to the free space loss of the path,which is given by

where fiis the carrier frequency and c is the speed of light.

Usually, the flying height of the UAV is very small compared with the GEO satellite so the difference between the path loss of link i-1 and i-2 is negligible. While the antenna gain of the MT is rather larger than the UAV, thus,the channel gain of the MT is much better than the UAV.

Consider the downlink transmission, in which each satellite transmits the superposed signal of the UAV and the MT and the receivers use the SIC technique to extract their own signal. Specifically, since the MT has better channel gain and is allocated less power, the received signal of the UAV has much better signal noise ratio (SNR) at the receivers. Thus,the UAV decodes its own signal first by recognizing the signal of the MT as noise, while the MT can first decode the signal of the UAV and remove the interference. Assume that two satellites use different frequency and there is no interference between the cells of satellite 1 and 2. Thus, the achievable rates of the two users belong to satellite i are

where pi,kis the power satellite i allocates to user k, N0denotes the power spectral density of the additive white Gaussian noise. Particularly, the rate of the MT, denoted by RMT, is the sum rate of the two links, given by

Since rain attenuation introduces slow fading, the appropriate performance metric is the outage probability. Given the target spectral efficiency r0, the outage probability of the UAV served by satellite i and the outage probability of the MT, denoted by P3, are respectively defined as

III. OUTAGE PROBABILITY ANALYSIS

Given the statistic rain attenuation distribution and power allocation, we will derive the outage probability of the UAVs and the MT.

Substituting (1) and (3) into (6), the outage probability of the UAV served by satellite i can be rewritten as

where variable αiwrites

Let the normalized normal variable of Ai,kbe

Leveraging (10), we can derive

whereis one minus the cumulative distribution function of the standard normal random variable.

When computing P3, the spatial correlation of the rain attenuations of the two links 1-2 and 2-2 must be considered, since the location of the two antennas of the MT is near.The correlation of rain attenuation of the two links will augment the outage probability of the MT. Denote ρ as the correlation coefficient between the normalized variable t1,2and t2,2. The correlation coefficient ρ lies in the interval (0,1) and is inversely proportional to the separation angle θ given the location,frequency, elevation angles [5]. Given ρ≠1(which is the case in practice), the joint probability density function (PDF) of random variable t1,2and t2,2is written as

f( t2,2|t1,2)f( t1,2) and the channel model (1),the outage probability of the MT writes

where f( t2,2|t1,2) is the conditional PDF of t2,2given t1,2and is a normal distribution represented by N(ρ t1,2,1 ? ρ2),f( t1,2) is the standard normal distribution. After some algebraic operations, the lower bounds u1and u2of the integral (13) are obtained

where variables β1and β2are given as

Lettingand changing the integral variable t2,2as t in (13), we obtain the analytical form for outage probability of the MT, written as

IV. POWER ALLOCATION ALGORITHM

In this section, we investigate the power allocation problem for the NOMA transmission,and then propose an efficient iteration algorithm to solve it.

The satellites allocate the limited power according to the statistic channel state information (CSI). Considering the fairness issue in NOMA, we formulate the min-max fairness problem, defined as

where P={pi,k}, constraints (17b) indicate the total power constraint of each satellite.

Problem (17) is not convex and is hard to solve directly using standard convex optimization tools. In [10], the problem of min-max outage probability with Rayleigh fading channel was proved to be quasi-convex and was solved by the bisection method, where in each iteration, it solves a power allocation solution for a specific outage probability and decide whether it is feasible. However, this method is infeasible for this problem since it is difficult to solve the solution inversely given the outage probability due to the complicated form of(16). In this section, we first analyze the characteristics of problem (17) and then propose an efficient algorithm to obtain the optimal solution. The following proposition sheds light on the optimal condition of the problem.

Proposition 1: At optimality, the constraints(17b) are tight and the optimal solution of problem (17) can be categorized into the two cases:

(a) UAV of satellite i (i=1 or 2) undergoes so bad channel that its outage performance is the worst although total power is allocated to it, i.e., pi,1= PT, pi,2=0, so only one link of the MT is available. The optimal power allocation of satellite,(∈ {1,2}i ) is achieved withPi= P3so that fairness is guaranteed.

(b) Both the two link of the MT are available. The optimal power allocation is achieved with P1= P2= P3.

Proof: Whenthe outage probability for all users can be decreased by increasing the power pi,kby a factor ofThus, the first part of the proposition is proved.We then prove the second part of the proposition. In case (a), the maximum outage probability is minimized with the maximum power and cannot be smaller. Solutions satisfying Pi≥and Pi≥ P3maintain the maximum outage probability minimized, thus they are optimal. For case (b), we prove by contradiction. Suppose that not all Pj, j∈ J={1,2,3}are equal at optimality. Denote J′ as the set of the indices such thatIf {1,3}∈J′, increase p1,1and p2,2, while decrease p2,1and p1,2until all Pjare equal,which leads to a lower.It is can be shown thatcan be further decreased for other user combinations in J′ in a similar way. This contradicts the assumption and the proposition is true.

Based on these characteristics, a simple iterative algorithm is proposed in Algorithm 1 to obtain the optimal solution stated in proposition 1. In line 4 and 5 of the algorithm, the current outage probability with zth power is used as the penalty coefficient of power update, since Pi,i=1,2 and P3decrease monotonically with pi,1and pi,2, respectively. This leads to the outage probability greater than the optimal to decrease and less to increase in the next iteration. Besides, line 6 guarantees that the updated power always satisfies constraints (17b) with equality and is in the feasible region. Setting reasonable step length by adjusting z, the gaps among outage probabilities gradually reduce and finally equal outage probability is achieved with a tolerable error ∈ between the greatest and smallest one.Then the update of power remains unchanged and according to proposition 1, the algorithm converges to the optimal solution.

V. SIMULATION RESULTS AND DISCUSSION

We evaluate the outage performance of the NOMA-based UAV satellite networks with some fixed parameters listed in table 1. We present the outage performance first with a fixed power allocation, and then with the minmax fairness power allocation.

5.1 Outage performance with fixed power allocation

This section simulates the outage performance of the system and verifies the analytical analysis, for a fixed power allocation. Define the nominal signal noise ratio (SNR) without rain attenuation of the link 2-1 as

which is used as the reference SNR for other links since their SNR varies proportionately. Set the fixed power allocation as pi,1=0.9 PT, pi,2= 0.1PT, i =1,2. Assume the correlation coefficient of rain attenuation is ρ=0.6 and the target for transmission is r0=1 bits/s/Hz.

?

Table I. System parameters for simulation.

Fig. 2. The outage probability versus nominal SNR.

Fig. 3. The outage probability of the MT versus correlation coefficient of rain attenuation.

Figure 2 plots the outage probability of the UAVs and the MT versus varying SNR. Together with the analytical results of outage probability in (11) and (16), Monte Carlo simulation results, obtained by calculating the percentage of the outage event based on 105realizations of the rain attenuation, are also plots for verifi-cation. They agree very well in the whole SNR regime. Despite only a small part of power allocated to the MT, the outage performance of the MT largely outperforms the UAV under low SNR for the following reasons: a) good channel condition due to high antenna gain, b) avoiding interference by SIC, c) multiplexing gain from the dual-satellite network. However, the superiority gradually disappears as SNR increases because rain attenuation is statistically stronger at the MT. Fixing SNR=10 dB, the in fluence of the rain attenuation correlation on the outage performance of the MT is illustrated in figure 3 under three configurations of target rate. The outage probability P3increases approximately linearly with the correlation coefficient ρ. In practice, we can improve the outage performance by enlarging the separation angle, which means satellites with large distance are preferred to cooperate.

5.2 Power allocation for min-max fairness

We will next provide the numerical results of the fair power allocation and the proposed algorithm. Setting SNR=10 dB, r0=1 bits/s/Hz,the error ∈=10?3and initializing the power allocation pi,1=0.9 PT, pi,2=0.1PT, i=1,2.Figure 4 shows the convergence process of the proposed iteration algorithm with the penalty power configured as z=1, z=2 and z=3, respectively. Comparing figure 4(a) with figure 4(b),it can be seen that with smaller z, the step length is smaller so the outage probabilities converge to the optimal value more smoothly, while the convergence rate slows down and the needed number of iteration is larger.However, when z increases to some extent, the oscillation encumbers the convergence process leading to inferior convergence rate, as observed from figure 4(c).

The maximum outage probability achieved by the fair power allocation scheme in NOMA and OMA is compared for target spectral efficiency r0=1 bits/s/Hz and r0=0.5 bits/s/Hz in figure 5. In OMA, equal bandwidth-split is used and the fair power allocation is per-formed by a similar algorithm to Algorithm 1.It can be observed that NOMA outperforms OMA in terms of the outage performance under the same power and spectrum resource.For higher target spectral efficiency, the performance superiority is larger. Meanwhile, the performance gap between NOMA and OMA narrows as SNR increases, since all outage probabilities approach 0 as SNR improves.

Fig. 4. The convergence process of the proposed iteration algorithm.

Fig. 5. The maximum outage probability achieved by the fair power allocation scheme in NOMA and OMA.

VI. CONCLUSIONS AND FUTURE WORK

In this paper, the NOMA based UAV satellite networks, where UAVs integrate the multi-satellite network by NOMA was investigated.Since Ku-band and above is used, we studied the outage performance in rain fading channel.The analytic outage probability was obtained for a fixed power allocation. On this basis, we proposed an efficient power allocation algorithm to address the issue of min-max fairness.Simulation results have verified the outage analysis and confirmed the convergence of the proposed algorithm. It showed that the outage performance deteriorates with rain attenuation correlation and NOMA outperforms OMA with the same fair power allocation.

This paper assumed the two satellites transmit on the same frequency. An important future work is considering spectrum sharing between satellites, which leads to inter-cell interference. Precoding at the transmitters and interference cancellation at the receivers are the key technologies to be researched.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China (No.91638205, 91438206, 61771286, 61621091).

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