Energy Efficient Multi-Antenna UAV-Enabled Mobile Relay

Qingheng Song, Fuchun Zheng＊

1 National Mobile Communications Research Laboratory, Southeast University, Nanjing 210093, China

2 Department of Mechanical engineering and Photoelectric Physics, Huaihua University, Huaihua 418008, China

I. INTRODUCTION

Unmanned aerial vehicles (UAVs) have many advantages, such as high mobility,cost-effectiveness, on-demand deployment,and high probability of line-of-sight (LoS)links between UAV and ground terminals [1].Therefore, UAV-enabled relay can be used as an effective complement to existing terrestrial communications networks, e.g., emergency communications recovery and hot spots offloading [2].

In the past few years, UAV-enabled relaying systems have attracted significant attention.Many works have studied how to maximize the end-to-end throughput of UAV-enabled relay, such as trajectory optimization, power allocation, beamforming, and time allocation[3]-[7]. The authors in [3] optimize the time allocation ratio between two hops for UAV mobile half-duplex relay to maximize the endto-end throughout. In [4], beamforming is adopted for UAV mobile relay, where UAV is equipped with single antenna and ground terminals are equipped with multi-antenna. The ergodic normalized transmission rate is maximized through the control of the UAV heading in [5]. In [6], beamforming and path planning are both optimized to maximize the end-to-end signal-to-noise ratio (SNR) for UAV relaying system under Rayleigh channels. The authors in [7] maximize the throughput by joint design of power allocation and UAV trajectory. However, all the aforementioned works mainly fo-cus on single-antenna UAV relaying systems.

On the other hand, since on-board energy of UAV is limited due to size and load capacity, energy efficient UAV-enabled relaying systems have attracted considerable attention recently [8]-[11]. The authors in [8] maximize the energy efficiency by joint design of trajectory and flight speed, and a propulsion energy consumption model of fixed-wing UAVs is developed, where straight-and-level flight, banked level turn, and general trajectories are all considered. In [9], a trade-off between transmit power of ground terminals and propulsion energy consumption of UAV was studied, where two practical UAV trajectories, namely circular flight and straight flight, were considered. The authors in [10]maximize the spectrum and energy efficiency of UAV-enabled mobile relaying system by joint optimization of time allocations, flight speed and trajectory. In [11], energy efficiency of UAV mobile relaying system is maximized by joint optimization of flight speed and load factor. However, no closed-form expression of energy efficiency has been derived and their results were all obtained by simulation. The optimal altitude is obtained for both static and mobile UAV relaying systems [12], but the results only focused on the single antenna relaying system and the energy efficiency was not considered. Finally, compared with rotary wing UAVs, fixed wing UAVs are more energy efficient [13].

Motivated by the above works, in this paper, we focus on the energy efficiency maximization problem for dual-hop half-duplex relaying system with multi-antenna fixed wing UAVs by jointly optimizing the beamforming,statistical power allocation, circular radius and flight speed, where the energy efficiency can be obtained in semi closed-form. We consider simple but energy efficient circular trajectory[10], [11] and adopt the time-division duplexing (TDD) based decode-and-forward (DF) relaying strategy [10]. We devide each time slot into two sub-slots with equal durations. In the first time slot, the source transmits data to the UAV relay, while the UAV relay transmits the decoded data to the destination in the second time slot. From [3], the end-to-end throughput can be improved by reducing circular radius.However, from [8], the propulsion energy consumption of fixed wing UAVs increases with the decreasing of circular radius. Therefore,there exists a trade-off between throughput and propulsion energy consumption, which affects the energy efficiency.

The remainder of this paper is organized as follows. The system model, channel model and problem formulation are given in Section II. Implementation of energy efficiency maximization is investigated in Section III. Numerical results are presented in Section IV and Section V concludes the paper.

Notation: In this paper, lower-case letters denote scalars, bold upper and lower case letters denote matrices, and vectors, respectively.(·)†, and trace(·) denote Frobenius norm, conjugate transpose, and the trace of a matrix, respectively. E[ x] stands for the expectation of the random variable x.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a UAV-enabled half-duplex relaying system as shown in figure 1, where fixed wing UAV U acts as a relay to help the information transmission from source node S to destination node D. We assume that there is no direct link between S and D due to blockage,and hence the information can only be transmitted through the UAV relay U. We also assume that both source node S and destination node D have single antenna, and the UAV relay has N antennas. Without loss of generality,we consider the three-dimensional Cartesian coordinate system, where S and D are located at (d , 0, 0) and (− d , 0, 0), respectively.The circular flight is of high practical signi ficance for fixed wing UAVs, since it provides a flexible trade-off between achievable rates and UAV’s propulsion energy consumption [8],the circular trajectory is energy efficient [10]-[11], and the UAV’s propulsion energy con-sumption can be modeled in closed-form as a function of the UAV’s flight speed and circular radius [8]. We assume that the UAV flies at the fixed altitude h following a circular trajectory with center O′(0, 0, h), circular radius r and flight speed v. In practice, h could correspond to the minimum altitude required for collision avoidance without frequent UAV ascending and descending [8]. Then, the circling period is T =2πr/ v . Without loss of generality, we assume the initial location of UAV is(r, 0, h). At t moment, the UAV’s location is given by (r cos(v t/ r) , r sin(v t/ r) , h), where θ(t)=vt/ r corresponds to the azimuth angle of the circle along which the UAV flies.

At time instant t, the distances between S(or D) and UAV relay U are respectively given by

and

We assume quasi-static block fading channels for the ground-UAV links, where the channel remains unchanged within each fading block and may change over different blocks [3], [14]. In addition, for simplicity, we assume that the channel links are composed of large-scale path loss and statistically independent small-scale quasi-static frequency non-selective fading [3]. Furthermore, the Doppler effect due to the circling operation of UAV is assumed to have been completely compensated [8], [15], and both links use the same frequency resource. Due to the high altitude of UAVs, there exists a LoS link between UAV and ground terminals with high probability.Therefore, the small-scale fading channels can be modeled as Rician fading channels [5]. We assume that both the S-U and U-D channels have the same Rician factor. From [16], the path loss exponent of UAV-ground links is stable and close to 2 for a certain flight altitude.Mathematically, we have

whereis the large-scale fading, β denotes the channel power loss at the reference distance d0=1 m, and hj(t) denotes the small-scale fading between UAV and ground terminals, which is modeled as

where hSU(t)∈CN×1and hUD(t)∈C1×N,hjmis the deterministic strong LoS components of the channel satisfyingK is the Rician factor, and hjsdenotes the scattered components of the channel, whose entries are independent and identically distributed circularly symmetric complex Gaussian random variables with zero mean and unit variance.

At time instance t, the received signal at the UAV relay U can be expressed as

where PS( t ) denotes the transmitting power of source node S, xS(t ) denotes the information-bearing symbols transmitted by S with unit power and nUis the zero-mean additive white Gaussian noise (AWGN) at the UAV relay U with

Assume that a linear receiver wrwithis employed at the UAV relay U for signal receiving, which gives

Fig. 1. A UAV relaying system where UAV circles above at the fixed altitude of h from SD plane with center O′(0, 0, h), where the circular radius is r and the flight speed is constant v.

With the TDD based DF half-duplex relay,the received signal at the destination node D is

where PU( t ) denotes the transmitting power of the UAV relay node U, wtis transmit beamforming vector at the UAV relay U withand nDis the zero-mean AWGN noise at the destination node D with variance. xU(t) is the information-bearing symbols transmitted by UAV with unit power. For ease of exploration, we assumeMoreover, it is assumed that the global channel state information (CSI) is available at the UAV, and the size of the data buffer at the UAV is assumed to be sufficiently large.

Therefore, the SNR at D and U can be respectively given by

and

The channels of S-U and U-D links are time-varying and change rapidly due to the circling operation of UAV. However, the statistical CSI is rather static over a long time scale at any location of UAV during the circling period. Therefore, we first take the expectation of SNRU(t) and SNRD(t) with respect to the square of norm, and then express the end-toend SNR as

which is an approximation of the end-to-end SNR of DF half-duplex relay [17]

For fixed wing UAVs on a steady circular flight, where the circular radius is r and the flight speed is constant v, it has been shown that the propulsion power consumption of UAV can be modeled as [8]

where g is the gravitational acceleration in m/s2, a1and a2are constants corresponding to aerodynamics and aircraft design, such as air density, zero-lift drag coefficient, wing area,wing span efficiency, aspect ratio of the wing,and total weight of aircraft. It is observed from(12) that the propulsion power consumption depends on the circular radius r and flight speed v. The propulsion power consumption increases with the decreasing of circular radius r. For the extreme case of r=0 or v=0,we have PC→∞, which reflects the fact that the fixed-wing UAV must maintain a forward speed in order to remain in the air.

For half-duplex relay, each interval is divided into two equal time slots, one for receiving and one for transmitting. The energy efficiency is defined as the ratio of end-to-end achievable rates to the UAV propulsion energy consumption over the circling period T. In practice, the total flight time is usually much longer than the circling period T, hence it is flexible to use the energy efficiency of a circling period T instead of the energy efficiency of the total flight time [8], [10]. In addition, the communications and signal processing-related energy is ignored in this paper as it is usually much smaller than the UAV’s propulsion energy [8].Then, we have

where B is communications bandwidth,T =2πr/ v , and 1/2 is the effect of half-duplex relay.

The objective of this paper is to maximize η by optimizing wt, wr, v and r, subject to the sum transmit power constraint over source node S and UAV relay node U. Thus, we formulate the energy efficiency maximization problem as

where PTis sum transmit power of S and U[18], vmaxis the maximum flight speed of UAV. The objective of this paper is to maximize the energy efficiency under sum transmit power constraint PT, which is equivalent to maximize the utilization of sum transmit power PT. Note that the denominator of η does not depend on beamforming and power allocation. Therefore, we can first maximize the numerator of η with respect to wt, wr,PS( t ) and PU( t), and then optimize the energy efficiency with respect to r and v. Thus,problem P1 can be decomposed as

and

III. PROPOSED SOLUTION FOR ENERGY EFFICIENCY MAXIMIZATION

In this section, we first maximize the energy efficiency with respect to beamforming vectors wr, wtand power allocation PS( t ) and PU( t). And then, with the help of the obtained wr, wt, PS( t ) and PU( t), we optimize flight speed v and circular radius r to maximize the energy efficiency.

As the denominator of η does not depends on beamforming and power allocation, we turn to maximize the numerator of η with respect to wr, wt, PS( t) and PU( t). In addition, if the end-to-end SNR at any time slot is maximized,the integral is also maximized. Therefore,problem P2 is equivalent to following problem

From (8) and (9), we observe that SNRU(t)only depends on wr, while SNRD(t) only depends on wt. Therefore, if N＞1, SNRU(t)can be maximized via maximum ratio combing, i.e.,and SNRD(t)can be maximized via maximum ratio transmission, i.e.,Substituting wrand wtinto problem P4, problem P4 reduces to an univariate optimization problem given by

Having obtained the beamforming results, we now optimize the power allocation PS( t ) and PU( t). It is obvious that the endto-end SNR depends on the minimum SNR value of the two hops. To maximize the endto-end SNR, the source should transmit with more power to improve performance when the UAV moves closer to the destination. In contrast, the UAV relay should transmit with more power when the UAV moves closer to the source. Over a long time scale, the statistical CSI is rather static, the power allocation should be adaptively adjusted to guarantee E[ SNRU(t) ] =E [SNRD(t)], which is optimal[20]. Then, we transfer the power allocation problem P5 to following equations

Using the fact thatthen, statistical power allocation problem further reduces to

By solving above equations, we have

and

We observe that the statistical power allocation depends on PTand the location of UAV via dSU(t) and dUD(t). Substituting PS( t ) and PU( t) into η, we have (23) shown in the bottom at this page, where (a) is due to the fact that[21]. As observed from (23), the flight speed v only affects the denominator of η. Therefore, from (23), we now optimize the circular radius and the flight speed of problem P3, by taking the derivation of the denominator of η with respect to v, and making it equal to

Fig. 2. Transmit power versus position of UAV relay U with different distance d,e.g., d=1000 m, d=500 m and d=200 m.

zero [22], combined with the maximum flight speed vmax, the optimal v*( r) can be readily obtained as

The optimal speed varies with circular radius. Substituting v*( r) into (23), we have

From (25), for fixed circular radius, it is obvious that the energy efficiency increases with the increasing of PTand N, while the energy efficiency degrades with the increasing of d and h. Optimal circular radius r is very difficult to obtain in closed-form due to the complicated expression of η. However, by taking derivation of η with respect to r, and making it equal to zero, optimal circular radius r can be efficiently obtained by numerical computation, and then the optimal flight speed can be computed via (24). After joint design of beamforming, power allocation, circular radius and flight speed, the energy efficiency of UAV-enabled mobile relay is maximized.

IV. NUMERICAL RESULTS

In this section, numerical results are presented to validate the performance of the proposed energy efficient UAV-enabled mobile relay.Unless otherwise specified, the flight altitude of UAV is fixed at h=100 m, the gravitational acceleration is 9.8 m/s2, the Rician factor is K=10, the bandwidth is B=1 MHz, and the noise power density is −174 dBm/Hz. Thus,the noise power is σ2=− 174 × B=−114 dBm. The channel power loss at the reference distance 1 m is β= −60 dB. For the propulsion power consumption model of UAV in (12), we assume that a1=9.26× 10−4and a2=2250 as in [8]. The sum transmit power PTof source and relay is 2 W. We investigate different configurations by assuming that the distance d between S (or D) and origin changes from 100 m to 1000 m.

Figure 2 depicts the transmit power of the proposed energy efficient UAV-enabled mobile relay over the circling operation with different distance d. We assume the antenna number of UAV is N=4. First, we obtain the optimal circular radius r*from (25) by numerical computation. And then, by substituting r*into(21), the optimal transmit power of source node S and UAV relay node U at different location is obtained. For different distance d,e.g., d=1000 m, d=500 m and d=200 m., the optimal circular radii are r*=476 m,r*=364 m and r*=276 m, respectively. For these configurations, the propulsion power consumption of UAV are 102.68 W, 104.47 W,and 107.43 W, respectively. Compared with PT, we have PC≫ PT. Therefore, the communications and signal processing-related power consumption can be neglected in energy efficiency. In addition, PS( t ) increases with θ from zero to 180°, which demonstrates that the source node S should transmit with more power when the UAV moves closer to the destination. In contrast, due to PU( t) =PT− PS( t ),the UAV should transmit with more power when the UAV moves closer to the source.This is because when the UAV moves closer to the destination, compared with the U-D link,the S-U link suffers more severe path loss.Thus, the source node S should transmit with more power to balance the SNR between two hops and vice versa.

Figure 3 shows the energy efficiency versus different circular radius with fixed distance d=1000 m and different antenna number,e.g. N=8, N=4, N=2 and N=1. It is clear that there exist an optimal circular radius where the energy efficiency is maximized. The maximum value of energy efficiency corresponds to the optimal circular radius. With the increasing of circular radius, the energy efficiency first increases monotonously until the maximum value, and then decreases monotonously. This is because with the increasing of circular radius, the effect of propulsion power consumption reduction is more beneficial at first which leads to the increase of energy efficiency, until achieving the maximum value,then the effect of path loss increase becomes dominant, and hence the energy efficiency degrades. In addition, the energy efficiency improves with the increasing of antenna num-ber. Furthermore, from the dotted line, it is interesting that the maximum energy efficiency increases approximately linearly with the number of antennas.

Fig. 3. Energy efficiency versus circular radius from 100 m to 1000 m for fixed d=1000 m with different number of antennas: N=8, N=4, N=2 and N=1,where “°” denotes the maximum energy efficiency corresponding to the optimal circular radius.

Fig. 4. Comparison of optimal circular radius versus distance d with different number of antennas: N=8, N=4, N=2 and N=1.

Figure 4 presents the optimal circular radius versus distance d with different number of antennas, e.g. N=8, N=4, N=2 and N=1. The optimal circular radius increases with the increasing of distance d, which is approximately linear. In addition, for any given distance d, the optimal circular radius increases with the increasing of number of antennas.This result further validates the insights obtained from Figure 3. Furthermore, the curves of optimal flight speed can be readily obtained according to (24), which is omitted here for brevity.

Figure 5 shows that the maximum energy efficiency can be achieved for different distance d and number of antennas N, where the energy efficiency is maximized by joint design of beamforming, power allocation, together with circular radius and flight speed. First, the maximum energy efficiency decreases with the distance d, and this is because the increasing of distance d brings about greater path loss.From Figure 4, the optimal circular radius increases with the distance d, but the beneficial effect of propulsion power consumption reduction by increasing circular radius becomes less noticeable compared with the incurred path loss. Hence, the energy efficiency decreases. In addition, the maximum energy effi-ciency improves with the number of antennas,and this is because greater number of antennas can provide greater array gain.

Fig. 5. Maximum energy efficiency versus distance d with different number of antennas: N=8, N=4, N=2 and N=1.

V. CONCLUSIONS

We have considered an energy efficient multi-antenna UAV-enabled half-duplex DF relaying system, subject to sum transmit power constraint over source node and UAV relay node. By assuming a simple circular UAV trajectory, the energy efficiency has been maximized by jointly optimizing the beamforming,statistical power allocation, flight speed and circular radius. A semi closed-form energy efficiency expression has also been obtained,where optimal circular radius can be obtained by numerical methods. Numerical results have revealed that the proposed scheme can signifi-cantly enhance the energy efficiency. However, this paper has only studied the simple circular trajectories, and extending this work to more general trajectories will be considered in our future study.

ACKNOWLEDGEMENT

This work was supported in part by the National Science Foundation (NSFC) for Distinguished Young Scholars of China with Grant 61625106 and the National Natural Science Foundation of China under Grant 61531011.

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