# UAV-Aided Wireless Communication Designs With Propulsion Energy Limitations

This paper studies unmanned aerial vehicle (UAV) aided wireless communication systems where a UAV supports uplink communications of multiple ground nodes (GNs) while flying over the area of the interest. In this system, the propulsion energy consumption at the UAV is taken into account so that the UAV's velocity and acceleration should not exceed a certain threshold. We formulate the minimum average rate maximization problem and the energy efficiency (EE) maximization problem by jointly optimizing the trajectory, velocity, and acceleration of the UAV and the uplink transmit power at the GNs. As these problems are non-convex in general, we employ the successive convex approximation (SCA) techniques. To this end, proper convex approximations for the non-convex constraints are derived, and iterative algorithms are proposed which converge to a local optimal point. Numerical results demonstrate that the proposed algorithms outperform baseline schemes for both problems. Especially for the EE maximization problem, the proposed algorithm exhibits about 109 gain over the baseline scheme.

## Authors

• 2 publications
• 11 publications
• 2 publications
• 10 publications
• ### Minimum Throughput Maximization in UAV-Aided Wireless Powered Communication Networks

This paper investigates unmanned aerial vehicle (UAV)-aided wireless pow...
01/09/2018 ∙ by Junhee Park, et al. ∙ 0

• ### Joint Worst Case Secrecy Rate and UAV Propulsion Energy Optimization via Robust Resource Allocation in Uncertain Adversarial Networks

The mobile relaying technique is a critical enhancing technology in wire...
06/28/2020 ∙ by S. Ahmed, et al. ∙ 0

• ### Trajectory Design in Rechargeable UAV Aided Wireless Networks: A Unified Framework for Energy Efficiency and Flight Time Minimization

This paper studies a rechargeable unmanned aerial vehicle (UAV) assisted...
04/28/2021 ∙ by Yuan Liao, et al. ∙ 0

• ### Throughput Maximization for Full-Duplex UAV Aided Small Cell Wireless Systems

This paper investigates full-duplex unmanned aerial vehicle (UAV) aided ...
12/10/2019 ∙ by Meng Hua, et al. ∙ 0

• ### Energy and Age Pareto Optimal Trajectories in UAV-assisted Wireless Data Collection

This paper studies an unmanned aerial vehicle (UAV)-assisted wireless ne...
06/07/2021 ∙ by Yuan Liao, et al. ∙ 0

• ### Efficiency Maximization for UAV-Enabled Mobile Relaying Systems with Laser Charging

This work studies the joint problem of power and trajectory optimization...
12/26/2019 ∙ by Ming-Min Zhao, et al. ∙ 0

• ### Throughput Maximization for Laser-Powered UAV Wireless Communication Systems

Laser power has become a viable solution to provide convenient and susta...
03/02/2018 ∙ by Jie Ouyang, et al. ∙ 0

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## I Introduction

Recently, unmanned aerial vehicles (UAVs) have received great attentions as a new communication entity in wireless networks [1]. Compared to conventional terrestrial communications where users are served by ground base stations (BSs) fixed at given position [2], UAV-aided systems could be dispatched to the field with various purposes such as disaster situations and military uses. Moreover, located high above users, UAVs are likely to have line-of-sight (LoS) communication links for air-to-ground channels.

Utilizing these advantages, UAVs have been considered to diverse wireless communication systems. The authors in [3] and [4] studied a mobile relaying system where a UAV helps the communication of ground nodes (GNs) without direct communication links. In this UAV-aided relaying system, compared to conventional static relay schemes [5],[6], the UAV can move closer to source and destination nodes in order to obtain good channel conditions, and thus the system throughput can be significantly improved. In [3], the throughput of mobile relaying channels was maximized by optimizing the transmit power at the source and the relay node as well as the trajectory of the mobile relay. For the fixed relay trajectory, the work [4] addressed the secrecy rate maximization problem for the UAV-based relaying system with an external eavesdropper.

In addition, UAVs have been adopted to assist conventional terrestrial communication infrastructures [7, 8, 9]. For the disaster situation, UAVs were employed in [7] to recover malfunctioned ground infrastructure. The work in [8] examined a system where the UAV serves cell-edge users by jointly optimizing UAV’s trajectory, bandwidth allocation, and user partitioning. Also, the flying computing cloudlets with UAVs were introduced to provide the offloading opportunities to multiple users [9].

Moreover, the UAVs could play the role of mobile BSs in wireless networks [10, 11, 12]. The authors in [10] derived mathematical expressions for the optimum altitude of the UAVs that maximizes the coverage of the cellular network. Also, the trajectory optimization methods for mobile BSs were presented in [11] and [12]. Assuming that the GNs are located in a line, the minimum throughput performance was maximized in [11] by optimizing the position of a UAV on a straight line. This result was extended in [12] to a general scenario where multiple UAVs fly three-dimensional space to communicate with GNs. The joint optimization algorithms for the UAV trajectory, transmit power, and time allocation were provided in [12] to maximize the minimum throughput performance. However, these works did not consider the propulsion energy consumption of the UAVs necessary for practical UAV designs under limited on-board energy situation [13].

By taking this issue into account, recent works [14, 15, 16] investigated energy efficiency (EE) of the UAV system. Different from conventional systems which consider only communication-related energy consumption [17, 18, 19], the EE of the UAV should addresses the propulsion energy at the UAV additionally. The authors in [14] maximized the EE by controlling the turning radius of a UAV for mobile relay systems. Also, by jointly optimizing the time allocation, speed, and trajectory, both the spectrum efficiency and the EE were maximized in [15]. In [16], the propulsion energy consumption of the UAV was theoretically modeled, and the EE of the UAV was maximized for a single GN system.

This paper studies UAV-aided wireless communications where a UAV with limited propulsion energy receives the data of multiple GNs in the uplink. It is assumed that all GNs and the UAV operate in the same frequency band and there are no direct communication links among GNs. Under these setup, we formulate the minimum rate maximization problem and the EE maximization problem by jointly optimizing the UAV trajectory, the velocity, the acceleration, and the uplink transmit power at the GNs. A similar approach for solving the minimum rate maximization was studied in [12], but the authors in [12] did not involve the propulsion energy consumption at the UAV. For the EE maximization problem, our work can be regarded as a generalization of the single GN system in [16] to the multi-GN scenario, and thus we need to deal with inter-node interference as well. Due to these issues, existing algorithms presented in [12] and [16] cannot be directly applied to our problems.

To tackle our problem of interest, we introduce auxiliary variables which couple the trajectory variables and the uplink transmit power in order to jointly optimize these variables. As the equivalent problem is still non-convex, we employ the successive convex approximation (SCA) technique which successively solves approximated convex problems of the original non-convex one. In order to apply the SCA to our optimization problems, we present new convex surrogate functions for the non-convex constraints. Then, we propose efficient algorithms for the minimum rate maximization problem and the EE maximization problem which yield local optimal solutions. Simulation results confirm that the proposed algorithms provide a significant performance gain over baseline schemes.

The rest of this paper is organized as follows: Section ii@ explains the system model and the problem formulations for the UAV-aided communication systems. In Section iii@, the minimum rate maximization and the EE maximization algorithms are proposed. We examine the circular trajectory case as baseline schemes in Section iv@. Section v@ presents the numerical results for the proposed algorithms and we conclude the paper in Section vi@.

: Throughout this paper, the bold lower-case and normal letters denote vectors and scalars, respectively. The space of

-dimensional real-valued vectors are represented by . For a vector , and indicate norm and transpose, respectively. The gradient of a function is defined as . For a time-dependent function , and stand for the first-order and second-order derivatives with respect to time , respectively.

## Ii System Model And Problem Formulation

As shown in Fig. 1, we consider UAV-aided wireless communications where a UAV receives uplink information transmitted from GNs. The UAV horizontally flies at a constant altitude with a time period , while the GNs are located at fixed positions, which are perfectly known to the UAV in advance. For the location of the GNs and the UAV, we employ a three-dimensional Cartesian coordinate system, and thus the horizontal coordinate of GN is denoted by . Also, we define the time-varying horizontal coordinate of the UAV at time instant as . Then, the instantaneous velocity and the acceleration of the UAV are expressed by and , respectively.

Continuous time expressions of variables make analysis and derivations in the UAV systems intractable. For ease of analysis, we discretize the time duration into time slots with the same time interval [3]. As a result, the trajectory of the UAV can be represented by vector sequences , , and for . When the discretized time interval is chosen as a small number, the velocity and the acceleration can be approximated by using Taylor expansions as [16]

 v[n]=v[n−1]+a[n−1]δt, for  n=1,...,N, (1) q[n]=q[n−1]+v[n−1]δt+12a[n−1]δ2t, for  n=1,...,N. (2)

Also, assuming the periodical operation at the UAV, we have [12]

 q[0]=q[N],v[0]=v[N],a[0]=a[N], (3)

which implies that after one period , the UAV returns to its starting location with the same velocity and acceleration.

In addition, the acceleration and the velocity of the practical UAV are subject to

 ∥a[n]∥≤amax, for  n=0,1,...N, (4) Vmin≤∥v[n]∥≤Vmax, for%   n=0,1,...,N, (5)

where indicates the maximum UAV acceleration in m/sec and and stand for the minimum and the maximum UAV speed constraints in m/sec, respectively. Notice that the minimum speed constraint is important for practical fixed-wing UAV designs which need to move forward to remain aloft and thus cannot hover over a fixed location [16].

For the power consumption at the UAV, we take into account the propulsion power utilized for maintaining the UAV aloft and supporting its mobility. The propulsion power of the UAV at time slot is given by [16]

 Pprop[n]=c1∥v[n]∥3+c2∥v[n]∥(1+∥a[n]∥2g2), for  n=0,1,...,N, (6)

where and are the parameters related to the aircraft design and = 9.8 m/sec equals the gravitational acceleration. Thus, the average propulsion power and the total consumed propulsion energy over time slots are obtained by and , respectively. The power consumed by signal processing circuits such as analog-to-digital converters and channel decoders are ignored since they are practically much smaller than the propulsion power [16].

Now, let us explain the channel model between the UAV and the GNs. We assume that the air-to-ground communication links are dominated by the LoS links. Moreover, the Doppler effect due to the UAV mobility is assumed to be well compensated. Then, the effective channel gain from GN to the UAV at time slot follows the free-space path loss model as [3]

 hk[n]=γ0d2k[n], (7)

where represents the reference signal-to-noise ratio (SNR) at 1 m with and being the channel power at 1 m and the white Gaussian noise power at the UAV, respectively, and the distance is written by

 dk[n]=√∥q[n]−wk∥2+H2. (8)

At time slot , GN transmits its data signal to the UAV with power , where is the peak transmission power constraint at the GNs. Accordingly, the instantaneous achievable rate can be expressed as

 Rk[n]=log2⎛⎝1+pk[n]hk[n]1+∑Kj=1,j≠kpj[n]hj[n]⎞⎠, (9)

where the term stands for interference from other GNs. Therefore, the achievable average rate of the GN and the total information bits transmitted from GN over time slots are denoted as and , respectively, where means the bandwidth.

In this paper, we jointly optimize the variables , , and and the uplink transmit power at the GNs so that the minimum average rate among multiple GNs and the EE are maximized, respectively. First, the minimum rate maximization problem can be formulated as

 (P1): max{q[n],v[n],a[n]}{pk[n],τ} τ (10a) s.t.1NN∑n=1Rk[n]≥τ,∀k, (10b) 0≤pk[n]≤P%peak,∀k,n, (10c) 1NN∑n=1Pprop[n]≤Plim, (10d) (1)−(5),

where in (10d) indicates the propulsion power constraint at the UAV.

Next, to support all of the individual GNs, the fairness based EE [20, 21, 22] is more suitable than the network-wise EE [18, 19]. Thus, we define the EE in the UAV-aided wireless communication systems as the ratio between the minimum information bits transmitted among the GNs and the total energy consumed at the UAV. Therefore, the EE maximization problem can be written by

 (P2): max{q[n],v[n],a[n]}{pk[n],η}   η∑Nn=1P% prop[n] (11a) WN∑n=1Rk[n]≥η,∀k, (11b) s.t.  (1)−(5),(10c).

In general, (P1) and (P2) are non-convex problems due to the constraints and the objective functions. Compared to [12], we additionally consider the propulsion power constraint (10d) in the minimum rate maximization problem (P1). Also, note that the EE maximization problem (P2) can be regarded as a generalization of [16] which investigated only a single GN scenario. From these respects, the works in [12] and [16] can be regarded as special cases of our problems (P1) and (P2), respectively. To solve the problems (P1) and (P2), we adopt the SCA framework [23] [24] which iteratively solves approximated convex problems for the original non-convex problems.

## Iii Proposed Algorithm

In this section, we propose iterative algorithms for efficiently solving (P1) and (P2) by applying the SCA method. First, the minimum rate maximization problem (P1) is considered in Section iii@-A, and then it is followed by the EE maximization problem (P2) in Section iii@-B.

### A Minimum Average Rate Maximization

Applying the change of variables as

 Gk[n]≜pk[n]hk[n]=pk[n]γ0∥q[n]−wk∥2+H2,∀k,n, (12)

where is a new optimization variable, the constraint (10c) becomes , where Then, we can rewrite the achievable rate in (9) as

 Rk[n]=log2(1+K∑m=1Gm[n])−^Rk[n], (13)

where

By introducing new auxiliary variables , (P1) can be recast to

 (P1.1): max{q[n],v[n],a[n]}{Gk[n],V1[n],τ} τ (14a) (14b) 0≤Gk[n]≤Gk,max[n],∀k,n, (14c) 1NN∑n=1c1∥v[n]∥3+c2V1[n]+c2∥a[n]∥2g2V1[n]≤Plim, (14d) Vmin≤V1[n],∀n, (14e) V21[n]≤∥v[n]∥2,∀n, (14f) ∥v[n]∥≤V%max,∀n, (14g) (1)−(4).

It can be shown that at the optimal point of (P1.1), the inequality constraint in (14f) holds with the equality, since otherwise we can enlarge the feasible region corresponding to (14d) by increasing . Therefore, we can conclude that (P1.1) is equivalent to (P1). Thanks to the new auxiliary variables , constraints (14d) and (14e) now become convex, while (14b), (14c), and (14f) are still non-convex in general.

To address these constraints, we employ the SCA methods. First, it can be checked that constraint (14b) is given by a difference of two concave functions. Hence, the convex surrogate function for can be computed from a first order Taylor approximation as

 ^Rubk[n]≜^Γk[n]K∑j=1,j≠k(Gj,l+1[n]−Gj,l[n])+log2⎛⎝1+K∑j=1,j≠kGj,l[n]⎞⎠≥^Rk[n], (15)

where indicates a solution of attained at the -th iteration of the SCA process and Next, to identify the surrogate functions of (14c) and (14f), we present the following lemmas.

###### Lemma 1

Denoting as a solution for calculated at the -th iteration, the concave surrogate function for can be expressed as

 Glbk,max[n] ≜ Ppeakγ0(−∥ql+1[n]−wk∥2H4+Bk[n](ql+1[n]−wk)T(ql[n]−wk)+Ck[n]) (16) ≤ Gk,max[n],

where the constants and are respectively given as

 Bk[n] ≜ 2⎛⎜ ⎜⎝1H4−1(∥ql[n]−wk∥2+H2)2⎞⎟ ⎟⎠, Ck[n] ≜ 1∥ql[n]−wk∥2+H2+2∥ql[n]−wk∥2(∥ql[n]−wk∥2+H2)2−∥ql[n]−wk∥2H4.

###### Lemma 2

From a solution obtained at the -th iteration, the concave surrogate function of can be computed as

 −∥vl+1[n]∥2+2vTl(2vl+1[n]−vl[n])≤∥vl+1[n]∥2. (17)
###### Proof:

Applying a similar process in Appendix A, we can conclude that the function in (17) satisfies the conditions for a concave surrogate function [23].

With the aid of Lemmas 1 and 2, at the -th iteration, the non-convex constraints in (14c) and (14f) can be approximated as

 0≤Gk[n]≤Glbk,max[n], (18) V21[n]≤−∥vl+1[n]∥2+2vTl(2vl+1[n]−vl[n]). (19)

As a result, with given solutions , , at the -th iteration, we solve the following problem at the -th iteration of the SCA procedure

 (P1.2): max{ql+1[n],vl+1[n],a[n]}{Gk,l+1[n],V1[n],τlb} τlb (20a) s.t.1NN∑n=1(log2(1+K∑m=1Gm,l+1[n])−^Rubk[n])≥τlb,∀k, (20b) (1)−(4),(14d),(14e),(14g),(18),(19),

where denotes the lower bound of in the original problem (P1). Since (P1.2) is a convex problem, it can be optimally solved via existing convex optimization solvers, e.g. CVX [25]. Based on these results, we summarize the proposed iterative procedure in Algorithm 1.

 Algorithm 1: Proposed algorithm for (P1) Initialize {q0[n],v0[n],Gk,0[n]},∀k,n and let l=0. Repeat Compute {ql+1[n],vl+1[n],Gk,l+1[n]} for (P1.2) with given {ql[n],vl[n],Gk,l[n]}. Update l←l+1. Until Convergence. Obtain pk[n]=Gk,l+1[n]hk,l+1[n].

For the convergence analysis of Algorithm 1, let us define the objective values of (P1) and (P1.2) at the -th iteration as and , respectively. Then we can express the relationship

 τl = τlbl ≤ τlbl+1≤ τl+1, (21)

where the first equation holds because the surrogate functions in (15), (16), and (17) are tight at the given local points, the second inequality is derived from the non-decreasing property of the optimal solution of (P1.2), and the third inequality follows from the fact that the approximation problem (P1.2) is a lower bound of the original problem (P1).

From (21), we can conclude that the objective value in (P1) is non-decreasing for every iterations of Algorithm 1. Since the objective value in (P1) has a finite upper bound value and at given local points, the surrogate functions in (15), (16), and (17) obtain the same gradients as their original functions, it can be verified that Algorithm 1 is guaranteed to converge to at least a local optimal solution for (P1) [23, 24].

### B Energy Efficiency Maximization

In this subsection, we consider the EE maximization problem (P2). First, by applying (12)-(13), and introducing an auxiliary variable , (P2) can be transformed as

 (P2.1): max{q[n],v[n],a[n]}{Gk[n],V1[n],η} η∑Nn=1c1∥v[n]∥3+c2V1[n]+c2∥a[n]∥2g2V1[n] (22a) s.t. WN∑n=1(log2(1+K∑m=1Gm[n])−^Rk[n])≥η,∀k, (22b) (1)−(4),(14c),(14e)−(14g).

Similar to (P1.1), we can see that (P2.1) is equivalent to (P2), but (P2.1) is still non-convex due to the constraints in (14c), (14f), and (22b).

To tackle this issue, we can employ the similar SCA process presented in Section iii@-A. By adopting (15) and Lemmas 1 and 2, a convex approximation of (P2.1) at the -th iteration is given by

 (P2.2): max{ql+1[n],vl+1[n],a[n]}{Gk,l+1[n],V1[n],ηlb} ηlb∑Nn=1c1∥v[n]∥3+c2V1[n]+c2∥a[n]∥2g2V1[n] (23a) s.t. WN∑n=1(log2(1+K∑m=1Gm,l+1[n])−^Rubk[n])≥ηlb,∀k (23b) (1)−(4),(14e),(14g),(18),(19),

where denotes the lower bound of in the original problem (P2).

It can be shown that (P2.2) is a concave-convex fractional problem, which can be optimally solved via the Dinkelbach’s method [26],[27]. Then, denoting with a given constant , (P2.2) can be converted to (P2.3) as

 (P2.3): max{ql+1[n],vl+1[n],a[n]}{Gk,l+1[n],V1[n],ηlb} ηlb−λmμ (24a) s.t.(1)−(4),(14e),(14g),(18),(19),(23b).

Based on (P2.3), we summarize the proposed iterative procedure in Algorithm 2. The convergence and the local optimality of Algorithm 2 can be verified similar to Algorithm 1, and thus the details are omitted for brevity.

 Algorithm 2: Proposed algorithm for (P2) Initialize {q0[n],v0[n],Gk,0[n]},∀k,n and let λ0=0, m=0, and l=0. Repeat Repeat Compute {ql+1[n],vl+1[n],Gk,l+1[n]} for (P2.3) with given {ql[n],vl[n],Gk,l[n]},∀k,n and λm. Update l←l+1. Until Convergence. Let F(λm)=ηlb−λmμ and λm+1=ηlb/μ. Update m←m+1. Let {q0[n],v0[n],Gk,0[n]} = {ql+1[n],vl+1[n],Gk,l+1[n]},∀k,n and l=0. Until Convergence. Obtain pk[n]=Gk,l+1[n]hk,l+1[n],∀k,n.

It is worthwhile to note that we need to initialize the trajectory variables for (P1) and (P2). However, it is not trivial to find such variables satisfying the UAV movement constraints (1)-(5) and the propulsion power constraint (10d). This will be clearly explained in Section iv@-C.

## Iv Circular trajectory system

Now, we examine the circular trajectory system which will be used as a baseline scheme. First, we choose the center of the circular trajectory

as the geometrical mean of the GNs

. Denoting as the radius of the trajectory and as the angle of the circle along which the UAV flies at time slot , the horizontal coordinate of the UAV can be obtained by . Also, the location of GN can be represented as , where and equal the distance and the angle between the geometric center and GN , respectively. Thus, the distance between the UAV and GN in (8) can be expressed as

By adopting the angular velocity and the angular acceleration , equations in (1)-(6) can be rewritten as

 ω[n]=ω[n−1]+α[n−1]δt, for  n=1,...,N, (25) θ[n]=θ[n−1]+ω[n−1]δt+12α[n−1]δ2t, for  n=1,...,N, (26) θ[N]=θ[0]+2π,ω[0]=ω[N],α[0]=α[N], (27) ∥a[n]∥2=∥a∥[n]∥2+∥a⊥[n]∥2=r2α2[n]+r2ω4[n]≤a2max, for  n=0,1,...N, (28) ωmin≤ω[n]≤ωmax, % for  n=0,1,...,N, (29) Pprop[n]=c1r3ω3[n]+c2rω[n]+c2rω3[n]g2+c2rα2[n]g2ω[n], for  n=0,1,...,N, (30)

where and are the tangential and centripetal accelerations, respectively, and and indicate the minimum and maximum angular velocity, respectively.

Similar to Section iii@, we address the minimum average rate maximization problem and the EE maximization problem for the circular trajectory, which are respectively formulated as

 (P3): max{θ[n],ω[n],α[n]}{r,pk[n],τ} τ (31a) s.t.rmin≤r≤rmax, (31b) (10b)−(10d),(25)−(29),
 (P4): max{θ[n],ω[n],α[n]}{r,pk[n],η}   η∑Nn=1Pprop[n] (32a) s.t.  (10c),(11b),(25)−(29),(31b),

where and denote the minimum and maximum radius of the circular trajectory, respectively. It is emphasized that (P3) and (P4) are difficult to solve because of the non-convex constraints and objective functions. To deal with the problems (P3) and (P4), similar SCA frameworks in Section iii@ are applied.

### A Minimum Average Rate Maximization and EE maximization

For the minimum average rate maximization problem (P3), we first find with given and then updates for a fixed . For given , we adopt the change of variable and as

 Sk[n]≜pk[n]hk[n]=pk[n]γ0(r−ζkcos(θ[n]−θk))2+ζ2ksin2(θ[n]−θk)+H2, (33)
 (34)

Similar to the method in Section iii@-A, we employ the SCA to . Based on Lemma 1, the concave surrogate function of with a solution at the -th iteration can be chosen as

 Slb1k,max[n] ≜ Ppeakγ0⎛⎜ ⎜⎝−(rl+1−ˇbk[n])2ˇA2k[n]+ˇBk[n](rl+1−ˇbk[n])(rl−ˇbk[n])+ˇCk[n]⎞⎟ ⎟⎠ (35) ≤ Sk,max[n],∀n,

where the constants , , , and are respectively given by

 ˇbk[n]≜ζkcos(θ[n]−θk), ˇAk[n]≜ζ2ksin2(θ[n]−θk)+H2,
 ˇBk[n]≜2⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1ˇA2k[n]−1((rl−ˇbk[n])2+ˇAk[n])2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,
 ˇCk[n]≜1(rl−ˇbk[n])2+ˇAk[n]+2(rl−ˇbk[n])2((rl−ˇbk[n])2+ˇAk[n])2−(rl−ˇbk[n])2ˇA2k[n].

By applying (15), (P3) for fixed can be reformulated as an approximated convex problem at the -th iteration of the SCA

 (P3.1): max{rl+1,Sk,l+1[n],τlb1} τlb1 (36a) s.t.  1NN∑n=1(log2(1+K∑m=1Sm,l+1[n])−˘R% ubk[n])≥τlb1,∀k, (36b) 0≤Sk[n]≤Slb1k,max[n],∀k,n, (36c) (10d),(31b),

where and (P3.1) can be successively solved by the CVX until convergence.

Next, we present a solution for (P3) with a given . To obtain the concave surrogate function of , we introduce the following lemma which identifies the surrogate function of the cosine function.

###### Lemma 3

For any given , the concave surrogate function of can be computed as

 −(ϕ−ϕl+sinϕl)22+cosϕl+sin2ϕl2≤cosϕ. (37)
###### Proof:

With a similar process in Appendix A, we can conclude that the function in (37) satisfies the conditions for a concave surrogate function [23].

By inspecting Lemmas 1 and 3, the concave surrogate function for can be identified as

 Slb2k,max[n]≜Ppeakγ0⎛⎜ ⎜⎝−rζk(θl+1[n]−^bk[n])2^A2k[n]+^Bk[n]sin(θl[n]−θk)(θl+1[n]−^bk[n])+^Ck[n]⎞⎟ ⎟⎠
 ≤Ppeakγ0rζk(θl+1[n]−^bk[n])2+^Ak[n]≤Sk,max[n], (38)

where , , , and