Outage Probability Minimization for UAV-Enabled Data Collection with Distributed Beamforming
aa r X i v : . [ c s . I T ] M a r Outage Probability Minimization for UAV-EnabledData Collection with Distributed Beamforming
Tianxin Feng ∗ , Lifeng Xie ∗ , Jianping Yao ∗ , and Jie Xu †∗ School of Information Engineering, Guangdong University of Technology † Future Network of Intelligence Institute (FNii) and School of Science and Engineering,The Chinese University of Hong Kong, ShenzhenE-mail: [email protected], [email protected], [email protected], [email protected]
Abstract —This paper studies an unmanned aerial vehicle(UAV)-enabled wireless sensor network, in which one UAV fliesin the sky to collect the data transmitted from a set of sensors viadistributed beamforming. We consider the delay-sensitive appli-cation scenario, in which the sensors transmit the common/sharedmessages by using fixed data rates and adaptive transmit powers.Under this setup, we jointly optimize the UAV’s trajectorydesign and the sensors’ transmit power allocation, in orderto minimize the transmission outage probability, subject to theUAV’s flight speed constraints and the sensors’ individual averagepower constraints. However, the formulated outage probabilityminimization problem is non-convex and thus difficult to beoptimally solved in general. To tackle this issue, we first considerthe special problem in the ideal case with the UAV’s flightspeed constraints ignored, for which the well-structured optimalsolution is obtained to reveal the fundamental performance upperbound. Next, for the general problem with the UAV’s flight speedconstraints considered, we propose an efficient algorithm to solveit sub-optimally by using the techniques of convex optimizationand approximation. Finally, numerical results show that ourproposed design achieves significantly reduced outage probabilitythan other benchmark schemes.
I. I
NTRODUCTION
Unmanned aerial vehicles (UAVs) or drones are expectedto have a lot of applications in beyond-fifth-generation (B5G)and sixth-generation (6G) wireless networks as dedicatedly de-ployed aerial wireless platforms and cellular-connected aerialusers (see, e.g., [1]–[5] and the references therein). Amongothers, there has been an upsurge of interest in using UAVsas aerial data collectors (or fusion centers) to collect datain large-scale wireless sensor networks. Different from theconventional design using on-ground fusion centers for datacollection, the UAVs in the sky can exploit the fully control-lable mobility in the three-dimensional (3D) space to fly closeto sensors for collecting data more efficiently, and can alsoleverage the strong line-of-sight (LoS) ground-to-air (G2A)channels for increasing the communication quality.In the literature, there are a handful of prior works studyingthe UAV-enabled data collection, in which the UAV trajectoryis designed for enhancing the system performance (see e.g.,[6]–[11]). For example, the authors in [6] and [7] jointlydesigned the UAV’s flight trajectory and wireless resourceallocation/scheduling to minimize the mission completiontime, in the scenarios when the sensors are deployed in
J. Yao is the corresponding author. one-dimensional (1D) and two-dimensional (2D) spaces, re-spectively. The authors in [8] and [9] optimized the UAVtrajectory and the sensors’ transmission/wakeup scheduling, inorder to maximize the energy efficiency of the wireless sensornetworks while ensuring the collected data amounts fromsensors. Furthermore, [10] exploited the UAV’s 3D trajectoryoptimization for maximizing the minimum average rate fordata collection, by considering angle-dependent Rician fadingchannels. In addition, [11] characterized the fundamental ratelimits of UAV-enabled multiple access channels (MAC) fordata collection in a simplified scenario with linearly deployedsensors on the ground. In these prior works, the on-ground de-vices (or sensors) were assumed to send independent messagesto the UAV under different multiple access techniques, andthe average data-rate throughput was used as the performancemetric by considering the adaptive-rate transmission.In contrast to communicating independently, distributedbeamforming has been recognized as another promising tech-nique to enhance the data rate and energy efficiency in wire-less sensor networks (see e.g., [12]–[14] and the referencestherein), in which a large number of sensors are enabled tocoordinate in transmitting common or shared messages to afusion center (the UAV of our interest). By properly controllingthe phases, the signals transmitted from different sensors canbe coherently combined at the fusion center, thus increasingthe communication range and enhancing the energy efficiencyvia exploiting the distributed beamforming gain. Under thistechnique, how to jointly design the UAV’s trajectory and thesensors’ wireless resource allocation for improving the datacollection performance is a new problem that has not beeninvestigated in the literature yet.Motivated by this, this paper focuses on a new UAV-enableddata collection system with distributed beamforming, in whichthe UAV collects data from multiple single-antenna sensorsvia the distributed beamforming. Different from prior worksconsidering the adaptive-rate transmission, we consider thedelay-sensitive application scenario (e.g., for real-time videodelivery) with adaptive-power but fixed-rate transmission. Inthis scenario, we aim to minimize the outage probabilityfor data collection by jointly optimizing the UAV’s trajec-tory and the sensors’ transmit power allocation over time,subject to the sensors’ individual average power constraintsand the UAV’s flight speed constraints. However, the outage ig. 1. Illustration of the UAV-enabled data collection system with distributedbeamforming. probability minimization problem is non-convex and generallydifficult to be optimally solved. To deal with this issue, wefirst consider the special problem in the ideal case withoutconsidering the UAV’s flight speed constraints, for whichthe well-structured optimal solution is obtained to reveal thefundamental performance upper bound. Then, motivated bythe obtained trajectory for the above special problem, wepropose an efficient approach to obtain a high-quality solutionto the general problem with the UAV’s flight speed constraintsconsidered, by using techniques from convex optimization andapproximation. Finally, numerical results show that our pro-posed design achieves significantly reduced outage probabilityas compared with other benchmark schemes.II. S
YSTEM M ODEL
As shown in Fig. 1, we consider a UAV-enabled datacollection system, in which one single-antenna UAV acts asa mobile date collector to periodically collect data from a setof K , { , . . . , K } single-antenna sensors on the ground. Weassume that all the sensors collaborate as a cluster to transmitcommon or shared sensing messages towards the UAV withdistributed beamforming employed. It is assumed that eachsensor k ∈ K is deployed at a fixed location ( x k , y k , on theground in the 3D Cartesian coordinate system. For notationalconvenience, let S k = ( x k , y k ) denote the horizontal locationof sensor k ∈ K , which is assumed to be known by the UAV a-priori to facilitate the trajectory design.We focus on one particular mission period of the UAV withfinite duration T in second (s), denoted by T , (0 , T ] . TheUAV is assumed to fly at a fixed altitude H , with the time-varying horizontal location q ( t ) = ( x ( t ) , y ( t )) for any timeinstant t ∈ T . Suppose that q I and q F denote the UAV’sinitial and final locations, respectively. Let V max denote theUAV’s maximum flying speed. Thus, we have ˙ x ( t ) + ˙ y ( t ) ≤ V , ∀ t ∈ T , (1) q (0) = q I , q ( T ) = q F , (2)where ˙ x ( t ) and ˙ y ( t ) denote the first-derivatives of x ( t ) and y ( t ) with respect to t , respectively. We also assume that theUAV’s mission duration T satisfies T ≥ k q F − q I k /V max , inorder for the trajectory from the initial to final locations tobe feasible. Accordingly, the distance between the UAV andsensor k ∈ K at any time instant t ∈ T is given by d k ( q ( t )) = p k q ( t ) − S k k + H . (3) As the G2A channels from sensors to UAVs are LoSdominated, we consider a channel model with LoS pathloss together with random phases. Consequently, the channelcoefficient between the UAV and sensor k ∈ K at any timeinstant t ∈ T is given by h k ( q ( t )) = q β d − αk ( q ( t )) e jψ k ( t ) , (4)where β denotes the channel power gain at the referencedistance of d = 1 m, j = √− denotes the imaginaryunit, ψ k ( t ) denotes the channel phase shift that is uniformlydistributed within the interval [ − π, π ] [13], and α ≥ denotesthe path loss exponent.In particular, we consider that all the sensors collaborate as acluster to transmit a common message s , which is a circularlysymmetric complex Gaussian (CSCG) random variable withzero mean and unit variance (i.e., s ∼ CN (0 , ). Suchcommon information can be obtained at different sensors eitherby their independent sensing (e.g., the common temperatureinformation) or via sharing with each other. At any timeinstant t ∈ T , the transmit signal of sensor k ∈ K is p P k ( t ) e jϕ k ( t ) s , where P k ( t ) ≥ and ϕ k ( t ) ∈ [ − π, π ] denotesensor k ’s transmit power and signal phase, respectively.Suppose that each sensor k ∈ K is subject to a maximumaverage transmit power P ave k . Therefore, the average transmitpower constraint for each sensor k is given by T Z T P k ( t )d t ≤ P ave k , ∀ k ∈ K . (5)Then, the received signal at the UAV at any time instant t ∈ T is given by y ( t ) = K X k =1 q P k ( t ) β d − αk ( q ( t )) e j ( ϕ k ( t )+ ψ k ( t )) s + v, (6)where v denotes the additive white gaussian noise (AWGN)at the UAV’s information receiver, which is a CSCG randomvariable with zero mean and variance σ (i.e., v ∼ CN (0 , σ ) ).In order to achieve the maximum received signal power at theUAV, we design the signal phase as ϕ k ( t ) = − ψ k ( t ) , ∀ k ∈K , t ∈ T . Thus, the received signal-to-noise ratio (SNR) atthe UAV at any time instant t ∈ T is given by SNR ( q ( t ) , { P k ( t ) } ) = E s K X k =1 q P k ( t ) β d − αk ( q ( t )) s ! /σ = K X k =1 q P k ( t ) β d − αk ( q ( t )) ! /σ , (7)where E s [ · ] denotes the stochastic expectation over the randomvariable s .In particular, we consider the delay-sensitive applicationscenario when the sensors use a fixed transmission rate. Inorder for the UAV to successfully decode the message at anygiven time instance, the received SNR must be no smallerthan a certain threshold γ min . In this case, the transmissionoutage occurs if the received SNR at the UAV falls below In order to realize the distributed beamforming, the UAV needs to transmitreference signals over time in order for the sensors to synchronize theirtransmissions [13]. min . Therefore, we use the following indicator function toindicate the transmission outage at any time instant t ∈ T . ( SNR ( q ( t ) , { P k ( t ) } )) = (cid:26) , SNR ( q ( t ) , { P k ( t ) } ) < γ min , , SNR ( q ( t ) , { P k ( t ) } ) ≥ γ min . Accordingly, we define the outage probability as the probabil-ity that the transmission is in outage over the whole duration T , which is expressed as O ( { q ( t ) , P k ( t ) } ) = 1 T Z T ( SNR ( q ( t ) , { P k ( t ) } )) d t. (8)Our objective is to minimize the outage probability O ( { q ( t ) , P k ( t ) } ) , by jointly optimizing the UAV’s trajectory { q ( t ) } and sensors’ power allocation { P k ( t ) } , subject to theUAV’s flight speed constraints in (1), the UAV’s initial andfinal locations constraints in (2), and the sensors’ averagetransmit power constraints in (5). Consequently, the outageprobability minimization problem of our interest is formulatedas ( P1 ) : min { q ( t ) ,P k ( t ) ≥ } O ( { q ( t ) , P k ( t ) } ) , s . t . (1) , (2) , and (5) . It is worth noting that the objective function of problem ( P1 ) isnon-convex and even non-smooth due to the indicator functionwith coupled variables q ( t ) ’s and P k ( t ) ’s. In addition, problem ( P1 ) contains an infinite number of optimization variables overcontinuous time. As a result, problem ( P1 ) is challenging tobe solved optimally.III. P ROPOSED S OLUTION TO P ROBLEM (P1)In this section, we first obtain the optimal solution to arelaxed problem of (P1) in the special case with T → ∞ to gain key engineering insights. Then, based on the optimalsolution under the special case, we propose an alternating-optimization-based algorithm to obtain an efficient solution tothe original problem ( P1 ) under any finite T . A. Optimal Solution to Relaxed Problem of ( P1 ) with T → ∞ First, we consider the special case that the UAV’s flightduration T is sufficiently large (i.e., T → ∞ ), such that wecan ignore the finite flight time of the UAV from one locationto another. As a result, the UAV’s flight speed constraints in(1) as well as the initial and final locations constraints in (2)can be neglected. Therefore, problem ( P1 ) can be relaxed as ( P1 . ) : min { q ( t ) } , { P k ( t ) ≥ } O ( { q ( t ) , P k ( t ) } ) , s . t . (5) . Though problem ( P1 . ) is still non-convex, it satisfies theso-called time-sharing condition [15]. Therefore, the strongduality holds between problem ( P1 . ) and its Lagrange dualproblem. As a result, we can optimally solve problem ( P1 . ) by using the Lagrange duality method [16] as follows.Let µ k ≥ denote the optimal dual variable associatedwith the k -th constraint in (5). For notational convenience, wedefine µ , [ µ , . . . , µ K ] . The Lagrangian of problem ( P1 . ) is given as ˜ L ( { q ( t ) } , { P k ( t ) } , µ ) = 1 T Z T ( SNR ( q ( t ) , { P k ( t ) } )) d t + Z T K X k =1 µ k P k ( t )d t − T K X k =1 µ k P ave k . (9) The dual function is ˜ g ( µ ) = min { q ( t ) } , { P k ( t ) ≥ } ˜ L ( { q ( t ) } , { P k ( t ) } , µ ) . (10)The dual problem of problem ( P1 . ) is given by ( D1 . ) : max { µ k ≥ } ˜ g ( µ ) . (11)In the following, we solve problem ( P1 . ) by first obtainingthe dual function ˜ g ( µ ) and then solving the dual problem( D1 . ). First, to obtain ˜ g ( µ ) , we solve problem (10) by solvingthe following subproblem, in which the index t is dropped forfacilitating the analysis. min q , { P k ≥ } ( SNR ( q , { P k } )) + K X k =1 µ k P k . (12)To solve problem (12), we consider the following two caseswhen ( SNR ( q , { P k } )) equals one and zero, respectively.First, consider that ( SNR ( q , { P k } )) = 1 . In this case, wehave P k = 0 , and q can be any arbitrary value. Accordingly,the optimal value for problem (12) is .Next, consider that ( SNR ( q , { P k } )) = 0 . In this case,we solve problem (12) by first deriving the sensors’ powerallocation under any given UAV’ location q and then searchover q via a 2D exhaustive search. Under given q and defining ρ k = √ P k , ∀ k ∈ K , problem (12) is reduced as min { ρ k ≥ } K X k =1 µ k ρ k (13) s . t . K X k =1 ρ k q β d − αk ( q ) ≥ √ γ min σ. Notice that problem (13) is a convex optimization problem.If µ k > , then we check the Karush-Kuhn-Tucker (KKT)conditions, and have the optimal solution as ρ ( µ , q ) k = q γ min β d − αk ( q ) σ (cid:18) P Kk =1 ( β d − αk ( q ) /µ k ) (cid:19) µ k . (14)If µ k = 0 , then problem (13) is a linear program, for which theoptimal solution of { ρ ( µ , q ) k } can be obtained via CVX [16].Furthermore, suppose that P ( µ , q ) k = ρ ( µ , q ) k . By substituting P ( µ , q ) k into problem (12), we can obtain the optimal UAVlocation q ( µ ) by using the 2D exhaustive search, given as q ( µ ) = arg min q ( SNR ( q , { P k } )) + K X k =1 µ k P ( µ , q ) k . Accordingly, the obtained power allocation is given by { P ( µ , q ( µ ) ) k } . In this case, the optimal value for problem (12)is P Kk =1 µ k P ( µ , q ( µ ) ) k .By comparing the corresponding optimal values under ( SNR ( q , { P k } )) = 1 and ( SNR ( q , { P k } )) = 0 , we can ob-tain the optimal solution to problem (12) as the one achievingthe smaller optimal value. Therefore, the dual function ˜ g ( µ ) is obtained.Next, we solve the dual problem ( D1 . ) by maximizing thedual function ˜ g ( µ ) . This is implemented via using subgradient-based methods, such as the ellipsoid method [17]. We denotethe optimal dual solution to ( D1 . ) as µ opt .inally, with the optimal µ opt obtained, it remains to findthe optimal primal solution to ( P1 . ) . Notice that under µ opt ,the optimal solution to problem (12) is non-unique in general.Suppose that there are ˜ V solutions, denoted by { q ( µ opt )˜ ν } and { P ( µ opt , q ( µ opt )˜ ν ) k } , ˜ ν = 1 , . . . , ˜ V . In this case, we need to timeshare among these UAV locations and the corresponding powerallocation strategies to construct the primal optimal solutionto ( P1 . ) as follows.Let ˜ τ ˜ ν denote the UAV’s hovering durations at the location q ( µ opt )˜ ν , ˜ ν = 1 , . . . , ˜ V . In the following, we solve the followingproblem to obtain the optimal hovering durations for timesharing. min { ˜ τ ˜ ν ≥ } T (cid:18) T − ˜ V X ˜ ν =1 ˜ τ ˜ ν (cid:19) s . t . ˜ V X ˜ ν =1 ˜ τ ˜ ν P ( µ opt , q ( µ opt )˜ ν ) k ≤ T P ave k , ∀ k ∈ K (15a) ˜ V X ˜ ν =1 ˜ τ ˜ ν ≤ T. (15b)As problem (15) is a linear program, the optimal hoveringdurations { ˜ τ opt ˜ ν } can be obtained by CVX. Therefore, problem( P1 . ) is finally solved. Note that at the optimal solution, theUAV hovers at multiple locations over time to collect data fromsensors, and the sensors adopt an on-off power allocation, i.e.,the sensors are active to send messages with properly designedpower allocation when no outage occurs, but inactive with zerotransmit power when outage occurs. Also note that if µ k =0 , ∀ k ∈ K , then the resulting outage probability is zero (i.e.,no outage occurs during the data collection); otherwise, theduration with outage occurring is given by ˜ τ opt = T − ˜ V P ˜ ν =1 ˜ τ opt ˜ ν ,with the resulting outage probability being ˜ τ opt /T . B. Proposed Solution to Problem ( P1 ) with Finite T In this subsection, we consider problem ( P1 ) in the generalcase with finite T . Motivated by the optimal solution to therelaxed problem ( P1 . ) in the previous subsection, we proposean efficient solution based on the techniques from convexoptimization and approximation. Towards this end, we firstdiscretize the whole duration T into a finite number of N time slots denoted by the set N , { , ..., N } , each with equalduration δ = T /N . Accordingly, problem ( P1 ) is re-expressedas ( P1 . ) : min { q [ n ] } , { P k [ n ] ≥ } N N X n =1 ( SNR ( q [ n ] , { P k [ n ] } ))s . t . N N X n =1 P k [ n ] ≤ P ave k , ∀ k ∈ K (16a) k q [ n ] − q [ n − k ≤ V δ , ∀ n ∈ N (16b) q [0] = q I , q [ N ] = q F . (16c)Notice that problem ( P1 . ) is still non-convex.To tackle this issue, define l n ( q [ n ] , { P k [ n ] } ) = SNR ( q [ n ] , { P k [ n ] } ) − γ min , ∀ n ∈ N and l ( { q [ n ] } , { P k [ n ] } ) = [ l ( q [1] , { P k [1] } ) , . . . , l N ( q [ N ] , { P k [ N ] } )] . As a result,problem ( P1 . ) is equivalently expressed as ( P1 . ) : min { q [ n ] } , { P k [ n ] ≥ } N k l ( { q [ n ] } , { P k [ n ] } ) k s . t . (16 a ) , (16 b ) , and (16 c ) , where k x k denotes the zero norm of a vector x returning thenumber of non-zero coordinates of x . To handle the zero-normfunction in problem ( P1 . ) , we use k l ( { q [ n ] } , { P k [ n ] } ) k to approximate k l ( { q [ n ] } , { P k [ n ] } ) k [18]. Note that to re-duce the outage probability with minimized energy consump-tion, the received SNR of each time slot should not belarger than γ min . Thus, we have the following constraints: SNR ( q [ n ] , { P k [ n ] } ) ≤ γ min , ∀ n ∈ N . By further introducingtwo sets of auxiliary variables { a k [ n ] } and { A k [ n ] } , k ∈K , n ∈ N , problem ( P1 . ) is approximated as ( P1 . ) : max { q [ n ] } , { P k [ n ] ≥ } , { A [ n ] } , { a k [ n ] } N N X n =1 A [ n ] /σ s . t . A [ n ] ≤ K X k =1 a k [ n ] ! , ∀ n ∈ N (17a) a k [ n ] ≤ s P k [ n ] β ( k q [ n ] − S k k + H ) α/ , ∀ k ∈ K , n ∈ N (17b) A [ n ] /σ ≤ γ min , ∀ n ∈ N (17c) (16 a ) , (16 b ) , and (16 c ) . Problem ( P1 . ) is still non-convex due to the non-convexconstraints in (17a) and (17b).Next, we solve the non-convex problem ( P1 . ) by optimiz-ing the UAV trajectory and the sensors’ power allocation inan alternating manner. First, under any given { P k [ n ] ≥ } , weoptimize the UAV trajectory by adopting the successive convexapproximation (SCA) technique. In particular, we update theUAV trajectory { q [ n ] } and { a k [ n ] } in an iterative manner byapproximating the non-convex problem into a convex problem.Let { q ( i ) [ n ] } and { a ( i ) k [ n ] } denote the local points at the i -thiteration. Under given UAV trajectory { q ( i ) [ n ] } and { a ( i ) k [ n ] } ,since any convex function is globally lower-bounded by it first-order Taylor expansion at any point, we have the lower boundsfor q P k [ n ] β ( k q [ n ] − S k k + H ) α/ and (cid:16)P Kk =1 a k [ n ] (cid:17) as follows. s P k [ n ] β ( k q [ n ] − S k k + H ) α/ ≥ p P k β (cid:18) ( k q ( i ) [ n ] − S k k + H ) − α/ − α ( k q [ n ] − S k k − k q ( i ) [ n ] − S k k )4( k q ( i ) [ n ] − S k k + H ) α/ (cid:19) , a low k ( i ) ( q [ n ]) , (18) (cid:18) K X k =1 a k [ n ] (cid:19) ≥ (cid:18) K X k =1 a ( i ) k [ n ] (cid:19) + 2 (cid:18) K X k =1 a ( i ) k [ n ] (cid:19) × (cid:18) K X k =1 a k [ n ] − K X k =1 a ( i ) k [ n ] (cid:19) , A low( i ) ( a k [ n ]) . (19)In each iteration i with given local point { q ( i ) [ n ] } nd { a ( i ) k [ n ] } , we replace q P k [ n ] β ( k q [ n ] − S k k + H ) α/ and (cid:18) P Kk =1 a k [ n ] (cid:19) as their lower bounds a low k ( i ) ( q [ n ]) and A low( i ) ( a k [ n ]) , respectively. As a result, the trajectoryoptimization problem becomes a convex optimizationproblem, which can be optimally solved by CVX.Next, under any given UAV trajectory, we optimize thesensors’ power allocation by using the SCA technique aswell. Similarly as for optimizing the UAV trajectory, weapproximate the non-convex terms into convex forms, so as tooptimize the UAV trajectory iteratively, for which the detailsare omitted for brevity. By alternately updating the UAVtrajectory and sensors’ power allocation, we can obtain aconverged solution to problem ( P1 . ) , which is denoted by { q ∗ [ n ] } and { P ∗ k [ n ] } .Finally, we use an additional step to obtain the sensors’power allocation { P k [ n ] } for problem ( P1 . ) under the ob-tained UAV trajectory { q ∗ [ n ] } , for which the problem is givenas ( P1 . ) : min { P k [ n ] ≥ } N N X n =1 ( SNR ( q ∗ [ n ] , { P k [ n ] } ))s . t . N N X n =1 P k [ n ] ≤ P ave k , ∀ k ∈ K . (20)To solve problem ( P1 . ) , we sort the timeslots based on the SNR { SNR ( q ∗ [ n ] , { P ∗ k [ n ] } ) } ,i.e., SNR ( q ∗ [ π (1)] , { P ∗ k [ π (1)] } ) ≥ · · · ≥ SNR ( q ∗ [ π ( N )] , { P ∗ k [ π ( N )] } ) , with π ( · ) denoting thepermutation over N . Then, we allocate the sensors’ transmitpower over a subset N ′ of time slots with the highest SNRvalues, i.e., N ′ = { π (1) , . . . , π ( N ′ ) } , where N ′ is a variableto be determined. To find N ′ and the corresponding powerallocation, we define the following feasibility problem. ( P1 . ) :find { P k [ n ] ≥ } s . t . SNR ( q ∗ [ π ( n )] , { P k [ π ( n )] } ) ≥ γ min , ∀ n ∈ N ′ (21a) N ′ N ′ X n =1 P k [ π ( n )] ≤ P ave k , ∀ k ∈ K . (21b)By letting ρ ′ k [ n ] = √ P k [ n ] , problem ( P1 . ) can be trans-formed into a convex form and thus be solved optimally viaCVX. By solving problem ( P1 . ) under given N ′ togetherwith a bisection search over N ′ , we can find a high-qualitysolution to problem ( P1 . ) . By combining this together with { q ∗ [ n ] } , an efficient solution of N ′ and the correspondingpower allocation at sensors to problem ( P1 ) is finally obtained.Note that in order to guarantee the performance of theobtained solution to problem ( P1 ) , we need an initial point foriteration. Here, we choose the successive hover-and-fly (SHF)trajectory as the initial point. In SHF trajectory, the UAV fliesat the maximum speed from the initial location to successivelyvisit these optimal hovering locations and finally flies to finallocation. During the flight, we choose the minimum flyingpath by solving the traveling salesman problem (TSP) (see,e.g., [3]). Suppose that the minimum flying duration amongthese locations is T fly . If T < T fly , we alternatively consider the direct flight as the initial point, i.e., the UAV flies frominitial location to final location directly at a constant speed k q I − q F k /T . IV. N UMERICAL R ESULTS
In the simulation, we consider the scenario with sensors,which are located at (20 , m, (30 , m, (46 , m, (56 , m, (94 , m, (100 , m, (112 , m, (162 , m, (178 , m, and (200 , m. We set β = − dB, σ = − dBm, K = 10 , α = 2 . , V max = 40 m/s, N = 128 , H = 50 m, q I = ( , ) m, q F = ( , ) m, and γ min = 550 .First, Fig. 2 shows the system setup and the obtainedtrajectories with T = 20 s. It is observed that there are ˜ V = 3 optimal hovering locations for problem ( P1 . ) .Next, we compare the performance of our proposed designversus the following three benchmark schemes. • Fly-hover-fly trajectory design : The UAV flies straightlyfrom the initial location to one optimized fixed location ( x fix , y fix , H ) , and then to the final location at the maxi-mum speed. The fixed location ( x fix , y fix , H ) is obtainedvia 2D exhaustive search to minimize the outage proba-bility. Under such trajectory, the sensors’ power allocationcan be obtained by solving problem ( P1 . ) . • Power design only : In this scheme, the UAV flies fromthe initial location to the final location with a constantflight speed. Under such trajectory, the power allocationsat sensors are obtained by solving problem ( P1 . ) . • Trajectory design only : In this scheme, the sensorsuse the uniform power allocation and accordingly theUAV’s trajectory is obtained by iteratively solving prob-lem ( P1 . ) . x (m) y ( m ) Initial locationFinal locationOptimal hovering locations for (P1.1)Sensors‘ locationsProposed designSHF trajectory
Fig. 2. System setup and the obtained trajectories with T = 20 s. Fig. 3 shows the outage probability of the system versusthe sensor’s maximum average power P ave k = P ave , ∀ k ∈K , where T = 20 s. It is observed that when P ave is lessthan 31 dBm, the outage probability achieved by the trajectorydesign only scheme is ; while that achieved by other schemesis less than . This shows that power optimization is quite Average power, P ave (dBm) O u t age p r obab ili t y Performance upper bound by (P1.1)Proposed designFly-hover-fly trajectory designPower design onlyTrajectory design only
Fig. 3. Outage probability versus the sensor’s maximum average transmitpower P ave . significantly in this case. It is also observed that our proposeddesign considerably outperforms other benchmark schemes inall regimes of transmit power, by jointly designing the UAV’strajectory and the sensors’ power allocation.
10 20 30 40 50 60 70 80 90 100
Duration, T (s) O u t age p r obab ili t y Performance upper bound by (P1.1)Proposed designFly-hover-fly trajectory designPower design only
Fig. 4. Outage probability versus the flight duration T . Fig. 4 shows the outage probability versus the flight duration T , where P ave k = 30 dBm, ∀ k ∈ K . Notice that the trajectorydesign only scheme always leads to the outage probability ofone, and therefore, this scheme is not shown in this figure.It is observed that the proposed design achieves much loweroutage probability than the other benchmark schemes, and theperformance gain becomes more substantial when T becomeslarge. Furthermore, with sufficiently large T , the proposeddesign is observed to lead to similar performance as theperformance upper bound achieved by problem ( P1 . ) .V. C ONCLUSION
In this paper, we considered the UAV-enabled data col-lection from multiple sensors with distributed beamforming.We minimized the transmission outage probability, by jointly optimizing the UAV’s trajectory and the sensors’ power alloca-tion. To deal with this challenging problem, we first optimallysolved the relaxed problem without considering the UAV’sflight speed constraints. Next, we used the techniques fromconvex optimization and approximation to find the sub-optimalsolutions to the general problem. Finally, we conducted simu-lations to show the effectiveness of our proposed design. Howto extend our results to other scenarios, e.g., with multipleUAVs and multi-antenna UAVs is an interesting directionworth further investigation.R
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