Cognitive UAV Communication via Joint Maneuver and Power Control
aa r X i v : . [ c s . I T ] J u l Cognitive UAV Communication via Joint Maneuverand Power Control
Yuwei Huang, Weidong Mei, Jie Xu,
Member, IEEE,
Ling Qiu,
Member, IEEE, and Rui Zhang,
Fellow, IEEE
Abstract —This paper investigates a new scenario of spectrumsharing between unmanned aerial vehicle (UAV) and terrestrialwireless communication, in which a cognitive/secondary UAVtransmitter communicates with a ground secondary receiver(SR), in the presence of a number of primary terrestrial com-munication links that operate over the same frequency band. Weexploit the UAV’s mobility in three-dimensional (3D) space to im-prove its cognitive communication performance while controllingthe co-channel interference at the primary receivers (PRs), suchthat the received interference power at each PR is below a pre-scribed threshold termed as interference temperature (IT). First,we consider the quasi-stationary UAV scenario, where the UAV isplaced at a static location during each communication period ofinterest. In this case, we jointly optimize the UAV’s 3D placementand power control to maximize the SR’s achievable rate, subjectto the UAV’s altitude and transmit power constraints, as well as aset of IT constraints at the PRs to protect their communications.Next, we consider the mobile UAV scenario, in which the UAVis dispatched to fly from an initial location to a final locationwithin a given task period. We propose an efficient algorithm tomaximize the SR’s average achievable rate over this period byjointly optimizing the UAV’s 3D trajectory and power control,subject to the additional constraints on UAV’s maximum flyingspeed and initial/final locations. Finally, numerical results areprovided to evaluate the performance of the proposed designs fordifferent scenarios, as compared to various benchmark schemes.It is shown that in the quasi-stationary scenario the UAV shouldbe placed at its minimum altitude while in the mobile scenario theUAV should adjust its altitude along with horizontal trajectory,so as to maximize the SR’s achievable rate in both scenarios.
Index Terms —UAV communication, spectrum sharing, 3Dplacement, 3D trajectory design, power control, interferencemanagement.
I. I
NTRODUCTION
With continuous technology advancement and cost reduc-tion, unmanned aerial vehicles (UAVs) or drones have beenmore widely used in various applications, such as cargodelivery, aerial photography, surveillance, search and rescue,etc [2]. It is projected by Federal Aviation Administration(FAA) [3] that there will be around seven million UAVs in
This work has been presented in part at the IEEE International Workshop onSignal Processing Advances in Wireless Communications (SPAWC), June 25–28, 2018,Kalamata, Greece [1].Y. Huang and L. Qiu are with the Key Laboratory of Wireless-Optical Communica-tions, Chinese Academy of Sciences, School of Information Science and Technology,University of Science and Technology of China, Hefei, Anhui, 230027, China (e-mail:[email protected], [email protected]).W. Mei is with the NUS Graduate School for Integrative Sciences and Engineering,National University of Singapore, and also with the Department of Electrical andComputer Engineering, National University of Singapore (e-mail: [email protected]).J. Xu is with the School of Information Engineering, Guangdong University ofTechnology, and also with the National Mobile Communications Research Laboratory,Southeast University (e-mail: [email protected]). J. Xu is the corresponding author.R. Zhang is with the Department of Electrical and Computer Engineering, NationalUniversity of Singapore (e-mail: [email protected]). the United States only in 2020. With the explosively increasingnumber of UAVs, how to integrate them into future wirelessnetworks to enable their bidirectional communications with theground users/pilots has become a critical task to be tackled. Onone hand, for emergency situations (e.g., after natural disaster)and temporary hotspots (e.g., stadium during a football match),UAVs can be employed as aerial wireless communicationplatforms (e.g., relays or base stations (BSs)) to provide dataaccess, enhance coverage, and improve communication ratesfor ground users [4], [5]. On the other hand, for UAVs in vari-ous missions (e.g., cargo delivery), it is crucial to enable themas aerial mobile users to access existing wireless networks(e.g., cellular networks), in order to support not only secure,reliable, and low-latency remote command and control, butalso high-capacity mission-related data transmission [6]–[8].Therefore, UAV-assisted terrestrial communications [4] andnetwork-connected UAV communications [7] have becometwo widely investigated paradigms for integrating UAVs intofuture wireless communication networks.UAV communications are different from conventional ter-restrial wireless communications in the following two mainaspects. First, UAVs normally have strong line-of-sight (LoS)links with ground nodes, thus offering better channel con-ditions than terrestrial fading channels and even makingit possible to predict channel state information (CSI) andhence communication performances at different UAV’s three-dimensional (3D) locations based on the ground nodes’ lo-cation information. Second, UAVs have fully controllablemobility in 3D, by exploiting which UAVs can adjust theiraltitude and horizontal location over time to optimize theircommunication performances with ground nodes.In the literature, there are generally two lines of researchthat exploit the UAV mobility for communication performanceoptimization, namely the quasi-stationary UAV with 3D place-ment optimization and the mobile UAV with 3D trajectoryoptimization, respectively. For the quasi-stationary scenario,the UAV is placed at a static location over each communicationperiod of interest, while its location can be changed from oneperiod to another. This may practically correspond to a UAVcommunication platform that is connected by a cable/wirewith a ground control platform (see, e.g., “flying cell-on-wings (COWs)” of AT&T [9] and “Air Masts” of Everything-Everywhere (EE: the UK’s largest mobile network operator)[10]). Substantial research efforts have been devoted to thisresearch paradigm. For example, the works [11] and [12]optimized the UAV-BS’s altitude to maximize the coverageprobability on the ground and minimize the system outageprobability, respectively; while [13] optimized the UAV-BS’s
3D location to maximize the number of served ground users.Furthermore, [14], [15] and [16] optimized multiple UAV-BSs’locations to minimize the number of required UAV-BSs tocover a given area and maximize the minimum throughputamong all ground users, respectively. In [17], the downlinkcoverage probability for a reference ground user was analyzedin the presence of multiple UAV-BSs, while [18] showed thatthe deployment of UAV-BSs at their optimized locations canimprove the coverage performance and spectral efficiency ofthe network. In addition, [19] investigated the optimal place-ment of a UAV-relay to maximize the end-to-end throughputfrom a source to a destination by using a new LoS map basedapproach.On the other hand, under the mobile UAV scenario, priorworks have designed the UAV trajectory (i.e., 3D locationsover time) jointly with communication scheduling and re-source allocation for performance optimization. For example,when the UAV is employed as a mobile relay, the authors in[4] and [20] optimized the UAV-relay’s trajectory to maximizethe end-to-end throughput and minimize the system outageprobability, respectively. When UAVs are employed as cellularBSs, the authors in [5], [21]–[24] optimized the UAVs’ trajec-tories to maximize the achievable rates under different setupssuch as broadcast channel [5], [21], multicast channel [22],[23], and interference channel [24]. Furthermore, the UAVtrajectory design was also investigated for other applicationswhen the UAV is employed as an access point (AP) forwireless power transfer [25], wireless powered communication[26], and mobile edge computing [27]. In addition, when UAVsact as cellular users that perform tasks in a long range, theworks [7] and [8] studied the UAV users’ trajectory designto minimize the mission completion time, subject to variouscommunication connectivity constraints with ground BSs. In[28], an interference-aware path planning design was proposedfor multiple UAV users, which aimed to achieve an optimaltrade-off between energy efficiency, latency and interferencecaused by the UAVs to the ground network. The work [29]proposed an energy-efficient path planning design to minimizethe energy consumption of UAV swarms, subject to individualenergy availability constraints at UAVs.Despite the above research progress, existing works havemostly assumed that the UAV communications are operatedover dedicated frequency bands. Nevertheless, due to thescarcity of wireless spectrum, it is practically difficult toallocate dedicated spectrum to new UAV communications.To address this challenge and motivated by the technicaladvancement of spectrum sharing in cognitive radio (CR) [30],a viable solution is to allow UAVs to operate as cognitive orsecondary communication nodes to access the spectrum thatis originally allocated to existing (primary) terrestrial wirelesscommunication networks (see, e.g., [31]). For instance, in thenetwork-connected UAV communication (as shown in Fig.1(a)), the UAV communicates with its associated ground BSby reusing the resource blocks (RBs) assigned to existingground users; whereas in the device-to-device (D2D)-enabledUAV-ground communication (as shown in Fig. 1(b)), the UAVcommunicates with its associated ground user via D2D com-munication by reusing the RBs in the uplink cellular commu- nications. In both the above two cases, a new and severe air-to-ground (A2G) interference issue needs to be tackled [6], [32]–[34], since the A2G channels are normally LoS-dominated.Specifically, the UAV may impose severe uplink interferenceto multiple co-channel non-associated ground BSs (primaryreceivers (PRs)) in network-connected UAV communication(Fig. 1(a)). Similarly, the D2D communication from the UAVto ground user may impose severe uplink interference at co-channel ground BSs (PRs) in D2D-enabled UAV-ground com-munication (Fig. 1(b)). As a result, how to maximize the UAVcommunication throughput while effectively mitigating theA2G co-channel interference to the primary communicationsystem is an important and yet challenging problem that callsfor innovative solutions. It is worth noting that there havebeen some initial studies on A2G interference mitigation fornetwork-connected UAV communication in the literature [35]–[37], which, however, only considered the case of a staticUAV user. By leveraging the UAV’s controllable mobility,in this paper, we propose a new approach to tackle thisproblem, which jointly optimizes the UAV’s 3D placement ortrajectory (for the quasi-stationary and mobile UAV scenarios,respectively) and CR-based interference-aware transmit powercontrol to achieve the maximum throughput of UAV-to-groundsecondary communication, while controlling the interferenceto existing primary ground receivers below a tolerable level.For the purpose of exposition, this paper considers aspectrum sharing system where a cognitive/secondary UAVtransmitter communicates with a ground secondary receiver(SR), in the presence of a number of primary terrestrialcommunication links that operate over the same frequencyband. Under this setup, we adopt the interference temperature (IT) technique in CR [41] to protect the primary communi-cations, so that the received power at each PR cannot exceeda prescribed IT threshold. The main results of this paper aresummarized as follows. • First, we consider the quasi-stationary UAV scenario, inwhich the UAV is placed at an optimized location thatis fixed during the communication period of interest. Wejointly optimize the UAV’s 3D placement and transmitpower to maximize the SR’s achievable rate, subject tothe UAV’s flight altitude and transmit power constraints,and a set of IT constraints at the PRs. The joint 3Dplacement and power optimization problem is non-convexand difficult to be optimally solved in general. To tacklethis challenge, we first prove that the UAV should beplaced at the lowest altitude at the optimality. Buildingupon this, we further use the semi-definite relaxation(SDR) technique to obtain the UAV’s optimal horizontallocation and transmit power. • Next, we consider the mobile UAV scenario, in which theUAV is dispatched to fly from an initial location to a finallocation during a particular task period. We maximizethe SR’s average achievable rate over this period byjointly optimizing the UAV’s 3D trajectory and transmitpower over time, subject to the UAV’s maximum flyingspeed, altitude, and transmit power constraints, as wellas the PRs’ IT constraints. Due to the time-dependentUAV trajectory variables, this problem is more involved
GroundBS GroundBS GroundBS . . .
Information flow Interference . . . (a) Network-connected UAV communication.
GroundBS GroundBS Ground user . . .
Information flow Interference . . . (b) D2D-enabled UAV-ground communication.Fig. 1. Illustration of the cognitive UAV communication systems. and thus more difficult to be solved as compared to thatin the quasi-stationary scenario. To tackle this problem,we propose an efficient algorithm that ensures a locallyoptimal solution by applying the technique of successiveconvex approximation (SCA). • Finally, numerical results are presented to validate theperformance of our proposed cognitive UAV communi-cation designs, as compared to other benchmark schemes,for both the quasi-stationary and mobile scenarios.Specifically, it is shown that in the mobile scenario,the UAV needs to adaptively adjust its altitude togetherwith horizontal location over time to balance the trade-off between maximizing the SR’s rate versus minimizingthe interference with PRs. This is in a sharp contrast tothe quasi-stationary scenario, where it is shown that theUAV should always be placed at its lowest altitude at theoptimality.The remainder of this paper is organized as follows. SectionII introduces the system model of the cognitive UAV com-munication system and formulates the optimization problemsof our interest. Section III presents the optimal solutionto the joint 3D placement and power control problem inthe quasi-stationary UAV scenario. Section IV proposes anefficient algorithm to obtain a locally optimal solution to thejoint 3D trajectory and power optimization problem in themobile UAV scenario. Section V provides numerical resultsto demonstrate the efficacy of our proposed designs versusbenchmark schemes. Finally, Section VI concludes this paper.
Notations:
In this paper, scalars are denoted by italic letters,vectors and matrices are denoted by bold-face lower-case andupper-case letters, respectively. R x × y denotes the space of x × y real-valued matrices. For a square matrix M , Tr ( M ) ,det ( M ) , and rank ( M ) represent its trace, determinant, andrank, respectively, while M (cid:23) ( M (cid:22) ) means that M ispositive (negative) semi-definite. I and denote an identitymatrix and an all-zero matrix with proper dimensions, respec-tively. For a vector a , k a k represents its Euclidean norm, a T denotes its transpose, and diag ( a ) denotes a diagonalmatrix whose diagonal elements are specified by a . For a time-dependent function x ( t ) , ˙ x ( t ) denotes its first derivative withrespect to time t . The notation log ( · ) denotes the logarithmfunction with base 2, e denotes the natural constant, and E ( · ) denotes the statistic expectation. II. S YSTEM M ODEL
As shown in Fig. 1, we consider a new spectrum sharing sce-nario for UAV communications, where a cognitive/secondaryUAV transmitter communicates with a ground SR, in thepresence of a set of K ≥ primary users that operate over thesame frequency band. Let K , { , . . . , K } denote the set ofground PRs. We focus on the cognitive UAV communicationover a particular mission period, denoted by T = [0 , T ] , withduration T > in second (s). In practice, the mission period T is generally prescribed, which is set based on the UAV’s maxi-mum endurance and the requirements in different applications.Without loss of generality, we consider a 3D coordinate systemwith the SR located at the origin (0 , , and each PR k ∈ K at a fixed location ( x k , y k , , where w k = ( x k , y k ) ∈ R × denotes the horizontal location of PR k . We consider offlineoptimization in this paper by assuming that the UAV perfectlyknows the locations of the ground SR and PRs , as well asthe channel propagation environments (channel parameters) a-priori to facilitate the joint maneuver and power controldesign. This provides key insights and the performance upperbound for practical designs with partial/imperfect knowledgeof location and channel information. In the following, weconsider the 3D placement optimization and 3D trajectoryoptimization (jointly with power control) for quasi-stationary and mobile UAV scenarios, respectively.
A. Quasi-Stationary UAV Scenario
First, we consider the quasi-stationary UAV scenario, inwhich the UAV is placed at a fixed location ( x, y, z ) (to be op-timized later) over the communication period T . For notationalconvenience, let q = ( x, y ) and z denote the UAV’s horizontallocation and altitude, respectively. Accordingly, the distancesfrom the UAV to the SR and each PR k ∈ K are given by d ( q , z ) = p z + k q k and d k ( q , z ) = p z + k q − w k k ,respectively. Furthermore, let H min > and H max > denote the minimum and maximum flight altitudes of the UAV,respectively. Then we have H min ≤ z ≤ H max . As shown in Fig. 1, as the PRs in both of our considered scenarios andthe SR in the network-connected UAV communication scenario are groundBSs at fixed locations, their location information can be easily obtained bythe UAV a priori . In the D2D-enabled UAV-ground communication scenario,the SR can obtain its location via global positioning system (GPS) and thenreports such information to the UAV.
In practice, A2G wireless channels are normally dominatedby the LoS links owing to the UAV’s high flight altitude [38]–[40]. Therefore, we consider the LoS channel model with path-loss exponent α ≥ for the wireless channels from the UAVto the SR and the PRs . As such, the UAV can easily obtainthe CSI with them over time based on its own as well as their(fixed) locations. As a result, the channel power gains from theUAV to the SR and each PR k ∈ K are respectively expressedas h ( q , z ) = β u d − α ( q , z ) = β u ( z + k q k ) α/ , (1) g k ( q , z ) = β g,k d − αk ( q , z ) = β g,k ( z + k q − w k k ) α/ , (2)where β u and β g,k denote the reference channel power gainsfrom the UAV to the SR and each PR k ∈ K , respectively,including the transmit and receive antenna gains of commu-nication nodes involved. In practice, the UAV may adjust itsantenna’s main lobe towards the SR to improve the cognitivecommunication rate, and the PRs (ground BSs) may adjusttheir main lobes downwards to better serve their respectiveprimary transmitters (ground users) by reducing the co-channelinterference from other primary transmitters. As a result, theA2G interference is generated and received via the side-lobesof the UAV’s and the PRs’ antennas, respectively. Therefore,we have β g,k ≤ β , ∀ k ∈ K , where β denotes the maximumreference channel power gain from the UAV to the PRs whenthey are all equipped with the omnidirectional antennas.Accordingly, by letting p ≥ denote the transmit power ofthe UAV, the maximum achievable rate from the UAV to theSR in bits/second/Hertz (bps/Hz) is given by R ( p, q , z ) = log (cid:18) h ( q , z ) pσ (cid:19) = log (cid:18) η u p ( z + k q k ) α/ (cid:19) , (3)where σ denotes the total power of receiver noise andterrestrial interference at the SR, and η u , β u /σ denotes thereference signal-to-interference-plus-noise ratio (SINR). Let P > denote the maximum transmit power at the UAV. Wethus have ≤ p ≤ P .Under spectrum sharing, the secondary UAV communica-tion introduces A2G co-channel interference to the groundPRs, and the resultant interference power at each PR k ∈ K isgiven by ˜ Q k ( p, q , z ) = g k ( q , z ) p = β g,k p ( z + k q − w k k ) α/ . As theUAV may not be able to know the exact receive antenna gain ateach PR (due to the unknown receive antenna direction), weconsider the worst-case A2G interference by replacing β g,k with β , ∀ k ∈ K . Thus, we have ˜ Q k ( p, q , z ) ≤ Q k ( p, q , z ) = β p ( z + k q − w k k ) α/ , ∀ k ∈ K . (4)In order to protect the primary communications, we apply theIT technique that is widely adopted in the CR literature (see, Notice that the proposed methods can also be extended to handle otherA2G channel models such as Rician fading and probabilistic LoS channelmodels. Please refer to Remark 3.2 in Section III and Remark 4.2 in Section IVfor details under the quasi-stationary and mobile UAV scenarios, respectively. e.g., [41]), such that the received (worst-case) interferencepower Q k ( p, q , z ) at each PR k cannot exceed a maximumthreshold, denoted by Γ ≥ , i.e., Q k ( p, q , z ) ≤ Γ , ∀ k ∈ K ,and thus we have β p/ ( z + k q − w k k ) α/ ≤ Γ , ∀ k ∈ K .In the quasi-stationary UAV scenario, our objective is tomaximize the SR’s achievable rate (i.e, R ( p, q , z ) ), by jointlyoptimizing the UAV’s 3D location q and z , and transmit power p . The problem is formulated as max p, q ,z log (cid:18) η u p ( z + k q k ) α/ (cid:19) s.t. H min ≤ z ≤ H max , (5) ≤ p ≤ P, (6) β p ( z + k q − w k k ) α/ ≤ Γ , ∀ k ∈ K . (7)Notice that the cognitive communication performance in thisscenario is regardless of the mission duration T . Due to themonotonic increasing property of the log ( · ) function, theabove problem is equivalent to maximizing the SR’s receivedSNR, i.e., (P1): max p, q ,z p ( z + k q k ) α/ s.t. (5)–(7),where the constant η u is omitted at the objective functionwithout loss of optimality. Note that problem (P1) is non-convex, as the objective function is non-concave and theconstraints in (7) are non-convex. Therefore, this problem isgenerally difficult to be solved optimally. We will tackle thisproblem in Section III. B. Mobile UAV Scenario
Next, we consider the mobile UAV scenario, in which theUAV flies freely in the 3D space during the mission period T ,subject to pre-determined initial and final locations. Supposethat the UAV has a time-varying 3D location (ˆ x ( t ) , ˆ y ( t ) , ˆ z ( t )) at time instant t ∈ T , where ˆ q ( t ) = (ˆ x ( t ) , ˆ y ( t )) denotesthe horizontal UAV location, and ˆ z ( t ) denotes the flightaltitude. Specifically, the UAV’s initial and final horizontallocations are given as ˆ q I = ( x I , y I ) and ˆ q F = ( x F , y F ) ,and the corresponding altitudes are ˆ z I and ˆ z F , respectively.Let ˆ V H , ˆ V A and ˆ V D denote the UAV’s maximum horizontalspeed, vertical ascending speed, and vertical descending speedin meters/second (m/s), respectively (e.g., ˆ V H =
26 m/s, ˆ V A = ˆ V D = q ˙ˆ x ( t ) + ˙ˆ y ( t ) ≤ ˆ V H , − ˆ V D ≤ ˙ˆ z ( t ) ≤ ˆ V A , ∀ t ∈ T . Inthis case, the minimum required duration for the UAV to fly In practice, each PR also suffers the terrestrial uplink interference fromother co-channel terrestrial users. However, due to the more severe path-loss,shadowing, and small-scale fading over terrestrial channels, as well as therelatively mature interference mitigation techniques for terrestrial networks[44], we assume that in this paper the terrestrial interference is much weakerthan the A2G interference from the UAV. As a result, under our consideredsetup, each PR’s rate performance is mainly limited by the A2G interferencegiven in (4). Notice that the IT constraint at each PR k ∈ K only depends on its location w k . Therefore, the UAV only needs to know the locations of PRs, but doesnot need to know the locations of primary transmitters (ground users). straightly from the initial location to the final location is givenby T min , max (cid:16) k ˆ q F − ˆ q I k / ˆ V H , | ˆ z F − ˆ z I | / ˆ V A (cid:17) , if ˆ z F ≥ ˆ z I , max (cid:16) k ˆ q F − ˆ q I k / ˆ V H , | ˆ z F − ˆ z I | / ˆ V D (cid:17) , if ˆ z F < ˆ z I . Therefore, we must have T ≥ T min in order for the UAVtrajectory design to be feasible. For ease of exposition, wediscretize the communication period T into N time slotseach with equal duration δ t = T /N , which is chosen to besufficiently small such that the UAV’s location can be assumedto be approximately constant within each time slot even at itsmaximum flying speed . Accordingly, let q [ n ] = ( x [ n ] , y [ n ]) and z [ n ] denote the UAV’s horizontal location and altitudeat time slot n ∈ N , { , . . . , N } . Define V H = ˆ V H δ t , V A = ˆ V A δ t , and V D = ˆ V D δ t . As a result, we have thefollowing constraints on the UAV trajectory: k q [ n ] − q [ n − k ≤ V H , ∀ n ∈ N \{ } , − V D ≤ z [ n ] − z [ n − ≤ V A , ∀ n ∈ N \{ } , q [1] = ˆ q I , q [ N ] = ˆ q F , z [1] = ˆ z I , z [ N ] = ˆ z F . Furthermore, let p [ n ] denote the transmit power of the UAVat time slot n , where ≤ p [ n ] ≤ P, ∀ n ∈ N . Assumingthat the Doppler effect due to the UAV’s mobility is perfectlycompensated at the receiver based on existing techniques [43],the achievable rate from the UAV to the SR in bps/Hz inthis slot is expressed as R ( p [ n ] , q [ n ] , z [ n ]) in (3). In addition,at each time slot n ∈ N , the UAV’s resultant (worst-case)interference power at each PR k cannot exceed the IT threshold Γ , i.e., β p [ n ]( z [ n ] + k q [ n ] − w k k ) α/ ≤ Γ , ∀ n ∈ N , k ∈ K . Our objective is to maximize the SR’s average achiev-able rate (i.e., N P Nn =1 R ( p [ n ] , q [ n ] , z [ n ]) ), by optimizing theUAV’s time-varying 3D locations (or trajectory) { q [ n ] , z [ n ] } ,and the transmit power allocation { p [ n ] } . Therefore, the prob-lem of our interest is formulated as(P2): max { p [ n ] , q [ n ] ,z [ n ] } N N X n =1 log (cid:18) η u p [ n ]( z [ n ] + k q [ n ] k ) α/ (cid:19) s.t. k q [ n ] − q [ n − k ≤ V H , ∀ n ∈ N \{ } , (8) − V D ≤ z [ n ] − z [ n − ≤ V A , ∀ n ∈ N \{ } , (9) q [1] = ˆ q I , q [ N ] = ˆ q F , z [1] = ˆ z I , z [ N ] = ˆ z F , (10) H min ≤ z [ n ] ≤ H max , ∀ n ∈ N , (11) ≤ p [ n ] ≤ P, ∀ n ∈ N , (12) β p [ n ]( z [ n ] + k q [ n ] − w k k ) α/ ≤ Γ , ∀ n ∈ N , k ∈ K . (13)Here, (8) denotes the UAV’s maximum horizontal speedconstraints, (9) denotes its maximum vertical ascending anddescending speed constraints, (10) specifies the constraints on However, if δ t is chosen too small, the number of time slots N will be-come excessively large, thus leading to prohibitive computational complexity.Therefore, δ t or N should be chosen to balance between the computationalaccuracy and complexity. its initial and final locations, (11) denotes its flight altitudeconstraints, (12) denotes its maximum transmit power con-straint, and (13) denotes the PRs’ stringent IT constraints. Notethat problem (P2) is non-convex, which is even more difficultto be solved than (P1) due to the involvement of time-varyingoptimization variables. We will propose an efficient algorithmto solve (P2) sub-optimally in Section IV.III. J OINT
3D P
LACEMENT AND T RANSMIT P OWER O PTIMIZATION IN Q UASI -S TATIONARY
UAV S
CENARIO
In this section, we derive the solution to the joint 3Dplacement and transmit power optimization problem (P1) inthe quasi-stationary UAV scenario. To start with, we introducethe following variable transformation for the UAV’s transmitpower p , i.e., p = ˆ p α/ , to change the objective function of(P1) into (ˆ p/ ( z + k q k )) α/ . Due to the monotonic increasingproperty of the function (˜ x ) α/ with ˜ x ≥ , the exponent α/ can be omitted without loss of optimality. Accordingly,with some manipulation, problem (P1) can be recast in a moretractable form, i.e.,(P1.1): max ˆ p, q ,z ˆ pz + k q k s.t. ˆ β ˆ pz + k q − w k k ≤ ˆΓ , ∀ k ∈ K , (14) ≤ ˆ p ≤ ˆ P , (15)(5) , where ˆ β = β /α , ˆΓ = Γ /α , and ˆ P = P /α . As a result, theoptimal solution to (P1) can be obtained by solving (P1.1) andthen obtaining the optimal p via the relation p = ˆ p α/ . In thefollowing, we first consider a simplified problem of (P1.1)with UAV’s horizontal location being given to draw someuseful insights. Next, we derive the UAV’s optimal altitude for(P1.1) and then apply the SDR technique to obtain the optimalsolution of q and ˆ p to (P1.1). Finally, we consider a specialcase of (P1.1) with only K = 1 PR, for which the closed-formoptimal solution is obtained to draw further insights.
A. Simplified Problem Given UAV’s Horizontal Location
First, in order to gain design insights, we consider asimplified case when the UAV’s horizontal location q is given a-priori . In this case, the original problem (P1.1) is simplifiedas (P3): max ˆ p,z ˆ pz + k q k s.t. ˆ β ˆ pz + k q − w k k ≤ ˆΓ , ∀ k ∈ K , (16)(5) and (15) . Let ˜ k ( q ) = arg min k ∈K k q − w k k denote the PR that is closestto the UAV in the horizontal direction. Then it is evident thatthe IT constraints for the K PRs in (16) are satisfied as longas that for the ˜ k ( q ) -th PR is ensured. Notice that if ˜ k ( q ) isnot unique, i.e., the UAV has the same shortest distance withtwo or more PRs, then we can simply choose any one of these PRs as ˜ k ( q ) without loss of optimality. Accordingly, the ITconstraints in (16) can be reduced to ˆ β ˆ pz + k q − w ˜ k ( q ) k ≤ ˆΓ . (17)Then, we have the following proposition. Proposition 3.1:
The optimal solution to (P3), denoted by z ∗ ( q ) and ˆ p ∗ ( q ) , is given as follows. • Case 1: If k q − w ˜ k ( q ) k < k q k (i.e.,the UAV is closer to PR ˜ k ( q ) than theSR), then the optimal altitude is z ∗ ( q ) =min( H max , max( q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k , H min )) ,and the optimal solution of ˆ p is ˆ p ∗ ( q ) =min( ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) , ˆ p ) . • Case 2: If k q − w ˜ k ( q ) k > k q k (i.e., the UAV is closerto the SR than PR ˜ k ( q ) ), then the optimal altitude is z ∗ ( q ) = H min , and the optimal solution of ˆ p is p ∗ ( q ) =min( ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) , ˆ p ) . • Case 3: If k q − w ˜ k ( q ) k = k q k (i.e., the UAVhas the same distance with the SR and PR ˜ k ( q ) ),then the optimal altitude z ∗ ( q ) is non-unique andcan be chosen as any value between H min and min( H max , max( q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k , H min )) .In this case, the optimal solution of ˆ p is ˆ p ∗ ( q ) = min( ˆΓˆ β ( z ∗ ( q ) + k q − w ˜ k ( q ) k ) , ˆ p ) . Proof:
See Appendix A.By combining z ∗ ( q ) together with p ∗ ( q ) = ˆ p ( ∗ ) α/ ( q ) , theoptimal solution to (P1) in the simplified case with the givenUAV’s horizontal location is finally obtained. B. Proposed Solution to (P1.1)
Next, we consider the original problem (P1.1), for whichwe use q ⋆ , z ⋆ , and ˆ p ⋆ to denote the optimal solution. We firstpresent the following lemma. Lemma 3.1:
At the optimal solution to (P1.1), the UAVmust be placed no closer to any of the PRs than the SR, i.e., k q ⋆ − w k k ≥ k q ⋆ k , ∀ k ∈ K . Proof:
See Appendix B.Then, we have the following proposition for the optimalUAV’s altitude.
Proposition 3.2: z ⋆ = H min is optimal for problem (P1.1),i.e., it is optimal to place the UAV at its lowest altitude. Proof:
Given UAV’s horizontal location q as q ⋆ , theoptimal altitude solution z ∗ ( q ⋆ ) to problem (P3) is identical to z ⋆ to problem (P1.1). It follows from Lemma 3.1 that k q ⋆ k ≤k q ⋆ − w ˜ k ( q ⋆ ) k must hold. By using this fact together withCases 2 and 3 in Proposition 3.1, we have z ∗ ( q ⋆ ) = H min . Asa result, it follows that z ⋆ = z ∗ ( q ⋆ ) = H min . This propositionis thus proved. Remark 3.1:
It is interesting to compare Proposition 3.2versus Proposition 3.1. It is observed from Proposition 3.2 Lemma 3.1 essentially reduces the set that contains the optimal horizontallocation q ⋆ from R × to a smaller convex set, which is the intersection of K half-spaces each specified by the inequality k q ⋆ − w k k ≥ k q ⋆ k for PR k . However, how to search q ⋆ in this convex set is still challenging as shownnext. that when the UAV’s horizontal location is at the optimalpoint, the UAV should accordingly stay at its lowest altitude.This is in a sharp contrast to Proposition 3.1, which showsthat if the UAV’s horizontal location is fixed at any givenpoint, then it is generally necessary for the UAV to adaptivelyadjust its altitude depending on its horizontal distances withthe SR and the PRs, in order to maximize the SR’s achievablerate subject to the PRs’ IT constraints. In particular, when theUAV’s horizontal location is closer to any PR than the SR,Proposition 3.1 shows that the UAV may need to ascend toa higher altitude to achieve the best cognitive communicationrate. This implies that if the UAV has to fly over an areawith distributed PRs (e.g., for certain long-range tasks), thenadjusting its flight altitude (together with horizontal location)becomes crucial for achieving the maximum cognitive UAVcommunication rate, especially when the UAV has to visitcertain locations closer to some PRs than the SR during theflight. This result will be exploited for the 3D trajectory designin the mobile UAV scenario later.Next, it remains for us to find the optimal solution of ˆ p and q to problem (P1.1). By substituting z = z ⋆ = H min and introducing an auxiliary value τ , problem (P1.1) is re-expressed as(P4): max τ, ˆ p, q τ s.t. k q k ≤ ˆ pτ − H , (18) k q − w k k ≥ ˆ β ˆ p ˆΓ − H , ∀ k ∈ K , (19) ≤ ˆ p ≤ ˆ p. (20)However, problem (P4) is still non-convex , as constraint (18)is non-convex due to the coupling between ˆ p and τ , and theconstraints in (19) are non-convex quadratic constraints.First, we deal with the non-convex quadratic constraintsin (19) by using the SDR technique. Towards this end, wefirst equivalently recast (P4) as the following problem (P4.1)with homogeneous quadratic terms by introducing an auxiliaryvariable θ .(P4.1): max τ, ˆ p, q ,θ τ s.t. ( q , θ ) T A ( q , θ ) ≤ ˆ pτ − H , (21) ( q , θ ) T B k ( q , θ ) ≥ ˆ β ˆ p ˆΓ − H , ∀ k ∈ K , (22) θ = 1 , (23)(20),where A , diag ((1 , , ∈ R × and B k , (cid:20) I − w Tk − w k k w k k (cid:21) ∈ R × , ∀ k ∈ K . Notice that given q , (P4) is a linear programming (LP) over τ and ˆ p , thuscan be optimally solved; nevertheless, the key challenge here is to jointlyoptimize all variables, which renders (P4) a non-convex problem. Then, by introducing s = ( q , θ ) ∈ R × and S = ss T ∈ R × , with S (cid:23) and rank ( S ) ≤ , problem (P4.1) is furtherreformulated as(P4.2): max τ, ˆ p, S τ s.t. Tr ( AS ) ≤ ˆ pτ − H , (24)Tr ( B k S ) ≥ ˆ β ˆ p ˆΓ − H , ∀ k ∈ K , (25)Tr ( CS ) = 1 , (26)rank ( S ) ≤ , (27) S (cid:23) , (28)(20),where C , diag ((0 , , . Notice that the rank constraint in(27) is non-convex. To address this issue, we relax problem(P4.2) by dropping this rank constraint, and denote the relaxedproblem of (P4.2) as (P4.3).Next, we consider problem (P4.3). Although constraint (24)is non-convex due to the coupling between ˆ p and τ , problem(P4.3) can be solved by equivalently solving the followingfeasibility problems (P4.4. τ ) under any given τ ≥ , togetherwith a bisection search over τ .(P4.4. τ ): Find ˆ p, S s.t. (20), (24), (25), (26), and (28).In particular, denote by τ ⋆ the optimal solution of τ to (P4.3).Then, under any given τ ≥ , we have τ ≤ τ ⋆ if problem(P4.4. τ ) is feasible; otherwise, we have τ > τ ⋆ . Therefore, wecan solve (P4.3) by using the bisection search over τ ≥ , andchecking the feasibility of problem (P4.4. τ ) under any given τ ≥ [45]. Notice that under any given τ ≥ , problem(P4.4. τ ) is a semi-definite program (SDP) that is convex,and thus can be optimally solved by using standard convexoptimization techniques, such as the interior point method[45]. With the obtained τ ⋆ , suppose that the correspondingfeasible/optimal solution to problem (P4.4. τ ) is ˆ p ⋆ ( τ ) and S ⋆ ( τ ) . Accordingly, they are also the optimal solution toproblem (P4.3), denoted by ˆ p ⋆ and S ⋆ .Now, it still remains to construct the optimal solutionto (P4.2), or equivalently (P4.1) and (P4). In particular, ifrank ( S ⋆ ) ≤ , then the SDR is tight. In this case, the solutionof ˆ p ⋆ and S ⋆ are also the optimal solution to (P4.2). Byperforming the eigenvalue decomposition (EVD) for S ⋆ , wecan obtain the optimal solution s ⋆ = ( q ⋆ , θ ⋆ ) to (P4.1) and(P4) as the dominant eigenvector of S ⋆ . Accordingly, ˆ p ⋆ isalso the optimal solution of ˆ p to (P4.1) and (P4). However, ifrank ( S ⋆ ) > , then we need to construct a rank-one solutionof S to (P4.2) via additional processing such as the Gaussianrandomization procedure that is widely adopted in the SDRliterature (see, e.g., [46]). Fortunately, in our simulations withrandomly generated PRs’ locations, the optimal solution of S ⋆ Suppose that the searching range of τ is an interval [0 , τ max ] . Assuch, the maximum number of iterations for bisection search is given by ⌈ log ( τ max /ǫ ) ⌉ , where ǫ is a positive constant that controls the accuracy, and ⌈ ˜ y ⌉ denotes the minimum integer that is no smaller than ˜ y . Since the numberof required iterations is a logarithmic function with respect to τ max /ǫ , theconvergence of the bisection search is exponentially fast. to problem (P4.3) is always rank-one. Therefore, the Gaussianrandomization procedure is not required in general. Morespecifically, we can rigorously prove that the optimal solutionof S ⋆ to (P4.3) is rank-one in the following special case,though our proposed SDR-based solution is applicable for thegeneral case with any PRs’ locations. Proposition 3.3:
When the K PRs are located at the sameside of the SR , it follows that the optimal solution S ⋆ to(P4.3) is always rank-one. Proof:
See Appendix C.Therefore, the solution to (P4) is finally obtained as ˆ p ⋆ and q ⋆ . By combining them together with z ⋆ = H min , problem(P1.1) is solved. As a result, the optimal solution to (P1) isfinally obtained as z ⋆ , q ⋆ , and p ⋆ = ˆ p ( ⋆ ) α/ . C. Special Case with K = 1 PR In this subsection, we consider the special case of (P1.1)with K = 1 PR and derive the closed-form optimal solutionto gain additional insights. In this case, by substituting z = z ⋆ = H min , problem (P1.1) is simplified as(P5): max ˆ p, q ˆ pH + k q k s.t. ˆ β ˆ pH + k q − w k ≤ ˆΓ , (29)(15) . For (P5), we first have the following lemma.
Lemma 3.2:
Under any feasible ˆ p , the optimal solution of q to problem (P5) is given by q = − a w k w k , where a ≥ denotes the horizontal distance between the UAV and the SR. Proof:
See Appendix D.From Lemma 3.2, it is evident that at the optimality, theUAV should be placed above a point at the PR’s oppositedirection along the line connecting the SR and the PR, to mini-mize the interference to the PR. By substituting q = − a w k w k ,problem (P5) is re-expressed as(P5.1): max ˆ p,a ≥ ˆ pH + a s.t. ˆ β ˆ pH + ( a + k w k ) ≤ ˆΓ , (30)(15) . For convenience, we define p , ˆΓˆ β ( k w k + H ) , whichdenotes the UAV’s maximally allowable value of ˆ p for the ITconstraint (29) to be feasible, when the UAV is located exactlyabove the SR at (0 , , H min ) . Then, we have the followingproposition. Proposition 3.4:
The optimal solution of ˆ p to problem (P5.1)is given by ˆ p ⋆ = min(ˆ p, ˜ p ⋆ ) , where ˜ p ⋆ , ˆΓˆ β (cid:16) k w k + p k w k + 4 H (cid:17) H . (31) There are in total four cases when the K PRs are located at the same sideof the SR: 1) x k ≥ , ∀ k ∈ K , and there exists at least one PR ¯ k ∈ K with x ¯ k > ; 2) x k ≤ , ∀ k ∈ K , and there exists at least one PR ¯ k ∈ K with x ¯ k < ; 3) y k ≥ , ∀ k ∈ K , and there exists at least one PR ¯ k ∈ K with y ¯ k > ; and 4) y k ≤ , ∀ k ∈ K , and there exists at least one PR ¯ k ∈ K with y ¯ k < . Accordingly, the optimal solution of a to problem (P5.1) isgiven as a ⋆ = ˜ a ⋆ , √ k w k +4 H −k w k , ˆ p > ˜ p ⋆ , q ˆ β P ˆΓ − H − k w k < ˜ a ⋆ , p ≤ ˆ p ≤ ˜ p ⋆ , , ˆ p < p , (32) Proof:
See Appendix E.By combining Proposition 3.4 and Lemma 3.2, the optimalsolution to problem (P5) is finally obtained as ˆ p ⋆ and q ⋆ = − a ⋆ w k w k . As a result, the optimal solution to (P1) in thespecial case with K = 1 PR is finally obtained as z ⋆ , q ⋆ , and p ⋆ = ˆ p ( ⋆ ) α/ .Proposition 3.4 provides interesting insights on the optimalhorizontal location and transmit power solution to the specialcase of (P1) with K = 1 PR. Firstly, when the UAV’smaximum transmit power P is sufficiently large, the UAVshould transmit at an optimized power p ⋆ = ˜ p ( ⋆ ) α/ (with ˜ p ⋆ given in (31)) and be placed at an optimized horizontallocation with distance ˜ a ⋆ given in (32) from the SR. Noticethat ˜ a ⋆ is only dependent on k w k and H min but irrelevantto P ; as H min increases and/or k w k decreases, ˜ a ⋆ becomeslarger and thus the UAV needs to be placed further away fromthe SR for maximizing the SR’s achievable rate subject tothe IT constraint. Furthermore, when P becomes smaller with ˆ p ≤ ˜ p ( ⋆ ) , the UAV should transmit at the full power P and beplaced at a horizontal location closer to the SR. In addition,if P becomes sufficiently small with ˆ p < p , then the UAVshould be placed exactly above the SR and transmit with fullpower P . Remark 3.2:
Notice that although in this paper we considerthe LoS channel model, the design principles are applicableto other stochastic A2G channel models such as Rician fadingand probabilistic LoS channels.Specifically, denote by ˜ h ( q , z ) and ˜ g k ( q , z ) the instanta-neous channel power gains from the UAV to the SR andto the PR k ∈ K , respectively, which are random vari-ables whose probability density functions generally dependon the elevation angles between the UAV and the groundnodes. In general, as the UAV altitude z increases, the K -factor becomes larger for Rician fading channel [40] andthe LoS probability increases for probabilistic LoS channel[47]. For convenience, we denote ˆ h ( q , z ) = E (˜ h ( q , z )) and ˆ g k ( q , z ) = E (˜ g k ( q , z )) as their mean values. Then, we canmaximize the average rate from the UAV to the SR, i.e., E (cid:16) log (cid:16) p ˜ h ( q ,z ) σ (cid:17)(cid:17) , subject to the average IT constraintsat all PRs, i.e., E ( p ˜ g k ( q , z )) = p ˆ g k ( q , z ) ≤ Γ , ∀ k ∈ K . Ingeneral, we can adopt the exhaustive search over the 3D spaceto find the optimal solution to this new problem.However, due to the concavity of the log( · ) function, itfollows from the Jensen’s inequality [47] that E log p ˜ h ( q , z ) σ !! ≤ log p E (˜ h ( q , z )) σ ! = log p ˆ h ( q , z ) σ ! . Since the channel power gains achieve their maximum un-der the LoS channel model, we have ˆ h ( q , z ) ≤ h ( q , z ) and ˆ g k ( q , z ) ≤ g k ( q , z ) , ∀ k ∈ K , which lead to log (cid:16) p ˆ h ( q ,z ) σ (cid:17) ≤ log (cid:16) ph ( q ,z ) σ (cid:17) , i.e., the achievablerate under the LoS channel serves as an upper bound for theaverage achievable rate at the SR under the stochastic channelmodels. Similarly, the average interference power at each PRsatisfies p ˆ g k ( q , z ) ≤ pg k ( q , z ) , ∀ k ∈ K . As a result, theoptimal solution to our considered problem (P1) can be viewedas an approximate solution to the problem in the stochasticchannel models. In particular, such approximations becomemore accurate when the K -factor is larger for Rician fadingchannel or the LoS probability is higher for probabilistic LoSchannel.IV. J OINT
3D T
RAJECTORY AND T RANSMIT P OWER O PTIMIZATION IN M OBILE
UAV S
CENARIO
In this section, we consider the joint UAV 3D trajectory andtransmit power optimization problem (P2) in the mobile UAVscenario. To tackle this problem, we use the SCA techniqueto obtain a locally optimal solution.To facilitate the implementation of SCA, we first obtain theoptimal transmit power levels under any given feasible UAVtrajectory { q [ n ] , z [ n ] } , for which the problem is expressed as(P6): max { p [ n ] } N N X n =1 log (cid:18) η u p [ n ]( z [ n ] + k q [ n ] k ) α/ (cid:19) s.t. ≤ p [ n ] ≤ P, ∀ n ∈ N , (33) β p [ n ]( z [ n ] + k q [ n ] − w k k ) α/ ≤ Γ , ∀ n ∈ N , k ∈ K . (34)It is easy to show that problem (P6) can be decomposed intothe following N subproblems each for one slot n ∈ N , forwhich the coefficient /N is ignored for brevity.(P6.1. n ): max p [ n ] ≥ log (cid:18) η u p [ n ]( z [ n ] + k q [ n ] k ) α/ (cid:19) s.t. p [ n ] ≤ min( P, min k ∈K Γ β (cid:0) z [ n ] + k q [ n ] − w k k (cid:1) α/ ) . (35)It is evident that the optimality of problem (P6.1. n ) is attainedwhen constraint (35) is tight. Therefore, we have the optimalsolution to (P6.1. n )’s and (P6) as p [ n ] = min (cid:18) P, min k ∈K Γ β (cid:0) z [ n ] + k q [ n ] − w k k (cid:1) α/ (cid:19) , ∀ n ∈ N . (36) By substituting (36) into the objective function of (P2), prob-lem (P2) is reformulated as(P7): max { q [ n ] ,z [ n ] } N N X n =1 ˆ R ( q [ n ] , z [ n ]) s.t. k q [ n ] − q [ n − k ≤ V H , ∀ n ∈ N \{ } , (37) − V D δ t ≤ z [ n ] − z [ n − ≤ V A , ∀ n ∈ N \{ } , (38) q [1] = ˆ q I , q [ N ] = ˆ q F , z [1] = ˆ z I , z [ N ] = ˆ z F , (39) H min ≤ z [ n ] ≤ H max , ∀ n ∈ N , (40) where ˆ R ( q [ n ] , z [ n ])= log η u min (cid:18) P, min k ∈K Γ β (cid:0) z [ n ] + k q [ n ] − w k k (cid:1) α/ (cid:19) ( z [ n ] + k q [ n ] k ) α/ . (41) Next, to solve problem (P7), we introduce two sets ofauxiliary variables { ζ [ n ] } Nn =1 and { ζ [ n ] } Nn =1 , and define ˜ R ( ζ [ n ] , ζ [ n ]) = log (1 + η u ζ [ n ] /ζ [ n ]) . Accordingly, wereformulate problem (P7) as(P7.1): max { q [ n ] ,z [ n ] ,ζ [ n ] ,ζ [ n ] } N N X n =1 ˜ R ( ζ [ n ] , ζ [ n ]) s.t. ≤ ζ [ n ] ≤ P, ∀ n ∈ N , (42) ζ [ n ] ≤ Γ β (cid:0) z [ n ] + k q [ n ] − w k k (cid:1) α/ , ∀ n ∈ N , k ∈ K , (43) ( k q [ n ] k + z [ n ]) α/ ≤ ζ [ n ] , ∀ n ∈ N , (44)(37)–(40).It is easy to verify that at the optimality of (P7.1), constraint ( k q [ n ] k + z [ n ]) α/ ≤ ζ [ n ] must hold with equality forany n ∈ N , since otherwise, we can decrease ζ [ n ] toachieve a higher objective value of (P7.1) without violatingthis constraint. Notice that problem (P7.1) is still non-convex,as the objective function is non-concave and the constraintsin (43) are non-convex. To tackle this problem, we adopt theSCA technique to obtain a locally optimal solution to (P7.1)in an iterative manner. The key idea of SCA is that given alocal point at each iteration, we approximate the non-concaveobjective function (or non-convex constraints) into a concaveobjective function (or convex constraints), in order to obtainan approximate convex optimization problem. By iterativelysolving a sequence of approximate convex problems, wecan obtain an efficient solution to the original non-convexoptimization problem (P7.1).Specifically, suppose that { q ( j ) [ n ] , z ( j ) [ n ] , ζ ( j )1 [ n ] , ζ ( j )2 [ n ] } corresponds to the local point at the j -th iteration with j ≥ , where { q (0) [ n ] , z (0) [ n ] , ζ (0)1 [ n ] , ζ (0)2 [ n ] } corresponds to theinitial point. In the following, we explain how to approximatethe objective function of (P7.1) and the constraints in (43),respectively. First, we rewrite the objective function of (P7.1)as ˜ R ( ζ [ n ] , ζ [ n ]) = log ( ζ [ n ] + η u ζ [ n ]) − log ( ζ [ n ]) . (45)Note that the objective function in (45) is still non-concave,as − log ( ζ [ n ]) is non-concave. However, − log ( ζ [ n ]) is convex with respect to { ζ [ n ] } . Notice that any convexfunction is globally lower-bounded by its first-order Taylorexpansion at any point [45]. Therefore, with given localpoint { ζ ( j )2 [ n ] } in the j -th iteration, j ≥ , it follows that ˜ R ( ζ [ n ] , ζ [ n ]) ≥ ˜ R lb ( ζ [ n ] , ζ [ n ]) , where ˜ R lb ( ζ [ n ] , ζ [ n ]) , log ( ζ [ n ] + η u ζ [ n ]) − log ( ζ ( j )2 [ n ]) − ( ζ [ n ] − ζ ( j )2 [ n ]) log ( e ) ζ ( j )2 [ n ] . (46) Next, we consider the non-convex constraints in (43). Since ( k q [ n ] − w k k + z [ n ]) α/ is a convex function with re-spect to { q [ n ] , z [ n ] } , we have the following inequalities byapplying the first-order Taylor expansion at any given point { q ( j ) [ n ] , z ( j ) [ n ] } : ( k q [ n ] − w k k + z [ n ]) α/ ≥ (cid:16) k q ( j ) [ n ] − w k k + z ( j )2 [ n ] (cid:17) α/ + α (cid:16) k q ( j ) [ n ] − w k k + z ( j )2 [ n ] (cid:17) α − (cid:16) ( q ( j ) [ n ] − w k ) T ( q [ n ] − q ( j ) [ n ]) + z ( j ) [ n ]( z [ n ] − z ( j ) [ n ]) (cid:17) , ∀ n ∈ N , k ∈ K . (47)By replacing ( k q [ n ] − w k k + z [ n ]) α/ in (43) as the right-hand-side (RHS) of (47), we approximate (43) as the followingconvex constraints: ζ [ n ] ≤ Γ β (( k q ( j ) [ n ] − w k k + z ( j )2 [ n ]) α/ + α ( k q ( j ) [ n ] − w k k + z ( j )2 [ n ]) α/ − (( q ( j ) [ n ] − w k ) T ( q [ n ] − q ( j ) [ n ]) + z ( j ) [ n ]( z [ n ] − z ( j ) [ n ]))) , ∀ n ∈ N , k ∈ K . (48)To summarize, by replacing ˜ R ( ζ [ n ] , ζ [ n ]) in the objectivefunction as ˜ R lb ( ζ [ n ] , ζ [ n ]) in (46), and replacing the con-straints in (43) as those in (48), problem (P7.1) is approxi-mated as the following convex optimization problem (P7.2) atany local point { q ( j ) [ n ] , z ( j ) [ n ] , ζ ( j )1 [ n ] , ζ ( j )2 [ n ] } , which canbe solved via standard convex optimization techniques suchas the interior point method [45], with the optimal solutiondenoted as { q ( j ) ∗ [ n ] } , { z ( j ) ∗ [ n ] } , { ζ ( j ) ∗ [ n ] } and { ζ ( j ) ∗ [ n ] } . ( P7.2 ) : max { q [ n ] ,z [ n ] ,ζ [ n ] ,ζ [ n ] } N N X n =1 ˜ R lb ( ζ [ n ] , ζ [ n ]) s.t. (37) , (38) , (39) , (40) , (42) , (44) , and (48) . With the convex optimization problem (P7.2) at hand,we can obtain an efficient iterative algorithm to solve(P7.1), explained as follows. In the j -th iteration, thealgorithm solves the convex optimization problem (P7.2)at the local point { q ( j ) [ n ] , z ( j ) [ n ] , ζ ( j )1 [ n ] , ζ ( j )2 [ n ] } , where { q ( j ) [ n ] , z ( j ) [ n ] , ζ ( j )1 [ n ] , ζ ( j )2 [ n ] } corresponds to the optimalsolution to (P7.2) obtained in the ( j − -th iteration, i.e., q ( j ) [ n ] = q ( j − ∗ [ n ] , z ( j ) [ n ] = z ( j − ∗ [ n ] , ζ ( j )1 [ n ] = ζ ( j − ∗ [ n ] , and ζ ( j )2 [ n ] = ζ ( j − ∗ [ n ] , ∀ n ∈ N . We summarizethis algorithm in Table I as Algorithm 1. Denote the obtainedsolution to (P7) as { q ∗ [ n ] , z ∗ [ n ] } . By substituting q ∗ [ n ] and z ∗ [ n ] into (36), the corresponding transmit power is p ∗ [ n ] =min( P, min k ∈K Γ β (cid:0) z ∗ [ n ] + k q ∗ [ n ] − w k k (cid:1) α/ ) , ∀ n ∈ N . Bycombining { p ∗ [ n ] } , { q ∗ [ n ] } , and { z ∗ [ n ] } , the solution to (P2)by SCA is finally obtained.Similarly as in [4], it can be shown that in Algorithm1, after each iteration j , the objective function of (P7.2)achieved by { q ( j ) [ n ] , z ( j ) [ n ] , ζ ( j )1 [ n ] , ζ ( j )2 [ n ] } is monotonicallynon-decreasing. As the optimal value of problem (P7.1) isupper-bounded by a finite value, it is evident that Algorithm 1can converge to a locally optimal solution to problem (P7.1)(and thus (P2)). TABLE IA
LGORITHM FOR S OLVING P ROBLEM (P7.1) a) Initialization:
Set the initial UAV trajectory as { q (0) [ n ] , z (0) [ n ] } Nn =1 , ζ (0)2 [ n ] = ( k q (0) [ n ] k + z (0)2 [ n ]) α/ , ∀ n ∈ N , and j = 0 .b) Repeat:
1) Solve problem (P7.2) to obtain the optimal solution as { q ( j ) ∗ [ n ] } Nn =1 , { z ( j ) ∗ [ n ] } Nn =1 , { ζ ( j ) ∗ [ n ] } Nn =1 , and { ζ ( j ) ∗ [ n ] } Nn =1 .2) Update the trajectory as q ( j +1) [ n ] = q ( j ) ∗ [ n ] and z ( j +1) [ n ] = z ( j ) ∗ [ n ] , ζ ( j +1)1 [ n ] = ζ ( j ) ∗ [ n ] , and ζ ( j +1)2 [ n ] = ζ ( j ) ∗ [ n ] , ∀ n ∈ N .3) Update j = j + 1 .c) Until the objective value of (P7.2) converges within a given accuracy ora maximum number of iterations is reached.
Denote by D the total number of iterations required inAlgorithm 1. In each iteration, the convex optimization prob-lem (P7.2) is solved via the standard interior-point methodwith the complexity of O ( N . K . ) [48]. As a result, theoverall complexity of Algorithm 1 is O ( DN . K . ) , which ispolynomial. Also note that we consider the offline optimizationfor the joint trajectory and power design. Thus, Algorithm 1only needs to be implemented in an offline manner prior tothe UAV flight. Remark 4.1:
In order to efficiently implement Algorithm 1for solving (P7.2) as well as (P2), we need to properly designan initial UAV trajectory. Notice that the optimal UAV location ( q ⋆ , z ⋆ ) to (P1) obtained in Section III is quite efficient formaximizing the SR’s communication rate while controlling theinterference at the PRs. Therefore, we intuitively design a fly-hover-fly (FHF) trajectory, where the UAV first flies straightlyfrom the initial location (ˆ q I , ˆ z I ) to the optimal UAV location ( q ⋆ , z ⋆ ) , then hovers at this point for a certain time duration,and finally flies straightly towards the final location (ˆ q F , ˆ z F ) .To prolong the hovering duration for improving the SR’sperformance, the UAV should fly at the maximum horizontalspeed ˆ V H and maximum ascending/descending speed ˆ V D / ˆ V A during the flight (notice that we have z ⋆ = H min ). As a result,we obtain the flight duration as T fly = max( | ˆ z I − z ⋆ | / ˆ V D , k q ⋆ − ˆ q I k / ˆ V H )+ max( | ˆ z F − z ⋆ | / ˆ V A , k ˆ q F − q ⋆ k / ˆ V H ) , and the hovering duration as T − T fly . Notice that the proposedinitial trajectory is only applicable when the mission duration T is no smaller than T fly . When T min ≤ T < T fly , we insteaduse the straight flight as the initial UAV trajectory, in whichthe UAV flies directly from the initial location to the finallocation at a constant horizontal speed ˜ V H = k ˆ q F − ˆ q I k /T and a constant vertical speed ˜ V L = | ˆ z F − ˆ z I | /T . Remark 4.2:
The design principles used in this section arealso applicable to other stochastic channel models such asRician fading and probabilistic LoS channels. Similarly as inthe quasi-stationary UAV scenario, we consider the averagerate performance of the considered system. Specifically, inthe objective function of (P2), the achievable rate under thedeterministic LoS channel for each slot n can be replacedwith the SR’s average rate over the same slot. Additionally,in the IT constraints in (13), the PR’s received interferencepower at each slot n is modified as the average interference power over the same slot. Our proposed solution under theLoS channel then provides an efficient approximate solutionto this new problem, while such approximations becomemore accurate when the K -factor is larger for Rician fadingchannel or the LoS probability is higher for probabilistic LoSchannel. Alternatively, we can also introduce a homogenousapproximation to the K -factor or LoS probability by assumingthat they are constant throughout the UAV’s flight in thestochastic channel models (see e.g., [47]). Accordingly, theresulting problem has the same form as (P2), for which wecan adopt a similar SCA-based algorithm to obtain a convergedsolution. V. N UMERICAL R ESULTS
In this section, we present numerical results to validate theperformance of our proposed joint design of UAV’s maneuverand transmit power. Unless otherwise stated, we set the noisepower at the SR (including the background interference andnoise) as σ = –80 dBm, the reference channel power gainat the SR as β u = –30 dB, the maximum reference channelpower gain from the UAV to PRs as β = –30 dB, the path-loss exponent as α = 2 , and the UAV’s minimum and max-imum flight altitudes as H min =
170 m and H max =
220 m[39], respectively.
A. Quasi-Stationary UAV Scenario
In this subsection, we evaluate the performance of ourproposed optimal solution to (P1) for the quasi-stationaryUAV scenario, as compared to the following two benchmarkschemes. • Power optimization only:
The UAV is placed ex-actly above the SR with the lowest altitude, i.e., ( q , z ) = (0 , , H min ) . In this case, analogous to (36),the UAV’s optimal transmit power is obtained as ¯ p ∗ =min( P, min k ∈K Γ β ( w k + H ) α/ ) . • Placement optimization only:
The UAV optimizes itslocation ( q , z ) with the maximum transmit power P used,i.e., p = P . This corresponds to solving problem (P1)under given p = P , for which the optimal solution canbe obtained by applying the SDR technique, which issimilar as in Section III-B.First, we consider the case with K = 1 PR with the PRlocated at ( x , , with x ≥ . Fig. 2 shows the SR’sachievable rate versus the distance x from the SR to thePR with Γ = –80 dBm and P =
23 dBm. It is observedthat as the distance x from the SR to the PR increases, theachievable rates by all schemes increase. This is due to the factthat when the PR is away from the SR, the IT constraint atthe PR becomes less stringent. It is also observed that whenthe PR is located very close to the SR (i.e., x → ), theperformance gap between the proposed design and the twobenchmark schemes is negligible. This is because in this case,the SR’s received signal power is fundamentally limited by thePR’s IT constraint, thus leading to the comparable performancefor the three schemes. By contrast, as the distance x increases,the performance gap between the proposed and benchmarkschemes is observed to be enlarged. Fig. 2. SR’s achievable rate versus the distance from SR to PR with K = 1 PR. -80 -70 -60 -50 -400246810 Proposed designPower optimization onlyPlacement optimization only
Fig. 3. SR’s achievable rate versus the IT threshold with K = 1 PR. -60 -40 -20 0 20 40 60-150-100-50050100
Fig. 4. UAV’s optimal horizontal locations with K = 3 PRs. -80 -70 -60 -50 -40246810 Proposed designPower optimization onlyPlacement optimization only
Fig. 5. SR’s achievable rate versus the IT threshold with K = 3 PRs.
Fig. 3 shows the SR’s achievable rate versus the IT threshold Γ with w = (100 m, 0 m) and P =
23 dBm. It is observedthat when Γ is sufficiently large (e.g., Γ ≥ –53 dBm), allschemes achieve the same rate performance. This is becausein this case, the transmit power constraint dominates the ITconstraints, and thus the three schemes become equivalent.However, when Γ becomes smaller (e.g., Γ < –53 dBm), ourproposed design is observed to outperform the two benchmarkschemes. In particular, when Γ is sufficiently small (e.g., Γ = –80 dBm), the SR’s achievable rate by our proposed design isapproximately more than that by the scheme with poweroptimization only, and more than that by the schemewith placement optimization only (as also shown in Fig. 2more clearly).Next, we consider the setup with K = 3 PRs as shown inFig. 4 . Based on Proposition 3.2, the UAV should alwaysbe placed at the lowest altitude with z ⋆ = H min =
170 m.Therefore, for simplicity, only the optimized horizontal loca-tions are shown in Fig. 4. It is observed that when the UAV’smaximum transmit power P increases and/or the IT threshold Γ decreases, the UAV needs to move further away from thePRs, so as to meet the interference constraints at the PRs.Fig. 5 shows the SR’s achievable rate versus the IT threshold Note that our proposed SDR-based solution is applicable to any locationsof PRs. In Fig. 4, we consider that the PRs are all located at the same sideof the SR, only for the purpose of showing the impacts of Γ and P on theUAV’s optimal location. Fig. 6. SR’s achievable rate of the proposed design versus the number ofPRs K . Γ with P =
23 dBm. Similar observations can be made as inFig. 3, where the performance gains over the two benchmarkschemes are more significant with smaller values of Γ .Furthermore, Fig. 6 shows the SR’s achievable rate of theproposed design versus the number of PRs, K . For each K , we randomly generate the PRs’ locations in an areaof × m . The results are obtained by averagingover 100 random realizations, where we set Γ = –90 dBmand P = 23 dBm. It is observed that as K increases, theSR’s achievable rate is non-increasing. This is because as K increases, the number of IT constraints increases, thus making -500 0 500 1000-1000-50005001000 PR 2 PR 3 PR 5 t=190 st=48 sPR 1t=18 s PR 4 SR t=152 s t=162 s t=178 sPR 10PR 9PR 8PR 7PR 6
Fig. 7. UAV’s horizontal trajectories.
50 100 150 200170180190200210220 t=18 s t=48 s t=152 s t=178 st=190 st=162 s
Fig. 8. UAV’s flight altitudes over time.
50 100 150 2000510152025 50 100 150 200-4-20246
Fig. 9. UAV’s horizontal and vertical speeds over time.
50 100 150 20068101214161820 t=18 s t=48 s t=152 st=162 st=178 st=190 s
Fig. 10. UAV’s transmit powers over time. the feasibility region of (P1) smaller. As a result, the SR’sachievable rate may decrease.
B. Mobile UAV Scenario
In this subsection, we evaluate the performance of ourproposed solution to (P2) under the mobile UAV scenario.Unless otherwise stated, we assume that there are K =10 ground PRs distributed in a 2D area of × km ,as shown in Fig. 7. The speed limits for the UAV areset according to DJI’s Inspire 2 drones [42], i.e., ˆ V H =
26 m/s, ˆ V A = ˆ V D = T min =
107 s.Figs. 7 and 8 show the UAV’s horizontal locations and flightaltitudes over time by the proposed design, under differentvalues of P and Γ with the communication duration set as T =
200 s. Figs. 9 and 10 show the corresponding UAV’sflying speeds and the optimized transmit powers over time,respectively. It is observed that when
Γ = –50 dBm and P =
20 dBm, the UAV flies simply following its initialFHF trajectory, along which the UAV always stays at theminimum altitude and transmits with the full power P . Thisis consistent with Proposition 3.1, which shows that when themaximum transmit power P is sufficiently small and/or theIT threshold Γ is sufficiently large, the UAV should stay atthe minimum altitude and transmit with the maximum power.However, when Γ decreases and P increases in the caseof Γ = –70 dBm and P =
23 dBm, the UAV trajectory is observed to deviate from the initial FHF trajectory. Inparticular, when the UAV approaches PRs 1–4 and PRs 7–10 (at time instants t =
18 s, t =
48 s, t =
152 s, t =
162 s, t =
178 s and t =
190 s, respectively), it increases the flightaltitude and reduces transmit power, in order to meet the ITconstraint at the nearest PR. Notice that this observation is alsoconsistent with Proposition 3.1 and Remark 3.1, which revealsthat the UAV should increase its altitude to maximize thecognitive communication rate when it moves closer to somePRs than the SR. Furthermore, it is observed that the UAVhovers above a point closer to the SR than all PRs at the lowestaltitude for a certain period of time to take advantage of thefavorable communication channel with the SR for enhancingthe SR’s achievable rate. This is consistent with Proposition3.2.Finally, Fig. 11 shows the SR’s average achievable rate bythe proposed design with
Γ = –80 dBm and P =
23 dBm, ascompared to the following two benchmark schemes. • Joint 2D trajectory and power optimization:
The UAVjointly optimizes its 2D trajectory { q [ n ] } and transmitpower { p [ n ] } , where the flight altitude is fixed as itsminimum flight altitude. This corresponds to solvingproblem (P2) under given z [ n ] = H min , ∀ n ∈ N . • Power optimization with proposed initial trajectory:
The UAV sets its trajectory as the proposed initial trajec-tory, as given in Remark 4.1. Under this trajectory, theUAV optimizes its power allocation based on (36).From Fig. 11, it is observed that as the flight duration T
120 140 160 180 200 220 2400.40.60.811.2
Joint trajectory and power optimizationJoint 2D trajectory and power optimizationPower optimization with proposed initial trajectory
Fig. 11. SR’s average achievable rate versus the flight duration T . increases, the average achievable rates by all the three schemesincrease. This is because for all cases with adaptive trajectorydesign, the UAV in general stays longer close to the SRwhen T increases, leading to a better channel condition onaverage and thus a higher average achievable rate. It is alsoobserved that the proposed joint 3D trajectory and powerdesign outperforms its 2D counterpart. This is because in theproposed design, the UAV can adjust its altitude more freely tocontrol the co-channel interference, especially when the UAVis close to the PRs. This is consistent with our observationsin Remark 3.1 and validates the importance of 3D trajectorydesign with altitude control.VI. C ONCLUDING R EMARKS
This paper studied a new spectrum sharing scenario forUAV-to-ground communications, where a cognitive/secondaryUAV transmitter communicates with a ground SR, in the pres-ence of co-channel primary terrestrial wireless communicationlinks. We exploited the UAV’s 3D mobility to improve the cog-nitive communication rate performance under two scenariosof quasi-stationary and mobile
UAVs, respectively. For bothscenarios, we proposed efficient algorithms to obtain high-quality solutions to the joint UAV maneuver and power controloptimization problems. It was shown via simulations thatthe proposed designs with joint 3D placement/trajectory andpower control optimization significantly outperform bench-mark schemes without such a joint design or with only 2Doptimization. Due to the space limitation, there are otherimportant issues that remain unaddressed yet in this paper,which are discussed in the following to motivate future work. • This paper considered the offline UAV maneuver designby assuming that the UAV perfectly knows the channelparameters in advance. Such offline design, however,may lead to sub-optimal performance in real-time imple-mentation. This is because the deterministic LoS chan-nel (or even stochastic channels) model may mismatchwith realistic radio propagation environments, due to theunevenly distributed obstacles (such as buildings andtrees) around. How to optimize UAV maneuver basedon the actual channel is thus an important problem tobe tackled in future work. In this case, a promisingsolution is by using the radio map technique [49] toobtain the location-dependent channel knowledge offline, or adopting reinforcement learning to adapt the UAVmaneuver to the actual channel in real time. Accordingly,our proposed solutions based on the a-priori known LoSchannel model can not only provide a performance upperbound for practical maneuver design, but also serve as aninitial input for the online design. • This paper considered the basic setup with one UAV andone SR. In practice, there may exist multiple coexistingSRs and UAVs within the same network. In the case withone UAV communicating with multiple SRs, the UAVneeds to properly schedule its transmission to the mul-tiple SRs based on the adopted multiple access scheme(e.g., time division multiple access (TDMA), orthogonalfrequency division multiple access (OFDMA), or non-orthogonal multiple-access (NOMA)). How to jointlydesign the UAV maneuver, SRs’ scheduling and resourceallocation is an interesting problem worth pursuing infuture work. Furthermore, when there are multiple UAVs,the co-channel interference from UAVs to SRs becomesa new issue to be dealt with. For instance, these UAVscan jointly design their maneuvers and power allocationsto maximize their weighted sum rates, while ensuringthat their caused aggregate interference power at eachPR does not exceed the prescribed IT constraint. More-over, different SRs may use the coordinated multi-point(CoMP) technique to jointly decode the messages frommultiple UAVs for better mitigating or even utilizing thestrong co-channel A2G interference. These problems areworthy of more in-depth investigation in future work.A
PPENDIX
A. Proof of Proposition 3.1
First, we recast constraint (17) as ˆ p ≤ ˆΓˆ β ( z + k q − w ˜ k ( q ) k ) . Obviously, at least one of the constraints (15) and(17) must be tight at the optimality of (P3). Notice that ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) ≤ ˆΓˆ β ( z + k q − w ˜ k ( q ) k ) ≤ ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) due to H min ≤ z ≤ H max . Basedon this, we consider the following three cases to obtain theoptimal solution to (P3).If ˆ p > ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) , then only constraint (17)is tight at the optimality of (P3), and thus we have ˆ p ( q ) = ˆΓˆ β ( z + k q − w ˜ k ( q ) k ) . (49)The corresponding objective value of (P3) is expressed as f ( z ) = ˆΓˆ β k q − w ˜ k ( q ) k + z k q k + z . (50)Then, we consider the following three cases to obtain themaximum value of f ( z ) . • In Case 1 (i.e., k q − w ˜ k ( q ) k < k q k ), it can be verified that f ( z ) monotonically increases with z ∈ [ H min , H max ] ,and thus we have z ∗ ( q ) = H max . By substituting thisinto (49), we have ˆ p ∗ ( q ) = ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) . • In Case 2 (i.e., k q k < k q − w ˜ k ( q ) k ), it can be verified that f ( z ) monotonically decreases with z ∈ [ H min , H max ] , and thus we have z ∗ ( q ) = H min . By substituting thisinto (49), we have ˆ p ∗ ( q ) = ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) . • In Case 3 (i.e., k q k = k q − w ˜ k ( q ) k ), we have f ( z ) =ˆΓ / ˆ β , which is regardless of the UAV’s flight altitude z .Thus, the optimal flight altitude z ∗ ( q ) can be an arbitraryvalue within the interval [ H min , H max ] . By substitutingthis into (49), we have ˆ p ∗ ( q ) = ˆΓˆ β ( z ∗ ( q ) + k q − w ˜ k ( q ) k ) .If ˆ p < ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) , then only the powerconstraint (15) is tight at the optimality of (P3). Thus, theUAV can be placed at the lowest altitude and transmit at themaximum power to maximize the received power at the SR,i.e., ˆ p ∗ ( q ) = P and z ∗ ( q ) = H min .Finally, if ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) < ˆ p < ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) , it follows that constraint (17) must betight at the optimality of (P3), since otherwise we can de-crease the UAV’s altitude and/or increase the UAV’s transmitpower to increase the SR’s achievable rate, without violatingthe PR’s IT constraint. Therefore, we still have ˆ p ( q ) in(49) and f ( z ) in (50). By substituting (49) into the powerconstraint (15), we have z ≤ q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k . With ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) < ˆ p < ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) ,it follows that H min < q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k < H max .Thus, we can obtain H min ≤ z ≤ q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k . Next,by checking the monotonicity of f ( z ) (similarly as the casewith ˆ P > ˆΓˆ β ( H + k q − w ˜ k ( q ) k ) ), we can obtain thefollowing results: • In Case 1 (i.e., k q − w ˜ k ( q ) k < k q k ), z ∗ ( q ) = q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k and ˆ p ∗ ( q ) = ˆ p . • In Case 2 (i.e., k q k < k q − w ˜ k ( q ) k ), z ∗ ( q ) = H min and ˆ p ∗ ( q ) = ˆΓˆ β ( k q − w ˜ k ( q ) k + H ) . • In Case 3 (i.e., k q k = k q − w ˜ k ( q ) k ), z ∗ ( q ) is an arbitraryvalue within the interval [ H min , q ˆ β ˆ p ˆΓ − k q − w ˜ k ( q ) k ] ,and ˆ p ∗ ( q ) = ˆΓˆ β ( z ∗ ( q ) + k q − w ˜ k ( q ) k ) By combining all the results above and with some manipula-tion, this proposition is proved.
B. Proof of Lemma 3.1
First, we define p , ˆΓˆ β (min k ∈K k w k k + H ) , whichdenotes the maximally allowable value of ˆ p for IT constraints(14) to be feasible, in the case when the UAV is located exactlyabove the SR at (0 , , H min ) . Notice that if ˆ p is sufficientlylow with ˆ p ≤ p , the UAV can be placed above the SR atthe lowest altitude. Then Lemma 3.1 holds accordingly. As aresult, it only remains to consider the case with ˆ p > p , forwhich we can prove Lemma 3.1 by contradiction. Specifically,suppose that, at the optimal solution, the UAV is placedcloser to a PR k ∈ K than the SR with k q ⋆ − w k k < k q ⋆ k , ∀ k ∈ K . By combining constraints (15) and (17), wehave ˆ p ⋆ ≤ min(ˆ p, ˆΓˆ β ( k q ⋆ − w k k + z )) , ∀ k ∈ K , and the corresponding objective value of (P1.1) can be obtained as ξ ≤ min(ˆ p, ˆΓˆ β ( k q ⋆ − w k k + z )) k q ⋆ k + z ≤ ˆΓˆ β k q ⋆ − w k k + z k q ⋆ k + z , ξ , ∀ k ∈ K . Due to k q ⋆ − w k k < k q ⋆ k , ∀ k ∈ K , it follows that ξ ≤ ξ < ˆΓ / ˆ β . Next, it is easy to verify that ( q , z, p ) = ( , H min , p ) isa feasible solution to (P1) (i.e., the UAV hovers right above theSR at the lowest altitude). The corresponding objective valueof (P1.1) can be obtained as ξ = ˆΓˆ β min k ∈K k w k k + H H . Itis evident that ξ > ˆΓ / ˆ β > ξ ≥ ξ , which contradicts theoptimality of q ⋆ . Therefore, this lemma is proved. C. Proof of Proposition 3.3
Suppose that the optimal solution to problem (P4.3) isdenoted by ˆ p ⋆ , τ ⋆ , and S ⋆ . Then we construct the followingproblem: (P4.5): max E, S E s.t. Tr ( B k S ) ≥ E, ∀ k ∈ K , (51)Tr ( AS ) ≤ ˆ p ⋆ τ ⋆ − H , (52)Tr ( CS ) = 1 , (53) S (cid:23) . (54)It is evident that problem (P4.5) and problem (P4.3) have thesame optimal solution of S . Therefore, S ⋆ is also optimalfor (P4.5). Hence, to prove this proposition, we only needto show that when the K PRs are located at the same sideof the SR, we have rank ( S ⋆ ) = 1 for problem (P4.5). Noticethat (P4.5) is a convex SDP and satisfies the Slater’s condition.Therefore, strong duality holds between problem (P4.5) and itsdual problem. Let γ k ≥ , ∀ k ∈ K , λ ≥ , and µ denote thedual variables associated with the constraints in (51), (52), and(53), respectively. Then the Lagrangian of (P4.5) is expressedas L ( E, S , λ, { γ k } , µ, G )= − X k ∈K γ k ! E − λ (cid:18) H − ˆ p ⋆ τ ⋆ (cid:19) + µ + Tr ( GS ) , (55)where G = − λ A − µ C + P k ∈K γ k B k . Accordingly, the dualproblem of (P4.5) is given by(D4.5) min λ ≥ , { γ k ≥ } ,µ − λ (cid:18) H − ˆ p ⋆ τ ⋆ (cid:19) + µ s.t. G (cid:22) , (56) X k ∈K γ k = 1 . (57)Denote the optimal solution of (D4.5) as λ ⋆ , γ ⋆k , ∀ k ∈ K , and µ ⋆ . Accordingly, the resultant G ⋆ can be explicitly expressedas G ⋆ = P k ∈K γ ⋆k − λ ⋆ − P k ∈K γ ⋆k x k P k ∈K γ ⋆k − λ ⋆ − P k ∈K γ ⋆k y k − P k ∈K γ ⋆k x k − P k ∈K γ ⋆k y k − µ ⋆ + P k ∈K ( γ ⋆k ( x k + y k )) . Then the optimal solution to (P4.5) and (D4.5) should sat-isfy the complementary slackness condition Tr ( G ⋆ S ⋆ ) = 0 ,or equivalently, G ⋆ S ⋆ = . Therefore, in order to showrank ( S ⋆ ) ≤ , we only need to show that rank ( G ⋆ ) ≥ .Towards this end, in the following we show that P k ∈K γ ⋆k − λ ⋆ = 0 must hold by contradiction. Suppose that P k ∈K γ ⋆k − λ ⋆ = 0 . Then we have G ⋆ = − P k ∈K γ ⋆k x k − P k ∈K γ ⋆k y k − P k ∈K γ ⋆k x k − P k ∈K γ ⋆k y k − µ ⋆ + P k ∈K ( γ ⋆k ( x k + y k )) . Denote by d an eigenvalue of the matrix G ⋆ . Then, we havedet ( G ⋆ − d I ) = 0 , which leads to d (cid:0) − d + ǫ d + ǫ (cid:1) = 0 , (58)where ǫ = (cid:0)P k ∈K γ ⋆k x k (cid:1) + (cid:0)P k ∈K γ ⋆k y k (cid:1) , and ǫ = − µ ⋆ + P k ∈K (cid:0) γ ⋆k (cid:0) x k + y k (cid:1)(cid:1) . Due to the fact that γ ⋆k ≥ , ∀ k ∈ K , P k ∈K γ ⋆k = 1 , and all PRs are located at the same side ofthe SR (e.g., x k ≥ , ∀ k ∈ K , and there exists at least onePR ¯ k ∈ K with x ¯ k > , among the four possible cases), wehave ǫ > . Therefore, there must exist a positive root toequation (58), i.e., the matrix G ⋆ has a positive eigenvalue.This contradicts G (cid:22) in (56). Hence, P k ∈K γ ⋆k − λ ⋆ mustbe non-zero.With P k ∈K γ ⋆k − λ ⋆ = 0 , it is easy to show thatrank ( G ⋆ ) ≥ via some simple elementary transformation.Based on G ⋆ S ⋆ = , it thus follows that rank ( S ⋆ ) ≤ .Thus, Proposition 3.3 is proved. D. Proof of Lemma 3.2
Without loss of generality, we denote q = a ˆ q with a ≥ and k ˆ q k = 1 . Accordingly, problem (P5) can be re-expressedas (P5.2): max ˆ p, ˆ q ,a ≥ ˆ pH + a (59)s.t. ˆ β ˆ pH + k a ˆ q − w k ≤ ˆΓ , (60) k ˆ q k = 1 , (61)(15).Under any given feasible ˆ p , optimizing a and ˆ q in (P5.2) isequivalent to solving(P5.3): min ˆ q ,a ≥ a s.t. k a ˆ q − w k ≥ ˆ β ˆ p ˆΓ − H , (62) k ˆ q k = 1 . (63)On one hand, if q ˆ β ˆ p ˆΓ − H ≤ k w k , then it is easyto verify that a = 0 is the optimal solution to (P5.3).Thus, Lemma 3.2 directly follows. On the other hand, if q ˆ β ˆ p ˆΓ − H > k w k , then a = 0 becomes infeasible andconstraint (62) should be tight. In this case, constraint (62)can be rewritten as a k ˆ q − w /a k = ˆ β ˆ p/ ˆΓ − H . Asa consequence, a is minimized only when ˆ q is chosen suchthat k ˆ q − w /a k is maximized. As a result, ˆ q = − w k w k must hold. By substituting ˆ q = − w k w k into q = a ˆ q , wehave q = − a w k w k with a > . By combining the two cases,Lemma 3.2 is proved. E. Proof of Proposition 3.4
First, we consider a relaxed problem of (P5.1) with themaximum transmit power constraint ˆ p ≤ ˆ p ignored, denotedas (P5.4). It is evident that with the optimal solution to (P5.4),the IT constraint (30) must be tight, and thus we have ˜ p = ˆΓˆ β (cid:0) ( a + k w k ) + H (cid:1) . (64)By substituting the above into the objective function of (P5.4),it can be recast as φ ( a ) = (( a + k w k ) + H ) / ( a + H ) . By checking the first-order derivative of φ ( a ) with respect to a , the optimal a can be obtained as ˜ a ⋆ , as given in (32). Bysubstituting this into (64), we have ˜ p ⋆ given in (31).Next, we consider the problem (P5.1) with the maximumtransmit power constraint ˆ p ≤ ˆ p considered. We prove thisproposition by considering the following three cases, respec-tively.If ˆ p > ˜ p ⋆ , the optimal solution to problem (P5.4) is alsofeasible to problem (P5.1). As the objective value of (P5.4)serves as an upper bound on that of (P5.1), it follows thatsuch a solution is also optimal to (P5.1).If ˆ p < p , it is evident that the UAV should hover exactlyabove the SR at the minimum altitude and transmit at themaximum power to maximize the received power at the SR,with the PR’s IT constraint satisfied. Therefore, we have ˆ p ⋆ =ˆ p and a ⋆ = 0 in this case.If p ≤ ˆ p ≤ ˜ p ⋆ , the IT constraint (30) must be tightat the optimality, since otherwise we can always move theUAV closer to the SR and/or increase the UAV’s transmitpower to increase the SR’s achievable rate, without violatingthe PR’s IT constraint. Therefore, we have ˆ p = ˜ p in (64),or equivalently, a = q ˆ β ˆ p/ ˆΓ − H − k w k . Given this,the objective function of (P5.1) can be recast as ˆ φ (ˆ p ) =ˆ p/ ( ˆ β ˆ p/ ˆΓ + k w k − k w k q ˆ β ˆ p/ ˆΓ − H ) . It is easy toverify that ˆ φ (ˆ p ) monotonically increases with ˆ p ∈ [0 , ˆ p ] . Thus,we have ˆ p ⋆ = ˆ p and a ⋆ = q ˆ β ˆ p/ ˆΓ − H − k w k .By combining the above three cases, Proposition 3.4 isproved. R EFERENCES[1] Y. Huang, J. Xu, L. Qiu, and R. Zhang, “Cognitive UAV communicationvia joint trajectory and power control,” in
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