Improving PHY-Security of UAV-Enabled Transmission with Wireless Energy Harvesting: Robust Trajectory Design and Communications Resource Allocation
aa r X i v : . [ c s . I T ] A p r Improving PHY-Security of UAV-EnabledTransmission with Wireless Energy Harvesting:Robust Trajectory Design and CommunicationsResource Allocation
Milad Tatar Mamaghani ∗ , Yi Hong ∗ , Senior Member, IEEE ∗ Electrical and Computer Systems Engineering Department, Monash University, Melbourne, Australia
Abstract —In this paper, we consider an unmanned aerialvehicle (UAV) assisted communications system, including twocooperative UAVs, a wireless-powered ground destination nodeleveraging simultaneous wireless information and power transfer(SWIPT) technique, and a terrestrial passive eavesdropper. OneUAV delivers confidential information to destination and the othersends jamming signals to against eavesdropping and assist desti-nation with energy harvesting. Assuming UAVs have partial infor-mation about eavesdropper’s location, we propose two transmis-sion schemes: friendly UAV jamming (FUJ) and Gaussian jam-ming transmission (GJT) for the cases when jamming signals areknown and unknown a priori at destination, respectively. Then,we formulate an average secrecy rate maximization problem tojointly optimize the transmission power and trajectory of UAVs,and the power splitting ratio of destination. Being non-convexand hence difficult to solve the formulated problem, we proposea computationally efficient iterative algorithm based on blockcoordinate descent and successive convex approximation to obtaina suboptimal solution. Finally, numerical results are providedto substantiate the effectiveness of our proposed multiple-UAVschemes, compared to other existing benchmarks. Specifically, wefind that the FUJ demonstrates significant secrecy performanceimprovement in terms of the optimal instantaneous and averagesecrecy rate compared to the GJT and the conventional single-UAV counterpart.
Index Terms —UAV communications, PHY-security, SWIPT,trajectory design, power control, cooperative mobile jammer,convex optimization.
I I
NTRODUCTION R ECENTLY , unmanned aerial vehicle (UAV) has beendeemed as a promising wireless service provider along-side with plethora of other civilian applications (see [1]–[4] and references therein). This is driven by advances inwireless equipment miniaturization as well as the economicease of deployment and flexibility of UAVs inasmuch asvarious Tech giants (e.g. Facebook and Google) [5] have beenfocusing on establishing massive UAV-assisted networks forubiquitous connectivity. As a matter of fact, the upsurge ofUAV applications in wireless communications is double-edgedsword; in that bringing new opportunities and facilitating noveltechnologies, while accompanying with undeniable criticalchallenges when employed in the real world. “This research was supported by the Australian Research Council underGrant DP160100528.”
On the one hand, with an increasing demand of Internet-of-things (IoT) applications, UAVs equipped with various typesof sensors, cameras, GPS, and so on, can be regarded as goodcandidates to serve as aerial base stations/legitimate termi-nals/mobile relaying and even power beacons for prolongingenergy-constraint IoT devices [6]–[10]. In such applications,a challenging issue is that how to prolong device lifetimedue to limited access to power resources and/or infrequentbattery replacements [7]. To tackle this problem, apart fromconventional energy harvesting techniques, simultaneous wire-less information and power transfer (SWIPT) has recentlyemerged [11]. To be specific, SWIPT captures both data andenergy from the same radio frequency (RF) signal and con-verts into direct current for battery recharging, which enablesenergy harvesting in a controllable manner. This characteristicis particularly important for UAV applications to guaranteereplenishable-energy ground nodes considering their dynamicadjustment capability [8], [10], [12], [13]. Specifically, aSWIPT-based UAV-aided relaying scenario to transmit powerand confidential information to an energy-constrained grounduser has been analyzed in terms of average achievable secrecyrate and energy coverage probability in [12], while the secrecyrate lower bound optimization problem of such setup hasbeen conducted in [10]. Aiming at minimization of the UAV’stotal power consumption, the authors in [13] also explored anon-security UAV-based wireless communications system withenergy harvesting to enable data transmission of ground usersin both half duplex and full duplex modes using the harvestedenergy.On the other hand, safeguarding such wireless communi-cations system is of the most paramount challenges due tothe broadcast nature of transmission and mobility of UAVs.To guarantee security of UAV communications, physical-layer (PHY) security [1], [14]–[22] have, providentially, beenascertained as a promising and computationally-efficient in-formation secrecy approach. For example, the resource al-location problem for a UAV-assisted secure SWIPT systemis investigated in [15]. The authors in [18] also consideredthe PLS of a four-node setup with UAV-enabled relayingwhere the eavesdroppers are distributed in a certain area withpartially known location information and then studied thepower allocation problem of the source and relay. Amongstvarious PHY-security techniques, cooperative jamming is one viable anti-eavesdropping strategy via collaboratively trans-mitting jamming signals to degrade wiretap channel quality.In [19], the authors have considered a mobile UAV servingas a flying base station delivering data to a ground nodein the presence of a passive eavesdropper. In [20], lever-aging the mobility of a UAV, the authors have studied theachievable secrecy rate via trajectory design and power controloptimization, and showed its improvement over conventionalstatic jammers. This is due to the fact that the mobility ofUAV-jammer allows an opportunistically jamming at a closerdistance to the eavesdropper. In [21], the authors have tackledmaximizing the minimum secrecy rate of jammer-incorporatedUAV communications via a joint optimization of trajectory andtransmit power of UAVs. In [22], the authors have studiedthe problem when a UAV is employed as friendly jammerto assist secure communication in the presence of unknowneavesdropper location, and they have examined the UAV-jammer displacement and power control to guarantee goodreliability and security.Motivated by above research, in this paper, we considertwo flying cooperative UAVs as well as a ground destinationnode equipped with wireless RF energy harvester, in thepresence of a passive ground eavesdropper. One UAV actsas source transmitting confidential information to destinationwhile the other UAV broadcasts jamming signals to assist anti-eavesdropping and energy harvesting of the destination node.Note that, different from [19], [23], [24], we here consider a
SWIPT-enabled receiver at destination for security and energyscavenging. Also, different from [24], [25] wherein UAVsknow the exact location of eavesdropper, we here assumethat UAVs have only partial information of eavesdropper’slocation. Following our setting, we make the following con-tributions in the paper. • We propose two cooperative UAV-jamming PHY-securityschemes: friendly UAV jamming (FUJ) and
Gaussianjamming transmission (GJT). In particular, in FUJ, UAVtransmits jamming signals that are known a priori atdestination, while in GJT, destination node has no priorinformation of the jamming signals from UAV. • Via trajectory discretization approach, we formulate anaverage secrecy rate (ASR) optimization problem, whichis challenging to solve due to non-smooth and non-concave objective function and non-convex feasible set. • To make the optimization problem tractable, we proposean efficient iterative algorithm based on block coordinatedescent (BCD) and successive convex approximation(SCA) methods in order to find a unique sub-optimalsolution to the problem with guaranteed convergence. • Via the proposed iterative algorithm, we conduct opti-mization of the following sub-problems: transmit powerof UAVs, power splitting ratio in SWIPT, as well as UAVstrajectory. • We compare by simulations secrecy and energy harvest-ing performance, transmit power of UAVs of our pro-posed schemes under various scenarios, demonstrating itssignificant performance improvement over conventionalwithout-jamming (WoJ) scheme, wherein there exists no (cid:460) (cid:454)(cid:455) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:104)(cid:4)(cid:115)(cid:3)(cid:58)(cid:258)(cid:373)(cid:373)(cid:286)(cid:396)(cid:894) (cid:45) (cid:895)(cid:104)(cid:4)(cid:115)(cid:3)(cid:94)(cid:381)(cid:437)(cid:396)(cid:272)(cid:286)(cid:894) (cid:54) (cid:895)(cid:24)(cid:286)(cid:400)(cid:410)(cid:349)(cid:374)(cid:258)(cid:410)(cid:349)(cid:381)(cid:374)(cid:3)(cid:894) (cid:39) (cid:895) (cid:28)(cid:258)(cid:448)(cid:286)(cid:400)(cid:282)(cid:396)(cid:381)(cid:393)(cid:393)(cid:286)(cid:396)(cid:3)(cid:894) (cid:40) (cid:895) (cid:116)(cid:349)(cid:396)(cid:286)(cid:410)(cid:258)(cid:393)(cid:3)(cid:367)(cid:349)(cid:374)(cid:364)(cid:58)(cid:258)(cid:373)(cid:373)(cid:349)(cid:374)(cid:336)(cid:3)(cid:367)(cid:349)(cid:374)(cid:364)(cid:68)(cid:258)(cid:349)(cid:374)(cid:3)(cid:367)(cid:349)(cid:374)(cid:364) (cid:28)(cid:400)(cid:410)(cid:349)(cid:373)(cid:258)(cid:410)(cid:286)(cid:282)(cid:3)(cid:396)(cid:286)(cid:336)(cid:349)(cid:381)(cid:374)
Fig. 1: System model of UAV-enabled secure information andpower transfer.UAV-jammer.The rest of the paper is organized as follows. Section IIintroduces system model. Section III presents two 2-UAVtransmission schemes via cooperative UAV jamming. In Sec-tion IV, we formulate ASR optimization problem via trajectorydiscretization approach and provide solutions in Section V.Simulation results are given in Section VI, followed by con-clusions in Section VII.II S
YSTEM M ODEL
We consider a UAV-enabled wireless communications sys-tem (see Fig. 1), where a UAV-source ( S ) flies from ini-tial to final locations to deliver confidential information toa legitimate ground destination ( D ) in the presence of a ground eavesdropper ( E ) with unknown location . Here, weconsider D to be an energy-limited IoT device that is capableof harvesting energy from ambient radio resources and itsreceiver adopts power splitting architecture for simultaneousenergy scavenging and data decoding with a power splittingratio (PSR) ζ ( ≤ ζ ≤ ) [7], [26]. Finally, a UAV-jammer ( J )is employed to transmit noise-like jamming signals coopera-tively to improve security and power the energy-constraint D .We consider that all nodes have single omnidirectionalantenna that operate in half-duplex mode . We define mainlink ( S - D ), wiretap link ( S - E ), jamming link ( J - D , J - E ),as shown in Fig. 1. II-A System Parameters
Without loss of generality, we assume that all the nodesare located in a three-dimensional Cartesian coordinate systemwith the following parameters: • D has the horizontal coordinate W D ∈ R × with zeroaltitude, • S and J ’s initial and final locations corresponding to theprespecified launching and landing sites of the UAVs are Q SI ∈ R × , Q JI ∈ R × , Q SF ∈ R × , and Q JF ∈ R × , with constant flying altitude H . • S and J have the same mission time T , and their horizon-tal location at time instant t ∈ [0 , T ] are Q S ( t ) ∈ R × and Q J ( t ) ∈ R × , • S and J have a safety distance ˜ D to avoid collision , • S and J have total transmission power P totS and jammingpower P totJ , respectively, whereas at t ∈ [0 , T ] , theirassociate instantaneous powers are P S ( t ) and P J ( t ) , • PSR, denoted by ζ ∈ (0 , , is the fraction of receivedpower for information processing , while (1- ζ ) is the fraction of which to be harvested and stored for futureuse . The instantaneous PSR is, therefore, denoted by ζ ( t ) .Further, we have the following assumptions on D and E ’slocations: • D ’s location is known to both UAVs (e.g. [28], [19]), • E ’s location ( W E ∈ R × ) is unknown, but both UAVscan approximately estimate it [29] in a collaborativemanner. As such, we assume that E ’s circular estimatedregion centered at ˆ W E ∈ R × (namely most-likelylocation of E ) with radius R E ≥ k W E − ˆ W E k (namely maximum estimation error ) are known to the UAVs,where k · k represents the L -norm (Euclidean norm). Remark:
Note that according to [16], the availability of theeavesdroppersâ ˘A ´Z location information can be classified intothree cases: I) full position information, II) partial positioninformation, and III) absence of position information. CaseI becomes possible, when the eavesdroppers stay stationary,and UAV is equipped with an optical camera or a syntheticaperture radar to detect the eavesdropper’s location, or itmight be the case when the ground nodes are part of thesame network with different roles; e.g., unscheduled users toreceive particular information compared to the intended ones.Other method is presented in [28] to obtain eavesdropper’slocation information from the local oscillator power whichis inadvertently leaked from its RF front-end, given coherentdetection is used. Case II occurs when the above detectionis non-accurate, or when eavesdroppers have moved slightlyso that the camera/aperture radar in UAV cannot perfectlyobtain their location information. Case III occurs when allthe above detection methods are failed and eavesdroppershide themselves physically very well. In this paper, we haveconsidered Case II.
II-B Channel Model
Motivated by literature (see [25], [27], [30]–[33]), in thiswork, we adopt a probabilistic line-of-sight (LOS) channelmodel that models both LoS and Non-LoS propagations bytaking into account their occurrence probabilities [30]. Av-eraging over surrounding environment and small-scale fading, Indeed, one justification from a practical viewpoint behind this constantUAVs’ flying altitude assumption is to guarantee the safety consideration likecollision avoidance with buildings or terrain, and also more importantly, forenergy consumption reduction when ascending or descending of UAVs, e.g.,[23], [27]. This is different from the traditional approach in [21], where differentflying altitudes are allocated to each UAV to avoid possible collision. the expected channel power of UAV-ground (UG) links at timeinstant t is [30] ˆ h ag ( t ) = ˆ β ( θ ag ( t )) d ag ( t ) − α , (1)with the regularized attenuation factor given by ˆ β ( θ ag ( t )) , β [ P LoS ( θ ag ( t )) + κ (1 − P LoS ( θ ag ( t )))] , (2)where d ag ( t ) = p k Q a ( t ) − W g k + H represents the timevarying distance between the aerial node a and the groundnode g . Moreover, θ ag ( t ) = tan − (cid:16) Hd ag ( t ) (cid:17) denotes the time-varing elevation-angle between those two, wherein a ∈ { S , J }and g ∈ { D , E }, α denotes the path-loss exponent ( ≤ α ≤ )[34], β is the path loss at reference distance d meter foromnidirectional antennas under LoS, i.e., β ,
20 log (cid:18) C4 πd f c (cid:19) , ( in dB ) where C = 3 × m / s is the speed of light and f c isthe carrier frequency [34]. The parameter κ is the additionalattenuation factor characterizing Non-LoS propagation (inpractice it is a random variable with log-normal distributiondenoting the shadowing effect); however, in (2), this parameteris regarded to be constant following homogeneous assumptionfor Non-LoS environment. Here, In consistent with [30], [35],we assume that for the area of interest the elevation angledependent probabilistic LoS function P LoS ( θ ( t )) = 11 + k exp( − k ( θ ( t ) − k )) , with environmental constants k , k > follow homogeneity,leading ultimately to ˆ β ( θ ( t )) ≈ ¯ β for the sake of simplicityof trajectory and resource allocation design .III P ROPOSED
PHY-
SECURITY S CHEMES AND I NSTANTANEOUS /A VERAGE S ECRECY C APACITY
In this work, we present two PHY-security schemes in-volving two UAVs. Major difference between our schemesand other known two-UAV schemes (e.g. [21], [24], [25])lies in that the additional cooperative UAV conducts not onlyjamming transmission but also powering D in a more practicalchannel modelling : • A FUJ scheme, wherein FUJ transmits jamming signalsthat are known a priori at D • A GJT scheme, wherein D has no prior knowledge ofthe noise-like jamming signal. It is worth pointing out that this approximated and simplified model istoo fruitful in some applications such as post-disaster area wherein it is non-trivial to categorize the environment based on which the probabilistic modelhas been developed. However, the minimum and maximum values of path-losscomponent α can be used for upper and lower bound performance [2]. Note that while the FUJ scheme requires a priori to generate jammingsignals at J and also costs a higher computational complexity at D tooperate jamming cancellation, it can be implemented via various approachessuch as key-assisted coding; i.e., an intelligent combination of conventionalcryptography with PHY-security [36]. Specially, when the location of theeavesdropper E is unknown to the legitimate nodes and the wiretap linkquality might experience a better channel condition compared to the mainlink, the former scheme is capable of PHY-security enhancement, while thelatter lacks such an undeniable performance advantage nonetheless providesa low complex implementation approach. To evaluate performance of above schemes (particularly inlater simulations), we consider a benchmark scheme : • No additional UAV-jammer (WoJ) scheme with SWIPT atdestination. Note that this setup is similar to [19], except[19] has no SWIPT.
III-A Instantaneous Secrecy Rate (ISR)
Recall system parameters in subsection II-A and assumenormalized bandwidth in all links.GJT has the achievable average rate over the randomchannel realizations at time instant t as I M ( t )=log γ S ( t ) ζ ( t ) (cid:0) k Q S ( t ) − W D k + H (cid:1) − α γ J ( t ) ζ ( t ) ( k Q J ( t ) − W D k + H ) − α +1 ! , (3)where γ S ( t ) ∆ = P S ( t ) ¯ βN and γ J ( t ) ∆ = P J ( t ) ¯ βN with N being thenoise power at the receiver of D . Since the UAV jammingsignal is known a priori by D as well as the channel stateinformation (CSI) is available, it can be removed from thereceived signals. Therefore, FUJ has the achievable instanta-neous ensumble rate from S to D as I M ( t )=log (cid:16) γ S ( t ) ζ ( t ) (cid:0) k Q S ( t ) − W D k + H (cid:1) − α (cid:17) . (4)Additionally, for both GJT and FUJ, the exact instantaneouswiretap channel capacity ˆ I E ( t ) at eavesdropper can be ob-tained as ˆ I E ( t )=log γ S ( t ) (cid:0) k Q S ( t ) − W E k + H (cid:1) − α γ J ( t ) ( k Q J ( t ) − W E k + H ) − α +1 ! , (5)where the AWGN noise power at E is considered identical tothat at D for the simplicity of exposition.The maximum achievable data rate by E , denoted by I maxE ( t ) , within the uncertainty region R E , which serves asan upper-bound for the case of exact location of E , can becalculated, by considering the worst-case estimation scenarioby two UAVs, as I maxE ( t )=log γ S ( t ) (cid:18)(cid:16) k Q S ( t ) − ˆ W E k− R E (cid:17) + H (cid:19) − α γ J ( t ) (cid:18)(cid:16) k Q J ( t ) − ˆ W E k + R E (cid:17) + H (cid:19) − α +1 . (6) Proof.
Please see Appendix A. (cid:4)
III-B Average Secrecy Capacity
The achievable ASR from S to D with normalized trans-mission bandwidth is defined in bits/s/Hz as [37] ¯ R sec = 1 T Z T [ I M ( t ) − I maxE ( t )] + dt, (7)where [ x ] + = max { x, } and I M ( t ) for GJT and FUJ schemesare given in (3) and (4), respectively, I maxE ( t ) is in (6) forcooperative jamming. Note that I maxE ( t ) for WoJ is identicalto (6) but with setting γ J ( t ) = 0 . IV P ROBLEM F ORMULATION FOR M AXIMIZING
ASRTo maximize (7), we need a joint design of UAV trajectory,transmission power allocations, and power splitting ratio. Tomake our design practically feasible, we consider the trajec-tory discretization approach dividing the mission time T into N equally-spaced time slots δ t ∆ = TN ,
Given δ t , assuming distance variation between any UAV andthe ground terminals is adequately small, we adopt constantaverage channel gains per slot. Other system design param-eters and definitions are quantized accordingly and beingconstant within each time slot. Hence, our problem of inter-est with variables P S ∆ = { P S [ n ] } Nn =1 , P J ∆ = { P J [ n ] } Nn =1 , ζζζ ∆ = { ζ [ n ] } Nn =1 , Q S ∆ = { Q S [ n ] } Nn =1 , and Q J ∆ = { Q J [ n ] } Nn =1 is formulated as ¯ R optsec ( P ⋆ S , P ⋆ J , ζζζ ⋆ , Q ⋆ S , Q ⋆ J ) = maximize 1 N N X n =1 h ˜ R sec [ n ] i + s.t. C1 − C15 , (8)with ˜ R sec [ n ] given by (9) (shown on top of the next page)wherein ˜ H = R E + H . The constraints C1-C4 are C1 : 1 N N X n =1 P S [ n ] ≤ ¯ P S , C2 : 0 ≤ P S [ n ] ≤ ˆ P S , (10) C3 : 1 N N X n =1 P J [ n ] ≤ ¯ P J , C4 : 0 ≤ P J [ n ] ≤ ˆ P J , (11)where (C1,C3) and (C2,C4) are constraints of averageand maximum transmission/jamming powers per time slotat S and J , i.e., ( ¯ P S , ¯ P J ) and ( ˆ P S , ˆ P J ), respectively, where ¯ P S ∆ = P totS /N, ¯ P J ∆ = P totJ /N. Additionally, these fixed powers are chosen subject to the peakto average power ratio (PAPR) constraint, i.e., ˆ P K ¯ P K is restricteddue to hardware limitations, where K ∈ { S, J } , and maximumnetwork transmission power per time slot as ˆ P R = ˆ P S + ˆ P J . (12)To ensure a sufficient discretization as well as valid assump-tions of invariant channel condition and unchanged distancebetween any UAV and ground nodes, we have mobility con-straints as C5 : Q S [1] = Q SI , C6 : k Q S [ n + 1] − Q S [ n ] k ∆ = V S [ n ] δ t ≤ ˜ d δ , n = 1 · · · N − k Q SF − Q S [ N ] k ∆ = V S [ N ] δ t ≤ ˜ d δ , (13)and C8 : Q J [1] = Q JI , C9 : k Q J [ n + 1] − Q J [ n ] k ∆ = V J [ n ] δ t ≤ ˜ d δ , n = 1 · · · N − k Q JF − Q J [ N ] k ∆ = V J [ N ] δ t ≤ ˜ d δ , (14) ˜ R sec [ n ] = log ζ [ n ] P S [ n ] ¯ β (cid:0) k Q S [ n ] − W D k + H (cid:1) − α ζ [ n ] P J [ n ] ¯ β ( k Q J [ n ] − W D k + H ) − α + N ! − log P S [ n ] ¯ β (cid:16) k Q S [ n ] − ˆ W E k − R E k Q S [ n ] − ˆ W E k + ˜ H (cid:17) − α P J [ n ] ¯ β (cid:16) k Q J [ n ] − ˆ W E k + 2 R E k Q J [ n ] − ˆ W E k + ˜ H (cid:17) − α + N , (9)where V S [ n ] and V J [ n ] are constant speeds of S and J in timeslot n , but the velocities may vary from one slot to next. In par-ticular, the maximum horizontal displacement of S and J perslot is bounded by threshold maximum distance ˜ d δ ≪ H .For the considered two-UAV system, collision avoidance isrepresented by C11 : k Q S [ n ] − Q J [ n ] k ≥ ˜ D, (15)where ˜ D is the safety distance between the two UAVs. Then,the permitted flying zone for UAVs is assumed to be a circularregion with radius ˜ R , i.e., C12 : k Q S [ n ] − W D k ≤ ˜ R, C13 : k Q J [ n ] − W D k ≤ ˜ R, (16)where ˜ R ≤ vuut ˆ P R ¯ β Ψ H ! α − H , (17)must be satisfied to avoid power outage and guarantee theviability of energy harvesting. In (17), Ψ H is the minimumrequired input power for energy harvesting , and ˆ P R is givenin (12). Finally, energy harvesting constraints are C14 : 0 ≤ ζ [ n ] < , C15 : ˜ E H [ n ] ≥ Ψ H , ∀ n (18)with harvested power in time slot n given by (19) (see top ofthe next page) where η is power conversion efficiency factor , ζ [ n ] represents the discretized PSR for information processingat D , and (1 − ζ [ n ]) for energy harvesting.V P ROBLEM SOLUTION TO M AXIMIZE
ASRNote that (8) is non-convex and challenging to solve due tonon-convex objective function, non-smooth operator, [ · ] + , andsome non-convex constraints. However, at the optimal point, ˜ R sec [ n ] in (9) should be non-negative; otherwise, by setting P S [ n ] = 0 yields ˜ R sec [ n ] = 0 (It should be pointed outthat due to inequality max { x, } ≥ x , the resultant smoothobjective function given by (9) always serves a lower-boundfor the objective function of the problem (8)). Thus, ouroptimization problem can be turned into a non-convex yetsmooth (differentiable) problem as ( P
1) : maximize P S , P J , ζζζ, Q S , Q J N N X n =1 ˜ R sec [ n ] s.t. C1 − C15 . (20) The facts that, ( P is non-convex and the optimizationparameters are tightly coupled due to C15, make the prob-lem intractable and motivates us to propose an alternatingoptimization approach: an efficient iterative algorithm basedon block coordinate descent (BCD) and successive convexapproximation (SCA) methods, where at each iteration asingle block of variables is optimized by convex optimizationapproach, while the remaining variables remain unchanged.By doing so, the convergence of the proposed approach to atleast a sub-optimal solution is guaranteed under a feasible set[38]. The remaining analysis are given as follows. V-A Optimal Transmit Power of UAV-source
In the following, we optimize the power allocation of S forGJT, FUJ, and WoJ, under the given feasible trajectories andPSRs. Thus, the sub-problem for optimal transmission of S forthe most general case (GJT) can be obtained by reformulating ( P equivalently as ( P
2) : maximize P S N X n =1 [log (1+ A n P S [ n ]) − log (1+ B n P S [ n ])] s.t. C1 and C2 , g C15 : C n P S [ n ] + D n ≥ Ψ H , ∀ n (21)where log( · ) represents natural logarithm, the auxiliary con-stants { A n } Nn =1 , { B n } Nn =1 , { C n } Nn =1 , and { D n } Nn =1 , are givenby A n = γ ζ [ n ] (cid:0) k Q S [ n ] − W D k + H (cid:1) − α ζ [ n ] γ J [ n ] ( k Q J [ n ] − W D k + H ) − α + 1 , (22) B n = γ (cid:16) k Q S [ n ] − ˆ W E k − R E k Q S [ n ] − ˆ W E k + ˜ H (cid:17) − α γ J [ n ] (cid:16) k Q J [ n ] − ˆ W E k + 2 R E k Q J [ n ] − ˆ W E k + ˜ H (cid:17) − α +1 , (23) C n = η ¯ β (1 − ζ [ n ]) (cid:0) k Q S [ n ] − W D k + H (cid:1) − α , (24) D n = η (1 − ζ [ n ]) h P J [ n ] ¯ β (cid:0) k Q J [ n ] − W D k + H (cid:1) − α + N i , (25)where γ = ¯ βN , γ J [ n ] ∆ = P J [ n ] ¯ βN , and ˜ H = p R E + H .The sub-problem ( P is still non-convex with respect to P S due to non-convex objective function. Since one can readilyverify that problem ( P satisfies the Slater’s condition, strongduality attains which enables us to obtain the optimal solutionby solving the corresponding Lagrange dual problem using ˜ E H [ n ] ∆ = η (1 − ζ [ n ]) h P S [ n ] ¯ β (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + P J [ n ] ¯ β (cid:0) k Q J [ n ] − W D k + H (cid:1) − α + N i , (19)Karush-Kuhn-Tucker (KKT) conditions. As such, by temporar-ily dropping C2 and g C15 , and also letting ˜ P S and ( ˜ P S , λ ) beany primal and dual optimal points with zero duality gap, theLagrangian function can be computed as L ( P S , λ ) = N X n =1 (cid:2) log(1+ B n P S [ n ]) − log(1+ A n P S [ n ])+ λ (cid:0) P S [ n ] − ¯ P S (cid:1)(cid:3) , (26)where λ ≥ is the Lagrange factor . Then, maximizing theLagrangian dual function defined as g ( λ ) ∆ = inf P S {L ( P S , λ ) } , one can reach the optimality condition as [37] A n A n P S [ n ] − B n B n P S [ n ] − λ = 0 , ∀ n (27)Solving the above equation with respect to P S [ n ] and also tak-ing into account constraints C2 and g C15 , leads to the closed-form analytical solution for optimal UAV-source’s power allo-cation as P ⋆S [ n ]= min (cid:26) max (cid:26)h Ψ H − D n C n i + , ˜ P S [ n ] (cid:27) , ˆ P S (cid:27) , A n ≥ B n h Ψ H − D n C n i + , A n
Under keeping other variables unchanged, we aim at op-timizing the jamming transmit power for GJT and FUJ. As such, the sub-problem for optimization of the transmit powerof J for GJT can be obtained by rewriting ( P as ( P
3) : maximize P J N X n =1 log (cid:18) A n B n P J [ n ]+1 (cid:19) − log (cid:18) C n D n P J [ n ]+1 (cid:19) s.t. C3 , ˜C4 : (cid:20) Ψ H − E n F n (cid:21) + ≤ P J [ n ] ≤ ˆ P J , ∀ n (30)where the auxiliary constants { A n } Nn =1 , { B n } Nn =1 , { C n } Nn =1 , { D n } Nn =1 , { E n } Nn =1 , { F n } Nn =1 are taken as A n = ζ [ n ] γ S [ n ] (cid:0) k Q S [ n ] − W D k + H (cid:1) − α , (31) B n = γ ζ [ n ] (cid:0) k Q J [ n ] − W D k + H (cid:1) − α , (32) C n = γ S [ n ] (cid:16) k Q S [ n ] − ˆ W E k − R E k Q S [ n ] − ˆ W E k + ˜ H (cid:17) − α , (33) D n = γ (cid:16) k Q J [ n ] − ˆ W E k + 2 R E k Q J [ n ] − ˆ W E k + ˜ H (cid:17) − α , (34) E n = η (1 − ζ [ n ]) h P S [ n ] ¯ β (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + N i , (35) F n = η ¯ β (1 − ζ [ n ]) (cid:0) k Q J [ n ] − W D k + H (cid:1) − α . (36)The sub-problem ( P is still non-convex with respect to P J due to non-convex objective function being in the form ofconvex-minus-convex based on Lemma 1 given below. Lemma 1.
Let x ∈ R N × be a vector of variables, { a n } Nn =1 and { b n } Nn =1 be all non-negative constants. Then, the vectorfunction defined as f ( x ) = N X n =1 log (cid:18) a n b n x [ n ] + 1 (cid:19) , (37) is convex.Proof. By calculating the gradient vector and also obtainingthe Hessian matrix of f ( x ) we have ∇ f (x) = (cid:26) − a n b n (1 + a n + b n x [ n ])(1 + b n x [ n ]) (cid:27) Nn =1 , (38) H f = diag (cid:18) a n b n ( a n + 2 b n x [ n ] + 2)( b n x [ n ] + 1) ( a n + b n x [ n ] + 1) (cid:19) , (39)where ∇ ( · ) and H represent gradient and Hessian operators,respectively. The convexity of f ( x ) follows from the fact thatthe Hessian matrix given by (39) is positive semi-definite.Since it is in a diagonal form with all non-negative elements,which further implies that all the eigenvalues correspondingto the Hessian matrix are non-negative. This completes theproof. (cid:4) Note that compared to S ’s optimal power allocation ( P ), which we couldsolve the non-convex but differentiable problem analytically via its Lagrangedual approach, the objective function of ( P ) is quite sophisticated inasmuchas the Lagrange method leads to a harder problem to solve analytically.Therefore, here we employ another technique. Since the first term of the objective function to be maxi-mized is convex, our approach is two-fold: approximating thisconvex term with its corresponding concave lower bound, andapplying SCA in an iterative manner. By doing so, we are ableto reach an approximate solution with guaranteed convergence.Specifically, we replace the first convex term ( P with its firstorder Taylor expansion at { P kJ [ n ] } Nn =1 , which is defined as thegiven transmit power of J at iteration k . It is worth mentioningthat based on first-order condition [39], the first order Taylorapproximation at the local point x ∈ R N × provides a globalunder-estimator of a convex function f ( x ) , i.e., f ( x ) ≥ f ( x ) + ∇ f ( x ) T ( x − x ) , (40)where ( · ) † represents transpose operator. Thus, for any givenlocal point at iteration k ; i.e., P k J = { p kJ [ n ] } Nn =1 , ( P turnsinto an approximated convex problem as ( P
4) : maximize P J N X n =1 ˆ B n + ˆ A n P J [ n ] − log (cid:18) C n D n P J [ n ]+1 (cid:19) s.t. C3 and ˜C4 , ∀ n (41)where ˆ A n = − A n B n (1+ A n + B n P kJ [ n ])(1 + B n P kJ [ n ]) , (42) ˆ B n = log (cid:18) A n B n P kJ [ n ] + 1 (cid:19) , (43)Note that ( P is a convex problem for which the Slater’sconditions can be readily verified, any points P ⋆ J and ( P ⋆ J , λ ⋆ ) satisfying the KKT conditions are primal and dual optimalwith zero duality gap, which implies that the dual optimum isattained. Although problem ( P can be numerically solvedby any standard convex optimization techniques such as theinterior-point method [39], we are going to step further andapply Lagrangian method to gain more insight into structureof the sub-optimal solution and also effectively reduce thecomplexity of the algorithm. As such, temporarily droppingthe constraint ˜ C , the Lagrange dual function is written as g ( P J , ν ) =inf P J ( N X n =1 (cid:20) − ˆ A n P J [ n ] − ˆ B n +log (cid:18) C n D n P J [ n ]+1 (cid:19) + ν (cid:0) P J [ n ] − ¯ P J (cid:1)(cid:21)) , (44)where the non-negative scalar ν is the Lagrange multipliercorresponding to ˜C3 in ( P . Then, solving ∇ g ( P J , ν ) = 0 results in the optimality condition as P J [ n ] + C n D n − P J [ n ] + D n + ν − ˆ A n = 0 , (45)which can be rewritten as P J [ n ]+ (cid:18) C n D n (cid:19) P J [ n ]+ " C n D n − C n D n ( ν − ˆ A n ) = 0 , (46) Finally, solving the equation above while considering con-straint ˜C4 ; we reach the optimal solution of P ⋆ J = { P ⋆J [ n ] } Nn =1 as P ⋆J [ n ] =min max q C n D n ν − ˆ A n + C n − ( C n +2)2 D n , (cid:20) Ψ H − E n F n (cid:21) + , ˆ P J . (47)where ν ≥ is the Lagrange multiplier at optimal point, satis-fying P Nn =1 P ⋆J [ n ] ≤ P totJ , which can be attained by a simplebisection search. We note that ( P is a lower-bound to ( P but with the same constraints, so the solution to ( P , i.e., P ⋆J ,is no less than that of ( P at the given point (cid:16) ˆ B n , ˆ A n , P k J (cid:17) .Similarly, for the FUJ scheme the optimal J power allocation P ⋆J is obtained as (47) but with setting ˆ A n = 0 . V-C Optimal power splitting ratio
We aim at designing an efficient power splitter at desti-nation D . For fixed P K and Q K , where K ∈ { S , J }, theequivalent sub-problem for optimizing PSR { ζ [ n ] } Nn =1 of bothGJT and FUJ, is recasted as ( P
6) : maximize ζζζ N X n =1 (cid:20) log (cid:18) A n ζ [ n ] B n ζ [ n ] + 1 (cid:19)(cid:21) s.t. ˜C14 : 0 ≤ ζ [ n ] ≤ (cid:20) − Ψ H ηC n (cid:21) + , ∀ n (48)where the auxiliary constants for n ∈ { , , · · · , N } aredefined as A n = γ S [ n ] (cid:0) k Q S [ n ] − W D k + H (cid:1) − α , (49) B n = γ J [ n ] (cid:0) k Q J [ n ] − W D k + H (cid:1) − α , (50) C n = P S [ n ] ¯ β (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + P J [ n ] ¯ β (cid:0) k Q J [ n ] − W D k + H (cid:1) − α + N , (51)It can be verified from Lemma 2 that the problem ( P isconcave and its objective function is monotonically increasing. Lemma 2.
Let x > be a scalar variable and a and b be positive constants. Define f ( x ) = log (cid:16) axbx +1 + 1 (cid:17) . Takingthe first and the second derivative of f ( x ) with respectto x results in D f ( x ) = a ( bx +1)( ax + bx +1) and D f ( x ) = − a (2 abx + a +2 b ( bx +1))( bx +1) ( ax + bx +1) respectively, where D is the differenti-ation operator. Since for any value of x in the domain of f we have D f ( x ) > and D f ( x ) < , this illustrates thatthe function is strictly concave being monotonic increasing.Besides, we know that the log-product function or equivalently h = P log( x ) where x = { x i } Ni =1 is concave and non-increasing with respect to each argument x i . Therefore, fromthe vector composition law [39] one can readily conclude that g (x) = hof ( x ) = h ( f ( x ) , f ( x ) , · · · , f ( x N )) is concave. Therefore, the analytical solution for ζ ⋆ for GJT scenariocan be readily obtained as ζ ⋆ [ n ] = (cid:20) − Ψ H ηC [ n ] (cid:21) + , (52) For FUJ and WoJ, replacing the constants B n = 0 and C n = P S [ n ] ¯ β (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + N with (50) and (51),respectively, one can apply similar approach in (52) to obtainthe optimal solution ζζζ ⋆ . V-D Optimal UAV-source trajectory design
We now aim at optimizing the approximated path of S of-fline for the three schemes in terms of ASR under giventhe other variables. The corresponding sub-problem of S -trajectory design for GJT is reformulated as ( P
7) : maximize Q S N X n =1 log Φ ( Q S [ n ]) s.t. C5 − C7 , C11 − C13 , ˜C15 : C n (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + D n ≥ Ψ H , (53)where Φ ( Q S [ n ]) =1+ A n (cid:0) k Q S [ n ] − W D k + H (cid:1) − α B n (cid:16) k Q S [ n ] − ˆ W E k − R E k Q S [ n ] − ˆ W E k + ˜ H (cid:17) − α , (54) A n = ζ [ n ] γ S [ n ] ζ [ n ] γ J [ n ] ( k Q J [ n ] − W D k + H ) − α + 1 , (55) B n = γ S [ n ] γ J [ n ] (cid:16) k Q J [ n ] − ˆ W E k +2 R E k Q J [ n ] − ˆ W E k + ˜ H (cid:17) − α +1 , (56) C n = η ¯ β (1 − ζ [ n ]) P S [ n ] , (57) D n = η (1 − ζ [ n ]) h P J [ n ] ¯ β (cid:0) k Q J [ n ] − W D k + H (cid:1) − α + N i , (58)The optimization problem ( P is non-convex due to the factthat the objective function is not concave with respect to Q S [ n ] and the constraints C11 and ˜C15 are not convex, therefore, it ishard to solve optimally. To simplify it, we reformulate ( P byintroducing the slack variables T = { T [1] , T [2] , · · · , T [ N ] } and U = { U [1] , U [2] , · · · , U [ N ] } and obtain ( P
8) : maximize Q S , T , U N X n =1 log 1 + A n T − α [ n ]1 + B n U − α [ n ] s.t. C5 − C7 , C11 − C13 , ˜C15 : C n T − α [ n ]+ D n ≥ Ψ H , C16 : k Q S [ n ] − W D k + H − T [ n ] ≤ , C17 : k Q S [ n ] − ˆ W E k − R E k Q J [ n ] − ˆ W E k + ˜ H − U [ n ] ≥ , (59)Note that C16 must hold with equality at the optimal point,otherwise by decreasing T [ n ] one can increase the value ofobjective function without violating any constraints, similarlyfor C17. Then ( P and ( P are equivalent and have thesame optimal points. Next, based on Lemma 3, we observethat the objective function of ( P is in the form of convex-minus-convex. Lemma 3.
Let define the function f ( x ; a, b ) = log(1 + ax − b ) with non-negative parameters a and b . Taking the first andsecond derivatives of the function with respect to x yields D f = − abx ( a + x b ) , D f = ab (cid:0) a + ( b + 1) x b (cid:1) x ( a + x b ) , (60) where f ( x ) is convex as D f ≥ . Note that we implicitlytake the extended-value extension of f ( x ) , i.e., e f ( x ) , whichis defined ∞ outside the domain of f ( x ) for the latter result.Thus, the summation of convex functions results in a convexfunction. This completes the proof. Lemma 4.
Let x be a vector of variables { x i } Ni =1 and a ∈ R N × be a constant vector . The function of negativenorm-squared of this two vectors; f ( x ) = −k x − a k , whichobviously is a concave function with respect to the vector x ,has a convex upper-bound given by −k x − a k ≤ k x k − x − a ) † x − k a k , (61) Proof.
See appendix B. (cid:4)
Using Lemmas 3 and 4, we reformulate ( P in an approxi-mated convex form by having concave objective function withconvex feasible set as ( P
9) : maximize Q S , T , U N X n =1 ˆ A n T [ n ] − log (cid:0) B [ n ] U − α [ n ] (cid:1) s.t. C5 − C7 , ˜C11 : ˜ D + k Q kS k − (cid:0) Q kS − Q kJ (cid:1) † Q S [ n ] −k Q kJ k ≤ − C13 , e C15 : (cid:20) Ψ H − D n C n (cid:21) + T α [ n ] ≤ , C16 , ˜C17 : 2 R E k Q S [ n ] − ˆ W E k− (cid:16) Q kS − ˆ W E (cid:17) † Q S [ n ] − U [ n ]+ e H ≤ , (62)where e H ∆ = k Q kS [ n ] k − k ˆ W E k − ˜ H . Besides, ˆ A n = − αA n T n (cid:16) A n + T α n (cid:17) , (63)Note that ˜C11 and ˜C17 follow from Lemma 4. Additionally,using C16 implies that T is non-negative such that T [ n ] ≥ H .Therefore, for H ≥ , ˜C15 is regarded as a convex constraint,and therefore, ( P being a convex problem can be optimallysolved by any known solvers, here, we use CVX [40]. Further,for FUJ, the corresponding sub-problem of S -trajectory designis similar to ( P by replacing (55) with A [ n ] = ζ [ n ] γ S [ n ] ,and following similar approach taken above, the solution ofthat problem can be obtained.Finally, for the conventional case WoJ, the sub-problemof S path planning with SWIPT at destination and partiallyknown E location is reformulated as ( P
10) : maximize Q S N X n =1 log Φ ( Q S [ n ]) s.t. C5 − C7 , C12 , e C15 : C n (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + D n ≥ Ψ H , (64) where A n = ζ [ n ] γ S [ n ] , B n = γ S [ n ] ,C n = η ¯ β (1 − ζ [ n ]) P S [ n ] , D n = η (1 − ζ [ n ]) N . (65)which is non-convex because of Lemma 3 or non-convex con-straint e C15 , and therefore we obtain a convex approximatedproblem of ( P as ( P
11) : maximize Q S , T , U N X n =1 h − ˆ A n T [ n ] − log (cid:0) B n U − α [ n ] (cid:1)i s.t. C5 − C7 , C12 , e C15 : (cid:20) Ψ H − D n C n (cid:21) + T α [ n ] ≤ , C16 : k Q S [ n ] − W D k + H − T [ n ] ≤ , ˜C17 : 2 R E k Q S [ n ] − ˆ W E k− (cid:16) Q kS [ n ] − ˆ W E (cid:17) † Q S [ n ] − U [ n ]+ e H ≤ , (66)Now ( P is convex. With an initial point (cid:0) Q k S , T k , U k (cid:1) ,we can solve it optimally with CVX. V-E Optimal UAV-jammer trajectory design
We are finally after designing an optimal trajectory of J ,provided that ( P S , P J , ζζζ, Q S ) are given. For GJT, we formu-late the sub-problem of J -trajectory design as ( P
12) : maximize Q J N X n =1 log Φ ( Q J [ n ]) s.t. C8 − C11 , C13 , ˜C15 : E n + F n (cid:0) k Q J [ n ] − W D k + H (cid:1) − α ≥ Ψ H , (67)where for ∀ n ∈ { , , · · · , N } , we have Φ ( Q J [ n ])= 1+ A n B n ( k Q J [ n ] − W D k + H ) − α +1 C n D n ( k Q J [ n ] − ˆ W E k +2 R E k Q J [ n ] − ˆ W E k + ˜ H ) − α +1 , (68) A n = ζ [ n ] γ S [ n ] (cid:0) k Q S [ n ] − W D k + H (cid:1) − α , B n = ζ [ n ] γ J [ n ] , (69) C n = γ S [ n ] (cid:16) k Q S [ n ] − ˆ W E k − R E k Q S [ n ] − ˆ W E k + ˜ H (cid:17) − α , (70) E n = η (1 − ζ [ n ]) h P S [ n ] ¯ β (cid:0) k Q S [ n ] − W D k + H (cid:1) − α + N i , (71) D n = γ J [ n ] , F n = η ¯ β (1 − ζ [ n ]) P J [ n ] . (72)Reformulating problem ( P by introducing theslack variables S = { S [1] , S [2] , · · · , S [ N ] } and V = { V [1] , V [2] , · · · , V [ N ] } yields ( P
13) : maximize Q J , S , V N X n =1 log A n B n S − α [ n ]+1 (1 + C n D n V − α [ n ]+1 s.t. C8 − C11 , C13 , e C15 : [Ψ H − E n ] + S [ n ] α ≤ F n , C16 : k Q J [ n ] − W D k + H − S [ n ] ≥ , C17 : S [ n ] ≥ , C18 : k Q J [ n ] − ˆ W E k +2 R E k Q J [ n ] − ˆ W E k + ˜ H − V [ n ] ≤ , (73) Lemma 5.
Define the bivariate function f ( x, y ) =log (1+ a exp( x ))+log (1+ a exp( y )) , x, y > with the non-negative parameters a and a and the constraint a ≥ .By taking the first and second derivative of the function withrespect to the variable x and obtaining the correspondinggradient and Hessian of f , one can reach at ∇ ( f ) = D f = (cid:20) a e x a exp( x ) , a e y a exp( y ) (cid:21) † , (74) H( f ) = D f = " a e x [1+ a exp( x )] a e y [1+ a exp( y )] , (75) Since matrix H is positive semidefinite for t > , thefunction f ( x, y ) is convex. Therefore, its first Taylor expansionproviding a global under-estimator of f ( x, y ) at point ( x , y ) is given by f ( x, y ) ≥ f ( x , y )+ (cid:20) a e x a exp( x ) , a e y a exp( y ) (cid:21) ( x − x , y − y ) † . (76)Based on Lemma 5, the objective function of ( P isin the form of convex-minus-convex with respect to ˜ V [ n ] = α log V [ n ] and ˜ S [ n ] = α log S [ n ] , i.e., it is still non-convex.Hence, the approximated convex problem corresponding to ( P can be obtained as ( P
14) : maximize Q J , e S , e V N X n =1 f LB [ n ] − log (cid:16) b e ˜ S [ n ] (cid:17) − log (cid:16) b e ˜ V [ n ] (cid:17) s.t. C8 − C10 , C13 , ˜C11 : ˜ D + k Q kJ k − (cid:0) Q kJ − Q kS (cid:1) † Q J [ n ] −k Q kS k ≤ , e C15 : [Ψ H − E n ] + exp( ˜ S [ n ]) ≤ F n , ˜C16 : 2 (cid:0) Q kJ [ n ] − W D (cid:1) † Q J [ n ] − exp (cid:18) α ˜ S [ n ] (cid:19) + H ≥ , ˜C18 : k Q J [ n ] − ˆ W E k +2 R E k Q J [ n ] − ˆ W E k + ˜ H ≤ I n + J n ˜ V [ n ] , (77)where H = −k Q kJ [ n ] k + k ˆ W D k + H and the concavelower-bound function f LB is given by f LB [ n ] ∆ = a exp (cid:16) e S k [ n ] (cid:17) a exp (cid:16) e S k [ n ] (cid:17) e S [ n ]+ a exp (cid:16) e V k [ n ] (cid:17) a exp (cid:16) e V k [ n ] (cid:17) e V [ n ] , (78)where a = A n B n , a = D n , b = B n , b = C n D n , I n = (cid:16) − α ˜ V k [ n ] (cid:17) exp (cid:16) α ˜ V k [ n ] (cid:17) , and J n = α exp (cid:16) α ˜ V k [ n ] (cid:17) .Note that constraints ˜C11 , ˜C16 , and ˜C18 are obtained bysubstituting the non-convex terms of the left hand side con-straints C11, C16, and C18 of ( P with their approximatedconvex expressions using Lemma 4. Since ( P is nowconvex, we use CVX and [41] to solve it, given an initial point ( Q k J , e S k , e V k ) , where the superscript k denotes iteration index.Further, to optimize J -trajectory for FUJ, we solve ( P byremoving the terms involving e S [ n ] from its objective function. Algorithm 1:
Proposed iterative algorithm1:
Initialize:
Set initial feasible points P (0) S , P (0) J , ζζζ (0) , Q (0) S , and Q (0) J , as well as put the initial values of slackvariables T (0) and U (0) , ˜S (0) and ˜V (0) , and let k = 0 ;2: Repeat: k ← k + 1;
4: Given P ( k − J , ζζζ ( k − , Q ( k − S , and Q ( k − J solve (P2) using (28) updating P ( k ) S ;5: Given P ( k ) S , P ( k − J , ζζζ ( k − , Q ( k − S , and Q ( k − J ,solve (P4) via updating P ( k ) J using (47);6: Given P ( k ) S , P ( k ) J , Q ( k − S , and Q ( k − J , update ζζζ ( k ) using (52);7: Given P ( k ) S , P ( k ) J , ζζζ ( k ) , Q ( k − S , Q ( k − J , T ( k − , and U ( k − solve (P9) for GJT/FUJ and (P11) for WoJ,updating Q ( k ) S , T ( k ) , and U ( k ) ;8: Given P ( k ) S , P ( k ) J , ζζζ ( k ) , Q ( k ) S , Q ( k − J , ˜S ( k − , and ˜V ( k − solve (P14) updating Q ( k ) J , ˜S ( k ) , and ˜V ( k ) ;9: Until the absolute increase of the objective function isbelow the threshold ǫ ;10: Return: P ⋆ S ← P ( k ) S , P ⋆ J ← P ( k ) J , ζζζ ⋆ ← ζζζ ( k ) , Q ⋆ S ← Q ( k ) S , Q ⋆ J ← Q ( k ) J ; V-F Overall algorithm
In order to solve problem (P1) by using BCD method forthe jammer-included scenarios, we jointly optimize UAVs’transmit power P S and P J , destination’s PSR factor ζζζ , as wellas UAV-source and UAV-jammer’s trajectories Q S and Q J alternatively via solving sub-problems (P2), (P3), (P6), (P7),and (P12), respectively. We summarize the detail of overalliterative solution in Algorithm 1.Now, aiming at convergence analysis of Algorithm 1let define the objective value of original problem; i.e.,(P1), at iteration k as ¯ R (cid:0) P k S , P k J , ζζζ k , Q k S , Q k J (cid:1) . Similar def-initions are taken for the objective values of problems(P4), (P9), and (P14) defined as Θ lb (cid:0) P k S , P k J , ζζζ k , Q k S , Q k J (cid:1) , Ξ lb (cid:0) P k S , P k J , ζζζ k , Q k S , Q k J (cid:1) , Ω lb (cid:0) P k S , P k J , ζζζ k , Q k S , Q k J (cid:1) , respec-tively. Now, we prove the convergence of Algorithm 1 in whatfollows. ¯ R (cid:0) P k S , P k J , ζζζ k , Q k S , Q k J (cid:1) ( a ) ≤ ¯ R (cid:16) P ( k +1) S , P k J , ζζζ k , Q k S , Q k J (cid:17) ( b ) = Θ lb (cid:16) P ( k +1) S , P k J , ζζζ k , Q k S , Q k J (cid:17) ( c ) ≤ Θ lb (cid:16) P ( k +1) S , P ( k +1) J , ζζζ k , Q k S , Q k J (cid:17) ( d ) ≤ ¯ R (cid:16) P ( k +1) S , P ( k +1) J , ζζζ k , Q k S , Q k J (cid:17) ( e ) ≤ ¯ R (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q k S , Q k J (cid:17) ( f ) = Ξ lb (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q k S , Q k J (cid:17) ( g ) ≤ Ξ lb (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q ( k +1) S , Q k J (cid:17) ( h ) ≤ ¯ R (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q ( k +1) S , Q k J (cid:17) ( i ) = Ω lb (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q ( k +1) S , Q k J (cid:17) ( j ) ≤ Ω lb (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q ( k +1) S , Q ( k +1) J (cid:17) ( k ) ≤ ¯ R (cid:16) P ( k +1) S , P ( k +1) J , ζζζ ( k +1) , Q ( k +1) S , Q ( k +1) J (cid:17) , (79)where the inequalities ( a ) , ( c ) , ( e ) , ( g ) , and ( j ) all followfrom the definition of the optimal solution to the problems(P2), (P4), (P6), (P9), and (P14), respectively. Besides, theequality ( b ) holds since the first order Taylor approximationis adopted and that the objective function of problems (P3)and (P4) share the same value at P k J . Similar justificationscan be explained for the equalities ( f ) and ( i ) at points Q k S and Q k J , respectively. Further, ( d ) , h , and ( k ) follow fromthe fact that the objective functions of problems (P4), (P9),and (P14) are tight lower-bounds to that of (P3), (P7), and(P12), respectively. The last inequality in (79) indicates thatthe objective value of (P1) is non-decreasing over the iterationindex. As well as that, the optimal value of (P1) is finite,i.e., the optimal ASR is upper bounded by a finite value,which means the proposed iterative Algorithm 1 is guaranteedto converge. Due to the convexity of the approximated sub-problems (P4), (P9), and (P14), the proposed algorithm isappropriate for UAV applications as it can be efficientlyimplemented in practice as having a complexity of O ( kN m ) ,where m is the number of variable blocks, which means thesolution can be obtained at worst-case in polynomial time.VI N UMERICAL R ESULTS
In simulations, unless otherwise stated, we adopt the fol-lowing parameters. The mission time duration, in consistencywith [42] is chosen T = 2 s which is discretized into N = 100 equal time slots to balance the accuracy and computationalcomplexity, the total power budget divided equally betweenUAVs is P totS + P totJ = 20 dBm with the maximum instanta-neous transmit power of mW and PAPR ratio of , leading toaverage transmission power . mW of each. Considering thenormalized transmission bandwidth, we set N = − dBm, Ψ H = − dBm with power conversion efficiency factor η = 0 . in (19). We set γ = 40 dB and path-loss exponent α = 2 . . We set H = 1 . with R = 2 . H (radius of permittedflying circular region centered at D ), W D = (0 , , L = R (distance between ground destination and geometric center ofthe eavesdropper), ˆ W E = ( L, , where the exact locationof E is a random point within the circular region centeredat ˆ W E with radius R E = H , and safety distance betweenUAVs ˜ D = H . ¯ β is obtained by averaging over channelrealizations over the area of interest. In all plots, we compareFUJ, GJT, and WoJ in terms of the following aspects: • convergence of the proposed iterative algorithm, demon-strated by variation of average secrecy rate with respect toiteration index wherein we utilized absolute error function f err ( k ) = k ¯ R opt,ksec − ¯ R opt,k − sec k as the termination criteriasimilar to [10], Iteration A v e r a g e S ec r ec y R a t e [ B it s / s / H z ] FUJGJTWoJ
Fig. 2: Average secrecy rate against iteration. • optimal UAVs’ trajectory, • instantaneous secrecy rate, • ASR and average harvested energy (AHE) at D , • instantaneous secrecy energy efficiency (ISEE) defined asthe ratio between ¯ R optsec [ n ] and P S [ n ] + P J [ n ] , • UAVs’ transmit power over flying horizon, • impact of estimated location of E on the ASR and AHE • harvested power efficiency defined as ˜ P H [ n ] P S [ n ]+ P J [ n ] .In particular, in optimal trajectory comparisons, we adopt theso-called baseline scheme for UAVs initial trajectory; i.e.,both S and J fly with their maximum speeds towards as closeas D and the geometric center of estimated location of E ,respectively. Then, both UAVs hover above the correspondingpoints as long as possible in order to send the data and conductjamming transmission, respectively, followed by heading withtheir maximum speeds towards final location, provided that themission time is sufficient. Otherwise, they turn from a midwayheading towards the final locations.Fig. 2 illustrates the convergence plot of the proposediterative algorithms for FUJ, GJT, and WoJ. We plot theASR as the number of iteration k varies. We see all schemesconverge with terminating threshold ǫ = 10 − , validatingour analysis in terms of convexity of the approximated sub-problems. It should be mentioned that Algorithm 1 for all thescenarios converges quite quickly in few iterations making itan efficient solution for the considered UAV application.Fig. 3 illustrates the optimal UAVs’ trajectory for FUJ,GJT, and WoJ using the proposed sequential algorithm. Notethe green-edge and black-edge circles denote the exact lo-cation of D and E , respectively. We observe that, for FUJscheme, S gets the closest to D among all, with substantiallyimproved ASR. For FUJ, the operation time and energyconstraints can make J head directly to the best possibleposition for jamming, which is much shorter than GJT.Fig. 4 compares ISR of FUJ, GJT, and WoJ using the pro-posed optimization methods and the aforementioned baselinescheme , and demonstrates our method leads to a significantperformance improvement. We also observe that FUJ bringsalways positive secrecy rate; nonetheless, WoJ provides zeroISR at the beginning and end of the mission. Note thatsince our objective function formulated to be optimized was -1 -0.5 0 0.5 1 1.5 2 2.5 3 x -4-3-2-101234 y
2D Horizental Coordinate
Final Location Re Initial Location
US - GJTUJ - GJTUS - FUJUJ - FUJUS - WoJ
Fig. 3: Optimal trajectory of UAVs for GJT, FUJ, and WoJscenarios.
Time [s] I n s t a n t a n e ou s S ec r ec y R a t e [ B it s / s / H z ] FUJ - OptimalFUJ - BaselineGJT - OptimalGJT - BaselineWoJ - OptimalWoJ - Baseline
Fig. 4: Instantaneous secrecy rate verses timethe average secrecy rate over the mission time, so the ISRperformance is not necessarily expected to be improved atall the mission time, though, we observe significant out-performance compared to the base-line curves on the whole.Particularly, it can be seen from the optimal curves belongto the FUJ, GJT, and WoJ schemes in Fig. 4 that by jointlyoptimizing transmit power of UAVs as well as their trajectoriesalongside with the PSR factor we could obtain approximately2, 1, and 0.5 bits/S/Hz ISR improvements during middle ofthe mission, respectively.Fig. 5 illustrates ASR and AHE at D vs R E (estimationerror of E ’s location) in FUJ, GJT, WoJ, and demonstrates theresultant ASRs decrease as R E increases. We observe AHEof WoJ decreases, AHE of FUJ remains approximately un-changed, AHE of GJT increases. This can be interpreted that,as the uncertainty of eavesdropper’s location increases (corre-sponding to a larger R E ), for FUJ and WoJ, UAV S flies fur-ther and has longer distance to D , resulting in decreased mainlink capacity and AHE. However, for GJT, since UAV J hasquite less impact on secrecy as the wiretap link might be betterthan the main link due to estimation erroneous, UAV S tries toget as close as possible to D in a straight way for improvingASR which, of course, makes AHE increased.Fig. 6 is presented to draw insight into the impact of Estimation Error (R E ) - normalized by H GJT - ASRFUJ - ASRWoJ -ASRGJT - AHEFUJ - AHEWoJ - AHE
Fig. 5: Secrecy and energy harvesting performance againstestimation error ( R E ). Destination-eavesdropper distance ratio (L/R) A v e r a g e S ec r ec y R a t e [ B it s / s / H z ] FUJGJTWoJ
Fig. 6: Average Secrecy Rate vs. destination-eavesdropperdistance. y-coordinate of the eavesdropper is set to zero.the location of the eavesdropper. As it can be clearly seenfrom the figure, the farther the eavesdropper’s location fromthe destination becomes, the higher the ASR performance isobtained, as expected, for all the scenarios. Notably, havingthe highest slop the curve belong to the GJT scheme is moresensitive to this parameter in comparison with the others,which means eavesdropper’s location has more impact on theASR performance of the GJT which should be considered insystem design. Further, when E gets closer to D the proposedjamming-included scenarios could obtain positive secrecy ratethough the WoJ scheme lacks. Particularly, the FUJ schemeregardless of the eve’s location provides the best secrecyperformance.Fig. 7 shows ISEE vs. mission time for FUJ, GJT, and WoJand demonstrates the significant performance improvement ofFUJ. This ISEE plot provides a trade-off between ASR and thecost of energy level for communications. We observe, for allcases, decreasing the distances between ( S , J ) and intendedground nodes ( D , E ) leads to higher ISEE.Fig. 8 shows UAVs’ transmit power over time horizon. ForFUJ, at the beginning S decreases its power to against infor-mation leakage while J increases power to satisfy the requiredminimum energy constraint at destination. When S and J fly Time [s] S ec r ec y E n e r gy E ff i c i e n c y [ B it s / H z / J ou l e ] FUJ-OptimalGJT-OptimalWoJ-Optimal
Fig. 7: Secrecy energy efficiency vs. mission time.
Time [s] T r a n s m it P o w e r [ m W ] US - FUJUJ - FUJUS - GJTUJ - GJTUS - WoJ
Fig. 8: Transmit power vs. time.to proper positions for data transmission and jamming, trans-mit power varies accordingly. For GJT, jamming power re-mains lowest to avoid degradation of ASR. Finally, WoJ keepsits power resource for the best use when having a better mainchannel quality with keeping S trajectory to be as far as possi-ble from the estimated location of E . Interestingly, we observethat even with a significantly lower transmission power ofUAV-jammer for the GJT compared to the UAV-source, thesecrecy performance of the jammer-included scenarios couldbe enhanced.Fig. 9 is provided to demonstrate how the PSR factor variesto make the adequate energy to be harvested by the EHcomponent of the destination for all the three scenarios. Weobserve that for the GJT the more fraction of the receivedsignals should be dedicated for energy scavenging to satisfythe energy requirement of the destination node over the timehorizon.Fig. 10 illustrates instantaneous harvested power efficiencyfor FUJ, GJT, and WoJ, with respective fraction of total powerbudget P S [ n ] + P J [ n ] and the ratio of total harvested powerto the transmit network power obtained as 6.8 % , 1.8 % and7.8 % , respectively. We see that, for all cases, energy harvestingconstraint is satisfied and the harvested power is well above theminimum requirement Ψ H , particularly WoJ. This indicatesthat how we can design secure as well as energy efficient UAV- Time [s] P o w e r S p litti ng R a ti o () FJUGUJWoJ
Fig. 9: Power splitting ratio vs. time.
Time [s] P o w e r E ff i c i e n c y F ac t o r FUJGJTWoJ
Fig. 10: Harvested power efficiencybased communications protocols which is a good direction forour future work. VII C
ONCLUSION
We have considered a 2-UAV based wireless communi-cation system. It consists of two flying cooperative UAVs,a ground destination node equipped with SWIPT technique,and a passive ground eavesdropper. One UAV acts as sourcetransmitting confidential information to destination, while theother UAV propagates jamming to assist destination with anti-eavesdropping and energy harvesting. Assuming that UAVshave imperfect channel estimation eavesdropper, we have pro-posed two transmission schemes: FUJ and GJT, transmittingjamming signals that are a priori known and unknown at des-tination, respectively. Under such setting, we have formulatedan average secrecy rate (ASR) maximization problem in termsof trajectory design and power controlling, and proposed aniterative algorithm based on the block coordinated descent andsuccessive convex approximation. Via this algorithm, we havefound the best transmit power and trajectory of both UAVs, aswell as the best power splitting ratio of destination. Finally, wehave evaluated the proposed schemes by simulations in termsof ASR, ISR, AHE, and demonstrated their effectiveness. Inparticular, FUJ provides by far the highest ASR improvementcompared to GJT and WoJ (the benchmark schemes). A
PPENDIX AA PPENDIX
A: D
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Lemma 6.
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