Gridded UAV Swarm for Secrecy Rate Maximization with Unknown Eavesdropper
Christantus O. Nnamani, Muhammad R. A. Khandaker, Mathini Sellathurai
aa r X i v : . [ c s . I T ] F e b Gridded UAV Swarm for Secrecy RateMaximization with Unknown Eavesdropper
Christantus O. Nnamani ∗ , Muhammad R. A. Khandaker † and Mathini Sellathurai ‡ School of Engineering and Physical Sciences, Heriot-Watt University,
Edinburgh EH14 4AS, United KingdomEmail: { ∗ con1, † m.khandaker, ‡ m.sellathurai } @hw.ac.uk Abstract — This paper considers grid formation of an un-manned aerial vehicle (UAV) swarm for maximizing the secrecyrate in the presence of an unknown eavesdropper. In particular,the UAV swarm performs coordinated beamforming onto the nullspace of the legitimate channel to jam the eavesdropper locatedat an unknown location. By nulling the channel between thelegitimate receiver and the UAV swarm, we obtain an optimaltrajectory and jamming power allocation for each UAV enablingwideband single ray beamforming to improve the secrecy rate.Results obtained demonstrate the effectiveness of the proposedUAV-aided jamming scheme as well as the optimal number ofUAVs in the swarm necessary to observe a saturation effect inthe secrecy rate. We also show the optimal radius of the unknownbut constrained location of the eavesdropper.
Index Terms —Secure communication, jamming, UAV, swarm,optimization, physical layer security.
I. I
NTRODUCTION
Due to aerial visibility, ease of maneuvering and costeffectiveness, the unmanned aerial vehicles (UAVs) is rapidlybecoming a preferred choice for on-demand wireless com-munication applications [1], [2]. The benefits accruing tothe use of a single UAV as observed in several wirelesscommunication applications can be extrapolated with the useof multiple UAVs, popularly termed as a UAV swarm [3]–[5].The control mechanism of the UAV swarm has been discussedin [6]–[8] for different applications in order to characterize thetrade-off between design complexity and performance. In thispaper, we explore this trade-off for the deployment of the UAVswarm for physical layer security (PLS).PLS uses the dynamic intrinsic properties of wirelesscommunication channels to support the legitimate receiver’ssignal while reducing the information content received bythe eavesdropper(s) [3]. Such intrinsic channel propertiesinclude fading, interference, multi-path, shadowing and noise.Although PLS can be traced to Shannon’s theorem [9], PLSoptimization methods have been recently broadened into theuse of UAVs [10]–[13]. Several use-cases can be found in theliterature deploying UAVs for PLS. The use of a single UAVas a security enabler and as a relay node have been exploredin [3], [14], [15]; delivering artificial noise interwoven withthe relayed information signal. It has also been used todeliver classified message to legitimate ground stations amidsteavesdroppers with restrictions of no-fly regions as discussed
This work was supported in part by the EPSRC Project EP/P009670/1,Petroleum Technology Development Fund and University of Nigeria Nsukka. in [16]. The results obtained in these works showed that thecombination of the deployment of UAV jamming and trans-mission power optimization improves the secrecy performanceof the respective models. However, a major bottleneck of thesemodels is the assumption that perfect channel state information(CSI) of the eavesdropper is known to the transmitter. This isimpractical especially for passive eavesdroppers that do nottransmit or register their locations.To alleviate this challenge, [17] developed algorithms usinga UAV as the information source and optimized its trajectoryand transmitting power constrained by sparse eavesdropperswithin an independent small uncertainty region. Nevertheless,the performance achieved cannot be guaranteed as the uncer-tainty region expands and overlaps with the certainty regionof the legitimate receiver. Similarly, [12] discussed PLS fora UAV network where the eavesdropper is mobile within thenetwork. In addition to the weak obscurity of the eavesdropperas tracking is enabled, the advantage of aerial visibility of theUAV was undermined. In continuation to [12], investigationof UAV-aided jamming technique for enabling PLS in groundstation scenarios where the exact eavesdropper location isunknown was pursued in [3]. The unknown eavesdropperlocation was assumed to be within an ellipse characterizingthe coverage region of the transmitter. Although this workguarantees positive secrecy rate for the unknown eavesdropperCSI, the secrecy rates reported were relatively low.Following positive results from single UAV use-case sce-narios for PLS, explorations of techniques to increase thesecrecy rate led to the deployment of multiple UAVs in formof a swarm for PLS. The UAV swarm is simply a collectionof independent flying UAVs that are autonomous but interactreactively to produce an aggregated behaviour [6], [8]. Thetheoretical framework for the relationship between the UAVswarm and the ground base station was proffered in [18]. In[5], the UAV swarm tracked the movement of the eavesdropperto jam its received signal while maximizing the secrecy rateof the main receivers. The location of the eavesdropper isassumed to be known in this work and the geometry of theUAV swarm was not considered contrary to the specificationssuggested in [7], [18]. However, without considering thegeometry of the UAV swarm, the optimized beamformingweights are not guaranteed to produce a beam pattern [18].Therefore, in this paper, we consider a grid formation for theUAV swarm with unknown location of the eavesdropper inrder to increase the secrecy rate via coordinated jamming.Simulation results demonstrate the superior performance ofthe proposed grid-structured UAV swarm approach comparedto conventional approaches. The technical contribution in thispaper can be characterized in terms of seeking the answers tothe following questions with respect to existing techniques:1) What is the position of the k -th member of the UAVswarm at a particular time instance?2) How to harness the properties of the UAV swarm toachieve maximum secrecy rates?These questions as enumerated are correlated and are answeredwithin the general principles of controlling the UAV swarmand beamforming design proposed in this paper.II. S YSTEM M ODEL AND P ROBLEM FORMULATION
We consider a scenario in which a transmitter (Alice) wantsto send a confidential message to a legitimate receiver (Bob) inthe presence of an eavesdropper (Eve). The primary objectiveis to ensure that Alice sends the confidential message toBob without prior knowledge of the presence or location ofEve. A group of K UAVs coordinate their jamming signalsto ensure worst channel state of Eve despite its unknownlocation without tampering the channel of Bob. A pictorialrepresentation is given in Fig. 1.We assume that Eve is located in a closed circular regionwith radius ε within the coverage region of Alice. Thisassumption is feasible given that a passive Eve will practicallybe within an area where it can easily purloin informationwithout revealing its location. As ε → , the closer we arriveat the exact location of Eve. However, since the exact positionof Eve is unknown, ε must always be greater than zero andpossibly can be increased to cover the entire coverage region ofAlice, thereby introducing the maximum uncertainty on Eve’slocation. Let the location of Alice, Bob, and the center ofthe region of Eve be denoted as Ω a , Ω b , Ω e , respectively.Consider that the entire flight time ( T ) of the UAVs is sampledat discrete time-stamps of N equal time slots, with duration δ = TN . Without loss of generality, we assume that the UAVsfly at constant altitude, H and with maximum speed of Z m/sfor each δ seconds giving rise to a trajectory represented as Q = { q k [ n ] , n ∈ N & k ∈ K } . We note that as N → ∞ , theUAVs are seen as following a continuous trajectory satisfyingtime-sharing conditions, thereby, delivering jamming signalscontinuously through its entire flight time [10].The received signal at Bob (b) and Eve (e) in the n -th time-slot is given by r i [ n ] = h ai w a s |{z} x a + K X k =1 ( h ki [ n ] w k [ n ]¯ s [ n ] | {z } x k [ n ] ) + n i [ n ] (1) for i ∈ { b , e } , where n i for i ∈ { b, e } is an additive independent and identicalwhite Gaussian noise signal received by Bob or Eve with ∼C (0 , σ i ) , h ai is the complex channel gain between Alice andBob or Eve while x a and x k represents the uncorrelated trans-mission symbols ( s ) and jamming signals (¯ s ) , respectively. Fig. 1. UAV Swarm interaction with ground stations { . } ∗ , { . } T and { . } H represent complex conjugate, transposeand Hermitian, respectively at the n th sample. We also assumethat E [ | x a [ n ] | ] = σ = 1 . Having prior knowledge of thechannel of Bob, we design the jamming signal such thatBob remains in a region free from it. This is achieved bynulling the jamming signal on the channel of Bob for each n ∈ { , . . . , N } , such that the beamforming coefficient ( w k )for each UAV is chosen as to ensure that h Tb w = 0 , where h b = [ h , . . . , h K b ] T and w = [ w , . . . , w K ] T . Thisensures that w lies in the null space of h b . Nevertheless, sincethe UAV swarm will continue to jam Eve at every possiblelocation within the described uncertainty region, it will implythat h Te w = 0 for h e = [ h , . . . , h K e ] T .Furthermore, we define the channel impulse response be-tween the k th UAV and the ground stations, i ∈ { b , e } as[18] h ki = c k e jφ k δ ( t − τ k ) , (2)where φ k = ( ω c + ω d ) t − ω d τ k is the phase shift due to Dopplereffect ω d and time delay τ k ; c k is the large scale channeleffect due to path loss and shadowing. We assume that thereis a line of sight (LoS) communication link between the UAVswarm and the ground stations (Bob and Eve), hence c k [ n ] = ρ ς k q k [ n ] − Ω i k − , where ρ represents the channel powergain at reference distance d = 1 m and ς is an exponentialrandom variable with unit mean [3], [10], [13].Accordingly, the signal to interference plus noise ratio ( γ ) at Bob and Eve is given respectively by γ b = | h ab | σ (3a) γ e [ n ] = | h ae | P Kk =1 ( | h ke [ n ]x k [ n ] | ) + σ , (3b)here h ai is Rayleigh distributed with gain of E [ | h ai [ n ] | ] = ρ ς k Ω a − Ω i k − µ for i ∈ { b , e } and µ is the pathloss [10].Since Alice is does not adjust its transmission rate because itis ignorant of Eve and the UAV swarm transmits in the nullspace of Bob, the channel between Alice and Bob is unaffectedby the samples of the UAV as reflected in (3).The average secrecy rate is defined as the difference in theinformation rate of Bob and Eve is given in (4) [10], [19]. R s = 1 N N X n =1 [log (1 + γ b ) − log (1 + γ e [ n ])] + , (4)where [ x ] + = max { , x } ensures that the information ratereceived by Eve is not greater than that received by Bob in avariable rate scheme [19]. This guarantee of positive secrecyis maintained by setting the power of the transmitter (Alice)when the CSI of Eve is greater than the CSI of Bob. However,if we consider a more realistic scenario where Alice is ignorantof Eve, then it cannot adjust its power based on the CSI ofEve, which may invariably lead to negative secrecy rate if Evehas better CSI than Bob. Hence, the objective of this work isto ensure that the negative secrecy rate is completely mitigatedeven for cases when Alice is ignorant of Eve. If we assumethat the uncorrelated jamming symbols have unity energy, weaim to maximize R s in (4) by finding the appropriate w and Q which represents the beamforming vectors and UAV swarmtrajectory respectively.To simplify the trajectory problem, we consider centralizedgrid swarm control where one element of the swarm isclassified as the head of the swarm q c = [ x c , y c , H ] [8]. Otherelements are distributed in a grid form within a predefinedwidth from the head. This enables the optimization of onlythe trajectory of the head of the UAV swarm through theoptimization process under the constraints in (5). k q c [ n + 1] − q c [ n ] k ≤ ( Zδ ) (5a) q c [ N ] = q f . (5b)Equation (5a) provides the upper bound for the maximumdistance covered by the UAV swarm head within a sampleperiod, while (5b) constrained its final destination. The directimplication of (5a) ensures that the time of flight of the UAVis lower bound by T ≥ k q f − q o k Z , where q f and q o representsthe final and initial trajectory of the UAV head. This meansthat the number of discrete time-stamps, N = Tδ , must besufficient to allow the UAV travel at least in a straight linefrom its initial to its final point. However, this trajectory is notguaranteed to be optimal in terms of maximizing the secrecyrate.The UAV swarm transmit power is bound by the peak andaverage power constraints given in (6) due to the limitedcapability of each individual UAV payload and power. Tr( w [ n ] w [ n ] H ) ≤ ¯ P tot , (6a) ≤ | w k | ≤ P max , (6b)where P max and ¯ P tot represents the maximum power trans-mitted by a single UAV in the swarm and the average powertransmitted by the UAV swarm respectively. (6b) constrains theminimum and maximum value of the jamming power while(6a) bounds the collective power radiated from the UAV swarm at each n ∈ { , . . . , N } to minimize external interference.Subsequently, we formulate the UAV swarm problem as max w , q R s (7a) s . t . h T b [ n ] w [ n ] = 0 , (7b) k q c [ n + 1] − q c [ n ] k ≤ ( Zδ ) , (7c) q c [ N ] = q f , (7d) Tr( w [ n ] w [ n ] H ) ≤ ¯ P tot , (7e) ≤ | w k [ n ] | ≤ P max . (7f)We note that the problem in (7) is easily solvable withperfect knowledge of both the channels of Bob and Eve bysolving the semi-definite program (SDP) of the successiveconvex approximation (SCA). Nevertheless, we consider thatthe location of Eve is unknown within a circular region boundby ε . III. P ROPOSED S OLUTION
Similar to [20], we define the exact location of Eve ( Ω e ) asa point on a circular uncertain region such that Ω e = ˆ Ω e ± ∆ Ω e (8a) k ± ∆ Ω e k = k Ω e − ˆ Ω e k ≤ ε, for ε ≥ (8b) k ∆ Ω e k ≤ ε, (8c)hold, where ˆ Ω e , ∆ Ω e and ε define the estimated locationof Eve, the error of the estimation and the radius of errorrespectively.To solve (7), we decompose the problem into a two (2)sub-problems describing the beamforming vectors and theUAV swarm trajectory. The original problem (7) can besolved iteratively between the sub-problems to obtain the sub-optimal/near optimal results that satisfies the constraints asused in [3], [10], [21]. A. Solving for beamforming vectors ( w ) Decomposing (7) into (9) and considering only constraintsthat relates to beamforming vectors, we have that, max W N X n =1 (cid:20) log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − log σ | h ae | σ P Kk =1 ( | h ke [ n ] w k [ n ] | ) + 1 (9a) s . t . h T b [ n ] w [ n ] = 0 , (9b) Tr( w [ n ] w [ n ] H ) ≤ ¯ P tot , (9c) ≤ | w k [ n ] | ≤ P max . (9d)Since the power transmitted by the UAV swarm at each n -thsample of the trajectory is independent of other samples, theproblem in (9) can be simplified by obtaining the values of w k for each n . Hence, we can rewrite (9) for each n ∈ { , . . . , N } s max W log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − log σ | h ae | σ h T e Wh e + 1 !! , ∀ n (10a) s . t . h T b w = 0 (10b) Tr( W ) ≤ ¯ P tot (10c) diag( W ) ≤ P max (10d) rank( W ) = 1 , (10e)where (10d) is a reformation of (9d) and (10e) is a corollary of W = ww H . Note that (10b) is a condition necessary to fulfilthe nulling of the channel of the main receiver, Bob. Hence,we desire to find some w such that (10b) will be satisfied. Forsimplicity, we satisfy this condition by obtaining a set of com-plex vectors, v , such that w = { v | h Hb v = 0 } . v becomes theprojection vector onto the subspace of w . Hence, w = H ⊥ v ;where H ⊥ is the transformation matrix for the projection of v on w . It is apparent that W = H ⊥ v ( H ⊥ v ) H = H ⊥ vv H H H ⊥ ( [22, eq. 1.9]). Let V = vv H , using [22, eq. 1.1.17], (10) canbe reformulated as max V log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − log | h ae | σ σ Tr( h e h T e ( H ⊥ VH H ⊥ )) + 1 , ∀ n (11a) s . t . Tr( H ⊥ VH H ⊥ ) ≤ ¯ P tot (11b) diag( H ⊥ VH H ⊥ ) ≤ P max (11c) rank( V ) = 1 . (11d)The problem in (11) is a non-convex SDP problem [23].Hence, using the method for solving an SDP problem obtainedin [24], we omit the rank constraint. We note that due to thegrid formation of the swarm, V is symmetric and representsthe tensors (outer product) of v and v H , the rank( V ) = 1 isguaranteed provided that the v is a non-zero vector [25]. SinceAlice is assumed ignorant of Eve and subsequently transmitcontinuously through the flight of the UAV swarm, the UAVswarm will continually send jamming signal through out theentire flight duration. This ensures that the vector, v is not zerofor each n -th sample. Hence, neglecting the constant terms of(11), we reformulate as (12). max V Tr( h e h T e ( H ⊥ VH H ⊥ )) + 1 , ∀ n (12a) s . t . Tr( H ⊥ VH H ⊥ ) ≤ ¯ P tot (12b) diag( H ⊥ VH H ⊥ ) ≤ P max . (12c)We note that the solution obtained in (12) is sufficient tocharacterize the suboptimal solution of (11). (12) is a convexSDP problem that can be efficiently solved using SDPT3solvers [26]. B. Solving for trajectory of the UAV swarm ( q ) To obtain the trajectory of the UAV swarm, we optimize thetrajectory of the head of the swarm and derive the trajectory of other members of the UAV swarm in relation to the head sincethey are in grid formation with fixed spacing. If we considerthe scenario when the location of Eve is unknown but can beestimated to exist within (8). Problem (7) can be reformulatedin terms of trajectory of the head UAV swarm as given in (13). max Q c N X n =1 (cid:20) log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − log γ k Ω a − ( ˆ Ω e − ∆ Ω e ) k − µγ p u [ n ] k q c [ n ] − ( ˆ Ω e − ∆ Ω e ) k + 1 (13a) s . t . Constraint (5) , (13b)where γ [ n ] = ρ σ [ n ] , and p u = | w k | represents the signalto interference noise ratio at reference distance d = 1 m andthe transmit power from swarm head. Since the location ofAlice and Bob are fixed, we assume that the noise variationis the same for each n ∈ N . Using triangular inequality for x ∈ { q c [ n ] , Ω a } , and substituting (8), we have that k x − ( ˆ Ω e − ∆ Ω e ) k ≤ k x − ˆ Ω e k + ε. (14)The right hand side is a lower bound to euclidean distancebetween the UAV swarm head and the center of the circularregion in which Eve is located. The lower bound representsthe best case scenario since the influence of the jamming signalof the swarm will be greater if it is close to Eve. On thecontrary, the lower bound represents the worst case scenariofor the transmitter (Alice), since it gives the closest euclideandistance between Alice and Eve. If Eve is close to Alice thelikelihood of it to purloin information increases. Thus, (13)can be rewritten with bounds as max Q c N X n =1 (cid:20) log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − log γ ( k Ω a − ˆ Ω e k + ε ) − µγ p u [ n ]( k q c [ n ] − ˆ Ω e k + ε ) + 1 (15a) s . t . Constraint (5) . (15b)Note that (15) is non-convex due to (15a) but it can be solvedby introducing slack variable, M = { m [ n ] = ( k q c [ n ] − ˆ Ω e k + ε ) , n ∈ { , . . . , N }} to characterize the separation betweenthe UAV and Eve. max Q c , M N X n =1 (cid:20) log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − log γ ( k Ω a − ˆ Ω e k + ε ) − µγ p u [ n ] m [ n ] + 1 (16a) s . t . ( k q c [ n ] − ˆ Ω e k + ε ) − m [ n ] ≤ , (16b) Constraint (5) . (16c)However, (16) is till non-convex due to the maximization ofa convex function of (16a). It can be solved using succes- Due to the approximation of (14), ε = 0 does not represent the case forwhen the exact location of Eve is known, however, it gives an insight intothe goodness of the estimator. If ε → , then Eve is bound to be located atthe center of the region of uncertainty. ive convex approximation (SCA) technique given in [27].This allows to solve a local tight approximation under tightconstraints and relax until the original problem is solved.Hence, given a predefined initial feasible trajectory, Q lc [ n ] = { q lc [ n ] , n ∈ { , . . . , N }} for the l − th iteration, the non-constant term of the objective of (16) can approximated withthe first order taylor expansion as given in (17) log γ ( k Ω a − ˆ Ω e k + ε ) − µγ p u [ n ] m [ n ] + 1 ≤ F l [ n ]( m [ n ] − m l [ n ]) + G l [ n ] (17)where F l [ n ] =[ γ ( k Ω a − ˆ Ω e k + ε ) − µ p u [ n ]] ∗ [( m l [ n ] + γ p u [ n ])(( γ ( k Ω a − ˆ Ω e k + ε ) − µ + 1) m l [ n ] + γ p u [ n ])] − ,m l [ n ] = ( k q lc [ n ] − ˆ Ω e k + ε ) ,G l [ n ] = log γ ( k Ω a − ˆ Ω e k + ε ) − µ m l [ n ] m l [ n ] + γ p u [ n ] ! . Neglecting all constant terms in (17), (16) can be reformulatedas max Q c , M log (cid:18) (cid:18) | h ab | σ (cid:19)(cid:19) − F l [ n ] m [ n ] (18a) s . t . ( k q c [ n ] − ˆ Ω e k + ε ) − m [ n ] ≤ (18b) Constraint (5) . (18c)Now, (18) is convex and can be efficiently solved using interiorpoint method in cvx [26]. Since (18) maximizes the lowerbound of (16), the objective value obtained is at least equalto (16) using the updated trajectory, Q l [ n ] . In the followingsection, we evaluate the performance of the UAV swarm withAlgorithm 1. Algorithm 1
Iterative algorithm for solving w , and Q Initialize w and Q c such that the constraints in (6) and(5) are satisfied. m ← . repeat Using the grid parameters, construct the location of allthe other K UAVs in the swarm, giving rise to Q . Determine the channel impulse response between theUAV swarm and the ground nodes from (2). Compute and update w with Q by solving (12). Compute R s as defined in (4). e = (cid:12)(cid:12)(cid:12)(cid:12) R news − R olds R news (cid:12)(cid:12)(cid:12)(cid:12) . m ← m + 1 . Using updated w , solve (18) and update Q c . until e < θ OR m ≥ m max . Output: w and Q .IV. R ESULTS AND A NALYSIS
We evaluate the performance of the UAV swarm withparameters set as follows unless otherwise stated: Ω a =[0 , , T , Ω b = [1000 , , T , ˆ Ω e = [500 , , T , Z = 3 m/s, γ = 90 dB, q o = [ − , , T , q f =[1500 , , T , δ = 1 s, ¯ P tot = 20 dBm, P max = 26 dBm, µ = 3 . (outdoor), Grid gutter = 10 λ , Grid cell = 3 .The initial values of the optimization parameters satisfyingrespective constraints were obtained via feasibility analysis.The UAV swarm beamforming vectors and its trajectory weresolved by iteratively optimizing each parameter with theknowledge of the others, until the error ( e ) between steps isless than θ (where θ = 10 − ) or the maximum number ofiterations is reached (where m max = 200 ). In the legends inFigs. 2 - 6; T , K , and e represents the UAV swarm flighttime, the number of UAVs and the radius ( ε ) of the locationof Eve respectively. While estEve and knownEve representsthe trajectory plots when Eve location is unknown and whenit is known respectively. Number of UAV in swarm (K) A v e r a g e s ec r ec y ( bp s / H z ) T= 600T= 700T= 800
Known EveLocationUnknown EveLocation, =300m
Fig. 2. Comparative performance of the UAV Swarm on average secrecy ratewhen the eavesdropper location is known and unknown. A v e r a g e s ec r ec y ( bp s / H z ) T= 600T= 700T= 800
K=9K=3
Fig. 3. Effect of radius of Eve region on average secrecy.
The performance of the UAV swarm in relation to PLS atdifferent flight times is given in Fig. 2. It shows that as thenumber of UAVs in the swarm increases, the improvementon the average secrecy rate converges. Hence, considering the
00 250 300 350 400 450 500T(s)1.522.533.544.5 A v e r a g e s ec r ec y r a t e ( bp s / H z ) Unknown Eve = 0Unknown Eve = 30Known EveJTP
Fig. 4. Comparative performance of the average secrecy rate of the UAVSwarm and single UAV jammer for K = 9 , Ω b = [200 , , T , ˆ Ω e =[200 , , T , q f = [150 , , T . power constraints of UAVs, determining the minimum numberof UAVs required to attain the secrecy rate convergence isnecessary for resource management purposes. However, theeffect of K is minimal when the exact position Eve is known.Furthermore, as the number of UAVs making up the swarmincreases, the flight time of the swarm does not influence thesecrecy rate. -100 -50 0 50 100 Azimuth Angle (degrees) -60-50-40-30-20-100 N o r m a li ze d P o w e r ( d B ) K= 3K= 4K= 5K= 6K= 7K= 8K= 9
Fig. 5. Beam Pattern at T = 600 and ε = 300 m . Further observation on Fig. 3 presents the limit to theuncertainty location area of Eve at different flight times. Asthe radius where Eve is located increases, the impact of theUAV swarm jamming signal becomes less significant as theaverage secrecy rate reduces. Negative average secrecy rateis obtained once Eve has better CSI than Bob and the UAVswarm do not provide enough jamming power as shown inFig. 3. The influence of the UAV swarm flight time is furtherincreased at larger radius. Nevertheless, the characterization -500 0 500 1000 1500 x(m) y ( m ) Alice BoBKnown Eve estEve = 50, T= 600estEve = 200, T= 600knownEve T= 600 e=200mInitial UAV swarm location e=50m Final UAVswarm location
Fig. 6. The UAV swarm trajectory. of the maximum tolerable error radius in the position of Evewill guide the design of its location estimators.In Fig. 4, the average secrecy rate performance analysis ofthe UAV swarm joint trajectory and beamforming optimizationfor known and unknown Eve’s location are compared withknown Eve location scenario considered in [10] (referred toas JTP in the legend) under similar power constraint. It isevident from the figure that the application of the UAV swarmout-performs the baseline scheme of a single UAV jammingmodel. It can be further observed that the longer time offlight of the UAVs ensures better secrecy performance. Thisis intuitively, and correlates with similar results presented in[10], [11], [16], since the UAVs delivers more jamming signalduring the communication with longer duration of flight time.When the radius of error ε is zero, the scenario where Evelocation is unknown relaxes to known Eve’s location sinceEve can only be located at a single point. However, as ε > ,the average secrecy rate reduces (as also presented in Fig. 3).Examining the radiation pattern generated by the UAVswarm, as presented in Fig. 5, it is observed that for K valuesthat did not complete a quadrilateral formation of the grid ( K = 4 , , , , the null depth is shallow compared to valueswhere the grid quadrilateral is complete ( K = 3 , , . Theimplication is that little spurious jamming signal from theUAV swarm has greater tendencies of leaking to Bob whenthe grid formation is incomplete. Despite the observation thathigher values of K gives higher power in the main lobe, itis apparent that even when multiple UAVs are available, aselection needs to be made to ensure the grid quadrilateral iscomplete with recourse to the minimum number required toachieve maximum average secrecy rate as shown in Fig. 2.However, for all values of K , side lobes with high powerlevels are observed. Although in conventional beamforming,the aim is to minimize the side lobes, however, since the exactposition of Eve is unknown and Bob is at the null of thejamming signal, the jamming power radiated from the sideobes will further reduce the information content received byEve, especially where multiple Eve exist.The trajectory of the swarm in Fig. 6, shows that the optimaltrajectory when Eve’s location is unknown follows a pathclose to Alice and Bob. Since the channel models are distancedependent, if Eve enjoys better channel quality than Bob thenthe zero or negative average secrecy rate ensues. The UAVprioritizes sending higher jamming signal to Eve when it iscloser to Alice. Since it cannot determine the exact locationof Eve, following this trajectory ensures that the channel forEve is always degraded despite its proximity to the transmitter(Alice). On the contrary, when the location of Eve is known(knownEve T=600 in Fig. 6), the swarm flies directly aboveit and jams the signal.V. C ONCLUSION
We have exploited the idea of grid-formed UAV swarm todemonstrate that the system can improve the secrecy rate ofground communications. Based on the results obtained, wedefined the optimal trajectory of the UAV swarm harnessingthe optimal number of UAVs in the grid to improve PLS.Furthermore, we examined the impact of the radius of theeavesdropper’s constrained location on the secrecy rate anddemonstrated the need to develop techniques to increase thedirectivity of the UAV swarm.R
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