Optimal Jammer Placement in UAV-assisted Relay Networks
OOptimal Jammer Placement in UAV-assisted RelayNetworks
Seyyedali Hosseinalipour
ECE DepartmentNorth Carolina State University
Raleigh, [email protected]
Ali Rahmati
ECE DepartmentNorth Carolina State University
Raleigh, [email protected]
Huaiyu Dai
ECE DepartmentNorth Carolina State University
Raleigh, [email protected]
Abstract —We consider the relaying application of unmannedaerial vehicles (UAVs), in which UAVs are placed between twotransceivers (TRs) to increase the throughput of the system.Instead of studying the placement of UAVs as pursued in existingliterature, we focus on investigating the placement of a jammer or a major source of interference on the ground to effectivelydegrade the performance of the system, which is measured bythe maximum achievable data rate of transmission between theTRs. We demonstrate that the optimal placement of the jammeris in general a non-convex optimization problem, for whichobtaining the solution directly is intractable. Afterward, using theinherent characteristics of the signal-to-interference ratio (SIR)expressions, we propose a tractable approach to find the optimalposition of the jammer. Based on the proposed approach, weinvestigate the optimal positioning of the jammer in both dual-hop and multi-hop UAV relaying settings. Numerical simulationsare provided to evaluate the performance of our proposed method.
I. I
NTRODUCTION
Applications of unmanned aerial vehicles (UAVs) in wirelesscommunication has attracted lots of attentions due to their easeof deployment and 3D movement capability, where one oftheir recent applications is data relaying [1]. On the other hand,jamming can degrade the performance of relay networks, whichshould be carefully addressed in practice. Although jammingand anti-jamming approaches have been investigated in wirelessnetworks [2]–[4], in the context of UAV-assisted networks, thecurrent state of the art still lacks maturity [5].Optimal jammer placement has been studied in the contextof network partitioning in wireless networks, e.g., [2], [3].In these works, the authors investigate the effective jammerpositioning to partition a wireless network into multiple residualsub-networks with a constraint on the number of jammers.It is shown that there is a trade-off between the number ofrequired jammers and the maximum order, i.e., the numberof functional nodes, of the residual sub-networks. Anotherapplication of jamming is providing a secure communication forsensitive information, where the usage of friendly jammers toprotect sensitive communications is common [4], [6]. In [4], theplacement and power consumption of jammers is optimized inspace and time to guarantee information-theoretic security for asecure communication. The aim is to prevent the eavesdroppersoutside the protected zone from having a knowledge about thetransmitted data. A similar problem is studied in [6], wherejamming via transmitting artificial noise is considered to protect the communication from eavesdroppers. More discussions on(anti-)jamming techniques can be found in [7]. Moreover, thereis a body of literature devoted to jammer localization , i.e.,detecting the location of jammers, in wireless networks [8].In the context of UAV relay networks, we were among thefirst to study the placement optimization and trajectory designfor UAV relays to evade the interference caused by the jammers[9]–[12]. Considering a major source of interference (MSI), theoptimal placement of the UAV relays along with identifyingthe minimum number of required UAVs to satisfy a desiredcommunication quality are studied in [9], [10]. A joint powerallocation and trajectory design is proposed in [11], [12] toevade the interference caused by another established wirelessnetwork. In [13], the optimal position and jamming power of alegitimate UAV monitor are obtained to maximize the averagesurveillance rate. In [14], a scenario is studied where a UAVtransmits artificial noise to confuse the ground eavesdropperfor protection of the transmitted data. In [15], an anti-jammingapproach is proposed in which the UAVs dynamically adjusttheir trajectory. Nevertheless, none of the aforementioned worksinvestigates efficient degradation of the communication qualityof UAV relay networks from the perspective of a jammer, whichis our main motivation.In this paper, we consider a terrestrial jammer or MSIthat aims to effectively deteriorate the communication qualityof a UAV-assisted relay network working in the decode-and-forward relaying mode. We consider a two-way communicationscenario, where the UAVs function as two-way relays betweentwo terrestrial transceivers. The goal is to obtain the optimalplacement of the terrestrial jammer to minimize the maximumachievable data rate of transmission between the terrestrialtransceivers. We note that the optimal jammer placementproblem belongs to the family of non-convex optimizationproblems, for which direct derivation of the solution is ingeneral intractable. Using the inherent characteristics of thesignal-to-interference ratio (SIR) expressions that result inpiece-wise convexity of the objective function, we propose twoefficient algorithms with polynomial complexity to obtain theoptimal position of the jammer in the dual-hop and multi-hopUAV relay networks. Numerical simulations are conducted toreveal the performance of our proposed approach. a r X i v : . [ ee ss . SP ] F e b I. P
RELIMINARIES AND P ROBLEM F ORMULATION
A. Preliminaries
We consider a two-way communication between a pair oftransceivers (TRs), named TR 1 and TR 2, both engagedin transmitting and receiving the data. We assume a left-handed coordination system ( x, y, h ) , and, without loss ofgenerality, TR 1 and TR 2 are assumed to be located at ( , , ) and ( D, , ) , respectively. To improve the qualityof communication, a set of UAV relays are placed between theTRs. We aim to effectively place a jammer/MSI on the groundto maximally deteriorate the communication performance ofthe system. Let ( x MSI , y
MSI , h
MSI = ) denote the position of theMSI. The transmission powers of TR 1, TR 2, and the MSIare denoted by p TR 1 , p TR 2 , and p MSI , respectively. We considerboth the line-of-sight (LoS) and the non-line-of-sight (NLoS)channel models, for which the path-loss is given by [16]: L LoS i,j = µ LoS d αi,j , L NLoS i,j = µ NLoS d αi,j , (1)where µ LoS ≜ C LoS ( πf c / c ) α , µ NLoS ≜ C NLoS ( πf c / c ) α , C LoS ( C NLoS ) is the excessive path loss factor incurred by shadowing,scattering, etc., in the LoS (NLoS) link, f c is the carrierfrequency, c is the speed of light, α = is the path-lossexponent, and d i,j is the Euclidean distance between node i and node j . The link between two UAVs (air-to-air) is modeledusing the LoS model, while the link between the MSI and aTR (ground-to-ground) is modeled based on the NLoS model.For the link between a UAV and the TRs or the MSI (air-to-ground and ground-to-air), we denote the path loss between aUAV i and terrestrial node j by η NLoS d ij (for more informationon η NLoS please refer to [10] and references therein). Due tothe geographical limitations, direct communication betweenthe TRs is not considered. While the above channel modelsare relatively simple, they represent the current art in UAVmodeling, and facilitate the derivation of many interestingresults in current literature, e.g., [1], [16].
B. Problem Formulation
As shown in Fig. 1, consider N UAVs between the TRs,where the location of UAV i is denoted by ( x i , y i , z i ) . Let usdefine Link 1 as the transmission link from TR 1 to TR 2(when TR 1 acts as a transmitter and TR 2 acts as a receiver),and Link 2 as the transmission link from TR 2 to TR 1. It isassumed that the UAVs utilize the same frequency but differenttime slots to avoid mutual interference among the UAVs. Weconsider an interference limited environment , where the powerof noise is negligible compared to that of interference causedby the MSI, and thus the SIR is used to describe the quality ofcommunication. For Link 1, let SIR i denote the SIR at UAV i , ≤ i ≤ N , and SIR N + denote the SIR at TR 2. Similarly, forLink 2, SIR N + + i denotes the SIR at UAV N − i , ≤ i ≤ N − ,and SIR N + denotes the SIR at TR 1. Assuming decode-and-forward relaying, the SIR of Link 1 and Link 2 are givenby: Considering the MSI to be a flying UAV with a fixed altitude h MSI = ˆ h MSI incurs minor modifications in the derivations. x h y (0,0,0) (D,0,0) (x MSI ,y MSI ,0) S I R S I R N + Jammer/MSI TR TR2 (x ,y ,h ) S I R N + S I R N + (x ,y ,h ) (x N ,y N ,h N ) S I R N S I R S I R N S I R N + L i nk_2 L i nk_1 Fig. 1: A jammer/MSI that aims to deteriorate the communication performancein multi-hop UAV relay setting.
SIR
Link 1 ( x MSI , y
MSI ) = min { SIR i ( x MSI , y
MSI ) ∶ ≤ i ≤ N + } , (2)SIR Link 2 ( x MSI , y
MSI ) = min { SIR N + i + ( x MSI , y
MSI ) ∶ ≤ i ≤ N } . (3) The goal of the jammer is to locate itself to effectively degradethe maximum achievable data rate of transmission betweenthe TRs . Assuming the same bandwidth for both links, this isequivalent to minimizing the maximum value of the SIR of thelinks denoted by SIR max = max ( SIR
Link 1 , SIR
Link 2 ) . Thus, ( x ∗ MSI , y ∗ MSI ) = arg min x MSI ,y MSI max { SIR
Link 1 ( x MSI , y
MSI ) , SIR
Link 2 ( x MSI , y
MSI )} . (4)The SIR expressions are convex functions with respect to (w.r.t) x MSI and y MSI (see (21)). However, since the minimum of convexfunctions is not necessary convex, SIR
Link 1 and SIR
Link 2 are,in general, non-convex functions w.r.t the position of the jammer.This results in non-convexity of our main problem in (4), whichmakes classic convex optimization techniques irrelevant andobtaining the solution non-trivial. In this work, we aim todevelop a tractable analytical approach to solve this problem.Given the fact that tackling the problem where x MSI , y
MSI are bothvariable is highly nontrivial, we fix one of those coordinates,which is y MSI in this work such that y MSI = ˆ y MSI , and obtain x ∗ MSI .Even with this assumption, the problem remains to be non-convex and non-trivial. Given the notable low complexity of ourproposed method, one can obtain x ∗ MSI for a set of given y MSI -sand choose the best solution among them. Also, the proposedmethodology can be easily applied to the case where x MSI isfixed and y MSI is variable. Thus, one can set x MSI = x ∗ MSI to obtainthe corresponding y ∗ MSI in a successive manner. Throughout,we assume that the MSI is mounted on a vehicle with thefeasible moving area confined by − x − jam ≤ x MSI ≤ x + jam , where x + jam ≥ D, − x − jam ≤ . To facilitate the discussion, we firstinvestigate the problem in the dual-hop setting, which itself isof interest, and then extend the study to the multi-hop setting.III. J AMMER P LACEMENT IN D UAL - HOP S ETTING
Consider the jammer placement in the dual-hop setting, wherethe data is relayed via a single UAV located at ( x u , y u , h u ) with transmission power p u (see Fig. 2). The SIR expressionsare given by: SIR ( x MSI , ˆ y MSI )= p TR 1 (( x u − x MSI ) + ( ˆ y MSI − y u ) + h u ) p MSI ( x u + y u + h u ) , (5) SIR ( x MSI , ˆ y MSI ) = p u ( ˆ y MSI + ( D − x MSI ) ) p MSI (( D − x u ) + y u + h u ) ( η NLoS µ NLoS ) , (6) SIR ( x MSI , ˆ y MSI ) = p TR 2 (( x u − x MSI ) + ( ˆ y MSI − y u ) + h u ) p MSI (( D − x u ) + y u + h u ) , (7) h y (0,0,0) (D,0,0) (x MSI ,y MSI ,0) S I R S I R [(x u -x MSI ) +(y u -y MSI ) +h u2 ] Jammer/MSI TR TR2
UAV (x u ,y u ,h u ) S I R S I R L i nk_2 L i nk_1 Fig. 2: A jammer/MSI that aims to deteriorate the communication performancein dual-hop UAV relay setting.
SIR ( x MSI , ˆ y MSI ) = p u ( ˆ y MSI + x MSI ) p MSI ( x u + y u + h u ) ( η NLoS µ NLoS ) . (8)Consequently, the SIR of Link 1 and Link 2 are given by: SIR
Link 1 ( x MSI , ˆ y MSI ) = min { SIR ( x MSI , ˆ y MSI ) , SIR ( x MSI , ˆ y MSI )} , (9) SIR
Link 2 ( x MSI , ˆ y MSI ) = min { SIR ( x MSI , ˆ y MSI ) , SIR ( x MSI , ˆ y MSI )} , (10)and the optimal position of the jammer is given by: x ∗ MSI = arg min − x − jam ≤ x MSI ≤ x + jam max { SIR
Link 1 ( x MSI , ˆ y MSI ) , SIR
Link 2 ( x MSI , ˆ y MSI )} . (11) As discussed earlier, SIR
Link 1 and SIR
Link 2 are, in general,non-convex functions w.r.t x MSI . This results in non-convexityof (11). The direct approach to solve (11) is to obtain themathematical expressions of SIR
Link 1 and SIR
Link 2 , andthen solve (11) using a non-convex optimization technique.However, functions SIR
Link 1 and SIR
Link 2 are piecewise-defined functions . This makes SIR max a piece-wise function,for which the detailed specification is tedious. Also, it canbe noticed that upon having multiple UAVs this approach isintractable. Considering this fact, we propose a systematicapproach to efficeintly obtain the solution of (11).
Definition . A function f ∶ R → R is called a piecewise convex function if it can be represented as f ( x ) = min { f j ( x ) ∶ j ∈ M} ,where f j ∶ R → R is convex ∀ j ∈ M ≜ { , , ⋯ , ∣M∣} .In other words, the domain of a piecewise convex functioncan be partitioned into multiple intervals such that at eachinterval the corresponding sub-function is convex. Note thatpiecewise convex functions are in general non-convex. In thefollowing, we present three lemmas, the proofs of which arestraightforward and omitted in the interest of space. All of thefunctions considered below are assumed to be continuous. Lemma . Let g i ∶ R → R , ≤ i ≤ M , be convex functions withthe set of critical points C g i , ∀ i . Function q = min ( g , ⋯ , g M ) is a piecewise convex function, for which the set of criticalpoints C q is given by: C q ⊂ ( ∪ i ∶ ≤ i ≤ M C g i )∪( ∪ ( i,j )∶ ≤ i < j ≤ M S g i ,g j ) ,where S g i ,g j ≜ {( x, g i ( x )) ∶ x ∈ R , g i ( x ) = g j ( x )} . Lemma . Let z i ∶ R → R , ≤ i ≤ M , be piecewiseconvex functions with the set of critical points C z i , ∀ i .Function w = max ( z , ⋯ , z M ) is a piecewise convex func-tion, for which the set of critical points C w is given by: C w ⊂ ( ∪ i ∶ ≤ i ≤ M C z i ) ∪ ( ∪ ( i,j )∶ ≤ i < j ≤ M S z i ,z j ) , where S z i ,z j ≜{( x, z i ( x )) ∶ x ∈ R , z i ( x ) = z j ( x )} . A piecewise-defined function is a function defined by multiple sub-functions,each of which applying to a certain interval of the original function’s domain. At any critical point such as ( x, g ( x )) , the derivative of the function g iseither zero or does not exist. Lemma . Let f ∶ R → R be a piecewise convex function with theset of critical points C f . The global minimum of the function ( x ∗ f , f ( x ∗ f )) , where x ∗ f = arg min x ∈ R f ( x ) , always belongs tothe set of critical points of the function, i.e., ( x ∗ f , f ( x ∗ f )) ∈ C f .In other words, in a special case where M = , Lemma 1asserts that the critical points of function q = min ( g , g ) ,where g and g are two convex functions, are either locatedat the intersections of g and g or coincide with those of g and g . Lemma 2 conveys a similar message for themaximum of two piecewise convex functions. Also, accordingto Lemma 3 the minimum of a piecewise convex function isalways among the critical points of the function. Given theconvexity (and continuity) of (5)-(8), SIR Link 1 and SIR
Link 2 are both piecewise continuous convex functions w.r.t x MSI , whichresults in the piecewise convexity of SIR max . According toLemma 3, the global minimum of SIR max , i.e., the solutionof (11), belongs to its set of critical points C SIR max , whichis a subset of the set of critical points of SIR
Link 1 andSIR
Link 2 , i.e., C SIR
Link 1 and C SIR
Link 2 , and the intersection pointsof SIR
Link 1 and SIR
Link 2 . However, direct derivation of theintersection points of SIR
Link 1 and SIR
Link 2 requires obtainingtheir expressions, which we aim to avoid. In the following, wepresent a corollary that alleviates this issue.
Corollary . Let v = min ( f , ⋯ , f N + ) and v = min ( f N + , ⋯ , f N + ) , where f i , ∀ i , is a single variable convexfunction with its domain and range defined on the set of realnumbers, and let v = max ( v , v ) . Then, C v ⊂ ( ∪ i ∶ ≤ i ≤ N + C f i ) ∪ ( ∪ ( i,j )∶ ≤ i < j ≤ N + S f i ,f j ) , (12)where S f i ,f j ≜ {( x, f i ( x )) ∶ x ∈ R , f i ( x ) = f j ( x )} . Let Ψ v =( ∪ i ∶ ≤ i ≤ N + C f i ) ∪ ( ∪ ( i,j )∶ ≤ i < j ≤ N + S f i ,f j ) denote the candidateset of critical points of function v . The global minimum ofthe piecewise convex function v , i.e., ( x ∗ v , v ( x ∗ v )) , where x ∗ v = arg min x ∈ R v ( x ) , can be found as follows: x ∗ v = arg min x { v ( x ) ∶ ( x, y ) ∈ Ψ v , v ( x ) = y } . (13)In Corollary 1, we reveal a fast method of obtaining theminimum of the piecewise function v as defined above, bysolely inspecting the points belonging to the candidate set ofcritical points. In the following, we first derive the candidateset of critical points of function SIR max and then proposean algorithm that implements Corollary 1 assuming N = with f i = SIR i ( x MSI , ˆ y MSI ) , i ∈ { , , , } , v = SIR
Link 1 and v = SIR
Link 2 to obtain the minimum of SIR max . Proposition . The critical point of function SIR ( x MSI , ˆ y MSI ) ,SIR ( x MSI , ˆ y MSI ) , SIR ( x MSI , ˆ y MSI ) , SIR ( x MSI , ˆ y MSI ) is x ( ) MSI = x u , x ( ) MSI = D , x ( ) MSI = x u , and x ( ) MSI = , respectively. Lemma . Consider the equality of two distinct quadratic curvesin the format of A [( x − B ) + C ] = D [( x − E ) + F ] . Define ∆ ≜ ( AB − DE ) + ( D − A ) [ A ( B + C ) − D ( F + E )] . If ∆ < , the quadratic equations have no intersection; otherwise,the intersecting points are given by: Throughout, we use sub-index + and − to denote the larger and the smallersolution, respectively. Note that if ∆ = , x − = x + . ± = AB − DE ± √ ∆ A − D if A ≠ D,x − = x + = ( B + C ) − ( E + F ) ( B − E ) O.W. (14)
Proposition . The intersection points of the two SIR curvesas a function of x MSI for each link in the dual-hop setting aregiven as follows, where x ( i,j )± denote the intersection pointsof SIR i and SIR j : ● For Link 1, replace A = p TR 1 /[ x u + y u + h u ] , B = x u , C =( ˆ y MSI − y u ) + h u , D = p u /[(( D − x u ) + y u + h u ) ( η NLoS µ NLoS ) ] , E = D , and F = ˆ y MSI in (14) to obtain x ( , )± . ● For Link 2, replace A = p TR 2 /[(( D − x u ) + y u + h u )] , B = x u , C = ( ˆ y MSI − y u ) + h u , D = p u /[( x u + y u + h u ) ( η NLoS µ NLoS ) ] , E = , and F = ˆ y MSI in (14) to obtain x ( , )± . Proposition . The four SIR curves in the dual-hop settingintersect with each other in the following points: ● If p TR 1 (( D − x x ) + y u + h u ) p TR 2 ( x u + y u + h u ) = , two functions SIR , SIR arealways equal; otherwise, they have no intersection. ● For SIR , SIR , replace A = p TR 1 x u + y u + h u , B = x u , C = ( ˆ y MSI − y u ) + h u , D = p u /[( x u + y u + h u ) ( η NLoS µ NLoS ) ] , E = , and F = ˆ y MSI in (14) to obtain x ( , )± . ● For SIR , SIR , replace A = p u /[(( D − x u ) + y u + h u ) ( η NLoS µ NLoS ) ] , B = D , C = ˆ y MSI , D = p TR 2 (( D − x u ) + y u + h u ) , E = x MSI , and F =( ˆ y MSI − y u ) + h u in (14) to obtain x ( , )± . ● For SIR , SIR , replace A = p u /[(( D − x u ) + y u + h u ) ( η NLoS µ NLoS ) ] , B = D , C = ˆ y MSI , D = p u /[( x u + y u + h u ) ( η NLoS µ NLoS ) , E = , and F = ˆ y MSI in (14) to obtain x ( , )± .The pseudo-code of our optimal jammer placement algorithmis given in Algorithm 1. The input ˆ y MSI is inherently assumed,and thus eliminated from the argument of the SIR functions forcompactness. The algorithm uses the candidate set of criticalpoints of function SIR max , which consists of the points obtainedin Proposition 1, 2, and 3. Note that in cases where x ( i,j )± doesnot exist according to Lemma 4, the algorithm automaticallyskips it. For each of the points, the algorithm first tests thefeasibility of the point, i.e., v ( x ) = y in (13). For instance,for ( x ( ) MSI , SIR ( x ( ) MSI )) , it checks that this point also belongsto SIR max in lines 4 and 5. Finally, it derives the minimumof function SIR max , i.e., the solution of (11), according toCorollary 1 by testing all the feasible candidates for the criticalpoints of the function in line 39. Note that our method reducesthe analysis of an intractable function to systematic calculationof values of the SIR expressions at points (c.f. Footnote 6).IV. J AMMER P LACEMENT IN M ULTI - HOP S ETTING
Consider the system model explained in Section II-B anddepicted in Fig. 1. Let δ x ( i,j ) = x i − x j , δ y ( i,j ) = y i − y j , δ h ( i,j ) = h i − h j , for UAV i and UAV j . In this case, the SIRexpressions for Link 1 and Link 2 are given as follows: When the two functions match, their critical points also match. Hence, wecan easily assume that x ( , )± do not exist without affecting the analysis. Algorithm 1:
Optimal jammer placement in dual-hop UAV-assisted relay networks The set of final candidates of exterma
P = {} Derive x ( ) MSI , x ( ) MSI , x ( ) MSI , and x ( ) MSI using Proposition 1. for i ∈ { , } do if min { SIR ( x ( i ) ) , SIR ( x ( i ) )} = SIR i ( x ( i ) ) then if SIR i ( x ( i ) ) ≥ min { SIR ( x ( i ) ) , SIR ( x ( i ) )} then P = P ∪ { [ x ( i ) , SIR i ( x ( i ) )] } end end end for i ∈ { , } do if min { SIR ( x ( i ) ) , SIR ( x ( i ) )} = SIR i ( x ( i ) ) then if SIR i ( x ( i ) ) ≥ min { SIR ( x ( i ) ) , SIR ( x ( i ) )} then P = P ∪ { [ x ( i ) , SIR i ( x ( i ) )] } end end end Derive x ( , )± and x ( , )± using Proposition 2. Define y ( ) = x ( , )− , y ( ) = x ( , )+ , z ( ) = x ( , )− , z ( ) = x ( , )+ . for i ∈ { , } do if SIR ( y ( i ) ) ≥ min { SIR ( y ( i ) ) , SIR ( y ( i ) )} then P = P ∪ { [ y ( i ) , SIR ( y ( i ) )] } end end for i ∈ { , } do if SIR ( z ( i ) ) ≥ min { SIR ( z ( i ) ) , SIR ( z ( i ) )} then P = P ∪ { [ z ( i ) , SIR ( z ( i ) )] } end end Derive x ( , )± , x ( , )± , and x ( , )± using Proposition 3. for ( i, j ) ∈ {( , ) , ( , ) , ( , )} do if min { SIR ( x ( i,j )− ) , SIR ( x ( i,j )− )} = SIR i ( x ( i,j )− ) and min { SIR ( x ( i,j )− ) , SIR ( x ( i,j )− )} = SIR j ( x ( i,j )− ) then P = P ∪ { [ x ( i,j )− , SIR i ( x ( i,j )− )] } end if min { SIR ( x ( i,j )+ ) , SIR ( x ( i,j )+ )} = SIR i ( x ( i,j )+ ) and min { SIR ( x ( i,j )+ ) , SIR ( x ( i,j )+ )} = SIR j ( x ( i,j )+ ) then P = P ∪ { [ x ( i,j )+ , SIR i ( x ( i,j )+ )] } end end Consider P in the following format: P = ∪ ∣P∣ i = {[ a i , b i ]} x ∗ MSI = a i ∗ , i ∗ = arg min i { b i ∶ [ a i , b i ] ∈ P , − x − jam ≤ a i ≤ x + jam } SIR ( x MSI , ˆ y MSI ) = p TR 1 (( x MSI − x ) + ( ˆ y MSI − y ) + h ) p MSI ( x + y + h ) , ⋮ SIR N ( x MSI , ˆ y MSI ) = p N − η NLoS (( x MSI − x N ) + ( ˆ y MSI − y N ) + h N ) p MSI µ LoS (∣ δ x ( N − ,N ) ∣ +∣ δ y ( N − ,N ) ∣ +∣ δ h ( N − ,N ) ∣ ) , SIR N + ( x MSI , ˆ y MSI ) = p N µ NLoS (( x MSI − D ) + ˆ y MSI ) p MSI η NLoS (( x N − D ) + y N + h N ) , SIR N + ( x MSI , ˆ y MSI ) = p TR 2 (( x MSI − x N ) + ( ˆ y MSI − y N ) + h N ) p MSI (( x N − D ) + y N + h N ) , ⋮ SIR N + ( x MSI , ˆ y MSI ) = p η NLoS (( x MSI − x ) + ( ˆ y MSI − y ) + h ) p MSI µ LoS (∣ δ x ( , ) ∣ + ∣ δ y ( , ) ∣ + ∣ δ h ( , ) ∣ ) , SIR N + ( d , h ) = p µ NLoS ( x MSI + ˆ y MSI ) p MSI η NLoS ( x + y + h ) . (21) imilar to Section III, our method is based on Corollary 1. Inthe following, we derive the candidate set of critical points offunction SIR max . Proposition . Define x = and x N + = D . For Link 1,the critical points of the functions SIR k ( x MSI , ˆ y MSI ) , ≤ k ≤ N + , are x ( k ) MSI = x i . Also, for Link 2, the critical points ofSIR N + k + ( x MSI , ˆ y MSI ) , ≤ k ≤ N , are x ( N + k + ) MSI = x N − k . Proposition . Consider the set of coefficients correspondingto Φ SIR , Φ SIR k , Φ SIR N + , Φ SIR N + , Φ SIR N + k + , and Φ SIR N + givenin (15)-(20). To obtain the intersections of the SIR curves ofLink 1, substitute Φ SIR j and Φ SIR j ′ , ≤ j < j ′ ≤ N + , in(14) to obtain x ( j,j ′ )± . For Link 2, substitute Φ SIR N + j + and Φ SIR N + j ′+ , ≤ j < j ′ ≤ N , in (14) to obtain x ( N + j + ,N + j ′ + )± . Proposition . Consider the set of coefficients given in (15)-(20). To obtain the intersections of the SIR curves of Link 1and Link 2, substitute Φ SIR j , ≤ j ≤ N + , and Φ SIR N + j ′+ , ≤ j ′ ≤ N , in (14) to obtain x ( j,N + j ′ + )± .The pseudo-code of our optimal jammer placement algorithmin the multi-hop relaying setting is given in Algorithm 2. Asbefore, the input ˆ y MSI is inherently assumed and eliminatedfrom the argument of the SIR functions for compactness. Thelogic and steps of the algorithm are similar to Algorithm 1,and thus we avoid further explanations. It is noteworthy tomentioned that, for N ≥ UAVs, using our method, obtainingthe position of the jammer is reduced to systematic calculationof values of SIR expressions at N + N + ∼ O ( N ) points,which is tractable even in large-scale networks. V. S
IMULATION R ESULTS
Similar to [16], we consider f c = GHz, C LoS = dB, C NLoS = dB, and η NLoS = µ LoS . Also, we assume p MSI = dBm, p TR 1 = dBm, and p TR 2 = dBm. Since, considering our networksetting, we are among the first to study the jammer placement,we propose the following baselines for performance comparison:i) Chasing a UAV : the jammer is placed directly under a UAVrelay. ii)
Random : the jammer is placed in a random positionbetween the TRs. iii)
Middle : The jammer is placed at themiddle of the line between the TRs. Considering the dual-hopsetting with p u = dBm, h u = m, D = m, and y u = m,Fig. 3 depicts SIR max upon moving the UAV from x u = m to x u = m. As can be seen, the best baseline method is chasing This is the sum of the points given by Proposition 4, which is N + ,Proposition 5, which is N ( N + ) , and Proposition 6, which is ( N + ) .In the dual-hop setting ( N = ), only points need to be examined. Thisdue to the reciprocity of the SIR expressions that eliminates two solutions(see the first case of Proposition 3). Algorithm 2:
Optimal jammer placement in multi-hopUAV-assisted relay networks The set of final candidates of exterma
P = {} Derive x ( k ) MSI , ≤ k ≤ N + using Proposition 4. for i ∈ { , , ⋯ , N + } do if min { SIR j ( x ( i ) ) ∶ ≤ j ≤ N + } = SIR i ( x ( i ) ) then if SIR i ( x ( i ) )≥ min { SIR N + + j ( x ( i ) ) ∶ ≤ j ≤ N + } then P = P ∪ { [ x ( i ) , SIR i ( x ( i ) )] } end end end for i ∈ { N + , N + , ⋯ , N + } do if min { SIR N + + j ( x ( i ) ) ∶ ≤ j ≤ N + } = SIR i ( x ( i ) ) then if SIR i ( x ( i ) ) ≥ min { SIR j ( x ( i ) ) ∶ ≤ j ≤ N + } then P = P ∪ { [ x ( i ) , SIR i ( x ( i ) )] } end end end Derive x ( n + N + ,n ′ + N + )± and x ( n,n ′ )± , ≤ n < n ′ ≤ N + ,using Proposition 5. for ind ∈ {+ , −} do for ( n, n ′ ) ∈ {( n, n ′ ) ∶ ≤ n < n ′ ≤ N + } do if SIR n ( x ( n,n ′ ) ind ) = min { SIR j ( x ( n,n ′ ) ind ) ∶ ≤ j ≤ N + } and SIR n ( x ( n,n ′ ) ind )≥ min { SIR N + + j ( x ( n,n ′ ) ind )∶ ≤ j ≤ N + } then P = P ∪ { [ x ( n,n ′ ) ind , SIR n ( x ( n,n ′ ) ind )] } end end for ( n, n ′ ) ∈ {( n, n ′ ) ∶ N + ≤ n < n ′ ≤ N + } do if SIR n ( x ( n,n ′ ) ind )= min { SIR N + + j ( x ( n,n ′ ) ind )∶ ≤ j ≤ N + } and SIR n ( x ( n,n ′ ) ind )≥ min { SIR j ( x ( n,n ′ ) ind )∶ ≤ j ≤ N + } then P = P ∪ { [ x ( n,n ′ ) ind , SIR n ( x ( n,n ′ ) ind )] } end end end Derive x ( n,n ′ + N + )± , ≤ n < n ′ ≤ N + using Proposition 6. for ind ∈ {+ , −} do for ( i, j ) ∈ {( n, n ′ + N + ) ∶ ≤ n < n ′ ≤ N + } do if min { SIR j ( x ( i,j ) ind ) ∶ ≤ j ≤ N + } = SIR i ( x ( i,j ) ind ) and min { SIR N + + j ( x ( i,j ) ind )∶ ≤ j ≤ N + } = SIR j ( x ( i,j ) ind ) then P = P ∪ { [ x ( i,j ) ind , SIR i ( x ( i,j ) ind )] } end end end Consider P in the following format: P = ∪ ∣P∣ i = {[ a i , b i ]} x ∗ MSI = a i ∗ , i ∗ = arg min i { b i ∶ [ a i , b i ] ∈ P , − x − jam ≤ a i ≤ x + jam } . the UAV; our method leads to considerably more (between . dB to . dB) reduction in SIR max . To better illustrate theperformance gain, the percentage of reduction in SIR max uponusing our method as compared to the baselines is depicted inFig. 4, which reveals around (average) SIR reduction ofour method.Considering the jammer placement in the multi-hop setting Φ SIR ∶ [ A = p TR 1 /[ x + y + h ] , B = x , C = ( ˆ y MSI − y ) + h ] (15) Φ SIR k ∶ [ A = p k − η NLoS /[ µ LoS (∣ δ x ( k − ,k ) ∣ + ∣ δ y ( k − ,k ) ∣ + ∣ δ h ( k − ,k ) ∣ ) ] , B = x k , C = ( ˆ y MSI − y k ) + h k ] if ≤ k ≤ N (16) Φ SIR N + ∶ [ A = p N µ NLoS / ( η NLoS (( x N − D ) + y N + h N )) , B = D, C = ˆ y MSI ] (17) Φ SIR N + ∶ [ A = p TR 2 /[ (( x N − D ) + y N + h N ) ] , B = x N , C = ( ˆ y MSI − y N ) + h N ] (18) Φ SIR N + k + ∶ [ A = p N − k + η NLoS µ LoS (∣ δ x ( N − k,N − k + ) ∣ + ∣ δ y ( N − k,N − k + ) ∣ + ∣ δ h ( N − k,N − k + ) ∣ ) , B = x N − k , C = ( ˆ y MSI − y N − k ) + h N − k ] if ≤ k ≤ N − (19) Φ SIR N + ∶ [ A = p µ NLoS /[ η NLoS ( x + y + h )] , B = , C = ˆ y MSI ] (20) x u (horizontal position of the UAV) -2 dB0 dB2 dB4 dB6 dB8 dB10 dB12 dB14 dB S I R m a x Optimal Chasing the UAV Random Middle
Fig. 3: Comparison between SIR max considering moving the UAV in theinterval x u ∈ [ , ] upon usingour optimal method as compared tothe baseline methods. Chasing the UAV vs. Optimal Random vs. Optimal Middle vs. Optimal
75 %76 %77 %78 %79 %80 %81 %82 %83 %84 %85 % P e r ce n t a g e o f d ec r ea s e i n S I R m a x Fig. 4: Average percentage of de-crease in SIR max considering movingthe UAV in the interval x u ∈ [ , ] upon using our optimal method ascompared to the baseline methods. Network Realization -20 dB-18 dB-16 dB-14 dB-12 dB-10 dB-8 dB-6 dB-4 dB-2 dB0 dB S I R m a x Optimal Chasing a UAV Random Middle
Fig. 5: Comparison between SIR max considering network realizationsupon using our optimal method ascompared to the baseline methods for UAV relays in the network.
Chasing a UAV vs. Optimal Random vs. Optimal Middle vs. Optimal
60 %65 %70 %75 %80 %85 %90 %95 %100 % P e r ce n t a g e o f d ec r ea s e i n S I R m a x N=10 N=20 N=30 N=40
Fig. 6: Average percentage of de-crease in SIR max considering network realizations upon using ouroptimal method as compared to thebaseline methods for different num-bers of UAV relays in the network N . with D = km, we choose the position and the transmittingpowers of UAVs randomly with respect to the followingintervals: x i ∈ ( , km ) , h i ∈ [ , ] m, y i ∈ [− , ] m,and p i ∈ [ , ] dBm, ≤ i ≤ N . Each random assignmentof the transmitting powers and positions of the UAVs isconsidered as one network realization . Upon using the chasinga UAV baseline, the jammer is placed underneath a randomlyselected UAV in each network realization. Considering UAVs in the system, Fig. 5 depicts SIR max for networkrealizations. As before, the best baseline method is chasinga UAV, which is considerably outperformed by our method.To reveal the performance gain, the average percentage ofreduction in SIR max considering different numbers of UAVsin the network for network realizations upon using ourmethod as compared to the baselines is depicted in Fig. 6,which shows a SIR reduction between to upon usingour method. Examining Fig. 6, it is noteworthy to mention thatas the number of UAVs increases, the performance gap betweenour method and the chasing a UAV baseline decreases, which isvice versa considering the other two baselines. That is because,in general, considering a fixed distance between the TRs, as thenumber of UAVs increases and they get closer to each other, theposition of the jammer becomes less important. Nevertheless,the chasing a UAV baseline significantly deteriorates the SIRat only one UAV, which is the UAV located above the jammer.This makes this baseline method less effective as the number ofUAVs increases since, considering (2) and (3), there is a smallerchance that deteriorating the SIR at only a UAV correspondsto the decrease of both SIR Link 1 and SIR
Link 2 .VI. C
ONCLUSION
We proposed an effective approach for jammer placementin UAV-assisted wireless networks aiming to minimize themaximum achievable data rate of transmission of the system.We studied the problem for both the dual-hop and multi-hoprelay settings. Given the non-convexity of the problem, weproposed a systematic tractable approach that can efficientlyfind the optimal placement of the jammer for both settings.As a future work, we suggest studying the problem when theUAVs can evade the interference by changing their locations.In this case, designing online adaptive algorithms for both thejammer and the UAVs is of particular interest. R
EFERENCES[1] Y. Chen, N. Zhao, Z. Ding, and M. Alouini, “Multiple UAVs as relays:Multi-hop single link versus multiple dual-hop links,”
IEEE Trans.Wireless Commun. , vol. 17, no. 9, pp. 6348–6359, Sep. 2018.[2] J. Feng, X. Li, E. L. Pasiliao, and J. M. Shea, “Jammer placement topartition wireless network,” in
IEEE Global Commun. Conf. Workshops(GC Wkshps) , 2014, pp. 1487–1492.[3] J. Feng, W. E. Dixon, and J. M. Shea, “Fast algorithms for jammerplacement to partition a wireless network,” in
Proc. IEEE Int. Conf.Commun. (ICC) , 2017, pp. 1–6.[4] Y. Allouche, E. M. Arkin, Y. Cassuto, A. Efrat, G. Grebla, J. S. Mitchell,S. Sankararaman, and M. Segal, “Secure communication through jammersjointly optimized in geography and time,”
Pervasive Mobile Comput. ,vol. 41, pp. 83–105, 2017.[5] C. L. Krishna and R. R. Murphy, “A review on cybersecurity vulnerabil-ities for unmanned aerial vehicles,” in
IEEE Int. Symp. Safety, SecurityRescue Robot. (SSRR) , 2017, pp. 194–199.[6] E. Arkin, Y. Cassuto, A. Efrat, G. Grebla, J. S. Mitchell, S. Sankararaman,and M. Segal, “Optimal placement of protective jammers for securingwireless transmissions in a geographic domain,” in
Proc. 14th Int. Conf.Inf. Process. Sensor Netw.
ACM, 2015, pp. 37–46.[7] K. Grover, A. Lim, and Q. Yang, “Jamming and anti-jamming techniquesin wireless networks: a survey,”
Int. J. Ad Hoc and Ubiquitous Comput. ,vol. 17, no. 4, pp. 197–215, 2014.[8] X. Wei, Q. Wang, T. Wang, and J. Fan, “Jammer localization in multi-hopwireless network: A comprehensive survey,”
IEEE Commun. SurveysTut. , vol. 19, no. 2, pp. 765–799, 2017.[9] S. Hosseinalipour, A. Rahmati, and H. Dai, “Interference avoidanceposition planning in dual-hop and multi-hop UAV relay networks,” arXivpreprint arXiv:1907.01930 , 2019.[10] S. Hosseinalipour, A. Rahmati, and H. Dai, “Interference avoidanceposition planning in UAV-assisted wireless communication,” in
Proc.IEEE Int. Conf. Commun. (ICC) , May 2019, pp. 1–6.[11] A. Rahmati, S. Hosseinalipour, Y. Yapici, X. He, I. Guvenc, H. Dai, andA. Bhuyan, “Interference avoidance in UAV-assisted networks: Joint 3Dtrajectory design and power allocation,” arXiv preprint arXiv:1904.07781 ,2019.[12] ——, “Dynamic interference management for UAV-assisted wirelessnetworks,” arXiv preprint arXiv:1909.12777 , 2019.[13] D. Hu, Q. Zhang, Q. Li, and J. Qin, “Proactive unmanned aerial vehiclesurveilling via jamming in decode-and-forward relay networks,”
IEEEAccess , vol. 7, pp. 90 465–90 475, 2019.[14] C. Zhong, J. Yao, and J. Xu, “Secure UAV communication withcooperative jamming and trajectory control,”
IEEE Commun. Lett. , vol. 23,no. 2, pp. 286–289, 2018.[15] H. Wang, J. Chen, G. Ding, and J. Sun, “Trajectory planning in UAVcommunication with jamming,” in , 2018, pp. 1–6.[16] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Mobile unmannedaerial vehicles (UAVs) for energy-efficient internet of things communica-tions,”