1

UAV-Enabled Cooperative Jamming for CovertCommunications

Hangmei Rao, Sa Xiao, Jianquan Wang, Wanbin Tang, Member, IEEE

Abstract—In this paper a novel unmanned aerial vehicleaided (UAV) cooperative jamming scheme is proposed for covertcommunications. We first analyze the detection performance ofthe system to obtain the minimum error detection probabilityof the eavesdropper and then determine the transmission rateas the objective function by analyzing the transmission outageprobability of the communication. The problem formulate is non-convex that is difficult to solve. To solve this, two efficient algo-rithms are proposed for general signal to interference plus noiseratio (SINR) and high SINR, respectively. The first algorithmapplying the block coordinate descent (BCD) to decompose theproblem into two subproblems and then solve them by successiveconvex approximation (SCA). For the second algorithm, we use ageometric method(GM) based on the Apollonius of Sphere to solvethe optimization problem. The proposed scheme can enhancethe covert performance significantly. Simulations verify that theproposed joint design can enhance the covert transmission rateof the considered system as compared to the benchmark schemes.

Index Terms—Covert communications, UAV communications,mobile jammer, trajectory optimization, transmit power control.

I. INTRODUCTION

Due to the broadcasting nature of wireless communications,both legitimate and illegitimate users can receive data signalsfrom wireless air interface. Therefore, how to guarantee thecommunication security is of paramount significance. Al-though various physical layer security schemes have beenproposed to prevent the information being extracted by ad-versaries [1]–[4]. There still exists various practical scenarios,such as in military application environments, where the exis-tence of communication is not allowed to be discovered byany eavesdropper. To avoid the data transmission from beingdiscovered, covert or low probability of detection (LPD) com-munications have been studied in both industry and academiarecently.

Fig. 1 illustrates a basic model of covert communications.Particularly, a transmitter Alice transmits data signals to areceiver Bob, while an eavesdropper Willie tries to detect theirtransmission. If the detection performance of Willie is close torandom guessing, the transmission from Alice to Bob is called“covert”.

There have been some studies aiming at analyzing theachievable rate of covert communications from the perspectiveof information theory. The covert rate was firstly analyzedover additive white Gaussian noise (AWGN) channels in [5].

H. Rao, S.Xiao, J.Wang and W. Tang are with the National Key Laboratoryof Science and Technology on Communications, University of ElectronicScience and Technology of China, Chengdu 611731, China.

Alice Bob

Secret

Willie

Fig. 1. Covert communications model.

Then the result was extended to discrete memoryless channels(DMC) in [6] and [7], binary symmetric channels (BSC) in [8],and multiple access channels (MAC) in [9], respectively. Theaforementioned studies all followed square root law (SRL),where the covert rate was O(1/

√n) and approached to 0

when channel use n went to infinite. To break SRL andachieve constant rate, some studies tried to exploit the channeluncertainties [11]–[15] or artificial noise from the friendlyjammer [17]–[22]. Since statistical features and powers ofartificial noise are easier to control, jamming schemes are moreimplementable in practice in comparison with the channeluncertainty based ones.

With the recent progress of unmanned aerial vehicle (UAV)techniques, it becomes possible to mount miniaturized commu-nication transceivers on UAVs to provide wireless connectionservices. Compared with traditional terrestrial communica-tions, UAV communications are in general more flexible andcost-effective to deploy and likely to have better communi-cation links due to the high chance of line-of-sight (LoS)links. Therefore, some studies exploited the UAV as mobilejammer. In [23]–[27], UAV-assisted jamming schemes wereproposed to realize the physical security. In [28], a jointtrajectory design and power control scheme was proposed forcovert communications, where an full-duplex UAV workedas a jammer and receiver simultaneously. However, in somepractical scenarios, the UAV only need to be a mobile jammerto protect the transmission of terrestrial users. Therefore, howto optimize its trajectory and transmitter power according tothe locations of the receiver and eavesdropper is worth furtherexploring.

In this paper, we investigate UAV-assisted covert commu-nications, where an UAV is employed as a dedicated mobilejammer. We develop a framework of joint trajectory designand transmit power control to optimize the covert rate of thesystem. The main contributions of the paper are summarizedas follows:

arX

iv:2

101.

0750

2v1

[cs

.IT

] 1

9 Ja

n 20

21

2

Alice

Willie

Bob

UAV

Fig. 2. An UAV-assisted covert communications system.

- First, this paper is one of the early attempts to investi-gate UAV-assisted covert communications. It considers ascenario with a dedicated UAV jammer and develops aframework of joint trajectory design and power control.

- Second, we propose an iterative method to solve thetrajectory design and power control problem by ap-plying block coordinated decent (BCD) and concave-convex procedure (CCCP). Simulation results show thatthe proposed iterative method can enhance the coverttransmission rate of the considered system with accept-able convergence rate in comparison with the benchmarkschemes.

- Third, we also propose a geometric method to solve theproblem for the high received signal-to-interference-and-noise ratio (SINR) case by analyzing the geometricalfeatures of the trajectory. Compared with the proposediterative method, the heuristic one can achieve a betterperformance in the high SINR region and does not requireany iteration to approach the solution.

The rest of this paper is organized as follows. In Section II,we describe the system model and formulate the optimizationproblem. To solve the problem, we develop a BCD joint CCCPbased iterative method and a geometrical method in SectionsIII and IV, respectively. Then we present simulation results inSection V. Finally, conclusions are drawn in Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

In this paper, we consider an UAV-enabled covert commu-nication system, where Alice, Bob and Willie are terrestrialusers, while an UAV acts as a mobile jammer to jam theeavesdropping of Willie. The UAV and all terrestrial usersare equipped with single antennas. In addition, the locationsof Alice, Bob and Willie are fixed to qa , [xa, ya]T ,qb , [xb, yb]

T and qw , [xw, yw]T respectively, while theUAV flies at a fixed altitude of H . To facilitate the UAVtrajectory design, we equally divide the flight period T intoN time slots with duration σt = T

N , and N is large enoughso that the distance between the UAV and the terrestrialusers is approximately constant within each slot. Hence, thecontinuous UAV’s trajectory can be replaced by N discretepositions qu[n] , [xn, yn]T , n ∈ N , 0, 1, 2, . . . , N.

We assume that the channel distribution information (CDI)of all involved channels and the locations of terrestrial usersare available to the UAV. The channels from Alice to Boband to Willie are defined as Rayleigh fading channels. Thus,

the corresponding channel power gains can be denoted byha,b[n] = β0d

−αa,b ζa,b[n] and ha,w[n] = β0d

−αa,wζa,w[n], respec-

tively, where β0 is the channel power gain at the referencedistance of 1 m, da,b and da,w are corresponding link length,α is the path loss exponent, and ζa,b[n] and ζa,w[n] areexponentially distributed random variables.

Analogous to [29] and [30], the UAV-ground channels aremainly dominated by line-of-sight (LoS) components. Accord-ingly, for the n-th time slot, the channel power gain from theUAV to Bob and to Willie can be expressed as

|hu,b[n]|2 =β0

‖qu[n]− qb‖2 +H2(1)

and|hu,w[n]|2 =

β0

‖qu[n]− qw‖2 +H2, (2)

respectively.Denote xa(i) ∼ CN (0, 1) and Pa[n] be the transmitted

symbol and transmit power of Alice for the i-th channel usein time slot n, respectively, and v

(i)b [n] ∼ CN (0, σ2

b ) be thecorresponding additive Gaussian white noise (AWGN) at Bob.Then the received signals at Bob can be written as

y(i)b [n] =

√Pa[n]ha,b[n]x(i)

a [n] + Φ + v(i)b [n], (3)

where Φ =√Pu[n]hu,b[n]x

(i)u [n], Pu[n] denotes the jamming

power of the UAV, and x(i)u [n] denotes the transmitted artificial

noise symbol of the UAV satisfying E[x(i)u x

(i)†u ] = 1. Similar

to [17], we assume that Pu[n] is an independent identicallydistributed (i.i.d.) uniform random variables on [0, Pu[n]].More specifically, the probability density function (PDF) ofPu[n] is given by

fPu[n](x) =

1

Pu[n], if 0 ≤ x ≤ Pu[n],

0, otherwise.(4)

Based on (3), the channel capacity from Alice to Bob isgiven by

C = log2

(1 +

Pa[n]β0d−αa,b ζa,b[n]

Pu[n]|hu,b[n]|2 + σ2b

). (5)

Since the UAV is a cooperative jammer, Bob is assumed toknow the maximum transmit power of the UAV at each timeslot. Thus the outage probability can be expressed as

Pout = PC ≤ Rb[n] = 1− exp (−Ξ) (6)

where

Ξ = − (2Rb[n] − 1)(Pu[n]|hu,b[n]|2 + σ2b )

Pa[n]β0d−αa,b

(7)

and Rb[n] is the transmit rate of Alice.From (6), we can see that the outage probability is increas-

ing with Pu[n]. Then, if the outage probability is guaranteedto be always lower than the threshold of ρb for all possiblePu[n], Rb[n] is given by

Rb[n] = log2

(1−

ln(1− ρb)Pa[n]β0d−αa,b

Pu[n]|hu,b[n]|2 + σ2b

). (8)

3

B. Detection Performance at Willie

The received signal at Willie can be written as

y(i)w [n] =

∆ + v(i)w [n], H0,√

Pa[n]ha,w[n]x(i)a [n] + ∆ + v(i)

w [n], H1,(9)

where ∆ =√Pu[n]hu,w[n]x

(i)u [n], H0 and H1 denote the

transmission of Alice is on and off, respectively, and v(i)w ∼

CN (0, σ2w) is AWGN at Willie.

In this paper, we assume that Willie uses a radiometerdetector at each time slot as in [17]. Therefore, we have

Pw[n] ,1

m

m∑i=1

∣∣∣yw(i)[n]∣∣∣2 H1

≷H0

ϑ, (10)

where Pw[n] is the average received power and ϑ is thethreshold of the detector.

Let PFA[n] and PMD[n] be the probabilities of false alarmand missed detection, respectively. Then we have

PFA[n] = P (Pw[n] > ϑ|H0)

= P((σ2

w + γ[n])X 2

2m

m> ϑ|H0

),

(11)

PMD[n] = P (Pw[n] < ϑ|H1)

= P((σ2

w + γ[n])X 2

2m

m< ϑ|H1

),

(12)

where X 22m represents the chi-squared random variable with

2m degrees of freedom, and

γ[n] =

Pu[n]|hu,w[n]|2, H0,

β0Pa[n]d−αa,wζa,w[n] + Pu[n]|hu,w[n]|2, H1.(13)

Meanwhile, based on the strong law of large numbers, we have

Pr

limm→∞

X 22m

m= 1

= 1. (14)

Therefore, the expression of PFA[n] and PMD[n] are given inthe following theorem, which is proved in Appendix A.

Theorem 1: Let Γ = Pu[n]|hu,w[n]|2 and φ = β0

dαa,w. Then

we have

PFA[n] =

1, if ϑ− σ2

w ≤ 0,

1− ϑ−σ2w

Γ , if 0 < ϑ− σ2w ≤ Γ,

0, if ϑ− σ2w > Γ,

(15)

and

PMD[n] =

0, if ϑ− σ2

w ≤ 0,

Λ, if 0 < ϑ− σ2w ≤ Γ,

Ω, if ϑ− σ2w > Γ,

(16)

where

Λ =φPa[n]

Γexp

(− (ϑ− σ2

w)

φPa[n]− 1

)+ϑ− σ2

w

Γ,

and

Ω = 1− φPa[n]

Γ

(exp

(σ2w + Γ− ϑφPa[n]

)− exp

(σ2w − ϑφPa[n]

)).

Based on the above theorem, the minimum value of ξ =PFA + PMD and corresponding threshold are given in thefollowing theorem, which is proved in Appendix B.

Theorem 2: For the radiometer detector considered in (10),the optimal detection threshold of Willie at the n-th time slotis ϑ∗ = Γ + σ2

w, and the minimum detection error rate is

ξ∗[n] = 1 +

(exp

(− Γ

φPa[n]

)− 1

)φPa[n]

Γ. (17)

C. Problem Formulation

In this paper, we aim to maximize the covert transmissionrate from Alice to Bob via joint trajectory design and transmitpower control. Let Q , q[n],∀n and P , Pa[n],∀n.Then the optimization problem is formulated as

maxQ,PL(Q,P) =

1

N

N∑n=1

Rb[n] (18)

s.t. ξ∗[n] ≥ 1− ε (18a)

1

N

N∑n=1

Pa[n] ≤ Pa,∀n (18b)

0 ≤ Pa[n] ≤ Pmaxa ,∀n, (18c)‖qu[n]− qu[n− 1]‖ ≤ Vmaxσt,∀n, (18d)qu[N ] = qu,F , (18e)qu[0] = qu,0, (18f)

where Vmax is the maximum speed of the UAV, qu,0 =[xu,0, yu,0] and qu,F = [xu,F , yu,F ] are the initial and finallocations of the UAV, respectively; and Pa and Pmaxa denotethe average and peak transmit powers of Alice, respectively.Constraint (18a) ensures the covertness of the transmission.Constraints (18b) and (18c) are the transmit power constraintsfor Alice at each time slot. Constraints (18d), (18e), and (18f)are the UAV’s mobility constraints.

Note that problem (18) is non-convex due to the non-convexstructures of both the objective functions and constraints.Therefore, it is difficult to solve it directly. In the followingsections, we propose an iterative and a geometrical methods tosolve the problem. The first method is a technique to handlethe problem in general cases, while the second one is anefficient technique for the scenarios with high SINR of thereceived signals at Bob.

III. ITERATIVE METHOD TO SOLVE THE OPTIMIZATIONPROBLEM

In this section, we employ the following iterative methodto solve problem (18), which applies the BCD to partition thevariables into two blocks, i.e., UAV trajectory variable Q andsource transmit power variable P.

Q(0)︸︷︷︸Initialization

→ P(1) → Q(1)︸ ︷︷ ︸First iteration

→ . . .→ P(l−1) → Q(l−1)︸ ︷︷ ︸(l−1)−thiteration

→ P(l) → Q(l)︸ ︷︷ ︸(l)−th iteration

→ . . .→ P∗ → Q∗︸ ︷︷ ︸(l−1)−th iteration

. (19)

4

Specially, the iterative method starts by find initial trajectoryof the UAV Q(0). Then, in each iteration l, it first derives Pl

from the trajectory Ql−1 from the last iteration and design Ql

based on the obtained Pl. Since the problem for each block isstill non-convex, we adopt CCCP to handle it. The above stepsare repeated until the convergence conditions are satisfied.

A. Feasibility Analysis and Trajectory Initialization

In order to find the initial solution Q(0), we first rewriteconstraint (18a) as

G (Pa [n] ,qu [n]) =−Pa [n] (du,w [n])

2

Pu [n] dαa,w exp

(Pu[n]dαa,w

Pa[n](du,w[n])2

)+Pa [n] (du,w [n])

2

Pu [n] dαa,w

≤ ε. (20)

where (du,b [n])2

= (‖qu[n] − qb‖2 + H2), (du,w [n])2

=(‖qu[n]− qw‖2 +H2).

In Appendix D, it is proved that G (Pa [n] ,qu [n]) is anincrease function for Pa [n] ≥ 0 when qu [n] is given.Meanwhile, G (Pa [n] ,qu [n]) = 0 for Pa [n] = 0. Therefore,we have the following proposition for the feasible solution tothe problem.

Proposition 1 : For any Q satisfying (18d), (18e), and (18f),there always exists P satisfying (18b) and (18c) rendering(18a) hold.

Based on the above proposition, Q(0) can be determinedonly by considering trajectory constraints (18d), (18e), and(18f). As in [25], we use the best-effort manner to initializethe trajectory Q(0) where the UAV tries its best to approachWillie. For different values of qu,0,qu,F , and qw,Q

(0) canbe obtained as follows.

Case1:⌈||qw−qu,0||Vmaxδt

⌉+⌈||qu,F−qw||Vmaxδt

⌉6 N

In this case, N is sufficient large so that the UAV can firstfly from the initial point to Willie, and then fly from Willieto the final point. The corresponding UAV flight trajectory isshown as the red line in Figure. 3(a). This is

qu[n] =

qu,0 + nVmaxδtθ1, n < N1,

qw, N1 6 n < N2,

qw + (n−N2)Vmaxδtθ2, N2 6 n < N,

qu,F , n = N,

(21)

where θ1 =qw−qu,0||qw−qu,0|| ,N1 =

⌈||qw−qu,0||Vmaxδt

⌉, N2 = N −⌈

||qu,F−qw||Vmaxδt

⌉, and θ2 =

qu,F−qw||qu,F−qw|| .

Case2:⌈||qw−qu,0||Vmaxδt

⌉+⌈||qu,F−qw||Vmaxδt

⌉> N

In this case, N is insufficient for the UAV to successivelyarrive at Willie and the final point. Thus, the initial flighttrajectory of the UAV is shown as the red line in Figure.3(b), where the UAV first flies from the initial point to thedirection of Willie, and then turns to the terminal direction at

D

(a),0uq ,Fuq

wq

E

(b) ,Fuq,0uq

wq

Willie

Final location of UAV

Initial location of UAV

UAV fly trajectory

Fig. 3. Initial flight trajectory of UAV.

intermediate point qE between the initial point and Willie. Asproved in Appendix C, the intermediate point qE is given by

qE = N3Vmaxδtθ1, (22)

where N3 =⌊

N2(Vmaxδt)2−||qu,F−qu,0||2

2Vmaxδt(NVmaxδt−||qu,F−qu,0|| cos(θ1−θ2))

⌋.

Then the initial trajectory of the UAV can be obtained from

qu[n] =

qu,0 + nVmaxδtθ1, n < N3,

qE + (n−N3)Vmaxδtθ3, N3 6 n ≤ N4,

qu,F , N4 < n ≤ N.(23)

where N4 =⌊||qE−qu,F ||(Vmaxδt)

⌋, and θ3 =

qE−qu,F||qE−qu,F || .

It should be noted that when N3 = 0, the intermediate pointcoincide with the initial point, i.e., qE = qu,0. In this case,the UAV has to fly directly from the initial point to the finalpoint.

B. Transmit Power Control

For any given Q, problem (18) can be reformulated as

maxP

1

N

N∑n=1

log2 (1 + ψPa [n]) , (24)

s.t.Pa [n]φ

Γ− Pa [n]φ

Γexp

(− Γ

Pa [n]φ

)≤ ε, (24a)

1

N

N∑n=1

Pa[n] ≤ Pa, (24b)

0 ≤ Pa[n] ≤ Pmaxa ,

where

ψ = −ln(1− ρb)β0d

−αa,b

Pu[n]|hu,b[n]|2 + σ2b

.

It is not difficult to find that the left hand side (LHS) of(24a) is the difference of two convex functions, which rendersthe problem a non-convex structure. To solve the problem, wefirst exploit the CCCP method to deal with the LHS of theconstraint.

Let P(l) =P

(l)a [n], ∀n

be the fixed point of Pa [n] at

the l-th iteration of the CCCP method. Then constraint (24a)can be rewritten as

P(l)a [n]φ

Γ− g1 (Pa[n]) ≤ ε. (25)

where g1 (Pa[n]) is given by (26) at the top of the next page,and it is obtained by approximating the second term of theLHS function with its first order Taylor expansion.

Substituting (25) into problem (24), P(l+1) can be obtainedby solving the following problem

5

g1 (Pa[n])∆=P

(l)a [n]φ

Γexp

(− Γ

P(l)a [n][n]φ

)− exp

(− Γ

P(l)a [n]φ

)(1

Γφ+

1

P(l)a [n]

)(Pa[n]− P (l)

a [n]). (26)

P(l+1) = arg maxP

1

N

N∑n=1

log2 (1 + ψPa[n]) , (27)

s.t.1

N

N∑n=1

Pa [n] ≤ Pa, (27a)

P(l)a [n]φ

Γ− g1 (Pa[n]) ≤ ε, (27b)

The above problem is a standard convex optimization prob-lem, which can be efficiently solved by convex optimizationsolver such as CVX [33]. Then the solution to problem (24)can be iteratively approached by solving a sequence of (27)as shown in Algorithm 1.

Algorithm 1 The CCCP method for transmit power control1: Initialize2: l = 0 and P(l), tolerance σ;3: repeat:4: Substitute P(l) into Problem (27), and then solve the

problem to obtain P(l+1);5: Update l = l + 1;6: Until: |P(l+1) −P(l)| < σ.

C. UAV Trajectory Design

After obtaining P = Pa [n] , ∀n from Algorithm 1,problem P1 can be simplified as

maxQ

1

N

N∑n=1

log2

1− ψPu[n]β0

‖qu[n]−qb‖2+H2 + σ2b

(28)

s.t. G (qu [n]) ≤ ε,∀n, (28a)‖qu[n]− qu[n− 1]‖ ≤ Vmaxσt,∀n, (28b)qu[N ] = qu,F , (28c)qu[0] = qu,0, (28d)

where ψ = ln(1− ρb)Pa [n]β0d−αa,b , and

G (qu [n]) =Pa [n]

(‖qu [n]− qw‖2 +H2

)Pu[n]dαa,w

−Pa [n]

(‖qu [n]− qw‖2 +H2

)Pu[n]dαa,w exp

(Pu[n]dαa,w

Pa[n](‖qu[n]−qw‖2+H2)

) .Notice that the objective function is a concave function

with ‖qu[n]− qb‖2, while the trajectory constraints (28b),(28c) and (28d) are also convex sets. However, the covertnessconstraint (28a) is non-convex with ‖qu[n]− qw‖2. Similar

to the power control problem, we use the CCCP method todeal with the non-convex constraint.

Let Q(m) =q

(m)u [n] ,∀n

be the fixed point of Q in

the m-th iteration of the CCCP method. Then (28a) can betransformed as(∥∥∥q(m)

u [n]− qb

∥∥∥2

+H2

)Pa[n]

Pu[n]dαa,w− g2 (qu[n]) ≤ ε, (29)

where g2 (qu[n]) is given by (30) at the top of nextpage, which is obtained by approximate the second term ofG (qu [n]) with its first order of Taylor expansion.

As a result, Q(m) can be obtained by solving the followingproblem

Q(m+1) = arg maxQ

1

N

N∑n=1

log

1− ψPu[n]β0

‖qu[n]−qb‖2+H2 + σ2b

(31)

s.t. ‖q(l+1)u [n+ 1]− q(l+1)

u [n]‖ ≤ Vmaxσt, (31a)

q(m+1)u [N ] = q

(m+1)u,F , (31b)

q(m+1)u [0] = q(m+1)

u [0], (31c)(∥∥∥q(m)u [n]− qb

∥∥∥2

+H2

)Pa[n]

Pu[n]dαa,w− g2 (qu[n]) ≤ ε,∀n.

(31d)

The objective function and constraints of the optimizationproblem are all convex structure, which is a standard convexoptimization problem and can be effectively solved by CVX,such as [33]. The detailed procedure of the proposed CCCPmethod are summarized in Algorithm 2.

Algorithm 2 The CCCP method for trajectory design.1: Initialize2: m = 0 and Q(m), tolerance σ;3: repeat:4: Substitute Q(m) into Problem (31), and then solve the

problem to obtain Q(m+1);5: Update m = m+ 1;6: Until: |Q(m+1) −Q(m)| < σ.

D. Overall Algorithm

Based on the solutions to sub-problems (27) and (31), weapply an iterative algorithm to solve the problem (18). Thewhole procedure is summarized in Algorithm 3.

Remark 1: Note that problems (27) and (31) are bothconvex , which can be optimally solve via CVX. Therefore,Algorithm 1 and Algorithm 2 are guaranteed to converge to

6

g2 (qu[n])∆= exp

− Pu[n]dαa,w

Pa[n]

(∥∥∥q(m)u [n]− qw

∥∥∥2

+H2

) Pa[n]

(∥∥∥q(m)u [n]− qw

∥∥∥2

+H2

)Pu[n]dαa,w

+

Pa[n]

Pu[n]dαa,w+ 1(∥∥∥q(m)

u [n]−qw∥∥∥2

+H2

)(‖qu[n]− qw‖2 −

∥∥∥q(m)u [n]− qw

∥∥∥2)

exp

Pu[n]dαa,w

Pa[n]

(∥∥∥q(m)u [n]−qw

∥∥∥2+H2

) . (30)

Algorithm 3 Iterative Algorithm for Trajectory and PowerOptimization

1: Initialize:2: Set l = 0, tolerance σ > 0;3: Initialize Q(0) based (21) or (23);4: repeat:5: l = l + 1;6: For given feasible Q(l−1), P(l) can be obtained by

Algorithm 1;7: Given feasible P(l), Q(l) is obtained by the Algorithm 2;8: Until : |L

(Q(l),P(l)

)− L

(Q(l−1),P(l−1)

)| ≤ σ;

9: Set Q∗,P∗ =Q(l),P(l)

;

10: Return: Q∗,P∗.

the local optimal solutions satisfying the Karush-Kuhn-Tucker(KKT) condition [34].

Remark 2: As proved in [31], Algorithm 3 is guaranteedto converge since the objective function in problem (18) isnon-decreasing over the iterations. In addition, the overallcomplexity is O(LN

72 ) according to the analyses in [25],

where L is the number of iterations. However, Algorithm 3is not guaranteed to converge to a global optimal or a localoptimal solution, since only a local optimal solution for powercontrol or trajectory design can be found in each iteration.Unfortunately, there is no better way to get the global optimal.

IV. GEOMETRIC METHOD TO SOLVE THE OPTIMIZATIONPROBLEM

In this section, we develop a low-complexity geometricmethod to strike a balance between the covert transmissionperformance and computational complexity. Specifically, weaim to design a UAV trajectory and transmit power of thesource at high SINR to maximize the transmission rate.

A. Optimization Problem Formulation

In this case the interference from the UAV is much higherthan the additive noise σ2

b but much smaller than the powerof the received signal at the destination Bob. In other words,1 σ2

b Pu[n]|hu,b[n]|2 − ln(1 − ρb)Pa[n]β0d−αa,b [35].

Therefore the objective function of (18) can be replaced as

maxQ,P

1

N

N∑n=1

log

(−

ln(1− ρb)Pa[n]β0d−αa,b

Pu[n]|hu,b[n]|2

). (32)

Further, the optimization problem (18) rewrite as

maxQ,P

1

N

N∑n=1

log

((du,b[n])

2

(du,w[n])2 ×

κPa[n] (du,w[n])2

Pu[n]dαa,w

)(33)

s.t.Pa[n](du,w [n])

2

Pu[n]dαa,w

(1− e

−Pu[n]dαa,w

Pa[n](du,w [n])2

)≤ ε, (33a)

1

N

N∑n=1

Pa[n] ≤ Pa[n], (33b)

0 ≤ Pa[n] ≤ Pmaxa , (33c)

‖qu[n]− qu[n− 1]‖ ≤ Vmaxσt, n = 0, 1, ..., N, (33d)qu[N ] = qu,F , (33e)qu[0] = qu,0, (33f)

where d2u,b[n] = ‖qu[n] − qb‖2 + H2 and d2

u,w[n] =

‖qu[n] − qw‖2 + H2 and κ = − ln(1 − ρb)d−αa,b d

αa,w. We

note that the objective function (33) first item represents thedistance relationship between UAV→ Willie and UAV→Bob,which is only related to constraints (33d), (33e) and (33f).We also note that the second term Pa[n](du,w[n])2

Pu[n]dαa,w, which is

consistent with the covertness constraint(33a). These factsenable us to decompose the optimization problem (33) intothe the following subproblems:

maxQ

1

N

N∑n=1

log

((du,b[n])

2

(du,w[n])2

)(34)

s.t. ‖qu[n]− qu[n− 1]‖ ≤ Vmaxσt, n = 0, 1, ..., N, (34a)qu[N ] = qu,F , (34b)qu[0] = qu,0, (34c)

and

maxP

1

N

N∑n=1

log ($Pa[n]τ) (35)

s.t. Pa[n]τ(

1− e−1

Pa[n]τ

)≤ ε, (35a)

1

N

N∑n=1

Pa[n] ≤ Pa[n], (35b)

7

0 ≤ Pa[n] ≤ Pmaxa , (35c)

where τ =(du,w[n])2

Pu[n]dαa,wand $ = − ln(1−ρb)d−αa,b . In the follow-

ing, we will prove that there is a unique optimal solution forsubproblems (34) and (35). As such, we propose a joint designbased on geometric method to the optimization problem (34)and (35), which can be balanced the computational complexityand the achievable covert communication performance.

B. Geometric scheme for Solving (34)

In this subsection, we develop a geometric method tosolve the the optimization problem (34). The central ideais transform the optimization problem (34) into a geometricproblem. To proceed, we first note that the objective functionof the optimization problem (34) is proportional to the distance(du,b[n])

2. Constraint in (33a) indicates that by introducingthe smaller distance between the UAV and Willie the moreseriously jamming will be induced on Willie. As a result,optimizing the trajectory requiring the larger (smaller) distancefrom the UAV to Bob (Willie). Following this fact, we divide

u,0

u,F

Willie

Bob

Fig. 4. The trajectory of UAV.

the problem into each slot for optimization as is shown inFigure. 4, and then the problem can be regarded as the positionq[n − 1] of the n − 1th time slot is know to find the nthslot optimal position q[n], as shown in Figure. 5. Thus, theoptimization problem (34) can be equivalently rewritten as

maxqu[n]

((du,b[n])

2

(du,w[n])2

)(36)

s.t. ‖qu[n]− qu[n− 1]‖ ≤ Vmaxσt, n = 0, 1, ..., N, (36a)qu[N ] = qu,F , (36b)qu[0] = qu,0. (36c)

W

B

[n 1]u

-q

[n]uq

Fig. 5. The trajectory point of UAV in each time slot.

However, according to the knowledge of geometry, we knowthat the ratio (du,b[n])

(du,w[n]) = k, k 6= 1 is a “Apollonius of Sphere”.To this end we have the following theorem, which is provenin Appendix E.

Theorem 3: [Apollonius of Sphere] According to geometry,the ratio of the distance between a moving point q[n] inspace and two fixed points B and W (i.e.,

((du,b[n])(du,w[n])

)=√

||q[n]−qb||+H2

||q[n]−qw||+H2 = k, k 6= 1) is a sphere, which satisfies thefollowing equation(

x[n] + x′)2

+(y[n] + y

′)2

+ (z[n])2 =(

kck2−1

)2, k 6= 1,

x[n](xw − xb) + y[n](yw − yb) = 0, k = 1.(37)

where x′

= k2xw−xb1−k2 , y

′= k2yw−xb

1−k2 and c = (xb − xw)2 +

(yb − yw)2 the center of the sphere, R = c kk2−1 is the radius

of the sphere, and when z[n] = H is the fixed altitude of theUAV.Following Theorem 3 the the optimization problem (34)problem is equivalent to finding a point qu[n] in the feasibleregion of UAV flight with the height of H to maximize theratio of the square of the distance from the two fixed points(Bob and Willie). From the above analysis, the essence ofoptimization problem (36) is to find the maximum k value.Thus, the optimization problem (36) can be rewritten as

maxk≥1

((du,b[n])

2

(du,w[n])2

)= k2 (38)

s.t. ‖qu[n]− qu[n− 1]‖ ≤ Vmaxσt, n = 0, 1, ..., N, (38a)qu[N ] = qu,F , (38b)qu[0] = qu,0. (38c)

In order to obtain the position of the UAV at the nexttime slot, we have the following theorem, which is provenin Appendix F.

Theorem 4: Let qu[n − 1] be the position of the n − 1thslot of the UAV, the optimal location point qu[n] is at thetangent point between UAV feasible area and the Apolloniusof Sphere, which satisfies the following equation

((du,b[n])2

(du,w[n])2

)=

k1+√k2

1−4k0k2

2k0, k > 1,

x[n](xw − xb) + y[n](yw − yb) = 0, k = 1,((du,b[n])2

(du,w[n])2

)=

k1−√k2

1−4k0k2

2k0, k < 1,

(39)

where k0 = (U + S)2 + 4(Vmaxδt)2H2, k1 = (2SV −

2U2 − 2US + 2UV − 4Vmaxδt)2(c2 + H2)), k2 = (U +

V )2 + 4(Vmaxδt)2H2, U = d2 − (Vmaxδt)

2 + H2, d =x2[n−1]+y2[n−1], S = x2

w−2x[n−1]xw+y2w−2y[n−1]yw

and V = x2b − 2x[n− 1]xb + y2

b − 2y[n− 1]yb.In the above process of solving the UAV trajectory, when we

obtain the current time slot q[n], the next time slot trajectoryq[n + 1] design needs to judge whether the UAV can reachthe destination in the remaining time, i.e.,

||q[n]− qu,F ||Vmaxδt

≤ N − n. (40)

If ||q[n]−qu,F ||Vmaxδt

> N − n, the UAV needs to return to theposition q[n− 1] of the previous time slot, and then fly to thedestination at the maximum speed.

In summary, the proposed low-complexity geometricmethod of Q is summarized in Algorithm 4.

8

Algorithm 4 The geometric method to problem (34)Require:

1: Given an initial position qu,0 = (xu,0, yu,0) of the UAV;2: Repeat

3: The objective function maxqu[n]

((du,b[n])2

(du,w[n])2

)can be trans-

formed into a geometric problem, which is to find thetangent point between the sphere with the smallest radiusand the boundary (x[n]−x[n−1])2 +(y[n]−y[n−1])2 =(Vmaxσt)

2 of the feasible region of UAV flight;4: If ||q[n]−qu,F ||

Vmaxδt> N −n, the optimal location point qu[n]

is satisfying

(

(du,b[n])2

(du,w[n])2

)=

k1+√k2

1−4k0k2

2k0, k > 1,

x[n](xw − xb) + y[n](yw − yb) = 0, k = 1,((du,b[n])2

(du,w[n])2

)=

k1−√k2

1−4k0k2

2k0, k < 1,

(41)

else qu[n] = qu[n− 1];5: Update n = n+ 1.

Note that our geometric method is dedicated to solving thetrajectory design of the UAV to reach the optimal hoveringposition. The trajectory from the optimal hovering position tothe UAV destination is flying with the maximum speed.

Remark 3: Note that our geometric method of Algorithm 4is a greedy algorithm and is approximately optimal. However,there is generally no optimal solution to the UAV trajectoryoptimization problem, and the block gradient method is used,to solve this problem we propose a geometric method, and itsperformance is better than previous methods.

In the following, we focus on proving the uniquenesssolution of optimization problem (35).

C. Geometric scheme for Solving the Optimization Problem(35)

We note that the optimization problem (35) is convexoptimization problem, since the objective function (35) mono-tonically increases with Pa[n] is highly concave and the thecovertness constraint (35a) is convex with Pa[n]. Thus, theoptimization problem (35) can be efficiently solved by convexoptimization solver such as CVX [33].

Following the above discussions, the geometric method canbe summarized as:

Algorithm 5 The geometric method for problem (33)Require:

1: Solving problem (34) and obtain the optimal solution isQ∗;

2: Substitute Q∗ from step 1 into problem (35), then solveproblem (35) by CVX, and obtain the optimal solution isP∗;

Remark 4: It is worth noting that the geometrical method toproblem (33) of Algorithm 5 only needs two steps to get thelocal optimal solution of the optimization problem. Compared

with the traditional BCD method, it has less computationalcomplexity as it converges more quickly.

V. SIMULATION RESULTS

x-100 0 100 200 300 400 500

y

100

150

200

250

300

350

JTP T=350sJTP T=210sJTP T=200sLTP T=350sLTP T=210sLTP T=200s

(a)x

-100 0 100 200 300 400 500

y

100

150

200

250JTP T=200sJTP T=210sJTP T=350sLTP T=350sLTP T=210sLTP T=200s

(b)

Fig. 6. UAV’s trajectories for different values of the flight period T , withε = 0.1, where (a) is the BCD method and (b) is the geometrical method.

0 50 100 150 200 250 300 350T(s)

0

0.05

0.1

0.15

0.2

0.25

Tra

nsm

it po

wer

(W

)

epsion=0.1epsion=0.2epsion=0.3

(a) BCD method.

0 50 100 150 200 250 300 350T(s)

0

0.02

0.04

0.06

0.08

0.1

0.12

Tra

nsm

it po

wer

(W

)

epsion=0.1epsion=0.2epsion=0.3

(b) Geometrical method.

Fig. 7. The Alice’s transmit power for different values of ε

0 50 100 150 200 250 300 350T(s)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Tra

nsm

it po

wer

(W

)

PJmax=0.1PJmax=0.5

(a) BCD method.

0 50 100 150 200 250 300 350T(s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Tra

nsm

it po

wer

(W

)

PJmax=0.1PJmax=0.5

(b) Geometrical method

Fig. 8. The Alice’s transmit power for different values of the maximum ANtransmit power by UAV.

In this section, simulation results are given to evaluatethe performance of the UAV aided covert communicationsachieved by our proposed joint trajectory and power controldesign, denoted as the JTP scheme, as compared to the bench-mark algorithm with line-segment trajectory and optimizedpower control, denoted as the LTP scheme. We assume thatthe UAV firstly flies towards the location above Willie, thenhovers above Willie, and finally flies with the maximumspeed to reach its final location by the last time slot. Overallsimulation parameters are given as below: q0 = [−100, 100]T ,qF = [500, 100]T , H = 100 m, V = 3 m/s, σt = 0.5 s,qa = [0, 0]T , qb = [300, 0]T , qw = [200, 200]T , Pu,max = 16dBm, Pa = 30 dBm, Pa,max = 36 dBm, ρb = 0.1 β0 = −60dB, and β0

σ2 = 80 dB. In addition, the Eb/N0 is defined as(Pa[n]σ2b

).

In Fig. 6, we plot the UAV’s trajectories versus the periodT by the BCD and geometrical methods, respectively, whereAlice, Bob, Willie, and the UAV’s initial and final locations

9

0 20 40 60 80 100 120 140 160 180 200T(s)

0

0.5

1

1.5

2

2.5

3

3.5

4

UA

V s

peed

(m

/s)

LTPJTP

(a)

0 20 40 60 80 100 120 140 160 180 200T(s)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Tra

nsm

it po

wer

(W

)

LTPJTP

(b)

0 50 100 150 200 250 300 350T(s)

0

0.5

1

1.5

2

2.5

3

3.5

UA

V s

peed

(m

/s)

LTPJTP

(c)

0 50 100 150 200 250 300 350T(s)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Tra

nsm

it po

wer

(W

)

LTPJTP

(d)

Fig. 9. The UAV’s speed for different values of the flight period T , where T =210s for (a), while T = 350s for (c), and (b) and (d) are the correspondingsource transmit power.

200 250 300 350 400 450 500

T(s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Ave

rage

cov

ert t

rans

mis

sion

rat

e R

(bps

/Hz)

0.1 LTP0.2 LTP0.3 LTP0.1 JTP0.2 JTP0.3JTP

(a) BCD method200 250 300 350 400 450 500

T (s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Ave

rage

cov

ert t

rans

mis

sion

rat

e R

(bps

/Hz)

0.1 LTP0.2 LTP0.3 LTP0.1 JTP0.2 JTP0.3 JTP

(b) Geometrical method.

Fig. 10. Covert rate versus of the flight period T for different of ε.

90 100 110 120 130 140 1500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Av

erag

e c

ov

ert

tran

smis

sion

rat

e R

(bp

s/H

z)

JTP- GM

JTP- BDC

GM- Nonoptimation Pa

BCD- Nonoptimation Pa

b

0

dBE

N

Fig. 11. Covert rate versus of the Eb/N0.

are marked with 2, ♦, ×, and 4, respectively. It is observedthat when T = 200 s, which is the minimum required timefor the UAV to fly from the initial to the final location atthe maximum speed, the trajectories of JTP and LTP are thesame outcome. However, their trajectories appear graduallydifferent as T increases. Specifically, when T = 350 s, itis observed that UAV flies to the near of Willie, rather thandirectly above Willie, then hovers around at this location aslong as possible, and finally choosing the route to its finalposition with the maximum speed at the end of the last timeslot. In addition, from the results, we observe that the above

trajectory is different from the two algorithms. The reasonis that through the proposed JTP scheme, when the UAV fliesaway from Bob, the jamming power can be increased, whereasfor LTP, it travels along the shortest path. It is worth notingthat for JTP, there is a hovering location, which is aiming totradeoff between the introduction of undesired interference toBob and average minimum error rate at Willie. It also showsthat this hovering location is optimal in the static scenario andhence to achieve the maximum covert rate.

Fig. 7, illustrates the obtained Alice’s transmit power fordifferent values of the covertness level parameter ε. In Fig. 7(a)and Fig. 7(b), we observe that the Alice transmit power to Bobincrease as the ε. we also observe that Alice’s transmit poweris first rapid-increasing stage until the maximum transmitpower stable stage and then decreases rapidly by increasingthe value of T in geometrical scheme and the BCD scheme,respectively. This coincide with the UAV’s flight trajectoryas shown in Fig. 6. In addition, we observe that the Alice’stransmit power achieved by the geometrical scheme is largerthan that achieved by the BCD scheme. This allows the Aliceto transmit signals with a higher transmit power to furtherimprove the average covert transmission rate in the geometricalscheme, while guaranteeing the same level covertness.

Fig. 8 shows the Alice’s transmit power for different valuesof the maximum jamming power transmitted by the UAV andversus different flight period T . As we expected, the Alice’s transmit power Pa[n] first increases and then approachesto a same constant value as T increase. The reason is thatwith the UAV moves to Willie, the artificial noise effectwill be better and better. Hence, the required jamming powerwill be reduced, while the transmission power of the sourcewill increase due to the good interference effect. When theoptimal position is reached, the jamming power transmitted byUAV is minimal, while the power transmitted by the sourceis maximal. Thus, the Alice’ s transmit power Pa[n] is themaximum at the optimal position. Therefore, the UAV startsto fly to the final point at the maximum speed away from theWillie, and the Pa[n] is opposite to the first stage.

Fig. 9 shows the obtained Alice’s transmit power and flightspeed for different values of the flight period T . In Fig.9(a), we can see that the UAV always flies at its maximumspeed, which is due to the fact that the flight period T is notsufficiently large for the UAV to arrive at Bob’s location. Thatis the reason why the Alice’s transmit power as shown in Fig.9(b) first increases and then decreases. In Fig. 7(c), we can seethat the UAV’s speed at the balancing location is zero, whichmeans that the UAV stays static at this location for a certaintime period. This exactly matches with the Alice’s transmitpower as shown in Fig. 9(c), i.e., the Alice has the highesttransmit power at the best position.

Fig. 10 gives the average achieve covert rate versus T underthe effect of varying covertness requirements of ε. We observethat the covert rate increases as T increases, as expected.This observation is consistent with our understanding of thepresented system, since Pa[n] increases with ε, while anincreasing Pa[n] also increases the received SINR at Bob.Moreover, we observe that the proposed JTP algorithm alwaysachieves the highest covert rate. Such results confirm the

10

validity and necessity of joint UAV trajectory and powercontrol design for mobile jamming in covert communications.

Fig. 14 shows the average covert rate of Bob achieved by allabove methods versus the Eb/N0. From the results, we observethat under the JTP scheme, the average covert rate obtainedby geometrical method is better than the BCD method. Wealso observe that for different method the average covert rateincreases as Eb/N0 increases. Moreover, from the results wenotice furthermore that, the proposed JTP scheme is betterthan LTP scheme in both Apollonius of Perga method and theBCD method.

VI. CONCLUSIONS

UAV as a friendly jammer to enhance the performance ofcover communications has been studied in this paper. A jointoptimization problem of the UAV’s trajectory and transmitpower and Alice’s transmit power has been formulated tomaximize the average covert transmission rate subject to thetransmission outage and covertness constraints. To solve thisnon-convex optimization problem, the BCD and geometricaltechniques algorithms are proposed. The first algorithm is suit-able for general communication scenarios, while the secondalgorithm is advantageous for high SINR. It is shown bysimulation that the effectiveness of the proposed method.

APPENDIX APROOF OF THEOREM 1

Since the

PFA[n] = P (Pw[n] > ϑ|H0) = P((σ2

w + γ[n])X 2

2m

m> ϑ|H0

),

PMD[n] = P (Pw[n] < ϑ|H1) = P((σ2

w + γ[n])X 2

2m

m< ϑ|H1

),

(42)according to [15], and from the Strong Law of Large Numbers,we have

Pr

limm→∞

X 22m

m= 1

= 1. (43)

Thus,

PFA[n] = P (Pw[n] > ϑ|H0) = P((σ2w + γ[n]) > ϑ|H0

),

PMD[n] = P (Pw[n] < ϑ|H1) = P((σ2w + γ[n]) < ϑ|H1

).

(44)By (9), we have

γ[n] =

Pu[n]|hu,w[n]|2, H0,

Pa[n]d−αa,wβ0|ha,w[n]|2 + Pu[n]|hu,w[n]|2, H1.(45)

To solve (44), the PDF of γ[n] is needed. To this end, we firstderive the PDF of the random variable γ.

From our hypothesis, Pu[n] is an i.i.d. sequence ofuniform random variables on [0, Pu[n]]. Thus, ξ =Pu[n]|hu,w[n]|2 follows a uniform distribution over the in-terval [0, Pu[n]|hu,w[n]|2], and ϕ = Pa[n]β0|ha,w[n]|2d−αa,wfollows an exponential distribution with the parameter λ =dαa,w

Pa[n]β0. Then, γ = ξ + ϕ. Hence, the PDF of ξ and η is

respectively given by

fξ[n](x) =

1

Pa[n]φe− 1Pa[n]φ

x, if x > 0,

0, otherwise(46)

and

fϕ[n](y) =

1Γ , if 0 ≤ x ≤ Γ,

0, otherwise,(47)

where Γ , Pu[n]|hu,w[n]|2, following (46) and (47), theCumulative Distribution Function (CDF) of γ is

Fγ(z) =

∫∫x+y≤z

fξ,ϕ(x, y)dxdy. (48)

According to our hypothesis, ξ and ϕ are independent of eachother. As such, the CDF of γ can be redefined as

Fγ(z) =

∫ ∞−∞

∫ z−x

−∞fξ(x)fϕ(y)dxdy

=

∫∫x+y≤z

1

Pa[n]φe−

1Pa[n]φ

x 1

Γdxdy.

(49)

For 0 < z < Γ, we have

Fγ(z) =1

Pa[n]φcPu[n]

∫ z

0

e−1

Pa[n]φxdx

∫ z−x

0

dy

=1

Pa[n]φcPu[n]

∫ z

0

e−1

Pa[n]φx(z − x)dx

=1

Γ

((e−

zPa[n]φ − 1

)(Pa[n]φ− z) + e−

zPa[n]φ

).

(50)

Derivation of the above formula, then, for 0 < z < Γ, wehave the PDF of γ is

fγ[n](z) =1− e−

zPa[n]φ

Γ. (51)

For z ≥ Γ, we have

Fγ(z) =1

Pa[n]φ

∫ z−cPu[n]

0

e−1

Pa[n]φxdx

∫ cPu[n]

0

1

Γdy

+1

Pa[n]φ

∫ z

z−cPu[n]

e−1

Pa[n]φxdx

∫ z−x

0

1

Γdy

=

(1 +

Pa[n]φ

Γe−

zPa[n]φ

(1− e

ΓPa[n]φ

)).

(52)

Similarly, the PDF of γ for z ≥ cPu[n] can be expressed as

fγ[n](z) =e−

(z−Γ)Pa[n]φ − e−

zPa[n]φ

Γ. (53)

Thus, the PDF of the random variable γ is

fγ[n](z) =

0, if z ≤ 0,

1−e− zPa[n]φ

Γ , if 0 < z ≤ Γ,

e− (z−Γ)Pa[n]φ−e

− zPa[n]φ

Γ , if z > Γ.

(54)

Combining (44), (45) and (54), we can obtain

PFA[n] =

1, if ϑ− σ2

w ≤ 0,

1− ϑ−σ2w

Γ , if 0 < ϑ− σ2w ≤ Γ,

0, if ϑ− σ2w > Γ

(55)

11

and

PMD[n] =

0, if ϑ− σ2

w ≤ 0,

Λ, if 0 < ϑ− σ2w ≤ Γ,

Ω, if ϑ− σ2w > Γ,

(56)

where

Λ =

∫ ϑ−σ2w

0

1− e−z

Pa[n]φ

Γdz

=ϑ− σ2

w

Γ+Pa[n]φ

Γ

(e−

(ϑ−σ2w)

Pa[n]φ − 1

)(57)

and

Ω =

∫ Γ

0

1− e−zdαa,wPa[n]β0

Γdz +

∫ ϑ−σ2w

0

1− e−zdαa,wPa[n]β0

Γdz

+

∫ ϑ−σ2w

Γ

e−(z−Γ)Pa[n]φ − e−

zPa[n]φ

Γdz

= 1−Pa[n]φ

Γ

(e−

(ϑ−σ2w−Γ)

Pa[n]φ −e−(ϑ−σ2

w)

Pa[n]φ

). (58)

This completes the proof of theorem 1.

APPENDIX BPROOF OF THEOREM 2

Proof 1: Willie’s error probability is defined as PFA+PMD,which can be obtained by substituting into (55) and (56) as

PMD[n] +PFA[n] =

1, if ϑ− σ2

w ≤ 0,

1− ϑ−σ2w

Γ + Λ, if 0 < ϑ− σ2w ≤ Γ,

Ω, if ϑ− σ2w > Γ.

(59)For obtaining optimal ϑ∗, we need to calculate∂(PMD[n]+PFA[n])

∂ϑ . Hence, when 0 < ϑ − σ2w ≤ Γ, we

have∂(PMD[n] + PFA[n])

∂ϑ= − 1

Γe−

(ϑ−σ2w−Γ)

Pa[n]φ (60)

is negative. And when ϑ− σ2w > Γ, we have

∂(PMD[n] + PFA[n])

∂ϑ=e−

(ϑ−σ2w)

Pa[n]φ

Γ

(e

ΓPa[n]φ − 1

)(61)

is positive. Thus, the optimal detection threshold is given byϑ∗ = Γ + σ2

w. Substituting ϑ∗ into (59), we can obtain ξ∗.

APPENDIX CPROOF OF qE

E

'E

,Fuq

wq

,0uq

Fig. 12.

Proof 2: We assume that the number of time slots requiredfor the UAV to fly from the initial point to point E is x. to flyto point E is x, thus, the remaining time slots are N − x, asshown by the red circle in Figure. 12. According to the Lawof cosines, we have

(xVmaxδt)2 + ||qu,F − qu,0||2−

2xVmaxδt||qu,F − qu,0|| cos(θ1 − θ2) = (N − x)2(Vmaxδt)2.

(62)After calculation, we have

x =N2(Vmaxδt)

2 − ||qu,F − qu,0||2

2Vmaxδt(NVmaxδt − ||qu,F − qu,0|| cos(θ1 − θ2)). (63)

Since lVmaxδt

< N , we have

x =N2(Vmaxδt)

2 − ||qu,F − qu,0||2

2Vmaxδt(NVmaxδt − ||qu,F − qu,0|| cos(θ1 − θ2)). (64)

Therefore, point E must exist and the corresponding UAVflight trajectory is shown in Figure. 3(c).

In addition, we set

x =

⌊N2(Vmaxδt)

2 − ||qu,F − qu,0||2

2Vmaxδt(NVmaxδt − ||qu,F − qu,0|| cos(θ1 − θ2))

⌋.

(65)This because if we set

x =

⌈N2(Vmaxδt)

2 − ||qu,F − qu,0||2

2Vmaxδt(NVmaxδt − ||qu,F − qu,0|| cos(θ1 − θ2))

⌉,

(66)there is another point E

′instead of E, which becomes the

turning point of UAV flight trajectory. Since the sum of thetwo sides of the triangle is greater than the third side, we have

dE,F < dE′ ,E + dE′ ,F , (67)

where di,j is the distance from i to j with i, j ∈ E,E′ , F.Futher, we can obtain

dE′ ,E + dE′ ,FVmaxδt

> N − x, (68)

this contradicts that there are only N−x remaining time slots.Thus, x =

⌊N2(Vmaxδt)

2−||qu,F−qu,0||22Vmaxδt(NVmaxδt−||qu,F−qu,0|| cos(θ1−θ2))

⌋.

Let N3 = x, then the coordinates of point E is qE =N3Vmaxδtθ1.

APPENDIX DPROOF OF G (Pa [n] ,qu [n])

Proof 3: Let f(x) = x(

1− e− 1x

), where x > 0. We first

determine the first-order derivative of f(x) with respect to xis given by

f ′(x) = 1− e− 1x (1 +

1

x). (69)

Since the x ≥ 0 and x = 0 is the singularity of f ′(x), wecan not determine at x = 0 the sign of f ′(x). Therefore, thesecond-order derivative of f(x) is given by

f ′′(x) = − 1

x3e−

1x < 0. (70)

12

Thus, the first derivative of f(x) is a monotonically decreasingfunction, and because of

limx→+∞

f ′(x) = limx→+∞

(1− e− 1x (1 +

1

x))

= limx→+∞

1− limx→+∞

e−1x (1 +

1

x)

= 1− limx→+∞

1

x3e1x

= 1.

(71)

So far, we have obtain that the f(x) is an increasing function.We see that the term

Pa[n](‖qu[n]− qw‖2 +H2)

Pu[n]dαa,we−

Pu[n]dαa,w

Pa[n](‖qu[n]−qw‖2+H2) (72)

on the LHS of (20) is the same form as f(x).Thus, G (Pa [n] ,qu [n]) is an increasing function for

Pa [n] ≥ 0.

APPENDIX EPROOF OF THEOREM 3

Proof 4: Let q[n] = (x[n], y[n], z[n]) represent the positionof the UAV in the n-th slot, when z[n] = H is the altitudeat which the UAV flies, and we assume that the coordinatesof Bob (B) and Willie (W) are qb = (xb, yb, 0) and qw =(xw, yw, 0), respectively.

Thus, when k2 = ||q[n]−qb||2+H2

||q[n]−qw||2+H2 , (k 6= 1), we have

(x[n]− xb)2 + (y[n]− yb)2 + (z[n])2 =

k2((x[n]− xw)2 + (y[n]− yw)2 + (z[n])2)(73)

After a simple mathematical simplification, (73) can be ex-pressed as(

x[n] + x′)2

+(y[n] + y

′)2

+ (z[n])2 =

(c

k

k2 − 1

)2

(74)where x

′= k2xw−xb

1−k2 , y′

= k2yw−xb1−k2 and c = (xb − xw)2 +

(yb−yw)2. We note that the (74) is the equation for a standardsphere which center and radius is (−k

2xw−xb1−k2 ,−k

2yw−xb1−k2 , 0)

and ckk2−1 , respectively.

When k = 1, we have

(x[n]− xb)2 + (y[n]− yb)2 + (z[n])2 =

((x[n]− xw)2 + (y[n]− yw)2 + (z[n])2)(75)

After a simple mathematical simplification, (75) can be ex-pressed as

x[n](xw − xb) + y[n](yw − yb) = 0. (76)

This completes the proof of Theorem 3.

APPENDIX FPROOF OF THEOREM 4

Proof 5:We first prove that the tangent point between the UAV and

the sphere is the best position for the next slot. We note that

H

q[n-1]

R B

W

r

UAV

H

C

XY

Z

'C

Fig. 13. Trajectory position of UAV in each time slot.

the radius of the sphere is R = ckk2−1 , and we observe that the

first derivative of R with respect to k is given by

(R(k))′ =

(ck

k2 − 1

)′= −c(k

2 + 1)

(k2 − 1)2< 0. (77)

Thus, R(k) is a monotonically decreasing function of k fork > 1. In order to be more intuitive, we give the top view ofthe radius change when k > 1, as shown in Figure. 14.

B

W

q[n-1]

O

max tV d

k is increasing

q[n]

2 2 2 222 2

22 2 2[n] [n]

1 1 1

b w b ww b w bk x x y yk x x k y y

x yk k k

2 2

b wy y2 2

b wb w

2 2 2 22 2x

2 2 2k x22 2k 2 22 22 22 2

2 22 22 2

2 22 2 xk x22 22 2k x2 22 2x2 22 2k y y2 22 2 xb wb wk xb wb wk xw bxw bw b k y y b wb wb wb wk xb wb wb ww bk y yw bw b[ ][ ]

k xw bw bw b w bw bw b b wb wb w

[ ]w b

[ ]w b

[ ]w bw b

[ ][n]x[n] [n][[n][[[n][[n][[x[n]x[n] [n][[n][[n][2

2kkkkk111111x[n]

111 k 2 22 22 22 22 2k1 k1 k1 k12 2k2 22 212 22 22 2[n]

2 22 2[

2 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2

( )22 2

max( [n] [n 1]) ( [n] [n 1]) tx x y y V d- - + - - =

1k >1k <

Fig. 14. Radius change process when k > 1.As a result, the smaller value of R, the greater value

of k. And from Figure. 14, we know that as R decreases,Apollonius of Sphere will be separated from UAV flightfeasible region. In addition, the sphere center coordinate is(k2xw−xb

(k2−1 , k2yw−xbk2−1 , 0

). Thus, when k → ∞, the sphere

center coordinate is to the point qw. On the contrary, withthe increasing of R, the Apollonius of Sphere is close to thefeasible region of UAV flight. Thus, the tangent point k valueis optimal, and the corresponding tangent point is the positionof the next time slot of UAV flight.

Following the above fact and according to the geometricproperties of tangency and the Pythagorean theorem, at thetangent point, we have√( kc

k2 − 1

)2

−H2 + Vmaxδt

2

=

(x[n− 1]− k2xw − xb

k2 − 1

)2

+

(y[n− 1]− k2yw − yb

k2 − 1

)2

.

(78)By simplifying the above formula, the (78) can be equiva-

lently rewritten as

k4(U2 + S2 + 2US)− 2k2(U + SV + 2(Vmaxδt)2c2+

d(V + S)) + U + V 2 + 2dV = 0(79)

13

where U = d2 − (Vmaxδt)2 +H2, d = x2[n− 1] + y2[n− 1],

S = x2w − 2x[n − 1]xw + y2

w − 2y[n − 1]yw and V = x2b −

2x[n− 1]xb + y2b − 2y[n− 1]yb.

Since the UAV can find the next flight point in the flightarea anyway, the solution of the above equation must exist.Thus, we have

k21 − 4k0k2 > 0 (80)

where k0 = (U + S)2 + 4(Vmaxδt)2H2, k1 = (2SV −

2U2 − 2US + 2UV − 4Vmaxδt)2(c2 + H2)), k2 = (U +

V )2 + 4(Vmaxδt)2H2, U = d2 − (Vmaxδt)

2 + H2, d =x2[n−1]+y2[n−1], S = x2

w−2x[n−1]xw+y2w−2y[n−1]yw

and V = x2b−2x[n−1]xb+y

2b−2y[n−1]yb. After complicated

mathematical calculation, we have

k2 =k1 +

√k2

1 − 4k0k2

2k0. (81)

Since our goal is to find the maximum k value, the optimalposition of the q[n] is satisfied(

(du,b[n])2

(du,w[n])2

)=k1 +

√k2

1 − 4k0k2

2k0. (82)

For k < 1, we note that the radius of the sphere is R =ck

1−k2 , and we observe that the first derivative of R with respectto k is given by

(R(k))′ =

(ck

1− k2

)′=c(k2 + 1)

(k2 − 1)2> 0. (83)

Thus, R(k) is a monotonically increasing function of k fork < 1. In order to be more intuitive, we give the top viewof the radius change when k < 1, as shown in Figure.15. From Figure. 15, we can see that k increases with theincreasing radius R, the Apollonius of Sphere will graduallybe inscribed with the UAV, and finally separated from the UAVflight feasible region. Therefore, when the UAV flight feasibleregion is inscribed with the Apollonius of Sphere, the k valueis optimal, and the corresponding inscribed point is the UAVflight position in the next slot.

B

W

q[n-1]

O

1k <

max tV d

k is decreasing

q[n]

2 2 2 222 2

22 2 2[n] [n]

1 1 1

b w b ww b w bk x x y yk x x k y y

x yk k k

2 22 2

b w b wy y2 22 2

b w b wb w b w

2 2 2 22 22 2 2k 22 2k 2 22 22 22 2

2 22 22 2

2 22 2 xk x22 22 2k x2 2x2 2k y y2 22 2 xk xk xw b wxw b ww b wk y y b w bb w bxb w bb w bk xb w bb w bbk y yw b w bw b w b[ ][ ]

k xw b ww b ww b w bw b w bw b w b b w bb w bb w bb w bb w bb w bb w bb w bb w bb w b

[ ]w b w b

[ ]w b w b

[ ]w b w bw b w b

[ ][n]x[n] [n][[[n][[[n][[[n][[[n][[x[n]x[n] [n][[[n][[[n][[2

2kkkkk111111x[n]

111 k 2 22 22 22 22 22 2k1 k1 k12 2k2 22 212 22 22 22 22 22 22 22 22 22 2[n]

2 22 2[[

2 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2

( )22 2

max( [n] [n 1]) ( [n] [n 1]) tx x y y V d- - + - - =

Fig. 15. Radius change process when k < 1.Similar to k > 1, when k < 1, at the inscribed point thefollowing equation is satisfied

√( kc

1− k2

)2

−H2 − Vmaxδt

2

=

(x[n− 1]− k2xw − xb

k2 − 1

)2

+

(y[n− 1]− k2yw − yb

k2 − 1

)2

.

(84)

After complicated mathematical calculation, we have

k2 =k1 −

√k2

1 − 4k0k2

2k0. (85)

Thus, the optimal position of the q[n] is satisfied((du,b[n])

2

(du,w[n])2

)=k1 −

√k2

1 − 4k0k2

2k0. (86)

This completes the proof of Theorem 4.

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