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Joint Optimization of a UAV's Trajectory and Transmit Power for Covert Communications

Xiaobo Zhou, Shihao Yan, Jinsong Hu, Jiande Sun, Jun Li, Feng Shu

arXiv:1812.00583v2cs.IT

TL;DR

The paper addresses covert UAV communication when Bob and Willie have location uncertainties and Willie monitors the UAV’s transmission. It jointly optimizes the UAV’s trajectory and transmit power using approximations and SCA, and reports significantly better covert communication performance than a benchmark scheme.

  • Problem

    Covert UAV communication must hide transmission or the UAV itself from Willie while delivering information to Bob under location uncertainty.

  • Method

    The paper jointly optimizes trajectory and transmit power, conservatively approximates the outage constraint, convexifies the problem with first-order restrictive approximation, and solves it iteratively using SCA.

  • Results

    The joint trajectory-and-power optimization scheme achieves significantly better covert communication performance than a benchmark scheme.

  • Takeaways & Limitations

    Jointly adapting the UAV’s trajectory and transmit power is presented as an effective design framework for improving covert communication performance.

Abstract

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This work considers covert communications in the context of unmanned aerial vehicle (UAV) networks, aiming to hide a UAV for transmitting critical information out of a scenario that is monitored and where communication is not allowed. Specifically, the UAV as a transmitter intends to transmit information to a legitimate receiver (Bob) covertly in order to avoid being detected by a warden (Willie) with location uncertainties at Bob and/or Willie. In order to enhance the considered covert communication performance, we prefer to jointly optimize the UAV's trajectory and transmit power in terms of maximizing the average covert transmission rate from the UAV to Bob subject to transmission outage constraint and covertness constraint. The formulated optimization problem is difficult to tackle directly due to the intractable constraints. As such, we first employ conservative approximation to transform a constraint into a deterministic form and then apply the first-order restrictive approximation to transform the optimization problem into a convex form. By applying the successive convex approximation (SCA) technique, an efficient iterative algorithm is developed to solve the optimization problem. Our examination shows that the developed joint trajectory and transmit power optimization scheme achieves significantly better covert communication performance as compared to a benchmark scheme.

I. INTRODUCTION

The paper extends covert communications from static settings to UAV networks, where a mobile UAV communicates with Bob while avoiding detection by Willie. It jointly optimizes trajectory and transmit power under outage, covertness, mobility, and power constraints.

  • Motivation: Prior UAV-security work focused on preventing interception, while hiding the UAV’s transmission or location remained overlooked.The paper motivates covert communication as important because detected transmission behavior can expose a UAV’s location and increase vulnerability to attack.
  • System and novelty: The proposed scenario uses a UAV as Alice, Bob and Willie on the ground, and extends covert communications from static to dynamic settings.The UAV’s mobility can extend covert-transmission range, while static covert communication is treated as a special case.
  • System and novelty: Both Bob’s and Willie’s locations are uncertain, producing random UAV-to-node distances approximated by Gaussian distributions.The approximation supports derivation of a lower bound on Willie’s average minimum total error rate over location uncertainty.
  • Optimization approach: The optimization maximizes average covert transmission rate subject to Bob’s transmission-outage, Willie’s covertness, UAV mobility, and transmit-power constraints.Because the constraints are non-convex, conservative approximation yields a deterministic outage constraint and first-order restrictive approximation yields a convex formulation.
  • Optimization approach: An SCA algorithm iteratively solves the convex problem and generates feasible UAV trajectories and transmit powers, outperforming a benchmark scheme.The resulting trajectory may hover around Bob and move closer to Willie as the covertness constraint becomes stricter.

C. Binary Hypothesis Testing at Willie

Willie performs binary hypothesis testing to decide whether the UAV transmitted to Bob, under assumptions of known UAV location, uncertain noise power, and location uncertainty. The paper derives Willie’s minimum total error rate and imposes a lower bound on its average as the covertness requirement.

  • Hypothesis testing: Willie is assumed to know the UAV’s location perfectly, representing the best case for detection and the worst case for covert communication.This assumption is justified because Willie can directly observe the UAV’s flight trajectory.
  • Hypothesis testing: Willie decides between H0, no UAV transmission, and H1, transmission to Bob, using received signals containing UAV transmission and AWGN.The UAV’s transmit power is fixed within each time slot but may vary across slots.
  • Uncertainty modeling: The model includes noise uncertainty because a transceiver may not know its AWGN power exactly in a dynamic environment.The noise power is modeled through a distribution parameterized by the size of noise uncertainty.
  • Detection performance: The paper defines false-alarm and miss-detection rates, combines them into Willie’s total error rate, and derives Willie’s optimal detection threshold minimizing that rate.The detection rule is evaluated at each time slot under the binary hypotheses.
  • Detection performance: UAV design enforces an average minimum total error rate of at least 1 − ρw at Willie to guarantee the required covertness.The average is taken over Willie’s prior location-estimation error distribution, with ρw determining the required covertness.

III. DETECTION PERFORMANCE

The section derives Willie’s optimal detection threshold and minimum total error rate, then averages this performance over Willie’s location uncertainty. Because the resulting expression is intractable, the paper approximates the relevant distribution and derives a lower bound used for covertness.

  • Detection Performance at Willie: The optimal detection threshold minimizes Willie’s total error rate, from which the corresponding minimum total error rate is derived.The section first establishes Willie’s false-alarm and miss-detection rates before obtaining the optimal threshold.
  • Detection Performance at Willie: Willie can detect covert communication without error when the received-signal condition involving E_w[n] and noise variance holds.The paper therefore focuses on the complementary case when this condition does not hold.
  • Detection Performance from UAV’s Point of View: The UAV evaluates Willie’s detection performance through the minimum total error rate averaged over Willie’s uncertain location.This perspective accounts for the difficulty of obtaining Willie’s exact location information.
  • Detection Performance from UAV’s Point of View: X[n] follows a noncentral chi-square distribution with two degrees of freedom, with expectation λ[n]+2 and variance 4λ[n]+4.Its noncentral parameter λ[n] is determined by the distance from the UAV to Willie’s estimated location.
  • Detection Performance from UAV’s Point of View: The paper approximates X[n] by a Gaussian distribution and uses a convex-function lemma to derive a lower bound on the average minimum total error rate.The approximation is stated as N(2 + λ[n], 4 + 4λ[n]) when λ[n] is usually large in the considered scenario.
  • Detection Performance from UAV’s Point of View: The derived lower bound replaces the original average minimum-error expression as the covertness constraint in subsequent optimization problems.The substituted constraint is ˇξ*[n] ≥ 1 − ρ_w.

IV. JOINT OPTIMIZATION OF UAV’S TRAJECTORY AND TRANSMIT POWER FOR COVERT COMMUNICATIONS

The paper designs the UAV’s trajectory and transmit power to maximize transmission rate to Bob under transmission-outage and covertness constraints.

  • The optimization jointly designs the UAV’s trajectory and transmit power to maximize transmission rate to Bob subject to outage and covertness constraints.

A. Optimization Problem Formulation

The optimization problem maximizes average transmission rate while enforcing Bob’s outage requirement, Willie’s covertness requirement, and UAV mobility and power constraints. Its outage and covertness constraints are non-convex, making the problem difficult to solve directly.

  • The formulation maximizes average transmission rate over time slots while constraining transmission outage probability at Bob and covertness at Willie.The outage constraint enforces a required QoS level between the UAV and Bob, while the covertness constraint protects the transmission from detection.
  • The problem also includes UAV mobility and transmit-power constraints alongside the outage and covertness requirements.
  • The outage and covertness constraints are non-convex, whereas the objective, mobility constraint, and power constraint have convex or linear forms.

B. On the Constraints of the Optimization Problem (P1)

The paper addresses the two difficult constraints separately: Bernstein-type inequalities produce a deterministic outage form, while restrictive first-order approximations convert the covertness formulation into convex constraints. SCA then iteratively solves the resulting convex problem.

  • Transmission Outage Probability Constraint (26b): The outage constraint is difficult because its probability expression is intractable, so a Bernstein-type inequality transforms it into deterministic constraints.
  • Transmission Outage Probability Constraint (26b): The deterministic outage reformulation uses slack variables and yields convex SOC and linear constraints, but one power-rate term remains non-convex.
  • Covertness Constraint (26c): The covertness constraint is difficult because its complex error-rate expression couples transmit power and UAV position.
  • Covertness Constraint (26c): Restrictive first-order approximations and slack variables convert the covertness subconstraints into convex forms.The transformations use concave-function upper bounds, convex-function lower bounds, and tractable reformulations.
  • After both transformations, the optimization problem is reformulated with auxiliary variables and solved iteratively using the SCA algorithm.The algorithm initializes feasible points, solves the convex problem, updates the feasible points, and checks objective convergence.

C. Overall Algorithm

The algorithm iteratively solves a convex restrictive approximation that remains feasible for the original problem, using carefully constructed initial feasible points. The initialization combines a line-segment trajectory with transmit-power adjustments to satisfy the relevant constraints.

  • P1.2 has a linear objective and convex constraint set, so it can be efficiently solved with a convex optimization solver.
  • Each P1.2 solution is feasible for the original P1.1, while successive iterations improve its objective value and approach P1.1.
  • Initial feasible points are critical because they determine P1.2 feasibility and the convergence speed of Algorithm 1.
  • The initialization uses a line-segment trajectory: fly to Bob along the shortest path, hover above Bob, then fly to the final location.
  • When the flight period is insufficient for reaching Bob and returning, the UAV turns at a midway point before flying to the final location.
  • Initial transmit-power values are randomly generated in a very small set, while initial auxiliary points are generated according to (39b).

V. NUMERICAL RESULTS

Numerical results compare joint trajectory-and-power optimization (JTP) with fixed-trajectory transmit-power optimization (STP). The results show how flight period, uncertainty, and covertness requirements shape trajectories, transmit power, and ACTR.

  • Trajectory and speed: For T = 200 s, JTP and STP produce identical trajectories because this is the minimum feasible flight period; at T = 300 s, JTP detours around Willie and hovers near Bob.The JTP trajectory reaches Bob at maximum speed, pauses near Bob, then follows a curved path avoiding Willie before reaching the final location.
  • Transmit power: JTP uses higher transmit power than STP because its trajectory remains farther from Willie while maintaining the same detection-error level.The optimized power exhibits rapid-decreasing, stable, and varying stages corresponding to the UAV’s changing locations.
  • Willie uncertainty and covertness: Stricter covertness constraints and lower Willie-location uncertainty alter the JTP trajectory and transmit power, with the UAV moving farther from Willie as ε2_w decreases.For ρ_w = 0.1, the UAV can choose a trajectory farther from Willie than for ρ_w = 0.05, enabling higher transmit power.
  • ACTR comparison: JTP always achieves a higher maximum ACTR than STP, and maximum ACTR increases with flight period T and Willie’s noise uncertainty d_B.Longer flight periods permit more hovering around Bob, while greater noise uncertainty makes Willie’s decisions harder.
  • Location uncertainty: Maximum ACTR decreases as Bob’s location uncertainty ε2_b increases, while Willie’s location uncertainty also substantially affects achieved maximum ACTR.The two uncertainty sources make the transmission-outage and covertness constraints harder to satisfy, respectively.

VI. CONCLUSION

The paper jointly optimizes the UAV’s trajectory and transmit power to maximize average covert transmission rate under outage and covertness constraints. The resulting scheme outperforms a benchmark, with trajectories hovering near Bob and moving closer to Willie as covertness requirements tighten.

  • The optimization maximizes ACTR from the UAV to Bob subject to transmission outage and covertness constraints.
  • Conservative and first-order restrictive approximations transform the problem into convex forms solvable through an SCA-based algorithm.
  • The joint trajectory-and-power scheme achieves significantly better covert communication performance than a benchmark scheme.
  • The UAV prefers hovering around Bob for a period and moves closer to Willie as covertness becomes stricter.

APPENDIX A PROOF OF LEMMA 1

The appendix derives the optimal detection threshold and minimum total error rate, then characterizes the uncertainty-related random variable using a noncentral chi-square distribution.

  • The total error rate is analyzed as a function of Willie’s detection threshold, with separate monotonic regimes around the noise-power bounds.
  • Continuity and monotonicity imply an optimal detection threshold and a corresponding minimum total error rate.
  • Under the Gaussian error model, location errors in the x and y coordinates follow N(0, ε2_w).
  • The random variable X[n] follows a noncentral chi-square distribution with two degrees of freedom and noncentral parameter λ[n].
  • The expectation and variance of X[n] are λ[n]+2 and 4λ[n]+4, respectively.

APPENDIX C PROOF OF LEMMA 4

The appendix establishes a Gaussian approximation for a noncentral chi-square variable when its degrees of freedom are small and its noncentral parameter is sufficiently large, then applies it to bound the error rate.

  • The noncentral chi-square variable is characterized through a sum of independent normally distributed components.
  • The approximation is extended beyond the Central Limit Theorem setting to small degrees of freedom with a sufficiently large noncentral parameter.
  • For small k and sufficiently large η, the noncentral chi-square distribution is approximated by N(k+η, 2k+4η).
  • Applying the approximation gives X[n] mean 2+λ[n] and variance 4+4λ[n].
  • The resulting bound produces an upper bound on E_X[n][g(X[n])] and a lower bound on the average minimum total error rate.

APPENDIX E PROOF OF LEMMA 6

The appendix uses convexity and concavity properties to construct first-order restrictive approximations for expressions involving transmit power, rate, and location uncertainty.

  • For positive variables, the relevant product-based function is shown to be convex through its Hessian matrix.
  • Because a convex function is lower-bounded by its first-order approximation, the approximation supports the inequality used in the derivation.
  • The expression involving (P_a[n])^-1 and 2R_b[n]−1 is jointly convex, enabling inequality (34).
  • A related product-based function is shown to be jointly concave for positive variables.
  • The first-order approximation of the concave function provides the upper bound used to obtain inequality (41).
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