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Optimal Transmit Power and Flying Location for UAV Covert Wireless Communications

Shihao Yan, Stephen V. Hanly, Iain B. Collings

arXiv:2010.09195v1cs.ITmath.IT

TL;DR

For UAV surveillance, communications carrying urgent results must remain undetected within the surveillance region. The paper jointly optimizes UAV location and transmit power using scenario-specific feasible-region searches. It finds that heuristics can be close to optimal in some cases, while the developed search methods can yield significant gains in others.

  • Problem

    UAVs reporting surveillance results need to communicate with a legitimate receiver without revealing their transmissions to people in the surveillance region.

    Source: §I · p. 1

  • Method

    The paper derives feasible flight regions across constraint scenarios, reducing the search for optimal location and power, with one-dimensional searches sufficient for some two-dimensional cases.

    Source: §I · p. 2

  • Results

    Heuristics are close to optimal in some cases, but the developed search techniques achieve significant gains in others.

    Source: §VI · p. 10

  • Takeaways & Limitations

    The nearest feasible point to Bob is not always the optimal location for covert communications.

    Source: §I · p. 2

Abstract

from arXiv · show

This paper jointly optimizes the flying location and wireless communication transmit power for an unmanned aerial vehicle (UAV) conducting covert operations. This is motivated by application scenarios such as military ground surveillance from airborne platforms, where it is vital for a UAV's signal transmission to be undetectable by those within the surveillance region. Specifically, we maximize the communication quality to a legitimate ground receiver outside the surveillance region, subject to: a covertness constraint, a maximum transmit power constraint, and a physical location constraint determined by the required surveillance quality. We provide an explicit solution to the optimization problem for one of the most practical constraint combinations. For other constraint combinations, we determine feasible regions for flight, that can then be searched to establish the UAV's optimal location. In many cases, the 2-dimensional optimal location is achieved by a 1-dimensional search. We discuss two heuristic approaches to UAV placement, and show that in some cases they are able to achieve close to optimal, but that in other cases significant gains can be achieved by employing our developed solutions.

I. INTRODUCTION

The paper formulates covert UAV communication for urgent surveillance reporting and jointly optimizes flight location and transmit power under covertness, power, and surveillance-quality constraints. It develops scenario-specific search reductions and channel assumptions for that problem.

I. INTRODUCTION

  • The UAV jointly selects its location and transmit power to maximize communication quality at Bob while meeting covertness, power, and surveillance-quality constraints.The angle and distance bounds preserve surveillance view and image resolution, while the minimum distance also limits how close the UAV gets to Willie.Source: §I · pp. 1–2
  • Earlier UAV security work addressed secrecy rather than covert communication, while cited UAV covert-trajectory work adopted infinite-symbol slots and LoS-channel assumptions.The paper motivates covert transmission as hiding the existence of surveillance-result communications from people in the surveillance region.Source: §I · pp. 1–2
  • Six constraint scenarios yield reduced feasible flight regions, and some two-dimensional optimal locations can be found by one-dimensional search.For the common case where Bob is far from Willie, the paper explicitly determines the optimal location and transmit power.Source: §I · p. 2
  • The nearest feasible location to Bob is not always optimal for covert communication, even in the paper’s most common scenario.The common scenario is when Bob is far from Willie relative to the UAV’s distance bounds.Source: §I · p. 2
  • For flight directly above the surveillance area, the paper solves the optimization explicitly and reports that Bob’s effective SNR increases with the UAV’s maximum height.Source: §I · p. 2

A. Channel Model

  • The channel model imposes a maximum blocklength for urgent reporting and approximates probabilistic air-to-ground channels as LoS channels to keep covertness analysis tractable.The authors state that probabilistic-channel Gaussian-mixture likelihoods make the covertness constraints analytically intractable.Source: A. Channel Model · pp. 2–3

B. Binary Hypothesis Testing at Willie

Willie tests whether the UAV transmitted by comparing two signal hypotheses. The paper measures detection performance by minimum total error and uses a KL-divergence bound to express the covertness constraint.

B. Binary Hypothesis Testing at Willie

C. Delay-Constrained Transmission from the UAV to Bob

This section uses effective SNR as the delay-constrained performance metric and characterizes UAV location and transmit-power optimization under covert-operation and surveillance constraints, reducing the location search to a line or arc in several geometric scenarios. In the green region, increasing θw lowers Bob’s objective and raises the covertness divergence, restricting the optimal location to GH; H is optimal if maximum power satisfies covertness there.

C. Delay-Constrained Transmission from the UAV to Bob

  • Because finite-blocklength decoding errors are non-negligible, the paper uses effective SNR as its delay-constrained metric, approximating Bob’s channel with the LoS component as a worst-case lower bound.Effective throughput is stated to increase monotonically with SNR.Source: C. Delay-Constrained Transmission from… · pp. 3–4

III. OPTIMAL LOCATION AND TRANSMIT POWER FOR A 2-DIMENSIONAL UAV

  • The UAV jointly chooses location and transmit power to maximize covert communication performance, subject to surveillance-quality and operational-location constraints.The stated location constraints include staying close enough to Willie with an acceptable viewing angle while avoiding visual observation.Source: §III · p. 4

A. Optimal Location without Constraints

  • For a fixed angle, the optimal distance from Willie is unique and lies between L cos(θw) and L/ cos(θw); the objective increases below and decreases above that interval.The interval expands with θw, from dw = L at θw = 0 to an unbounded range at θw = π/2.Source: A. Optimal Location without Constraints · p. 4
  • For fixed dw, the optimal angle is unique: it lies in [0, arccos(dw/L)] when dw ≤ L and [0, arccos(L/dw)] when dw > L.For dw = L, the optimal location is Bob’s location; for dw > L, it is on Bob’s left side.Source: A. Optimal Location without Constraints · pp. 4–5
  • In Scenario 1, the optimal location lies on arc GN regardless of the maximum transmit power constraint; the joint optimum is characterized by a proposition and can use a one-dimensional angle search.When the covertness constraint is active, transmit power is determined by that constraint, leaving the angle as the one-dimensional search variable.Source: A. Optimal Location without Constraints · pp. 5–6
  • In Scenario 2, the optimal location lies on segment GM; in Scenario 3, a location on segment GH can outperform any feasible point in the specified green region to its right.The paper also states that the remaining five scenarios have feasible regions of significantly reduced size for locating the optimum.Source: A. Optimal Location without Constraints · pp. 6–7
  • On GH, the UAV’s objective and covertness divergence both decrease with dw, and if maximum power meets covertness at H, H is optimal.For fixed dw in the green region, increasing θw increases db, decreases θb, and lowers the objective; it also raises D01 because the LoS probability toward Willie increases.Source: A. Optimal Location without Constraints · p. 7

E. Scenario 4 with d

Decreasing L from Scenario 2 can produce Scenario 4 instead of Scenario 3; the optimal UAV location is then restricted to a specified region or line segment. Further decreases in L lead to Scenario 5, with a different stated feasible location set.

E. Scenario 4 with d

G. Scenario 6 with L cos(θ

Following Scenario 5, continuing to reduce L yields Scenario 6; the optimal UAV location is on Bob’s left side, within closed region GEFH.

G. Scenario 6 with L cos(θ

H. Heuristic Approaches

The paper compares two UAV-placement heuristics with its optimization approach: choosing the feasible point closest to Bob or the point with the largest angle to Bob. Their performance varies by scenario.

H. Heuristic Approaches

I. Summary of Results

The summary identifies a geometric condition under which location optimization becomes one-dimensional and notes Scenario 1’s practical relevance among six scenarios.

I. Summary of Results

  • When the entire feasible region lies upper right of Bob, 2D location optimization reduces to a 1D problem, as in Scenarios 1–3.Source: §I · p. 7
  • Scenario 1 is described as most common, with Bob far from Willie and urgent surveillance outcomes motivating communication to Bob.Source: §I · p. 7
  • The paper considers all scenarios to present a whole picture of UAV location optimization for covert wireless communications.Source: §I · p. 7

IV. OPTIMAL HEIGHT AND TRANSMIT POWER FOR A VERTICAL UAV

The vertical-UAV case restricts flight to height adjustment, as when surveillance equipment must be directly above its target. The section determines the optimal height and transmit power under this restriction.

IV. OPTIMAL HEIGHT AND TRANSMIT POWER FOR A VERTICAL UAV

  • The vertical-UAV scenario applies when surveillance equipment can perform its task only directly above the target.Source: §IV · p. 7
  • In this special case, the UAV adjusts only its flying height.Source: §IV · p. 7
  • The section determines the optimal height h and transmit power P for covert transmission.Source: §IV · p. 7

A. Optimal Height without Any Constraint

For a vertical UAV, the unconstrained height maximizing Bob’s effective SNR is uniquely specified and depends on environmental parameters. With covertness, transmit-power, and surveillance constraints, the paper gives a case-based joint solution for height and power.

A. Optimal Height without Any Constraint

  • The unconstrained optimal height h† is the unique solution of a fixed-point equation and is independent of transmit power and Bob’s noise variance.It is determined by environmental parameters including ξL, a, b, and L.Source: A. Optimal Height without Any Constraint · p. 8

B. Optimal Height with Covertness, Transmit Power, and Surveillance Constraints

  • With covertness, transmit-power, and surveillance constraints, Theorem 2 gives the jointly optimal height and transmit power in Table II.The solution uses thresholds defined through D01(Pm, h) = 2ε² and D01(P, h).Source: B. Optimal Height with Covertness,… · p. 8, Table II · p. 8
  • With a covertness constraint, UAV optimal height h* depends on transmit power and covertness ε, whereas unconstrained optimal height is transmit-power independent; sufficiently large maximum transmit power places the solution in Case 3.Source: B. Optimal Height with Covertness,… · p. 8

V. NUMERICAL RESULTS

The numerical results examine how optimal UAV placement and effective SNR vary across scenarios and constraints. They show that feasible-region search can outperform the heuristics in some cases, while optimal height generally decreases as the covertness parameter increases.

V. NUMERICAL RESULTS

  • As Bob–Willie distance L falls from 10 km to 0.5 km, the optimal location moves from maximum to minimum distance from Willie and remains on the feasible-region boundary in these simulations.In Scenarios 4 and 5 it lies almost directly above Bob; in Scenario 6 it is the nearest feasible location to Bob.Source: §V · pp. 8–9, Figure 3 · p. 9
  • In Scenario 1, Bob’s optimal effective SNR increases with ε, and neither placement heuristic is generally optimal.The better heuristic depends on the upper distance bound to Willie: nearest-to-Bob for 3.5 km, highest-angle-to-Bob for 2.5 km.Source: §V · p. 9, Figure 4 · p. 9
  • The optimal effective SNR decreases as the lower angle constraint θw increases, and neither heuristic is optimal across the full θw range.Source: §V · p. 9, Figure 5 · p. 9
  • The optimal height generally decreases as ε increases, with solutions moving from Case 3 to Case 2 and then Case 4.In Case 2 the covertness constraint is tight; in Cases 3 and 4, D01 < 2ε².Source: §V · p. 9, Figure 6 · p. 10

VI. CONCLUSION

  • The six scenario-specific feasible regions enable efficient search for optimal UAV placement; the two heuristics are close to optimal in some cases, while the developed search yields gains in others.Source: §VI · p. 10

APPENDIX A: PROOF OF LEMMA 1

The proof brackets the distance that optimizes Bob’s effective SNR for a fixed angle, then establishes uniqueness by monotonicity within that interval.

APPENDIX A: PROOF OF LEMMA 1

  • For a fixed angle, the proof studies the derivative of f(dw, θw) with respect to UAV distance dw.Source: Appendix A · p. 10
  • The derivative’s sign outside [L cos(θw), L / cos(θw)] places its zero within that interval.Source: Appendix A · p. 10
  • Within the interval, monotonic decrease of f̄(dw) makes its zero unique, so the SNR-maximizing distance is unique.Source: Appendix A · p. 10

APPENDIX B: PROOF OF LEMMA 2

The proof characterizes the stationary point by a scalar equation, bounds its location, and establishes uniqueness under a stated condition. It also shows that the objective decreases beyond the upper bound.

APPENDIX B: PROOF OF LEMMA 2

  • The stationary condition ∂f/∂θw = 0 is equivalent to solving ḡ(θw) = 0.This equivalence reduces the search for the solution to the scalar equation.Source: Appendix B · p. 11
  • The solution to ḡ(θw) = 0 lies in [0, arccos(dw/L)].The sign of G(θw) requires θw < arccos(dw/L), placing the solution within the stated closed interval.Source: Appendix B · p. 11
  • When L − dw cos(θw) > 0, ḡ(θw) decreases throughout that interval, so its solution there is unique.The proof derives this monotonicity from the behavior of pb(dw, θw), G(θw), and the remaining factors in ḡ(θw).Source: Appendix B · p. 11
  • For arccos(dw/L) ≤ θw ≤ π, f(dw, θw) decreases with θw.Here G(θw) < 0, which yields ∂f(dw, θw)/∂θw < 0.Source: Appendix B · p. 11
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