Capacity Characterization of UAV-Enabled Two-User Broadcast Channel

Qingqing Wu; Jie Xu; Rui Zhang

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Capacity Characterization of UAV-Enabled Two-User Broadcast Channel

Qingqing Wu, Jie Xu, Rui Zhang

arXiv:1801.00443v2Jan 1, 2018cs.ITmath.HOmath.OC

TL;DR

The paper addresses the largely unknown capacity limits of UAV-enabled multiuser communication by characterizing the capacity region of a UAV-enabled broadcast channel. It shows that optimal transmission structures depend on flight duration and speed, including HFH with TDMA for long duration and fixed-location hovering with generally required SC for low speed.

Problem
The fundamental capacity limits of UAV-enabled multiuser communication systems remain largely unknown.
Method
The paper characterizes the capacity region of a UAV-enabled broadcast channel.
Results
For T →∞, HFH with hovering above the two GUs and TDMA is capacity-achieving, while for V →0, the UAV hovers nearer the GU with larger achievable rate and generally requires SC.
Takeaways & Limitations
The capacity-achieving transmission structure changes with UAV flight duration and speed, spanning orthogonal TDMA and generally non-orthogonal SC operation.
Takeaways & Limitations
The analysis assumes equal noise power for the two users and a simplified LoS link, requiring further investigation.

Abstract

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Although prior works have exploited the UAV's mobility to enhance the wireless communication performance under different setups, the fundamental capacity limits of UAV-enabled/aided multiuser communication systems have not yet been characterized. To fill this gap, we consider in this paper a UAV-enabled two-user broadcast channel (BC), where a UAV flying at a constant altitude is deployed to send independent information to two users at different fixed locations on the ground. We aim to characterize the capacity region of this new type of BC over a given UAV flight duration, by jointly optimizing the UAV's trajectory and transmit power/rate allocations over time, subject to the UAV's maximum speed and maximum transmit power constraints. First, to draw essential insights, we consider two special cases with asymptotically large/low UAV flight duration/speed, respectively. For the former case, it is shown that a simple hover-fly-hover (HFH) UAV trajectory with time division multiple access (TDMA) based orthogonal multiuser transmission is capacity-achieving, while in the latter case, the UAV should hover at a fixed location that is nearer to the user with larger achievable rate and in general superposition coding (SC) based non-orthogonal transmission with interference cancellation at the receiver of the nearer user is required. Next, we consider the general case with finite UAV speed and flight duration. We show that the optimal UAV trajectory should follow a general HFH structure, i.e., the UAV successively hovers at a pair of initial and final locations above the line segment of the two users each with a certain amount of time and flies unidirectionally between them at the maximum speed, and SC is generally needed.

I. INTRODUCTION

The paper characterizes capacity limits for a UAV-enabled two-user broadcast channel by jointly designing trajectory and communication. It identifies capacity-achieving hover-fly-hover structures and transmission strategies across asymptotic and finite flight regimes.

The work addresses largely unknown fundamental capacity limits for UAV-enabled multiuser communication systems.
The capacity region is characterized by jointly optimizing UAV trajectory and transmit power/rate allocations over a given flight duration.
For asymptotically large flight duration, hovering above the two ground users with TDMA and an HFH trajectory is capacity-achieving.
For asymptotically low speed, the UAV hovers nearer the user with larger achievable rate, while SC-based transmission with interference cancellation is generally required.
With finite speed and duration, the optimal trajectory generally flies at maximum speed between two hovering locations above the users’ connecting line, with SC generally needed.
Increasing maximum speed or flight duration enlarges the capacity region, especially at low SNR, while mobility becomes less effective as SNR increases.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models a UAV-enabled two-user broadcast channel and formulates capacity-region characterization as joint optimization of trajectory and power allocation under speed and power constraints.

System model: The system has one UAV transmitting independent information to two ground users at fixed locations.The UAV flies at constant altitude H in a two-dimensional coordinate system.
System model: The UAV-to-user channels are modeled using free-space path loss with Doppler compensation and AWGN receivers.The channel is treated as constant within each symbol interval because UAV movement during a symbol period is negligible relative to altitude.
Capacity-region formulation: The capacity region contains all achievable average rate-pairs over flight duration T subject to maximum speed and maximum transmit power constraints.For a given trajectory and power allocation, achievable rates are measured in bits per second per Hertz.
Capacity-region formulation: The objective is to characterize the Pareto boundary by jointly optimizing the UAV trajectory and power allocation.The rate-profile technique is used because the capacity region may be non-convex, unlike weighted-sum optimization that can miss boundary points in non-convex regions.
Optimization challenge: The resulting optimization is highly non-convex because rate constraints are non-concave in trajectory variables and the trajectory is defined over continuous time.These features prevent a standard efficient solution method in general.

A. Capacity Region Properties and HFH Trajectory

The capacity region is symmetric, and an optimal trajectory can remain over the users’ line segment and move unidirectionally; this motivates a hover-fly-hover structure.

Capacity-region properties: The capacity region is symmetric with respect to the line r1 = r2.Reflecting the trajectory and swapping the users’ power allocations maps any achievable rate-pair to its symmetric counterpart.
Capacity-region properties: An optimal UAV trajectory stays above the line segment between the two ground users.Restricting the UAV to x*(t) ∈[-D/2, D/2] can reduce its distances to both users and increase the rate-pair componentwise.
Capacity-region properties: There always exists an optimal trajectory that is unidirectional over time.A non-unidirectional trajectory can be replaced by a feasible unidirectional trajectory with the same objective value.
HFH trajectory: The HFH trajectory lets the UAV hover at initial and final locations and fly between them at maximum speed.The hover durations satisfy tI ≥0, tF ≥0, and tI+tF ≤T, with both hover locations within the users’ line segment.
HFH trajectory: Under HFH, the UAV hovers at at most two locations; when xI = xF, it remains fixed throughout the flight.The paper then analyzes the special cases T →∞ and V →0.

III. CAPACITY CHARACTERIZATION WITH LARGE FLIGHT DURATION

With asymptotically large flight duration, the UAV-enabled two-user BC has a triangular capacity region achieved by an HFH trajectory and TDMA transmission. Mobility can significantly improve capacity over a static UAV, but large flight durations introduce a throughput-delay trade-off.

Without the maximum-speed constraint, the capacity region is an equilateral triangle.
As T →∞, the capacity region equals the unconstrained region for every V > 0.
The capacity-achieving trajectory is unidirectional HFH, hovering above the two GUs at xI = −D/2 and xF = D/2, with TDMA transmission.
For a static UAV at unequal-channel locations, SC achieves boundary points that TDMA cannot, except at the two extreme points.
High mobility yields significant capacity gains over a static UAV but can require one GU to wait about T/2 for transmission.

IV. CAPACITY CHARACTERIZATION WITH LIMITED UAV MOBILITY

With limited UAV mobility, the UAV effectively hovers at a fixed location selected for the users’ rate requirements, while finite-duration mobility can enlarge the capacity region through location time-sharing. In the general finite-speed, finite-duration setting, maximum-speed HFH motion connects superior hovering locations.

When V →0, the UAV’s horizontal movement has negligible impact and it should hover at one fixed location throughout T.
For α2 ≥ α1, an optimal hovering location satisfies 0 ≤ x∗ ≤ D/2, placing the UAV closer to GU 2.
At the selected fixed location, GU 2 decodes GU 1’s signal first and then cancels its interference before decoding its own signal.
As V →0, the capacity region is non-convex and larger than the fixed-location AWGN BC region at every location.
Superior locations can dominate inferior ones componentwise, but finite-speed travel may require crossing inferior locations between distant hovering points.
The general finite-speed, finite-duration optimum therefore uses maximum-speed HFH motion, and increasing speed or duration can significantly improve capacity.

V. CAPACITY CHARACTERIZATION FOR FINITE UAV SPEED AND FLIGHT DURATION

For finite UAV speed and flight duration, the capacity-achieving trajectory has a hover-fly-hover structure governed by two endpoint locations and hovering time, with superposition coding generally required. The resulting capacity boundary can be non-convex and its benefit from movement depends on speed, duration, and rate regime.

Optimal trajectory: Endpoint hovering locations have rate superiority over intermediate locations on the line segment between them.A boundary rate-pair can be componentwise no smaller than any rate-pair from a fixed intermediate location.
Optimal trajectory: The optimal trajectory is hover-fly-hover, with initial and final hovering locations joined by unidirectional maximum-speed flight.The trajectory is determined by xI, xF, and tI, with tF = T − tI − (xF−xI)/V.
Transmission design: Superposition coding is generally required to achieve the capacity boundary in the finite-speed, finite-duration case.For a fixed HFH trajectory, power allocation is optimized first, followed by a search over xI, xF, and tI.
Capacity-region geometry: The finite-parameter capacity region is generally non-convex, with a concave-convex-concave boundary.For V = 0, the boundary is convex over r2 ∈ [1, 3] bps/Hz, whereas it is concave over r2 ∈ [3.5, 6] bps/Hz in the cited regime.
Mobility effects: Increasing speed or duration enlarges the capacity region when movement lets the UAV reach superior locations, but movement can be unhelpful when it is locally constrained.With V = 30 m/s and T = 20 s, one cited boundary segment remains unchanged from V = 0; increasing T to 60 s shifts part of it upward-right.

C. High SNR Case

In the high-SNR case, the optimal low-mobility placement is directly above the user requiring the larger rate, with superposition coding needed. The resulting capacity region is independent of UAV speed and flight duration under the stated approximation.

High-SNR characterization: The high-SNR capacity region is independent of UAV flight duration T and maximum speed V.The result follows from the stated high-SNR approximation and the simplified endpoint-hovering trajectory.
High-SNR characterization: At V = 0, the UAV hovers above the user requiring the larger rate for all rate-pairs satisfying r2 ≥ r1, and superposition coding is needed.For α2 ≥ α1, the simplified trajectory is x*(t) = D/2 for all t; for α2 < α1, it is x*(t) = −D/2.
High-SNR characterization: Hovering midway between the users causes significant capacity loss in the high-SNR regime, even when maximizing equal rate.At x = 0, the two ground users have equal channel gains.
High-SNR characterization: In this regime, increasing UAV speed or flight duration provides very limited capacity improvement because the far user’s SNR is already very small.The cited comparison concerns the gap between C(0, T, P̄) and C(V, ∞, P̄).

B. Numerical Results

Numerical results compare the SC-based capacity region with TDMA-based achievable regions across UAV mobility and rate regimes. SC can provide substantial gains at low mobility, while TDMA approaches capacity as speed or duration increases.

TDMA mobility: TDMA’s achievable-region boundary is always convex, so increasing UAV speed or duration is always beneficial for enlarging it.With TDMA, greater mobility lets the UAV fly closer to the scheduled user.
SC versus TDMA: SC generally yields a larger rate region than TDMA under the same V, T, and P̄.At V = 0 and r1 = 1 bps/Hz, GU 2’s achievable rate improves by about 80% with SC.
SC versus TDMA: The SC gain comes from both superposition coding and the associated UAV-location optimization.The comparison is between C(V, T, P̄) and CTD(V, T, P̄).
SC versus TDMA: As UAV speed or flight duration increases, TDMA becomes closer to optimal and coincides with capacity as T →∞.The two boundaries touch at more points as V and/or T increases.
SC versus TDMA: For V = 30 m/s and T = 60 s, the TDMA and capacity boundaries overlap for r1 ∈ [1, 4] bps/Hz.In this regime, the optimal TDMA trajectory is also capacity-achieving when the users’ rates are relatively comparable.
TDMA mobility: For rate-pairs where one user’s rate is small but nonzero, TDMA power allocation can avoid moving toward that user, making greedy scheduled-user flight strictly suboptimal versus SC.This distinction holds even when VT > D.

APPENDIX A: PROOF OF LEMMA 4

The appendix proves the fixed-location high-mobility solution using weighted optimization and the broadcast-channel polymatroid structure. Depending on the rate weights, all power is assigned to one user at the corresponding endpoint, while equal weights require time-sharing.

Weighted optimization: The weighted optimization is decoupled over time after relaxing the maximum-speed constraint and invoking the broadcast-channel polymatroid structure.For μ1 > μ2, the resulting per-time problem optimizes location and both users’ powers.
Unequal weights: When μ2 > μ1, the symmetric optimum assigns p2*(t) = P̄ and p1*(t) = 0 at x*(t) = D/2, yielding r1 = 0.This is the counterpart of the μ1 > μ2 solution.
Equal weights: When μ1 = μ2, the optimum is non-unique and time-sharing between the two endpoint solutions achieves different rate pairs.The appendix identifies the two endpoint trajectory and power-allocation solutions as optimal in this case.
Interference cancellation: For a rate pair with h1(x*) > h2(x*), GU 1 can cancel interference before decoding its own signal.The corresponding rates are expressed using the users’ transmit powers and channel gains at x*.

APPENDIX C: PROOF OF PROPOSITION 3

The proof constructs a modified feasible trajectory and power allocation that improves the assumed optimum whenever an intermediate rate pair violates the relevant convex-hull condition.

Construction: The argument begins by assuming an optimal rate pair and constructing an alternative trajectory and power allocation.The flight duration is partitioned into initial, middle, and final portions, with sufficiently short endpoint intervals approximating locations xI and xF.
Rate-pair comparison: The endpoint rate pairs must lie at common-tangent points of Cf(xI) and Cf(xF), otherwise a larger objective value is achievable through rate reallocation.This condition supports expressing the average rate pair as a weighted combination of endpoint and intermediate rate pairs.
Construction: The constructed solution adds hovering at xI, xF, and xA for selected time proportions while retaining the original solution for the remaining duration.The proportions satisfy ˆβI + ˆβF + ˆβA = 1.
Rate-pair comparison: When the intermediate rate pair lies outside the triangle region CIF, suitable time proportions can make the constructed average rate pair componentwise larger than the assumed optimal pair.The proof treats three cases according to rA 1 and concludes by selecting parameters that ensure inequality (59).
Geometric argument: The proof addresses the difficulty that no explicit relation directly links the intermediate and endpoint rate pairs by introducing an equivalent interpretation based on the capacity region CIF.The triangle characterization enables the required convex-combination comparison.
Conclusion: The resulting feasible construction achieves a larger objective value, contradicting optimality and completing Proposition 3.The contradiction follows after establishing the existence of suitable time proportions.

APPENDIX D: PROOF OF THEOREM 2

The proof establishes the high-SNR optimal solution by upper-bounding the objective and showing that the bound is attained by hovering at a fixed location.

Upper bound: Under the high-SNR assumption, the proof first derives an upper bound on the optimal objective value of problem (P1).The bound follows from the rate constraints and the total power constraint.
Achievability: The upper bound is tight because a feasible fixed-location solution reduces problem (P1) to problem (P3) and attains the bound.At the resulting operating point, GU 2 performs interference cancellation before decoding its own signal when r2 ≥ r1.
Location optimization: The achievable rate of GU 1 is independent of the UAV location, whereas GU 2's rate increases monotonically with x over [−D/2, D/2].Consequently, the sum-rate objective increases with x.
Location optimization: The optimal UAV location is x∗ = D/2, where the maximum objective value is achieved under the high-SNR assumption.The corresponding fixed trajectory is x∗(t) = D/2 for all t ∈ T.
Achievability: The resulting high-SNR capacity region is obtained from the fixed-location solution.The proof concludes after establishing the optimal objective and corresponding region.

APPENDIX F: PROOF OF LEMMA 5

The proof of Lemma 5 characterizes the relation between capacity regions at two UAV locations using rate-function intersections and a common tangent.

Rate-function analysis: The proof analyzes two UAV locations xB and xC and derives the corresponding achievable-rate functions under successive interference cancellation.Depending on channel-gain ordering, either GU 2 or GU 1 decodes the other user's signal first.
Intersection property: The rate functions rB 2(r1) and rC 2(r1) have one unique intersection, denoted by (¯rBC 1, ¯rBC 2).Uniqueness is established by considering the possible orderings of xB and xC and the associated channel gains.
Boundary ordering: The proof uses monotonicity and the unique intersection to establish the ordering of boundary rates before and after the intersection point.For rates below the intersection, one location yields the larger second-user rate; above it, the ordering reverses.
Common-tangent geometry: The upper-right common tangent of the convex hull of Cf(xB) and Cf(xC) touches the two capacity-region boundaries at designated rate pairs.The tangent is used to compare the regions and identify the relevant boundary points.
Convex-hull characterization: The geometric argument handles the special case xB = 0 separately because Cf(xB) becomes an equilateral triangle rather than a strictly convex set.For xB ≠ 0, strict convexity rules out the proposed common-tangent configuration.
Convex-hull characterization: If an intermediate capacity region lies inside CIF but outside the convex hull of the endpoint regions, a rate pair from that region contradicts the endpoint inclusion assumption.The contradiction is obtained by constructing a rate pair that belongs to an endpoint capacity region but not to CIF.

1. Thus, the rate-pairs that lie on and below the line

The proof continues by comparing an intermediate rate pair with endpoint capacity regions and deriving contradictions from the assumed inclusion in CIF.

Case analysis: For a rate pair in CIF with r1 between the relevant endpoint rates, the intermediate pair must fall into one of two positional cases.The cases are 0 < ¯rA 1 < ¯rF 1 and ¯rI 1 < ¯rA 1.
Case analysis: When the intermediate pair is outside Cf(xF), its second component exceeds the corresponding boundary value at xF.This produces a rate pair that dominates the boundary pair at the same first-user rate.
Contradiction: The resulting intersection-point ordering places the endpoint pair below the constructed intermediate-rate pair.The proof uses this ordering to continue the contradiction argument.
Contradiction: The symmetric case is handled similarly and also contradicts the assumption that the endpoint capacity region is contained in CIF.Thus, the considered rate pairs cannot violate the stated convex-hull relation.