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CoMP in the Sky: UAV Placement and Movement Optimization for Multi-User Communications

Liang Liu, Shuowen Zhang, Rui Zhang

arXiv:1802.10371v1cs.IT

TL;DR

The paper addresses UAV placement for multi-user communications when ground users move and fixed deployments cannot continuously maintain high-quality service. It proposes CoMP in the sky, derives closed-form rate approximations, and optimizes UAV movement using them. The resulting mobile-UAV locations are weighted averages of current user locations and neighboring-episode UAV positions.

  • Problem

    Moving ground users make fixed RAU/RRH deployments difficult for continuously high-quality service, while existing approaches considered static users or omitted CoMP cooperation.

  • Method

    The paper models UAVs as flying RAUs/RRHs forwarding received signals to a CP, derives LoS random-phase rate bounds, and optimizes placement and movement using iterative approximation.

  • Results

    Optimized mobile-UAV locations are weighted averages of current user locations and the UAV’s previous and/or next locations.

  • Takeaways & Limitations

    CoMP in the sky combines coordinated interference mitigation with UAV mobility for multi-UAV communications serving moving ground users.

Abstract

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Driven by the recent advancement in unmanned aerial vehicle (UAV) technology, this paper proposes a new wireless network architecture of \emph{coordinate multipoint (CoMP) in the sky} to harness both the benefits of interference mitigation via CoMP and high mobility of UAVs. Specifically, we consider uplink communications in a multi-UAV enabled multi-user system, where each UAV forwards its received signals from all ground users to a central processor (CP) for joint decoding. Moreover, we consider the case where the users may move on the ground, thus the UAVs need to adjust their locations in accordance with the user locations over time to maximize the network throughput. Utilizing random matrix theory, we first characterize in closed-form a set of approximated upper and lower bounds of the user's achievable rate in each time episode under a realistic line-of-sight (LoS) channel model with random phase, which are shown very tight both analytically and numerically. UAV placement and movement over different episodes are then optimized based on the derived bounds to maximize the minimum of user average achievable rates over all episodes for both cases of full information (of current and future episodes) and current information on the user's movement. Interestingly, it is shown that the optimized location of each UAV at any particular episode is the weighted average of the ground user locations at the current episode as well as its own location at the previous and/or next episode. Finally, simulation results are provided to validate and compare the performance of the proposed UAV placement and movement designs under different practical application scenarios.

I. INTRODUCTION

The paper motivates CoMP in the sky by combining coordinated interference mitigation with UAV mobility, enabling UAV-mounted RAUs/RRHs to adapt to moving users. This addresses limitations of static deployments and prior UAV work that omits moving users or CoMP cooperation.

  • CoMP and UAV motivation: CoMP coordinates multiple transmission/reception points to mitigate inter-cell interference and exploit distributed antenna systems.
  • CoMP and UAV motivation: Static RAU/RRH deployments struggle to provide continuously high-quality services when users move over time.
  • CoMP and UAV motivation: UAVs can dynamically adjust their locations to provide flexible services and shorten communication distances under favorable LoS channels.
  • Research gap: The proposed CoMP-based system places UAV-mounted RAUs/RRHs in the sky to combine UAV mobility with coordinated communications.
  • Research gap: Prior UAV trajectory studies considered static users or lacked CoMP-based cooperation for mitigating inter-user interference.

B. Main Contributions

The paper introduces CoMP in the sky for uplink multi-UAV communications with moving users, then derives rate approximations and optimizes UAV placement and movement. The resulting mobile-UAV locations combine current user positions with neighboring-episode UAV locations.

  • Network architecture: CoMP in the sky uses UAVs as flying RAUs/RRHs that forward received user signals to a CP for joint decoding.
  • Transmission protocol: The transmission protocol divides time into episodes, treating user locations as static within each episode and variable across episodes.
  • Channel model: A random-phase LoS channel model preserves constant amplitude within an episode while capturing fast phase variation across coherence intervals.
  • Optimization problems: The placement problem maximizes the minimum average user rate across episodes, including dynamic and static UAV scenarios with different movement information.
  • Algorithms and results: Random matrix theory yields tight closed-form rate bounds, while successive convex approximation produces locally optimal solutions satisfying KKT conditions.
  • Algorithms and results: At each episode, an optimized mobile-UAV location is a weighted average of current user locations and the UAV’s previous and/or next location.

C. Organization

The paper is organized around the UAV-enabled CoMP system model, placement and movement optimization, rate bounds, algorithms, simulations, and conclusions. The system assumes multiple single-antenna UAVs connected to a ground CP and operating in episodic time.

  • Paper organization: The paper presents the system model, optimization formulations, closed-form rate bounds, solution algorithms, simulations, and conclusion in Sections II–VII.
  • System model: The system contains multiple single-antenna UAVs and ground users, with UAVs connected to one ground CP through high-speed fronthaul links.
  • System model: Users are divided into groups, with same-group users served simultaneously by SDMA and different groups assigned orthogonal time or frequency dimensions.
  • Episodic model: Communication is divided into N equal-duration episodes, with nominal user and UAV horizontal locations specified for each episode.
  • Episodic model: All UAVs use fixed altitude H in the model, while the distance to each user is determined by horizontal separation and H.
  • Episodic model: Within an episode, horizontal positions are quasi-static and distance can be approximated using nominal locations when episode duration is sufficiently small.

A. Channel Model

The model describes episode-based uplink CoMP with LoS channels whose amplitudes are distance-dependent and whose phases vary randomly across coherence intervals and UAV–user pairs.

  • LoS channel model: The LoS channel with random phase models each phase as uniform on [0, 2π) and independent across coherence intervals and UAV–user pairs.The model captures sensitivity of phase to small location variations.
  • Channel evolution: Each episode contains many coherence intervals, with fixed distance-based amplitudes across intervals and independent random channel phases.Distance changes between episodes as users and UAVs move.
  • CoMP reception: At the CP, signals forwarded by all UAVs are jointly processed using linear beamforming with unit-norm beamforming vectors.The CP is assumed to know all channel coefficients perfectly through training and fronthaul transmission.
  • ZF-based transmission: ZF beamforming nulls inter-user interference by enforcing wk,l[n]Hhj,l[n] = 0 for distinct users.The resulting SNR and ergodic rate are defined per coherence interval and averaged over random phase variations.
  • Signal model: User symbols are modeled as i.i.d. CSCG variables with zero mean and unit variance, while UAV noise is AWGN with variance σ2.The average achievable rate is evaluated over the N episodes.

III. PROBLEM FORMULATION

The paper formulates UAV placement and movement as a max–min optimization of users’ average achievable rates across episodes, then replaces difficult rate expressions with tractable bounds.

  • III. PROBLEM FORMULATION: The optimization chooses each UAV’s horizontal locations over N episodes to maximize the minimum average achievable rate among the users.The decision variables are the coordinates xm[n] and ym[n].
  • III. PROBLEM FORMULATION: UAV movement is constrained by a per-episode displacement limit Dm[n] determined by the UAV speed limit.The constraint limits squared horizontal displacement between consecutive episodes.
  • III. PROBLEM FORMULATION: The formulation covers static, high-mobility, and semi-dynamic UAV operation by setting displacement limits to zero or positive values across episodes.Static UAVs have fixed locations, while semi-dynamic UAVs move only during selected episodes.
  • III. PROBLEM FORMULATION: The original problem is challenging because each episode’s ergodic rate lacks a closed-form expression without an expectation operation.The paper addresses this by developing efficient bounds and approximations.
  • IV. USER ERGODIC RATE CHARACTERIZATION: The rate-characterization section derives approximated upper and lower bounds under independent Rayleigh fading.These bounds are then transferred to the LoS channel with random phase and numerically validated as tight.
  • A. Rayleigh Fading Channel: The derivation uses expectations involving ZF-based beamforming, where non-identically distributed channel elements create a main analytical difficulty.The bounds are presented through theorems under the isotropic-vector assumption.
  • A. Rayleigh Fading Channel: The upper and lower bounds are very tight because their only stated difference is the denominator term, particularly when M is much larger than K.Theorem proofs are referred to Appendices B and C.

B. LoS Channel with Random Phase

The paper justifies using Rayleigh-fading rate bounds for its LoS random-phase channel by showing that ZF projects both models into the same beamforming-space dimension.

  • Channel comparison: The LoS random-phase model differs from Rayleigh fading primarily in amplitude: LoS amplitudes are fixed by distance, whereas Rayleigh amplitudes are random.Both models retain channel-vector randomness relevant to the projection analysis.
  • Numerical verification: Either bound can be used as a tight approximation of the LoS user rate in the paper’s subsequent optimization.This conclusion follows the numerical agreement reported for the considered channel model.
  • ZF projection: ZF makes wk,l[n] orthogonal to the other users’ channel span, so the desired channel power is a projection onto an (M−K+1)-dimensional space.For K < M, the channel matrix has rank K with probability one.
  • Rank property: The considered LoS channel has rank K with probability one because independently selected vectors avoid any particular hyperplane through the origin with probability one.This extends the linear-independence argument beyond Rayleigh fading.
  • Rate approximation: Although the projected vectors have different distributions, the projected random space is the same under Rayleigh fading and the considered LoS model.The resulting desired-channel-power distributions are therefore expected to be close.
  • Numerical verification: Fig. 3 compares simulated LoS ergodic rates with Rayleigh rates and the analytical approximations.The reported comparison shows the bounds are very tight and the two channel-model rates are close.
  • Rate approximation: The Rayleigh-derived upper and lower bounds are proposed as good approximations for the LoS random-phase user rates.The paper states Rk,l[n] ≈ ˜Rk,l[n] and supports this with a numerical example.

C. Numerical Example

The numerical example evaluates the analytical rate bounds under a single-episode UAV scenario and reports close agreement among the LoS rates, Rayleigh rates, and approximations.

  • C. Numerical Example: Fig. 3 compares simulated ergodic rates under the LoS random-phase model and Rayleigh fading with the analytical upper and lower bounds.The figure is used to validate the rate approximations.
  • C. Numerical Example: The example uses M = 10 UAVs and K = 6 users in one episode with random locations in a 100 m × 100 m square.All UAVs have height H = 100 m.
  • C. Numerical Example: The simulation sets user transmit power to 23 dBm, AWGN power to −169 dBm/Hz, bandwidth to 10 MHz, and reference channel power to τ0 = −40 dBm.The reference distance is 1 m.
  • C. Numerical Example: The Rayleigh-fading upper and lower bounds are reported as very tight for the approximated rates ˜Rk,l[n].This also verifies the stated isotropic-vector assumption numerically.
  • C. Numerical Example: The rates under Rayleigh fading are very close to those under the considered LoS random-phase channel, supporting use of either bound.The paper explicitly states that either expression can approximate Rk,l[n].
  • C. Numerical Example: Each user’s bounded rate depends only on its own distance to all M UAVs because ZF nulls inter-user interference.Consequently, the user grouping does not affect rates in this setup.
  • C. Numerical Example: The placement and movement solution to problem (P1) therefore applies to all user groupings in the considered protocol.The protocol uses quasi-static UAVs within each episode and orthogonal blocks for groups.
  • C. Numerical Example: The paper studies full-information dynamic placement, current-information dynamic placement, and static UAV placement as practical scenarios.Full information enables joint optimization across episodes, whereas current information permits separate episode-wise optimization.

A. Dynamic UAV Placement With Full User Location Information

With full current and future user-location information, UAV locations are jointly optimized across episodes by reformulating the rate problem and solving successive convex approximations. The resulting algorithm converges monotonically, and each UAV location combines current users’ locations with neighboring-episode UAV locations.

  • Optimization formulation: Problem (P2) is equivalently reformulated with auxiliary variables, while its non-concave constraint is approximated by a concave lower bound.The resulting convex problem can be solved efficiently by CVX within successive convex approximation.
  • Optimization algorithm: Algorithm 1 iteratively updates auxiliary variables and solves the approximated convex problem until the objective improvement is at most ǫ.The iteration index is q, and each update solves problem (29).
  • Convergence: Monotonic convergence is guaranteed: R(q) ≥ R(q−1) for all q ≥ 2, and the converged solution satisfies all KKT conditions of problem (P2-eqv).This guarantee is stated in Theorem 4.
  • Optimal placement structure: The x-axis optimal locations across episodes are characterized by N linear equations, with analogous equations for the y-axis locations.The coordinate solutions are obtained after solving problem (P2-eqv) locally optimally.
  • Optimal placement structure: For each UAV and episode, the optimal horizontal location is the weighted average of current user locations and the UAV’s locations in the previous and next episodes.The weights are the corresponding optimal dual variables, and neighboring locations enter through the maximum-displacement constraint.

B. Dynamic UAV Placement With Current User Location Information

With only current user-location information, UAV locations are optimized episode by episode using a successive convex approximation of the reformulated problem. Each location depends on current users and the previous UAV location, but not the next one because future user information is unavailable.

  • Problem formulation: At episode n, the design optimizes UAV locations using current user positions to maximize the minimum current user ergodic rate.Previous UAV locations are treated as known, and future effects are not considered.
  • Solution method: The current-information problem is transformed with auxiliary variables and solved locally optimally using the same successive convex approximation technique.The algorithm is similar to Algorithm 1 and is omitted in the paper.
  • Optimal placement structure: Each UAV location is the weighted average of current user locations and its previous-episode location.The weights are optimal dual variables associated with the reformulated constraints.
  • Static comparison: Static UAV placement removes the movement constraint and substantially reduces the number of variables and constraints, lowering solution complexity.Static locations can be represented without episode-dependent coordinates.
  • Static comparison: With full user-location information and static UAVs, each UAV location is the weighted average of all user locations across the N episodes.This static case can be optimized by setting Dm[n] = 0 and applying the same approximation approach.

VI. NUMERICAL EXAMPLES

Numerical experiments evaluate convergence and compare dynamic UAV placement with full or current user-location information against static deployment. The results show close agreement between the approximation and original problem, substantial gains over random initialization, and strong dependence on available movement information when UAVs are slower than users.

  • Convergence evaluation: About 70% rate improvement is obtained after convergence compared with the random UAV placement used for initialization.The comparison is made using the actual achievable minimum user rate after each iteration.
  • Speed comparison: With full user-location information, minimum-rate performance is not very sensitive to UAV maximum speed; static deployment loses about 0.1 bps/Hz relative to dynamic deployment at vuav = 20 m/s.The comparison is among dynamic full-information, dynamic current-information, and static full-information scenarios.
  • Speed comparison: When only current user locations are known, minimum rate increases rapidly with UAV speed because faster movement compensates for the lack of future information.When vuav ≤ 15 m/s, partial information performs much worse than full information; at higher speeds, the loss is much smaller.
  • Placement behavior: Across scenarios, isolated users receive dedicated UAV coverage, whereas nearby users are served by a UAV positioned between them with larger weights for nearby users.Full-information UAVs tend to move in one direction, while current-information UAVs often move back and forth.

B. Effect of User Grouping on User Minimum Rate

The number of user groups creates a trade-off in minimum-rate maximization: fewer groups provide more scheduling resources but reduce the ZF gain. In the reported example, three groups are optimal.

  • Effect of User Grouping: Adding users to one group does not affect the resulting user achievable rate because of ZF-based beamforming.The grouping trade-off therefore concerns the number of groups and users per group rather than an effect from adding users to one group alone.
  • Effect of User Grouping: Grouping users into fewer groups gives each group more transmission time or bandwidth, but produces a smaller ZF gain during scheduled intervals.The paper identifies these opposing effects as the reason an intermediate grouping can be preferable.
  • Effect of User Grouping: The user minimum rate first increases and then decreases as the number of groups L grows across the evaluated UAV-information settings.The comparison covers dynamic UAVs with full or current user-location information and static UAVs with full information.
  • Effect of User Grouping: L = 3 groups, equivalently K = 6 users per group in the example, is optimal for user minimum-rate maximization.The evaluated cases are L = 2, 3, 6, 9, corresponding to K = 9, 6, 3, 2 users per group.
  • System Context: The paper studies minimum-rate maximization in uplink multi-UAV, multi-user communications with UAV placement and movement designed over time.UAVs serve as flying RAUs or RRHs that relay user messages to a central processor for joint decoding.
  • Analysis and Design: Closed-form ergodic-rate characterizations from random matrix theory support an efficient successive-convex-approximation algorithm for UAV placement and movement.Numerical results evaluate the proposed algorithm under various practical scenarios.

APPENDIX

The appendix derives rate bounds and reformulates the UAV placement problem for optimization. It also uses ZF geometry and dual methods to characterize feasible and optimal solutions.

  • Rate Bounds: Jensen’s inequality supplies upper and lower bounds for the approximated achievable rate expressions.The derivation uses concavity of log2(1 + ax) and convexity of log2(1 + b/x) over x > 0.
  • ZF Structure: ZF beamforming makes each user’s beamforming vector orthogonal to the other K − 1 users’ channel vectors in the same group.Under the stated Gaussian assumption, the projected channel power is treated through an (M − K + 1)-dimensional beamforming space.
  • Channel Analysis: Modeling the channel matrix with a Gaussian factor and a diagonal phase-related matrix leads to Wishart-matrix analysis of the inverse Gram matrix.This result is substituted into the rate expressions to prove the corresponding theorem.
  • Problem Reformulation: Introducing a minimum-rate variable R converts the placement formulation into an equivalent max-min problem over UAV coordinates and R.The appendix establishes equality of the optimal values through feasible-solution mappings.
  • Problem Reformulation: The optimal reformulated solution must satisfy the distance relation in (60), because the objective increases with the associated channel coefficients.The appendix uses this property to show the reformulated problem has the same optimum as the original formulation.
  • Dual Derivation: The Lagrangian dual formulation associates nonnegative dual variables with the placement constraints and derives optimal coordinates by setting Lagrangian derivatives to zero.The dual function and dual problem are formed before obtaining the coordinate conditions.
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