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UAV Trajectory and Beamforming Optimization for Integrated Periodic Sensing and Communication

Kaitao Meng, Qingqing Wu, Shaodan Ma, Wen Chen, Tony Q. S. Quek

arXiv:2203.10223v1cs.ITeess.SP

TL;DR

The paper addresses the difficulty of balancing communication quality and sensing timeliness in UAV-enabled ISAC with shared spectrum. It introduces periodic sensing and jointly optimizes trajectory, transmit precoding, and sensing timing, deriving closed-form rate expressions and exploiting inter-frame structure. Simulations show a more flexible sensing–communication trade-off than benchmark schemes.

  • Problem

    Shared spectrum and asymmetric sensing-frequency requirements make it difficult to balance high-quality communication with timely sensing in UAV-enabled ISAC.

  • Method

    The paper proposes periodic ISAC and jointly optimizes UAV trajectory, transmit beamforming, and sensing start time under sensing and beam-pattern constraints.

  • Results

    The method derives an optimal precoder and closed-form achievable rate for each UAV location, proves unconstrained inter-frame structural symmetry, and provides a high-quality location-constrained trajectory solution.

  • Takeaways & Limitations

    Periodic sensing enlarges the available sensing–communication trade-off by allowing standalone communication and always-sensing operation as special cases.

Abstract

from arXiv · show

Unmanned aerial vehicle (UAV) is expected to bring transformative improvement to the integrated sensing and communication (ISAC) system. However, due to shared spectrum resources, it is challenging to achieve a critical trade-off between these two integrated functionalities. To address this issue, we propose in this paper a new integrated \emph{periodic} sensing and communication mechanism for the UAV-enable ISAC system. Specifically, the user achievable rate is maximized via jointly optimizing UAV trajectory, transmit precoder, and sensing start instant, subject to the sensing frequency and beam pattern gain constraints. Despite that this problem is highly non-convex and involves an infinite number of variables, we obtain the optimal transmit precoder and derive the optimal achievable rate in closed-form for any given UAV location to facilitate the UAV trajectory design. Furthermore, we first prove the structural symmetry between optimal solutions in different ISAC frames without location constraints and then propose a high-quality UAV trajectory and sensing optimization algorithm for the general location-constrained case. Simulation results corroborate the effectiveness of the proposed design and also unveil a more flexible trade-off in ISAC systems over benchmark schemes.

I. INTRODUCTION

The paper introduces periodic sensing for UAV-enabled ISAC to accommodate asymmetric sensing-frequency and communication requirements. It jointly optimizes trajectory, beamforming, and sensing timing to obtain a more flexible sensing–communication trade-off.

  • I. INTRODUCTION: Periodic sensing addresses practical cases where sensing frequency differs from communication needs, rather than forcing both functions to operate simultaneously throughout the considered period.The framework targets requirements such as low- or high-frequency sensing for different tracking tasks.
  • I. INTRODUCTION: The proposed IPSAC mechanism jointly optimizes UAV trajectory, transmit beamforming, and sensing instant to maximize communication performance under sensing requirements.The initial study considers one ground target and one user cluster in a UAV-enabled multicast downlink system.
  • I. INTRODUCTION: IPSAC provides a flexible trade-off over time because standalone communication and always-sensing operation are both special cases of the proposed framework.This expands the operating choices beyond schemes that always transmit data and sense simultaneously.
  • I. INTRODUCTION: The paper proves structural symmetry between optimal solutions across unconstrained ISAC frames, reducing the complexity of the resulting optimization algorithm.The symmetry result concerns the case without location constraints.
  • I. INTRODUCTION: The authors derive a closed-form optimal achievable rate for a given UAV location and provide both an unconstrained trajectory and a high-quality location-constrained solution.The closed-form rate and trajectory results support the joint design of beamforming and UAV motion.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models a UAV with a linear antenna array that periodically senses a ground target while serving single-antenna users over LoS channels. The formulation maximizes achievable communication rate while enforcing sensing, power, speed, and endpoint constraints.

  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The system uses a UAV-mounted ULA to sense one ground target and provide multicast downlink service to a cluster of single-antenna users modeled as one user.The geometry is represented in two-dimensional Cartesian coordinates with the user and target separated along the x-axis.
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: Sensing occurs at least once per ISAC frame, with frame length determined by the required sensing frequency and sensing duration fixed by the sensing period.The sensing interval in frame l is [t_l, t_l + τ_0].
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The UAV is assumed to hover during sensing because the sensing period is short and a fixed location avoids introducing complex Doppler shifts from UAV motion.The sensing period is practically determined by radar bandwidth, waveform, and related factors.
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: Communication is modeled through a free-space path-loss LoS channel, with linear precoding producing the received signal, user SNR, and achievable rate.The user rate is R(t) = log2(1 + γ(t)), while the same communication waveform is also exploited for sensing.
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The optimization maximizes achievable rate subject to beam-pattern gain, total transmit-power, UAV-speed, and initial/final-location constraints.The problem is difficult because the trajectory has infinitely many variables and both the objective and sensing-gain constraint are non-convex.

III. PROPOSED SOLUTION TO (P1)

The solution first obtains an optimal transmit precoder and a closed-form user SNR for each UAV location. This reduction facilitates subsequent UAV trajectory optimization.

  • III. PROPOSED SOLUTION TO (P1): For any fixed UAV location, the proposed solution uses semidefinite relaxation and eigenvalue decomposition to obtain the optimal transmit precoder and a closed-form user SNR.The closed-form SNR is derived specifically to support trajectory optimization.

A. Optimal Transmit Precoder to (P1)

For a fixed UAV location, the paper derives the optimal sensing-period precoder and a closed-form user SNR, reducing the trajectory problem's complexity.

  • A. Optimal Transmit Precoder to (P1): Outside sensing intervals, maximum-ratio transmission is the optimal transmit precoder.The communication objective is optimized directly when sensing is not active.
  • A. Optimal Transmit Precoder to (P1): The optimal sensing-period transmit precoder is obtained by semidefinite relaxation and recovered from a rank-one solution using eigenvalue decomposition.The relaxed problem can be solved by convex optimization, and an optimal rank-one solution always exists.
  • A. Optimal Transmit Precoder to (P1): During sensing, the optimal user SNR is available in closed form for any given UAV location.The expression depends on the user-target channel correlation and the sensing beam-pattern constraint.
  • A. Optimal Transmit Precoder to (P1): The closed-form rate characterization enables direct evaluation of achievable rate at each UAV location, simplifying subsequent trajectory design.This removes the need to repeatedly solve the precoder optimization during trajectory optimization.

B. Optimal Solution Without Location Constraints

Without initial and final location constraints, the paper exploits trajectory structure to simplify periodic UAV optimization and characterize an optimal sensing-location search.

  • B. Optimal Solution Without Location Constraints: For the location-constrained-free special case, an optimal trajectory in each frame can be chosen unidirectional between the user and target positions.Thus, only monotonic motion over [0, D] needs to be considered.
  • B. Optimal Solution Without Location Constraints: Optimal trajectories in adjacent ISAC frames can be constructed by reversing the preceding frame's trajectory, establishing structural symmetry across frames.The sensing time and transmit precoder are matched at corresponding locations, and the resulting solution remains feasible with no lower total rate.
  • B. Optimal Solution Without Location Constraints: The resulting trajectory uses a hover-then-fly structure, with sensing at xr followed by maximum-speed flight toward the user's direction.Its sum-rate combines the sensing-location rate and the communication-only rate after sensing.
  • B. Optimal Solution Without Location Constraints: An optimal trajectory can be found by one-dimensional search over the sensing location while checking the stated optimality condition.When Tf = τ0, the optimal trajectory hovers at a location satisfying g′(xr) = 0.

C. Location Constrained Trajectory and Sensing Optimization

With initial and final location constraints, the paper constructs a high-quality trajectory and sensing optimization using optimal sensing-location conditions and maximum-speed motion.

  • C. Location Constrained Trajectory and Sensing Optimization: Consecutive sensing locations on the same side of the user require maximum-speed flight between them, while otherwise the UAV may hover above the user between sensing movements.These rules determine the intermediate motion among frame-specific sensing locations.
  • C. Location Constrained Trajectory and Sensing Optimization: At an interior optimal sensing location, the achievable-rate derivative satisfies g′(x*_l) = 0; otherwise, speed limits can force sensing to a frame boundary.The optimal sensing start instant may therefore occur at the start or end of an ISAC frame.
  • C. Location Constrained Trajectory and Sensing Optimization: The constrained solution searches sensing locations in one dimension and checks the optimality conditions for each ISAC frame.The sensing locations and durations are selected to satisfy at least one condition from Lemma 3.
  • C. Location Constrained Trajectory and Sensing Optimization: The UAV tends to fly at maximum speed toward the selected sensing location and then uses hover-fly-hover subtrajectories before returning toward the final location.This construction adapts the unconstrained trajectory structure to the initial and final position requirements.

IV. NUMERICAL RESULTS

The numerical results characterize trade-offs among sensing requirements, achievable rate, and flight speed, showing that the proposed scheme approaches the upper bound and outperforms benchmarks under the reported settings.

  • The proposed scheme is very close to the P1 upper bound, demonstrating near-optimality of the location-constrained solution.It also improves achievable rate over the time-division and optimal-precoder-only benchmarks in the reported comparisons.
  • As sensing frequency decreases, the proposed scheme gains more rate over the optimal-precoder-only benchmark because more non-sensing time is available for trajectory adjustment.
  • Higher beam pattern gain thresholds cause faster achievable-rate degradation as sensing frequency increases.The threshold forces sensing closer to the target, increasing path loss during pure communication periods.
  • The achievable-rate gain over the optimal-precoder-only scheme increases with maximum flight speed.The reported explanation is that higher speed enables better channel gain within a shorter flying time.
  • For the proposed scheme, rates under different beam pattern gain thresholds are almost equal when the distance D is below 100 m.This indicates better communication performance when the target is closer to the user in that reported range.

V. CONCLUSION AND FUTURE WORKS

The paper concludes that periodic UAV-enabled ISAC supports a structural symmetry across unconstrained ISAC frames and a near-optimal solution under location constraints, while identifying multi-UAV 3D optimization as future work.

  • The proposed method achieves a near-optimal location-constrained solution using a derived closed-form achievable rate.
  • Optimal solutions in different ISAC frames exhibit a novel structure-symmetry characteristic when location constraints are absent.
  • Numerical results show that the proposed scheme enlarges the sensing–communication performance trade-off of UAV-enabled ISAC systems.
  • More general 3D trajectory optimization for multi-UAV ISAC scenarios is identified as future work.

APPENDIX A: PROOF OF PROPOSITION 1

The appendix establishes optimal-precoder conditions through KKT analysis and relates the sensing constraint to the transmit covariance solution and resulting user SNR.

  • The transmit covariance is constrained to use full power at the optimum, allowing the power constraint to be written as tr(W) = Pmax.
  • The optimal solution satisfies the KKT conditions for the transmit-covariance formulation.
  • The sensing constraint multiplier is nonzero only when the achieved sensing gain is below the threshold.
  • When the sensing multiplier is nonzero, maximum-ratio transmission is optimal for the transmit-precoder subproblem.
  • The derivation combines the channel matrix and covariance relations to obtain the optimal user SNR for nonzero multipliers.
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