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Throughput Maximization for UAV-enabled Integrated Periodic Sensing and Communication

Kaitao Meng, Qingqing Wu, Shaodan Ma, Wen Chen, Kunlun Wang, Jun Li

arXiv:2203.06358v3cs.ITeess.SP

TL;DR

Existing UAV-enabled ISAC approaches often force sensing and communication throughout the entire period, despite asymmetric sensing requirements. This paper introduces IPSAC, jointly optimizes trajectory and transmission-related decisions under sensing constraints, and develops closed-form and algorithmic solutions that support a more flexible trade-off.

  • Problem

    Existing ISAC works mainly perform sensing with communication throughout the entire period, overlooking practical differences between sensing frequency and data-frame rate.

  • Method

    The paper jointly optimizes UAV trajectory, user association, target sensing selection, sensing time, and transmit beamforming under sensing frequency and beam pattern gain constraints.

  • Results

    The proposed IPSAC framework provides a more flexible sensing–communication trade-off than benchmark schemes, while closed-form beamforming and a penalty-based algorithm reduce solution complexity.

  • Takeaways & Limitations

    Periodic sensing allows sensing tasks to be scheduled according to practical timeliness requirements while communication services continue over time.

Abstract

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Unmanned aerial vehicle (UAV) is expected to revolutionize the existing integrated sensing and communication (ISAC) system and promise a more flexible joint design. Nevertheless, the existing works on ISAC mainly focus on exploring the performance of both functionalities simultaneously during the entire considered period, which may ignore the practical asymmetric sensing and communication requirements. In particular, always forcing sensing along with communication may make it is harder to balance between these two functionalities due to shared spectrum resources and limited transmit power. To address this issue, we propose a new integrated periodic sensing and communication mechanism for the UAV-enabled ISAC system to provide a more flexible trade-off between two integrated functionalities. Specifically, the system achievable rate is maximized via jointly optimizing UAV trajectory, user association, target sensing selection, and transmit beamforming, while meeting the sensing frequency and beam pattern gain requirement for the given targets. Despite that this problem is highly non-convex and involves closely coupled integer variables, we derive the closed-form optimal beamforming vector to dramatically reduce the complexity of beamforming design, and present a tight lower bound of the achievable rate to facilitate UAV trajectory design. Based on the above results, we propose a penalty-based algorithm to efficiently solve the considered problem. The optimal achievable rate and the optimal UAV location are analyzed under a special case of infinity number of antennas. Furthermore, we prove the structural symmetry between the optimal solutions in different ISAC frames without location constraints and propose an efficient algorithm for solving the problem with location constraints.

I. INTRODUCTION

The paper addresses the mismatch between continuously coupled sensing and communication and practical asymmetric sensing needs by introducing periodic sensing for UAV-enabled ISAC. It jointly designs system decisions to maximize communication rate while satisfying sensing requirements and derives structural and algorithmic solutions.

  • Motivation: UAV mobility and strong line-of-sight links provide flexible observation, communication quality, and service coverage for integrated sensing and communication.The UAV is positioned as an aerial platform for enhanced ISAC service.
  • Motivation: Prior ISAC studies generally perform sensing together with communication throughout the entire considered period, overlooking differing sensing frequencies.The paper motivates sensing frequency according to target motion and task timeliness.
  • Proposed mechanism: The proposed IPSAC mechanism periodically executes sensing tasks while providing communication, jointly optimizing beamforming, user association, sensing time, and UAV trajectory.The optimization maximizes achievable rate subject to sensing frequency and beam pattern gain requirements.
  • Problem and solution: The optimization is highly non-convex because integer decisions are closely coupled with UAV trajectory and beamforming vectors.This coupling makes conventional trajectory discretization increasingly difficult for long mission periods.
  • Problem and solution: A closed-form beamforming vector, a tight achievable-rate lower bound, and a penalty-based algorithm reduce the complexity of jointly optimizing the system decisions.The paper also analyzes infinite-antenna special cases and structural symmetry across unconstrained ISAC frames.
  • Results: Simulations show a more flexible sensing–communication trade-off than benchmark schemes, with UAV trajectory design important for balancing both functions.The UAV tends to communicate with users closer to the sensed target.

C. Problem Formulation

The formulation maximizes achievable rate under sensing, communication, power, mobility, and location-related requirements. It models target selection and user association with coupled binary decisions and uses a two-layer solution based on closed-form beamforming and a rate lower bound.

  • Optimization formulation: The objective maximizes achievable rate by optimizing beamforming, user association, sensing time selection, and UAV trajectory.The formulation includes sensing frequency, sensing power, and per-frame quality-of-service requirements.
  • Modeling assumptions: Interference from multiple targets is approximately ignored because targets are sensed in separate time slots and beam power is concentrated toward the intended target.The paper also states that clutter effects can be mitigated using prior clutter knowledge and precoding.
  • Sensing constraints: The sensing constraints can represent either reflected-signal SNR requirements with pathloss exponent 4 or target-location signal-power requirements with exponent 2.The exponent-4 case corresponds to monostatic sensing, while exponent 2 corresponds to bistatic sensing with a dedicated echo receiver.
  • Optimization formulation: Target selection and user association are represented by binary variables over time slots, with at most one target sensed in each slot.The sensing schedule is organized across ISAC frames.
  • Solution approach: Because the problem is non-convex with coupled integer variables, the proposed solution derives closed-form beamforming and a tight rate lower bound before applying a two-layer penalty-based algorithm.Ignoring initial and final location constraints enables a symmetry-based low-complexity algorithm for long flight periods.

III. PENALTY-BASED ALGORITHM TO (P1)

This section derives a closed-form beamforming solution and uses it, together with rate bounds, to reduce the joint optimization complexity. It also characterizes beamforming effects and antenna-rich special cases.

  • A. Closed-form Optimal Beamforming: The beamforming design is jointly coupled with user association, sensing selection, and UAV trajectory through the received signal-strength maximization problem.The coupling is especially present when a user is served while the UAV senses a target.
  • A. Closed-form Optimal Beamforming: The optimal beamforming vector is derived in closed form for any given UAV location and can facilitate subsequent trajectory optimization.The vector is interpreted as two linearly superimposed beams directed toward the associated user and sensing target.
  • A. Closed-form Optimal Beamforming: The optimal user SNR depends on the distance-based channel power and the channel correlation coefficient cos φk,j between communication and target channels.When cos φk,j = 1, the channels are linearly related; when cos φk,j = 0, they are orthogonal and the user channel power is reduced.
  • A. Closed-form Optimal Beamforming: When Mx and My approach infinity, the optimal user SNR during sensing can be simplified, and the corresponding optimal UAV location is characterized by the user-target distance.The special-case expressions distinguish whether the user and sensed target locations coincide.
  • A. Closed-form Optimal Beamforming: The closed-form beamforming vector provides the basis for deriving a tight achievable-rate lower bound and jointly optimizing the remaining design variables.The remaining variables include user association, sensing time selection, and UAV trajectory.

B. Lower Bound of Achievable Rate

This section replaces the complicated sensing-period achievable rate with a tight lower bound that separates communication-only and sensing-time contributions. The resulting problem removes beamforming from the remaining joint optimization.

  • B. Lower Bound of Achievable Rate: The user’s achievable rate is decomposed into communication-only and sensing-time terms before constructing the lower-bound optimization problem.The communication-only and sensing-time rates are denoted separately, with the latter based on the sensing-target configuration.
  • B. Lower Bound of Achievable Rate: The lower-bound formulation remains challenging because its objective contains a piece-wise non-concave function.This motivates the subsequent penalty-based transformation and iterative optimization procedure.
  • B. Lower Bound of Achievable Rate: Lemma 2 establishes a condition-based lower bound for the optimal achievable rate during sensing target j.The bound is used to replace the original sensing-period rate in the achievable-rate maximization problem.
  • B. Lower Bound of Achievable Rate: The lower bound is tight as the antenna count M approaches infinity.This tightness follows from the large-antenna SNR characterization established earlier.
  • B. Lower Bound of Achievable Rate: Using the lower bound, achievable-rate maximization only jointly optimizes user association, sensing time selection, and UAV trajectory.The beamforming vector is obtained from the closed-form result rather than optimized as an independent remaining variable.

C. Penalty-based Problem Transformation

This section transforms coupled binary variables and their constraints into a penalty-based continuous optimization formulation. The penalty coefficient is controlled so that feasible binary solutions are eventually recovered.

  • C. Penalty-based Problem Transformation: The auxiliary variable ek,j[n] = αk[n]cj[n] decouples the user-association and sensing-selection integer variables in the lower-bound problem.The reformulated rate uses ek,j[n] ∈ {0, 1} and adds consistency constraints.
  • C. Penalty-based Problem Transformation: The introduced consistency constraints ensure that ek,j[n] = 1 if and only if αk[n] = 1, preserving equivalence with the original formulation.The replacement constraints also preserve the sensing-selection and association relationships.
  • C. Penalty-based Problem Transformation: Continuous relaxation followed by rounding may violate QoS and beam-pattern-gain constraints, so equivalent equality constraints are introduced instead.Slack matrices are used to transform the binary constraints into equality constraints.
  • C. Penalty-based Problem Transformation: Penalty terms are added for the equality constraints, with coefficient η > 0 controlling the penalty for constraint violations.The construction targets binary-valued association and sensing-selection variables.
  • C. Penalty-based Problem Transformation: As η approaches infinity, the relaxed solutions satisfy the equality constraints; the algorithm therefore updates η progressively while applying alternating optimization.The coefficient is initialized large and gradually reduced according to the described procedure.

D. Inner and Outer layer Iteration

The proposed algorithm alternates among auxiliary variables, binary-selection variables, and UAV trajectory while an outer penalty update enforces equality constraints. Convex subproblems are solved iteratively until objective change and constraint violation meet thresholds.

  • D. Inner and Outer layer Iteration: A two-layer penalty-based algorithm iteratively optimizes auxiliary variables, association and sensing variables, and the UAV trajectory.The outer layer updates the penalty coefficient, while the inner layer cycles through the variable blocks.
  • D. Inner and Outer layer Iteration: The auxiliary-variable subproblem is solved by setting derivatives of the objective with respect to the slack variables to zero.This yields the optimal slack variables for fixed association, sensing, and trajectory variables.
  • D. Inner and Outer layer Iteration: The association and sensing-variable subproblem is convex with a quadratic objective and linear inequality constraints, so standard solvers such as CVX can solve it.The relevant constraints include the problem’s stated feasibility and penalty constraints.
  • D. Inner and Outer layer Iteration: The trajectory subproblem is handled by successive convex optimization after introducing slack variables and linearizing logarithmic terms.The transformed constraints are convex and lead to a convex optimization problem solvable by CVX.
  • D. Inner and Outer layer Iteration: The transformed trajectory problem P2.5 has convex constraints and can be efficiently solved using convex optimization solvers such as CVX.This follows the successive-convexification steps applied to the non-concave subproblem.
  • D. Inner and Outer layer Iteration: Algorithm 1 updates the variable blocks and objective value repeatedly, then increases the penalty coefficient until the equality-constraint violation falls below threshold ϵ2.The inner iterations stop when the objective change is at most ϵ1.

4) Outer layer Iteration:

The outer layer progressively decreases the penalty coefficient, while the inner alternating-optimization iterations monotonically reduce the objective toward convergence. The resulting procedure has complexity determined by the variable dimensions and inner and outer iteration counts.

  • Outer-layer update: The outer layer updates the penalty coefficient as η = zη with 0 < z < 1; larger z improves performance but requires more iterations.
  • Convergence criterion: The terminal criterion requires the maximum binary-constraint residual to be at most the predefined accuracy ξ.
  • Convergence criterion: The inner-layer objective is non-increasing under alternating optimization and is upper bounded by limited flying time and transmit power.
  • Complexity: Algorithm 1 has complexity O(L_outerL_inner((KN + JKN)^3.5 + (2N + KN + JN)^3.5)).L_inner and L_outer denote the inner- and outer-layer iteration counts required for convergence.
  • Structural analysis: Without initial and final location constraints, problem (P3) is introduced to derive structural insights for the periodic design.
  • Structural analysis: The structural characteristics of optimal solutions across ISAC frames support a low-complexity algorithm for problem (P1).

A. Analysis of Optimal Solution to (P3)

Without location constraints, optimal solutions across ISAC frames exhibit an alternating equality or reversal symmetry. Consequently, solving one frame can generate solutions for the others, while increasing frame length cannot reduce the maximum achievable rate.

  • Structural symmetry: Lemma 3 guarantees an optimal solution to (P3) satisfying a parity-dependent relation between solutions in different ISAC frames.
  • Structural symmetry: For odd frame differences, the optimal sequence is reversed; for even differences, the same sequence remains feasible.
  • Structural symmetry: Adjacent frames therefore have equal or oppositely ordered optimal solutions, with symmetry around the relevant frame midpoint.
  • Algorithmic implication: Only the first ISAC frame needs to be solved for (P3), because other frame solutions can be generated using the structural relation and Algorithm 1.
  • Frame-length effect: The maximum achievable rate in (P3) increases monotonically as the frame length T_L increases.
  • Frame-length effect: The proof constructs longer-frame solutions by extending the UAV dwell at a location associated with the maximum rate.
  • Implications: The symmetry result reveals a sensing-frequency and communication-rate trade-off while also supporting efficient solutions for the constrained problem (P1).

B. Low-Complexity Algorithm for solving (P1)

A low-complexity method for (P1) uses structural properties from the unconstrained problem to reduce trajectory-design burden. Simulations show that sensing requirements reshape trajectories and create a rate trade-off relative to benchmark schemes.

  • Algorithm motivation: A long mission period can create many trajectory points and prohibitive computational complexity for direct UAV trajectory design.
  • Algorithm construction: Problem (P3.1) removes the initial and final location constraint, enabling a reduced-complexity trajectory construction for (P1).
  • Algorithm construction: The constructed trajectory combines straight flight from the initial location, the optimized periodic trajectory, and straight flight to the final location.
  • Algorithm construction: The low-complexity algorithm is preferred when the number of ISAC frames L is relatively large.
  • Trajectory behavior: As the beam pattern gain threshold Γ_th increases, the proposed UAV trajectory contracts from larger user-oriented arcs toward smaller arcs between targets and users.
  • Rate comparison: The achievable rate decreases as Γ_th increases, while the proposed scheme gains more over SF when the sensing power requirement is lower.
  • Rate comparison: Under high-frequency sensing, SF becomes infeasible when Γ_th exceeds 4 × 10^-5, whereas the proposed trajectory remains usable under the stated comparisons.
  • Rate comparison: At lower sensing frequency, the proposed scheme significantly improves over FHF because more communication-only time slots are available.

B. Comparison Versus Sensing Frequency

Increasing sensing frequency increasingly restricts UAV trajectories and reduces achievable communication rate. The proposed method retains performance advantages, while its lower-complexity approximation approaches the penalty-based algorithm for longer flight periods.

  • Trajectory response: As sensing frequency increases, UAV trajectories include more turn-backs between targets and users and become more restricted around targets.With T = TL, the UAV can nearly fly above each user; with T = 8TL, trajectories contain multiple nearly overlapping segments between targets and one user.
  • Rate response: Achievable rates for all considered mechanisms decrease as sensing frequency increases.The proposed scheme’s gain over the FHF and SF benchmarks increases when sensing frequency decreases because more non-sensing time remains for trajectory adjustment.
  • Rate response: A higher beam pattern gain threshold further accelerates achievable-rate degradation under higher sensing frequency.Higher thresholds force sensing closer to targets, increasing communication path loss during communication-only periods.
  • Association and beam pattern: At sensing time slots, beams concentrate mainly toward the selected target and associated user, and the UAV tends to serve users closer to the selected target.The reported setting is T = TL = 40 s and Γth = 10^-3.
  • Lower-bound accuracy: For M larger than 16, the average achievable rate from the original objective is approximated by the lower-bound objective with less than 1% difference.This supports the accuracy of the lower bound used for trajectory design.
  • Low-complexity method: The low-complexity algorithm incurs no more than 5% performance loss relative to the penalty-based algorithm when the flight period exceeds 200 s.Its relative advantage over the benchmarks increases with flight period, while the penalty-based algorithm’s gain over it decreases.

APPENDIX A: PROOF OF PROPOSITION 1

The proof derives the optimal beamforming structure by showing the communication-power constraint is active and analyzing the associated KKT conditions. It then characterizes cases that yield maximum-ratio transmission or the sensing-constrained optimum.

  • KKT setup: The communication beamforming norm satisfies ∥wc∥^2 = Pmax at the optimum because increasing it improves the objective until the power constraint is active.The proof then forms the corresponding Lagrangian and KKT conditions.
  • KKT analysis: The KKT analysis combines the communication and sensing channel conditions to determine the optimal beamforming coefficients.The proof introduces H = [hc,k, hr,j] and derives conditions by multiplying the stationarity equation with wc and related channel vectors.
  • Special cases: When the sensing multiplier λ2 equals zero, maximum-ratio transmission is optimal.In this case, the communication coefficient satisfies βc,k = Pmax∥hc,k∥ under the stated condition.
  • Special cases: The sensing constraint determines the beamforming solution in the complementary case, with phase conditions distinguishing the relevant coefficient expressions.The proof substitutes the phase cases into the preceding relations and concludes Proposition 1.

APPENDIX B: PROOF OF LEMMA 1

The proof analyzes the infinite-antenna limit by examining channel-angle differences and shows that the optimal UAV location lies on the line connecting the user and target. The location is then obtained from a scalar equation.

  • Infinite-antenna limit: When the user and target coincide, the angular differences vanish and the limiting communication-rate expression follows directly.The proof separately examines nonzero angular differences and their limiting cosine behavior.
  • Infinite-antenna limit: As Mx or My tends to infinity, the cosine of the user-target phase difference approaches zero in the analyzed noncoincident cases.The proof treats cases where both angular differences are nonzero, one is zero, or both are zero.
  • Optimal location: The optimal horizontal UAV coordinate lies on the line formed by the user and target locations.The proof parameterizes the horizontal distance from the UAV to the user along this line and differentiates the resulting rate expression.
  • Optimal location: The maximum-achievable-rate location is obtained by solving the scalar stationarity equation for the line-coordinate variable x.The resulting coordinate is expressed relative to the user-target distance.
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