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UAV-Assisted Intelligent Reflecting Surface Symbiotic Radio System

Meng Hua, Luxi Yang, Qingqing Wu, Cunhua Pan, Chunguo Li, A. Lee Swindlehurst

arXiv:2007.14029v3eess.SPcs.IT

TL;DR

The paper addresses BER optimization in a UAV-assisted IRS symbiotic radio system, where IRSs transmit their own data while also enhancing UAV communication. It jointly optimizes trajectory, IRS phase shifts, and scheduling, using relaxation-based and penalty-based algorithms for weighted-sum and fairness objectives. The relaxation solution yields binary scheduling without reconstruction for the weighted-sum problem, while the fairness problem requires a penalty-based treatment of its constraints.

  • Problem

    The paper studies how to optimize UAV trajectory, IRS phase shifts, and scheduling for weighted-sum and fairness BER objectives under minimum primary-rate requirements.

  • Method

    It relaxes binary scheduling and uses alternating optimization for weighted-sum BER, then transforms binary constraints into equivalent equalities and applies a penalty-based algorithm for fairness BER.

  • Results

    The relaxed weighted-sum formulation produces binary scheduling without reconstruction, automatically satisfying UAV rate constraints; the penalty-based fairness algorithm is reported effective.

  • Takeaways & Limitations

    UAV mobility and IRS phase tuning provide effective design variables for improving system performance in both weighted-sum and fairness scenarios.

Abstract

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This paper investigates a symbiotic unmanned aerial vehicle (UAV)-assisted intelligent reflecting surface (IRS) radio system, where the UAV is leveraged to help the IRS reflect its own signals to the base station, and meanwhile enhance the UAV transmission by passive beamforming at the IRS. First, we consider the weighted sum bit error rate (BER) minimization problem among all IRSs by jointly optimizing the UAV trajectory, IRS phase shift matrix, and IRS scheduling, subject to the minimum primary rate requirements. To tackle this complicated problem, a relaxation-based algorithm is proposed. We prove that the converged relaxation scheduling variables are binary, which means that no reconstruct strategy is needed, and thus the UAV rate constraints are automatically satisfied. Second, we consider the fairness BER optimization problem. We find that the relaxation-based method cannot solve this fairness BER problem since the minimum primary rate requirements may not be satisfied by the binary reconstruction operation. To address this issue, we first transform the binary constraints into a series of equivalent equality constraints. Then, a penalty-based algorithm is proposed to obtain a suboptimal solution. Numerical results are provided to evaluate the performance of the proposed designs under different setups, as compared with benchmarks.

I. INTRODUCTION

This paper studies a UAV-assisted IRS symbiotic radio system that jointly supports IRS data transmission and enhanced UAV communication. It formulates weighted-sum and fairness BER optimization problems and develops relaxation- and penalty-based algorithms for them.

  • System model: The UAV assists IRS data transmission while IRS phase tuning coherently combines UAV-IRS-BS and UAV-BS signals at the base station.IRS on/off states carry IRS information, while passive beamforming enhances UAV communication.
  • Weighted-sum BER optimization: The weighted-sum BER problem jointly optimizes UAV trajectory, IRS phase shifts, and scheduling subject to the UAV minimum data-rate requirement.The problem is mixed-integer and non-convex, motivating a relaxation-based alternating optimization method.
  • Weighted-sum BER optimization: Relaxed scheduling variables converge to binary values, so no reconstruction strategy is needed and UAV rate constraints remain automatically satisfied.The proposed relaxation-based method converges within only a few iterations.
  • Fairness BER optimization: The fairness BER problem is also mixed-integer and non-convex, but relaxation followed by binary reconstruction may violate the UAV rate requirements.The paper therefore transforms binary constraints into equivalent equality constraints and applies a penalty-based two-layer algorithm.
  • Numerical evaluation: Simulations show that optimized UAV trajectories and tunable IRS phase shifts significantly improve system performance across both optimization scenarios.Higher weighting factors place the optimized UAV trajectory closer to the corresponding IRS, while fairness utility depends strongly on the IRS phase-shift matrix.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system comprises a UAV, fixed-position IRSs, and a base station, with the UAV assisting IRS transmission while receiving passive reflection assistance. A periodic, discretized UAV trajectory and IRS phase-control model define the system design space.

  • A. System Model: The system contains a single-antenna base station, a single-antenna UAV, and K IRSs assisted by the UAV.
  • A. System Model: The base station and IRSs remain fixed, whereas the UAV freely adjusts its heading during flight.
  • A. System Model: The UAV and base station form the primary network, while the UAV helps each IRS transmit its own data to the base station.
  • A. System Model: The UAV flies periodically at fixed altitude Hu over period T, which is divided into N slots of duration δ = T/N.
  • A. System Model: The UAV trajectory is represented by N two-dimensional position sequences, with sufficiently small δ ensuring approximately unchanged position within each slot.
  • A. System Model: The UAV mobility model includes maximum speed Vmax, initial and final locations qI and qF, and corresponding slot-to-slot mobility constraints.
  • A. System Model: Each IRS has M reflecting elements, modeled by a diagonal phase-shift matrix Θk[n] whose mth element applies phase shift θk,m[n].
  • A. System Model: The model uses UAV–IRS, IRS–base-station, and UAV–base-station channels, with all channels modeled as Rician to capture large- and small-scale fading.

K1 K1 + 1hLoS

The paper models Rician channels, frame timing, wake-up scheduling, composite signal detection, and achievable primary rates for the symbiotic radio link. IRS symbols modulate the reflected channel while passive beamforming strengthens UAV transmission.

  • The UAV–IRS and IRS–base-station channels include deterministic LoS and random non-LoS components, with large-scale fading determined by communication distance and path-loss exponents.
  • For the ULA-based IRS, LoS channel vectors use array geometry through element spacing, wavelength, and AoA or AoD cosine terms.
  • Non-LoS channel elements are modeled as independent circularly symmetric complex Gaussian variables with zero mean and unit variance.
  • IRS phase shifts enhance UAV transmission through passive beamforming, while channel acquisition can estimate the concatenated UAV–IRS–base-station and UAV–base-station channels.
  • The frame assumes IRS symbols span L primary symbols, with Ts = LTu and L ≫1, while each UAV slot equals the channel coherence time δ = Tc.
  • The UAV can communicate with at most one IRS per slot; ak[n] indicates whether IRS k is served, and the scheduling variables satisfy corresponding scheduling constraints.
  • With IRS assistance, the received primary signal contains direct and IRS-aided links, while the IRS uses on-off keying with xs,k[n,n1] ∈ {0,1}.
  • The base station applies SIC-based detection because the UAV signal is generally stronger, then uses joint-energy detection for IRS symbols after primary-signal subtraction.

B. Problem Formulation

The paper formulates weighted-sum and fairness BER minimization problems by jointly optimizing UAV trajectory, IRS phase shifts, and IRS scheduling under primary-rate requirements. These problems are difficult because scheduling is binary, design variables are coupled non-convexly, and BER has an implicit integral expression.

  • The weighted-sum objective minimizes BER across IRSs over time while jointly optimizing UAV trajectory, IRS phase shifts, and scheduling.
  • Binary scheduling variables create integer constraints in the weighted-sum formulation.
  • The trajectory, IRS phase-shift, and scheduling variables are intricately coupled, making the optimization problem non-convex.
  • The fairness objective minimizes the maximum BER among IRSs over time using the same jointly optimized design variables.
  • The BER objective is difficult to analyze directly because its expression contains an integral and depends implicitly on the optimization variables.
  • The design replaces BER with a concave, increasing utility of average SNR and uses F(E{γ̄_k[n]}) as a design metric.

III. RELAXATION-BASED ALGORITHM FOR WEIGHTED SUM BER OPTIMIZATION PROBLEM

The weighted-sum BER problem is simplified through optimal IRS phase shifts and solved with alternating relaxation-based optimization of scheduling and UAV trajectory. The relaxation yields binary scheduling variables at convergence, eliminating reconstruction and automatically satisfying UAV rate constraints.

  • The proposed relaxation-based algorithm alternates IRS scheduling optimization with UAV trajectory optimization.
  • For fixed UAV trajectory and scheduling, a closed-form optimal IRS phase-shift matrix is derived and substituted into the original problem.
  • The reformulated problem involves only UAV trajectory and scheduling variables, removing the cosine function and simplifying solution.
  • The relaxation-based algorithm does not require scheduling reconstruction because converged relaxation variables are binary.
  • Binary convergence automatically satisfies the UAV rate constraints in the weighted-sum problem.

A. IRS Scheduling Optimization

With a fixed UAV trajectory, IRS scheduling becomes a linear optimization problem. Its relaxed optimum remains binary, so the scheduling solution requires no reconstruction and preserves the stated constraints.

  • For a given UAV trajectory, the IRS scheduling subproblem is formulated separately from the full weighted-sum BER problem.
  • The scheduling objective and constraints are linear in the scheduling variables, making the subproblem a linear optimization problem.
  • The optimal relaxed scheduling solution is binary despite relaxing the original binary constraint.
  • Because the relaxed solution is already binary, no reconstruction operation is needed.
  • The scheduling subproblem has very low computational complexity because it is a linear optimization problem.

B. UAV Trajectory Optimization

For fixed IRS scheduling, the UAV trajectory problem is reformulated with slack variables and solved using successive convex optimization. First-order approximations convert key non-convex constraints into convex subproblems iteratively.

  • For any fixed IRS scheduling, the UAV trajectory optimization problem is solved as a separate subproblem.
  • The original trajectory problem is difficult because its objective is non-convex and lacks an efficient general optimal solution method.
  • Successive convex optimization is adopted to solve the reformulated trajectory problem.
  • Slack variables z1,k[n] and z3[n] are introduced to recast the non-convex objective and enforce equality at optimality.
  • First-order Taylor expansions provide lower bounds for convex terms, enabling convex approximations of trajectory constraints.
  • Algorithm 1 alternates solving the scheduling and trajectory subproblems until objective improvement falls below a threshold or rmax is reached.

C. Convergence Analysis and Computational Complexity

The proposed alternating-optimization procedures iteratively solve decomposed subproblems, while the fairness problem requires a penalty-based treatment because binary reconstruction can violate primary-rate constraints.

  • Convergence Analysis: Algorithm 1 alternates between solving problems (32) and (40), using each subproblem's solution to initialize the other.The procedure is summarized in Algorithm 1.
  • Fairness BER Optimization: The relaxation-based method cannot solve the fairness problem because binary reconstruction may violate the primary rate requirement.Continuous scheduling solutions can lose constraint (41c) after conversion to binary.
  • Penalty-Based Algorithm: A two-layer penalty-based algorithm addresses fairness optimization by transforming binary constraints into equivalent equalities and penalizing their violations.Slack variables are introduced, and the resulting equality constraints are added to the objective through a penalty term.
  • Penalty Update: The penalty coefficient is initialized relatively large and gradually decreased so the equality constraints are satisfied within a predefined accuracy.A very small initial coefficient would cause the penalty terms to dominate the objective.
  • Inner Optimization: For fixed penalty coefficient, the penalty problem remains non-convex and is solved by alternating optimization over the primary variable blocks.The inner procedure iteratively optimizes {¯a_k[n]}, {a_k[n]}, and {q_u[n]}.

A. Inner layer iteration

The inner layer iteratively optimizes slack variables, scheduling variables, and UAV trajectory, using convex subproblems or successive convex optimization within the penalty framework.

  • Inner layer iteration: The a_k[n] subproblem is convex with a quadratic objective and linear inequality constraints.Standard convex optimization techniques, including interior-point methods, can solve it numerically.
  • Inner layer iteration: The UAV trajectory subproblem is non-concave because of non-convex constraints and is addressed using successive convex optimization.Slack variables and local points yield an equivalent convex formulation.
  • Inner layer iteration: The ¯a_k[n] subproblem is transformed into an equivalent convex optimization problem and can be solved by an interior-point method.Slack variables and local points are introduced for the trajectory-related formulation.
  • Outer layer iteration: The outer layer updates the penalty coefficient according to η = cη, where 0 < c < 1.A larger c can achieve better performance but requires more outer-layer iterations.
  • Convergence: The inner objective is non-increasing and bounded, enabling a stationary point, while decreasing the penalty coefficient ultimately satisfies the equality constraints.The algorithm terminates using objective-decrease or maximum-iteration criteria.
  • Inner layer iteration: The inner layer alternates optimization of ¯a_k[n], a_k[n], and q_u[n] while holding the other variables fixed.These blocks correspond to slack variables, scheduling variables, and UAV trajectory.

Appendix B], this penalty-based framework is guaranteed to converge.

Numerical experiments evaluate convergence, trajectories, scheduling, and utility against benchmark schemes under specified UAV-IRS system settings.

  • Computational Complexity: Algorithm 2 complexity depends on inner and outer iteration counts and the variable dimensions of its scheduling and trajectory subproblems.The main inner-layer costs are O(KN + 2N + 1)^3.5 and O(2KN + 3N + 1)^3.5.
  • Simulation Setup: The simulations use five IRSs and a 755 MHz carrier with 1 MHz bandwidth, UAV altitude 30 m, transmit power 20 dBm, and maximum speed 10 m/s.The UAV starts and ends at q_I = q_F = [15m 0]^T.
  • Weighted Sum BER Optimization: The weighted-sum utility increases quickly and converges within only 3 iterations in both tested periods.This evaluates Algorithm 1's convergence behavior.
  • Weighted Sum BER Optimization: The optimized trajectory visits all IRSs for w1 but only closely flies by IRS 3 for w2, which assigns IRS 3 a lower weight.The UAV spends less time hovering above IRS 3 under w2.
  • Weighted Sum BER Optimization: The IRS scheduling results are binary when optimizing weighted-sum utility under T = 40 s.The result verifies the effectiveness of Algorithm 1.
  • Weighted Sum BER Optimization: The proposed approach substantially outperforms the benchmark methods in average weighted-sum utility value.The reported comparison includes circular trajectories and fixed IRS phase shifts.
  • Weighted Sum BER Optimization: Performance gains increase with the number of reflecting elements because more elements provide higher passive beamforming gain.The proposed approach also outperforms the circular trajectory by leveraging UAV mobility.
  • Weighted Sum BER Optimization: The IRS significantly affects system performance, requiring finely tuned phase shifts in system design.Phase alignment increases the SNR of the UAV-IRS-BS link.

B. Fairness BER Optimization

The fairness BER design uses a penalty-based algorithm whose constraint violation converges rapidly, while optimized UAV mobility, IRS scheduling, and phase shifts improve fairness performance.

  • Algorithm convergence: The penalty-based algorithm reduces constraint violation to 10^-10 after 34 iterations for T = 20 s and satisfies the predefined accuracy for T = 40 s.These results indicate that the penalty-based method effectively handles the binary scheduling constraints.
  • Algorithm convergence: The fairness utility value increases quickly with outer-layer iterations for both T = 20 s and T = 40 s, reaching a fraction of its final value within 4 iterations.
  • UAV trajectory and speed: As T increases, the UAV enlarges and adjusts its trajectory to move closer to each IRS, eventually visiting and hovering above all IRSs when T = 40 s.When the UAV is close to an IRS, the double-channel fading propagation length is reduced, improving the IRS transmission SNR.
  • UAV trajectory and speed: For T = 40 s, the UAV alternates between maximum and zero speed, moving rapidly toward each IRS and then remaining stationary above it.
  • IRS scheduling: The IRS sequentially communicates with each UAV, and the resulting binary scheduling satisfies constraints (42) and (43).
  • Performance evaluation: The proposed fairness scheme improves as T and the number of IRS reflecting elements increase, whereas circular trajectories remain constant with T and fixed phase shifts perform poorly.The fixed-phase scheme's fairness utility nearly approaches zero because the relevant link angles are unaligned, highlighting the need for carefully tuned IRS phase shifts.

APPENDIX A PROOF OF THEOREM 1

The appendix establishes analytical properties used to characterize the system's rate and phase-shift optimization, including a Jensen-based bound and closed-form phase alignment.

  • Theorem 1 proof: The function f(z) = log2(1 + z), z ≥ 0, is concave, so Jensen's inequality gives E{f(z)} ≤ log2(1 + E{z}).
  • Theorem 1 proof: The proof evaluates the required expectations using the channel terms and obtains the corresponding closed-form expression involving M, β1,k[n], β2,k, K1, and K2.
  • Phase-shift characterization: The IRS phase shifts are tuned so the UAV-IRS-BS signal phase aligns with the UAV-BS direct-link phase, enabling coherent combining at the BS.The resulting phase-shift expression is also optimal for maximizing the IRS reflecting rate.
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