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Resource Allocation for IRS-assisted Full-Duplex Cognitive Radio Systems

Dongfang Xu, Xianghao Yu, Yan Sun, Derrick Wing Kwan Ng, Robert Schober

arXiv:2003.07467v1cs.IT

TL;DR

The paper tackles resource allocation for IRS-assisted FD cognitive-radio networks, where simultaneous UL/DL operation creates coupled optimization and PU-interference challenges under imperfect CSI. It jointly designs secondary beamforming, UL powers and reception, and IRS phases using a safe approximation and iterative BCD framework. The method converges to a stationary point of the approximated problem, while simulations report higher secondary sum rates, robustness to PU-CSI uncertainty, and effective interference mitigation.

  • Problem

    Resource allocation is challenging in IRS-assisted FD cognitive-radio networks because simultaneous UL/DL transmissions, direct and reflected paths, and coupled design variables interact under PU-interference constraints and imperfect CSI.

  • Method

    The paper jointly optimizes DL transmit beamformers, UL receive beamformers and powers, and IRS phase shifts using safe approximation, semidefinite relaxation, penalty methods, SCA, and iterative BCD.

  • Results

    The proposed scheme achieves substantially higher secondary-system sum rates than three baseline schemes, remains robust to imperfect PU CSI, and effectively mitigates FD-network interference.

  • Takeaways & Limitations

    IRSs show strong potential for improving secondary-network performance while managing the various interference forms in FD cognitive-radio networks.

Abstract

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In this paper, we investigate the resource allocation design for intelligent reflecting surface (IRS)-assisted full-duplex (FD) cognitive radio systems. In particular, a secondary network employs an FD base station (BS) for serving multiple half-duplex downlink (DL) and uplink (UL) users simultaneously. An IRS is deployed to enhance the performance of the secondary network while helping to mitigate the interference caused to the primary users (PUs). The DL transmit beamforming vectors and the UL receive beamforming vectors at the FD BS, the transmit power of the UL users, and the phase shift matrix at the IRS are jointly optimized for maximization of the total sum rate of the secondary system. The design task is formulated as a non-convex optimization problem taking into account the imperfect knowledge of the PUs' channel state information (CSI) and their maximum interference tolerance. Since the maximum interference tolerance constraint is intractable, we apply a safe approximation to transform it into a convex constraint. To efficiently handle the resulting approximated optimization problem, which is still non-convex, we develop an iterative block coordinate descent (BCD)-based algorithm. This algorithm exploits semidefinite relaxation, a penalty method, and successive convex approximation and is guaranteed to converge to a stationary point of the approximated optimization problem. Our simulation results do not only reveal that the proposed scheme yields a substantially higher system sum rate for the secondary system than several baseline schemes, but also confirm its robustness against CSI uncertainty. Besides, our results illustrate the tremendous potential of IRS for managing the various types of interference arising in FD cognitive radio networks.

I. INTRODUCTION

The paper addresses spectrum-efficient IRS-assisted FD cognitive radio networks, where simultaneous transmissions intensify interference and existing IRS designs do not directly apply. It jointly optimizes secondary-network resources and IRS phase shifts to improve secondary sum rate while mitigating PU interference.

  • Motivation: FD communication can potentially double spectral efficiency versus traditional HD cognitive radio networks, but simultaneous UL and DL transmissions increase interference to PUs and within the secondary system.The cited interference includes PU interference, self-interference, and co-channel interference.
  • Motivation: Existing FD cognitive-radio resource-allocation designs may fail to satisfy PU QoS requirements in unfavorable and uncontrollable propagation environments.Such failures can severely restrict secondary communication because PUs have higher spectrum-utilization priority.
  • IRS rationale: IRSs provide additional resource-allocation degrees of freedom by adaptively shaping propagation and combining reflected and direct signals constructively or destructively.These mechanisms can enhance desired-signal power or suppress detrimental interference.
  • Research gap: Prior IRS-assisted designs considered HD systems and therefore cannot exploit the full potential of IRSs or apply directly to IRS-assisted FD cognitive-radio networks.The paper identifies the coupled UL/DL transmissions, direct and reflected paths, and jointly optimized variables as making the FD design challenging.
  • Contribution: The paper integrates IRSs into FD cognitive radio and jointly optimizes DL beamformers, UL transmit power, UL receive beamformers, and IRS phase shifts to maximize secondary sum rate.The design accounts for concurrent spectrum sharing while protecting PU QoS.
  • Contribution: The proposed BCD-based approach converges to a stationary point of the approximated optimization problem, while simulations show improved secondary performance and effective PU-interference mitigation.The reported results also indicate robustness to imperfect PU CSI.

II. SYSTEM MODEL

The system is an underlay IRS-assisted FD cognitive-radio network in which a multi-antenna secondary BS simultaneously serves HD UL and DL users while sharing spectrum with PUs. IRS phase shifts shape direct and reflected links, and the model includes SI, CCI, and related channel components.

  • Spectrum sharing: The underlay model permits concurrent spectrum use provided interference leakage to PUs remains below a specified threshold.PU QoS is impaired by spectrum sharing, motivating explicit interference management.
  • Network configuration: The network contains one primary transmitter, I PUs, one secondary FD BS, J UL users, and K DL users.The PUs and secondary users are single-antenna HD devices, while the secondary BS has N_T > 1 antennas.
  • Full-duplex operation: The secondary FD BS simultaneously transmits to DL users and receives UL signals in the same frequency band.The model assumes N_T ≥ J to facilitate reliable UL signal detection.
  • IRS model: The IRS uses M programmable phase shifters controlled through a diagonal phase-shift matrix Ψ.Each element introduces an independently tunable phase shift ψ_m within [−π, π].
  • Signal and channel model: DL beamformers and UL-user transmit powers determine the secondary transmissions, while the received signals include direct, reflected, SI, and co-channel interference components.The channel definitions cover BS–IRS, IRS–user, BS–user, BS–PU, IRS–PU, and UL-user links.

B. Channel State Information

The paper assumes slowly time-varying channels and perfect secondary-network CSI but models PU-channel uncertainty explicitly. Bounded estimation-error sets support a worst-case resource-allocation formulation.

  • CSI assumptions: The secondary FD BS is assumed to have perfect CSI for all links within the secondary network.This CSI is obtained with assistance from secondary users and the IRS.
  • PU-CSI uncertainty: PU CSI may be outdated because PUs cannot directly interact with the secondary BS and may remain idle for long periods.PU channel information is acquired only occasionally when PUs are active.
  • Robust formulation: A worst-case optimization framework is developed to account for PU-CSI uncertainty during resource allocation.The resulting interference constraints are subsequently approximated before applying the iterative solution method.
  • Uncertainty model: The model represents PU-channel estimates as the true channels plus unknown estimation errors for BS–PU, IRS–PU, and PU–UL-user links.The error sets are bounded by ε_D,i, ε_R,i, and ε_i,j.

III. RESOURCE ALLOCATION PROBLEM FORMULATION

The paper formulates joint resource allocation for an IRS-assisted full-duplex cognitive radio secondary network, maximizing its sum rate while limiting interference to primary users under imperfect CSI. The resulting problem is highly non-convex, motivating a polynomial-time iterative BCD solution that alternates optimization blocks using SCA, SDR, and closed-form receive beamforming.

  • A. Performance Metrics: Residual self-interference remains after cancellation because of the FD receiver’s limited dynamic range and is modeled as Gaussian distortion noise.Its variance is proportional to transmit power, with residual-interference impact captured by η.
  • B. Optimization Problem Formulation: The design jointly optimizes secondary DL beamforming, UL receive beamforming, UL transmit powers, and the IRS phase-shift matrix.The objective is the secondary system sum rate.
  • B. Optimization Problem Formulation: The formulation constrains PU interference leakage not to exceed each PU’s maximum tolerance despite imperfect CSI.Transmit-power limits constrain the FD BS and each secondary UL user, while the IRS phase matrix has unit-modulus diagonal components.
  • B. Optimization Problem Formulation: The resource-allocation problem is highly non-convex because variables are coupled and the model includes a non-convex objective, unit-modulus constraint, and semi-infinite constraint.The authors state that a globally optimal solution is generally intractable.
  • IV. SOLUTION OF THE OPTIMIZATION PROBLEM: A safe approximation converts the PU-interference constraint into convex constraints, after which the approximated problem remains highly non-convex.The approximation is conservative: every feasible solution of the approximated problem is feasible for the original problem.
  • A. Performance Metrics: The reflected interference term is approximately 10^-8 of the direct residual self-interference term under the stated 100-meter IRS and path-loss example.This supports neglecting reflected interference relative to self-interference in the performance model.
  • IV. SOLUTION OF THE OPTIMIZATION PROBLEM: BCD divides the variables into {w_k, p_j}, {v_j}, and {Ψ}, optimizing one block while fixing the others.SCA and SDR handle transmit beamforming and UL power, closed-form optimization handles receive beamforming, and penalty method plus SCA handles Ψ.

A. Transformation of the Semi-Infinite Constraints

The paper replaces the semi-infinite PU-interference constraint with a conservative tractable formulation. It first upper-bounds the coupled direct-and-reflected interference, then applies the S-Procedure and block-specific convexification, while solving remaining non-convex subproblems iteratively.

  • A. Transformation of the Semi-Infinite Constraints: After transformation, constraints are convex with respect to selected variable blocks but remain non-convex in Ψ because of the quadratic term ΨBΨ^H.The reformulated constraints are convex in p_j, W_k, or jointly in p_j and W_k where specified.
  • A. Transformation of the Semi-Infinite Constraints: Coupling between direct and reflected paths makes it difficult to obtain an LMI jointly convex in the beamforming vector and IRS phase matrix.This motivates the safe approximation of the interference constraint.
  • A. Transformation of the Semi-Infinite Constraints: The inequality |a + b + c|^2 ≤ 3|a|^2 + 3|b|^2 + 3|c|^2 produces a tractable upper bound and a subset of feasible interference constraints.The resulting approximated problem is then used for the remainder of the algorithm.
  • A. Transformation of the Semi-Infinite Constraints: The safe approximation preserves feasibility: every feasible solution of the approximated problem is feasible for the original problem.Thus, a stationary point of the approximated problem is a feasible suboptimal solution of the original problem.
  • A. Transformation of the Semi-Infinite Constraints: The S-Procedure transforms implications involving quadratic uncertainty constraints into linear matrix inequality constraints under a stated strict-feasibility condition.The paper introduces quadratic functions and requires a point satisfying the strict inequality condition.
  • B. Optimizing {W_k, p_j} for Given Ψ and v_j: For fixed Ψ and v_j, the method rewrites SINR-related terms and applies SCA to obtain iterative suboptimal solutions for transmit beamforming and UL power.The negative objective is decomposed into convex components and concave components are replaced by global underestimators.
  • B. Optimizing {W_k, p_j} for Given Ψ and v_j: The SCA procedure initializes a feasible point, repeatedly solves the convexified problem, stores intermediate solutions, and terminates on convergence.The algorithm is guaranteed to converge to a locally optimal solution of the addressed subproblem.
  • B. Optimizing {W_k, p_j} for Given Ψ and v_j: SDR removes the rank-one constraint C6, allowing the relaxed problem to be solved by standard convex solvers such as CVX.Theorem 1 states that when P_DL^max > 0, an optimal beamforming matrix satisfying Rank(W_k) ≤ 1 can always be obtained.

C. Optimizing vj for Given Ψ, Wk, and pj

For fixed IRS phases, DL beamforming, and UL powers, the paper optimizes each FD base-station receive beamformer to maximize the corresponding UL receive SINR. The resulting problem has an equivalent convex form with a closed-form solution.

  • C. Optimizing v_j for Given Ψ, W_k, and p_j: For fixed Ψ, W_k, and p_j, each receive beamforming vector v_j is selected to maximize the corresponding UL receive SINR.The receive beamformer is obtained by solving an optimization problem for each uplink user.
  • C. Optimizing v_j for Given Ψ, W_k, and p_j: The UL sum-rate optimization is equivalent to maximizing each user’s receive SINR, with v_j chosen so equality constraint C7 is satisfied.The scalar normalization parameter is selected so v_j^H b_hj = 1.
  • C. Optimizing v_j for Given Ψ, W_k, and p_j: The receive-beamforming subproblem can be recast as an equivalent convex optimization problem.Its optimal solution is given in closed form using the defined channel-related quantities.

D. Optimizing Ψ for Given Wk, pj, and vj

For fixed transmit and receive variables, the IRS phase-shift design is reformulated using semidefinite lifting, penalty optimization, and successive convex approximation. The resulting iterative procedure converges to a stationary point of the penalized problem, while the overall BCD method converges to a stationary feasible suboptimal solution of the original problem's safe approximation.

  • Non-convexity: Non-convexity arises from the objective, constraints, and rank-one requirement on Θ.The rank-one constraint is needed to recover Θ = eθeθ^H, while the objective and selected constraints remain non-convex.
  • Rank-one relaxation: Semidefinite relaxation removes the rank-one constraint, but the relaxed solution may not be rank one and Gaussian randomization does not guarantee overall BCD convergence.The method therefore replaces the rank-one condition with a difference-of-convex formulation and a penalty approach.
  • Penalty formulation: The penalty method penalizes matrices with rank greater than one, and increasing χq toward infinity makes the penalized problem equivalent to the rank-constrained problem.The penalty factor χ is increased through a sequence χq.
  • Successive convex approximation: Successive convex approximation constructs global underestimators and solves convex subproblems whose objective values are monotonically non-increasing.Each convex subproblem can be solved using CVX, progressively tightening the upper bound.
  • Convergence: The overall BCD solution is a feasible suboptimal solution of the original problem because the maximum-interference constraint was safely approximated.The BCD iterations update downlink beamformers and uplink powers, receive beamformers, and IRS phases in sequence.
  • Convergence: The phase-shift algorithm converges to a stationary point, and the overall BCD sequence converges to a stationary value in polynomial time.Because the interference constraint uses a safe approximation, the resulting stationary point is feasible but potentially suboptimal for the original problem.

V. SIMULATION RESULTS

The simulation section evaluates the proposed resource-allocation scheme for the IRS-assisted full-duplex cognitive-radio network. The study uses the schematic system model shown in Figure 2.

  • Simulation study: The simulations assess the proposed resource-allocation scheme's secondary-network performance using the IRS-assisted full-duplex cognitive-radio system model.The simulated network is represented schematically in Figure 2.

A. Simulation Setup

The simulations model one secondary-network sector with randomly distributed primary and secondary users, bounded PU-CSI errors, and IRS-assisted reflected channels. Three baselines provide comparisons against fixed beamforming, no IRS, and half-duplex operation.

  • Network configuration: The default network contains I = 2 PUs, K = 2 secondary DL users, and J = 3 secondary UL users distributed uniformly and randomly.The IRS is 50 meters from the secondary FD BS.
  • CSI uncertainty: PU CSI errors are bounded, and the maximum normalized estimation error is represented by υ2.The simulations study performance under imperfect PU channel knowledge.
  • Channel model: The model uses Rayleigh fading for direct paths and Rician fading for reflected paths, with separate direct- and reflected-path loss models.The reflected-path model uses αBR = 2.1 and αRU = 2.3, while the direct-path exponent is αBU = 3.99.
  • Baseline schemes: Baseline scheme 3 operates the secondary BS in half-duplex mode, separating DL and UL transmission into equal-duration orthogonal time slots.Its total sum rate is multiplied by one half for the time-sharing loss, while CCI and SI are absent.

C. Convergence of Algorithm 3

The simulations show monotonic convergence of the proposed BCD algorithm and higher average system sum rates than the baselines across power, user, antenna, IRS-element, and CSI-uncertainty settings. Increasing antennas and IRS elements improves performance, with diminishing growth as antenna count rises.

  • C. Convergence of Algorithm 3: The proposed BCD algorithm monotonically converges to a stationary point in all three tested configurations.It converges within 10 iterations for Case 1 and requires roughly 30 iterations for Case 2.
  • C. Convergence of Algorithm 3: The convergence iteration count is more sensitive to the number of users than to the number of antennas and reflecting elements.Larger antenna and IRS dimensions require about 10 extra iterations, whereas more users substantially enlarge the optimization problem.
  • D. Average System Sum Rate versus Maximum DL Transmit Power: The proposed scheme outperforms all baseline schemes in average system sum rate as maximum DL transmit power increases.The gain is attributed to jointly optimizing IRS phases, DL beamformers, UL powers, and UL receive beamformers.
  • D. Average System Sum Rate versus Maximum DL Transmit Power: Baseline performance is lower because fixed beamforming, absent IRS degrees of freedom, or half-duplex scheduling limits interference management or spectral efficiency.Half-duplex operation avoids CCI and SI but loses efficiency through orthogonal DL and UL time resources.
  • E. Average System Sum Rate versus Number of Secondary Users: Average system sum rates increase with the numbers of DL and UL users, while the proposed scheme remains above the baselines.The increase is associated with multiuser diversity; baseline limitations become more pronounced as user counts grow.
  • E. Average System Sum Rate versus Number of Secondary Users: Average system sum rates improve with more FD-BS antennas, but channel hardening produces a diminishing growth rate as NT increases.The proposed scheme increases faster with NT than the baseline schemes.
  • E. Average System Sum Rate versus Number of Secondary Users: Increasing IRS elements with fixed NT yields a larger performance gain than increasing NT with fixed IRS size.Additional IRS elements provide more degrees of freedom for BS-IRS-user channels and improve both DL and UL beamforming gain.

G. Average System Sum Rate versus Maximum Normalized Channel Estimation Error

The proposed scheme’s average system sum rate decreases as CSI uncertainty increases but remains superior to the baseline schemes across the considered error range. Its robust design also reduces primary-user outage probability compared with a non-robust scheme.

  • Average system sum rate: The average system sum rate decreases as the maximum normalized channel estimation error υ2 increases.The secondary BS becomes more conservative and allocates more degrees of freedom to satisfy the interference-leakage constraint.
  • Average system sum rate: Across the entire υ2 range, the proposed scheme significantly outperforms the three baseline schemes.Jointly optimizing the available degrees of freedom mitigates interference leakage more effectively under CSI uncertainty.
  • Baseline comparisons: Baseline scheme 2 is less sensitive because it does not deploy an IRS, so only imperfect CSI of the direct paths affects performance.This still leads to a performance loss as channel-estimation error increases.
  • Primary-user outage: For target interference leakage tolerances p^tar_i ≤ −90 dBm, the proposed and baseline schemes’ primary-user outage probabilities decrease to zero, whereas the non-robust scheme still has outages.The outage probability is the probability that secondary-network interference leakage to the i-th PU exceeds p^tar_i.
  • Proposed design: The proposed resource allocation jointly optimizes DL beamforming, UL receive beamforming, UL transmit power, and IRS phase shifts in the robust FD cognitive-radio design.The intractable interference-leakage constraint is safely approximated before optimization.
  • Proposed design: The BCD algorithm combines SCA, SDR, closed-form UL receive beamformers, and a penalty method, and converges to a stationary point of the approximated problem.The IRS unit-modulus constraint is transformed into a rank-constrained problem before penalty-based SCA processing.
  • Overall findings: Simulation results show significant system sum-rate improvement over three baselines, robustness to imperfect PU CSI, and IRS-based mitigation of interference in FD cognitive radio networks.These findings support IRS integration for improving secondary-network performance while mitigating interference to PUs.

APPENDIX

The appendix develops optimality and convergence arguments for the relaxed and penalized subproblems. It uses convexity, strong duality, KKT conditions, matrix-rank structure, and limiting arguments to establish the stated solution properties.

  • Proof of Theorem 1: The relaxed problem is jointly convex in its optimization variables, satisfies Slater’s condition, and therefore has zero duality gap.The appendix introduces the Lagrangian and associated semidefinite and scalar multipliers for the constraints.
  • Proof of Theorem 1: KKT analysis with respect to each DL beamforming matrix investigates the structure of the optimal solution.The proof uses the optimal Lagrange multipliers and the activity of the DL transmit-power constraint.
  • Proof of Theorem 1: The optimal DL beamforming matrix satisfies a rank property derived from the structure of the dual matrix and channel randomness.The proof constructs a unit-norm principal eigenvector and expresses the optimal beamforming solution through it.
  • Proof of Theorem 1: The beamforming solution is scaled so that the DL transmit-power constraint is satisfied.The appendix states that the relevant parameter can be tuned to enforce constraint C1.
  • Penalty-method convergence: The convergence proof defines objective functions for the original and penalized problems and compares their values along solutions indexed by the penalty factor.Continuity and norm nonnegativity support the limiting inequalities.
  • Penalty-method convergence: For a limit point of the generated sequence, an infinite convergent subsequence is selected and the inequalities are taken to the limit.The resulting relation establishes the claimed optimality conclusion for the penalized formulation.
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