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Intelligent Reflecting Surface-Assisted Cognitive Radio System

Jie Yuan, Ying-Chang Liang, Jingon Joung, Gang Feng, Erik G. Larsson

arXiv:1912.10678v1eess.SP

TL;DR

The paper addresses how to improve SU transmission in a spectrum-sharing CR network while controlling PU interference and accounting for imperfect CSI. It introduces multiple IRSs and jointly optimizes SU-TX beamforming with IRS reflecting coefficients under power and interference constraints. Simulations report significant SU achievable-rate improvements for both perfect and imperfect CSI.

  • Problem

    CR improves spectral efficiency through spectrum sharing, while practical CSI errors complicate beamforming under PU interference constraints; IRS offers a route toward improving energy efficiency.

  • Method

    The paper formulates SU-rate maximization for a multiple-IRS downlink MISO CR system and jointly optimizes SU-TX beamforming and IRS reflecting coefficients for perfect and imperfect CSI.

  • Results

    Simulations verify that multiple IRSs can significantly improve the SU achievable rate under both perfect and imperfect CSI.

  • Takeaways & Limitations

    The proposed IRS-assisted CR scheme combines passive IRS assistance with CR spectrum sharing while maintaining the specified SU power and PU interference constraints.

Abstract

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Cognitive radio (CR) is an effective solution to improve the spectral efficiency (SE) of wireless communications by allowing the secondary users (SUs) to share spectrum with primary users. Meanwhile, intelligent reflecting surface (IRS), also known as reconfigurable intelligent surface (RIS), has been recently proposed as a promising approach to enhance energy efficiency (EE) of wireless communication systems through intelligently reconfiguring the channel environment. To improve both SE and EE, in this paper, we introduce multiple IRSs to a downlink multiple-input single-output (MISO) CR system, in which a single SU coexists with a primary network with multiple primary user receivers (PU-RXs). Our design objective is to maximize the achievable rate of SU subject to a total transmit power constraint on the SU transmitter (SU-TX) and interference temperature constraints on the PU-RXs, by jointly optimizing the beamforming at SU-TX and the reflecting coefficients at each IRS. Both perfect and imperfect channel state information (CSI) cases are considered in the optimization. Numerical results demonstrate that the introduction of IRS can significantly improve the achievable rate of SU under both perfect and imperfect CSI cases.

I. INTRODUCTION

The paper combines cognitive radio and multiple IRSs in a downlink MISO system to improve SU rate while respecting SU power and PU interference constraints. It addresses both perfect and imperfect CSI through joint beamforming and reflection optimization.

  • I. INTRODUCTION: The motivation combines CR's spectral-efficiency benefits with IRS's energy-efficiency benefits and low-power passive reflection.IRS elements introduce phase shifts and amplitude variations without active radio-frequency components or substantial additional power consumption.
  • I. INTRODUCTION: The proposed network uses multiple IRSs to assist one SU-TX/SU-RX pair sharing spectrum with multiple PU-RXs.The SU-TX has multiple antennas, while the IRSs provide passive reflecting elements for transmission assistance.
  • I. INTRODUCTION: The design maximizes the SU achievable rate under a total SU-TX power constraint and PU-RX interference temperature constraints.Beamforming and reflecting coefficients are optimized jointly.
  • I. INTRODUCTION: For perfect CSI, a BCD algorithm alternates SOCP beamforming optimization with SDR-based reflection optimization and Gaussian randomization.The relaxation handles rank-1 constraints on reflecting-coefficient matrices.
  • I. INTRODUCTION: For imperfect CSI, ellipsoidal channel uncertainty yields a worst-case robust maxmin problem solved using SDR, the S-Procedure, and alternating variable updates.The formulation targets robustness against CSI uncertainty.
  • I. INTRODUCTION: The work fills a stated gap by introducing multiple IRSs into CR networks while considering both perfect and imperfect CSI.The paper identifies no related work covering both aspects together.

A. Channel Model

The channel model represents direct links with i.i.d. complex Gaussian fading and IRS-related links through composite channels, with SU and PU received signals expressed using the reflecting coefficients.

  • A. Channel Model: Direct SU-TX-to-SU-RX and SU-TX-to-PU-RX channels are modeled as i.i.d. complex Gaussian vectors with zero mean and unit variance.The direct channels are denoted hd,s and hd,p,k.
  • A. Channel Model: SU-TX-to-IRS channels use Rician fading because the deployment assumption provides a line-of-sight path on the reflecting link.The model separates fixed LoS and random NLoS components, with κ1 denoting the Rician factor.
  • A. Channel Model: IRS array responses encode angles of arrival and departure using uniform linear-array vectors with element spacing set to d/λ = 1/2.The angle-dependent response is used to model the LoS component.
  • B. Signal Model: The reflecting coefficients determine phase shifts and amplitude gains, while θ concatenates all IRS coefficients with a final direct-link component.The resulting composite channels determine the received SU signal and PU interference temperature.
  • B. Signal Model: The SU instantaneous SNR and each PU-RX's interference power are derived from the received-signal expressions.The latter quantity is identified as interference temperature in the CR network.

C. Discussion on Channel Estimation

The paper estimates composite downlink channels through TDD-based uplink pilots and uses these estimates for joint beamforming and reflection optimization.

  • C. Discussion on Channel Estimation: Joint optimization requires channel estimation for the composite SU and PU channel matrices.The SU-TX estimates Hs and {Hp,k} rather than every individual direct and reflected channel.
  • C. Discussion on Channel Estimation: Channel reciprocity enables downlink-channel estimation from uplink pilot signals under a time division duplex protocol.The protocol exploits the passive nature of the IRS.
  • C. Discussion on Channel Estimation: Estimating the secondary composite channel requires NL + 1 orthogonal pilots transmitted by the SU-RX over NL + 1 time slots.The first slot estimates the direct channel with all IRS elements off.
  • C. Discussion on Channel Estimation: Each subsequent pilot slot activates one IRS element while the others remain off, and the channel estimates minimize mean square error.The same protocol is used for the primary composite channels, whose details are omitted.
  • C. Discussion on Channel Estimation: For perfect CSI, the joint design is decomposed into SOCP and SDP subproblems within an iterative BCD procedure.This perfect-CSI design provides a basis for the robust imperfect-CSI design.

A. Problem Formulation and Relaxation

The SU rate-maximization problem jointly designs SU-TX beamforming and IRS reflections under interference-temperature and transmit-power constraints. Because the variables are coupled non-convexly, alternating optimization decomposes the task into tractable beamforming and reflecting-coefficient subproblems.

  • A. Problem Formulation and Relaxation: The objective maximizes SU-RX achievable rate by jointly optimizing the IRS reflecting vector θ and SU-TX beamforming vector w.
  • A. Problem Formulation and Relaxation: The formulation constrains PU-RX interference temperature, passive IRS gain, and SU-TX transmit power.The interference threshold is Γ_k and the maximum SU-TX power is P.
  • A. Problem Formulation and Relaxation: Because log(x) is monotonic, maximizing rate is equivalent to maximizing received signal power, but the coupled objective over w and θ remains non-concave.
  • B. Alternating Optimization Algorithm Based on BCD: The original problem is decoupled into two subproblems, one optimizing beamforming and the other optimizing reflecting coefficients.
  • 1) Transmit Beamforming Optimization:: For fixed θ, the beamforming subproblem is reformulated as an SOCP because phase rotation preserves feasibility and objective value.
  • 2) Reflecting Coefficients Optimization:: For fixed w, introducing Θ = θθ† yields an SDP after relaxing its non-convex rank-1 constraint.The relaxed SDP has linear objective and constraints over a convex symmetric positive semidefinite matrix set.

3) Gaussian Randomization for Rank-1 Condition:

The reflecting-coefficient subproblem uses Gaussian randomization to recover feasible rank-1 solutions after semidefinite relaxation, followed by phase quantization and convergence analysis.

  • 3) Gaussian Randomization for Rank-1 Condition:: Gaussian randomization generates rank-1 candidate reflecting solutions from the relaxed matrix Θ using random vectors z drawn from CN(0_NL, I_NL).
  • 3) Gaussian Randomization for Rank-1 Condition:: The matrix decomposition uses the left singular matrix U and singular-value matrix Σ of Θ to construct randomized solutions.
  • 3) Gaussian Randomization for Rank-1 Condition:: After feasibility testing, the feasible randomized candidate with the largest objective value is selected and partitioned into reflecting coefficients for each IRS.
  • 3) Gaussian Randomization for Rank-1 Condition:: Hardware limits require reflecting phases to be quantized uniformly over Q discrete levels, using a log2(Q)-bit quantizer.
  • 3) Gaussian Randomization for Rank-1 Condition:: The BCD algorithm alternately optimizes w and θ, feeds each iteration’s solution into the next, and quantizes the converged phase shifts.
  • 4) Overall Algorithm:: The algorithm is guaranteed to converge when the iteration updates satisfy the stated condition, because the objective is monotonically nondecreasing.
  • 4) Overall Algorithm:: The overall time complexity is O(T1(M^2K^1.5+M^3K^0.5+KN^2L^2M+N^4.5L^4.5 log(1/ε)+N^3L^3+GKNL)).T1 is the number of iterations required to converge for tolerance factor ϵ and interior-point solution accuracy ε.

1 Input: Hs and {Hp,k}

The algorithm initializes the reflecting coefficients and iteratively updates beamforming, randomized reflecting solutions, convergence checks, and final quantized outputs.

  • 1 Input: Hs and {Hp,k}: The procedure initializes θ as an all-one matrix and sets η_a, η_0, δ, and iteration index i before optimization.
  • 1 Input: Hs and {Hp,k}: At each iteration, the updated beamforming vector is obtained by solving SOCP subproblem (P1.1′).
  • 1 Input: Hs and {Hp,k}: A random vector z is generated to obtain the next reflecting solution from the relaxed matrix.
  • 1 Input: Hs and {Hp,k}: The stopping criterion uses the relative change δ = |η_i+1 − η_i| / η_i.
  • 1 Input: Hs and {Hp,k}: After convergence, the algorithm records the optimized beamforming vector and reflecting solution, then extracts each IRS’s coefficient block.
  • 1 Input: Hs and {Hp,k}: The final IRS coefficients are obtained by uniformly quantizing the converged phase values.

20 Obtain

The imperfect-CSI design models channel errors within ellipsoidal uncertainty regions and optimizes robust beamforming and phase shifts against worst-case channels. Semidefinite relaxation and alternating optimization produce an efficient suboptimal solution.

  • A. Problem Formulation with CSI Uncertainty: Introducing Θ = θθ† and W = ww† enables semidefinite relaxation of the rank-1 constraints on reflecting and beamforming matrices.
  • A. Problem Formulation with CSI Uncertainty: The perfect channels are represented as estimated channels plus CSI errors, with errors bounded by ellipsoidal uncertainty regions.Positive-definite scaled inverse covariance matrices determine the uncertainty-region boundaries.
  • A. Problem Formulation with CSI Uncertainty: Robust beamforming and phase shifts are designed by solving a minmax problem over the deterministic channel uncertainty region.
  • A. Problem Formulation with CSI Uncertainty: The robust objective is the worst-case, or minimum, received SNR at the SU-RX.
  • A. Problem Formulation with CSI Uncertainty: The objective is biconvex in Θ and W for fixed channel errors and concave in the errors for fixed Θ and W.
  • A. Problem Formulation with CSI Uncertainty: The transformed problem is mixed-integer and non-convex, so alternating optimization of Θ and W yields an efficient suboptimal solution.

B. Alternating Optimization Based on BCD

For imperfect CSI, the paper converts uncertainty-constrained subproblems into tractable SDPs and alternately optimizes reflecting and beamforming variables using BCD. The resulting Algorithm 2 has an explicit complexity expression and uses Gaussian randomization to recover approximate rank-1 solutions.

  • Imperfect CSI reformulation: S-Procedure reformulates the infinite uncertainty constraints into equivalent semidefinite constraints for the SNR and interference-temperature conditions.This converts the semi-infinite subproblems caused by channel uncertainty into tractable convex SDP formulations.
  • Alternating optimization: BCD decouples the imperfect-CSI problem into two SDP subproblems, which alternately optimize the reflecting-coefficient matrix and beamforming matrix.The converged matrices are followed by Gaussian randomization to obtain approximate reflecting coefficients and beamforming vectors.
  • Complexity: Algorithm 2 has time complexity O(T2((N^4.5L^4.5+M) log(1/ε)+KN^2L^2M)+N^3L^3+M^3+GKNLM).T2 denotes the BCD iterations required for convergence; the expression includes SDP, matrix-decomposition, and Gaussian-randomization costs.

VI. SIMULATIONS RESULTS AND DISCUSSIONS

Simulations evaluate convergence and runtime for the proposed algorithms under perfect and imperfect CSI. The reported BCD procedures converge in fewer than seven iterations, while Algorithm 2 requires higher computational cost than Algorithm 1.

  • Convergence: Algorithms 1 and 2 converge with fewer than seven BCD iterations across the tested numbers of reflecting elements.The evaluation uses M = 4, K = 2, N = 2, ε = 10^-2, ǫ = 10^-4, P = 10 dB, and Γ = 5 dB.
  • Runtime evaluation: For L = 10, 20, 30, and 40, the tested average T2 values are 4.78, 5.46, 5.88, and 6.29, respectively, while T1 is 2.5 for all L.These values are used to evaluate the algorithms’ numerical runtime.

11 Gaussian randomization

The simulations examine Gaussian-randomization outputs, computational complexity, and achievable-rate behavior under varying quantization, power, and IRS configurations. Five phase-quantization bits are sufficient in the tested setting, and the imperfect-CSI algorithm’s complexity is higher.

  • Gaussian randomization: Gaussian randomization produces the approximate beamforming vector and reflecting coefficients from the converged matrix solutions.The procedure extracts the approximate w and phase-shift vectors after convergence.
  • Complexity comparison: The imperfect-CSI algorithm has higher computational complexity than the perfect-CSI algorithm because its optimization uses SDP rather than SOCP for the beamforming subproblem.This comparison is reported together with runtime trends over the number of reflecting elements L.
  • Simulation setup: The achievable-rate evaluation uses a four-antenna SU-TX, a single-antenna SU-RX, two single-antenna PU-RXs, and two IRSs unless otherwise stated.The IRSs are configured with channel and interference-temperature assumptions described in the simulation setup.
  • Quantization: Five bits are sufficient for uniform IRS phase-shift quantization when N = 2, P = 6 dB, and Γ = 5 dB.The paper therefore sets Q = 2^5 for discrete phase shifts in the simulations.

A. Perfect CSI Case

Under perfect CSI, the proposed IRS-assisted CR system is evaluated against random-IRS and no-IRS benchmarks, showing gains from optimized reflections and larger IRS configurations.

  • Benchmark schemes: The perfect-CSI evaluation compares the proposed IRS-assisted CR system with random-IRS and no-IRS benchmark schemes.The random-IRS scheme uses randomly selected reflecting coefficients, while the no-IRS scheme removes the reflect-link channels.
  • Performance versus transmit power: As transmit power P increases, achievable rates increase for all evaluated schemes.The comparison considers Γ = 5 dB, N ∈{1, 2, 3}, and L = 10.
  • Performance versus transmit power: The proposed IRS scheme outperforms the random-IRS and no-IRS references by optimally directing reflected signals toward the SU-RX instead of PU-RXs.Deploying more IRSs further increases the achievable SU-RX rate.
  • Effect of interference limits and IRS size: As the interference temperature limit Γ increases, the achievable rate of all schemes increases.Figure 6 evaluates rate versus Γ for L ∈{10, 30, 60} at P = 10 dB.
  • Effect of interference limits and IRS size: Increasing the number of reflecting elements L significantly improves the proposed IRS scheme, while random-IRS and no-IRS rates remain unchanged or stable.This behavior is attributed to more flexible IRS direction focusing and stronger reflection with larger L; Figure 7 fixes Γ = 7 dB and P = 10 dB.
  • CSI uncertainty: With imperfect CSI, achievable rate decreases because IRS-reflected power does not accurately focus on the intended SU-RX, although degradation becomes trivial as Γ increases.The imperfect-CSI evaluation uses τ = 0.01 and considers Γ ∈{−7, 0, 7, 15} dB.
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