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Energy Efficient Robust Beamforming and Cooperative Jamming Design for IRS-Assisted MISO Networks
Qun Wang, Fuhui Zhou, Rose Qingyang Hu, Yi Qian
TL;DR
Energy-efficient secure transmission is difficult when channel conditions limit secrecy and eavesdropper CSI is uncertain. The paper jointly designs transmit and jamming beamforming with IRS phases under perfect and imperfect CSI, and reports improved energy efficiency over benchmark schemes, including with imperfect CSI.
Problem
Secrecy rates depend on the channel-condition difference between legitimate users and eavesdroppers, and perfect CSI for IRS-to-eavesdropper links is difficult to obtain.
Source: §I · pp. 1–2Method
The paper jointly designs information and jamming beamforming with IRS phase shifts for energy efficiency under perfect and imperfect CSI.
Source: §I · p. 2, §VI · p. 14Results
The proposed design outperforms benchmark schemes in energy efficiency, and IRS improves energy efficiency even under imperfect CSI.
Source: §V · p. 11, §VI · p. 14Takeaways & Limitations
The IRS-assisted MISO network exhibits a tradeoff between energy efficiency and secrecy rate.
Source: §V · p. 12, §VI · p. 14Under imperfect CSI, energy efficiency degrades relative to the perfect-CSI case because of CSI uncertainty.
Source: §V · p. 11
Abstract
from arXiv · showhide
Energy-efficient design and secure communications are of crucial importance in wireless communication networks. However, the energy efficiency achieved by using physical layer security can be limited by the channel conditions. In order to tackle this problem, an intelligent reflecting surface (IRS) assisted multiple input single output (MISO) network with independent cooperative jamming is studied. The energy efficiency is maximized by jointly designing the transmit and jamming beamforming and IRS phase-shift matrix under both the perfect channel state information (CSI) and the imperfect CSI. In order to tackle the challenging non-convex fractional problems, an algorithm based on semidefinite programming (SDP) relaxation is proposed for solving energy efficiency maximization problem under the perfect CSI case while an alternate optimization algorithm based on $\mathcal{S}$-procedure is used for solving the problem under the imperfect CSI case. Simulation results demonstrate that the proposed design outperforms the benchmark schemes in term of energy efficiency. Moreover, the tradeoff between energy efficiency and the secrecy rate is found in the IRS-assisted MISO network. Furthermore, it is shown that IRS can help improve energy efficiency even with the uncertainty of the CSI.
I. INTRODUCTION
The paper studies energy-efficient secure transmission in an IRS-assisted MISO network, with robust design motivated by uncertain eavesdropper CSI. It jointly designs information and jamming beamforming with IRS phases under perfect and imperfect CSI.
I. INTRODUCTION
- Energy efficiency and secrecy are both important goals, while physical-layer secrecy depends on the channel-condition difference between the legitimate user and eavesdroppers.The paper notes that artificial noise and cooperative jamming are approaches considered to address limited secrecy rates.Source: §I · p. 1
A. Related Work and Motivation
- Prior work leaves joint beamforming, friendly jamming, and IRS phase design under imperfect CSI in IRS-assisted MISO networks insufficiently studied.The coupled variables and uncertainty constraints make these optimization problems challenging.Source: A. Related Work and Motivation · p. 3
B. Contribution and Organization
- The paper jointly optimizes information beamforming, cooperative jamming, and IRS phases for energy efficiency under both perfect and imperfect CSI.For perfect CSI it uses alternating optimization with semidefinite relaxation; for bounded channel errors it applies the S-procedure to robust design.Source: B. Contribution and Organization · p. 4
- The authors report that their design outperforms benchmark schemes in energy efficiency, while identifying a tradeoff between energy efficiency and secrecy rate.They also report that IRS can improve energy efficiency under imperfect CSI.Source: B. Contribution and Organization · p. 4
II. SYSTEM MODEL
- The network combines an N-antenna base station and friendly jammer, an M-element IRS, one legitimate user, and K single-antenna eavesdroppers.The base station sends confidential information while the jammer transmits jamming signals.Source: §II · pp. 4–5
- Under imperfect CSI, the model treats IRS-to-eavesdropper channel knowledge as uncertain and represents estimation errors with bounded regions.The uncertainty radii specify the sizes of those regions.Source: §II · p. 4
III. SYSTEM DESIGN WITH PERFECT CSI
The perfect-CSI problem maximizes energy efficiency by jointly choosing the transmit beamformer, jammer vector, and IRS phase shifts. Because the formulation is non-convex, the design alternates between optimizing the beamforming variables and the IRS phases.
III. SYSTEM DESIGN WITH PERFECT CSI
- The perfect-CSI design jointly optimizes the information beamformer, jamming vector, and IRS phase-shift matrix for energy efficiency.The section introduces an alternating algorithm for this non-convex problem.Source: §III · p. 5, A. Problem Formulation · p. 5
A. Problem Formulation
- The formulation is non-convex because its objective is fractional and its constraints are non-convex.The minimum required secure rate is denoted by Rth.Source: A. Problem Formulation · p. 5
- For a fixed IRS phase vector, the beamforming and jamming vectors are optimized; the phase vector is then updated, and the process repeats until convergence.The design variables are coupled, which contributes to the problem's non-convexity.Source: A. Problem Formulation · p. 5
B. Optimizing the Beamforming for a Given w
For a fixed IRS phase vector, the method optimizes the transmit and jamming beamforming matrices. It relaxes rank constraints and uses auxiliary variables and fractional optimization to obtain a tractable iterative solution.
B. Optimizing the Beamforming for a Given w
- For a given IRS phase vector, the method solves for the optimal secure transmit beamformer and jammer vector.The vectors are represented through positive semidefinite matrices, with rank-one constraints corresponding to vector solutions.Source: B. Optimizing the Beamforming for a… · pp. 5–6
- Semidefinite relaxation removes the rank-one constraints that make the beamforming-matrix problem difficult.The resulting problem remains non-convex because of its objective and a constraint involving the transmit and jamming matrices.Source: B. Optimizing the Beamforming for a… · p. 6
- Auxiliary variables transform logarithmic terms into bounds whose optimal values are specified by the method's lemma.The resulting objective is a single-ratio maximization problem solved iteratively using Dinkelbach's method.Source: B. Optimizing the Beamforming for a… · p. 6
- The optimization is recast with a slack variable bounding the eavesdropper terms, yielding the problem in the auxiliary variables and beamforming matrices.The objective denominator includes transmit power and the base-station, jammer, and IRS circuit-power terms.Source: B. Optimizing the Beamforming for a… · p. 6
+ PBS + PG + PIRS)
The final steps solve the convex subproblem and recover beamforming vectors from the optimized matrices. When the matrices are not rank one, Gaussian randomization provides approximate vectors.
+ PBS + PG + PIRS)
- The convex subproblem can be solved with a standard convex optimization tool.The passage identifies this as the final convex problem in the optimization procedure.Source: + PBS + PG + PIRS) · p. 6
- When both optimized matrices are rank one, eigenvalue decomposition recovers the transmit and jamming vectors.For non-rank-one matrices, the method instead uses Gaussian randomization to obtain approximate vectors.Source: + PBS + PG + PIRS) · p. 6
- The fixed-phase beamforming problem alternates updates of the auxiliary variables and the transmit and jamming vectors.The procedure is summarized as Algorithm 1.Source: + PBS + PG + PIRS) · p. 6
C. Optimizing w with (f1, f2)
With the beamforming vectors fixed, the section reformulates IRS phase optimization using semidefinite relaxation and solves the resulting fractional problem iteratively. It then recovers reflection coefficients and an approximate beamformer when needed.
C. Optimizing w with (f1, f2)
- After the transmit and jamming beamformers are obtained, the user and eavesdropper SINRs are expressed using the corresponding effective channels.Source: C. Optimizing w with (f1, f2) · p. 6
- The IRS phase problem is transformed using semidefinite relaxation and an auxiliary variable that bounds the maximum eavesdropper term.Source: C. Optimizing w with (f1, f2) · p. 7
- Dinkelbach’s method solves the concave-convex fractional problem by iteratively solving a convex subproblem.Source: C. Optimizing w with (f1, f2) · p. 7
- The beamformer is recovered by eigenvalue decomposition for rank-one W, or by Gaussian randomization otherwise; reflection coefficients are then obtained from the resulting phase ratios.Source: C. Optimizing w with (f1, f2) · p. 7
D. Convergence Analysis
The proposed iterative method has a non-decreasing objective and converges to a stationary point. When semidefinite relaxation returns non-rank-one matrices, Gaussian randomization yields a sub-optimal solution.
D. Convergence Analysis
- The objective of P1.4 is non-decreasing across iterations and finite under limited system resources, so the method converges to a stationary point.Source: D. Convergence Analysis · p. 7
- For non-rank-one solutions, Gaussian randomization generates candidate vectors, whose feasibility and monotonicity are checked against the previous iteration.Source: D. Convergence Analysis · p. 7
- Gaussian randomization can make the proposed method sub-optimal; the section states that energy efficiency will be compared with MRT beamforming.Source: D. Convergence Analysis · p. 7
IV. SYSTEM DESIGN WITH IMPERFECT CSI
The design extends energy-efficiency maximization to imperfect CSI on the IRS-to-eavesdropper links, jointly optimizing beamforming and IRS phase shifts.
IV. SYSTEM DESIGN WITH IMPERFECT CSI
- The section extends the energy-efficiency maximization problem to imperfect CSI for links from the IRS to the eavesdroppers.Source: §IV · p. 7
- Beamforming vectors and the IRS phase-shift matrix are jointly optimized in the imperfect-CSI setting.Source: §IV · p. 7
- The optimization objective in this setting is to maximize energy efficiency.Source: §IV · p. 7
A. Problem Formulation
The section formulates energy-efficiency maximization under imperfect CSI and solves the resulting problem using alternating optimization.
A. Problem Formulation
- The problem formulation accounts for imperfect CSI between the eavesdroppers and the IRS.Source: A. Problem Formulation · p. 7
- The formulated objective is energy-efficiency maximization.Source: A. Problem Formulation · p. 7
- Problem P3 can be solved using alternating optimization, following the approach used for P1.Source: A. Problem Formulation · p. 7
B. Optimizing the Beamforming with a Given Θ
With the IRS phase-shift matrix Θ fixed, the section optimizes the secure transmit and jamming beamforming vectors. It uses semidefinite relaxation for the rank-one constraints and applies the S-procedure to handle uncertainty in the IRS-to-eavesdropper channels.
B. Optimizing the Beamforming with a Given Θ
- The section optimizes the secure transmit beamforming vector f1 and jamming vector f2 for a given Θ.Source: B. Optimizing the Beamforming with a… · p. 7
- Semidefinite relaxation removes the rank-one constraints on the beamforming matrices; eigenvalue decomposition or Gaussian randomization recovers beamformers afterward.Eigenvalue decomposition is used when both matrices have rank one; otherwise, Gaussian randomization can recover approximate vectors.Source: B. Optimizing the Beamforming with a… · pp. 8–9
- For uncertain IRS-to-eavesdropper channels, slack variables and the S-procedure transform the constraints, and the method alternates between rate-related variables and beamforming matrices.The S-procedure is applied when solving for the beamforming matrices with the rate-related variables fixed.Source: B. Optimizing the Beamforming with a… · pp. 8–9
- The resulting P3.6 problem is convex and can be solved with a standard convex optimization tool.Source: B. Optimizing the Beamforming with a… · p. 9
C. Optimizing w with Given (f1, f2)
The section reformulates the beamformer design using a matrix variable and semidefinite relaxation, then solves convex subproblems within an alternating algorithm. When the recovered matrix is not rank one, Gaussian randomization provides an approximate beamformer.
C. Optimizing w with Given (f1, f2)
- The design uses W = wwH with a rank-one constraint, then applies semidefinite relaxation to reformulate the optimization problem.The relaxed problem is optimized over W while other variables are fixed.Source: C. Optimizing w with Given (f1, f2) · pp. 9–10
- The resulting problem is convex in W when the other variables are fixed, and a standard convex optimization tool can solve it.If the solution W is rank one, eigenvalue decomposition recovers w; otherwise, Gaussian randomization provides an approximate w.Source: C. Optimizing w with Given (f1, f2) · p. 10
- Algorithm 2 alternates optimization for up to T iterations and stops when successive energy-efficiency values differ by at most δ.The algorithm outputs the beamforming variables and the resulting energy efficiency.Source: C. Optimizing w with Given (f1, f2) · p. 10
V. SIMULATION RESULTS
Across the simulations, the perfect-CSI IRS design achieves the highest energy efficiency, while the robust IRS design can still outperform the no-IRS benchmark despite channel uncertainty. The results also show energy-efficiency/secrecy-rate tradeoffs and gains from adding IRS elements.
V. SIMULATION RESULTS
- The simulations compare Efficiency-IRS and Robust-IRS with rate- and power-oriented IRS designs, no IRS, no phase adjustment, and MRT beamforming.The perfect-CSI design is Efficiency-IRS; the imperfect-CSI design is Robust-IRS.Source: §V · p. 11
- The perfect-CSI Efficiency-IRS design achieves the highest energy efficiency, while Robust-IRS still exceeds Efficiency-NoIRS despite CSI uncertainty.The perfect- and imperfect-CSI designs have almost the same, relatively low power consumption.Source: §V · pp. 11–12
- Energy efficiency and secrecy rate trade off: beyond 23 dBm in this setting, further secrecy-rate gains can increase energy use enough to reduce efficiency.Efficiency-IRS maintains a stable secrecy rate to maximize energy efficiency, whereas Rate-IRS continues using available power to raise secrecy rate.Source: §V · p. 12
- As the minimum secrecy-rate threshold rises past the optimal rate, energy efficiency declines, and several benchmark curves disappear where no feasible solution meets the threshold.The cited feasibility cutoffs are 1.5, 2, 2.5, and 3 Bits/Hz/s for Efficiency-NoIRS, Robust-IRS, MRT-IRS, and Efficiency-NoAngle, respectively.Source: §V · p. 12
- Increasing the user–IRS distance raises path loss and reduces reflected-path gain; energy efficiency and secure rate fall, although Efficiency-IRS remains the strongest method.Robust-IRS has lower secrecy rate but higher energy efficiency than Efficiency-NoIRS in the distance comparison.Source: §V · p. 13
- Adding IRS reflecting elements improves energy efficiency, with larger gains for the proposed method than for the benchmarks; Efficiency-IRS and Rate-IRS also improve secure rate.The paper attributes these gains to improved desired-signal enhancement and diversity gain.Source: §V · p. 13
VI. CONCLUSIONS
The proposed IRS-assisted MISO design with independent cooperative jamming outperforms other schemes in energy efficiency. The paper also reports an energy-efficiency/secrecy-rate tradeoff and improved energy efficiency with IRS under imperfect CSI.
VI. CONCLUSIONS
- The proposed method outperforms other schemes in energy efficiency, and IRS can improve energy efficiency even under imperfect CSI.The paper also reports a tradeoff between secrecy rate and energy efficiency.Source: §VI · p. 14
- The design jointly optimizes beamforming, jamming precoders, and the IRS phase-shift matrix under perfect and imperfect CSI.Source: §VI · p. 14
- Two alternating algorithms are proposed for the non-convex fractional optimization problems.Source: §VI · p. 14
- The proposed alternating algorithm could be extended to multi-IRS, multi-UE MIMO networks, which the authors leave for future work.Source: §VI · p. 14