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Intelligent Reflecting Surface (IRS)-Aided Covert Wireless Communications with Delay Constraint

Xiaobo Zhou, Shihao Yan, Qingqing Wu, Feng Shu, Derrick Wing Kwan Ng

arXiv:2011.03726v2cs.ITeess.SP

TL;DR

Covert communication remains constrained by the need to hide transmission from a warden, motivating IRS assistance. The paper jointly designs transmit power and IRS reflection coefficients under global CSI and without Willie’s instantaneous CSI, showing that IRS deployment can substantially improve covert communication performance.

  • Problem

    Covert communication must hide transmission existence under stringent covertness requirements, which can limit achievable covert rates.

  • Method

    The paper jointly designs Alice’s transmit power and IRS reflection coefficients for global CSI and for the case without Willie’s instantaneous CSI, using PSCA and a low-complexity two-stage algorithm in the global-CSI case.

  • Results

    With global CSI, perfect covertness with non-zero transmit power is achievable for a single-antenna transmitter when the reflected path to Willie is better than the direct path; IRS-assisted schemes also outperform the no-IRS system.

  • Takeaways & Limitations

    Properly designed IRS reflection coefficients can enhance Bob’s received signal while deteriorating Willie’s detection performance in covert communications.

Abstract

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This work examines the performance gain achieved by deploying an intelligent reflecting surface (IRS) in covert communications. To this end, we formulate the joint design of the transmit power and the IRS reflection coefficients by taking into account the communication covertness for the cases with global channel state information (CSI) and without a warden's instantaneous CSI. For the case of global CSI, we first prove that perfect covertness is achievable with the aid of the IRS even for a single-antenna transmitter, which is impossible without an IRS. Then, we develop a penalty successive convex approximation (PSCA) algorithm to tackle the design problem. Considering the high complexity of the PSCA algorithm, we further propose a low-complexity two-stage algorithm, where analytical expressions for the transmit power and the IRS's reflection coefficients are derived. For the case without the warden's instantaneous CSI, we first derive the covertness constraint analytically facilitating the optimal phase shift design. Then, we consider three hardware-related constraints on the IRS's reflection amplitudes and determine their optimal designs together with the optimal transmit power. Our examination shows that significant performance gain can be achieved by deploying an IRS into covert communications.

I. INTRODUCTION

IRSs are introduced as low-cost, passive surfaces that can reshape wireless propagation, motivating their use to improve covert communication, where hiding transmission existence complements content protection. This work jointly designs transmit power and IRS reflection coefficients under global CSI and without Willie’s instantaneous CSI.

  • IRSs use reconfigurable passive elements with controllable amplitudes and phase shifts to customize propagation for signal enhancement or interference suppression.
  • Covert communication hides the existence of a wireless transmission, addressing a privacy goal that physical-layer secrecy alone does not provide.
  • Stringent covertness requirements can keep achievable covert rates low, while IRSs can enhance legitimate reception and deteriorate warden reception.
  • The study jointly designs Alice’s transmit power and IRS reflection amplitudes and phase shifts for global CSI and for the case without Willie’s instantaneous CSI.For the latter case, the covertness constraint is derived analytically and three reflection-amplitude constraints are considered.
  • With global CSI, perfect covertness with non-zero transmit power is achievable for a single-antenna Alice when the reflected Alice-IRS-Willie channel is better than the direct Alice-Willie channel.The IRS-reflected signal can cancel Alice’s direct signal at Willie; without an IRS, single-antenna Alice cannot achieve this perfect covertness.
  • The considered system uses single-antenna Alice, Bob, and Willie, with an IRS of N passive elements whose reflection coefficients are dynamically adjustable.The work focuses on delay-constrained covert communication with a finite number L of channel uses.

C. Transmission from Alice to Bob

The paper evaluates Alice-to-Bob communication quality through Bob’s SNR while accounting for finite-blocklength decoding and Willie’s detection constraint. It maximizes Bob’s SNR by jointly choosing transmit power and IRS reflection beamforming subject to covertness and practical constraints.

  • Finite channel uses make Bob’s decoding error probability non-negligible, so effective throughput is modeled as LR(1 − δ).Effective throughput increases with L, but additional observations also improve Willie’s detection performance.
  • The KL-divergence covertness measure increases with Willie’s received power, motivating transmit-power and reflection-beamforming designs that limit that received power.
  • The optimization maximizes Bob’s SNR γb subject to D(P0|P1) ≤ 2ε^2 and other practical constraints.The formulation uses γb to evaluate communication quality for a fixed transmission rate.
  • The global-CSI setting assumes all channel state information is publicly available to obtain an upper bound on the performance gain from introducing the IRS.The paper notes that accurate CSI acquisition is practically challenging because IRS deployment increases the number of channel coefficients requiring estimation.

A. Optimization Problem and Perfect Covertness Condition

The paper jointly optimizes Alice’s transmit power and IRS reflection beamforming under covertness, power, and reflection-coefficient constraints. It establishes when perfect covertness is feasible and develops PSCA-based and lower-complexity solution approaches.

  • Optimization formulation: The optimization maximizes Bob’s received SNR by jointly designing transmit power and IRS reflection coefficients under covertness and hardware constraints.The reflection coefficients satisfy |v_n| ≤ 1, reflecting independently adjustable amplitudes and phase shifts in the considered model.
  • Perfect covertness condition: Perfect covertness with non-zero transmit power is achievable when the IRS-assisted Alice-to-Willie reflected channel can cancel the direct Alice-to-Willie channel.The condition is expressed through the reflected-path coefficients and the direct-path coefficient, with feasibility requiring Σ_n|a_n| ≥ |h_aw|.
  • Perfect covertness condition: When the reflected-path channel quality exceeds the direct-path channel quality, properly designed IRS beamforming can reduce Willie’s received covert information energy.The single-antenna setup demonstrates this effect and contrasts with finite-blocklength systems without an IRS, where perfect covertness is not achievable.
  • Joint design solution: The general joint-design problem is non-convex because of its non-concave objective, non-convex covertness constraint, and rank-one constraint after reformulation.The paper introduces W = ww^H, identifies a convex reformulation of the covertness constraint, and handles the remaining rank-one condition with successive approximation and penalty methods.
  • Joint design solution: The PSCA algorithm iteratively applies convex approximations and an exact penalty term, with convergence to a locally optimal solution.The penalty formulation enlarges the feasible set using a slack variable and ultimately drives the slack to zero when the penalty parameter reaches its upper bound.

C. Low-Complexity Algorithm

The paper proposes a low-complexity two-stage algorithm to balance covert-transmission performance and computational complexity. IRS reflection beamforming is designed first, followed by Alice’s transmit-power optimization.

  • Two-stage design: The two-stage algorithm designs the IRS reflection beamforming first and determines Alice’s transmit power second.This decomposition is intended to balance covert transmission performance against the computational burden of the joint optimization.
  • IRS beamforming design: The beamforming stage uses the ratio of received energy at Bob to received energy at Willie because Willie’s covertness constraint is governed by received energy.The divergence constraint is described as monotonically increasing with the relevant received energy at Willie.

1) IRS Beamforming Design:

The IRS beamforming subproblem is difficult because its objective is highly non-concave and its unit-modulus constraints are non-convex. The paper therefore derives bounds and an analytic phase-alignment design.

  • IRS beamforming design: The IRS beamforming objective is highly non-concave and includes non-convex reflection-coefficient constraints, preventing direct solution of the subproblem.The problem also differs from a generalized Rayleigh quotient because multiple reflection-amplitude and phase limits must be enforced.
  • Bound construction: The paper constructs a concave lower bound and an upper bound on u^H A u to obtain a tractable lower-bound optimization problem.The upper bound uses the maximum eigenvalue of A and the bounded norm of the augmented reflection vector.
  • Analytic phase design: For a fixed feasible solution, the analytic design aligns each IRS phase with the corresponding element of f and sets every reflection amplitude to one.The resulting vector is u = e^{j arg(f)}, and the construction is identified as sub-optimal relative to the more general design.

2) Transmit Power Design:

The section derives transmit-power solutions under covertness constraints, using approximation for analytical tractability and a lower-complexity two-stage design. It also identifies when that approximation cannot achieve perfect covertness.

  • Transmit-power approximation: The conservative approximation replaces the transcendental covertness equation with an analytically tractable transmit-power expression.The exact optimum satisfies the original covertness constraint with equality, whereas the approximation is introduced for lower complexity.
  • Transmit-power approximation: The approximate optimal transmit power decreases as the required covertness level ǫ decreases.This reflects the more stringent covertness requirement at smaller ǫ.
  • Low-complexity algorithm: The two-stage algorithm designs IRS reflection beamforming first and then determines Alice’s transmit power to explicitly satisfy covertness.Its main computational cost comes from calculating f and Pa, and its complexity is much lower than Algorithm 1.
  • Algorithm limitation: The low-complexity two-stage algorithm cannot achieve perfect covertness with non-zero transmit power in problem P1′.This limitation results from multiple lower-bound approximations that make Alice’s transmit power approach 0 as ǫ → 0.
  • Perfect-covertness design: For perfect-covertness design, phase choices cancel Alice’s direct signal at Willie when κ = 1, while the corresponding scaled coefficients remain feasible for κ > 1.The construction uses v_n = e^j(arg(a_n)−π−arg(h_aw)) with P_a = P_max for κ = 1.

IV. COVERT COMMUNICATION DESIGN WITHOUT WILLIE’S INSTANTANEOUS CSI

Without Willie’s instantaneous CSI, the paper replaces the expected KL-divergence constraint with a closed-form equivalent constraint and designs IRS phases, amplitudes, and transmit power accordingly. The design covers three practical amplitude-control scenarios.

  • Covertness formulation: When Willie’s instantaneous CSI is unavailable, the model assumes Alice and the IRS know channel distributions while Willie knows instantaneous channel realizations.Covertness is measured using the expected KL divergence over channel realizations.
  • Covertness formulation: Theorem 2 transforms the mathematically intractable expected KL-divergence constraint into the analytical constraint (35).The resulting form facilitates joint design of the IRS reflection beamforming and Alice’s transmit power.
  • IRS phase design: The covertness constraint is independent of IRS phase shifts, so optimal phases maximize Bob’s SNR.The phase design sets θ_n + arg(b_n) = arg(h_ab) for every IRS element.
  • IRS amplitude design: The design considers fixed unit amplitudes, a common amplitude coefficient, and independently bounded amplitudes 0 ≤ ρ_n ≤ 1.The third case is reformulated as a convex optimization problem solvable with CVX.
  • Joint design trade-off: Increasing transmit power or the common IRS amplitude can improve Alice-to-Bob communication quality but may reduce covertness.The optimal solution therefore balances communication quality against the covertness constraint, which holds with equality at optimum.
  • Joint design trade-off: The optimal common IRS amplitude decreases as χ_aw or χ_rw increases because stronger channels toward Willie make detection easier.The IRS reduces its amplitudes to maintain the same covertness level.

V. NUMERICAL RESULTS

The numerical evaluation uses a three-dimensional Alice–IRS–Bob–Willie setup with a uniform rectangular IRS array. Figure 2 reports Bob’s SNR against the number of IRS elements for different covertness levels.

  • Simulation setup: The numerical setup places Alice, IRS, Bob, and Willie at specified three-dimensional coordinates and uses an IRS with N = N_xN_z reflecting elements.The IRS is modeled as a uniform rectangular array.
  • Simulation setup: The default parameters include β_0 = −30 dB, P_max = 36 dBm, L = 100, σ_w^2 = −80 dBm, and N_x = 5.The path-loss exponents are α_ar = 2.4, α_ab = 4.2, α_aw = 4.2, α_rb = 3, and α_rw = 3.
  • Figure 2: Figure 2 plots Bob’s SNR versus the number of IRS reflecting elements for different covertness levels ǫ.The figure compares how the SNR changes with IRS size under varying covertness requirements.

A. With Global CSI

With global CSI, the study evaluates IRS-assisted covert communication through benchmark comparisons and numerical designs of reflection coefficients, IRS location, and transmit power. The results show that IRS improves Bob’s performance while reducing Willie’s received channel power, and can enable perfect covertness as the number of reflecting elements grows.

  • Numerical evaluation: The evaluated schemes comprise PSCA variants, a low-complexity algorithm, an upper bound, and a no-IRS baseline.The PSCA variants either jointly optimize reflection amplitudes and phase shifts or fix amplitudes to one; the low-complexity method is Algorithm 2.
  • Numerical evaluation: IRS-assisted schemes achieve higher Bob SNR than the no-IRS scheme, and Bob’s SNR increases with the number of reflecting elements.Stricter covertness levels reduce Bob’s SNR, while PSCA approaches the upper bound.
  • Covertness mechanism: IRS lowers Willie’s received channel power relative to the no-IRS scheme, allowing Alice to transmit with higher power under the same covertness constraint.The reflected path can enhance Bob’s signal while deteriorating Willie’s detection performance.
  • Algorithm comparison: The PSCA algorithms slightly outperform the low-complexity algorithm because they achieve a larger received channel-power reduction at Willie.The comparison is attributed to the larger value of |hH rwΘhar +haw|2 achieved by PSCA.
  • IRS placement: The optimal IRS horizontal location balances Alice–Bob communication quality and the covertness constraint, shifting toward Bob as N increases and toward Willie as N decreases.Transmit power increases as the IRS moves closer to Willie, consistent with reduced Willie channel power gain.
  • Perfect covertness: Perfect covertness becomes easier to satisfy as the number of reflecting elements increases, and the simulations confirm perfect covertness with an IRS.The reflected Alice–IRS–Willie channel quality grows with N, facilitating the condition for perfect covertness.

B. Without Willie’s Instantaneous CSI

Without Willie’s instantaneous CSI, the paper analytically designs IRS phases, reflection amplitudes, and transmit power under several amplitude constraints. IRS amplitude control improves Bob’s SNR, while IRS placement balances Bob’s transmission quality against covertness.

  • Reflection-amplitude designs: The three considered designs fix all amplitudes to 1, share a common amplitude ρ0, or optimize each element’s amplitude, while optimizing transmit power Pa in every case.
  • Performance results: Bob’s SNR decreases as the covertness requirement becomes harsher, and optimized amplitudes achieve higher SNR than common or unit amplitudes.
  • Performance results: As the number of IRS elements N increases, the required transmit power Pa decreases because the no-instantaneous-CSI covertness constraint depends on amplitudes and power, not phase shifts.
  • Performance results: The system with an IRS requires lower Pa than the system without an IRS, unlike the corresponding case with Willie’s instantaneous CSI.
  • IRS placement: The best IRS horizontal location is close to Bob’s left-hand side, balancing a short IRS-to-Bob path with an easier covertness constraint.
  • IRS placement: As the IRS moves closer to Willie, Pa decreases because the stronger reflected path to Willie makes the covertness constraint harder to satisfy.
  • Design implication: For no instantaneous CSI, IRS phase shifts are independent of the covertness constraint, enabling joint determination of optimal phases, amplitudes, and Alice’s transmit power.

APPENDIX A PROOF OF THEOREM 1

The proof characterizes perfect covertness by requiring zero received energy at Willie and reduces feasibility to whether IRS-reflected contributions can cancel the direct channel.

  • Perfect covertness requires ǫ = 0 and zero received energy at Willie, so the original problem is reformulated with a cancellation condition.
  • Coherent IRS combining maximizes the reflected-channel magnitude, while adjustable amplitudes and phases can orient the synthesized reflection to cancel the direct signal.
  • Nonzero-power perfect covertness is achievable if and only if the maximum reflected-channel magnitude satisfies Σ_n |a_n| ≥ |h_aw|.
  • The convexity argument for the associated optimization uses the convexity of x ln(x) for x > 0 and affine transformation under positive-semidefinite matrices.

APPENDIX C PROOF OF THEOREM 2

The proof derives the distribution needed for the expected covertness constraint, expresses it through an exponential-integral function, and transforms the constraint into an analytically usable form.

  • The reflected-plus-direct channel is modeled as a zero-mean complex Gaussian variable, making X exponentially distributed with parameter δ^-1.
  • The expected divergence constraint EX[D(P0|P1)] ≤ 2ǫ^2 is rewritten using the derived distribution and an exact analytical expression.
  • The resulting expression is mathematically intractable because it involves an exponential integral function.
  • Defining g(x) = (1 + x)e^xE1(x) enables differentiation-based analysis, including showing that g(x) is monotonically decreasing for x > 0.
  • The transformed expression yields the equivalent covertness constraint used for subsequent IRS beamforming and transmit-power design.
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