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Polarforming for MIMO Covert Communications

Xin Xie, Jinpeng Xu, Yihang Huang, Li Zhou, Zhaolong Ning

arXiv:2610.10466v1cs.ITmath.IT

TL;DR

Conventional multi-antenna covert designs rely mainly on spatial beamforming, which can be constrained by limited spatial degrees of freedom, and coordination with polarforming for finite-length detection remains insufficiently understood. This paper jointly designs MIMO covariance and transmit/receive polarforming while characterizing Willie’s detection performance. At 10 dBm, the joint design’s average achievable rate is 53.8% higher than LPA’s.

  • Problem

    Spatial-only covert beamforming may be constrained by limited spatial degrees of freedom, and coordination with polarforming and finite-length warden detection remains insufficiently understood.

    Source: §I · p. 1

  • Method

    The paper jointly optimizes MIMO transmit covariance and transmit/receive polarforming, derives finite-length exact DEP and asymptotic covertness conditions, and develops an alternating optimization algorithm.

    Source: §I · pp. 1–2

  • Results

    At P = 10 dBm, the joint design’s average achievable rate is 53.8% higher than LPA’s.

    Source: B. Algorithm Convergence and Rate… · p. 11

  • Takeaways & Limitations

    The joint design improves achievable rate and adapts effectively to changes in the covertness requirement and polarized channel composition.

    Source: §VII · p. 12

Abstract

from arXiv · show

Covert communication conceals wireless transmission activity, but conventional multi-antenna designs rely mainly on spatial beamforming and can be constrained by limited spatial degrees of freedom. This paper investigates polarization-reconfigurable antenna (PRA)-aided multiple-input multiple-output (MIMO) covert communication, where Alice and Bob perform transmit and receive polarforming, respectively, and Willie uses a vertically polarized antenna for radiometric detection. We jointly optimize the transmit covariance matrix and the transmit and receive phase shift vectors to maximize Bob's achievable rate under the covertness and transmit power constraints. Under Gaussian signaling, perfect polarized channel state information, and known noise power at Willie, we derive his optimal radiometric threshold and exact minimum detection error probability (DEP) for a finite observation length, and convert the DEP requirement into a deterministic constraint on his received signal power. We further prove that, for a fixed transmission design, positive signal leakage becomes detectable as the observation length tends to infinity. To solve the design problem for a finite observation length, we develop an alternating optimization (AO) algorithm with a generalized water-filling covariance update and closed-form phase updates. Numerical results validate the exact DEP analysis and the rapid convergence of the proposed algorithm. Compared with the sufficient divergence-based condition, the exact covertness constraint permits a $0.99$--$1.12$~dB higher received signal-to-noise ratio (SNR) at Willie, while joint transmit and receive polarforming provides a rate gain of up to $53.8\%$ over the considered benchmark schemes.

I. INTRODUCTION

Spatial beamforming can be constrained by limited spatial degrees of freedom, while transmission unobservability remains distinct from protecting message content. The paper combines transmit and receive polarforming with MIMO covariance design and characterizes covertness for finite and asymptotically long observations.

I. INTRODUCTION

  • Existing covert beamforming enhances Bob’s reception while limiting Willie’s observation, but spatial-only designs may be constrained when spatial degrees of freedom are limited.Cryptographic and secrecy-oriented techniques protect message content but do not necessarily conceal that a transmission occurred.Source: §I · p. 1
  • The framework jointly optimizes transmit covariance and Alice’s and Bob’s phase-shift vectors to maximize Bob’s achievable rate.It models antenna coupling, channel depolarization, cross-polar leakage, MIMO covariance signaling, and Willie’s vertically polarized observation.Source: §I · p. 1
  • For finite observations, the paper derives Willie’s optimal radiometric threshold and exact minimum DEP; for a fixed design, positive leakage becomes detectable as observation length grows.The finite-length DEP requirement is converted into a received-signal-power constraint; asymptotic nontrivial covertness requires zero leakage.Source: §I · pp. 1–2
  • The finite-observation design uses alternating optimization with generalized water-filling covariance updates and closed-form transmit and receive phase updates.The resulting objective sequence is nondecreasing and convergent.Source: §I · p. 2
  • Numerical results validate the exact DEP analysis and rapid AO convergence, and show less-conservative exact covertness and rate gains from joint polarforming over the considered benchmarks.The results also show adaptation to changes in the covertness requirement and polarized channel composition.Source: §I · p. 2

A. Polarforming for Communication and Security

  • Prior polarforming work covers communication and sensing applications, while cited security work includes secrecy-rate optimization and a transmit-only MISO covert design with a sufficient divergence-based constraint.The latter does not provide exact finite-observation DEP analysis or consider receive polarforming and general MIMO covariance signaling.Source: A. Polarforming for Communication and… · p. 2

B. Physical-Layer Covert Communication

  • Prior covert communication research spans multiple domains and antenna architectures, while existing covert polarforming leaves receive polarforming, MIMO covariance signaling, and exact finite-observation DEP unaddressed.The stated gap is specific to prior covert polarforming, which uses transmit polarization optimization in a MISO finite-blocklength model under a sufficient divergence-based condition.Source: B. Physical-Layer Covert Communication · pp. 2–3

III. SYSTEM MODEL

  • The system model pairs Alice and Bob equipped with PRAs with Willie, who uses a single vertically polarized antenna for radiometric detection.Each phase-shifter-based PRA combines orthogonally polarized elements through one RF chain; its phase shift controls their relative phase and the synthesized polarization state.Source: §III · p. 3, A. PRA Model Based on Phase Shifters · p. 3

2) Polarized MIMO Channel:

The channel and detection models account for polarized Alice–Bob propagation, Gaussian covariance signaling, and Willie’s vertically polarized observation. Bob’s effective channel includes transmit and receive polarforming, while Willie’s observation includes co-polar propagation and cross-polar leakage.

2) Polarized MIMO Channel:

  • Each polarized subchannel connects one transmit PRA to one receive PRA and is modeled under Rayleigh fading with entries distributed as CN(0, βb).βb accounts for large-scale attenuation of the Alice–Bob link [20].Source: 2) Polarized MIMO Channel · p. 4

3) Achievable Rate:

  • Alice uses independent Gaussian transmit vectors with covariance W, and Bob’s received signal depends on the effective channel formed with Alice’s and Bob’s polarforming.The passage identifies Bob’s effective noise as additive white Gaussian noise and gives the achievable-rate expression after the received-signal model.Source: 3) Achievable Rate · p. 4

C. Covert Detection

  • The detection model assumes perfect instantaneous CSI at Alice and that Willie knows his noise power and the signal variance under transmission.Willie therefore implements the optimal likelihood-ratio test for the stated model.Source: C. Covert Detection · p. 4

1) Alice-Willie Channel:

  • Willie’s fixed vertical antenna observes only the vertical projection, combining co-polar V-to-V propagation with effective H-to-V leakage.The leakage arises from Alice’s transmit-side XPI and channel depolarization, not receive-port crosstalk at Willie.Source: 1) Alice-Willie Channel · p. 4

2) Binary Hypothesis:

The section models Willie’s detection of Alice’s transmission and defines covertness through his minimum total detection error probability. It uses this finite-observation-length requirement to formulate joint transmit-covariance and phase-shift optimization.

2) Binary Hypothesis:

  • Willie’s hypotheses are that Alice is silent (H0) or active (H1), and his radiometric statistic uses sample energy because the hypotheses differ only in variance.The observations include receiver noise and, under Gaussian random signaling, Alice’s transmitted signal.Source: 2) Binary Hypothesis · pp. 4–5, 3) Radiometric Detection · p. 5

4) Covertness Metric:

  • The minimum total detection error probability weights false alarms and missed detections equally, and larger observation length L indicates stronger covertness.The requirement is set by a prescribed covertness parameter ϵ.Source: 4) Covertness Metric · p. 5
  • The finite-observation-length DEP requirement is converted into an equivalent deterministic constraint on Willie’s received signal power.The paper uses this equivalent constraint in the subsequent joint optimization.Source: 4) Covertness Metric · p. 5

IV. COVERTNESS ANALYSIS AND PROBLEM FORMULATION

  • The resulting design jointly optimizes the transmit covariance matrix and Alice’s and Bob’s phase shift vectors.The objective is Bob’s achievable rate under the finite-observation-length covertness characterization.Source: §IV · p. 5

A. Finite-Observation-Length Covertness Analysis

For finite observation length, the paper derives Willie’s exact minimum detection error probability and optimal radiometric threshold. The DEP’s monotonicity makes the covertness requirement equivalent to a received-signal-power constraint.

A. Finite-Observation-Length Covertness Analysis

  • At ρ = 0, Willie’s two hypotheses induce identical distributions and are statistically indistinguishable.The exact DEP is obtained by substituting the optimal threshold into the false-alarm and missed-detection probabilities.Source: A. Finite-Observation-Length… · p. 5
  • The exact minimum DEP is a function of Willie’s received SNR ρ, and it is continuous and strictly decreasing for ρ > 0.The derivation uses the likelihood-ratio test for the two hypotheses and the Gamma-distributed normalized test statistic.Source: A. Finite-Observation-Length… · pp. 5–6
  • For a prescribed covertness parameter ϵ, the finite-length DEP requirement is equivalent to a deterministic constraint on Willie’s received signal power.The corresponding threshold can be computed by one-dimensional bisection for fixed observation length, ϵ, and noise power.Source: A. Finite-Observation-Length… · p. 6

B. Asymptotic Covertness Analysis for Infinite Observation Length

For a fixed design, the paper characterizes covertness as observation length grows and finds that positive signal leakage becomes detectable. It then uses the finite-length constraint to formulate and solve the joint design problem with alternating optimization.

B. Asymptotic Covertness Analysis for Infinite Observation Length

  • For a fixed design with known noise power, the asymptotic covertness constraint holds if and only if Willie’s received signal power Sw is zero.When Sw is positive, the minimum DEP tends to zero as L increases; when Sw is zero, the hypotheses remain identical and the DEP is one.Source: B. Asymptotic Covertness Analysis for… · p. 6
  • Polarforming adds a degree of freedom for reducing Willie’s received signal power beyond spatial covariance design alone, while simultaneous zero leakage and nonzero rate at Bob depend on the Alice-Bob and Alice-Willie channels.The asymptotic result holds with the transmission design fixed as L increases.Source: B. Asymptotic Covertness Analysis for… · p. 6

C. Problem Formulation

  • The finite-length problem maximizes Bob’s achievable rate subject to the transmit-power and covertness constraints; its covertness constraint is expressed as a received-signal-power bound.The asymptotic zero-leakage case is used only to characterize asymptotic covertness, not solved by the proposed AO algorithm.Source: C. Problem Formulation · p. 6

V. PROPOSED JOINT OPTIMIZATION ALGORITHM

  • The proposed algorithm cyclically optimizes the transmit covariance matrix and the receive and transmit phase shift vectors to address their coupling.The resulting problem remains non-convex despite reformulating the DEP constraint as a received-power constraint.Source: §V · p. 7

A. Transmit Covariance Optimization

With the phase vectors fixed, the transmit-covariance subproblem is convex and its optimal solution has a generalized water-filling structure. The associated dual variables satisfy the KKT conditions.

A. Transmit Covariance Optimization

  • For fixed phase vectors, the covariance subproblem is convex because its objective is concave in W and its inequality constraints are affine in W.For P > 0 and τε,L > 0, Slater’s condition holds, making the KKT conditions necessary and sufficient for optimality.Source: A. Transmit Covariance Optimization · p. 7
  • The optimal transformed covariance shares the effective channel matrix’s eigenvectors, and its power coefficients yield a generalized water-filling structure.The dual variables λ and μ satisfy the KKT conditions; the optimal dual variables are found by minimizing a two-dimensional convex dual function.Source: A. Transmit Covariance Optimization · p. 7

B. Receive Polarforming Optimization

Receive phases can be optimized one coordinate at a time without changing Willie’s received signal power. Transmit-phase updates instead maximize Bob’s rate subject to preserving covertness feasibility.

B. Receive Polarforming Optimization

  • Each receive-phase update maximizes Bob’s rate for one coordinate while holding the other phases fixed, and cannot decrease the objective.For fixed transmit covariance and transmit phases, the receive phase does not affect Willie’s received signal power.Source: B. Receive Polarforming Optimization · pp. 7–8

C. Transmit Polarforming Optimization

  • Transmit phases affect both Bob’s rate and Willie’s received signal power, so each coordinate update maximizes rate subject to the covertness constraint.If the unconstrained rate-maximizing phase is infeasible, the optimum lies on the covertness boundary; a feasible current iterate guarantees boundary points exist.Source: C. Transmit Polarforming Optimization · pp. 8–9
  • Cyclic coordinate-ascent sweeps over the transmit phases preserve feasibility and cannot decrease Bob’s achievable rate.When Willie’s received signal power is independent of the selected phase, the unconstrained rate-maximizing phase is feasible whenever the current iterate is feasible.Source: C. Transmit Polarforming Optimization · p. 9

D. Overall Algorithm, Convergence, and Complexity

The proposed AO procedure alternates optimal covariance updates with feasible scalar phase updates, producing a convergent objective sequence. This monotonicity does not guarantee global optimality.

D. Overall Algorithm, Convergence, and Complexity

  • The AO algorithm starts from a feasible zero covariance and alternates optimal covariance updates with feasible phase-coordinate updates, so no update decreases Bob’s rate.The initialization is feasible for any phase vectors when the transmit-power and received-signal-power limits are nonnegative.Source: D. Overall Algorithm, Convergence, and… · p. 9
  • The objective sequence converges because the rate is nondecreasing and bounded above, but this argument does not establish global optimality.The rate is bounded above because transmit power is bounded and both phase domains are compact.Source: D. Overall Algorithm, Convergence, and… · p. 9
  • The stated direct-implementation cost comes from covariance and phase updates; matrix-inverse updates can further reduce computational cost.The covariance update uses a two-dimensional dual-function search.Source: D. Overall Algorithm, Convergence, and… · pp. 9–10

VI. NUMERICAL RESULTS

The numerical study evaluates finite-length DEP analysis, AO convergence, and achievable rates under Rayleigh-fading channels. It compares joint polarforming with fixed-polarization and restricted-design benchmarks.

VI. NUMERICAL RESULTS

  • The numerical study evaluates exact finite-observation-length DEP analysis, AO convergence, achievable rate, and the effects of covertness and polarized-channel parameters.Default simulation parameters are listed in Table I.Source: §VI · p. 10
  • Alice–Willie channels use independent Rayleigh coefficients, with total channel power gain held at −90 dB as χw varies.The coefficients are u_n ∼ CN(0, βw,V) and v_n ∼ CN(0, βw,H).Source: §VI · p. 10
  • Benchmarks [44] include an achievable-rate upper bound, transmit-only polarforming, fixed-zero phases, vertical LPA, left-hand CPA, and random phases.The upper bound removes the covertness constraint; the other schemes optimize the covariance under their stated phase or polarization restrictions.Source: §VI · p. 10

A. Validation of the Exact Finite-Observation-Length DEP

The numerical study validates the exact finite-observation-length DEP analysis and shows that joint polarforming improves rate across the considered benchmarks and polarized-channel conditions.

A. Validation of the Exact Finite-Observation-Length DEP

  • Exact minimum DEP closely matches Monte Carlo results, decreases as Willie’s received SNR rises, and is lower at longer observation lengths.Longer observation lets Willie accumulate more evidence, so allowable leakage depends on observation length.Source: Figure 2 · p. 10, A. Validation of the Exact… · p. 10
  • The exact DEP constraint permits 0.99–1.12 dB higher received SNR at Willie than the sufficient KL condition for ϵ ∈ {0.05, 0.1}.Both constraints’ maximum allowable received SNR decreases with observation length.Source: Figure 3 · p. 10, A. Validation of the Exact… · p. 10

B. Algorithm Convergence and Rate Comparison

C. Rate and Covertness Tradeoffs

  • Bob’s average rate rises as ϵ increases but falls as Willie’s observation length grows; joint spatial and polarization adaptation accommodates much of the tighter covertness requirement.Rate gains diminish as the Alice–Bob channel and maximum transmit power become limiting after relaxing covertness.Source: Figure 6 · p. 11, C. Rate and Covertness Tradeoffs · p. 11

D. Impact of Polarization Coupling and Cross-Polar Leakage

VII. CONCLUSION

  • For finite-observation design, AO combines generalized water-filling covariance updates with closed-form transmit and receive phase updates; under a fixed design, asymptotic covertness requires zero leakage.The conclusion identifies imperfect channel knowledge, noise uncertainty, and adaptive-polarization or multi-antenna wardens as future-work settings.Source: §VII · p. 12

APPENDIX A PROOF OF PROPOSITION 1

The proof establishes that the exact minimum DEP decreases continuously with Willie’s received SNR, from 1 at zero SNR to 0 as SNR grows without bound. It also characterizes the likelihood ratio and density intersection used in deriving that DEP.

APPENDIX A PROOF OF PROPOSITION 1

  • The exact minimum DEP is obtained from the densities under H0 and H1 and identities evaluated at their intersection.The proof states the DEP expression and substitutes the intersection identities into it.Source: Appendix A · p. 12
  • For positive SNR, the likelihood ratio is strictly increasing, and the two densities intersect uniquely at the point where they are equal.The proof uses the intersection to express the exact DEP.Source: Appendix A · p. 12
  • The exact minimum DEP is strictly decreasing for positive received SNR, equals 1 as SNR approaches zero, and tends to 0 as SNR approaches infinity.The proof also establishes continuity over nonnegative SNR.Source: Appendix A · p. 12
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