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Covert Digital Semantic Communication With Movable Antennas: Constellation Shaping, Spatial Optimization, and Semantic Mapping

Yunfan Bai, Yuwen Qian, Hao Jiang, Yiyang Ni, Feng Shu, Jun Li, Shuning Zhang

arXiv:2610.23076v1eess.SPcs.ITmath.IT

TL;DR

Covert digital semantic communication must handle finite constellations and an unknown warden noise level, which challenge conventional Gaussian-based covert designs. The paper jointly designs the constellation, movable-antenna placement, and semantic transceiver; its simulations report improved warden-SNR values for Gauss–Laguerre constellations relative to the classical law.

  • Problem

    Covert design criteria for finite-constellation transmitters with movable antennas under warden noise uncertainty remain insufficiently characterized.

    Source: §I · pp. 1–2

  • Method

    The paper derives a closed-form covert-rate characterization and jointly develops a Gauss–Laguerre ring constellation, matching-order-aware antenna placement, and covert semantic mapping and transceiver.

    Source: §I · p. 2

  • Results

    The Gauss–Laguerre constellations with m = 2 and m = 4 admit warden SNRs of −6.5 dB and −0.3 dB compared with −37 dB under the classical law.

    Source: B. Verification of Theoretical Results · p. 10

  • Takeaways & Limitations

    The proposed system combines a moment-matching-based covert-rate characterization with semantic mapping and an invertible flow to convey semantics under a derived distortion bound.

    Source: §VI · p. 11

Abstract

from arXiv · show

Digital semantic communication (SemCom) is emerging as a bandwidth-efficient paradigm for next-generation wireless networks by delivering the meaning of a source over finite constellations. However, the broadcast nature of wireless channels exposes transmissions to adversarial wardens, while discrete symbols deviate from the Gaussian signaling assumed by conventional covert communication. In this paper, we propose a covert digital SemCom system assisted by movable antennas (MAs), where the constellation, the antenna positions, and the semantic transceiver are jointly designed to conceal the transmission while preserving the semantic fidelity. Specifically, the maximum covert rate against the optimal detector with unknown noise level is derived in closed-form, which reveals that the legitimate signal-to-noise ratio (SNR) scales with the moment-matching order of the constellation and the spatial gain provided by the MA array. A Gauss-Laguerre ring constellation is then designed to raise the moment-matching order under the peak power constraints, and a covert symbol mapping with rank-to-ring assignment and scrambling is developed to carry the semantics in the ordering of the latent magnitudes. Guided by the derived semantic distortion bound, the semantic encoder and decoder are trained end-to-end through an invertible flow over the designed constellation and MA array. Simulation results demonstrate that the proposed scheme supports a higher achievable covert rate and improves the reconstruction quality by 24% compared to state-of-the-art SemCom schemes.

I. INTRODUCTION

Digital SemCom offers compact semantic transmission, but covert communication with finite constellations and movable antennas lacks formal design criteria under warden noise uncertainty. The paper jointly designs the constellation, antenna positions, and semantic transceiver, with system and channel models specifying their roles and constraints.

I. INTRODUCTION

  • Existing MA-assisted covert designs use Gaussian signaling, while finite-alphabet analyses fix the constellation and antennas; formal power and placement criteria under warden noise uncertainty remain lacking.Existing semantic covert schemes also insufficiently characterize the joint effect of MA placement and finite-constellation mappings under covertness and block-statistics constraints.Source: §I · p. 2
  • The proposed framework jointly designs a finite constellation, antenna positions, and semantic transceiver for covert digital SemCom with a warden whose noise level is unknown.Its contributions include a closed-form maximum covert rate, a Gauss–Laguerre ring constellation, matching-order-aware antenna placement, and a covert mapping with an invertible-flow semantic transceiver.Source: §I · p. 2

II. SYSTEM MODEL

  • Each covert session spans n channel uses and carries J = n/L semantic objects, each represented by L symbols, from Alice to Bob while concealing transmission from Willie.Alice has M movable antennas, Bob has N fixed antennas, and Willie has Mw fixed antennas.Source: §II · p. 2

A. Digital SemCom Framework

  • The semantic transmitter compresses an image into a latent vector, transforms it through an invertible mapping, and maps its coordinates to finite-constellation symbols; Bob decodes the image and latent task label.The weighted semantic distortion balances image reconstruction and task accuracy, and source-independent scrambling and interleaving are refreshed per block and shared by Alice and Bob, but unknown to Willie.Source: A. Digital SemCom Framework · pp. 2–3

B. Transmission Model

  • Movable antennas are repositioned within a two-dimensional region, changing path phases under the far-field model, while Alice knows the legitimate-link channel and only the wiretap departure-angle distribution.Because the wiretap gain is random, power control reserves an outage margin set by a quantile for a given outage tolerance.Source: B. Transmission Model · p. 3

C. Binary Hypothesis Testing at Willie

Willie tests whether Alice is transmitting despite an unknown noise level, so the paper formulates covertness against a composite null rather than a detector tied to one noise value. Its leakage and risk constraint express the required detection guarantee.

C. Binary Hypothesis Testing at Willie

  • Willie’s binary test has a composite null hypothesis because the noise level is unknown, and the guarantee is set against the optimal test rather than a specific detector.The null consists of noise distributions CN(0, sI_n) over s > 0.Source: C. Binary Hypothesis Testing at Willie · p. 4
  • Session leakage quantifies detectability through the joint observation distributions under transmission and silence.The worst-case sum of false-alarm and missed-detection probabilities is bounded using the minimax and Pinsker inequalities.Source: C. Binary Hypothesis Testing at Willie · p. 4
  • The covertness constraint guarantees Willie’s minimum risk ξ⋆ ≥ 1 − ϵ, and must hold with probability at least 1 − δ because the normalized fading factor is random.Source: C. Binary Hypothesis Testing at Willie · p. 4

III. PROBLEM FORMULATION AND COVERT RATE OPTIMIZATION

The section formulates semantic-distortion minimization under covertness and derives how finite-constellation statistics determine admissible power and covert rate. Its closed-form characterization links moment matching to rate scaling and separates constellation and antenna contributions to legitimate SNR.

A. Problem Formulation

  • The design minimizes end-to-end semantic distortion subject to covertness, whose constraint couples the symbol distribution with antenna positions through Willie’s detectability.The formulation also imposes transmit-power and peak-power constraints.Source: A. Problem Formulation · p. 4

B. Covert Rate Characterization

  • The structured mapping produces i.i.d. symbols with distribution µ, reducing session leakage to a single-letter expression while finite constellations yield non-Gaussian observations at Willie.The composite null is invariant to common observation scaling, motivating analysis of a normalized equivalent scalar channel.Source: B. Covert Rate Characterization · p. 4
  • Output non-Gaussianness Ψµ captures both Willie’s per-symbol leakage and Bob’s achievable-rate loss under input distribution µ.For non-Gaussian µ, Ψµ is strictly increasing with SNR and is zero at zero SNR.Source: B. Covert Rate Characterization · p. 4
  • Moment matching order t means the constellation’s moments through order t match those of CN(0, 1), governing low-SNR output non-Gaussianness.The vanishing per-use leakage budget makes this low-SNR behavior central to design.Source: B. Covert Rate Characterization · p. 4
  • Per-symbol leakage scales with α^(t + 1) at low SNR, so higher moment matching order permits greater transmit power under the same leakage budget.The expansion characterizes leakage through the first nonzero Hermite moments.Source: B. Covert Rate Characterization · p. 5
  • Theorem 1 gives a closed-form admissible power limit under the covertness constraint when antenna positions and beamforming remain fixed and symbols are i.i.d. with moment matching order t.The resulting legitimate SNR and covert rate are evaluated at that power limit.Source: B. Covert Rate Characterization · p. 5
  • As session length n grows, covert rate decays as n^−1/(t+1) while total delivered information grows as n^t/(t+1); legitimate SNR factorizes into constellation and antenna terms.For t = 1, total information growth reduces to the classic square-root law; the factorization permits separate constellation and antenna-position design.Source: B. Covert Rate Characterization · p. 5

C. Gaussian-Matched Constellation Design

The constellation uses discrete rotational symmetry to match phase and amplitude moments separately, then selects ring radii and probabilities through Gauss–Laguerre quadrature. Under the peak-power constraint, this construction attains the maximum matching order available for its ring count.

C. Gaussian-Matched Constellation Design

  • Discrete rotational symmetry makes off-diagonal moments vanish, allowing phase and amplitude moments to be matched separately.The phase sum over equispaced phases vanishes for the relevant nonzero differences.Source: C. Gaussian-Matched Constellation Design · p. 5
  • The m-ring constellation uses Mph equispaced phases per ring, with equal probabilities for points on each ring and ring probabilities ωk.Its design variables reduce to the squared ring radii ρk and probabilities ωk, subject to unit-average-power and peak-power constraints.Source: C. Gaussian-Matched Constellation Design · p. 5
  • For the e^−ρ weight, the moment-matching condition becomes an m-node quadrature rule whose nodes and weights determine the ring radii and probabilities.The unit-power constraint is included as the j = 1 case.Source: C. Gaussian-Matched Constellation Design · p. 5
  • With Mph ≥ 4m and the peak-power constraint, the matching order is at most 4m − 1, attained by m-point Gauss–Laguerre quadrature when the outermost node is admissible.The peak-power bound limits the number of rings through the largest quadrature node.Source: C. Gaussian-Matched Constellation Design · p. 5
  • The Gauss–Laguerre constellation raises matching order from t = 3 for uniform 64-QAM to t = 15 at m = 4.Its rings spread outward along quantile circles as m increases.Source: C. Gaussian-Matched Constellation Design · p. 6

D. MA Placement and Power Control

The MA placement and beamforming design balances legitimate-link gain against transmit array gain toward an unknown warden direction. A matching-order-aware fading norm makes the placement objective differentiable, while final power is set using the exact covertness quantile.

D. MA Placement and Power Control

  • The placement objective maximizes the legitimate channel gain while suppressing transmit array gain across all angles because Willie’s direction is unknown.The average warden array gain is represented using a transmit correlation matrix whose entries depend on inter-antenna distances.Source: D. MA Placement and Power Control · p. 6
  • The admissible average warden SNR is divided by the (t + 1)-th order fading norm, which is non-decreasing with matching order.This couples MA placement to constellation design; the differentiable norm approximates the upper-tail behavior of the non-differentiable quantile.Source: D. MA Placement and Power Control · p. 6
  • The placement optimization replaces a non-differentiable quantile objective with the fading norm ς_t(T), enabling gradient-based updates over feasible antenna positions.The reported update projects each step onto the feasible set and uses backtracking; the objective sequence converges to a stationary point.Source: D. MA Placement and Power Control · p. 6
  • The optimized placement uses the exact quantile u_{1−δ}(T⋆) to set transmit power, preserving Theorem 1’s covertness guarantee after differentiable fading-norm optimization.The MA elements can be relocated between sessions as the legitimate channel changes.Source: D. MA Placement and Power Control · p. 6

A. Semantic Distortion Bound

A lower bound relates covertness-constrained session distortion to a covert semantic rate. The symbol-mapping layer enforces a source-independent block distribution at Willie while conveying semantics through scrambled symbol structure.

A. Semantic Distortion Bound

  • The distortion theorem bounds session-averaged weighted distortion under a covertness constraint with probability at least 1−δ, for arbitrary joint coding across J independent semantic objects.The bound uses a covert semantic rate optimized over K-point constellations and distributions subject to a peak-power constraint.Source: A. Semantic Distortion Bound · p. 7
  • The equality conditions require transmitted symbols in each block to be i.i.d. with distribution μ regardless of the source, so semantics are conveyed conditioned on Bob’s scrambling sequence.The resulting design separates source-independent randomization from the learning layer that minimizes weighted distortion.Source: A. Semantic Distortion Bound · p. 7

B. Covert Symbol Mapping

  • The covert mapping assigns latent-coordinate magnitude ranks to constellation rings, quantizes phases, then scrambles phases and interleaves symbols.The rank-to-ring assignment allocates smaller magnitudes to smaller-radius rings; random occupancy, phase offsets, and permutation produce the required source-independent block distribution.Source: B. Covert Symbol Mapping · pp. 7–8

C. Digital Semantic Encoder and Invertible Flow

An invertible flow spreads latent energy across coordinates before rank-based constellation mapping. At the receiver, inverse mapping and flow operations recover latent estimates for semantic decoding, trained under a fixed covert configuration.

C. Digital Semantic Encoder and Invertible Flow

  • The encoder produces a normalized latent vector, then cascaded sign flips, Hadamard mixing, and permutations form an invertible flow before rank-based symbol mapping.Orthogonal flow factors distribute energy uniformly over coordinates, making the encoder latent’s elliptical scatter isotropic after the flow.Source: C. Digital Semantic Encoder and… · p. 8

D. Receiver Design and End-to-End Training

  • At Bob, de-interleaving and de-rotation recover symbols, posterior estimation and the inverse flow reconstruct the latent, and a semantic decoder produces class and image outputs.The decoder is conditioned on operating SNR, while training minimizes weighted distortion with the covert configuration fixed.Source: D. Receiver Design and End-to-End… · p. 8
  • The covert mapping and invertible flow have no trainable parameters; gradients reach the encoder through a straight-through estimator and soft constellation assignment.Training samples operating SNRs around the design value to cover different covertness, session-length, and placement settings.Source: D. Receiver Design and End-to-End… · p. 8

V. NUMERICAL AND SIMULATION RESULTS

The evaluation uses a 100-class ImageNet subset, compares the proposed scheme with digital, analog, and covert baselines, and specifies shared operating conditions and training settings.

1) Dataset:

  • The evaluation uses 1,000 training, 50 validation, and 50 test images per class from a 100-class ImageNet subset, cropped to 128 × 128.Performance is measured by PSNR, SSIM, and top-1 accuracy of a frozen ResNet-50 classifier.Source: 1) Dataset · p. 9

2) Training Parameters:

  • All networks are trained for 100 epochs with AdamW, a 3 × 10−4 learning rate, batch size 64, and loss weight β = 0.02.Source: 2) Training Parameters · p. 9

3) Benchmarks:

  • The benchmarks are DeepJSCC-Q with 64-QAM, masked VQ-VAE with a learned codebook, SwinJSCC with analog symbols, and MA-covert with Gaussian signaling.The schemes use the same block length, MA array, peak-to-average power constraint, and scrambling, and transmit at their respective maximum powers admitted by Theorem 1.Source: 3) Benchmarks · p. 9

4) System Parameters and Network Settings:

  • The default setup uses 16 movable antennas in an 8λ-square region, with minimum spacing λ/2, and four antennas each at Bob and Willie.The covertness parameters are ϵ = 0.1, n = 10^6, and δ = 10−2, with a 6 dB peak-to-average power constraint.Source: 4) System Parameters and Network… · p. 9

B. Verification of Theoretical Results

The simulations validate the covert-rate predictions and show that the proposed constellation and movable-antenna design improves semantic reconstruction under covertness constraints. The reported gains arise across covertness budgets, antenna placements, and reconstruction comparisons.

B. Verification of Theoretical Results

  • The Monte Carlo estimates validate Lemma 1 and Lemma 2; at Ψµ = 10−6, admissible α rises from 0.06 for 64-QAM to 0.72 for GL m = 4.The matching-order gain is confined to the low-SNR regime enforced by the budget 2ϵ2/n.Source: B. Verification of Theoretical Results · p. 9
  • At n = 10^6, GL constellations with m = 2 and m = 4 admit warden SNRs of −6.5 dB and −0.3 dB, versus −37 dB under the classical law.The BPSK curve with t = 1 coincides with the classical square-root law 2ϵ/√n.Source: B. Verification of Theoretical Results · pp. 9–10, Figure 5 · p. 10

C. Performance Analysis and Comparison

  • Under admissible power, the optimal-test ROC remains within the Pinsker band PD ≤ PFA + ϵ for all four covertness budgets.The passage states that this bounds any arbitrary detector's gain over ϵ.Source: C. Performance Analysis and Comparison · p. 10
  • Compared with fixed arrays, movable-antenna placement yields PSNR gains of 1.6 to 2.2 dB at CBR 1/3 and gains of 0.8 to 1.1 dB over a uniform array.Against the half-wavelength array the gain reaches 2.2 dB; the movable-array PSNR increases with antenna count and region size, with diminishing returns.Source: C. Performance Analysis and Comparison · p. 10, Figure 7 · p. 10
  • At ϵ = 0.02, the proposed scheme improves PSNR by over 3.8 dB and SSIM by 24% over the strongest benchmark, while semantic accuracy stays above 95% across ϵ.Each scheme operates at its own admissible power under Theorem 1.Source: C. Performance Analysis and Comparison · p. 10, Figure 8 · p. 11
  • The proposed scheme attains the highest PSNR above −6 dB and a gap over 2.8 dB at the respective covert operating points.Its operating point is 2.9 to 9.1 dB higher than the benchmarks; reconstructed vehicle and dispenser images show sharper contours and textures.Source: C. Performance Analysis and Comparison · p. 11, Figure 9 · p. 11

VI. CONCLUSION

  • The conclusion identifies the proposed constellation, matching-order-aware antenna placement, covert mapping, and invertible flow as the design for semantic delivery under the derived distortion bound.The stated practical route is to conceal semantic transmission on digital transceivers under covertness constraints.Source: §VI · p. 11

APPENDIX A PROOFS OF LEMMA 1 AND LEMMA 2

The appendices derive the divergence and low-SNR behavior used to characterize covert leakage, then use those results to establish the session-level rate and legitimate-SNR bounds.

APPENDIX A PROOFS OF LEMMA 1 AND LEMMA 2

  • The composite-null divergence is minimized at s⋆ = 1 + γ, yielding log(1 + γ) − Iµ(γ) for the constellation-dependent divergence.The proof uses the output variance and the mutual-information identity for h(Y).Source: Appendix A · p. 11
  • Normalizing the observation gives Ψµ(γ) = D(PZ∥CN(0, 1)); its derivative is positive, establishing monotonicity from Ψµ(0) = 0.The derivative uses the MMSE relation and the unit-power input bound.Source: Appendix A · pp. 11–12
  • The proof expands the normalized output density around a Gaussian using Hermite polynomials, with the first nonzero degree set by the moment-matching order.The Taylor expansion and Gaussian-chaos bound establish the low-SNR divergence order.Source: Appendix A · p. 12

APPENDIX B PROOF OF THEOREM 1

  • For fixed antenna state and warden gain, session leakage is nΨµ(γ), using additivity of divergence across the n independent symbols and noise samples.The proof applies the session covertness probability condition through the quantile of the channel-state variable.Source: Appendix B · p. 12
  • The legitimate SNR bound follows by rearranging the covertness quantile condition and multiplying the admissible power by the spatial gain ∆.The covert-rate expansion uses x = 2ϵ2/n and yields an asymptotic order O(n^−1/(t+1)).Source: Appendix B · p. 12

APPENDIX C PROOF OF THEOREM 2

The proof bounds Willie’s leakage using marginal distributions, then bounds the semantic information delivered per object and converts that bound to session-averaged distortion.

APPENDIX C PROOF OF THEOREM 2

  • The divergence against the product Gaussian is decomposed into marginal divergences and total correlation, then bounded by dropping total correlation and applying convexity in the symbol distribution.The resulting legitimate-SNR bound uses the covertness quantile condition.Source: Appendix C · p. 12
  • For each object, data processing and the memoryless channel bound delivered information through the channel mutual information.The proof divides by the number of objects and applies concavity of mutual information over the common constellation.Source: Appendix C · pp. 12–13
  • The session-averaged distortion bound follows from convexity and monotonicity of Dβ; equality conditions include independent observations with common symbol distributions and the admissible power.The proof states that these conditions are guaranteed by (39).Source: Appendix C · p. 13
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