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Multi-Antenna Covert Communications in Random Wireless Networks

Tong-Xing Zheng, Hui-Ming Wang, Derrick Wing Kwan Ng, Jinhong Yuan

arXiv:1902.06936v1cs.IT

TL;DR

The paper studies how multi-antenna covert communication performs in random wireless networks with randomly located wardens and interferers, comparing CAS and DAS. It uses stochastic geometry to derive covertness and reliability metrics, then optimizes worst-case detection thresholds, transmit power, and rate. The maximal covert throughput is invariant to interferer density and interfering power for both systems, while CAS is more favorable than DAS, especially with many antennas.

  • Problem

    Whether CAS or DAS provides a higher covert communication rate in fading random networks, especially with multiple wardens at uncertain locations, is unclear.

  • Method

    The paper uses stochastic geometry to analyze covert outage and connectivity probabilities, then jointly optimizes worst-case detection thresholds, transmit power, and transmission rate.

  • Results

    Maximal covert throughput for both CAS and DAS is invariant to interferer density and interfering power, while CAS outperforms DAS and the gap enlarges with more transmit antennas.

  • Takeaways & Limitations

    CAS is the more favorable antenna architecture for covert communications in random networks, particularly when many transmit antennas are deployed.

Abstract

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This paper studies multi-antenna-aided covert communications coexisting with randomly located wardens and interferers, considering both centralized and distributed antenna systems (CAS/DAS). The throughput performance of the covert communication is analyzed and optimized under a stochastic geometry framework, where the joint impact of the small-scale channel fading and the large-scale path loss is examined. To be specific, two probabilistic metrics, namely, the covert outage probability and the connectivity probability, are adopted to characterize the covertness and reliability of the transmission, respectively, and analytically tractable expressions for the two metrics are derived. The worst-case covert communication scenario is then investigated, {where the wardens invariably can maximize the covert outage probability by adjusting the detection thresholds for their detectors}. Afterwards, the optimal transmit power and transmission rate are jointly designed to maximize the covert throughput subject to a covertness constraint. Interestingly, it is found that the maximal covert throughput for both the CAS and DAS is invariant to the density of interferers and the interfering power, regardless of the number of transmit antennas. Numerical results demonstrate that the CAS outperforms the DAS in terms of the covert throughput for the random network of interest, and the throughput gap between the two systems increases dramatically when the number of transmit antennas becomes larger.

I. INTRODUCTION

The paper addresses multi-antenna covert communication in random wireless networks, where the relative performance of centralized and distributed antenna systems under randomly located wardens remains unclear. It develops a stochastic-geometry analysis and optimization framework for covertness, reliability, and covert throughput.

  • I. INTRODUCTION: Multi-antenna covert communication remains less investigated than single-antenna communication, despite CAS and DAS offering distinct spatial-processing benefits.CAS antennas are co-located, whereas DAS antennas are geographically distributed for spatial diversity and coverage.
  • I. INTRODUCTION: The paper examines whether CAS or DAS provides a higher covert communication rate and how their performance gap behaves in fading channels.It also considers multiple wardens whose spatial locations are random to the monitored entity.
  • I. INTRODUCTION: The system comprises a multi-antenna Alice, a single-antenna Bob, randomly distributed single-antenna wardens, and randomly located interferers.CAS and DAS are analyzed with MRT and distributed beamforming, respectively.
  • I. INTRODUCTION: Stochastic geometry captures the joint impact of channel fading and path loss on covert communication performance.The framework derives analytical expressions for covert outage probability and connectivity probability.
  • I. INTRODUCTION: The paper optimizes detection thresholds, transmit power, and transmission rate to maximize covert throughput under a covertness constraint.The wardens’ thresholds are selected from their viewpoint in the worst-case scenario.
  • I. INTRODUCTION: For both CAS and DAS, maximal covert throughput is invariant to interferer density and interfering power, while CAS gains over DAS and the gap enlarges with more antennas.These conclusions are stated for the interference-limited system studied in the paper.

B. Detection of Covert Communications

The detection model treats wardens as radiometer-based hypothesis testers distinguishing silence from transmission using received average power. With infinitely many samples, signal and noise uncertainties vanish, leaving fading and node-location randomness in the detection probability.

  • B. Detection of Covert Communications: Wardens distinguish H0, no transmission, from H1, ongoing transmission, using an optimal statistical hypothesis test.The paper assumes radiometers as the practical detectors.
  • B. Detection of Covert Communications: A radiometer compares Willie’s average received power with a predefined threshold ξ to decide between H0 and H1.The decisions are denoted D0 and D1.
  • B. Detection of Covert Communications: The detection probability is the probability of a correct decision under equal prior probabilities for H0 and H1.Perfect detection has probability 1, while random guessing has probability 0.5.
  • B. Detection of Covert Communications: As K approaches infinity, uncertainties from transmitted signals and receiver noise vanish, while fading and node-position uncertainty remain.The average received power is then used in the detection analysis.
  • B. Detection of Covert Communications: For fixed ξ, the detection probability is Bernoulli distributed after accounting for randomness in the received signal and interference powers.It takes values 1 or 0.5 depending on the threshold setting and channel realization.

1) Covert Outage Probability:

The paper defines covertness through the probability that at least one random Willie detects Alice, and reliability through Bob’s connectivity probability. It derives tractable CAS expressions using stochastic-geometry interference analysis and approximations validated by simulation.

  • 1) Covert Outage Probability:: Covert outage probability O is the probability that at least one Willie has detection probability equal to one.The inner probability averages over signal and interference randomness, while the outer expectation averages over Willie locations.
  • 2) Connectivity Probability:: Connectivity probability C is the probability that Bob’s SINR γo exceeds β = e^R −1, enabling successful message recovery at rate R.It measures transmission reliability.
  • 2) Connectivity Probability:: Covert throughput T equals C multiplied by R when O ≤ ǫ, and is zero when the covertness constraint is violated.The threshold ǫ is the maximal acceptable covert outage probability.
  • 1) Covert Outage Probability:: The analysis focuses on an interference-limited network by ignoring thermal noise because aggregate interference typically dominates it.The authors state that including thermal noise complicates the analysis without significant qualitative difference.
  • 1) Covert Outage Probability:: For CAS, MRT makes the desired received-power distribution statistically unchanged when the number of antennas varies at fixed total transmit power.Consequently, adding transmit antennas does not affect covert outage probability under this condition.
  • 1) Covert Outage Probability:: The CAS covert-outage analysis derives a Laplace transform for aggregate interference, an approximate interference CDF, and an analytically tractable average detection probability.The approximation is reported to coincide well with the exact expression for L = 5 and is confirmed by Monte Carlo simulation.
  • 1) Covert Outage Probability:: The CAS covert outage probability is then obtained by averaging Willie detection over the binomial point process of random warden locations.Its nested finite-interval integrals are described as practically numerically evaluable.

B. Optimal Detection Threshold from Willie’s Viewpoint

The worst-case design lets each Willie choose a location-dependent detection threshold that maximizes covert outage. The average detection probability is unimodal in the threshold, and its worst-case maximum increases with Alice’s transmit power.

  • B. Optimal Detection Threshold from Willie’s Viewpoint: The worst-case scenario allows Willie to adjust ξ according to his distance from Alice to maximize detection accuracy and covert outage probability.The threshold is designed from Willie’s viewpoint rather than fixed by Alice.
  • B. Optimal Detection Threshold from Willie’s Viewpoint: For α = 4, average detection probability first increases and then decreases with ξ, yielding a unique optimal threshold ξo.ξo can be found efficiently by bisection using the stated root equation.
  • B. Optimal Detection Threshold from Willie’s Viewpoint: The optimal threshold ξo increases with interferer density λJ, interfering power PJ, and Alice’s transmit power PA.The paper interprets this adjustment as helping Willie distinguish Alice’s signal from interference more accurately.
  • B. Optimal Detection Threshold from Willie’s Viewpoint: The maximal average detection probability ¯pw,max monotonically increases with Alice’s transmit power PA.Higher transmit power enlarges the feasible received-power region for detection.
  • B. Optimal Detection Threshold from Willie’s Viewpoint: With the optimal threshold, ¯pw increases with PA and decreases with λJ in the illustrated results.The optimal threshold substantially improves detection probability compared with a constant threshold.

C. Connectivity Probability

The CAS connectivity probability is given in closed form, with the expression separating single-antenna and multi-antenna contributions. Adding antennas improves reliability, but the incremental gain becomes insignificant for sufficiently large antenna counts.

  • A closed-form expression for the CAS connectivity probability C is provided.
  • The exact expression uses subset-indexing terms and normalized gamma fading with shape parameter M.
  • The first term in C arises from a single antenna, while the second term captures the contribution of multiple antennas.
  • Adding one more antenna increases C, but the increment becomes insignificant when M is sufficiently large.

D. Covert Throughput Maximization

The covert-throughput problem maximizes T = CR under a covert-outage constraint by separately optimizing transmit power and transmission rate. The resulting design reveals how antennas, wardens, interference, and system parameters affect throughput.

  • The optimization maximizes covert throughput T = CR subject to the constraint O ≤ ǫ.
  • Because C depends on PA and R while O is independent of R, the problem decomposes into power optimization followed by rate optimization.
  • 1) Optimal PA:: Pmax is independent of M, increases with ǫ, decreases with N, and is proportional to λJ^(α/2)PJ.
  • 1) Optimal PA:: To(β) first increases and then decreases with β, with the optimum βo obtained as the unique root of Q(β) = 0.
  • 1) Optimal PA:: The optimal βo decreases with φo and increases with Kα,M.
  • 1) Optimal PA:: The optimal rate Ro and maximal throughput To increase with M and ǫ, decrease with ra,o and N, and are invariant to λJ and PJ.
  • 1) Optimal PA:: The CAS throughput gain over prior single-antenna work is significant, while the throughput degrades with more wardens and improves with more transmit antennas.
  • 1) Optimal PA:: With a transmit-power budget, the interference invariance can fail, and To decreases with λJ and PJ when Pmax exceeds Pbud.

IV. DISTRIBUTED ANTENNA SYSTEM

The DAS distributes Alice’s antennas geographically while connecting them to a central processor. Distributed beamforming enables simultaneous transmission of the same message to Bob to enhance reliability.

  • In the DAS, Alice’s antennas are geographically spread and connected to a central processor.
  • Distributed beamforming allows the antennas to transmit the same message simultaneously to Bob and enhance transmission reliability.

A. Worst-case Covert Outage Probability

For the DAS, the covert-outage analysis modifies the CAS expressions to account for geographically distributed antenna powers and derives exact and approximate connectivity probabilities. The distributed configuration complicates reliability analysis and can worsen covertness as antennas are added.

  • The DAS covert outage probability retains the CAS form but replaces the received Alice-signal power with the distributed-antenna power expression.
  • For α = 4, the worst-case detection thresholds are obtained analogously to the CAS by modifying the parameter B.
  • Unlike the CAS, DAS covert outage probability depends on M, and adding antennas can exacerbate covertness because distributed antennas are more vulnerable to randomly located Willies.
  • As M becomes sufficiently large, the DAS covert-outage deterioration gradually vanishes and O approaches a constant.
  • The DAS connectivity expression involves a squared sum of independent, nonidentically distributed Rayleigh variables rather than a CAS gamma variable, complicating computation.
  • An exact multiple-integral expression and a large-connectivity approximation are provided for DAS connectivity probability C.
  • In the considered DAS regime, C decreases linearly with λJ, PJ^δ, and β^δ, simplifying subsequent transmit-power and rate design.

C. Covert Throughput Maximization

The paper maximizes DAS covert throughput under a covertness constraint using a tractable equal-distance, equal-power antenna configuration and procedures analogous to the CAS optimization. It also studies a large-connectivity approximation and finds that the optimal transmission rate and maximal covert throughput are invariant to interferer density and interfering power.

  • DAS optimization: The DAS optimization maximizes covert throughput T = CR subject to the covertness constraint O ≤ ǫ.The problem is resolved through the same two-step process used for the CAS.
  • DAS optimization: Equal antenna-to-Bob distances and equal transmit powers are adopted because jointly optimizing location-dependent antenna powers is intractable.The special case sets PDm = PD for m = 1, · · · , M.
  • Large-connectivity approximation: The large-connectivity regime provides an easy-to-compute suboptimal transmission rate Ro using the connectivity probability expression in (34).
  • Large-connectivity approximation: To(β) first increases and then decreases with β, attaining its maximum at the unique zero-crossing βo of its derivative.The optimum βo can be rapidly searched by bisection because the derivative changes from positive to negative.
  • Throughput invariance: The optimal transmission rate R∗ and maximal covert throughput T∗ are invariant to interferer density λJ and interfering power PJ, regardless of the antenna count M.

V. SIMULATION RESULTS

The simulations compare CAS and DAS covertness, connectivity, and covert throughput under varying antenna counts, interferer densities, detection settings, and transmission rates. They show that CAS generally achieves higher covert throughput, with the advantage increasing as more antennas are deployed, while maximal throughput remains invariant to interferer density.

  • Covertness: Willie’s location determines which antenna architecture provides lower average detection probability: DAS is better near co-located antennas, whereas CAS is better near distributed antennas.The comparison reverses as Willie’s position changes relative to the antenna deployments.
  • Covertness: CAS invariably achieves lower covert outage probability than DAS when wardens’ locations are completely uninformed.Distributed antennas reduce per-antenna power but can expose some wardens to larger aggregate power.
  • Antenna scaling: For CAS, covert outage probability remains constant with M, whereas for DAS it increases with M and eventually reaches a plateau.Under MRT, the power perceived at Willie is equivalent to that from a single-antenna transmitter; distributed deployment behaves differently.
  • Reliability: Connectivity probability increases with M for both systems, and the CAS–DAS gap is nearly negligible.Both architectures exploit spatial degrees of freedom and coherent signal superposition at the destination.
  • Throughput optimization: Optimal transmission rate produces a peak covert throughput because increasing rate eventually makes connectivity probability too small.The optimized rate increases with the covert outage threshold and antenna count.
  • Throughput comparison: CAS gains substantially more throughput from additional antennas than DAS, and maximal covert throughput remains unchanged with interferer density.Interference can enable higher transmit power that offsets its adverse throughput effect while preserving covertness.

APPENDIX

The appendix establishes optimality properties for detector thresholds, covert outage, and throughput through derivative and monotonicity arguments. It also derives the invariance of the covertness-constrained transmit-power relationship with respect to interference parameters.

  • Detector threshold: The optimal detection threshold ξo is the unique zero-crossing of d¯pw(ξ)/dξ and maximizes average detection probability.The derivative is initially positive and becomes negative after ξo.
  • Covert outage: Covert outage probability is constant with M, increases with PA, and decreases with N.These monotonicity properties provide the basis for the subsequent proposition proof.
  • Invariance: When A2 = A0, maintaining the same maximal detection probability requires PA,2 = 2P0.The proof uses monotonicity of the maximal average detection probability with respect to transmit power.
  • Throughput optimization: The throughput objective To(β) is quasi-concave: it first increases with β, then decreases, and is maximized at β = βo.The result follows from the derivative changing sign and the negative second derivative at the optimum.

D. Proof of Proposition 1

The proof of Proposition 1 combines parameter monotonicity with the throughput derivative condition. It concludes that optimal throughput is unaffected by interference density and interfering power while varying predictably with system parameters.

  • Parameter dependence: The optimal throughput parameter βo increases with M and ǫ, and decreases with ra,o and N.These dependencies follow from the monotonicity properties of the connectivity probability and the maximal transmit power.
  • Interference invariance: The optimal βo is independent of λJ and PJ because φo does not change with either interference parameter.The proof uses Pmax ∝ λJ^(α/2)PJ to establish the cancellation.
  • Connectivity: Connectivity probability increases with M and ǫ, decreases with ra,o and N, and remains unchanged with λJ and PJ.These monotonicity results complete the proof of Proposition 1’s invariance property.
  • Optimization condition: Positive throughput requires β < W^(-1/δ), and To(β) is maximized at the interior point β = βo.The derivative is positive at β = 0 and negative as β approaches the upper bound.
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