PaperlayerView exact revision on arXiv

Source-linked AI summary

Covert Wireless Communication with a Poisson Field of Interferers

Biao He, Shihao Yan, Xiangyun Zhou, Hamid Jafarkhani

arXiv:1712.07062v3cs.IT

TL;DR

The paper studies how covert communication can operate when Alice and Bob communicate amid uncertain interference while Willie detects Alice’s transmission. Using stochastic geometry, it analyzes covert throughput under covertness and decoding-reliability requirements for non-fading and fading channels, finding that interferer density and power matter differently depending on whether receiver noise is negligible.

  • Problem

    The paper examines covert communication between Alice and Bob when Bob and Willie are subject to uncertain shot noise from a Poisson field of interferers.

  • Method

    Using stochastic geometry, the paper analytically evaluates covert throughput under covertness requirements against Willie and decoding-reliability requirements at Bob for non-fading and fading channels.

  • Results

    In the interference-limited regime, interferer density and transmit power do not affect covert throughput for either non-fading or fading channels; with non-negligible AWGN, throughput increases as either quantity increases.

  • Takeaways & Limitations

    The impact of concurrent interferers on covert throughput depends on whether interference dominates receiver noise or is comparable to it.

Abstract

from arXiv · show

In this paper, we study covert communication in wireless networks consisting of a transmitter, Alice, an intended receiver, Bob, a warden, Willie, and a Poisson field of interferers. Bob and Willie are subject to uncertain shot noise due to the ambient signals from interferers in the network. With the aid of stochastic geometry, we analyze the throughput of the covert communication between Alice and Bob subject to given requirements on the covertness against Willie and the reliability of decoding at Bob. We consider non-fading and fading channels. We analytically obtain interesting findings on the impacts of the density and the transmit power of the concurrent interferers on the covert throughput. That is, the density and the transmit power of the interferers have no impact on the covert throughput as long as the network stays in the interference-limited regime, for both the non-fading and the fading cases. When the interference is sufficiently small and comparable with the receiver noise, the covert throughput increases as the density or the transmit power of the concurrent interferers increases.

I. INTRODUCTION

The paper studies covert communication in wireless networks, where uncertain ambient interference may help hide transmissions from a warden. Using stochastic geometry, it analyzes covertness, reliability, and throughput under Poisson-distributed interferers.

  • Positive covert throughput is motivated by uncertainty in a warden’s receiver noise or interference.
  • Unlike prior friendly-helper models, the paper considers randomly transmitting network nodes that do not intend to assist covert communication.
  • The scenario uses time-varying interference from a large-scale wireless network to hide communication between a transmitter and receiver.
  • The analysis evaluates average covert probability, connection outage probability, and covert throughput as measures of covertness, reliability, and overall rate.
  • In interference-limited networks, covert throughput is affected by neither interferer density nor transmit power for non-fading and fading channels.
  • With AWGN, covert probability is unchanged, while covert throughput decreases with noise and increases with interferer density or transmit power.

II. SYSTEM MODEL

The system models covert communication from Alice to Bob while Willie detects transmission in a two-dimensional wireless network with concurrent interferers distributed according to a homogeneous PPP. Bob and Willie receive Alice’s signal, aggregate interference, and AWGN, with interference power treated as random and partially uncertain to Willie.

  • Network and roles: The network contains Alice, Bob, Willie, and a Poisson field of concurrent interferers in a two-dimensional wireless setting.Alice transmits to Bob while Willie attempts to detect whether Alice is transmitting.
  • Interferer field: Concurrent interferers follow a homogeneous PPP with node density λI, and their locations remain static within a time slot.The model also assumes that interferer locations change between time slots, as in mobile or randomly accessing networks.
  • Signal model: Bob and Willie receive Alice’s channel signal together with aggregate interference and additive white Gaussian noise.The received-signal expressions use separate Alice-to-receiver channel coefficients and noise terms for Bob and Willie.
  • Interference model: Random interferer locations and channel coefficients make the aggregate received-interference powers random variables.The interference powers at Bob and Willie share a distribution under the stationary property of the PPP.
  • Knowledge and assumptions: Willie knows the distribution of aggregate interference but not its instantaneous power in a slot, which remains static across the N samples.The paper adopts average covert probability as its covertness measure and mainly analyzes the interference-limited case with zero AWGN.

A. Willie’s Hypothesis Test

Willie tests whether Alice is transmitting by applying a radiometer to received interference or interference-plus-Alice-signal samples. Covertness is evaluated through false-alarm and misdetection errors, with uncertain aggregate interference treated probabilistically.

  • Willie distinguishes interference alone from interference plus Alice’s signal using a binary hypothesis test.
  • The radiometer uses the average received-signal power as its test statistic.
  • Willie decides D0 when T(yw) ≤ γ and D1 when T(yw) > γ, making γ central to detector performance.
  • Covertness errors are quantified through false alarms and misdetections, whose combined performance is represented by ξ.
  • The model assumes infinitely many Willie samples and that Bob knows Alice’s transmission times through a shared secret timetable.

1) Average Covert Probability:

The paper defines covertness through average covert probability and reliability through connection outage, then optimizes Alice’s power and rate under both constraints. In the interference-limited regime, covert throughput is independent of interferer density and transmit power for non-fading channels.

  • 1) Average Covert Probability:: Average covert probability measures covertness from a Bayesian perspective using Willie’s optimal detection threshold.
  • C. Reliability Measure: Connection outage probability captures the chance that communication fails because Bob’s instantaneous aggregate interference is unknown.
  • D. Covert Throughput: Covert throughput is obtained by first selecting Alice’s maximum power satisfying covertness, then selecting the highest rate satisfying reliability.
  • D. Covert Throughput: The non-fading throughput derivation is analytically intractable because both constrained solutions may require complicated infinite series.
  • D. Covert Throughput: Theorem 1 states that non-fading covert throughput is unaffected by concurrent-interferer density λI or transmit power PI in an interference-limited network.
  • D. Covert Throughput: The density and power of concurrent interferers have balanced positive and negative effects on covert throughput.

IV. COVERT COMMUNICATION IN INTERFERENCE-LIMITED NETWORK WITH FADING

For fading channels, the paper applies the same constrained power-and-rate optimization while modeling fading and aggregate interference distributions. In the interference-limited regime, fading-channel covert throughput likewise does not depend on interferer density or transmit power.

  • IV. COVERT COMMUNICATION IN INTERFERENCE-LIMITED NETWORK WITH FADING: The fading-channel analysis uses the same two-step optimization: constrain Alice’s power by covertness, then constrain rate by connection outage.
  • IV. COVERT COMMUNICATION IN INTERFERENCE-LIMITED NETWORK WITH FADING: The fading-channel covert-throughput derivation is analytically intractable because the constrained power and rate solutions are not available in simple form.
  • IV. COVERT COMMUNICATION IN INTERFERENCE-LIMITED NETWORK WITH FADING: Theorem 2 states that fading-channel covert throughput is unaffected by concurrent-interferer density λI or transmit power PI.
  • IV. COVERT COMMUNICATION IN INTERFERENCE-LIMITED NETWORK WITH FADING: The density and transmit power of concurrent interferers affect neither covert throughput in fading channels.
  • IV. COVERT COMMUNICATION IN INTERFERENCE-LIMITED NETWORK WITH FADING: For Rayleigh fading with α = 4, the paper gives a closed-form average covert probability and obtains throughput with a similar algorithm.

V. COVERT COMMUNICATION IN INTERFERENCE NETWORK WITH AWGN

This section focuses on an interference-limited network, where the network’s interference dominates the relevant noise contribution.

  • V. COVERT COMMUNICATION IN INTERFERENCE NETWORK WITH AWGN: The analysis considers an interference-limited network in which the interference dominates the receiver-noise setting.

A. Impact of AWGN on System Performance

The analysis shows that non-zero AWGN does not affect Willie’s average covert probability, but it increases Bob’s connection outage probability and reduces achievable covert throughput.

  • Willie’s detection: Willie uses a radiometer to decide between interference-plus-noise and Alice’s signal plus interference and noise.The test statistic is the received-signal energy over the observation interval.
  • Willie’s detection: As N →∞, Willie’s hypothesis-test performance is characterized through the false-alarm and misdetection probabilities.
  • Willie’s detection: Non-zero AWGN at Willie does not affect the average covert probability for any a > 0.The optimal threshold can be shifted relative to the no-AWGN case while preserving the resulting average covert probability.
  • Bob’s reliability: Non-zero AWGN at Bob increases the connection outage probability because the corresponding noise term increases.The outage expressions are obtained for both AWGN and fading channels with AWGN.
  • Covert throughput: Non-zero AWGN leaves the covertness constraint unchanged but requires a lower transmission rate to satisfy the reliability constraint.Consequently, achievable covert throughput decreases compared with the interference-limited network.

B. Impact of Network Parameters with AWGN Consideration

With AWGN, increasing interferer density or transmit power improves covert throughput when interference is small, but the benefit vanishes as the network becomes interference-limited.

  • Analytical findings: In AWGN channels, covert throughput increases as the density or transmit power of concurrent interferers increases.
  • Analytical findings: The same increase in density or transmit power also applies to fading channels with AWGN.
  • Trade-off: Increasing interference improves average covert probability but worsens reliability, creating opposing effects on covert throughput.The connection outage probability increases as λI or PI increases.
  • Interference-limited regime: When interferer density becomes relatively large, covert throughput remains almost constant as density increases because the network becomes interference-limited.
  • Interference-limited regime: When interferer transmit power becomes relatively large, covert throughput likewise remains almost constant as power increases.
  • Other parameters: Covert throughput increases as the covertness requirement becomes looser and decreases as AWGN power increases.The former is shown by η increasing with ϵ, while the latter is consistent with the AWGN analysis.

VII. CONCLUSION AND FUTURE WORK

The paper analyzes covert communication in Poisson wireless networks for non-fading and fading channels, deriving expressions for covert probability, connection outage, and throughput. It finds that interferer density and transmit power do not affect throughput in interference-limited networks, while throughput increases with either when AWGN is not negligible; the study focuses on a single-link covertness constraint.

  • The study analyzes average covert probability, connection outage probability, and covert throughput for non-fading and fading channels.
  • The derived average covert and connection-outage expressions may be complicated for general path-loss exponents because of stochastic geometry.
  • The covert throughput is unaffected by concurrent-interferer density or transmit power in interference-limited networks for both channel types.
  • When AWGN is not negligible, covert throughput increases as concurrent-interferer density or transmit power increases.
  • Future work: The paper studies a covertness constraint imposed on a single link and identifies extending it to all network links as future work.

APPENDIX A

Appendix A establishes Willie’s radiometer statistic as sufficient for the hypothesis test and proves that covert throughput is invariant to interferer density and transmit power under the considered setting. The proof compares systems with scaled interferer density and uses the corresponding optimal transmission quantities.

  • Willie’s hypothesis test determines whether Alice transmits from his received observations using a radiometer statistic.
  • The Neyman factorization argument shows that T(yw) is a sufficient statistic for Willie’s hypothesis test.
  • Systems with interferer densities λI1 and λI2 = uλI1 have the same covert throughput for positive u.
  • The proof states that covert throughput is likewise unaffected by concurrent-interferer transmit power, with the analogous derivation omitted.

APPENDIX C

Appendix C derives the optimal detector threshold and average covert probability for the relevant case, using quasiconcavity of the threshold-dependent function. It also states that the fading-case covert throughput is unaffected by interferer density.

  • The appendix determines Willie’s optimal detector threshold before rewriting the average covert probability.
  • T1(γ) is strictly quasiconcave for γ > Pw, so γo is obtained by solving T′1(γ) = 0 in that domain.
  • Substituting γo into the threshold expression yields the average covert probability.
  • The proof states that systems with λI1 and uλI1 have the same covert throughput for any u > 0.

APPENDIX E

Appendix E derives the fading-case threshold properties and proves that covert throughput increases with concurrent-interferer density and transmit power when AWGN is present. It also notes that the optimal Alice transmit power is unchanged from the no-AWGN case.

  • The appendix determines Willie’s optimal threshold and rewrites the average covert probability for the fading case.
  • T2(γ) is strictly quasiconcave for γ > Pw, with γo obtained by solving T′2(γ) = 0.
  • Covert throughput increases when the concurrent-interferer density increases, and the proof states that the same holds for interferer transmit power.
  • The optimal Alice transmit power has the same expression as in the no-AWGN case because average covert probability is unrelated to non-zero AWGN.
Loading 1712.07062v3…