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Covert Wireless Communications under Quasi-Static Fading with Channel Uncertainty

Khurram Shahzad, Xiangyun Zhou

arXiv:2009.13023v1cs.ITmath.IT

TL;DR

Finite-blocklength covert communication must account for channel uncertainty at both the intended receiver and the adversary, whose improved channel estimate can aid detection. The paper analyzes Willie’s detection with perfect CSI and CDI only, then optimizes Alice’s transmission; in the large detection error regime, the two CSI cases are essentially indistinguishable, and fading favors a different data-symbol design than AWGN.

  • Problem

    Finite-blocklength covert communication with imperfect channel knowledge at both Bob and Willie is less explored, and pilot training can improve Bob’s estimate while also helping Willie detect transmission.

    Source: B. Scenario, Approach and Contributions · p. 2, A. Background · p. 2

  • Method

    The paper derives Willie’s optimal detection performance under perfect CSI and CDI only, then optimizes Alice’s data power and symbol count under covertness constraints.

    Source: §III · p. 4, §V · p. 8

  • Results

    In the large detection error regime, Willie’s detection performances with perfect CSI and CDI only are essentially indistinguishable.

    Source: B. Scenario, Approach and Contributions · p. 2, §V · p. 8

  • Takeaways & Limitations

    The design need not account for the accuracy of Willie’s channel knowledge in the large detection error regime, and quasi-static fading favors fewer data symbols than AWGN when enough symbols are used.

    Source: B. Scenario, Approach and Contributions · p. 2

Abstract

from arXiv · show

Covert communications enable a transmitter to send information reliably in the presence of an adversary, who looks to detect whether the transmission took place or not. We consider covert communications over quasi-static block fading channels, where users suffer from channel uncertainty. We investigate the adversary Willie's optimal detection performance in two extreme cases, i.e., the case of perfect channel state information (CSI) and the case of channel distribution information (CDI) only. It is shown that in the large detection error regime, Willie's detection performances of these two cases are essentially indistinguishable, which implies that the quality of CSI does not help Willie in improving his detection performance. This result enables us to study the covert transmission design without the need to factor in the exact amount of channel uncertainty at Willie. We then obtain the optimal and suboptimal closed-form solution to the covert transmission design. Our result reveals fundamental difference in the design between the case of quasi-static fading channel and the previously studied case of non-fading AWGN channel.

I. INTRODUCTION

The paper studies finite-blocklength covert communication when channel uncertainty affects both the intended receiver and the adversary. It derives Willie’s detection performance and Alice’s transmission design, finding that channel fading changes which data-symbol count maximizes covert throughput.

A. Background

  • Prior finite-blocklength studies include AWGN, strict-delay, multi-antenna Willie, and slow-fading settings, while the paper focuses on quasi-static fading with channel uncertainty.The introduction then outlines the paper’s scenario, detection analysis, transmission design, numerical results, and conclusion.Source: A. Background · p. 2, C. Paper Organization · p. 3

B. Scenario, Approach and Contributions

  • Finite-blocklength channel uncertainty creates a training trade-off: pilots improve Bob’s channel estimate but also help Willie detect covert data transmission.The impact of channel uncertainty on Willie’s detection performance is described as less explored than its impact on throughput.Source: B. Scenario, Approach and Contributions · p. 2
  • In the large detection error regime, Willie’s detection performance is extremely insensitive to his channel-knowledge accuracy, so the design need not depend on his channel estimate.This regime corresponds to strict covertness requirements, with Willie’s detection error probability desired to be close to 1.Source: B. Scenario, Approach and Contributions · p. 2
  • For quasi-static fading, using fewer data symbols yields higher covert throughput when the transmission has a sufficient number of symbols, unlike the non-fading AWGN result.The paper also provides a suboptimal closed-form design; the optimal problem requires numerical search.Source: B. Scenario, Approach and Contributions · p. 3

II. SYSTEM MODEL

  • Alice sends a covert codeword to Bob while Willie tries to detect whether transmission occurred; Alice and Bob share a secret unknown to Willie.The model assumes single antennas and independently drawn zero-mean complex Gaussian codeword symbols.Source: §II · p. 3

A. Channel Model

The paper models Alice’s links to Bob and Willie as quasi-static Rayleigh fading and examines Willie’s detection under different channel knowledge. It also describes pilot-based transmission and Willie’s covertness criterion.

A. Channel Model

  • In quasi-static Rayleigh fading, each channel coefficient remains constant within a slot and changes independently between slots; CSI may be perfect or limited to CDI.The fading coefficients are modeled as zero-mean, unit-variance CSCG variables.Source: A. Channel Model · p. 3

B. Training-Based Transmission

C. Willie’s Detection and Covertness Criterion

  • With equally probable transmission and no-transmission hypotheses, covertness requires Willie’s false-alarm and missed-detection probabilities to sum to at least 1 − ϵ.A sufficiently small ϵ makes any detector employed by Willie ineffective.Source: C. Willie’s Detection and Covertness… · p. 4

A. Detection under Perfect CSI Knowledge

  • With perfect CSI, Willie’s optimal detector is a radiometer that thresholds average received power, and its optimal threshold minimizes detection error for the given channel realization.The likelihood ratio test uses threshold ν∗ = 1.Source: A. Detection under Perfect CSI Knowledge · pp. 4–5

B. Detection under Knowledge of CDI only

Under CDI-only knowledge, Willie lacks the instantaneous channel coefficient and is considered using a radiometer. His optimal threshold accounts for the channel distribution.

B. Detection under Knowledge of CDI only

C. Performance Comparison between CSI and CDI Cases

The paper compares Willie’s detection performance with perfect CSI and CDI only in the low-power regime. It finds their linear approximations equivalent, while qualifying the result by the radiometer detector assumption.

C. Performance Comparison between CSI and CDI Cases

  • In the asymptotically low-power regime, Willie’s perfect-CSI and CDI-only detection error probabilities have equivalent linear approximations.The regime is around PD → 0.Source: C. Performance Comparison between CSI… · p. 5
  • When detection error is large, Willie’s minimum detection error probabilities under perfect CSI and CDI only are asymptotically indistinguishable.The paper uses the perfect-CSI detection error as a worst-case design assumption, describing the resulting approach as more robust and accurate.Source: C. Performance Comparison between CSI… · p. 5
  • The equivalence is based on radiometer detection; under CDI only, it is not established that the radiometer is generally optimal.Under perfect CSI, the radiometer is optimal; the paper says the CDI-only radiometer already approaches that performance in the large-error regime.Source: C. Performance Comparison between CSI… · pp. 5–6

IV. COVERTNESS UNDER CHANNEL UNCERTAINTY

The covertness design optimizes data power and the number of data symbols under a covertness constraint. Bob estimates his channel during training and uses that estimate for data detection.

IV. COVERTNESS UNDER CHANNEL UNCERTAINTY

  • Alice’s design problem is to choose data transmit power and data-symbol count to maximize covert throughput under a covertness constraint.The setup fixes training to one symbol and sets pilot power to Pmax.Source: §IV · p. 6

A. Channel Estimation at Bob

  • Bob uses LMMSE channel estimation during training, with the channel estimate and estimation error denoted by ˆhb and ˜hb.Their distributions are zero-mean CSCG, and βb is the variance of the channel estimation error.Source: A. Channel Estimation at Bob · p. 6
  • Bob estimates the channel only in a slot where Alice transmits and uses that estimate for data detection in the same slot.Source: A. Channel Estimation at Bob · p. 6

B. Covert Connection Probability

The section defines covert connection probability as Bob’s probability of reliable decoding at a fixed rate under quasi-static fading and channel uncertainty. Alice’s design chooses data power and symbol count subject to covertness and transmission constraints; the optimal solution requires numerical search.

B. Covert Connection Probability

  • A transmission outage occurs when channel capacity C ≤ R, so covert connection probability is the probability Bob reliably decodes Alice’s message at fixed rate R.The section uses the complement of outage probability to characterize communication performance under quasi-static fading.Source: B. Covert Connection Probability · p. 6
  • Lemma 4 gives the covert connection probability for fixed-rate Alice-to-Bob transmission under channel uncertainty at Bob.The passage identifies data power P_D and Bob’s channel-estimation error variance β_b as relevant quantities.Source: B. Covert Connection Probability · p. 6
  • For a fixed rate, the note states that making R a design parameter would not change the main conclusion because Willie’s detection depends on data power and data-symbol count, not directly on rate.The transmitter is assumed to have no freedom to change R.Source: B. Covert Connection Probability · p. 6

C. Optimal Transmit Power and Number Of Transmit Symbols

  • Alice optimizes data transmit power P_D and data-symbol count N_D to maximize N_D R P_cc under a covertness constraint, with pilot power set to P_max.The design also imposes maximum and minimum symbol counts and a maximum transmit-power constraint.Source: C. Optimal Transmit Power and Number… · p. 7
  • The optimal data power is specified as a function of N_D, but finding the optimal symbol count requires numerical search.The solution incorporates the allowed minimum and maximum values of P_D and N_D.Source: C. Optimal Transmit Power and Number… · p. 7

D. Suboptimal Solution

The section uses a linear approximation in the asymptotically low-power, small-ϵ regime to obtain a suboptimal closed-form design. It concludes that the throughput-maximizing number of data symbols is the minimum allowed count.

D. Suboptimal Solution

  • In the asymptotically small ϵ regime, a linear approximation yields a suboptimal closed-form solution for Alice’s data-transmission design.The approximation is developed for the low-power regime.Source: D. Suboptimal Solution · p. 7
  • After using the covertness constraint at equality, the section expresses P_cc as a function of N_D for the remaining optimization.The proof notes that P_cc increases with P_D while covertness bounds P_D from above.Source: D. Suboptimal Solution · p. 8
  • The throughput-maximizing data-symbol count is the minimum allowed value, N_D,min.This conclusion follows in the proof of the suboptimal solution.Source: D. Suboptimal Solution · pp. 7–8

V. NUMERICAL RESULTS AND DISCUSSIONS

The numerical results show that Willie’s detection performance is nearly insensitive to CSI accuracy when detection error is high. Alice’s optimal design uses the minimum number of data symbols, and the optimal scheme substantially outperforms using the maximum number.

V. NUMERICAL RESULTS AND DISCUSSIONS

  • The numerical study uses rate R = 1, noise variances σ²_w = 0.05 and σ²_b = 0.01, maximum power Pmax = 1, and data-symbol bounds 50–100.The paper notes that these blocklengths fall within ranges used in short-packet communications and practical error-correcting codes.Source: §V · p. 8
  • Willie’s detection performance is indistinguishable with perfect CSI and CDI-only knowledge in the large detection-error regime, and remains close even at detection error 0.8.As Alice’s data power or number of data symbols increases, Willie’s detection performance improves.Source: §V · p. 8, Figure 2 · p. 8
  • Higher noise power at Willie allows Alice to transmit at higher data power, while the suboptimal power design performs close to the optimal design, especially at low transmit power.The power designs diverge more as covertness is relaxed, when Willie is no longer in the large detection-error regime.Source: §V · pp. 8–9, Figure 3 · p. 8
  • 110-fold more throughput is achieved by the optimal and suboptimal designs than by using the maximum number of data symbols; the suboptimal design is roughly 20% below optimal.The moderate reduction is the price of the closed-form suboptimal design.Source: Figure 5 · p. 9, §V · p. 9
  • Both optimal and suboptimal designs use the minimum possible number of data symbols, in contrast with the maximum-symbol choice for non-fading AWGN channels.The paper attributes the difference to fading dominating decoding errors under quasi-static fading, making blocklength effects negligible.Source: §V · p. 9

VI. CONCLUSION

The paper analyzes covert communication with channel uncertainty and pilot-assisted channel estimation, comparing Willie’s detection under complete CSI and CDI-only information. It finds these cases indistinguishable at low transmit power in the large-detection-error regime and derives covert-throughput design choices.

VI. CONCLUSION

  • At low transmit power, Willie’s detection performance with complete CSI and CDI only is indistinguishable in the large-detection-error regime.Thus, the quality of Willie’s channel knowledge does not improve his detection performance when detection error is constrained to remain large.Source: §VI · p. 9
  • The paper derives the optimal data transmit power and number of data symbols that maximize covert throughput.The setting includes channel uncertainty and pilot symbols that help the intended receiver estimate its channel.Source: §VI · p. 9
  • With enough data symbols for outage-based analysis, using fewer data symbols achieves higher covert throughput.Source: §VI · p. 9

APPENDIX A PROOF OF LEMMA 2

The proof approximates Willie’s CSI-based detection error near zero data transmit power, evaluating the value and slope of the approximation through limits of derivative terms.

APPENDIX A PROOF OF LEMMA 2

  • At zero data transmit power, Willie’s CSI-based minimum detection error is 1, giving the linear approximation its intercept.The proof then calculates the slope by differentiating the detection-error expression and taking the limit as transmit power approaches zero.Source: Appendix A · p. 10
  • The CSI-based derivative is evaluated term by term using incomplete-Gamma-function properties, Euler’s identity, and L’Hôpital’s rule.Source: Appendix A · pp. 10–11
  • Combining the limits of the derivative’s two terms yields the slope and completes the first-order approximation proof.Source: Appendix A · p. 11

APPENDIX B PROOF OF LEMMA 3

The proof linearly approximates Willie’s CDI-based detection error around zero data transmit power, then derives the threshold that optimizes detection under that approximation.

APPENDIX B PROOF OF LEMMA 3

  • The CDI-based approximation uses the first two Taylor-series terms around zero transmit power, with the function value at zero equal to 1.Source: Appendix B · p. 11
  • The proof finds Willie’s best threshold under the CDI-based linear approximation by setting its derivative to zero and simplifying.Source: Appendix B · pp. 11–12
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