PaperlayerView exact revision on arXiv

Source-linked AI summary

Gaussian Signalling for Covert Communications

Shihao Yan, Yirui Cong, Stephen Hanly, Xiangyun Zhou

arXiv:1807.00719v3cs.ITcs.CR

TL;DR

The paper asks whether Gaussian signalling is optimal under two asymmetric KL-divergence covertness constraints in AWGN covert communications. It analyzes mutual-information maximization under each constraint and finds Gaussian optimal for D(p1||p0), but not for D(p0||p1), where skew-normal signalling can perform better. For Gaussian signalling, D(p1||p0) is the tighter constraint and produces lower mutual information.

  • Problem

    The paper examines how the asymmetric constraints D(p1||p0)≤2ε^2 and D(p0||p1)≤2ε^2 affect signalling optimality and covert information.

  • Method

    Under AWGN at Bob and Willie, the paper compares signalling distributions by maximizing I(x;z) subject to each KL-divergence covertness constraint.

  • Results

    Gaussian signalling is optimal under D(p1||p0)≤2ε^2, while skew-normal signalling can achieve higher mutual information than Gaussian signalling under D(p0||p1)≤2ε^2.

  • Takeaways & Limitations

    For Gaussian signalling, D(p1||p0) is the tighter constraint because D(p0||p1)≤D(p1||p0), and it yields lower mutual information.

Abstract

from arXiv · show

In this work, we examine the optimality of Gaussian signalling for covert communications with an upper bound on $\mathcal{D}(p_{_1}||p_{_0})$ or $\mathcal{D}(p_{_0}||p_{_1})$ as the covertness constraint, where $\mathcal{D}(p_{_1}||p_{_0})$ and $\mathcal{D}(p_{_0}||p_{_1})$ are different due to the asymmetry of Kullback-Leibler divergence, $p_{_0}(y)$ and $p_{_1}(y)$ are the likelihood functions of the observation ${y}$ at the warden under the null hypothesis (no covert transmission) and alternative hypothesis (a covert transmission occurs), respectively. Considering additive white Gaussian noise at both the receiver and the warden, we prove that Gaussian signalling is optimal in terms of maximizing the mutual information of transmitted and received signals for covert communications with an upper bound on $\mathcal{D}(p_{_1}||p_{_0})$ as the constraint. More interestingly, we also prove that Gaussian signalling is not optimal for covert communications with an upper bound on $\mathcal{D}(p_{_0}||p_{_1})$ as the constraint, for which as we explicitly show skew-normal signalling can outperform Gaussian signalling in terms of achieving higher mutual information. Finally, we prove that, for Gaussian signalling, an upper bound on $\mathcal{D}(p_{_1}||p_{_0})$ is a tighter covertness constraint in terms of leading to lower mutual information than the same upper bound on $\mathcal{D}(p_{_0}||p_{_1})$, by proving $\mathcal{D}(p_{_0}||p_{_1}) \leq \mathcal{D}(p_{_1}||p_{_0})$.

I. INTRODUCTION

The paper studies whether Gaussian signalling is optimal under two asymmetric KL-divergence covertness constraints in AWGN covert communications. It shows that the choice of divergence affects achievable mutual information, with Gaussian signalling optimal for one constraint but not the other.

  • KL-divergence constraints are conservative sufficient conditions for the operational detection-error covertness requirement, but the gap in optimality under these constraints remains unbounded.
  • The two KL divergences differ because KL divergence is asymmetric, and their effects on covert information had not previously been examined.
  • Gaussian signalling is optimal for maximizing mutual information subject to D(p1||p0) ≤2ε^2.
  • Skew-normal signalling can achieve higher mutual information than Gaussian signalling subject to D(p0||p1) ≤2ε^2.
  • Gaussian signalling minimizes D(p1||p0) under an average transmit-power constraint but cannot minimize D(p0||p1).
  • For Gaussian signalling, D(p0||p1) ≤D(p1||p0), so the D(p1||p0) constraint is tighter and yields lower mutual information.

II. SYSTEM MODEL

The system models Alice transmitting over AWGN to Bob while Willie observes the channel under transmission and no-transmission hypotheses. Covertness is expressed through Willie’s minimum detection error and bounded using either direction of KL divergence.

  • A. Channel Model: Alice, Bob, and Willie use single antennas, with independent AWGN channels from Alice to Bob and Willie.
  • B. Binary Hypothesis Testing at Willie: Willie distinguishes H0: y[i]=nw[i] from H1: y[i]=x[i]+nw[i] to detect whether Alice transmitted.
  • B. Binary Hypothesis Testing at Willie: The covertness requirement is ξ*≥1−ε, where ξ* is Willie’s minimum detection error probability under an optimal detector.
  • B. Binary Hypothesis Testing at Willie: Because total variation is generally intractable analytically, Pinsker’s inequality motivates sufficient constraints D(p1||p0)≤2ε^2 and D(p0||p1)≤2ε^2.
  • B. Binary Hypothesis Testing at Willie: The two KL constraints are both valid but differ because KL divergence is asymmetric.

C. Mutual Information

The paper evaluates mutual information I(x;z) for signalling distributions under power and covertness constraints. Gaussian inputs maximize mutual information at fixed power, and the paper uses this fact within its optimization under D(p1||p0).

  • C. Mutual Information: Mutual information I(x;z) measures the communication performance between Alice’s input x and Bob’s received signal z.
  • C. Mutual Information: A zero-mean Gaussian input maximizes I(x;z) subject to a fixed average power constraint.
  • C. Mutual Information: The paper therefore studies whether Gaussian signalling remains optimal when mutual information is maximized under alternative covert-communication constraints.
  • C. Mutual Information: For the D(p1||p0) constraint, the optimization selects a zero-mean Gaussian signalling distribution with an appropriate variance.

A. Zero-Mean Gaussian Signalling is Optimal

The paper proves that zero-mean Gaussian signalling solves the mutual-information optimization under the D(p1||p0) covertness constraint. The proof first identifies the optimal received distribution and then establishes that the corresponding Gaussian input is optimal.

  • A. Zero-Mean Gaussian Signalling is Optimal: Theorem 2 identifies p1(y) = N(0, Py) as the solution minimizing D(p1||p0) under the received-power, normalization, and nonnegativity constraints.The solution is obtained using calculus of variations and is shown to satisfy the required constraints.
  • A. Zero-Mean Gaussian Signalling is Optimal: The calculus-of-variations proof derives the candidate distribution by setting the functional derivative of the constrained objective to zero.Lagrange multipliers enforce the received-power and normalization constraints, after which the candidate is verified as a valid solution.
  • A. Zero-Mean Gaussian Signalling is Optimal: The candidate received distribution is Gaussian with zero mean and variance Py after determining the Lagrange-multiplier parameters.The resulting density also satisfies the nonnegativity condition and the sufficient optimality condition.
  • A. Zero-Mean Gaussian Signalling is Optimal: The optimal transmit distribution is zero-mean Gaussian because Gaussian signalling maximizes mutual information under the relevant average-power constraint.This combines the divergence-minimizing received distribution with the Gaussian mutual-information optimality result.
  • A. Zero-Mean Gaussian Signalling is Optimal: The proof concludes that Gaussian signalling simultaneously maximizes mutual information and satisfies the minimum-divergence requirement under D(p1||p0) ≤ 2ǫ2.This establishes Gaussian signalling as optimal for the specified covertness constraint.

B. Optimal Transmit Power

The optimal transmit power is determined by the divergence constraint because D(p1||p0) increases monotonically with received and transmit power. The resulting Gaussian signalling therefore uses the largest power consistent with D(p1||p0) = 2ǫ2.

  • B. Optimal Transmit Power: The proof establishes the transmit-power choice by differentiating D(p1||p0) with respect to Py and showing the derivative is positive.This links the received variance Py to the transmit variance Px through the additive-noise model.
  • B. Optimal Transmit Power: D(p1||p0) increases monotonically with Py and therefore with Px, so the optimal transmit power is the value attaining D(p1||p0) = 2ǫ2.The divergence constraint is active at the optimum because unconstrained Gaussian mutual information also increases with Px.
  • B. Optimal Transmit Power: Theorem 1 uses the zero-mean Gaussian input with variance P_x* selected to satisfy the covertness constraint while maximizing I(x, z).The selected variance is the largest feasible value under the monotone divergence relationship.

IV. WITH D(p0||p1) ≤2ǫ2 AS THE COVERTNESS

Under the reverse divergence constraint D(p0||p1) ≤ 2ǫ2, the paper proves that Gaussian signalling is not generally optimal. A calculus-of-variations argument shows that the Gaussian candidate cannot satisfy the necessary optimality conditions in a special equal-noise case.

  • IV. WITH D(p0||p1) ≤2ǫ2 AS THE COVERTNESS: The paper proves that Gaussian signalling is not optimal when D(p0||p1) ≤ 2ǫ2 is the covertness constraint.The result concerns maximizing mutual information under the reverse KL-divergence constraint.
  • IV. WITH D(p0||p1) ≤2ǫ2 AS THE COVERTNESS: The section introduces skew-normal signalling as a benchmark for demonstrating higher mutual information than Gaussian signalling under the reverse divergence constraint.The numerical comparison is described as part of the paper’s later evaluation.
  • IV. WITH D(p0||p1) ≤2ǫ2 AS THE COVERTNESS: The proof specializes to identical AWGN at Bob and Willie, making the distributions of the received signals z and y under H1 identical.This reduction lets the optimization be expressed directly in terms of p1(y).
  • IV. WITH D(p0||p1) ≤2ǫ2 AS THE COVERTNESS: Calculus of variations supplies necessary conditions involving Lagrange multipliers for the candidate optimal p1(y).The constrained functional and its derivative are used to test whether a Gaussian density can solve the optimization problem.
  • IV. WITH D(p0||p1) ≤2ǫ2 AS THE COVERTNESS: The Gaussian density p1(y) = N(0, Py) cannot satisfy the resulting conditions, so the optimal signalling distribution, if it exists, is not Gaussian.The contradiction is completed by showing the relevant multiplier cannot be zero while simultaneously meeting the pdf and power constraints.

B. A Benchmark p(x): Skew-Normal Distribution

The paper uses skew-normal signalling as a non-Gaussian benchmark under D(p0||p1) ≤ 2ǫ2. It derives the corresponding received density and varies skewness while preserving zero mean and variance to search for higher mutual information.

  • B. A Benchmark p(x): Skew-Normal Distribution: Skew-normal signalling is evaluated because it can outperform Gaussian signalling in mutual information under D(p0||p1) ≤ 2ǫ2.The benchmark assumes identical AWGN at Bob and Willie so the received distributions coincide.
  • B. A Benchmark p(x): Skew-Normal Distribution: The skew-normal family is parameterized by location, scale, and skew parameters, with the normal distribution recovered when θ = 0.Positive and negative θ produce right- and left-skewed distributions, respectively.
  • B. A Benchmark p(x): Skew-Normal Distribution: The construction adjusts µ and ω for each θ so that the skew-normal input retains zero mean and variance Px.Varying θ then produces candidate input distributions with the same first two moments as the Gaussian benchmark.
  • B. A Benchmark p(x): Skew-Normal Distribution: Proposition 1 derives the received density p1(y) for a zero-mean, variance-Px skew-normal input with nonzero skew parameter θ.The expression also gives p(z) when Bob’s and Willie’s noises are identically distributed.
  • B. A Benchmark p(x): Skew-Normal Distribution: The derived density enables calculation of both D(p0||p1) and I(x, z) for numerical comparison with Gaussian signalling.The paper obtains p(z) from the same expression under the i.i.d.-noise assumption.

V. COVERT COMMUNICATIONS WITH GAUSSIAN SIGNALLING

This section evaluates Willie’s minimum detection error under Gaussian signalling and compares covertness constraints based on the two KL divergences.

  • Gaussian signalling is used to characterize Willie’s detection performance and assess the relative tightness of the two KL-divergence-based covertness constraints.

A. Willie’s Detection Performance

The section derives Willie’s optimal detection performance for Gaussian signalling, including the likelihood-ratio test, error probabilities, and mutual-information evaluation under covertness constraints.

  • Gaussian signalling defines the likelihood functions under the null and alternative hypotheses used to analyze Willie’s detector.
  • With equal prior probabilities, the likelihood-ratio test using threshold 1 minimizes Willie’s detection error.
  • The derived cumulative distributions and error expressions determine Willie’s false-positive and miss-detection rates.
  • These error rates are used to evaluate the maximum mutual information subject to ξ* = α* + β* ≥ 1 − ε.

B. Mutual Information with Gaussian Signalling

This section relates Gaussian signalling, mutual information, and the two asymmetric KL constraints, showing that D(p0||p1) yields a looser Gaussian signalling constraint than D(p1||p0).

  • B. Mutual Information with Gaussian Signalling: Gaussian signalling models x as zero-mean with variance P_x, and its mutual information is evaluated as a function of P_x.
  • B. Mutual Information with Gaussian Signalling: The exact-detection-error optimization is mathematically intractable and therefore requires a numerical search over the optimal transmit power.
  • C. Difference between D(p0||p1) and D(p1||p0): For Gaussian signalling, D(p0||p1) ≤ D(p1||p0), so the former imposes a tighter lower bound on Willie’s detection error.
  • B. Mutual Information with Gaussian Signalling: The constraint D(p0||p1) ≤ 2ε^2 permits higher transmit power and higher maximum mutual information than D(p1||p0) ≤ 2ε^2 for Gaussian signalling.
  • C. Difference between D(p0||p1) and D(p1||p0): Figure 2 compares KL divergence and mutual information for skew-normal signalling across skew-parameter values.

VI. NUMERICAL RESULTS

Numerical results compare skew-normal and Gaussian signalling under KL-divergence, total-variation, and detection-error constraints. They show skew-normal signalling can improve mutual information under D(p0||p1) and total-variation constraints, while D(p1||p0) is stricter for Gaussian signalling.

  • KL-divergence constraint: Skew-normal signalling achieves higher mutual information than Gaussian signalling under D(p0||p1) ≤ 2ǫ^2, showing Gaussian signalling is not optimal for that constraint.The comparison varies the skew parameter for skew-normal signalling and transmit power for Gaussian signalling to obtain comparable mutual-information and divergence values.
  • Total-variation constraint: Skew-normal signalling also achieves higher mutual information than Gaussian signalling for some total-variation values, so Gaussian signalling is not optimal under ξ* ≥ 1 − ǫ.The total variation satisfies ξ* = 1 − VT(p0, p1), linking the numerical comparison to the exact detection-error constraint.
  • Detection-error bounds: For Gaussian signalling, the D(p0||p1)-based lower bound on ξ* is tighter than the D(p1||p0)-based lower bound.The two bounds are close when ξ* is near one, which is the usual covert-communication regime, but they lead to different signalling-optimality conclusions.
  • Comparison of covertness constraints: Under Gaussian signalling, the ξ* constraint permits higher maximum transmit power and mutual information than either KL-divergence constraint.The numerical comparison considers ξ* ≥ 1 − ǫ, D(p0||p1) ≤ 2ǫ^2, and D(p1||p0) ≤ 2ǫ^2.
  • Comparison of covertness constraints: The D(p1||p0) ≤ 2ǫ^2 constraint is stricter than D(p0||p1) ≤ 2ǫ^2 because it yields lower allowable transmit power and mutual information.The paper states that this strictness conclusion also holds for optimal signalling strategies, not only Gaussian signalling.
  • Summary of results: The numerical section summarizes that Gaussian signalling is optimal for D(p1||p0) but not for D(p0||p1) or ξ* constraints.These conclusions are respectively established analytically for the KL constraint or numerically for the total-variation constraint.

VII. CONCLUSION

The paper establishes Gaussian signalling as optimal under one KL-divergence constraint but not the other, where skew-normal signalling can achieve higher mutual information. For Gaussian signalling, the D(p1||p0) constraint is stricter and yields lower mutual information than D(p0||p1).

  • VII. CONCLUSION: Gaussian signalling is optimal under D(p1||p0) ≤ 2ε^2, while skew-normal signalling can outperform it under D(p0||p1) ≤ 2ε^2.The latter constraint’s optimal signalling strategy remains unresolved.
  • VII. CONCLUSION: D(p1||p0) ≤ 2ε^2 is stricter than D(p0||p1) ≤ 2ε^2 for Gaussian signalling, producing lower mutual information.This follows from D(p0||p1) ≤ D(p1||p0).
Loading 1807.00719v3…