Source-linked AI summary
Covert Communication in Fading Channels under Channel Uncertainty
Khurram Shahzad, Xiangyun Zhou, Shihao Yan
TL;DR
The paper studies covert communication over uncertain block-fading channels, where public transmission to a legitimate user covers private transmission to a covert user. It derives Willie’s optimal radiometer threshold and analyzes achievable rates under outage and covertness constraints, finding that channel uncertainty and relaxed covertness requirements expand feasible rates while also affecting legitimate decoding.
Problem
Exact achievable covert rates were not quantified, motivating analysis of public transmission as cover for covert communication in wireless fading channels.
Method
The paper exploits channel uncertainty under block fading, models Willie’s radiometer detection, and analyzes legitimate and covert rates under outage and covertness constraints.
Results
Channel uncertainty at Willie permits more power and higher achievable rate for Bob while affecting decoding, and relaxing ϵ from 0.1 to 0.3 expands the rate region for β = 0.2.
Takeaways & Limitations
Channel uncertainty can provide covertness, while the achievable rates depend on both channel uncertainty and the required level of covertness.
Abstract
from arXiv · showhide
A covert communication system under block fading channels is considered where users experience uncertainty about their channel knowledge. The transmitter seeks to hide the covert communication to a private user by exploiting a legitimate public communication link while the warden tries to detect this covert communication by using a radiometer. We derive the exact expression for the radiometers optimal threshold which determines the performance limit of the wardens detector. Furthermore for given transmission outage constraints the achievable rates for legitimate and covert users are analyzed while maintaining a specific level of covertness. Our numerical results illustrate how the achievable performance is affected by the channel uncertainty and required level of covertness.
I. INTRODUCTION
The paper studies covert communication in wireless fading channels when channel knowledge is uncertain, using legitimate public transmission as cover for communication with a covert user. It derives the warden’s optimal detection threshold and analyzes feasible rates under covertness and outage constraints.
- Covert communication hides the communication itself rather than only protecting message content.
- Prior work established square-root scaling, but exact achievable covert rates remained unquantified.Earlier studies extended covert communication to BSCs, DMCs, and MACs, while one study obtained positive rate under noise uncertainty.
- The paper exploits channel-knowledge uncertainty, derives Willie’s exact optimal threshold, and analyzes legitimate and covert rates under outage constraints.
- The system hides Alice’s transmission to Bob behind her ongoing public transmission to Carol while Willie attempts detection with a radiometer.Alice, Carol, Bob, and Willie each use a single antenna, with distances denoted d_ac, d_ab, and d_aw.
- Alice uses zero-mean Gaussian signaling with transmit powers P_ac and P_ab for Carol and Bob, respectively.The average symbol powers are normalized to 1, and Alice keeps Carol’s transmit power constant because Carol is unaware of the covert transmission.
A. Channel Model
The model uses quasi-static block-fading channels with uncertain channel coefficients and defines Willie’s binary detection problem for Alice’s optional transmission to Bob. Covertness requires the sum of false-alarm and missed-detection probabilities to remain near one.
- Channels are block fading: each coefficient remains constant within a block and changes independently between blocks.Messages are assumed to complete within one block, corresponding to quasi-static fading.
- Alice continuously transmits to Carol and may transmit covertly to Bob in a preshared-secret-selected block.Willie tests whether Bob’s transmission occurred while knowing Alice’s transmit power.
- Under H0 Alice does not transmit to Bob, while under H1 she does; Willie decides using observations from the selected block.
- Channel uncertainty decomposes each coefficient into known and uncertain independent CSCG parts, with β_k measuring uncertainty variance.The known-part variance is 1 − β_k because the total channel variance is 1.
- Covertness is achieved when P_FA + P_MD ≥ 1 − ϵ as n → ∞, with smaller ϵ making Willie’s detector ineffective.
III. DETECTION STRATEGY AT WILLIE
Willie applies likelihood-ratio hypothesis testing to distinguish whether Alice transmitted covertly to Bob. The detector’s ambiguity arises from both receiver noise and channel uncertainty.
- Willie’s observations are modeled through their i.i.d. per-symbol distributions under H0 and H1.
- The Neyman–Pearson criterion gives Willie a likelihood-ratio test comparing the observation likelihoods under H1 and H0.Equal prior probabilities imply Υ = 1 in the test.
- Receiver noise and channel uncertainty jointly create ambiguity between the two hypotheses.
A. Detection using a Radiometer
For the considered block-fading model, a radiometer is Willie’s optimal detector, and its decision is implemented as a threshold test on total received power. The paper then derives the threshold setting that minimizes detection error.
- The radiometer is optimal for Willie under the considered system model.
- The paper evaluates the optimal radiometer threshold after establishing the detector’s optimality.
- Willie compares the total received block power P_w = ∑|y_w|^2 with a chosen threshold λ.The threshold test determines whether Willie decides for H1 or H0.
B. Optimal Threshold for Willie’s Radiometer
The paper establishes the radiometer as Willie’s optimal detector and derives the threshold that minimizes his false-alarm plus missed-detection probability. The optimum is determined through case analysis and derivative conditions.
- The radiometer is shown to be Willie’s optimal detector for identifying Alice-Bob covert transmission.The detector uses a sufficient statistic derived from Willie’s observation vector and classical hypothesis testing.
- Willie’s threshold is selected by minimizing the sum of false-alarm and missed-detection probabilities.The analysis evaluates the possible threshold cases and minimizes PF A + PMD.
- λ† is independent of Willie’s channel realization and represents the inflection point of PF A + PMD.The derivative is negative below λ†, while the second derivative is positive under the stated condition.
- λ† is the optimal threshold when it satisfies λ† > |ˆhaw|2ζ1 + σ2.This condition ensures the interior optimum applies to the relevant threshold range.
- When λ† does not satisfy the condition, Willie selects the minimum threshold satisfying λ ≥ |ˆhaw|2ζ1 + σ2.This follows from the monotonic increase of PF A + PMD for λ > λ†.
IV. PERFORMANCE OF COVERT COMMUNICATION
The performance analysis first evaluates Willie’s average detection error to quantify covertness, then derives outage probabilities that determine feasible communication rates.
- Willie’s average detection error quantifies covertness, while Carol’s and Bob’s outage probabilities determine feasible transmission rates.
A. Average Detection Error Probability
Because Alice does not know Willie’s unknown channel component, she assesses covertness using Willie’s detection performance averaged over channel realizations. Alice then selects transmit powers to meet the covertness requirement.
- Alice uses the average of Willie’s false-alarm plus missed-detection probability because ˆhaw is unknown to her.
- The paper derives Willie’s average detection error probability using the law of total expectation.
- Alice chooses her transmit power levels to Carol and Bob to achieve covertness.
B. Outage Probabilities at Carol and Bob
The paper derives outage probabilities for Carol and Bob and uses them to evaluate achievable rates under the covert-transmission hypothesis. Carol’s performance is degraded by interference from Alice’s transmission to Bob.
- Carol: The paper derives Carol’s outage probability under H1 for a target rate Rc.
- Carol: Carol’s outage probability under H0 is lower than under H1 because Alice-Bob transmission causes interference under H1.
- Bob: The paper derives Bob’s outage probability under H1 for a target rate Rb.
- Achievable rates: For δc ≤0.1 and δb ≤0.1, achievable rates for Carol and Bob under H1 are calculated using the outage expressions.
- Achievable rates: Fig. 2 presents Carol’s and Bob’s achievable rate region as channel uncertainty β varies, with ϵ = 0.2 and α = 3.
V. NUMERICAL RESULTS AND DISCUSSION
The numerical results show that channel uncertainty and the required covertness level reshape the achievable rate region for Carol and Bob. Greater uncertainty favors Bob while imposing a rate loss on Carol, whereas relaxing covertness expands the feasible region.
- Increasing β increases Bob’s achievable rate by allowing more transmit power for Bob under a fixed covert requirement.Greater uncertainty at Willie deteriorates his detection performance, increasing feasible Pab.
- Increasing β also reduces Carol’s rate because channel uncertainty affects Carol’s decoding performance.
- Relaxing ϵ from 0.1 to 0.3 expands the achievable rate region for Carol and Bob at fixed β = 0.2.The expansion favors Bob because relaxing ϵ directly increases feasible Pab for a given Pac.
VI. CONCLUSION
The paper studies covert communication over a public legitimate link when users have channel uncertainty. It derives Willie’s optimal detection threshold and quantifies outage-rate regions, showing that channel uncertainty enables covertness while affecting Carol’s and Bob’s achievable rates.
- The system examines covert communication embedded in a public legitimate communication link under user channel uncertainty.
- β = 0.3 and ϵ = 0.3 are considered quite large from a practical perspective in the numerical results.These values are included to illustrate their effects on the achievable rate region.
- The paper derives a closed-form expression for Willie’s optimal detector threshold and quantifies the achievable outage-rate region for Carol and Bob.
- Channel uncertainty at Willie allows Alice to achieve a certain amount of covertness while also affecting Carol’s and Bob’s achievable rates.