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Covert Beamforming Design for Integrated Radar Sensing and Communication Systems

Shuai Ma, Haihong Sheng, Ruixin Yang, Hang Li, Youlong Wu, Chao Shen, Naofal Al-Dhahir, Shiyin Li

arXiv:2208.05878v1cs.ITeess.SPmath.IT

TL;DR

IRSC beamforming must balance radar detection and covert communication while meeting covertness and power constraints. The paper develops designs for perfect and imperfect eavesdropper CSI, and simulations show they realize radar detection and covert communication in both scenarios.

  • Problem

    The design problem is to maximize radar detection MI or communication rate while satisfying covertness, counterpart-performance, and total-power constraints.

    Source: §I · p. 2, 2) Rate Maximization · p. 4

  • Method

    For perfect WCSI, the paper uses semidefinite relaxation, bisection, and a zero-forcing scheme; for imperfect WCSI, it develops robust relaxation and restriction methods.

    Source: §I · p. 2

  • Results

    Simulations show the proposed framework can simultaneously realize radar detection and covert communication with perfect and imperfect WCSI.

    Source: §VI · p. 12

  • Takeaways & Limitations

    The framework addresses both accurately known WCSI and estimation-error settings, with robust optimization targeting worst-case covert IRSC performance in the latter.

    Source: §VI · p. 12

Abstract

from arXiv · show

We propose covert beamforming design frameworks for integrated radar sensing and communication (IRSC) systems, where the radar can covertly communicate with legitimate users under the cover of the probing waveforms without being detected by the eavesdropper. Specifically, by jointly designing the target detection beamformer and communication beamformer, we aim to maximize the radar detection mutual information (MI) (or the communication rate) subject to the covert constraint, the communication rate constraint (or the radar detection MI constraint), and the total power constraint. For the perfect eavesdropper's channel state information (CSI) scenario, we transform the covert beamforming design problems into a series of convex subproblems, by exploiting semidefinite relaxation, which can be solved via the bisection search method. Considering the high complexity of iterative optimization, we further propose a single-iterative covert beamformer design scheme based on the zero-forcing criterion. For the imperfect eavesdropper's CSI scenario, we develop a relaxation and restriction method to tackle the robust covert beamforming design problems. Simulation results demonstrate the effectiveness of the proposed covert beamforming schemes for perfect and imperfect CSI scenarios.

I. INTRODUCTION … 1) Detection Only:

The paper develops covert beamforming frameworks for IRSC systems that hide communication within radar probing waveforms, addressing the risk that open, broadcasting waveforms expose information. It studies perfect and imperfect eavesdropper CSI and introduces system and detection-only models.

I. INTRODUCTION

  • Unlike physical-layer security, which protects information content, covert communication aims to hide communication behavior; prior tag-based IRSC approaches rely on multipath conditions that may be unavailable.Earlier IRSC secrecy studies focused on physical-layer security, while the paper identifies covert IRSC beamforming as not yet rigorously established.Source: §I · p. 1
  • The paper establishes covert beamforming frameworks for maximizing radar detection mutual information or covert communication rate under covert and total-power constraints, for perfect and imperfect Willie CSI.For perfect CSI, semidefinite relaxation yields convex subproblems solved by bisection; for imperfect CSI, a relaxation-and-restriction method tackles robust design.Source: §I · p. 2
  • To reduce iterative-optimization complexity, the proposed zero-forcing scheme uses the target-detection beamformer as cover and projects communication signals into the null space of Willie’s channel.It designs the communication beamformer to be orthogonal to Willie’s channel state information.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The IRSC model has an N-antenna radar that detects a target and transmits to single-antenna Bob, while single-antenna Willie monitors whether information is being sent.The target-reflected signals are ignored at Bob.Source: §II · p. 2

A. Signal Model

  • The model distinguishes radar detection and communication signals and specifies Radar-Target, Radar-Bob, and Radar-Willie channel state information.The channels are assumed to follow a Rayleigh flat-fading model.Source: §II · p. 2

1) Detection Only:

  • In the detection-only case, the radar transmits a detection signal through a beamformer, and the target-reflected signal is received at the radar.The model assumes the target’s azimuth and path-loss parameters can be estimated; Bob can also receive the detection signal because transmissions may be omnidirectional.Source: 1) Detection Only · pp. 2–3

2) Detection and Communication: … 2) Detection and Communication:

The section describes simultaneous radar detection and communication and defines radar mutual information (MI) and Bob’s achievable rate as performance measures. It characterizes detection MI as information about the target learned from the radar observation.

2) Detection and Communication:

  • The radar’s omni-directional communication signals can also be captured in target echoes when detection and communication operate simultaneously.Source: 2) Detection and Communication · p. 3
  • The beamformer vector wR,1 is used for sC when the radar sends detection and communication signals, and the passage introduces Bob’s received signal.Source: 2) Detection and Communication · p. 3

B. Performance Metrics

1) Detection Only:

  • The radar uses information-theoretic mutual information to quantify target-characteristic learning, building on its use for communication-channel information transmission and radar transmission design –.Source: 1) Detection Only · p. 3
  • Radar MI measures how much information the observation yR provides about target channel hT, conditioned on transmission signal sR.Source: 1) Detection Only · p. 3

2) Detection and Communication:

C. Covert Constraints · 1) Detection by Willie:

Willie tests whether the radar sends only detection signals or also communicates with Bob, and the paper quantifies covertness through Willie’s detection errors. It defines the test using likelihoods under both hypotheses and a Neyman–Pearson decision rule.

C. Covert Constraints

  • Willie distinguishes H0, where the radar sends only detection signals, from H1, where it sends detection and communication signals.The received-signal model is written for each hypothesis in (9).Source: C. Covert Constraints · p. 3
  • For tracking and communication, the radar assumes Willie exists and tries to conceal its communication with Bob.The radar and communication receivers use deterministic detection signal sR to facilitate signal extraction.Source: C. Covert Constraints · p. 3, 1) Detection by Willie · p. 3
  • The paper defines KL divergences in both directions between Willie’s likelihoods under H0 and H1.These are D(p0 ∥ p1) and D(p1 ∥ p0).Source: C. Covert Constraints · p. 3

1) Detection by Willie:

  • The covertness measure ξ is the sum of Willie’s false-alarm and missed-detection probabilities.ξ = P(D1 | H0) + P(D0 | H1).Source: 1) Detection by Willie · p. 4
  • With equal hypothesis probabilities, the Neyman–Pearson likelihood-ratio test minimizes ξ.The paper introduces p0(yW) and p1(yW) as the likelihood functions under H0 and H1, respectively.Source: C. Covert Constraints · p. 3, 1) Detection by Willie · p. 4
  • Willie’s optimal detection threshold φ* is specified for the average-power statistic |yW|^2.The paper also introduces the CDFs of |yW|^2 under H0 and H1.Source: 1) Detection by Willie · p. 4

2) Definition of Covert Constraint: … III. BEAMFORMING DESIGN BASED ON PERFECT WCSI

The paper formulates integrated radar sensing and communication around covert transmission, optimizing either radar detection mutual information or communication rate under covertness and other performance and power constraints. It also considers perfect eavesdropper CSI, where the radar uses Willie’s channel information to help Bob hide.

2) Definition of Covert Constraint:

  • The paper converts Willie’s optimal-detector error requirement ξ* ≥ 1−ε into tractable covert constraints using total variation distance and Pinsker’s inequality.The resulting constraints depend on the transmission variables and receiver noise.Source: 2) Definition of Covert Constraint · p. 4

D. Problem Formulation

  • The radar maximizes either detection mutual information or communication rate when performing sensing and communication simultaneously.These are presented as two separate optimization problems.Source: D. Problem Formulation · p. 4

1) MI Maximization:

  • The MI-maximization problem designs beamformers to maximize radar detection MI subject to covert communication, covert-rate, and total transmit-power constraints.The total transmit power is bounded by Ptotal.Source: 1) MI Maximization · p. 4

2) Rate Maximization:

  • The rate-maximization problem designs beamformers to maximize Bob’s achievable rate subject to covertness, a radar MI requirement, and total transmit-power constraints.The beamformers split available power between the two metrics, creating an MI–covert-rate tradeoff.Source: 2) Rate Maximization · p. 4
  • The paper focuses on integrated sensing and communication, omitting the detection-only case as a simplified version of the MI-maximization problem.It numerically discusses the tradeoff between MI and covert rate.Source: 2) Rate Maximization · p. 4

III. BEAMFORMING DESIGN BASED ON PERFECT WCSI

  • With full CSI of Willie’s channel hW, the radar can help Bob hide under H1; the resulting covert constraint implies perfect covert transmission.Source: §III · pp. 4–5

A. MI Maximization · 1) Covert Beamformer:

For MI maximization, the paper reformulates the problem and proposes covert and zero-forcing beamformers; the covert-beamformer method uses semidefinite relaxation and bisection to obtain beamformers.

A. MI Maximization

  • The MI-maximization problem is reformulated, and covert and zero-forcing beamformers are proposed as two solution methods.The supplied passage identifies these methods for solving problem (24).Source: A. MI Maximization · p. 5

1) Covert Beamformer:

  • An auxiliary variable IR is introduced to express the covert-beamformer problem in an equivalent form.Source: 1) Covert Beamformer · p. 5
  • The method applies semidefinite relaxation (SDR), ignoring rank-1 constraints to obtain a relaxed version of problem (25).Source: 1) Covert Beamformer · p. 5
  • For any IR ≥ 0, the relaxed problem is a convex SDP, so feasibility testing at a fixed IR can find the optimal solution.Source: 1) Covert Beamformer · p. 5
  • The method solves a series of convex subproblems over IR ≥ 0 and uses bisection to obtain the optimal covert beamformers.A standard convex solver such as CVX can solve the subproblems.Source: 1) Covert Beamformer · p. 5
  • Because SDR rank relaxation can produce optimal solutions with rank greater than 1, the method applies Gaussian randomization to problem (25) to obtain high-quality rank-1 beamformers.Source: 1) Covert Beamformer · p. 5

2) Zero-Forcing Beamformer:

The zero-forcing design nulls interference by jointly choosing radar beamformers, then minimizes the power of one beamformer to support the objective. It uses semidefinite relaxation and convex optimization to solve the design.

Zero-Forcing Beamformer

  • The design sets wR,1 to eliminate interference to Willie and the radar, while wR,0 eliminates Bob’s interference.Source: 2) Zero-Forcing Beamformer · p. 5
  • Because the objective increases with wR,0 power but does not depend on wR,1, the design minimizes wR,1 transmission power subject to constraints (28b), (28c), and (28e).Source: 2) Zero-Forcing Beamformer · p. 5
  • The non-convex wR,1 design is relaxed using semidefinite relaxation by dropping the rank-one constraint.Source: 2) Zero-Forcing Beamformer · p. 6
  • If the relaxed solution is not rank one, Gaussian randomization produces a high-quality rank-one solution; the resulting problem is an SOCP solvable with standard convex optimization solvers.Source: 2) Zero-Forcing Beamformer · p. 6

B. Rate Maximization · IV. BEAMFORMING DESIGN BASED ON IMPERFECT WCSI · A. MI Maximization

The paper develops rate-maximizing beamforming for perfect WCSI and robust MI-maximizing beamforming for imperfect WCSI. The imperfect-CSI method approximates infinitely many error-dependent constraints with tractable LMIs and solves the resulting design by bisection.

B. Rate Maximization

  • The perfect-WCSI rate design maximizes communication rate subject to zero covertness divergence and a minimum radar detection MI.The authors apply binary search and zero-forcing beamforming to solve the design.Source: B. Rate Maximization · p. 6
  • Binary search and zero-forcing beamforming are used to solve the perfect-WCSI rate-maximization problem.The derivation details are omitted for brevity.Source: B. Rate Maximization · p. 6

IV. BEAMFORMING DESIGN BASED ON IMPERFECT WCSI

  • Imperfect WCSI is considered because limited cooperation with Willie may prevent the radar from accessing his full channel state information, –.The channel error is modeled around an estimated CSI using an ellipsoidal uncertainty region.Source: §IV · p. 6
  • Because full WCSI is unavailable, the design adopts constraint (20) rather than requiring perfect covert transmission under constraint (23).The ellipsoid matrix controls its axes, while the positive parameter υW determines its volume.Source: §IV · p. 6

A. MI Maximization

  • The imperfect-WCSI MI-maximization problem is non-convex, so the method reformulates its covertness constraint using properties of a logarithmic function.The reformulation uses roots of an equation involving the covertness divergence.Source: A. MI Maximization · p. 6
  • Semidefinite relaxation removes rank-one constraints, and the S-lemma converts infinitely many channel-error constraints into LMIs as a tractable safe approximation.The reformulated error constraints are represented by a finite set of LMIs.Source: A. MI Maximization · p. 7
  • For fixed target MI, the conservative approximation is a convex SDP that can be solved efficiently with off-the-shelf solvers and the proposed bisection method.The bisection procedure is summarized in Algorithm 2.Source: A. MI Maximization · p. 7

2) Case of D (p1 ∥p0 ) ≤2ε2:

This subsection considers robust covert-rate maximization under D(p1 ∥ p0) ≤ 2ε2 and applies the relaxation-and-restriction approach used in the previous subsection. It notes that achievable covert rates differ substantially between the two signal constraints, with the comparison deferred to the next section.

2) Case of D (p1 ∥p0 ) ≤2ε2:

  • Under D(p1 ∥ p0) ≤ 2ε2, the robust covert-rate problem is solved using the relaxation-and-restriction approach from the previous subsection.The detailed derivations are omitted for brevity.Source: 2) Case of D (p1 ∥p0 ) ≤2ε2 · p. 7
  • Achievable covert rates differ substantially under the two signal constraints, and their comparison is deferred to the next section.Source: 2) Case of D (p1 ∥p0 ) ≤2ε2 · p. 7

B. Rate Maximization

For rate maximization with imperfect eavesdropper CSI, the paper uses a relaxation-and-restriction approach for robust beamforming design under two hidden constraints.

B. Rate Maximization

  • With estimation errors in the eavesdropper’s WCSI, robust design for problem (33) imposes D(p0 ∥ p1) ≤ 2ε^2 and D(p1 ∥ p0) ≤ 2ε^2, then applies relaxation and restriction.The paper notes that perfect covert transmission, D(p0 ∥ p1) = 0, is difficult in this case; numerical results are discussed in the next section.Source: B. Rate Maximization · p. 8

V. NUMERICAL RESULTS … 1) MI Maximization:

The numerical results show that the covert beamformer outperforms the ZF design in the reported perfect-WCSI comparisons, while robust design satisfies the reported covertness requirements under CSI errors. They also show how radar MI and Willie’s detection probabilities respond to covertness, CSI error, antenna count, and covert-rate thresholds.

V. NUMERICAL RESULTS

  • The numerical section presents simulations to demonstrate the effectiveness of the proposed beamforming design schemes.The section introduction also begins a general simulation setup, but the supplied passage ends before its parameter details.Source: §V · p. 8

B. Imperfect WCSI — 1) MI Maximization:

  • With CSI errors vw = 0.005, about 50% and 57% of non-robust designs exceed the 0.005 thresholds for D (p0 ∥p1 ) and D (p1 ∥p0 ); the robust design satisfies both requirements.Both designs use covertness thresholds 2ε2 = 0.005; the non-robust design uses ˆhW rather than hW.Source: 1) MI Maximization · p. 8, Figure 4 · p. 9

A. Perfect WCSI

  • With N = 5, the covert beamformer achieves higher MI and RB than the ZF design, and both increase almost linearly with total transmit power.The comparison is reported for mutual information I (yR; hT |sR ) and RB versus Ptotal.Source: Figure 2 · p. 8, A. Perfect WCSI · p. 8
  • At Ptotal = 10dBm, MI and RB increase logarithmically with antenna count, while the gap between covert and ZF designs widens.The covert design achieves larger MI and RB than the ZF design in the reported comparisons.Source: Figure 3 · p. 8, A. Perfect WCSI · p. 8

B. Imperfect WCSI — 1) MI Maximization:

  • As ε increases, MI rises while FA and MD probabilities fall; FA remains below MD, and MI is higher under D (p0 ∥p1 ) ≤2ε2 than under D (p1 ∥p0 ) ≤2ε2.The passages report these trends for CSI errors vw = 0.001 and attribute the MI difference to the stricter D (p1 ∥p0 ) constraint.Source: 1) MI Maximization · pp. 8–10, Figure 5 · p. 9
  • Under both covertness constraints, MI increases with antenna count N for vw = 0.005 and ε = 0.05.The passage describes this trend as similar to the perfect-WCSI antenna-count result.Source: 1) MI Maximization · p. 9
  • As CSI error vw increases, MI rises while radar performance worsens, and FA and MD probabilities increase with FA remaining below MD.The reported results also indicate that larger CSI errors can degrade Willie’s detection, potentially benefiting Bob.Source: 1) MI Maximization · p. 9, Figure 6 · p. 10
  • As covert-rate threshold β increases, MI decreases; MI is higher under D (p0 ∥p1 ) ≤2ε2 than under D (p1 ∥p0 ) ≤ 2ε2.The passage states that the latter constraint is stricter, consistent with.Source: 1) MI Maximization · p. 10

2) Rate Maximization: · VI. CONCLUSIONS

The rate-maximization results show that robust beamforming satisfies the covert constraint under eavesdropper CSI errors, while covert rates and Willie’s detection performance vary with the covertness constraint and system conditions. The conclusion summarizes a framework for covert radar-communication under perfect and imperfect WCSI.

2) Rate Maximization:

  • The robust design satisfies Willie’s error-detection requirement, whereas the non-robust design fails the KL-divergence constraints under CSI errors.The comparison uses the same covertness threshold, 2ε^2 = 0.02; the non-robust design uses the estimated channel ĥW.Source: 2) Rate Maximization · p. 10, Figure 9 · p. 11
  • With imperfect CSI, the empirical CDF of the non-robust design’s KL divergence fails the covertness constraints, while the robust design meets them.The comparison is under threshold 2ε^2 = 0.02 and CSI errors vw = 0.001.Source: 2) Rate Maximization · p. 10, Figure 9 · p. 11
  • As ε increases, covert rate RB rises while false-alarm and missed-detection probabilities fall; the D(p1 ∥ p0) constraint yields lower rates than D(p0 ∥ p1).The latter constraint is stricter, and the false-alarm probability remains below the missed-detection probability in both cases.Source: 2) Rate Maximization · pp. 10–11, Figure 10 · p. 11
  • Across both KL-divergence constraints, looser covertness allows higher rates but improves Willie’s detection performance.For both constraints, false-alarm and missed-detection probabilities decrease as ε increases.Source: 2) Rate Maximization · pp. 10–11
  • Higher CSI error vw lowers covert rate RB but increases both Willie’s false-alarm and missed-detection probabilities, with false alarms remaining lower.The rate comparison uses ε = 0.20; the detection-error comparison uses ε = 0.05.Source: 2) Rate Maximization · p. 11, Figure 11 · p. 12
  • Increasing antenna count N raises covert rate RB, while increasing the mutual-information threshold γ lowers it.These trends are reported for vw = 0.001 and ε = 0.20; rates under D(p0 ∥ p1) ≤ 2ε^2 exceed those under D(p1 ∥ p0) ≤ 2ε^2.Source: 2) Rate Maximization · pp. 11–12, Figure 12 · p. 12, Figure 13 · p. 12

VI. CONCLUSIONS

  • The framework proposes zero-forcing single-iterative beamforming for accurately known WCSI and robust optimization for estimated WCSI, enabling covert communication alongside radar detection.The robust method is designed to ensure worst-case covert IRSC performance; simulations demonstrate the framework for perfect and imperfect WCSI.Source: §VI · p. 12

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