PaperlayerView exact revision on arXiv

Source-linked AI summary

Robust and Secure Resource Allocation for ISAC Systems: A Novel Optimization Framework for Variable-Length Snapshots

Dongfang Xu, Xianghao Yu, Derrick Wing Kwan Ng, Anke Schmeink, Robert Schober

arXiv:2206.13307v2cs.ITeess.SP

TL;DR

The paper develops a variable-length snapshot optimization framework for ISAC resource allocation, addressing the non-convexity of eavesdroppers’ uncertainty regions. It jointly optimizes key communication and sensing resources, and simulations report higher sum secrecy rate and physical-layer security than the single-snapshot framework.

  • Problem

    The non-convexity of the joint uncertainty region obstructs robust resource allocation for eavesdroppers’ small-scale fading.

  • Method

    The proposed variable-length snapshot framework jointly optimizes the snapshot resources, including beamforming and AN covariance, while using a bound for the eavesdroppers’ uncertainty region.

  • Results

    Simulation results show a higher sum secrecy rate and significantly higher physical-layer security than the widely adopted single-snapshot optimization framework.

  • Takeaways & Limitations

    Variable-length snapshot optimization facilitates highly directional beam patterns for sensing and allows communication resources to be prioritized.

Abstract

from arXiv · show

In this paper, we investigate the robust resource allocation design for secure communication in an integrated sensing and communication (ISAC) system. A multi-antenna dual-functional radar-communication (DFRC) base station (BS) serves multiple single-antenna legitimate users and senses for targets simultaneously, where already identified targets are treated as potential single-antenna eavesdroppers. The DFRC BS scans a sector with a sequence of dedicated beams, and the ISAC system takes a snapshot of the environment during the transmission of each beam. Based on the sensing information, the DFRC BS can acquire the channel state information (CSI) of the potential eavesdroppers. Different from existing works that focused on the resource allocation design for a single snapshot, in this paper, we propose a novel optimization framework that jointly optimizes the communication and sensing resources over a sequence of snapshots with adjustable durations. To this end, we jointly optimize the duration of each snapshot, the beamforming vector, and the covariance matrix of the AN for maximization of the system sum secrecy rate over a sequence of snapshots while guaranteeing a minimum required average achievable rate and a maximum information leakage constraint for each legitimate user. The resource allocation algorithm design is formulated as a non-convex optimization problem, where we account for the imperfect CSI of both the legitimate users and the potential eavesdroppers. To make the problem tractable, we derive a bound for the uncertainty region of the potential eavesdroppers' small-scale fading based on a safe approximation, which facilitates the development of a block coordinate descent-based iterative algorithm for obtaining an efficient suboptimal solution.

I. INTRODUCTION

ISAC improves spectrum use by combining sensing and communication, but its directional sensing signals can expose information to malicious targets. The paper addresses limitations of multi-stage, single-snapshot designs through robust resource allocation over variable-length snapshots.

  • ISAC jointly supports sensing and communication, improving spectral efficiency while using shared DFRC hardware.
  • Directional sensing beams illuminate targets strongly, but their information-carrying signals can leak and be decoded by malicious eavesdroppers.
  • Artificial noise can degrade eavesdropper channels while preserving legitimate-user SINR when designed using CSI.
  • Existing multi-stage designs use coarse initial sensing, which can cause clutter, inaccurate target information, ineffective subsequent beam patterns, and degraded sensing quality.
  • The paper jointly optimizes resources over variable-length snapshots, flexibly adjusting durations and sequentially serving sector slices with highly directional beams.
  • The robust non-convex formulation accounts for imperfect CSI and targets secure multiuser ISAC resource allocation.
  • Simulation results report higher sum secrecy rates than three baselines and show that CSI uncertainty matters for ISAC design.

II. SYSTEM MODEL

The system uses a DFRC base station to scan a sector through sequential directional-beam snapshots while jointly supporting communication and sensing. Snapshot durations are adjustable so sensing coverage and communication priorities can be coordinated.

  • The DFRC BS scans a sector with a sequence of sensing beams to detect new targets and update known-target information during one scanning period.
  • Known targets are treated as potential eavesdroppers, while legitimate users actively interact with the DFRC BS at each scanning-period start.
  • Each beam covers a sector slice with a desired power gain, using directional-beam requirements determined by slice geometry.
  • During snapshot m, the DFRC BS transmits the corresponding beam pattern and captures the environment for duration t_m > 0.
  • The BS synthesizes pre-designed sensing patterns using information-carrying beamformers and artificial noise, with snapshot durations flexibly adjusted to application requirements.
  • A snapshot can be extended when its sensing pattern favors secure communication, allowing communication or sensing to be prioritized according to the application.

B. Signal Model

The signal model combines user-directed data beams with artificial noise and represents target channels using Ricean fading. Robustness is incorporated through bounded uncertainty models for outdated legitimate-user CSI and imperfect eavesdropper location and multipath information.

  • In each snapshot, the DFRC BS transmits independent data streams to legitimate users together with artificial noise.
  • The user beamformer w_k^m serves user k, while the artificial-noise signal v^m follows a zero-mean complex Gaussian model with covariance V^m.
  • Potential-eavesdropper channels include a deterministic line-of-sight component and multipath fading, forming a Ricean channel model for terrestrial scattering conditions.
  • Legitimate-user CSI is modeled as an initial channel estimate plus a bounded snapshot-dependent uncertainty caused by outdated CSI and estimation errors.
  • Eavesdropper distance and angle errors are bounded, while unavailable multipath information is represented as bounded elementwise channel uncertainty.
  • The resulting eavesdropper uncertainty contains distance, angle, and multipath terms; angle and multipath uncertainty are identified as challenging for robust allocation.
  • A safe approximation is used to derive a tractable bound on the joint angle and multipath uncertainty for robust resource allocation.

III. PROBLEM FORMULATION

The formulation maximizes sum secrecy rate across snapshots while enforcing sensing fidelity, user-rate, leakage, power, timing, and beamforming constraints. Its coupled robust structure is non-convex and semi-infinite, motivating a computationally efficient iterative solution.

  • B. Problem Formulation: The optimization maximizes system sum secrecy rate over M snapshots within the scanning period.
  • B. Problem Formulation: The formulation constrains average achievable rates for legitimate users and limits each user's information leakage to potential eavesdroppers.
  • B. Problem Formulation: The variables jointly designed are snapshot durations, beamforming matrices, and artificial-noise covariance matrices.
  • B. Problem Formulation: Transmit power, total scanning duration, and each snapshot duration are bounded, while sensing fidelity is controlled through a threshold on actual-versus-desired beam covariance mismatch.
  • B. Problem Formulation: The rank constraint preserves rank-one beamforming structure, and the formulation also includes implementation bounds for data-symbol transmission and minimum snapshot duration.
  • B. Problem Formulation: A smaller sensing-mismatch threshold favors target sensing, whereas a larger threshold gives more beamforming flexibility that can facilitate higher secrecy rates.
  • B. Problem Formulation: The problem is non-convex, semi-infinite, and difficult because variables are coupled, SINR expressions are fractional, uncertainty sets are continuous, and rank constraints remain.
  • B. Problem Formulation: The proposed approach applies transformations and a block coordinate descent-based iterative algorithm to obtain a computationally efficient solution.

IV. SOLUTION OF THE OPTIMIZATION PROBLEM

The solution approach first bounds the difficult joint eavesdropper uncertainty using a safe approximation, then transforms and decomposes the problem for iterative optimization. Block coordinate descent, inner approximation, and semidefinite relaxation are used for the resulting subproblems.

  • A safe approximation is used to derive a bound for the eavesdroppers' uncertainty region induced by angle and multipath fading uncertainty.
  • The bound addresses the semi-infinite difficulty associated with continuous uncertainty sets in the robust optimization problem.
  • The geometric analysis compares the actual joint uncertainty region with its proposed safe-approximation bound.
  • The resulting optimization is transformed into an equivalent form before its coupled variables are divided into two blocks using block coordinate descent.
  • Inner approximation and semidefinite relaxation are applied to solve the block coordinate descent subproblems.

A. Bound for Uncertainty Region of Eavesdroppers’ Small-Scale Fading

The paper derives a tractable circular bound for the joint angle and multi-path fading uncertainty of potential eavesdroppers, enabling robust resource allocation despite a non-convex uncertainty region. The bound is particularly suitable for the proposed framework because sensing beams and adjustable scanning periods can update target information and potentially reduce angle uncertainty.

  • Geometric uncertainty model: The joint angle and multi-path fading uncertainty region is non-convex, preventing direct use of methods such as the S-procedure that require convex uncertainty regions.The region is formed by a union of circles whose centers lie on a common circle.
  • Safe circular approximation: The authors bound the joint uncertainty region by a tractable circle, although the convex hull would provide a tighter but harder-to-model bound.The circle is centered at the zero-angle-uncertainty point and is derived using the law of cosines.
  • Bound accuracy: The bound is reported as relatively accurate, especially when the angle uncertainty is small and sensing quality is high.Its accuracy is evaluated using the ratio of the actual uncertainty-region area to the corresponding bounding-area measure.
  • Framework suitability: The proposed sensing framework supports offline design of directional beam patterns and adjustable scanning periods that can frequently update target information.These updates can potentially mitigate eavesdroppers’ angle uncertainty, making the derived bound well suited to the optimization framework.
  • CSI uncertainty representation: The bound yields an upper bound for the CSI uncertainty of each potential eavesdropper and represents all possible small-scale fading uncertainties with bounded norm.The uncertainty components are collected into a vector for joint robust optimization.

B. Problem Reformulation

The original robust optimization problem is reformulated with slack variables and transformations that remove semi-infinite objective terms and replace some semi-infinite constraints with more manageable forms. The resulting formulation prepares the problem for subsequent robust optimization techniques.

  • Slack-variable reformulation: The reformulation introduces slack variables to express the objective and constraints in a tractable hypograph form.The variables include ξ_k^rms, λ_k^rms, η_k^rms, κ_k^rms, and ζ_k,j^rms.
  • Preparation for BCD: The reformulated problem remains suitable for BCD after replacing the original problem with a lower-bound version and introducing additional semi-infinite constraints.The next step applies the S-procedure to convert those constraints into linear matrix inequalities.
  • Uncertainty decoupling: Multiplying the SINR numerator and denominator by α^-1(1+ρ_j) decouples the uncertainty variables.An additional slack variable ζ_k,j^rms is introduced for constraint C12.
  • Resulting formulation: The reformulations eliminate the semi-infinite programming terms from the objective function, making it more manageable for robust resource allocation.The semi-infinite constraints in C4 and C5 are replaced by bilinear constraints.

C. Handling Semi-Infinite Constraints C10b, C12a, and C12b

The paper applies the S-procedure, semidefinite relaxation, inner approximation, and block coordinate descent to handle the reformulated robust problem. The resulting iterative methods have convergence guarantees, while the uncertainty-region bound yields a feasible suboptimal solution to the original problem.

  • Block 1: The first BCD block jointly optimizes snapshot durations, beamforming matrices, artificial-noise covariance matrices, and selected slack variables.The rank constraint is removed through SDP relaxation, yielding a convex problem solvable by standard convex optimization solvers.
  • Rank recovery: The relaxed first-block problem admits a rank-constrained optimal solution, allowing recovery of an optimal beamforming vector from the beamforming matrix.The theorem establishes tightness of the SDP relaxation for this block.
  • Block 2: The second BCD block uses inner approximation and first-order Taylor underestimation to convexify the non-convex logarithmic constraint.Each inner-approximation iteration solves a convex optimization problem with a monotonically non-decreasing objective.
  • Convergence and solution quality: The inner-approximation algorithm converges to a locally optimal solution, while the overall BCD algorithm converges to a stationary point.Because the uncertainty bound is used, the resulting solution is feasible but suboptimal for the original problem.

V. SIMULATION RESULTS

The section evaluates the system performance of the proposed resource allocation scheme through simulations.

  • Simulation setup: The proposed resource allocation scheme is evaluated using system-performance simulations.

A. Simulation Setup

The simulations model a sector-scanning DFRC system with multiple users and detected targets, using imperfect CSI and robust resource-allocation baselines. The evaluation compares joint optimization of snapshot durations, beamforming, and AN against fixed-duration, non-robust, and multi-stage designs.

  • System model: The ISAC system covers a 120-degree sector with M sequential snapshots and serves K legitimate users alongside J potential eavesdroppers.The DFRC BS scans the sector using dedicated beams, with each snapshot associated with one scanning direction.
  • Channel and array model: The DFRC BS uses NT antennas with half-wavelength spacing, while legitimate-user small-scale fading follows independent identically distributed Rayleigh distributions.Potential-eavesdropper channels include estimated line-of-sight components and Ricean fading.
  • Uncertainty model: The simulations incorporate uncertainty in eavesdropper distance, angle, and multipath fading, including a 5 m distance bound and 5-degree maximum angle uncertainty.The eavesdropper-channel uncertainty bound is obtained using the paper’s safe approximation, while legitimate-user CSI uncertainty is also parameterized.
  • Compared resource-allocation schemes: The proposed scheme jointly optimizes snapshot durations, beamforming vectors, and AN covariance to maximize system sum secrecy rate within the scanning period.The study compares this design with fixed-duration, non-robust zero-forcing, and single-snapshot multi-stage baselines.
  • Compared resource-allocation schemes: Baseline scheme 3 first designs a multi-beam pattern toward potential eavesdroppers, then solves a single-snapshot optimization with M = 1 and t_1^s = T_tot.The proposed and other schemes are evaluated using averaged results, with feasibility assessed separately.

C. Convergence of the Proposed BCD Algorithm

The proposed BCD algorithm converges monotonically and quickly across tested system configurations. The simulations also show higher secrecy rates than baseline schemes, with robustness to uncertainty and trade-offs from sensing constraints and user CSI errors.

  • Convergence: The proposed BCD algorithm monotonically converges quickly to a stationary point across all four tested cases.The cases vary the numbers of snapshots, antennas, users, eavesdroppers, and legitimate-user CSI uncertainty.
  • Convergence: Case 1 converges within 10 Algorithm 2 iterations on average, while Case 2 requires roughly 5 additional iterations.Case 2 reaches a slightly larger objective value, attributed to expanded feasible resources from larger M and NT.
  • Secrecy-rate performance: The proposed scheme achieves substantially higher system sum secrecy rates than baseline schemes 1 and 2 as maximum transmit power increases.Variable snapshot durations provide extra resource-allocation degrees of freedom, whereas fixed durations and zero-forcing constrain the baselines.
  • Robustness: Small distance, angle, and multipath uncertainties for potential eavesdroppers cause only a small degradation in the proposed scheme’s system performance.Under eavesdropper CSI uncertainty, the proposed scheme and baseline scheme 1 allocate more power to AN to impair eavesdropper channels.
  • Robustness: The proposed scheme significantly outperforms the three baselines across the considered legitimate-user CSI-uncertainty range.Increasing uncertainty makes the design more conservative and can reduce legitimate-user SINRs, while the proposed framework retains the strongest reported secrecy-rate performance.
  • Sensing–communication trade-off: Tighter beam-pattern mismatch requirements reduce secrecy rate because more degrees of freedom are consumed by sensing-oriented beam synthesis.Larger ς_d leaves more freedom for communication-oriented beamforming, so the tolerance factor must be selected carefully.

F. Infeasibility rate versus Average User Achievable Rate

The infeasibility analysis evaluates whether the rate and other constraints can be jointly satisfied over random channel realizations. The proposed robust framework reduces infeasibility and uses variable snapshot durations to prioritize users with unfavorable channels.

  • Evaluation procedure: The infeasibility rate is estimated from 100 random channel realizations by counting feasible solutions and dividing the infeasible cases by the total.A realization is infeasible when no solution satisfies the desired achievable-rate requirement and the remaining constraints.
  • Infeasibility results: The proposed scheme significantly reduces infeasibility compared with the two baseline schemes as the average achievable-rate requirement increases.The proposed and baseline infeasibility rates are generally non-decreasing with the required average rate.
  • Infeasibility results: For larger R_req_k, variable snapshot durations let the proposed scheme prioritize unfavorable legitimate-user channels by extending their snapshot durations.This flexibility supports the average achievable-rate constraints while retaining the joint resource-allocation structure.
  • Robustness requirement: The non-robust scheme has roughly 10% infeasibility even under a low average achievable-rate constraint because it ignores CSI uncertainty.Its infeasibility becomes much higher than the proposed scheme as R_req_k increases.
  • Variable-duration allocation: In the illustrated scanning policy, remaining time is assigned to the snapshot contributing most to sum secrecy rate after all users’ average-rate requirements are met.Snapshot 5 receives the extra time because it jointly serves three legitimate users while eavesdroppers are far away, increasing user rates without severe leakage.
  • Beam and power allocation: Sequential scanning preserves highly directional synthesized beams for sensing, while the framework flexibly allocates power between beamforming and AN across snapshots.Some snapshots may synthesize AN or transmit only AN, depending on the resource-allocation solution.
  • Variable-duration allocation: Increasing user 1’s requirement from 0.5 bits/s/Hz to 1 bits/s/Hz reduces total sum secrecy rate by taking time from other snapshots, especially snapshot 5.The example shows the trade-off between satisfying stricter QoS requirements and preserving time for high-secrecy-rate snapshots.

VI. CONCLUSIONS

The paper develops a robust resource-allocation framework for ISAC over variable-length snapshots, jointly designing durations, beamforming, and artificial-noise covariance under CSI uncertainty. Simulations report stronger physical-layer security than three baselines and show that variable-length snapshots support both directional sensing and application-dependent prioritization.

  • Framework: The framework jointly optimizes snapshot durations, beamforming vectors, and artificial-noise covariance over a series of snapshots to maximize the sum secrecy rate.The design uses the eavesdroppers’ current-period CSI and treats resource allocation as a non-convex optimization problem.
  • Robust optimization: A bounded uncertainty model and tractable fading-uncertainty bound support robust design for legitimate-user and potential-eavesdropper CSI uncertainty.The algorithm uses a BCD-based method with IA and SDR and is reported to have guaranteed convergence.
  • Results: The proposed scheme achieves significantly higher physical-layer security performance than three baseline schemes in simulations.The results also indicate that CSI uncertainty for both legitimate users and potential eavesdroppers should be considered in robust ISAC design.
  • Implications: Compared with single-snapshot optimization, variable-length snapshots facilitate offline highly directional beam design for high-quality sensing.The framework also enables flexible prioritization of communication and sensing according to the application scenario.
  • Future work: The framework can be extended by incorporating target tracking, which the paper identifies as future work.This extension would produce a more general resource-allocation problem.
Loading 2206.13307v2…