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Joint Active and Passive Beamforming Design for IRS-Aided Radar-Communication

Meng Hua, Qingqing Wu, Chong He, Shaodan Ma, Wen Chen

arXiv:2203.14532v1cs.ITeess.SP

TL;DR

The paper studies IRS-aided Radcom beamforming under SINR, interference, and cross-correlation constraints. It proposes penalty-based and SDR-based AO methods, showing that dedicated radar signals are unnecessary in one case but generally required when cross-correlation design is considered.

  • Problem

    The paper addresses joint beamforming and IRS phase-shift design for IRS-aided Radcom under SINR and cross-correlation constraints.

  • Method

    The paper proposes a penalty-based algorithm and an SDR-based alternating optimization algorithm for the two considered cases.

  • Results

    Dedicated radar signals are not required in case I but are generally required to enhance system performance when cross-correlation design is considered.

  • Takeaways & Limitations

    Whether dedicated radar signals are needed depends on the presence of cross-correlation design and the considered interference conditions.

  • Takeaways & Limitations

    The optimality conclusion may not hold because of the limited degrees of freedom of the transmitted signals.

Abstract

from arXiv · show

In this paper, we study an intelligent reflecting surface (IRS)-aided radar-communication (Radcom) system, where the IRS is leveraged to help Radcom base station (BS) transmit the joint of communication signals and radar signals for serving communication users and tracking targets simultaneously. The objective of this paper is to minimize the total transmit power at the Radcom BS by jointly optimizing the active beamformers, including communication beamformers and radar beamformers, at the Radcom BS and the phase shifts at the IRS, subject to the minimum signal-to-interference-plus-noise ratio (SINR) required by communication users, the minimum SINR required by the radar, and the cross-correlation pattern design. In particular, we consider two cases, namely, case I and case II, based on the presence or absence of the radar cross-correlation design and the interference introduced by the IRS on the Radcom BS. For case I where the cross correlation design and the interference are not considered, we prove that the dedicated radar signals are not needed, which significantly reduces implementation complexity and simplifies algorithm design. Then, a penalty-based algorithm is proposed to solve the resulting non-convex optimization problem. Whereas for case II considering the cross-correlation design and the interference, we unveil that the dedicated radar signals are needed in general to enhance the system performance. Since the resulting optimization problem is more challenging to solve as compared with the case I, the semidefinite relaxation (SDR) based alternating optimization (AO) algorithm is proposed. Simulation results demonstrate the effectiveness of proposed algorithms and also show the superiority of the proposed scheme over various benchmark schemes.

I. INTRODUCTION

The paper studies IRS-aided Radcom to address propagation-control limits by jointly designing active beamforming and IRS phase shifts for multi-user, multi-target operation. It distinguishes two cases and shows that dedicated radar signals are unnecessary in case I but generally useful in case II, which requires different algorithms.

  • Related background: Radcom integrates radar and communication on one hardware platform, while earlier information-embedding approaches can limit data rate through radar pulse repetition frequency.Joint beamforming and waveform design provides additional design degrees of freedom compared with communication-only beamforming.
  • Motivation: IRSs provide passive control over electromagnetic propagation, extending Radcom beamforming beyond active waveform or encoding design.Each IRS element independently adjusts phase shift and/or amplitude, enabling propagation toward directions of interest.
  • Problem and scope: The paper formulates a multi-user, multi-target IRS-aided Radcom problem around whether dedicated radar signals are needed in two operating cases.The cases differ by the presence or absence of radar cross-correlation design and IRS-introduced interference.
  • Case I: Case I proves dedicated radar signals are unnecessary, reducing implementation complexity and simplifying subsequent algorithm design.A penalty-based algorithm is then proposed for the resulting optimization problem.
  • Case II: Case II generally requires dedicated radar signals to enhance system performance and uses an SDR-based alternating-optimization algorithm because the problem is more challenging.The proposed reconstruction strategy achieves tightness instead of relying on Gaussian randomization.
  • Case II: In case II, an IRS can reduce transmit power when deployed far from the Radcom BS but may deteriorate performance when deployed nearby because of interference.Dedicated radar signals can also significantly reduce outage probability relative to omitting them in case II.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system comprises a Radcom BS, communication users, radar targets, and an IRS. Its transmit waveform combines communication and radar components, with separate beamformers and statistical independence assumptions.

  • A. System Model: The system includes a Radcom BS, K single-antenna users, L radar targets, and an IRS with M reflecting elements.These components define the multi-user, multi-target IRS-aided Radcom setting.
  • A. System Model: The BS has N_t + N_r antennas: N_t transmit antennas serve users and targets, while N_r receive antennas collect target echoes.The transmit and receive antenna roles are separated as specified in the system model.
  • 1) Transmit Waveform Design:: The transmitted signal is modeled through a dedicated transmit-waveform design at the Radcom BS.The waveform combines communication and radar signal components.
  • 1) Transmit Waveform Design:: Communication symbols x_c are modeled as circularly symmetric complex Gaussian signals, with W_c as their communication beamformer.The communication signal vector has dimension K, and W_c has dimensions N_t × K.
  • 1) Transmit Waveform Design:: Radar signals x_r are zero-mean and are transmitted through radar beamformer W_r.The radar signal vector has dimension N_t, and W_r is an N_t × N_t matrix.
  • 1) Transmit Waveform Design:: Communication and radar signals are assumed statistically independent and uncorrelated.This assumption separates the two signal components in the transmit-waveform model.

2) Communication Model:

The communication model uses quasi-static flat-fading channels and IRS reflection coefficients to support joint communication and radar tracking. The design minimizes Radcom BS transmit power while satisfying communication and radar SINR and cross-correlation requirements.

  • The system assumes quasi-static flat-fading channels and perfect communication-channel state information at the Radcom BS.
  • The IRS reflection matrix is diagonal, with each element represented by a unit-modulus phase shift.
  • The received radar echo comprises radar-target-radar and radar-IRS-radar components after highly attenuated three-hop paths are neglected.
  • Known communication signals are not interference for target tracking because the Radcom BS knows them.
  • The optimization minimizes total Radcom BS transmit power while enforcing communication-user SINR, radar SINR, cross-correlation, and unit-modulus constraints.

III. PROPOSED SOLUTION TO CASE I

Case I assumes IRS interference is perfectly canceled and omits cross-correlation design. Under these assumptions, dedicated radar beams are unnecessary, and a penalty-based alternating procedure solves the remaining non-convex problem.

  • Case I structure: Under independently distributed target amplitudes and uncorrelated target and user channels, the optimal Case I solution does not require dedicated radar beams.
  • Case I structure: Removing dedicated radar beams reduces Radcom BS implementation complexity and algorithm-design complexity.
  • Penalty-based algorithm: The penalty-based algorithm decouples variables across blocks by alternately optimizing communication beamformers, IRS phase shifts, and auxiliary variables.
  • Penalty-based algorithm: With fixed penalty coefficient, the inner procedure updates the blocks using closed-form, element-wise, dual, or bisection-based subproblem solutions.
  • Penalty-based algorithm: The algorithm’s inner-layer subproblems can be separated and solved in parallel when their variables are decoupled.

B. Outer Layer Update

The outer layer gradually decreases the penalty coefficient to enforce the equality constraint. The update factor trades performance against iteration count and divergence risk.

  • The penalty coefficient ρ_t is gradually decreased across outer-layer iterations.
  • A larger update factor c can improve performance but requires more outer-layer iterations.
  • A smaller c reduces outer-layer iterations but makes the penalty algorithm more prone to divergence.
  • Empirical tests suggest choosing c from 0.7 to 0.9 to balance system performance and computational complexity.

C. Overall Algorithm

The overall procedures use inner-layer block optimization and termination checks, with Case II requiring an SDR-based alternating optimization method because of interference and cross-correlation constraints.

  • Case I: The inner procedure terminates when its fractional objective decrease falls below a threshold, while the outer procedure monitors a termination indicator.
  • The inner-layer blocks are optimally solved without coupling between variables in different blocks.
  • Algorithm 1 is guaranteed to converge to a stationary point.
  • Case II: Case II includes uncanceled IRS interference and cross-correlation design, making the Case I penalty algorithm inapplicable.
  • Case II: For Case II, an SDR-based alternating optimization algorithm partitions transmit covariance matrices and IRS phase shifts into two iteratively optimized blocks.

A. Optimization of Transmit Covariance Matrices

The transmit-covariance subproblem is convex for fixed IRS phase shifts, while the IRS phase-shift update is convexified through lower bounds and first-order Taylor expansion. Auxiliary variables then produce an explicit convex objective for the phase-shift update.

  • For fixed IRS phase shifts, the transmit-covariance subproblem is convex and solvable by an interior-point method.
  • For fixed transmit covariance matrices, the IRS phase-shift update uses convex lower bounds and Taylor expansion to handle nonconvex constraints.
  • The quadratic terms in the IRS phase-shift formulation are convex after establishing positive semidefiniteness of the relevant matrices.
  • Introducing auxiliary non-negative variables converts the IRS phase-shift problem into one with an explicit objective function.
  • The resulting auxiliary-variable problem is convex and can be solved using convex optimization techniques.

C. Overall Algorithm

The overall method alternates between transmit-covariance and IRS phase-shift updates, with rank-one recovery and convergence checks. The special-case analysis shows when dedicated radar signals are unnecessary and when cross-correlation constraints can make them necessary.

  • Overall Algorithm: The algorithm alternately updates transmit covariance matrices and IRS phase shifts until the objective decrease falls below a threshold.
  • Overall Algorithm: Rank-one communication beamformers are recovered from covariance solutions, while radar beamformers are reconstructed accordingly.
  • Overall Algorithm: Gaussian randomization can incur performance loss and high computational complexity, but Theorem 2 provides a converged rank-one communication-beamformer solution without it in general.
  • D. Special Case Discussion: For the special case without the cross-correlation constraint, dedicated radar signals are not needed regardless of IRS interference.
  • D. Special Case Discussion: With cross-correlation design, SDR tightness may fail because of limited transmitted-signal degrees of freedom, so randomization and performance loss may be required.
  • Numerical Results: The numerical study evaluates the joint active and passive beamforming design under specified antenna, target, channel, and SINR settings.

A. Convergence Behavior of the Proposed Two Algorithms

Algorithm 1 reaches the prescribed constraint accuracy in roughly 55–57 outer iterations and ultimately converges despite intermediate oscillations. Algorithm 2 exhibits monotonically decreasing transmit power and converges after about 58 iterations, while simulations show IRS-assisted schemes reduce required transmit power.

  • Algorithm 1: Algorithm 1 reduces constraint violation to the predefined accuracy 10^-7 after about 55 iterations for M = 50.For M = 100, approximately 57 outer iterations are required.
  • Algorithm 1: Algorithm 1 may exhibit nonmonotonic objective values because large penalty coefficients temporarily violate the equality constraint.As the penalty coefficient decreases, constraint violation approaches the predefined accuracy and the algorithm converges.
  • Algorithm 2: Algorithm 2 has monotonically decreasing transmit power and converges in about 58 iterations across the tested setups.The behavior follows from solving each subproblem optimally or locally while the objective remains lower-bounded by the minimum SINR requirements.
  • IRS-assisted performance: With IRS phase-shift optimization, required transmit power decreases monotonically as the number of IRS elements increases, including when interference exists.The IRS provides higher passive beamforming gain toward desired users, reducing transmit power.
  • IRS-assisted performance: In case I, communication-only transmission matches joint communication-and-radar transmission, indicating dedicated radar signals are unnecessary.This equality is observed for both user-number and IRS-element comparisons in the no-interference setting.
  • IRS-assisted performance: Required transmit power increases with the number of users, while IRS-equipped schemes outperform the no-IRS scheme, especially for larger user counts.The power requirement is governed strongly by the user with the worst channel quality.

3) Effect of Radar SINR:

Increasing the radar SINR requirement raises transmit-power demands, especially when IRS-induced interference is present. IRS placement and cross-correlation constraints further shape performance, while dedicated radar signals improve case-II outage behavior.

  • 3) Effect of Radar SINR: Required transmit power with IRS interference increases markedly as the radar SINR requirement becomes high.The IRS-induced interference becomes more prominent at larger radar SINR thresholds, requiring additional transmit power.
  • 3) Effect of Radar SINR: When IRS interference is perfectly canceled, IRS-assisted transmission requires much lower transmit power than transmission without IRS.The reduction is attributed to the IRS's high passive beamforming gains.
  • 3) Effect of IRS deployment: IRS-induced interference is significant when the IRS is close to the Radcom BS but becomes smaller as deployment distance increases beyond dx ≥ 45.The performance gap between interference and no-interference schemes narrows as dx increases.
  • 3) Effect of cross-correlation constraint: A stringent cross-correlation threshold keeps reflected-signal correlations small, whereas larger thresholds produce high correlations between adjacent reflected signals.High correlation can significantly degrade adaptive multi-target radar detection.
  • 3) Effect of dedicated radar signals: The paper formulates two IRS-aided Radcom power-minimization cases and uses penalty-based optimization for case I and SDR-based alternating optimization for case II.Case I excludes radar cross-correlation design and IRS interference; case II includes them and generally requires dedicated radar signals.
  • 3) Effect of dedicated radar signals: In case II, dedicated radar signals significantly reduce outage probability compared with omitting dedicated radar signals.The comparison uses Communication & Radar and Communication only transmission schemes.

APPENDIX A: PROOF OF THEOREM 1

The proof establishes that the SDR relaxation is tight and that dedicated radar beams are unnecessary under the theorem’s conditions. It analyzes dual optimality and shows that rank-one communication solutions satisfy the original problem.

  • SDR tightness: The SDR is formulated as an SDP whose optimal solutions satisfy positive-semidefiniteness and rank-one constraints for communication beamformers.The proof uses duality and rank properties to connect the relaxed problem to the original formulation.
  • Dual analysis: When all relevant dual variables vanish, the optimal dual multiplier is determined by the maximum eigenvalue of A^HA.The proof further uses the target-amplitude assumption to characterize the associated eigenspaces.
  • Dual analysis: All communication beams would then point toward the targets rather than users, so the minimum user SINR requirements cannot be satisfied.This contradiction excludes the case in which all relevant dual variables vanish.
  • Conclusion: The remaining case proves that the relaxed problem is equivalent to the original problem and requires no dedicated radar beams.The conclusion follows after establishing the required rank condition for the communication covariance matrices.
  • Conclusion: The proof concludes that radar beams are not required.This completes the theorem’s claim under the stated assumptions.

APPENDIX B: PROOF OF LEMMA 2

The proof of Lemma 2 constructs rank-one feasible solutions with unchanged objective value and shows equivalence between the original formulation and its zero-radar-covariance version.

  • Constructed solutions: The constructed communication covariance matrices are rank-one and positive semidefinite.The proof verifies the rank and definiteness properties of the newly constructed solutions.
  • Constructed solutions: The transformed radar covariance can be represented as a sum of K + 1 positive semidefinite matrices.This establishes its positive semidefiniteness within the construction.
  • Feasibility and objective: The newly constructed solutions preserve the objective value while satisfying the relevant constraints.The proof checks constraints (34b)-(34d) after the transformation.
  • Problem equivalence: Setting Zr = 0 defines a reduced problem whose feasible solutions are also feasible for the original problem.The proof establishes one direction of feasibility between the two formulations.
  • Problem equivalence: The original and reduced problems are equivalent because their feasible solutions can be transformed with the same objective value.This completes the proof of Lemma 2.
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