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SNR/CRB-Constrained Joint Beamforming and Reflection Designs for RIS-ISAC Systems
Rang Liu, Ming Li, Qian Liu, A. Lee Swindlehurst
TL;DR
The paper addresses joint communication and sensing design in RIS-assisted ISAC systems, where challenging propagation conditions motivate RIS deployment and beamforming optimization. It formulates constrained sum-rate maximization for target detection and DoA estimation, develops alternating algorithms, and reports higher sum rates than other schemes.
Problem
ISAC beamforming and waveform designs must support communication and sensing simultaneously, while harsh propagation conditions can degrade both functions.
Method
The paper jointly designs BS transmit/receive beamforming and RIS reflection coefficients through two constrained non-convex sum-rate optimization problems solved with FP, MM, ADMM, and related transformations.
Results
The proposed designs achieve notably higher sum rates than other schemes, and the developed alternating procedures converge to stationary points or locally optimal solutions.
Takeaways & Limitations
The study provides joint RIS-assisted ISAC designs that accommodate both target detection and DoA estimation constraints alongside multi-user communication performance.
Abstract
from arXiv · showhide
In this paper, we investigate the integration of integrated sensing and communication (ISAC) and reconfigurable intelligent surfaces (RIS) for providing wide-coverage and ultra-reliable communication and high-accuracy sensing functions. In particular, we consider an RIS-assisted ISAC system in which a multi-antenna base station (BS) simultaneously performs multi-user multi-input single-output (MU-MISO) communications and radar sensing with the assistance of an RIS. We focus on both target detection and parameter estimation performance in terms of the signal-to-noise ratio (SNR) and Cramer-Rao bound (CRB), respectively. Two optimization problems are formulated for maximizing the achievable sum-rate of the multi-user communications under an SNR constraint for target detection or a CRB constraint for parameter estimation, the transmit power budget, and the unit-modulus constraint of the RIS reflection coefficients. Efficient algorithms are developed to solve these two complicated non-convex problems. Extensive simulation results demonstrate the advantages of the proposed joint beamforming and reflection designs compared with other schemes. In addition, it is shown that more RIS reflection elements bring larger performance gains for direct-of-arrival (DoA) estimation than for target detection.
I. INTRODUCTION
The paper motivates RIS-assisted ISAC as a way to combine communication and radar sensing while improving coverage and performance in difficult propagation environments. It studies joint BS beamforming and RIS reflection design for multi-user communications, target detection, and DoA estimation.
- Motivation: ISAC combines communication and radar sensing on a shared platform, improving spectral, energy, and hardware efficiency.
- Motivation: MIMO spatial degrees of freedom improve radar waveform diversity, communication beamforming gains, and spatial multiplexing, making beamforming design central to ISAC.
- RIS background: RIS elements can be independently adjusted to shape propagation, provide passive beamforming gains, establish NLoS links, and expand coverage.
- System model: The considered system uses a multi-antenna BS, an N-element RIS, multiple single-antenna users, and one point-like target, with direct and reflected links contributing to communications and sensing.
- Performance metrics: The paper evaluates communications by sum rate, target detection by worst-case radar SNR, and DoA estimation by a CRB defined for DoAs relative to the BS and RIS.
- Optimization: Two non-convex problems maximize communication sum rate under sensing, transmit-power, and unit-modulus RIS constraints using FP, MM, ADMM, and related transformations.
III. PERFORMANCE METRICS
The paper evaluates multi-user communications by achievable sum-rate and radar sensing by SNR-based target detection performance. It models matched-filtered radar observations, hypothesis testing, and a worst-case radar SNR for optimization.
- Sum-rate for Multi-user Communications: The achievable sum-rate evaluates multi-user communications and is computed from the users’ SINRs as R = sum_k log2(1 + SINR_k).
- SNR for Target Detection in Sensing: Matched filtering uses the transmitted symbols to improve the radar output SNR and target detection probability.
- SNR for Target Detection in Sensing: The radar receiver applies a receive filter or beamformer to the vectorized received signal before hypothesis testing.
- SNR for Target Detection in Sensing: For a fixed false-alarm probability, the target detection probability is positively proportional to the radar SNR.
- SNR for Target Detection in Sensing: The optimization uses a lower bound representing the worst-case achieved radar SNR as the target-detection performance metric.
C. CRB for Parameter Estimation in Sensing
Parameter-estimation accuracy is measured using the CRB, which the paper derives for estimating the target’s DoAs relative to the BS and RIS. The resulting CRB depends on both the BS beamforming matrix and RIS reflection coefficients.
- The paper measures parameter-estimation accuracy using the Cramér-Rao bound, a lower bound for any unbiased estimator.
- DoA estimation targets θ = [θ1, θ2]^T, while the unknown parameter vector also includes the real and imaginary parts of the target reflection coefficient.
- The Fisher information matrix is formed from the vectorized received signal under a complex Gaussian observation model.
- The CRB matrix is the inverse of the Fisher information matrix, and its diagonal entries provide CRBs for the unknown parameters.
- Each Fisher-information-matrix element depends on both the transmit beamforming matrix W and RIS reflection coefficients φ.
A. FP-based Transformation
The proposed optimization reformulates the sum-rate objective and decomposes the resulting non-convex design into iterative updates. FP, MM, and ADMM are used to obtain tractable beamforming and reflection-coefficient subproblems.
- FP-based Transformation: Fractional programming transforms the logarithmic sum-rate objective into a polynomial expression using auxiliary variable r.
- FP-based Transformation: Introducing auxiliary variable c further converts the transformed objective after expanding its quadratic terms.
- FP-based Transformation: The reformulated objective is conditionally concave in each variable when the others are fixed, enabling iterative block updates.
- Block Update: With other variables fixed, the r update is an unconstrained convex problem, while the receive filter u is updated by maximizing a radar-SNR lower bound.
- Block Update: The transmit-beamforming update is a convex problem, whereas the RIS reflection update is formulated separately under fixed auxiliary variables and beamforming.
- Reflection Update: The SNR constraint is transformed using MM, and ADMM introduces an auxiliary reflection variable ϕ to handle the transformed constraint and unit-modulus condition.
- Reflection Update: The ADMM procedure alternates updates of φ, ϕ, and the dual variable μ, with decreasing penalty parameter ρ to enforce equality constraints.
C. Summary and Initialization
The SNR-constrained alternating design converges to a stationary point, while initialization and complexity are addressed through channel-gain-based starting points and explicit update costs.
- Convergence: The alternating updates produce a non-decreasing objective, and every limit point is a stationary point of the original problem.The objective is upper bounded by the transmit power budget, yielding convergence to a stationary point and locally optimal solution.
- Initialization: The RIS phase and beamforming variables are initialized by maximizing channel gains toward the target and communication users.The channel gain is used as a proxy for propagation-channel quality.
- Complexity: The overall SNR-constrained algorithm has complexity O{M^3.5(M + K)^3.5 + N^3.5}.The beamforming and RIS updates contribute O{M^3.5(M + K)^3.5} and O{N^3.5}, respectively, under interior-point solution of the convex subproblems.
V. CRB-CONSTRAINED JOINT BEAMFORMING AND REFLECTION DESIGN
The CRB-constrained design maximizes communication sum-rate while enforcing estimation accuracy, then decomposes the resulting non-convex problem into tractable alternating updates using auxiliary variables, MM surrogates, and penalty methods.
- Problem formulation: The design maximizes sum-rate subject to a CRB threshold, transmit-power budget, and unit-modulus RIS coefficients.The CRB threshold is denoted by ε.
- CRB reformulation: An auxiliary positive-semidefinite matrix J and variable f transform the CRB constraint and decouple W and φ from the matrix constraint.The transformation uses monotonicity of Tr{A^-1}, the Schur complement, and auxiliary representations of the matrix functions.
- Block updates: The algorithm alternately updates FP variables, J and f, beamforming w, RIS phases φ, auxiliary variables, and dual variables.Penalty parameters are reduced iteratively to enforce the introduced equality constraints.
- Majorization-minimization: MM constructs tractable surrogate functions for the non-convex quartic beamforming term and the RIS phase subproblem.The beamforming subproblem becomes convex, while the RIS update uses a closed-form solution under the unit-modulus constraint.
- Subproblem solution: Each subproblem is solved through alternating optimization, with convex SDP or standard convex-optimization steps used where stated.The SDP subproblem can be solved by standard algorithms.
- RIS update: The RIS-phase subproblem is handled with MM because SDR and alternating methods can incur high complexity or additional iterative loops.The stated motivation is especially relevant for large N, where a direct closed-form update is preferred.
- Algorithm 2: Algorithm 2 is initialized with RCG and iteratively updates all variables and penalty parameters before returning W⋆ and φ⋆.The update sequence includes J and f, w, φ, auxiliary variables, and dual variables.
C. Summary
The CRB-constrained alternating procedure tightens surrogate bounds while increasing the original objective, and its boundedness yields convergence to a stationary point and locally optimal solution.
- Penalty updates: Penalty parameters are shrunk during iterations to accelerate satisfaction of the equality constraints.This penalty update is part of the alternating procedure.
- Convergence: The objective of the auxiliary problem is monotonically non-increasing, while the associated lower bound for the original problem is tightened over iterations.The objective of the original problem is non-decreasing during the alternating updates.
- Guarantee: The transmit power budget upper bounds the original objective, so every limit point is a stationary point of the original CRB-constrained problem.Algorithm 2 therefore guarantees a stationary point and a locally optimal solution.
VI. EXTENSIONS TO IMPERFECT SELF-INTERFERENCE CANCELLATION SCENARIO
With imperfect self-interference cancellation, residual interference enters the sensing models and complicates both performance constraints and beamforming design, requiring modified SNR and CRB procedures.
- Residual interference: Residual self-interference is modeled as H_SI x and interferes with echo processing at the BS receive antenna array.The residual channel is represented by H_SI ∈ C^{M×M}.
- Target detection: The radar SINR becomes more complex because the self-interference term depends jointly on transmit beamforming w and receive filter u.The SNR-constrained design therefore requires modifications under imperfect SIC.
- Target detection: The optimal receive filter for worst-case radar SINR is obtained using a Rayleigh-quotient formulation.Substituting this filter yields a reformulated worst-case radar SINR for subsequent optimization.
- Algorithm modification: MM constructs a lower bound for the inverse term in the worst-case SINR using a first-order Taylor expansion.This produces a tractable surrogate for the SNR-constrained update.
- Parameter estimation: For parameter estimation, residual self-interference adds H_SI W W^H H_SI^H to the FIM-related covariance expression.The resulting dependence on W makes the CRB optimization more complicated.
- Parameter estimation: The CRB algorithm handles the interference-dependent covariance by fixing R(W) during each W update and recalculating it afterward.Algorithm 2 is then used for the imperfect-SIC CRB-constrained design.
VII. SIMULATION RESULTS
The simulations use a six-antenna BS, four users, an RIS-assisted target-sensing setup, specified link distances and path-loss exponents, and residual self-interference assumptions.
- The BS-RIS, RIS-target, and RIS-user distances are 50m, 3m, and 8m, respectively.
- The geometry sets θBR = θ2 = π/4, with θ1 calculated from the specified BS-RIS and RIS-target geometry.
- The path-loss exponents are 2.2, 2.2, 2.3, 2.4, and 3.5 for the BS-RIS, RIS-target, RIS-user, BS-target, and BS-user links.
- Residual self-interference uses αSI = −110dB with random unit-modulus phase, and αt = 1 is assumed for target DoA estimation.
A. Illustration of Radar Sensing Performance
The enhanced beampattern illustrates simultaneous communication and radar sensing: the BS beams toward the RIS, target, and users, while the RIS forms passive beams toward the target and users.
- The BS generates strong beams toward the RIS, target, and users, while the RIS directs multiple passive beams toward the target and users.
B. Convergence Performance of the Proposed Algorithms
The proposed algorithms converge quickly and jointly designed schemes improve sum-rate under sensing constraints, while revealing resource-dependent communication–sensing trade-offs.
- B. Convergence Performance of the Proposed Algorithms: Algorithm 1 and Algorithm 2 both converge within 100 iterations, with only small sum-rate increases after 30 iterations.
- C. Impact of Transmit Power: Joint radar–communication beamforming outperforms the “Separate” scheme, while “Comm only” achieves the highest sum-rate and the proposed scheme exceeds “BF only.”
- C. Impact of Transmit Power: Residual self-interference causes performance losses for all ISAC schemes and affects parameter estimation more strongly than target detection.Parameter estimation is sensitive to both received-signal power and phase, whereas target detection depends on interference and useful-signal power.
- D. Impact of the Number of Reflecting Elements: 40% versus 76%: increasing RIS elements raises achievable sum-rate by 40% for target detection and 76% for target DoA estimation.The larger gain in the CRB-constrained scenario is associated with improved DoA-estimation accuracy.
- C. Impact of Transmit Power: Tighter radar sensing requirements create a communication–sensing trade-off by allocating more power to radar beamforming and less to communication beamforming.For the “Separate” scheme, changing radar SNR from 0dB to 2dB causes huge performance degradation.
- VII. SIMULATION RESULTS: The proposed designs optimize sum-rate subject to radar SNR or CRB constraints, transmit power, and unit-modulus RIS coefficients using efficient alternating algorithms.
APPENDIX A
The appendix derives explicit Fisher information matrix terms by differentiating the sensing model, vectorizing matrix expressions, and defining functions of W and φ.
- APPENDIX A: Derivatives of η with respect to each parameter are calculated, and derivatives of Ht(φ) with respect to θ1 and θ2 are used to form FIM elements.
- APPENDIX A: The derivation assumes SSH = LIK because sufficient samples are usually collected for parameter estimation.
- APPENDIX A: The identity vec{ABC} = (C^T ⊗ A)vec{B} re-formulates the expressions so FIM elements are explicit in W and φ.
- APPENDIX A: Functions F1(W, φ) through F6(W, φ) are defined and rearranged, with detailed derivation shown for F1 and analogous details omitted.