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Intelligent Reflecting Surface Enabled Sensing: Cramér-Rao Bound Optimization

Xianxin Song, Jie Xu, Fan Liu, Tony Xiao Han, Yonina C. Eldar

arXiv:2207.05611v1eess.SPcs.IT

TL;DR

The paper tackles NLoS wireless sensing when AP–target LoS links are blocked by obstructions. It derives CRBs for point and extended targets and designs beamformers to minimize them. The proposed CRB-minimization designs improve sensing performance over traditional schemes in the reported numerical results.

  • Problem

    Conventional wireless sensing relies on AP–target LoS links, which may be unavailable when targets lie in obstructed NLoS regions.

  • Method

    The paper derives closed-form CRBs and optimizes AP transmit and IRS reflective beamforming, using AO, SDR, and SCA for point targets and a closed-form transmit design for extended targets.

  • Results

    The proposed CRB-minimization designs achieve improved sensing performance in MSE versus traditional schemes for both point and extended target cases.

  • Takeaways & Limitations

    CRB depends on both beamformers for point-target DoA estimation but only on AP transmit beamforming for extended-target response-matrix estimation.

Abstract

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This paper investigates intelligent reflecting surface (IRS) enabled non-line-of-sight (NLoS) wireless sensing, in which an IRS is dedicatedly deployed to assist an access point (AP) to sense a target at its NLoS region. It is assumed that the AP is equipped with multiple antennas and the IRS is equipped with a uniform linear array. We consider two types of target models, namely the point and extended targets, for which the AP aims to estimate the target's direction-of-arrival (DoA) and the target response matrix with respect to the IRS, respectively, based on the echo signals from the AP-IRS-target-IRS-AP link. Under this setup, we jointly design the transmit beamforming at the AP and the reflective beamforming at the IRS to minimize the Cramér-Rao bound (CRB) on the estimation error. Towards this end, we first obtain the CRB expressions for the two target models in closed form. It is shown that in the point target case, the CRB for estimating the DoA depends on both the transmit and reflective beamformers; while in the extended target case, the CRB for estimating the target response matrix only depends on the transmit beamformers. Next, for the point target case, we optimize the joint beamforming design to minimize the CRB, via alternating optimization, semi-definite relaxation, and successive convex approximation. For the extended target case, we obtain the optimal transmit beamforming solution to minimize the CRB in closed form. Finally, numerical results show that for both cases, the proposed designs based on CRB minimization achieve improved sensing performance in terms of mean squared error, as compared to other traditional schemes.

I. INTRODUCTION

The paper addresses NLoS sensing when obstructions block conventional AP–target LoS links, using an IRS to create a virtual LoS path. It develops CRB-based beamforming designs for point and extended target estimation.

  • Motivation: Dense obstructions can place sensing targets in APs’ NLoS regions, where conventional LoS sensing may not apply.The resulting NLoS sensing problem is identified as challenging.
  • IRS-enabled sensing: An IRS can establish virtual LoS links through the AP-IRS-target-IRS-AP path and steer reflections toward desired target directions.Adaptive element phase shifts help compensate for severe propagation loss over the triple-reflected link.
  • Design objective: Unlike prior IRS sensing studies that commonly optimize SNR, beampattern gain, or detection probability, this paper uses CRB for target-estimation beamforming.CRB provides a lower bound on the variance of unbiased parameter estimators.
  • System model: The system contains a multi-antenna AP, a ULA-based IRS, and an NLoS target whose parameters are estimated from AP-IRS-target-IRS-AP echoes.The AP is assumed to know the AP-IRS channel state information.
  • Contributions: For point targets, the paper derives CRB expressions and jointly optimizes AP transmit and IRS reflective beamforming using alternating optimization, SDR, and SCA.The point-target optimization is non-convex and includes a maximum AP transmit-power constraint.
  • Contributions: For extended targets, the paper derives a CRB expression and obtains a closed-form optimal transmit beamforming solution using SVD-based subchannel diagonalization and amplitude-inversion power allocation.The extended-target CRB minimization is convex in the AP transmit beamforming variables.

2) Extended Target Model:

The paper models extended targets as distributed scatterers producing multiple angle-dependent paths, then derives the estimation CRB and examines identifiability conditions for point-target DoA estimation.

  • Extended Target Model: Extended targets are modeled as multiple scatterers with different angles, without assuming prior knowledge of their spatial distribution.The point-target model is therefore not accurate for a target with substantial spatial extent.
  • Signal model: The AP receives echoes formed by transmission through G, IRS reflection with Φ, target response H, and the return AP-IRS path.The target-dependent echo at the IRS is HΦGx(t).
  • Estimation objectives: The AP estimates point-target DoA θ or extended-target response matrix H from received echoes, assuming known AP-IRS CSI and transmitted signals.The AP-IRS channel is assumed slowly varying because the AP and IRS are fixed.
  • A. Point Target Case: For point targets, the CRB for DoA is obtained from the first diagonal element of the inverse Fisher information matrix.The unknown parameter vector includes θ and the real and imaginary parts of the complex channel coefficient α.
  • A. Point Target Case: The point-target CRB is re-expressed using b = G^TAv, making its dependence on the IRS reflective beamforming vector explicit.The derivative of b with respect to θ is then used to form the CRB-related matrix expressions.
  • A. Point Target Case: When rank(G) = 1, the Fisher information matrix is singular and DoA estimation is impossible; when rank(G) > 1, the CRB is bounded.With one AP-IRS path, α and the angle-dependent factor remain coupled in a single recoverable observation.

B. Extended Target Case

For extended targets, the paper derives the CRB for estimating the complete target response matrix and characterizes when that matrix is estimable. The resulting CRB depends on transmit beamforming but not reflective beamforming.

  • CRB Derivation: The extended-target model estimates the real and imaginary parts of the vectorized target response matrix H from the received echo signal.The Fisher information matrix is formed for the unknown parameters associated with H.
  • CRB Derivation: Lemma 2 gives a closed-form CRB for estimating the target response matrix H.
  • CRB Dependence: The extended-target CRB depends only on the transmit beamformers or sample coherence matrix Rx, regardless of the reflective beamformer Φ.
  • Estimability: If rank(G) < N, the Fisher information matrix is singular and CRB(H) = ∞; otherwise, it is invertible and the CRB is bounded.
  • Estimability: The target response matrix H is estimable only when rank(G) = N.

IV. JOINT BEAMFORMING OPTIMIZATION FOR CRB MINIMIZATION WITH POINT TARGET

For point targets, the paper jointly optimizes AP transmit and IRS reflective beamforming to minimize the DoA CRB under an AP power constraint. The non-convex joint problem is handled by alternating optimization with convex reformulations for the transmit update.

  • Problem Formulation: The design jointly optimizes AP transmit beamforming and IRS reflective beamforming to minimize the point-target DoA CRB under a maximum AP transmit-power constraint.
  • Problem Formulation: The joint CRB-minimization problem is non-convex because of its objective function and the IRS unit-modulus constraint.
  • Alternating Optimization: Alternating optimization updates the transmit beamformers Rx and reflective beamformer v in turn.
  • Transmit Beamforming Update: With fixed reflective beamforming, minimizing the DoA CRB is transformed into a transmit-beamforming optimization problem involving trace-based matrix expressions.
  • Transmit Beamforming Update: An auxiliary variable and the Schur complement convert the transmit update into a semidefinite program solvable optimally by convex solvers such as CVX.

B. Reflective Beamforming Optimization

The reflective-beamforming subproblem remains non-convex because of the objective and unit-modulus constraints. The paper applies SDR and SCA, then uses Gaussian randomization to recover a feasible rank-one beamformer while ensuring monotonic progress under sufficient randomizations.

  • Subproblem Formulation: For a fixed transmit covariance Rx, the paper optimizes the IRS reflective beamformer v using the point-target CRB formula.
  • Subproblem Formulation: The reflective-beamforming problem is non-convex because its objective is non-concave and v has a unit-modulus constraint.
  • Semidefinite Relaxation: Lifting V = vvH expresses the quadratic beamforming terms as traces and introduces auxiliary variables t1 and t2.
  • Semidefinite Relaxation: Dropping the rank-one constraint yields a relaxed problem whose constraints are convex, while its objective remains non-convex because f1 is non-concave.
  • Successive Convex Approximation: SCA replaces the non-concave objective term with a sequence of convex approximations solved iteratively using first-order Taylor lower bounds and convex solvers.
  • Convergence: With sufficient randomizations, the alternating algorithm has a monotonically non-increasing CRB and its convergence is ensured.
  • Rank-One Recovery: Gaussian randomization constructs feasible rank-one candidates from the relaxed solution and selects the candidate with the maximum objective value.

C. Complete Algorithm of Alternating Optimization

The point-target algorithm alternates between transmit and reflective beamforming updates, while the extended-target case admits a closed-form transmit design based on channel decomposition and power allocation.

  • Point-target alternating optimization: Each outer iteration updates the AP transmit beamformers and IRS reflective beamformer alternately, producing a non-increasing CRB and ensuring convergence.The transmit update is solved optimally; Gaussian randomization supports monotonic improvement for the reflective update.
  • Extended-target transmit optimization: For the extended-target case, the CRB design assumes rank(G) = N and M ≥ N, so the target response matrix is estimable and its CRB is bounded.The optimization minimizes tr((GR_xG^H)^-1) under the AP transmit-power constraint.
  • Extended-target transmit optimization: The optimal transmit covariance uses only the first N SVD subchannels, with the remaining covariance blocks set to zero and the active block diagonal.This structure follows from the optimality conditions and a diagonalization argument.
  • Extended-target transmit optimization: The resulting convex problem has a closed-form optimum obtained from KKT conditions, yielding the optimal solution stated in Propositions 4 and 5.Proposition 4 gives the reduced problem’s optimum, while Proposition 5 gives the corresponding optimum for the original transmit-design problem.
  • Extended-target transmit optimization: SVD decomposes the AP–IRS channel into parallel sensing subchannels, followed by channel-amplitude-inversion power allocation across them.The design can also be interpreted through the eigendecomposition of the sample coherence matrix and N transmitted sensing beams.
  • Extended-target transmit optimization: The isotropic-transmission solution is optimal when M = N and all singular values are equal, matching conventional IRS-free sensing in that special case.Under these equality conditions, the IRS-reflected signals are isotropic.

VI. NUMERICAL RESULTS

The numerical study evaluates CRB-minimizing beamforming under distance-dependent path loss, Rician AP–IRS fading, and a specified IRS sensing geometry.

  • Simulation setup: The simulations use distance-dependent path loss with reference loss K0 = −30 dB, reference distance d0 = 1 m, and exponent α0 = 2.5.The exponent applies to the AP–IRS and IRS–target links.
  • Simulation setup: The AP is placed at (0, 0), the IRS at (5 m, 5 m), and the point target at (5 m, 0), corresponding to DoA θ = 0° relative to the IRS.The point target has unit radar cross section; the extended target is modeled separately using distributed scatterers.
  • Simulation setup: The AP–IRS channel follows a Rician model with factor βAI = 0.5, combining LoS and NLoS Rayleigh-fading components.The model explicitly includes both line-of-sight and non-line-of-sight components.
  • Simulation setup: The radar dwell time is T = 256, IRS element spacing is dIRS = λR/2, and AP noise power is σ_R^2 = −120 dBm.These settings define the sensing and array configurations used in the numerical evaluation.
  • Convergence evaluation: Figure 3 examines convergence for P0 = 30 dBm, M = 8, and N = 8.The caption identifies the operating point and dimensions for the convergence experiment.

A. Point Target Case

For the point-target case, the proposed alternating-optimization algorithm converges quickly, and the study compares its CRB-based joint beamforming design with benchmark schemes.

  • Point target case: The proposed alternating-optimization algorithm converges within around 5 outer iterations for the point-target problem.The reported setting is P0 = 30 dBm with M = 8 and N = 8.
  • Point target case: The evaluation compares the proposed CRB-minimizing joint beamforming design against benchmark schemes for IRS-enabled sensing estimation performance.The passage introduces the comparison but does not report the benchmark identities or numerical outcomes.

1) SNR maximization:

The paper compares SNR-oriented and CRB-oriented sensing designs across point and extended target cases, using MSE and CRB evaluations under varying transmit power and AP antenna counts.

  • SNR maximization:: Maximum-ratio transmission is optimal for any fixed IRS reflective beamformer, while IRS optimization maximizes the effective channel norm under unit-modulus constraints.The SNR design jointly optimizes AP transmission and IRS reflection, following a standard MISO-style IRS optimization formulation.
  • Point target case: The proposed CRB minimization scheme achieves the lowest CRB across transmit-power regimes for point-target DoA estimation.The comparison evaluates CRB alongside MSE obtained with a practical maximum-likelihood estimator.
  • Point target case: At P0 > 25 dBm, CRB minimization yields lower DoA MSE than three benchmark schemes, whereas at P0 < 25 dBm, SNR maximization performs better with MLE.At low SNR, the CRB is not achievable using MLE, making MLE particularly sensitive to echo-signal power.
  • Point target case: The proposed joint design has the lowest DoA CRB across AP antenna counts, with its advantage over reflective-beamforming-only increasing as M grows.The result indicates that joint beamforming becomes particularly important for larger AP antenna arrays.
  • Extended target case: For extended-target response-matrix estimation, the CRB decreases with transmit power, matches MLE MSE, and CRB minimization outperforms isotropic transmission across antenna counts.The performance gap over isotropic transmission becomes more significant as the number of AP antennas increases.
  • VII. CONCLUSION: The overall numerical results report the lowest CRB and MLE MSE for the proposed CRB-minimization scheme compared with traditional benchmarks.The study covers both point and extended target models in IRS-enabled NLoS sensing.

APPENDIX A THE DERIVATION OF THE FIM IN (10)

The appendix derives the Fisher information matrix and establishes when the point-target DoA CRB is bounded or unbounded based on the AP–IRS channel rank.

  • APPENDIX A THE DERIVATION OF THE FIM IN (10): The appendix derives the FIM by differentiating the likelihood structure while using the noise covariance matrix's parameter independence.The derivation proceeds through expressions associated with the FIM in (10).
  • APPENDIX A THE DERIVATION OF THE FIM IN (10): When rank(G) = 1, the FIM is non-invertible and the DoA CRB is unbounded.The rank-one channel is represented using its dominant singular vectors, which leads to aligned derivative terms.
  • APPENDIX A THE DERIVATION OF THE FIM IN (10): When rank(G) > 1, the FIM is invertible and the DoA CRB is bounded.The proof uses non-alignment of the relevant vectors and the Cauchy–Schwarz inequality.

APPENDIX C THE DERIVATION OF THE FIM IN (22)

The appendix develops the FIM-related derivation and closed-form optimization steps for the extended-target response-matrix estimation problem.

  • APPENDIX C THE DERIVATION OF THE FIM IN (22): Because the noise covariance is independent of ζ, its derivatives with respect to ζ_i vanish in the FIM calculation.This simplifies the Fisher-information derivation for the extended-target parameterization.
  • APPENDIX C THE DERIVATION OF THE FIM IN (22): The appendix formulates the relevant subproblem through its Lagrangian and KKT conditions, including dual variables and complementary slackness.The conditions include η_i p_i = 0, η_i ≥ 0, and p_i > 0.
  • APPENDIX C THE DERIVATION OF THE FIM IN (22): A closed-form solution is then obtained for the constrained subproblem.The solution follows from the stated KKT conditions.

APPENDIX F MLE FOR ESTIMATING TARGET RESPONSE MATRIX WITH

The appendix derives maximum-likelihood estimators from vectorized received-signal models for point-target DoA and extended-target response-matrix estimation.

  • APPENDIX F MLE FOR ESTIMATING TARGET RESPONSE MATRIX WITH: For the extended-target model, vectorization gives a linear Gaussian estimation problem with E = XTGTΦT ⊗ GTΦT, and MLE minimizes the squared residual norm.The resulting estimator is obtained directly from the linear observation model.
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