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Cooperative Target Localization in RIS-Enabled ISAC Systems

Rouhollah Amiri, Abdollah Ajorloo, Mohammad Mahdi Mojahedian, Ahmad Reza Hassanshahi, Fabiola Colone

arXiv:2609.39708v1eess.SP

TL;DR

The paper considers target localization in ISAC when high-frequency links face path loss and blockage, and RISs can reshape propagation and create virtual LoS paths. It derives localization information and designs a cooperative RIS-assisted system; coherent phases yield quadratic Fisher-information scaling, while random phases yield linear scaling in expectation.

  • Problem

    High-frequency links enable sensing and communication but suffer severe path loss and blockage, motivating RIS-assisted paths for localization in non-line-of-sight scenarios.

    Source: §I · p. 1

  • Method

    The paper derives target-position information for cooperative RIS paths and formulates joint BS beamforming and RIS-phase design, solved through iterative SCA.

    Source: §I · p. 2, A. Equivalent FIM of Target Position · p. 5, §VII · p. 12

  • Results

    RIS-assisted Fisher information scales as N_s^2 with coherent phase alignment and as N_s in expectation with independent random phases.

    Source: §I · p. 2, §VII · p. 12

  • Takeaways & Limitations

    The analysis and simulations support cooperative RIS-equipped users as a means to improve target localization while serving multiple communication users.

    Source: §VII · p. 12

    The EFIM approximation assumes paths are sufficiently separated in delay, with |τ_i−τ_k| ≳ 1/B as a practical resolution rule.

    Source: A. Equivalent FIM of Target Position · p. 5

Abstract

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This paper develops a cooperative integrated sensing and communication (ISAC) framework in which a multi-antenna base station (BS) simultaneously localizes a target and serves multiple communication users, a subset of which is equipped with reconfigurable intelligent surfaces (RISs). The Fisher information matrix (FIM) for target positioning is derived, explicitly characterizing its dependence on the BS beamforming coefficients, RIS phase profiles, and bistatic sensing geometry. Under the stated scaling assumptions, coherent RIS phase alignment yields a Fisher-information gain that scales quadratically with the number of RIS elements, whereas independent random phases provide a linear gain in expectation. We formulate a sensing-centric joint active and passive beamforming problem that minimizes the position error bound (PEB) subject to per-user signal-to-interference-plus-noise ratio (SINR) and transmit-power constraints. The resulting non-convex problem is addressed through an iterative successive convex approximation (SCA) procedure that solves a sequence of convex subproblems. We further develop a target localization estimator that fuses one direct time-of-arrival (ToA) measurement, one angle-of-arrival (AoA) measurement, and multiple RIS-assisted indirect ToA measurements. Under small measurement errors and asymptotically efficient first-stage ToA/AoA estimation, the estimator covariance approaches the Cram\'er--Rao bound (CRB) to first order. Numerical results validate the analytical scaling laws and demonstrate the localization gains enabled by cooperative RIS-equipped users.

I. INTRODUCTION

The paper studies cooperative ISAC in which RIS-equipped users add sensing paths while the base station serves communication users, and analyzes localization information, joint beamforming, and estimation. Its main contributions include RIS-size scaling results, a multi-measurement estimator, and an SCA-based design under communication constraints.

I. INTRODUCTION

  • RIS-equipped downlink terminals act as bistatic illumination nodes, adding BS–RIS–target–BS sensing paths while the BS remains the sensing receiver.Other users operate purely in communication mode, and the joint design minimizes PEB subject to per-user SINR constraints.Source: §I · pp. 1–2
  • The non-convex joint design minimizes target-position error through iterative SCA, successively solving convex subproblems while meeting users’ SINR thresholds.Source: §I · p. 2
  • Coherent RIS phase alignment gives an RIS-assisted Fisher-information contribution scaling as N_s^2, while independent random phases give an expected scaling of N_s.With average RIS-assisted information bounded and bounded away from zero as cooperative-user count grows, PEB scales as 1/(N_s K) for coherent alignment and 1/√(K N_s) for random phases.Source: §I · p. 2
  • The proposed estimator fuses one AoA, one direct ToA, and K RIS-assisted indirect ToAs, with covariance approaching the CRB to first order under small errors and asymptotically efficient first-stage estimates.Source: §I · p. 2

A. System Configuration

  • A multi-antenna BS localizes one target and serves multiple single-antenna users; a subset have RISs with known positions and orientations.The RIS-equipped users have N_s reflecting elements and serve as cooperative terminals.Source: A. System Configuration · p. 2

B. Transmitter Model

  • The transmitted signal is modeled as X[n] = W S[n], with the BS beamforming matrix W designed for sensing and communication requirements.The frame-averaged transmit covariance is W W^H under the stated unit-power, mutually uncorrelated user-stream model.Source: B. Transmitter Model · p. 2

C. Radar Channel

  • The direct echo follows BS–target–BS and RIS-assisted echoes follow BS–RIS–target–BS; all share the target-to-BS receive AoA but differ in delay and illumination geometry.RIS phase profiles are design variables that control the coherent response along each indirect path.Source: C. Radar Channel · pp. 2–3

D. Radar Receiver

The radar receiver models direct and RIS-assisted echoes with path-specific delays and noise, while the communications receiver uses a known channel model. Performance is assessed through a position CRB criterion and per-user SINR, with explicit assumptions about communication-path isolation and the sensing environment.

D. Radar Receiver

  • The radar receiver observes the target echoes across OFDM subcarriers with path-specific ToAs and additive complex Gaussian noise.The subcarrier spacing is Δf = B/(N+1), and the model includes target reflection coefficients and path scaling.Source: D. Radar Receiver · p. 3

E. Communications Receiver

  • The communication-side model uses a known BS channel matrix with independently distributed entries and white complex Gaussian receiver noise.Source: E. Communications Receiver · p. 3

F. Performance Metrics

  • The design uses A-optimality, minimizing the trace of the position CRB, alongside per-user SINR as its communication metric.RISs are aimed at the sensing target rather than the communication receiver; RIS-mediated user paths are neglected under an isolation and conventional-channel-dominance assumption.Source: F. Performance Metrics · p. 4
  • The analysis assumes known BS channel and RIS geometry, adequate synchronization and calibration, resolvable specular paths, known Swerling-I RCS statistics, and ideal continuous RIS phases.Diffuse clutter, unmodeled multipath, and non-dominant LoS links are outside the stated assumptions.Source: F. Performance Metrics · p. 4

III. ESTIMATION-THEORETIC FUNDAMENTAL LIMITS

This section derives the target-position information bound by mapping the position and nuisance parameters to channel parameters and forming an equivalent FIM. Its approximation and interpretation depend on path resolvability and the geometry of direct and bistatic measurements.

A. Equivalent FIM of Target Position

  • The position-related channel vector η collects path delays, an angle, and nuisance reflection parameters, and its differentiable mapping from position supports the FIM chain rule.The EFIM accounts for information loss due to the nuisance reflection parameter β.Source: A. Equivalent FIM of Target Position · p. 4
  • Ranging information points along the monostatic or bistatic ranging direction, while angular information points orthogonally; their geometry-weighted combination forms the position EFIM.A bistatic ranging direction depends on the BS–target and user–target directions.Source: A. Equivalent FIM of Target Position · pp. 4–5
  • When the K+1 paths are sufficiently separated in delay so cross-delay correlation terms are negligible, the EFIM of target position is well approximated; |τ_i − τ_k| ≳ 1/B serves as a practical resolution rule.When paths are not resolvable, additional terms with semi-random phases can cause a severe fading-like effect and highly degrade the overall EFIM.Source: A. Equivalent FIM of Target Position · p. 5
  • The PEB, derived from the EFIM, bounds the best achievable accuracy of any unbiased position estimator.The paper uses a ranging-dominant approximation for subsequent scaling analysis and beamforming design, while retaining the full delay-and-angle model in its estimator.Source: A. Equivalent FIM of Target Position · p. 5

B. User-Aided Target Localization Accuracy Gain

This section analyzes average localization accuracy under RIS-assisted paths and characterizes how the PEB scales with RIS elements and cooperative users. The scaling depends on phase choice and bounded, nonvanishing average RIS information, while a strong direct path can delay its finite-user onset.

B. User-Aided Target Localization Accuracy Gain

  • Independent random RIS phases provide an average information gain ξ = N_s, so RIS-assisted sensing can contribute even without cooperative phase adjustment.The analysis treats these phases as independently and uniformly distributed over [0, 2π].Source: B. User-Aided Target Localization… · pp. 6–7
  • Under (52), the quantity ¯ΓRIS,K remains bounded above and bounded away from zero as K increases, assuming rotationally invariant geometry, Swerling-I RCS fluctuations, and isotropic average transmit covariance.Source: B. User-Aided Target Localization… · pp. 6–7
  • A strong direct path can make gains from a few RIS-enabled users modest and delay the asymptotic K-scaling until their aggregate information is significant.The scaling exponent is unchanged if the direct-path average contribution remains finite and independent of K.Source: B. User-Aided Target Localization… · p. 7
  • The RIS scaling analysis permits different propagation gains across paths, provided their average information contribution remains finite and nonvanishing as cooperative-user count increases.Individual gains may differ with BS–RIS and RIS–target distances, loss factors, and bistatic geometries.Source: B. User-Aided Target Localization… · pp. 6–7
  • The optimal phase adjustment requires BS–radar cooperation and prior target-position knowledge; the paper notes uncoordinated or random profiles still provide a gain.The required target position may be available from a prior tracking estimate.Source: B. User-Aided Target Localization… · p. 7

IV. DUAL-FUNCTIONAL JOINT ACTIVE AND PASSIVE BEAMFORMING

The beamforming design minimizes an A-optimal sensing objective subject to user SINR and transmit-power constraints, using RIS gains and active beamforming variables. Its solution is a local SCA procedure, and its nominal-position requirement and lack of global convergence guarantees bound its scope.

IV. DUAL-FUNCTIONAL JOINT ACTIVE AND PASSIVE BEAMFORMING

  • The joint design uses the A-optimality criterion to optimize active and passive beamforming subject to per-user SINR and transmit-power constraints.The design requires a nominal target position, which may come from the previous coherent processing interval or be replaced by a worst-case or Bayesian design.Source: §IV · p. 7
  • Because the communication SINR and power constraints are independent of RIS phases, the passive phase design decouples from active beamforming.The phase choice maximizes each RIS-assisted path gain through coherent alignment.Source: §IV · p. 7
  • The active-beamforming relaxation lifts beamforming vectors to semidefinite matrices, relaxes rank-one constraints, and applies rank-one recovery when needed.Recovered candidates are retained only if they satisfy the original SINR and power constraints.Source: §IV · p. 8
  • The nonconvex constraint is linearized locally to form convex subproblems that are solved iteratively.Termination uses stabilization of the original A-optimal objective, followed by explicit rechecking of the original constraints after rank-one recovery.Source: §IV · p. 8
  • The procedure is local and does not claim guaranteed MM/KKT convergence because the first-order approximation has not been shown to globally majorize the nonconvex constraint.A convergence-guaranteed SCA variant is left for future work.Source: §IV · p. 8

V. EFFICIENT ESTIMATOR DESIGN

This section aims to develop an efficient estimator for target localization.

V. EFFICIENT ESTIMATOR DESIGN

  • The section develops an efficient estimator for the target localization problem.Source: §V · p. 8

A. Problem Formulation

The target is localized from direct and RIS-assisted ToAs together with an AoA extracted from the radar receiver signal. The measurement model includes Gaussian noise, and the resulting measurements motivate a weighted pseudo-linear approximation to the localization problem.

A. Problem Formulation

  • The target position is estimated from ToA and AoA parameters extracted from the received radar signal.Source: A. Problem Formulation · p. 8
  • The true measurement vector contains the direct and RIS-assisted ToAs together with the AoA, and its observed version includes measurement noise.Source: A. Problem Formulation · pp. 8–9
  • The measurement noise is modeled as a zero-mean Gaussian vector with covariance C_m.Source: A. Problem Formulation · p. 9
  • The direct ToA, indirect ToAs, and AoA constrain the target to circular, elliptical, and linear loci, respectively, motivating a weighted pseudo-linear approximation solved as a GTRS.The associated maximum-likelihood problem is highly nonlinear and nonconvex; the approximated GTRS has a globally solvable solution.Source: A. Problem Formulation · p. 9

B. Proposed Estimator

The proposed estimator converts the noisy range and angle measurements into a first-order pseudo-linear model, then solves a weighted constrained least-squares problem. This approximation is a GTRS with a globally obtainable solution; user-location uncertainty can be incorporated into its weighting.

B. Proposed Estimator

  • The estimator uses a first-order pseudo-linear model of the noisy measurement equations and applies weighted least squares.Source: B. Proposed Estimator · p. 9
  • The weighting matrix uses measured or coarse range estimates because its ideal form depends on unknown ranges, and it converges to ideal weighting in the small-noise regime.Source: B. Proposed Estimator · p. 9
  • The constrained optimization is a GTRS with necessary and sufficient optimality conditions, so its global solution can be found using efficient algorithms.Source: B. Proposed Estimator · p. 9
  • The final target position is obtained from the first two components of the optimized parameter vector.Source: B. Proposed Estimator · p. 10
  • User-position uncertainty can be incorporated by replacing the weighting matrix, although further analysis is omitted.The replacement weighting accounts for both measurement noise and user-location uncertainty distributions.Source: B. Proposed Estimator · p. 10

C. Performance Analysis

The proposed constrained weighted least-squares estimator is characterized in the small-noise regime through the constrained CRB for its linearized model. Under small measurement errors and asymptotically efficient first-stage ToA/AoA estimation, its covariance approaches the position CRB to first order.

C. Performance Analysis

  • The covariance analysis concerns the constrained weighted least-squares estimator from first-order pseudo-linearization, not exact maximum-likelihood equivalence for the original nonlinear measurements.Source: C. Performance Analysis · p. 10
  • Under small-noise conditions, the proposed constrained weighted least-squares estimator is asymptotically unbiased to first order and has the covariance characterized by the constrained CRB.Source: C. Performance Analysis · p. 10
  • With small measurement errors and asymptotically efficient first-stage ToA/AoA estimation, the estimator covariance approaches the position CRB to first order.Source: C. Performance Analysis · p. 10

VI. NUMERICAL SIMULATIONS

The numerical evaluation uses the simulation settings in Table I unless otherwise specified. Figure 2 presents optimized BS beampatterns under optimized and random RIS phase shifts.

VI. NUMERICAL SIMULATIONS

  • The framework is evaluated using the simulation parameters reported in Table I.Source: §VI · p. 10
  • The Table I simulation settings are used throughout the section unless otherwise specified.Source: §VI · p. 10
  • Figure 2 compares optimized BS beampatterns with optimized versus random RIS phase shifts.Source: Figure 2 · p. 11

A. Optimal Beampattern

The results show how RIS phase design, bistatic geometry, and user configuration shape localization accuracy. Cooperative RIS-enabled users improve localization relative to ordinary-user additions, while performance also varies with communication requirements and system parameters.

A. Optimal Beampattern

  • PEB increases from 1.58 m with optimized RIS phases to 14.44 m with random phases in the two-RIS scenario.The optimized beampattern allocates more transmit gain toward the RISs to strengthen weak but geometrically useful indirect paths.Source: A. Optimal Beampattern · p. 11
  • With a target at 0° and ten RIS-enabled users, the optimized beampattern shows distinct beams directed toward the users’ angular positions.The users span azimuths from −75° to −15° and 15° to 75°.Source: A. Optimal Beampattern · p. 11, Figure 3 · p. 11

B. Geometric Analysis

  • With one RIS, localization accuracy deteriorates along a halfline where direct- and RIS-assisted ranging directions become nearly dependent; a second RIS substantially reduces this low-accuracy region.The additional bistatic measurement adds another information direction, so the gain depends on BS–RIS–target geometry as well as received power.Source: B. Geometric Analysis · p. 11

C. ISAC Trade-off Analysis

  • Adding ordinary users degrades localization, whereas adding RIS-enabled users improves it; both comparison cases already include one RIS-enabled user.Ordinary-user communication constraints consume transmit resources without adding sensing information, while RIS-enabled users create additional bistatic paths.Source: C. ISAC Trade-off Analysis · p. 11
  • Increasing the minimum SINR requirement causes considerably greater PEB degradation in the ordinary-user configuration than in the cooperative RIS-enabled configuration.The comparison uses eight users, with the first RIS-enabled and the remaining users either all ordinary or all RIS-enabled.Source: C. ISAC Trade-off Analysis · p. 12

D. Impact of System Parameters

  • Localization improves with bandwidth and transmit power, follows approximate 1/Ns and 1/√Ns PEB trends for optimized and random RIS phases, and estimator RMSE approaches CRB as measurement quality improves.The two RIS-element trends correspond to coherently optimized phases and independent random phases, respectively.Source: D. Impact of System Parameters · p. 12

VII. CONCLUSION

  • The conclusion reports Fisher-information gains of Ns for random RIS phases and Ns^2 for optimized phases, and first-order CRB approach by the estimator under stated conditions.The estimator combines AoA, direct ToA, and RIS-assisted indirect ToA measurements; the condition includes small errors and asymptotically efficient first-stage estimates.Source: §VII · p. 12

APPENDIX A CALCULATION OF THE FIM

The appendix derives Fisher-information matrix terms from the received-signal likelihood and proves the stated information form under path-resolution and array-steering conditions.

APPENDIX A CALCULATION OF THE FIM

  • The FIM entries are obtained from the complex-Gaussian received-signal model, treating scalar path gains as locally fixed nuisance amplitudes.The appendix identifies the conditioned likelihood and gives derivatives of the channel matrices with respect to target angle.Source: Appendix A · p. 13
  • For RIS-assisted paths, the channel-matrix angle derivative is expressed as γk times the derivative of c(θ) and the Hermitian transpose of a(φk).Source: Appendix A · p. 13

APPENDIX B PROOF OF THEOREM 1

  • When path delays are separated by more than 1/B, overlapping terms become negligible and same-path terms dominate the approximation.Source: Appendix B · p. 13
  • For the centered ULA, each steering vector is orthogonal to its own derivative, but steering vectors at different angles are not assumed orthogonal.Source: Appendix B · p. 14
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