Chapter Summary

Chapter Summary

Key Points

• 1.

  Near-field RIS enables single-panel 3D positioning. In the near-field regime ($d < d_F$), the RIS-side phase pattern has both linear (angle) and quadratic (distance) components. A single RIS panel in the near-field thus provides a full 3D position estimate — no triangulation needed. In the far-field, only angle is observable; triangulation of multiple panels is required.

• 2.

  Fisher Information Matrix quantifies accuracy. FIM = expected signal sensitivity to position. CRB = FIM⁻¹ = lower bound on estimator covariance. RIS optimization increases FIM (tighter CRB), directly improving accuracy. Position CRB scales as $1/N^2$ — linear in elements, unlike comm's logarithmic rate scaling.

• 3.

  Multi-RIS fusion is additive in FIM. $M$ independent RIS panels contribute $\sum_m \mathbf{J}_{m}$ of total information. CRB decreases as $1/M$ under equivalent panel strength. Geometric diversity gives additional conditioning benefits. Typical 4-panel deployment: 2-3× better than equivalent single-panel.

• 4.

  Practical algorithms: MLE with Newton method. Maximum likelihood estimation over position is the gold standard. Newton iterations converge in $\sim 5$-$10$ steps from a good initial guess. Gradient descent is the robust fallback. Initialization via coarse grid search or previous estimate (Kalman tracking).

• 5.

  mm-level accuracy feasible at sub-THz. Key deployment scenarios: AR/VR indoor positioning, robotic manipulation, precision industrial tracking. At 140 GHz with $N = 512$ RIS: theoretical CRB < 1 mm per coordinate. Real deployments achieve $\sim 0.1$-$1$ mm with careful calibration.