Summary

Chapter Summary

Key Points

• 1.

  Four measurement models, four geometries. TOA maps to range circles ($\hat{d}_i = c\hat{\tau}_i$), TDOA to hyperbolas ($\hat{d}_{ij} = c(\hat{\tau}_i - \hat{\tau}_j)$), AOA to bearing lines ($\hat{\theta}_i = \arctan((y-y_i)/(x-x_i)) + \eta_i$), and RSSI to coarse range circles via path-loss inversion. Each model has distinct synchronisation requirements and accuracy characteristics: TOA and TDOA achieve metre-level accuracy with sufficient bandwidth, AOA accuracy scales with array size, and RSSI is limited to room-level (3--10 m) by shadow fading.

• 2.

  Position estimation is a nonlinear problem with elegant structure. Linearised LS (via reference-BS subtraction) provides a fast closed-form initialiser. Gauss-Newton iteration refines the estimate by repeatedly linearising the range equations, converging quadratically near the solution. The ML estimator minimises the weighted sum of squared range residuals and is asymptotically efficient, achieving the CRB at high SNR.

• 3.

  The PEB quantifies fundamental accuracy limits. The position error bound $\mathrm{PEB} = \sqrt{\mathrm{tr}(\mathbf{J}^{-1})}$, derived from the Fisher information matrix $\mathbf{J} = \sum_i \sigma_{r,i}^{-2} \mathbf{u}_i \mathbf{u}_i^T$, captures the joint effect of measurement accuracy and BS geometry. Good geometry (angularly diverse BS placement) is as important as low noise: the PEB depends on both eigenvalues of $\mathbf{J}$, and is minimised when the BS directions are uniformly spread around the UE.

• 4.

  5G NR provides a comprehensive positioning framework. Dedicated reference signals (PRS with comb-staggered design, SRS for positioning), five RAT-dependent methods (DL-TDOA, DL-AoD, UL-TDOA, UL-AoA, Multi-RTT), and a positioning architecture centred on the LMF enable sub-metre accuracy. Wide bandwidth ($\leq$ 400 MHz) reduces ranging error, massive arrays ($\leq$ 256 elements) sharpen angle estimates, and hybrid methods fuse complementary information for robust performance.

• 5.

  NLOS is the dominant practical challenge. Non-line-of-sight propagation introduces positive range biases that can cause tens-of-metres positioning errors. Three mitigation strategies exist: detect-and-exclude (residual analysis, hypothesis testing), robust estimation (M-estimators, RANSAC), and NLOS modelling (Bayesian frameworks with bias priors). Effective NLOS handling is the single largest factor separating laboratory accuracy from real-world performance.

• 6.

  Radio-SLAM turns multipath from foe to friend. By treating each resolvable multipath component as a virtual anchor, radio-SLAM jointly estimates the UE trajectory and the environment map using factor-graph inference. Channel-SLAM achieves sub-metre accuracy with a single base station in multipath-rich environments --- a capability impossible with classical single-BS methods. This paradigm shift, enabled by wideband signals and massive arrays at mmWave/sub-THz, transforms the propagation environment into an active positioning resource.