Chapter Summary

Chapter Summary

Key Points

• 1.

  Four ranging observables — one infrastructure. TOA, TDOA, AOA, and RSSI extract complementary information from the same uplink signal. Their variance under the CRB scales as $1/(\beta_{\text{rms}}^2 \text{SNR})$ for TOA and as $1/(N (D/\lambda)^2 \text{SNR})$ for AOA. Wide bandwidth buys time resolution; large arrays buy angular resolution.

• 2.

  Fisher information aggregates across distributed APs. Each cell-free AP contributes a rank-1 (TOA-only) or rank-2 (TOA + AOA) piece of position information. The total is their sum: $\mathbf{J}_\mathbf{p} = \sum_l (\lambda_{\tau,l}/c^2) \mathbf{u}_l \mathbf{u}_l^T + (\lambda_{\phi,l}/d_l^2) \mathbf{u}_l^\perp (\mathbf{u}_l^\perp)^T$. The Position Error Bound is $\text{PEB} = \sqrt{\text{tr}(\mathbf{J}_\mathbf{p}^{-1})}$.

• 3.

  Geometry matters as much as SNR. High SNR alone does not guarantee good positioning; the anchor geometry (GDOP) enters multiplicatively. A cell-free deployment with users inside the convex hull of the APs achieves near-optimal GDOP; users near or outside the boundary degrade rapidly. PEB heatmaps are essential for deployment planning.

• 4.

  UL-TDOA and multi-RTT are the two dominant cell-free techniques. Multi-RTT requires user-AP synchronization but uses all $L$ TOAs independently; UL-TDOA uses only $L-1$ differenced measurements but eliminates user-side clock error. Multi-RTT is typically 10-25% more informative per anchor; UL-TDOA is preferred for low-power IoT.

• 5.

  Joint detection-positioning beats decoupled processing. Iterative (decoupled) schemes commit to hard symbol decisions before using them as timing pilots, which propagates symbol errors into the position estimate. EM-based joint schemes weight all symbol hypotheses by their likelihood and close a 3-5 dB gap to the joint CRB at moderate SNR. Both converge asymptotically to the bound.

• 6.

  The rate-PEB tradeoff is a convex Pareto frontier. Under a power constraint, the achievable pairs $(R, \text{PEB})$ form a convex region whose boundary is parameterized by a sensing-communication weight $\mu \geq 0$. The CommIT result of Liu et al. (2023) establishes that Gaussian inputs are optimal and derives the water-filling-like form of the boundary.

• 7.

  Cell-free ISAC generalizes positioning to environment sensing. The same distributed infrastructure detects and localizes passive targets via multi-static bistatic range measurements. Coherent combining at the CPU delivers $L_t \cdot L_r$ detection gain. Direct-path cancellation is the critical engineering hurdle.

• 8.

  CommIT contributions anchor the theory. Liu, Caire, and collaborators established the fundamental rate-distortion tradeoff for ISAC (IT 2023), the joint positioning- channel estimation framework for cell-free (TSP 2023), and the unified ISAC tutorial (JSAC 2022) that frames 6G standardization efforts.