Prerequisites & Notation

Before You Begin

This chapter merges two historically separate fields — communication and positioning — onto a single cell-free infrastructure. Readers should be comfortable with the cell-free MIMO system model from earlier chapters and with the basics of statistical estimation. If any of the items below feel unfamiliar, revisit the listed chapter first.

Cell-free massive MIMO system model with $L$ distributed APs(Review ch11)
Self-check: Can you write the uplink received signal at AP $l$ and explain what macro-diversity provides beyond co-located MIMO?
User-centric clustering and cooperation levels(Review ch13)
Self-check: Can you compare Level 2 (local estimation + central combining) vs Level 4 (fully centralized MMSE) in terms of fronthaul load?
Use-and-then-forget (UatF) bound and achievable rates(Review ch15)
Self-check: Can you state the UatF SINR expression and explain what the coherent combining gain $(\sum_l \gamma_{lk})^2$ represents?
Channel estimation with orthogonal pilots(Review ch03)
Self-check: Can you describe the MMSE channel estimator and the pilot overhead $\tau_p/\tau_c$ ?
Cramer-Rao bound and Fisher information
Self-check: Can you define the Fisher information matrix $\mathbf{J}(\boldsymbol{\theta})$ and state the CRB inequality $\text{Cov}(\hat{\boldsymbol{\theta}}) \succeq \mathbf{J}^{-1}$ ?
Time and frequency synchronization in distributed systems
Self-check: Do you know the difference between carrier-phase synchronization (needed for coherent beamforming) and timing synchronization (needed for TOA-based ranging)?

Notation for This Chapter

Symbols introduced or specialized in this chapter. See also the NGlobal Notation Table master table for general conventions.

Symbol	Meaning	Introduced
$L$	Number of cell-free access points acting as position anchors	s02
$\mathbf{p} = (p_x, p_y) \in \mathbb{R}^2$	User position to be estimated (2D)	s01
$\mathbf{q}_l \in \mathbb{R}^2$	Known position of AP $l$ (anchor)	s02
$d_l = \\|\mathbf{p} - \mathbf{q}_l\\|$	True distance from user to AP $l$	s01
$\tau_l$	Time-of-arrival at AP $l$ : $\tau_l = d_l/c$	s01
$\Delta\tau_{l,1} = \tau_l - \tau_1$	Time-difference-of-arrival between AP $l$ and reference AP 1	s01
$\phi_l$	Angle-of-arrival (azimuth) at AP $l$	s01
$\mathbf{J}(\boldsymbol{\theta})$	Fisher information matrix for parameter vector $\boldsymbol{\theta}$	s01
$\text{PEB}(\mathbf{p})$	Position Error Bound: $\sqrt{\text{tr}(\mathbf{J}_{\mathbf{p}}^{-1})}$	s04
$\text{AEB}(\phi)$	Angle Error Bound: CRB on angle estimation	s04
$\boldsymbol{\theta} = (\mathbf{p}, \mathbf{s}, \ldots)$	Joint parameter vector containing position and data symbols	s03
$c$	Speed of light, $c \approx 3 \times 10^8$ m/s	s01
$\beta_{\text{rms}}$	Effective (root-mean-square) bandwidth of the transmitted waveform	s01
$\text{SNR}_{l}$	Per-AP SNR: $\text{SNR}_{l} = P_t \beta_{l} / \sigma^2$	s02
$\mathbf{J}_{\mathbf{p}}$	Equivalent Fisher information matrix (EFIM) on position after nuisance-parameter reduction	s04
$\mathcal{L}(\mathbf{s}, \mathbf{p})$	Joint log-likelihood of data $\mathbf{s}$ and position $\mathbf{p}$	s03
$\text{GDOP}$	Geometric Dilution of Precision	s02

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