Prerequisites & Notation

Before You Begin

Chapter 14 specializes the sensing framework (Chapter 13) to localization — estimating UE positions from signals that pass through one or more RIS panels. The key mathematical object is the Fisher Information Matrix $\mathbf{J}$ , which lower-bounds position-estimation errors via the Cramér-Rao bound.

Estimation theory: MLE, FIM, CRB(Review ch06)
Self-check: For observations $\mathbf{y} = \mathbf{h}(\boldsymbol{\theta}) + \mathbf{w}$ , write the FIM and the resulting CRB on $\boldsymbol{\theta}$ .
Classical localization (TOA, TDOA, AOA)(Review ch30)
Self-check: What is the key distinction between TOA-based and AOA-based positioning?
Near-field channel models (Ch. 3)(Review ch03)
Self-check: Recall the quadratic phase term in near-field channels; why is it essential for localization?
Sensing with RIS (Ch. 13)(Review ch13)
Self-check: Recall the $N^4$ sensing SNR gain.
Joint optimization framework (Ch. 5-7)(Review ch05)
Self-check: Recall AO for joint precoder and RIS phase optimization.

Notation for This Chapter

Localization-specific notation. We use $\mathbf{p} \in \mathbb{R}^3$ or $\mathbb{R}^2$ for position, and the FIM token $\mathbf{J}$ .

Symbol	Meaning	Introduced
$\mathbf{p} \in \mathbb{R}^3$	UE 3D position coordinates	s01
$\hat{\mathbf{p}}$	Position estimate	s01
$\mathbf{J}$	Fisher Information Matrix for position estimation	s02
$\text{CRB}(\mathbf{p}) = \mathbf{J}^{-1}$	Cramér-Rao bound on position estimation covariance	s02
$d_F = 2 D^2/\lambda$	Fraunhofer distance (near-/far-field boundary)	s01
$\mathbf{h}_{k,2}(\mathbf{p})$	RIS-to-UE channel as a function of UE position $\mathbf{p}$	s01
$\mathbf{J}_p$	Jacobian of signal with respect to position $\mathbf{p}$	s02
$M$	Number of RIS panels in multi-RIS fusion	s03

← Ch 13 RIS Localization Signal Model