Prerequisites

This chapter draws on several earlier chapters. If any item below feels unfamiliar, revisit the linked material before proceeding.

Linear algebra: least-squares, pseudo-inverses, matrix inversion lemma(Review ch01)
Self-check: Can you set up and solve an overdetermined system $\mathbf{A}\mathbf{x} \approx \mathbf{b}$ via the normal equations $\hat{\mathbf{x}} = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T\mathbf{b}$ ?
Probability: Gaussian distributions, maximum likelihood estimation, Fisher information(Review ch02)
Self-check: Can you write the log-likelihood for a vector of independent Gaussian observations and derive the Fisher information matrix?
Propagation: path loss models, multipath, shadowing(Review ch05)
Self-check: Can you relate received power to distance via the log-distance path loss model and explain the effect of log-normal shadowing?
Antenna arrays: steering vectors, angle-of-arrival estimation(Review ch07)
Self-check: Can you write the ULA steering vector $\mathbf{a}(\theta) = [1, e^{j 2\pi d\sin\theta/\lambda}, \ldots]^T$ and explain how phase differences across elements encode angle?
Estimation theory: MVUE, Cramer-Rao bound, iterative estimators(Review ch09)
Self-check: Can you state the Cramer-Rao inequality $\mathrm{Cov}(\hat{\boldsymbol{\theta}}) \succeq \mathbf{J}^{-1}$ and explain when equality is achieved?

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{p} = [x, y]^T$	Unknown 2D position of the user equipment (UE)	s01
$\mathbf{p}_i = [x_i, y_i]^T$	Known position of the $i$ -th base station (BS) or anchor node	s01
$d_i = \\|\mathbf{p} - \mathbf{p}_i\\|$	True geometric distance from UE to BS $i$	s01
$\hat{\tau}_i$	Estimated time-of-arrival from BS $i$	s01
$\hat{d}_i = c\hat{\tau}_i$	Range measurement derived from TOA	s01
$\hat{d}_{ij}$	TDOA-derived range difference between BSs $i$ and $j$	s01
$\hat{\theta}_i$	Estimated angle-of-arrival at BS $i$	s01
$P_{r,i}$	Received signal strength (RSS) from BS $i$ in dBm	s01
$\mathbf{J}(\mathbf{p})$	Fisher information matrix for position estimation	s02
$\mathrm{PEB}$	Position error bound: $\sqrt{\mathrm{tr}(\mathbf{J}^{-1})}$	s02
$\sigma_r$	Standard deviation of range measurement error (metres)	s02
$\sigma_\theta$	Standard deviation of angle measurement error (radians)	s02
$B$	Signal bandwidth (Hz), determines ranging resolution	s03