Prerequisites & Notation

Before You Begin

This chapter addresses a physical reality that the previous seventeen chapters systematically ignored: when the base-station aperture becomes comparable to the propagation range, the channel is no longer spatially stationary. A user no longer illuminates every antenna of an extra-large array (XL-MIMO); instead it reaches only a subset we will call a visibility region (VR). Estimating the channel now requires first detecting where the user lives on the array and then estimating the channel restricted to that region. We assume familiarity with the following prior material.

Near-field propagation: Fraunhofer distance, spherical wavefront, beam focusing(Review ch17)
Self-check: Can you compute $d_F = 2D^2/\lambda$ for a given aperture $D$ and wavelength $\lambda$ , and explain when a user at range $r$ is in the near or far field?
LS and MMSE channel estimation; pilot contamination(Review ch03)
Self-check: Can you write the LS and MMSE estimators for a MIMO pilot observation model $\mathbf{Y}_p = \mathbf{H}\mathbf{S}_{i,k} + \mathbf{w}$ and state the MSE of each?
Spatial correlation matrices and the one-ring model(Review ch02)
Self-check: Can you explain how $\mathbf{R}_k$ encodes the angular support of the user's multipath cluster?
Sparse recovery: LASSO, OMP, compressed sensing guarantees(Review ch12)
Self-check: Can you state when $\ell_1$ minimization recovers a $s$ -sparse vector from $m$ noisy linear measurements?
Markov random fields, Ising model, belief propagation / loopy BP
Self-check: Can you write the joint distribution of a 2D Ising model with nearest-neighbor coupling $J$ and external field $h$ ?
EM algorithm for latent-variable estimation
Self-check: Can you write the E- and M-steps of EM for a Gaussian observation model with a discrete latent state?

Notation for This Chapter

Symbols introduced or specialized in this chapter. Throughout the chapter we use $\mathbf{R}_k$ for the per-user spatial covariance matrix (local to this chapter, not a global \ntn{} symbol) and $R_k$ (plain italic) for achievable rate. See NGlobal Notation Table for the master table.

Symbol	Meaning	Introduced
$N_t$	Number of BS antennas (XL-MIMO regime, typically $N_t \geq 256$ )	s01
$N_1, N_2$	Horizontal and vertical antenna counts for a UPA, $N_t = N_1 N_2$	s01
$\mathbf{m}_k \in \{0,1\}^{N_t}$	Binary visibility-region mask for user $k$ : $m_{k,n} = 1$ iff antenna $n$ is illuminated	s01
$\mathcal{V}_k \subseteq \{1,\ldots,N_t\}$	Visibility-region support: $\mathcal{V}_k = \{n : m_{k,n} = 1\}$	s01
$\|\mathcal{V}_k\|$	Visibility-region cardinality (number of illuminated antennas)	s01
$J$	Coupling strength of the 2D Markov (Ising) prior on $\mathbf{m}_k$	s02
$h$	External field of the 2D Markov prior (biases toward active / inactive)	s02
$\mathbf{R}_k \in \mathbb{C}^{N_t \times N_t}$	Spatial covariance of the unmasked channel of user $k$ (chapter-local)	s01
$S$	Number of subarrays partitioning the XL-MIMO array	s03
$M_s = N_t / S$	Antennas per subarray	s03
$d_F = 2D^2/\lambda$	Fraunhofer (far-field) distance of an aperture $D$	s04
$\mathbf{A}_{\text{polar}}$	Polar-domain (angle $\times$ range) dictionary for near-field sparse recovery	s04
$(\psi_q, r_q)$	$q$ -th polar grid point: azimuth cosine and range	s04
$\mathcal{L}(\mathbf{m}_k, \mathbf{H}_{k})$	Joint log-posterior of VR mask and channel given pilot observations	s05
$q_n$	Variational marginal $q_n = \Pr[m_{k,n} = 1 \mid \mathbf{Y}_p]$	s05
$\mathbf{S}_{i,k}$	Pilot matrix (uplink training), $\mathbf{S}_{i,k} \in \mathbb{C}^{K \times \tau_p}$	s01
$\tau_p$	Pilot length in symbols	s01

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