Prerequisites & Notation

Before You Begin

This is the final chapter of the book. Unlike earlier chapters, which prove what is known, this chapter catalogs what remains open. The prerequisites are therefore not a specific mathematical toolbox but the whole of the book that preceded it. The items below list the concepts we will assume the reader has already internalized; if any feels unfamiliar, consult the cross-reference before reading the corresponding open-problem section.

Uplink/downlink massive MIMO signal model $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ , with MRC/ZF/MMSE combining and favorable propagation(Review ch01)
Self-check: Can you state the ergodic sum-rate scaling $K\log_2 N_t$ and explain why it is the starting point for every open problem in this chapter?
Spatial-correlation models and the 3GPP TR 38.901 wide-sense stationary (WSS) assumption(Review ch02)
Self-check: Can you write the Kronecker correlation model $\mathbf{R} = \mathbf{R}_t \otimes \mathbf{R}_r$ and state the WSS-US assumption that Section 27.1 will put into question?
Cell-free massive MIMO: distributed APs, centralized vs distributed processing, fronthaul capacity limits(Review ch11)
Self-check: Can you recall the $\mathcal{O}(K^{3})$ and $\mathcal{O}(N_{\text{AP}}^3)$ complexity terms in centralized cell-free and explain which grows faster as the network densifies?
Near-field (Fresnel) propagation and XL-MIMO visibility regions(Review ch17)
Self-check: Can you state the Fraunhofer distance $d_F = 2L^2/\lambda$ and explain why XL-MIMO arrays with $L > 1\,\mathrm{m}$ routinely have users in the near field at 3.5 GHz?
RIS fundamentals and the cascaded Tx-RIS-Rx channel(Review ch21)
Self-check: Can you write the cascaded effective channel $\mathbf{h}_{\text{eff}} = \mathbf{H}_{2} \,\mathbf{\Theta}\, \mathbf{H}_{1}$ and identify which factors are known and which must be estimated?
Full-duplex concept and self-interference cancellation hierarchy
Self-check: Can you state, in order, the three cancellation stages (passive antenna isolation, analog/RF cancellation, digital cancellation) and rough budget per stage?

Notation for This Chapter

This chapter revisits symbols from across the book but uses them in settings where our certainties break down. All customizable symbols use $\ntn{}$ tokens; the displayed values are the defaults from the notation registry. A few new quantities are introduced for holographic and distributed-processing analyses.

Symbol	Meaning	Introduced
$N_t$	Number of BS or AP antennas	s01
$N_r$	Number of receive antennas (user or remote AP)	s01
$K$	Number of active users	s01
$N_{\text{AP}}$	Number of access points in a cell-free network	s02
$\mathbf{H}$	Uplink channel matrix (context determines whether WSS or spatially non-stationary)	s01
$\mathbf{R}_k$	Spatial covariance matrix of user $k$ ; in Section 27.1 this is element-dependent	s01
$\mathcal{V}_k$	Visibility region of user $k$ : the subset of antenna elements that see non-negligible power from user $k$	s01
$\mathbf{v}$	Per-user beamforming or combining vector	s02
$\mathbf{W}$	Joint precoding or combining matrix	s02
$\mathbf{w}$	Additive receiver noise, $\sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$	s01
$\sigma^2$	Noise variance per receive antenna	s01
$\text{SNR}$	Operating SNR	s01
$P_t$	Transmit power per device	s03
$f_0$	Carrier frequency	s03
$\lambda$	Carrier wavelength, $\lambda = c/f_0$	s04
$W$	Signal bandwidth	s02
$L$	Physical aperture length of a holographic surface (meters)	s04
$L_F$	Effective Fresnel length, $\sqrt{\lambda d}$ for communication distance $d$	s04
$\mathbf{\Theta}$	Diagonal RIS reflection matrix with entries $e^{j\phi_m}$	s05
$\gamma_{\text{SI}}$	Residual self-interference-to-noise ratio after the cancellation cascade	s03
$\xi$	Total self-interference cancellation depth (dB)	s03
$N_{\text{iter}}$	Number of consensus/message-passing iterations in distributed processing	s02

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