Prerequisites & Notation

Before You Begin

This chapter is the first of Part IV (Near-Field, XL-MIMO and Hardware-Aware Design). It is physics-first: we go back to the wave equation, derive the exact path length from each array element to a point in space, and identify the regime where the far-field plane-wave model breaks down. The reader should be comfortable with the classical array model from earlier chapters — everything here is an exact refinement of those expressions.

Plane-wave steering vector $\mathbf{a}(\theta) = [1, e^{-j\pi\sin\theta}, \ldots, e^{-j\pi(N-1)\sin\theta}]^T$ for a half-wavelength ULA(Review ch02)
Self-check: Can you derive the phase progression $e^{-j\kappa n d \sin\theta}$ from the assumption that the incoming wave is a single plane wave?
MIMO capacity formula $C = \log_2\det(\mathbf{I} + \text{SNR}\,\mathbf{H}\mathbf{H}^{H})$ and its SVD interpretation(Review ch01)
Self-check: Can you explain why the rank of $\mathbf{H}$ bounds the number of usable spatial streams, and what 'rank' means for a LoS channel?
Wave propagation: Helmholtz equation, Green's function $G(\mathbf{r}) = e^{-j\kappa r}/(4\pi r)$ , Fresnel vs Fraunhofer diffraction regions(Review ch06)
Self-check: Can you write the field at a point $\mathbf{p}$ due to a point source at $\mathbf{p}_0$ in terms of the distance $\|\mathbf{p} - \mathbf{p}_0\|$ ?
Taylor expansion of $\sqrt{1 + x}$ and binomial approximations
Self-check: Do you remember that $\sqrt{1+x} \approx 1 + x/2 - x^2/8 + \mathcal{O}(x^3)$ and the conditions under which the quadratic term dominates an error?
Concept of an 'extremely large' array (XL-MIMO): aperture on the order of one meter or more at mmWave/sub-THz(Review ch01)
Self-check: Can you estimate how many $\lambda/2$ -spaced elements fit along a 1 m aperture at 28 GHz?

Notation for This Chapter

Symbols used throughout this chapter. The customizable symbols appear as $\ntn{}$ tokens; the rendered shape reflects the default in the notation registry and can be changed site-wide from the notation preferences panel.

Symbol	Meaning	Introduced
$D$	Array aperture (largest dimension), in metres	s01
$d_F$	Fraunhofer (far-field) distance, $d_F = 2 D^2 / \lambda$	s01
$d_R$	Reactive near-field boundary, $d_R \approx 0.62 \sqrt{D^3/\lambda}$	s01
$\lambda$	Carrier wavelength, $\lambda = c/f_0$	s01
$\kappa$	Wavenumber, $\kappa = 2\pi/\lambda$	s01
$\mathbf{p}_n$	3-D position of the $n$ -th array element (metres)	s02
$\mathbf{p}_k$	3-D position of the $k$ -th user / scatterer	s02
$r_{k,n}$	Exact path length from user $k$ to antenna $n$ , $r_{k,n} = \\|\mathbf{p}_k - \mathbf{p}_n\\|$	s02
$\mathbf{a}_{\text{NF}}(\mathbf{p}_k)$	Near-field array response vector at focal point $\mathbf{p}_k$	s02
$\mathbf{v}(\mathbf{p}_k)$	Near-field focusing (matched-filter) beamformer aimed at $\mathbf{p}_k$	s03
$\Delta r$	Depth of focus (range extent of the focused beam) at the $-3$ dB level	s03
$\mathcal{V}_k$	Visibility region: subset of array elements with non-negligible channel gain to user $k$	s04
$\eta_{\text{DoF}}$	Effective spatial degrees of freedom of a near-field link (can exceed $\min(N_t,N_r)$ )	s05
$N_t$	Number of base-station (XL-MIMO) transmit antennas	s01
$N_r$	Number of user-side receive antennas (often 1, but $\geq 1$ for holographic studies)	s05

← Ch 16 From Far Field to Near Field