Chapter Summary

Chapter 17 Summary: Near-Field Communications

Key Points

• 1.

  Far field is a statement about $d/d_F$, not absolute distance. The Fraunhofer distance $d_F = 2 D^2/\lambda$ scales quadratically in aperture and linearly in carrier frequency. For 6G XL-MIMO at sub-THz, $d_F$ is on the order of hundreds of metres — every user in the cell is in the near field. Plane-wave steering vectors are systematically wrong in this regime.

• 2.

  The near-field array response is position-parameterised, not direction-parameterised. The exact steering vector has entries $e^{-j\kappa\|\mathbf{p}_k - \mathbf{p}_n\|}/\sqrt{N_t}$; its Fresnel expansion contains the classical linear-phase term plus a quadratic phase $-\kappa y_n^2\cos^2\theta_k/(2 r_k)$ that encodes range. Dropping the quadratic term is exactly the plane-wave approximation, valid only for $d \geq d_F$.

• 3.

  Beam focusing replaces beam steering. The matched-filter near-field beamformer $\mathbf{v}(\mathbf{p}_k)$ delivers the full $N_t$-fold coherent gain to a spot at position $\mathbf{p}_k$, not to a ray. The half-power depth of focus is $\Delta r = 2 d_F d_k^2/(d_F^2 - d_k^2)$, which shrinks at short ranges and opens up to infinity as $d_k \to d_F$. Two co-directional users can be spatially multiplexed once their range separation exceeds $\Delta r$.

• 4.

  XL-MIMO channels are spatially non-stationary. The visibility region $\mathcal{V}_k$ is the subset of antennas that actually see user $k$; the visibility ratio $\rho_k = |\mathcal{V}_k|/N_t$ is typically well below $1$. Non-stationarity-blind combiners lose $10\log_{10}(1/\rho_k)$ dB of array gain. Systems cope with sub-array processing — effectively, cell-free massive MIMO inside a single physical aperture.

• 5.

  Near-field degrees of freedom can exceed $\min(N_t, N_r)$. For planar apertures of size $L$ at range $d < d_F$, the Pizzo– Marzetta–Sanguinetti formula gives $\eta_{\text{DoF}} \sim (L/\sqrt{\lambda\,d})^4 = (L^2/(\lambda\,d))^2$, a fourth-power law in the number of Fresnel zones that fit on the aperture. This is the quantitative basis of "holographic MIMO" and the reason sub-THz LoS links can spatially multiplex tens to thousands of streams.

• 6.

  Polar-domain codebooks replace angular codebooks. The near-field channel is sparse in a chirp-Fourier basis, not in a DFT basis. Near- field codebooks grid angle $\sin\theta$ uniformly and grid $1/r$ with step $\lambda/D^2$. The codebook size grows as $\mathcal{O}(N_t^{3/2})$, which is the price paid for the extra range dimension.