Chapter Summary

Chapter Summary: Channel Models for Large Arrays

Key Points

• 1.

  The i.i.d. Rayleigh baseline. The simplest MIMO channel model has $\mathbf{H} = \sqrt{\beta}\,\mathbf{G}$ with i.i.d. $\mathcal{CN}(0,1)$ entries. As $N_t \to \infty$, the strong law of large numbers yields channel hardening ($\|\mathbf{h}_k\|^2/N_t \to 1$ a.s.) and favorable propagation ($\mathbf{h}_k^H\mathbf{h}_j/N_t \to 0$ a.s., $k \neq j$). The empirical eigenvalue distribution of $\mathbf{H}\mathbf{H}^{H}/N_t$ follows the Marchenko–Pastur law.

• 2.

  Spatial correlation is the norm in outdoor deployments. When the BS is elevated above local scatterers, channels from the same user arrive from a restricted angular window, making all antenna elements see correlated versions of the same channel. The one-ring model (two parameters: $\theta_0$, $\Delta\theta$) captures this via the Bessel integral; the Kronecker model ($\mathbf{R}_t \otimes \mathbf{R}_r$) separates TX and RX correlations; Weichselberger's model allows arbitrary TX-RX coupling via the matrix $\mathbf{W}_W \odot \tilde{\mathbf{G}}$ in the eigenbasis.

• 3.

  The angular domain reveals sparsity. The DFT transformation $\tilde{\mathbf{H}} = \mathbf{U}_r^H \mathbf{H} \mathbf{U}_t$ maps the physical channel to the angle domain, where each propagation path appears as a localized "bright spot" at angle-bin pair $(\text{AoA bin}, \text{AoD bin})$. For $L$ paths, $\tilde{\mathbf{H}}$ is approximately $L$-sparse. This sparsity is the foundation for compressed-sensing channel estimation, JSDM precoding, and hybrid beamforming.

• 4.

  3GPP TR 38.901 is the standard MIMO channel model. It defines a geometry-based stochastic channel model (GBSM) with large-scale parameters (delay spread, angular spreads, $K$-factor, shadowing) modeled as correlated log-normal random variables. The CDL and TDL fixed profiles enable reproducible link-level testing. QuaDRiGa implements TR 38.901 with spatial consistency for research simulations.

• 5.

  Sub-6 GHz and mmWave are fundamentally different regimes. At sub-6 GHz, moderate spatial correlation ($L \approx 3$–10 clusters, ASD $\approx$ 5–30°) makes the Kronecker/one-ring models adequate. At mmWave, extreme sparsity ($L \leq 3$ paths, ASD $\approx$ 1–10°) makes hybrid beamforming essential and compressed sensing feasible. The coherence time at mmWave with vehicular mobility ($< 1$ ms) mandates beam-level tracking rather than element-level CSI.

• 6.

  Channel model choice drives system design. The one-ring covariance model informs pilot design (how many orthogonal pilots suffice), precoder structure (JSDM groups users by angular separation), and capacity estimates (effective rank of $\mathbf{R}_t$ limits multiplexing gain). Using the wrong model can lead to 15–40% overestimate of achievable rates in outdoor deployments.