Chapter Summary

Chapter Summary

Key Points

• 1.

  The cascaded channel model. For a BS with $N_t$ antennas, an RIS with $N$ elements, and a UE with $N_r$ antennas, the effective end-to-end channel is $\mathbf{H}_{\text{eff}} = \mathbf{H}_d + \mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$, a $N_r \times N_t$ matrix linear in the RIS phase vector $\boldsymbol{\phi}$. The Khatri-Rao identity $\text{vec}(\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1) = \big(\mathbf{H}_1^T \odot \mathbf{H}_2\big) \boldsymbol{\phi}$ is the workhorse that every optimization algorithm exploits.

• 2.

  Rank and keyhole. The rank of $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ is bounded by $\min(\text{rank}(\mathbf{H}_1), N, \text{rank}(\mathbf{H}_2))$. Pure LoS on both hops gives a rank-1 cascade — the RIS keyhole, which eliminates multi-stream multiplexing regardless of BS or UE antenna count. Rich scattering on either (or both) sides is required for multi-stream operation.

• 3.

  LoS RIS phases are a spatial sawtooth. Under LoS on both hops, the optimal RIS phase is a linear (for planar arrays, bilinear) function of element index: $\theta_{n_1, n_2}^\star = \pi[n_1 \Delta u + n_2 \Delta v]$, where $\Delta u, \Delta v$ are determined by the BS and UE directions. This is the phased-array "generalized reflection" pattern.

• 4.

  Cascaded Rayleigh has fatter tails. For small $N$, cascaded Rayleigh fading yields higher outage probability than direct Rayleigh of the same mean SNR. At $N \geq 256$, the CLT approximates the cascaded distribution back to Gaussian, and the "keyhole penalty" in outage vanishes. Another reason to design with hundreds of elements.

• 5.

  Near-field at sub-THz is the rule, not the exception. The Fraunhofer distance $d_F = 2D^2/\lambda$ can exceed a kilometer for 1-m apertures at 140 GHz. Inside $d_F$, wavefront curvature across the aperture enables distance-specific focusing and adds extra degrees of freedom absent in the far-field. Near-field modeling is essential for sub-THz RIS design; it is the technical foundation of the array-fed RIS architecture (Chapter 11).