Chapter Summary

Chapter Summary

Key Points

• 1.

  A passive RIS pays double fading. An $N_{\text{RIS}}$-element reconfigurable surface boosts link SNR by a factor of $N_{\text{RIS}}^2$ (the aperture gain in the reflected link), but the end-to-end path loss is $(d_1 d_2)^{-2}$, so the nominal benefit is nearly erased at realistic geometries. A 1024-element RIS only decisively beats a direct link when the direct link is blocked or when one of the two ranges is small.

• 2.

  The array-fed RIS collapses $d_1 \to 0$. Placing a small active array of $N_a$ elements only a few wavelengths in front of the passive RIS eliminates the first path-loss factor and preserves the aperture gain. The resulting system is a hybrid architecture with only $N_a$ RF chains driving an effective aperture of $N_{\text{RIS}}$ elements — the CommIT engineering contribution of Caire and collaborators at mmWave/sub-THz.

• 3.

  Rank is set by $N_a$, gain is set by $N_{\text{RIS}}$. The cascaded channel $\mathbf{H}_{\text{eff}} = \mathbf{H}_{\text{RIS-Rx}} \text{diag}(\boldsymbol{\phi}) \mathbf{G}_f$ has rank at most $\min(N_r, N_a)$ no matter how many tiles $N_{\text{RIS}}$ the surface contains. A diagonal phase matrix cannot raise rank. Within this rank limit, each of the $N_a$ non-zero singular values scales like $\sigma_r N_{\text{RIS}}$, so the array gain is concentrated on a small number of high-quality eigenmodes.

• 4.

  Multiuser beamforming uses alternating optimization. Joint design of $\mathbf{W}$ (active-array precoder) and $\boldsymbol{\phi}$ (RIS phases) is bilinear and non-convex, but each subproblem is tractable: ZF/MMSE precoder in closed form for fixed $\boldsymbol{\phi}$, and a closed-form single-sinusoid maximization for each $\phi_n$ individually. Five to fifteen outer iterations suffice in practice.

• 5.

  The equivalent digital chain count $N_{\text{eq}}$ is close to $N_a$. Caire's equivalent-chain theorem shows that an array-fed RIS with $(N_a, N_{\text{RIS}})$ matches the sum rate of a fully digital array with $N_{\text{eq}} \approx N_a (1 + c \log N_{\text{RIS}} / N_a)$ chains. Adding more RIS tiles yields logarithmically diminishing rate returns once per-stream SNR saturates.

• 6.

  Power rather than rate is the practical win. At matched aperture and mmWave carriers, the array-fed RIS typically achieves 75–85% of the fully digital sum rate at 15–30% of the DC power — a 10–30x improvement in rate-per-watt. The advantage is largest at high frequencies where wavelengths are small and RF-chain DC power dominates the hardware budget.

• 7.

  Spatial DoF, not aperture, limits multiuser multiplexing. Whenever $K > N_a$, users must be scheduled across time or frequency resources, with at most $N_a$ users served simultaneously on orthogonal spatial modes. This is a hard constraint that holds regardless of $N_{\text{RIS}}$, because a passive RIS cannot create new degrees of freedom.

• 8.

  Engineering reality imposes phase quantization and mutual coupling. Real RIS tiles offer 1–3 bits of phase resolution (loss of $0.2$–$1$ dB) and non-negligible element coupling. The array-fed RIS tolerates these non-idealities gracefully because most of the rate comes from the digital precoder, not from sub-wavelength phase precision. This is a robustness advantage over a purely passive RIS.