Exercises

$\mathbf{H}_1 \in \mathbb{C}^{8 \times 4}$, $\boldsymbol{\Phi} \in \mathbb{C}^{8 \times 8}$, $\mathbf{H}_2 \in \mathbb{C}^{2 \times 8}$, $\mathbf{H}_d \in \mathbb{C}^{2 \times 4}$.

$\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 \in \mathbb{C}^{2 \times 4}$, so $\mathbf{H}_{\text{eff}} = \mathbf{H}_d + \mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 \in \mathbb{C}^{2 \times 4}$. $\blacksquare$

From Theorem 3.2, $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 = \alpha_1\alpha_2\,s(\boldsymbol{\phi})\,\mathbf{a}_{\text{UE}}\mathbf{a}_{\text{BS}}^H$.

The outer product $\mathbf{a}_{\text{UE}}\mathbf{a}_{\text{BS}}^H$ is rank-1. Multiplying by a scalar preserves rank. So the cascaded matrix is rank-$\leq 1$ (rank 1 when $s(\boldsymbol{\phi}) \neq 0$, else 0). $\blacksquare$

$\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1 = \mathbf{I}_2 \text{diag}(1, j) \mathbf{I}_2 = \text{diag}(1, j)$. $\text{vec}(\text{diag}(1, j)) = [1, 0, 0, j]^T$ (column-major vec).

Khatri-Rao product column $n$: $(\mathbf{H}_1)_{n,:}^T \otimes (\mathbf{H}_2)_{:,n}$. $n = 1$: $[1, 0]^T \otimes [1, 0]^T = [1, 0, 0, 0]^T$. $n = 2$: $[0, 1]^T \otimes [0, 1]^T = [0, 0, 0, 1]^T$. Matrix: $\begin{pmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{pmatrix}$. Times $\boldsymbol{\phi} = [1, j]^T$: $[1, 0, 0, j]^T$. ✓

Three linear gains combine multiplicatively: $N_t$ at the BS (matched filter on $N_t$ antennas), $N_r$ at the UE (aperture on $N_r$ antennas), and $N^2$ at the RIS (coherent combining of $N$ elements).

The BS and UE gains are linear in antenna count because each side uses matched filtering of an $N_t$-dim or $N_r$-dim vector. The $N^2$ at the RIS comes from the two-in-one nature of the RIS (re-transmission + aperture), as explained in Chapter 1.

$D = \sqrt{1.44 + 0.64} = \sqrt{2.08} \approx 1.44\text{ m}$.

$\lambda = c/f = 3 \cdot 10^8 / 6 \cdot 10^{10} = 5\text{ mm}$.

$d_F = 2 D^2 / \lambda = 2 \cdot 2.08 / 0.005 = 832\text{ m}$. Most practical links would be in the near-field of this RIS at 60 GHz.

$d_F = 2 \cdot 1 / (3 \cdot 10^{-3}) \approx 667\text{ m}$. Both UEs are well within near-field.

Each UE's near-field focal spot has size $\sim \lambda d_s / D$. UE 1: $0.003 \cdot 20 / 1 = 60\text{ mm}$. UE 2: $0.003 \cdot 50 / 1 = 150\text{ mm}$. Both are at $\sim \text{cm}$ scale, well-separated in distance ($30\text{ m}$ apart), so the RIS can focus independently at each.

Yes — the near-field enables distance-specific focusing, so the RIS can spatially multiplex these two UEs even though they share the same angular direction. This is impossible in the far-field.

$|z|^2 \sim \text{Exp}(1/N) = \text{Exp}(1/4)$ with mean 4. $\Pr[|z|^2 < 0.5] = 1 - e^{-0.5 \cdot 0.25} = 1 - e^{-0.125} \approx 0.118$.

The exact distribution has a modified Bessel $K_0$ form. Numerically, for $N = 4$: $\Pr[|z|^2 < 0.5] \approx 0.21$ — nearly twice the CLT prediction. The heavier tail near $|z| = 0$ means the outage is materially worse than CLT suggests.

Small-$N$ RIS cannot be analyzed by CLT; use the exact double-Rayleigh (or simulate). Large-$N$ ($\geq 64$ typically) is safe for CLT. $\blacksquare$

$\text{tr}(\mathbf{R}) = \sum_k \lambda_k$; $\|\mathbf{R}\|_F^2 = \text{tr}(\mathbf{R}^2) = \sum_k \lambda_k^2$. Ratio: $(\sum_k \lambda_k)^2 / \sum_k \lambda_k^2$.

By Cauchy-Schwarz: $(\sum_k \lambda_k)^2 \leq N \sum_k \lambda_k^2$. So $N_{\text{eff}} \leq N$, with equality iff all $\lambda_k$ are equal.

$\lambda_k \geq 0$ for PSD $\mathbf{R}$, so $(\sum_k \lambda_k)^2 \geq \max_k \lambda_k^2 \geq \sum_k \lambda_k^2 / N \cdot N = \ldots$ More directly: if one eigenvalue dominates, $N_{\text{eff}} \to 1$.

Cauchy-Schwarz equality requires all $\lambda_k$ proportional — i.e., all equal (since Cauchy-Schwarz is applied to the constant sequence vs. $\lambda_k$). So $N_{\text{eff}} = N$ iff $\mathbf{R}$ is a multiple of $\mathbf{I}$ — the i.i.d. case. $\blacksquare$

$\|\Delta\mathbf{r}\| = 7\sqrt{2} \cdot \lambda/2 = 3.5\sqrt{2}\lambda$.

$[\mathbf{R}]_{1,64} = \text{sinc}(2 \cdot 3.5\sqrt{2}) = \text{sinc}(7\sqrt{2}) = \text{sinc}(9.899)$. Since $\text{sinc}(x) = \sin(\pi x)/(\pi x)$: $\text{sinc}(9.899) = \sin(31.1)/31.1 \approx 0.032$. Nearly zero.

Corner-to-corner correlation is essentially zero — the two elements see independent fields in isotropic scattering. Adjacent-element correlation is also zero ($\text{sinc}(1) = 0$), confirming that half-wavelength spacing gives approximately i.i.d. elements. $\blacksquare$

Under the rank-1 cascaded structure, the end-to-end BS→UE beamformer-combiner gain is $\mathbf{v}_{\text{UE}}^H (\alpha_1 \alpha_2 s(\boldsymbol{\phi}) \mathbf{a}_{\text{UE}} \mathbf{a}_{\text{BS}}^H) \mathbf{v}$. Maximizing over $\mathbf{v}, \mathbf{v}_{\text{UE}}$ decouples as: $\mathbf{v}^\star \propto \mathbf{a}_{\text{BS}}$ and $\mathbf{v}_{\text{UE}}^\star \propto \mathbf{a}_{\text{UE}}$, regardless of $\boldsymbol{\phi}$.

The LoS RIS separates the optimization: BS/UE sides are independent of the RIS phases (they only depend on angles), and the RIS phases are chosen to maximize $|s(\boldsymbol{\phi})|$. Alternating optimization converges in two steps for this special case. For Rayleigh/mixed channels this separation breaks and AO needs more iterations. $\blacksquare$

Hop 1: $K_1 = 10 \text{ dB} = 10$, so LoS power fraction is $10/11 \approx 91\%$. Hop 2: $K_2 = 5 \text{ dB} \approx 3.16$, so LoS power fraction is $3.16/4.16 \approx 76\%$.

LoS-LoS end-to-end fraction: $0.91 \cdot 0.76 = 0.69$ (69%). Remaining 31% is some combination of LoS-NLoS, NLoS-LoS, and NLoS-NLoS terms. The cascaded K-factor is typically smaller than the smaller of the two hops — you can think of cascading as "mixing down" the strength of the deterministic path.

Even with moderately strong per-hop LoS, the cascaded channel has a sizable random component (~31%) that cannot be predicted from geometry alone. This is why practical RIS systems need channel estimation (Chapter 4) even in LoS-dominated deployments. $\blacksquare$

In the near-field, $\mathbf{H}_1$ has rank bounded by the number of distinct quadratic phase profiles fitting in the RIS aperture, which scales roughly as $(D/\lambda) \cdot (D/\lambda)$ for a 2D aperture at distance $d_{\text{AR}}$. For $M$ active array elements, the effective rank is $\min(M, (D/\lambda)^2)$.

The SVD of $\mathbf{H}_1$ decomposes the BS-RIS channel into parallel eigenmodes. The top $r_1 = \text{rank}(\mathbf{H}_1)$ eigenmodes can carry independent streams; each stream is a specific active-array $\to$ RIS beam pattern. The RIS then focuses each beam to a different UE direction, giving multi-user multiplexing.

By carefully choosing $d_{\text{AR}}$ and $M$, Caire et al. engineer $\mathbf{H}_1$ to have a rich high-rank structure — exactly $K$ useful eigenmodes for a $K$-user system. The RIS aperture then provides $K$ independent beams toward the UEs. This is the architectural insight behind Chapter 11.