Exercises

$B$ bits require $B$ PIN diodes (each toggling a different fraction of the cell's impedance to yield distinct phase states).

1-bit: 1 diode, 2 phase states. 2-bit: 2 diodes, 4 states. 3-bit: 3 diodes, 8 states. Note: this is the straightforward binary encoding; alternative designs (varactor + DAC) use a single tunable element.

$B=1: \text{sinc}^2(\pi/2) = 0.405 \to 3.92\text{ dB}$. $B=2: 0.811 \to 0.91\text{ dB}$. $B=3: 0.949 \to 0.22\text{ dB}$. $B=4: 0.987 \to 0.056\text{ dB}$. $B=5: 0.997 \to 0.014\text{ dB}$.

The return on additional bits diminishes sharply after 3 bits. Going from 2 → 3 bits saves $0.7\text{ dB}$; 4 → 5 bits saves only $0.04\text{ dB}$ — not worth the doubled diode count.

The error $\epsilon_n = \theta_n^\star - \text{nearest}(\theta_n^\star, \Theta_B)$ is uniform on $[-\Delta\theta/2, \Delta\theta/2]$ iff $\theta_n^\star \bmod \Delta\theta$ is uniform on $[0, \Delta\theta)$. This holds, e.g., if the optimal phases themselves are i.i.d. uniform on $[0, 2\pi)$ — which is the case when the channels $\mathbf{h}_1, \mathbf{h}_2$ have uniform random phases (generic Rayleigh).

In pure LoS scenarios with a regular array geometry, the optimal phases follow a quadratic spatial pattern (steering vector), and their values-mod-$\Delta\theta$ may cluster unfavorably. In this case the quantization loss can be slightly better or worse than the sinc prediction; on average the two bracket agree.

$\bar{a} = \frac{1}{2\pi}\int_0^{2\pi} (0.5 + 0.5 \cos^2 \theta) d\theta = 0.5 + 0.5 \cdot (1/2) = 0.75$.

$\eta_{\text{APC}} = 0.75^2 = 0.5625$, loss $= -10\log_{10}(0.5625) = 2.5\text{ dB}$.

Quantization ($B=2$): $\eta_2 = 0.811$, $\,-0.91\text{ dB}$. APC: $\eta_{\text{APC}} = 0.49$, $\,-3.1\text{ dB}$.

Total SNR factor $\approx 0.811 \cdot 0.49 = 0.397$, $\,-4.0\text{ dB}$. Or additively in dB: $0.91 + 3.1 = 4.0\text{ dB}$. $\blacksquare$

When element spacing is substantially smaller than half-wavelength (e.g., $\lambda/4$ or less), mutual coupling becomes strong, off-diagonal terms in $\boldsymbol{\Phi}_{\text{full}}$ cannot be ignored, and beam-pattern predictions from the diagonal model can be $\geq 3\text{ dB}$ off in main-lobe gain.

Continuous: $\text{SNR}^\star = (P_t/\sigma^2) N^2 \alpha^2 \beta^2$. 1-bit: $\text{SNR}^{(1)} = (P_t/\sigma^2) (N')^2 \cdot 0.405 \alpha^2 \beta^2$. Equal: $N^2 = 0.405 (N')^2$, so $N' = N / \sqrt{0.405} \approx 1.57 N$.

The 1-bit RIS needs about $57\%$ more elements to match a continuous-phase RIS. A $256$-continuous becomes $\sim 400$ 1-bit. Cost-wise, 400 1-bit elements (each 1 diode) vs. 256 continuous elements (each DAC + varactor) can go either way — depending on whether the DAC or the extra elements are cheaper. $\blacksquare$

For purely imaginary $Z_s = jX$, $\Gamma = (jX - Z_0)/(jX + Z_0)$. $\arg \Gamma = \arctan(X / (-Z_0)) - \arctan(X / Z_0) + \pi$. Simplifying, $\arg \Gamma = -2\arctan(Z_0/X) + \pi$ (modulo branch choices; in practice one sweeps $X$ and tracks $\arg \Gamma$).

As $X \to -\infty$ (capacitive-dominated, below resonance), $\arg \Gamma \to \pi + 2\cdot 0 = \pi$. As $X \to +\infty$ (inductive, above resonance), $\arg \Gamma \to \pi - 2\cdot 0 = \pi$. But passing through $X = 0$ makes $\arctan(Z_0/X)$ jump by $\pi$, so $\arg \Gamma$ sweeps continuously from $\pi$ down through $0$ and back to $\pi$ — covering $2\pi$ total.

The resonance provides the only way to achieve a full $2\pi$ phase range with a single reactive element. Off-resonance circuits (no inductor/capacitor cancellation) span only $\pi$ or less of phase. $\blacksquare$

$3 \cdot 1024 = 3072$ bits per full reconfiguration.

$3072 \times 100 = 307\,200$ bps $\approx 307$ kbps. Easily carried by even the cheapest out-of-band control link (Bluetooth LE, Zigbee, or a wired serial bus). Control overhead is not a bottleneck for low-rate update. At $10$ kHz update rates (Chapter 18) the bandwidth becomes $30$ Mbps, which starts to demand a dedicated link.

For a symmetric tridiagonal Toeplitz matrix with diagonal $a$ and off-diagonal $b$, eigenvalues are $\lambda_k = a + 2b\cos(k\pi/(N+1))$, $k = 1, \ldots, N$.

With $a = 2$ (from $\mathbf{Z}_s + \mathbf{Z}$ diagonal) and $b = -0.3 e^{j\pi/4}$ (if we take the magnitude for a symmetric model), eigenvalues spread around $2$ by $\pm 0.6 \cos(k\pi/(N+1))$.

As $N \to \infty$, the eigenvalues densely fill the interval $[2 - 0.6, 2 + 0.6] = [1.4, 2.6]$. The diagonal approximation $\lambda = 2$ replaces this $30\%$-spread distribution with a single value; the inversion error in $(\mathbf{Z}_s + \mathbf{Z})^{-1}$ is of order $0.6/2 = 30\%$ in the worst case. This motivates either explicit coupling modeling or design-for-isolation (spacing choices). $\blacksquare$

For Gaussian $\epsilon \sim \mathcal{N}(0, \sigma_\phi^2)$ (in radians), $\mathbb{E}[e^{j\epsilon}] = e^{-\sigma_\phi^2 / 2}$. $\sigma_\phi = 5^\circ = 0.0873\,\text{rad}$, so $e^{-0.00381} \approx 0.9962$.

Power is amplitude squared: $(0.9962)^2 = 0.9925$, $-0.033\text{ dB}$. Negligible.

Manufacturing variance of $5^\circ$ is much smaller than the $22.5^\circ$ step of a 4-bit quantizer. The loss is accordingly smaller. Manufacturing tolerances at the few-degree level are not a first-order concern for modern designs. $\blacksquare$

Half the $\theta_n^\star$ land in $[0, \pi)$ with $a = 1$ and half in $[\pi, 2\pi)$ with $a = 0.5$. $\bar{a} = 0.5 \cdot 1 + 0.5 \cdot 0.5 = 0.75$.

$\eta_{\text{APC}} = 0.75^2 = 0.5625$, $-2.5\text{ dB}$. $\blacksquare$

Half-circle reflectors are cheap (one diode, one reactance switch) but cost the equivalent of a 1-bit quantizer on top of the amplitude loss — around $2.5$-$3\text{ dB}$ total.

Passivity dictates $|\phi_n| \leq 1$; equality requires a perfectly lossless reflection.

To rotate the phase by up to $2\pi$ with a single tunable element, the element must pass through resonance. Near resonance, the small (but non-zero) loss resistance of any real diode causes $|\phi_n|$ to dip below 1. The only way to achieve exact unit modulus across all phases is to add an active amplifier — but then the element is no longer passive.

$|\phi_n| = 1$ across all $\theta_n$ is a strictly unreachable design target for passive hardware. Real designs accept a few-dB amplitude dip near resonance as the price of reconfigurability. $\blacksquare$