Exercises

$\mathbf{G}$: $N N_t$ complex entries. Separated: $\mathbf{H}_1$ has $N N_t$, $\mathbf{h}_2$ has $N$, total $NN_t + N$.

For any nonzero $\alpha_n \in \mathbb{C}$, replacing $(\mathbf{h}_2)_n \mapsto \alpha_n (\mathbf{h}_2)_n$ and $(\mathbf{H}_1)_{n,:} \mapsto \alpha_n^{-1*} (\mathbf{H}_1)_{n,:}$ leaves $\mathbf{G}$ unchanged. This gives $N$ complex degrees of freedom not recoverable from passive-RIS observations.

$N = 128$: $\tau_p/T = 12.9\%$ — acceptable. $N = 512$: $\tau_p/T = 51.3\%$ — marginal. $N = 2048$: $\tau_p/T = 204.9\%$ — infeasible, pilot alone exceeds coherence.

ON/OFF scaling imposes a hard cap on the useful $N$ for any given coherence budget. For fast-moving users with $T < N$, ON/OFF estimation is impossible; one must use CS or accept statistical beamforming.

$\hat{\mathbf{G}}^H = (1/\sqrt{N})(\mathbf{G}^H \sqrt{N}\mathbf{F} + \mathbf{W}) \mathbf{F}^H = \mathbf{G}^H + (1/\sqrt{N})\mathbf{W}\mathbf{F}^H$.

Writing $\tilde{\mathbf{W}} = (1/\sqrt{N})\mathbf{W}\mathbf{F}^H$, $\mathbb{E}[\tilde{\mathbf{W}} \tilde{\mathbf{W}}^H] = (1/N) \mathbf{F}\mathbb{E}[\mathbf{W}^H\mathbf{W}]\mathbf{F}^H \cdot \ldots$ — more carefully: $\text{vec}(\tilde{\mathbf{W}}) = ((\mathbf{F}^H)^T \otimes \mathbf{I}) \text{vec}(\mathbf{W})/\sqrt{N}$. Since $\mathbf{F}$ is unitary and $\text{vec}(\mathbf{W})$ is i.i.d. $\mathcal{CN}(0, \sigma^2)$, $\text{vec}(\tilde{\mathbf{W}})$ is also i.i.d. $\mathcal{CN}(0, \sigma^2/N)$.

Each row of $\hat{\mathbf{G}}^H$ has $N_t$ entries with per-entry variance $\sigma^2/N$, giving total row MSE $N_t \sigma^2/N$ (normalized by $1/P_t$ for pilot power): $\text{MSE} = N_t\sigma^2/(N\,P_t)$. $\blacksquare$

$\text{MSE}^{\text{ON/OFF}}/\text{MSE}^{\text{DFT}} = 2N = 512$.

$10\log_{10}(512) = 27.1\text{ dB}$. DFT is $> 27\text{ dB}$ better than ON/OFF in MSE at $N = 256$, at identical pilot length. A drastic argument for adopting DFT in any practical deployment. $\blacksquare$

$\phi^{(1)} = [1, 1, 1, 1, 1, 1, 1, 1]^T$. All unit-modulus.

$\phi^{(2)} = [e^{-j \pi n/4}]_{n=0}^{7} = [1, e^{-j\pi/4}, e^{-j\pi/2}, e^{-j3\pi/4}, -1, e^{-j5\pi/4}, e^{-j3\pi/2}, e^{-j7\pi/4}]^T$. All unit-modulus.

$\langle \phi^{(1)}, \phi^{(2)} \rangle = \sum_{n=0}^{7} e^{-j\pi n/4} = \frac{1 - e^{-j\pi \cdot 8/4}}{1 - e^{-j\pi/4}} = 0$. Orthogonal. ✓

$\tau_p = 4 \cdot 8 \cdot \log_2(512) / 16 = 32 \cdot 9/16 = 18$ slots.

DFT codebook requires $\tau_p = N = 512$. CS saves a factor of $\sim 28\times$ in pilot time. The savings come from exploiting angular sparsity that DFT ignores.

$\mathbb{E}[\text{SNR}] = 10^3 \cdot 0.9 = 900$, i.e., $29.5\text{ dB}$. A modest $0.5\text{ dB}$ power penalty.

$\Delta C = \log_2(1 + 1000) - \log_2(1 + 900) \approx 0.15$ bits/s/Hz — barely perceptible at high SNR.

At $\text{SNR}^{\text{ideal}} = 0\text{ dB}$: $\mathbb{E}[\text{SNR}] = 0.9$, capacity $= \log_2(1.9) = 0.93$ vs. ideal $\log_2(2) = 1.00$ — a $7\%$ penalty. CSI error is more costly at low SNR, where beamforming gain is most needed. $\blacksquare$

$R_{\text{eff}} \approx (1 - \tau_p/T) [\log_2 \text{SNR}^{\text{ideal}} + \log_2(1 - c/\tau_p)]$.

Let $R_1 = (1 - \tau_p/T)$ and $R_2 = [\log_2 \text{SNR}^{\text{ideal}} + \log_2(1 - c/\tau_p)]$. $dR_{\text{eff}}/d\tau_p = -1/T \cdot R_2 + R_1 \cdot (c/(\tau_p^2 \ln 2))/(1 - c/\tau_p)$. Setting to zero at the optimum.

At high SNR, $(1 - c/\tau_p) \approx 1$ and the balance gives $\tau_p^{\star 2} \approx c T \ln 2 / \log_2 \text{SNR}$, or $\tau_p^\star \approx \sqrt{c T \ln 2 / \log_2 \text{SNR}}$ — closely matching the $\sqrt{T/\text{SNR}}$ scaling. $\blacksquare$

In slot 0 ("all OFF"), the received signal is not just $\mathbf{h}_d x$ but $\mathbf{h}_d x + |\phi|_{\text{off}} \mathbf{G}^H \mathbf{1} x$. The subtraction $\mathbf{y}_n - \mathbf{y}_0$ is contaminated by this additional term, which does not scale with noise.

The bias is deterministic: it is a linear function of the true cascaded channel. More pilot power makes the bias larger in absolute terms, not smaller. Only hardware improvements (better OFF state) reduce the bias.

(i) Calibrate the OFF state and subtract the expected bias; (ii) use DFT codebook, where the bias cancels by orthogonality; (iii) average across multiple ON/OFF cycles to randomize phase of the bias. $\blacksquare$

$\epsilon_{\text{CSI}}^2 \equiv \text{MSE} = N_t\sigma^2/(\tau_p P_t) = N_t/(\tau_p \cdot \text{SNR})$ where $\text{SNR} = P_t/\sigma^2$.

$\tau_p = N_t/(\delta \cdot \text{SNR})$.

To halve the CSI error, double the pilot length. To get a $10\times$ smaller CSI error, spend $10\times$ more pilots — linear pilot-vs-accuracy tradeoff. $\blacksquare$

OMP selects the best-correlated atom at each iteration, adds it to the support, and solves a small least-squares on the selected support.

OMP needs $L$ iterations to recover an $L$-sparse signal (assuming the support is correctly identified). For $L = 2$: 2 iterations. Per iteration cost: $\mathcal{O}(\tau_p N_t \cdot N N_t) = \mathcal{O}(18 \cdot 8 \cdot 256 \cdot 8) \approx 3 \times 10^5$ flops. Total: $\sim 6 \times 10^5$ flops, well under 1 ms on a modern CPU. Much cheaper than LASSO.

$\text{MSE} = N_t\sigma^2/(\tau_p P_t) = N_t\sigma^2/E_{\text{pilot}}$.

MSE depends only on total pilot energy, not on the split between length and power. For fixed energy, DFT estimation is indifferent to the tradeoff. In practice, other factors (hardware linearity at high per-slot power, feedback latency at long pilot) tip the balance.

For compressed sensing, MSE at fixed energy depends on the specific recovery algorithm; LASSO favors moderate-length probing (more measurements) over high-power short probing, because the RIP condition improves with more measurements. So CS and DFT respond differently to length/power tradeoffs. $\blacksquare$

$\tau_p \geq \mathcal{O}(1 \cdot \log N / N_t)$; for $N = 256, N_t = 8$: $\tau_p \sim \log_2(256)/8 = 1$ slot.

In pure-LoS conditions, a single well-designed pilot slot suffices to identify the one nonzero scattering path and extract the full $\mathbf{G}$. This is the theoretical limit; in practice, $\tau_p \sim 4$-$8$ for robustness against off-grid effects and noise.

DFT codebook still requires $\tau_p = N = 256$. The CS advantage grows as $N/L$: for $L = 1, N = 256$, CS uses $256\times$ fewer pilots. $\blacksquare$

With $\tau_0$ slots of all-OFF state and $N$ slots for the individual elements (total $\tau_0 + N$ pilots), the direct channel MSE is $N_t\sigma^2/(\tau_0 P_t)$ and each element's MSE is $N_t\sigma^2(1/\tau_0 + 1)/P_t$ (noise adds from both slots, but only one slot for element $n$).

Total MSE $\propto 1/\tau_0 + N \cdot (1/\tau_0 + 1)$, minimizing over $\tau_0 + N = \tau_p$ fixed. The optimum is $\tau_0 = \sqrt{N+1}$ — allocate $\sqrt{N}$ slots to slot 0, which is a small fraction of the pilot budget for large $N$.

For large $N$, the $\mathbf{h}_d$ estimation dominates total MSE unless we allocate $\mathcal{O}(\sqrt{N})$ pilots to it. Practical implementations use $\tau_0 = \sqrt{N}$ or a fixed-fraction $\tau_0 = 0.1 N$. $\blacksquare$

$T = 10^5$, $\tau_p^\star \sim \sqrt{T/\text{SNR}}$ is very small. DFT codebook with $\tau_p = 30$-$100$ pilots achieves $\epsilon_{\text{CSI}} < 1\%$ at negligible overhead. Computational budget is generous. CS offers little benefit.

$T = 200$, pilot budget is tiny. DFT with $\tau_p = 200$ exhausts the coherence block. CS with $L = 4$ sparsity and $\tau_p \sim 20$ is the only feasible option. Must accept $\epsilon_{\text{CSI}} \sim 10\%$. Real-time LASSO solve becomes a compute bottleneck.

Fixed deployments: DFT codebook for simplicity and robustness. Mobile deployments: CS with an aggressive prior on sparsity and a compute-efficient solver (e.g., OMP). The two regimes demand genuinely different approaches. $\blacksquare$