Exercises

$\mathbf{y} = \boldsymbol{\Phi}\mathbf{h}_{\mathrm{ad}} + \mathbf{w}$ where $\boldsymbol{\Phi} = \mathbf{P} \otimes \mathbf{F}_R^H \in \mathbb{C}^{M \times 32}$, $\mathbf{h}_{\mathrm{ad}} \in \mathbb{C}^{32}$ is 3-sparse.

$\boldsymbol{\Phi} \leftrightarrow \mathbf{A}$ (sensing matrix), $\mathbf{h}_{\mathrm{ad}} \leftrightarrow \mathbf{c}$ (reflectivity), $\mathbf{w} \leftrightarrow \mathbf{w}$ (noise). The two models are structurally identical. $\blacksquare$

$M_{\mathrm{LASSO}} \geq 2 \times 5 \times 6.7 = 67$ pilots.

$M_{\mathrm{LS}} = N = 4096$ pilots.

$M_{\mathrm{LASSO}}/M_{\mathrm{LS}} = 67/4096 \approx 1.6\%$. LASSO reduces pilot overhead by $\sim 60\times$. $\blacksquare$

Each column of the spatial DFT matrix $\mathbf{F}_R$ is a steering vector $\mathbf{a}(\theta_k)$ for a plane wave arriving at angle $\theta_k = \arcsin(2k/N_R)$. The DFT of the channel across antennas projects onto these steering vectors.

At mmWave, each physical propagation path arrives from a specific angle, contributing energy to one (or a few, due to grid mismatch) DFT coefficient(s). With $K \ll N_R$ paths, the channel is sparse in the DFT (angular) domain. $\blacksquare$

$d = \lambda/2 = 2.5$ mm. $D = 128 \times 2.5$ mm $= 0.32$ m.

$d_F = 2D^2/\lambda = 2 \times 0.32^2/0.005 = 40.96$ m. Near-field estimation is needed for scatterers within $\sim 41$ m of the array. $\blacksquare$

$N = G \times 10 = 2560$. $M_{\mathrm{LASSO}} \geq 2 \times 24 \times \log(2560/24) = 48 \times 4.67 = 224$ pilots.

$M_{\mathrm{GL}} \geq 2(4 \times \log(256) + 24) = 2(4 \times 5.54 + 24) = 2(22.2 + 24) = 92$ pilots.

$M_{\mathrm{GL}}/M_{\mathrm{LASSO}} = 92/224 = 0.41$. Group LASSO requires $\sim 2.4\times$ fewer pilots by exploiting the cluster structure. $\blacksquare$

By the LMMSE formula (Chapter FSI): $\hat{\mathbf{h}} = \mathbf{C}_{h,y}\mathbf{C}_y^{-1}\mathbf{y}$ where $\mathbf{C}_{h,y} = \mathbf{C}_h\boldsymbol{\Phi}^H$ and $\mathbf{C}_y = \boldsymbol{\Phi}\mathbf{C}_h\boldsymbol{\Phi}^H + \sigma^2\mathbf{I}$.

If $\mathbf{C}_h$ has low rank (sparse channel in angular domain), then $\mathbf{C}_h = \mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^H$ with $\mathrm{rank} \approx K$. The LMMSE estimate projects onto the $K$-dimensional subspace spanned by $\mathbf{U}$, effectively performing subspace-based estimation. This connects LMMSE to the sparsity-based approaches. $\blacksquare$

Node 1 covers a k-space wedge of angular width $\sim 2/N_a = 2/64 \approx 3°$ in the direction perpendicular to its array. The radial extent is set by the bandwidth: $\Delta k = 4\pi W/c = 4\pi \times 400 \times 10^6/(3 \times 10^8) \approx 16.8$ rad/m.

Three nodes at $0°$, $120°$, $240°$ cover three k-space wedges at $120°$ spacing. The combined coverage is approximately isotropic (circular ring in k-space) with some gaps.

Single node: good resolution perpendicular to array ($\sim 0.6$ m at 20 m range), poor along array ($\gg 1$ m). Three nodes: $\sim 0.6$ m in all directions (isotropic). Improvement: from $\sim 4:1$ aspect ratio to $\sim 1:1$. $\blacksquare$

$(\mathbf{A}_{g}^{H}\mathbf{A}_{g} + \rho\mathbf{I})^{-1} \approx \rho^{-1}\mathbf{I}$. The local update becomes $\boldsymbol{\sigma}_g^{(t+1)} \approx \boldsymbol{\sigma}^{(t)} - \mathbf{u}_g^{(t)}$: each node copies the consensus, ignoring its own measurements. The algorithm converges instantly to the average initialisation.

$(\mathbf{A}_{g}^{H}\mathbf{A}_{g} + \rho\mathbf{I})^{-1} \approx (\mathbf{A}_{g}^{H}\mathbf{A}_{g})^{-1}$ (if invertible). Each node solves its local problem independently, ignoring the consensus. The dual variables $\mathbf{u}_g$ grow without bound — the algorithm diverges.

The optimal $\rho$ balances local data fidelity and consensus: too large ignores data, too small ignores consensus. Typically $\rho \sim \|\mathbf{A}_{g}^{H}\mathbf{A}_{g}\|$ works well. $\blacksquare$

Each node sends one complex image: $256^2 \times 8 = 524{,}288$ bytes. 6 nodes: $6 \times 0.5 = 3$ MB per ADMM iteration.

With 20 ADMM iterations per snapshot: $20 \times 3 = 60$ MB. At 50 Hz: $60 \times 50 = 3{,}000$ MB/s $= 24$ Gbps.

This exceeds 5G backhaul ($\sim 5$ Gbps practical). Options: (1) reduce image size to $64 \times 64$: $24/16 = 1.5$ Gbps (feasible); (2) reduce update rate to 10 Hz: $24/5 = 4.8$ Gbps (borderline); (3) use feature-level fusion for dynamic updates. $\blacksquare$

$\eta_{\mathrm{pilot}} = 32/200 = 16\%$. 32 pilot slots out of 200.

Residual rank: $P - L = 8 - 6 = 2$. $\eta_{\mathrm{DT}} = 16\% \times 2/8 = 4\%$. 8 pilot slots.

Data slots: without DT: $200 - 32 = 168$. With DT: $200 - 8 = 192$. Throughput gain: $192/168 = 1.14\times$ ($14\%$ increase). $\blacksquare$

128 pilot slots. Overhead: $128/150 = 85.3\%$. Data: 22 slots.

Expected slots: $0.95 \times 5 + 0.05 \times 128 = 4.75 + 6.4 = 11.15$. Overhead: $11.15/150 = 7.4\%$. Data: $\sim 139$ slots.

$139/22 = 6.3\times$ throughput increase. $\blacksquare$

Maximise $\sum_k f_k^2 |W_k|^2$ subject to $\sum_k |W_k|^2 = P_t$. Solution: $|W_k|^2 = P_t/2$ for $k = 1$ and $k = N_f$, zero elsewhere. $\mathrm{CRB}(\tau) = 1/(\text{SNR} \cdot P_t \cdot 2\pi^2(f_1^2 + f_{N_f}^2))$.

Uniform: $|W_k|^2 = P_t/N_f$ for all $k$. The sensing matrix has lower coherence (better RIP) when all frequencies are illuminated. NMSE $\propto K/(N_f \cdot \text{SNR})$ for LASSO reconstruction.

CRB-optimal uses 2 frequencies: effective measurements $= 2N_a$. Imaging-optimal uses 128 frequencies: effective measurements $= 128 N_a$. NMSE ratio: $\sim N_f/2 = 64\times$ ($\sim 18$ dB) worse for the CRB-optimal waveform. $\blacksquare$

Outer: $\min_{\mathbf{W}} \mathcal{L}(\mathbf{W}) = \mathbb{E}_{\mathbf{c},\mathbf{w}}[\|\hat{\mathbf{c}}(\mathbf{W}) - \mathbf{c}\|^2]$

Inner: $\hat{\mathbf{c}}(\mathbf{W}) = \arg\min_{\mathbf{c}} \frac{1}{2}\|\mathbf{y} - \mathbf{A}(\mathbf{W})\mathbf{c}\|^2 + \lambda\|\mathbf{c}\|_1$

$\nabla_\mathbf{W}\mathcal{L} = \mathbb{E}\left[\frac{\partial\hat{\mathbf{c}}}{\partial\mathbf{W}}^T \cdot 2(\hat{\mathbf{c}} - \mathbf{c})\right]$.

The Jacobian $\partial\hat{\mathbf{c}}/\partial\mathbf{W}$ requires implicit differentiation through the LASSO KKT conditions, or explicit differentiation through an unrolled ISTA/OAMP solver.

Replace the LASSO solver with $L$-layer unrolled ISTA. Each layer is differentiable (soft thresholding is differentiable a.e.). The gradient $\nabla_\mathbf{W}\mathcal{L}$ is computed via backpropagation through the $L$ layers. $\blacksquare$

The set of achievable $(R, \mathcal{J})$ pairs is $\{(R(\mathbf{W}), \mathcal{J}(\mathbf{W})) : \|\mathbf{W}\|_F^2 \leq P_t\}$. Since $R$ is concave and $\mathcal{J}$ is convex, the set $\{(R, -\mathcal{J}) : \exists\, \mathbf{W}\}$ is a concave set (superlevel set of a concave function in the epigraph).

The Pareto frontier is the upper boundary of the achievable region in the $(R, -\mathcal{J})$ plane. The upper boundary of a concave set is a concave curve.

Concavity means that time-sharing between two operating points achieves any point on the line segment between them. The interior of the frontier is achievable via randomised waveform design. $\blacksquare$

With 4 users and ZF precoding: $R = 4 \times \log_2(1 + 0.8 \times \text{SNR} \times 64/4)$. At $\text{SNR} = 20$ dB ($= 100$): $R = 4 \times \log_2(1 + 0.8 \times 100 \times 16) = 4 \times \log_2(1281) \approx 4 \times 10.3 = 41.2$ bps/Hz.

64 pilot subcarriers with $0.2P_t/64$ power each. Effective sensing SNR: $0.2 \times 100 \times 64/64 = 20$ (13 dB). NMSE $\approx K_s/(N_{\mathrm{pilot}} \cdot N_a \cdot \mathrm{SNR}_s) = 10/(64 \times 64 \times 20) \approx 1.2 \times 10^{-4}$ ($-39$ dB).

Using only 64 pilot subcarriers: $\mathrm{CRB}(\tau) = 1/(\text{SNR}_{s} \sum_k (2\pi f_k)^2) \approx 1/(20 \times 64 \times (2\pi W)^2/12)$. For $W = 400$ MHz: $\mathrm{CRB}(\tau) \approx 2.5 \times 10^{-20}$ s$^2$, giving $\sqrt{\mathrm{CRB}} \approx 0.16$ ns (range accuracy $\sim 4.8$ cm). $\blacksquare$

For a ring graph, the adjacency matrix has eigenvalues $\mu_k = 2\cos(2\pi k/N)$. The doubly stochastic averaging matrix has eigenvalues $\mu_k/(N\text{ max degree}) = \cos(2\pi k/N)$. The spectral gap is $\gamma = 1 - |\mu_1| = 1 - \cos(2\pi/N)$. For large $N$: $\gamma \approx 2\pi^2/N^2$.

We need $(1 - \gamma)^t < 10^{-0.1}$ (1 dB). $t > -0.1\ln 10 / \ln(1 - \gamma) \approx 0.23/\gamma = 0.23N^2/(2\pi^2)$. For $N = 8$: $t > 0.23 \times 64/19.7 \approx 0.75$, so $t = 1$. For $N = 20$: $t > 0.23 \times 400/19.7 \approx 4.7$, so $t = 5$. For $N = 100$: $t > 0.23 \times 10000/19.7 \approx 117$. $\blacksquare$

Under RIP, the LASSO NMSE satisfies $\mathrm{NMSE} \leq C K \sigma^2 / (\sum_k |W_k|^2 g_k)$ where $g_k$ is the "information gain" from frequency $k$.

$\mathrm{NMSE}(\{|W_k|^2\}) \leq C K \sigma^2 / (\sum_k p_k g_k)$ where $p_k = |W_k|^2$. This is a convex function of $\{p_k\}$ (reciprocal of a linear function is convex on the positive orthant).

Convexity means the optimal power allocation can be found efficiently via convex optimisation. Combined with the concave rate function, the rate-NMSE tradeoff is a standard multi-objective convex problem. $\blacksquare$

$\min_{\{\boldsymbol{\sigma}_g\}, \boldsymbol{\sigma}} \sum_g [f_g(\boldsymbol{\sigma}_g) + R_g(\boldsymbol{\sigma}_g)/N]$ s.t. $\boldsymbol{\sigma}_g = \boldsymbol{\sigma}$.

Local: $\boldsymbol{\sigma}_g^{(t+1)} = \operatorname{prox}_{R_g/(N\rho)}((\mathbf{A}_{g}^{H}\mathbf{A}_{g} + \rho\mathbf{I})^{-1}(\mathbf{A}_{g}^{H}\mathbf{y}_{g} + \rho(\boldsymbol{\sigma}^{(t)} - \mathbf{u}_g^{(t)})))$.

Global: $\boldsymbol{\sigma}^{(t+1)} = \frac{1}{N}\sum_g(\boldsymbol{\sigma}_g^{(t+1)} + \mathbf{u}_g^{(t)})$ (simple averaging since regularisation is in local steps).

Each $f_g + R_g/N$ is convex. The consensus ADMM converges to the minimiser of $\sum_g (f_g + R_g/N)$ subject to consensus, which is equivalent to $\sum_g f_g + \sum_g R_g/N = \sum_g f_g + R_{\mathrm{avg}}$. $\blacksquare$

1. Generate sensing matrix: $\mathbf{A}(\mathbf{p}) = \mathrm{diag}(\sqrt{\mathbf{p}}) \otimes \mathbf{F}_R^H$.
2. Generate measurements: $\mathbf{y} = \mathbf{A}(\mathbf{p})\mathbf{c} + \mathbf{w}$.
3. Reconstruct: $\hat{\mathbf{c}} = \mathrm{OAMP}_\theta(\mathbf{y}, \mathbf{A}(\mathbf{p}))$ (10 layers).
4. Compute rate: $R = \sum_{k=1}^{K_u} \log_2(1 + \mathrm{SINR}_k(\mathbf{p}))$.

$\mathcal{L} = -\mu R + (1-\mu)\|\hat{\mathbf{c}} - \mathbf{c}\|^2/\|\mathbf{c}\|^2$.

$\nabla_{\mathbf{p}}\mathcal{L}$ and $\nabla_\theta\mathcal{L}$ via backpropagation through the 10 OAMP layers and the rate expression. The soft thresholding and Onsager correction are differentiable.

After each Adam step: $\mathbf{p} \leftarrow \mathrm{Proj}_{\Delta_{P_t}}(\mathbf{p})$ (project onto the scaled simplex $\{p_k \geq 0, \sum p_k \leq P_t\}$). Alternatively, parameterise $p_k = P_t \cdot \mathrm{softmax}(\mathbf{z})_k$. $\blacksquare$

Local step: $\boldsymbol{\sigma}_g^{(t+1)} = (\mathbf{A}_{g}^{H}\mathbf{A}_{g} + \rho\mathbf{I})^{-1}(\mathbf{A}_{g}^{H}\mathbf{y}_{g} + \rho(\boldsymbol{\sigma}^{(t)} - \mathbf{u}_g^{(t)}))$.

Global step: $\boldsymbol{\sigma}^{(t+1)} = \mathcal{D}_g(\bar{\boldsymbol{\sigma}}^{(t+1)} + \bar{\mathbf{u}}^{(t)})$ where $\mathcal{D}_g$ is the denoiser for node $g$ (or an average denoiser).

ADMM convergence with PnP requires the denoiser to be the proximal operator of some convex function, or at minimum, firmly non-expansive: $\|\mathcal{D}(\mathbf{x}) - \mathcal{D}(\mathbf{y})\|^2 \leq \langle \mathbf{x} - \mathbf{y}, \mathcal{D}(\mathbf{x}) - \mathcal{D}(\mathbf{y})\rangle$.

If one node's denoiser is expansive ($\|\mathcal{D}(\mathbf{x})\| > \|\mathbf{x}\|$ for some inputs), the ADMM iteration may oscillate or diverge. The expansive node amplifies errors, which propagate through the consensus to all other nodes. Remedy: spectral normalisation of all denoisers to enforce Lipschitz constant $\leq 1$. $\blacksquare$