Exercises

$T_1 = \exp(-1.2 \times 0.15) = \exp(-0.18) = 0.835$.

$T_2 = T_1 \cdot \exp(-0 \times 3) = 0.835$.

$T_3 = 0.835 \cdot \exp(-1.2 \times 0.15) = 0.835 \times 0.835 = 0.697$. In dB: $10\log_{10}(0.697) = -1.57$ dB total power attenuation. $\square$

$\gamma(0.5) = [\sin(\pi/2), \cos(\pi/2), \sin(\pi), \cos(\pi), \sin(2\pi), \cos(2\pi)]$ $= [1, 0, 0, -1, 0, 1]$.

Each scalar produces $2L = 6$ components. For $\mathbf{x} \in \mathbb{R}^3$: dimension $= 3 \times 2L = 18$. $\square$

$T_1 = 1$, $T_2 = 1 \times (1 - 0.2) = 0.8$, $T_3 = 0.8 \times (1 - 0.7) = 0.24$.

$w_1 = 1 \times 0.2 = 0.2$, $w_2 = 0.8 \times 0.7 = 0.56$, $w_3 = 0.24 \times 0.5 = 0.12$.

$\sum_i w_i = 0.2 + 0.56 + 0.12 = 0.88 < 1$. The remaining $0.12$ is the transmittance past the last sample: $T_4 = 0.24 \times 0.5 = 0.12$. $\square$

$\mathcal{F} = 2^2 / (0.005 \times 10) = 4 / 0.05 = 80 \gg 1$. Ray optics is valid.

$\mathcal{F} = 0.1^2 / (0.06 \times 5) = 0.01 / 0.3 = 0.033 \ll 1$. Wave optics required --- diffraction dominates. $\square$

$\delta_1 = 1.0$, $\delta_2 = 0.2$, $\delta_3 = 1.8$, $\delta_4 = 0$ (last). $\alpha_1 = 1 - e^{-0.1} = 0.095$, $\alpha_2 = 1 - e^{-1.0} = 0.632$, $\alpha_3 = 1 - e^{-9.0} \approx 1.0$, $\alpha_4 = 0$.

$T_1 = 1$, $T_2 = 1 - 0.095 = 0.905$, $T_3 = 0.905 \times (1 - 0.632) = 0.333$, $T_4 \approx 0$.

Sample 2: $T_2 \alpha_2 |\rho_2| = 0.905 \times 0.632 \times 0.8 = 0.457$. Sample 3: $T_3 \alpha_3 |\rho_3| = 0.333 \times 1.0 \times 0.3 = 0.100$. Sample 2 (wall front face at 3 m) dominates. $\square$

The model overfits after $\sim 50$k iterations. With 1,000 measurements and a large MLP ($\sim 530$k parameters), the model memorises training data.

(1) Early stopping at 50k iterations. (2) Increase weight decay $\lambda$ from $10^{-4}$ to $10^{-3}$. (3) Reduce MLP size to 4 layers $\times$ 128 units ($\sim 130$k parameters). Additional options: hash encoding with smaller table, data augmentation. $\square$

$\kappa = 2\pi \times 5 \times 10^9 / (3 \times 10^8) = 104.72$ rad/m. $\Delta\phi = 2 \times 104.72 \times 0.015 = 3.14$ rad $\approx \pi$.

$\hat{S} = 0.5 \times 0.5 \times e^{j\phi_1} + 0.5 \times 0.5 \times e^{j(\phi_1 + \pi)}$. $|\hat{S}| = |0.25(1 + e^{j\pi})| = |0.25 \times 0| = 0$. Destructive interference: $|\hat{S}|^2 = 0$.

$\hat{C} = 0.5 \times 0.5 \times 0.5 + 0.5 \times 0.5 \times 0.5 = 0.25$. Incoherent sum always adds constructively.

The 1.5 cm spacing is exactly $\lambda/2$ at 5 GHz, causing perfect cancellation in coherent rendering. Ignoring phase (optical-style) misses this physical effect entirely. $\square$

$\ell_{\mathrm{LOS}} = 8.0$ m. Delay: $\tau_0 = 8.0/c = 26.7$ ns.

North wall ($y = 8$): image Tx at $(1, 12)$. $\ell = \sqrt{64 + 64} = 11.31$ m. Extra delay: 11.1 ns. South wall ($y = 0$): image Tx at $(1, -4)$. $\ell = \sqrt{64 + 64} = 11.31$ m. Extra delay: 11.1 ns. West wall ($x = 0$): image Tx at $(-1, 4)$. $\ell = 10.0$ m. Extra delay: 6.7 ns. East wall ($x = 10$): image Tx at $(19, 4)$. $\ell = 10.0$ m. Extra delay: 6.7 ns.

A single-ray NeRF$^2$ misses all 4 reflections. WiNeRT traces these secondary rays to capture multipath energy. $\square$

$16 \times 2^{19} \times 2 = 16{,}777{,}216$ parameters.

$16{,}777{,}216 \times 4$ bytes $= 67.1$ MB.

Input: $16 \times 2 = 32$. MLP: $32 \times 256 + 7 \times 256^2 + 256 = 467{,}200$ parameters ($\sim 1.8$ MB). Hash table is $36\times$ larger in memory but enables $10\times$--$50\times$ faster convergence. $\square$

$P_1 = 10^{-3}$ mW, $\hat{P}_1 = 1.585 \times 10^{-3}$ mW. $P_2 = 10^{-8}$ mW, $\hat{P}_2 = 1.585 \times 10^{-8}$ mW. Error 1: $(0.585 \times 10^{-3})^2 = 3.42 \times 10^{-7}$. Error 2: $(0.585 \times 10^{-8})^2 = 3.42 \times 10^{-17}$. MSE $\approx 1.71 \times 10^{-7}$ --- completely dominated by measurement 1.

Error 1: $(2)^2 = 4$. Error 2: $(2)^2 = 4$. MSE $= 4$ dB$^2$. Both contribute equally. This is why NeRF$^2$ uses dB-domain loss. $\square$

$r = 10 \times \tan(2°) = 0.349$ m $\approx 35$ cm.

The cross-section spans $\sim 6\lambda$. Point sampling misses this spatial averaging.

Mip-NeRF's integrated positional encoding naturally models antenna beamwidth averaging, relevant for directional mmWave antennas. $\square$

$\lambda = 3.9$ mm. $f_D = 2 \times 15 \cos(30°) / 0.0039 \approx 6.66$ kHz.

Building has $f_D = 0$. DART's velocity MLP assigns $\mathbf{v} \approx (15,0,0)$ at car locations. Incorrect velocity produces Doppler mismatch in the loss, driving correct assignment via gradients. $\square$

Range: $\Delta R = c/(2B) = 0.3$ m. Cross-range: $\Delta x = \lambda R / (2 L_{\mathrm{sa}}) = 0.75$ m.

Grating lobes produce ghost targets. ISAR-NeRF's spectral bias suppresses high-frequency artifacts; TV regulariser further cleans the image. Achieves 5--10 dB sidelobe suppression over back-projection. $\square$

$\mathcal{L} = \sum_{f \in \{2.4, 5.8\}} \sum_{\mathcal{D}_f} |\hat{P}^{(\mathrm{dB})}(f) - P^{(\mathrm{dB})}(f)|^2 + \lambda_1 \|\theta\|_2^2 + \lambda_2 \mathrm{ReLU}(-\alpha_0) + \lambda_3 \mathrm{ReLU}(-\alpha_1)$.

Two unknowns need two equations. Single frequency gives one constraint per ray: $\alpha_0, \alpha_1$ are not separable. $\square$

(1) $T = 1$ (no attenuation). (2) $\rho(\mathbf{x}) = c(\mathbf{x})$ (isotropic). (3) $\sigma = \sum_q \delta(\mathbf{x} - \mathbf{p}_q)$.

$\hat{S} = \sum_q c_q e^{-j\kappa(\|\mathbf{p}_q - \mathbf{s}\| + \|\mathbf{p}_q - \mathbf{r}\|)} = \sum_q c_q [\mathbf{A}]_{(s,r),q}$, which is the Born model $\mathbf{y} = \mathbf{A}\mathbf{c}$. $\square$

MLPs learn low-frequency components first (Rahaman et al., 2019). The target has frequency $50$ Hz, beyond the MLP's effective bandwidth.

Max frequency $2^7\pi = 128\pi > 100\pi$. The MLP can express $g$ via the encoding features.

Max frequency $4\pi \approx 12.6 \ll 100\pi$. Severe under-resolution. $\square$

$\kappa = 104.72$ rad/m. LOS: $e^{-j523.6}$. Reflected: $e^{-j837.8}$.

Over random phase realisations, $|\hat{S}|^2 \approx 0.3^2 + 0.075^2 = 0.096$, i.e., $-10.2$ dBm. The reflected path adds $\sim 0.3$ dB. $\square$

6 APs, 80 MHz bandwidth, 2,000 Rx locations = 12,000 CSI measurements.

Hash grid ($L=16$, $T=2^{19}$, $F=2$) + geometry MLP (4 layers $\times$ 128) + signal MLP (3 layers $\times$ 64). $\sim$17M parameters total.

Loss: $\|\hat{H} - H\|_F^2 + 0.01\|\sigma\|_1$. Adam, lr $10^{-3}$ cosine to $10^{-5}$. 100k iterations, $\sim$30 min on A100.

Threshold $\sigma$, skeletonise. Metrics: wall RMSE, IoU, Hausdorff distance. Baselines: ray-tracing inversion, back-projection, RSS-only NeRF$^2$. $\square$

Conditional hash encoding with scene embedding $\mathbf{z}_k \in \mathbb{R}^{64}$. Pre-train on all 50 environments jointly with RSS MSE loss.

Shared: hash encoder + 4-layer geometry MLP. Per-scene: embedding + 2-layer signal MLP. Shared: $\sim$17M params; per-scene: $\sim$30k.

Freeze backbone, learn new $\mathbf{z}$ + signal MLP. Expected: $\sim$5 dB RMSE after 5k iterations (vs $\sim$8 dB from scratch). $\square$

Born ignores attenuation: full $|\rho| = 0.95$ contribution from the metal reflector.

Round-trip wall transmittance: $T = e^{-2 \times 2 \times 0.3} = e^{-1.2} = 0.301$. Two-way: $0.301^2 = 0.091$. Metal contribution attenuated by $-10.4$ dB compared to Born.

Born overestimates the metal reflector by $10.4$ dB. Acceptable only for free-space scenes without intervening materials. $\square$