Exercises

(1) Model mismatch (Born vs multipath): $\sim 5$ dB. Multipath creates ghost targets not in the model. (2) Off-grid targets (continuous positions vs grid): $\sim 2$ dB. Basis mismatch in sparse recovery. (3) Phase noise (oscillator imperfections): $\sim 1$ dB. Broadens the PSF slightly. (4) Mutual coupling (unmodelled antenna interaction): $\sim 2$ dB. Distorts the array pattern. (5) Clutter (furniture, walls not modelled as targets): $\sim 3$ dB. Elevates the noise floor. Total: $\sim 13$ dB, consistent with the typical 10--15 dB sim-to-real gap. $\square$

A walking person shifts by 0.15 m/frame. This is a coherent rigid-body motion best modelled by optical flow: $\boldsymbol{\gamma}_t(\mathbf{x}) \approx \boldsymbol{\gamma}_{t-1}(\mathbf{x} - \mathbf{v}\Delta t)$ with $\mathbf{v} = 1.5$ m/s.

Smoothness: inappropriate because the change is spatially localised. Sparse innovation: partially appropriate (few pixels change) but ignores the motion structure. Optical flow: most appropriate because it directly models the translational displacement. $\square$

Primitives: 4 walls + 1 table + 8 chairs + 1 screen + 1 pillar = 15 primitives. Parameters: $15 \times 7 = 105$.

Voxels: $(8/0.05) \times (6/0.05) \times (3/0.05) = 160 \times 120 \times 60 = 1{,}152{,}000$.

$\rho = 1{,}152{,}000 / 105 \approx 10{,}971\times$. The primitive representation is nearly four orders of magnitude more compact. $\square$

(1) What are the "existing methods" compared? Are they fairly tuned, or default parameters? (2) Is the 8 dB consistent across different scenes, or driven by easy cases? Are confidence intervals reported? (3) What is the train/test split? Is there scene overlap? (4) Is RadarScenes a standard benchmark, or curated by the authors? Is it publicly available? $\square$

Pre-train on 50k simulations. Fine-tune on 10 real scenes with reduced learning rate ($0.1\times$) for 1,000 iterations. Risk: overfitting. Mitigate: early stopping, weight decay. Expected: 5--7 dB recovery.

Domain discriminator distinguishing sim from real features. Train imaging network to fool it. Expected: 3--5 dB recovery (unstable with few real examples).

Measurement consistency: hold out 20% of measurements per scene, reconstruct from 80%, predict held-out. No labels needed. Expected: 7--10 dB recovery (best).

Strategy 3 (self-supervised) likely best because it avoids label bottleneck and adversarial instability. Combining pre-train $\to$ self-supervised fine-tune may be optimal. $\square$

Input: $(\gamma(\mathbf{x}), \gamma(t)) \to (\sigma, \rho)$ where $\gamma(\cdot)$ is positional encoding, with $L_t = 6$ frequency bands for time. Alternatively, use a per-frame latent code $\mathbf{z}_t$ (auto-decoder).

Option A: single MLP conditioned on $t$ (smooth changes). Option B: deformation field $\mathbf{d}(\mathbf{x}, t)$ warping a canonical frame (better for rigid motion).

3D RF-NeRF: $V$ viewpoints at one time. 4D: $V$ viewpoints at $T$ time steps $= VT$ measurement sets. Typical: $T = 50$, $V = 10$ $\to$ 500 CSI snapshots.

(1) $50\times$ more data. (2) Temporal aliasing if motion exceeds frame rate. (3) $50\times$ training cost. (4) Ambiguity: new object vs moved object. $\square$

(1) Collect RF measurements in the new building. (2) Encode into shared embedding: $\mathbf{z} = f_{\mathrm{RF}}(\mathbf{y})$. (3) Condition decoder on embedding: $\hat{\boldsymbol{\gamma}} = g(\mathbf{z}, \mathbf{y})$. The embedding encodes scene-level features learned cross-modally.

The embedding captures scene types (open plan, partitioned, few/many scatterers) common across modalities. Even without optical images, the RF embedding provides a useful prior.

Novel scene types absent from pre-training may produce misleading embeddings. The model should report uncertainty (distance from nearest training embedding). $\square$

$P/\sigma^2 = 1000$. Key points: $\alpha = 0$: $R = 0$, $Q = 30$ dB. $\alpha = 0.1$: $R = 6.66$, $Q = 29.5$ dB. $\alpha = 0.5$: $R = 8.97$, $Q = 27.0$ dB. $\alpha = 0.9$: $R = 9.82$, $Q = 20.0$ dB. $\alpha = 1$: $R = 9.97$, $Q = -\infty$.

The frontier is convex. The "knee" at $\alpha \approx 0.1$--$0.3$: allocating 10--30% to communication costs only 0.5--3 dB imaging quality while providing 7--9 bits/s/Hz. $\square$

Campus surface area $\sim 500{,}000$ m$^2$. At 1 Gaussian per 10 m$^2$ surface: $50{,}000$ Gaussians. Memory: $50{,}000 \times 60$ bytes $= 3$ MB. Actually feasible for storage, but rendering against all is expensive for many queries.

Divide into $50 \times 50 = 2{,}500$ blocks of $10 \times 10$ m. Each building block: $\sim 2{,}000$ Gaussians. Active zone (100 m radius): $\sim 300$ blocks $\times 2{,}000 = 600{,}000$ Gaussians $= 36$ MB. Distant zone: path-loss model (no Gaussians). Stream blocks in/out as UEs move. Fits within 32 GB. $\square$

(1) Point targets. (2) Far field. (3) Known $\mathbf{A}$. (4) Sparse scene ($K$ targets). (5) AWGN noise. (6) Known $\sigma^2$.

(7) $\boldsymbol{\gamma} \in \mathbb{R}^N$: real-valued reflectivity (no phase). (8) On-grid targets. (9) Born approximation (linear model). (10) Stationary scene. (11) Narrowband (single $\mathbf{A}$). (12) Perfect calibration. $\square$

Fair as a lower bound but too weak for meaningful comparison. Keep but add stronger methods.

Unfair: $\lambda = 0.1$ is arbitrary. Performance is highly sensitive to $\lambda$. Fix: 5-fold cross-validation.

Potentially unfair: 50 may be insufficient. Fix: run to convergence ($\|\boldsymbol{\gamma}^{(k+1)} - \boldsymbol{\gamma}^{(k)}\| < 10^{-6}$) or max 500 iterations.

Add: (4) ADMM (different optimiser), (5) LISTA (learned unrolling), (6) U-Net (learned non-physics). This spans classical to learned. $\square$

Yes, for sparse scenes. The Rayleigh limit assumes no prior; with $K$ point targets, the CRB on separation can be much smaller at sufficient SNR.

Occurs when the prior dominates: (1) low SNR ($< 10$ dB); (2) scene types not in training; (3) two targets closer than the limit may merge or split incorrectly.

(1) Resolution chart: vary target separation, plot $P_d$. (2) Noise sensitivity: genuine degrades with SNR; hallucination does not. (3) OOD test: test on extended targets (not in training). (4) Uncertainty map: high uncertainty near resolution limit $\to$ prior-dominated. $\square$

Range resolution: $\Delta r = c/(2W) = 3 \times 10^8 / (2 \times 200 \times 10^6) = 0.75$ m. The CRB for range estimation of a single target: $\sigma_r^2 \geq \Delta r^2 / (8\pi^2 \text{SNR})$. At SNR $= 20$ dB (100): $\sigma_r \geq 0.75 / (8\pi^2 \times 100)^{1/2} \approx 0.0084$ m $= 8.4$ mm.

Array aperture: $D = (N_r - 1) \times \lambda/2 = 15 \times 0.0107/2 = 0.080$ m. Cross-range resolution: $\Delta x = \lambda R / D = 0.0107 \times 10 / 0.080 = 1.34$ m. CRB: $\sigma_x \geq 1.34 / (8\pi^2 \times 100)^{1/2} \approx 0.015$ m $= 1.5$ cm.

Range resolution is limited by bandwidth (0.75 m); cross-range by aperture (1.34 m). The CRBs show that at 20 dB SNR, we can localise a single target to cm accuracy -- much finer than the resolution cell. For multiple targets within one resolution cell, the CRB degrades and requires super-resolution. $\square$

The measurement response of box $k$ at measurement point $m$ (wavenumber $\mathbf{k}_m$): $[\mathbf{A}[\mathcal{P}_k]]_m = \alpha_k \, e^{-j\mathbf{k}_m \cdot \mathbf{p}_k} \prod_{d=1}^{3} \text{sinc}(k_{m,d} s_{k,d})$ where the sinc arises from the Fourier transform of a rectangular function.

Loss: $\mathcal{L} = \|\mathbf{y} - \sum_k \mathbf{a}_k\|^2$ where $\mathbf{a}_k = \mathbf{A}[\mathcal{P}_k]$. Residual: $\mathbf{r} = \mathbf{y} - \sum_k \mathbf{a}_k$. $\frac{\partial \mathcal{L}}{\partial \mathbf{p}_k} = -2\text{Re}\left[\sum_m r_m^* \frac{\partial a_{k,m}}{\partial \mathbf{p}_k}\right]$. $\frac{\partial a_{k,m}}{\partial \mathbf{p}_k} = -j\mathbf{k}_m \cdot a_{k,m}$. Therefore: $\frac{\partial \mathcal{L}}{\partial \mathbf{p}_k} = 2\text{Im}\left[\sum_m r_m^* \mathbf{k}_m a_{k,m}\right]$. $\square$

Model: $\mathbf{y} = \mathbf{A}\boldsymbol{\gamma} + \mathbf{w}$ with $\boldsymbol{\gamma} \sim \mathcal{CN}(\mathbf{0}, P/N \cdot \mathbf{I}_N)$ and $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I}_M)$. $I(\boldsymbol{\gamma}; \mathbf{y}) = h(\mathbf{y}) - h(\mathbf{y}|\boldsymbol{\gamma}) = \log\det(\mathbf{R}_y) - \log\det(\sigma^2 \mathbf{I})$.

$\mathbf{R}_y = P/N \cdot \mathbf{A}\mathbf{A}^H + \sigma^2 \mathbf{I}$. Using SVD $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$: $\mathbf{R}_y = \mathbf{U}(P/N \cdot \boldsymbol{\Sigma}\boldsymbol{\Sigma}^H + \sigma^2 \mathbf{I})\mathbf{U}^H$.

$C = \log_2\det(\mathbf{I} + \text{SNR}/N \cdot \boldsymbol{\Sigma}\boldsymbol{\Sigma}^H) = \sum_{k=1}^{r} \log_2(1 + \text{SNR} \cdot \sigma_k^2 / N)$ where $\text{SNR} = P/\sigma^2$. This is identical to the MIMO capacity formula, confirming the deep connection. $\square$

MC Dropout: run the network $M = 10$ times with random dropout at inference. Per-pixel variance $\sigma^2(\mathbf{x}) = \text{Var}[\hat{\gamma}_m(\mathbf{x})]$ estimates epistemic uncertainty. Overhead: $10\times$ inference ($\sim 100$ ms).

$R = |\{\mathbf{x} : \sigma^2(\mathbf{x}) < \tau\}| / N_{\text{pixels}}$. If $R < 0.8$, the reconstruction is unreliable. Threshold $\tau$ calibrated on validation set.

Conformal prediction: conformity score $s = \|\mathbf{y} - \mathbf{A}\hat{\boldsymbol{\gamma}}\|^2 / \sigma_n^2$. If $s > q_{1-\alpha}$ (calibrated quantile), flag as failure. Distribution-free guarantee: failure rate $\leq \alpha$. Total overhead: $\sim 100$ ms, acceptable for 10 Hz radar. $\square$

(1) Inverse crime: same forward model for training and testing; 30 dB PSNR is inflated. (2) Sim-to-real gap: no quantitative real-data metrics. (3) Single real measurement: proves nothing; could be cherry-picked. (4) No ground truth for real data. (5) Point model: real scenes have extended targets.

(1) Different forward models for train/test (ray tracing for test). (2) Collect $\geq 10$ real scenes with ground truth (laser scan). (3) Report quantitative real-data metrics. (4) Fine-tune on real examples. (5) Use extended target models in training. (6) Add ray-tracing data alongside point models. $\square$

Comparing 1 vs 2: if wall removal contributes $> 3$ dB, pre-processing dominates. 1 vs 3: complex convolutions capture phase (expected for coherent imaging). 1 vs 5: value of physics-informed architecture. Variant 6: lower bound. $\square$

(1) Signal processing: resolution limits, sparse recovery. Papers: Candes et al. (2014, super-resolution); Potter et al. (2010, CS-SAR). Perspective: guarantees, worst-case. (2) Machine learning: architectures, generalisation. Papers: Monga et al. (2021, unrolling); Ongie et al. (2020, deep inverse). Perspective: data-driven, empirical. (3) Wireless communications: channel models, ISAC. Papers: Liu et al. (2022, ISAC survey); Caire (2026, illumination model). Perspective: system-level, standards. (4) Computational imaging: physics-based reconstruction. Papers: Mildenhall et al. (2021, NeRF); Tulsiani et al. (2017, primitives). Perspective: novel view synthesis.

Missing any community risks reinventing techniques, wrong baselines, or ignoring constraints. Reviewers at TSP/JSAC span all four communities. $\square$

Problem: Current RF imaging methods use millions of parameters (voxels, neural fields) to represent scenes with inherently low-dimensional geometric structure, leading to high computational cost and poor physical interpretability.

Research questions: (Q1) Can indoor RF scenes be decomposed into $\leq 30$ geometric primitives with reconstruction quality matching voxel-based methods? (Q2) What is the optimal primitive dictionary (box, cylinder, superquadric) for indoor RF imaging at sub-6 GHz and mmWave? (Q3) Can BIM-initialised primitive representations improve reconstruction accuracy and reduce the data requirement?

Approach: Year 1: Differentiable primitive rendering for RF. Greedy decomposition algorithm. Validation on 2D simulated scenes. Year 2: Extension to 3D. Neural primitive predictor from matched-filter images. Real-data validation on 5 rooms. Year 3: BIM integration. Multi-frequency primitive sharing. Campus-scale demonstration.

Expected contributions: (1) First primitive-based reconstruction for RF imaging with $>10{,}000\times$ compression. (2) Differentiable RF primitive rendering library (open source). (3) BIM-integrated digital twin pipeline. (4) Dataset: 20 rooms with ground truth + BIM.

Timeline: M1-M6: literature, 2D implementation. M7-M12: first paper (2D results). M13-M18: 3D extension. M19-M24: real data, second paper. M25-M30: BIM integration. M31-M36: thesis, third paper. $\square$