Exercises

TI IWR6843ISK evaluation board: \$200. Provides 77 GHz FMCW radar with$N_t = 3$,$N_r = 4$ MIMO, 4 GHz bandwidth. Remaining budget for USB cables, mounting hardware, and corner reflectors.

Range resolution: $c/(2W) = 3.75$ cm. Virtual array: 12 elements at $\lambda/2 = 1.95$ mm spacing. Angular resolution: $\sim 14^\circ$ (12-element ULA). Maximum range: $\sim 10$ m (limited by transmit power and small antenna gain). Sufficient for indoor imaging demonstrations. $\square$

$\mathrm{PSNR} = 10\log_{10}(1/0.01) = 20$ dB. This is moderate quality.

SSIM $= 0.85$ is decent structural similarity. They may disagree because PSNR counts all errors equally (including background noise), while SSIM focuses on structural features. A reconstruction that is noisy everywhere but preserves edges and targets will have low PSNR but high SSIM. Conversely, a smooth but blurred reconstruction has high PSNR but low SSIM. $\square$

(a) Point scatterer: fast ($\mu$s per realisation), generates millions of training samples. Sufficient for sparse targets (Born approximation valid). But use ray tracing for the test set to avoid the inverse crime. (b) Ray tracing: captures wall reflections and furniture multipath. Moderate speed (ms per scene). Sionna with a 3D room model provides realistic channels. (c) FDTD: diffraction requires solving Maxwell's equations; geometric optics (ray tracing) cannot model it accurately. Slow but necessary for this scenario. $\square$

$P_n = 1.38 \times 10^{-23} \times 290 \times 10^8 \times 10^{0.8} = 4 \times 10^{-13} \times 6.31 = 2.52 \times 10^{-12}$ W $= -86$ dBm.

$\text{SNR} = P_s - P_n = -90 - (-86) = -4$ dB. The target is below the noise floor; coherent integration or CS reconstruction is needed for detection. $\square$

Forward: $\mathbf{y} = \mathbf{F}\mathbf{c} + \mathbf{w}$ (DFT on $64 \times 64$ grid). Reconstruction: $\hat{\mathbf{c}} = \arg\min \|\mathbf{F}\mathbf{c} - \mathbf{y}\|^2 + \lambda\|\mathbf{c}\|_1$.

Generate data on a $256 \times 256$ grid using the exact Green's function (near-field model with range-dependent phase). Add mutual coupling ($\mathbf{C}$ matrix with 5% coupling coefficient). Reconstruct on the $64 \times 64$ grid using the DFT model.

Crime PSNR: $\sim 35$ dB. Honest PSNR: $\sim 22$--28 dB (7--13 dB drop). The deep network may degrade more than ISTA because it overfit to the crime model during training. Solution: retrain the network on data from the honest forward model. $\square$

$N_{\mathrm{MC}} \geq (1.96 \times 3 / 0.5)^2 = (11.76)^2 = 138.3$. Round up: $N_{\mathrm{MC}} = 140$.

Per algorithm: $140 \times 5$ SNR levels $= 700$ trials. MF: $700 \times 10 = 7{,}000$ s ($\sim 2$ h). LASSO: $700 \times 60 = 42{,}000$ s ($\sim 12$ h). ISTA-Net: $700 \times 5 = 3{,}500$ s ($\sim 1$ h). U-Net: $700 \times 2 = 1{,}400$ s ($\sim 0.4$ h). Total: $\sim 15$ hours. Parallelisable across the 5 SNR levels: $\sim 3$ hours on 5 GPUs. $\square$

$\bar{d} = 28.3 - 27.5 = 0.8$ dB. Assume $s_d \approx \sqrt{s_1^2 + s_2^2 - 2\rho s_1 s_2} \approx 1.5$ dB (with $\rho = 0.7$ correlation). $t = \bar{d} \sqrt{N} / s_d = 0.8 \times 10 / 1.5 = 5.33$. $p < 0.001$: statistically significant.

$d = |\bar{d}| / s_d = 0.8 / 1.5 = 0.53$: medium effect size.

A 0.8 dB improvement is statistically significant but modest in practice (barely perceptible visually). The medium effect size suggests the improvement is real but the practical relevance depends on the application: for target detection near the noise floor, 0.8 dB can be meaningful; for image display, it is negligible. $\square$

Source: 500 ShapeNet models from 5 categories (chair, table, person, vehicle, box). Voxelise each onto a $64 \times 64$ grid. 10 random orientations per model $= 5{,}000$ scenes.

$f_0 = 77$ GHz, $W = 4$ GHz, $K = 64$ subcarriers. UPA with $N_t = 4$, $N_r = 8$, virtual array $= 32$ elements. SNR range: $[5, 30]$ dB (uniform).

Training/val: Born model on $64 \times 64$ grid (fast, allows large dataset). Test: Born model on $256 \times 256$ grid with off-grid targets and mutual coupling (avoids inverse crime). Split: 4000 train, 500 val, 500 test. $\square$

(1) LASSO is not tuned: the default $\lambda$ is likely suboptimal. (2) No validation set mentioned. (3) DL has 10,000 samples for learning; LASSO uses none. (4) Computational cost not reported. (5) PSNR alone may not capture task-relevant quality.

(1) Tune $\lambda$ via 5-fold cross-validation. (2) Fixed train/val/test split. (3) Report PSNR, SSIM, NMSE, and $P_D$. (4) Report inference time and memory. (5) Include ISTA-Net as an intermediate baseline. $\square$

All 5 targets found. $d_C = \frac{1}{5}(4 + 9 + 1 + 16 + 4) + \frac{1}{5}(4 + 9 + 1 + 16 + 4) = \frac{34}{5} + \frac{34}{5} = 13.6$ cm$^2$ (assuming symmetric nearest-neighbour).

4 found, 1 missed. Pred-to-truth: $\frac{1}{4}(1 + 1 + 1 + 1) = 1$ cm$^2$. Truth-to-pred: $\frac{1}{5}(1 + 1 + 1 + 1 + D_{\mathrm{miss}}^2)$ where $D_{\mathrm{miss}}$ is the distance from the missed target to the nearest detected target. If $D_{\mathrm{miss}} = 20$ cm, then $\frac{1}{5}(4 + 400) = 80.8$ cm$^2$, total $= 81.8$ cm$^2$.

Algorithm A is better by Chamfer distance despite larger individual errors, because it detects all targets. The missed target in B incurs a large penalty. This illustrates that Chamfer distance rewards coverage (detecting all targets) over per-target precision. $\square$

Place the corner reflector on boresight at $R = 2$ m. Measure the complex response $h_m$ for each virtual channel $m$. Expected phase: $\phi_m^{\mathrm{exp}} = -4\pi f_0 \|\mathbf{a}_m - \mathbf{p}_{\mathrm{ref}}\|/c$. Calibration vector: $c_m = e^{j(\phi_m^{\mathrm{exp}} - \angle h_m)}$.

If calibration reduces phase error from $15^\circ$ to $3^\circ$ RMS residual: sidelobe degradation is negligible for 32 elements. Without calibration ($15^\circ$ RMS): sidelobes degrade by $\sim 3$ dB, significant for fine angular imaging. $\square$

120 dB dynamic range: all 4 targets visible. The $-60$ dB target has a margin of 60 dB above the noise floor.

45 dB dynamic range: targets at $0$ and $-20$ dB are visible. Target at $-40$ dB is marginal. Target at $-60$ dB is invisible.

For CS-SAR: the VNA data supports recovery of all 4 targets via sparsity; the TI data supports only 2--3. For learned methods: training on VNA data produces networks that expect high dynamic range; deploying on TI hardware causes performance degradation. Train (or fine-tune) on data matching the deployment hardware. $\square$

1. Inverse crime (simulation uses same model): test by using different forward/inverse models. If PSNR drops to $\sim 22$ dB, the crime accounts for 10 dB.
2. Calibration errors (uncorrected phase/amplitude): test by imaging a single known reflector.
3. Model mismatch (Born vs multipath reality): test by adding ray-traced multipath to simulation.
4. Hardware impairments (IQ imbalance, phase noise, ADC quantisation): test by adding these to simulation.
5. Environmental factors (clutter, interference): test by measuring in an anechoic chamber.

Start with (2) calibration (easiest to check), then (1) inverse crime, then (4) hardware, then (3) model mismatch, finally (5) environment. $\square$

Easy: 5 well-separated targets, $\text{SNR} = 25$ dB, no clutter. Medium: 15 targets (some closely spaced), $\text{SNR} = 15$ dB, static clutter. Hard: 30 targets (dense), $\text{SNR} = 5$ dB, dynamic clutter. Each: 100 test images.

Primary: NMSE (dB), $P_D$ at $P_{\mathrm{FA}} = 10^{-4}$, SSIM. Secondary: inference time, memory, parameters. Composite: $S = 0.4 P_D + 0.3(1 - \mathrm{NMSE/NMSE_{MF}}) + 0.2\,\mathrm{SSIM} + 0.1(1 - t/t_{\max})$.

MF (reference), LASSO (tuned), ISTA-Net-plus (16 layers), U-Net post-processing. All provided in benchmark package. Automated leaderboard with evaluation server. $\square$

RTK-GPS: $\sim 2$ cm horizontal, $\sim 5$ cm vertical. Cost: $5k. Limitation: measures point positions only, not continuous surfaces.

Accuracy: $\sim 3$--5 cm. Cost: $15k--$50k. Dense 3D point cloud. Limitation: vegetation; metal multipath in LiDAR.

Accuracy: $\sim 5$--10 cm. Cost: $2k. Textured 3D mesh. Limitation: fails on textureless surfaces.

For 15 cm SAR: all three adequate ($3\times$--$8\times$ finer). Best: LiDAR for scene, GPS for reference points. Photogrammetry as low-cost supplement. $\square$

Define a custom LinearPhysics subclass. Implement A(x) as the Kronecker-structured forward and A_adjoint(y) as the adjoint. Use PyTorch tensors for GPU acceleration.

Use DeepInverse's pretrained DRUNet (handles multiple noise levels via $\sigma$-conditioning). Alternatively, train a denoiser on RF reflectivity maps if domain-specific performance is needed.

Penalty $\rho$: start with $\rho = 0.1$ and tune on validation set. Denoiser noise level $\sigma_d = \sqrt{\sigma^2/\rho}$. Number of ADMM iterations: 20--50 (monitor convergence).

Compare against MF, LASSO, and U-Net. Use $N_{\mathrm{MC}} = 100$ trials, report PSNR, SSIM, $P_D$, and inference time. Use honest forward model (different grid) for test data. $\square$

MF to LASSO: $P_D$ increase from 0.72 to 0.88 ($+0.16$) at a lower $P_{\mathrm{FA}}$ ($10^{-4}$ vs $10^{-3}$), making the comparison even more favourable for LASSO. MF to U-Net: $P_D$ increase from 0.72 to 0.93 ($+0.21$).

$\bar{d} = 0.93 - 0.88 = 0.05$. $s_d \approx \sqrt{0.03^2 + 0.03^2 - 2 \times 0.7 \times 0.03^2} \approx 0.023$. $t = 0.05 \times 10 / 0.023 = 21.7$. $p \ll 0.001$: highly significant. Cohen's $d = 0.05/0.023 = 2.2$: large effect size. The improvement is both statistically and practically significant. $\square$

TI IWR6843ISK on a turntable at room centre. Rotation: $360^\circ$ in 60 s ($6^\circ$/s). FMCW chirp: $f_0 = 77$ GHz, $W = 4$ GHz, 256 chirps per frame, 30 frames/s. Total data: 1800 frames $\times$ $3\text{TX} \times 4\text{RX}$ channels.

Calibrate with corner reflector at known position. Collect data during one full rotation. Record turntable angle (encoder) synchronised to radar frames. Export raw ADC data via DCA1000EVM. Repeat 3 times.

1. Range compression: FFT across fast-time ($\Delta R = 3.75$ cm). 2. Virtual array: 12 elements. 3. Circular SAR back-projection: coherently sum from all angles. 4. LASSO on $4\times$ oversampled grid. 5. CFAR detection.

Ground truth: laser measurements ($\pm 1$ cm). Overlay SAR image on floor plan. Metrics: wall RMSE, furniture $P_D$, false alarm rate. Expected: wall RMSE $\sim 5$ cm, furniture $P_D \sim 70\%$. $\square$

Forward model (generation): Sionna ray tracing at 28 GHz, indoor scene, $256 \times 256$ grid. Reconstruction: Born approximation on $64 \times 64$ grid. Off-grid targets. Noise: AWGN at $\text{SNR} \in \{0, 5, 10, 15, 20, 25, 30\}$ dB. Scenes: 3 rooms of increasing complexity.

$N_{\mathrm{MC}} = 200$ trials per SNR $\times$ scene ($200 \times 7 \times 3 = 4{,}200$ total). Metrics: PSNR, SSIM, NMSE, $P_D$. Report: mean $\pm$ 95% CI.

TI IWR6843. Calibrate at 3 positions. Scenes: empty room, office, corridor. 3 runs each. Ground truth: laser.

Paired $t$-test for each SNR and scene. Bonferroni correction for 21 tests. Cohen's $d$. Friedman test for overall ranking.

Table 1: PSNR/SSIM $\pm$ CI. Figure 1: PSNR vs SNR with error bars. Figure 2: example reconstructions (best, median, worst). Figure 3: measurement images + metrics. Table 2: compute cost. Table 3: statistical test $p$-values and effect sizes. $\square$

Let $\mathcal{T}$ be target voxels ($|\mathbf{c}_{q}| > \tau$) and $\mathcal{B}$ the background. Define: $$\mathrm{RFIQ} = \alpha \cdot \mathrm{PSNR}_\mathcal{T} + \beta \cdot \frac{1}{|\mathcal{T}|}\sum_{q \in \mathcal{T}} \frac{|\hat{\mathbf{c}}_q|}{|\mathbf{c}_{q}|} + \gamma \cdot 10\log_{10}\frac{\bar{|\hat{\mathbf{c}}|}_\mathcal{T}}{\bar{|\hat{\mathbf{c}}|}_\mathcal{B}}$$ where the three terms measure: (a) target region PSNR; (b) amplitude fidelity (ratio $\to 1$ is perfect); (c) contrast ratio (higher is better).

When $\mathcal{T}$ is the entire image ($\alpha = 1, \beta = \gamma = 0$), RFIQ reduces to standard PSNR.

RFIQ is threshold-dependent (via $\tau$). It separately penalises target errors and background leakage. The contrast term rewards methods that suppress background clutter even at the cost of overall PSNR. Limitations: requires known target locations (from ground truth) and threshold selection. $\square$