Exercises

At $f = 10$ GHz, $\lambda = 3$ cm, $\kappa = 2\pi/0.03 \approx 209.4$ rad/m. For target $k$ at distance $d_k = \|\mathbf{p}_{k}\|$: $y_k = \frac{\alpha_k}{d_k^2}\,e^{-j2\kappa d_k}$.

$y = \sum_{k=1}^{4} \frac{\alpha_k}{d_k^2}\,e^{-j2\kappa d_k} + w = [\mathbf{A}]_{1,:}\boldsymbol{\alpha} + w$, where $[\mathbf{A}]_{1,k} = \frac{e^{-j2\kappa d_k}}{d_k^2}$.

For each frequency $f_l$, compute $\kappa_{l} = 2\pi f_l / c$ and $[\mathbf{A}]_{l,k} = \frac{e^{-j2\kappa_{l} d_k}}{d_k^2}$. The resulting $\mathbf{A} \in \mathbb{C}^{16 \times 4}$.

$Q_x = Q_y = 1/0.05 + 1 = 21$, so $Q = 21^2 = 441$ grid points.

Nearest: $(0.35, 0.60)$ m. Offset: $\delta\mathbf{p} = (0.02, 0.02)$ m.

At 24 GHz: $\lambda = 12.5$ mm, $\kappa = 502.7$ rad/m. The direction from radar to target: $\hat{u} = (0.37-0.5, 0.62+1)/\|(0.37-0.5,1.62)\|$. Round-trip wavenumber component along $\delta\mathbf{p}$: $\Delta\phi \approx 2\kappa\,\hat{u}^T\delta\mathbf{p}$. With $\|\delta\mathbf{p}\| = 0.028$ m: $|\Delta\phi| \leq 2 \times 502.7 \times 0.028 \approx 28.2$ rad $\approx 4.5$ cycles.

With $\Delta = 0.025$ m, the maximum offset drops to $\|\delta\mathbf{p}\| \leq \sqrt{2} \times 0.0125 \approx 0.018$ m, and $|\Delta\phi| \leq 18.1$ rad $\approx 2.9$ cycles. Still significant at 24 GHz. Even 4x oversampling may be insufficient for coherent imaging at high frequencies.

$P(\sigma > 2\bar{\sigma}) = e^{-2} \approx 0.135$ or 13.5%.

$P(\sigma < 0.1\bar{\sigma}) = 1 - e^{-0.1} \approx 0.095$ or 9.5%. Nearly 10% of measurements see the target at less than 10% of its average strength.

$P(\sigma > 0.5\bar{\sigma}) = e^{-0.5} \approx 0.607$ or 60.7%. Only about 61% of measurements exceed the threshold. Multi-look integration is essential for reliable detection of Swerling I targets.

1st order: Image of $(3, 2, 1.5)$ in floor ($z=0$) is $(3, 2, -1.5)$. 2nd order: Image of $(3, 2, -1.5)$ in floor again is $(3, 2, 1.5)$ -- this coincides with the original but with additional path length and phase. More precisely, the 2nd-order path goes Tx $\to$ floor $\to$ target $\to$ floor $\to$ Rx, so the ghost is at the original position but with amplitude $\Gamma^2$ and additional round-trip delay.

1st order: $|\Gamma| = 0.6$ relative to direct. 2nd order: $|\Gamma|^2 = 0.36$ relative to direct.

$\mathbf{A}^{(0)}$ uses distances to $(3, 2, 1.5)$. $\mathbf{A}^{(1)}$ uses distances to $(3, 2, -1.5)$ (ghost). $\mathbf{A}_{\text{full}} = [\mathbf{A}^{(0)} \;|\; 0.6\mathbf{A}^{(1)} \;|\; 0.36\mathbf{A}^{(2)}]$.

With only $\mathbf{A}^{(0)}$, back-projection $\hat{\mathbf{c}} = \mathbf{A}^{(0)H}\mathbf{y}$ maps the multipath energy to incorrect positions (below floor). The extended model separates the $\mathbf{c}_{\text{ext}}$ components, correctly attributing multipath energy to ghosts.

$|\Gamma|^2 = \left(\frac{\sqrt{4.5}-1}{\sqrt{4.5}+1}\right)^2 = \left(\frac{1.12}{3.12}\right)^2 \approx 0.129$. $|T|^2 = (1-|\Gamma|^2)^2 \approx 0.871^2 \approx 0.759$, i.e., $-1.2$ dB per wall traversal.

$\Delta R = d_w(\sqrt{\varepsilon_r} - 1) = 0.25 \times (2.12 - 1) = 0.28$ m. The target appears 28 cm deeper than its true position.

Round-trip wall loss: $|T|^4 \approx 0.576$, i.e., $-2.4$ dB. Effective RCS: $\sigma_{\text{eff}} = 0.5 \times 0.576 = 0.288$ m$^2$.

At $\theta = 45°$: $|\Gamma_{\text{TE}}|^2 = \left|\frac{\cos 45° - \sqrt{4.5 - \sin^2 45°}}{\cos 45° + \sqrt{4.5 - \sin^2 45°}}\right|^2 = \left|\frac{0.707 - \sqrt{4.0}}{0.707 + \sqrt{4.0}}\right|^2 = \left|\frac{0.707 - 2.0}{0.707 + 2.0}\right|^2 = \left|\frac{-1.293}{2.707}\right|^2 \approx 0.228$. The reflection increases from 12.9% to 22.8%, making through-wall imaging harder at oblique angles.

$c_q = \sum_{n=1}^N e^{j\phi_n}$. By independence: $\mathbb{E}[|c_q|^2] = \sum_{n,m} \mathbb{E}[e^{j(\phi_n - \phi_m)}] = N$ (only $n=m$ terms survive).

For fully developed speckle ($N \gg 1$), $|c_q|^2 \sim \text{Exp}(N)$. $\text{Var}(|c_q|^2) = N^2$.

$\mathcal{C} = \sqrt{N^2}/N = 1$. This is the hallmark of fully developed Rayleigh speckle: the standard deviation of intensity equals the mean.

Averaging $L$ independent images: $\bar{I} = \frac{1}{L}\sum_{l=1}^L I_l$. $\text{Var}(\bar{I}) = N^2/L$, so $\mathcal{C}_L = 1/\sqrt{L}$.

$\mathcal{C}_L = 1/\sqrt{L} = 0.1 \implies L = 100$. One hundred independent looks are needed to reduce speckle contrast to 10%, which is expensive in terms of measurement diversity.

$D = 31 \times 2.5\text{ mm} = 77.5$ mm. $d_{\text{FF}} = 2 \times 0.0775^2 / 0.005 = 2.4$ m.

$\Delta\phi = \kappa\tilde{p}_\perp^2/(2R) = (2\pi/0.005) \times 0.25^2/(2 \times 1) = 1256.6 \times 0.0625/2 = 39.3$ rad $\approx 6.3$ cycles.

$5 > 2.4 = d_{\text{FF}}$, so we are in the far field. The Fresnel error at 5 m: $\Delta\phi = 1256.6 \times 0.0625/10 = 7.85$ rad $\approx 1.25$ cycles. Still non-negligible, suggesting the Kronecker structure is marginal. A factor of 2-3 beyond $d_{\text{FF}}$ is typically needed for good accuracy.

Far-field (Kronecker): $O(M + Q) \sim O(6144)$ operations. Near-field (explicit): $O(MQ) = O(8.4 \times 10^6)$ operations. Ratio: $\sim 1370\times$ more expensive.

(i) Foam: $|\chi| = 0.03$, $\kappa a = 20\pi \approx 62.8$, product $= 1.88$. Above threshold.

(ii) Wood: $|\chi| = 1$, $\kappa a = 4\pi \approx 12.6$, product $= 12.6$. Well above.

(iii) Concrete: $|\chi| = 5$, $\kappa a = 2\pi \approx 6.28$, product $= 31.4$. Far above.

(iv) Metal: $|\chi| \to \infty$, product $= \infty$. Born fails completely.

None of these cases satisfy $|\chi|\kappa a < 0.5$. Even foam at $10\lambda$ violates the criterion because the electrical size is large.

(i) Foam: Rytov approximation (good for smoothly varying, weakly scattering media). (ii) Wood: DBIM (moderate contrast, iterative refinement). (iii) Concrete: Full-wave (FDTD) or contrast source inversion. (iv) Metal: Physical optics or method of moments (surface currents, no volume integral needed for PEC).

Generate $\mathbf{A} \in \mathbb{C}^{32 \times 64}$ with entries $\sim \mathcal{CN}(0, 1/32)$. Scene: $\mathbf{c}^*$ has 5 entries of unit magnitude at random positions.

$\Delta\mathbf{A}$ has entries $\sim \mathcal{CN}(0, \sigma_\Delta^2)$. Vary $\sigma_\Delta \in [0, 0.5]$. Noise: $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, 0.01\mathbf{I})$.

$\hat{\mathbf{c}} = (\mathbf{A}^{H}\mathbf{A} + \lambda\mathbf{I})^{-1}\mathbf{A}^{H}\mathbf{y}$. Compute RMSE $= \|\hat{\mathbf{c}} - \mathbf{c}^*\|_2 / \|\mathbf{c}^*\|_2$ for each $(\lambda, \sigma_\Delta)$ pair, averaged over 100 trials.

At low mismatch ($\sigma_\Delta \approx 0$), $\lambda_{\text{opt}}$ is best. As $\sigma_\Delta$ increases, $10\lambda_{\text{opt}}$ becomes more robust: it has higher bias but lower mismatch amplification. At $\sigma_\Delta = 0.3$, the RMSE of over-regularized Tikhonov can be 2-3x lower than optimal Tikhonov.

For a flat plate, the PO scattered field is proportional to $\text{sinc}(\kappa a (\sin\theta_i + \sin\theta_r))$. At specular: argument $\approx 0$, $\text{sinc} \approx 1$. At $\theta_r = 60°$: argument $= \kappa \times 2\lambda \times \sin 60° = 4\pi \times 0.866 \approx 10.9$, so $\text{sinc} \approx 0.029$. RCS ratio: $(0.029)^2 / 1 \approx 8.4 \times 10^{-4}$, i.e., the off-specular RCS is $\sim 31$ dB below the specular RCS.

Only Tx-Rx pairs near the specular direction will see a strong return. With 6 pairs uniformly spaced in angle, typically 1-2 pairs will be near specular. The other 4-5 pairs see $> 20$ dB lower returns.

If $c_q$ is fit to the strong specular return, the predicted signal for off-specular pairs will be $\sim 30$ dB too high. If fit to the average, the specular pair will be under-predicted by $\sim 15$ dB.

Replace the scalar $c_q$ with an angle-dependent coefficient $c_q(\hat{\mathbf{s}}, \hat{\mathbf{r}})$, expanding in a basis (e.g., spherical harmonics): $c_q(\hat{\mathbf{s}}, \hat{\mathbf{r}}) = \sum_{l} \beta_{q,l}\,Y_l(\hat{\mathbf{s}}, \hat{\mathbf{r}})$. This increases the unknowns by the number of basis functions per pixel.

With targets on grid points, $\ell_1$ minimization recovers the sparse scene exactly (up to noise). Typical RMSE $\sim 10^{-2}$ at 20 dB SNR.

With off-grid targets, basis mismatch spreads each target across multiple grid points. $\ell_1$ fails to exploit sparsity, and RMSE $\sim 0.1$-$0.3$ (10-30x worse).

The inverse crime overstates performance by $\sim 20$ dB in RMSE. This is because basis mismatch is the dominant error source in practice, and the crime eliminates it entirely.

A $128 \times 128$ grid reduces the maximum offset from $\Delta/\sqrt{2}$ to $\Delta/(2\sqrt{2})$, roughly halving the mismatch. The RMSE gap narrows but does not close: even at 4x oversampling, some basis mismatch remains for high-frequency components.

Generate $\mathbf{c}_{c} \sim \mathcal{CN}(\mathbf{0}, \sigma_c^2\mathbf{I})$ with $\sigma_c^2$ chosen so that SCR = 15 dB. Back-projection: targets are visible above the speckled background.

$[\boldsymbol{\Sigma}_{c}]_{q,q'} = \sigma_c^2\,e^{-\|\mathbf{p}_{q} - \mathbf{p}_{q'}\|/\ell_c}$. Correlated clutter has smoother spatial structure, making it harder to distinguish from extended targets.

Compute $\boldsymbol{\Sigma}_{c}^{1/2}$ via Cholesky. Transform: $\tilde{\mathbf{y}} = \boldsymbol{\Sigma}_{c}^{-1/2}\mathbf{y}$, $\tilde{\mathbf{A}} = \boldsymbol{\Sigma}_{c}^{-1/2}\mathbf{A}$. The whitened problem has i.i.d. noise, and back-projection $\tilde{\mathbf{A}}^H\tilde{\mathbf{y}}$ has improved SCR by approximately $\ell_c/\Delta = 5$, i.e., $\sim 7$ dB.

$\lambda = 10.7$ mm, $D = 63 \times 5.35\text{ mm} = 0.337$ m. $d_{\text{FF}} = 2 \times 0.337^2/0.0107 = 21.2$ m. At $R = 3$ m, we are well inside the Fresnel region.

Compute $\mathbf{A}_{\text{FF}}$ using the Taylor approximation (Kronecker). The PSF has a sinc-like mainlobe with width $\delta_x \approx \lambda R/(2D) = 0.0107 \times 3/(2 \times 0.337) = 0.048$ m.

Compute $\mathbf{A}_{\text{NF}}$ using exact distances. The PSF remains focused with the correct mainlobe width. The far-field PSF is defocused due to the uncompensated quadratic phase, widening the mainlobe by a factor of $\sim 2$-$3$ at $R = 3$ m.

The 10% mainlobe width discrepancy typically occurs around $R \approx 0.3 d_{\text{FF}} \approx 6.4$ m. Beyond this range, the far-field model is adequate for most applications.

$Q = (5/0.02)^2 = 250^2 = 62{,}500$ pixels. This is a large but manageable problem.

$M = N_t \times N_r \times N_f = 8 \times 16 \times 64 = 8192$ measurements. Direct: $\mathbf{A}^{(0)} \in \mathbb{C}^{8192 \times 62500}$. With 3 wall reflections: $\mathbf{A}_{\text{ext}} \in \mathbb{C}^{8192 \times 4 \times 62500}$.

With $D = 40$ mm at 60 GHz: $d_{\text{FF}} = 0.64$ m. Room diagonal: $\sim 7$ m. Most of the room is in the far field. Only targets within $\sim 0.6$ m of the array need near-field correction.

Direct-only back-projection: ghost targets from wall reflections. Extended back-projection: reduced ghosts but low resolution. $\ell_1$ with extended $\mathbf{A}$: sharp reconstruction with ghost separation, but computational cost is high due to the large $\mathbf{A}_{\text{ext}}$.

Construct $\mathbf{A} \in \mathbb{C}^{M \times Q}$ with $M = 16 \times 32 = 512$ (16 elements, 32 frequencies) and $Q = 1024$.

Apply diagonal phase matrix: $\mathbf{A}_{\text{cal}} = \text{diag}(e^{j\boldsymbol{\phi}})\mathbf{A}$.

At $\sigma_\phi = 5°$: peak sidelobe rises by $\sim 3$ dB. At $\sigma_\phi = 20°$: sidelobes comparable to mainlobe; targets become unresolvable.

Estimate $\phi_n$ from the strongest target: $\hat{\phi}_n = \arg(y_n / [\mathbf{A}]_{n,q_0})$. Correct: $\hat{\mathbf{A}} = \text{diag}(e^{-j\hat{\boldsymbol{\phi}}})\mathbf{A}_{\text{cal}}$. This reduces residual errors to $\sim 1$-$2°$ even when the true errors are $20°$.