Exercises

(a) Starting from the forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ with $\mathbf{w} \sim \mathcal{CN}(0, \sigma^2\mathbf{I})$, derive the matched filter estimator by maximizing the per-pixel SNR.

(b) Show that $\mathbb{E}[\hat{\mathbf{c}}^{\text{BP}}] = \mathbf{A}^{H} \mathbf{A} \mathbf{c}$ (biased estimator).

(c) Compute the covariance of the filtered noise: $\text{Cov}[\mathbf{A}^{H} \mathbf{w}] = \sigma^2 \mathbf{A}^{H} \mathbf{A}$.

(d) For colored noise with covariance $\mathbf{R}_n$, derive the whitened matched filter: $\hat{\mathbf{c}} = \mathbf{A}^{H} \mathbf{R}_n^{-1} \mathbf{y}$.

Implement DAS beamforming for a monostatic radar with $N_r = 16$ elements (ULA, $d = \lambda/2$) and $N_f = 64$ frequencies (9 to 11 GHz).

(a) Generate measurements for 3 point targets at ranges 3, 4, 5 m and angles $-20°, 0°, 25°$. SNR $= 25$ dB.

(b) Implement DAS on a $32 \times 32$ grid (range: 2--6 m, angle: $-45°$ to $+45°$).

(c) Plot the DAS image and measure the mainlobe widths. Compare with the theoretical $\Delta_r$ and $\Delta_{\text{cr}}$.

(d) Add Hamming windowing. How do the sidelobes and mainlobe widths change?

Compute and compare the PSF for three array geometries, all with $M = 32$ total virtual elements and $N_f = 64$ frequencies at 10 GHz:

(a) ULA: 32 elements with $\lambda/2$ spacing.

(b) Sparse random array: 32 elements randomly placed in an aperture of $16\lambda$.

(c) MIMO virtual array: $N_t = 4$, $N_r = 8$ with $\lambda/2$-spaced RX and $4\lambda$-spaced TX.

For each, plot the 2D PSF, measure mainlobe width and peak sidelobe level.

Simulate a stripmap SAR scenario:

• Platform velocity: $v = 100$ m/s.
• Carrier frequency: $f_c = 10$ GHz, bandwidth $B = 200$ MHz.
• $L = 256$ slow-time pulses, PRF $= 1$ kHz.
• Scene: $128 \times 128$ grid at 5 km range.

(a) Implement the range-Doppler algorithm (FFT-based).

(b) Implement the time-domain backpropagation algorithm.

(c) Compare the images for 5 point targets.

(d) Add a curved flight path (circular arc, 50 km radius). Show that backpropagation handles it correctly while range-Doppler produces defocused images.

Implement filtered backpropagation for a circular aperture:

(a) Generate k-space data for a $64 \times 64$ scene with 5 point targets, using $L = 120$ angles and $N_f = 64$ frequencies ($B = 1$ GHz, $f_c = 10$ GHz).

(b) Implement plain backpropagation and compute the PSF for a central point scatterer.

(c) Implement the Ram-Lak filter: apply $|f|$ weighting in the frequency domain before backprojection.

(d) Compare the PSFs and measure the mainlobe widths.

(e) Implement the Hamming-windowed ramp filter. How does it affect the noise level vs. the Ram-Lak filter?

(a) Generate a radar scene with 4 scatterers: two separated by $0.6\Delta$ (sub-resolution) and two at $2\Delta$ spacing. $M = 32$, $N_f = 64$, SNR $= 25$ dB.

(b) Compute the Capon image using the sample covariance from $L = 50$ snapshots.

(c) Compare the Capon and DAS images. Can Capon resolve the sub-resolution pair?

(d) Reduce to $L = 1$ (single snapshot). Apply diagonal loading ($\gamma = 10\sigma^2$) and compare.

(a) Generate a 1D scene with $K = 5$ scatterers using a 16-element ULA. Estimate the covariance from $L = 100$ snapshots.

(b) Apply MUSIC: eigendecompose $\hat{\mathbf{R}}_y$, identify the noise subspace, and compute the pseudospectrum.

(c) Compare the MUSIC spectrum with the DAS power spectrum.

(d) Vary the assumed model order $\hat{K}$ from $K-2$ to $K+3$. How does MUSIC performance change?

Systematically study the resolution--dynamic range tradeoff:

(a) Generate a scene with 2 scatterers separated by distance $d$ with amplitude ratio $r = |c_1|/|c_2|$.

(b) For separations $d/\Delta \in \{0.5, 0.75, 1.0, 1.5, 2.0\}$ and amplitude ratios $r \in \{1, 2, 5, 10, 20, 50\}$, compute: (i) MF image, (ii) Capon image, (iii) LASSO image.

(c) For each method, determine whether the weak scatterer is detected (peak $> 3\sigma^2$ above background).

(d) Plot the detection boundary in the $(d/\Delta, r)$ plane.

Show how the matched filter connects to iterative optimization:

(a) Write the least-squares cost $J(\mathbf{c}) = \frac{1}{2}\|\mathbf{y} - \mathbf{A}\mathbf{c}\|^2$. Compute $\nabla J$ and verify that $\nabla J(\mathbf{0}) = -\mathbf{A}^{H} \mathbf{y}$.

(b) Implement gradient descent from $\mathbf{c}^{(0)} = \mathbf{0}$ for 100 iterations. Show that the image converges to $\mathbf{A}^\dagger \mathbf{y}$ (pseudoinverse).

(c) Add $\ell_1$ regularization: implement one step of ISTA starting from the MF image. Compare with the pure MF image.

For a multi-static radar with $N_t = 2$ transmitters at positions $\mathbf{s}_{1} = [-0.5, 0]^T$ m, $\mathbf{s}_{2} = [0.5, 0]^T$ m, and $N_r = 4$ receivers at positions $\mathbf{r}_{j} = [-0.3 + 0.2j, 0]^T$ m ($j = 0, \ldots, 3$), operating at frequencies 9--11 GHz:

(a) Plot the k-space coverage (all Tx-Rx-frequency triples).

(b) Identify the k-space extent and predict the range and cross-range resolution.

(c) Compute the backpropagation PSF and verify your resolution prediction.

(d) Is the k-space coverage uniform? If not, design density compensation weights for filtered backpropagation.

Implement filtered backpropagation using the NUFFT:

(a) For a non-uniform k-space sampling pattern (multistatic radar), compute the density compensation factors via Voronoi tessellation.

(b) Implement FBP using the direct summation $\hat{c}(\mathbf{p}) = \sum_m W_m y_m e^{j {\kappa_{\mathbf{s},\mathbf{r}}}_{m} \cdot \mathbf{p}}$.

(c) Implement FBP using the NUFFT (gridding + FFT).

(d) Compare the computation times and image quality for a $128 \times 128$ grid.

Experimentally verify the CommIT observation about physical vs. random sensing matrices:

(a) Generate a scene with 10 point scatterers. Compute $\mathbf{A}$ for a MIMO radar (physical) and a random Gaussian matrix (same dimensions).

(b) Compute the Gram matrix $\mathbf{A}^{H} \mathbf{A}$ for both. Visualize the off-diagonal structure.

(c) Compute the MF image for both and compare the sidelobe patterns.

(d) Train a simple 3-layer CNN to denoise the MF image (supervised, using 1000 training scenes). Compare the test-set NMSE for physical vs. random $\mathbf{A}$.

(e) Explain why the CNN performs better for random $\mathbf{A}$.

Design an apodization window that achieves a target peak sidelobe level of $-30$ dB with minimum mainlobe broadening:

(a) Implement the Dolph-Chebyshev window for 32 elements with $-30$ dB sidelobes.

(b) Implement the Taylor window ($\bar{n} = 5$, $-30$ dB).

(c) Compare the PSFs (mainlobe width, sidelobe pattern, integrated sidelobe energy).

(d) Which window is better for RF imaging? Justify your choice.

Build a comprehensive comparison of all imaging methods on a realistic indoor RF imaging scene:

(a) System: MIMO radar with $N_t = 8$, $N_r = 16$, 77 GHz, 4 GHz bandwidth, $N_f = 128$.

(b) Scene: Indoor scene with 5 strong scatterers (0 dB), 15 moderate ($-20$ dB), 10 weak ($-35$ dB). Grid: $64 \times 64$.

(c) Methods: MF, MF + Taylor window, Capon ($L=1$, diagonal loading), MUSIC, FBP with Ram-Lak, LASSO (FISTA).

(d) Metrics: NMSE (dB), SSIM, detection probability ($P_{\text{fa}} = 10^{-4}$), computation time.

(e) Create a summary table and discuss which method gives the best tradeoff between quality and computation.

Investigate filtered backpropagation under limited angular coverage:

(a) Simulate a scenario with only 60 degrees of angular coverage (instead of 360 degrees). Use $L = 60$ angles, $N_f = 64$.

(b) Compute the k-space coverage and identify the missing region.

(c) Apply plain BP and FBP. Compare the images and discuss artifacts due to the missing k-space data.

(d) Show that the FBP density compensation factors become unstable near the edges of the k-space coverage.

(e) Propose a strategy to handle the limited-angle case (e.g., iterative reconstruction with TV regularization).