Exercises

$N_v = 6 \times 10 = 60$ virtual elements.

$\eta = 60 / 16 = 3.75$.

$N_v = 8 \times 8 = 64$ virtual elements. $\eta = 64 / 16 = 4$. The equal split $N_t = N_r$ produces more virtual elements (64 vs. 60) and higher efficiency, confirming the AM-GM optimality.

$\lambda = c / f_0 = 12.5$ mm, so $d = 6.25$ mm. Rx spacing: $d = 6.25$ mm. Tx spacing: $N_r d = 37.5$ mm.

$N_v = 24$ elements. Length: $(N_v - 1)d = 23 \times 6.25 = 143.75$ mm.

$\Delta\theta = \lambda / (N_v \, d) = 12.5 / (24 \times 6.25) = 0.083$ rad $\approx 4.8°$.

Divides time. Advantage: simplest implementation. Disadvantage: reduces effective CPI by $N_t$, degrading Doppler resolution.

Divides frequency. Advantage: no cross-talk between Tx. Disadvantage: reduces per-Tx bandwidth, degrading range resolution.

Divides code space. Advantage: each Tx uses full bandwidth and full CPI. Disadvantage: requires good code design; residual cross-correlation creates sidelobes.

TDM is most common in automotive radar due to its simplicity and low hardware cost. The reduced CPI is acceptable because automotive scenarios have short coherence times.

$P_D = 1 - (1 - 0.5)^6 = 1 - 0.5^6 = 1 - 0.0156 = 0.984$.

Need $(1 - P_{D,1})^K \leq 0.01$, so $K \geq \log(0.01) / \log(0.5) = -4.605 / (-0.693) = 6.64$. Hence $K = 7$ pairs.

Correlated scattering cross-sections reduce the effective diversity order below $K$. In the extreme (fully correlated), the effective diversity is 1 regardless of $K$ --- this is the Swerling-0 (constant cross-section) case. The actual gain depends on the correlation structure between bistatic angles.

Each sub-array has $N_t/L$ elements. Coherent gain: $G(L) = (N_t/L)^2 = 256/L^2$.

$N_v(L) = L \times N_r = 16L$.

Metric: $G(L) \times N_v(L) = 256/L^2 \times 16L = 4096/L$. This is monotonically decreasing in $L$, so the metric favours small $L$ (more gain, fewer virtual elements).

A more balanced metric is $\sqrt{G(L)} \times N_v(L) = (16/L) \times 16L = 256$ --- constant! This shows the fundamental phased-MIMO trade-off: the product of beamwidth and number of beams is constant. The choice of $L$ depends on the specific ISAC requirements.

$d_{\mathrm{tx}} = \sqrt{(3-(-5))^2 + 8^2} = \sqrt{64 + 64} = 8\sqrt{2} \approx 11.31$ m. $d_{\mathrm{rx}} = \sqrt{(3-5)^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} \approx 8.25$ m.

$R_b = 11.31 + 8.25 = 19.56$ m.

Using the law of cosines in the Tx-target-Rx triangle: $L^2 = d_{\mathrm{tx}}^2 + d_{\mathrm{rx}}^2 - 2 d_{\mathrm{tx}} d_{\mathrm{rx}} \cos\beta$. $100 = 128 + 68 - 2(11.31)(8.25)\cos\beta = 196 - 186.6\cos\beta$. $\cos\beta = 96/186.6 = 0.514$. $\beta = 59.0°$.

$\Delta R_b = c / (2W\cos(\beta/2)) = 3 \times 10^8 / (2 \times 5 \times 10^8 \times \cos(29.5°)) = 0.3 / (1.0 \times 0.870) = 0.345$ m.

With $N_tN_r = 4 \times 4 = 16$ virtual elements (monostatic: Tx = Rx array) and 2 frequencies: $2 \times 16 = 32$ total measurements, but all at the same angular direction. Per direction, there are 2 k-space samples (one per frequency) with different radial positions.

At frequency $f_1$: an arc at radius $|\boldsymbol{\kappa}| = 2 \times 2\pi f_1/c$ spanning $\theta \in [-30°, 30°]$ with 16 samples. At frequency $f_2$: a similar arc at slightly larger radius. The coverage is two concentric arcs.

Effective bandwidth: $\Delta f = f_2 - f_1 = 1$ GHz. Range resolution: $\Delta R = c / (2\Delta f) = 0.15$ m.

$\kappa(\mathbf{A}) = \kappa(\mathbf{A}_f) \cdot \kappa(\mathbf{A}_V) = 3 \times 8 = 24$.

$\kappa(\mathbf{A}_V) = \kappa(\mathbf{A}_{\mathrm{tx}}) \cdot \kappa(\mathbf{A}_{\mathrm{rx}}) = 4 \times 2 = 8$. Consistent.

Currently $\kappa = 3 \times 4 \times 2 = 24$. Target: $< 12$. Improving $\mathbf{A}_{\mathrm{tx}}$ from $\kappa = 4$ to $\kappa = 2$ gives $3 \times 2 \times 2 = 12$. Alternatively, improving $\mathbf{A}_f$ from 3 to 1.5 gives $1.5 \times 8 = 12$. The largest factor ($\mathbf{A}_{\mathrm{tx}}$) offers the most leverage.

All 15 sums $d_i^{\mathrm{tx}} + d_j^{\mathrm{rx}}$: $\{0, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 21, 22\}d$. (Some values appear only once; no duplicates in this case.) There are 15 unique positions out of 15 pairs.

The virtual positions span $\{0, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 21, 22\}d$. This is not a filled ULA --- positions $\{1, 2, 4, 7, 17, 18, 20\}d$ are missing. However, the aperture extends to $22d$, much larger than $14d$ for the standard design.

Standard design: filled ULA, 15 elements, aperture $14d$. Coprime design: 15 elements (no redundancy), aperture $22d$, but with 7 holes.

The coprime design achieves $\sim 57\%$ larger aperture (better angular resolution) at the cost of missing virtual positions (higher sidelobes, worse conditioning).

From diffraction tomography: $\Delta x_\perp = \lambda / (2\sin(\Theta/2))$.

$\lambda = c / (28 \times 10^9) = 10.7$ mm. Need $\Delta x_\perp < 10$ mm: $10.7 / (2\sin(\Theta/2)) < 10$, so $\sin(\Theta/2) > 0.535$, $\Theta/2 > 32.3°$, $\Theta > 64.7°$.

Per-terminal: $N_v = 16$, $d = \lambda/2$. $\Delta x_\perp^{\text{per}} = R \times \lambda / (16 \times \lambda/2) = R/8$. Multi-view (at $\Theta = 65°$): $\Delta x_\perp = 10.7 / (2 \times 0.537) = 9.96$ mm. Crossover: $R/8 = 9.96$ mm, $R = 79.7$ mm $\approx 8$ cm. For $R > 8$ cm, multi-view is better.

$|\boldsymbol{\kappa}|$ ranges from $4\pi f_0 / c$ to $4\pi(f_0 + W)/c$. Extent: $\Delta k_r = 4\pi W / c$.

At fixed $|\boldsymbol{\kappa}|$, the angular span is determined by $\Delta\theta$. The transverse extent in k-space is $\Delta k_\perp \approx |\boldsymbol{\kappa}| \sin(\Delta\theta/2) \approx (4\pi f_0/c)\sin(\Delta\theta/2)$.

The image is the inverse Fourier transform of the k-space coverage (indicator function). For a rectangular k-space aperture (linearised polar $\to$ Cartesian):

$$\text{PSF}(x, y) \propto \text{sinc}\!\Bigl(\frac{\Delta k_r \, x}{2\pi}\Bigr) \cdot \text{sinc}\!\Bigl(\frac{\Delta k_\perp \, y}{2\pi}\Bigr).$$

The 3 dB widths are: down-range $\delta_x = c / (2W)$, cross-range $\delta_y = c / (4 f_0 \sin(\Delta\theta/2)) = \lambda / (2\sin(\Delta\theta/2))$.

By the mixed-product property of Kronecker products: $(\mathbf{A}_f \otimes \mathbf{A}_V)\text{vec}(\mathbf{C}) = \text{vec}(\mathbf{A}_V \mathbf{C} \mathbf{A}_f^T)$. So $\mathbf{Y} = \mathbf{A}_V \mathbf{C} \mathbf{A}_f^T \in \mathbb{C}^{N_V \times N_f}$ and $\mathbf{y} = \text{vec}(\mathbf{Y})$.

Direct: $O(MQ) = O(N_f N_V Q_r Q_\theta)$. Factored: $\mathbf{A}_V \mathbf{C}$ costs $O(N_V Q_\theta Q_r)$, then $(\cdot)\mathbf{A}_f^T$ costs $O(N_V Q_r N_f)$. Total: $O(N_V Q_r (Q_\theta + N_f))$ --- much less when $Q_\theta, N_f \ll Q_r Q_\theta$.

If $\mathbf{A}_V$ is a partial DFT (ULA steering vectors), $\mathbf{A}_V \mathbf{C}$ is a column-wise FFT: $O(Q_r N_V \log Q_\theta)$. Similarly $(\cdot)\mathbf{A}_f^T$ is a row-wise FFT: $O(N_V N_f \log Q_r)$. Total: $O(Q_r N_V \log Q_\theta + N_V N_f \log Q_r) = O(M \log Q)$.

$R_{\mathrm{FF}} = 2 \times 0.25 / 0.0039 = 128$ m. At $R = 20$ m: near-field (need spherical wavefront model). At $R = 2$ m: deep near-field.

$R_{\mathrm{FF}} = 2 \times 2500 / 0.0039 \approx 1.3 \times 10^6$ m = 1300 km! Any practical scene is in the extreme near-field of the distributed array. Planar wavefront assumptions are invalid.

Virtual array: 4 elements at spacing $d$. $\mathbf{a}_V(\theta) = [1,\; e^{j\pi\sin\theta},\; e^{j2\pi\sin\theta},\; e^{j3\pi\sin\theta}]^T$.

$G_{pq} = \mathbf{a}_V^H(\theta_p) \mathbf{a}_V(\theta_q) = \sum_{m=0}^{3} e^{j\pi m(\sin\theta_q - \sin\theta_p)} = \frac{\sin(2\pi\Delta_s)}{\sin(\pi\Delta_s/2)}$ where $\Delta_s = \sin\theta_q - \sin\theta_p$.

Compute $|\sin\theta_q - \sin\theta_p|$ for all pairs. $\theta = 0°$ vs. $20°$: $|\Delta_s| = 0.342$. $\theta = 20°$ vs. $40°$: $|\Delta_s| = |0.643 - 0.342| = 0.301$. The pair $(20°, 40°)$ has the smallest $|\Delta_s|$ and hence the highest coherence --- these are the hardest to resolve.

$\Delta R = c \Delta t = 3 \times 10^8 \times 10^{-9} = 0.3$ m.

$\Delta\phi = 2\pi \times 5 \times 10^9 \times 10^{-9} = 10\pi$ rad. This is 5 full cycles --- completely destroys phase coherence.

Phase slope across frequency: $\Delta\phi(f) = 2\pi f \Delta t$. Phase change per subcarrier: $2\pi \Delta f \Delta t = 2\pi \times 120 \times 10^3 \times 10^{-9} = 7.54 \times 10^{-4}$ rad $\approx 0.04°$. Across 1000 subcarriers: $0.754$ rad $\approx 43°$. The timing error manifests as a linear phase ramp across subcarriers, which can be estimated and removed.

For a filled virtual ULA, $\kappa \approx 1$ (DFT matrix is unitary). The question reduces to maximising the virtual aperture length $(N_t-1)N_rd + (N_r-1)d$ subject to $N_t + N_r = N$. The aperture is $(N_tN_r - 1)d$, maximised at $N_t = N_r = N/2$ by AM-GM.

Given $N$ element positions $\{p_1, \ldots, p_N\}$, find partition $\mathcal{T} \cup \mathcal{R} = \{1, \ldots, N\}$ minimising $\kappa(\mathbf{A}(\mathcal{T}, \mathcal{R}))$. This is NP-hard in general (related to minimum condition number column subset selection).

Start with all elements as Rx ($\mathcal{R} = \{1, \ldots, N\}$). Iteratively move the element to Tx that most reduces $\kappa$, stopping when $|\mathcal{T}| = N/2$. Complexity: $O(N^2)$ condition number evaluations, each $O(N_v^3)$ for small arrays. Total: $O(N^5)$ --- feasible for $N \leq 100$.

In the distributed case, $\boldsymbol{\kappa}_{ij}(f) = \kappa(f)(\hat{\mathbf{e}}_{\mathrm{tx},i} + \hat{\mathbf{e}}_{\mathrm{rx},j})$ depends on the direction to the target from each antenna. Since the antennas are at arbitrary positions, the resulting k-space samples are non-uniformly distributed --- unlike the co-located case where they lie on a regular grid.

Direct: $O(MQ)$. NUFFT: $O(Q\log Q + M)$. NUFFT is faster when $MQ > Q\log Q + M$, i.e., $M(Q - 1) > Q\log Q$, so $M > Q\log Q / (Q-1) \approx \log Q$. For $Q = 10^4$: $M > 13$. The NUFFT is faster for essentially all practical configurations.

The NUFFT approximation error depends on the oversampling factor $\sigma$ and the kernel width $W$. Typical choices ($\sigma = 2$, $W = 6$) give relative error $< 10^{-6}$, which is negligible compared to measurement noise. However, the adjoint $\mathbf{A}^{H} \mathbf{y}$ must use the same NUFFT approximation to preserve the self-adjoint structure needed for iterative algorithms.