Exercises

$W = 64 \times 312.5\;\text{kHz} = 20$ MHz. $\Delta R = c/(2W) = 3 \times 10^8 / (2 \times 20 \times 10^6) = 7.5$ m.

$R_{\max} = c T_{\mathrm{cp}} / 2 = 3 \times 10^8 \times 0.8 \times 10^{-6} / 2 = 120$ m.

$\lambda = c/f_0 = 0.0517$ m. $T_{\mathrm{sym}} = 1/\Delta f + T_{\mathrm{cp}} = 3.2 + 0.8 = 4\;\mu\text{s}$. $\Delta v = \lambda/(2 M T_{\mathrm{sym}}) = 0.0517/(2 \times 20 \times 4 \times 10^{-6}) \approx 323$ m/s. The poor Doppler resolution is due to the very short CPI.

The received signal includes the data modulation $d[n,m]$ as a multiplicative factor. Without compensation, the data acts as random phase rotations that destroy the 2D sinusoidal structure needed for coherent FFT processing. Division by $d[n,m]$ restores the clean delay-Doppler phase structure.

Null subcarriers ($d[n,m] = 0$) provide no illumination and cannot be compensated. These entries are excluded from the measurement vector, creating a partial Fourier sensing matrix with missing rows. Compressed sensing methods can recover the missing information if the scene is sparse.

$R_{\max} = c T_{\mathrm{cp}}/2 = 3 \times 10^8 \times 2.34 \times 10^{-6}/2 \approx 351$ m.

$R_{\max} = 3 \times 10^8 \times 0.59 \times 10^{-6}/2 \approx 88$ m.

Smaller $\Delta f$ gives longer CP and larger $R_{\max}$, but the OFDM symbol duration is longer ($T_{\mathrm{sym}} = 1/\Delta f + T_{\mathrm{cp}}$), leading to worse Doppler resolution for a fixed number of symbols. Range resolution depends only on total bandwidth $W = N_c \Delta f$, which can be the same for both.

Hamming window: peak sidelobe $\approx -43$ dB, main-lobe width $\approx 1.5\times$ that of the rectangular window.

Applying Hamming reduces sidelobes by about 30 dB but degrades range/Doppler resolution by a factor of $1.5\times$. For a system with $\Delta R = 0.5$ m (rectangular), the effective resolution becomes $\approx 0.75$ m with Hamming. This is the classic resolution-sidelobe trade-off.

In the OTFS delay-Doppler domain, each point target produces exactly one non-zero channel entry regardless of velocity, while in OFDM the high Doppler spreads each target's energy across many subcarriers via ICI. This sparsity makes OTFS robust to high-Doppler scenarios where OFDM fails.

Converting from delay-Doppler to time-frequency (or vice versa) requires an ISFFT/SFFT, which is a 2D-DFT of size $N_c \times M$. The cost is $O(N_c M (\log N_c + \log M))$ --- negligible compared to the baseband processing.

The $k$-th column of $\mathbf{A}$ is $\mathbf{a}_k = \mathbf{a}_{\mathrm{Dopp},k} \otimes \mathbf{a}_{\mathrm{range},k}$ where $[\mathbf{a}_{\mathrm{range},k}]_n = e^{-j2\pi n \Delta f \tau_k}$ ($N_c \times 1$) and $[\mathbf{a}_{\mathrm{Dopp},k}]_m = e^{j2\pi m T_{\mathrm{sym}} \nu_k}$ ($M \times 1$).

$\mathbf{A} = [\mathbf{a}_1, \ldots, \mathbf{a}_K]$. If the targets are on a grid ($\tau_k = p_k/(N_c \Delta f)$ and $\nu_k = q_k/(M T_{\mathrm{sym}})$), then $\mathbf{a}_{\mathrm{range},k}$ is the $p_k$-th column of $\mathbf{F}_{N_c}^H$ and $\mathbf{a}_{\mathrm{Dopp},k}$ is the $q_k$-th column of $\mathbf{F}_M$. Dimensions: $\mathbf{F}_{N_c} \in \mathbb{C}^{N_c \times N_c}$ (or $N_c \times G_\tau$ for an oversampled grid) and $\mathbf{F}_M \in \mathbb{C}^{M \times M}$ (or $M \times G_\nu$).

The DFT output on subcarrier $n$ for a target at normalised Doppler $\epsilon$ is $H[n] \propto \frac{\sin(\pi(\epsilon - n))}{N_c \sin(\pi(\epsilon - n)/N_c)}$. The desired signal power is $|H[0]|^2 \approx (\sin(\pi\epsilon)/(\pi\epsilon))^2 N_c^2$ (for $\epsilon \ll 1$). The ICI power sums to $\sum_{n \neq 0} |H[n]|^2 \approx N_c^2 (\pi\epsilon)^2/3$ for small $\epsilon$.

$\mathrm{SIR} \approx \frac{1}{(\pi\epsilon)^2/3} = \frac{3}{\pi^2 \epsilon^2}$.

$10 \log_{10}(3/(\pi^2 \epsilon^2)) = 10$ dB gives $\epsilon^2 = 3/(10\pi^2) \approx 0.0304$, so $\epsilon \approx 0.174$. At $\nu/\Delta f \approx 0.17$, the ICI-induced SIR drops below 10 dB.

$\mathcal{Z}_x(\tau + T, \nu) = \sqrt{T} \sum_{k} x(\tau + T + kT) e^{-j2\pi k T \nu}$. Let $k' = k + 1$, so $k = k' - 1$ and $x(\tau + T + kT) = x(\tau + k'T)$: $$= \sqrt{T} \sum_{k'} x(\tau + k'T) e^{-j2\pi (k'-1) T \nu} = e^{j2\pi T \nu} \sqrt{T} \sum_{k'} x(\tau + k'T) e^{-j2\pi k' T \nu}$$

The remaining sum is exactly $\mathcal{Z}_x(\tau, \nu)$, proving $\mathcal{Z}_x(\tau + T, \nu) = e^{j2\pi T \nu} \mathcal{Z}_x(\tau, \nu)$. $\blacksquare$

The cyclic prefix of each OFDM symbol ensures that the delay-domain convolution between the channel and the transmitted signal is circular (modulo $N_c$ delay bins). Without CP, the convolution would be linear and the OFDM symbols would suffer ISI.

The SFFT operates over a finite window of $M$ OFDM symbols. The Doppler processing via DFT implicitly assumes periodicity in the slow-time dimension, making the Doppler convolution circular (modulo $M$ Doppler bins). This is the standard DFT circular convolution property.

Together, the CP (delay) and finite DFT window (Doppler) make the full 2D channel convolution circular. This circularity is essential for the block-circulant structure of the OTFS sensing matrix and the efficiency of FFT-based algorithms.

The energy in the nearest bin ($k = k_0$) is proportional to $|\sin(\pi M \kappa)/(M \sin(\pi \kappa / M))|^2 \approx |\sin(0.3\pi \times 64)/(64 \sin(0.3\pi/64))|^2$. For large $M$: $\approx |\sin(0.3\pi)/(0.3\pi)|^2 = |\text{sinc}(0.3)|^2 \approx 0.724^2 \approx 0.524$. About 52% of energy is in the main bin.

The $k$-th adjacent bin has energy $\propto |\text{sinc}(k - \kappa)|^2$ for large $M$. For $k = \pm 1$: $|\text{sinc}(0.7)|^2 \approx 0.158$ and $|\text{sinc}(1.3)|^2 \approx 0.048$. Bins with $\geq 1\%$ energy: computing $|\text{sinc}(k - 0.3)|^2 \geq 0.01$ gives roughly $|k - 0.3| \leq 3.5$, so about 7 bins.

Since the total volume is fixed at 1, any reduction in main-lobe width (better resolution) must be compensated by redistributing volume to the sidelobes. A thumbtack ambiguity function (like OFDM with random data) spreads the sidelobes thinly across the entire delay-Doppler plane.

No waveform can have both arbitrarily narrow main lobe and arbitrarily low sidelobes. OFDM with random data achieves a good compromise: the sidelobes are at $-13$ dB but spread uniformly (like white noise). Windowing reduces peak sidelobes but widens the main lobe, keeping the volume constant. This is the Woodward ambiguity principle.

Both: $\Delta R = c/(2 \times 10^9) = 0.15$ m. The targets are separated by 0.1 m $< \Delta R$, so neither waveform resolves them in range. However, they differ in velocity.

$\nu_2 = 2 \times 22.2 \times 77 \times 10^9 / (3 \times 10^8) \approx 11.4$ kHz. Normalized: $\nu_2/\Delta f = 11.4/976.6 \approx 0.012$.

FMCW has no ICI issue. The two targets overlap in range but are separated in Doppler. With $M = 128$ chirps and PRI $\approx 1\;\mu\text{s}$, $\Delta v = \lambda/(2 \times 128 \times 10^{-6}) \approx 15.2$ m/s. Target 2 is at 22.2 m/s. Both targets are visible, separated by $\sim 1.5$ Doppler bins.

With $\nu/\Delta f \approx 0.012$, ICI is negligible ($\text{SIR} > 40$ dB). OFDM also resolves both targets via the Doppler dimension. Performance is comparable to FMCW in this low-Doppler regime.

Define $\mathbf{y}[l + kN_c] = Y_{\mathrm{DD}}[l,k]$ and similarly for $\mathbf{x}$ and $\mathbf{h}$. The index mapping is $(l,k) \to l + kN_c$ for $l \in \{0,\ldots,N_c-1\}$, $k \in \{0,\ldots,M-1\}$.

Partition $\mathbf{H}_{\mathrm{circ}}$ into $M \times M$ blocks of size $N_c \times N_c$. Block $(k, k')$ of $\mathbf{H}_{\mathrm{circ}}$ is $\mathbf{H}^{(k-k')_M}$ where $\mathbf{H}^{(j)}$ is an $N_c \times N_c$ circulant matrix with first column given by $h_{\mathrm{DD}}[\cdot, j]$. This is a block-circulant matrix (the block index depends only on $k - k'$).

Each block $\mathbf{H}^{(j)}$ is itself circulant because the delay convolution is circular (modulo $N_c$). Hence $\mathbf{H}_{\mathrm{circ}}$ is block-circulant with circulant blocks (BCCB), which is diagonalised by $\mathbf{F}_M \otimes \mathbf{F}_{N_c}$ --- the 2D-DFT. $\blacksquare$

$\text{PAPR} \approx 10 \log_{10}(1024) = 30$ dB. In practice, with clipping at $10^{-3}$ CCDF, PAPR $\approx 10$--$12$ dB.

With 10 dB PAPR, the PA must operate at 10 dB back-off. If the PA has 1 W saturated output, the average output is 0.1 W. For FMCW (PAPR $\approx 0$ dB), the full 1 W is available.

The radar range equation gives $R_{\max} \propto P_t^{1/4}$. A 10 dB reduction in average power reduces maximum detection range by factor $10^{10/(4 \times 10)} = 10^{0.25} \approx 1.78$. FMCW can detect targets $\sim 1.8\times$ farther than OFDM with the same PA. This is a significant disadvantage for OFDM/OTFS in automotive radar.

$\langle \mathbf{a}_i, \mathbf{a}_j \rangle = (\mathbf{a}_{\mathrm{Dopp},i}^H \mathbf{a}_{\mathrm{Dopp},j}) \cdot (\mathbf{a}_{\mathrm{range},i}^H \mathbf{a}_{\mathrm{range},j})$. Each factor is a Dirichlet kernel evaluated at the delay or Doppler separation.

If $\delta\tau = 1/W$ (one range bin) and $\delta\nu = 0$, then $|\mathbf{a}_{\mathrm{range},i}^H \mathbf{a}_{\mathrm{range},j}| = |D_{N_c}(1/N_c)| = 0$ (orthogonal) and $|\mathbf{a}_{\mathrm{Dopp},i}^H \mathbf{a}_{\mathrm{Dopp},j}| = M$. So $\mu = 0$ --- targets separated by exactly one grid cell are orthogonal.

For targets separated by a fraction $\alpha < 1$ of the resolution cell, $\mu \approx |D_{N_c}(\alpha/N_c)| / N_c = |\text{sinc}(\alpha)|$, which approaches 1 as $\alpha \to 0$. High coherence makes sparse recovery difficult for closely spaced targets.

The periodic autocorrelation of an m-sequence of length $L$ is $R[k] = L$ for $k = 0$ and $R[k] = -1$ for $k \neq 0$. The peak sidelobe level is $1/L = 1/1023 \approx -30$ dB.

OFDM with rectangular window: Dirichlet kernel with peak sidelobe $-13.2$ dB.

PMCW with m-sequence has $\sim 17$ dB better sidelobe suppression than unwindowed OFDM. For detecting a weak target ($-25$ dB below a strong target), the OFDM sidelobes at $-13$ dB would mask it, but PMCW sidelobes at $-30$ dB would not. The OFDM system would need Hamming windowing ($-43$ dB) to match PMCW performance.

With $\alpha N_c$ pilot subcarriers, the sensing observation has $\alpha N_c M$ measurements. The per-measurement SNR is $\rho = P_t |h|^2 / (N_c \sigma^2)$ (power per subcarrier). The effective sensing SNR after matched filtering is $\text{SNR}_{\text{sens}} = \alpha N_c M \rho$.

With $(1-\alpha) N_c$ data subcarriers, the ergodic rate is $R = (1-\alpha) N_c \Delta f \log_2(1 + \rho)$ bits/s.

The pair $(\text{SNR}_{\text{sens}}, R)$ is parametrised by $\alpha \in [0, 1]$. As $\alpha$ increases, sensing SNR increases linearly while communication rate decreases linearly. The Pareto frontier is the line segment connecting $(0, N_c \Delta f \log_2(1+\rho))$ to $(N_c M \rho, 0)$. The optimal operating point depends on the relative importance of sensing and communication.

After OTFS demodulation, the observation for a single target is $Y_{\mathrm{DD}}[l,k] = \alpha h_{\mathrm{DD}}[l,k] + W[l,k]$ where $h_{\mathrm{DD}}$ is concentrated near $(\tau, \nu)$. In the time-frequency domain (equivalent), $Z[n,m] = \alpha e^{-j2\pi n \Delta f \tau} e^{j2\pi m T_{\mathrm{sym}} \nu} + \tilde{W}[n,m]$.

The FIM for $\boldsymbol{\theta} = (\tau, \nu)$ is $[\mathbf{J}]_{ij} = \frac{2}{\sigma^2} \text{Re}\left(\frac{\partial \boldsymbol{\mu}^H}{\partial \theta_i} \frac{\partial \boldsymbol{\mu}}{\partial \theta_j}\right)$ where $\boldsymbol{\mu}$ is the noise-free signal. $J_{\tau\tau} = \frac{2|\alpha|^2}{\sigma^2} (2\pi \Delta f)^2 \sum_{n=0}^{N_c-1} n^2 \cdot M = \frac{8\pi^2 |\alpha|^2 M \Delta f^2}{\sigma^2} \frac{N_c(N_c-1)(2N_c-1)}{6}$.

For large $N_c$ and $M$: $\text{CRB}(\tau) \approx \frac{3}{(2\pi)^2 \text{SNR} \cdot M \cdot N_c^3 \Delta f^2} = \frac{3}{(2\pi)^2 \text{SNR} \cdot M \cdot W^{2} / N_c}$. Similarly, $\text{CRB}(\nu) \approx \frac{3}{(2\pi)^2 \text{SNR} \cdot N_c \cdot M^3 T_{\mathrm{sym}}^2}$. These are the standard CRBs for frequency estimation from $N_c M$ samples of a complex sinusoid --- identical to what a dedicated radar waveform achieves with the same bandwidth and CPI.

$\mathbf{y} = \mathbf{A} \mathbf{c} + \mathbf{w}$ where $\mathbf{y} \in \mathbb{C}^{256 \times 64}$ (vectorised to $\mathbb{C}^{16384}$), $\mathbf{c} \in \mathbb{C}^{1024 \times 256}$ (vectorised to $\mathbb{C}^{262144}$) is $K$-sparse with $K = 20$, and $\mathbf{A} \in \mathbb{C}^{16384 \times 262144}$ is a partial 2D DFT evaluated on the $4\times$ oversampled grid.

The sensing matrix $\mathbf{A} = \mathbf{P}_\Omega (\mathbf{F}_{G_\nu} \otimes \mathbf{F}_{G_\tau})$ where $\mathbf{P}_\Omega$ selects the $N_c M = 16384$ measurement rows. Partial Fourier matrices satisfy $2K$-RIP with constant $\delta_{2K} < 0.5$ when $N_c M \geq C K \log^4(G_\tau G_\nu) = C \times 20 \times \log^4(262144)$. Since $\log^4(262144) \approx 18^4 \approx 10^5$, we need $N_c M \gg 2 \times 10^6$, which our $16384$ measurements do NOT satisfy.

The RIP bound is pessimistic for structured (Fourier) matrices. In practice, OFDM sensing with $N_c M / K = 16384/20 = 819$ measurements per target works well for sparse recovery (LASSO, OAMP) even though the theoretical RIP bound is not met. The oversampled grid allows sub-resolution target localisation.

$\mathbf{A}_{\mathrm{FMCW}} = \mathbf{F}_M \otimes \mathbf{F}_{N_s}$ (two DFT factors). Cost: $O(N_s M \log(N_s) + N_s M \log(M)) = O(P \log P)$ where $P = N_s M$.

$\mathbf{A}_{\mathrm{OFDM}} = \mathbf{P}_\Omega (\mathbf{F}_M \otimes \mathbf{F}_{N_c})$ (partial Kronecker DFT). Cost: $O(N_c M (\log N_c + \log M))$ via row-column FFT. $\mathbf{P}_\Omega$ is a row selection (free).

$\mathbf{A}_{\mathrm{OTFS}} = \mathrm{diag}(\mathbf{x}) \cdot \mathbf{C}_{\mathrm{2D}}$ where $\mathbf{C}_{\mathrm{2D}}$ is BCCB. Cost: $O(N_c M \log(N_c M))$ via 2D-FFT to diagonalise the BCCB part, plus element-wise multiplication with $\mathbf{x}$.

$\mathbf{A}_{\mathrm{PMCW}} = \mathbf{F}_M \otimes \mathbf{S}_L$ where $\mathbf{S}_L$ is circulant (code shifts). Cost: $O(L M \log L + L M \log M)$ since circulant products use FFT. All four waveforms support $O(P \log P)$ matrix-vector products, making iterative sparse recovery feasible for scenes with $Q \gg P$.