Exercises

$\Delta R = 3 \times 10^8/(2 \cdot 5 \times 10^7) = 3$ m.

$\Delta v = 3 \times 10^8/(2 \cdot 2 \times 10^{-3} \cdot 2.4 \times 10^{10}) = 3.125$ m/s.

Vehicle-scale range resolution, walking-scale velocity resolution. Appropriate for automotive ISAC.

$1/(TW) = 1/(10^{-5} \cdot 10^8) = 1/10^3 = 10^{-3}$ (unitless in $\tau\cdot\nu$ space).

$\tau$ main-lobe width: $1/W = 10$ ns. $\nu$ main-lobe width: $1/T = 100$ kHz. Product: $10^{-6}$ s·Hz = $10^{-3}$ unitless.

The time-bandwidth product is $10^3$ — "large" $TW$. Many resolution cells in the unambiguous region.

$R_{\max} = 3 \times 10^8 \cdot 64/(2 \cdot 2 \times 10^7) = 480$ m.

$v_{\max} = 3 \times 10^8 \cdot 32/(4 \cdot 4 \times 10^{-3} \cdot 5 \times 10^9) = 120$ m/s.

Covers $\pm$120 m/s highway speeds and 480 m ranging — excellent for mid-range automotive applications.

PSL = $-13$ dB (Dirichlet first sidelobe).

PSL = $-43$ dB. Suppresses by $\sim 30$ dB relative to unwindowed.

PSL = $-58$ dB. Even better.

Hamming: 36% wider main lobe. Blackman: 73% wider. Typical choice depends on application's sidelobe tolerance.

$\rho = 10^3 = 1000$. $\sigma_R^2 = (3 \times 10^8)^2/(8\pi^2 (10^8)^2 \cdot 10^3) = 9 \times 10^{16}/(8 \pi^2 \cdot 10^{19}) = 1.14 \times 10^{-4}$. $\sigma_R = 1.07$ cm.

$\Delta R = c/(2W) = 1.5$ m. Accuracy 1.07 cm = $\Delta R/140$. Sub-resolution accuracy; high-SNR target tracking is possible.

$\Delta v \leq 4$ m/s (velocity separation).

$c/(2Tf_0) = 4$ m/s. $T = c/(8f_0) = 3 \times 10^8/(8 \cdot 2.8 \times 10^{10}) = 1.34 \times 10^{-3}$ s. $T = 1.34$ ms. Or shorter — e.g., 1 ms.

1.34 ms is shorter than typical OTFS frame (5-10 ms). Easy to achieve. Cyclist-pedestrian separation is resolvable at mmWave.

$A_s(\tau, \nu) = \int s(t) s^*(t-\tau) e^{-j 2\pi\nu t} dt$. $s(t) s^*(t-\tau) = \sqrt{2/T^2} \cdot e^{-(t^2 + (t-\tau)^2)/T^2}$.

$(t^2 + (t-\tau)^2)/T^2 = (2t^2 - 2t\tau + \tau^2)/T^2$ $= (2/T^2)(t - \tau/2)^2 + \tau^2/(2T^2)$.

$A_s(\tau, \nu) = \sqrt{2/T^2} e^{-\tau^2/(2T^2)} \int e^{-2(t-\tau/2)^2/T^2 - j 2\pi\nu t} dt$. Complete the square in the exponent — this is a standard Gaussian integral with an imaginary shift.

$A_s(\tau, \nu) = e^{-\tau^2/(2T^2)} e^{-\pi^2 T^2 \nu^2/2}$. Jointly Gaussian in $(\tau, \nu)$. First nulls at $|\tau| \sim T$ and $|\nu| \sim 1/(T\sqrt{\pi})$.

The Gaussian pulse saturates Woodward's uncertainty principle: main-lobe area is exactly $1/(TW)$. It's the "most concentrated" ambiguity function at fixed $TW$. $\blacksquare$

$T \geq c/(2 \cdot 0.5 \cdot f_0) = 3 \times 10^8/(0.5 \cdot 1.54 \times 10^{11}) = 3.9$ ms.

Need $R_{\max} = cM/(2W) \geq 80$ m. With $M = 100$: $W \geq cM/(2 \cdot 80) = 1.875 \times 10^8 = 188$ MHz. With $M = 200$: $W \geq 375$ MHz. Choose $W = 200$ MHz (mid-range automotive).

$(W, T) = (200 \text{MHz}, 4 \text{ms})$. $\Delta R = 0.75$ m (sub-vehicle), $\Delta v = 0.5$ m/s (targets met). Data rate: $MN \cdot 2/T = 100 \cdot 14 \cdot 2/4\text{ms} = 700$ kbps QPSK (modest — but sensing is primary).

$(50, 10)$ vs $(80, 5)$: $\Delta R = 30$ m >> 1.5 m (resolved in range). Velocities 10 vs 5: $\Delta v = 5 \gg 0.65$ m/s (resolved).

$(50, 10)$ vs $(50, 25)$: same range. Velocities 10 vs 25: $\Delta v = 15$ m/s (resolved in velocity).

$(80, 5)$ vs $(50, 25)$: $\Delta R = 30$ m (resolved). $\Delta v = 20$ m/s (resolved).

All three targets are resolvable. OTFS radar's thumbtack ambiguity handles 2D range-velocity separation cleanly.

Full OTFS ambiguity has terms like $\sum_{\ell, k, \ell', k'} X[\ell, k] X^*[\ell', k']$ times coupling phases. Data-dependent cross-terms exist.

With random QAM data: $\mathbb{E}[X X^*] = \delta$. Cross-terms vanish in expectation.

For specific data realizations, off-diagonal terms are random noise of order $1/\sqrt{MN}$. The main-lobe is deterministic.

At small offsets (within main lobe), the deterministic part dominates: $A_s \approx \text{sinc}(W\tau) \text{sinc}(T\nu)$. Far from main lobe: random cross-terms dominate, giving noise-like sidelobes at $-10\log_{10}(MN)$ dB below main peak.

For detection purposes, the ambiguity is effectively separable. Sidelobes contain data-specific information but are suppressed by the $1/\sqrt{MN}$ factor. At $MN = 10^4$: $-40$ dB sidelobes — comparable to Hamming windowing.

$10^{-5} = 3 \times 10^8/(2 T \cdot 6 \times 10^{10})$. $T = 3 \times 10^8/(2 \cdot 10^{-5} \cdot 6 \times 10^{10}) = 0.25$ s.

$T = 250$ ms — one-quarter of the heartbeat period. Would provide 4 measurements per heartbeat, enough for resolution of fine chest-wall motion (heartbeat produces $\sim 500\,\mu$m chest-wall displacement).

$T = 250$ ms frame is long. Multiple frames needed for continuous monitoring. Also: the environment must be static (block fading valid) over 250 ms — reasonable for medical monitoring.

LFM waveform $s(t) = e^{j\pi\alpha t^2}$ has ambiguity $A(\tau, \nu) \sim$ diagonal ridge: peaks at $\nu = \alpha\tau$.

OTFS ambiguity is thumbtack (separable $\tau, \nu$).

• Pulse compression: good range resolution with long pulse.
• Common in radar systems (simple to generate).
• Problem: range-Doppler coupling (targets at different $(\tau, \nu)$ can produce the same ridge peak).

• Clean separation of range and Doppler (no coupling).
• Simultaneously carries data.
• Standard 5G NR-compatible waveform.

For ISAC: OTFS. For dedicated radar at $T \ll T_c$ (channel coherent): chirp or OTFS — similar performance. For dedicated radar with long dwell ($T \sim$ seconds): OTFS's Doppler structure wins.

Recall that OTFS ambiguity has cross-terms $X[\ell, k] X^*[\ell', k']$. If data is i.i.d., these average to zero at $(\ell, k) \neq (\ell', k')$.

Correlated data: $\mathbb{E}[X[\ell, k] X^*[\ell', k']] = \rho[\ell - \ell', k - k']$ — non-zero correlations at some lags.

Non-zero correlations contribute deterministic sidelobes at specific $(\tau, \nu)$ locations. Instead of $-13$ dB floor, sidelobes elevate to $-13 + 10\log_{10}(\rho)$ dB.

LDPC/Turbo coding introduces small correlations; sidelobe floor typically rises by 1-2 dB. Scrambling (data randomization) restores i.i.d. behavior and the theoretical sidelobe floor.

For ISAC, always apply a scrambling/interleaving step before transmission to whiten data statistics. Standard in 5G NR (interleaver follows the LDPC encoder).

$T = c/(2 \cdot 0.05 \cdot f_0) = 3 \times 10^8/(10^{-1} \cdot 1.2 \times 10^{11}) = 2.5 \times 10^{-2}$ s = 25 ms.

$\Delta R = 0.1$ m (1 cm at 1 m). $W = c/(2 \cdot 0.1) = 1.5$ GHz.

$W = 1.5$ GHz, $T = 25$ ms. mmWave + wide frame. $MN = WT \cdot \Delta f / \Delta f = MN$ — let's use $M = 1000, N = 50$ (standard numerology). Data rate: QPSK at $MN/T = 2{\text{Mbps}}$. Minimal, but sensing is primary.

Medical monitoring, gesture UI, fine-motion analysis. OTFS ISAC at mmWave provides simultaneous data + precision radar.

Raised-cosine: $g_{tx}(t) = \text{sinc}(t/T_s)\cos(\pi\alpha t/T_s)/(1 - (2\alpha t/T_s)^2)$. As $\alpha$ increases: wider main lobe, lower sidelobes.

OTFS ambiguity factors as $A_g(\tau, \nu) \cdot \text{sinc}(W\tau)\text{sinc}(T\nu)$. Raised-cosine $g$ contributes a rolloff factor to $A_g$. At $\tau = 0$: A_g(0, \nu) = \text{pulse spectrum at\nu}.

At rolloff $\alpha = 0.25$: first sidelobe of pulse's autocorrelation is reduced by $\sim 8$ dB vs rectangular. Overall OTFS sidelobe floor improves by similar amount: rectangular PSL $-13$ dB; $\alpha = 0.25$ PSL $\sim -21$ dB.

Rolloff widens main lobe by $\sim \alpha$ factor, reducing effective resolution. Optimal $\alpha$: balance resolution vs sidelobe. Research: Hsieh-Şahin-Arslan (2022) on OTFS PAPR + rolloff codesign.

$M \geq 2W \cdot 100/c = 2 \cdot 2 \times 10^7 \cdot 100/(3 \times 10^8) = 13.3$. Choose $M = 14$ (or 16 for power-of-2 convenience).

$v_{\max} = cN/(4Tf_0) \geq 30$. Need to choose $T$ first. At 5G NR numerology-0, $T_s = 33\,\mu$s, so $T = N T_s$. For $v_{\max} = 30$ m/s: $N/(T N_{\text{subcarriers}}) = ?$ Let's re-derive: $v_{\max} = cN/(4Tf_0) = c/(4T_s f_0)$. At $T_s = 33\,\mu$s: $v_{\max} = 3 \times 10^8/(4 \cdot 33 \times 10^{-6} \cdot 10^{10}) = 227$ m/s. Already >> 30 m/s. Any $N \geq 4$ works.

$(M, N) = (16, 14)$ — 5G-aligned, adequate for scene extent. $R_{\max} = 120$ m, $v_{\max} \gg 30$ m/s. Extra margin on velocity; range just meets requirement.