Exercises

ISAC: one waveform serves both data transmission and sensing simultaneously.

(1) Data throughput (bits/s). (2) Sensing accuracy (CRLB). (3) Complexity/latency (real-time feasibility).

OTFS as the natural ISAC waveform. Its thumbtack ambiguity and DD-native signaling enable the simultaneous optimization of objectives 1-2, within the constraint of (3).

Data symbols live on the DD grid. Target scene = DD-domain channel. Both use the same grid.

One waveform, one receiver, dual tasks solved simultaneously. Joint estimation-detection on the same grid. No mode switching.

$MN = 8192$. $\log(MN) = 13$. $\log(M) = 9$. OTFS: $5 \cdot 8192 \cdot 13 = 5.3 \times 10^5$. OFDM time-mult: $1.5 \cdot 8192 \cdot 9 + 8192 \cdot 13/2 = 1.1 \times 10^5 + 5.3 \times 10^4 = 1.6 \times 10^5$. Ratio OTFS/OFDM: 3.3.

OTFS delivers both data and sensing per operation. OFDM delivers one at a time (time-multiplexed). Normalized: OTFS is $\sim 1.7\times$ per output.

Naive: OTFS is more expensive. Normalized for output: OTFS is nearly comparable — and it runs in continuous mode without switching penalty. Superior for time-critical applications.

$\Delta v = 3 \times 10^8/(2 \cdot 3 \times 10^{-3} \cdot 7.7 \times 10^{10}) = 0.65$ m/s.

Ped (2 m/s) vs Bike (5 m/s): $\Delta v_{\text{pair}} = 3$ m/s > $0.65$. Resolved. Ped vs Car: $13$ m/s > $0.65$. Resolved. Bike vs Car: $10$ m/s > $0.65$. Resolved.

All three targets resolvable in velocity. Also resolvable in range if they are at different distances. Full scene characterization possible.

$\partial A_s / \partial \tau$ and $\partial A_s / \partial \nu$ for a random-data OTFS signal. Each derivative involves sums over random data terms.

$\mathbb{E}[\partial_\tau A_s \cdot (\partial_\nu A_s)^*] =$ double sum over data. For i.i.d. data with $\mathbb{E}[X X^*] = \delta$: only diagonal terms survive in each dimension; cross-dimension terms average out.

$F_{\tau\tau} = \rho W^2$, $F_{\nu\nu} = \rho T^2$, $F_{\tau\nu} = 0$. Diagonal. Decoupled CRLB.

No range-velocity coupling in OTFS sensing — unlike chirp waveforms where the ambiguity is a diagonal ridge creating coupling. OTFS is coupling-free by construction.

$MN = 32768$. $\text{PAPR} \approx \log(32768) \approx 10.4$ dB.

$\log(M) = \log(1024) \approx 10$. $\text{PAPR}_{\text{OFDM}} \approx 10$ dB.

OTFS 0.4 dB worse at this ratio. Less than the standard 4 dB gap because $MN/M = 32$ is relatively small.

0.4 dB PA back-off: effective $\sim 15\%$ transmit power reduction vs OFDM. Small but measurable.

$\ell(\Theta, X; Y) = \log p(Y | X, \Theta) + \log p(X)$. $(Y, p(X)$ are data; $\Theta, X)$ are estimates.

$X^{(t)} = \arg\max_X \ell(\Theta^{(t-1)}, X; Y)$. By definition: $\ell(\Theta^{(t-1)}, X^{(t)}; Y) \geq \ell(\Theta^{(t-1)}, X^{(t-1)}; Y)$.

$\Theta^{(t)} = \arg\max_\Theta \ell(\Theta, X^{(t)}; Y)$. $\ell(\Theta^{(t)}, X^{(t)}; Y) \geq \ell(\Theta^{(t-1)}, X^{(t)}; Y)$.

Combining: $\ell(\Theta^{(t)}, X^{(t)}; Y) \geq \ell(\Theta^{(t-1)}, X^{(t-1)}; Y)$. Monotonic increase.

Bounded above (likelihood is bounded), monotone: converges. Limit is a local maximum. $\blacksquare$

$\Delta v = 3 \times 10^8/(2 \cdot 10^{-2} \cdot 2.8 \times 10^{10}) = 0.54$ m/s.

$0.54 < 1$ m/s. Requirement met with margin.

$MN = 500 \times 10^6 / 30 \text{kHz} \cdot 10\text{ms}/33\mu s = 16667 \cdot 303 = 5 \times 10^6$. QPSK at rate 3/4: $5 \times 10^6 \cdot 1.5/10\text{ms} = 750$ Mbps. Ample data capacity.

Velocity resolution: 0.54 m/s (< 1 m/s required). Data: 750 Mbps. 10 ms latency: acceptable for UAV systems. OTFS-ISAC meets all requirements.

With fraction $\eta$ of power to sensing:

• Sensing SNR: $\rho \cdot \eta$
• Data SNR: $\rho \cdot (1 - \eta)$

$D_{\text{sens}}(\eta) \sim 1/(\rho \eta)$.

$R_{\text{comm}}(\eta) \sim \log(1 + \rho(1 - \eta))$ (at high SNR, $\log(\rho(1-\eta))$).

Pareto frontier: $\{(R(\eta), D(\eta)^{-1})\}_{\eta \in [0, 1]}$. At $\eta = 0$: pure data, $D^{-1} = 0$ (infinite sensing error). At $\eta = 1$: pure sensing, $R = 0$. At intermediate: joint.

For typical ISAC: $\eta \sim 0.05$–$0.1$ (5-10% to sensing). Data rate $\sim 95\%$ of pure-data; sensing accuracy usable. Moderate $\eta$ is the sweet spot — OTFS naturally operates here because random data is inherently good for sensing (Chapter 12.2 Theorem).

$\Delta v = c/(2Tf_0) = 10^{-3}$. $T = c/(2 \cdot 10^{-3} \cdot 6 \times 10^{10}) = 2.5$ s.

$T = 2.5$ s is very long. Fine for steady-state monitoring; not real-time.

$T_{\text{frame}} = 10$ ms, $N_{\text{frames}} = 250$. Total integration: 2.5 s. Same effective resolution. Real-time frame processing.

$(T_{\text{frame}}, N_{\text{frames}}) = (10 \text{ ms}, 250)$. Per-frame: $\Delta v = 0.25$ m/s. Aggregated: 1 mm/s. Heart rate detectable.

$3 \times 10^6 \cdot 100 = 3 \times 10^8$ ops/sec.

Modern 5G SoC: $\sim 10^{10}$ ops/sec single core. OTFS-ISAC at $\sim 3\%$ of a single-core budget. Readily feasible; room for other processing.

$F_{\nu\nu} = \rho \cdot (2f_0/c)^2 T^2$.

$\sigma_\nu^2 \geq 1/F = c^2/(\rho T^2 (2f_0)^2)$. $\sigma_\nu = c/(2Tf_0\sqrt{\rho})$.

$\sigma_v = c\sigma_\nu/(2f_0) = c^2/(4Tf_0^{2}\sqrt{\rho}) \cdot 2 = c/(2Tf_0\sqrt{\rho})$. Same formula.

Velocity accuracy = resolution / SNR^{1/2}. At 20 dB: 1/10th resolution. At 40 dB: 1/100th resolution. Super-resolution enables tracking much finer than $\Delta v$.

Resources $(W, T, P_t)$ split 50-50. Each task gets half. Radar SNR: $\rho/2$. Data SNR: $\rho/2$. Sensing CRLB: $\sigma^2 \sim 2/\rho$. Data rate: $\log(1 + \rho/2)$.

Both tasks use full resources with $\eta = 0.05$ (say). Sensing SNR: $\rho \cdot 0.05$. Data SNR: $\rho \cdot 0.95$. Sensing CRLB: $\sigma^2 \sim 20/\rho$. Data rate: $\log(1 + 0.95\rho)$ $\approx \log(\rho)$.

OTFS-ISAC: data rate $\sim \log(\rho)$ (vs $\log(\rho/2) = \log(\rho) - 1$ for dedicated). OTFS-ISAC: sensing CRLB $20/\rho$ vs $2/\rho$ for dedicated. OTFS-ISAC trades 10× sensing CRLB (6 dB worse) for $\sim 0.5$ dB data rate gain.

Depends on the application. Dedicated: better for pure radar

• decent comms. OTFS-ISAC: better when real-time concurrent operation matters. The Pareto frontier is application-specific.

$\sigma_R = 0.01$ m, $W = 200$ MHz, $c = 3 \times 10^8$. $0.01 = 3 \times 10^8/(2 \cdot 2 \times 10^8 \cdot \sqrt{2\pi^2 \rho}) = 0.75/\sqrt{2\pi^2\rho}$. $\sqrt{2\pi^2\rho} = 75$. $\rho = 75^2/(2\pi^2) = 285$ (24.5 dB).

Path loss at 50 m, 77 GHz: FSPL $= 20\log(50) + 20\log(77 \times 10^9) - 147$ $= 34 + 158 - 147 = 45$ dB. With 10 dBi antenna gains each side: net $45 - 20 = 25$ dB path loss. For $\rho = 24.5$ dB receive SNR with noise floor $-85$ dBm: $P_t = 24.5 + 25 - 85 = -35.5$ dBm = $280$ nW. Absurdly low.

1-cm accuracy at 50 m is easily achievable at even modest transmit power. OTFS-ISAC's sensing accuracy is typically limited by scene clutter, not SNR.

Optimize: $(R_{\text{comm}}, \sigma_R^{-1}, \text{PAPR}^{-1}, L^{-1})$. 4-dimensional point in a 4D Pareto set.

Fix $(W, T, P_t, f_0)$. Waveform choice $s(t)$ controls tradeoffs. PAPR depends on prototype pulse + precoder. Latency = $T + \text{processing}$.

Non-convex in general. OTFS with Hamming windowing occupies a specific region; optimized pulses (Chapter 20) push further.

Active research: Zadoff-Chu-precoded OTFS reduces PAPR; DPSS pulses optimize simultaneously for frequency containment and sensing. Pareto frontier in 4D is a research-level problem. Real deployments select specific operating points via constrained optimization.

Established OTFS's thumbtack ambiguity and CRLB-matching. Technical foundation for all subsequent OTFS-ISAC work.

Synthesized the ISAC framework, positioning OTFS as 6G ISAC waveform. Applications, signal processing, standardization considerations.

These two papers define OTFS-ISAC as a field. Chapter 12 of this book is the textbook summary of their framework.