Exercises

V2X = Vehicle-to-Everything: an umbrella for vehicular wireless links.

V2I: vehicle-to-infrastructure. Example: traffic signal coordination, RSU-based tolling. V2V: vehicle-to-vehicle. Example: platooning, cooperative braking. V2P: vehicle-to-pedestrian. Example: collision warning to smartphone. V2N: vehicle-to-network. Example: cellular OTA software updates, streaming.

$v_{\text{rel}} = 100 + 100 = 200$ km/h = 55.6 m/s.

$\lambda = c/f_0 = 3 \times 10^8 / 77 \times 10^9 = 3.9$ mm.

$\nu = 55.6 / 0.0039 = 14{,}300$ Hz = 14.3 kHz.

14.3 kHz vs 120 kHz (5G NR mmWave subcarrier): $\sim 12\%$ of subcarrier. Significant ICI for OFDM.

$v_{\text{rel}} = v_1 + v_2$: Doppler doubles (relative to V2I where BS is stationary).

Either vehicle can change trajectory; prediction harder.

Vehicles pass each other, adjacent trucks, signs — rapid LOS-NLOS transitions.

$T_{\text{loop}} \cdot p^{-N} \leq T_{\text{stable}}$.

$0.8 \times 10^{-3} \cdot (10^{-4})^{-5} = 0.8 \times 10^{-3} \cdot 10^{20} = 8 \times 10^{16}$ s.

$T_{\text{stable}} = 1$ s. Condition: $8 \times 10^{16} \leq 1$? FALSE. Platoon is NOT string-stable for $p = 10^{-4}$ and $N = 5$.

Need tighter reliability. For stability: $p \geq (T_{\text{loop}}/T_{\text{stable}})^{1/N} = (0.8 \cdot 10^{-3})^{1/5} = 0.2$, i.e. $p \leq 1$. Trivially satisfied at $p = 10^{-4}$. Actually the inequality is reversed: smaller $p$ is worse. Condition is that $T_{\text{loop}} \cdot p^{-N}$ is bounded — which for a safe aero constant requires $p \geq p^* = (T_{\text{loop}}/T_{\text{stable}})^{1/N}$. For our case: need $p \geq 0.0002$. Our $p = 10^{-4} > 10^{-4}$ holds marginally. Actually the physical stability requires that packet losses do not accumulate — bound $p^{-N} \leq T_{\text{stable}}/T_{\text{loop}} = 1250$, i.e. $-N \log p \leq \log 1250$: $5 \log(10^4) = 20$ vs $\log 1250 = 3.1$. Violated.

$p = 10^{-4}$ is too loose for 5-vehicle platoon. Need $p = 10^{-1}$ or better loss-handling (retransmission).

$L = (4\pi \cdot 20 / 0.004)^2 = (6.28 \times 10^4)^2 = 4 \times 10^9$ = 96 dB.

Tx power: 20 dBm. Tx antenna gain (8-element): 12 dB. Rx antenna gain: 12 dB. Pathloss: -96 dB. Noise floor (100 MHz): -174 + 80 + 3 = -91 dBm (incl. 3 dB NF). SNR = 20 + 12 + 12 - 96 - (-91) = 39 dB.

Ample SNR for high-order modulation (64-QAM at 20+ dB), supporting 1 Gbps data rate at 100 MHz bandwidth.

Range is set not by noise but by obstacles. At 20 m, LOS probability > 95%. Beyond 100 m: NLOS probability rises sharply, link may degrade.

Each vehicle: 7 kbps.

10 vehicles: 70 kbps.

With SE 4 bps/Hz: 70 kbps / 4 = 17.5 kHz.

Tiny compared to 100 MHz V2V allocation. CP has negligible bandwidth overhead; constraint is latency and reliability.

Vehicle B's OTFS-ISAC detects pedestrian with standard algorithm: position, velocity, class.

B broadcasts DD-scene with pedestrian entry.

A's V2V receiver decodes B's scene. Transforms pedestrian position to A's coordinate system using B's location + timestamp. Fuses with A's own sensing (which shows no pedestrian). Flags pedestrian as "remote-only" / "unconfirmed".

A slows or changes lane to avoid potential pedestrian. Does not rely on unconfirmed target for aggressive maneuvers. Full confirmation when truck clears and A sees pedestrian.

Without CP: A sees pedestrian only at LOS range (~10 m), emergency braking required. With CP: A's decision happens at ~50 m range, smooth avoidance.

V2V safety messages: 100 Hz frame rate. Each vehicle sends 100 messages per second.

3600 seconds × 100 = 360{,}000 messages per vehicle per hour.

Collision risk tolerable if $< 1$ in $10^6$ hours of driving. Conservatively: $10^{-11}$ per message failure.

Per-message: $10^{-11}/\text{msg}$. Factoring in multiple messages needed for a collision event (several frames), per-message reliability of $10^{-5}$ is the practical engineering target.

OFDM at high mobility achieves $10^{-2}$-$10^{-3}$ reliability → unsafe. OTFS at $10^{-5}$-$10^{-6}$ → safe.

$P_r = P_t G_t G_r (\lambda/(4\pi d))^2 / L$, where $L$ is environmental loss factor.

$\mathrm{SNR} = P_r/(kTW) = P_t G_t G_r \lambda^2 / (16\pi^2 d^2 kTW L)$.

For target SNR = $\mathrm{SNR}_0$: $d^2 = \text{const}/\lambda^{-2}/W$. $d = \text{const} \cdot \lambda/\sqrt{W}$.

$d \propto 1/f_0 \cdot W^{-1/2}$. Doubling frequency halves range; doubling bandwidth reduces range by $\sqrt{2}$.

• 5.9 GHz, 10 MHz: range 1 km.
• 77 GHz, 1 GHz: range 90 m.
• 28 GHz, 100 MHz: range 400 m. Each band has specific range regime.

Drag reduction $= D_0 \cdot e^{-d/d_0}$, where $D_0 \approx 0.25$ (25% at zero spacing), $d_0 \approx 10$ m (characteristic scale).

Truck 1 (lead): no savings. Truck 2: $D_0 (1 - e^{-d/d_0})$ = 0.25 · 0.86 = 22% (if d = 5 m). Trucks 3, 4: similar.

Average savings: (0 + 22 + 22 + 22)/4 = 16.5%.

Matches industry reports of ~15-20% fuel savings for 4-truck platoons at 5-10 m spacing.

For $N = 5$ platoon, $T_{\text{stable}} = 1$ s, $T_{\text{loop}} = 2$ ms (baseline OFDM): $p^5 \geq T_{\text{loop}}/T_{\text{stable}} = 2 \times 10^{-3}$, i.e. $p \geq 0.29$.

At 200 km/h closing, OFDM BER floor ~$10^{-2}$ at 20 dB SNR. Packet reliability (100-byte packet, uncoded): $p \sim (1 - 10^{-2})^{800} = 0.0003$. Way below 0.29.

BER $\sim 10^{-10}$ at same SNR (due to DD diversity). $p \sim (1 - 10^{-10})^{800} = 1 - 8 \times 10^{-8} \approx 1$.

OFDM cannot achieve required platoon reliability at 200+ km/h. OTFS can. Hence OTFS is necessary.

Each vehicle broadcasts: position, velocity, intended path (straight/turn), ETA at intersection center.

Protocol: if two vehicles' intended paths conflict (cross at same time within tolerance), resolution needed. Detect via geometric check: do trajectories intersect within 1-2 s window?

First-come-first-served baseline. Override: (i) Emergency vehicle (lights/siren): immediate priority. (ii) Pedestrian in crosswalk: all vehicles yield. (iii) Larger vehicle: smaller yields (for safety).

Prioritized vehicle accelerates normally. Yielding vehicles slow to give $\geq 2$-second separation. RSU monitors compliance; overrides if vehicles violate.

If V2V messages lost or ambiguous: all vehicles default to stop-sign behavior. Safety margin: $\sim 1$ second.

100-byte packet = 800 bits. Assuming uncoded BPSK, packet error rate = $1 - (1 - \text{BER})^{800} \approx 800 \cdot \text{BER}$ for small BER.

For $p = 10^{-5}$ packet reliability: BER $\leq 10^{-5}/800 = 1.25 \times 10^{-8}$.

$\text{BER} \leq \binom{2P-1}{P}/(4 \text{SNR})^P$. For $P = 8$: $\binom{15}{8}/(4\rho)^8 \leq 1.25 \times 10^{-8}$. $6435/(4\rho)^8 \leq 10^{-8}$. $(4\rho)^8 \geq 6.4 \times 10^{11}$. $4\rho \geq (6.4 \times 10^{11})^{1/8} = 10^{1.46}$. $\rho \geq 7.2$.

10 log(7.2) = 8.6 dB. Required SNR: ~9 dB. Coding (rate 0.5): relaxes to ~5 dB. Comfortable margin at typical V2V SNR of 20-30 dB.

OFDM diversity = 1: BER $= 1/(4\rho)$. For same packet reliability: $\rho \geq 10^7 = 70$ dB. Impossible at V2V link margin. Hence OFDM cannot deliver $p = 10^{-5}$ at high mobility.

BS serving V2I at 28 GHz. Vehicle transmits V2V sidelink to another vehicle on adjacent channel or overlapping band.

V2V Tx power $\sim 20$ dBm. BS receives V2V signal at distance $d$: $P_r \sim 20 + 30 - 30 \log d$ dBm. For $d = 100$ m: $P_r \sim -40$ dBm — well above BS noise floor (-91 dBm/MHz).

Without isolation, V2V interferes severely with V2I. Mitigation: frequency division (V2V on separate channel), code division (different scrambling), spatial separation (BS beamforming nulls toward V2V).

In shared-spectrum deployments, adjacent-channel isolation must exceed $\sim 50$ dB. Achievable with proper filter design. Or use separate bands: V2V on 77 GHz (automotive), V2I on 28 GHz (cellular mmWave) — frequency-isolated by band design.

OFDM: $\sim 10^8$ ops/s at mmWave. OTFS: $\sim 3 \times 10^8$ ops/s (3× more due to DD-grid processing). Baseband SoC: 1-2 W for OFDM; 2-4 W for OTFS.

Target radiated power 20 dBm (100 mW) average. OTFS PAPR ~7 dB; OFDM PAPR ~8-10 dB. OFDM requires 10-12 dB back-off; OTFS requires 7 dB. PA efficiency: 30% at back-off. OTFS: 100/0.3 × 5 dB = ~500 mW PA input. OFDM: ~800 mW PA input.

OFDM: 1-2 W (BB) + 0.8 W (PA) = 1.8-2.8 W. OTFS: 2-4 W (BB) + 0.5 W (PA) = 2.5-4.5 W.

OTFS uses $\sim 30\%$ more power than OFDM. However: OTFS requires no re-transmission (high reliability), while OFDM at high mobility requires $\sim 50\%$ re-transmissions. Net energy per delivered packet: OTFS is roughly equal or less.

Tier 1 (safety): V2V BSM, EEBL, platooning control. Latency $\leq 1$ ms, reliability $10^{-5}$. Tier 2 (cooperative perception): V2V scene sharing. Latency 10 ms, reliability $10^{-3}$. Tier 3 (infotainment): V2N streaming, software updates. Latency 100 ms, reliability $10^{-2}$.

Tier 1: 77 GHz OTFS sidelink. Tier 2: 77 GHz OTFS sidelink or cellular OTFS-V2X. Tier 3: 28 GHz 5G NR OFDM uplink/downlink.

PHY: per-tier waveform. MAC: 5G NR-V2X sidelink for T1/T2, 5G NR uplink for T3. Network: IPv6 for all. Transport: UDP for T1 (low overhead), TCP for T3. Application: DENM/BSM (ETSI ITS) for T1, HTTP for T3.

Vehicle software selects tier per application. Runtime reallocation if link quality changes. Fallback: if OTFS sidelink fails, T1 drops to C-V2X OFDM (with latency penalty but graceful degradation).