Exercises

Peak rate, user rate, latency, connection density, mobility, energy efficiency. Plus new: sensing resolution, AI integration, NTN.

Mobility (OFDM ceiling ~400 km/h, OTFS unlimited) and ISAC (OTFS natural, OFDM requires heavy adaptation). Both point toward OTFS.

Peak rate, latency, density are achievable by both with proper numerology. Energy efficiency depends on mobility scenario.

DFT-s-OFDM = OFDM preceded by DFT precoding. Converts multicarrier signal to single-carrier-like in time domain.

CP-OFDM: ~10 dB PAPR. DFT-s-OFDM: ~5 dB. Reduction: ~5 dB.

UE PAs are battery-limited. Lower PAPR allows less PA back-off, more efficient Tx power use. Extends battery life and range.

BS PAs not battery-limited. CP-OFDM used for downlink (simpler structure). DFT-s only for uplink.

Users separated across multiple orthogonal dimensions beyond 2D OFDMA.

Time, frequency, code, spatial (beams/antennas), delay, Doppler. Total 6 dimensions.

OFDMA uses only time + frequency. MDMA uses all.

Orthogonal resources = product of per-dimension resources. $D$ dimensions → $64\times$ more users over OFDMA (factor $n_{\text{delay}} n_{\text{Doppler}} \approx 64$).

For OFDM usability: $\nu = v f_0/c \leq 0.1 \Delta f$.

$v \leq 0.1 \Delta f c / f_0 = 0.1 \cdot 120 \cdot 10^3 \cdot 3 \cdot 10^8 / (28 \cdot 10^9)$ $= 3.6 \cdot 10^{12} / 2.8 \cdot 10^{10} = 128$ m/s $\approx 460$ km/h.

Textbook bound: 460 km/h. Real with fading + UE constraints: ~100-150 km/h for acceptable BER at realistic SNR.

Automotive (100-150 km/h): OFDM works at 28 GHz $\mu = 3$. HST (200-350 km/h): marginal. LEO (>7000 km/h): OTFS mandatory.

CP-OFDM: 10-11 dB. DFT-s-OFDM: 5-6 dB. OTFS: 7-8 dB.

OTFS PAPR is between CP-OFDM and DFT-s. Compared to DFT-s: penalty ~2 dB. Compared to CP-OFDM: advantage ~3 dB.

Uplink (UE): DFT-s-OFDM wins PAPR but OTFS wins mobility. Choice depends on scenario.

BS PAs not PAPR-critical. OTFS has mobility advantage.

Static/vehicular: 1 Gbps each (full rate). HST: 200 Mbps (degraded). Aggregate: 900 × 1 + 100 × 0.2 = 920 Gbps.

Static: 900 Mbps (PAPR penalty ~10%). Vehicular: 980 Mbps. HST: 1 Gbps. Aggregate: 900 + 294 + 100 = 1294 Gbps. Hmm — actually this seems high. Let me reconsider: Static: 900 × 0.9 = 810 Gbps. Vehicular: 300 × 0.98 = 294 Gbps. HST: 100 × 1.0 = 100 Gbps. Sum: 1204 Gbps.

Static/Vehicular: OFDM at 1 Gbps × 900 = 900 Gbps. HST: OTFS at 1 Gbps × 100 = 100 Gbps. Sum: 1000 Gbps.

Dual-waveform: 1000 Gbps. OFDM-only: 920. OTFS-only: ~1200 (if PAPR penalty less than 10%). Closer analysis needed; dual-waveform is generally optimal in practice.

$K_{\max} = 100 \cdot 1000 = 10^5$ users.

$K_{\max} = 100 \cdot 1000 \cdot 8 \cdot 8 = 6.4 \cdot 10^6$ users.

64× more users with MDMA than OFDMA. Matches ITU's $10^7$-devices/km² target.

Orthogonal resources: each user gets one resource slot. Rate per user: log(1 + SNR) per resource × resource per user. Aggregate: K × log(1 + SNR). MDMA aggregate: 64x OFDMA. Matches 6G peak-rate target.

3 GFLOPS. Silicon: ~50 mm² at 3nm. Power: ~0.5 W.

5 GFLOPS (5/3 = 1.67× of 5G NR). Silicon: ~50 × 1.67 = 83 mm² (scales roughly linearly at fixed node). Power: ~0.83 W.

Cannot just add — shared FFT and resources. Net: ~75 mm², ~0.75 W. +50% vs OFDM-only baseline.

Acceptable for 6G UEs. 3nm node (2024+) absorbs the cost. Consumer UE price increment: ~$20-30.

After ISFFT + Heisenberg: $x(t) = \sum_{\ell, m} s[\ell, m] g(t - \ell \Delta\tau) e^{j 2\pi m \Delta\nu t}$ = 2D sum of QAM symbols.

Large $MN$: $x(t)$ is approximately complex Gaussian by CLT. Envelope: Rayleigh distribution.

$P_{\text{peak}}/P_{\text{avg}} = -\log(\alpha)$ for target exceedance prob $\alpha$. For $\alpha = 10^{-4}$: PAPR $\leq 9.2$ dB.

PAPR $\leq 8$ dB. Matches empirical measurements.

Tone reservation, clipping, partial transmit sequence. Can reduce by 2-3 dB with modest complexity.

SE $= D \log_2(1 + \mathrm{SNR})$. Aggregate = $D \times \log(1 + \text{SNR})$.

Real channels have coupling $\rho$ between domains. Effective SINR per domain: $\mathrm{SNR} / (1 + \rho \cdot (D-1) \cdot \text{noise-enhancement})$.

SINR reduction: 4 × 0.1 = 40% noise enhancement. SINR_eff = SNR/1.4 = 0.71 × SNR. SE per domain: log(1 + 0.71 SNR) ≈ 0.9 × log(1 + SNR). Aggregate: 4.5 × log(1 + SNR). 4.5× gain over OFDMA.

Expected 3-4× gain in practice (not full 5×). Still dramatic.

Per UE: estimate mobility, channel quality per-waveform, service type. Pick waveform (OFDM or OTFS) that maximizes expected per-UE rate. Complexity: O(K).

OFDMA across UEs assigned to OFDM. Time-frequency resource allocation with fairness. Complexity: O(K × RB) per slot.

Multi-domain allocation across UEs assigned to OTFS. DD grid

• spatial beams. Complexity: O(K × D) per slot.

Ensure OFDM and OTFS UEs don't overlap in time-frequency resources. Share spectrum efficiently.

Per-slot complexity: O(K × (RB + D)). For K = 200: 10⁵ ops. Real-time on server CPU.

FRAND rate: ~0.5%. Total license pool: ~$5B over 10 years.

FRAND rate: ~1%. Total: $10B+ over decade.

Expected FRAND: 0.5-1%. Total license pool: ~$5-10B for 6G commercial decade.

Chip vendor margins: ~5-10%. OTFS license fee: 0.5-1% = 5-10% of margin. Significant but absorbable. Chip prices may absorb the cost.

Academic alternative to Cohere IPR (CommIT contributions). 3GPP may standardize hybrid with minimal Cohere lock-in.

Rel. 20 study item on "Beyond-OFDM Waveforms for 6G". Multiple proposals (OTFS, enhanced-OFDM, hybrid). 3GPP technical evaluation framework established.

Companies submit technical contributions. Simulations demonstrate per-use-case performance. IPR disclosures by Cohere and academic affiliates.

Third-party evaluation. Performance comparisons. Implementation assessments.

Working group consensus on waveform choice. Likely: dual- waveform (OTFS + OFDM) for 6G.

Final report. Recommendation for Work Item in Rel. 21.

Detailed specification begins. Expected completion 2029-2030.

Existing 5G NR infrastructure. Enhanced NR NTN. OTFS in pilot trials only. Capex: $10M for OTFS testbed. Regulatory: no changes.

Participate in 3GPP study item. Partner with chip vendors (Qualcomm, MediaTek) for OTFS-capable UEs. Infrastructure vendor partnerships. Capex: $50M for pre-prod OTFS.

Initial dual-mode BSs in select cities. V2X on 77 GHz. LEO integration (via Starlink partnership). Capex: $500M.

Mass deployment of dual-mode. Urban densification. Cell-free OTFS in high-density sites. Capex: $2B.

OTFS in all new UE models. Legacy 5G maintained. Annual capex amortization: $500M/year over 5-10 years.

1. Enhanced OFDM proponents win study item. Risk: 20%.
2. IPR disputes. Risk: 10%.
3. Vendor indecision. Risk: 15%.
4. Geopolitical (US-China-EU tensions). Risk: 15%. Aggregate: 40-50% probability of delay.

CommIT contributions (Chapters 17-18) present irrefutable use cases (cell-free + LEO). IPR: Cohere FRAND + academic alternatives minimize friction. Vendor: Qualcomm commitment signals industry momentum. Geopolitical: decoupled standards (NR vs beyond) as backup.

50-60% probability Rel. 21 (2028-2030) on time. 40-50% probability Rel. 22 (2030-2032). Either way: commercial by 2032.

Dual-mode hardware platforms with software-definable PHY enable flexibility. No stranded investment.

4 billion 6G UE shipments (2035): each with OTFS chip. $10 per chip, 60$10 = $24B/year chip revenue.

10M BS units (global), with OTFS silicon. $500 per chip. Refresh every 5 years: 2M ×$500 = $1B/year.

6G premium services (mobility, NTN, V2X): 500M users at $20/month premium =$120B/year.

6G BS deployment: $500k per site × 10k new sites/year =$5B/year. License fees, testing, other: +30%.

Chip: $25B/year. Infrastructure:$6B/year. Service: $120B/year. Aggregate 2035 revenue related to OTFS-enabled 6G: ~$150B/year. 10-year cumulative (2030-2040): $1-1.5T. Significant economic stake.