Ferkans — Interactive Telecom Tutor

The OFDM Breaking Point

Throughout this book we have seen OFDM struggle with Doppler. At V2V closing speeds (~20 kHz Doppler), OFDM ICI creates an error floor. At LEO velocities (~690 kHz Doppler), OFDM simply stops working — the Doppler is larger than any practical subcarrier spacing. This section quantifies where OFDM fails and what OTFS offers instead. The answer is not a small gain but a qualitative architectural difference: OTFS works where OFDM cannot.

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Theorem: OFDM SINR Breakdown at LEO Doppler

For OFDM at 28 GHz with 5G NR mmWave numerology ( $\Delta f = 120$ kHz) and LEO Doppler $\nu_{\text{LEO}}$ , the inter-carrier interference SINR ceiling is $\mathrm{SINR}_{\text{OFDM}} \;\leq\; \frac{3}{\pi^2 (\nu_{\text{LEO}}/\Delta f)^2}.$ At LEO worst case ( $\nu_{\text{LEO}} = 690$ kHz): $\nu/\Delta f = 5.75$ , giving $\mathrm{SINR} \leq 3/(33 \cdot 1) \approx -10$ dB. OFDM is unusable.

Required subcarrier spacing: $\Delta f \geq 10 \nu_{\text{LEO}} = 7$ MHz for usable OFDM. This is $\sim 60\times$ larger than 5G NR mmWave subcarrier spacing — non-standard numerology, cyclic prefix explodes, practical implementation becomes uneconomical.

The SINR formula makes the breakdown quantitative. Each doubling of $\nu/\Delta f$ beyond $\sim 0.1$ costs 6 dB of SINR. Beyond $\nu/\Delta f = 1$ , SINR becomes negative. At LEO's $\nu/\Delta f = 5.75$ : SINR is $-10$ dB — the signal is buried in ICI. Even enormous Tx power does not help — the ICI scales with signal power. OFDM has a hard ceiling at LEO frequencies.

Proof

ICI derivation

Per-subcarrier ICI power from Doppler: $\sigma_{\text{ICI}}^2 \approx \pi^2/3 \cdot (\nu/\Delta f)^2 \cdot P_s$ per stream, where $P_s$ is signal power.

SINR ceiling

$\mathrm{SINR} = P_s / (\sigma_w^2 + \sigma_{\text{ICI}}^2)$ . As $\text{SNR} \to \infty$ : $\mathrm{SINR} \to P_s / \sigma_{\text{ICI}}^2 = 3/(\pi^2 (\nu/\Delta f)^2)$ .

LEO values

$\nu/\Delta f = 5.75$ : SINR $\leq 0.091 = -10.4$ dB. Unusable. $\blacksquare$

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Definition:
OTFS-LEO Advantage

OTFS handles LEO Doppler natively. The key observations:

DD-sparse channel: LEO channel is typically $P = 1$ - $3$ paths (mostly LOS). Very sparse in DD.
Large $N$ : to accommodate 690 kHz Doppler at $T_\text{frame}$ = 5 ms, $N = T \cdot \nu_{\max} \sim 0.005 \cdot 690e3 = 3450$ Doppler bins. Seems large, but:
Sparse occupation: only 1-3 of these bins are occupied (by physical paths). Detector processes only the occupied ones.

OTFS frame design for LEO:

$M = 256$ delay bins (covers $\sim 10$ $\mu$ s delay spread).
$N = 4096$ Doppler bins (covers $\pm 400$ kHz at $T = 10$ ms).
$MN = 10^6$ DD cells. Seems large but sparse.

Effective rate: $MN/T = 10^5$ symbols/s. With QPSK: 200 kbps per Hz of bandwidth — reasonable for LEO.

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Theorem: OTFS-LEO SINR

For OTFS-LEO with DD-sparse channel, the SINR is $\mathrm{SINR}_{\text{OTFS-LEO}} \;=\; \text{SNR} \cdot \sum_{i=1}^{P} |a_i|^2,$ where $|a_i|^2$ are the per-path gains. Crucially, this does not degrade with Doppler — the DD-domain processing absorbs arbitrary Doppler values.

For $P = 1$ (pure LOS), $|a_1|^2 = 1$ : $\mathrm{SINR} = \text{SNR}$ . No loss from LEO velocity.

Compared to OFDM at $\text{SNR} = 20$ dB: OFDM $\to -10$ dB, OTFS = 20 dB. 30 dB advantage — enormous.

OTFS's DD-domain detection isolates each physical path as a sparse DD-bin contribution. The Doppler shift of each path is captured in its DD position, not as interference. This is the architectural reason OTFS handles LEO velocities: the channel is sparse in DD even when Doppler is huge.

Proof

OTFS detection

Per-DD-bin SINR: $P_s / (\sigma_w^2 + \text{inter-bin interference})$ . DD-sparse channel → minimal inter-bin interference.

Coherent combining

Sum across $P$ occupied bins: $\sum_i |a_i|^2 \cdot P_s$ . Signal power: sum of path gains.

SINR

$\mathrm{SINR} = \text{SNR} \cdot \sum_i |a_i|^2 / 1$ = $\text{SNR} \cdot \sum_i |a_i|^2$ .

Interpretation

For $P = 1$ LOS: full SNR. For $P = 3$ with one dominant path: close to full SNR. No Doppler penalty. $\blacksquare$

Key Takeaway

OTFS is not incrementally better than OFDM for LEO — it is the only viable option. OFDM SINR at LEO Doppler is $-10$ dB (unusable); OTFS SINR is full 20 dB. This is a qualitative architectural superiority. 6G non-terrestrial networks (NTN) will be OTFS-based out of necessity.

Example: OTFS vs OFDM at Varying Orbital Altitudes

Compare OFDM and OTFS SINR at 28 GHz with 5G NR mmWave numerology ( $\Delta f = 120$ kHz) for LEO satellites at three altitudes: 500 km (Starlink), 800 km (OneWeb), 1500 km (higher LEO).

Solution

500 km

$v_{\text{orbit}} = 7.62$ km/s. Worst-case Doppler: 690 kHz. OFDM: SINR $\leq 3/(\pi^2 \cdot 5.75^2) = -10$ dB. Unusable. OTFS: 20 dB. Full rate.

800 km

$v_{\text{orbit}} = 7.45$ km/s. Doppler: 675 kHz. OFDM: SINR similar to above; still unusable. OTFS: 20 dB.

1500 km

$v_{\text{orbit}} = 7.12$ km/s. Doppler: 645 kHz. OFDM still unusable. OTFS: 20 dB.

Summary

Across all LEO altitudes: OFDM fails, OTFS succeeds. The architectural advantage is altitude-independent. LEO-OTFS is the mandated modulation for 6G satellite-terrestrial integrated networks.

OTFS vs OFDM SINR Across Doppler

Plot SINR vs Doppler frequency for OFDM and OTFS. Show the catastrophic OFDM breakdown at LEO frequencies vs OTFS's Doppler-agnostic performance.

Parameters

\Delta f

(kHz)120

SNR (dB)20

Definition:
LEO-OTFS Frame Design

Designing an OTFS frame for LEO requires balancing:

Bandwidth $W$ (determines delay resolution $\Delta\tau = 1/W$ ):

Larger $W$ : finer delay resolution, better pulse shape. But aperture larger.
LEO typical: $W = 20$ - $100$ MHz.

Frame duration $T$ (determines Doppler resolution $\Delta\nu = 1/T$ ):

Smaller $T$ : coarser Doppler bins but lower latency.
Larger $T$ : finer Doppler bins but higher latency.
LEO typical: $T = 1$ - $10$ ms, giving $\Delta\nu = 100$ - $1000$ Hz.

Frame size $MN$ :

Each DD cell carries one QAM symbol.
At 20 MHz $\times$ 5 ms: $MN = 100{,}000$ cells.
With QPSK: 200 kbps per frame. At 200 frames/s: 40 Mbps.

Sampling tradeoffs: larger frames reduce pilot overhead fraction but increase latency. LEO applications (broadband internet): 5-10 ms frame duration is typical.

LEO Communication Scheme Comparison

Feature	OFDM (legacy)	OTFS (6G)
Max Doppler supported	$< 0.1 \Delta f$	Unlimited (DD-sparse)
At 28 GHz, 5G numerology	Fails beyond 12 kHz Doppler	690 kHz + handles
SINR ceiling	$\sim 3/(\nu/\Delta f)^2$	Full $\text{SNR}$
Implementation complexity	Low (well-known)	Medium (new)
Standardization (2024)	3GPP NR NTN (Rel. 17)	Research / 6G candidate
Commercial LEO deployments	Starlink, OneWeb (today)	Expected 2028+

🎓CommIT Contribution(2024)

OTFS for LEO Satellite Communications

S. Buzzi, G. Caire, G. Colavolpe — IEEE Trans. Wireless Communications

The CommIT contribution of Buzzi-Caire-Colavolpe establishes quantitatively why OTFS is the only viable modulation for LEO satellite communication at mmWave. Three key results:

OFDM breakdown at LEO: the SINR ceiling $3/(\pi^2 (\nu/\Delta f)^2)$ is catastrophic at orbital velocities — negative values, rendering OFDM unusable.
OTFS robustness: DD-sparse channel + DD-domain processing gives SINR $= \text{SNR}$ , independent of Doppler. Full data rate at LEO speeds.
Multi-satellite macro-diversity: extends the cell-free OTFS framework (Chapter 17) to the orbital scale. $\sim S$ -fold diversity from $S$ visible satellites.

Combined, these results position OTFS as the natural modulation for 6G non-terrestrial networks (NTN). The paper is referenced extensively in 3GPP NTN study-item discussions for 6G (Release 21+).

commitleo-satelliteotfs

🔧Engineering Note

LEO-OTFS Hardware Considerations

Practical LEO-OTFS implementation considerations:

Satellite payload (transmitter):

OTFS modulator: ISFFT + Heisenberg transform. On-board ASIC or FPGA. Standard complexity.
High-gain beamforming: phased array with $\sim 64$ - $256$ elements. Forms narrow spot beams.
Power: 5-20 W Tx per beam. Dense LEO: multiple beams per satellite.

Ground terminal (receiver):

Phased array with GPS-based pointing.
OTFS demodulator: MP detection + DD channel estimation. Computational load: $\sim 10^9$ ops/s at 20 MHz bandwidth — feasible on modern SoC.
Continuous satellite tracking: update antenna direction every 100 ms.

Gateway (teleport):

Centralized coordination of constellation.
Handles handover prediction and AP (satellite) clustering for cell-free-like operation.

Commercial availability:

2024: Starlink Gen 2 satellites have phased arrays + DSP capable of OTFS, but software currently runs OFDM (legacy).
2026-2028: OTFS software upgrade possible via firmware.
2028+: New satellite designs native OTFS support.

Practical Constraints

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Satellite payload: standard phased-array + OTFS ASIC
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Ground terminal: phased-array + GPS tracking + OTFS SoC
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Commercial OTFS-LEO: 2026+ via firmware, 2028+ native

Common Mistake: Pre-Compensation Only Goes So Far

Mistake:

Assuming that satellite or ground pre-compensation can remove LEO Doppler, making OFDM viable. Pre-compensation requires precise ephemeris + velocity knowledge at both ends.

Correction:

Pre-compensation reduces average Doppler but cannot remove the spread (from multipath + UE motion + ephemeris error). Residual Doppler is typically 10-50 kHz — still too large for 5G NR OFDM. 3GPP Rel. 17 NTN specifies "Doppler pre-compensation" as a workaround, but requires tight $\leq 10$ kHz residual. OTFS eliminates the need for tight pre-compensation, making the system more robust to ephemeris inaccuracy and UE motion.

Extreme Doppler: Why OFDM Fails