Ferkans — Interactive Telecom Tutor

A Waveform Question the Terrestrial Case Never Had to Ask

5G NR is an OFDM system. So was 4G. So was Wi-Fi, and so is $99\%$ of anything you touch with a radio. For the terrestrial use case this is the right answer: OFDM's complexity is linear in the number of subcarriers and its one-tap equalizer is irresistible. But OFDM breaks in the presence of large Doppler — specifically, when $f_D/\Delta f$ exceeds a few percent, inter-carrier interference (ICI) starts to dominate.

The LEO Ka band raw Doppler of $\approx 700$ kHz sits well above any reasonable OFDM numerology. After ephemeris pre-compensation the residual drops to $O(100)$ Hz, which does fit comfortably inside a $\Delta f = 120$ kHz numerology — but not by a huge margin, and not at all uniformly across a wide beam where the elevation angle varies.

OTFS (Orthogonal Time-Frequency-Space modulation), introduced by Hadani et al. in 2017, provides an alternative. It represents the signal in the delay-Doppler domain, where a LOS channel with a single Doppler shift becomes a single sparse tap rather than a broadband inter-carrier mess. This section compares the two options on the LEO channel and explains why OTFS is the default choice in many 6G NTN research proposals. Book OTFS develops the formal theory; here we focus on the LEO-specific design consequences.

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Definition:
OFDM Inter-Carrier Interference from Doppler

Consider an OFDM system with $N$ subcarriers and spacing $\Delta f$ . If the channel has a single complex tone at frequency offset $\Delta f_D$ (the LEO Doppler case), the received subcarrier $k$ is

$Y_k = H_k X_k + \sum_{k' \neq k} I_{k, k'} X_{k'} + W_k, \qquad I_{k, k'} = H \cdot \frac{\sin(\pi (\Delta f_D / \Delta f + k - k'))}{N \sin(\pi (\Delta f_D / \Delta f + k - k') / N)} e^{j \pi \cdot \text{const}}.$

The ICI term $I_{k, k'}$ is the Dirichlet-kernel leak from subcarrier $k'$ into subcarrier $k$ . Its peak (summed over all $k' \neq k$ ) decays with $|\Delta f_D / \Delta f|$ and vanishes only at $\Delta f_D = 0$ . A useful rule-of-thumb bound, accurate for small $\Delta f_D/\Delta f$ , is

$\text{SIR}_{\text{ICI}}^{-1} \approx \left(\frac{\pi \Delta f_D}{\sqrt{3}\, \Delta f}\right)^2,$

so a residual Doppler of $1\%$ of the subcarrier spacing costs about $-34$ dB in SIR, and $10\%$ costs about $-14$ dB — already below the working point of most MCS levels.

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Theorem: OTFS Localizes the LEO Channel to One Delay-Doppler Tap

Consider a pure LOS channel with a single path of delay $\tau_0$ and Doppler shift $\Delta f_D$ , $h(\tau, \nu) = \alpha\, \delta(\tau - \tau_0)\, \delta(\nu - \Delta f_D)$ . Under an OTFS modulation with $N$ Doppler bins and $M$ delay bins on a delay-Doppler lattice of resolution $\Delta\tau = 1/W$ , $\Delta\nu = 1/(N \cdot T_{\text{sym}})$ , the received symbol at lattice point $(l, k)$ is

$\tilde{y}[l, k] = \alpha\, e^{j \phi}\, \tilde{x}[l - l_0, k - k_0] + \tilde{w}[l, k],$

where $l_0 = \tau_0 / \Delta\tau$ and $k_0 = \Delta f_D / \Delta\nu$ are integers (assuming lattice-aligned delay and Doppler). The effective channel is a single tap with a cyclic-shift structure, and per-symbol equalization requires only knowledge of $(l_0, k_0, \alpha)$ — three scalars per satellite path.

OFDM is efficient when the channel is time-invariant (then each subcarrier is a scalar multiplication). OTFS is efficient when the channel is delay-Doppler sparse — a condition the LEO LOS channel satisfies exactly after pre-compensation, and approximately even before. The win is that in the delay-Doppler domain the Doppler of the LOS path shows up as a fixed lattice shift, not as a broadband inter-carrier smear. Tracking three scalars ( $l_0, k_0, \alpha$ ) per path is drastically easier than maintaining a full ICI-corrected OFDM equalizer over a channel that varies within a symbol.

Proof

Delay-Doppler representation of the OTFS signal

OTFS transforms a time-frequency lattice of size $M \times N$ to a delay-Doppler lattice of the same size via the symplectic finite Fourier transform (SFFT). The transmitted signal is the SFFT-preimage of the information symbols $\tilde{x}[l, k]$ ; the receiver applies the forward SFFT to recover $\tilde{y}[l, k]$ .

Channel in the delay-Doppler domain

For a time-invariant channel $h(\tau)$ the effect is a cyclic convolution in the $l$ -index only. For a channel with a single Doppler shift $\Delta f_D$ , the effect is a cyclic convolution in the $k$ -index only. A single delay-Doppler impulse gives a two-dimensional cyclic shift $(l_0, k_0)$ .

Per-lattice-point equalization

Because the shift is integer and the tap is a single impulse, recovering $\tilde{x}[l, k]$ from $\tilde{y}[l, k]$ requires one complex multiplication per lattice point, with the divisor $\alpha e^{j \phi}$ and the same $(l_0, k_0)$ for every lattice point. Total complexity: $M N$ complex multiplications, matching OFDM on a flat channel but with the Doppler tolerance built in. $\blacksquare$

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OFDM vs OTFS on the LEO Channel

Axis	OFDM	OTFS
Native domain	time-frequency	delay-Doppler
LOS channel representation	broadband ICI smear	one sparse tap
Doppler tolerance	$f_D/\Delta f \ll 1$	$f_D$ set by lattice resolution only
Pre-compensation need	mandatory (large, ephemeris-based)	useful but not critical
Pilot design	time-frequency grid	embedded delay-Doppler impulse
Equalizer complexity	one tap per subcarrier (with ICI correction)	one tap per lattice point
5G NR support	native	no
6G NTN research proposals	with aggressive pre-compensation	default in many proposals
Peak-to-average power	moderate	comparable to OFDM
Channel estimation overhead	dense pilot grid needed	sparse delay-Doppler pilot

Analytical BER: OFDM vs OTFS in High Doppler

Compare the approximate BER of OFDM and OTFS as a function of SNR on a pure-LOS channel with varying residual Doppler $|\Delta f_D / \Delta f|$ . OFDM's BER floor grows rapidly with the Doppler ratio as ICI dominates; OTFS achieves the AWGN bound because the Doppler collapses into one lattice tap and is invisible to the equalizer. The two curves coincide at $\Delta f_D = 0$ (static channel) and diverge as the ratio grows. This is the engineering argument for OTFS in NTN.

Parameters

|\Delta f_D| / \Delta f

0.05

modulation

Example: OFDM CP Budget Across a Wide LEO Beam

A LEO satellite at $h = 600$ km serves a footprint of radius $500$ km using a single wide beam. The differential slant range between the beam center and edge is $\Delta d_{\text{slant}} = d_{\text{edge}} - d_{\text{center}} \approx 45$ km. This translates to a differential propagation delay across the footprint of $\Delta \tau \approx 150$ $\mu$ s. Compare this to the cyclic prefix of 5G NR numerology $\mu = 3$ (Ka-band FR2), where $\Delta f = 120$ kHz and $T_{\text{CP}} \approx 0.58$ $\mu$ s.

Solution

Differential delay

The differential slant range across the footprint induces a differential delay of $150$ $\mu$ s. This is the delay spread that OFDM sees if the satellite illuminates the full footprint with a common waveform: different users in the footprint receive the same signal at different times.

CP comparison

The 5G NR FR2 CP is only $0.58$ $\mu$ s — about $260\times$ smaller than the differential delay. Standard OFDM cannot absorb the spread; every terminal would see ISI from its neighbours.

Mitigation

Three options. (a) Per-user timing advance pre-compensation — the terminal aligns its own reception to its own slant range, but this assumes each user decodes only the signal addressed to it. (b) Narrower beams, so the differential delay within a beam is small ( $<$ CP). Starlink and OneWeb use this, at the cost of spot-beam count growing fast. (c) OTFS, where delay shows up as an integer lattice index and is handled by the equalizer without a CP budget. The OTFS option is conceptually simpler at the cost of being non-standard. $\blacksquare$

🔧Engineering Note

OTFS Deployment Status and Implementation Cost

OTFS is a 2017-era research waveform. It is not in 3GPP 5G NR (which is strictly OFDM) and is not expected to be in Release 18. Research proposals for 6G NTN frequently use OTFS as the baseline, and some academic and industrial prototypes exist (Cohere Technologies, the company founded by Hadani). From an implementation standpoint OTFS adds:

One extra 2D FFT at each end. Transmitter applies the inverse SFFT before the standard IFFT; receiver applies the SFFT after the FFT. Both are $\mathcal{O}(M N \log(MN))$ and can be pipelined on a standard modem ASIC.
Delay-Doppler pilot grid. Pilots are embedded as sparse impulses in the delay-Doppler lattice instead of a dense time-frequency grid. Channel estimation is then a two-dimensional sparse recovery problem.
Lattice-aligned channel estimates. The advantage of OTFS depends on the channel being delay-Doppler sparse. For LEO with a single LOS path this is automatically satisfied; for rich-scatter terrestrial channels the advantage over OFDM is much smaller.
Outer-code compatibility. OTFS passes bit-level data to the same LDPC / Polar outer codes used in 5G NR, so the coding layer is unaffected.

Bottom line: OTFS is a reasonable candidate for 6G NTN once the standards process opens up to non-OFDM waveforms, but until then real LEO systems will use heavily pre-compensated OFDM with narrow spot beams to keep the differential-delay and residual-Doppler budgets under control.

Practical Constraints

•
OTFS is not in 5G NR; deploying it requires a non-NR modem
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2D SFFT cost is additive to FFT, pipelineable
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Delay-Doppler pilot grid differs from NR DM-RS patterns

📋 Ref: No normative reference; prototype systems only as of 2024

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Common Mistake: OTFS Is Not Always Better Than OFDM

Mistake:

A common NTN research claim is that OTFS always outperforms OFDM in high-Doppler channels. The enthusiasm is understandable given the BER curves of Plot 📊Analytical BER: OFDM vs OTFS in High Doppler — but it is only half the story.

Correction:

OTFS wins only when the channel is delay-Doppler sparse, i.e. a small number of dominant paths with well-separated Doppler shifts. A LOS LEO channel satisfies this. A channel with rich scatter (urban canyon, indoor) violates it, and OTFS's delay- Doppler tap count grows until the representation is no more compact than OFDM. For the LEO case specifically, the OTFS advantage is real and substantial; for generic NTN with mixed urban users it is smaller. And OFDM has the overwhelming practical advantage of being native to 5G NR, meaning the ecosystem is there and the standards compliance is free. For the foreseeable future, OFDM with aggressive ephemeris-based pre-compensation and narrow-beam coverage is what actually flies.

Why This Matters: Book OTFS Goes Deeper

This section takes the engineering view of OTFS: what does it buy us in LEO, and what does it cost to implement? Book OTFS treats the waveform as a first-class subject, starting from the Zak transform and the Heisenberg-Weyl representation, deriving the SFFT from first principles, and quantifying the delay-Doppler sparsity of common channel models. Readers who want to go beyond the single-tap LEO case — to rich-scatter NTN, to high-speed rail, to underwater acoustic — should branch to Book OTFS after this chapter.

Quick Check

An OFDM system has subcarrier spacing $\Delta f = 60$ kHz. After Doppler pre-compensation, the residual shift is $|\Delta f_D| = 3$ kHz. Using the rule-of-thumb $\text{SIR}_{\text{ICI}}^{-1} \approx (\pi \Delta f_D / (\sqrt{3}\, \Delta f))^2$ , estimate the SIR floor induced by ICI.

$\approx 0$ dB (ICI fully destroys the signal)

$\approx 15$ dB

$\approx 21$ dB

$\approx 40$ dB

Correction:

\approx 21

dB

$\Delta f_D / \Delta f = 3/60 = 0.05$ . The ratio $(\pi \cdot 0.05 / \sqrt{3})^2 \approx (0.0907)^2 \approx 8.22 \cdot 10^{-3}$ . Inverting, $\text{SIR} \approx 122 \approx 21$ dB. This is good enough for QPSK but drops margin for 64QAM (which needs $\approx 25$ dB).

Key Takeaway

OFDM works in LEO only with aggressive pre-compensation and narrow beams; OTFS sidesteps both constraints. The residual Doppler after ephemeris pre-compensation is $O(100)$ Hz, within 5G NR FR2 tolerance, but the differential delay across a wide beam would exceed the CP by orders of magnitude, forcing spot- beam architectures. OTFS represents the LOS LEO channel as a single delay-Doppler tap and avoids both issues, at the cost of not being a 3GPP standard. For 5G NTN (today) the answer is OFDM; for 6G NTN (research) OTFS is a serious candidate.

OFDM vs OTFS for the LEO Channel

A Waveform Question the Terrestrial Case Never Had to Ask

Definition: OFDM Inter-Carrier Interference from Doppler

Theorem: OTFS Localizes the LEO Channel to One Delay-Doppler Tap

Delay-Doppler representation of the OTFS signal

Channel in the delay-Doppler domain

Per-lattice-point equalization

OFDM vs OTFS on the LEO Channel

Analytical BER: OFDM vs OTFS in High Doppler

Parameters

Example: OFDM CP Budget Across a Wide LEO Beam

Differential delay

CP comparison

Mitigation

OTFS Deployment Status and Implementation Cost

Common Mistake: OTFS Is Not Always Better Than OFDM

Why This Matters: Book OTFS Goes Deeper

Quick Check

Key Takeaway

Definition:
OFDM Inter-Carrier Interference from Doppler