Ferkans — Interactive Telecom Tutor

Doppler as the Enemy of OFDM

The point is that OFDM's parallel-channel decomposition (Ch 21 §1) assumes subcarrier orthogonality. That orthogonality relies on the channel being static within one OFDM symbol. When the terminal is moving, the channel varies: different paths have different Doppler shifts, and the composite channel is time-varying. The result is inter-carrier interference (ICI) — subcarriers leak into each other, proportional to the normalised Doppler. Understanding this ceiling is essential for V2X, HSR, and NTN systems.

Definition:
Inter-Carrier Interference (ICI)

Inter-carrier interference is the leakage of signal energy from one OFDM subcarrier into its neighbours due to time-variation of the channel within an OFDM symbol. For a subcarrier spacing $\Delta f$ and Doppler $\nu$ , the interference coefficient between subcarriers $k$ and $k + \Delta k$ is approximately $I[\Delta k] \approx \frac{\sin(\pi \nu / \Delta f)}{\pi (\Delta k - \nu/\Delta f)}.$ ICI manifests as an additional noise-like term with power growing quadratically in $\nu / \Delta f$ .

Theorem: OFDM SNR Loss Under Doppler

For an OFDM system with subcarrier spacing $\Delta f$ operating on a channel with Doppler spread $\nu_{\max}$ , the effective SNR loss due to ICI is approximately $\Delta\text{SNR} \approx \frac{\nu_{\max}^2}{2 \Delta f^2} \quad \text{for } \nu_{\max}/\Delta f \ll 1.$ The ICI noise power rises as the square of the normalised Doppler.

Proof

ICI coefficient variance

The per-subcarrier ICI coefficient has variance $\mathbb{E}[|I[\Delta k]|^2] \approx \frac{\sin^2(\pi \nu/\Delta f)}{\pi^2 (\Delta k - \nu/\Delta f)^2}$ . Summing over all non-zero $\Delta k$ : $\sigma^2_{\rm ICI} \approx \frac{\nu_{\max}^2}{3 \Delta f^2}$ (for small Doppler).

Effective SNR

The total noise now has power $\sigma^2 + P_{\rm sig}\,\sigma^2_{\rm ICI}$ . In dB, the loss for large SNR is $\Delta\text{SNR} \approx 10 \log_{10}(1 + \text{SNR}\,\sigma^2_{\rm ICI})$ . At very high SNR (ICI-limited), the effective SNR saturates at $1/\sigma^2_{\rm ICI}$ — an absolute ceiling.

Quadratic law

For $\nu_{\max} \ll \Delta f$ , the effective SNR loss in linear terms is $\approx \nu^2_{\max} / (2 \Delta f^2)$ . $\blacksquare$

Example: LTE-V2X: What Mobility Can OFDM Handle?

LTE uses $\Delta f = 15$ kHz. For a carrier at 5.9 GHz (V2X band), what terminal velocities give < 1 dB SNR loss from ICI?

Solution

Doppler at given velocity

$\nu = v f_c / c$ . At $v = 100$ km/h $= 27.8$ m/s, $f_c = 5.9$ GHz: $\nu = 27.8 \cdot 5.9 \cdot 10^9 / (3 \cdot 10^8) \approx 547$ Hz.

SNR loss estimate

$\nu / \Delta f = 547 / 15000 = 0.036$ . Loss $\approx 10 \log_{10}(1 + \text{SNR} \cdot 0.036^2 / 3)$ . For $\text{SNR} = 100$ (20 dB), $\Delta\text{SNR} \approx 0.19$ dB — acceptable.

Mobility ceiling

For 1 dB loss we tolerate $\text{SNR} \cdot \nu^2 / (3 \Delta f^2) \approx 0.26$ , i.e. $\nu/\Delta f \le 0.05$ . At 5.9 GHz this caps the velocity at ~140 km/h. For 500 km/h V2X on 30 GHz mmWave, OFDM (with $\Delta f = 15$ kHz) would see 5 dB loss — impractical. 5G NR uses larger $\Delta f$ (120 kHz) for mmWave, restoring headroom.

Mobility Impact: OFDM Breaks, OTFS Doesn't

BER vs velocity at fixed SNR. BICM-OFDM degrades quadratically (and eventually floors) as Doppler grows. OTFS remains flat because the delay-Doppler channel is static within the OTFS frame.

Parameters

SNR [dB]15

⚠️Engineering Note

5G NR Numerology for Mobility

5G NR supports multiple subcarrier spacings (15/30/60/120/240 kHz) — a departure from LTE's fixed 15 kHz. The rationale is that each numerology trades OFDM symbol length (and thus CP overhead) for Doppler tolerance:

15 kHz: long symbols, suitable for sub-6 GHz and low mobility.
120 kHz: 8× shorter symbols, suitable for mmWave FR2 and high mobility. At 28 GHz and 120 kHz spacing, the Doppler-to-spacing ratio at 500 km/h is $\approx 0.1$ — manageable with 1-2 dB loss. Mobility budget is the main driver of numerology selection.

Practical Constraints

•
120 kHz spacing → 4.5 μs symbol duration
•
CP overhead ~7% for 120 kHz (same fraction as 15 kHz)
•
Pilot density must increase with mobility

Common Mistake: CFO Is Different from Doppler

Mistake:

Treating Carrier Frequency Offset (CFO) and Doppler as the same impairment: "both cause subcarrier rotation, so they're equivalent."

Correction:

CFO is a static frequency offset between Tx and Rx oscillators — once estimated (from pilots), it is compensated exactly. Doppler is time-varying per path — different paths have different Doppler shifts, and the composite ICI cannot be removed by a single frequency correction. Simple CFO compensation + delay-Doppler processing is the modern approach.

Key Takeaway

OFDM subcarrier orthogonality requires the channel to be static within one symbol. Doppler breaks that assumption, causing ICI with power $\propto (\nu/\Delta f)^2$ . The workarounds: larger $\Delta f$ (5G NR flexible numerology), longer CP, or a completely different waveform (OTFS) that absorbs Doppler into a sparse 2D grid.

Doppler, ICI, and the OFDM Mobility Ceiling