Ferkans — Interactive Telecom Tutor

Why Revisit OFDM?

OFDM is the most successful physical-layer modulation in history — every 4G and 5G link uses it. When the channel is LTI over each OFDM symbol, OFDM is beautiful: the channel becomes a diagonal matrix with one complex scalar per subcarrier, and equalization reduces to a single complex division per cell. This assumption underlies all of Telecom Chapter 14.

But when the channel is time-varying — either because the receiver moves or a reflector does — the LTI assumption fails. Each OFDM symbol experiences a slightly different channel from its neighbors, which manifests as inter-carrier interference. The point is that the failure mode is not mysterious; it is the signature of OFDM's mismatch with the underlying DD-domain physics.

In this chapter we express OFDM in the DD language, make its time-variation failure precise, and argue that the remedy is not better equalization — it is a different waveform altogether. That waveform is OTFS.

Definition:
OFDM Transmit Signal (Block-by-Block)

An OFDM system transmits $N$ symbols in time and $M$ subcarriers per symbol. The data symbols $X_{TF}[n, m]$ for $n = 0, \ldots, N - 1$ and $m = 0, \ldots, M - 1$ are modulated as $s(t) \;=\; \sum_{n = 0}^{N - 1}\sum_{m = 0}^{M - 1} X_{TF}[n, m]\,g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)},$ where $g_{tx}(t)$ is a rectangular pulse of width $T_s$ (plus a cyclic prefix of length $L_{CP}$ ), $\Delta f = 1/T_s$ , and the total frame duration is $T = N T_s$ . The signal bandwidth is $W = M\Delta f$ .

The $(n, m)$ cell of the TF grid carries one QAM symbol. The total number of symbols per frame is $MN$ , identical to an OTFS frame on the same $(M, N)$ grid.

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Theorem: OFDM Under LTI Channels: The Clean Case

Suppose the channel is LTI with frequency response $H(f)$ constant over each OFDM symbol, and the cyclic prefix is longer than the channel's memory. Then the received symbol on subcarrier $m$ of OFDM symbol $n$ , after OFDM demodulation, is $Y_{TF}[n, m] \;=\; H(m\Delta f)\,X_{TF}[n, m] \;+\; W_{TF}[n, m],$ where $W_{TF}[n, m] \sim \mathcal{CN}(0, \sigma^2)$ .

Equivalently, the TF channel matrix is a diagonal matrix with $H$ samples on its diagonal.

LTI plus CP makes the linear convolution circular, which is diagonalized by the DFT that implements OFDM. Each subcarrier becomes an independent scalar channel. This is the single most important engineering fact about OFDM; it is why OFDM dominates 4G/5G.

Proof

Circular convolution

With CP length $L_{CP} \geq L_h$ (channel memory), the linear convolution $h * x$ within each OFDM symbol is effectively circular modulo $M$ .

DFT diagonalization

The $M$ -point circular convolution is diagonalized by the $M$ -point DFT. Applying the DFT to the received samples yields $\hat{Y}[m] = H(m\Delta f)\hat{X}[m]$ per subcarrier.

Symbol-by-symbol

Under the LTI assumption, every OFDM symbol experiences the same $H$ (no time variation), so the relation holds for all $n$ with the same $H(m\Delta f)$ . $\blacksquare$

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OFDM's Three Wins Under LTI

Under the LTI assumption, OFDM achieves three properties that are hard to match:

Diagonal channel matrix: per-subcarrier equalization is a single complex division.
ISI-free reception: CP removes inter-block interference.
Cheap transceiver: IFFT/FFT at $O(M \log M)$ is the only non-trivial DSP operation per symbol.

Under time-varying channels, all three degrade simultaneously. The question is whether the degradation is mild (OFDM + tracking) or fundamental (need a new waveform). Section 2 makes this quantitative.

Definition:
Time-Frequency Channel Matrix

For a time-varying channel with Bello function $H(f, t)$ (OTFS Ch. 1), the TF channel matrix seen by OFDM is $H[n, m] \;=\; H(m\Delta f, n T_s),$ i.e., the transfer function sampled at the center of the $(n, m)$ TF cell. Under LTI, $H[n, m] = H(m\Delta f)$ is constant in $n$ ; under mobility, $H[n, m]$ drifts across $n$ — this drift is what generates ICI.

TF Channel Matrix Under Varying Mobility

For a multipath channel with adjustable Doppler, plot the magnitude of the TF channel matrix $|H[n, m]|$ across OFDM symbol index $n$ and subcarrier $m$ . At zero velocity, the matrix is constant in $n$ (LTI). As velocity grows, horizontal stripes appear — the channel drifts across symbols within one frame. When the drift becomes comparable to the coherence time, per-subcarrier equalization no longer works.

Parameters

Velocity (km/h)60

Number of paths

P

3

Subcarriers

M

128

OFDM symbols

N

16

Carrier

f_0

(GHz)6

Seed1

Example: OFDM Works: Pedestrian Indoor

A WiFi link at 5 GHz, $\Delta f = 312.5$ kHz, $T_s = 3.2\,\mu$ s, sees pedestrian-induced Doppler with $f_D = 5$ Hz. Is OFDM an adequate waveform?

Solution

Doppler-to-subcarrier ratio

$f_D/\Delta f = 5 / 312{,}500 \approx 1.6 \times 10^{-5}$ .

Coherence time

$T_c \approx 1/f_D = 200$ ms. Coherence over $\sim 10^5$ OFDM symbols.

Conclusion

Doppler is far below one subcarrier; the LTI assumption holds to five decimal places. OFDM is optimal here — no reason to consider anything else. This is the regime 4G/5G WiFi was designed for.

Example: OFDM Struggles: Vehicular at mmWave

A V2X link at $f_0 = 28$ GHz, $\Delta f = 120$ kHz (5G NR numerology 3), $T_s = 8.33\,\mu$ s, sees vehicular mobility at $v = 120$ km/h. Is OFDM still adequate?

Solution

Max Doppler

$f_D = (33.3/3 \times 10^8) \cdot 28 \times 10^9 = 3110$ Hz.

Doppler-to-subcarrier ratio

$3110/120{,}000 = 2.6\%$ . ICI begins to leak significantly — roughly 3% of the symbol energy goes to neighboring subcarriers.

Effective SINR ceiling

ICI acts as effective noise. At $\rho_{\text{ICI}} \approx 0.03$ , the SINR ceiling is $1/0.03 \approx 33$ ( $\sim 15$ dB), even at zero AWGN. High-order QAM (64-QAM or higher) is impossible — the error floor is set by Doppler, not noise.

Conclusion

OFDM is stressed here. Either wider subcarriers (reducing spectral efficiency via more CP overhead) or a different waveform is needed. OTFS directly targets this regime.

OFDM Performance Across Mobility Regimes

Scenario	$f_D/\Delta f$	ICI-to-signal	OFDM adequacy
Indoor pedestrian (5 GHz, 5 Hz Doppler)	$10^{-5}$	negligible	Optimal
Urban vehicular (3.5 GHz, 300 Hz)	$10^{-2}$	$\sim 10^{-3}$	Adequate
HST (3.5 GHz, 1 kHz)	$3 \cdot 10^{-2}$	$\sim 10^{-2}$	Stressed
V2X mmWave (28 GHz, 3 kHz)	$3 \cdot 10^{-2}$	$\sim 10^{-2}$	Stressed
HST mmWave (28 GHz, 7 kHz)	$6 \cdot 10^{-2}$	$\sim 5 \cdot 10^{-2}$	Poor
LEO satellite (10 GHz, 30 kHz)	$\sim 1$	complete loss	Fails

⚠️Engineering Note

5G NR Numerologies Set by Mobility

5G NR defines multiple numerologies with increasing subcarrier spacing: $\Delta f \in \{15, 30, 60, 120, 240\}$ kHz. The choice is driven in large part by Doppler tolerance:

$\Delta f = 15$ kHz: sub-6 GHz, low mobility
$\Delta f = 30$ kHz: sub-6 GHz, moderate mobility
$\Delta f = 120$ kHz: mmWave, high mobility (V2X, HST)

The rule of thumb: $\Delta f \geq 20 \cdot f_D$ to keep ICI-to-signal below $-20$ dB. For $f_D = 7$ kHz (HST mmWave), this requires $\Delta f \geq 140$ kHz — pushing numerology 3. Wider subcarriers reduce symbol duration and so worsen CP overhead (longer CP needed to cover the same delay spread), creating a fundamental trade-off.

OTFS sidesteps this trade-off: the DD-domain signal is Doppler-robust regardless of subcarrier spacing. The OTFS Doppler budget is limited only by the grid's Doppler resolution $1/T$ , not by ICI.

Practical Constraints

•
$\Delta f \geq 20 \cdot f_D$ for ICI below $-20$ dB
•
CP overhead $= L_{CP}/(T_s + L_{CP})$ — grows as $T_s$ shrinks
•
Doubling $\Delta f$ halves $T_s$ but requires doubled CP overhead for the same $\tau_{\max}$

📋 Ref: 3GPP TS 38.211, §4.2

Historical Note: OFDM's Path to 4G/5G

1966–2018

OFDM's origins trace to R. Chang (Bell Labs, 1966) and B. Weinstein & P. Ebert (Bell Labs, 1971), who used the DFT to implement the frequency-multiplexed modulation at practical cost. For two decades OFDM was a laboratory curiosity — it found its first deployment in ADSL wireline modems in the 1990s. The wireless breakthrough came with 802.11a WiFi (1999) and the adoption of OFDM in LTE (2009, release 8).

The transition from LTE to 5G NR introduced multiple numerologies precisely to handle the mobility regimes OFDM was not originally designed for. The engineering compromise — wider subcarriers at higher Doppler — is what pushed the community to consider fundamentally different waveforms. OTFS, first presented in 2017, is the most prominent of these proposals.

Bingham's 1990 survey Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come is the canonical engineering reference. It anticipates essentially every subsequent deployment issue.

OFDM Signaling in the Time-Frequency Domain

Why Revisit OFDM?

Definition: OFDM Transmit Signal (Block-by-Block)

Theorem: OFDM Under LTI Channels: The Clean Case

Circular convolution

DFT diagonalization

Symbol-by-symbol

OFDM's Three Wins Under LTI

Definition: Time-Frequency Channel Matrix

TF Channel Matrix Under Varying Mobility

Parameters

Example: OFDM Works: Pedestrian Indoor

Doppler-to-subcarrier ratio

Coherence time

Conclusion

Example: OFDM Struggles: Vehicular at mmWave

Max Doppler

Doppler-to-subcarrier ratio

Effective SINR ceiling

Conclusion

OFDM Performance Across Mobility Regimes

5G NR Numerologies Set by Mobility

Historical Note: OFDM's Path to 4G/5G

Definition:
OFDM Transmit Signal (Block-by-Block)

Definition:
Time-Frequency Channel Matrix