Ferkans — Interactive Telecom Tutor

When OFDM Breaks Down: Doubly-Selective Channels

The point is that OFDM's parallel-channel decomposition assumes a STATIC channel over the OFDM symbol duration. For a vehicle at 500 km/h and a 30 GHz carrier, the Doppler spread $\nu_{\max}$ can exceed $\Delta f$ , which breaks subcarrier orthogonality and causes inter-carrier interference (ICI). OTFS (Orthogonal Time Frequency Space) modulation works in the delay-Doppler domain, where the channel is SPARSE and QUASI-STATIC — ideal for high-mobility scenarios. BICM over the delay-Doppler grid harvests diversity analogously to BICM-OFDM over subcarriers.

Definition:
Delay-Doppler Grid

The delay-Doppler domain represents a doubly-selective channel as a 2D function $h(\tau, \nu)$ of delay $\tau$ and Doppler $\nu$ . For a channel with $P$ distinct physical paths, each path $p$ contributes a Dirac impulse: $h(\tau, \nu) = \sum_{p=1}^P g_p\, \delta(\tau - \tau_p)\, \delta(\nu - \nu_p).$ OTFS symbols $X[\ell, k]$ are placed on an $N_\tau \times N_\nu$ delay-Doppler grid. The channel $h(\tau, \nu)$ acts on this grid as a 2D convolution — much simpler than the time-varying convolution that a receiver sees in the time-frequency domain.

Definition:
Zak Transform

The Zak transform converts a time-domain signal $x(t)$ to the delay-Doppler domain: $Z_x(\tau, \nu) = \sqrt{T} \sum_{n=-\infty}^\infty x(\tau + nT)\, e^{-j 2\pi \nu n T}, \quad \tau \in [0, T), \nu \in [0, 1/T).$ It is invertible: $x(t) = \int_0^{1/T} Z_x(t, \nu)\, d\nu$ . OTFS places data on the delay-Doppler grid, applies the inverse Zak transform to produce a time-domain signal, and inverts at the receiver. The key property: the channel, which is time-varying in the time-frequency domain, becomes a static 2D convolution in the delay-Doppler domain.

Theorem: OTFS Absorbs Doppler into a Sparse 2D Channel

Let a doubly-selective channel have $P$ physical paths with delays $\{\tau_p\}$ and Dopplers $\{\nu_p\}$ . Then OTFS produces a 2D delay-Doppler channel with support concentrated on $P$ grid points (plus a small spread from pulse-shaping windowing). The received OTFS symbols satisfy $Y[\ell, k] = \sum_p g_p\, X[\ell - \ell_p, k - k_p] + W[\ell, k]$ with $(\ell_p, k_p)$ the discretised delay-Doppler position of path $p$ .

Proof

Time-frequency response of a multipath channel

A single path with delay $\tau_p$ and Doppler $\nu_p$ produces a time-frequency response $H(t, f) = g_p e^{j 2\pi(\nu_p t - f \tau_p)}$ — a linear chirp in $(t, f)$ .

Zak transform of the channel

The symplectic Fourier transform of $H(t, f)$ is concentrated at the single grid point $(\tau_p, \nu_p)$ . Each path maps to ONE point; the full channel is a sum of $P$ sparse contributions.

OTFS input-output relation

With OTFS symbols $X[\ell, k]$ on the delay-Doppler grid, the received symbols after matched filtering are a 2D circular convolution with the delay-Doppler channel impulse response.

Static channel in delay-Doppler

Over one OTFS frame, the channel grid coefficients $g_p, \ell_p, k_p$ do NOT change — even if the terminal is moving at high speed. The time-varying behaviour is captured in the DOPPLER SHIFT, which is a coordinate, not a time-varying gain. $\blacksquare$

OTFS Delay-Doppler Grid for a Sparse Channel

Visualise a 3-5 path physical channel mapped onto the delay-Doppler grid. As terminal velocity increases, paths spread along the Doppler axis but remain distinct grid points.

Parameters

Number of paths

P

3

Terminal velocity [km/h]300

BICM Over the Delay-Doppler Grid

BICM on OTFS works just like BICM on OFDM, with the grid dimension replaced. Each coded bit is mapped to a delay-Doppler grid point; an interleaver spreads successive bits over different $(\ell, k)$ positions. The diversity harvested is $\min(d_H,\, P)$ where $P$ is the number of delay-Doppler resolvable paths — typically much larger than the OFDM $L$ at high mobility, because each physical path contributes a SEPARATE grid point rather than collapsing into the $L$ delay taps.

Example: OTFS Diversity on a 4G/5G High-Speed Rail Channel

A high-speed rail scenario at 350 km/h, carrier 3.5 GHz, has $P = 6$ physical multipath components spread across delays $\tau \in [0, 3\,\mu\mathrm{s}]$ and Dopplers $|\nu| \le 1100\,\mathrm{Hz}$ . Compare the diversity achievable by BICM-OFDM (subcarrier spacing 15 kHz) and BICM-OTFS (delay-Doppler grid $64 \times 8$ ) using an outer code with $d_H = 8$ .

Solution

BICM-OFDM diversity

OFDM resolvable paths $L \approx \tau_{\max} \cdot B_{\rm sys} = 3\,\mu\mathrm{s} \cdot 20\,\mathrm{MHz} \approx 60$ . But at 1100 Hz Doppler, OFDM suffers ICI: effective SNR drops ~3 dB. The paths beyond $d_H$ don't improve diversity, so $d = \min(8, 60) = 8$ , further reduced by ICI.

BICM-OTFS diversity

OTFS resolvable paths $= P = 6$ (sparse grid). The delay-Doppler grid is STATIC, so no ICI. Diversity $= \min(d_H, P) = \min(8, 6) = 6$ .

Apparent paradox?

OTFS shows LOWER raw diversity! But OTFS has zero ICI loss, whereas OFDM suffers several dB at 1100 Hz Doppler. The net effect: OTFS BER curves have a smaller slope but a much lower floor. In practice, OTFS wins at high velocities.

Common Mistake: OTFS Has Higher Receiver Complexity

Mistake:

Assuming OTFS is a "drop-in replacement" for OFDM: "we just swap the OFDM modulator for OTFS and everything else stays the same."

Correction:

OTFS receiver is 2D: symbols are detected on a delay-Doppler grid, and the channel matrix couples them through a 2D convolution. ML detection is exponential in the grid size; message-passing or iterative detectors are required for practical implementations. OTFS memory and processing grow by $N_{\rm grid}$ vs OFDM's $N_{\rm sc}$ — typically 4-8× more expensive per symbol.

🔧Engineering Note

OTFS in Research and Early Deployment

OTFS was proposed by Hadani and Rakib (2017) and is currently in the research-and-prototype stage:

5G NR: NOT standardised (OFDM with flexible numerology is used instead); OTFS proposed for 6G.
3GPP study items: OTFS discussed for non-terrestrial networks (NTN) and high-speed rail in Rel-17 onward.
Prototypes: Cohere Technologies, Qualcomm, and others have demonstrated OTFS at 500+ km/h and in airborne NTN links.
Standards path: likely 6G around 2030 for specific mobility use cases (V2X, HSR, NTN).

Key Takeaway

OTFS converts a doubly-selective channel into a sparse, quasi-static 2D channel on the delay-Doppler grid. BICM on the OTFS grid achieves $\min(d_H, P)$ diversity, where $P$ is the number of physical multipath components — smaller than OFDM's $L$ but without ICI losses. OTFS wins at high mobility (500+ km/h), where OFDM breaks down.

Coded Modulation for OTFS: The Delay-Doppler Grid