Ferkans — Interactive Telecom Tutor

An OFDM Symbol, Through the DD Lens

We now translate an OFDM symbol into the DD language. This is an exercise in applying the Zak transform (Chapter 2) and the SFFT (Chapter 3) to the OFDM transmit signal. The result is surprising — and diagnostic: an OFDM symbol, viewed in the DD domain, is spread uniformly across every Doppler bin. This is the structural reason OFDM is fragile to mobility: every OFDM symbol "fights" every DD Doppler cell of the channel, while OTFS symbols touch only the ones they need.

Theorem: DD-Domain Image of an OFDM Subcarrier

Consider an OFDM system with $M$ subcarriers, $N$ symbols per frame, and rectangular pulses. A single OFDM data symbol $X_{TF}[n_0, m_0] = 1$ at symbol position $n_0$ and subcarrier $m_0$ corresponds, in the DD domain after the SFFT, to $X_{DD}[\ell, k] \;=\; \frac{1}{\sqrt{MN}}\,\delta_{\ell, -m_0 \bmod M}\,e^{-j 2\pi k n_0 / N},$ where $\delta$ is the Kronecker delta. That is, a single OFDM cell maps to a full column of the DD grid at fixed delay $\ell = -m_0 \bmod M$ , with uniform-magnitude entries across all $N$ Doppler bins.

The OFDM subcarrier is a complex exponential in time — it has a single frequency. In the DD plane it corresponds to a particular delay (the group delay of the subcarrier's complex exponential) and a uniform Doppler content (because the subcarrier is a pure tone, it has no Doppler structure that distinguishes bins). Spreading across all $N$ Doppler bins is the DD-space manifestation of "one frequency, all times."

Contrast this with an OTFS symbol (Chapter 3, Example EISFFT of a Single DD-Grid Symbol): that is one DD cell, spread uniformly in TF. The roles of DD and TF are swapped. OFDM signals on the TF grid; OTFS signals on the DD grid.

Proof

Write the OFDM symbol

Single subcarrier $m_0$ at symbol $n_0$ : $s(t) = g_{tx}(t - n_0 T_s)\,e^{j 2\pi m_0 \Delta f(t - n_0 T_s)}$ with $g_{tx}$ the rectangular pulse of width $T_s$ .

Apply the Zak transform

Within one OFDM symbol, the Zak transform (period $T_0 = T_s$ ) of $e^{j 2\pi m_0\Delta f t}$ is (by the tone-Zak result of Exercise Eex-otfs-ch02-04): a Dirac at Doppler $\nu = m_0\Delta f$ , which on the DD grid lands at delay index $\ell = -m_0 \bmod M$ (modular because the Zak transform identifies subcarrier frequency with delay).

Across $N$ symbols

The $N$ -dimensional time axis is Fourier-transformed to $N$ Doppler bins. The phase $e^{-j 2\pi k n_0 / N}$ encodes the symbol position $n_0$ as a linear chirp across the Doppler axis. Uniform magnitude $1/\sqrt{MN}$ across all $k$ .

Combine

Total mapping: $X_{DD}[\ell, k] = \delta_{\ell, -m_0 \bmod M}\,e^{-j 2\pi k n_0/N}/\sqrt{MN}$ . A single-OFDM-cell input produces a single-delay, all-Doppler column in the DD grid. $\blacksquare$

,

Key Takeaway

An OFDM symbol is a full Doppler column in the DD plane. One OFDM data symbol lives at a fixed delay $-m_0 \bmod M$ and at every Doppler value from $0$ to $N - 1$ . Equivalently, each OFDM symbol "experiences" every Doppler tap of the channel; there is no notion of "the Doppler of this symbol." This is the DD-geometric reason for ICI: when the channel shifts a subcarrier in Doppler, it smears across all the Doppler cells that subcarrier was already living in.

OFDM Symbol vs OTFS Symbol in the DD Plane

Two side-by-side DD grids. Left: an OFDM symbol at TF cell $(n_0, m_0)$ mapped to DD — a full Doppler column at delay $-m_0 \bmod M$ . Right: an OTFS symbol at DD cell $(\ell_0, k_0)$ — a single point. Move the sliders; OFDM always shows a stripe; OTFS always shows a point. This is the visual distinction between the two signaling lattices.

Parameters

OFDM symbol

n_0

4

OFDM subcarrier

m_0

10

OTFS delay

\ell_0

10

OTFS Doppler

k_0

4

Delay bins

M

32

Doppler bins

N

16

Theorem: OFDM Plus Channel, Viewed in DD

Let an OFDM frame with data $X_{TF}$ be transmitted through a $P$ -path channel with spreading function $h(\tau, \nu)$ . The received signal, viewed in the DD domain (via the SFFT), satisfies $Y_{DD}[\ell, k] \;=\; h \star\star_{DD} \text{SFFT}^{-1}(X_{TF}) [\ell, k] \;+\; W_{DD}[\ell, k].$ Because each OFDM symbol maps to an entire Doppler column in the DD domain, the DD-convolution mixes the contributions of every OFDM cell, spread uniformly along Doppler. The result is that OFDM data symbols, which should be resolvable per subcarrier, instead appear in the DD domain as overlapping Doppler-spread patterns.

The DD channel acts by 2D convolution — a pointwise operation on the DD grid. When the DD image of the input is sparse (one cell per datum, as in OTFS), the convolution is efficient and localized. When the DD image is spread (a full column per datum, as in OFDM), the convolution smears data from different cells together, producing the ICI we saw in Section 2.

In other words, ICI is not a bug in OFDM — it is the DD-domain signature of using a non-DD signaling lattice.

Proof

DD image of OFDM transmit

$X_{DD} = \text{SFFT}(X_{TF}) = \sum_{n, m} X_{TF}[n, m]\,\phi_{n, m}^{DD}$ , where $\phi_{n, m}^{DD}[\ell, k] = \delta_{\ell, -m \bmod M}\,e^{-j 2\pi k n/N}/\sqrt{MN}$ . The superposition yields a DD pattern spread along Doppler.

Apply the DD channel

$Y_{DD} = h \star\star X_{DD}$ is the 2D convolution. Each path shifts the entire input DD pattern.

Per-cell received symbol

The SFFT of $Y_{DD}$ returns to the TF grid. Careful bookkeeping reveals that the TF-domain equivalent is a matrix-vector product with a now-dense TF channel matrix (because the OFDM-DD spread turns the clean DD convolution into a dense TF coupling). This is ICI. $\blacksquare$

OFDM vs OTFS Signaling Lattices: DD Footprint

Property	OFDM signal	OTFS signal
Lattice where data lives	TF grid ( $N \times M$ )	DD grid ( $M \times N$ )
DD footprint of one symbol	Full Doppler column ( $N$ cells)	Single point
Channel action (DD view)	2D conv. of spread pattern	2D conv. of sparse pattern
ICI under Doppler	Significant at $\delta \gtrsim 0.05$	Essentially zero
Diversity order per symbol	1 (single subcarrier)	Up to $P$ (paths)
PAPR	$\sim \log MN$	$\sim \log MN$ (similar)

The DD Lens Is Diagnostic

The point of this section is not that OFDM works badly — it is that OFDM's weakness has a geometric origin. In the DD plane where the channel lives naturally, OFDM symbols take up a whole column per datum; OTFS symbols take up a single point. The channel's 2D convolution is efficient on sparse inputs and inefficient on spread inputs.

This is not a subtle argument. It is the equivalent of saying: "if your system has an eigenstructure, signal in that eigenstructure." The DD domain is the channel's natural coordinate system. OFDM ignores it, signals on the TF grid, and pays the ICI penalty. OTFS respects it, signals on the DD grid, and reaps the sparsity benefit.

One OFDM Cell Becomes One DD Column

Animation: place a single OFDM data symbol at TF cell

(n_0, m_0)

and apply the SFFT to view it in the DD plane. The output is a vertical stripe at delay

-m_0 \bmod M

, with uniform magnitude across all

N

Doppler bins. Compare with an OTFS symbol at DD cell

(\ell_0, k_0)

: a single point. Two radically different DD footprints.

🔧Engineering Note

Windowed OFDM and Its Effect in DD

Practical OFDM uses a windowed rather than rectangular pulse to reduce side-lobe leakage in the TF domain. In the DD domain, this manifests as a softer Doppler column — the DD image of one OFDM symbol is no longer a uniform stripe but a roll-off profile matching the window's Fourier transform.

Benefits: slightly reduced ICI, slightly cleaner spectral containment. Cost: CP overhead increases; pulse mis-match at the receiver (if unwindowed) introduces ISI.

In the DD view, this is analogous to using a non-rectangular prototype pulse for OTFS: both trade off spectral containment against DD-grid "cleanness." OTFS with a raised-cosine pulse is a closer analog. See Chapter 20 for the full treatment of pulse-shaping trade-offs.

Practical Constraints

•
Typical windowed OFDM reduces side-lobe leakage by 3-6 dB at the cost of $\sim 10\%$ CP overhead
•
Receiver window mismatch adds residual ICI/ISI unless jointly designed

An OFDM Symbol in the Delay-Doppler Plane