Why the Delay-Doppler Representation Is Natural

The Golden Thread

We have seen that the channel is a small collection of $(h_i, \tau_i, \nu_i)$ triples. We have seen that the spreading function $h(\tau, \nu)$ encodes those triples as impulses in a 2D plane. What we have not yet seen is why this matters for communication: why a modulation that signals on the DD plane should outperform one that signals on the TF plane.

The answer, which will be made fully rigorous in Chapters 4 and 5, comes in two parts:

Sparsity — the DD representation has $\sim P$ non-zero points; the TF representation has $\sim MN \gg P$ non-zero samples. Sparsity means cheap estimation, low pilot overhead, and resilience to channel-estimation error.
Time-invariance — the DD input-output relation is a 2D convolution with a static kernel. The TF input-output relation is a time-varying multiplication that scrambles data when the channel moves. Time-invariance means a single uniform detector works across the whole frame.

These two properties together — sparsity and time-invariance — are the golden thread of the book. Every subsequent chapter either (a) builds the DD domain more rigorously, (b) exploits these properties for modulation and detection, (c) applies them to sensing and ISAC, or (d) deploys them in networks (cell-free, LEO). This section frames the argument at a high level; the formalism follows in Chapters 2–6.

Time-Frequency vs Delay-Doppler Representations

Property	Time-Frequency $H(f, t)$	Delay-Doppler $h(\tau, \nu)$
Time-invariance	No — channel visibly changes with $t$	Yes — no $t$ argument at all
Sparsity	Dense: $MN$ complex samples	Sparse: $P$ Dirac impulses
Input-output map	Multiplication on TF grid, with ICI/ISI	2D convolution with a sparse kernel
Channel estimation	$O(MN / (\tau_{\max} f_D \,TW))$ pilots	One pilot impulse + guard region
Diversity order	1 per subcarrier	Up to $P$ (full DD diversity)
Robustness to mobility	Degrades with $f_D$	Invariant over one OTFS frame
Native waveform	OFDM, DFT-s-OFDM	OTFS, Zak-OTFS

Theorem: Time-Invariance of the DD Input-Output Relation (Informal)

Let $x_{DD}(\tau, \nu)$ and $y_{DD}(\tau, \nu)$ denote the DD-domain representations of the transmitted and received signals, respectively. Then the channel acts as a 2D convolution: $y_{DD}(\tau, \nu) \;=\; \iint h(\tau', \nu')\,x_{DD}(\tau - \tau', \nu - \nu')\,d\tau'\,d\nu' \;+\; \mathbf{w}_{DD}(\tau, \nu),$ where $h(\tau, \nu)$ is the spreading function and $\mathbf{w}_{DD}$ is a DD-domain AWGN vector. In particular, $h(\tau, \nu)$ does not depend on the running time $t$ — the input-output map is time-invariant in the DD sense.

A path $(\tau_i, \nu_i, h_i)$ takes a DD impulse at $(\tau_0, \nu_0)$ and produces an echo at $(\tau_0 + \tau_i, \nu_0 + \nu_i)$ with complex amplitude $h_i$ . The channel shifts and scales: it is a 2D translation kernel. The formal proof uses the Zak transform (Chapter 2) and the symplectic Fourier transform (Chapter 3); we postpone it until those tools are available.

Proof

A preview of the proof

The full derivation requires the Zak and symplectic Fourier machinery of the next two chapters. Intuitively: (i) the physical channel is $y(t) = \sum_i h_i\,x(t - \tau_i)\,e^{j 2\pi \nu_i t}$ ; (ii) the delay-and-Doppler-shift operator $\pi_{\tau_i, \nu_i}$ acts on the Zak transform as a translation in the DD plane. Summing over $i$ yields the claimed 2D convolution. Chapter 4 provides the rigorous argument. $\blacksquare$

Key Takeaway

The DD domain is not a choice of basis — it is the channel's own coordinate system. In the DD representation, a time-varying multipath channel becomes a static, sparse 2D impulse response. Every subsequent difficulty in wireless communication — equalization, channel estimation, diversity exploitation, joint sensing and communication — has a cleaner formulation in the DD domain than in the TF domain, precisely because the DD representation matches the underlying physics. We will make this operational in Chapters 6 (modulation), 7 (estimation), and 8 (detection).

OFDM vs OTFS: What Each Symbol Looks Like in the DD Plane

A single OFDM symbol is a pulse $e^{j 2\pi f_k t}$ at subcarrier $k$ over symbol duration $T_{\text{sym}}$ . In the DD plane, it spreads across all Doppler bins (because it is localized in time). A single OTFS symbol is a delta at one point $(\ell_0, k_0)$ of the DD grid — it is delay- and Doppler-localized by construction. This is the fundamental structural difference.

Parameters

OFDM subcarrier

k

OTFS delay bin

\ell_0

OTFS Doppler bin

k_0

Grid size

M

An OFDM Symbol Under Growing Doppler

As the receiver's velocity increases from

0

300

km/h, an OFDM symbol spreads more and more across Doppler bins — the inter-carrier interference picture. An OTFS symbol at the same velocity picks up the corresponding Doppler shift (it translates along

\nu

) but remains a sharp impulse. The DD-domain convolution commutes with Doppler translation; TF multiplication does not.

Path Taps as Velocity Increases

For a fixed three-path channel, increase the receiver velocity and watch the DD impulses slide along the Doppler axis. The delay axis is fixed by the scene geometry; the Doppler axis is modulated by mobility. This plot is a concrete visualization of the time-invariance claim: the topology of $h(\tau, \nu)$ (three isolated impulses) is unchanged; only their Doppler coordinates scale.

Parameters

Velocity (km/h)120

Carrier

f_0

(GHz)5

Scene type

Historical Note: The Birth of OTFS (2017)

2017

Ronny Hadani, Shlomo Rakib, and their collaborators at Cohere Technologies presented OTFS at the IEEE Wireless Communications and Networking Conference in March 2017. Their paper, "Orthogonal Time Frequency Space Modulation," did not discover the delay-Doppler domain — Bello had that in 1963, and Ottersten and collaborators had used it for radar since the 1980s. What Hadani and Rakib did was pose a simple question: what if we signal directly in the DD plane instead of the TF plane? The answer — a well-defined modulation with a tractable receiver and remarkable robustness to mobility — caught the 5G community's attention and triggered the body of work that makes OTFS a 6G candidate today.

The elegance of their 2017 paper is that OTFS is, operationally, a precoded OFDM: the ISFFT precoder (Chapter 6) takes DD-domain data symbols to a TF grid, and a standard OFDM transmitter produces the time-domain waveform. The DD-domain picture and the TF-domain implementation are related by a 2D DFT. This implementation-level equivalence is what made OTFS deployable.

🎓CommIT Contribution(2024)

OTFS as a Natural ISAC Waveform (Part III)

W. Yuan, R. Schober, G. Caire — IEEE Communications Magazine

In a three-part tutorial published in 2024, Yuan, Schober, and Caire argued that OTFS's native signaling on the delay-Doppler plane makes it the natural waveform for Integrated Sensing and Communications (ISAC). Their Part III paper presented the key quantitative comparison: for the same time-bandwidth product, OTFS offers a "thumbtack" ambiguity function (localized in both $\tau$ and $\nu$ ) while OFDM produces a ridge (localized only in $\tau$ ). This makes OTFS superior for joint range-velocity estimation.

The CommIT group's contributions to ISAC-OTFS (covered in detail in Chapter 12 of this book) build on this framing. The Part III paper is a foundational reference for the sensing-communication dual use of OTFS.

isacotfswaveform-design

The DD Lens Also Works for Sensing

The scatterers whose delays and Dopplers define $h(\tau, \nu)$ are the targets in a radar sense. Every physical target a radar wants to detect contributes a Dirac impulse to the spreading function. This is why OTFS, designed as a communications waveform, doubles naturally as a delay-Doppler radar. Chapter 11 and Chapter 12 (where we will see CommIT's ISAC-OTFS contributions) make this precise. For now, note the principle: the channel the communicator wants to undo is the same channel the sensor wants to measure.

Common Mistake: DD Is Not "Just Another Basis"

Mistake:

A common misstatement in survey-level treatments: "OTFS is just precoded OFDM on a different basis; it can't do anything OFDM can't." This is misleading.

Correction:

Every linear transformation, viewed in isolation, is "just a basis change." What matters is whether the basis matches the structure of the channel. OFDM is efficient because the channel is diagonal in the TF basis when it is time-invariant. OTFS is efficient because the channel is sparse and its input-output map is a 2D convolution in the DD basis under the full time-varying model. The DD basis is not an arbitrary choice — it is the eigenbasis of the time-varying multipath operator. The performance difference shows up the moment mobility exceeds the coherence-time assumption of OFDM; Chapter 9 quantifies the gap.

Delay-Doppler domain

The two-dimensional space with coordinates $(\tau, \nu)$ on which the spreading function $h(\tau, \nu)$ is defined. In OTFS, information symbols are placed directly on a discrete $M \times N$ grid in this domain. The key properties are sparsity (channel has $P \ll MN$ support points) and time-invariance (channel does not vary over one OTFS frame).

Why This Matters: Where This Leads

Chapter 2 formalizes the Zak transform, the mathematical machinery that maps a time-domain signal to the delay-Doppler plane. Chapter 3 introduces the symplectic Fourier transform (SFT), the 2D DFT on the DD grid that relates the DD and TF pictures. Together, Chapters 2 and 3 provide the tools we need to prove the time-invariance claim and to construct the OTFS modulator in Chapter 6. The payoff will be a complete OTFS transceiver that exploits both the sparsity and the time-invariance advertised in this section.

The Scattering Function and WSSUS Channels Chapter Summary