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Why OFDM Struggles with High Doppler

We saw in Section 10.2 that OFDM sensing works well when the Doppler shift is small compared to the subcarrier spacing ( $\nu \ll \Delta f$ ). In high-mobility scenarios --- automotive radar at highway speeds, airborne imaging, LEO satellite sensing --- the Doppler can span a significant fraction of $\Delta f$ , destroying subcarrier orthogonality via ICI.

Orthogonal Time-Frequency Space (OTFS) modulation addresses this by operating directly in the delay-Doppler domain, where the channel is sparse and quasi-static even at high speeds. The key idea: instead of fighting the Doppler by making $\Delta f$ large (which hurts range coverage via shorter CP), embrace it by working in the domain where each target is a single point $(\tau, \nu)$ .

Definition:
The Zak Transform

The Zak transform of a signal $x(t)$ with period parameter $T$ is

$\mathcal{Z}_x(\tau, \nu) = \sqrt{T} \sum_{k=-\infty}^{\infty} x(\tau + kT) \, e^{-j2\pi k T \nu}$

where $\tau \in [0, T)$ is the delay variable and $\nu \in [0, 1/T)$ is the Doppler variable. The Zak transform maps a 1D signal to a 2D function on the delay-Doppler plane.

Key properties:

Quasi-periodicity: $\mathcal{Z}_x(\tau + T, \nu) = e^{j2\pi T \nu} \mathcal{Z}_x(\tau, \nu)$
Periodicity in Doppler: $\mathcal{Z}_x(\tau, \nu + 1/T) = \mathcal{Z}_x(\tau, \nu)$
Unitarity: $\|\mathcal{Z}_x\|^2_{L^2([0,T)\times[0,1/T))} = \|x\|^2_{L^2}$
Convolution: Time-domain convolution becomes twisted convolution in the delay-Doppler domain

The Zak transform is the mathematical backbone of OTFS. It provides a rigorous way to represent signals on the delay-Doppler plane, just as the Fourier transform represents signals in frequency. For finite OFDM frames, the discrete version of the Zak transform is the symplectic finite Fourier transform (SFFT).

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Definition:
OTFS Modulation via Symplectic Fourier Transform

OTFS places data symbols on a delay-Doppler grid rather than the time-frequency grid of OFDM. Let $X_{\mathrm{DD}}[l, k]$ denote the symbol at delay bin $l \in \{0, \ldots, N_c - 1\}$ and Doppler bin $k \in \{0, \ldots, M - 1\}$ .

The OTFS transmit signal is generated in two steps:

Step 1: Inverse Symplectic Finite Fourier Transform (ISFFT)

$X_{\mathrm{TF}}[n, m] = \frac{1}{\sqrt{N_c M}} \sum_{l=0}^{N_c-1}\sum_{k=0}^{M-1} X_{\mathrm{DD}}[l, k] \, e^{j2\pi\left(\frac{nl}{N_c} - \frac{mk}{M}\right)}$

This maps delay-Doppler symbols to time-frequency symbols.

Step 2: Heisenberg Transform (standard OFDM modulation)

The time-frequency grid $X_{\mathrm{TF}}[n, m]$ is transmitted using a standard OFDM transmitter: IDFT across subcarriers, add CP, and concatenate symbols.

OTFS can be implemented as a pre-processing layer on top of OFDM: apply the ISFFT before the OFDM modulator, and the SFFT after the OFDM demodulator. This makes OTFS backward-compatible with existing OFDM hardware --- a crucial advantage for ISAC systems.

Theorem: OTFS Channel in the Delay-Doppler Domain

For a channel with $K$ point targets at delays $\{\tau_k\}$ and Doppler shifts $\{\nu_k\}$ with reflectivities $\{\alpha_k\}$ , the delay-Doppler channel impulse response is

$h(\tau, \nu) = \sum_{k=1}^{K} \alpha_k \, \delta(\tau - \tau_k) \, \delta(\nu - \nu_k)$

and the input-output relation in the delay-Doppler domain is a 2D circular convolution:

$Y_{\mathrm{DD}}[l, k] = \sum_{l'}\sum_{k'} h_{\mathrm{DD}}[l', k'] \, X_{\mathrm{DD}}\bigl[(l - l')_{N_c}, (k - k')_M\bigr] + W_{\mathrm{DD}}[l, k]$

where $h_{\mathrm{DD}}[l, k]$ is the sampled channel with at most $K$ non-zero entries and $(\cdot)_N$ denotes modulo- $N$ operation.

This is the crucial property: in the delay-Doppler domain, the channel has only $K$ non-zero entries regardless of the target velocities. In the OFDM time-frequency domain, high Doppler spreads each target across many subcarriers. In OTFS, the channel remains sparse --- one point per target --- no matter how fast the targets move.

Proof

Start from the time-varying channel

The time-varying channel impulse response is $h(t, \tau) = \sum_{k=1}^K \alpha_k \delta(\tau - \tau_k) e^{j2\pi \nu_k t}$ . The Wigner representation in the delay-Doppler domain is the 2D Fourier transform of $h(t, \tau)$ with respect to $t$ , yielding $h(\tau, \nu) = \sum_k \alpha_k \delta(\tau - \tau_k)\delta(\nu - \nu_k)$ .

Apply the Zak transform

The OTFS modulator applies the Zak transform to map symbols from the delay-Doppler grid to the time domain. The channel acts on the transmitted signal via convolution in time and multiplication in Doppler. In the Zak domain, this becomes a 2D twisted convolution.

Discretise to obtain 2D circular convolution

For a finite grid of $N_c \times M$ points, the twisted convolution reduces to a 2D circular convolution with kernel $h_{\mathrm{DD}}[l, k] = \alpha_p$ when $(l, k)$ corresponds to the delay-Doppler bin containing $(\tau_p, \nu_p)$ , and zero otherwise. The circularity is due to the CP in each OFDM symbol (delay) and the finite SFFT aperture (Doppler). $\blacksquare$

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Definition:
Fractional Doppler and Its Effects

When the Doppler shift $\nu_k$ does not fall exactly on the Doppler grid (i.e., $\nu_k T_{\mathrm{sym}} M$ is not an integer), we have fractional Doppler. Let

$\nu_k = \frac{k_0 + \kappa}{M T_{\mathrm{sym}}}$

where $k_0$ is the integer part and $\kappa \in (-0.5, 0.5]$ is the fractional part. The discrete channel response is no longer a single point but spreads across adjacent Doppler bins with a Dirichlet-like envelope:

$h_{\mathrm{DD}}[l, k] \propto \frac{\sin(\pi M \kappa)}{M \sin(\pi \kappa / M)} \cdot e^{-j2\pi (k - k_0) \kappa / M}$

For small $|\kappa|$ , most energy remains in the nearest bin, but for $|\kappa| \to 0.5$ the leakage is significant.

Fractional Doppler is the OTFS analogue of spectral leakage in FFT processing. Mitigation strategies include: (1) oversampling the Doppler grid, (2) windowing in the Doppler dimension, and (3) estimating $\kappa$ as a continuous parameter via interpolation or ML estimation.

Example: OTFS vs OFDM for High-Mobility Sensing

An automotive radar at $f_0 = 77$ GHz uses $N_c = 1024$ , $M = 128$ , and $\Delta f = 120$ kHz. A vehicle moves at $v = 200$ km/h. Compare the channel sparsity in the OFDM time-frequency domain versus the OTFS delay-Doppler domain.

Solution

Doppler shift

$\nu = \frac{2vf_0}{c} = \frac{2 \times 55.6 \times 77 \times 10^9}{3 \times 10^8} \approx 28.5\;\text{kHz}$ $

OFDM ICI spreading

Normalized Doppler: $\nu / \Delta f \approx 0.24$ . Over $M = 128$ symbols, the accumulated phase is $2\pi \nu M T_{\mathrm{sym}} \approx 2\pi \times 32.6$ radians. This spreads each target's energy across $\sim 30$ subcarriers in the frequency domain, making the channel highly non-sparse.

OTFS delay-Doppler sparsity

In the delay-Doppler domain, the same target occupies a single bin at $(\tau, \nu)$ . For $K = 5$ targets, the OTFS channel has 5 non-zero entries out of $1024 \times 128 = 131{,}072$ bins --- a sparsity ratio of $3.8 \times 10^{-5}$ . The sparsity does not depend on velocity.

Theorem: OTFS Sensing Matrix Structure

The OTFS sensing problem can be formulated as

$\mathbf{y} = \mathbf{A}_{\mathrm{OTFS}} \, \mathbf{c} + \mathbf{w}$

where $\mathbf{y} \in \mathbb{C}^{N_c M}$ is the vectorised received delay-Doppler signal, $\mathbf{c} \in \mathbb{C}^{N_c M}$ is the sparse delay-Doppler channel vector (with at most $K$ non-zero entries), and the sensing matrix $\mathbf{A}_{\mathrm{OTFS}}$ has a block-circulant structure:

$\mathbf{A}_{\mathrm{OTFS}} = \mathrm{diag}(\mathbf{x}) \cdot \mathbf{P}_{\mathrm{circ}}$

where $\mathbf{x}$ contains the transmitted symbols and $\mathbf{P}_{\mathrm{circ}}$ is the 2D circular permutation matrix.

Crucially, matrix-vector products with $\mathbf{A}_{\mathrm{OTFS}}$ can be computed in $O(N_c M \log(N_c M))$ time via 2D-FFT, making iterative recovery algorithms (ISTA, ADMM, OAMP) computationally tractable.

The OTFS sensing matrix inherits the circulant structure from the 2D circular convolution. This is analogous to how convolution with a known signal in classical radar becomes a circulant matrix. The FFT structure makes OTFS sensing particularly well-suited for large-scale sparse recovery.

Proof

Vectorise the 2D circular convolution

Stack $Y_{\mathrm{DD}}[l, k]$ into a vector using lexicographic ordering. The 2D circular convolution becomes a matrix-vector product with a block-circulant matrix.

Factor into pilot and permutation

Since the input-output relation involves the transmitted symbols $X_{\mathrm{DD}}[l, k]$ , we can separate the matrix into a diagonal (pilot) part and a circulant (channel) part. The circulant part is diagonalised by the 2D-DFT. $\blacksquare$

OTFS Delay-Doppler Channel Response

Visualise the OTFS delay-Doppler channel for a multi-target scene. Compare the response with and without fractional Doppler. Observe how the channel remains sparse regardless of target velocity, unlike the OFDM time-frequency channel that smears at high Doppler.

Parameters

Number of targets3

Max velocity (km/h)200

Include fractional Doppler

N_c

256

Common Mistake: Assuming Integer Doppler in OTFS

Mistake:

Treating all Doppler shifts as falling exactly on the delay-Doppler grid, ignoring fractional Doppler leakage.

Correction:

In practice, Doppler shifts are continuous and almost never align with the grid. Fractional Doppler causes energy leakage across adjacent Doppler bins, reducing the effective sparsity. Robust estimators must account for this: (1) use a finer Doppler grid (oversampled SFFT), (2) apply parametric estimation after coarse detection, or (3) use atomic norm minimisation for gridless recovery.

Historical Note: The Invention of OTFS

2017--present

OTFS was introduced by Hadani, Rakib, and collaborators in 2017 at Cohere Technologies, a startup co-founded by Shlomo Rakib and Ronny Hadani. The key insight --- placing data in the delay-Doppler domain --- drew on the mathematical theory of the Zak transform, introduced by the physicist Joshua Zak in 1967 for solid-state physics. The connection to symplectic geometry and the Heisenberg group gave OTFS a rich mathematical foundation that has attracted extensive research interest, particularly from the ISAC community. Gaudio, Kobayashi, Caire, and Colavolpe (2020) were among the first to rigorously analyse OTFS for joint radar-communication.

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Quick Check

An OTFS system with $N_c = 512$ and $M = 64$ observes $K = 10$ point targets. How many non-zero entries does the delay-Doppler channel have (assuming integer Doppler)?

$512 \times 64 = 32{,}768$

10

$10 \times 64 = 640$

512

Correction:

10

Each target produces exactly one non-zero entry in the delay-Doppler channel, regardless of velocity. The sparsity ratio is $10/32{,}768 \approx 3 \times 10^{-4}$ .

Why This Matters: OTFS for Integrated Sensing and Communications

The delay-Doppler domain offers a dual advantage for ISAC: the channel is sparse for sensing (enabling compressed-sensing recovery with fewer pilots), and it is quasi-static for communications (enabling long coherence intervals without re-estimation). An OTFS frame can simultaneously carry data symbols for communication users and serve as a sensing waveform, with the same $\mathbf{A}$ operator governing both tasks. This duality is the foundation of the delay-Doppler ISAC framework ([?ch34:ISAC]).

See full treatment in Chapter 34، Section 1

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Zak Transform

A unitary transform that maps a 1D signal $x(t)$ to a 2D function $\mathcal{Z}_x(\tau, \nu)$ on the delay-Doppler plane. The discrete analogue for finite OFDM frames is the Symplectic Finite Fourier Transform (SFFT). The Zak transform diagonalises the time-varying channel operator, making it the natural mathematical tool for delay-Doppler processing.

Related: Range-Doppler Map (RDM)

OTFS (Orthogonal Time Frequency Space)

A modulation scheme that places data symbols in the delay-Doppler domain rather than the time-frequency domain of OFDM. OTFS is related to OFDM by a pre/post-processing step (ISFFT/SFFT) and is backward-compatible with OFDM hardware. The delay-Doppler channel has at most $K$ non-zero entries for $K$ targets, regardless of their velocities.

Key Takeaway

OTFS operates in the delay-Doppler domain where the channel is inherently sparse --- one entry per target, independent of velocity. It can be implemented as a pre/post-processing layer on standard OFDM hardware. The OTFS sensing matrix $\mathbf{A}_{\mathrm{OTFS}}$ has block-circulant structure enabling $O(N \log N)$ matrix-vector products via FFT, making large-scale sparse recovery tractable. Fractional Doppler is the main practical challenge.

OTFS Modulation for Sensing