Delay–Doppler Domain Transformation

• Delay–Doppler Domain Transformation is a mapping of time-frequency signals into delay and Doppler indices that directly reflect the channel's physical propagation characteristics.
• It employs mathematical tools like the symplectic finite Fourier transform, Zak transform, and discrete Zak transform to enable robust modulation, channel estimation, and equalization in high-mobility environments.
• This transformation yields sparse, quasi-stationary channel representations that enhance radar sensing, reduce pilot overhead, and support integrated communication systems in modern wireless networks.

The delay–Doppler domain transformation is a foundational operator in contemporary wireless communications and radar sensing. It enables the mapping of time- and frequency-domain signals into a domain indexed by physical propagation delay and Doppler shift, directly reflecting the underlying channel dynamics in high-mobility scenarios. Modern approaches, including OTFS, DDMC, and corresponding Zak-based frameworks, structure modulation, detection, and channel estimation tasks around the delay–Doppler (DD) representation, which supports sparse, quasi-stationary modeling even under severe doubly-selective fading. This transformation, realized mathematically via the Zak transform, symplectic finite Fourier transform, or DDT variant, is central to OTFS modem design, DD-domain channel sounding, and integrated communication/radar systems (Zhang et al., 2023, Surabhi et al., 2019, Ma et al., 6 Aug 2025, Mohammed et al., 2023, Lin et al., 2023).

1. Mathematical Formalism of Delay–Doppler Mapping

The DD transformation leverages the joint time–frequency structure of signals and channels, enabling a direct mapping from sampled time-frequency (TF) grids to delay–Doppler indices. For finite blocklengths, the standard transformation employs the symplectic finite Fourier transform (SFFT) and its inverse (ISFFT) (Surabhi et al., 2019, Arous et al., 14 Oct 2025):

$$X_{\rm DD}[k, \ell] = \frac{1}{\sqrt{NM}} \sum_{n=0}^{N-1} \sum_{m=0}^{M-1} X_{\rm TF}[n, m] \exp\Bigl\{ -j2\pi \left( \tfrac{n k}{N} - \tfrac{m \ell}{M} \right) \Bigr\}$$

Conversely, DD–TF mapping is:

$$X_{\rm TF}[n, m] = \frac{1}{\sqrt{NM}} \sum_{k=0}^{N-1} \sum_{\ell=0}^{M-1} X_{\rm DD}[k, \ell] \exp\Bigl\{ +j2\pi \left( \tfrac{n k}{N} - \tfrac{m \ell}{M} \right) \Bigr\}$$

The Zak transform provides the continuous equivalent:

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

It possesses quasi-periodicity and allows direct definition of DD-domain basis functions, each corresponding to a grid of delay and Doppler values (Li et al., 2024, Li et al., 2023).

2. Channel Characterization in the Delay–Doppler Domain

Wireless channels with time and frequency selectivity are naturally modeled in the DD domain using the spreading function:

$$h(\tau, \nu) = \sum_{i=1}^P h_i\, \delta(\tau - \tau_i)\, \delta(\nu - \nu_i)$$

Here, $\tau_i$ and $\nu_i$ are the physical delay and Doppler associated with each resolvable path. The DD representation is compact and typically sparse, directly mapping to scatterer parameters, offering quasi-stationarity over frame durations. The channel input–output law in DD is a 2D (twisted) convolution:

$$y[k', \ell'] = \sum_{k=0}^{N-1} \sum_{\ell=0}^{M-1} h_{\rm DD}[k, \ell]\, x[(k'-k)_N,\,(\ell'-\ell)_M]\, e^{j2\pi \tfrac{(k'-k)\ell}{MN}}$$

where $h_{\rm DD}$ represents the filtered, sampled channel on the DD grid (Mohammed et al., 2023, Li et al., 2024, Surabhi et al., 2019).

3. Modulation, Demodulation, and DD-Domain Pulse Design

OTFS and related schemes utilize DD domain symbol mapping, with modulation executed via the ISFFT followed by a Heisenberg modulator:

$$s(t) = \sum_{n=0}^{N-1} \sum_{m=0}^{M-1} X_{\rm TF}[n, m]\, g_{tx}(t-nT)\, e^{j2\pi m\Delta f (t-nT)}$$

At the receiver, the signal undergoes matched filtering and is mapped back to DD via SFFT.

DD-domain orthogonality is governed by the choice of transmit and receive pulses, with DDOPs (delay–Doppler orthogonal pulses) structured as (Lin et al., 2023, Lin et al., 2023):

$$u_{m, n}(t) = u(t - m \Delta_\tau) e^{j2\pi n \Delta_\nu (t - m\Delta_\tau)}$$

These pulses admit local orthogonality per grid position, a property leveraged for efficient multiplexing and interference control even at high pulse densities (Lin et al., 2023).

4. Practical Implementations: Zak, DZT, and DD-domain Equalization

Modern modems use Zak or Discrete Zak Transforms (DZT) for efficient DD-domain mapping. The DZT for a time-domain sequence $u[n]$ yields:

$$V_{l, k} = \frac{1}{\sqrt{K}} \sum_{m=0}^{K-1} u_{l + m L} e^{-j2\pi (k/K) m}$$

with inverse:

$$u_{l + k L} = \frac{1}{\sqrt{K}} \sum_{m=0}^{K-1} V_{l, m} e^{+j2\pi (m/K) k}$$

The DZT is unitary, agnostic to noise whitening, and achieves efficient implementation via parallel FFTs (Hama et al., 2024, Lampel et al., 2021).

DD-domain equalization exploits the sparse, quasi-diagonal structure of the channel matrix. Linear MMSE or message-passing methods operate on the DD grid for data recovery, supporting high-mobility robustness, full TF diversity, and simplified pilot-aided channel estimation (Hama et al., 2024, Ma et al., 6 Aug 2025).

5. Equivalence and Interoperability with Other Waveforms

OTFS shares DD-domain processing structures with DFT-precoded OFDM (SC-FDMA); both are equivalent up to known linear phase rotations absorbed into the channel matrix (Farhang et al., 2024). Conventional OFDM systems can be augmented with DD-aided processing (DD-a-OFDM) by incorporating DD-domain channel estimation and TF-domain equalization, yielding lower pilot overhead, improved BER, and enhanced Doppler resilience in 6G (Ma et al., 6 Aug 2025, Surabhi et al., 2019).

Pulse–Doppler radar range–Doppler processing is mathematically isomorphic to OTFS demodulation under rectangular pulses: the formation of fast-time/slow-time matrices and N-point DFTs directly implements the DD transformation (Zhang et al., 2023).

6. Algorithmic Procedures and Estimation in DD Domain

DD-domain algorithms for estimation of channel parameters—including fractional delay and Doppler—utilize 2D correlators and leakage analysis. For radar and sensing applications, DD-domain peak detection, nearest-neighbor ratioing, and iterative cancellation allow sub-bin resolution of range and velocity (Zhang et al., 2023, Bao et al., 22 Oct 2025). Pilot-aided channel sounding adapts PN sequences and guard regions in DD grids, supporting high-resolution, robust CSF (channel spreading function) estimation even in urban vehicular scenarios (Bao et al., 22 Oct 2025).

Table: Key Operators and Their Formulas

Operator	Formula	Context
SFFT	$$X_{\rm DD}[k, \ell] = \frac{1}{\sqrt{NM}} \sum X_{\rm TF}[\cdot] e^{-j2\pi(\cdot)}$$	DD–TF mapping
ISFFT	$$X_{\rm TF}[n, m] = \frac{1}{\sqrt{NM}} \sum X_{\rm DD}[\cdot] e^{+j2\pi(\cdot)}$$	TF–DD mapping
DZT (forward)	$$V_{l, k} = \frac{1}{\sqrt{K}} \sum u_{l + mL} e^{-j2\pi(k/K)m}$$	Time–DD mapping
Channel Model	$$h(\tau, \nu) = \sum h_i\, \delta(\tau-\tau_i)\,\delta(\nu-\nu_i)$$	DD domain channel
Modulation	$$s(t) = \sum X_{\rm TF}[n, m]\, g_{tx}(t-nT)\, e^{j2\pi m\Delta f(t-nT)}$$	OTFS transmit

7. Impact, Limitations, and Open Directions

The DD transformation yields sparse, physically interpretable channel representations, quasi-stationarity, full time-frequency diversity, enhanced Doppler resolution, and efficient equalization. It underpins advanced multiple access schemes, grant-free signatures, integrated sensing and communications, and robust waveform processing for vehicular, satellite, and mmWave systems (Lin et al., 2023, Ma et al., 6 Aug 2025, Arous et al., 14 Oct 2025).

Limitations include fractional tap leakage, error floors under rapid birth–death of scatterers, hardware burden from large 2D transforms, and the requirement of finely tuned DD pulses for optimal localization. Continued research focuses on waveform-domain NOMA, real-time adaptation, efficient channel tracking in fractional or overspread scenarios, and hardware-optimized DD processing for massive MIMO and AI-driven wireless (Arous et al., 14 Oct 2025, Ma et al., 6 Aug 2025).

• "Radar Sensing via OTFS Signaling: A Delay Doppler Signal Processing Perspective" (Zhang et al., 2023)
• "Multiple Access in the Delay-Doppler Domain using OTFS modulation" (Surabhi et al., 2019)
• "Delay-Doppler Domain Signal Processing Aided OFDM (DD-a-OFDM) for 6G and Beyond" (Ma et al., 6 Aug 2025)
• "OTFS -- A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain" (Mohammed et al., 2023)
• "Multi-Carrier Modulation: An Evolution from Time-Frequency Domain to Delay-Doppler Domain" (Lin et al., 2023)
• "Fundamentals of Delay-Doppler Communications: Practical Implementation and Extensions to OTFS" (Li et al., 2024)
• "SC-FDMA as a Delay-Doppler Domain Modulation Technique" (Farhang et al., 2024)
• "On the Pulse Shaping for Delay-Doppler Communications" (Li et al., 2023)
• "Single-Carrier Delay-Doppler Domain Equalization" (Hama et al., 2024)
• "A Unified Framework for Adaptive Waveform Processing in Next Generation Wireless Networks" (Arous et al., 14 Oct 2025)
• "Delay Doppler Transform" (Xia, 2023)