Zak-OTFS: Delay-Doppler Modulation

• Zak-OTFS is a delay–Doppler modulation framework that leverages the Zak transform to directly map quasi-periodic DD-domain signals onto time-domain waveforms.
• It employs the discrete Zak transform (DZT) to bridge rigorous continuous-time analysis with practical OTFS implementations, ensuring structured signal localization.
• The framework enables robust channel estimation and equalization under high Doppler spreads, optimizing multiuser and high-mobility communications.

Zak-Orthogonal Time Frequency Space (Zak-OTFS) is a delay–Doppler (DD) domain modulation framework in which the Zak transform, or its discrete counterpart, is used to map quasi-periodic DD-domain signals to time-domain waveforms and back. In this formulation, information symbols are mounted directly on DD-localized carriers rather than first being routed through a time–frequency grid, and the channel action is expressed through a DD-domain convolutional structure that is predictable under suitable support conditions. The foundational literature presents Zak-OTFS both as a first-principles derivation of OTFS from Zak analysis and as an implementation-oriented discrete Zak transform (DZT) architecture, thereby linking rigorous continuous-time DD localization, finite-dimensional orthonormal signaling, and practical digital transceiver constructions (Mohammed, 2020, Lampel et al., 2021).

1. Foundational formulation and Zak-domain structure

The continuous Zak representation of a time-domain signal $x(t)$, for period parameter $T>0$, is defined as

$$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$

Its two central structural properties are periodicity along Doppler with period $\Delta f=1/T$ and quasi-periodicity along delay with period $T$:

$$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$

This quasi-periodic geometry is what makes the Zak domain a natural coordinate system for DD signaling rather than merely a convenient transform pair (Mohammed, 2020).

A single-path time-domain channel action of delay $\tau_0$ and Doppler $\nu_0$ becomes a shift in Zak coordinates:

$$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$

The direct interpretive consequence is that physical propagation parameters map to translations in the DD plane. This is the conceptual basis for the recurrent claim in Zak-OTFS work that the DD domain is physically aligned with scattering geometry (Mohammed, 2020).

The same literature also gives exact inversion formulas. The time-domain signal is recovered by integrating over one Doppler period, and the Fourier-domain signal is obtained by integrating over one delay period. Later discrete formulations replace these continuous integrals by finite DZT/IDZT transforms over an $M\times N$ DD tile, preserving the same periodic/quasi-periodic logic in finite dimensions. In the DZT formulation, a length-$T>0$0 discrete-time sequence is reorganized into an $T>0$1 DD array, with periodicity in one index and quasi-periodicity in the other, thereby furnishing a finite-dimensional Zak-domain signal space suitable for digital communication (Lampel et al., 2021).

2. Basis construction, pulsones, and modulation law

The first-principles derivation constructs ideally DD-localized Zak impulses and then derives their time-domain realizations. A DD impulse centered at $T>0$2 has time-domain counterpart

$T>0$3

namely an impulse train modulated by a tone. Later Zak-OTFS papers call such waveforms “pulsones,” emphasizing that a DD point maps to a pulse train in time rather than to a single localized pulse (Mohammed, 2020).

Because such ideal objects are not physically realizable, the construction proceeds by imposing approximate time and bandwidth limitation. With a rectangular time window $T>0$4 over $T>0$5 and a band-limiting filter $T>0$6 supported over $T>0$7 in frequency, one obtains

$T>0$8

Sampling the centers on the lattice

$T>0$9

yields an $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$0-dimensional orthonormal basis $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$1, with exact orthonormality and no loss of dimension relative to the time–bandwidth product (Mohammed, 2020).

The corresponding modulation rule is

$$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$2

with DD grid spacings

$$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$3

Equivalently, in the period-based notation used elsewhere in the literature, one writes $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$4, $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$5, $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$6, and samples at $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$7 on the DD lattice (Khan et al., 21 Jul 2025).

A notable result is that, for sufficiently large $$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$8, the DD-domain modulation derived from Zak analysis coincides with the standard OTFS Heisenberg form with a rectangular transmit pulse. This establishes that Zak-OTFS is not a different modulation in the narrow sense, but a DD-first derivation and realization of OTFS whose carrier geometry and localization properties are made explicit at the Zak level (Mohammed, 2020). Lampel et al. later showed that the DZT/IDZT formulation supplies an efficient discrete implementation and a simplified derivation of the DD input–output relation relative to the conventional ISFFT/Heisenberg two-step chain (Lampel et al., 2021).

3. Channel model, DD-domain input–output relation, and equalization

The standard continuous-time multipath model used throughout the Zak-OTFS literature is

$$\mathcal Z_x(\tau,\nu)=\sqrt{T}\sum_{n=-\infty}^{\infty} x(\tau+nT)e^{-j2\pi n\nu T}.$$9

Under Zak processing, this becomes a DD-domain relation whose effective coupling is localized in both delay and Doppler. In the first-principles derivation, the sampled DD input–output law contains separable sinc kernels in the offset indices, and the received response to a DD basis signal remains concentrated within a main lobe of width approximately $\Delta f=1/T$0 in Doppler and $\Delta f=1/T$1 in delay, irrespective of Doppler magnitude (Mohammed, 2020).

This localization leads directly to the standard Zak-OTFS robustness interpretation. The main-lobe widths shrink as total frame duration $\Delta f=1/T$2 and total bandwidth $\Delta f=1/T$3 grow, so the interference neighborhood occupies only a small fraction of the DD frame. One quantitative expression given for the fraction of significantly interfered DD symbols is

$\Delta f=1/T$4

which decreases for large $\Delta f=1/T$5 and $\Delta f=1/T$6 (Mohammed, 2020).

A complementary modern formulation expresses the continuous-time received Zak-domain signal as a twisted convolution:

$\Delta f=1/T$7

with

$\Delta f=1/T$8

After sampling on the DD lattice, this yields a discrete twisted convolution

$\Delta f=1/T$9

A central property of this formulation is that every DD-domain symbol undergoes the same effective channel response, up to the twisted-convolution phase. This is what makes single-pilot estimation structurally plausible in Zak-OTFS and distinguishes it from implementations where pulse-dependent symbol-to-symbol variations remain pronounced (Zhang et al., 7 Feb 2026).

From the receiver perspective, equalization is therefore naturally joint and DD-domain. The effective coupling matrix is dense in principle but banded or approximately banded in practice because the received DD support is localized. The original derivation interprets this as the operational reason for improved Doppler robustness relative to OFDM, while DZT-based performance analysis later showed that both DZT-OTFS and two-step OTFS outperform OFDM and that DZT-OTFS achieves better performance than two-step OTFS over a wide range of Doppler spreads (Mohammed, 2020, Yogesh et al., 2023).

The same localization also encodes a direct system trade-off: “The degree of localization of the DD domain basis signals is inversely related to the time duration of the transmit signal, which explains the trade-off between robustness to Doppler shift and latency.” Increasing $T$0 improves Doppler localization but increases frame duration; increasing $T$1 improves delay localization by enlarging bandwidth (Mohammed, 2020).

4. Discrete realizations and compatibility with OFDM infrastructures

The DZT formulation makes Zak-OTFS explicitly digital. For a length-$T$2 sequence $T$3, one form of the DZT is

$T$4

with inverse

$T$5

This amounts to a bank of $T$6-point transforms over cosets of the time index and gives a direct DD-to-time mapping without an intermediate TF grid (Mattu et al., 4 Aug 2025). Lampel et al. further related this DZT view to the conventional OTFS overlay and showed that, with rectangular pulses, the Zak realization collapses to the same discrete-time modulation law while reducing transform redundancy (Lampel et al., 2021).

This discrete perspective is what enables recent “Zak-OTFS over CP-OFDM” architectures. In that construction, Zak-OTFS with sinc filtering and rectangular time windowing is realized as a low-complexity precoder before a standard CP-OFDM modulator and as a low-complexity post-processing stage after a CP-OFDM demodulator. The DD-to-frequency precoder is an inverse discrete frequency Zak transform, and CP-OFDM appears as a special case when the delay period takes its minimum value $T$7, i.e. when $T$8 (Mohammed et al., 5 Aug 2025). A related line, “Zak-OFDM,” keeps DD-domain channel estimation but performs joint equalization in the frequency domain, reducing naive $T$9 DD equalization to an $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$0 method by reconstructing the full banded ICI matrix from compact DD taps (Mohammed et al., 29 Jun 2025). Another frequency-domain approach shows that the Zak-OTFS FD model is unitarily equivalent to the DD model and that the FD channel matrix is banded, which enables conjugate-gradient equalization with complexity linear in frame dimension, $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$1, rather than cubic (Mattu et al., 10 Aug 2025).

Practical viability has also been demonstrated over the air. A mmWave implementation at 28 GHz on the COSMOS testbed used $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$2, bandwidth $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$3 MHz, sampling rate $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$4 Msps, a 256-sample Zadoff–Chu preamble, and RRC pulse shaping with $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$5. The reported system treated carrier-frequency offset and timing offset as part of the effective DD channel, showing that these impairments can be jointly absorbed into DD-domain estimation rather than requiring a separate TF-domain compensation chain (Ramachandran et al., 10 Nov 2025).

5. Pilot design, pulse shaping, sensing, and low-PAPR waveform engineering

A recurrent Zak-OTFS theme is that predictability is conditional rather than unconditional. The decisive regime is often called the crystallization condition: the effective delay spread must lie within the delay period and the effective Doppler spread within the Doppler period, so that lattice replicas of the DD response do not alias (Ubadah et al., 2024). Under this condition, the filter taps can be read off from the response to a single Zak-OTFS point pilot, and the entire DD I/O relation can be reconstructed model-free.

This basic pilot idea has led to several waveform and sensing extensions. One line uses spread pilots rather than point pulsones. A point pulsone has time-domain PAPR of about $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$6 dB, whereas a spread pulsone obtained by a discrete chirp filter was reported to reduce the PAPR to about $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$7 dB and to move the self-ambiguity support from the period lattice $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$8 to a rotated lattice $$\mathcal Z_x(\tau+T,\nu)=e^{j2\pi \nu T}\mathcal Z_x(\tau,\nu),\qquad \mathcal Z_x(\tau,\nu+\Delta f)=\mathcal Z_x(\tau,\nu).$$9, enabling integrated sensing and communication under appropriate crystallization conditions with respect to both lattices (Ubadah et al., 2024).

A distinct low-PAPR construction uses a unitary transform based on the discrete affine Fourier transform to convert the orthonormal pulsone basis into an orthonormal basis of spread carrier waveforms. In that work, a pulsone basis element has reported PAPR $\tau_0$0 dB, while the spread-carrier basis element has reported PAPR $\tau_0$1 dB, with “only $\tau_0$2 dB” stated for the spread carrier waveform class in the abstract and full spectral efficiency preserved because the transform is unitary (Mehrotra et al., 12 May 2025). These differing values reflect different constructions and measurement conventions across the literature rather than a contradiction in principle.

Pulse shaping itself is another active axis. In oversampled discrete Zak-OTFS, the effective channel localization is governed by the ambiguity functions of the transmit and receive windows. High sidelobes broaden the effective DD support and degrade single-pilot estimation. To address this, one paper synthesizes Prolate Spheroidal Wave Functions inside the IOTA framework, then orthogonalizes the sampled pulse set. Reported numerical results show average PSD bandwidths of approximately $\tau_0$3 arbitrary units for PSWF, $\tau_0$4 for IOTA-PSWF, and $\tau_0$5 for RRC, while IOTA-PSWF yields superior channel-estimation accuracy and BER performance compared with conventional RRC and rectangular windowing in the high-SNR regime (Zhang et al., 7 Feb 2026).

Zak-OTFS has also become a platform for sensing-oriented waveform design. DZT-based CAZAC constructions produce DD waveforms whose self-ambiguity is supported on a discrete line and whose pairwise cross-ambiguity magnitude is constant at $\tau_0$6 when the algebraic coprimality conditions are met. These properties underwrite mutually unbiased pilots, integrated sensing and communication, and simultaneous multi-preamble detection in grant-free random access (Mehrotra et al., 30 Mar 2025). More recently, a semi-blind atomic norm denoising scheme for bistatic Zak-OTFS ISAC formulated fractional delay–Doppler estimation and data detection as a joint atomic-norm problem and reported super-resolution sensing accuracy together with communication performance approaching the perfect-CSI lower bound (Zhang et al., 7 Jan 2026).

Theme	Core mechanism	Representative result
Point or spread pilot acquisition	Cross-ambiguity in the Zak/DD domain	Single-pilot model-free channel extraction under crystallization (Ubadah et al., 2024)
PSWF/IOTA pulse shaping	Window design with low ambiguity sidelobes	Better channel estimation and BER than RRC/rectangular at high SNR (Zhang et al., 7 Feb 2026)
CAZAC DD waveforms	DZT mapping with mutually unbiased ambiguity structure	Cross-ambiguity magnitude $\tau_0$7 and discrete-line self-ambiguity (Mehrotra et al., 30 Mar 2025)
Spread carriers	Unitary GDAFT/DAFT basis transform	Low-PAPR orthonormal carriers with full spectral efficiency (Mehrotra et al., 12 May 2025)
Semi-blind ISAC	Atomic norm denoising with discrete constraints	Super-resolution sensing with near-perfect-CSI communication performance (Zhang et al., 7 Jan 2026)

6. Multiuser operation, coding, MIMO, FTN, and open issues

Zak-OTFS research has increasingly moved from single-user SISO theory to system-level architectures. In multiuser uplink operation, one proposal shapes each user’s DD pulse so that the corresponding waveform occupies a chosen non-overlapping time–frequency rectangle, thereby reproducing OFDM-like scheduling flexibility while retaining DD predictability. In the reported Veh-A study, interference leakage between users remained below $\tau_0$8 dB with sinc pulses and below $\tau_0$9 dB with RRC pulses under very high Doppler spreads, and multiuser BER and NMSE matched single-user performance without guard bands between adjacent allocations (Khan et al., 21 Jul 2025).

Grant-free access is another prominent extension. A Zak-OTFS-based coded random access scheme for uplink mMTC uses one pilot tile per slot and exploits the near-invariance of the effective DD channel across slots to perform reliable successive interference cancellation over replicas. In the reported high-mobility frame-level setting with Gaussian pulses, the packet loss rate stayed below $\nu_0$0 up to about $\nu_0$1 active users at $\nu_0$2 dB SNR, whereas the OFDM baseline saturated at much lower user density because inter-slot channel prediction degraded under mobility (Mirri et al., 29 Jul 2025).

Coding interacts strongly with the DD reliability structure. In a Veh-A model with $\nu_0$3 MHz, $\nu_0$4 ms, $\nu_0$5, $\nu_0$6, and a single pilot at the DD-frame center, LDPC coding with reliability-aware bin allocation extended the usable Doppler range for BER near $\nu_0$7 from approximately $\nu_0$8 kHz in the uncoded case to approximately $\nu_0$9 kHz in the coded case. The same work found that strip allocation—placing information symbols in the reliable DD strip around the pilot—performed nearly the same as explicit RPE-based allocation and that coding reduced sensitivity to whether the DD windows were sinc or RRC (Dabak et al., 2024).

Decision-directed operation has also been formalized. A differential Zak-OTFS scheme treats detected data as pilots: the cross-ambiguity between the received frame and the detected data yields a channel estimate for the next frame. In the reported setup, one true pilot frame every $$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$0 frames was sufficient to mitigate error propagation, and the resulting data-only differential scheme achieved better BER than a spread-pilot full-spectral-efficiency baseline at lower complexity because it avoided pilot-removal processing and reassigned pilot energy to data (Mattu et al., 16 Jul 2025).

MIMO generalization has now been derived directly from the physical multipath channel. In the 2026 MIMO Zak-OTFS formulation, structured DD pilot placement supports channel estimation under CDL-C, and the reported performance exhibits a crossover phenomenon: CP-OFDM is slightly better at low SNR and low Doppler, while Zak-OTFS is better at higher SNR or under more severe Doppler dispersion. The same study reports an optimal pilot-to-data power ratio around $$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$1 dB for both Zak-OTFS and CP-OFDM, while noting that Zak-OTFS is more sensitive to overly large PDR because of pilot concentration on DD bins (Barati et al., 24 Jun 2026).

An even more aggressive direction is faster-than-Nyquist signaling via mutually unbiased bases. Rather than shrinking time or frequency spacings, the proposed Zak-OTFS FTN scheme superimposes data on two mutually unbiased orthonormal bases in the same $$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$2-dimensional space. Simulations with $$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$3, corresponding to $$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$4 and $$\mathcal Z_r(\tau,\nu)=e^{j2\pi \nu_0(\tau-\tau_0)}\mathcal Z_x(\tau-\tau_0,\nu-\nu_0).$$5 symbols per frame, showed uncoded performance similar to Nyquist signaling and, with trellis-coded modulation, better performance than Nyquist signaling at high SNRs (Mattu et al., 4 Aug 2025).

Two common misconceptions recur in this broader literature. The first is that Zak-OTFS is automatically low-PAPR; the basic pulsone construction is not, and low-PAPR operation requires spread pilots, spread carriers, or related unitary waveform shaping. The second is that single-pilot predictability is universal; in fact it is conditional on support localization, pilot-region design, and window ambiguity. Non-root-Nyquist windows induce colored effective DD noise, finite pilot regions truncate channel tails, and heavy MIMO or extreme Doppler may require more elaborate pilot layouts or iterative estimation (Zhang et al., 7 Feb 2026, Barati et al., 24 Jun 2026).

Across these developments, the unifying feature remains unchanged: Zak-OTFS treats the DD plane as the native signaling domain and uses Zak-structured carriers, transforms, and convolutions to make propagation-induced delay and Doppler coupling explicit. The resulting framework has evolved from a first-principles derivation of OTFS to a family of modulation, estimation, sensing, coding, and multiuser techniques whose common currency is predictable DD-domain structure (Mohammed, 2020).