Zak-OTFS Modulation

• Zak-OTFS modulation is defined as a delay-Doppler domain technique that constructs approximately time- and bandwidth-limited orthonormal bases using the Zak transform.
• The method enables robust transmission by diagonalizing time delays and Doppler shifts and supports efficient joint equalization over an MN-dimension symbol block.
• Its design offers a tunable trade-off between Doppler resilience and transmission latency, making it a promising alternative to conventional multicarrier schemes like OFDM.

Zak-OTFS modulation is a class of delay–Doppler (DD) domain modulation techniques that constructs orthonormal bases of approximately time- and bandwidth-limited signals with simultaneous localization in the DD domain. This modulation leverages the Zak (or ZAK) transform, which diagonalizes time delays and Doppler shifts, enabling robust transmission over highly time-varying wireless channels, particularly those encountered in high-mobility scenarios. The Zak-OTFS framework underpins a broad range of recent advances in next-generation wireless transceivers, offering improved resilience against Doppler-induced impairments, tractable DD domain equalization, and an explicit trade-off between robustness and latency. The rigorous mathematical derivation and engineering implications of Zak-OTFS mark a significant departure from conventional multicarrier schemes such as OFDM, especially regarding Doppler resilience and channel predictability (Mohammed, 2020, Lampel et al., 2021).

1. Zak Representation and Delay–Doppler Basis Construction

The Zak transform of a time-domain (TD) signal $x(t)$, for period $T$, is given by

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

where $\tau$ and $\nu$ act respectively as "delay" and "Doppler" variables. A fundamental property is that TD shifts and modulations (i.e., delays and frequency offsets) map to simple shifts in the $(\tau, \nu)$ Zak domain, which diagonalizes the multipath channel in the DD domain.

To construct a DD impulse basis, the Zak representation is set to a Dirac delta at a particular DD location $(\tau_0, \nu_0)$. The corresponding time-domain basis function is an impulse train with spacing $T$ modulated by $e^{j2\pi\nu_0 nT}$. These ideal basis functions lack time and bandwidth limitation, which is necessary for physical realizability and spectral efficiency.

Approximate DD–localized, time– and band–limited basis functions are synthesized by windowing in time (using $q(t)$, e.g., indicator on $[0, NT)$), and convolving with a band–limiting pulse $s(t)$ (e.g., a sinc pulse shaped to $[0, M\Delta f)$): $$\psi_{(q,s),(\tau_0,\nu_0)}(t) = [p_{(\tau_0,\nu_0)}(t) q(t)] * s(t)$$ Discretizing $(\tau_0, \nu_0)$ to $\tau_0 = lT/M, \; \nu_0 = k/(NT)$ provides an orthonormal set $\{ \alpha_{k,l}(t) \}$ spanning the $MN$-dimensional signal space bandlimited to $M\Delta f$ and time-limited to $NT$.

2. Modulation, Symbol Embedding, and Signal Construction

Given a set of information symbols $x[k,l]$ defined over the DD grid, the transmit signal is formed as

$$x(t) = \sum_{k=0}^{N-1}\sum_{l=0}^{M-1} x[k,l] \, \alpha_{k,l}(t)$$

Alternatively, the modulation can be decomposed into two stages. The DD-to–time–frequency (TF) conversion is realized via a finite 2D unitary transform

$$X_{TF}[n, m] = \sum_{k=0}^{N-1} \sum_{l=0}^{M-1} x[k,l] e^{j2\pi \left( \frac{nk}{N} - \frac{ml}{M} \right)}$$

and then mapped to time via multicarrier modulation (e.g., OFDM with appropriately constructed transmit pulse $g(t)$).

This construction ensures that each information symbol is "spread" over the TF domain within a tile whose width and height are controlled by the time and bandwidth constraints, while the underlying DD basis functions retain strong localization due to the Zak approach.

3. Channel Action and Robustness to Mobility

The action of a channel with delay $\tau_i$ and Doppler $\nu_i$ manifests in the Zak domain as

$$\mathcal{Z}_r(\tau, \nu) = e^{j2\pi \nu_i (\tau - \tau_i)} \mathcal{Z}_x(\tau - \tau_i, \nu - \nu_i)$$

This property indicates that a propagating symbol’s energy primarily shifts within its DD tile and only bleeds into a small neighborhood, even for large Doppler shifts, provided the basis is localized in DD.

Interference leakage mathematically occupies at most a span of roughly two grid slots along each dimension: only a fraction $(2M + 2N - 5)/(MN - 1)$ of DD symbols are significantly interfered. In contrast, in OFDM, Doppler causes inter-carrier interference that can sharply degrade orthogonality, often resulting in 40–50% or higher TF symbol interference for high Doppler environments.

The inherent diagonal and sparse behavior of the channel in the DD domain enables joint equalization over the full $MN$ symbol block, leveraging the predictable pattern of symbol coupling.

4. Trade-offs: Localization, Robustness, and Latency

The degree of DD localization is fundamentally limited by the finite product of signal duration and bandwidth due to uncertainty principles (as formalized by the Balian-Low theorem). The size of the DD tile—width $\sim 1/(M\Delta f)$ and height $\sim \Delta f/N$—determines both the interference leakage and the achievable localization.

Increasing $NT$ (i.e., longer time duration) allows for finer localization in Doppler ($\sim \Delta f / N$), which improves robustness to Doppler spread but necessarily increases the latency per transmission block. Conversely, tighter time localization (smaller $NT$) increases DD "spread" and mutual interference. This trade-off is fundamental in DD domain modulation and must be considered in system design.

5. Mathematical Formulation and Key Relationships

The following tabulates the core mathematical relationships central to Zak-OTFS:

Quantity	Mathematical Formulation	Description
Zak transform	$$\mathcal{Z}_x(\tau, \nu) = \sqrt{T} \sum_n x(\tau+nT) e^{-j2\pi\nu nT}$$	Maps TD signals to DD domain
Ideal DD impulse basis	$$p_{(\tau_0, \nu_0)}(t) = \sqrt{T} \sum_n e^{j2\pi \nu_0 nT} \delta(t-\tau_0-nT)$$	Localized but not physically realizable
Time-/band-limited DD basis	$$\psi_{(q,s),(\tau_0, \nu_0)}(t) = [p_{(\tau_0, \nu_0)}(t) q(t)]*s(t)$$	Physically usable, localized basis
Orthonormal basis (discrete grid)	$$\alpha_{k,l}(t) = \frac{1}{\sqrt{MN}} \psi_{(q,s)}^{(lT/M, k/NT)}(t)$$	Spanning NT in time, $M\Delta f$ in freq.
DD to TF domain encoding	$$X_{TF}[n,m] = \sum_{k,l} x[k,l] e^{j2\pi (nk/N - ml/M)}$$	Precedes multicarrier modulation
Channel action in Zak domain	$$\mathcal{Z}_r(\tau, \nu) = e^{j2\pi \nu_i (\tau - \tau_i)} \mathcal{Z}_x(\tau-\tau_i, \nu-\nu_i)$$	Illustrates shift/diagonalization

6. Comparison with OFDM and Implications for System Design

Zak-OTFS demonstrates substantial robustness against channel-induced Doppler shift compared to OFDM, which is critical in high-mobility environments typical of future wireless networks. The sparsity in DD domain interference enables low-complexity joint equalization, and the basis "tails" (i.e., leakage) are mathematically and practically controllable via time and bandwidth assignment.

OFDM, by contrast, suffers from non-negligible inter-carrier interference as Doppler increases, since its basis is strictly localized in frequency but not in Doppler. The Zak-OTFS formulation is particularly beneficial in environments where the channel time-variation (Doppler) exceeds the OFDM coherence time, making strict frequency grid allocation infeasible.

The Zak-OTFS first-principles derivation clarifies the latency-robustness trade-off, showing that higher Doppler resilience must be "paid for" via increased transmission duration. This transparency provides system designers explicit control over the localization/latency design space unavailable in conventional TF-based approaches.

7. Engineering Consequences, Extensions, and Research Directions

The mathematical rigor underlying Zak-OTFS yields several practical consequences. First, the basis design allows a systematic trade-off between orthogonality, localization, and implementation cost. The formal equivalence to multicarrier schemes (e.g., OTFS via OFDM overlay (Lampel et al., 2021)) exists only for certain pulse shapes; with more general pulses, Zak-OTFS provides greater design freedom.

Recent advances, such as the extension to discrete Zak transform (DZT) frameworks (Lampel et al., 2021), exploit digital implementations for efficient modulation/demodulation and further simplify channel analysis. The framework is extensible to pulse-shaping strategies, pilot/resource placement design, and robust equalization in non-integer and time-varying DD channels, as well as integration with advanced coding and detection methods.

A continuing research direction is the optimal selection and adaptation of DD basis functions under hardware constraints (e.g., windowing, quantization), as well as the investigation of extensions to multiuser, massive MIMO, and sensing–communication co-design contexts.

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Zak-OTFS constitutes a robust, theoretically grounded approach to wireless communication over doubly dispersive channels, with a foundation in Zak transform–based DD domain basis design, explicit channel diagonalization properties, and a rigorously quantifiable robustness/latency trade-off (Mohammed, 2020).