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arxiv: 2604.16781 · v1 · submitted 2026-04-18 · 💻 cs.IT · eess.SP· math.IT

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Zak-OTFS: A Predictable Physical Layer for Communications and Sensing

Sandesh Rao Mattu , Nishant Mehrotra , Venkatesh Khammammetti , Robert Calderbank
Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:28 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords Zak-OTFSdelay-Doppler domainpredictable I/O relationHeisenberg-Weyl groupphysical layer design6G waveformschannel estimationsensing
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The pith

Zak-OTFS carriers make the wireless channel input-output relation predictable and non-selective when spreads are bounded.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Zak-OTFS uses carrier waveforms that are pulses in the delay-Doppler domain, appearing as pulse trains modulated by tones in time. The paper shows that when a channel's delay spread is less than the delay period and its Doppler spread is less than the Doppler period, the input-output relation becomes predictable. This means the response at every delay-Doppler point can be determined from the response at just one point. No mathematical model of the channel is required for this prediction. The same structure appears in other proposed 6G waveforms such as AFDM and OTSM.

Core claim

The central claim is that under the condition that the channel delay spread is less than the delay period and the channel Doppler spread is less than the Doppler period, the Zak-OTFS input-output relation is predictable and non-selective. Given the I/O response at one DD point in a frame, it is possible to predict the I/O response at all other points, without recourse to some mathematical model of the channel. This follows from the ambiguity properties of the basis of carrier waveforms, which share a common pulse train modulated by a Hadamard matrix structure.

What carries the argument

The Zak-OTFS carrier waveform, defined as a quasi-periodic localized function in the delay-Doppler domain and realized as a pulsone in the time domain, serving as geometric modes or common eigenvectors of a maximal commutative subgroup of the discrete Heisenberg-Weyl group.

If this is right

  • The full input-output relation across all delay-Doppler points can be determined from a single measurement.
  • The predictability holds without needing any specific channel model.
  • Similar predictability applies to other waveforms sharing the pulse train modulated by Hadamard matrix, such as AFDM, OTSM, and ODDM.
  • This enables a physical layer suitable for both communications and sensing applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could simplify receiver design in environments with limited mobility by reducing the need for continuous channel tracking.
  • Extensions might include adapting the periods to match expected channel conditions in different scenarios.
  • Testing in real-world channels with known spread limits would verify if the prediction holds as expected.
  • Neighbouring problems in integrated sensing and communications could benefit from this non-selective property.

Load-bearing premise

The channel delay spread must not exceed the delay period and the Doppler spread must not exceed the Doppler period, and the carrier waveforms must satisfy the required ambiguity properties.

What would settle it

Observing that the input-output responses at different delay-Doppler points cannot be predicted from one another in a channel where the spreads are within the periods would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.16781 by Nishant Mehrotra, Robert Calderbank, Sandesh Rao Mattu, Venkatesh Khammammetti.

Figure 1
Figure 1. Figure 1: A pictorial representation of the difference between OFDM and Zak-OTFS. OFDM prevents ISI/ICI, that [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: For predictability (Lemma 1) with a rectangular channel support S, the self-ambiguity Aϕi [k ′ , l′ ] should have only 1 non-zero value at (k ′ , l′ ) = (0, 0) within the region KS (gray). We are now ready to formalize the notion of predictabil￾ity. Definition 2 ([21]): A modulation is predictable if all its constituent modulation carriers estimate the same chan￾nel spreading function via the cross-ambigui… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of self-ambiguity function magnitudes corresponding to different modulation schemes in the waveform [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The action of symplectic transformations on [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Unitary maps between various signal domains. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A DD domain pulse and its TD/FD realizations referred to as TD/FD pulsone. The TD pulsone comprises of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: DD domain information grid. There are M delay bins and N Doppler bins. The spacing along the Doppler axis is νp/N and along delay axis is τp/M. The DD grid can carry MN information symbols. pulse is a quasi-periodic localized function defined by the delay period τp and the Doppler period νp = 1/τp. The fundamental period in the DD domain is defined as: {(τ, ν) : 0 ≤ τ < τp, 0 ≤ ν < νp}, (17) where τ and ν … view at source ↗
Figure 8
Figure 8. Figure 8: Zak-OTFS transceiver signal processing. where the effective channel filter heff[k, l] and filtered noise samples ndd [k, l] are respectively given by heff[k, l] = heff  τ = kτp M , ν = lνp N  , (33) ndd [k, l] = n wrx dd  τ = kτp M , ν = lνp N  . (34) Because of the quasi-periodicity in the DD domain, it is sufficient to consider the received samples ydd [k, l] within the fundamental period D0. Writing… view at source ↗
Figure 9
Figure 9. Figure 9: BER versus SNR at different frequency bands [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Pulse shaping filter variation and energy concentration. (a): The proposed IOTA pulse shapes have a sharper [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the bit-error performance obtained with different pulse shaping filters with perfect I/O knowledge at the receiver. The performance of the Gaussian filter is far inferior compared to all the other filters, owing to its non-orthogonality. The best performance is achieved with IOTA PSWF, IOTA Gaussian, and sinc filters, thanks to their orthogonality. Since Gaussian-sinc is approximately orthogonal, th… view at source ↗
Figure 12
Figure 12. Figure 12: Block diagram of the differential communication [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Performance curves with differential communica [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison between (a) conventional coded random access in the time-frequency domain and (b) proposed [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Single slot structures for OFDM and Zak-OTFS in the contention window illustrated in Fig. 1. (a) Time [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: Two ways of improving spectral efficiency. Con [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Effective rate as a function of α and δ for 40 dB SNR. M = 31, N = 37, dfree = √ 20. For a given δ, increasing α increases the effective rate, but dips when α = 1, which corresponds to single frame transmission. For a given α, the effective rate increases with δ. The plot is agnostic to domain. TABLE III: Complexity of the MUB approach. Operation Complexity Mount on basis O(M2N2 ) QR-precoding O(M2N2 ) MM… view at source ↗
Figure 19
Figure 19. Figure 19: Throughput of the proposed MUB scheme as [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Modulo banded structure in the FD channel [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: BER performance comparing the DD and FD equalization (FDE) of Zak-OTFS. FDE of Zak-OTFS via CGM algorithm with k = 250, ϵ = 10−6 , b = ⌈νmaxT⌉ + 1. Simulation parameters: M = 31, N = 37, νp = 30 kHz, Veh-A channel with νmax = 815 Hz, RRC pulse shaping filter with βτ = βν = 0.6. channel matrix. This is characterized by a parameter called the “spread width” (denoted by b), which denotes the number of non-ze… view at source ↗
Figure 22
Figure 22. Figure 22: Block diagrams showing different radar architectures. Figure adapted from [19]. [PITH_FULL_IMAGE:figures/full_fig_p029_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Continuous & discrete radar architectures generate radar images with significant sidelobes around ground [PITH_FULL_IMAGE:figures/full_fig_p030_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The proposed discrete radar architecture in Fig. 22(c) achieves perfectly localized “bed-of-nails” ambiguity [PITH_FULL_IMAGE:figures/full_fig_p030_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Example illustrating the choice of an appropriate maximal commutative subgroup [PITH_FULL_IMAGE:figures/full_fig_p030_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: The GDAFT (Definition 8) reduces the PAPR of the pulsone basis element (Example 1) by about [PITH_FULL_IMAGE:figures/full_fig_p032_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Comparison of different approaches for polarimetry. (a) Sequential polarimetry with FMCW transmits polarized [PITH_FULL_IMAGE:figures/full_fig_p032_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Single target detection and estimation performance. (a) Receiver operating characteristic (ROC) curve [PITH_FULL_IMAGE:figures/full_fig_p035_28.png] view at source ↗
read the original abstract

This tutorial derives the mathematical foundations of what it means for a carrier waveform to be predictable and non-selective. We focus on Zak-OTFS, where each carrier waveform is a pulse in the delay-Doppler (DD) domain, formally a quasi-periodic localized function with specific periods along delay and Doppler. Viewed in the time domain, the Zak-OTFS carrier is realized as a pulse train modulated by a tone (termed a pulsone). We start by providing physical intuition, describing what it means for the Zak-OTFS carrier waveforms to be geometric modes of the Heisenberg-Weyl (HW) group of discrete delay and Doppler shifts that define the discrete-time communication model. In fact, we show that these geometric modes are common eigenvectors of a maximal commutative subgroup of our discrete HW group. When the channel delay spread is less than the delay period, and the channel Doppler spread is less than the Doppler period, we show that the Zak-OTFS input-output (I/O) relation is predictable and non-selective. Given the I/O response at one DD point in a frame, it is possible to predict the I/O response at all other points, without recourse to some mathematical model of the channel. While it may be intuitive that geometric modes of the HW group are predictable and non-selective wireless carriers, this is not a requirement. We provide a necessary and sufficient condition that depends on the ambiguity properties of the basis of carrier waveforms. In fact, we show that the structure of a pulse train modulated by a Hadamard matrix is common to several families of waveforms proposed for 6G, including Zak-OTFS, AFDM, OTSM and ODDM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript is a tutorial deriving the mathematical foundations of predictable and non-selective carrier waveforms for Zak-OTFS. It establishes that Zak-OTFS carriers are quasi-periodic localized pulse trains in the delay-Doppler domain (pulsone in time domain) that serve as common eigenvectors of a maximal commutative subgroup of the discrete Heisenberg-Weyl group. Under the explicit conditions that the channel delay spread is less than the delay period and the Doppler spread is less than the Doppler period, the input-output relation is shown to be predictable and non-selective: the response at one DD point determines the responses at all other points without requiring a channel model. A necessary and sufficient condition on the ambiguity function of the carrier basis is provided, and the same pulse-train modulated by Hadamard structure is shown to appear in AFDM, OTSM, and ODDM.

Significance. If the central derivations hold, the work supplies a rigorous group-theoretic and ambiguity-function basis for predictability in DD-domain modulations, which could simplify channel estimation, equalization, and integrated sensing in high-mobility 6G scenarios by making the I/O map deterministic under bounded-spread conditions. The explicit unification with other proposed waveforms adds cross-family insight.

major comments (2)
  1. [Ambiguity-function argument (around the predictability condition)] The section deriving the necessary and sufficient ambiguity-function condition for predictability: the manuscript states that the Zak-OTFS basis satisfies the condition when the spread bounds hold, but the explicit computation showing that the ambiguity function vanishes outside the relevant delay-Doppler support (thereby making the I/O relation non-selective) is not carried out in sufficient detail to verify the claim that prediction requires no channel model.
  2. [Heisenberg-Weyl eigenvector discussion] The paragraph linking the common-eigenvector property to the I/O predictability: while the HW-group eigenvectors are correctly identified, the step from this algebraic property to the concrete statement that one DD-point response predicts all others is asserted rather than derived from the quasi-periodic structure and the spread inequalities; an intermediate equation relating the received DD symbol to the transmitted one via the ambiguity kernel would make the argument load-bearing.
minor comments (2)
  1. Notation for the delay and Doppler periods is introduced without an explicit symbol table or consistent use of subscripts throughout the derivations.
  2. The tutorial would benefit from a short table comparing the ambiguity-function supports of Zak-OTFS versus AFDM/OTSM/ODDM to make the unification claim immediately verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our tutorial manuscript. The comments highlight opportunities to strengthen the explicit derivations of the ambiguity-function condition and the link from the Heisenberg-Weyl eigenvector property to I/O predictability. We address each major comment below and will revise the manuscript to incorporate the suggested details.

read point-by-point responses

  1. Referee: The section deriving the necessary and sufficient ambiguity-function condition for predictability: the manuscript states that the Zak-OTFS basis satisfies the condition when the spread bounds hold, but the explicit computation showing that the ambiguity function vanishes outside the relevant delay-Doppler support (thereby making the I/O relation non-selective) is not carried out in sufficient detail to verify the claim that prediction requires no channel model.

    Authors: We agree that the explicit computation was insufficiently detailed. In the revised version, we will expand this section with a step-by-step calculation of the ambiguity function for the Zak-OTFS pulsone basis. Starting from the quasi-periodic localized pulse-train structure in the delay-Doppler domain, we will show that under the conditions (channel delay spread less than the delay period and Doppler spread less than the Doppler period), the ambiguity function is identically zero outside the relevant bounded support. This vanishing directly implies that the input-output relation is non-selective, so that the response at any single DD point determines all others without invoking a channel model. The added equations will make the necessary-and-sufficient condition fully verifiable. revision: yes

  2. Referee: The paragraph linking the common-eigenvector property to the I/O predictability: while the HW-group eigenvectors are correctly identified, the step from this algebraic property to the concrete statement that one DD-point response predicts all others is asserted rather than derived from the quasi-periodic structure and the spread inequalities; an intermediate equation relating the received DD symbol to the transmitted one via the ambiguity kernel would make the argument load-bearing.

    Authors: We accept that the transition requires an explicit intermediate step. In the revision, we will insert a new equation immediately after identifying the common eigenvectors of the maximal commutative subgroup. This equation will express the received DD symbol as the transmitted symbol multiplied by the ambiguity kernel evaluated at the channel spread offsets. Applying the spread inequalities to the quasi-periodic structure then shows that the kernel reduces to a constant (or zero) factor independent of the specific DD point, thereby proving that the response at one point predicts all others. This derivation will connect the algebraic eigenvector property directly to the predictability claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the predictability and non-selectivity of Zak-OTFS carriers directly from the quasi-periodic pulse-train structure as common eigenvectors of a maximal commutative subgroup of the discrete Heisenberg-Weyl group, together with the explicit spread conditions (delay spread < delay period, Doppler spread < Doppler period) and the necessary-and-sufficient ambiguity-function condition on the basis. These steps are conditional on stated assumptions that are independent of the target conclusion and rest on standard group-theoretic and ambiguity-function properties rather than any fitted parameter, self-referential definition, or load-bearing self-citation. The extension to AFDM/OTSM/ODDM is presented as an observation of shared structure, not as a derivation that reduces to the Zak-OTFS result. No step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on two design parameters (the delay and Doppler periods) chosen to exceed channel spreads, plus standard mathematical properties of the Heisenberg-Weyl group and ambiguity functions.

free parameters (2)
  • delay period
    Design choice required to be larger than channel delay spread to guarantee predictability; value is not derived but selected by system designer.
  • Doppler period
    Design choice required to be larger than channel Doppler spread to guarantee predictability; value is not derived but selected by system designer.
axioms (2)
  • domain assumption Zak-OTFS carriers are common eigenvectors of a maximal commutative subgroup of the discrete Heisenberg-Weyl group
    Invoked in the abstract to establish that the waveforms are geometric modes yielding predictable I/O.
  • domain assumption The ambiguity properties of the carrier basis provide a necessary and sufficient condition for predictability and non-selectivity
    Used to generalize beyond Zak-OTFS to other waveforms and to derive the spread-bounded condition.

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