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OTFS -- Predictability in the Delay-Doppler Domain and its Value to Communication and Radar Sensing

Saif Khan Mohammed, Ronny Hadani, Ananthanarayanan Chockalingam, Robert Calderbank

arXiv:2302.08705v2eess.SPcs.ITmath.IT

TL;DR

The paper studies how to design pulsones for communication and sensing across environments with different delay and Doppler spreads. It reports that operation in the crystalline regime improves communication performance and sensing resolution, while supporting direct learning of the I/O relation without estimating the channel.

  • Problem

    Communication and sensing environments have delay and Doppler spreads that can exceed pulsone periods, a mismatch associated with non-predictability, fading, and radar ambiguity.

    Source: §VIII · p. 18

  • Method

    The paper designs a parametric family of pulsone waveforms matched to propagation-environment spreads and analyzes operation when both pulsone periods exceed the corresponding channel spreads.

    Source: §VIII · p. 18

  • Results

    In the crystalline regime, communication performance is superior through reduced DD aliasing and increased diversity exploitation, while radar sensing benefits through reduced ambiguity and increased resolution.

    Source: §VIII · p. 19

  • Takeaways & Limitations

    Choosing pulsone periods greater than environment spreads is beneficial for both communication and radar sensing, and crystalline-regime predictability permits learning the sampled I/O relation without knowing channel parameters.

    Source: §VIII · p. 19

    The stated benefits are bounded by the crystallization condition: exceeding a corresponding pulsone period with either propagation spread is associated with non-predictability and fading in communication and ambiguity in radar sensing.

    Source: §VIII · p. 18

Abstract

from arXiv · show

In our first paper [2] we explained why the Zak-OTFS input-output (I/O) relation is predictable and non-fading when the delay and Doppler periods are greater than the effective channel delay and Doppler spreads, a condition which we refer to as the crystallization condition. We argued that a communication system should operate within the crystalline regime. It is well known that it is possible to identify a linear time varying (LTV) channel if and only if it is under-spread. The crystallization condition is more restrictive than the under-spread condition, so identification is always possible. In the crystalline regime, we show that Zak-OTFS pilot sequences minimize the complexity of identifying the effective DD domain channel filter. We demonstrate that the filter taps can simply be read off from the response to a single Zak-OTFS pilot. In general, we provide an explicit formula for reconstructing the Zak-OTFS I/O relation from a finite number of received pilot symbols in the delay-Doppler (DD) domain. This reconstruction formula makes it possible to study predictability of the Zak-OTFS I/O relation for a sampled system that operates under finite duration and bandwidth constraints. We analyze reconstruction accuracy for different choices of the delay and Doppler periods, and of the pulse shaping filter. Reconstruction accuracy is high when the crystallization condition is satisfied, implying that it is possible to learn directly the I/O relation without needing to estimate the underlying channel. This opens up the possibility of a model-free mode of operation, which is especially useful when a traditional model-dependent mode of operation (reliant on estimation of the underlying physical channel) is out of reach (for example, when the channel comprises of unresolvable paths, or exhibits a continuous delay-Doppler profile such as in presence of acceleration). Our study clarifies the

I. INTRODUCTION

The introduction frames Zak-OTFS predictability around delay-Doppler periods matched to channel spreads, then develops a sampled-system model and outlines communication and radar contributions.

I. INTRODUCTION

  • In the crystalline regime, Zak-OTFS is predictable and non-fading when delay and Doppler periods exceed the effective channel spreads.The paper presents this regime as a basis for robust communication and explains that the I/O relation can be determined from a single Zak-OTFS pulsone response.Source: §I · p. 2
  • For finite-duration, finite-bandwidth systems, the paper studies reconstruction of the I/O relation from finite DD pilot samples, with accuracy depending on periods and pulse-shaping filters.Accuracy is high in the crystalline regime, enabling model-free operation when channel estimation is out of reach.Source: §I · p. 2
  • DD-domain aliasing causes non-predictability when a channel delay or Doppler spread exceeds its corresponding period.The paper introduces a crystalline decomposition of the effective channel filter into predictable and nonpredictable components; the nonpredictable component vanishes exactly under the crystallization condition.Source: §I · p. 2
  • Zak-OTFS is reported to outperform MC-OTFS as Doppler spread increases, because MC-OTFS has a less predictable I/O relation.The paper attributes Zak-OTFS predictability to discrete quasi-periodic information encoding and twisted-convolution filtering,, [7],,,,,,.Source: §I · p. 2
  • The paper derives a radar ambiguity function for the Zak pulsone and states that unambiguous delay-Doppler estimation is possible when both periods exceed the radar-scene spreads.It also notes a structural similarity between the Zak pulsone carrier and Woodward's waveform, while the concatenated radar signal's ambiguity function depends on both signal and carrier and.Source: §I · p. 2

A. The Zak-OTFS I/O Relation for a Sampled System

  • The sampled Zak-OTFS output is expressed as a discrete twisted convolution of the input with an effective DD channel filter.The symbols are encoded as a quasi-periodic DD signal and the continuous output is sampled on an information grid.Source: A. The Zak-OTFS I/O Relation for a… · pp. 3–4

B. Predictability of the Zak-OTFS I/O relation in the Crystalline Regime

In the crystalline regime, separated local pilot responses allow Zak-OTFS to predict responses at other DD points. This predictability is accompanied by a flat energy profile and non-fading I/O relation.

B. Predictability of the Zak-OTFS I/O relation in the Crystalline Regime

  • When delay and Doppler periods satisfy the crystallization condition, a response to one DD pilot can be accurately estimated from the response to another.The paper describes this for blue and green pilots at different DD-grid locations.Source: B. Predictability of the Zak-OTFS I/O… · p. 5
  • In the crystalline regime, non-overlapping local responses let the total response to a green pilot be predicted from the total response to a blue pilot.Without separation, overlapping local responses are superimposed and individual filter taps cannot generally be recovered.Source: B. Predictability of the Zak-OTFS I/O… · p. 5
  • The energy profile of local responses is independent of pilot position, so the Zak-OTFS I/O relation is non-fading in the crystalline regime.The paper states that the local-response energy is flat across pilot locations.Source: B. Predictability of the Zak-OTFS I/O… · p. 5

C. Non-predictability of the Zak-OTFS I/O relation in the Non-crystalline Regime

In the non-crystalline regime, overlapping local responses create delay-Doppler aliasing, so a pilot response cannot generally predict the response at another pilot location. Prediction remains possible for response regions free of overlap, aliasing, or self-interaction.

C. Non-predictability of the Zak-OTFS I/O relation in the Non-crystalline Regime

  • Only the part of the green-pilot response free from overlap, aliasing, or self-interaction can be predicted from the blue-pilot response.Source: Figure 3 · p. 6
  • When local responses overlap, the received value combines multiple responses, so the green-pilot response cannot be recovered from the total blue-pilot response unless the individual terms are known.In the example, interactions occur between adjacent Doppler-shifted responses.Source: C. Non-predictability of the Zak-OTFS… · p. 6
  • Because phase coefficients can make the response combinations differ by pilot location, the received energy profile generally depends on the DD location of the pilot.Source: C. Non-predictability of the Zak-OTFS… · p. 6

D. The Crystalline Decomposition

The crystalline decomposition separates the effective channel filter into taps whose responses can be estimated and taps obscured by overlapping local responses. The crystallization condition holds when the non-predictable component vanishes.

D. The Crystalline Decomposition

  • A tap is non-predictable when its response overlaps a shifted local response, making the received value a superposition that prevents estimating the tap from that response alone.Other taps in the support are predictable; their support is P, while the non-predictable support is Pc.Source: D. The Crystalline Decomposition · p. 6
  • The crystalline decomposition splits the effective channel filter into predictable and non-predictable components using the supports P and Pc.The indicator functions select the taps in each support.Source: D. The Crystalline Decomposition · p. 6
  • The exact crystallization condition is that the non-predictable component vanishes; this condition is weaker than the paper’s stricter delay- and Doppler-spread condition.The stricter condition is the focus because it directly relates to channel characteristics important to wireless systems engineering.Source: D. The Crystalline Decomposition · p. 6

E. Identification of Linear Time-varying Channels

Zak-OTFS pulsone probing simplifies channel identification in the crystalline regime, where effective channel taps can be read directly from the received response. In finite-duration and bandwidth simulations, prediction error is small in that regime and depends strongly on pulse shaping.

E. Identification of Linear Time-varying Channels

  • Prior LTV channel-identification methods solve large linear systems with roughly BT A unknown channel parameters, making practical implementation challenging.The cited methods use weighted Dirac impulse probes and Zak-transform receiver processing.Source: E. Identification of Linear… · pp. 6–7
  • A single Zak-OTFS pulsone probe lets the effective channel taps be read directly from the received response when crystallization holds, simplifying acquisition relative to earlier probing methods.The crystallization condition is more restrictive than the under-spread condition, but depends on the channel in relation to the modulation periods.Source: E. Identification of Linear… · p. 7

F. Error in prediction of the Zak-OTFS I/O relation

  • RRC filters decay faster than sinc filters, reducing DD-domain aliasing and increasing predictability, but the simulated settings increase bandwidth by 10% and duration by 20%.Source: F. Error in prediction of the Zak-OTFS… · p. 7
  • At νp = 30 KHz, the crystallization condition holds and relative prediction error is small; at νp = 1.25 KHz and 240 KHz, undersized Doppler or delay periods cause aliasing and error growth along the other dimension.Source: F. Error in prediction of the Zak-OTFS… · p. 8
  • Roughly −50 dB with root raised cosine filters versus roughly −20 dB with sinc filters for relative prediction error.The RRC improvement comes at the cost of increased frame duration and bandwidth.Source: F. Error in prediction of the Zak-OTFS… · pp. 7–8

III. MATRIX-VECTOR DESCRIPTION OF THE I/O RELATION

The paper describes the channel in the delay-Doppler domain, where the Zak-OTFS I/O relation uses a discrete effective channel filter. It also establishes matrix formulations for Zak-OTFS, TDM, and FDM.

III. MATRIX-VECTOR DESCRIPTION OF THE I/O RELATION

  • The DD representation expresses the Zak-OTFS I/O relation through twisted convolution with a discrete DD-domain channel filter.Source: §III · p. 8
  • In the crystalline regime, a single pulsone probe can acquire the effective channel filter efficiently, enabling prediction of the response to any Zak-OTFS input.Source: §III · p. 8
  • The section establishes matrix formulations for the I/O relations induced by the channel equation for Zak-OTFS, TDM, and FDM.Source: §III · p. 8

A. Zak-OTFS

The Zak-OTFS DD input-output relation can be represented using a finite set of samples in its fundamental period, while remaining valid inside or outside the crystalline regime. The section also derives sampled matrix-form relations for Zak-OTFS and TDM.

A. Zak-OTFS

  • The Zak-OTFS DD I/O relation uses an effective channel filter and a sampled noise term, and its matrix form relates MN received samples to MN transmitted symbols.The sequences can be reconstructed from samples within the fundamental period.Source: A. Zak-OTFS · pp. 8–9
  • The MN received DD samples in the fundamental period are sufficient to represent the transmitted symbols because the received signal is quasi-periodic.The I/O relation is valid whether or not the crystallization condition holds.Source: A. Zak-OTFS · p. 9

B. TDM

  • The TDM section derives a sampled time-domain I/O relation and arranges transmitted symbols, received samples, noise, and the effective channel filter in matrix form.The received vector includes K1 and K2 extra samples, determined by the channel delay spread.Source: B. TDM · p. 9

C. FDM

With perfect knowledge of the I/O relation, Zak-OTFS performs best in the crystalline regime and approaches TDM or FDM as its Doppler or delay period shrinks. The section also specifies sampled FDM relations and compares the schemes under doubly spread and single-axis channels.

C. FDM

  • The sampled FDM I/O relation uses a discrete frequency-domain effective channel filter and noise, then forms a matrix relation from the arranged symbol and received-sample vectors.The received-vector extension L1 and L2 is determined by the channel Doppler spread.Source: C. FDM · pp. 9–10

IV. IMPACT OF FADING IN THE CRYSTALLINE REGIME

  • At νp = 15 KHz, Zak-OTFS BER is superior to TDM and FDM; outside the crystalline regime, it approaches FDM at νp = 960 KHz and almost coincides with TDM at νp = 0.625 KHz.Fig. 7 compares BER along τp · νp = 1; outside the crystalline regime, aliasing produces fading.Source: §IV · pp. 10–11, Figure 7 · p. 11
  • For delay-only channels, crystalline-regime Zak-OTFS has no fading and BER changes little with νp, while BER degrades considerably when the delay period falls below the delay spread.In the delay-only case, TDM does not fade, whereas FDM does.Source: §IV · p. 11, Figure 8 · p. 11
  • For Doppler-only channels, crystalline-regime Zak-OTFS has no fading and BER changes little with νp, while BER degrades outside that regime.In the Doppler-only case, FDM does not fade, whereas TDM does.Source: §IV · p. 11, Figure 9 · p. 11

V. MODEL-FREE OPERATION IN THE CRYSTALLINE REGIME

The section compares model-dependent and model-free channel acquisition and evaluates model-free Zak-OTFS across channel conditions, Doppler spreads, and pulse shapes. In the crystalline regime, its performance can remain close to perfect-I/O knowledge, while aliasing and pulse-shaping trade-offs delimit reliable operation.

V. MODEL-FREE OPERATION IN THE CRYSTALLINE REGIME

  • Model-free operation estimates effective DD channel-filter taps directly, and in the crystalline regime those taps can be estimated from a single pilot response.The model-free receiver does not use a model of the physical delay-Doppler spreading function.Source: §V · p. 12
  • Model-free Zak-OTFS BER is considerably better than model-dependent performance and only slightly worse than perfect I/O knowledge when Veh-A paths cannot be accurately estimated.Insufficient frame bandwidth and duration prevent accurate estimation of path gains, delays, and Dopplers; Fig. 10 reports the model-free comparison.Source: §V · p. 12, Figure 10 · p. 12
  • For a resolvable five-path channel, model-dependent performance matches perfect I/O knowledge, while model-free performance is only slightly inferior, including at a 4.375 KHz Doppler spread.Fig. 11 reports this comparison for Zak-OTFS in the crystalline regime.Source: §V · pp. 12–13, Figure 11 · p. 13
  • With νp = 15 or 30 KHz, BER remains excellent and nearly invariant as Doppler spread increases; at smaller Doppler periods, BER degrades as spread approaches or exceeds the period.At νp = 7.5 KHz BER rises beyond a 4 KHz spread, and at νp = 3.75 KHz performance degrades completely when spread equals the period.Source: §V · p. 13, Figure 12 · p. 13
  • Better-localized RRC pulse shaping reduces DD aliasing and extends reliable model-free operation to higher Doppler spreads, at the cost of increased frame duration and bandwidth.Fig. 13 compares sinc and RRC pulses with βτ = 0.1 and βν values from 0.3 to 0.9.Source: §V · pp. 13–14, Figure 13 · p. 14

VI. MULTICARRIER APPROXIMATIONS TO ZAK-OTFS

The paper contrasts Zak-OTFS with the two-step multicarrier approximation MC-OTFS, whose I/O relation is less predictable. Under imperfect I/O knowledge, model-free Zak-OTFS performs better, particularly as Doppler spread increases.

VI. MULTICARRIER APPROXIMATIONS TO ZAK-OTFS

  • MC-OTFS uses linear convolution, multiplication by a quasi-periodic generator, and twisted convolution, unlike Zak-OTFS’s associative twisted-convolution chain and effective-filter I/O relation.The two-step modulation first maps DD symbols to TF symbols using inverse SFFT, then produces the TD signal through the Heisenberg transform.Source: §VI · p. 14
  • As the Doppler period shrinks toward zero, Zak-OTFS converges to TDM with a predictable linear-convolution I/O relation, whereas MC-OTFS does not converge to TDM.The MC-OTFS limiting I/O relation cannot be expressed as simple linear convolution with an effective time-domain channel filter.Source: §VI · p. 14
  • In the crystalline regime, MC-OTFS has larger relative prediction error than Zak-OTFS for the compared two-path channel and sinc pulses.Fig. 16 presents the MC-OTFS error heatmap; the comparison is at νp = 30 KHz and τp = 33.3 µs.Source: §VI · pp. 15–16, Figure 16 · p. 15
  • With imperfect I/O knowledge, model-free Zak-OTFS BER is greatly superior to MC-OTFS, whose BER degrades steadily as Doppler spread increases while Zak-OTFS remains almost constant.With perfect I/O knowledge, their BER performance is similar; Fig. 17 and Fig. 18 report the comparisons.Source: §VI · p. 16, Figure 17 · p. 16, Figure 18 · p. 16

VII. THE RADAR AMBUGUITY FUNCTION

The radar ambiguity function describes scene blurriness and constrains target estimation. Separating multiple targets requires limiting overlap between their ambiguity-function responses.

VII. THE RADAR AMBUGUITY FUNCTION

  • Matched filtering estimates target delay and Doppler by maximizing the cross-ambiguity function between received and transmitted waveforms.For multiple targets, the cross-ambiguity function relates the channel spreading function to the transmitted waveform's ambiguity function through twisted convolution.Source: §VII · pp. 16–17
  • Moyal’s identity bounds the volume under the squared ambiguity surface, expressing a fundamental limit on waveform-controlled radar blur.The ambiguity function constrains estimates of target distance and velocity.Source: §VII · p. 17
  • Two targets can be resolved by limiting response overlap, requiring ambiguity-function spreads below their separations in both delay and Doppler.Source: §VII · p. 17

A. Ambiguity functions for TDM and FDM waveforms

TDM and FDM waveforms localize ambiguity differently, leaving one target dimension poorly resolved in each case. Redistributing ambiguity energy can improve resolution in both delay and Doppler.

A. Ambiguity functions for TDM and FDM waveforms

  • TDM separates targets whose delays differ by more than 1/B, but its Doppler spread is B, limiting Doppler resolution in most scenarios.Its time-domain localization, without frequency-domain localization, explains the limitation; Fig. 19 illustrates the ambiguity-function contrast.Source: A. Ambiguity functions for TDM and FDM… · p. 17, Figure 19 · p. 17
  • FDM has a Doppler spread of about 1/T and a delay spread of T, so it cannot separate targets close in delay in most scenarios.Its frequency-domain localization, without time-domain localization, explains the delay-resolution limitation; Fig. 19 illustrates the contrast.Source: A. Ambiguity functions for TDM and FDM… · p. 17, Figure 19 · p. 17
  • Modulating a train of narrow time-domain Gaussian pulses with a broad Gaussian envelope redistributes ambiguity energy to better separate targets in delay and Doppler.The paper relates this waveform construction to Woodward’s 1953 description; Fig. 20 depicts the redistribution.Source: A. Ambiguity functions for TDM and FDM… · p. 17, Figure 20 · p. 18

B. Ambiguity function of Zak-OTFS pulsone

The Zak-OTFS pulsone produces a structured delay-Doppler ambiguity function that supports simultaneous target resolvability under the crystallization condition. The conclusions connect this condition to communication performance, sensing, and model-free operation.

B. Ambiguity function of Zak-OTFS pulsone

  • The pulsone ambiguity function comprises narrow DD impulses spaced by the delay and Doppler periods, each spread by 1/B in delay and 1/T in Doppler.Source: B. Ambiguity function of Zak-OTFS… · p. 18
  • Choosing delay and Doppler periods greater than the corresponding channel spreads guarantees unambiguous delay-Doppler estimation in the crystalline regime.These are the same crystallization conditions that guarantee predictability of the Zak-OTFS communication I/O relation; Fig. 21 illustrates the pulsone ambiguity function.Source: B. Ambiguity function of Zak-OTFS… · p. 18, Figure 21 · p. 19
  • For arbitrary pulse shaping, each ambiguity-function pulse within the fundamental period is the twisted convolution of the shaping filter with its complex conjugate.This structure enables precise ambiguity-function design.Source: B. Ambiguity function of Zak-OTFS… · p. 18

VIII. CONCLUSIONS

  • Pulsones form a parametric waveform family matched to channel delay and Doppler spreads by choosing periods greater than those spreads.A pulsone is a time-domain signal realizing a quasi-periodic localized DD function, with a prototypical structure of pulses modulated by a tone.Source: §VIII · p. 18
  • TDM and FDM are special cases of pulsones: TDM matches delay-only environments, while FDM matches Doppler-only environments.Source: §VIII · p. 18
  • When pulsone periods do not match channel spreads, DD aliasing makes communication I/O non-predictable and fading, while radar reflections can become ambiguous.Within the crystalline regime, minimizing aliasing and resolving reflections supports improved communication performance and increased radar resolution.Source: §VIII · pp. 18–19
  • In the crystalline regime, communication I/O can be learned directly without knowing underlying channel parameters, supporting model-free operation when channel estimation is out of reach.Source: §VIII · p. 19
  • MC-OTFS is less predictable than Zak-OTFS, with inferior BER performance as Doppler spread increases, making it less adapted to model-free operation.Source: §VIII · p. 19

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