Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters

Ananthanarayanan Chockalingam; Arpan Das; Fathima Jesbin

REVIEW 3 major objections 4 minor 38 references

Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Zak-OTFS delay-Doppler I/O relations become closed-form sums for sinc and Gaussian filters.

desk verdict A solid, gap-filling paper that derives exact closed-form Zak-OTFS I/O and noise covariance for most filtering configurations, with one approximate case whose validation is thinner than the authors' wording suggests. read the letter →

arxiv 2504.18887 v1 pith:OOHG72UV submitted 2025-04-26 cs.IT eess.SPmath.IT

Arpan Das , Fathima Jesbin , Ananthanarayanan Chockalingam This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords Zak-OTFSdelay-Dopplerdomainclosed-formI/OrelationnoisecovariancetwistedconvolutionsincfilterGaussianmatchedfiltering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Zak-OTFS transceivers shape information in the delay-Doppler (DD) domain with filters whose cascade is a twisted convolution, so the end-to-end input-output relation is a stack of integrals that previously had to be evaluated numerically. This paper derives closed-form expressions for the effective discrete DD channel taps $h_{\mathrm{eff}}[k,l]$ and the filtered-noise covariance when the transmit filter is sinc or Gaussian and the receive filter is identical, matched, or channel-matched. The formulas are exact for every combination except sinc with identical filtering, where the paper supplies an approximate closed form and demonstrates accuracy on a Vehicular-A channel with fractional delay-Doppler spreads. The payoff is that system matrices and noise statistics for Zak-OTFS can be assembled directly from sums of elementary functions, cutting simulation times from minutes to seconds and enabling noise whitening for detection.

What carries the argument

The central object is the twisted convolution, a variant of convolution between two DD-domain functions with an extra phase factor $e^{j2\pi \nu'(\tau-\tau')}$; the whole transceiver filter cascade collapses into the effective channel $h_{\mathrm{eff}}(\tau,\nu)=w_{\mathrm{rx}} *_{\sigma} h_{\mathrm{phy}} *_{\sigma} w_{\mathrm{tx}}$. The argument exploits the separable form $w_{\mathrm{tx}}(\tau,\nu)=w_1(\tau)w_2(\nu)$ and the fact that for sinc and Gaussian filters the inner integrals are Fourier transforms of windowed sincs or Gaussians. Those transforms turn the multiple integrals into products such as $\frac{B-|f|}{B^2}\operatorname{sinc}((B-|f|)\Delta\tau)$ or, for Gaussian filters, exponentials obtained by completing the square; sampling on the lattice $\Lambda_{\mathrm{dd}}$ yields the discrete taps. For sinc identical filtering, the derivation additionally drops the $\beta_i(x)$ correction in Eq. (92), arguing it is small for large $M,N$ in the crystalline regime, which produces the approximate closed form in Theorem 1.

What would settle it

Evaluate the exact tap in Eq. (28) by numerical quadrature and compare it with the approximate formula (29) for a channel whose delay spread approaches $\tau_p$ or whose Doppler spread approaches $\nu_p/2$, at small frame sizes such as $M=N=8$; a tap-level or BER deviation well beyond the Fig. 2 agreement would show the approximation's domain is narrower than claimed.

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Extended reading notes

Core claim

The paper's central claim is that the cascade $w_{\mathrm{rx}} *_{\sigma} h_{\mathrm{phy}} *_{\sigma} w_{\mathrm{tx}}$ can be evaluated in closed form on the Zak-OTFS information lattice for sinc and Gaussian transmit filters and for three receive-filter choices. For identical, matched, and channel-matched filtering, the effective channel taps $h_{\mathrm{eff}}[k,l]$ and the covariance $\mathbb{E}[n_{\mathrm{dd}}[k_1,l_1]n_{\mathrm{dd}}^*[k_2,l_2]]$ reduce to finite sums of windowed sinc products or Gaussians per path (or per path pair), with no numerical integration over the twisted-convolution kernel. The exception is sinc with identical filtering, where a correction term is dropped in Appendix A to obtain an approximate closed form that the paper validates against the exact numerical expression. Using these expressions, the paper evaluates bit error performance for BPSK and 8-QAM with MMSE detection on a Vehicular-A channel with fractional DDs and reports three performance findings: matched filtering essentially ties identical filtering, channel-matched filtering gives the best BER, and sinc generally beats Gaussian, while Gaussian improves with time and bandwidth expansion.

Load-bearing premise

The load-bearing premise is the Appendix A approximation that the $\beta_i(x)$ correction is negligible in the sinc-identical derivation for large frames in the crystalline regime (delay spread far below $\tau_p$ and Doppler spread far below $\nu_p/2$); the paper verifies this only for the tested Vehicular-A settings.

Editorial extensions

If this is right

Assembling the Zak-OTFS channel matrix and noise covariance directly from the closed-form sums removes the numerical-integration bottleneck; the paper reports noise-covariance computation falling from 26 minutes to 0.23 s and per-realization channel-tap computation from 92 minutes to 0.42 s in the tested setting.
Channel-matched filtering, which maximizes SNR by matching to the channel-Tx cascade, delivers the best BER of the three schemes, while matched and identical filtering perform nearly the same, so it is the benchmark receiver for Zak-OTFS.
Sinc filtering outperforms Gaussian filtering and confines the transmitted spectrum to $(-B/2,B/2)$, whereas the Gaussian leaks about 1 percent of energy outside that band; expanding the Gaussian to $B'=1.12B$, $T'=1.25T$ buys about 2 dB at high SNR.
Because the closed-form noise covariances permit noise whitening before detection, maximum-likelihood detection becomes implementable on small frames; the paper shows channel-matched Gaussian filtering gains 5-6 dB over the other schemes at $10^{-4}$ BER for $M=N=2$.
Since the derivations place no integrality constraint on the path delays or Dopplers, the closed forms apply to fractional delay-Doppler channels rather than only lattice-aligned ones.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Fourier-transform product technique should extend to other separable DD filters, including root raised cosine, although the paper leaves that case open.
The sinc-identical approximation is validated at one operating point; stressing it with delay spread near $\tau_p$, Doppler near $\nu_p/2$, or very small $M,N$ would likely reveal where the dropped correction term matters.
For sinc identical filtering the approximate noise covariance collapses to $N_0$ times the identity at large $M,N$, which the paper does not exploit; that configuration may need no whitening before equalization.
Channel-matched filtering sums over path pairs and therefore costs more nonzero taps than identical or matched filtering; the roughly 1 dB gain for sinc and 5-6 dB gain for Gaussian at small frame sizes leaves a complexity-performance tradeoff that the paper does not quantify.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth reading: Zak-OTFS is a niche but active area, and this paper fills concrete gaps. Only the Gaussian matched-filtering closed form existed in the literature; here they provide exact closed forms for sinc and Gaussian under identical, matched, and channel matched filtering, except for sinc identical filtering which is approximate. That is a real contribution. The derivations are detailed enough to check, and the expressions are practically useful: the reported speedups (noise covariance in 0.23 s versus 26 min) are entirely plausible for replacing nested numerical integrals.

The strongest parts are the exact results for Gaussian and for sinc matched/channel matched. Appendix A is the load-bearing weak point: the sinc identical filtering approximation drops the beta_i integral in Eq. (92) with an "it is observed" assertion and no bound. The stress-test note is fair. The split into alpha and beta terms is not controlled; beta_i contains |x| sinc(pi |x| nu'_i), which can grow when delay/Doppler spreads approach the crystalline boundary. The authors validate only one Veh-A realization (M=12, N=14, nu_max=815 Hz, BPSK/8-QAM). That is enough to show the approximation works somewhere, not enough to establish where it fails. Since that approximation feeds the sinc identical BER curves, the comparative claims for identical filtering at high Doppler rest on a shakier foundation than the rest of the paper. A serious referee should ask for either a bound or a second validation at a different operating point. The notation slip in Eq. (91) (sinc(pi |x| nu'_i) versus sinc(|x| nu'_i)) looks like a harmless typo but should be cleaned up.

Everything else holds up. The noise covariance derivations come from first principles, the truncation of infinite q-sums is standard and checked by heuristics, and the citation pattern is honest: the Gaussian matched result is explicitly credited to [29]. No shipped code, but this paper does not need it.

Bottom line: a well-executed gap-filler, not a paradigm shift. The approximate case needs more care but does not sink the paper. Send it to peer review with a request for stronger validation of Theorem 1.

Referee Report

3 major / 4 minor

Summary. The paper derives discrete delay-Doppler-domain closed-form expressions for the end-to-end I/O relation and noise covariance in Zak-OTFS. Sinc and Gaussian transmit filters are considered, together with identical, matched, and channel-matched receive filtering. For all cases except sinc identical filtering, the authors present exact closed-form expressions. For sinc identical filtering, they derive approximate closed-form expressions by neglecting a term in an integral in Appendix A and then use these expressions in BER simulations. The paper also presents BER and SNR comparisons among the filtering schemes, reports substantial simulation speedups, and validates the sinc identical approximation against numerical integration in one Vehicular-A setting.

Significance. If the results are correct, the paper provides a useful toolbox: it replaces numerically evaluated twisted-convolution integrals with compact formulas for most of the considered Zak-OTFS filter configurations, which should accelerate performance evaluation and simplify the derivation of detection and estimation algorithms. The exact formulas for matched and channel-matched filtering are derived carefully and from first principles, with detailed appendices, and the Gaussian matched-filtering result is properly credited to prior work. The main caveat is that the approximate sinc identical-filtering case rests on a heuristic term-drop that is not analytically quantified and is validated numerically only in a single configuration; since the comparative claims in Figs. 3, 6, and 7 use this approximation, the strength of those claims is currently tied to the validity of that unquantified approximation.

major comments (3)

[Appendix A, Eqs. (92)-(96)] The central approximation of Theorem 1 is the neglect of the β_i integral in Eq. (92). The paper states that this term is 'much small' for large M,N and the crystalline regime, but no bound or quantitative condition is provided. For ν'_i not close to zero, the retained term T sinc(Tν'_i)∫α_i(x)dx and the neglected term cos(πTν'_i)∫β_i(x)dx can have comparable magnitudes (both are roughly of order 1/(B|ν'_i|) before cancellations), so the assertion is not self-evident. Please supply an analytical error bound (for example, showing that the ratio of the neglected term to the retained term is O(1/(MN)) or controlled by a stated small parameter in the crystalline regime), or provide a systematic numerical comparison between the exact integral in Eq. (28) and the closed-form Eq. (29) over a range of fractional delays, Dopplers, and lattice sizes. Because Theorem 1 and the associated noise-covariance approximation underpin the sinc identical-filtering BER results in Figs. 3, 6, and 7, this issue is load-bearing for the paper's approximate closed-form claim.
[Section III-B, Eqs. (37)-(39)] The noise covariance for sinc identical filtering is approximated as an identity matrix by replacing the finite-interval integral in Eq. (37) with the infinite-interval orthogonality result in Eq. (38). This ignores finite-T leakage of the sinc products. The approximation is plausible for large BT, but no error bound or leakage estimate is given. Since the BER simulations and any noise-whitening procedure developed from these expressions depend on this approximate covariance, please either bound the off-diagonal terms or demonstrate numerically that the approximation error is negligible over a wider range of M,N and delay/Doppler spreads than the single setting shown in Fig. 2.
[Section VI, Fig. 2] The numerical validation of the sinc identical-filtering approximation is limited to one Veh-A realization with M=12, N=14, νmax=815 Hz, and BPSK/8-QAM constellations. The crystalline regime is much broader than this single point, and the paper does not state the intended validity region of the approximation or show how the error behaves as νmax/νp and τmax/τp approach the crystalline boundary. Please add validation at additional operating points, or explicitly restrict the accuracy claim to the tested regime and state that the comparative conclusions for sinc identical filtering in Figs. 3, 6, and 7 are demonstrated only there.

minor comments (4)

[Appendix A, Eq. (91)] There appears to be a notation inconsistency in the identity: the last term writes |x|sinc(π|x|ν'_i), whereas the expansion of (T-|x|)sinc((T-|x|)ν'_i) would require |x|sinc(|x|ν'_i) under the standard normalized-sinc convention. Since this term is dropped, the final result is unaffected, but the convention should be stated or corrected.
[Section II and Section VI, Fig. 4] For Gaussian filtering with time/bandwidth expansion (B'=1.12B, T'=1.25T), the paper does not explain how the derived formulas should be modified. Please clarify whether B,T in Eqs. (42)-(48), (62)-(63), and (79)-(85) are simply replaced by B',T', or whether α_τ and α_ν are rescaled.
[Theorems 3 and Eq. (63), (85)] The infinite sums over q_1,q_2 are truncated at ±20, with the statement that this range is 'found to be adequate'. Please provide a convergence criterion or a short numerical demonstration of convergence, since the required truncation range may vary with system parameters.
[Appendix A] The sentence 'the contribution of the integral of β_i(x) ... is much small' contains a grammatical error; it should read 'much smaller'. There is also a typo in the author affiliation block ('Chockalin gam').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the closed-form I/O and noise-covariance derivations are self-contained evaluations of the stated twisted-convolution integrals, with no fitted input renamed as a prediction.

full rationale

The paper's central claims are derived from the Zak-OTFS system model of Sec. II by explicit integration. Theorems 1-3 and the matched/channel-matched closed forms follow from manipulating the twisted-convolution integrals (e.g., Appendix A, B, C), not from fitting parameters to the target outputs. The sinc identical-filtering result is explicitly approximate: Appendix A drops the beta_i integral in Eq. (92) under a stated large-B/T, crystalline-regime assumption, and the paper validates this approximation against exact numerical evaluation of the original integral in Fig. 2. That validation is external to the derivation and is a robustness/correctness check, not circularity. The Gaussian matched-filtering closed forms are reproduced from [29] with explicit attribution ('we reproduce them here for immediate reference'), and they are not load-bearing for the paper's new contributions: the identical, channel-matched, and sinc matched forms are derived independently. The channel-matched-optimal claim relies on [28], which is not authored by this paper's authors, so no uniqueness or optimality result is imported from a self-citation chain. The paper does contain a minor internal notation inconsistency around sinc(pi |x| nu'_i) in Eq. (91) versus the implied sinc(|x| nu'_i), and the dropped beta_i term is only validated for one Veh-A scenario; both are correctness and generalization concerns, not instances where a prediction reduces by construction to its inputs. Overall, no self-definitional, fitted-input-as-prediction, or self-citation-load-bearing circularity is present, so the appropriate score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivations rest on the standard Zak-OTFS twisted-convolution system model and on the evaluation of sinc/Gaussian integrals. No free parameters are fitted to the target closed-form results; the listed numerical parameters (alpha, truncation ranges) are simulation choices. One ad-hoc approximation is introduced for the sinc identical case.

free parameters (3)

alpha_tau, alpha_nu Gaussian pulse parameters = alpha_tau = alpha_nu = 1.584 for no time/bandwidth expansion
Defines the Gaussian Tx filter in Eq. (24). Set to 1.584 to keep 99% of frame energy within T and B. Not fitted to the target result; the closed-form expressions hold for general alpha.
Truncation range for m,n in channel matrix (19) = m,n in {-2,...,2}
Finite support approximation for the effective channel in simulations. The paper states this range is adequate to capture channel spread.
Truncation range for q1,q2 in Gaussian noise covariance sums = q1,q2 in {-20,...,20}
The infinite sums in (48), (63), (85) are truncated; the paper reports this range is adequate.

assumptions (5)

domain assumption The physical channel is a sum of P discrete DD paths with Dirac-delta delays and Dopplers (Eq. 8).
Standard doubly-sparse channel model used throughout; the derivations assume this form.
domain assumption The end-to-end transceiver is modeled by twisted convolution of Tx filter, channel, and Rx filter, with the Zak transform properties (Eqs. 10-14).
Core system model; all derivations rely on the twisted convolution cascade and quasi-periodicity.
domain assumption Sinc and Gaussian filters in (23)-(24) have ideal infinite support properties; for Gaussian, a 99% energy localization assumption is used for simulation.
The closed-form derivations use the ideal infinite-support Gaussian; simulations set alpha to approximate time/bandwidth limits.
ad hoc to paper In Appendix A, the integral of beta_i(x) is approximated as negligible for large M,N and operation in the crystalline regime.
This is the key approximation enabling the approximate closed-form in Theorem 1 for sinc identical filtering.
standard math The noise is AWGN with autocorrelation N0*delta(t'), and the integrals over infinite intervals can be evaluated via standard sinc/Gaussian identities.
Used throughout to evaluate the noise covariance and effective channel integrals.

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Cite this review

Pith. "Pith review of Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters." pith.science (2026). https://pith.science/paper/OOHG72UV

@misc{pith2026250418887,
  author       = {Pith},
  title        = {Pith review of: Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OOHG72UV}},
  note         = {Machine review of arXiv:2504.18887}
}

abstract

The transceiver operations in the delay-Doppler (DD) domain in Zak-OTFS modulation, including DD domain filtering at the transmitter and receiver, involve twisted convolution operation. The twisted convolution operations give rise to multiple integrals in the end-to-end DD domain input-output (I/O) relation. The I/O relation plays a crucial role in performance evaluation and algorithm development for transceiver implementation. In this paper, we derive discrete DD domain closed-form expressions for the I/O relation and noise covariance in Zak-OTFS. We derive these expressions for sinc and Gaussian pulse shaping DD filters at the transmitter (Tx). On the receiver (Rx) side, three types of DD filters are considered, viz., $(i)$ Rx filter identical to Tx filter (referred to as `identical filtering'), $(ii)$ Rx filter matched to the Tx filter (referred to as `matched filtering'), and $(iii)$ Rx filter matched to both Tx filter and channel response (referred to as `channel matched filtering'). For all the above cases, except for the case of sinc identical filtering, we derive exact I/O relation and noise covariance expressions in closed-form. For the sinc identical filtering case, we derive approximate closed-form expressions which are shown to be accurate. Using the derived closed-form expressions, we evaluate the bit error performance of Zak-OTFS for different Tx/Rx filter configurations. Our results using Vehicular-A (Veh-A) channel model with fractional DDs show that, while matched filtering achieves slightly better or almost same performance as identical filtering, channel matched filtering achieves the best performance among the three.

Figures

Figures reproduced from arXiv: 2504.18887 by the authors.

**Figure 1.** Transceiver signal processing in Zak-OTFS. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** BER performance of Zak-OTFS for identical filterin [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: BER performance of Zak-OTFS for identical filter [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]

**Figure 5.** Figure 5: SNR performance of identical, matched, and channe [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 7.** Figure 7: BER performance comparison between sinc and [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: BER performance comparison between different [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: PSD plots of the DD domain signal after Tx filter for ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 29 canonical work pages

[29]

Zak-OTFS: pulse shaping and the tradeoff between time/bandwidth expansion and predictabi lity,

J. Jayachandran, R. K. Jaiswal, S. K. Mohammed, R. Hadan i, A. Chockalingam, and R. Calderbank, “Zak-OTFS: pulse shaping and the tradeoff between time/bandwidth expansion and predictabi lity,” online arxiv.org/abs/2405.02718, 4 May 2024

arXiv 2024
[1]

On the road to 6G: visions, requiremen ts, key technologies, and testbeds,

C. -X. Wang et al., “On the road to 6G: visions, requiremen ts, key technologies, and testbeds,” IEEE Commun. Surveys & Tuts. , vol. 25, no. 2, pp. 905-974, 2023

work page 2023
[2]

Toward integrated sensing and commun ications for 6G: key enabling technologies, standardization, and chall enges,

A. Kaushik et al., “Toward integrated sensing and commun ications for 6G: key enabling technologies, standardization, and chall enges,” IEEE Commun. Standards Mag. , vol. 8, no. 2, pp. 52-59, Jun. 2024

work page 2024
[3]

Characterization of randomly time-variant l inear channels,

P . Bello, “Characterization of randomly time-variant l inear channels,” IEEE Trans. Commun. Syst. , vol. CS-11, no. 4, pp. 360–393, Dec. 1963

work page 1963
[4]

Orthogonal time frequency space modul ation,

R. Hadani et al., “Orthogonal time frequency space modul ation,” Proc. IEEE WCNC’2017 , pp. 1-6, Mar. 2017

2017
[5]

OTFS: a new generation of modulati on addressing the challenges of 5G,

R. Hadani and A. Monk, “OTFS: a new generation of modulati on addressing the challenges of 5G,” online arxiv.org/abs/18 02.02623, Feb. 2018

work page 2018
[6]

On OTFS modulation for high- Doppler fading channels,

K. R. Murali and A. Chockalingam, “On OTFS modulation for high- Doppler fading channels,” Proc. ITA W orkshop, pp. 1-10, Feb. 2018

work page 2018
[7]

Interfe rence can- cellation and iterative detection for orthogonal time freq uency space modulation,

P . Raviteja, K. T. Phan, Y . Hong, and E. Viterbo, “Interfe rence can- cellation and iterative detection for orthogonal time freq uency space modulation,” IEEE Trans. Wireless Commun., vol. 17, no. 10, pp. 6501- 6515, Aug. 2018

work page 2018

Show all 38 references

[8]

Wei et al., ”Orthogonal time-frequency space modulat ion: a promising next-generation waveform,” IEEE Wireless Commun., vol

Z. Wei et al., ”Orthogonal time-frequency space modulat ion: a promising next-generation waveform,” IEEE Wireless Commun., vol. 28, no. 4, pp. 136-144, Aug. 2021

2021
[9]

On t he effective- ness of OTFS for joint radar parameter estimation and commun ication,

L. Gaudio, M. Kobayashi, G. Caire, and G. Colavolpe, “On t he effective- ness of OTFS for joint radar parameter estimation and commun ication,” IEEE Trans. Wireless Comm. , vol. 19, no. 9, pp. 5951-5965, Sep. 2020

2020
[10]

On t he diversity of uncoded OTFS modulation in doubly-dispersive channels,

G. D. Surabhi, R. M. Augustine and A. Chockalingam, “On t he diversity of uncoded OTFS modulation in doubly-dispersive channels, ” IEEE Trans. Wireless Commun. , vol. 18, no. 6, pp. 3049-3063, Jun. 2019

2019
[11]

MIMO-OTFS in h igh- Doppler fading channels: signal detection and channel esti mation,

M. K. Ramachandran and A. Chockalingam, “MIMO-OTFS in h igh- Doppler fading channels: signal detection and channel esti mation,” Proc. IEEE GLOBECOM’2018 , pp. 206-212, Dec. 2018

2018
[12]

Low-complexity lin ear equaliza- tion for OTFS modulation,

G. D. Surabhi and A. Chockalingam, “Low-complexity lin ear equaliza- tion for OTFS modulation,” IEEE Commun. Lett. , vol. 24, no. 2, pp. 330-334, Feb. 2020

2020
[13]

Low complexity LMMSE receiver for OTFS,

S. Tiwari, S. S. Das, and V . Rangamgari, “Low complexity LMMSE receiver for OTFS,” IEEE Commun. Lett. , vol. 23, no. 12, pp. 2205- 2209, Dec. 2019

2019
[14]

A simple variati onal Bayes detector for orthogonal time frequency space (OTFS) modula tion,

W. Y uan, Z. Wei, J. Y uan, and D. W. K. Ng, “A simple variati onal Bayes detector for orthogonal time frequency space (OTFS) modula tion,” IEEE Trans. V eh. Tech., vol. 69, no. 7, pp. 7976-7980, Jul. 2020

2020
[15]

Gaussian appr oximate mes- sage passing detection of orthogonal time frequency space m odulation,

L. Xiang, Y . Liu, L-L. Y ang, and L. Hanzo, “Gaussian appr oximate mes- sage passing detection of orthogonal time frequency space m odulation,” IEEE Trans. V eh. Tech., vol. 70, no. 10, pp. 10999-11004, Oct. 2021. 16

2021
[16]

Hybrid MAP and PIC detection for OTFS modulation,

S. Li, W. Y uan, Z. Wei, J. Y uan, B. Bai, D. W. Kwan, and Y . Xi e, “Hybrid MAP and PIC detection for OTFS modulation,” IEEE Trans. V eh. Tech., vol. 70, no. 7, pp. 7193-7198, Jul. 2021

2021
[17]

Embedd ed pilot-aided channel estimation for OTFS in delay-Doppler channels,

P . Raviteja, K. T. Phan, Y . Hong, and E. Viterbo, “Embedd ed pilot-aided channel estimation for OTFS in delay-Doppler channels,” IEEE Trans. V eh. Tech., vol. 68, no. 5, pp. 4906-4917, May 2019

2019
[18]

Sparse Bayesian learning of delay- Doppler channel for OTFS system,

L. Zhao, W. J. Gao, and W. Guo, “Sparse Bayesian learning of delay- Doppler channel for OTFS system,” IEEE Commun. Lett. , vol. 24, no. 12, pp. 2766-2769, Dec. 2020

2020
[19]

Off-grid c hannel estimation with sparse Bayesian learning for OTFS systems,

Z. Wei, W. Y uan, S. Li, J. Y uan, and D. W. K. Ng, “Off-grid c hannel estimation with sparse Bayesian learning for OTFS systems, ” IEEE Trans. Wireless Commun. , vol. 21, no. 9, pp. 7407-7426, Sep. 2022

2022
[20]

O TFS channel estimation and data detection designs with superimposed pi lots,

H. B. Mishra, P . Singh, A. K. Prasad, and R. Budhiraja, “O TFS channel estimation and data detection designs with superimposed pi lots,” IEEE Trans. Wireless Commun. , vol. 21, no. 4, pp. 2258-2274, Apr. 2022

2022
[21]

Low complexity precoding and detection in multi-user mass ive MIMO OTFS downlink,

B. C. Pandey, S. K. Mohammed, P . Raviteja, Y . Hong, and E. Viterbo, “Low complexity precoding and detection in multi-user mass ive MIMO OTFS downlink,” IEEE Trans. V eh. Tech., vol. 70, no. 5, pp. 4389-4405, May 2021

2021
[22]

Low complexity detection of spatial modulation aided OTFS in do ubly- selective channels,

Z. Sui, H. Zhang, Y . Xin, T. Bao, L-L. Y ang, and L. Hanzo, “ Low complexity detection of spatial modulation aided OTFS in do ubly- selective channels,” IEEE Trans. V eh. Tech., vol. 72, no. 10, pp. 13746- 13751, Oct. 2023

2023
[23]

Space- time shift keying aided OTFS modulation for orthogonal multiple acces s,

Z. Sui, H. Zhang, S. Sun, L-L. Y ang, and L. Hanzo, “Space- time shift keying aided OTFS modulation for orthogonal multiple acces s,” IEEE Trans. Commun., vol. 71, no. 12, pp. 7393-7408, Dec. 2023

2023
[24]

OTFS − a mathematical foundation for communication and radar sensing in the delay-Doppler domain,

S. K. Mohammed, R. Hadani, A. Chockalingam, and R. Calde rbank, “OTFS − a mathematical foundation for communication and radar sensing in the delay-Doppler domain,” IEEE BITS the Inform. Theory Mag., vol. 2, no. 2, pp. 36-55, 1 Nov. 2022

2022
[25]

OTFS − predictability in the delay-Doppler domain and its value to communication and radar sensing,

S. K. Mohammed, R. Hadani, A. Chockalingam, and R. Calde rbank, “OTFS − predictability in the delay-Doppler domain and its value to communication and radar sensing,” IEEE BITS the Inform. Theory Mag. , vol. 3, no. 2, pp. 7-31, Jun. 2023

2023
[26]

Zak-OTFS implementation via time and frequ ency windowing,

S. Gopalam, I. B. Collings, S. V . Hanly, H. Inaltekin, S. R. B. Pillai, and P . Whiting, “Zak-OTFS implementation via time and frequ ency windowing,” IEEE Trans. Commun. , vol. 72, no. 7, pp. 3873-3889, Jul. 2024

2024
[27]

Zak-OTFS for integration of sensing and commu- nication,

M. Ubadah, S. K. Mohammed, R. Hadani, S. Kons, A. Chockal ingam, and R. Calderbank, “Zak-OTFS for integration of sensing and commu- nication,” online arxiv.org/abs/2404.04182, 5 Apr 2024

2024 arXiv
[28]

Optimal Zak-OTFS receiver and its relation to the radar matched ﬁlte r,

S. Gopalam, H. Inaltekin, I. B. Collings, and S. V . Hanly , “Optimal Zak-OTFS receiver and its relation to the radar matched ﬁlte r,” IEEE Open J. of the Commun. Soc. , vol. 5, pp. 4462-4482, 2024

2024
[30]

Near-optimal detectio n of Zak-OTFS signals,

F. Jesbin and A. Chockalingam, “Near-optimal detectio n of Zak-OTFS signals,” Proc. IEEE ICC’2024 , pp. 4476-4481, Jun. 2024

2024
[31]

Zak-OTFS and LDPC codes,

B. Dabak, V . Khammammetti, S. K. Mohammed, and R. Calder bank, “Zak-OTFS and LDPC codes,” Proc. IEEE ICC’2024 , pp. 3785-3790, Jun. 2024

2024
[32]

Zak-OTFS and turbo signal processing for jo int sensing and communication,

J. Jayachandran, M. Ubadah, S. K. Mohammed, R. Hadani, a nd A. Chockalingam, “Zak-OTFS and turbo signal processing for jo int sensing and communication,” online: arXiv:2406.06024 10 Jun 2024

2024 arXiv
[33]

Zak-OTFS with interleaved pilots to extend the region of predictable operation,

J. Jayachandran, I. A. Khan, S. K. Mohammed, R. Hadani, a nd A. Chockalingam, “Zak-OTFS with interleaved pilots to extend the region of predictable operation,” online: arXiv:2408.09379 18 Au g 2024

2024 arXiv
[34]

On th e pulse shaping for delay-Doppler communications,

S. Li, W. Y uan, Z. Wei, J. Y uan, B. Bai, and G. Caire, “On th e pulse shaping for delay-Doppler communications,” Proc. IEEE GLOBE- COM’2024, pp. 4909-4914, Dec. 2023

2024
[35]

Fun- damentals of delay-Doppler communications: practical imp lementation and extensions to OTFS,

S. Li, P . Jung, W. Y uan, Z. Wei, J. Y uan, B. Bai, and G. Cair e, “Fun- damentals of delay-Doppler communications: practical imp lementation and extensions to OTFS,” online: arXiv:2403.14192 21 Mar 20 24

arXiv
[36]

S. K. Mohammed, R. Hadani, and A. Chockalingam, OTFS Modulation − Theory and Applications , IEEE-Wiley, 2024

2024
[37]

Bracewell, The F ourier Transform and Its Applications , 3rd ed

R. Bracewell, The F ourier Transform and Its Applications , 3rd ed. New Y ork: McGraw-Hill, 1999

1999
[38]

Guidelines for evaluation of radio tran smission tech- nologies for IMT-2000,

ITU-R M.1225, “Guidelines for evaluation of radio tran smission tech- nologies for IMT-2000,” Int. Telecommun. Union Radio Commu n., 1997

2000

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [29]

Zak-OTFS: pulse shaping and the tradeoff between time/bandwidth expansion and predictabi lity,

J. Jayachandran, R. K. Jaiswal, S. K. Mohammed, R. Hadan i, A. Chockalingam, and R. Calderbank, “Zak-OTFS: pulse shaping and the tradeoff between time/bandwidth expansion and predictabi lity,” online arxiv.org/abs/2405.02718, 4 May 2024

arXiv 2024

[2] [1]

On the road to 6G: visions, requiremen ts, key technologies, and testbeds,

C. -X. Wang et al., “On the road to 6G: visions, requiremen ts, key technologies, and testbeds,” IEEE Commun. Surveys & Tuts. , vol. 25, no. 2, pp. 905-974, 2023

work page 2023

[3] [2]

Toward integrated sensing and commun ications for 6G: key enabling technologies, standardization, and chall enges,

A. Kaushik et al., “Toward integrated sensing and commun ications for 6G: key enabling technologies, standardization, and chall enges,” IEEE Commun. Standards Mag. , vol. 8, no. 2, pp. 52-59, Jun. 2024

work page 2024

[4] [3]

Characterization of randomly time-variant l inear channels,

P . Bello, “Characterization of randomly time-variant l inear channels,” IEEE Trans. Commun. Syst. , vol. CS-11, no. 4, pp. 360–393, Dec. 1963

work page 1963

[5] [4]

Orthogonal time frequency space modul ation,

R. Hadani et al., “Orthogonal time frequency space modul ation,” Proc. IEEE WCNC’2017 , pp. 1-6, Mar. 2017

2017

[6] [5]

OTFS: a new generation of modulati on addressing the challenges of 5G,

R. Hadani and A. Monk, “OTFS: a new generation of modulati on addressing the challenges of 5G,” online arxiv.org/abs/18 02.02623, Feb. 2018

work page 2018

[7] [6]

On OTFS modulation for high- Doppler fading channels,

K. R. Murali and A. Chockalingam, “On OTFS modulation for high- Doppler fading channels,” Proc. ITA W orkshop, pp. 1-10, Feb. 2018

work page 2018

[8] [7]

Interfe rence can- cellation and iterative detection for orthogonal time freq uency space modulation,

P . Raviteja, K. T. Phan, Y . Hong, and E. Viterbo, “Interfe rence can- cellation and iterative detection for orthogonal time freq uency space modulation,” IEEE Trans. Wireless Commun., vol. 17, no. 10, pp. 6501- 6515, Aug. 2018

work page 2018