Zak-OTFS with Spread Carrier Waveforms

Nishant Mehrotra; Robert Calderbank; Sandesh Rao Mattu

REVIEW 3 major objections 3 minor 1 cited by

Zak-OTFS with Spread Carrier Waveforms

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A unitary change of basis turns Zak-OTFS pulsones into constant-envelope carriers with 5.6 dB less peak power.

desk verdict A clean, correct-in-example unitary construction that turns Zak-OTFS pulsones into CAZAC waveforms; the low-PAPR claim holds, but the channel-estimation equivalence rests on a crystallization condition that is checked by grid search rather than proven. read the letter →

arxiv 2505.08079 v3 pith:RZJLEBNS submitted 2025-05-12 eess.SP cs.ITmath.IT

Nishant Mehrotra , Sandesh Rao Mattu , Robert Calderbank This is my paper · ORCID

classification eess.SPcs.ITmath.IT

keywords Zak-OTFSpeak-to-averagepowerratiospreadcarrierwaveformsCAZACsequencesgeneralizeddiscreteaffineFouriertransformdelay-Dopplerdomainchannelestimationfullspectralefficiency

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 normally transmits on pulsones—narrow pulses repeated in time—which are simple and predictable but have a high peak-to-average power ratio (PAPR) around 12.2 dB. This paper proposes a unitary change of basis, the generalized discrete affine Fourier transform (GDAFT), that turns each pulsone into a constant-amplitude spread carrier with CAZAC (constant amplitude zero autocorrelation) structure and a per-basis PAPR of only 6.58 dB. Because the change of basis is unitary, the new carriers still span the full $MN$-dimensional signal space, so spectral efficiency is not sacrificed. The paper further claims that, under a 'crystallization' condition on the delay-Doppler channel support, channel estimation and uncoded BER match pulsone-based Zak-OTFS exactly, while beating OFDM, OTFS, and DFT-spread-OFDM in simulated high-mobility Veh-A channels. The significance is that low-PAPR waveforms and full-rate delay-Doppler signaling need not be tradeoffs.

What carries the argument

The GDAFT is the unitary transform $(\mathcal{F}_a x)[n] = \frac{1}{\sqrt{MN}} \sum_{m=0}^{MN-1} e^{j 2\pi (A n^2 + B n m + C m^2)/(MN)} x[m]$, with $A,B,C$ coprime to $MN$ and $MN$ the frame size. Its quadratic phase factor is what carries the argument: applied to a time-localized pulsone, the phase makes the inverse Zak-domain delta spread into a constant-envelope generalized Zadoff–Chu sequence (Theorem 1), and unitarity guarantees the spread carriers inherit orthonormality. For channel estimation, the GDAFT's action on the cross-ambiguity function (Theorem 2) maps the pulsone alias lattice to a rotated lattice $(k'_{n,m},l'_{n,m})$; Lemma 2 then says the channel estimate is exact exactly when the translated copies of the true channel support do not overlap—the crystallization condition—so $A,B,C$ must be chosen accordingly.

What would settle it

Run the crystallization-condition check of Lemma 2 for a channel whose delay-Doppler support is larger than the spacing between the shifted copies produced by the reported parameters $A=3,B=5,C=7$—for instance, a Veh-A channel with delay spread exceeding the example's $k$ range, or a support spanning several aliases. If any translated support intersects the original, the theory predicts an aliased channel estimate; if one finds such an $(M,N,S,A,B,C)$ and the simulation still matches pulsone-based NMSE/BER, the claimed equivalence fails. Conversely, a constructive search for parameters for random $(M,N,S)$ would test how generally the condition holds.

Watch

Extended reading notes

Core claim

Corollary 1 states the central discovery: applying the GDAFT to the orthonormal pulsone basis of Zak-OTFS produces another orthonormal basis whose elements are spread CAZAC waveforms, each a generalized Zadoff–Chu sequence. Since the GDAFT is unitary (Lemma 1), inner products and noise statistics are preserved; hence $MN$ information symbols can still be carried at full spectral efficiency. Theorem 1 gives the formula for the spread carrier $x^{(c)}_{(k_0,l_0)}[n]$, and Theorem 2 describes how the GDAFT rotates the cross-ambiguity lattice. The paper's simulations show the spread-carrier system achieving the same channel-estimation NMSE and BER as pulsone-based Zak-OTFS, while each basis element has $5.6$ dB lower PAPR ($6.58$ dB versus $12.2$ dB).

Load-bearing premise

The load-bearing premise is that the delay-Doppler region occupied by the channel can always be kept from overlapping its shifted copies, which the paper demonstrates for one example and then handles by grid search rather than a guarantee for all frame sizes and channels.

Editorial extensions

If this is right

Spread-carrier Zak-OTFS reaches full spectral efficiency with per-basis PAPR $6.58$ dB, eliminating the bandwidth penalty that DFT-spread-OFDM pays for low PAPR.
Channel estimation and BER equal pulsone-based Zak-OTFS in the simulated six-path Veh-A channel, so changing the basis costs no accuracy.
At $15$ dB SNR and above, the scheme beats OFDM, DFT-spread-OFDM, and OTFS in high-mobility BER, extending the non-fading predictability of Zak-OTFS to a low-PAPR waveform.
The result holds for both narrowband ($0.51$ MHz) and wideband ($19.9$ MHz, 5G-NR-like) settings.

Reading between the lines

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

A natural next step the authors do not take: because the GDAFT parameters $A,B,C$ generate a family of unitary bases, one could view the 'good' parameters as a code over the symplectic group and combinatorially characterize which frame sizes admit the crystallization condition.
Per-basis PAPR of 6.58 dB does not automatically mean the summed data signal has that PAPR; the paper's own data signals show 7.83 dB versus 7.95 dB, so the practical amplifier gain may be smaller than 5.6 dB once random data superpose carriers.
The CAZAC structure could enable alternatives beyond communication, such as low-power sensing or secure signaling, though the paper only lists these as future work.
The grid-search step suggests a testable extension: a constructive algorithm for $A,B,C$ from coarse channel-support bounds would turn the method from a demonstrated example into a general design rule.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Short version: this is a solid, incremental paper. The new piece is the GDAFT mapping from the pulsone basis to an orthonormal CAZAC basis, plus the ambiguity covariance in Theorem 2 and the crystallization condition in Lemma 2. The math is checkable and the simulations back the claims. It deserves a serious referee; it also deserves one revision to tighten the conditions.

What it does well: Theorem 1 is a direct calculation using quadratic Gauss sums, and Corollary 1 follows because the GDAFT is unitary. That is real. The PAPR reduction numbers are plausible and the BER/NMSE equivalence with pulsone-based Zak-OTFS is demonstrated for the tested settings. The comparison with DFT-spread-OFDM is fair about spectral efficiency. The citation pattern is honest; the reliance on [7] is explicit and the paper does not oversell the difference.

Soft spots, in order: 1. The Gauss-sum application needs N odd and gcd(CM,N)=1, and the theorem statement does not say so. Also, if M and N are not coprime, it is unclear whether the construction holds. This is a minor fix, but it matters. 2. The crystallization condition in Lemma 2 is load-bearing. The paper gives one good example and says grid search is performed. No theorem or counting argument says such A,B,C exist for general M,N or channel support. The stress-test note is right: if the condition fails, the spread-carrier channel estimate aliases and the claimed equivalence breaks. This is not a fatal flaw for the tested configurations, but it should be stated as an assumption or proved. 3. The reported PAPR numbers need care. "6.58 dB per basis element" is not the PAPR of a modulated data signal; the data-signal number is 7.83 dB and the improvement over pulsones is smaller. The abstract's framing could mislead a casual reader.

None of this undermines the central construction. For a Wireless Communications Letters submission, this is appropriate scope. The paper is not reorganizing the field, but it is a useful tool for low-PAPR Zak-OTFS, and the equivalence result is strong enough to be worth citing.

Recommendation: send it to review. A good referee will ask for the stated conditions and a constructive or probabilistic existence result for the crystallization condition, but the paper's core is sound and the authors are capable of addressing those.

Referee Report

3 major / 3 minor

Summary. The paper proposes a low-PAPR spread-carrier implementation of Zak-OTFS. It defines a generalized discrete affine Fourier transform (GDAFT) that maps the orthonormal pulsone basis to another orthonormal basis of constant-envelope (CAZAC) waveforms, proves the unitarity of the GDAFT, derives a closed-form CAZAC expression for the image of a pulsone, and derives the transformation rule for cross-ambiguity functions under the GDAFT. The channel-estimation analysis shows that spread-carrier and pulsone-based systems are equivalent provided a crystallization condition on the GDAFT parameters and the channel support holds. Numerical results compare PAPR, channel-estimation NMSE, and BER for the two Zak-OTFS variants and compare against OTFS, OFDM, and DFT-spread-OFDM in Veh-A channels.

Significance. If the construction is fully validated, it is a useful contribution: it gives a simple unitary basis change that reduces per-basis-element PAPR by about 5.6 dB without sacrificing spectral efficiency, and the authors are honest that the data-signal PAPR gain is much smaller (Section V.A). The derivations are grounded in standard identities, and the simulations cover both narrowband and wideband regimes with a doubly selective channel. The NMSE and BER comparisons between pulsone- and spread-carrier Zak-OTFS are a good falsifiable check of the unitary-basis claim. The main unresolved issue is the general existence of GDAFT parameters satisfying the crystallization condition; this limits the generality of the claims as stated.

major comments (3)

[Section IV.C, Lemma 2] The equivalence between spread-carrier and pulsone-based Zak-OTFS is conditional on the crystallization condition S ∩ (∪_{(n,m)≠(0,0)} (S + (k'_{n,m}, l'_{n,m}))) = ∅. The paper verifies this condition for one favorable example (M=17, N=19, A=3, B=5, C=7) and then states that 'grid search is performed' in practice. No theorem, counting argument, or constructive algorithm establishes that such a triple exists for general M, N, and channel support S, nor is the grid search shown to be exhaustive or its complexity quantified. Because equation (8) and Lemma 2 are the mechanism that connects the pilot stage to the NMSE/BER equivalence in Section V and to the 'Predictable' entry in Table I, this missing existence result is load-bearing. The wideband simulation in Section V.E also appears to use the same A=3, B=5, C=7 without displaying a verification of the crystallization condition for M=83, N=13. Please provide a sufficient condition or counting argument for the existence of admissible (A,B,C), or explicitly scope the claims to parameter regimes in which the condition is verified, and report the search procedure.
[Section III, Theorem 1 and Identity 2] The closed-form expression for x^{(c)}_{(k0,l0)}[n] is obtained by applying the quadratic Gauss sum Identity 2 to the sum over d of e^{j2π/N (CM d^2 + (Bn+l0+2Ck0)d)}. Identity 2 requires N odd and gcd(CM,N)=1, but Definition 1 imposes only gcd(A,MN)=gcd(B,MN)=gcd(C,MN)=1. Consequently, for N even or gcd(M,N)>1, Theorem 1 as stated is not proved and the 'CAZAC' / '6.58 dB PAPR' claims do not follow. Please add the hypotheses N odd and gcd(M,N)=1 (or the appropriate Gauss-sum conditions) to Theorem 1 and Corollary 1, and note that the numerical examples satisfy them.
[Section IV.C, Lemma 2] The crystallization condition is not fully well-defined as written because the expressions for k'_{n,m} and l'_{n,m} use B^{-1}_{MN}, i.e., residues modulo MN, while the channel support S is a set of integer pairs that may include negative values. The example appears to use a particular integer lift of these residue classes when drawing the dashed red rectangles, but this lift is not specified. Please state explicitly that the condition is checked for the unique representatives in a fixed fundamental domain (or equivalently formulated modulo MN), otherwise the verification in the example is ambiguous.

minor comments (3)

[Section III, Identity 2] The statement of Identity 2 says 'a, b ∈ Z+', but the identity is valid for b ∈ Z (and for a an integer coprime to N). The argument b = Bn + l0 + 2Ck0 in Theorem 1 is nonnegative in the intended regime, but the hypothesis should be stated for all integers to avoid confusion.
[Section II] The paper does not state whether M and N are required to be coprime or whether N is odd, even though these assumptions appear in the Gauss-sum evaluation in Theorem 1 and are satisfied by all simulations. Please state these assumptions in the preliminaries.
[Section IV.C, Example] The sentence 'prior knowledge of channel support S does not need to be exact' is used to justify approximating S by a rectangle, but no bound on the estimation error of kmin, kmax, lmin, lmax is given. A short statement on how much margin is needed would make the practical claim more precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence is a direct unitary-transform derivation, and the channel-estimation equivalence is conditional on an explicitly stated support condition rather than on fitted data.

full rationale

The derivation chain is self-contained. Lemma 1 proves that the GDAFT is unitary using the roots-of-unity identity; Theorem 1 evaluates the GDAFT on a point pulsone and obtains the CAZAC form via the quadratic Gauss sum; Theorem 2 is a direct computation of the cross-ambiguity function under the unitary GDAFT. Lemma 2 is a conditional statement: the spread-carrier channel estimate equals the true effective channel only if the crystallization condition on the channel support and the GDAFT parameters holds. The parameters A, B, C are chosen by grid search to satisfy that sufficient condition, with one explicit favorable example, and they are not fitted to the simulated BER or NMSE outcomes. The simulations are therefore confirmatory rather than circular. The paper cites prior work by overlapping authors for the pulsone ambiguity expression and for orthonormality of the pulsone basis, but these are standard mathematical identities with independent content; they serve as inputs to the new derivation, not as the conclusion. No prediction reduces by construction to a fitted parameter or to a self-referential definition. The only notable weakness is that no general existence proof for the crystallization condition is provided, but that is a completeness risk, not circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the GDAFT design parameters and the crystallization condition; these are chosen by grid search rather than derived. No new physical entities are introduced.

free parameters (1)

GDAFT parameters A, B, C = A=3, B=5, C=7 in simulations; chosen by grid search
The transform requires integers coprime to MN that satisfy the crystallization condition in Lemma 2. No constructive algorithm is given, and a bad choice (A=2, B=5, C=7) causes overlapping channel aliases.

assumptions (3)

standard math Quadratic Gauss sum Identity 2 holds, requiring N odd and gcd(CM,N)=1
Invoked in Theorem 1 to evaluate the GDAFT of a pulsone. The paper does not state the parity and coprimality conditions as system requirements.
domain assumption The channel support S is compact and the crystallization condition in Lemma 2 is satisfied by the chosen GDAFT parameters
Channel estimation equals the true channel only if aliases do not overlap S. The paper assumes this holds and verifies one example.
domain assumption The channel is constant across pilot and data stages, and the physical channel is a sum of P resolvable paths sampled via the effective channel in equations (11) and (12)
Two-stage pilot/data model in Section IV.B and channel generation in Section V define the simulation and the estimation problem.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Zak-OTFS with Spread Carrier Waveforms." pith.science (2026). https://pith.science/paper/RZJLEBNS

@misc{pith2026250508079,
  author       = {Pith},
  title        = {Pith review of: Zak-OTFS with Spread Carrier Waveforms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RZJLEBNS}},
  note         = {Machine review of arXiv:2505.08079}
}

abstract

Zak-OTFS (orthogonal time frequency space) modulation is a communication framework that parameterizes the wireless channel in the delay-Doppler (DD) domain, where the parameters map directly to physical attributes of the scatterers that comprise the scattering environment. As a consequence, the channel can be efficiently acquired and equalized. The Zak-OTFS carrier is a pulse in the DD domain, and the Zak transform converts it to a pulse train modulated by a tone (pulsone) in the time domain. The pulsone waveform is localized rather than spread, and it suffers from high peak-to-average power ratio (PAPR). We describe how to transform the orthonormal basis of Zak-OTFS pulsones into an orthonormal basis of spread carrier waveforms with low PAPR (only $6.58$ dB) that support communication in the presence of mobility and delay spread. This transformation is realized by a unitary transform based on the discrete affine Fourier transform. Unlike other spread modulations that achieve low PAPR by spreading information across a wider bandwidth (thus reducing the spectral efficiency), the proposed spread carrier-based Zak-OTFS achieves full spectral efficiency like pulsone-based Zak-OTFS, with $5.6$ dB lower PAPR per basis element. We demonstrate uncoded bit error rate (BER) similar to pulsone-based Zak-OTFS, and improved BER performance over competing methods based on OFDM and OTFS in high mobility & delay spread environments.

Figures

Figures reproduced from arXiv: 2505.08079 by the authors.

**Figure 1.** Example illustrating choice of GDAFT parameters. For spread carrier-based Zak-OTFS, from Theorem 2: Ax (c) (kp,lp) [k, l]=e j2π MN (−Ak2+lk+CB−2 MN (2Ak−l) 2 )Ax (p) (kp,lp) [k, ¯ ¯l], where ¯k = −B−1 MN (2Ak − l), ¯l = 2CB−1 MN (2Ak − l) − Bk. On substituting the expression for Ax (p) (kp,lp) [k, l], Ax (c) (kp,lp) [k, l]=e j2π MN (−Ak2+lk+CB−2 MN (2Ak−l) 2 ) X n,m∈Z e j2π N nlp × e − j2π M mkp δ[ ¯k − nM]δ[ ¯l − m… view at source ↗

**Figure 2.** plots the complementary cumulative distribution function (CCDF) of the PAPR for pulsone- and spread carrierbased Zak-OTFS for system parameters: M = 17, N = 19, BW = 0.51 MHz and an oversampling factor of 4. Due to the CAZAC property of the spread carrier basis (see Theorem 1), each spread carrier basis element has 5.6 dB lower PAPR than each pulsone basis element. When modulated by information symbols, the spread … view at source ↗

**Figure 3.** Normalized mean squared error in channel estimation is the same for pulsone- and spread carrier-based Zak-OTFS. B. Channel Estimation Performance [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 6.** Figure 6: Bit error rate curves showing similar data detection performance for narrowband and wideband systems. E. Narrowband vs Wideband Performance [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 5.** Figure 5: Bit error rate curves comparing the performance of the proposed spread Zak-OTFS scheme with existing approaches. D. Comparison with Existing Approaches [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Differential Communication in Channels with Mobility and Delay Spread using Zak-OTFS
eess.SP 2025-07 conditional novelty 5.0 of 10

In Zak-OTFS, the cross-ambiguity of received and transmitted random data is approximately the channel, so detected data can replace periodic pilots and enable pilot-free differential detection.

Reference graph

Works this paper leans on

16 extracted references · 10 canonical work pages · cited by 1 Pith paper

[7]

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

M. Ubadah, S. K. Mohammed, R. Hadani, S. Kons, A. Chockali ngam, and R. Calderbank, “Zak-OTFS for integration of sensing and commu- nication,” arXiv preprint arXiv:2404.04182 , 2024

arXiv 2024
[1]

OTFS—a mathematical foundation for communication and rad ar sens- ing in the delay-Doppler domain,

S. K. Mohammed, R. Hadani, A. Chockalingam, and R. Calder bank, “OTFS—a mathematical foundation for communication and rad ar sens- ing in the delay-Doppler domain,” IEEE BITS the Information Theory Magazine, vol. 2, no. 2, pp. 36–55, 2022

work page 2022
[2]

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

——, “OTFS—predictability in the delay-Doppler domain a nd its value to communication and radar sensing,” IEEE BITS the Information Theory Magazine , vol. 3, no. 2, pp. 7–31, 2023

work page 2023
[3]

Sensing Integrated D FT- Spread OFDM Waveform and Deep Learning-Powered Receiver De sign for Terahertz Integrated Sensing and Communication System s,

Y . Wu, F. Lemic, C. Han, and Z. Chen, “Sensing Integrated D FT- Spread OFDM Waveform and Deep Learning-Powered Receiver De sign for Terahertz Integrated Sensing and Communication System s,” IEEE Transactions on Communications , vol. 71, no. 1, pp. 595–610, 2023

work page 2023
[4]

Orthogonal Time Frequency Spa ce Modulation,

R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmit h, A. F. Molisch, and R. Calderbank, “Orthogonal Time Frequency Spa ce Modulation,” in 2017 IEEE Wireless Communications and Networking Conference (WCNC), 2017, pp. 1–6

work page 2017
[5]

Space- Time Shift Keying Aided OTFS Modulation for Orthogonal Multiple Access,

Z. Sui, H. Zhang, S. Sun, L.-L. Y ang, and L. Hanzo, “Space- Time Shift Keying Aided OTFS Modulation for Orthogonal Multiple Access,” IEEE Transactions on Communications, vol. 71, no. 12, pp. 7393–7408, 2023

work page 2023
[6]

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 Transactions on V ehicular Technology , vol. 72, no. 10, pp. 13 746–13 751, 2023. 7

work page 2023
[8]

A Primer on Zadoff Chu Sequences,

J. G. Andrews, “A Primer on Zadoff Chu Sequences,” arXiv preprint arXiv:2211.05702, 2022

arXiv 2022

Show all 16 references

[9]

Zak-OTFS fo r Mutually Unbiased Sensing and Communication,

N. Mehrotra, S. R. Mattu, and R. Calderbank, “Zak-OTFS fo r Mutually Unbiased Sensing and Communication,” 2025. [Online]. Avai lable: https://arxiv.org/abs/2503.23540

2025 arXiv
[10]

Phase-Coded Waveforms and Their Design,

J. J. Benedetto, I. Konstantinidis, and M. Rangaswamy, “Phase-Coded Waveforms and Their Design,” IEEE Signal Processing Magazine , vol. 26, no. 1, pp. 22–31, 2009

2009
[11]

Closed-Form Discrete Fracti onal and Afﬁne Fourier Transforms,

S.-C. Pei and J.-J. Ding, “Closed-Form Discrete Fracti onal and Afﬁne Fourier Transforms,” IEEE Transactions on Signal Processing , vol. 48, no. 5, pp. 1338–1353, 2000

2000
[12]

Discrete Zak transforms , polyphase transforms, and applications,

H. Bolcskei and F. Hlawatsch, “Discrete Zak transforms , polyphase transforms, and applications,” IEEE Transactions on Signal Processing , vol. 45, no. 4, pp. 851–866, 1997

1997
[13]

Evaluation of the quadratic Gau ss sum,

M. Murty and S. Pathak, “Evaluation of the quadratic Gau ss sum,” Evaluation, vol. 86, no. 1-2, 2017

2017
[14]

Decimation generator of zadoff-chu seque nces,

S. Budisin, “Decimation generator of zadoff-chu seque nces,” in Se- quences and Their Applications – SETA 2010 , C. Carlet and A. Pott, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010 , pp. 30–40

2010
[15]

Tse and P

D. Tse and P . Viswanath, Fundamentals of wireless communication . Cambridge university press, 2005

2005
[16]

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,” International Telecommunication Union Radio communication, 1997

2000

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [7]

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

M. Ubadah, S. K. Mohammed, R. Hadani, S. Kons, A. Chockali ngam, and R. Calderbank, “Zak-OTFS for integration of sensing and commu- nication,” arXiv preprint arXiv:2404.04182 , 2024

arXiv 2024

[2] [1]

OTFS—a mathematical foundation for communication and rad ar sens- ing in the delay-Doppler domain,

S. K. Mohammed, R. Hadani, A. Chockalingam, and R. Calder bank, “OTFS—a mathematical foundation for communication and rad ar sens- ing in the delay-Doppler domain,” IEEE BITS the Information Theory Magazine, vol. 2, no. 2, pp. 36–55, 2022

work page 2022

[3] [2]

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

——, “OTFS—predictability in the delay-Doppler domain a nd its value to communication and radar sensing,” IEEE BITS the Information Theory Magazine , vol. 3, no. 2, pp. 7–31, 2023

work page 2023

[4] [3]

Sensing Integrated D FT- Spread OFDM Waveform and Deep Learning-Powered Receiver De sign for Terahertz Integrated Sensing and Communication System s,

Y . Wu, F. Lemic, C. Han, and Z. Chen, “Sensing Integrated D FT- Spread OFDM Waveform and Deep Learning-Powered Receiver De sign for Terahertz Integrated Sensing and Communication System s,” IEEE Transactions on Communications , vol. 71, no. 1, pp. 595–610, 2023

work page 2023

[5] [4]

Orthogonal Time Frequency Spa ce Modulation,

R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmit h, A. F. Molisch, and R. Calderbank, “Orthogonal Time Frequency Spa ce Modulation,” in 2017 IEEE Wireless Communications and Networking Conference (WCNC), 2017, pp. 1–6

work page 2017

[6] [5]

Space- Time Shift Keying Aided OTFS Modulation for Orthogonal Multiple Access,

Z. Sui, H. Zhang, S. Sun, L.-L. Y ang, and L. Hanzo, “Space- Time Shift Keying Aided OTFS Modulation for Orthogonal Multiple Access,” IEEE Transactions on Communications, vol. 71, no. 12, pp. 7393–7408, 2023

work page 2023

[7] [6]

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 Transactions on V ehicular Technology , vol. 72, no. 10, pp. 13 746–13 751, 2023. 7

work page 2023

[8] [8]

A Primer on Zadoff Chu Sequences,

J. G. Andrews, “A Primer on Zadoff Chu Sequences,” arXiv preprint arXiv:2211.05702, 2022

arXiv 2022