Differential Communication in Channels with Mobility and Delay Spread using Zak-OTFS

Nishant Mehrotra; Robert Calderbank; Sandesh Rao Mattu

REVIEW 4 major objections 5 minor 14 references

Differential Communication in Channels with Mobility and Delay Spread using Zak-OTFS

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Zak-OTFS can estimate the channel from its own detected data, so periodic pilot symbols become optional resets only.

desk verdict A plausible specialization of decision-directed channel estimation to Zak-OTFS, but the pilot-free claim rests on an unanalyzed bootstrap and an unbounded finite-frame residual. read the letter →

arxiv 2507.12593 v2 pith:LFSEHRLM submitted 2025-07-16 eess.SP cs.ITmath.IT

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

classification eess.SPcs.ITmath.IT

keywords Zak-OTFSdelay-Dopplerdomaindifferentialcommunicationchannelpredictabilitycross-ambiguityfunctionpilot-freeestimationspectralefficiencydoubly-dispersive

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

The paper's aim is to make a Zak-OTFS wireless link work without regularly sending pilot symbols. It shows that the receiver can estimate the delay-Doppler channel from the data it has already detected, then use that estimate to detect the next frame, and repeat. The key analytical result is that the cross-ambiguity between a received frame and the transmitted data frame is, for large frames and random data, just the effective channel scaled by the data-symbol energy. Because the delay-Doppler channel changes slowly, the estimate stays valid across consecutive frames. The payoff is that all frame energy goes to data, spectral efficiency is full, and the receiver is simpler than a spread-pilot receiver while achieving better bit-error rate in simulation.

What carries the argument

The load-bearing object is the DD-domain cross-ambiguity function $$A_{y,x}[k',l'] = \frac{1}{MN}\sum_{n=0}^{MN-1} y[n]x^*[n-k']$e^{{-j\frac{2\pi}}${MN}l'(n-k')},$$ which is the maximum-likelihood, model-free estimate of the effective channel. The sum in (13) is a twisted convolution of $h_{\mathrm{eff}}$ with $B[k,l]$, a convolution variant that carries the quasi-periodic phase structure of the DD pulsones. The key identity is (16): with random data symbols and a large frame, the self-interference term $B[k,l]$ collapses to $e_d\delta[k]\delta[l]$, so the cross-ambiguity of a received data frame with the transmitted data frame is approximately $e_d h_{\mathrm{eff}}[k',l']$. The differential recursion then alternates between detecting data from the current channel estimate and refreshing the estimate from the detected data.

What would settle it

Measure the instantaneous NMSE of the channel estimate from a data-only frame at a fixed SNR while increasing frame size $MN$; Eq. (16) predicts convergence to the scaled effective channel, so if the NMSE stops improving or stays well above the spread-pilot estimate at large frames, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that in Zak-OTFS a data frame can serve as its own pilot. The DD-domain cross-ambiguity $A_{y,x}[k',l']$ between the received frame $y$ and the transmitted data frame $x$ is shown to satisfy $$A_{y,x}[k',l'] \approx e_d\,h_{\mathrm{eff}}[k',l'],$$ where $e_d$ is the average energy of the information symbols and $h_{\mathrm{eff}}$ is the effective delay-Doppler channel. The identity follows because the interference term $B[k,l]$ in (14)--(15) averages to a scaled delta, $e_d\,\delta[k]\delta[l]$, when data symbols are random and the frame is large. Hence the receiver can start from a pilot-assisted channel estimate, detect data, treat the detected frame as a pilot to update the estimate, and carry that estimate forward through the predictable DD channel. Periodic pilots are retained only as occasional resets to stop error propagation.

Load-bearing premise

The scheme assumes that the symbols the receiver detects are close enough to what was sent that re-estimating the channel from them is as good as using pilots; if decision errors or frame-size effects are too large, the estimate degrades and periodic pilot resets become necessary.

Editorial extensions

If this is right

Pilot energy can be reallocated to data: a data-only frame carries the full frame energy in its information symbols, so the effective data SNR is higher than in point-pilot or embedded-pilot frames.
Spectral efficiency becomes full, matching the spread-pilot scheme while removing the separate pilot-removal stage, so receiver complexity is roughly halved relative to spread-pilot reception.
At SNR high enough that detection errors are rare, the bootstrap does not accumulate errors, and an occasional pilot frame every 30 frames is enough to reset the channel estimate.
Channel estimates obtained from data improve with frame size and are essentially independent of constellation size up to 256-QAM, matching the asymptotic form of the cross-ambiguity result.

Reading between the lines

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

The paper does not discuss joint sensing, but if the bootstrap is stable the same detected-data channel estimate could simultaneously feed radar sensing in a Zak-OTFS sensing-and-communication system, yielding a continuous sensing stream without dedicated pilot frames.
A threshold-triggered pilot reset, based on observed NMSE or decoder confidence, could replace the fixed every-30-frames schedule and reduce pilot overhead further at low SNR, where the paper's Figure 2 shows error propagation building between resets.
Because the analysis assumes true data symbols in the cross-ambiguity, the practical robustness of the scheme hinges on the decision-error rate; one could test this by feeding the receiver correlated or coded data and measuring how quickly the estimate drifts.
Since Eq. (16) is asymptotic, the residual $B[k,l]$ is the quantity to watch for finite frames; quantifying its norm in terms of $M$, $N$, and constellation statistics would predict exactly when the pilot-free bootstrap starts to fail.

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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

The headline is that this paper is a credible, mostly honest application of classical decision-directed channel estimation to Zak-OTFS, but the central claim that detected data can replace pilots is not fully proven: the asymptotic result (16) is derived for the true transmitted frame, and the replacement by detected symbols is only simulated. That said, the paper is not sloppy. It explicitly identifies the residual B[k,l], shows the undulations in Fig. 1c, and admits that pilot frames are still inserted every 30 transmissions at low SNR. The sign error in Eq. (15) looks like a typo rather than a fatal flaw, since the intended autocorrelation interpretation is clear.

What is actually new: applying differential/decision-directed communication to Zak-OTFS and showing that, for random data, the DD-domain cross-ambiguity between received and transmitted frames converges to the effective channel up to a scale. The derivation is short but clean, reusing known ambiguity-function identities. The complexity comparison is fair: DO avoids the pilot-removal step of spread pilot, which is a genuine saving. The numerical results back the qualitative claims, especially the effect of frame size on estimation accuracy in Fig. 3. No parameters are fitted, and the asymptotic statement is not circular by construction.

The soft spots: first, the detected-data step. The paper proves A_{y,x} ≈ e_d h_eff, then uses A_{y,\hat{x}}. Decision errors change the residual B from a data autocorrelation to a mixed data/decision-error term, and no bound or analysis is given. Error propagation is visible at 0 dB in Fig. 2, and the fix is a periodic pilot frame. That does not kill the pilot-free claim at high SNR, but it makes the alleviation conditional. Second, the finite-frame residual B[k,l] is asserted to be small without a non-asymptotic bound. The undulations in Fig. 1c show it is present; the paper does not show it decays fast enough. Third, the comparison against spread pilot is fair only under the chosen equal-energy split; the α sweep helps, but the comparison does not cover all operating points. The index sign in Eq. (15) is minor but should be corrected.

This paper is for researchers working on OTFS receiver design and integrated sensing and communication. It deserves a serious referee, but the referee should push for an analysis of the detected-data bootstrap and a bound on the finite-frame residual. If those are added, the central claim becomes defensible as stated. My verdict: conditional accept, with revision.

Referee Report

4 major / 5 minor

Summary. The paper proposes a decision-directed differential communication scheme for Zak-OTFS. It argues that the cross-ambiguity between the received frame and the data frame is approximately a scaled version of the effective delay-Doppler channel, so that detected data symbols can serve as implicit pilots. The receiver carries the channel estimate across frames, reducing pilot overhead while retaining full spectral efficiency. The paper provides an analytic derivation for known data, simulation results for 4-QAM and 16-QAM, comparisons with spread-pilot schemes, and a complexity analysis.

Significance. If the central claim holds, the scheme offers a practical way to reduce pilot overhead in Zak-OTFS without sacrificing spectral efficiency, with lower complexity than spread-pilot receivers. The paper identifies a meaningful problem, provides a plausible asymptotic mechanism, and demonstrates empirically that the data-only estimate improves with frame size. It also usefully displays the finite-frame residual in Fig. 1 and diagnoses error propagation in Fig. 2. However, the advertised analytic justification is incomplete because it is derived for known transmitted data rather than for detected symbols, and the finite-frame residual is not bounded; these gaps currently limit the strength of the contribution.

major comments (4)

[Section II-C, Eq. (4)] In the displayed definition of A_{X_p,k0,l0; X_p,k1,l1}[k,l], the indices (k1,l1) of the second pulsone appear both as parameters and as summation variables. The extra summation over k1,l1 makes the expression ill-defined as a cross-ambiguity between two specific pulsones; as written, the reduction to Eq. (6) does not follow from Eq. (4). Since Eq. (14) and the asymptotic estimate (16) rely on Eq. (6), this part of the derivation must be corrected.
[Section III, Eq. (15)] The index substitution is inconsistent: from the condition 1{k=(k0-k1) mod M} in Eq. (14), the correct second summation index is k1 = (k0-k)_M, not (k-k0)_M, and similarly l1 = (l0-l)_N. As written, Eq. (15) describes a mirrored autocorrelation and is not the quantity arising from Eq. (14). The authors should correct the indices and re-verify that the asymptotic claim (16) remains valid after the correction.
[Section III, Eqs. (11)-(16)] The analytic derivation computes A_{y,x} with the true transmitted symbols X. In the proposed differential scheme, the receiver only has the detected frame \hat{X}, so the actual residual in Eq. (14) becomes a mixed data/decision-error term. The paper does not analyze how decision errors affect B or the subsequent channel estimate, and the statement that Eq. (16) holds 'asymptotically' is not accompanied by a formal limit or concentration argument. Because the estimated channel is used to detect the next frame, error propagation is a first-order concern; this omission leaves the central 'detected data as pilots' claim unproven.
[Section IV-A, Fig. 2] The results show that at SNR=0 dB the instantaneous NMSE grows between pilot insertions and that the scheme requires a pilot frame every 30 transmissions; at 25 dB the error propagation is small but no analytical threshold is given. The abstract's claim that the scheme 'alleviates' the need for periodic pilots is compatible with these results, but the paper should state precisely under what conditions the reduced pilot rate is reliable and how the 30-frame period is chosen.

minor comments (5)

[Section II-A, Eq. (1)] The vector x is described as the vector of transmitted symbols in the time-domain, while the surrounding text discusses DD-domain data symbols; the notation should be aligned.
[Section II-C, Eq. (4)] Even after fixing the summation-variable issue, the notation should clearly distinguish the indices of the two pulsones from the summation indices.
[Section III, Eq. (16)] Please specify whether the approximation is in expectation or with high probability, and over what asymptotic regime (frame size, constellation size) it is intended to hold.
[Section IV-C] The phrase 'the performance with DO frame is the lower bound for the performance with SP frame' is ambiguous; since lower BER is better, the intended ordering should be stated explicitly.
[Section IV, Fig. 3] The nine-subfigure layout is hard to read at print size; consider presenting a subset of the cases or using larger panels.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: Eq. (16) is an in-paper asymptotic concentration result, not an identity-by-construction; minor same-author citations are standard and non-load-bearing, while the detected-symbol bootstrap and Eq. (15) indexing are soundness gaps, not circularity.

full rationale

The paper's central claim is that the cross-ambiguity between the received and transmitted data frames satisfies A_{y,x}[k',l'] is approximately e_d times h_eff[k',l'] (Eq. 16), so detected data can act as pilots. The derivation (Eqs. 11-16) shows A_{y,x} = h_eff twisted-convolved with B, where B[k,l] is the autocorrelation of the transmitted frame (Eq. 15); the step to Eq. (16) is a law-of-large-numbers concentration (i.i.d. zero-mean constellation symbols make B approximate e_d times a delta), which is a genuine asymptotic statement and not an identity-by-construction, and no parameter is fitted to make it hold. The self-citations in the chain ([6],[7] for DD-domain predictability, [9] for the cross-ambiguity being the ML channel estimate, [11] for the time-domain input-output relation Eq. (10), and [12] for the spread-pilot comparison) are standard, parameter-free model facts rather than an unverified uniqueness theorem; notably, Eq. (10) is cited to [11], a preprint by the exact same three authors, but it is the standard Zak-OTFS superposition relation, so the derivation's substantive content remains in this paper. Two genuine deficiencies exist but neither is circular: (a) Eqs. (11)-(16) compute the cross-ambiguity with the true transmitted frame X, whereas the receiver substitutes detected symbols, and the paper gives no bound for the resulting data/decision-error residual, conceding in Fig. 2 that at 0 dB error propagation forces a pilot frame every 30 transmissions (Sec. IV), so the pilot-free claim rests on an unanalyzed decision-directed bootstrap, a soundness or evidence gap, not a reduction of the output to the input; (b) the substitution k1 = (k-k0)_M in Eq. (15) is inconsistent with the indicator 1{k=(k0-k1) mod M} in Eq. (14), which requires k1 = k0 - k, so the displayed finite-frame B[k,l] and the undulations in Fig. 1(c) are asserted small without a validated expression. Because the asymptotic estimate is not statistically forced by fitted inputs and Eq. (16) does not collapse to its assumptions by definition, no specific circular step can be exhibited with quote-and-reduction evidence; the low score of 2 reflects only the minor same-author citations that bookend an otherwise in-paper derivation.

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

The central claim rests on known Zak-OTFS ambiguity-function identities, on an asymptotic concentration assumption for random data, and on the unproven substitution of detected symbols for true symbols. No free parameters or invented physical entities are introduced; the load-bearing burden is analytic, not parametric.

assumptions (5)

domain assumption Information symbols are zero-mean, iid, and uniformly drawn from a constellation, so the data self-interference term B[k,l] concentrates to e_d delta[k] delta[l].
Invoked just before Eq (16); no rate or finite-frame concentration bound is supplied, and Fig 1 shows residual undulations for finite frames.
standard math Time-domain pulsone ambiguity equals DD-domain ambiguity and is supported on the lattice (nM,mN).
Used to pass from (13) to (14); this is a known Zak-OTFS property from [9] and is not rederived.
domain assumption Pulsone samples and noise samples are uncorrelated, allowing step (a) in Eq (11).
Needed to drop the noise cross-term in the cross-ambiguity; standard for iid noise.
domain assumption The DD channel is predictable across consecutive frames, so a channel estimate from the previous instant remains valid for the next data frame.
This is the enabling predictability premise stated in the Introduction and Sec III; its timescale validity is not quantified.
ad hoc to paper Detected data symbols can be substituted for true data symbols in the cross-ambiguity (11) without breaking the estimate.
Not derived; the paper only simulates error propagation (Fig 2) and periodically resets with a pilot frame every 30 transmissions, indicating the assumption fails at low SNR.

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Cite this review

Pith. "Pith review of Differential Communication in Channels with Mobility and Delay Spread using Zak-OTFS." pith.science (2026). https://pith.science/paper/LFSEHRLM

@misc{pith2026250712593,
  author       = {Pith},
  title        = {Pith review of: Differential Communication in Channels with Mobility and Delay Spread using Zak-OTFS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LFSEHRLM}},
  note         = {Machine review of arXiv:2507.12593}
}

read the original abstract

Zak-transform based orthogonal time frequency space (Zak-OTFS) is a delay-Doppler (DD) domain modulation scheme in which the signal processing is carried out in the DD domain. The channel when viewed in the DD domain is predictable. However, even with Zak-OTFS, pilots need to be sent periodically, albeit at a lower rate. In this paper, we propose a differential communication scheme for Zak-OTFS systems that alleviates the need for periodic pilot transmission. Towards this, we analytically show that the detected data can be used as a pilot and that the channel estimate obtained from the detected data can enable further detection enabling the "differential" aspect of the communication. Specifically, we leverage the prediction capability of the DD channel in Zak-OTFS to use the channel estimate (obtained from detected data symbols treated as pilots) in the previous instant to detect data in the next instant and propagate this forward. The advantages are two fold. First, it allows the data symbols to enjoy higher energy since the energy that would otherwise be required for pilot symbols can also be allocated to data symbols. Second, it allows for full spectral efficiency compared to point or embedded pilots. Comparison with the full spectral efficiency achieving spread pilot scheme shows that the proposed method achieves better bit-error rate at lower complexity.

Figures

Figures reproduced from arXiv: 2507.12593 by the authors.

**Figure 2.** Instantaneous NMSE as a function of number of frames. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Demonstrating the effect of size of frame and constel [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Performance curves with proposed differential comm [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Performance curves with proposed differential comm [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Tse and P

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

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Joint energy and co rrelation detection assisted non-coherent OFDM-DCSK system for unde rwa- ter acoustic communications,

X. Cai, W. Xu, L. Wang, and G. Kaddoum, “Joint energy and co rrelation detection assisted non-coherent OFDM-DCSK system for unde rwa- ter acoustic communications,” IEEE Transactions on Communications , vol. 70, no. 6, pp. 3742–3759, 2022

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S. N. Diggavi, N. Al-Dhahir, A. Stamoulis, and A. R. Calde rbank, “Great expectations: The value of spatial diversity in wire less networks,” Proceedings of the IEEE , vol. 92, no. 2, pp. 219–270, 2004

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S. K. Mohammed, R. Hadani, A. Chockalingam, and R. Calder bank, “OTFS—A Mathematical Foundation for Communication and Rad ar Sensing in the Delay-Doppler Domain,” IEEE BITS the Information Theory Magazine, vol. 2, no. 2, pp. 36–55, 2022

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Zak-OTFS: Pulse Shaping and th e Tradeoff between Time/Bandwidth Expansion and Predictabi lity,

J. Jayachandran, R. K. Jaiswal, S. K. Mohammed, R. Hadani , A. Chock- alingam, and R. Calderbank, “Zak-OTFS: Pulse Shaping and th e Tradeoff between Time/Bandwidth Expansion and Predictabi lity,” arXiv preprint arXiv:2405.02718, 2024

arXiv 2024