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A Gaussian-Sinc Pulse Shaping Filter for Zak-OTFS

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper proposes a Gaussian-sinc pulse shaping filter for Zak-OTFS that keeps the sinc filter's nulls while cutting side lobes, and reports about 4 dB SNR gain at 10^-2 uncoded BER and more than 6 dB at 10^-4 coded BER over Gaussian and…

desk verdict A solid incremental Zak-OTFS pulse-shaping paper with substantial closed-form derivations, but the headline 'no expansion' claim is criterion-dependent and needs matched-bandwidth comparison. read the letter →

arxiv 2502.03904 v1 pith:5X5URTDR submitted 2025-02-06 cs.IT eess.SPmath.IT

Arpan Das , Fathima Jesbin , Ananthanarayanan Chockalingam This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords Zak-OTFSdelay-DopplerdomainpulseshapingfilterGaussian-sincI/Orelationestimationnoisecovarianceequalizationanddetectionembeddedpilot
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 sets out to establish that a new delay-Doppler pulse shape, the Gaussian-sinc (GS) filter, gives Zak-OTFS the best of both existing filters: the sinc filter's nulls at information grid points, which help equalization and detection, and the Gaussian filter's low side lobes, which help model-free input-output (I/O) relation estimation. It claims this combination is achieved without the time and bandwidth expansion that root-raised-cosine filters require for side-lobe reduction. The paper derives closed-form expressions for the effective channel and noise covariance of GS-filtered Zak-OTFS, and supports the claim with simulations on a fractional delay-Doppler Veh-A channel using exclusive and embedded pilots. If the claim holds, replacing the pulse shape alone improves Zak-OTFS reliability by the reported margins while keeping spectral efficiency unchanged.

What carries the argument

The load-bearing object is the Gaussian-sinc (GS) filter, a separable delay-Doppler pulse built as $g_{tx}(\tau,\nu)=g_1(\tau)g_2(\nu)$ with $g_1(\tau)=\Omega_\tau\sqrt{B}\,\mathrm{sinc}(B\tau)e^{-\alpha_\tau B^2\tau^2}$ and $g_2(\nu)=\Omega_\nu\sqrt{T}\,\mathrm{sinc}(T\nu)e^{-\alpha_\nu T^2\nu^2}$. The sinc factor provides nulls at the Nyquist sampling points on the information lattice, which the paper identifies as the property that makes sinc good for equalization and detection, while the Gaussian factor provides fast decay and low side lobes, the property that makes Gaussian good for I/O relation estimation; the energy constants $\Omega_\tau,\Omega_\nu$ normalize each factor to unit energy, and at $\alpha_\tau=\alpha_\nu=0.044$ the 99% energy-containment duration and bandwidth equal the unexpanded $T$ and $B$. With the matched receiver $g_{rx}(\tau,\nu)=g_{tx}^*(-\tau,-\nu)e^{j2\pi\nu\tau}$, Theorem 1 reduces the effective channel to a sum over physical channel paths of closed-form integrals of the filter's delay and Doppler components, with indicator-function cases separating zero and nonzero delay and Doppler offsets, and Theorem 2 gives the noise covariance as a double sum over quasi-periodic replicas with closed-form erf-based integral terms; these expressions constitute the paper's analytic basis for evaluating GS-filtered Zak-OTFS performance.

What would settle it

Reproduce the paper's setup (M=32, N=48, Veh-A fractional channel, embedded pilot at 0 dB PDR, 8-QAM, MMSE detection) and compare GS to sinc and Gaussian: the central claim predicts about 4 dB SNR gap at $10^{-2}$ uncoded BER and more than 6 dB at $10^{-4}$ coded BER, so a materially smaller gap would falsify the practical claim; separately, computing the 99.9% energy-containment time-bandwidth product of the GS filter and comparing it with RRC's expanded product would test how far the no-expansion advantage extends beyond the 99% convention.

Watch

Extended reading notes

Core claim

The central claim is that the Gaussian-sinc (GS) filter, defined as a separable product of sinc and Gaussian pulses in delay and Doppler, inherits the complementary strengths of its two parents. For the delay variable it takes $g_1(\tau)=\Omega_\tau\sqrt{B}\,\mathrm{sinc}(B\tau)e^{-\alpha_\tau B^2\tau^2}$, with the analogous form $g_2(\nu)$ for Doppler, so the transmitted pulse is $g_{tx}(\tau,\nu)=\Omega_\tau\Omega_\nu\sqrt{BT}\,\mathrm{sinc}(B\tau)\mathrm{sinc}(T\nu)e^{-\alpha_\tau B^2\tau^2}e^{-\alpha_\nu T^2\nu^2}$. The paper shows that with $\alpha_\tau=\alpha_\nu=0.044$ the filter keeps 99% of its energy within the unexpanded frame bandwidth $B$ and duration $T$, and that its delay/Doppler magnitude profile retains the sinc nulls while lowering side lobes. It then derives closed-form expressions (Theorems 1 and 2) for the matched-filter effective channel and the noise covariance, and reports that in Veh-A channels with model-free I/O estimation using an embedded pilot and 8-QAM, GS achieves about 4 dB SNR gain over both Gaussian and sinc filters at $10^{-2}$ uncoded BER and more than 6 dB at $10^{-4}$ coded BER with rate-1/2 coding.

Load-bearing premise

The load-bearing premise is that 'no time or bandwidth expansion' is measured by the 99% energy-containment convention, so an infinitely supported GS filter counts as unexpanded only under that threshold, and that the effective channel spreads stay within the delay and Doppler periods (the crystallization condition) so model-free estimation can read a single local response.

Editorial extensions

If this is right

  • If the claim holds, Zak-OTFS can be made more reliable at no spectral-efficiency cost: the GS filter uses the same time and bandwidth as sinc or Gaussian while reporting lower BER in the simulated Veh-A channel.
  • The closed-form effective channel and noise covariance remove the need to compute full twisted convolutions for GS-filtered Zak-OTFS, making analysis and optimization of the filter parameters $\alpha_\tau,\alpha_\nu$ feasible.
  • Because the GS filter keeps the sinc nulls, its detection performance with perfect channel knowledge stays close to sinc's; because it cuts side lobes, its I/O estimation MSE stays close to Gaussian's, so the two receiver functions no longer force opposite filter choices.
  • The reported gains appear in both exclusive and embedded pilot setups, with the embedded pilot case showing the larger improvement, meaning the filter is compatible with the practical pilot-and-guard frame structure.
  • The U-shaped BER-versus-PDR behavior in embedded-pilot frames is shallower for GS than for non-Gaussian filters, indicating less sensitivity to strong pilot power.

Reading between the lines

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

  • One could apply the same product construction to other pulse shapes or to the time-frequency windowing of windowed Zak-OTFS, since the conflict between main-lobe nulls and side-lobe leakage is not specific to the separable sinc and Gaussian pair.
  • The single operating point $\alpha_\tau=\alpha_\nu=0.044$ is one choice on a continuous family interpolating between sinc ($\alpha=0$) and increasingly Gaussian shapes; the closed forms in the paper make it possible to search this family per channel profile, a step the paper does not take.
  • The crossover between Gaussian and sinc BER curves suggests an adaptive rule: use more Gaussian-like shaping when estimation noise dominates and more sinc-like shaping when detection dominates; the GS filter is a fixed compromise that could be outperformed by a channel-aware switch.
  • The paper's own future-work note points to superimposed or spread pilots; a testable extension is to check whether the GS advantage persists when pilot and guard regions are removed and throughput is higher.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper proposes a separable delay-Doppler pulse shaping filter for Zak-OTFS, the Gaussian-sinc (GS) filter, defined as the product of a sinc and a Gaussian in each axis (Eq. (23)). The authors derive closed-form expressions for the effective channel (Theorem 1, Eq. (25)) and the noise covariance (Theorem 2, Eq. (27)) under matched filtering. They evaluate MSE and BER for model-free I/O relation estimation with exclusive and embedded pilot frames over a Veh-A channel, reporting, for example, about 4 dB SNR gain at 10^-2 uncoded BER and more than 6 dB at 10^-4 coded BER with 8-QAM compared to Gaussian and sinc filters (Figs. 11-12). The paper argues that the GS filter inherits the sinc filter's nulls at information grid points and the Gaussian filter's low side lobes without incurring time or bandwidth expansion under a 99% energy-containment definition.

Significance. If the claims hold, the GS filter is a simple and attractive option for Zak-OTFS: it has no parameters fitted to BER data (alpha = 0.044 is fixed by a 99% energy-containment criterion), the separable construction is easy to state, and the closed-form I/O and noise-covariance expressions could be useful for analysis and faster simulation. The paper's strengths include self-contained derivations in the appendices, evaluation under both exclusive and embedded pilot schemes, and results for both uncoded and coded BER. The performance advantage is, however, less clean than advertised: the 'no expansion' property is tied to the 99% energy-containment definition and is not compared against RRC at equal occupied bandwidth, the closed forms are not numerically verified against direct simulation, and Theorem 1 contains a typo. These issues are fixable, so the contribution warrants revision rather than rejection.

major comments (4)
  1. [§IV-C, Eq. (25)] The theorem statement repeats C^(2)_{i,1}(tau,nu) in both Doppler indicator branches, but the Appendix B derivation (Eqs. (43)-(44) and the subsequent combination) uses C^(2)_{i,2}(tau,nu) for the case nu = nu_i. As written, Theorem 1 is incorrect. Please correct the second branch and verify that the simulation code uses the corrected expression.
  2. [§IV-B and Appendix A] The 'no time or bandwidth expansion' claim is made only with respect to the 99% energy-containment criterion. The GS filter in Eq. (23) is the product of a bandlimited sinc and a Gaussian, so its spectrum is the convolution of a rectangular mask with a Gaussian of standard deviation B sqrt(alpha/(2*pi^2)) ≈ 0.047B at alpha = 0.044. Under a stricter occupied-bandwidth threshold (e.g., 99.9% or -40 dB), the occupied bandwidth grows by several times this width, which is comparable to or larger than the 5% and 10% expansion allowed for the RRC filter in Section V. The paper should quantify occupied bandwidth as a function of the energy threshold and, ideally, provide a BER comparison against RRC at matched occupied bandwidth; otherwise the advertised advantage over RRC is criterion-dependent.
  3. [§IV-C and §V] The closed-form expressions for the effective channel and noise covariance are not validated against direct simulation (e.g., numerical Zak transform or time-domain I/O) on any of the reported MSE/BER plots. Given the algebraic complexity of Appendices B-C and the typo in Theorem 1, the reader cannot tell whether the simulated curves actually use the claimed closed forms or whether those forms are correct. Please include at least one validation figure or table comparing closed-form and direct-simulation effective-channel coefficients, MSE, or BER curves.
  4. [§V, paragraph after Eq. (10)] The replica indices a and b in Eq. (10) are truncated to -1, 0, 1 with the statement that this 'is found to ensure an adequate support set.' This truncation is load-bearing for the model-free channel-matrix construction, especially for the high side lobes of the sinc filter and the infinite tails of the GS filter. Please justify the truncation numerically, for example by showing MSE or BER sensitivity to the truncation range for the filters and channel parameters considered.
minor comments (4)
  1. [Abstract and Introduction] The phrase 'does not incur time or bandwidth expansion' should be qualified as 'under the 99% energy-containment definition' in the abstract and introduction to avoid overstatement.
  2. [§V-B] The coded BER results use a rate-1/2 convolutional code with constraint length 7, but the generator polynomials are not given; please provide them for reproducibility.
  3. [§V] No Monte Carlo run counts or confidence intervals are reported for the BER/MSE curves; at least the number of channel realizations and frames should be stated.
  4. [Fig. 3 and §IV-A] The figure caption does not specify the roll-off factors for RRC, the alpha parameters for Gaussian and GS, or the normalization used; please state these values in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GS filter is defined first, its parameters are chosen to meet a stated energy-containment design target rather than fitted to BER, and the reported gains are simulation outcomes.

full rationale

The paper's derivation chain is self-contained. The GS filter is defined in Eq. (23) as a product of sinc and Gaussian pulses; the normalization constants Omega are derived from unit-energy constraints (Appendix A), and the values alpha_tau = alpha_nu = 0.044 are selected to satisfy the stated 99% energy-containment/no-expansion design target. That parameter choice is not fitted to the BER results, so the 4-6 dB SNR gains reported in Section V are genuine simulation outcomes for the chosen filter configuration. The closed-form effective-channel and noise-covariance expressions in Theorems 1 and 2 are algebraic specializations of known matched-filter integral forms to the GS pulse, not assumptions that force the performance comparison. The paper's self-citations to prior Zak-OTFS work supply background, the crystalline-regime condition, and the baseline Gaussian filter parameter, but none is used as a uniqueness argument or as a substitute for the present simulations. The 'no time or bandwidth expansion' claim is definition-dependent on the 99% energy criterion, which is a robustness/fairness caveat for the comparison with an expanded RRC filter, but it is not a circular derivation: the claim is a design specification made explicit in the text, not a prediction obtained from fitted constants.

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

The central BER results rest primarily on simulation parameters rather than on fitted physical constants. The GS filter itself is a design object, not a new physical entity. The only hand-set parameters are the Gaussian exponents alpha_tau = alpha_nu = 0.044, chosen by an energy-containment convention, plus baseline RRC roll-offs. The truncation of replica indices is an ad hoc simulation choice.

free parameters (3)
  • alpha_tau = 0.044
    Gaussian exponent for the delay shaping component in the GS filter, chosen so that 99 percent of pulse energy lies within bandwidth B and time T; a hand-set design parameter, not fitted to BER.
  • alpha_nu = 0.044
    Same choice for the Doppler shaping component; sets the no-expansion operating point for the GS filter.
  • RRC roll-off beta_tau, beta_nu = 0.05, 0.1
    Used for the RRC baseline in comparisons, causing 5 and 10 percent bandwidth and time expansion; not part of the proposed filter.
assumptions (5)
  • domain assumption Effective channel spread satisfies the crystallization condition tau_max < tau_p and nu_max < nu_p, so the local (0,0) response is sufficient for model-free I/O estimation.
    Invoked in Sec. III-A and satisfied by simulation parameters: Veh-A max delay 2.51 microseconds and max Doppler 815 Hz against periods 66.66 microseconds and 15 kHz.
  • domain assumption Time duration and bandwidth are measured by 99 percent energy containment rather than strict support; the infinite-support GS filter is considered to have no expansion under this criterion.
    Uses the convention from prior Gaussian-filter work and Sec. IV-B; load-bearing for the no-expansion claim.
  • domain assumption The physical channel is a sum of P discrete paths with delays and Dopplers (Eq. 3), and the receive filter is matched to the transmit filter (Eq. 20).
    Standard Zak-OTFS model; follows prior work [10], [13]-[15].
  • standard math Parseval's theorem, Fourier transform pairs, and erf approximations are valid for the integral manipulations in Appendices A through C.
    Standard background used without proof.
  • ad hoc to paper The discrete effective channel matrix is adequately captured by truncating quasi-periodic replica indices a and b to -1, 0, 1 in Eq. (10).
    Simulation choice in Sec. V; the paper states this is found to ensure adequate support but provides no quantitative validation.

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Cite this review

Pith. "Pith review of A Gaussian-Sinc Pulse Shaping Filter for Zak-OTFS." pith.science (2026). https://pith.science/paper/5X5URTDR

@misc{pith2026250203904,
  author       = {Pith},
  title        = {Pith review of: A Gaussian-Sinc Pulse Shaping Filter for Zak-OTFS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5X5URTDR}},
  note         = {Machine review of arXiv:2502.03904}
}
abstract

The choice of delay-Doppler domain (DD) pulse shaping filter plays an important role in determining the performance of Zak-OTFS. Sinc filter has good main lobe characteristics (with nulls at information grid points) which is good for equalization/detection, but has high side lobes which are detrimental for input-output (I/O) relation estimation. Whereas, Gaussian filter is highly localized with very low side lobes which is good for I/O relation estimation, but has poor main lobe characteristics which is not good for equalization/detection. In this paper, we propose a new filter, termed as {\em Gaussian-sinc (GS) filter}, which inherits the complementary strengths of both Gaussian and sinc filters. The proposed filter does not incur time or bandwidth expansion. We derive closed-form expressions for the I/O relation and noise covariance of Zak-OTFS with the proposed GS filter. We evaluate the Zak-OTFS performance for different pulse shaping filters with I/O relation estimated using exclusive and embedded pilots. Our results show that the proposed GS filter achieves better bit error rate (BER) performance compared to other filters reported in the literature. For example, with model-free I/O relation estimation using embedded pilot and 8-QAM, the proposed GS filter achieves an SNR gain of about 4 dB at $10^{-2}$ uncoded BER compared to Gaussian and sinc filters, and the SNR gain becomes more than 6 dB at a coded BER of $10^{-4}$ with rate-1/2 coding.

Figures

Figures reproduced from arXiv: 2502.03904 by the authors.

Figure 1
Figure 1. Block diagram of Zak-OTFS transceiver. OTFS with the proposed GS filter. We evaluate the Zak￾OTFS performance for different pulse shaping filters with I/O relation estimated using exclusive and embedded pilots. We consider ITU Vehicular-A (Veh-A) channel model [20] with fractional delays and Dopplers in performance evaluation. Our simulation results show that the proposed GS filter achieves better bit error rate (BE… view at source ↗
Figure 2
Figure 2. Embedded pilot frame with pilot symbol, pilot [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Delay pulse magnitude |1 ()| (in dB) as a function of the normalized delay . Path index () 1 2 3 4 5 6 Delay () 0 0.31 0.71 1.09 1.73 2.51 Relative power (dB) 0 -1 -9 -10 -15 -20 Table I: Power delay profile of Veh-A channel model. IV. PROPOSED GAUSSIAN-SINC DD FILTER In the Zak-OTFS literature, sinc, RRC, and Gaussian pulse shaping filters have been considered. In this section, we present the rationale for a new pu… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Performance of sinc and Gaussian filters (a) with pe [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: MSE vs pilot SNR performance for different filters [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 8
Figure 8. Figure 8: MSE vs data SNR performance of different filters [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: BER vs PDR performance of different filters with [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: BER vs data SNR performance of different filters [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: BER vs data SNR performance of different filters [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

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Forward citations

Cited by 2 Pith papers

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

  1. Zak-OTFS Based Coded Random Access for Uplink mMTC

    eess.SP 2025-07 conditional novelty 6.0 of 10

    A delay-Doppler Zak-OTFS coded random access scheme achieves lower packet loss than OFDM for uplink massive IoT under high mobility in simulations.

  2. Waveform for Next Generation Communication Systems: Comparing Zak-OTFS with OFDM

    eess.SP 2025-05 conditional novelty 4.0 of 10

    Zak-OTFS outperforms CP-OFDM in effective spectral efficiency for doubly-spread channels, with the largest gains (more than 2x) in high mobility plus large cell scenarios.

Reference graph

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