Design of OTFS Signals with Pulse Shaping and Window Function for OTFS-Based Radar

Liangchen Sun; Yutaka Jitsumatsu

REVIEW 5 major objections 4 minor 16 references

Design of OTFS Signals with Pulse Shaping and Window Function for OTFS-Based Radar

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A root raised cosine window on OTFS pilots makes fractional Doppler estimation more accurate than linear interpolation.

desk verdict Plausible new waveform idea for OTFS radar, but the headline claim is ahead of the evidence: the evaluation is noise-free, known-grid, and key derivations are omitted. read the letter →

arxiv 2506.04861 v1 pith:2MZBJWXA submitted 2025-06-05 eess.SP

Liangchen Sun , Yutaka Jitsumatsu This is my paper · ORCID

classification eess.SP

keywords OTFSrootraisedcosinewindowfunctionpulseshapingfractionalDopplerestimationambiguityjointcommunicationandsensingradar

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

This paper proposes an OTFS-based pulse radar that replaces the usual rectangular window with a root raised cosine (RRC) window whose values are allowed to go negative or exceed one. When combined with a rectangular transmit pulse, this window concentrates the pilot's ambiguity function in delay and Doppler and suppresses the sidelobe oscillations that a rectangular window produces. The authors also introduce an interpolation step for fractional delay and Doppler that uses the exact autocorrelation shape of the RRC window instead of assuming a linear triangular shape. They report that this RRC-autocorrelation interpolation clearly outperforms conventional linear interpolation for fractional Doppler estimation, while matching it for attenuation and fractional delay. If the result holds, finer velocity estimates are obtainable without changing bandwidth, modulation, or transmit power.

What carries the argument

The load-bearing object is the windowed OTFS waveform $h_{k,\ell}(t) = \{w(t)x_{k,\ell}(t)\} * p(t)$, where $w(t)$ is the RRC window with roll-off $\beta_w$, $p(t)$ is a rectangular pulse of width $T_s$, and $x_{k,\ell}(t)$ is a train of shifted Dirac deltas. This construction lets the window shape the comb spectrum in the frequency domain. The second mechanism is the interpolation in Algorithm 2, which fits the measured $2\times 2$ ambiguity values to the product $p*\hat p(\tau)\,\cdot\, W*\hat W(\nu)$ using the explicit piecewise-trigonometric autocorrelation of the RRC window.

What would settle it

Run the complete pipeline (Algorithm 1 detection under noise followed by Algorithm 2 interpolation) on simulated echoes with fractional Doppler values spread over $[0,1)$ and compare the RMSE of RRC-autocorrelation interpolation against linear interpolation. If the gap disappears once peaks are found by an actual detector rather than given as ground truth, the paper's comparative fractional-Doppler claim fails in the full system.

Watch

Extended reading notes

Core claim

The central claim is that choosing a generalized RRC window, one that is allowed to take negative values and values above 1, together with a rectangular pulse, shapes the transmitted OTFS pilot so that its cross-ambiguity function has a sharp, low-oscillation peak. This sharpness lets the receiver estimate fractional Doppler shifts more accurately when interpolation uses the closed-form RRC autocorrelation function rather than linear interpolation, because the RRC autocorrelation is triangular-like but nonlinear. The paper derives a discrete-time input-output relation for fractional delay and Doppler using a Toeplitz (not circulant) structure, and shows in simulation that the proposed interpolation reduces fractional Doppler RMSE across one to five propagation paths.

Load-bearing premise

The fractional estimation step is tested with the true $2\times 2$ delay-Doppler grid around each peak handed to it as input, so the coarse detection stage is bypassed; if the full detection chain must locate those peaks from noisy data, the reported fractional Doppler improvement may shrink or vanish.

Editorial extensions

If this is right

With the RRC window plus rectangular pulse, the ambiguity function of the pilot shows no Doppler-direction oscillations, simplifying delay-Doppler processing.
The proposed input-output relation captures fractional delay and Doppler without assuming a periodic transmitted signal, avoiding the circulant-matrix approximation used in earlier OTFS radar models.
RRC-autocorrelation interpolation estimates fractional Doppler with lower RMSE than linear interpolation across one to five propagation paths.
Theorem 1 gives design rules: an RRC pulse and an RRC window satisfying Nyquist conditions in time and frequency make the basis waveforms $h_{k,\ell}(t)$ orthonormal.

Reading between the lines

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

Inference: Because the RRC window is allowed outside $[0,1]$, the transmit waveform can carry more energy in the pilot peak without raising peak power, which may improve detection range; the paper does not quantify this.
Inference: The autocorrelation-matched interpolation idea should extend to other smooth windows, not only RRC, and to fractional delay when the pulse autocorrelation is known; a natural test is raised-cosine or Kaiser windows.
Inference: The Toeplitz non-circulant model reintroduces edge effects that ODDM and circulant designs remove; a testable extension is whether the RRC window also reduces those edge terms.

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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

Colleague,

Here's the quick read on Sun and Jitsumatsu. The new combination is real: an RRC window (relaxed to take values outside [0,1]) paired with a rectangular pulse for OTFS radar, plus an interpolation scheme that uses the RRC autocorrelation instead of the usual triangular (linear) kernel. That specific setup is not in the OTFS-radar papers they cite, which stick to rectangular windows and linear interpolation. The ambiguity-function plots in Fig. 4 are genuinely informative — the RRC window suppresses the Doppler sidelobe oscillations that the rectangular window produces.

The soft spots are concentrated where the claims are strongest. Lemma 1, Theorem 1, and the piecewise interpolation formula (26) are all asserted without proof. For a methods paper that's a real gap; eq. (26) is the mechanism behind the claimed improvement, and it just appears. The simulation in Sec. IV-B is the load-bearing evidence, and it is idealized: the 2x2 grid at each ambiguity peak is assumed known, so Algorithm 1 (detection) is bypassed, and no noise model is given. The stress-test note on this is on target — in a full receive chain the coarse detection and noise will likely dominate the fractional-Doppler RMSE, so the reported advantage of RRC interpolation could shrink or disappear. The separability approximation (24) is also used without error bounds on the 2x2 grid, and if it is off, both interpolators are fitting the wrong target.

That said, the paper is not a toy. The Toeplitz-versus-circulant distinction in Lemma 2 is a useful correction to [9] and [10], and the idea of designing the window as a free parameter for sensing is worth pursuing. The writing is readable, and the authors are honest about what they skip — they explicitly say proofs are omitted for space.

Who gets value: people working on OTFS joint communication and sensing, specifically fractional delay-Doppler estimation. If I were refereeing, I'd want the missing proofs (or at least a sketch of (26)), a noise model with SNR, and an evaluation that runs Algorithm 1 before Algorithm 2. That is a major revision, not a desk reject.

My recommendation: send it to peer review, but with a clear expectation that the evaluation be rebuilt. The idea is new enough to merit referee time; the current evidence is not strong enough to support the abstract's "significantly outperforms" claim.

Referee Report

5 major / 4 minor

Summary. The manuscript proposes an OTFS-based pulse radar waveform in which the transmit window is a generalized root-raised-cosine (RRC) function allowed to take values outside [0,1], together with a rectangular pulse. It claims that this combination concentrates the ambiguity function in both delay and Doppler, and that a fractional delay/Doppler estimator using the RRC autocorrelation as an interpolation kernel outperforms conventional linear interpolation. The paper derives a discrete-time input-output relation with fractional delay and Doppler, states an orthogonality theorem, gives an explicit piecewise formula for the RRC autocorrelation, and reports Monte Carlo RMSE comparisons.

Significance. If the claims hold, the paper offers a simple and low-cost modification to OTFS radar: choosing an RRC window and matching the interpolation kernel to its autocorrelation improves fractional Doppler estimation without extra bandwidth or changes to the modulation. The manuscript is clearly organized and provides explicit closed-form interpolation formulas and a simple simulation setup that is easy to follow. However, the central mathematical results are stated without proofs, and the evaluation is model-matched and idealized, so the current evidence is insufficient to support the journal-level claim of significant improvement.

major comments (5)

[III-A, Lemma 1] Equations (15)-(16) define the discrete input-output relation that underpins the entire radar model, but the proof is omitted with only 'Due to space limitations, we omit the proof.' The phase term e^{jπ f_D (ℓ+ℓ')T_s} and the factorization H(t_D, f_D) = D(f_D/2) Toep(t_D) D(f_D/2) need to be derived or supported by a citation; otherwise the subsequent estimation algorithm rests on an unverified model.
[III-B, Theorem 1] The orthonormality of the basis waveforms h_{k,ℓ}(t) is asserted, with the proof omitted. The sufficiency of the two Nyquist conditions in (20) and (21) is not obvious because h_{k,ℓ}(t) is defined as a convolution of a windowed Dirac train with the pulse p(t). A proof or a precise reference is needed, since orthonormality is load-bearing for the claimed ambiguity-function concentration and for the interpolation error model in Section IV-A.
[IV-A, Eq. (26)] The explicit piecewise expression for W*Ŵ(ν) is the core of the proposed interpolation method, but no derivation is provided. I could not verify the breakpoints (such as 2a and a+1/2) directly from the RRC definition. Please derive the expression or validate it against a numerical autocorrelation of the RRC window, and also study the sensitivity to the roll-off factor β, which is fixed to 0.25 in the paper.
[IV-B] The Monte Carlo evaluation assumes that the 2×2 ambiguity grid centered at each peak is known and feeds that grid directly into Algorithm 2, thereby bypassing the coarse detection stage of Algorithm 1. No additive noise model is specified for the results in Fig. 5; only Fig. 6 is explicitly labeled noise-free. In a real radar chain the coarse peak is noisy and may be off by a bin, so the reported fractional-Doppler RMSE improvement may shrink or disappear. The comparison is also model-matched, because the interpolation function in Eq. (26) is the exact autocorrelation of the same RRC window used to generate the data. An end-to-end simulation with noise, CFAR detection, and unknown peak locations is required before the abstract's 'significantly outperforms conventional linear interpolation' claim can stand.
[IV-A, Eq. (24)] The separability approximation A_ss(τ,ν) ≈ p*p̂(τ) · W*Ŵ(ν) is used to define the cost function in Eq. (23), but no error bound or numerical validation is given for the region |τ| ≤ T_s, |ν| ≤ 1/(NT). If the pilot ambiguity function is not actually separable on the 2×2 grid, both interpolation methods are fitting the wrong target, and the comparison in Fig. 5 is not direct evidence about real estimation performance.

minor comments (4)

[Abstract] There are two typos in the front matter: 'Index T erms' should be 'Index Terms', and 'Deptartment' in the author affiliation should be 'Department'.
[II-B, Fig. 3] The axis label 'H_{k,l}(ℓ)' in Fig. 3 appears inconsistent with the text, which uses h_{k,ℓ}(t) for the time-domain waveform and H_{k,ℓ}(f) for its Fourier transform. Please redraw the figure and use one consistent notation.
[II-B, footnote 2] Footnote 2 states that 'the difference between the two design approaches is minimal' for the two possible waveform definitions, but this claim is not quantified. Either provide a bound or remove the statement.
[IV-A, Eq. (24)] The symbol for the Doppler variable is written as 'v' in A_ss(τ,v) but as 'ν' in the surrounding text of Section IV-A. Please unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RRC autocorrelation interpolation kernel is derived from the defined window, and the performance comparison is a matched-model evaluation rather than a fitted input renamed as a prediction.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The RRC window is explicitly defined in Section III-B, and its autocorrelation W*Ŵ(ν) is computed analytically and given in closed form in Eq. (26). This function is then used as the interpolation kernel in the least-squares cost function of Eq. (23). No parameter is fitted to the simulation outcomes or to the RMSE curves; the unknown quantities α, ε_t, and ε_f are estimated from the 2×2 ambiguity samples using this fixed kernel. The simulation generates echoes from the same OTFS model with the RRC window and rectangular pulse, so the estimator is model-matched. That is a standard matched-estimator evaluation, not a tautology: the estimator would not have been expected to win if the true ambiguity shape were, say, sinc-like, and linear interpolation is evaluated under the same data. The paper's admission that the 2×2 grid is assumed known (Section IV-B) is an evaluation simplification that bypasses coarse detection, but it does not make the fractional estimation circular; it only limits the end-to-end claim. There are no load-bearing self-citations, no imported uniqueness theorem, and no ansatz smuggled in via citation. The omitted proofs of Lemma 1 and Theorem 1 are gaps in exposition rather than circular dependencies. The central comparison is therefore an honest, if idealized, simulation study, and no specific step reduces to its inputs by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a standard echo superposition model, small-Doppler approximations, unproved orthogonality and Toeplitz results, and a simulation setup that assumes known peak grids. Only the roll-off factor beta=0.25 is explicitly chosen by hand as a free parameter. No new physical entities are introduced.

free parameters (1)

RRC roll-off factor beta = 0.25
The roll-off factor for the RRC window, the RRC pulse, and the interpolation formula is chosen as 0.25 without optimization. The reported performance comparison is demonstrated only for this value.

assumptions (5)

domain assumption The received signal is a superposition of P attenuated, delayed, and Doppler-shifted copies of the transmitted signal, as in Eq. (1).
Invoked in Section II-A as the starting model for radar returns. All subsequent derivations and simulations depend on this linear echo model.
domain assumption The Doppler shift is small compared with the sampling rate, |f_D,i| << 1/T_s, so the pulse autocorrelation can be separated from the Doppler phase.
Used after Eq. (14) to replace the ambiguity function A_{p,p} by p * \hat p, which is necessary for the Toeplitz form in Lemma 1. The approximation is stated but not quantified.
ad hoc to paper The pulse and window satisfy the Nyquist conditions in Theorem 1, so the basis waveforms are orthonormal.
Theorem 1 in Section III-B asserts this orthogonality, but the proof is omitted. The signal design and the ambiguity function model rely on this orthogonality.
domain assumption The delay is bounded by T_B < t_D < (U-1)T_B so that transmission and reception do not overlap.
Section II-A introduces this range assumption to justify the pulse structure with a silent interval and to determine which received samples are observable.
ad hoc to paper The 2x2 grid centered at each ambiguity peak is known when evaluating the fractional parameter estimator.
Section IV-B states that this grid is assumed known and is used as input to Algorithm 2. This bypasses the coarse detection stage and is a load-bearing simplification for the performance claim.

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Cite this review

Pith. "Pith review of Design of OTFS Signals with Pulse Shaping and Window Function for OTFS-Based Radar." pith.science (2026). https://pith.science/paper/2MZBJWXA

@misc{pith2026250604861,
  author       = {Pith},
  title        = {Pith review of: Design of OTFS Signals with Pulse Shaping and Window Function for OTFS-Based Radar},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2MZBJWXA}},
  note         = {Machine review of arXiv:2506.04861}
}

read the original abstract

We propose a pulse radar system that employs a generalized window function derived from the root raised cosine (RRC), which relaxes the conventional constraint that the window values are within the range [0, 1]. The proposed window allows both negative values and values exceeding 1, enabling greater flexibility in signal design. The system transmits orthogonal time frequency space (OTFS) signals intermittently, establishing a flexible input-output relationship that captures both fractional delays and Doppler shifts. By combining the generalized RRC window with a rectangular pulse, the resulting pilot signal achieves a sharp concentration in the ambiguity function over both the delay and Doppler domains. To enhance the estimation accuracy of fractional parameters, we apply frequency-domain interpolation based on the autocorrelation of the RRC window, which outperforms conventional linear interpolation by preserving the signal structure more effectively.

Figures

Figures reproduced from arXiv: 2506.04861 by the authors.

**Figure 2.** Structure of the Transmitted Signal IV provides the simulation results, and Section V concludes the entire paper. II. Background and Proposed Model A. Assumptions for environment and transmitted signal We examine coexistence challenges between radar and wireless communications. A car, equipped with an oscillator and a millimeter-wave antenna, transmits and receives data while using the same antenna to detect reflec… view at source ↗

**Figure 3.** Construction of ℎ𝑘,ℓ (𝑡) with the shape of 𝑝(𝑡) and 𝑤ˆ [ℓ ] and their spectra (𝑁 = 5, 𝑀 = 8) where 𝑥˜TD [ℓ] = 𝑤ˆ [ℓ]𝑥TD [ℓ], and 𝑥TD can be expressed as 𝑥TD [𝑛𝑀 + ℓ] = 1 √ 𝑁 𝑁 ∑︁−1 𝑘=0 𝑋DD [𝑘, ℓ]𝑊−𝑛𝑘 𝑁 . (10) It is worth noting that when the window function 𝑤ˆ [ℓ] in (9) is rectangular, it appears to have the same form as (3). However, the pulse width of 𝑤ˆ [ℓ] is 𝑁𝑇, while the pulse width of 𝑔TX is 𝑇, highlighting … view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The fine ambiguity function 𝐴𝑠,𝑠 (𝜏, 𝜈) for −5𝑇𝑠 < 𝜏 < 5𝑇𝑠 and −5/𝑇𝐵 < 𝜈 < 5/𝑇𝐵. (a) Rectangular pulse, (b) sinc pulse and (c) rrc pulse. The roll off factor is 25%. (x1) denotes a rectangular window, while (x2) means an rrc window. as 𝐴00 𝐴01 𝐴10 𝐴11 . The step has…

**Figure 5.** Figure 5: Comparison of estimation accuracy between linear interpolation [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Estimation error analysis of fractional Doppler shifts using linear [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [9]

DFT-spread orthogonal time frequency space system with superimposed pilots for terahertz integrated sensing and communication,

Y. Wu, C. Han, and Z. Chen, “DFT-spread orthogonal time frequency space system with superimposed pilots for terahertz integrated sensing and communication,”IEEE Trans. Wireless Comm., vol. 22, no. 11, pp. 7361–7376, 2023

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work page 2023

[3] [1]

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W. Hong, Z. H. Jiang, C. Yu, D. Hou, H. Wang, C. Guo, Y. Hu, L. Kuai, Y.Yu,Z.Jiang etal.,“Theroleofmillimeter-wavetechnologiesin5g/6g wireless communications,”IEEE Journal of Microwaves, vol. 1, no. 1, pp. 101–122, 2021

work page 2021

[4] [2]

Enabling joint communication and radar sensing in mo- bile networks—a survey,

J. A. Zhang, M. L. Rahman, K. Wu, X. Huang, Y. J. Guo, S. Chen, and J. Yuan, “Enabling joint communication and radar sensing in mo- bile networks—a survey,”IEEE Communications Surveys & Tutorials, vol. 24, no. 1, pp. 306–345, 2022

work page 2022

[5] [3]

Joint radar and communication design: Applications, state-of-the-art, and the road ahead,

F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” IEEE Trans. Commun., vol. 68, no. 6, pp. 3834–3862, 2020

2020

[6] [4]

Joint state sensing and communication: Optimal tradeoff for a memoryless case,

M. Kobayashi, G. Caire, and G. Kramer, “Joint state sensing and communication: Optimal tradeoff for a memoryless case,” in2018 IEEE Int. Symp. Inform. Theory (ISIT), 2018, pp. 111–115

work page 2018

[7] [5]

An information-theoretic approach to joint sensing and communication,

M. Ahmadipour, M. Kobayashi, M. Wigger, and G. Caire, “An information-theoretic approach to joint sensing and communication,” IEEE Trans. Inform. Theory, vol. 70, no. 2, pp. 1124–1146, 2022. Fig. 5. Comparison of estimation accuracy between linear interpolation and the proposed RRC autocorrelation-based interpolation: (a) attenuation coefficients, (b) fra...

work page 2022

[8] [6]

On the effective- ness of OTFS for joint radar parameter estimation and communication,

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

work page 2020