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REVIEW 3 major objections 4 minor 34 references

On the Time-Frequency Localization Characteristics of the Delay-Doppler Plane Orthogonal Pulse

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Pith's one-line read This paper derives closed-form time-frequency localization metrics for the delay-Doppler plane orthogonal pulse, showing that it spreads energy widely in time, frequency, and jointly while still obeying the Gabor limit.

desk verdict Useful closed forms for the DDOP's TF spread, but the frequency dispersion in Theorem 1 is an unspecified essential-bandwidth quantity, not the exact RMS width, and Eq. (16) has a minor misprint. read the letter →

arxiv 2412.13216 v1 pith:GN2IINSH submitted 2024-12-14 eess.SP

Akram Shafie , Jinhong Yuan , Nan Yang , Hai Lin This is my paper · ORCID
classification eess.SP
keywords delay-DopplerplaneorthogonalpulseODDMtime-frequencylocalizationGaborlimitHeisenberguncertaintyprincipleroot-raised-cosinebetter-than-RRCOTFSbasispulses
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 quantify how the delay-Doppler plane orthogonal pulse (DDOP), the prototype pulse of orthogonal delay-Doppler division multiplexing (ODDM), spreads its energy across time and frequency. It derives closed-form expressions for the pulse's time dispersion, frequency dispersion, joint time-frequency area, and direction parameter. The central result is that the DDOP is spread out in both dimensions at once: its joint time-frequency area sits far above the Gabor lower bound, while each of the roughly $MN$ small scattering areas behaves locally like a well-localized pulse. The paper argues this scattered spread is useful, since it lets a single waveform harvest both time and frequency diversity and provide fine delay and Doppler resolution for sensing. It also shows that the dispersion formulas can be read off from the envelope functions of the pulse's time and frequency representations, which extends the calculation to generalized DDOPs and to the effective basis pulses of OTFS.

What carries the argument

The carrier of the argument is the concatenation construction $u(t)=\sum_{n=0}^{N-1} a(t-nT-T_a/2)$, together with its frequency-domain counterpart, a train of sinc-shaped tones $\mathrm{sinc}(NTf-mN)$ spaced $1/T$ apart. This structure makes the second-moment integrals separable: the time dispersion of the DDOP is essentially the time dispersion of its rectangular time envelope, while the frequency dispersion is essentially the frequency dispersion of its sub-pulse envelope. A supporting lemma evaluates shifted second-moment integrals of even functions, and the derivation treats the truncated RRC spectrum as essentially the ideal RRC spectrum, so the sinc-train sums can be approximated as integrals for large $M$ and $N$.

What would settle it

Compute $\Delta F$ by numerically integrating (10) with the exact truncated RRC spectrum $A(f)=\tilde{A}(f)\star\mathrm{sinc}(T_a f)$ over a wide band, say $\pm 5M/T$, for small $Q$ and low $\beta$; if the result departs from (26c) by more than the sub-one-percent margin reported for the simulated parameters, then the essential-bandwidth assumption is the point of failure.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the time-frequency localization of the DDOP is governed by two independent envelopes: the rectangular time window of length $NT$ sets the time dispersion, and the sub-pulse spectrum sets the frequency dispersion. For a DDOP built from $N$ truncated root-raised-cosine sub-pulses with roll-off $\beta$, the resulting metrics are $\Delta T \approx NT/\sqrt{12}$, $\Delta F \approx (M/T)\sqrt{1/12+(\pi^2-8)\beta^2/(4\pi^2)}$, $\Delta A \approx (MN/12)\sqrt{1+3(\pi^2-8)\beta^2/\pi^2}$, and $\kappa \approx (NT^2/M)\sqrt{\pi^2/(\pi^2+3(\pi^2-8)\beta^2)}$. Because both $\Delta T$ and $\Delta F$ are large, the joint area lies orders of magnitude above the Gabor limit, yet the pulse obeys the Heisenberg uncertainty bound and behaves locally like a narrow pulse in small tiles of the time-frequency plane.

Load-bearing premise

The closed forms depend on treating the truncated RRC sub-pulse's spectrum as essentially the untruncated RRC spectrum and on $M$ and $N$ being large enough that the sinc-tone sums behave like integrals; if the small frequency tails carry significant energy or the block sizes are small, the stated $\Delta F$ and $\Delta A$ formulas shift.

Editorial extensions

If this is right

  • The DDOP is physically realizable as a prototype pulse: its time-frequency area respects the Gabor limit, so the fine delay-Doppler resolutions of ODDM do not force a pulse that violates the uncertainty principle.
  • Compared with TDM and FDM benchmark pulses, the DDOP has a larger joint time-frequency area because it spreads widely in both dimensions instead of being narrow in one; its direction parameter also falls between the TDM and FDM values.
  • Locally, each of the roughly $MN$ scattered tiles has a time-frequency area on the order of $1/(4\pi MN)$, so the pulse behaves like a well-localized pulse in small regions of the time-frequency plane.
  • For the generalized DDOP with cyclic prefix and suffix, the time dispersion and time-frequency area grow in steps with the sub-pulse duration $T_a$, through the parameter $D=\lceil T_a/T\rceil$.
  • Choosing the RRC sub-pulse instead of the better-than-RRC sub-pulse yields a lower frequency dispersion and a lower joint time-frequency area, especially at high roll-off $\beta$.

Reading between the lines

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

  • The envelope shortcut of reading $\Delta T$ from the time envelope and $\Delta F$ from the frequency envelope should extend to any ODDM-like pulse whose sub-pulse is spectrally concentrated, letting future DD-domain waveforms be compared by evaluating only their envelopes.
  • The same formulas give a design handle: increasing $N$ or $M$ raises both dispersions and the joint area, so system designers could tune block sizes to trade diversity gain against sensing resolution or out-of-band constraints.
  • Because the DDOP is locally narrow and globally spread, it suggests a single-waveform joint communication-and-sensing system in which the same pulse provides data throughput, delay resolution, and Doppler resolution; a direct experiment would measure delay-Doppler ambiguity sidelobes against the formulas in Theorem 1.
  • One might test whether a sub-pulse with a more rectangular spectrum than RRC directly lowers $\Delta A$ through the same formula, turning the time-frequency metric into an optimization target.
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Editorial analysis

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

Referee Report

3 major / 4 minor

Summary. The paper derives closed-form approximations for the time-frequency (TF) localization metrics of the delay-Doppler plane orthogonal pulse (DDOP): the TF area ΔA, time dispersion ΔT, frequency dispersion ΔF, and direction parameter κ. The derivation covers the DDOP with truncated root-raised-cosine and better-than-RRC sub-pulses, as well as a generalized DDOP with cyclic prefix/suffix extensions. The paper interprets the resulting large TF area as a consequence of the pulse energy being scattered across approximately MN small TF regions, discusses implications for diversity exploitation and sensing, and validates the analytical formulas numerically.

Significance. If the formulas stand, they give a compact analytical description of the DDOP's energy spread and clarify why the jointly large ΔT and ΔF can coexist with local fine-resolution behavior. The numerical validation is a genuine strength: the closed forms are compared with direct numerical integration for a range of M, N, and β and agree to about 1% for the intended operating region, and no parameter is fitted to the target metrics. The envelope-function relationships in Section V are also useful for extending the results to variants of the DDOP. However, the exact-frequency-dispersion issue described below affects the theorem's statement and must be resolved before the results can be taken at face value.

major comments (3)
  1. [Section III-A, Eq. (16)] Computing t̄ directly from (11) and (3) gives t̄ = T(N−1)/2 + Ta/2 for a sub-pulse centered at t = 0, not the value T(N−1)+Ta/2 stated in Eq. (16). With the erroneous value, the cancellation of the TTa and Ta² terms that leads from (15b) to (17) does not hold as written. The final approximation ΔT ≈ NT/√12 is still the correct large-N variance of the N pulse positions, so the theorem's conclusion can be repaired, but Eq. (16) and the intervening algebra must be corrected.
  2. [Section II-B, Eq. (10) and Footnote 7; Theorem 1, Eq. (26c)] Because a(t) in (4) is time-limited and does not vanish at ±Ta/2, the convolution A(f) = A~(f) ⋆ Ta·sinc(Ta f) in (6) decays only as O(1/f). Consequently the exact RMS bandwidth defined in (10) is infinite: f²|A(f)|² has a non-integrable tail. The finite closed form (26c) is obtained only after replacing A(f) by A~(f) (Footnote 5) and adopting an 'essential' bandwidth convention (Footnote 7), and the numerical validation in Section VII integrates only up to ±5M/T. Since the central quantity ΔA in (26a) is the product of ΔT and this ΔF, the theorem needs either an explicit, testable definition of the essential-bandwidth cutoff or a quantitative bound showing that the discarded tail does not affect (26c) over a specified cutoff range.
  3. [Section III-A, Eqs. (22b)–(23)] The parameter K introduced in (22b) is the number of sinc(NTf) zero-crossings retained, and the term ΔF2² = K/(π²T²N²) is then dropped because it is asserted to be negligible compared with ΔF1². For the truncated RRC sub-pulse, however, the 1/f tail of A(f) contributes a roughly constant density to f²|A(f)|², so the ratio ΔF2²/ΔF1² is not obviously negligible for all large cutoffs. Please quantify this ratio, or fold it into the essential-bandwidth convention requested in the preceding major comment.
minor comments (4)
  1. [Section V-A] The sentence '∆F1 and ∆F1 are same as ∆F of a(t) and b(t)' should read '∆F1 and ∆F2 are the same as ∆F of a(t) and b(t), respectively.'
  2. [Section I, last paragraph] The organization paragraph says 'followed by the conclusion in Section VII', but the conclusion is in Section VIII.
  3. [Section IV-B, Remark 2] 'bellow' should be 'below'.
  4. [Footnote 5] Footnote 5 states that the truncation sidelobes of A(f) are 'negligibly small'; given the O(1/f) decay, a quantitative statement of how small and over which frequency range would help the reader assess the approximation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the DDOP TF-localization metrics follow from the DDOP's defining sub-pulse and standard Fourier analysis; self-citations provide structural context, not the derived result.

full rationale

The paper's central quantities ΔT, ΔF, ΔA, and κ are computed from the standard RMS definitions (9)–(13) applied to the DDOP constructed in (3) and its Fourier transform (5). The derivation chain is self-contained: ΔT follows from the sub-pulse support condition Ta≪T and the rectangular envelope NT/√12 (Eqs. (15)–(19)); ΔF follows from the sampled comb structure of U(f), the approximation that A(f)≈Ã(f), and the finite-band integral (24). No parameter is fitted to the target metrics, and the numerical validation simulates the DDOP from its definition and compares with the closed forms without optimizing any free parameter. The self-citations [2],[5],[8],[9] supply the DDOP definition, orthogonality statements, and the U(f) representation; the paper also sketches the derivation of U(f) via (29), and the cited prior work does not assert or presuppose the TF-localization theorem. The main caveat is declared rather than concealed: Footnotes 5 and 7 state that truncation sidelobes are ignored and that ΔF is evaluated in an essential-bandwidth sense, and Section VII adopts the same convention by integrating only to ±5M/T. That is an approximation with an explicit cutoff and an unstated tail bound, so it is a correctness/robustness risk rather than a circular reduction; nothing is defined in terms of the claimed result and no fitted input is relabeled as a prediction.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central formulas depend only on system parameters (M,N,T,β,Q) and on the standard deviation definitions in (9)-(13). One intermediate truncation index K is introduced by hand but cancels from the final expressions. The derivations additionally assume: (i) the standard TF localization definitions are the right measures; (ii) the Gabor limit applies; (iii) the DDOP definition and orthogonality from self-authored prior work are valid; (iv) the truncated-RRC spectrum can be approximated by the untruncated one; (v) the essential-bandwidth convention is acceptable; (vi) A(f) is locally constant across sinc replicas; and (vii) the envelope-function relationships extend to pulse variants. No new physical entities are introduced.

free parameters (1)
  • K = intermediate; cancels
    Truncation index for the sinc replica sum in (21)-(23). It is chosen by hand but does not appear in the final closed forms because the term it controls is negligible; no fitting to data.
assumptions (7)
  • standard math The TF localization metrics in (9)-(12) (standard deviation based ΔT, ΔF, ΔA, κ) are the appropriate measures of energy spread.
    Section II-B adopts classical definitions from Sahin et al. [10] and Gabor [12]; the paper's claims are expressed in terms of these.
  • standard math Heisenberg/Gabor limit ΔA ≥ 1/(4π) applies to any physical pulse.
    Invoked in Section IV-A and Remark 1 to conclude that the DDOP does not violate uncertainty; no proof needed.
  • domain assumption The DDOP is constructed as u(t)=Σ_{n=0}^{N-1} a(t-nT-Ta/2) with a truncated square-root-Nyquist sub-pulse and the orthogonality conditions from [2,5,8,9].
    Section II-A; the present paper does not re-derive these properties but uses them to define u(t).
  • ad hoc to paper Frequency sidelobes of the truncated RRC sub-pulse can be neglected, i.e., A(f)≈Ã(f).
    Stated in Section II-B footnote 5 and used in Sections III and V; the authors assert the sidelobes are negligibly small and numerically validate within ~1%.
  • domain assumption The frequency dispersion can be computed in the 'essential' sense, ignoring very small frequency tails.
    Stated in Section II-B footnote 7 as necessary due to Balian-Low; standard practice in communications pulse design.
  • ad hoc to paper A(f) is approximately constant over the significant support of each sinc replica, justifying Eq. (21).
    Used to pass from (20) to (21); not proven formally, only supported by numerical examples.
  • ad hoc to paper The envelope-function relationships (30)-(31) hold for DDOP variants.
    Used to obtain metrics for generalized DDOP, BTRRC sub-pulse, and OTFS basis functions; supported numerically for the generalized and BTRRC cases, but not for OTFS.

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Pith. "Pith review of On the Time-Frequency Localization Characteristics of the Delay-Doppler Plane Orthogonal Pulse." pith.science (2026). https://pith.science/paper/GN2IINSH

@misc{pith2026241213216,
  author       = {Pith},
  title        = {Pith review of: On the Time-Frequency Localization Characteristics of the Delay-Doppler Plane Orthogonal Pulse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GN2IINSH}},
  note         = {Machine review of arXiv:2412.13216}
}
read the original abstract

In this work, we study the time-frequency (TF) localization characteristics of the prototype pulse of orthogonal delay-Doppler (DD) division multiplexing modulation, namely, the DD plane orthogonal pulse (DDOP). The TF localization characteristics examine how concentrated or spread out the energy of a pulse is in the joint TF domain, the time domain (TD), and the frequency domain (FD). We first derive the TF localization metrics of the DDOP, including its TF area, its time and frequency dispersions, and its direction parameter. Based on these results, we demonstrate that the DDOP exhibits a high energy spread in the TD, FD, and the joint TF domain, while adhering to the Heisenberg uncertainty principle. Thereafter, we discuss the potential advantages brought by the energy spread of the DDOP, especially with regard to harnessing both time and frequency diversities and enabling fine-resolution sensing. Subsequently, we examine the relationships between the time and frequency dispersions of the DDOP and those of the envelope functions of DDOP's TD and FD representations, paving the way for simplified determination of the TF localization metrics for more generalized variants of the DDOP and the pulses used in other DD domain modulation schemes. Finally, using numerical results, we validate our analysis and find further insights.

Figures

Figures reproduced from arXiv: 2412.13216 by the authors.

Figure 2
Figure 2. The frequency response of the DDOP, where phase terms are ignored [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Illustration of the metrics ∆T and ∆F of a pulse in comparison to its duration Tg and bandwidth Bg. As a consequence of the Heisenberg uncertainty principle, ∆A obeys a lower bound known as the Gabor limit, given by ∆A ≥ 1 4π . The Gabor limit is attained by the Gaussian pulse [12], [23]. Typically, a pulse is considered to have minimal energy spread in the joint TF domain (or to be well￾localized in the joint TF do… view at source ↗
Figure 4
Figure 4. Simplified TF occupancy of the DDOP, the pulse used in the TDM scheme, and the pulse used in the FDM scheme. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: Illustration of the overall TF regions of the DDOP and a staggered [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: The TF localization metrics of the DDOP considered in (3) versus the roll-off factor, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Illustration of the approximation error of ∆A for different values of M and N. In [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: A comparison of the TF localization metrics of the DDOP with those of the pulses used in TDM and FDM schemes. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: The TF localization metrics of the generalized design of the DDOP given in (32) versus [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: The TF localization metrics of the DDOP with different sub-pulses versus the roll-off factor, [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

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