{"id":"9947c02a-6704-4313-bb1c-20e7808a85c2","arxiv_id":"2412.13216","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The delay-Doppler orthogonal pulse has closed-form time and frequency dispersions that are both large, giving a time-frequency area far above the Gabor limit, with numerical validation.","lead":"This paper derives the time-frequency localization metrics of the orthogonal delay-Doppler pulse used in ODDM modulation. It finds the pulse spreads energy widely in both time and frequency, keeping its joint time-frequency area far above the Gabor limit while behaving locally like a well-localized pulse.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The closed-form ΔF in Theorem 1 is computed after discarding RRC truncation sidelobes and adopting an 'essential bandwidth' definition; for a time-limited RRC sub-pulse the true RMS bandwidth diverges, so (26c) is cutoff-dependent unless the essential-bandwidth rule is specified and validated.","rationale":"I read the paper as making its central quantitative claim in Theorem 1: the DDOP has the TF localization metrics (26a)-(26d). The most load-bearing condition for that claim is that ΔF, and hence ΔA and κ, are well-defined and accurately captured by the closed forms. The paper's own footnotes 5 and 7 and Section VII reveal that this condition is not exactly met: the actual truncated RRC spectrum has algebraic sidelobes, the true RMS bandwidth is formally divergent, and the numerical verification uses a finite cutoff of ±5M/T. The reader's weakest assumption identified this same issue, and I agree with it. The numerical agreement for M=256, N=64, β=0.1 is genuine supporting evidence, and the approximation is likely acceptable for the intended ODDM parameter regime, which is why I do not move the verdict to REJECT. However, the concern is load-bearing because the theorem states a specific numerical value for ΔF without defining the essential-bandwidth cutoff on which that value depends; at β=0 the tail contribution does not vanish as the cutoff grows. The concrete cutoff-sweep test would settle whether the stated formula is a well-defined physical quantity or an artifact of a particular truncation choice. Since this matches the reader's conditional verdict, I keep the verdict unchanged with the same condition: the authors should specify the essential-bandwidth convention and validate the cutoff insensitivity, especially at small β.","tokens_in":22510,"tokens_out":18443,"duration_ms":166574,"concrete_test":"With M=256, N=64, T=1, and Q=0.05M, compute the simulated ΔF of Eq. (10) using the exact truncated-RRC spectrum A(f)=A~(f)⋆Ta·sinc(Ta f) over |f|≤c·M/T for c=1.5, 2, 5, 10, and 20, at β=0 and β=0.1. If ΔF_sim(c) changes by more than about 1% as c increases, Eq. (26c) is cutoff-dependent and Theorem 1 must be restated with an explicit essential-bandwidth convention; if ΔF_sim(c) stabilizes, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central object of Theorem 1 is the frequency dispersion ΔF defined in Eq. (10) as the RMS width of |U(f)|². For the actual truncated RRC sub-pulse, A(f)=A~(f)⋆Ta·sinc(Ta f) per Eq. (6); because a(t) is time-limited and does not vanish at ±Ta/2, A(f) decays only as 1/f, so ∫ f²|A(f)|² df diverges. The exact ΔF of the DDOP is therefore infinite, and the finite value in (26c) is an 'essential' bandwidth quantity. The derivation reaches (26c) by (i) Footnote 5 replacing A(f) with the bandlimited A~(f), (ii) Footnote 7 declaring that very small frequency tails are ignored, and (iii) Eq. (24) integrating only over the nominal RRC band. No bound is provided for the discarded tails, and the numerical validation in Section VII integrates only up to ±5M/T. Because the theorem does not specify the essential-bandwidth cutoff, (26c) depends on a free parameter; at β=0 the 1/f tail contributes to f²|A(f)|² at a roughly constant density, so increasing the cutoff from 5M/T to larger values changes ΔF, ΔA, and κ. The practical parameter set M=256, N=64, β=0.1 gives <1% error, but Fig. 7 varies M and N only and does not test cutoff dependence or small β.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22865,"tokens_out":6852,"duration_ms":63525,"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":[{"comment":"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.","section":"Section III-A, Eq. (16)"},{"comment":"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.","section":"Section II-B, Eq. (10) and Footnote 7; Theorem 1, Eq. (26c)"},{"comment":"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.","section":"Section III-A, Eqs. (22b)–(23)"}],"minor_comments":[{"comment":"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.'","section":"Section V-A"},{"comment":"The organization paragraph says 'followed by the conclusion in Section VII', but the conclusion is in Section VIII.","section":"Section I, last paragraph"},{"comment":"'bellow' should be 'below'.","section":"Section IV-B, Remark 2"},{"comment":"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.","section":"Footnote 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript builds on the authors' own prior work on the DDOP, but the TF localization analysis itself is new and not circular. The two substantive issues—the incorrect mean-time expression in Eq. (16) and the unspecified essential-bandwidth cutoff behind Eq. (26c)—are repairable within the manuscript's scope. The paper's central qualitative claim that the DDOP has large joint TF energy spread is likely robust, but the theorem's statement needs qualification and the derivation needs correction before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the useful part: the paper gives closed-form expressions for the time and frequency dispersions, joint TF area, and direction parameter of the DDOP, and they check out numerically to about 1% for practical M,N. That fills a real gap in the ODDM literature, and the envelope-function shortcut in Sections V-VI is a neat way to extend those results to generalized DDOP variants and OTFS basis functions without redoing the integrals. The numerical validation is honest and covers a reasonable parameter range.\n\nThe soft spots are real but mostly fixable. The biggest one concerns the frequency dispersion. As defined in (10), ΔF is the RMS width of |U(f)|^2. For a time-limited sub-pulse, A(f) decays only as 1/f, so the exact integral diverges; the true ΔF is infinite. The paper acknowledges this in footnote 7 by invoking an \"essential\" bandwidth, but Theorem 1 presents (26c) as if it were the exact result, and no cutoff is specified. The numerical validation integrates only up to ±5M/T, so the <1% agreement is relative to that arbitrary window, not to a well-defined metric. A referee should ask for either a precise definition of the essential bandwidth with a justification that the neglected tail is negligible in the intended operating regime, or a reformulation as a finite-bandwidth measure. This does not sink the practical usefulness, but it makes the theorem less precise than it appears.\n\nSecond, Eq. (16) has an off-by-two in the mean time: direct computation gives t̄ = T(N−1)/2 + Ta/2. The final result (17) is nonetheless correct because the mistake cancels when substituted, so it is a cosmetic typo, but it should still be fixed.\n\nThird, the OTFS results in Section VI are asserted from the envelope analogy rather than derived. They may be right, but the derivation steps are missing. The discussion of diversity and sensing benefits is qualitative, which is fine as motivation.\n\nWho is this for? Anyone working on ODDM or comparing it with OTFS, TDM, or FDM. The central formulas will be useful. I would send it to review, but with a referee who insists on clarifying the essential-bandwidth definition and correcting (16). My own verdict is conditional: accept after minor revisions that tighten the metric definition.","headline":"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.","tokens_in":23388,"tokens_out":3802,"would_cite":false,"duration_ms":34474,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["delay-Doppler plane orthogonal pulse","ODDM","time-frequency localization","Gabor limit","Heisenberg uncertainty principle","root-raised-cosine pulse","better-than-RRC pulse","OTFS basis pulses"],"falsifier":"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.","tokens_in":22265,"feed_emoji":"📡","tokens_out":8316,"duration_ms":70329,"temperature":0.7,"pith_summary":"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.","feed_headline":"Delay-Doppler pulse's time-frequency spread exceeds the Gabor limit","feed_subtitle":"Closed-form dispersions show it spreads broadly in time and frequency, enabling diversity and fine sensing.","key_machinery":"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 $N$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\beta$."],"supporting_citations":[{"why":"Defines the DDOP as a concatenation of sub-pulses and establishes the orthogonality properties of ODDM that motivate the pulse.","marker":"[2]"},{"why":"Proves the fine delay-Doppler orthogonality of the DDOP that underlies its role as the ODDM prototype pulse.","marker":"[5]"},{"why":"Extends orthogonality to the case without the strict sub-pulse duration constraint and provides the DDOP frequency response used in the derivations.","marker":"[8]"},{"why":"Derives the DDOP frequency response as a train of sinc tones and describes the scattered time-frequency occupancy that the metrics quantify.","marker":"[9]"},{"why":"Supplies the definitions of time dispersion, frequency dispersion, time-frequency area, and direction parameter used throughout the analysis.","marker":"[10]"},{"why":"Establishes the Gabor limit against which the DDOP's time-frequency area is compared.","marker":"[12]"},{"why":"Provides the comparative time-frequency localization framework and the direction-parameter interpretation used for benchmark comparisons.","marker":"[14]"},{"why":"Introduces the better-than-RRC sub-pulse whose frequency response is used for the alternative DDOP design analyzed in the paper.","marker":"[24]"}],"fun_headline_variants":["Delay-Doppler pulse spreads wide, beats Gabor, keeps Heisenberg","Wide time-frequency spread from one pulse powers dual diversity","Pulse's big TF footprint enables fine sensing and diversity","DDOP: broad in time and frequency, yet Heisenberg-compliant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Delay-Doppler pulse spreads wide, beats Gabor, keeps Heisenberg","Wide time-frequency spread from one pulse powers dual diversity","Pulse's big TF footprint enables fine sensing and diversity","DDOP: broad in time and frequency, yet Heisenberg-compliant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1501,"prompt_tokens":1011,"completion_tokens":490,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":415}},"tokens_in":627,"tokens_out":490,"duration_ms":4635,"temperature":1.0,"reasoning_tokens":415,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:29:56.430730+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Multicarrier modulation on delay-Doppler plane: Achieving orthogonality with fine resolutions,","cited_arxiv_id":null,"evidence_quote":"Proves the fine delay-Doppler orthogonality of the DDOP that underlies its role as the ODDM prototype pulse."},{"cited_title":"On delay-Doppler plane orthogonal pulse,","cited_arxiv_id":null,"evidence_quote":"Extends orthogonality to the case without the strict sub-pulse duration constraint and provides the DDOP frequency response used in the derivations."},{"cited_title":"A survey on multicarrier communi- cations: Prototype filters, lattice structures, and implementation aspects,","cited_arxiv_id":null,"evidence_quote":"Supplies the definitions of time dispersion, frequency dispersion, time-frequency area, and direction parameter used throughout the analysis."},{"cited_title":"Theory of communication,","cited_arxiv_id":null,"evidence_quote":"Establishes the Gabor limit against which the DDOP's time-frequency area is compared."},{"cited_title":"Filter bank multicarrier modu- lation schemes for future mobile communications,","cited_arxiv_id":null,"evidence_quote":"Provides the comparative time-frequency localization framework and the direction-parameter interpretation used for benchmark comparisons."},{"cited_title":"A better than Nyquist pulse,","cited_arxiv_id":null,"evidence_quote":"Introduces the better-than-RRC sub-pulse whose frequency response is used for the alternative DDOP design analyzed in the paper."}],"review_version":1}