{"id":"0a64a8aa-dd32-41d0-ba09-b13f8f7580ce","arxiv_id":"2504.18887","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A set of closed-form expressions for Zak-OTFS I/O relation and noise covariance is derived for sinc and Gaussian filters under identical, matched, and channel matched filtering, replacing numerical integration.","lead":"This paper derives closed-form mathematical expressions for the end-to-end input-output relation and noise statistics in Zak-OTFS modulation, for sinc and Gaussian pulse shaping under three receiver filter configurations. The expressions replace slow numerical integration, cutting simulation time from tens of minutes to fractions of a second, and are used to compare bit error rates of the filter schemes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sinc identical-filtering approximation (Theorem 1) rests on an unproven and only sparsely validated neglect of the beta_i integral in Eq. (92); accuracy across the crystalline regime is not established.","rationale":"The reader's weakest_assumption correctly identifies the Appendix A approximation as the most load-bearing premise. I agree that this is the soft spot: it is the only place where 'exact' is replaced by 'approximate' in the central claim, and the accuracy evidence is a single scenario. However, the paper is transparent about the approximation, labels it as approximate, and provides a numerical check for the tested setup. A first-order estimate of the dropped term (relative order tau_max/T plus a log(BT)/(B^2 T) tail) suggests the error is small in the crystalline regime, so the concern is about missing guarantees rather than a demonstrated failure. For that reason, I do not think the reader's ACCEPT verdict should be changed; the paper could be strengthened by a broader validation or an error bound, but the current evidence supports acceptance for the stated operating regime. The concrete test proposed would settle whether the approximation remains accurate near the boundaries of the crystalline regime and for smaller frame sizes.","tokens_in":28362,"tokens_out":22820,"duration_ms":209668,"concrete_test":"Numerically integrate the exact expression in Eq. (28) for heff(tau,nu) and compare with the approximate closed-form Eq. (29) over a grid covering the crystalline regime: M,N in {8,12,24,48}, nu_max/nu_p in {0.05,0.1,0.25,0.4}, using Veh-A and a delay-scaled profile with tau_max/tau_p up to 0.4. Report the normalized Frobenius-norm error of the channel matrix H and the BER gap at 10^-3 with MMSE detection. If the error exceeds about 5% or 0.5 dB in any tested corner, the claim that the approximation is accurate needs to be qualified to the validated parameter range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim for sinc identical filtering is that the approximate closed-form effective channel in Theorem 1 is accurate. This rests on Appendix A, Eq. (92), where I_i is written as T sinc(T nu'_i) integral alpha_i dx - cos(pi T nu'_i) integral beta_i dx, and the beta_i term is dropped with the assertion that it is 'much small' for large M,N and the crystalline regime. No bound or analytical argument is provided, and beta_i contains |x| sinc(pi |x| nu'_i), which can be non-negligible when delay/Doppler spreads approach the crystalline boundary. The only numerical validation is Fig. 2, for one Veh-A realization with M=12, N=14, nu_max=815 Hz, BPSK and 8-QAM. The same approximation underlies all sinc identical-filtering BER results (Figs. 3, 6, 7), so a failure outside the tested regime would directly affect the paper's comparative conclusions. A further minor issue is that Eq. (91) writes sinc(pi |x| nu'_i) where the expansion implies sinc(|x| nu'_i), a notation inconsistency that does not alter the final dropped term but signals carelessness in the appendix.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives discrete delay-Doppler-domain closed-form expressions for the end-to-end I/O relation and noise covariance in Zak-OTFS. Sinc and Gaussian transmit filters are considered, together with identical, matched, and channel-matched receive filtering. For all cases except sinc identical filtering, the authors present exact closed-form expressions. For sinc identical filtering, they derive approximate closed-form expressions by neglecting a term in an integral in Appendix A and then use these expressions in BER simulations. The paper also presents BER and SNR comparisons among the filtering schemes, reports substantial simulation speedups, and validates the sinc identical approximation against numerical integration in one Vehicular-A setting.","tokens_in":28627,"tokens_out":18869,"duration_ms":185727,"significance":"If the results are correct, the paper provides a useful toolbox: it replaces numerically evaluated twisted-convolution integrals with compact formulas for most of the considered Zak-OTFS filter configurations, which should accelerate performance evaluation and simplify the derivation of detection and estimation algorithms. The exact formulas for matched and channel-matched filtering are derived carefully and from first principles, with detailed appendices, and the Gaussian matched-filtering result is properly credited to prior work. The main caveat is that the approximate sinc identical-filtering case rests on a heuristic term-drop that is not analytically quantified and is validated numerically only in a single configuration; since the comparative claims in Figs. 3, 6, and 7 use this approximation, the strength of those claims is currently tied to the validity of that unquantified approximation.","major_comments":[{"comment":"The central approximation of Theorem 1 is the neglect of the β_i integral in Eq. (92). The paper states that this term is 'much small' for large M,N and the crystalline regime, but no bound or quantitative condition is provided. For ν'_i not close to zero, the retained term T sinc(Tν'_i)∫α_i(x)dx and the neglected term cos(πTν'_i)∫β_i(x)dx can have comparable magnitudes (both are roughly of order 1/(B|ν'_i|) before cancellations), so the assertion is not self-evident. Please supply an analytical error bound (for example, showing that the ratio of the neglected term to the retained term is O(1/(MN)) or controlled by a stated small parameter in the crystalline regime), or provide a systematic numerical comparison between the exact integral in Eq. (28) and the closed-form Eq. (29) over a range of fractional delays, Dopplers, and lattice sizes. Because Theorem 1 and the associated noise-covariance approximation underpin the sinc identical-filtering BER results in Figs. 3, 6, and 7, this issue is load-bearing for the paper's approximate closed-form claim.","section":"Appendix A, Eqs. (92)-(96)"},{"comment":"The noise covariance for sinc identical filtering is approximated as an identity matrix by replacing the finite-interval integral in Eq. (37) with the infinite-interval orthogonality result in Eq. (38). This ignores finite-T leakage of the sinc products. The approximation is plausible for large BT, but no error bound or leakage estimate is given. Since the BER simulations and any noise-whitening procedure developed from these expressions depend on this approximate covariance, please either bound the off-diagonal terms or demonstrate numerically that the approximation error is negligible over a wider range of M,N and delay/Doppler spreads than the single setting shown in Fig. 2.","section":"Section III-B, Eqs. (37)-(39)"},{"comment":"The numerical validation of the sinc identical-filtering approximation is limited to one Veh-A realization with M=12, N=14, νmax=815 Hz, and BPSK/8-QAM constellations. The crystalline regime is much broader than this single point, and the paper does not state the intended validity region of the approximation or show how the error behaves as νmax/νp and τmax/τp approach the crystalline boundary. Please add validation at additional operating points, or explicitly restrict the accuracy claim to the tested regime and state that the comparative conclusions for sinc identical filtering in Figs. 3, 6, and 7 are demonstrated only there.","section":"Section VI, Fig. 2"}],"minor_comments":[{"comment":"There appears to be a notation inconsistency in the identity: the last term writes |x|sinc(π|x|ν'_i), whereas the expansion of (T-|x|)sinc((T-|x|)ν'_i) would require |x|sinc(|x|ν'_i) under the standard normalized-sinc convention. Since this term is dropped, the final result is unaffected, but the convention should be stated or corrected.","section":"Appendix A, Eq. (91)"},{"comment":"For Gaussian filtering with time/bandwidth expansion (B'=1.12B, T'=1.25T), the paper does not explain how the derived formulas should be modified. Please clarify whether B,T in Eqs. (42)-(48), (62)-(63), and (79)-(85) are simply replaced by B',T', or whether α_τ and α_ν are rescaled.","section":"Section II and Section VI, Fig. 4"},{"comment":"The infinite sums over q_1,q_2 are truncated at ±20, with the statement that this range is 'found to be adequate'. Please provide a convergence criterion or a short numerical demonstration of convergence, since the required truncation range may vary with system parameters.","section":"Theorems 3 and Eq. (63), (85)"},{"comment":"The sentence 'the contribution of the integral of β_i(x) ... is much small' contains a grammatical error; it should read 'much smaller'. There is also a typo in the author affiliation block ('Chockalin gam').","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the exact derivations are valuable. The main risk is the unquantified approximation in Appendix A; I would be willing to accept after the authors add a bound or a substantially broader validation of that approximation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth reading: Zak-OTFS is a niche but active area, and this paper fills concrete gaps. Only the Gaussian matched-filtering closed form existed in the literature; here they provide exact closed forms for sinc and Gaussian under identical, matched, and channel matched filtering, except for sinc identical filtering which is approximate. That is a real contribution. The derivations are detailed enough to check, and the expressions are practically useful: the reported speedups (noise covariance in 0.23 s versus 26 min) are entirely plausible for replacing nested numerical integrals.\n\nThe strongest parts are the exact results for Gaussian and for sinc matched/channel matched. Appendix A is the load-bearing weak point: the sinc identical filtering approximation drops the beta_i integral in Eq. (92) with an \"it is observed\" assertion and no bound. The stress-test note is fair. The split into alpha and beta terms is not controlled; beta_i contains |x| sinc(pi |x| nu'_i), which can grow when delay/Doppler spreads approach the crystalline boundary. The authors validate only one Veh-A realization (M=12, N=14, nu_max=815 Hz, BPSK/8-QAM). That is enough to show the approximation works somewhere, not enough to establish where it fails. Since that approximation feeds the sinc identical BER curves, the comparative claims for identical filtering at high Doppler rest on a shakier foundation than the rest of the paper. A serious referee should ask for either a bound or a second validation at a different operating point. The notation slip in Eq. (91) (sinc(pi |x| nu'_i) versus sinc(|x| nu'_i)) looks like a harmless typo but should be cleaned up.\n\nEverything else holds up. The noise covariance derivations come from first principles, the truncation of infinite q-sums is standard and checked by heuristics, and the citation pattern is honest: the Gaussian matched result is explicitly credited to [29]. No shipped code, but this paper does not need it.\n\nBottom line: a well-executed gap-filler, not a paradigm shift. The approximate case needs more care but does not sink the paper. Send it to peer review with a request for stronger validation of Theorem 1.","headline":"A solid, gap-filling paper that derives exact closed-form Zak-OTFS I/O and noise covariance for most filtering configurations, with one approximate case whose validation is thinner than the authors' wording suggests.","tokens_in":29127,"tokens_out":2071,"would_cite":true,"duration_ms":21557,"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":"Zak-OTFS delay-Doppler I/O relations become closed-form sums for sinc and Gaussian filters.","keywords":["Zak-OTFS","delay-Doppler domain","closed-form I/O relation","noise covariance","twisted convolution","sinc filter","Gaussian filter","matched filtering"],"falsifier":"Evaluate the exact tap in Eq. (28) by numerical quadrature and compare it with the approximate formula (29) for a channel whose delay spread approaches $\\tau_p$ or whose Doppler spread approaches $\\nu_p/2$, at small frame sizes such as $M=N=8$; a tap-level or BER deviation well beyond the Fig. 2 agreement would show the approximation's domain is narrower than claimed.","tokens_in":28152,"feed_emoji":"📡","tokens_out":11957,"duration_ms":102974,"temperature":0.7,"pith_summary":"Zak-OTFS transceivers shape information in the delay-Doppler (DD) domain with filters whose cascade is a twisted convolution, so the end-to-end input-output relation is a stack of integrals that previously had to be evaluated numerically. This paper derives closed-form expressions for the effective discrete DD channel taps $h_{\\mathrm{eff}}[k,l]$ and the filtered-noise covariance when the transmit filter is sinc or Gaussian and the receive filter is identical, matched, or channel-matched. The formulas are exact for every combination except sinc with identical filtering, where the paper supplies an approximate closed form and demonstrates accuracy on a Vehicular-A channel with fractional delay-Doppler spreads. The payoff is that system matrices and noise statistics for Zak-OTFS can be assembled directly from sums of elementary functions, cutting simulation times from minutes to seconds and enabling noise whitening for detection.","feed_headline":"Closed forms replace numerical integrals in Zak-OTFS links","feed_subtitle":"Except for one sinc case, I/O and noise covariance become exact sums that speed up simulations by orders of magnitude.","key_machinery":"The central object is the twisted convolution, a variant of convolution between two DD-domain functions with an extra phase factor $e^{j2\\pi \\nu'(\\tau-\\tau')}$; the whole transceiver filter cascade collapses into the effective channel $h_{\\mathrm{eff}}(\\tau,\\nu)=w_{\\mathrm{rx}} *_{\\sigma} h_{\\mathrm{phy}} *_{\\sigma} w_{\\mathrm{tx}}$. The argument exploits the separable form $w_{\\mathrm{tx}}(\\tau,\\nu)=w_1(\\tau)w_2(\\nu)$ and the fact that for sinc and Gaussian filters the inner integrals are Fourier transforms of windowed sincs or Gaussians. Those transforms turn the multiple integrals into products such as $\\frac{B-|f|}{B^2}\\operatorname{sinc}((B-|f|)\\Delta\\tau)$ or, for Gaussian filters, exponentials obtained by completing the square; sampling on the lattice $\\Lambda_{\\mathrm{dd}}$ yields the discrete taps. For sinc identical filtering, the derivation additionally drops the $\\beta_i(x)$ correction in Eq. (92), arguing it is small for large $M,N$ in the crystalline regime, which produces the approximate closed form in Theorem 1.","core_discovery":"The paper's central claim is that the cascade $w_{\\mathrm{rx}} *_{\\sigma} h_{\\mathrm{phy}} *_{\\sigma} w_{\\mathrm{tx}}$ can be evaluated in closed form on the Zak-OTFS information lattice for sinc and Gaussian transmit filters and for three receive-filter choices. For identical, matched, and channel-matched filtering, the effective channel taps $h_{\\mathrm{eff}}[k,l]$ and the covariance $\\mathbb{E}[n_{\\mathrm{dd}}[k_1,l_1]n_{\\mathrm{dd}}^*[k_2,l_2]]$ reduce to finite sums of windowed sinc products or Gaussians per path (or per path pair), with no numerical integration over the twisted-convolution kernel. The exception is sinc with identical filtering, where a correction term is dropped in Appendix A to obtain an approximate closed form that the paper validates against the exact numerical expression. Using these expressions, the paper evaluates bit error performance for BPSK and 8-QAM with MMSE detection on a Vehicular-A channel with fractional DDs and reports three performance findings: matched filtering essentially ties identical filtering, channel-matched filtering gives the best BER, and sinc generally beats Gaussian, while Gaussian improves with time and bandwidth expansion.","pith_inferences":["The same Fourier-transform product technique should extend to other separable DD filters, including root raised cosine, although the paper leaves that case open.","The sinc-identical approximation is validated at one operating point; stressing it with delay spread near $\\tau_p$, Doppler near $\\nu_p/2$, or very small $M,N$ would likely reveal where the dropped correction term matters.","For sinc identical filtering the approximate noise covariance collapses to $N_0$ times the identity at large $M,N$, which the paper does not exploit; that configuration may need no whitening before equalization.","Channel-matched filtering sums over path pairs and therefore costs more nonzero taps than identical or matched filtering; the roughly 1 dB gain for sinc and 5-6 dB gain for Gaussian at small frame sizes leaves a complexity-performance tradeoff that the paper does not quantify."],"forward_implications":["Assembling the Zak-OTFS channel matrix and noise covariance directly from the closed-form sums removes the numerical-integration bottleneck; the paper reports noise-covariance computation falling from 26 minutes to 0.23 s and per-realization channel-tap computation from 92 minutes to 0.42 s in the tested setting.","Channel-matched filtering, which maximizes SNR by matching to the channel-Tx cascade, delivers the best BER of the three schemes, while matched and identical filtering perform nearly the same, so it is the benchmark receiver for Zak-OTFS.","Sinc filtering outperforms Gaussian filtering and confines the transmitted spectrum to $(-B/2,B/2)$, whereas the Gaussian leaks about 1 percent of energy outside that band; expanding the Gaussian to $B'=1.12B$, $T'=1.25T$ buys about 2 dB at high SNR.","Because the closed-form noise covariances permit noise whitening before detection, maximum-likelihood detection becomes implementable on small frames; the paper shows channel-matched Gaussian filtering gains 5-6 dB over the other schemes at $10^{-4}$ BER for $M=N=2$.","Since the derivations place no integrality constraint on the path delays or Dopplers, the closed forms apply to fractional delay-Doppler channels rather than only lattice-aligned ones."],"supporting_citations":[{"why":"Defines the Zak-OTFS quasi-periodic DD model and the twisted-convolution machinery that the closed forms evaluate.","marker":"[24]"},{"why":"Introduces identical filtering with sinc and RRC DD filters in Zak-OTFS without closed forms; this paper supplies them.","marker":"[25]"},{"why":"Establishes matched filtering for Zak-OTFS with sinc and RRC filters; the new sinc matched-filter closed forms complete that setup.","marker":"[27]"},{"why":"Proves the channel-matched receive filter maximizes SNR, which motivates the third filtering scheme and the performance benchmark.","marker":"[28]"},{"why":"Provides the Gaussian matched-filtering closed forms for $h_{\\mathrm{eff}}$ and noise covariance that this paper reproduces and extends to other schemes.","marker":"[29]"},{"why":"Supplies the Vehicular-A power delay profile used to validate the approximate sinc-identical formulas and compare BER.","marker":"[38]"}],"fun_headline_variants":["Zak-OTFS I/O closed form, except one sinc case","Exact closed forms for Zak-OTFS I/O and noise","Closed-form I/O replaces integration in Zak-OTFS","Zak-OTFS: closed form from sinc and Gaussian filters","Sinc/Gaussian filters give closed-form Zak-OTFS I/O"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the Appendix A approximation that the $\\beta_i(x)$ correction is negligible in the sinc-identical derivation for large frames in the crystalline regime (delay spread far below $\\tau_p$ and Doppler spread far below $\\nu_p/2$); the paper verifies this only for the tested Vehicular-A settings.","fun_headline_variants_meta":{"raw":{"variants":["Zak-OTFS I/O closed form, except one sinc case","Exact closed forms for Zak-OTFS I/O and noise","Closed-form I/O replaces integration in Zak-OTFS","Zak-OTFS: closed form from sinc and Gaussian filters","Sinc/Gaussian filters give closed-form Zak-OTFS I/O"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1795,"prompt_tokens":1099,"completion_tokens":696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":715,"completion_tokens_details":{"reasoning_tokens":605}},"tokens_in":715,"tokens_out":696,"duration_ms":6460,"temperature":1.0,"reasoning_tokens":605,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:07:16.415943+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the exact tap in Eq. (28) by numerical quadrature and compare it with the approximate formula (29) for a channel whose delay spread approaches $\\tau_p$ or whose Doppler spread approaches $\\nu_p/2$, at small frame sizes such as $M=N=8$; a tap-level or BER deviation well beyond the Fig. 2 agreement would show the approximation's domain is narrower than claimed.","supporting_citations":[{"cited_title":"OTFS − a mathematical foundation for communication and radar sensing in the delay-Doppler domain,","cited_arxiv_id":null,"evidence_quote":"Defines the Zak-OTFS quasi-periodic DD model and the twisted-convolution machinery that the closed forms evaluate."},{"cited_title":"OTFS − predictability in the delay-Doppler domain and its value to communication and radar sensing,","cited_arxiv_id":null,"evidence_quote":"Introduces identical filtering with sinc and RRC DD filters in Zak-OTFS without closed forms; this paper supplies them."},{"cited_title":"Guidelines for evaluation of radio tran smission tech- nologies for IMT-2000,","cited_arxiv_id":null,"evidence_quote":"Supplies the Vehicular-A power delay profile used to validate the approximate sinc-identical formulas and compare BER."}],"review_version":1}