{"id":"d3d3f6b8-5c44-4692-a4ef-f81e725b3d1c","arxiv_id":"2507.15621","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A delay-Doppler pulse-shaping phase ramp shifts each Zak-OTFS user's signal into its own time-frequency slot, and simulations show multiuser uplink performance matching single-user performance without guard bands.","lead":"This paper designs a way for wireless users using Zak-OTFS modulation to transmit in separate, non-overlapping slices of time and frequency, with a base station separating them by matched filters. If the scheme works as simulated, users with very different mobility and delay profiles can share uplink spectrum without guard zones and still reach single-user performance.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Different users choose different DD periods (Table I), while Eqs. (20)-(22) apply a single Zak transform to the received superposition; these identities are invalid when periods differ, so the theoretical MUI expressions and the ~-30 dB claim are not established.","rationale":"The reader's weakest_assumption matches the load-bearing gap. I considered other candidate concerns: Theorem 1's approximate TF confinement is well-supported by the product-window argument in Appendix D; the matched-filter definition (37) is consistent with the transmit filter; and the single-user performance claim is backed by Monte Carlo simulations, though code and error bars are absent. The period compatibility issue is the one that can invalidate the theoretical scaffolding: (20)-(22) are presented as an identity following from the Zak transform, but for different user periods no such single Zak transform exists. The paper does not flag this 'common period' assumption, and it is not a matter of disagreement with consensus—it is an internal mathematical gap. The test I propose is a direct verification of (20); if it fails, the interference ratios (46) are not trustworthy, even if the time-domain implementation happens to work. Since the simulations may still be right, CONDITIONAL remains appropriate; the gap is fixable by adding per-user Zak transforms or a common period with a corrected derivation.","tokens_in":23653,"tokens_out":14132,"duration_ms":141566,"concrete_test":"For two users with the Table I periods (τ_p,1=1/15 kHz, τ_p,2=1/30 kHz), take user 2's transmitted signal x_2(t) on an AWGN channel and compute its Zak transform with user 1's period τ_p,1; show whether Z_{τ_p,1}[x_2(t)] equals x_wtx_dd,2(τ,ν). If it does not, Eq. (20) fails. Then implement the receiver as a time-domain projection ydd,q[k,l]=⟨y(t),ψ_q^{k,l}(t)⟩ with ψ_q^{k,l} from (15), for the four-user scenario of Table I, and compare the resulting per-symbol MUI and BER with the expressions (41)-(46) and Fig. 8. Agreement would rescue the claim; disagreement would show the theoretical interference measures are not the true receiver outputs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—single-user performance in a multiuser uplink without guard TF resources—rests on the DD-domain superposition model in Eqs. (20)-(22) and the interference measures (41)-(46) built on it. Eq. (7) defines the Zak transform with a fixed delay period τ_p; Eq. (2) makes every DD carrier quasi-periodic with that user's period (τ_p,s, ν_p,s=1/τ_p,s). The text states that users may choose different periods independently (Eq. (28), Table I: UT-1/2 have ν_p=15 kHz, UT-3/4 have ν_p=30 kHz). But (20) claims that one Zak transform of y(t) equals Σ_s h_s *σ x_wtx_dd,s. Each term h_s *σ x_wtx_dd,s is quasi-periodic with period τ_p,s; the sum of functions with different periods is not quasi-periodic with any single period, so no single Zak transform can satisfy (20). Consequently the matched-filter step (21), the effective channel decomposition (22), and the sampling in (24) are not defined for the multi-period setting. The formulas (40)-(46) are therefore not derived for the actual receiver, which would need either a common commensurate period (not provided) or per-user Zak transforms. If the simulations computed (41)-(46) directly rather than implementing true time-domain projections, the reported -30 dB MUI and the matched BER curves may be artifacts of the undefined algebra.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a multiuser uplink scheme based on Zak-OTFS, where each user's transmit pulse is shaped in the delay-Doppler (DD) domain by a phase factor exp(j2π(ν_s τ − ν τ_s)) so that its time-frequency (TF) occupancy is shifted to an allocated, non-overlapping TF region. Users are allowed to choose independent DD periods (τ_p,s, ν_p,s) to satisfy the crystallization condition for their own channels. The base station applies a DD-domain matched filter per user, and the paper derives expressions for the effective channel and for interference-to-useful-signal ratios, claims an O(1/K^3) leakage bound, and presents simulations for a four-user vehicular-A uplink showing that the multiuser BER of UT-1 coincides with the single-user BER without guard TF resources. The central claim is thus that single-user performance can be achieved in a multiuser Zak-OTFS uplink with independently chosen numerologies and no TF guards.","tokens_in":24031,"tokens_out":6772,"duration_ms":70498,"significance":"If the claimed result holds, it is a substantial contribution: it would give Zak-OTFS the TF-resource allocation flexibility of OFDMA while preserving the predictable DD I/O relation, and would allow users with different delay/Doppler spreads to choose independent numerologies. The TF-shifting mechanism of Theorem 1 is elegant and the single-user derivation in Appendices D and E is a strength, as are the self-contained proofs of supporting lemmas. The paper also ships a detailed simulation setup (Table I, Table II, pilot/data frame structure) that would allow reproduction. However, the analytical multiuser model in Section II is not valid for the multi-period setting, because a single Zak transform of the received superposition cannot represent users with different periods simultaneously; this undermines the derived MUI measures. The O(1/K^3) leakage bound is heuristic. The numerical results may well be correct, but the theoretical framework as written does not yet support them, so the central claim is not established in the present form.","major_comments":[{"comment":"The derivation of the DD-domain received signal assumes that a single Zak transform of y(t) yields y_dd(τ,ν) equal to the sum of per-user twisted convolutions, but users are explicitly allowed different periods (Eq. (28), Table I: UT-1/2 have ν_p=15 kHz, UT-3/4 have ν_p=30 kHz). The Zak transform in Eq. (7) is defined for a fixed delay period τ_p, and each x_wtx_dd,s is quasi-periodic with period (τ_p,s, ν_p,s). A sum of functions with different quasi-periodicities is not quasi-periodic with any single period, so Eq. (20) cannot hold as an identity in the multi-period setting. Consequently the matched-filter step (21), the effective channel decomposition (22), and the sampling on the lattice Λ_q in (24) are not well-defined for the actual receiver. The paper must either impose a common commensurate period for all users (which would restrict the independent-choice claim made in Section III) or define per-user Zak transforms y_dd,q = Z_{τ_p,q}(y) and re-derive the effective channel and interference expressions accordingly.","section":"Section II, Eqs. (20)-(24)"},{"comment":"The O(1/K^3) leakage bound is heuristic. The passage from the support and energy properties of h_eff,q,s to the statement that the DD-domain energy of h~_{q,s}^{k,l} is O(1/K^2) over an area O(1/K) neglects the aliasing arising from the infinite Dirac comb in Eq. (4) and gives no rigorous treatment of overlapping shifted supports; no constants or error terms are provided. This bound is used in the text to assert that the interference-to-useful-signal ratio is at most about 10^-6 for moderate/large frames, so it should either be proved or explicitly identified as an approximation. The numerical claims in Section IV do not rest on this bound, but the presentation should not present a heuristic estimate as a theorem.","section":"Section III-C, after Eq. (46)"},{"comment":"The paper does not state whether the leakage ratios plotted in Figs. 5-7 are computed from the analytical expression (46) together with (41), or by direct time-domain projection of the received signal onto each user's matched-filter prototype. If the former, the simulations inherit the period-mismatch problem of Eqs. (20)-(24); if the latter, the simulation methodology should be described explicitly and its consistency with (46) justified. This clarification is necessary to confirm that the reported -30 dB MUI level and the single-user BER curves are not artifacts of the undefined algebraic framework.","section":"Section IV, Figs. 5-7"}],"minor_comments":[{"comment":"The text contains a typo: \"pulse shaping fillters\" should be \"pulse shaping filters\".","section":"Section III-A"},{"comment":"The body text states \"ν_max = 815 KHz for all UTs\" in two places; the figures and the crystallization-condition calculation in Fig. 12 indicate the intended value is 815 Hz.","section":"Section IV, after Fig. 8"},{"comment":"The phrase \"Doppler spread of 6 kHz\" is used inconsistently: Eq. (28) and the discussion in Fig. 12 imply the Doppler spread is 2ν_max (so 6 kHz spread corresponds to ν_max = 3 kHz), whereas Fig. 5 shows ratios at ν_max up to 6 kHz. Please define the term \"Doppler spread\" explicitly and use it consistently.","section":"Abstract and Section IV"},{"comment":"The summation limits in Eq. (24) are written as M-1 and N-1 without subscripts; they should be M_s-1 and N_s-1 to match the per-user notation.","section":"Eq. (24)"},{"comment":"The double sum in Eq. (41) is missing summation limits; it should be over n,m ∈ Z, and the phase factor should be displayed in a way that makes the quasi-periodicity of the sampled function explicit.","section":"Eq. (41)"}],"recommendation":"major_revision","confidential_remarks":"The period-mismatch issue in Eqs. (20)-(24) is the central technical problem and must be addressed before the paper can be accepted. The fix is within scope: the authors can either adopt a common commensurate period (at the cost of restricting the independent-numerology claim) or, preferably, re-derive the multiuser matched-filter output using per-user Zak transforms and time-domain projections. The single-user derivations and the TF-shifting idea appear sound; the simulations are extensive but need to be tied to a valid analytical model. The heavy reliance on the authors' prior Zak-OTFS results is expected in this subfield and, in my view, is not a reason for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the phase-ramp DD pulse in Theorem 1 is a legitimately new and useful tool, and the single-user BER evidence is encouraging. But the multiuser theory has a load-bearing gap: Eqs. (20)–(22) apply one Zak transform to the received sum while letting each user choose its own DD period. That needs fixing before the MUI claims can be trusted.\n\nWhat I liked: Theorem 1's proof in Appendix D is clean. Shifting a Zak-OTFS carrier to an arbitrary TF region by the phase factor exp(j2π(ν_s τ − ν τ_s)) is simple and correct, and it is genuinely absent from the cited prior work including [20] and [24]. The paper also derives the effective channel and the leakage ratio in appendices rather than hand-waving. The four-user Monte Carlo results (Figs. 5, 6, 8) show leakage below −30 dB at 6 kHz Doppler and BER matching single-user without guard bands, which is an interesting result if it survives scrutiny.\n\nThe soft spots: the period-compatibility problem. Table I gives UT-1/2 ν_p = 15 kHz and UT-3/4 ν_p = 30 kHz. Eq. (7) defines the Zak transform with one fixed τ_p. The received signal is a sum of contributions each quasi-periodic with its own τ_p,s. The sum is not quasi-periodic with any single period, so Eq. (20) and the matched-filter algebra that follows are not well-defined as written. This is not a cosmetic issue; (41)–(46) are built on it. The O(1/K^3) leakage bound in Section III-C is also heuristic. The simulations may well be doing true time-domain projections, in which case the BER curves can stand on their own, but the theoretical framework has to be reworked (e.g., a common commensurate period or per-user Zak transforms).\n\nThe paper leans on the same group's earlier Zak-OTFS work, but that's acceptable because the new pieces are derived independently. No code or data is shipped, and there are no error bars, so the numerical claims are hard to verify.\n\nWho it's for: people working on OTFS-based multiple access and 6G waveform design. I'd send it to a serious referee, but the referee should push hard on the period issue. The idea is promising and the flaw looks fixable, but the current version is not trustworthy in its analytical claims.","headline":"The TF-shifting DD pulse in Theorem 1 is a genuinely new and useful tool, but the multiuser superposition analysis assumes a common Zak period that the paper itself lets each user choose differently—this gap needs to be fixed before the MUI claims can be trusted.","tokens_in":24587,"tokens_out":4398,"would_cite":true,"duration_ms":47088,"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":"A phase-twisted delay-Doppler pulse gives each Zak-OTFS user its own time-frequency tile, so a four-user uplink can match single-user bit error rates without any guard bands.","keywords":["Zak-OTFS","multiuser uplink","delay-Doppler domain","pulse shaping","time-frequency resource allocation","doubly-spread channels","matched filtering","orthogonal multiple access"],"falsifier":"Simulate two users with identical TF allocations but different delay and Doppler periods, then compare the measured leakage ratio $I_{q,s}/S_{s,s}$ against the paper's formula; if the measured leakage departs from the predicted value once the periods differ, the shared-superposition equation (20) needs a per-user Zak basis. A simpler check is to numerically compute the Zak transform of a sum of two signals with different periods and verify the energy relations used in (42)-(45).","tokens_in":1876,"feed_emoji":"📡","tokens_out":2408,"duration_ms":93537,"temperature":0.7,"pith_summary":"The paper tries to establish that the predictable input-output relation of Zak-OTFS can be kept in a multiuser uplink by shaping each user's delay-Doppler transmit pulse with a phase twist that shifts the signal's time-frequency location. If true, users with very different mobility can share spectrum without forcing everyone onto one subcarrier spacing, and without paying for guard bands between adjacent allocations. The key quantitative claim is that the interference-to-useful-signal ratio stays below about -30 dB even at 6 kHz Doppler spread, so at practical signal-to-noise ratios the multiuser interference is buried under noise. The result matters because it removes a standard cost of orthogonal multiple access and keeps each user's channel acquisition predictable through its own pilot.","feed_headline":"No guard bands: four-user Zak-OTFS uplink matches single-user BER","feed_subtitle":"A phase twist in the delay-Doppler pulse lets each user pick its own time-frequency tile while interference stays below -30 dB.","key_machinery":"The central object is the DD-domain transmit pulse-shaping filter carrying the phase factor $e^{j2\\pi(\\nu_s\\tau - \\nu\\tau_s)}$, applied through twisted convolution to quasi-periodic Dirac-delta DD pulses that carry information symbols. Twisted convolution is the DD-domain operation that combines a filter with a quasi-periodic DD function while preserving quasi-periodicity, and it is what makes the Zak-transform input-output relation associative and tractable. The receiver uses the matched filter in (37), the conjugate time- and frequency-reversed version of the shifted pulse, and Theorem 2 expresses the effective channel as a sum over paths of two separated integrals, one governing frequency-domain overlap and one governing time-domain overlap. That separation is what turns the no-guard-band claim into a calculable interference ratio.","core_discovery":"The paper's central claim is that a transmit delay-Doppler filter of the form $\\tilde{w}_{tx,s}(\\tau,\\nu) = w_{tx,s}(\\tau,\\nu)e^{j2\\pi(\\nu_s\\tau - \\nu\\tau_s)}$ shifts the TF occupancy of a Zak-OTFS signal to approximately $[\\tau_s - T_s/2, \\tau_s + T_s/2] \\times [\\nu_s - B_s/2, \\nu_s + B_s/2]$, while leaving each user free to choose its own delay and Doppler periods. At the base station, the received superposition is match-filtered separately for each user, and the paper derives the effective continuous DD-domain channel between every transmit-receive pair as a sum over propagation paths of two overlap integrals. For an ideal channel the leakage from user $s$ into user $q$'s matched-filter output is bounded by $O(1/K^3)$ with $K = \\min(M_s N_s, M_q N_q)$. In simulations on a vehicular-A doubly-spread channel the leakage ratio stays below about $-30$ dB at 6 kHz Doppler spread, and the bit error rate and channel-estimation NMSE for UT-1 in a four-user uplink coincide with the single-user curves, even without guard resources.","pith_inferences":["A testable extension is to push the frame size down: the paper's $O(1/K^3)$ bound implies that for frames with $M_s N_s$ below about a hundred, leakage rises above the -60 dB ideal-channel level, so the no-guard claim should be rechecked in the short-frame regime.","The same phase-twist shaping could be reused in a downlink broadcast or relay setting, where the base station transmits a superposition and each user's own matched filter extracts its stream without joint decoding.","Because the residual interference is nearly uniform in the DD domain, per-carrier power allocation or simple interference cancellation at the BS could push the operating point even further below noise-limited performance."],"forward_implications":["Users with very different Doppler spreads can be assigned different delay and Doppler periods, so the numerology of one fast-moving user no longer dictates the subcarrier spacing of all users.","Because the interference-to-useful-signal ratio is below about -30 dB at 6 kHz Doppler spread, at typical SNRs below 30 dB the BER and channel-estimation NMSE are limited by noise rather than by multiuser interference.","Adjacent time-frequency allocations need no guard bands: interference that is localized at TF boundaries spreads almost uniformly across DD carriers, so no single carrier is hit disproportionately hard.","The rectangular per-user TF allocations line up with 3GPP-style resource blocks, allowing mixed terrestrial and non-terrestrial users to share one uplink frame.","Each user's crystallization condition can be satisfied independently, preserving predictable single-pilot channel acquisition for every user."],"supporting_citations":[{"why":"Supplies the Zak-OTFS framework, twisted convolution, quasi-periodic DD pulses, and the single-user predictable I/O relation that the multiuser derivation builds on.","marker":"[12]"},{"why":"Provides the embedded pilot-data frame structure and the channel estimation method used to estimate the effective discrete DD channel taps.","marker":"[15]"},{"why":"Defines the vehicular-A power-delay profile that the numerical BER, NMSE, and leakage simulations use as the doubly-spread channel.","marker":"[19]"},{"why":"Supplies the pulse-shaping relations, including the time/bandwidth-limited filter design and the twisted-convolution-to-time-domain conversion used in the paper's transmitter derivation.","marker":"[20]"},{"why":"Justifies the receive matched filter $w_{rx,q}(\\tau,\\nu) = w_{tx,q}^*(-\\tau,-\\nu)e^{j2\\pi\\nu\\tau}$ as the SNR-optimal Zak-OTFS receiver.","marker":"[23]"},{"why":"Provides the LSMR equalizer used to detect 4-QAM symbols after DD-domain channel estimation in the BER simulations.","marker":"[25]"},{"why":"Supplies the continuous delay-Doppler channel spreading representation that the superposition model in (19) adopts.","marker":"[21]"}],"fun_headline_variants":["Zak-OTFS uplink: no guard bands, multiuser BER equals single-user","Phase-twisted pulses let Zak-OTFS users share spectrum without guard bands","Four users, zero guard bands: Zak-OTFS matches single-user performance","Zak-OTFS multiuser uplink: predictable channel, no guard-band penalty","Interference below -30 dB: Zak-OTFS uplink scales to multiple users"],"cache_read_input_tokens":26496,"weakest_assumption_plain":"The receiver-side derivation assumes that one Zak transform of the received signal is enough to produce a single DD-domain superposition on which every user's matched filter can act, even though each user is allowed a different delay and Doppler period; the paper never specifies a common period for that single transform.","fun_headline_variants_meta":{"raw":{"variants":["Zak-OTFS uplink: no guard bands, multiuser BER equals single-user","Phase-twisted pulses let Zak-OTFS users share spectrum without guard bands","Four users, zero guard bands: Zak-OTFS matches single-user performance","Zak-OTFS multiuser uplink: predictable channel, no guard-band penalty","Interference below -30 dB: Zak-OTFS uplink scales to multiple users"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000303,"raw_usage":{"total_tokens":1824,"prompt_tokens":1107,"completion_tokens":717,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":723,"completion_tokens_details":{"reasoning_tokens":607}},"tokens_in":723,"tokens_out":717,"duration_ms":6858,"temperature":1.0,"reasoning_tokens":607,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:28:59.785067+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate two users with identical TF allocations but different delay and Doppler periods, then compare the measured leakage ratio $I_{q,s}/S_{s,s}$ against the paper's formula; if the measured leakage departs from the predicted value once the periods differ, the shared-superposition equation (20) needs a per-user Zak basis. A simpler check is to numerically compute the Zak transform of a sum of two signals with different periods and verify the energy relations used in (42)-(45).","supporting_citations":[{"cited_title":"OTFS Modulation: Theory and Applications,","cited_arxiv_id":null,"evidence_quote":"Supplies the Zak-OTFS framework, twisted convolution, quasi-periodic DD pulses, and the single-user predictable I/O relation that the multiuser derivation builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the embedded pilot-data frame structure and the channel estimation method used to estimate the effective discrete DD channel taps."},{"cited_title":"Guidelines for evaluation of radio transmission tech- nologies for IMT-2000,","cited_arxiv_id":null,"evidence_quote":"Defines the vehicular-A power-delay profile that the numerical BER, NMSE, and leakage simulations use as the doubly-spread channel."},{"cited_title":"Optimal Zak- OTFS Receiver and Its Relation to the Radar Matched Filter,","cited_arxiv_id":null,"evidence_quote":"Justifies the receive matched filter $w_{rx,q}(\\tau,\\nu) = w_{tx,q}^*(-\\tau,-\\nu)e^{j2\\pi\\nu\\tau}$ as the SNR-optimal Zak-OTFS receiver."},{"cited_title":"Low-Complexity Symbol Detection and Interference Cancellation for OTFS System,","cited_arxiv_id":null,"evidence_quote":"Provides the LSMR equalizer used to detect 4-QAM symbols after DD-domain channel estimation in the BER simulations."},{"cited_title":"Characterization of Randomly Time-Variant Linear Chan- nels,","cited_arxiv_id":null,"evidence_quote":"Supplies the continuous delay-Doppler channel spreading representation that the superposition model in (19) adopts."}],"review_version":1}