{"id":"195ad714-33ac-4c6f-81c4-69821b192e62","arxiv_id":"2506.23045","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Equalizing Zak-OTFS in the frequency domain exploits the banded channel structure, cutting complexity from O(M^3 N^3) to O(M^2 N^2).","lead":"A new receiver algorithm for the wireless modulation scheme Zak-OTFS cuts the cost of equalization from cubic to quadratic in frame size by exploiting the banded structure of the frequency-domain channel. The result could make high-mobility wireless links more practical.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's O(M^2 N^2) complexity claim omits the band-width factor b=4lmax+1; under the paper's own crystallization constraint this factor grows as Θ(N), so the FD equalizer is really Ω(M^2 N^3) once N is allowed to scale.","rationale":"The reader correctly identified band truncation leakage as a weak assumption, and the paper's own admission that out-of-band energy is forced to zero without a quantitative error bound is a genuine limitation. However, the more load-bearing problem for the headline complexity claim is that the band-width factor b is not independent of N. The paper states the complexity as O(b M^2 N^2) in Section IV but drops the b factor in the abstract and conclusion, claiming O(M^2 N^2). Because tau_p = 1/nu_p = T/N and the crystallization condition requires tau_p > tau_max, any scaling with N going to infinity forces T = Θ(N) and hence lmax = Θ(N), so b = Θ(N). The complexity in N is therefore cubic, not quadratic, for the advertised asymptotic regime. This concern does not require simulation and follows from the paper's own constraints. The method may still be practical in finite regimes, such as the Veh-A example with lmax = 3, and the derivation of Theorem 1 appears sound. The appropriate outcome is a conditional acceptance: either revise the abstract and conclusion to state the complexity as O(b M^2 N^2) with explicit b = Θ(N) scaling, or specify a regime with fixed T and bounded N where O(M^2 N^2) is meaningful. The proposed scaling check would settle the matter.","tokens_in":8557,"tokens_out":24106,"duration_ms":259841,"concrete_test":"Analytic scaling check: fix the Veh-A channel parameters (tau_max = 2.51 us, nu_max = 815 Hz) and a fixed Doppler period nu_p = 30 kHz, so T = N/nu_p. Compute b = 4 ceil(T nu_max) + 1 for N = 37, 100, 300, 1000 and verify the crystallization constraint tau_p = T/N > tau_max. If b grows linearly with N, then substituting b = Θ(N) into the paper's complexity expression O(b M^2 N^2) yields Ω(M^2 N^3), contradicting the abstract's O(M^2 N^2). This single calculation settles whether the complexity claim is stated with the correct asymptotic dependence.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim, as stated in the abstract and conclusion, is that FD equalization reduces complexity from O(M^3 N^3) to O(M^2 N^2). Section IV is more careful: the complexity is O(b M^2 N^2), where b = 4 lmax + 1 and lmax = O(ceil(T nu_max)). The paper treats b as a constant independent of M and N. This is not consistent with the Zak-OTFS design constraints used elsewhere in the paper. The delay and Doppler periods satisfy tau_p nu_p = 1, with nu_p = N/T and tau_p = 1/nu_p = T/N. The crystallization condition requires tau_p > tau_max, hence T > N tau_max. Since the Doppler period constraint also gives T < N/(2 nu_max), T = Θ(N) for any unbounded sequence of frames with fixed channel parameters. Therefore lmax = ceil(T nu_max) = Θ(N), so b = Θ(N). Substituting into the paper's own complexity expression gives Ω(M^2 N^3), not O(M^2 N^2). The advertised saving relative to DD-domain inversion is a factor M, rather than the claimed factor MN. If, alternatively, T is kept fixed so that b is constant, then N is bounded above by T/tau_max and the asymptotic statement in N is vacuous. This is load-bearing because the headline result is specifically the complexity reduction, and it follows from the paper's own parameters rather than from an external assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to perform linear equalization of Zak-OTFS in the frequency domain (FD). It derives an exact FD input-output relation from the DD-domain effective channel via the inverse discrete frequency Zak transform (Theorem 1), shows that the FD channel matrix is banded with width determined by lmax = O(ceil(T*nu_max)), and proposes a low-complexity conversion of the estimated DD channel to the FD channel. The LMMSE equalizer then inverts a banded matrix, which is claimed to have complexity O(bM^2N^2) with b = 4*lmax+1, compared to O(M^3N^3) for DD-domain equalization. Simulations with a Veh-A channel show that the proposed FD equalizer achieves the same uncoded 4-QAM BER as DD-domain equalization.","tokens_in":8847,"tokens_out":10061,"duration_ms":113492,"significance":"The derivation of the FD I/O relation in Theorem 1 is a clean, self-contained contribution, and the observation that the FD channel matrix is banded is a genuine structural insight that could enable practical equalization for Zak-OTFS. The conversion formula in (27) is novel and potentially useful. However, the headline complexity reduction is overstated as O(M^2N^2) without the band-width factor b, and the performance parity is demonstrated in only a single simulation scenario. If the complexity claim is corrected and the out-of-band leakage issue is addressed, the paper would be a solid contribution to the OTFS literature.","major_comments":[{"comment":"The abstract and conclusion claim that FD equalization reduces complexity from O(M^3N^3) to O(M^2N^2), but Section IV gives the complexity as O(bM^2N^2) with b = 4*lmax+1, where lmax = O(ceil(T*nu_max)). Under the crystallization conditions stated in Section II-C, namely tau_p = T/N > tau_max and nu_p = N/T > nu_max, we have N*tau_max < T < N/nu_max for fixed channel parameters. Hence T = Theta(N) for an unbounded frame size, so lmax = ceil(T*nu_max) = Theta(N) and b = Theta(N). Substituting into the paper's own complexity expression gives Omega(M^2N^3) rather than O(M^2N^2) once N is allowed to scale. The advertised saving relative to DD-domain inversion is therefore a factor M, not MN, and the abstract and conclusion must be corrected, or the scaling regime in which b is constant must be explicitly stated.","section":"Abstract, Section II-C, Section IV"},{"comment":"The claim that FD equalization achieves the same performance as DD equalization rests on a single simulation run with the Veh-A channel (Fig. 3) and on the approximation that all energy outside the band b is negligible and can be zeroed. This approximation is load-bearing for both the BER parity and the complexity, because significant out-of-band energy would force a larger b and erode the complexity gain. The paper should provide quantitative evidence on the leakage, for example the fraction of channel energy lying outside the band as a function of channel spread and pulse shape, or demonstrate performance parity across a range of channel conditions. Without such support, the generality of the 'same performance' claim is not established.","section":"Section IV, Section V"}],"minor_comments":[{"comment":"The word 'comlete' in the last sentence should be 'complete'.","section":"Section II-C"},{"comment":"It would be helpful to state explicitly that the sum over k' is modulo MN, relying on the MN-periodicity of h[k,l], rather than leaving the index range implicit.","section":"Section III, equation (27)"},{"comment":"The text states MN = BT, but with the given parameters M=31, N=37, B=930 kHz, and T=1.23 ms, BT = 1143.9 while MN = 1147; please clarify the rounding or adjust the values for consistency.","section":"Section V"},{"comment":"The abbreviation 'FD' is used both for 'frequency domain' and for 'FD realization'; consider using 'frequency-domain' in full in places where ambiguity might arise, or define the two uses explicitly.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The paper's core derivation appears sound, and the FD I/O relation is a nice result. The main concern is the asymptotic complexity claim, which is overstated because the band width b = 4*lmax+1 grows linearly with N under the crystallization constraints imposed by the channel spreads. This is a headline issue that must be fixed before publication. The performance parity is also only supported by one simulation, so the authors should either add more simulation scenarios or an analysis of the out-of-band leakage. The paper is concise and well written, but these issues warrant a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the useful thing: Theorem 1 gives a clean frequency-domain I/O relation for Zak-OTFS and a direct way (27) to convert an estimated DD channel into the FD channel. That is genuinely new as far as I know, and the derivation is straightforward once you see it. The banded structure of the FD channel is real and the complexity reduction to banded inversion is a practical idea.\n\nThe paper is honest about its limits: perfect CSI, one Veh-A simulation, no rigorous bound on the truncation of out-of-band entries. Those are minor.\n\nThe soft spot is the complexity claim. The abstract and conclusion say the complexity is O(M^2N^2), but Section IV gives O(bM^2N^2) with b = 4lmax+1. The paper never discusses how b scales with N. Under the paper's own crystallization constraint, T must grow with N (since τ_p = T/N > τ_max), so lmax = O(T ν_max) = Θ(N). That makes b Θ(N), so the actual complexity is Ω(M^2N^3), not O(M^2N^2). The advertised saving relative to DD-domain inversion is a factor M, not MN. The simulation uses parameters where T ν_max ≈ 1, so b is small (≈13) and the saving is real, but the asymptotic claim in the abstract is overstated. This is load-bearing because the whole point is the complexity reduction.\n\nThe fix is straightforward: either state the complexity as O(bM^2N^2) and discuss the b = Θ(N) scaling under crystallization, or define the asymptotic regime differently (e.g., fixed T and growing M). But as written, a referee should push on this.\n\nThe math itself checks out. The conversion formula and the proof of Theorem 1 are sound. The bandedness argument is correct. The citation pattern is fine—most self-citations are to the authors' prior work on Zak-OTFS predictability, which is the established basis.\n\nWho is this for? People working on OTFS receiver design, especially those interested in low-complexity equalization. It deserves a serious referee; the core result is worth publishing, but the complexity claim needs correction before acceptance. I would send it back for a revision that fixes the scaling claim.","headline":"A genuinely new FD I/O result for Zak-OTFS, but the O(M^2N^2) complexity claim overstates the saving once b = Θ(N) is accounted for.","tokens_in":9398,"tokens_out":4816,"would_cite":true,"duration_ms":43211,"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":"Frequency-domain equalization of Zak-OTFS cuts receiver complexity from cubic to quadratic while matching delay-Doppler equalizer error rates.","keywords":["Zak-OTFS","frequency-domain equalization","delay-Doppler domain","doubly-spread channels","banded channel matrix","low-complexity receiver","orthogonal time frequency space"],"falsifier":"For a given channel profile, compute the fraction of the Frobenius energy of $\\mathbf{H}$ that lies outside the band $b=4l_{\\max}+1$ and compare the banded equalizer BER against full delay-Doppler equalization; if that fraction is not negligible at the operating SNR, the claimed equal performance and complexity saving will not both hold.","tokens_in":8352,"feed_emoji":"📡","tokens_out":6860,"duration_ms":92291,"temperature":0.7,"pith_summary":"Zak-OTFS sends information on quasi-periodic delay-Doppler pulses, and one pilot carrier can reveal the full scattering response of a doubly-spread channel, so channel estimation is inexpensive. The costly step is equalization: the delay-Doppler channel matrix is dense, and inverting it costs ${\\mathcal O}(M^3N^3)$ for a frame with $MN$ information symbols. This paper shows that equalization can instead be done in the frequency domain, where the effective channel matrix is banded, reducing the cost to ${\\mathcal O}(M^2N^2)$ without a loss in error-rate performance in the reported Veh-A channel simulations. It also derives a low-complexity formula to convert the estimated delay-Doppler channel into the frequency-domain channel that the equalizer needs.","feed_headline":"Banded channel lets Zak-OTFS equalization scale quadratically","feed_subtitle":"Frequency-domain equalization exploits a banded channel matrix, matching delay-Doppler error rates at far lower cost.","key_machinery":"The load-bearing object is the banded effective frequency-domain channel matrix $\\mathbf{H}$, together with the new conversion formula (27) that computes it from the delay-Doppler channel by a discrete Fourier Zak transform along the delay dimension. Because the delay-Doppler channel has significant taps only for $|l| \\le l_{\\max}$, each row of $\\mathbf{H}$ is nonzero only within a window of width $2l_{\\max}+1$, and this bandedness carries through to $\\mathbf{H}^H\\mathbf{H} + \\mathbf{I}/\\rho$, whose band width is $b=4l_{\\max}+1$. The receiver moves the received symbols to the frequency domain with an inverse discrete Fourier Zak transform, equalizes by fast banded inversion, and returns to the delay-Doppler domain with a discrete Fourier Zak transform.","core_discovery":"The central claim is that the Zak-OTFS frequency-domain input-output relation is sparse: the frequency-domain channel coefficient $h_f[i,l]$ is significant only when $|i-l| \\le l_{\\max}$, where $l_{\\max} = {\\mathcal O}(T\\nu_{\\max})$ is the channel Doppler spread measured in bins. Consequently the $MN \\times (MN+2l_{\\max})$ frequency-domain matrix $\\mathbf{H}$ is banded, and the linear MMSE estimate $\\hat{\\mathbf{S}} = \\mathbf{H}^H(\\mathbf{H}^H\\mathbf{H} + \\mathbf{I}/\\rho)^{-1}\\mathbf{Y}$ costs only ${\\mathcal O}(b M^2N^2)$ with band width $b = 4l_{\\max}+1$, versus ${\\mathcal O}(M^3N^3)$ for inverting the dense delay-Doppler matrix. Simulations with the Veh-A channel show the same uncoded 4-QAM bit error rate as delay-Doppler domain equalization.","pith_inferences":["For very large Doppler-delay products, the band width $b$ could grow with $N$ and begin to erode the complexity gain; the paper's strongest regime is therefore channels with modest $T\\nu_{\\max}$, and the crossover point could be quantified by computing the out-of-band energy fraction as a function of $T\\nu_{\\max}$.","The conversion formula is a discrete Fourier transform along the delay dimension, so the same banded structure likely appears for other quasi-periodic delay-Doppler waveforms, not only for Gaussian-pulsed Zak-OTFS; this could be tested by evaluating $\\mathbf{H}$ for different pulse shapes.","A natural extension is to replace the perfect-channel-knowledge assumption with an estimated delay-Doppler channel from a spread pilot and measure how estimation error interacts with band truncation, since the current paper defers joint channel estimation and equalization to future work."],"forward_implications":["Frequency-domain equalization of Zak-OTFS reduces the equalization complexity from ${\\mathcal O}(M^3N^3)$ to ${\\mathcal O}(M^2N^2)$ while matching the error-rate performance of delay-Doppler equalization.","Because the band width is set by $T\\nu_{\\max}$, receivers for channels with small normalized Doppler spread can use narrow bands, making large $M,N$ frames practical at low cost.","Channel estimation can remain in the delay-Doppler domain, where a single pilot or a spread pilot suffices, while equalization runs in the frequency domain on the converted channel.","Truncating the band below $4\\lceil T\\nu_{\\max}\\rceil + 1$ gives an additional complexity saving at the price of slight performance degradation at high SNR.","The proposed approach applies to doubly-spread channels where the channel delay and Doppler spreads fit within the Zak-OTFS periodicity, the regime in which Zak-OTFS channel prediction is already known to work."],"supporting_citations":[{"why":"Supplies the Zak-OTFS foundation: quasi-periodic delay-Doppler pulses and the continuous Zak transform formalism used throughout the system model.","marker":"[3]"},{"why":"Establishes predictability of the delay-Doppler channel response, which justifies acquiring the full input-output relation from a single pilot carrier.","marker":"[4]"},{"why":"Provides the Zak transform, inverse discrete Zak transform, twisted convolution, and the dense delay-Doppler input-output relation whose equalization complexity is the baseline to be reduced.","marker":"[5]"},{"why":"Shows how a spread pilot acquires the delay-Doppler input-output relation without pilot overhead, referenced as the estimation method feeding the proposed frequency-domain conversion.","marker":"[6]"},{"why":"Supplies the fast banded matrix inversion algorithm that gives the ${\\mathcal O}(bM^2N^2)$ complexity for equalizing the banded frequency-domain channel.","marker":"[9]"},{"why":"Defines the Veh-A channel model whose power-delay profile is used in the simulations demonstrating equal error-rate performance.","marker":"[10]"}],"fun_headline_variants":["Banded channel cuts Zak-OTFS equalization to quadratic cost","Frequency-domain equalization turns Zak-OTFS cubic into quadratic","Zak-OTFS FD equalization matches DD accuracy at quadratic cost","Banded FD channel enables quadratic-complexity Zak-OTFS equalization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that almost all of the channel's frequency-domain energy lies in a narrow diagonal band, so the energy outside that band can be forced to zero without noticeably hurting accuracy.","fun_headline_variants_meta":{"raw":{"variants":["Banded channel cuts Zak-OTFS equalization to quadratic cost","Frequency-domain equalization turns Zak-OTFS cubic into quadratic","Zak-OTFS FD equalization matches DD accuracy at quadratic cost","Banded FD channel enables quadratic-complexity Zak-OTFS equalization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000786,"raw_usage":{"total_tokens":3495,"prompt_tokens":1002,"completion_tokens":2493,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":2416}},"tokens_in":618,"tokens_out":2493,"duration_ms":21270,"temperature":1.0,"reasoning_tokens":2416,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:52:11.098854+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a given channel profile, compute the fraction of the Frobenius energy of $\\mathbf{H}$ that lies outside the band $b=4l_{\\max}+1$ and compare the banded equalizer BER against full delay-Doppler equalization; if that fraction is not negligible at the operating SNR, the claimed equal performance and complexity saving will not both hold.","supporting_citations":[{"cited_title":"Performance degradation of OFDM systems due to Doppler spreading,","cited_arxiv_id":null,"evidence_quote":"Supplies the Zak-OTFS foundation: quasi-periodic delay-Doppler pulses and the continuous Zak transform formalism used throughout the system model."},{"cited_title":"OTFS - A mathematical foundation for communication and radar sensing in the delay-Doppler domain,","cited_arxiv_id":null,"evidence_quote":"Establishes predictability of the delay-Doppler channel response, which justifies acquiring the full input-output relation from a single pilot carrier."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Zak transform, inverse discrete Zak transform, twisted convolution, and the dense delay-Doppler input-output relation whose equalization complexity is the baseline to be reduced."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows how a spread pilot acquires the delay-Doppler input-output relation without pilot overhead, referenced as the estimation method feeding the proposed frequency-domain conversion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fast banded matrix inversion algorithm that gives the ${\\mathcal O}(bM^2N^2)$ complexity for equalizing the banded frequency-domain channel."},{"cited_title":"Mahmood, D","cited_arxiv_id":null,"evidence_quote":"Defines the Veh-A channel model whose power-delay profile is used in the simulations demonstrating equal error-rate performance."}],"review_version":1}