{"id":"32efda1b-3a09-47f6-9151-a96bd57097ff","arxiv_id":"2505.19843","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Closed-form BER formulas for OTFS over Nakagami-m fading are claimed, but the derivation drops terms and misuses incomplete Gamma functions.","lead":"This paper derives closed-form bit error rate formulas for OTFS modulation over Nakagami-m fading channels in single-user and multi-user setups, and compares OTFS with OFDM. A generalist might read it to gauge whether OTFS's promised high-mobility gains come with tractable error-rate math for 6G system design.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central closed-form BER (Eq. 31) is derived from MRC over P Nakagami paths but validated against ML detection on the OTFS effective channel H_eff; the paper never shows these receivers are equivalent, so the headline OTFS-specific claim rests on an unproved identity.","rationale":"The paper's stated contribution is a BER law 'specifically for OTFS' under Nakagami-m fading. That contribution stands or falls on the claim that ML detection over H_eff behaves like the MRC statistic used in the derivation. This is the single point where the OTFS-specific part enters; all other ingredients (Erlang sum, Gamma moments, Meijer-G) are generic fading statistics. The reader's weakest_assumption pinpoints the same step. I examined the equations and find no derivation bridging Eq. (14) to Eq. (21)/(23): the effective channel is a sum of shifted, Doppler-rotated path matrices, so each entry and each quadratic form mixes the h_p; the distribution of such a quadratic form is not automatically Erlang. The multi-user analysis has an independent red flag (Eq. 32 vs Fig. 3) that reinforces the mismatch but does not replace the main concern. Because this is a falsifiable modeling assumption and the paper provides no code or data, the rejection verdict is not weakened by my pass. If the authors supplied a step proving equivalence or a simulation showing Eq. (31) matches exact ML pairwise-error averaging, the paper would need re-review; absent that, the rejection verdict remains appropriate.","tokens_in":17462,"tokens_out":5761,"duration_ms":64681,"concrete_test":"Re-run the Fig. 2 setup (P=2, m1=1, m2=2) with the same Monte Carlo channel realizations and compute the exact ML pairwise-error probability averaged over the fading, using ||H_eff(e_i - e_j)||² from Eq. (14), not the Erlang sum of Eq. (21). If the averaged curve does not coincide with Eq. (31), the MLD/MRC equivalence fails and the central OTFS-specific claim is unsupported. Also report the empirical distribution of ||H_eff v||² for a unit error vector; if it is not Erlang with the parameters used in Eq. (21), the closed form is the wrong statistic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III-A builds the SISO analysis on the Erlang sum of P independent squared Nakagami path gains (Eq. 21) and the MRC SER integral (Eq. 23), then presents Eq. (31) as the OTFS BER. The simulations, however, use ML detection on the OTFS effective channel H_eff = (F_N⊗G_rx)(Σ_p h_p Π^{l_p}Δ^{k_p})(F_N†⊗G_tx) from Eq. (14). For ML detection, the relevant quantity for a decision error v is ||H_eff v||², a quadratic form in the h_p with cross terms and phase rotations from the delay-Doppler structure; it is not generally a sum of P independent Erlang variables. The paper supplies no argument that MLD on H_eff has the same pairwise error probability as MRC over the paths, nor that entries of H_eff remain Nakagami/Erlang after the symplectic transforms. Without that equivalence, Eq. (31) describes legacy MRC reception, not OTFS MLD, and the claimed simulation validation of an OTFS-specific formula would be accidental. The same model mismatch appears in the multi-user part: Eq. (32) places interfering users in the denominator, predicting worse BER as K_u grows, while Fig. 3 shows K_u=2 outperforming K_u=1; that is consistent only if the 'users' are actually diversity branches, not the interference model used in the derivation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to provide the first comprehensive analytical and simulation-validated BER analysis of OTFS over Nakagami-m fading channels. For the SISO case it derives a closed-form BER by summing P independent squared-Nakagami path gains, whose sum is modeled as Erlang, and then applying the standard MRC SER integral. For the SIMO/multi-user case it approximates the aggregate interference by a Gamma distribution via moment matching and expresses the resulting SER through Meijer-G functions. The analytical expressions are compared with ML-detection-based Monte Carlo simulations and with OFDM under the same channel, and the paper also reports empirical diversity-gain comparisons.","tokens_in":17802,"tokens_out":7102,"duration_ms":73785,"significance":"If correct, the closed-form error-rate expressions would be a useful addition to OTFS performance analysis and would support the claim that OTFS outperforms OFDM in high-mobility Nakagami-m fading. The paper has a clearly stated system model, cites standard statistical results, and includes Monte Carlo validation. However, the central analytical derivation for the SISO case is a legacy MRC result applied to path gains, not a BER derivation for ML detection on the OTFS effective channel; the multi-user derivation is internally inconsistent with the reported simulation trends; and the closed forms contain several algebraic errors. The load-bearing claims therefore do not currently hold.","major_comments":[{"comment":"The Erlang CDF is mis-expanded. Eq. (18) starts the inner sum at i=1 instead of at ℓ=0, so it omits the term exp(-z/μ_i). Eq. (24) correctly includes ℓ=0, but Eq. (25) silently changes the summation to ℓ=1,...,m_i-1 and Eq. (31) inherits this omission. The missing ℓ=0 term contributes a nonzero integral D with ℓ=0 in Eq. (27), so Eq. (31) is not the closed-form BER of the stated Erlang sum model. This error is load-bearing for the main SISO result.","section":"Section III-A, Eqs. (18), (25), (31)"},{"comment":"The analytical derivation models the SNR as the sum of P independent squared Nakagami path gains and uses the MRC SER integral of Eq. (23). The simulations, however, apply ML detection to the effective channel matrix H_eff of Eq. (14). The paper provides no argument that ML detection on H_eff has the same pairwise error probability as MRC over the P paths, nor that the entries of H_eff, which are linear combinations of the time-domain path gains, remain Nakagami-m distributed. As written, Eq. (31) describes MRC reception over P fading paths, not OTFS MLD, so the claimed validation in Figs. 1-2 does not establish an OTFS-specific analytical BER.","section":"Section III-A, Eqs. (21)-(23) vs. Eq. (14) and Section IV-A"},{"comment":"The CDF expression is incorrect. Since Υ_k = (E_s/N0)/Z_k with Z_k ≥ 1, the correct CDF is F_Υ(y) = P(Z_k ≥ E_s/(N0 y)) = 1 - γ(m_z,(E_s/(N0 y)-1)/Ω_z)/Γ(m_z) = Γ(m_z,(E_s/(N0 y)-1)/Ω_z)/Γ(m_z). Eq. (44) instead states F_Υ(y) = γ(m_z,·)/Γ(m_z), which has the wrong monotonicity and is therefore not a valid CDF. This error propagates into the Meijer-G SER expression.","section":"Section III-B, Eq. (44)"},{"comment":"The Meijer-G derivation contains two algebraic errors. First, the exponential in Eq. (47) should be exp(-(E_s/(N0 y)-1)/Ω_z), not exp(-(E_s/(N0 y)-1)); the factor 1/Ω_z is missing. Second, the final closed form in Eq. (49) omits the 1/Γ(m_z) normalization that the CDF in Eq. (44) requires. Consequently Eq. (49) does not follow from the preceding equations and is not a valid BER expression for the stated model.","section":"Section III-B, Eqs. (47)-(49)"},{"comment":"The multi-user derivation treats additional users as interference in the denominator of the SINR in Eq. (32), so the analytical model predicts that increasing K_u degrades performance. Fig. 3 shows the opposite: K_u=2 substantially outperforms K_u=1, and the text attributes this to diversity gain. This contradiction indicates that the simulated configuration adds diversity branches rather than realizing the interference model used in the derivation, so the claimed agreement between theory and simulation for the multi-user case is not meaningful.","section":"Section III-B, Eqs. (32)-(35) vs. Fig. 3"}],"minor_comments":[{"comment":"Eq. (15) defines μ_i = E[h_p^2]/m_i, but for a complex channel coefficient the second moment should be E[|h_p|^2]/m_i; the notation should be corrected for consistency with the subsequent Erlang model.","section":"Section II-B, Eq. (15)"},{"comment":"The coefficient Ξ is very hard to parse because P is used both for the number of paths and as a summation index, and the unit-step function U(a) is defined but never used. A cleaner notation would improve reproducibility.","section":"Section III-A, Eqs. (19)-(20)"},{"comment":"The summation index l in Eq. (31) collides with the delay-domain index l used earlier in the paper; renaming the summation index would avoid confusion.","section":"Section III-A, Eq. (31)"},{"comment":"The diversity-gain formulas are presented without derivation, and the cited source [53] concerns N-Nakagami relaying rather than OTFS. These formulas should be derived or clearly labeled as heuristic approximations.","section":"Section IV-C, Eqs. (53)-(54)"},{"comment":"The claim of being the first comprehensive analytical and simulation-validated BER analysis specifically for OTFS over Nakagami-m fading is overstated, given that the SISO derivation reduces to a standard MRC result and the paper itself cites earlier OTFS error-performance studies [16]-[18], [22].","section":"Abstract and Section I"},{"comment":"There are several typographical and formatting issues, including the footnote 'TÜB˙ITAK', the inconsistent spacing in 'E V A', and the use of 'z' in the denominator of Eq. (43) where Ω_z^{m_z} Γ(m_z) is intended. These do not affect the technical conclusions but should be cleaned up.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The algebraic errors and the receiver-model mismatch are not localized presentation issues. The SISO closed-form BER describes MRC over path gains rather than ML detection on the OTFS effective channel, and the multi-user analysis contradicts its own simulation trend. These problems concern the central claimed contribution and would require a substantially different derivation or simulation setup to resolve, so I cannot recommend major revision within the paper's current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You asked about the OTFS/Nakagami-m BER paper. I read it so you don't have to. The idea is sensible: take the known Erlang sum PDF for squared Nakagami variables, plug it into the standard MRC SER integral, and call it OTFS. The writing is clean, the OTFS system model is standard, and they do provide Monte Carlo validation for tiny grids (M=N=2). The OFDM comparison is plausible. But the central derivations have load-bearing errors, and the analysis doesn't actually describe the simulated receiver.\n\nThe math problems are concrete. Eq. (31) drops the ell=0 term in the Erlang CDF expansion; the CDF in Eq. (25) is wrong as written. Eq. (44) uses the lower incomplete Gamma where the tail probability needs the upper incomplete Gamma. Eq. (49) loses the 1/Gamma(m_z) normalization. These are not cosmetic typos; they change the values of the claimed closed forms.\n\nMore fundamentally, the SISO analysis is for MRC combining over P Nakagami paths, while the simulations use maximum-likelihood detection on the OTFS effective channel H_eff. The paper never shows that MLD's pairwise error probability equals the MRC statistic. H_eff entries are linear combinations of the path gains with delay-Doppler phases, so ||H_eff v||^2 is a quadratic form with cross terms, not a sum of independent Erlangs. Without that equivalence, Eq. (31) describes legacy MRC reception, not OTFS-MLD.\n\nThe multi-user part has a second contradiction. Eq. (32) puts interfering users in the denominator, so BER should worsen as K_u grows. But Fig. 3 shows K_u=2 outperforming K_u=1, and the text calls that diversity gain. That is consistent only if the 'users' are actually receive branches, not interferers. The diversity-gain formulas (53)-(54) look ad hoc and are inconsistent with the worked numbers in Section IV-C. No code or data is provided.\n\nWho is this for? Someone wanting a quick OTFS-vs-OFDM simulation comparison under Nakagami-m might skim it, but the analytical results are not usable. I would not cite it. I would not bring it to reading group. My recommendation: desk-reject. If the authors fix the incomplete Gamma errors and either prove the MRC-MLD equivalence or redo the analysis for the actual receiver, this could be worth another look, but as written it does not support the claims.","headline":"The paper's OTFS-specific BER claims rest on an unproved MRC-MLD equivalence and the closed forms contain concrete errors, so the headline result does not stand.","tokens_in":18263,"tokens_out":4416,"would_cite":false,"duration_ms":44424,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives closed-form bit error rate formulas for OTFS over Nakagami-m fading and shows, by simulation, that OTFS outperforms OFDM in high mobility.","keywords":["OTFS modulation","Nakagami-m fading","bit error rate analysis","Erlang distribution","Meijer-G function","moment matching","diversity gain","multi-user interference"],"falsifier":"Simulate ML-detected OTFS on a larger delay-Doppler grid, such as M=N=4, with Nakagami-m path gains and compare the empirical BER with Eq. (31); if the gap grows systematically with grid size rather than vanishing with more simulation runs, the assumed ML-to-MRC equivalence does not hold. Alternatively, estimate the distribution of an entry of the effective channel matrix from many channel realizations and test whether it is Nakagami-m; a rejection would break the derivation's foundation.","tokens_in":17248,"feed_emoji":"📡","tokens_out":5225,"duration_ms":47309,"temperature":0.7,"pith_summary":"This paper sets out to give closed-form bit error rate formulas for OTFS modulation when the channel is Nakagami-m distributed, covering both a single-user SISO link and a multi-user SIMO link. It claims these are the first analytical and simulation-validated BER expressions specifically for OTFS over Nakagami-m fading. The SISO result uses the Erlang density of the sum of squared path gains to produce a finite closed form, and the multi-user result approximates co-channel interference by a Gamma variable via moment matching and expresses the error rate through a Meijer-G function. Monte Carlo simulations with maximum likelihood detection show that the formulas track the simulated BER, and comparisons with OFDM show OTFS achieving lower BER and higher diversity at the same SNR. If these formulas are right, they give system designers a direct way to predict OTFS reliability across fading severities and user counts without running link-level simulations.","feed_headline":"OTFS beats OFDM in high-mobility Nakagami-m fading","feed_subtitle":"New closed-form BER formulas for single- and multi-user OTFS match ML simulations and show higher diversity gain.","key_machinery":"The load-bearing object is the sum of squared Nakagami-m path gains, whose Erlang-mixture PDF given in Eqs. (19)-(20) converts the fading statistics into an error-probability integral. For SISO, the machinery is that Erlang CDF inserted into the MRC symbol-error integral of Eq. (23), yielding the closed form in Eq. (31). For multi-user SIMO, the machinery is moment matching: the interference power is replaced by a Gamma variable with matched mean and variance, and the resulting SINR CDF is written as incomplete Gamma functions, which are recast as Meijer-G functions and integrated to produce Eq. (49). The Meijer-G product-integration identity is what turns the nested integral into a closed form.","core_discovery":"On its own terms, the paper's central claim is that the bit error rate of OTFS over Nakagami-m fading collapses to a small set of closed-form expressions: Eq. (31) for SISO, built from the Erlang PDF of squared-Nakagami path gains, and Eqs. (49)-(50) for multi-user SIMO, built from a Gamma approximation of interference and Meijer-G functions. The paper treats the ML-detected OTFS system as equivalent, for error analysis, to maximal-ratio combining over the P propagation paths, so the standard MRC symbol-error integral with modulation constants A and B applies. Against that model, the derived curves match ML simulations for BPSK and QPSK, single- and two-path channels, and m = 1 and m = 2 Nakagami parameters. The same simulations show OTFS reaching lower BER than OFDM under identical settings, with steeper BER slopes, which the paper reads as a diversity advantage arising from delay-Doppler dispersion.","pith_inferences":["Editorial inference: because the derivation never proves that entries of the effective delay-Doppler channel matrix stay Nakagami-m, the formulas' validity on larger OTFS grids, such as M,N greater than 2, is an open testable question rather than an established fact.","Editorial inference: the Erlang-plus-Meijer-G template could plausibly be carried over to other delay-Doppler waveforms, but the paper does not make that claim.","Editorial inference: the diversity-gain expressions suggest concrete resource-allocation rules, such as assigning more delay-Doppler bins to users with low Nakagami-m parameters, but the paper does not develop that application."],"forward_implications":["For a SISO OTFS link over Nakagami-m fading, BER can be evaluated directly from Eq. (31) for any modulation covered by Table I, without Monte Carlo simulation.","For multi-user SIMO OTFS, Eq. (50) gives BER as a function of user count, path count, and Nakagami shape parameters, so interference-limited performance can be predicted analytically.","Under the paper's settings, OTFS achieves lower BER than OFDM at the same SNR in high-mobility EVA channels, with the gap widening as fading becomes milder.","Empirical diversity gain grows with path count P and Nakagami parameter m, and for SIMO with receive users, OTFS holds a consistent diversity advantage over OFDM.","Because the A and B parameters in Table I cover BPSK, QPSK, M-PSK, M-QAM, and FSK families, the same formulas extend beyond the two modulations simulated."],"supporting_citations":[{"why":"Supplies the Nakagami-m fading model and the path-gain PDF used throughout the analysis.","marker":"[37]"},{"why":"Provides the closed-form statistics for the sum of squared Nakagami-m variates, which become the Erlang-mixture PDF in Eq. (19).","marker":"[38]"},{"why":"Prior OTFS outage analysis that supplies the form of the sum-density expression used in Eq. (22).","marker":"[41]"},{"why":"Gives the CDF and SER integral structure used to convert fading statistics into bit error probability.","marker":"[39]"},{"why":"Provides the Meijer-G identities and product-integration formula that produce the closed-form multi-user BER in Eq. (49).","marker":"[46]"},{"why":"Defines the SINR model for OTFS-based single-user and multi-user transmissions used in Eq. (32).","marker":"[47]"},{"why":"Supplies the moment-matching Gamma approximation for Nakagami-m interference used in Eqs. (40)-(41).","marker":"[49]"},{"why":"Defines the EVA channel model used in the simulations that benchmark the theoretical curves.","marker":"[51]"}],"fun_headline_variants":["OTFS closed-form BER matches ML in Nakagami-m fading","Multi-user OTFS BER derived under Nakagami-m fading","Closed-form BER for OTFS shows diversity gain","OTFS beats OFDM with closed-form BER model","New BER formulas for OTFS in Nakagami-m fading"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The closed-form BER expressions assume that maximum-likelihood detection over the OTFS effective channel behaves like maximal-ratio combining over the original Nakagami-m paths, but the paper does not prove that equivalence.","fun_headline_variants_meta":{"raw":{"variants":["OTFS closed-form BER matches ML in Nakagami-m fading","Multi-user OTFS BER derived under Nakagami-m fading","Closed-form BER for OTFS shows diversity gain","OTFS beats OFDM with closed-form BER model","New BER formulas for OTFS in Nakagami-m fading"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001399,"raw_usage":{"total_tokens":5695,"prompt_tokens":1024,"completion_tokens":4671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":4589}},"tokens_in":640,"tokens_out":4671,"duration_ms":28168,"temperature":1.0,"reasoning_tokens":4589,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:05:29.509052+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate ML-detected OTFS on a larger delay-Doppler grid, such as M=N=4, with Nakagami-m path gains and compare the empirical BER with Eq. (31); if the gap grows systematically with grid size rather than vanishing with more simulation runs, the assumed ML-to-MRC equivalence does not hold. Alternatively, estimate the distribution of an entry of the effective channel matrix from many channel realizations and test whether it is Nakagami-m; a rejection would break the derivation's foundation.","supporting_citations":[{"cited_title":"The m-distribution—a general formula of intensity distribution of rapid fading,","cited_arxiv_id":null,"evidence_quote":"Supplies the Nakagami-m fading model and the path-gain PDF used throughout the analysis."},{"cited_title":"Closed-form statistics for the sum of squared Nakagami-m variates and its applications,","cited_arxiv_id":null,"evidence_quote":"Provides the closed-form statistics for the sum of squared Nakagami-m variates, which become the Erlang-mixture PDF in Eq. (19)."},{"cited_title":"Outage Analysis of Orthogonal Time- Frequency Space Modulation in Nakagami-m Fading Channel,","cited_arxiv_id":null,"evidence_quote":"Prior OTFS outage analysis that supplies the form of the sum-density expression used in Eq. (22)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the CDF and SER integral structure used to convert fading statistics into bit error probability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Meijer-G identities and product-integration formula that produce the closed-form multi-user BER in Eq. (49)."},{"cited_title":"Outage Analysis for OTFS- based Single User and Multi-User Transmissions,","cited_arxiv_id":null,"evidence_quote":"Defines the SINR model for OTFS-based single-user and multi-user transmissions used in Eq. (32)."},{"cited_title":"Analysis of Uplink IRS-Assisted NOMA Under Nakagami-m Fading via Moments Matching,","cited_arxiv_id":null,"evidence_quote":"Supplies the moment-matching Gamma approximation for Nakagami-m interference used in Eqs. (40)-(41)."},{"cited_title":"Evolved Universal Terrestrial Radio Access (E-UTRA); User Equip- ment (UE) Radio Transmission and Reception,","cited_arxiv_id":null,"evidence_quote":"Defines the EVA channel model used in the simulations that benchmark the theoretical curves."}],"review_version":1}