{"id":"9a909a26-43fb-4348-9ce0-3aeed3e57799","arxiv_id":"2504.18901","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A low-complexity MMSE channel estimator for AFDM is derived by modeling the doubly selective channel with GCE-BEM, with closed-form NMSE and BER analysis that match simulations.","lead":"This paper applies a known mathematical tool, the generalized complex exponential basis expansion model, to estimate wireless channels in affine frequency division multiplexing (AFDM) systems with fast-moving users. The method reduces the number of channel parameters to estimate and is shown by simulation to handle fractional Doppler shifts that degrade earlier schemes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's R_g uses the inverse of a non-square Θ=B⊗I_L; the MMSE estimator and NMSE (14) are therefore not well-defined as stated.","rationale":"The central claim is a low-complexity MMSE estimator whose optimality and error analysis rest on R_g. A non-invertible Θ makes the estimator ill-defined. Even if the authors intended the pseudo-inverse, the text is incorrect and the subsequent NMSE expression (14) would change. This is more fundamental than the reader's concern about statistics mismatch because it affects the derivation even under perfect statistics. Correcting this requires re-deriving the estimator and the NMSE; the current manuscript cannot be accepted as is. The BER convexity issue noted by the reader remains secondary but does not alleviate the primary flaw.","tokens_in":8432,"tokens_out":13015,"duration_ms":120751,"concrete_test":"For the N=256, Q=4, L=2 Jakes scenario, compute R_g correctly as R_g = (Θ^HΘ)^{-1}Θ^H R_hh Θ(Θ^HΘ)^{-1} (or via sample covariance of g = (Θ^HΘ)^{-1}Θ^H h over many channel realizations). Evaluate the analytical NMSE in Eq. (14) with this correct R_g and with the paper's expression replaced by the pseudo-inverse Θ^† R_hh (Θ^†)^H. Compare both curves to the Monte Carlo NMSE in Fig. 2. If the curve from the correct R_g matches the simulation but the paper's stated formula does not, the derivation is wrong and must be corrected; if neither matches, the estimator itself is not the MMSE estimator described.","verdict_should_be":"REJECT","load_bearing_attack":"In Eq. (6), the MMSE estimator requires the covariance R_g of the BEM coefficient vector g. The paper defines Θ = B ⊗ I_L with B ∈ C^{N×(Q+1)} (basis functions as columns), so Θ is N(L+1) × (Q+1)(L+1). For the chosen parameters N=256, Q=4, L=2, Θ is 768×15 and has no inverse. The stated relation R_g = Θ^{-1} R_hh (Θ^H)^{-1} is therefore undefined. The correct mapping from the true channel h to the projected BEM coefficients is g = (Θ^H Θ)^{-1} Θ^H h, giving R_g = (Θ^H Θ)^{-1} Θ^H R_hh Θ (Θ^H Θ)^{-1}. This reduces to the paper's expression only when Θ is square (N=Q+1), the opposite of the claimed low-complexity regime. Since Eq. (6) and the NMSE in Eq. (14), which uses trace{Θ R_˜g Θ^H}, depend on R_g, the central derivation of the MMSE estimator is invalid as written. The Monte Carlo match in Fig. 2 cannot validate Eq. (14) unless the simulations used either the same incorrect R_g (which would not be the MMSE estimator) or a corrected R_g not stated in the paper.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a GCE-BEM-based channel estimation scheme for AFDM over doubly selective channels. It represents the time-varying channel with (Q+1)(L+1) BEM coefficients, inserts two embedded pilots with guard bands, derives an MMSE estimator for the BEM coefficients, and gives closed-form NMSE and BER expressions. Simulations compare the analytical NMSE and BER against Monte Carlo results and against EPA-AML and BEM-OTFS benchmarks, claiming lower complexity and robustness to fractional Doppler.","tokens_in":8699,"tokens_out":13399,"duration_ms":133007,"significance":"If the derivations are corrected, the paper offers a useful reduction in channel-estimation complexity for AFDM (from O(N^3) to O(N^2QL)) and provides analytical NMSE and BER tools that can guide system design. The use of GCE-BEM to handle fractional Doppler is timely, and the comparison with EPA-AML and OTFS is relevant. However, the central MMSE derivation currently contains an undefined matrix inverse, and the BER lower bound relies on an unverified convexity interval, so the theoretical support needs substantial repair before the claims can be accepted.","major_comments":[{"comment":"The definition R_g = Θ^{-1} R_hh (Θ^H)^{-1} is not well defined because Θ = B ⊗ I_L is an N(L+1) × (Q+1)(L+1) matrix with N ≫ Q+1 in the operating regime (N=256, Q=4, L=2), so Θ^{-1} does not exist. Since R_g enters the MMSE estimator (6), the data covariance (11), the estimation-error covariance (13), and the NMSE (14), the central derivation is invalid as written. The projection relation should be g = (Θ^H Θ)^{-1} Θ^H h, leading to R_g = (Θ^H Θ)^{-1} Θ^H R_hh Θ (Θ^H Θ)^{-1}; the authors need to re-derive (6), (11), (13), and (14) with this corrected covariance and confirm that the simulation code used the corrected expression.","section":"Section III-B, Theorem 1, Eq. (6)"},{"comment":"The Jensen lower bound in (10) requires φ(x) = erfc( sqrt( b_M x/(1-x) ) ) to be convex over the realized values of x = T(i,i). The proof only cites [16] and states that convexity holds in the high-SINR region without specifying this region. The simulations include SNR_d from 0 to 24 dB, which likely includes low-SINR subcarriers; if the convexity range is not verified for all simulated operating points, (10) is not established as a lower bound. Please provide the valid SINR interval from [16] and either verify that it covers the simulation range or replace Jensen with a numerically verified bound.","section":"Section III-C, Appendix B, Eq. (10)"},{"comment":"The Monte Carlo validation uses channels generated from the same Jakes model and the same covariance matrices R_hh,l and N0 assumed in the estimator. Consequently, the agreement between analysis and simulation confirms the algebra but does not test the estimator under statistics mismatch (e.g., imperfect knowledge of R_hh,l, N0, or non-Jakes Doppler spectra). Since the practical claim is low-complexity estimation for real receivers, the paper should include a mismatch experiment or explicitly state this as a limitation.","section":"Section IV, Figs. 2-5"}],"minor_comments":[{"comment":"The text contains several typos, including 'adopt' for 'adopted' and 'embeded' for 'embedded'; please proofread the manuscript.","section":"Abstract and Section I"},{"comment":"The vector h in e_mod = h − ħ is never defined; please define the stacking of time-domain channel taps and its covariance R_hh.","section":"Section III, Eq. (14)"},{"comment":"The 'AFDM, Analysis' curve is not identified in the caption; state that it is the Jensen lower bound from Theorem 2.","section":"Section IV, Fig. 5"},{"comment":"Reference [14] has an incomplete author list ('D. G. G.'); please complete the citation.","section":"References"},{"comment":"The notation in the erfc argument is ambiguous; introduce (1/N)Σ_i T(i,i) as a single variable for clarity.","section":"Eq. (10)"}],"recommendation":"major_revision","confidential_remarks":"The R_g inverse error appears to be a fixable slip, but it sits at the very center of the paper. I would urge the editor to require the corrected derivation and a robustness experiment before publication. The topic is timely and within the journal's scope, and the complexity reduction claim is attractive, but the theoretical core needs to be repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a genuine extension of GCE-BEM to AFDM: the two-pilot MMSE estimator, the closed-form NMSE, and the BER lower bound. The complexity reduction from N(L+1) to (Q+1)(L+1) parameters is real, and the simulation work is sensible—comparing against OTFS GCE-BEM and the EPA-AML baseline under fractional Doppler is the right test.\n\nBut the central Theorem 1 contains a linear-algebra error that the authors need to fix. They define R_g = Θ^{-1} R_hh (Θ^H)^{-1}, with Θ = B⊗I_L. Here B is N×(Q+1), so Θ is N(L+1) × (Q+1)(L+1). For their own simulation parameters (N=256, Q=4, L=2), Θ is 768×15. That matrix has no inverse. The correct mapping between the channel vector h and the BEM coefficients is a least-squares projection, g = (Θ^H Θ)^{-1} Θ^H h, which gives R_g = (Θ^H Θ)^{-1} Θ^H R_hh Θ (Θ^H Θ)^{-1}. The paper's expression only holds when N=Q+1, which is exactly the regime they are not targeting. As written, Eq. (6), the MMSE estimator, and Eq. (14), the NMSE, are not well-defined. The Monte-Carlo match in Fig. 2 cannot validate those formulas unless the simulations used a corrected R_g that isn't stated. This is load-bearing, not a typo.\n\nOther soft spots are smaller. Theorem 2's BER lower bound uses Jensen's inequality with a convexity claim from [16], but the valid SINR region isn't stated or checked in the simulations. The validation also assumes the receiver knows the exact channel statistics—the Jakes model and covariance—so model mismatch isn't tested; that's a common gap but worth acknowledging. No code is shipped, and 'perfectly match' appears without error bars or a numerical discrepancy measure.\n\nWho gets value from this: AFDM and 6G waveform researchers, especially people working on receiver design for doubly selective channels. The idea is useful and the intent is clear. With the R_g fix and a stated convexity region, the paper would be a solid contribution. As it stands, I'd send it to review because the underlying work is serious, but I'd demand a major revision and a corrected derivation.\n\nFor peer review: send it, but flag the rectangular-matrix issue as blocking.","headline":"Useful GCE-BEM-for-AFDM extension, but Theorem 1's R_g definition is invalid and needs correction before the results stand.","tokens_in":9265,"tokens_out":5050,"would_cite":false,"duration_ms":47552,"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 GCE-BEM representation reduces AFDM channel estimation to $(Q+1)(L+1)$ coefficients and yields closed-form error and BER expressions that match simulations.","keywords":["AFDM","GCE-BEM","channel estimation","doubly selective channels","fractional Doppler","inter-Doppler interference","MMSE estimator","basis expansion model"],"falsifier":"Fix the estimator to the Doppler-spectrum covariance used in the paper, then generate test channels from a different Doppler spectrum or from a measured channel and compare the empirical NMSE with Eq. (14); a large gap would show the closed-form analysis depends on exact statistical knowledge.","tokens_in":8217,"feed_emoji":"📡","tokens_out":7940,"duration_ms":69773,"temperature":0.7,"pith_summary":"This paper tries to show that generalized complex exponential basis expansion model (GCE-BEM) makes channel estimation in affine frequency division multiplexing (AFDM) affordable over doubly selective channels. The authors represent the time-varying channel as a short sum of $Q+1$ complex exponential basis functions per delay tap, cutting unknown coefficients from $N(L+1)$ to $(Q+1)(L+1)$. With two embedded pilots and a minimum mean square error estimator, they derive closed-form channel estimation error and a bit error rate lower bound. The claim is that this removes the inter-Doppler interference caused by fractional Doppler shifts and that the analytic formulas match Monte Carlo simulations.","feed_headline":"Two-pilot BEM estimator slashes AFDM channel estimation cost","feed_subtitle":"Basis expansion compresses the time-varying channel and keeps AFDM robust under fractional Doppler and high mobility.","key_machinery":"The load-bearing object is the GCE-BEM decomposition $H=\\sum_{q=0}^{Q}\\mathrm{diag}\\{b_q\\}F^H\\mathrm{diag}\\{F_L g_q\\}F+E_{\\mathrm{mod}}$, which expresses the $N(L+1)$ time-domain channel coefficients through $(Q+1)(L+1)$ BEM coefficients $g_q$. The basis functions sample the Doppler spectrum at intervals of $1/(RN)$, so fractional Doppler shifts fall inside the modeled support instead of causing inter-Doppler interference. This decomposition feeds a two-pilot embedded pilot structure with null guard bands, a linear MMSE estimator for $g$, and closed-form expressions for model error, estimation error, and BER.","core_discovery":"The central discovery is that GCE-BEM sampling of the Doppler spectrum, with basis $b_q(n)=e^{j2\\pi(q-\\lceil Q/2\\rceil)n/(RN)}$, converts the fractional-Doppler channel estimation problem in AFDM into a small linear estimation problem. The estimator in Eq. (6) uses two pilots surrounded by null guards; its error covariance is Eq. (13), the total NMSE is Eq. (14), and the BER lower bound is Eq. (10), obtained by applying convexity to the per-subcarrier SINR expression $\\zeta_i=T(i,i)/(1-T(i,i))$. On the paper's evidence, the derived NMSE and BER curves match Monte Carlo results over the simulated three-path channel at up to 675 km/h, and the scheme keeps working even when paths share the same delay shift.","pith_inferences":["A natural next test is statistics mismatch: if the true Doppler spectrum differs from the covariance assumed by the MMSE estimator, the closed-form NMSE may stop matching; the paper's Monte Carlo setup uses the same statistics for generation and estimation.","The model-error covariance terms in the estimator suggest that BEM order $Q$ and pilot guard width $Q_B$ could be jointly optimized for a target Doppler spread.","The same GCE-BEM-plus-two-pilot structure could extend to MIMO-AFDM or integrated sensing and communications, where per-antenna channel estimation cost multiplies."],"forward_implications":["Channel estimation complexity falls from $O(N^3)$ for full MMSE to $O(N^2 Q L)$, since $QL \\ll N$.","AFDM receivers remain usable under fractional Doppler shifts, including cases where multipath components have identical delays.","The derived NMSE and BER expressions can predict performance without running link-level simulations.","GCE-BEM AFDM reaches BER comparable to GCE-BEM OTFS with lower pilot overhead.","Against the EPA-AML benchmark, the proposed estimator is more robust when the channel delay profile is not known."],"supporting_citations":[{"why":"Supplies the AFDM signal model and the EPA-AML channel estimation scheme used as the main baseline.","marker":"[7]"},{"why":"Introduces the GCE-BEM channel representation, basis order condition, and model-error covariance used throughout.","marker":"[14]"},{"why":"Provides the superimposed-pilot AFDM estimator that the paper contrasts by handling fractional Doppler.","marker":"[13]"},{"why":"Gives the diagonally reconstructed low-complexity AFDM estimator that motivates the complexity comparison.","marker":"[12]"},{"why":"Establishes convexity of the erfc-based BER function used to derive the lower bound.","marker":"[16]"},{"why":"Provides the constellation-dependent BER coefficients $a_M$ and $b_M$ used in Eq. (10).","marker":"[15]"}],"fun_headline_variants":["GCE-BEM turns AFDM channel estimation into a small linear problem","Two-pilot BEM estimator tames fractional Doppler for AFDM","Low-complexity BEM channel estimation for AFDM matches simulations","Basis expansion model cuts AFDM channel estimation cost","Two-pilot GCE-BEM: accurate AFDM channel estimation at high speed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The estimator needs the exact per-path channel covariance matrices and noise variance, and the simulations generate channels from exactly those statistics, so the reported match does not test what happens when the receiver's assumed statistics are wrong.","fun_headline_variants_meta":{"raw":{"variants":["GCE-BEM turns AFDM channel estimation into a small linear problem","Two-pilot BEM estimator tames fractional Doppler for AFDM","Low-complexity BEM channel estimation for AFDM matches simulations","Basis expansion model cuts AFDM channel estimation cost","Two-pilot GCE-BEM: accurate AFDM channel estimation at high speed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000581,"raw_usage":{"total_tokens":2694,"prompt_tokens":860,"completion_tokens":1834,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":1742}},"tokens_in":476,"tokens_out":1834,"duration_ms":14828,"temperature":1.0,"reasoning_tokens":1742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:06:53.547089+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix the estimator to the Doppler-spectrum covariance used in the paper, then generate test channels from a different Doppler spectrum or from a measured channel and compare the empirical NMSE with Eq. (14); a large gap would show the closed-form analysis depends on exact statistical knowledge.","supporting_citations":[{"cited_title":"Affine Frequency Division Multiplexing for Next Generation Wireless Communications,","cited_arxiv_id":null,"evidence_quote":"Supplies the AFDM signal model and the EPA-AML channel estimation scheme used as the main baseline."},{"cited_title":"Near-Optimal BEM OTFS Receiver With Low Pilot Overhead for High-Mobility Communications,","cited_arxiv_id":null,"evidence_quote":"Introduces the GCE-BEM channel representation, basis order condition, and model-error covariance used throughout."},{"cited_title":"Channel Estimation for AFDM With Superimposed Pilots,","cited_arxiv_id":null,"evidence_quote":"Provides the superimposed-pilot AFDM estimator that the paper contrasts by handling fractional Doppler."},{"cited_title":"Diagonally Reconstructed Channel Estimation for MIMO-AFDM With Inter-Doppler Interference in Doubly Selective Channels,","cited_arxiv_id":null,"evidence_quote":"Gives the diagonally reconstructed low-complexity AFDM estimator that motivates the complexity comparison."},{"cited_title":"Chirp Parameter Selection for Affine Frequency Division Multiplexing with MMSE Equalization,","cited_arxiv_id":null,"evidence_quote":"Establishes convexity of the erfc-based BER function used to derive the lower bound."},{"cited_title":"Goldsmith, Wireless communications.Cambridge University Press, 2005","cited_arxiv_id":null,"evidence_quote":"Provides the constellation-dependent BER coefficients $a_M$ and $b_M$ used in Eq. (10)."}],"review_version":1}