BEM-Assisted Low-Complexity Channel Estimation for AFDM Systems over Doubly Selective Channels

Haowei Wu; Limin Liu; Pei Xiao; Qihao Peng; Qu Luo; Zhe Li

REVIEW 3 major objections 5 minor 16 references

BEM-Assisted Low-Complexity Channel Estimation for AFDM Systems over Doubly Selective Channels

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Useful GCE-BEM-for-AFDM extension, but Theorem 1's R_g definition is invalid and needs correction before the results stand. read the letter →

arxiv 2504.18901 v2 pith:M4JAZHWL submitted 2025-04-26 eess.SP

Limin Liu , Zhe Li , Qihao Peng , Qu Luo , Pei Xiao , Haowei Wu This is my paper · ORCID

classification eess.SP

keywords AFDMGCE-BEMchannelestimationdoublyselectivechannelsfractionalDopplerinter-DopplerinterferenceMMSEestimatorbasisexpansionmodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

But 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.

Other 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.

Who 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.

For peer review: send it, but flag the rectangular-matrix issue as blocking.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section III-B, Theorem 1, Eq. (6)] 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 III-C, Appendix B, Eq. (10)] 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 IV, Figs. 2-5] 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.

minor comments (5)

[Abstract and Section I] The text contains several typos, including 'adopt' for 'adopted' and 'embeded' for 'embedded'; please proofread the manuscript.
[Section III, Eq. (14)] 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 IV, Fig. 5] The 'AFDM, Analysis' curve is not identified in the caption; state that it is the Jensen lower bound from Theorem 2.
[References] Reference [14] has an incomplete author list ('D. G. G.'); please complete the citation.
[Eq. (10)] The notation in the erfc argument is ambiguous; introduce (1/N)Σ_i T(i,i) as a single variable for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derived NMSE and BER expressions follow from the stated statistical model and standard MMSE/Jensen arguments, not from fitted inputs; the closed simulation loop is a consistency check, and the serious Theta-invertibility gap is a correctness issue, not circularity.

full rationale

The paper's derivation chain is self-contained rather than self-referential. Eq. (6) is the standard Wiener filter for the linear model (5); Eqs. (13)-(14) are algebraic consequences of the MMSE error covariance, and Eq. (10) follows from the per-subcarrier SINR (17) plus Jensen's inequality with the convexity condition taken from [16]. The GCE-BEM representation and its modeling-error covariances are imported from [14], an external prior work by different authors, not from the present paper's own conclusions. No parameter is fitted and then renamed a prediction: the estimator assumes channel statistics R_hh and N0, and the Monte Carlo channels are drawn from the same Jakes statistics, so the agreement in Figs. 2-5 is a consistency check of the algebra, not evidence that the model is circular. Self-citations [3], [11], [16] are background or mathematical lemmas and do not carry the central claim. Separately, I flag a serious omitted/incorrect proof that is not a circularity: Theorem 1 defines R_g = Theta^{-1} R_hh (Theta^H)^{-1} with Theta = B kron I_L, while for the cited parameters (N=256, Q=4, L=2) Theta is 768x15 and has no inverse, so Eq. (6) and Eq. (14) are not well-defined as stated and need the least-squares projection (Theta^H Theta)^{-1} Theta^H. This should be corrected, but it does not make the predictions equivalent to their inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central analysis rests on standard linear algebra, the GCE-BEM model from [14], and the assumption that the receiver has exact channel statistics. No new physical entities or fitted constants are introduced; Q and R are design hyperparameters rather than fitted data parameters. The closed-form NMSE and BER are derived under these assumptions, and the simulations use the same statistics, so the match is a consistency check rather than a test against mismatched models.

free parameters (2)

BEM order Q = Q = 4 (simulation)
The receiver models the channel with Q+1 basis functions; Q constrains the maximum Doppler that can be captured and controls the complexity O(N^2 Q L). It is chosen by hand in the simulations, not fitted to the results, but the performance and complexity claims depend on it.
Oversampling factor R = R = 2 (simulation)
GCE-BEM oversampling controls how finely the Doppler spectrum is sampled; R=2 is selected to satisfy the pilot guard design Q_B = Q + 2 N c1 l_max. It affects the model error and the required BEM order.

assumptions (6)

domain assumption A doubly selective channel is a sum of P discrete paths each with a single complex gain, Doppler shift, and delay.
Eq. (2) in Section II-A models h_n(l)=sum_i h_i e^{-j2 pi f_i n} delta(l-l_i); this excludes diffuse scattering and continuous Doppler spectra.
domain assumption The GCE-BEM representation in Eq. (3) accurately approximates the true time-varying channel with a known, small model error E_mod.
Borrowed from [14]; the MMSE estimator, NMSE expression (14), and BER analysis all treat E_mod as zero-mean noise with covariance Phi R_hh,l Phi^H.
domain assumption The receiver knows the channel covariance R_hh (and per-path R_hh,l) and the noise variance N0.
Theorem 1 (6) and equations (11)-(14) require these statistics; the Monte Carlo simulations supply exactly the statistics used to generate the channel.
ad hoc to paper The function phi(x)=erfc(sqrt(b_M x/(1-x))) is convex over the SINR range of interest.
Theorem 2's Jensen lower bound (10) requires this convexity; the paper cites [16] for convexity only in a high-SINR region but does not quantify the region or check it for all simulated points.
domain assumption Data symbols are independent, zero-mean, and Gray-coded QAM, with covariance R_xd.
Used in the data covariance R_d (11) and in the BER mapping in Theorem 2; standard but stated only implicitly.
standard math AWGN w is zero-mean complex Gaussian with covariance N0 I.
Used throughout the MMSE and SINR derivations; standard assumption.

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Cite this review

Pith. "Pith review of BEM-Assisted Low-Complexity Channel Estimation for AFDM Systems over Doubly Selective Channels." pith.science (2026). https://pith.science/paper/M4JAZHWL

@misc{pith2026250418901,
  author       = {Pith},
  title        = {Pith review of: BEM-Assisted Low-Complexity Channel Estimation for AFDM Systems over Doubly Selective Channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M4JAZHWL}},
  note         = {Machine review of arXiv:2504.18901}
}

read the original abstract

In this paper, we propose a low-complexity channel estimation scheme of affine frequency division multiplexing (AFDM) based on generalized complex exponential basis expansion model (GCE-BEM) over doubly selective channels. The GCE-BEM is used to solve fractional Doppler dispersion.Then, the closed-form expression of channel estimation error is derived for the minimum mean square error (MMSE) estimation algorithm. Based on the estimated channel, the MMSE detection is adopt to characterize the impacts of estimated channel on bit error rate (BER) by deriving the theoretical lower bound. Finally, numerical results demonstrate that the proposed scheme effectively mitigates severe inter-Doppler interference (IDoI). Our theoretical performance analysis can perfectly match the Monte-Carlo results, validating the effectiveness of our proposed channel estimation based on GCE-BEM.

Figures

Figures reproduced from arXiv: 2504.18901 by the authors.

**Figure 2.** NMSE for channel estimation at different speeds. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 5.** BER performance of AFDM and OTFS systems based on GCE-BEM, [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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