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REVIEW 3 major objections 6 minor 45 references

Channel Estimation in Massive MIMO Systems with Orthogonal Delay-Doppler Division Multiplexing

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that a two-stage MAMP-plus-DFT estimator can approach the Bayes-optimal channel estimation accuracy in massive MIMO-ODDM systems as the antenna count grows, with about 30% NMSE improvement over OMP and 3D-OMP baselines.

desk verdict A useful effective model and a plausible low-complexity estimator, but the Bayes-optimality claim rests on an unproven randomness premise and a mismatched real-valued NLE; the engineering gains over OMP likely survive. read the letter →

arxiv 2507.19812 v1 pith:TXVJ74QA submitted 2025-07-26 eess.SP

Dezhi Wang , Chongwen Huang , Xiaojun Yuan , Sami Muhaidat , Lei Liu , Xiaoming Chen , Zhaoyang Zhang , Chau Yuen
show 1 more author
Mérouane Debbah
This is my paper · ORCID
classification eess.SP
keywords massiveMIMOODDMmodulationchannelestimationmemoryapproximatemessagepassingDFTangleBernoulli-Gaussianpriornormalizedmeansquareerror
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

The paper claims that channel estimation in massive MIMO-ODDM systems can be both accurate and low-complexity: a two-stage estimator first recovers sparse delay, Doppler, and gain information with memory approximate message passing (MAMP), then recovers angles with a discrete Fourier transform and a phase-rotation refinement. The argument rests on an effective model in which the equivalent channel vector has Bernoulli-Gaussian magnitudes and the equivalent coefficient matrix becomes effectively random as the number of antennas grows. On that basis the paper claims the MAMP stage approaches the Bayesian-optimal mean square error, and reports about 30% normalized-MSE improvement over orthogonal matching pursuit baselines. A sympathetic reader would care because high-mobility links change quickly, so an estimator that is linear-complexity and near-optimal at the same time would make ODDM practical for fast-moving terminals.

What carries the argument

The load-bearing object is the equivalent coefficient matrix $\tilde{\Phi}$ formed by concatenating antenna-specific circulant-block matrices $\Phi_{n_t}$ in Eq. (12). The paper's premise is that, as the antenna count grows, the entries of $\tilde{\Phi}$ approach completely random, which is exactly the regime in which MAMP is known to be Bayes-optimal. The machinery then consists of MAMP's long-memory linear estimator with recursively chosen relaxation and damping parameters, a nonlinear estimator derived for the Bernoulli-Gaussian prior, and a DFT-based angle estimator with a rotation matrix $F_{N_t}^{\Delta\theta_p}$ that counters power leakage.

What would settle it

Simulate the MAMP stage on the block-circulant matrices of Eq. (12) at $N_t = 64, 128, 256, 512$ and compare the steady-state NMSE against an exact MMSE reference at fixed SNR: if the gap does not shrink with $N_t$, or if a statistical-distance measure between the entries of $\tilde{\Phi}$ and a Gaussian random matrix does not decrease toward zero, the 'completely random' premise is falsified.

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Extended reading notes

Core claim

The central claim is that the massive MIMO-ODDM channel estimation problem, written as $y = \tilde{\Phi}\tilde{h} + n_c$, can be solved by a two-stage algorithm that separates sparse recovery from angular recovery. The first stage applies MAMP to the effective channel vector, whose nonzero entries' magnitudes follow a Bernoulli-Gaussian law, to obtain the gains, delays, and Doppler shifts; the second stage applies a normalized DFT with an optimized phase rotation to read off the angles from the angularly sparse received vector. The paper further claims that as $N_t \to \infty$, the symbols carried by independent antennas randomize the effective coefficient matrix enough for MAMP's long-memory orthogonalization to reach the Bayes-optimal MSE, and that numerical results show the estimator approaching that limit and improving NMSE by about 30% over OMP and 3D-OMP.

Load-bearing premise

The load-bearing premise is Remark 1's claim that the structured coefficient matrix becomes completely random as antennas grow; if that does not happen at practical array sizes, the near-Bayes-optimality guarantee falls away, even if the gains over OMP persist.

Editorial extensions

If this is right

  • If correct, massive MIMO-ODDM channel estimation can run at complexity linear in the problem dimension while meeting the MMSE limit, which is what fast CSI refresh at high mobility needs.
  • The paper's BER results indicate that with the estimated CSI, ODDM stays close to the perfect-CSI bound and beats OTFS, especially as speed and subcarrier count vary.
  • The NMSE gap between the proposed method and traditional MAMP with a fully random matrix closes as antennas grow, which the paper presents as empirical evidence for the randomness premise.
  • Across the tested antenna counts, path counts, speeds, and SNRs, the proposed estimator keeps an about 30% NMSE advantage over both OMP baselines.

Reading between the lines

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

  • Editorial inference: the randomness premise is testable directly by measuring how fast the empirical entry distribution or spectral statistics of $\tilde{\Phi}$ converge to those of a random matrix as $N_t$ increases; the paper does not quantify this rate.
  • Editorial inference: because the pilot symbols are described as randomly generated within a fixed set, pilot design is a hidden control knob; if pilots are chosen adversarially or with structure, the Bernoulli-Gaussian and randomness assumptions could degrade.
  • Editorial inference: the DFT angle stage's validity is tied to the far-field ULA assumption and $d_{BS} \le \lambda_c/2$; applying the same split to near-field or spherical-wavefront massive MIMO would need a different angular basis.
  • Editorial inference: the single-user formulation omits multi-user interference, so an extension to multiple users would need to justify why the effective prior remains Bernoulli-Gaussian after interference is modeled.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper studies downlink channel estimation for massive MIMO-ODDM systems. The authors construct an effective sparse channel model whose element magnitudes are Bernoulli-Gaussian and propose a two-stage estimator: a memory approximate message passing (MAMP) stage for gains, delays, and Doppler, and a DFT-based stage for angle estimation. The paper claims that the proposed algorithm approaches Bayesian optimality as the number of antennas tends to infinity and reports roughly 30% NMSE improvement over OMP and 3D-OMP baselines, together with BER comparisons of ODDM against OTFS and OFDM.

Significance. If established, the main contribution would be a low-complexity channel estimator that approaches the MMSE limit for a structured massive MIMO-ODDM problem, which is a nontrivial extension of MAMP to a non-i.i.d. sensing model. The paper includes a useful complexity analysis, extensive NMSE and BER simulations across SNR, user speed, antenna count, number of paths, modulation order, and channel models, and comparisons against OMP, 3D-OMP, and a random-matrix MAMP benchmark. However, the advertised Bayesian-optimality result is currently supported only by an unproven random-matrix assertion and a scalar real-Gaussian NLE that does not match the stated complex model; these are load-bearing issues for the headline claim rather than presentation defects.

major comments (3)
  1. [Section III, Remark 1] The premise that the elements of the equivalent coefficient matrix Phi_tilde 'approach completely random' as Nt grows is asserted without proof, and it is load-bearing because MAMP's Bayesian optimality is established for i.i.d. or right-unitarily invariant sensing matrices, not for the structured blocks in Eq. (12). The reasoning in Remark 1 counts independent pilot symbols per antenna, but footnote 4 states that these pilots are fixed and belong to a defined set, and the symbols enter through cyclic-shift and DFT-structured blocks; independence of the generating symbols does not by itself imply i.i.d. or asymptotically free entries of the resulting matrix. Please either prove a quantitative randomness property (e.g., convergence of the empirical singular value distribution, or asymptotic freeness of the relevant block-circulant operators) or remove/weaken the optimality claim.
  2. [Section IV-A, Theorem 1 and Remark 2; Appendix B] The NLE in Theorem 1 is derived for a real scalar observation y = b g + n with real Gaussian g and n, as shown in Appendix B, which uses (y-u_g)^2, y^2, and real densities. The system model (9)-(10) and the effective model (16) are complex: y, Phi_tilde, and h_tilde are complex, and steering phases enter h_tilde through F_nt in Eq. (11); Remark 2 itself concedes that only the magnitudes of the elements are Bernoulli-Gaussian, not the elements. Therefore the posterior update (52)-(53) is not the posterior mean of the stated model, so the algorithm does not compute the MMSE estimate of h_tilde even in the random-matrix limit. A complex-valued derivation or an explicit real-valued model with the correct likelihood is required.
  3. [Section IV, Eq. (20); Section V-B, Fig. 8] The Bayesian-optimality comparison is partly internal: Eq. (20) defines the target MSE under the same Bernoulli-Gaussian prior used to construct the NLE, and Fig. 8 compares the proposed estimator with 'traditional MAMP' running on a fully random matrix rather than with an oracle MMSE bound for the actual structured matrix. The observed convergence in Fig. 8 shows that the structured matrix begins to behave like a random matrix in this test, but it does not establish that the algorithm achieves the MMSE of the true model, nor does it quantify the gap at finite Nt. Please add an explicit MMSE baseline (or a state-evolution prediction) for the actual matrix and report the gap.
minor comments (6)
  1. [Section III] The paragraph beginning 'Further, let Phi_tilde = ...' and Eqs. (16)-(17) are duplicated verbatim; remove one copy.
  2. [Section IV, Theorem 1 and Section V] The Bernoulli-Gaussian prior parameters (p, u_g, v_g) and the noise variance sigma^2 are not specified in the numerical setup; state how these values are set or estimated, since the NLE is sensitive to them.
  3. [Section V, Fig. 3] The sentence 'The NMSE of all algorithms shows an increased trend with higher SNRs' appears to contradict the plotted decrease of NMSE; correct the wording and the garbled axis labels in Fig. 3.
  4. [Section IV-C, Algorithm 1] The complexity comparison should clarify that O(MN(2K+1)LNtTt) is per iteration, state whether Phi_tilde Phi_tilde^H is precomputed, and include the cost of the one-dimensional angle-rotation search in Algorithm 1.
  5. [Section IV-A, Lemma 1] Lemma 1 is stated for an arbitrary matrix C, but the proof uses unitary diagonalization, which is only valid for normal matrices; since the lemma is applied to the Hermitian matrix theta_t B, state that assumption explicitly or prove the Neumann series directly.
  6. [References and figures] References [13] and [35] appear to cite the same paper; merge them and ensure the year and venue are consistent. Also fix the typo 'prefect' in Fig. 9 and check whether the last legend entry in Fig. 12 should be 'OTFS 350 km/h ETU'.

Circularity Check

1 steps flagged · score 4.0 of 10

The 'Bayesian optimal' claim is partly self-referential: the target is defined as MAMP on a random matrix, and the randomness of the actual ODDM matrix is inferred from convergence to that same target; the external OMP/3D-OMP comparisons are non-circular.

  1. other [Section V-B (Fig. 8) and Remark 1 (Section III)]
    "In the traditional MAMP scheme, the elements in the effective channel matrix are entirely randomly generated and can achieve Bayesian optimal results. ... It is evident that as the number of antennas increases, the NMSE performance of the proposed algorithm converges towards that of the traditional MAMP scheme. This convergence trend implies that, as the number of antennas approaches infinity, the effective channel matrix can be approximated as being completely random."

    The paper operationalizes 'Bayesian optimal results' as the NMSE of MAMP on an entirely random matrix, then observes that the proposed algorithm's NMSE converges to that same MAMP-on-random curve, and finally concludes that the ODDM coefficient matrix becomes completely random and that the algorithm approaches Bayesian optimality. Because the target is the same algorithm applied to a different, fully random sensing matrix, the 'approaches Bayesian optimal' claim is a self-referential benchmark comparison rather than a check against the true MMSE of the actual structured matrix. The randomness premise of Remark 1 is inferred from the very convergence it is invoked to explain.

full rationale

Most of the paper is not circular: the ODDM system model, the effective channel formulation, and the NMSE/BER comparisons against OMP, 3D-OMP, and OTFS are independent of the algorithm's construction and provide external evidence of practical gains. The circularity concern is concentrated in the 'Bayesian optimality' claim. Equation (20) defines the target MMSE as the posterior mean under an assumed Bernoulli-Gaussian prior, and Theorem 1 constructs the NLE as that posterior mean, so the optimality claim is partly a definitional restatement under the assumed model. More significantly, the asymptotic conclusion that the algorithm 'approaches the Bayesian optimal results' is validated in Fig. 8 by comparison with traditional MAMP on a completely random matrix; the paper then uses convergence to that self-defined target to infer that the actual ODDM coefficient matrix becomes random. This makes the central optimality claim partially self-referential. The unsupported nature of Remark 1 and the mismatch between the real-valued scalar NLE and the stated complex model are correctness risks rather than additional circularity. The external baseline improvements are genuine and keep the overall circularity burden moderate.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The algorithm rests on a set of modeling assumptions: real Gaussian channel gains, a Bernoulli-Gaussian prior with known parameters, known noise variance, integer delay-Doppler grid, a single dominant path per grid point, and an unproven asymptotic randomness of the equivalent coefficient matrix. No new physical entities are introduced; the effective channel vector and angle rotation matrix are mathematical constructs.

free parameters (3)
  • Bernoulli-Gaussian prior parameters (p, u_g, v_g) = not specified
    The NLE in Theorem 1 assumes h_i = b g with b ~ B(p) and g ~ N(u_g, v_g). The paper does not state how p, u_g, and v_g are obtained; in simulations they must be set from channel statistics, and in practice they would need estimation.
  • Noise variance sigma^2 = assumed known
    Used in the LLE via rho_t = sigma^2 / nu and in the NLE formulas of Theorem 1; no estimation procedure is given.
  • Angle rotation parameters Delta_theta_p = obtained via 1D search in Eq. (63)
    The angle refinement step searches for Delta_theta_p per path to minimize power leakage; this is an optimization parameter rather than a physical constant.
assumptions (6)
  • ad hoc to paper Channel gains are real Gaussian rather than complex Gaussian (footnote 1).
    Invoked to obtain closed-form posterior mean and variance in Theorem 1; excludes complex gains used in physical wireless channel models.
  • domain assumption Equivalent coefficient matrix elements become completely random as Nt approaches infinity (Remark 1).
    No proof is given; the matrix has repeated circulant-like blocks, so the claim is heuristic and underpins the Bayes-optimality conclusion.
  • domain assumption Pilot symbols across antennas are independent and randomly generated (footnote 4).
    Needed for the randomness claim in Remark 1; pilots are described as fixed but randomly generated within a set, which is not the same as independent random sensing in MAMP theory.
  • domain assumption Each delay-Doppler grid point maps to a single dominant path with an angle assigned from alpha_p after quantization (footnote 3).
    Assumes a one-to-one path-to-grid mapping and ignores off-grid leakage and unresolved multipath within one grid cell.
  • domain assumption Integer delay and Doppler shifts on the resolution grid (Eq. (1)).
    The model assumes tau_p = k_p T/M and nu_p = l_p 1/(NT) with integers; fractional delay and Doppler are not modeled.
  • domain assumption Pulse orthogonality condition in Eq. (3), borrowed from prior ODDM work.
    Required for the simplified input-output relationship; the validity depends on the square-root Nyquist pulse design from [11].

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Cite this review

Pith. "Pith review of Channel Estimation in Massive MIMO Systems with Orthogonal Delay-Doppler Division Multiplexing." pith.science (2026). https://pith.science/paper/TXVJ74QA

@misc{pith2026250719812,
  author       = {Pith},
  title        = {Pith review of: Channel Estimation in Massive MIMO Systems with Orthogonal Delay-Doppler Division Multiplexing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TXVJ74QA}},
  note         = {Machine review of arXiv:2507.19812}
}
read the original abstract

Orthogonal delay-Doppler division multiplexing~(ODDM) modulation has recently been regarded as a promising technology to provide reliable communications in high-mobility situations. Accurate and low-complexity channel estimation is one of the most critical challenges for massive multiple input multiple output~(MIMO) ODDM systems, mainly due to the extremely large antenna arrays and high-mobility environments. To overcome these challenges, this paper addresses the issue of channel estimation in downlink massive MIMO-ODDM systems and proposes a low-complexity algorithm based on memory approximate message passing~(MAMP) to estimate the channel state information~(CSI). Specifically, we first establish the effective channel model of the massive MIMO-ODDM systems, where the magnitudes of the elements in the equivalent channel vector follow a Bernoulli-Gaussian distribution. Further, as the number of antennas grows, the elements in the equivalent coefficient matrix tend to become completely random. Leveraging these characteristics, we utilize the MAMP method to determine the gains, delays, and Doppler effects of the multi-path channel, while the channel angles are estimated through the discrete Fourier transform method. Finally, numerical results show that the proposed channel estimation algorithm approaches the Bayesian optimal results when the number of antennas tends to infinity and improves the channel estimation accuracy by about 30% compared with the existing algorithms in terms of the normalized mean square error.

Figures

Figures reproduced from arXiv: 2507.19812 by the authors.

Figure 1
Figure 1. Graphic illustration of ODDM modulation and demodulation [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The illustration about the phase rotation of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 5
Figure 5. The NMSE performance for different number of antennas [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The NMSE performance for different speeds [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 9
Figure 9. Figure 9: The BER performance versus SNRs for different speeds [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: The BER performance versus SNRs for different numbers of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: The BER performance versus SNRs for different modulations [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: The BER performance versus SNRs for different channel models [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.