{"id":"09e22e57-c1ff-4bcd-8fb2-b81593ad7e62","arxiv_id":"2501.07724","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A doubly-dispersive MIMO channel model parametrized by stacked intelligent metasurfaces and RISs, with optimized surface phases improving BER for OFDM, OTFS, and AFDM.","lead":"This paper introduces a mathematical model of MIMO wireless channels that combine time dispersion and Doppler shift with programmable stacked intelligent metasurfaces at transmitter and receiver and extra reflecting surfaces in the environment. Simulations of OFDM, OTFS, and AFDM then show that optimizing the metasurfaces lowers bit error rates, especially narrowing the gap between classic OFDM and newer waveforms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The SIM-optimization surrogate in eq. (44) omits delay/Doppler structure, so the Fig. 3 BER gains are not yet supported.","rationale":"The reader’s weakest assumption and my strongest concern coincide: the SIM optimization objective in eq. (44) is a power-only, delay-Doppler-blind surrogate, and the paper’s BER improvements are obtained with that objective. This is load-bearing because the central claim goes beyond proposing a channel model; it asserts that the programmable SIM functionality can boost detection performance in DD channels. The purported demonstration of that claim is Fig. 3, and the only optimized quantity feeding that figure is eq. (44), which footnote 11 explicitly says is unaffected by DD effects. The subsequent Frobenius-norm equalization in Section V-D removes a pure total-power advantage, so the reported BER gains would have to come from a reshaping of the delay-Doppler channel that eq. (44) does not directly target. That connection is never tested or argued. I therefore agree with the CONDITIONAL verdict: the modeling and effective-channel derivations are plausible contributions, but the numerical evidence for the performance claim is conditional on an unverified surrogate. I do not rely on the reader’s additional concern that eq. (18) misses a square root, since the coefficient in eq. (18) appears to match the product of the normalizations in eqs. (15) and (16). The unoptimized-SIM assertion in Section IV-E is a separate inconsistency and further supports caution, but the surrogate-validation gap is the decisive issue.","tokens_in":27395,"tokens_out":6507,"duration_ms":68889,"concrete_test":"Run the Fig. 3 SISO and SIMO setups with three SIM phase configurations: (i) phases optimized via eq. (44); (ii) phases optimized via a DD-aware objective, e.g., replacing B_p in eq. (44) with the waveform-dependent G_p factors from eqs. (30), (36), and (42), or maximizing a channel-power/SINR surrogate that includes delay and Doppler; and (iii) unoptimized phases. Apply the same channel-power normalization used in Section V-D in all cases. If configuration (i) does not consistently outperform (iii), or performs worse than (ii), at the SNR values of Fig. 3, then the claimed BER gains are not attributable to the proposed optimization. As an additional direct check, compute the Spearman rank correlation between O(Z, \\tilde{Z}) in eq. (44) and BER over many random phase draws; a nonnegative correlation would falsify the surrogate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The numerical demonstration of SIM-enabled BER gains rests on the surrogate objective in eq. (44): a sum over paths of Frobenius norms of the static spatial matrices formed from \\tilde{h}_p, \\Upsilon_R, \\Upsilon_T, and B_p, with no delay, Doppler, or waveform-dependent factors such as \\Theta_p, \\Omega^{f_p}, or \\Pi^{\\ell_p}. Footnote 11 states explicitly that this objective is “not impacted by DD effects,” and Section V-D then rescales all effective channels so that \\|\\bar{H}_{OFDM}\\|_F^2 = \\|\\bar{H}_{OTFS}\\|_F^2 = \\|\\bar{H}_{AFDM}\\|_F^2 = \\|\\bar{H}_{MIMO}\\|_F^2. The paper’s central performance claim is that optimizing phase shifts with this power-only objective reduces BER for OFDM, OTFS, and AFDM in a doubly-dispersive channel. That claim requires an unstated assumption: that concentrating per-path power also improves the delay-Doppler structure of the effective channel in a way that helps the GaBP detector, despite the fact that the waveform-specific channel-building matrices in eqs. (30), (36), and (42) are exactly the objects omitted from eq. (44). This is not established. A power-focused phase alignment could, for example, strengthen a few delay-Doppler taps while changing inter-carrier or inter-symbol interference patterns, and the resulting BER effect need not be beneficial. Because the optimization objective is the only mechanism producing the reported performance gains, the validity of the Fig. 3 conclusion is directly tied to whether eq. (44) is a faithful surrogate for BER in a DD channel.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a metasurface-parametrized doubly-dispersive (MPDD) MIMO channel model in which a transmit ULA is preceded by a stacked intelligent metasurface (SIM), a receive SIM is placed before the receive ULA, and K reconfigurable intelligent surfaces (RISs) are placed in the propagation environment. The end-to-end channel is written as H = Υ_R R_RX^{1/2} (H_d + Σ_k H_RX,k Φ_k H_k,TX) R_TX^{1/2} Υ_T, where each constituent channel is a sum of delay-Doppler paths with UPA steering matrices. The authors derive a discrete-time input-output relation, then specialize it to OFDM, OTFS, and AFDM effective channel matrices. As an application, they optimize the TX/RX SIM phase shifts by gradient ascent on a sum of per-path Frobenius norms, propose a GaBP detector, and report BER simulations showing performance gains with optimized SIMs.","tokens_in":27682,"tokens_out":13520,"duration_ms":135723,"significance":"If the modeling part is correct, the paper offers a useful unified framework for analyzing delay-Doppler MIMO channels with programmable metasurfaces, and the closed-form gradient expressions plus the linear-complexity GaBP detector are attractive practical ingredients. The derivation is detailed and builds on independently published SISO and SIM models, so the central channel-modeling contribution is credible under the stated hardware assumptions. However, the numerical demonstration that optimized SIMs reduce BER rests on a surrogate objective that is explicitly independent of delay-Doppler effects, and the connection from that objective to the reported BER gains is not established. The modeling contribution can stand alone, but the headline performance claim needs additional support before the paper can be accepted.","major_comments":[{"comment":"The optimization objective in Eq. (44) is a sum of per-path Frobenius norms of the static spatial matrices, i.e., it contains only the factors multiplying the waveform-specific delay-Doppler matrices G_p and G_{k,bar p,tilde p} in Eqs. (30), (36), and (42). It omits the CP-phase, Doppler, and cyclic-delay matrices Θ_p, Ω^{f_p}, and Π^{ell_p}. Footnote 11 explicitly states that the objective is \"not impacted by DD effects.\" Section V-D then normalizes all effective channels so that their Frobenius norms are equal. Therefore the BER gains in Fig. 3 can only come from reshaping the delay-Doppler structure or the relative weighting of paths, yet no analysis or ablation shows that the power-only objective is a reliable surrogate for such reshaping. Please add a waveform-aware objective (e.g., one that includes the G_p factors) or a control experiment with random or fixed phase configurations under the same Frobenius-norm normalization, and explain the mechanism by which per-path spatial power focusing improves GaBP detection in a doubly dispersive channel.","section":"V-A, V-D, Fig. 3"},{"comment":"The index ranges in the gradient derivations are difficult to follow and should be verified. For example, the product in Eq. (49) runs over q' = 1 to q+1 with terms Ψ_{Q-q'+1}Γ_{Q-q'+1}, which, for small Q, does not obviously isolate the layer Ψ_q that is being differentiated; the definitions of S_q and S_tilde_q in Eqs. (50) and (54) are likewise non-obvious. Since the gradient updates in Algorithm 1 drive the headline performance results, a worked small-Q example or a corrected index range would remove any doubt about the correctness of the optimization procedure.","section":"V-A, Eqs. (49)-(54)"}],"minor_comments":[{"comment":"The statement that \"unoptimized SIMs have no effect onto the DD channels\" should be qualified: even with Ψ_q = I, the transfer matrices Υ_T and Υ_R in Eqs. (6)-(7) contain the diffraction matrices Γ_q, which are not identity and therefore change the channel relative to a system with no SIMs. If the comparison with [8] is made after some normalization or calibration, that procedure should be stated explicitly.","section":"IV-E"},{"comment":"There is a typo in the sentence \"It be shown here that SIMs can significantly lower the performance gap\" and a grammatical error in \"the design of waveforms suitable to mitigating\" later in the same paragraph; both should be corrected.","section":"V-B"},{"comment":"The simulation setup for Fig. 3 is not fully reproducible: the text specifies N, K, K', P, N_T, and N_R, but not the CP length, the maximum delay and Doppler values used to generate the random paths, the GaBP iteration count, the gradient-ascent iteration count, or the learning-rate schedule. Please add these settings either in the text or in the figure captions.","section":"V-D"},{"comment":"Footnote 11 contains grammatical errors (\"significantly improvement\" and \"described by in Section IV\") and, more importantly, asserts that the DD-agnostic nature of the optimization \"further validates the overall contribution\" without explaining why that follows; the sentence should be rewritten to state the intended implication clearly.","section":"Footnote 11"},{"comment":"For the record, I checked the cascaded-path normalization in Eq. (18): the factor J√(M M̃/(P̄ P̃)) is the product of the normalization constants of Eqs. (15) and (16), so I do not find a missing square root there.","section":"Eq. (18)"}],"recommendation":"major_revision","confidential_remarks":"The modeling contribution is solid and will likely interest the SIM/RIS and OTFS/AFDM communities. My reservation concerns the headline performance claim: the objective in Eq. (44) is explicitly delay-Doppler-agnostic, yet the paper uses it to draw conclusions about BER in doubly dispersive channels. I do not see grounds for rejection—the central model equations are internally consistent and there is no data-fitting circularity—but the performance demonstration needs either a waveform-aware objective, a control experiment under the same normalization, or a clear argument for why per-path power concentration should help GaBP detection. If the authors provide that, I expect the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Carl,\n\nThe useful part of this paper is the channel model. It takes the unified SISO delay-Doppler waveform framework from [8] and pushes it to MIMO with TX/RX SIMs and ambient RISs, giving explicit effective channels for OFDM, OTFS, and AFDM in eqs. (30), (36), and (42). The Kronecker decomposition is clean, the special cases in Remark 1 are genuinely helpful, and the derivation is reproducible algebra from published external results rather than a fitting exercise. If you work on metasurface-parametrized DD channels, this is likely the reference you will want to cite.\n\nThe soft spot is the performance claim. The SIM optimization in eq. (44) maximizes the sum of per-path Frobenius norms of the static spatial matrices, with no delay, Doppler, or waveform-dependent factors. Footnote 11 says this outright. Then Section V-D normalizes all effective channels to the same total Frobenius power. So the BER gains in Fig. 3 cannot come from total received power; they must come from the optimization reshaping the channel in a way that happens to help the GaBP detector. That is plausible, but it is not argued or tested. A power-only objective could just as well concentrate energy on taps that make inter-carrier or inter-symbol interference worse. Unless the authors show that the optimized channel improves the delay-Doppler structure directly, or they optimize a metric that includes the G_p matrices, the Fig. 3 headline is not supported. This is a load-bearing gap for the numerical section, though not for the modeling contribution.\n\nThere are also two smaller things. The claim that unoptimized SIMs have no effect on the channel sits oddly with the model, since the diffraction matrices Γ_q are not identity even with zero phase shifts; it probably needs a clarifying sentence. And eq. (18) looks like it has a normalization typo for the cascaded RIS path – worth checking against eqs. (15) and (16).\n\nThe central modeling equations hold together. The paper is honest about limitations (footnote 11, footnote 15) and the citation practice is normal. I would send it to review: the model deserves to be in the literature, and the authors should be pushed to fix the surrogate problem or soften the BER claims.\n\n— [Your name]","headline":"A solid modeling extension of SISO DD waveforms to SIM/RIS MIMO, but the SIM-optimization surrogate ignores delay-Doppler structure, so the reported BER gains need an additional argument.","tokens_in":28346,"tokens_out":7835,"would_cite":true,"duration_ms":66355,"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":"The paper introduces a unified metasurfaces-parametrized doubly-dispersive MIMO channel model that covers OFDM, OTFS, and AFDM, and shows that optimized stacked intelligent metasurfaces reduce bit-error rates and narrow the performance…","keywords":["doubly-dispersive channels","stacked intelligent metasurfaces","reconfigurable intelligent surfaces","MIMO channel modeling","OFDM","OTFS","AFDM","Gaussian belief propagation"],"falsifier":"Take a fixed MPDD channel realization, run the SIM phase optimization, then randomly permute the delay and Doppler phases of the paths while keeping their amplitudes and the total channel power fixed, and measure BER for OFDM, OTFS, and AFDM; if BER follows the optimized objective rather than the permutation, the power surrogate is supported, and if not, the reported gains depend on an artifact of the objective.","tokens_in":27160,"feed_emoji":"📡","tokens_out":6683,"duration_ms":59224,"temperature":0.7,"pith_summary":"This paper proposes a metasurfaces-parametrized doubly-dispersive (MPDD) MIMO channel model that places stacked intelligent metasurfaces (SIMs) at both link ends and any number of reconfigurable intelligent surfaces (RISs) in the environment. The model yields explicit end-to-end input-output relationships for arbitrary time-domain signals and for OFDM, OTFS, and AFDM waveforms, expressed through effective channel matrices with the same Kronecker structure. The paper's central claim is that this one parametrized model lets a designer program SIM and RIS phase configurations and then design detection and estimation algorithms once for all three waveforms. As an application, the paper optimizes transmit and receive SIM phases by gradient ascent and detects data with a Gaussian belief propagation receiver, reporting bit-error-rate gains that bring OFDM close to OTFS and AFDM in high-mobility channels. A sympathetic reader would care because it offers a path from separately modeled metasurface links to a unified, mobility-aware MIMO framework for next-generation systems.","feed_headline":"Metasurfaces tame Doppler-spread channels with one model","feed_subtitle":"A single MIMO channel model covers OFDM, OTFS, and AFDM, and optimized surfaces close the BER gap between them.","key_machinery":"The load-bearing machinery is the Kronecker-sum factorization of the effective channel into a small spatial matrix $\\check{\\mathbf{H}}_p$, built from SIM transfer functions, diffraction coefficients, array responses, and beamformers, and a per-path time-frequency matrix $\\mathbf{G}_p = \\boldsymbol{\\Theta}_p \\boldsymbol{\\Omega}^{f_p} \\boldsymbol{\\Pi}^{\\ell_p}$ that encodes the delay and Doppler of each path. Each waveform's effective matrix is obtained by conjugating $\\mathbf{G}_p$ with its own transform: the DFT for OFDM, the discrete Zak transform for OTFS, and the discrete affine Fourier transform for AFDM, yielding the same structural form in the effective channel equations. The same factorization lets the SIM phase gradients factor through the SIM layers, giving closed-form gradients that drive the steepest-ascent optimization of the SIM phase configurations.","core_discovery":"The paper's central discovery is an end-to-end channel model in which the doubly dispersive MIMO channel factorizes as $\\mathbf{H}(Z, \\tilde{Z}, \\mathcal{F}, t, \\tau) = \\boldsymbol{\\Upsilon}_R(\\tilde{Z}) \\mathbf{R}_{RX}^{1/2} \\tilde{\\mathbf{H}}(\\mathcal{F}, t, \\tau) \\mathbf{R}_{TX}^{1/2} \\boldsymbol{\\Upsilon}_T(Z)$, with $\\boldsymbol{\\Upsilon}_T$ and $\\boldsymbol{\\Upsilon}_R$ the diffraction-based transfer matrices of the transmit and receive SIMs, $\\mathbf{R}_{TX}$ and $\\mathbf{R}_{RX}$ spatial correlation matrices, and $\\tilde{\\mathbf{H}}$ the RIS-parametrized delay-Doppler channel containing direct and reflected paths. From that factorization, the effective OFDM, OTFS, and AFDM channel matrices all take the form $\\bar{\\mathbf{H}} = \\sum_p \\check{\\mathbf{H}}_p^d \\otimes \\mathbf{G}_p^{\\text{waveform}} + \\sum_{k,\\bar{p},\\tilde{p}} \\check{\\mathbf{H}}_{k,\\bar{p},\\tilde{p}}^{\\text{RIS}} \\otimes \\mathbf{G}_{k,\\bar{p},\\tilde{p}}^{\\text{waveform}}$, so the waveform-specific linear transforms, namely the DFT, the discrete Zak transform, and the discrete affine Fourier transform, act only on the per-path delay-Doppler matrices. The paper argues this structure makes waveform comparison and unified receiver design straightforward, and its numerical study shows that optimizing the SIM phase layers with the proposed gradient ascent, followed by its Gaussian belief propagation detector, lowers bit-error rate for all three waveforms and narrows the gap between OFDM and the more sophisticated OTFS and AFDM.","pith_inferences":["The paper's own footnote notes that the optimization objective is blind to delay and Doppler phases; a natural extension is to insert those phases into the objective and test whether dispersion-aware SIM tuning yields larger gains than the power-based tuning reported here.","Because the delay-Doppler blocks carry radar-relevant parameters, the MPDD model is a ready-made input-output model for metasurface-aided integrated sensing and communications, even though sensing-specific estimation is left for future work.","The same factorization likely applies to other doubly dispersive-friendly waveforms such as OCDM and ODDM by substituting their transform for $\\mathbf{G}_p$, so the model may serve as a generic DD-waveform testing platform.","The reported SISO and SIMO bit-error gains suggest a testable field experiment: with fixed total channel power, compare BER for OFDM with and without optimized SIM phases on the same measured doubly dispersive link; if the gains persist, the passive lensing effect is real."],"forward_implications":["If the model is correct, channel estimation, equalization, and detection blocks designed for the MPDD matrix $\\bar{\\mathbf{H}}$ transfer directly between OFDM, OTFS, and AFDM, since only the $\\mathbf{G}_p$ block changes.","Programmable SIMs become a physical-layer resource that can be optimized separately from digital beamformers, augmenting rather than replacing conventional MIMO processing.","With optimized SIMs, OFDM's bit-error rate approaches that of OTFS and AFDM in doubly dispersive channels, so waveform choice matters less when the propagation environment can be programmed.","Increasing the number of SIM layers or receive antennas both improve bit-error rate, and the gains come from passive wave-domain processing rather than increased transmit power, since the compared channels are power-normalized.","The proposed receiver achieves near-LMMSE bit-error rate with per-iteration complexity linear in the number of channel coefficients, avoiding matrix inversion."],"supporting_citations":[{"why":"Supplies the unified SISO doubly dispersive channel model and the per-waveform G matrices that this paper extends to MIMO with SIMs and RISs.","marker":"[8]"},{"why":"Provides the Rayleigh-Sommerfeld diffraction coefficients, SIM transfer-matrix construction, and sinc spatial correlation model used in the end-to-end channel equation.","marker":"[54]"},{"why":"Establishes the electromagnetic-compliant beamforming framework for RISs and SIMs that the channel parametrization follows.","marker":"[52]"},{"why":"Defines the AFDM chirp parameter and waveform model that yields the effective AFDM channel.","marker":"[33]"},{"why":"Supplies the inverse discrete Zak transform formulation used to build the OTFS transmit signal and its effective channel.","marker":"[63]"},{"why":"Provides the gradient ascent technique for updating SIM phase shifts adopted in the SIM optimization algorithm.","marker":"[70]"}],"fun_headline_variants":["Metasurface model unifies OFDM, OTFS, and AFDM on doubly-dispersive links","Stacked metasurfaces close BER gap between OFDM and OTFS","Single channel model covers RIS and SIM for high-mobility MIMO","Metasurfaces unify Doppler-spread MIMO for OFDM, OTFS, AFDM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that maximizing the sum of per-path channel-matrix powers in the optimization objective, an objective the paper notes is not affected by delay or Doppler phases, is a valid proxy for lowering bit error rate in a doubly dispersive channel.","fun_headline_variants_meta":{"raw":{"variants":["Metasurface model unifies OFDM, OTFS, and AFDM on doubly-dispersive links","Stacked metasurfaces close BER gap between OFDM and OTFS","Single channel model covers RIS and SIM for high-mobility MIMO","Metasurfaces unify Doppler-spread MIMO for OFDM, OTFS, AFDM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000689,"raw_usage":{"total_tokens":3256,"prompt_tokens":1217,"completion_tokens":2039,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":833,"completion_tokens_details":{"reasoning_tokens":1948}},"tokens_in":833,"tokens_out":2039,"duration_ms":12891,"temperature":1.0,"reasoning_tokens":1948,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:37:50.739380+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed MPDD channel realization, run the SIM phase optimization, then randomly permute the delay and Doppler phases of the paths while keeping their amplitudes and the total channel power fixed, and measure BER for OFDM, OTFS, and AFDM; if BER follows the optimized objective rather than the permutation, the power surrogate is supported, and if not, the reported gains depend on an artifact of the objective.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the unified SISO doubly dispersive channel model and the per-waveform G matrices that this paper extends to MIMO with SIMs and RISs."},{"cited_title":"Stacked Intelligent Metasurfaces for Efficient Holo- graphic MIMO Communications in 6G,","cited_arxiv_id":null,"evidence_quote":"Provides the Rayleigh-Sommerfeld diffraction coefficients, SIM transfer-matrix construction, and sinc spatial correlation model used in the end-to-end channel equation."},{"cited_title":"Stacked Intelligent Metasurface-aided MIMO Transceiver Design,","cited_arxiv_id":null,"evidence_quote":"Establishes the electromagnetic-compliant beamforming framework for RISs and SIMs that the channel parametrization follows."},{"cited_title":"Orthogonal Time Frequency Space Mod- ulation,","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse discrete Zak transform formulation used to build the OTFS transmit signal and its effective channel."},{"cited_title":"Stacked Intelligent Metasurface performs a 2D DFT in the Wave Domain for DOA Estimation,","cited_arxiv_id":null,"evidence_quote":"Provides the gradient ascent technique for updating SIM phase shifts adopted in the SIM optimization algorithm."}],"review_version":1}