{"id":"3ca36c6c-0573-4460-a1e5-f32b9336bf53","arxiv_id":"2504.15042","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A two-layer variational Bayesian inference method estimates delay and Doppler simultaneously in time-varying ISAC channels, improving MSE over prior art.","lead":"This paper proposes a Bayesian method to estimate the distance and speed of multiple fast-moving targets from wireless channel measurements in integrated sensing and communication systems. It uses a two-layer variational inference approach to avoid parameter mismatch, and simulations show lower estimation error than existing methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central MSE-vs-CRB claim rests on an idealized H_fd input; Section II-B promises an error-impact analysis that never appears, so the method is unverified under realistic channel-estimation error.","rationale":"The reader's weakest assumption identifies the same gap that I consider most load-bearing: the algorithm operates on an estimated channel matrix whose error model is promised but never delivered. The two-layer VBI derivation itself is internally coherent as a sparse-recovery procedure on the re-aligned channel matrix, and the complexity reduction argument is reasonable, so I do not see a separate flaw that would invalidate the algorithm when its input is exactly the idealized noisy tilde-H_fd. The CRB derivation, though abbreviated, follows a standard FIM structure. The missing channel-estimation-error analysis is more serious than omitted grid sizes or hyperparameters, because every simulation comparison and the CRB-matching statement inherit the assumption that tilde-H_fd is available as a direct observation. Since the reader already conditioned acceptance on this point, my stress-test does not move the verdict; it reinforces the condition under which the central claim should be accepted.","tokens_in":24087,"tokens_out":8883,"duration_ms":89846,"concrete_test":"Re-run Figs. 3, 4, and 7 replacing the idealized tilde-H_fd input with an estimate H_fd_est produced by a concrete frequency-domain channel estimator (e.g., LS with N orthogonal pilots per OFDM block, or the estimator of [30]), keeping all other simulation settings identical. If the two-layer VBI MSE no longer tracks the CRB within the original margin at SNR=15 dB, or if the gap widens with increasing Doppler, the unmodeled channel-estimation error is the dominant factor and the central claim requires qualification. Also report the MSE of H_fd_est relative to the true H_fd to quantify the error level at which the proposed sensing advantage survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section II-B explicitly says: 'we assume that the estimated channel matrix H_fd is known for developing sensing algorithms, and will discuss the impact of H_fd estimation error on proposed designs.' No such discussion appears in Sections III-VII. This is load-bearing because the strongest claim, that two-layer VBI 'is able to approach the perfect-VBI and CRB' (discussion of Figs. 3-7), is demonstrated only on the re-aligned matrix tilde-H_fd of Eq. (7), which is treated as a direct observation with additive Gaussian noise. In a real ISAC receiver, tilde-H_fd is not observed directly; it must be estimated from pilots under the same ICI that motivates the paper, and that estimation error is neither characterized nor propagated. The CRB in Section V is derived from the received-signal model r(k) = H_fd(k)s(k) + w_1(k) in Eq. (55), not from the noisy channel estimate fed to Algorithm 1, so matching it does not validate the full sensing chain. If realistic channel-estimation errors are correlated across subcarriers or Doppler slices, or are non-Gaussian, the sparse-support recovery in Eqs. (8)-(10) and the MUSIC delay stage in Eqs. (47)-(52) may fail in a way not captured by the current simulations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a frequency-domain sensing method for OFDM-based ISAC under fast time-varying channels. It formulates joint delay, integer Doppler, and fractional Doppler estimation as a 3D multiple-measurement sparse signal recovery problem using the re-aligned channel matrix in Eq. (7). The main contribution is a two-layer variational Bayesian inference (VBI) algorithm that alternates between estimating the row-sparse matrix C(n) and the sparse matrix D(n), thereby avoiding the parameter mismatch of separately estimated integer and fractional Dopplers. A lower-complexity two-stage MUSIC-based VBI variant is also proposed, and a Cramér-Rao bound is derived for the sensing parameters. Simulation results compare the MSE of the proposed methods against FFT, OTFS-based, two-stage VBI, and EM-VB baselines, and show the two-layer VBI approaching the perfect-VBI benchmark and the CRB.","tokens_in":24387,"tokens_out":8721,"duration_ms":78791,"significance":"The contribution is potentially useful: estimating integer and fractional Doppler jointly in the same sparse Bayesian framework is a genuine step beyond the two-step estimation in [31], the two-layer decomposition is an interesting way to avoid vectorizing a 3D dictionary, and the factor-graph presentation with appendices makes the algorithm structure transparent. The paper also provides concrete MSE comparisons over SNR, target number, and velocity, and the simplified MUSIC-VBI variant offers a clear complexity-performance trade-off. The CRB derivation from the frequency-domain input-output model is a useful reference point, although it is not directly the bound for the estimator's actual input. Overall, if the technical errors below are fixed and the missing error analysis is supplied, the work would be a solid contribution to ISAC sensing in time-varying channels.","major_comments":[{"comment":"Section II-B explicitly states that the estimated channel matrix H_hat_fd is assumed known and that the impact of its estimation error 'will be discussed', but no such discussion appears in Sections III-VII. The entire sensing pipeline (Algorithms 1 and 2) consumes the re-aligned channel matrix tilde_H_fd of Eq. (7) as a direct noisy observation, and the CRB in Section V is derived from r(k)=H_fd(k)s(k)+w_1(k) in Eq. (55), not from an estimated channel matrix. Consequently, the claimed 'approach the perfect-VBI and CRB' behavior reported in Section VI-A is demonstrated only for an idealized input. A realistic channel-estimation error, especially correlated or non-Gaussian error, is neither characterized nor propagated. I request either an explicit error analysis for the sensing estimators under channel-estimation error, an end-to-end simulation that estimates H_hat_fd from pilots under the same ICI, or a revised claim that restricts the results to the case where the channel matrix is available.","section":"II-B and Section V"},{"comment":"The derivation of q(alpha) in Eq. (75) contains '+ N ln p(alpha)', which multiplies the Gamma prior by N and is inconsistent with the joint distribution in Eq. (28), where p(alpha) appears once. This leads to the posterior shape parameter a+MNQ in Lemma 1; the shape parameter should reflect the number of scalar observations in the likelihood, and the displayed formula introduces a Q-dependent term that is not present in the likelihood. The same error pattern appears in Lemma 3, where the shape is written as a+N P Q. Since these shape parameters enter every VBI iteration, the update equations as printed are not correct as written. Please correct the derivations or justify the additional factors.","section":"III-D, Lemma 1 and Appendix B"},{"comment":"The pseudo-inverse in Eq. (54) is written as hat_A_tau^dagger = hat_A_tau (hat_A_tau^H hat_A_tau), which is dimensionally N by L and misses the inverse. Right-multiplying Eq. (53) to remove hat_A_tau^H requires the right inverse (hat_A_tau^H)^dagger = hat_A_tau (hat_A_tau^H hat_A_tau)^{-1}. As written, Eq. (54) does not follow from Eq. (53), and the derivation of the simplified two-stage MUSIC-VBI method is therefore incomplete.","section":"IV, Eq. (54)"},{"comment":"The delay grid size P, the Doppler grid size Q, the grid spacings, and the Gamma hyperparameters a, b, c, d, e, f of Section III-B are never specified in Section VI. Without these values the MM-SSR formulation, the complexity expressions, and the simulation results cannot be reproduced. In addition, Algorithm 1, Step 12, computes the outer-loop convergence test by dividing a change in Gamma_c by ||Gamma_x2(n)||^2, mixing two different precision matrices; this appears to be a typo and should be corrected to a consistent norm.","section":"VI and Algorithm 1"}],"minor_comments":[{"comment":"The sentence 'It is evident that the MSE degrades as the SNR increases' states the opposite of the expected and plotted trend; it should say that the MSE improves (decreases) as the SNR increases.","section":"VI-A, Fig. 6 discussion"},{"comment":"The convergence criterion divides the Gamma_c residual by ||Gamma_x2(n)||^2, which is inconsistent; the denominator should involve the same matrix as the numerator.","section":"Algorithm 1, Step 12"},{"comment":"The noise term in Eq. (50) is written as sigma^2 I_M, but M is not defined; from context it should be sigma^2 I_N.","section":"IV, Eq. (50)"},{"comment":"The symbol M is used for the number of columns of C(n) and Y(n), which equals N in this model; this conflicts with the use of N as the number of subcarriers and should be clarified or renamed.","section":"III-B and III-D"},{"comment":"Reference [45] is incomplete: it lacks the article title, volume, and page numbers, which prevents the reader from locating it.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the algorithmic idea is sound, but the advertised system-level validation currently rests on an idealized channel-estimate assumption, and the update equations contain derivation errors that need correction. I recommend major revision rather than rejection; the missing channel-estimation-error analysis is the key risk. The authors should also be required to state all grid sizes, spacings, and prior hyperparameters so that the simulations are reproducible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look if you care about high-mobility ISAC sensing. The genuinely new thing here is the 3D MM-SSR formulation and the two-layer VBI decomposition: prior SBL-ISAC methods were time-invariant, and the frequency-domain baseline [31] estimates integer and fractional Doppler separately, creating a mismatch problem. This paper estimates delay, integer Doppler, and fractional Doppler simultaneously from the re-aligned frequency-domain channel matrix. The update equations in Lemmas 1–5 are internally consistent, the two-layer structure avoids vectorizing the full dictionary, and the simplified MUSIC-based variant is a sensible complexity/performance trade. The CRB derivation is standard and looks correct for the assumed model.\n\nThe soft spot is the one the stress-test flags, and it is real. Section II-B explicitly says the impact of H_fd estimation error on the proposed designs will be discussed. It never is. The algorithm ingests the re-aligned estimated channel matrix as if it were a direct observation with i.i.d. Gaussian noise, but in practice that matrix comes from pilots under the same ICI that motivates the paper. Correlated or non-Gaussian channel-estimation errors could break the sparse-support recovery, and the CRB is derived from the received-signal model, not from the noisy channel estimate fed to Algorithm 1. Matching the CRB therefore validates only the sensing algorithm on an idealized input, not the full sensing chain. This is load-bearing, but it is fixable: a proper error-propagation analysis or at least simulations with realistic channel-estimation front-ends would address it.\n\nAlso minor but annoying: the delay/Doppler grid sizes P and Q and the Gamma hyperparameters a–f are never specified, N=8, K=8 for the main simulations, and no code is provided. That makes the central MSE-vs-CRB results hard to reproduce. The Fig. 4 sentence about degrading \"about 6 dB\" is confusing as written. The authors lean heavily on their own [31], but that is appropriate because the signal model is inherited from it.\n\nWho is this for? People working on frequency-domain sensing for time-varying ISAC channels. It is not a new physical mechanism and the impact is bounded by the importance of the parameter-mismatch problem, but within that niche it is a credible algorithmic advance. The paper deserves a serious referee. My recommendation: send it out, and require the authors to either deliver the promised error analysis or honestly scope the claims to the perfect-channel-estimate setting, plus specify all implementation parameters.","headline":"Real algorithmic idea in a crowded subfield, but the central claim is validated only under a perfect-channel-estimate assumption the paper promises to relax and never does.","tokens_in":24932,"tokens_out":1861,"would_cite":false,"duration_ms":19883,"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 claims a two-layer variational Bayesian estimator can recover delay, integer Doppler, and fractional Doppler simultaneously from the re-aligned frequency-domain channel matrix, avoiding the integer–fractional Doppler mismatch of…","keywords":["integrated sensing and communication","time-varying channels","sparse Bayesian learning","variational Bayesian inference","Doppler estimation","delay estimation","multiple measurement sparse signal recovery","Cramér-Rao bound"],"falsifier":"Run the same Doppler MSE experiment but feed $\\hat{\\mathbf H}_{fd}$ from an actual least-squares channel estimator (for example, with known pilot symbols) instead of the perfect matrix assumed in Section II-B; if the two-layer VBI's MSE no longer approaches the Cramér-Rao bound at high SNR, the paper's central performance claim depends on an unmodeled error source.","tokens_in":23896,"feed_emoji":"📡","tokens_out":9731,"duration_ms":80064,"temperature":0.7,"pith_summary":"This paper sets out to show that in a fast-fading OFDM-based ISAC system, the delay, integer Doppler, and fractional Doppler of multiple targets can be recovered at once from the re-aligned frequency-domain channel matrix by solving a three-dimensional multiple-measurement sparse-signal-recovery problem. The proposed two-layer variational Bayesian inference alternates between a Doppler layer and a delay layer, so the three parameters are read from the same nonzero entries of a sparse tensor and are matched by construction. A simplified two-stage variant first estimates delays with MUSIC and then estimates Doppler with VBI at lower complexity. The authors derive the Cramér-Rao bound for these parameters and their simulations show the two-layer estimator approaching that bound and outperforming two-stage and OTFS-based benchmarks. If this is right, ISAC receivers in high-mobility links could sense targets without a separate Doppler-association step.","feed_headline":"Two-layer Bayesian method estimates Doppler and delay jointly","feed_subtitle":"It avoids the integer–fractional Doppler mismatch of prior two-stage ISAC sensing and nears the Cramér–Rao bound.","key_machinery":"The load-bearing object is the re-aligned frequency-domain channel matrix $\\tilde{\\mathbf H}_{fd}$, formed by circularly shifting the $m$-th column of the estimated channel matrix $\\hat{\\mathbf H}_{fd}$ upward by $(m-1)$ positions so that each row corresponds to one quantized Doppler shift. Its factored form $\\tilde{\\mathbf H}_{fd} = \\mathbf A_\\tau \\mathcal{D}(\\mathbf I_N \\otimes \\mathbf A_\\nu^H) + \\tilde{\\mathbf W}_{fd}$ recasts sensing as a 3D multiple-measurement sparse-signal-recovery problem, where $\\mathcal{D} \\in \\mathbb{C}^{P\\times Q\\times N}$ has nonzero entries only at delay-fractional-Doppler-integer-Doppler grid points. The two-layer variational Bayesian inference is the mechanism that solves this problem without Kronecker-style vectorization: a first layer estimates the intermediate row-sparse matrix $\\mathbf C(n)=\\mathbf X(n)\\mathbf A_\\tau^H$, and a second layer estimates $\\mathbf D(n)$ and its precision matrix $\\Gamma_d(n)$, whose column statistics feed back into the precision $\\Gamma_c(n)$ of the first layer. This feedback loop couples the integer Doppler, fractional Doppler, and delay estimates in one solution.","core_discovery":"At the center of the paper is the claim that the frequency-domain channel matrix of a time-varying OFDM link, after a column-wise circular realignment, has the structure $\\tilde{\\mathbf H}_{fd}(n) = \\mathbf A_\\tau \\mathbf D(n) \\mathbf A_\\nu^H + \\tilde{\\mathbf W}_{fd}(n)$, in which the sparse tensor $\\mathcal{D}$ carries all sensing information. Estimating this tensor solves the sensing problem: the row, column, and slice indices of its nonzero entries are the delay, fractional Doppler, and integer Doppler of the targets, so the three parameters come out matched by construction. The proposed two-layer variational Bayesian inference estimates the sparse tensor without vectorizing the full 3D dictionary; a Doppler layer estimates an intermediate row-sparse matrix and a delay layer refines the precision matrix that the Doppler layer needs. A simplified two-stage variant estimates delays by MUSIC first and then uses VBI on a smaller dictionary for the Doppler, trading a little accuracy for lower complexity. The paper also derives the Cramér-Rao bound for $\\tau_l$, $\\nu_l$, and the complex path coefficients, and reports Doppler mean-square error close to that bound in simulation.","pith_inferences":["Editorial inference: if the alternating precision-feedback idea is waveform-agnostic, it may transfer to joint angle-delay-Doppler estimation by replacing the delay dictionary with a joint angle-delay dictionary.","Editorial inference: the practical accuracy ceiling is likely set by the quality of $\\hat{\\mathbf H}_{fd}$; a natural extension is to feed the sensing stage with an explicit channel estimator and quantify how the gap to the CRB grows.","Editorial inference: the stacking construction behind the MUSIC stage may also resolve ambiguities in other multiple-measurement MUSIC settings where summed correlation matrices smear closely spaced targets; the paper does not claim this generalization."],"forward_implications":["Multi-target sensing in fast-fading OFDM is reduced to a single sparse recovery problem, so integer Doppler, fractional Doppler, and delay are associated by construction rather than by a post-hoc matching step.","Because the two-layer scheme avoids Kronecker-style vectorization of the full 3D dictionary, its complexity is substantially lower than a direct large sparse-recovery solution, at the price of a nested iteration.","The MUSIC-based variant cuts the iteration count by estimating delays non-iteratively and reduces the Doppler dictionary dimension from $N$ to $L$, which is suited to low-complexity receivers.","The derived Cramér-Rao bound gives a target for any frequency-domain time-varying channel estimator; the reported MSE close to the CRB suggests the realignment preserves the information needed for sensing.","The method remains effective over a range of velocities up to normalized Doppler $2.8f_0$ and across different target counts, supporting the paper's claim of wide applicability."],"supporting_citations":[{"why":"Provides the frequency-domain sensing formulation in time-varying channels and the FFT baseline; its separate intra- and inter-block estimation is the parameter-mismatch problem this paper targets.","marker":"[31]"},{"why":"Supplies the closed-form frequency-domain input-output relationship for OTFS as a precoded OFDM system, on which the channel model is built.","marker":"[29]"},{"why":"Provides the DFT/IDFT channel representations used to bridge domains and to re-align the channel matrix.","marker":"[30]"},{"why":"Foundational sparse Bayesian learning framework used for the hierarchical Gaussian-Gamma priors and dictionary-based recovery.","marker":"[32]"},{"why":"Establishes the Gaussian-Gamma sparse prior and Student-t marginal that produces sparsity in the precision-matrix updates.","marker":"[33]"},{"why":"Gives the variational Bayesian inference tutorial used for the mean-field factorizations and alternating updates.","marker":"[34]"},{"why":"The variational approximation method used to derive the VBI update rules and also appears as the two-stage VBI and perfect-VBI benchmarks.","marker":"[38]"},{"why":"OTFS-based radar sensing benchmark using 2D correlation peak picking that the proposed estimator is compared against.","marker":"[44]"},{"why":"Two-stage EM-VB benchmark against which the proposed VBI methods are compared.","marker":"[45]"}],"fun_headline_variants":["Two-layer Bayesian sensing matches delay and Doppler by construction","Sparse Bayesian ISAC sensing avoids integer-fractional Doppler mismatch","Variational Bayesian ISAC sensing nears Cramer-Rao bound for Doppler","Doppler and delay estimated jointly via two-layer sparse Bayesian learning","ISAC sensing without Doppler mismatch: two-layer Bayesian estimator"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the re-aligned frequency-domain channel matrix $\\hat{\\mathbf H}_{fd}$ is already known accurately: the paper explicitly assumes this in Section II-B and promises but never delivers an analysis of the impact of channel estimation error, so an inaccurate channel estimate would flow unmodeled into the sparse recovery, the Doppler and delay estimates, and the CRB comparison.","fun_headline_variants_meta":{"raw":{"variants":["Two-layer Bayesian sensing matches delay and Doppler by construction","Sparse Bayesian ISAC sensing avoids integer-fractional Doppler mismatch","Variational Bayesian ISAC sensing nears Cramer-Rao bound for Doppler","Doppler and delay estimated jointly via two-layer sparse Bayesian learning","ISAC sensing without Doppler mismatch: two-layer Bayesian estimator"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000806,"raw_usage":{"total_tokens":3591,"prompt_tokens":1045,"completion_tokens":2546,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":2459}},"tokens_in":661,"tokens_out":2546,"duration_ms":16349,"temperature":1.0,"reasoning_tokens":2459,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:34:39.316985+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same Doppler MSE experiment but feed $\\hat{\\mathbf H}_{fd}$ from an actual least-squares channel estimator (for example, with known pilot symbols) instead of the perfect matrix assumed in Section II-B; if the two-layer VBI's MSE no longer approaches the Cramér-Rao bound at high SNR, the paper's central performance claim depends on an unmodeled error source.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Two-stage EM-VB benchmark against which the proposed VBI methods are compared."},{"cited_title":"Frequency-doma in sensing in time-varying channels,","cited_arxiv_id":null,"evidence_quote":"Provides the frequency-domain sensing formulation in time-varying channels and the FFT baseline; its separate intra- and inter-block estimation is the parameter-mismatch problem this paper targets."},{"cited_title":"Low-overhead OTFS transmission with frequency or time domain channel estimation,","cited_arxiv_id":null,"evidence_quote":"Provides the DFT/IDFT channel representations used to bridge domains and to re-align the channel matrix."},{"cited_title":"Sparse Bayesian learning for ba sis selection,","cited_arxiv_id":null,"evidence_quote":"Foundational sparse Bayesian learning framework used for the hierarchical Gaussian-Gamma priors and dictionary-based recovery."},{"cited_title":"Sparse Bayesian learning and the releva nce vector machine,","cited_arxiv_id":null,"evidence_quote":"Establishes the Gaussian-Gamma sparse prior and Student-t marginal that produces sparsity in the precision-matrix updates."},{"cited_title":"A tutorial on variational Ba yesian inference,","cited_arxiv_id":null,"evidence_quote":"Gives the variational Bayesian inference tutorial used for the mean-field factorizations and alternating updates."},{"cited_title":"The vari ational approximation for Bayesian inference,","cited_arxiv_id":null,"evidence_quote":"The variational approximation method used to derive the VBI update rules and also appears as the two-stage VBI and perfect-VBI benchmarks."},{"cited_title":"Radar sensing via OTFS signaling: A delay Doppler signal processi ng perspec- tive,","cited_arxiv_id":null,"evidence_quote":"OTFS-based radar sensing benchmark using 2D correlation peak picking that the proposed estimator is compared against."}],"review_version":1}