{"id":"77d9edd8-46d1-4dda-968c-59b45a0ab35f","arxiv_id":"2501.01721","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For random ISAC signals with Nyquist pulse shaping, OFDM minimizes the average ranging sidelobe among sub-Gaussian constellations, and a convex iceberg shaping design cuts sidelobes versus root-raised cosine pulses.","lead":"This paper derives a closed-form formula for the average squared auto-correlation of random communication signals used for radar sensing, and shows which modulation and pulse shape keep sidelobes lowest. It then designs Nyquist pulses to suppress sidelobes in a chosen delay window, improving multi-target range estimation for 6G ISAC systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's uniqueness proof is invalid: the equality condition in Appendix B fails when the vector \\(\\tilde{g}_k\\odot\\tilde{f}^*_{k+1}\\) has repeated entries, so the claim that OFDM is the only optimal basis is not established.","rationale":"The paper has real independent value: Theorem 1 gives a closed-form decomposition that is not present in the cited preprints, the numerical curves match the formula, and the convex pulse-shaping formulation is concrete and falsifiable. The reader's CONDITIONAL verdict is appropriate. I do not find an error in the main derivation of Theorem 1 itself; the normalization of \\(\\tilde{f}_{k+1}\\) is stated loosely (the first entries of an \\(LN\\)-point DFT column are used with an \\(N\\)-point normalization), but the final displayed formulas are consistent with the numerics, so this appears to be a notational issue rather than a load-bearing flaw. The most serious weakness is the uniqueness part of Theorem 2. The majorization argument in Appendix B gives a valid upper bound, and permutations achieve it, but the claimed equality condition is false when \\(\\tilde{g}_k\\odot\\tilde{f}^*_{k+1}\\) has repeated entries. This happens for concrete parameter choices, so the proof's assertion that the only optimal \\(V\\) is a complex permutation matrix is incorrect. The theorem may still be true in the simultaneous-lag sense, because a non-permutation matrix tied at one lag need not be optimal at all lags, but the manuscript does not prove that. I agree with the reader's secondary concern about Assumption 2 excluding BPSK and 8-QAM despite the abstract's unrestricted QAM/PSK claim; that is a scope issue, not a discovered contradiction. Neither issue warrants rejection, but both justify the conditional verdict and the request for a repaired proof and a corrected scope statement.","tokens_in":19951,"tokens_out":27201,"duration_ms":277968,"concrete_test":"Compute the sea-level term \\(\\|\\tilde{V}(\\tilde{g}_k\\odot\\tilde{f}^*_{k+1})\\|^2\\) for \\(N=8\\), \\(L=2\\), \\(k=4\\), any Nyquist pulse, and (a) the OFDM basis \\(U=F^H\\), versus (b) the basis \\(U=F^H V^H\\) where \\(V\\) is a block-Hadamard unitary with \\(2\\times2\\) Hadamard blocks acting on the repeated-value pairs. If the values coincide, the Appendix B equality claim is demonstrably false. To test the actual joint uniqueness statement of Theorem 2, then enumerate or randomly sample unitary matrices and check whether any non-OFDM basis achieves the OFDM value of \\(E(|R_k|^2)\\) for all \\(k=1,\\ldots,LN-1\\) under a fixed RRC or optimized Nyquist pulse; if none does, Theorem 2 may survive but its proof still needs repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 2 rests on the proof in Appendix B. The proof relaxes the maximization of \\(\\|\\tilde{V}(\\tilde{g}_k\\odot\\tilde{f}^*_{k+1})\\|^2\\) over unistochastic matrices to the Birkhoff polytope of bistochastic matrices and invokes majorization to conclude \\(\\|\\tilde{V}b_R\\|^2\\le\\|b_R\\|^2\\) and \\(\\|\\tilde{V}b_I\\|^2\\le\\|b_I\\|^2\\). It then asserts that all equalities hold if and only if \\(\\tilde{V}=V\\odot V^*\\) is a permutation matrix. This equality condition is false in general. Equality holds whenever \\(\\tilde{V}\\) maps the vector to a permutation of itself; if the vector has repeated entries, any bistochastic matrix that averages within groups of equal entries also preserves the norm. Such matrices can be unistochastic. For example, with \\(N=8\\), \\(L=2\\), and lag \\(k=4\\), we have \\(\\tilde{g}_{n,4}=1\\) for every Nyquist pulse and the entries of \\(\\tilde{f}^*_{4}\\) are \\(\\{1,-1,j,-j\\}\\), each repeated twice. A block-Hadamard unitary mixing each repeated pair gives a non-permutation unistochastic \\(\\tilde{V}\\) that maps the vector to itself, so a non-OFDM modulation basis achieves exactly the same minimum sea level at this lag. Thus the per-lag uniqueness statement used to prove Theorem 2 is false as written, and the 'only OFDM' conclusion is not rigorously supported. The joint statement -- no other basis is optimal at every lag simultaneously -- may still be true, but the proof does not establish it. A secondary but real issue is the abstract's blanket QAM/PSK claim, which conflicts with Assumption 2's explicit exclusion of BPSK and 8-QAM; this is a scope overclaim but is less damaging than the proof gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes the sensing performance of random ISAC signals, characterized by the expected squared periodic ACF under an arbitrary orthonormal modulation basis and a Nyquist pulse-shaping filter. The main result, Theorem 1, expresses E(|R_k|^2) as an 'iceberg' term (the squared ACF of the pulse) plus a 'sea-level' term governed by the constellation kurtosis and the entrywise-squared unitary modulation matrix. Corollaries 1-3 give the mainlobe level and the M-fold variance reduction under coherent integration. The paper then claims in Theorem 2 that for sub-Gaussian constellations (e.g., QAM/PSK satisfying the zero-pseudo-variance assumption), OFDM is the unique modulation basis that minimizes the ranging sidelobe at every lag, and in Theorem 3 that SC is optimal for super-Gaussian constellations. Based on the iceberg geometry, the authors propose a convex 'iceberg shaping' pulse design. Numerical examples with RRC pulses and ranging simulations are used to validate the analysis.","tokens_in":20257,"tokens_out":15233,"duration_ms":130354,"significance":"If the optimality result were rigorously established, Theorem 2 would be an important and somewhat surprising design guideline: within the considered class of unitarily equivalent modulation bases, OFDM is singled out as the best (and only) choice for ranging sidelobe suppression, independent of the specific Nyquist pulse. The closed-form decomposition in Theorem 1 is the paper's strongest contribution; it is plausible, clearly structured, and consistent with the numerical curves. The convex pulse-shaping formulation is a useful practical outcome. However, the uniqueness proof for Theorem 2 rests on an equality condition in Appendix B that is false in general (see Major Comment 1), so the central optimality claim is not yet established. The paper also contains a normalization inconsistency in the statement of Theorem 1 that must be corrected.","major_comments":[{"comment":"The claim that equality in ||\\tilde{V}b_R||^2 <= ||b_R||^2 and ||\\tilde{V}b_I||^2 <= ||b_I||^2 holds if and only if \\tilde{V} is a permutation matrix is false when the vector being transformed has repeated entries. For example, with N=8, L=2 and lag k=4, \\tilde{g}_{n,4}=1 for every Nyquist pulse and \\tilde{f}^*_{k+1} has entries {1,-j,-1,j} each repeated twice; both b_R and b_I then contain repeated values. A block-Hadamard unitary that averages within each repeated-value pair produces a unistochastic matrix \\tilde{V} that is not a permutation matrix but preserves both b_R and b_I, so equality holds in (68). Thus the per-lag uniqueness assertion used to prove Theorem 2 is invalid. The authors need to repair the proof, for example by establishing that a non-permutation \\tilde{V} that attains the bound for some k must fail to attain it for some other k, or by qualifying the theorem's uniqueness claim.","section":"Appendix B, Eq. (68)"},{"comment":"The definition of \\tilde{f}_{k+1} in (27) as 'the first N entries of f_{k+1}' is inconsistent with the scaling in (26). Since f_{k+1} is a column of F_{LN}, its entries are normalized by 1/\\sqrt{LN}; with this definition one obtains N|\\tilde{f}_{k+1}^H \\tilde{g}_k|^2 = (1/L)|\\Sigma_n \\tilde{g}_{n,k} e^{j2\\pi k(n-1)/(LN)}|^2, whereas the derivation in (59)-(60) and Corollary 1 require this term to equal |\\Sigma_n \\tilde{g}_{n,k} e^{j2\\pi k(n-1)/(LN)}|^2. The definition should be corrected, e.g., \\tilde{f}_{k+1,n} = e^{-j2\\pi k(n-1)/(LN)}/\\sqrt{N}, or the coefficient in (26) should be changed to LN. This is a notational error in the central formula that should be fixed.","section":"Theorem 1, Eqs. (26)-(27)"},{"comment":"The statement that the result applies to 'QAM/PSK constellations' overstates the scope actually proven. Assumption 2 (E(s)=0 and E(s^2)=0) excludes BPSK and 8-QAM, as the authors themselves note after (2). Since Lemma 2 and Theorem 1 rely on this assumption, the abstract, Theorem 2, and the conclusion should explicitly restrict to constellations satisfying Assumption 2 (e.g., proper QAM/PSK, excluding BPSK and 8-QAM).","section":"Abstract and Theorem 2"}],"minor_comments":[{"comment":"There are several typos, including 'staitionary' (Sec. III-C), 'ainticipated' and 'perofrmance' (Introduction), 'Sqaured' and 'Integartion' in Fig. 1, and 'Integartion' in Fig. 3.","section":"Throughout"},{"comment":"The optimization problem (47) cites constraints '(71)-(74)', but those equation numbers belong to Appendix C; the intended constraints are (43)-(46).","section":"Sec. IV-C3, Eq. (47)"},{"comment":"References to subfigures appear as 'Fig. ??' twice; the authors should insert the correct figure references.","section":"Sec. V-C"},{"comment":"Lemma 2 is stated without proof, citing the authors' unreviewed arXiv preprint [26]. Given the paper's reliance on this lemma, including its proof or a peer-reviewed reference would improve self-containedness.","section":"Appendix A, Lemma 2"},{"comment":"The sentence following Theorem 2 claims that uniqueness is 'guaranteed, as being detailed in the proof'; this is not supported once the Appendix B equality condition is corrected, so the sentence should be revised to match the actual proof.","section":"Sec. IV-A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript leans heavily on the authors' own arXiv preprint [26], including the fourth-moment lemma and the proof technique for Theorems 2 and 3. The false equality condition in Appendix B appears to be inherited from that work; the authors should verify whether the same issue affects [26]. The numerical validation of Theorem 1 is solid, but the central optimality theorem is not yet rigorously proven. The paper is likely to be of interest to the ISAC community if the proof is repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing you should know: the iceberg/sea decomposition in Theorem 1 is real, clean, and useful. The paper extends the authors' earlier discrete-time result to pulse-shaped, oversampled signals, and the closed-form expression for E(|R_k|^2) checks out against their numerics. The convex pulse-shaping design is a practical contribution and the simulations are convincing. If you work on ISAC or random-waveform sensing, this is worth a serious look.\n\nThe soft spot is Theorem 2. The appendix proves that OFDM achieves the lowest sea level by a majorization bound, which is fine. But the claim that OFDM is the only basis achieving that minimum at every lag rests on an equality condition that is false. As the stress-test note points out, when the vector g_k ⊙ f*_{k+1} has repeated entries, there are non-permutation unistochastic matrices that preserve it exactly. The N=8, L=2, k=4 example is a concrete counterexample to the per-lag uniqueness statement. The joint claim — no other basis is optimal at all lags simultaneously — might still be true, but the proof given does not establish it. This is a load-bearing gap for the \"only\" part of the theorem, though the optimality part (OFDM attains the minimum) appears sound.\n\nA smaller issue: the abstract says \"QAM/PSK\" but Assumption 2 excludes BPSK and 8-QAM. The authors acknowledge this in the body, so they know; the abstract should match. Also, releasing the solver settings or code for the pulse design would help reproducibility, but that's minor.\n\nI don't see a circularity problem. The heavy self-citation to [26] is legitimate because the continuous-time results are genuinely new and the imported Lemma 2 is a known tool. The main formula is verified numerically and the derivation is transparent.\n\nBottom line: this deserves a serious referee, but not as-is. The authors should either close the equality-condition gap in Appendix B or soften Theorem 2 to \"OFDM is optimal\" and add a separate conjecture or partial result about uniqueness. I'd be happy to see it in a good signal-processing journal after that revision.","headline":"Solid new decomposition and pulse-design tool for random ISAC waveforms, but the proof that OFDM is the unique optimal modulation basis has a real gap in Appendix B that should be fixed before publication.","tokens_in":20899,"tokens_out":2836,"would_cite":true,"duration_ms":28784,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Random ISAC signals have a hidden deterministic 'iceberg' in their autocorrelation, and OFDM is the unique modulation that keeps it lowest for QAM/PSK under Nyquist pulse shaping.","keywords":["integrated sensing and communications","random ISAC signals","auto-correlation function","ranging sidelobe","OFDM","pulse shaping","coherent integration","kurtosis"],"falsifier":"Simulate the average squared autocorrelation for OFDM and single-carrier signals with BPSK (or 8-QAM) symbols, using $N=128$, oversampling $L=10$, and an RRC pulse with $\\alpha=0.35$: if at any nonzero lag the single-carrier sidelobe is below the OFDM sidelobe, the 'OFDM is uniquely optimal for QAM/PSK' claim fails for those constellations; the same simulation with 16-QAM should reproduce the theorem and validate the pipeline.","tokens_in":19679,"feed_emoji":"📡","tokens_out":7661,"duration_ms":70374,"temperature":0.7,"pith_summary":"This paper tries to establish that the ranging performance of a communication signal pressed into sensing service is not random at all in expectation: the average squared autocorrelation decomposes into a deterministic 'iceberg' set by the pulse-shaping filter and a 'sea level' set by the randomness of the data symbols. For zero-mean, zero-pseudo-variance constellations (which covers most QAM/PSK but not BPSK or 8-QAM) under Nyquist pulse shaping, the paper proves that OFDM is the unique modulation basis achieving the lowest ranging sidelobe at every lag, while super-Gaussian constellations make single-carrier modulation optimal. Because coherent integration suppresses the sea level by a factor of M, the paper argues the long-run sensing limit is set by the pulse's own autocorrelation, and it proposes a convex 'iceberg shaping' design for Nyquist pulses that lowers sidelobes in a chosen delay region. The practical stake is that 6G systems can reuse OFDM data payloads for radar-like ranging without a dedicated sensing waveform, and can tune the pulse to where sidelobes matter.","feed_headline":"OFDM gives the lowest ranging sidelobes among random ISAC waveforms","feed_subtitle":"For QAM/PSK payloads, OFDM is uniquely optimal; pulse shaping can cut ranging sidelobes by tens of dB.","key_machinery":"The load-bearing object is the frequency-domain representation of the autocorrelation: after oversampling by $L$, the pulse enters through its squared spectrum $g_n$, and the Nyquist (folded-spectrum) condition pins $g_{(L-1)N+n}=1-g_n$. The modulation basis enters only through the unistochastic matrix $\\tilde{\\mathbf{V}}=\\mathbf{V}\\odot \\mathbf{V}^*$ (entrywise squared modulus of a unitary), and the Iceberg Theorem expresses the sea level in terms of how $\\tilde{\\mathbf{V}}$ mixes the vector $\\tilde{\\mathbf{g}}_k\\odot\\tilde{\\mathbf{f}}_{k+1}^*$. The optimality proofs use majorization: because the $\\ell^2$ norm is Schur-convex, any bistochastic averaging can only shrink (sub-Gaussian) or grow (super-Gaussian) the sea level, and equality holds only for permutation-like $\\tilde{\\mathbf{V}}$, which is exactly OFDM up to phase and permutation.","core_discovery":"The central discovery is the Iceberg Theorem: for i.i.d. symbols drawn from a unit-power, zero-mean, zero-pseudo-variance constellation, sent through an orthonormal modulation basis and a Nyquist pulse, the expected squared autocorrelation at lag $k$ is $$\\mathbb{E}(|R_k|^2)=N|\\tilde{\\mathbf{f}}_{k+1}^H \\tilde{\\mathbf{g}}_k|^2+\\|\\tilde{\\mathbf{g}}_k\\|^2+(\\mu_4-2)N\\|\\tilde{\\mathbf{V}}(\\tilde{\\mathbf{g}}_k\\odot \\tilde{\\mathbf{f}}_{k+1}^*)\\|^2,$$ where $\\tilde{\\mathbf{g}}_k$ encodes the folded spectrum of the pulse, $\\tilde{\\mathbf{V}}$ is the entrywise-squared unitary that carries the modulation basis, and $\\mu_4$ is the constellation kurtosis. The first term is the squared mean of $R_k$ and equals the squared autocorrelation of the pulse alone, the iceberg; the remaining terms are the variance from random data, the sea level. From this identity the paper proves that for sub-Gaussian constellations (kurtosis below 2, which includes PSK and QAM), OFDM is the only basis that minimizes the sea level at every lag, hence the lowest ranging sidelobes; for super-Gaussian constellations the single-carrier basis is optimal. Coherent integration over $M$ independent symbol blocks reduces the sea level by $1/M$, so after enough integrations the iceberg alone governs ranging, which motivates shaping the pulse's autocorrelation directly.","pith_inferences":["The theorem's domain excludes BPSK and 8-QAM because they violate the zero-pseudo-variance assumption; a direct corollary of the proof structure is that one should derive a generalized fourth-moment formula for these constellations before claiming OFDM optimality for all PSK/QAM.","The iceberg-shaping optimization moves sidelobe energy out of a chosen delay window, so a natural extension is a region-adaptive pulse that trades a protected range interval against an allowed sidelobe budget elsewhere.","Because the sea level is the only component that depends on symbol randomness, another testable extension is to randomize only the phases of a constant-envelope waveform: the theorem predicts this should behave like PSK and drain the sea entirely under OFDM.","If a future standard uses a modulation basis that is only approximately orthogonal, the unistochastic structure of $\\tilde{\\mathbf{V}}$ suggests the sea level will be raised by the amount of cross-talk among basis vectors; quantifying that sensitivity is outside this paper but follows from the same expression."],"forward_implications":["For QAM/PSK payloads and any Nyquist pulse, OFDM gives the lowest average ranging sidelobe among all orthonormal bases, so existing OFDM-based 6G waveforms need no change to get the best sensing sidelobe floor.","Coherently combining M matched-filter outputs cuts the variance ('sea level') by a factor M, so the ranging sidelobe floor after many transmissions is set by the pulse autocorrelation, not the random data.","When OFDM carries PSK symbols the sea level vanishes, and the average squared autocorrelation is exactly the autocorrelation of the pulse, meaning the sensing response is fully deterministic.","For super-Gaussian constellations the ordering reverses, and single-carrier modulation becomes the best basis, so the optimal waveform depends only on whether the excess kurtosis is negative or positive.","Nyquist pulses with larger roll-off generate stronger periodic ripples in the sea level, while a sinc pulse gives a flat sea level, and the mainlobe width is $(1+\\alpha)/B$."],"supporting_citations":[{"why":"Supplies Lemma 2 (the second- and fourth-moment structure of the symbol vector) and the prior discrete-time framework this paper extends to pulse-shaped signals.","marker":"[26]"},{"why":"Introduces the randomness-aware pulse shaping design for single-carrier signals that the proposed iceberg shaping generalizes to arbitrary modulation bases.","marker":"[28]"},{"why":"Provides the folded-spectrum criterion that defines Nyquist pulses and constrains the squared spectrum throughout the derivation.","marker":"[32]"},{"why":"Establishes the kurtosis-controlled probabilistic constellation shaping tradeoff the paper invokes for choosing low-kurtosis constellations.","marker":"[14]"},{"why":"Shows OFDM signals can estimate delay and Doppler, the baseline application that motivates the optimal modulation basis question.","marker":"[15]"},{"why":"Makes the point that fractional delays matter for sensing in continuous time, justifying the oversampled analysis that brings pulse shaping into the picture.","marker":"[27]"}],"fun_headline_variants":["OFDM optimal for QAM/PSK random ISAC ranging","Iceberg theorem: pulse shaping rules ISAC sidelobes","Sub-Gaussian constellations favor OFDM for ranging","Random ISAC: pulse shaping cuts sidelobes by tens of dB"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof rests on the data symbols having zero mean and zero pseudo-variance, $\\mathbb{E}(s)=\\mathbb{E}(s^2)=0$, which is false for BPSK and 8-QAM; without that assumption the closed-form and the OFDM optimality claim are not established.","fun_headline_variants_meta":{"raw":{"variants":["OFDM optimal for QAM/PSK random ISAC ranging","Iceberg theorem: pulse shaping rules ISAC sidelobes","Sub-Gaussian constellations favor OFDM for ranging","Random ISAC: pulse shaping cuts sidelobes by tens of dB"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000355,"raw_usage":{"total_tokens":2041,"prompt_tokens":1171,"completion_tokens":870,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":787,"completion_tokens_details":{"reasoning_tokens":799}},"tokens_in":787,"tokens_out":870,"duration_ms":8091,"temperature":1.0,"reasoning_tokens":799,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:23:52.868238+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the average squared autocorrelation for OFDM and single-carrier signals with BPSK (or 8-QAM) symbols, using $N=128$, oversampling $L=10$, and an RRC pulse with $\\alpha=0.35$: if at any nonzero lag the single-carrier sidelobe is below the OFDM sidelobe, the 'OFDM is uniquely optimal for QAM/PSK' claim fails for those constellations; the same simulation with 16-QAM should reproduce the theorem and validate the pipeline.","supporting_citations":[{"cited_title":"Pulse Shaping for Random ISAC Signals: The Ambiguity Function Between Symbols Matters","cited_arxiv_id":"2407.15530","evidence_quote":"Introduces the randomness-aware pulse shaping design for single-carrier signals that the proposed iceberg shaping generalizes to arbitrary modulation bases."},{"cited_title":"Reshaping the ISAC tradeoff under OFDM signaling: A probabilistic constellation shaping approach,","cited_arxiv_id":null,"evidence_quote":"Establishes the kurtosis-controlled probabilistic constellation shaping tradeoff the paper invokes for choosing low-kurtosis constellations."},{"cited_title":"Fractional delay and doppler estimation for OTFS based ISAC systems,","cited_arxiv_id":null,"evidence_quote":"Makes the point that fractional delays matter for sensing in continuous time, justifying the oversampled analysis that brings pulse shaping into the picture."}],"review_version":1}