{"id":"f4ef9ec1-50db-4d3d-af0a-674dde546674","arxiv_id":"2607.17663","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For OFDM signals containing both deterministic unit-modulus pilots and random data, closed-form mean-squared discrete ambiguity functions are derived: DP-AF sidelobes depend on pilot pattern and symbols, while FST-AF sidelobes depend only on pilot count.","lead":"This paper derives closed-form formulas for the expected radar ambiguity sidelobes of OFDM signals that mix known pilot symbols with random data payloads. The formulas show which sidelobe features depend on pilot placement and which depend only on the number of pilots, which matters for ISAC pilot design.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the DP-AF/FST-AF derivations are consistent under stated assumptions, and the flagged third-moment assumption is a scope limitation rather than a flaw.","rationale":"The reader's weakest_assumption about Assumption 2 is a legitimate scope restriction, but it is not load-bearing for the paper's central claims because the analysis is explicitly conditional on that assumption and on Assumption 1. BPSK, which the reader cites, is already excluded by Assumption 1, so the specific counterexample does not land. My independent verification of the algebra in Appendix A, consistency checks at L=0 and L=N, and a manual N=4 example all support the correctness of the DP-AF formula. The FST-AF formula is also correct and pattern-independent due to the unimodular DFT entries. The paper's main limitation—the small-Doppler approximation—is openly acknowledged and does not invalidate the derived expressions. Since the central claim is well-supported within its stated scope, the ACCEPT verdict stands.","tokens_in":8982,"tokens_out":35368,"duration_ms":270078,"concrete_test":"Independently reproduce the DP-AF Monte Carlo simulation for the comb-type pilot pattern with N=64, L=16, ZC pilot sequence, 16-QAM data, and 1000 trials (as in Figure 2). Compare the empirical mean squared DP-AF at a set of (k,q) with q≠0 to the closed-form f_I(k,q) from Proposition 1. If the error exceeds 5% across all tested points, the derivation has a hidden assumption; otherwise the central claim is empirically confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined the proofs of Propositions 1 and 2 in detail. The DP-AF expectation in Proposition 1 is derived through a careful expansion of fourth-order moments; the cross terms vanish only under Assumption 2 (Eq. (2)). This is an explicit assumption, and it is not guaranteed by Assumption 1 alone. However, the paper scopes its results to constellations satisfying both assumptions, and standard ISAC constellations (QAM, PSK with M≥4) do satisfy them. The reader's example of BPSK is not a valid counterexample because BPSK already violates Assumption 1's zero-pseudo-variance condition (E(s^2)=0). The FST-AF result (Proposition 2) is robust: it relies only on the constant-magnitude entries of the DFT matrix (|F_N(i,j)|=1/√N) and the number of unit-modulus pilots, so pilot pattern and symbols genuinely drop out under the small-Doppler approximation. This limitation is explicitly acknowledged in Section IV-B. I found no internal inconsistency or missing step in the derivations; the formulas reduce correctly to the all-random (L=0) and all-pilot (L=N) limits, and spot checks with small N confirm the stated expectations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies OFDM signals composed of deterministic unit-modulus pilots and random data symbols, and derives closed-form expressions for the mean squared discrete periodic ambiguity function (DP-AF) and fast-slow-time ambiguity function (FST-AF). Proposition 1 (Eq. (21)) gives a piecewise expression for E[|A_DP(k,q)|^2]: N^2+(κ−1)(N−L) at the mainlobe, (κ−1)(N−L) on the zero-Doppler sidelobe, and a pilot-pattern/pilot-symbol-dependent term f_I(k,q) elsewhere. Proposition 2 (Eq. (26)) gives E[|A_FST(k,q)|^2] = L+(MN−L)κ + M^2N^2δ_{k,0}δ_{q,0} − MN, depending only on the number of pilots. The formulas are validated with 1000-trial Monte Carlo simulations using 16-QAM and ZC pilots, and the paper discusses implications for pilot design in communication-centric ISAC systems.","tokens_in":9225,"tokens_out":20947,"duration_ms":170721,"significance":"If correct, the paper is a useful and nontrivial extension of the random-waveform ambiguity analysis in [8] to practical hybrid pilot-data frames. The derivations are self-contained, and the results are parameter-free in the sense that they depend only on the known constellation kurtosis κ, the pilot count L, and the given pilot pattern/symbols; no fitting is involved. The L=0 and L=N limits are consistent, and the DP-AF/FST-AF distinction is practically relevant for ISAC pilot design. The explicit assumptions—Eq. (2) for the third-order data moments and the small-Doppler approximation before Eq. (14)—appropriately scope the claims. The paper deserves publication after minor presentation fixes.","major_comments":[],"minor_comments":[{"comment":"The additional assumption E(|s_c|^2 s_c)=E(|s_c|^2 s_c^*)=0 is stated to hold for 'most symmetric constellations'. Please state the precise symmetry condition, or at least verify it explicitly for the constellations used (16-QAM and M-PSK with M≥4). This will prevent misapplication to asymmetric constellations that satisfy Assumption 1 but not Eq. (2).","section":"Section II-A, Eq. (2)"},{"comment":"The caption reads 'N=64, M=20, L=16M'. This is ambiguous: is L=320 total pilots (16 per OFDM symbol), or should it be L=16 with M=20? Please clarify the exact pilot count used in the FST-AF simulation.","section":"Fig. 3 caption"},{"comment":"The transition from the g/h sum to the cosine sum is terse. Adding a one-line identity, e.g., Σ_{n,l} e^{j2πk(n−l)/N}=N^2δ_{k,0} and similarly for the M-dimension, would improve readability and make the M^2N^2δ_{k,0}δ_{q,0} term transparent.","section":"Appendix B, Eq. (42)"},{"comment":"The phrase 'binary valued' for the no-pilot DP-AF could be made precise: the theoretical mean squared sidelobe equals N for q≠0 and (κ−1)N for q=0, k≠0. This would help the reader map the qualitative description to Eqs. (21)–(23).","section":"Section IV-A"},{"comment":"Reference [8] is cited as an arXiv preprint. If a peer-reviewed version is now available, please update the citation.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid incremental extension of the authors' prior work [8], with clean derivations and matching simulations. No concerns about novelty or fit. The main risk is that Proposition 1 relies on Eq. (2), but that assumption is explicit and satisfied by standard constellations; the minor comments above should resolve any residual ambiguity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does one real thing: it takes the statistical ambiguity-function framework from Zhang et al. [8] and derives closed-form mean-squared DP-AF and FST-AF expressions for OFDM frames that contain a mix of deterministic unit-modulus pilots and random data. The DP-AF result is genuinely new — the pilot-data mixture produces a pattern-dependent term f_I(k,q) that does not appear in the all-random treatment. The FST-AF result, showing that pilot pattern and symbols drop out and only the pilot count matters, is a clean and somewhat surprising consequence. I checked the limits L=0 and L=N against the all-random and all-pilot cases, and the algebra is consistent. The simulations match, and it is a real plus that nothing is fitted: kappa is a known constellation statistic and the ZC roots are not optimized retroactively.\n\nThe soft spots are real but not disqualifying. The proof of Proposition 1 relies on Assumption 2 (third-order moments vanish), which fails for BPSK and asymmetric constellations. The stress-test note is right that BPSK already violates Assumption 1, so this is a scope limitation more than a hidden flaw, but it does mean the DP-AF formula is not universal. The FST-AF result depends on a small-Doppler approximation that the authors acknowledge; it is a useful asymptotic, not a general statement. The paper also leans heavily on [8] for definitions and metrics, which is fine given the authors' own prior work, but it limits the paper's standalone value for outsiders. No code or data is provided, which is a minor inconvenience for a simulation-verification paper.\n\nThe central argument holds up. The derivations are careful, the assumptions are explicit, and the conclusions are appropriately scoped. The paper will be useful to ISAC waveform designers who want to evaluate pilot-embedded frames without Monte-Carlo runs, and to anyone building on the random-waveform AF literature. It deserves a serious referee — the math is verifiable and the novelty, while incremental, is real. I would not put it at the top of a reading group list, but it is a solid, honest contribution.\n\nRecommendation: send to peer review. It will likely need minor revisions (clarify the scope of Assumption 2, maybe add a note on BPSK), but it is publishable in a good venue.","headline":"A clean analytical extension of the random-OFDM AF framework to pilot-embedded frames; the DP-AF formula is new and the proofs hold up under stated assumptions, though the scope is deliberately narrow.","tokens_in":9747,"tokens_out":606,"would_cite":true,"duration_ms":7973,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A12","94A13"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives exact formulas for the mean squared ambiguity function of OFDM signals with deterministic pilots embedded among random data, showing that pilot placement and symbols shape delay-Doppler sidelobes in one formulation while o","keywords":["OFDM","Integrated sensing and communication","Ambiguity function","Discrete periodic ambiguity function","Fast-slow-time ambiguity function","Pilot design","Expected sidelobe level","Random waveforms"],"falsifier":"Transmit a 1D OFDM signal with BPSK data symbols and L pilots, average |A_DP(k,q)|^2 over many trials for q≠0, and compare with the paper's formula: for BPSK the moment E(|s|^2 s) is nonzero, so the pilot-data cross terms the proof sets to zero will appear as a systematic discrepancy. The same test can be run analytically by inserting the measured fourth-order moments into the expansion before Eq. (37) and checking that the residual matches the simulation.","tokens_in":8845,"feed_emoji":"📡","tokens_out":4548,"duration_ms":41539,"temperature":0.7,"pith_summary":"In integrated sensing and communication, the same OFDM waveform carries both known pilot symbols and random data, and the paper asks how those pilots change the waveform's sensing ambiguity function—the delay-Doppler response that determines how well targets can be separated. The authors derive closed-form expressions for the mean squared discrete periodic ambiguity function (DP-AF) and the fast-slow-time ambiguity function (FST-AF) of pilot-embedded random OFDM signals. The central result is a split: under the DP-AF formulation, sidelobes away from zero Doppler depend on exactly where pilots sit and what symbols they carry, while under the FST-AF formulation every off-mainlobe sidelobe equals the same constant, determined only by the number of pilots, not their pattern or values. Giving designers a formula for this split matters because it turns pilot placement into a quantitative tool for shaping sensing sidelobes without changing the communication payload.","feed_headline":"Pilot patterns change OFDM sensing sidelobes","feed_subtitle":"New closed-form formulas show which pilot choices shape delay-Doppler response—and which do not.","key_machinery":"The argument rests on splitting the transmitted symbol vector into a deterministic pilot component p and a random data component d, encoded by a binary indicator vector w, and then expanding the fourth-order moment E(s_l^* s_r s_m s_n^*) of the mixed signal. Using constellation symmetry assumptions that force odd and certain cross moments to vanish, the expansion collapses to diagonal terms controlled by the kurtosis κ plus pilot-pilot terms controlled by the filtered pilot sequence. This is what produces the DP-AF formula whose off-Doppler part depends on pilot pattern and symbols. For the FST-AF, the key simplification is the identity A_FST = √(MN) F_N^H |S|^2 F_M, which shows the ambiguit","core_discovery":"For a length-N OFDM symbol containing L unit-modulus pilots and N−L i.i.d. data symbols drawn from a constellation with kurtosis κ, the paper proves E[|A_DP(k,q)|^2] equals N^2 + (κ−1)(N−L) at the mainlobe (k,q)=(0,0), equals (κ−1)(N−L) along the zero-Doppler axis q=0, and for q≠0 equals a pilot-dependent term: the squared ambiguity of the pilot sub-sequence alone, |p^H F_N D_{N,q} J_{N,k} F_N^H p|^2, plus a residual N − w^H F_N D_{N,q} F_N^H w, where w marks pilot locations and p holds pilot symbols. Thus off-Doppler DP-AF sidelobes are shaped by pilot pattern and pilot symbols. For the two-dimensional fast-slow-time formulation with M OFDM symbols, the paper proves E[|A_FST(k,q)|^2] = L +","pith_inferences":["The DP-AF formula supplies a ready-made cost function for pilot design: one could optimize the indicator vector w and pilot symbols p to push sidelobes into delay-Doppler regions that least interfere with known targets or clutter, something the paper demonstrates qualitatively but does not optimize.","Because the off-Doppler DP-AF term is an ambiguity of the pilot sequence itself, familiar complementary-sequence or ambiguity-shaping ideas could be imported to design pilot sets that null or suppress specific sidelobe ridges; the paper stops short of testing such designs.","The FST-AF's complete independence from pilot pattern is a consequence of the small-Doppler block-constant approximation; a natural extension is to ask where pattern dependence re-emerges as Doppler grows, by evaluating the exact DP-AF of the full MN-length signal rather than the block-wise approximation.","The stated DP-AF formula relies on the symmetry assumption E(|s|^2 s)=E(|s|^2 s*)=0, which fails for BPSK and asymmetric constellations; extending the derivation to those cases would require carrying the extra fourth-order cross moments instead of dropping them."],"forward_implications":["For a fixed OFDM frame and constellation, adding pilots does not change the total expected integrated sidelobe power of the DP-AF—it redistributes sidelobes, lowering some delay-Doppler regions at the cost of raising others.","The zero-Doppler sidelobe level of the DP-AF is (κ−1)(N−L), independent of pilot pattern or pilot symbols, so only the number of pilots and the constellation kurtosis govern that cut.","In the off-Doppler region q≠0, the DP-AF sidelobes are governed by a pilot-pattern term plus a residual, so choosing pilot positions and pilot symbols is a direct design degree of freedom for shaping the delay-Doppler response.","Under the small-Doppler FST-AF approximation, no pilot pattern or pilot symbol choice can shape the sidelobes; only the pilot count matters, so pattern-dependent sensing improvements should be sought in the exact DP-AF domain.","Comb-type pilot patterns produce periodic peaks and troughs along delay and Doppler, while block-type patterns create clustered high- and low-sidelobe regions, offering qualitatively different trade-offs in practice."],"fun_headline_variants":["Pilot placement reshapes OFDM radar sidelobes","OFDM sensing: pilots dictate delay-Doppler sidelobes","Closed-form proof: pilots control OFDM ambiguity","Which pilots sharpen OFDM sensing response","Pilot design tunes OFDM ambiguity sidelobes"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The DP-AF derivation holds only when the random data constellation is symmetric enough that E(|s|^2 s)=E(|s|^2 s*)=0, and the FST-AF result additionally assumes Doppler is small enough that phase is constant within each OFDM block; if either condition fails, the corresponding formula is incomplete or approximate.","fun_headline_variants_meta":{"raw":{"variants":["Pilot placement reshapes OFDM radar sidelobes","OFDM sensing: pilots dictate delay-Doppler sidelobes","Closed-form proof: pilots control OFDM ambiguity","Which pilots sharpen OFDM sensing response","Pilot design tunes OFDM ambiguity sidelobes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000142,"raw_usage":{"total_tokens":1012,"prompt_tokens":756,"completion_tokens":256,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":180}},"tokens_in":500,"tokens_out":256,"duration_ms":2867,"temperature":1.0,"reasoning_tokens":180,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T17:22:46.455344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Transmit a 1D OFDM signal with BPSK data symbols and L pilots, average |A_DP(k,q)|^2 over many trials for q≠0, and compare with the paper's formula: for BPSK the moment E(|s|^2 s) is nonzero, so the pilot-data cross terms the proof sets to zero will appear as a systematic discrepancy. The same test can be run analytically by inserting the measured fourth-order moments into the expansion before Eq. (37) and checking that the residual matches the simulation.","supporting_citations":[],"review_version":1}