{"id":"07e4a60b-16fd-471a-9659-2fb22572cfef","arxiv_id":"2504.18016","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Uniform power allocation is the unique global minimizer of the expected integrated sidelobe level for OFDM-based random ISAC ranging.","lead":"This paper proves that spreading power evenly across OFDM subcarriers gives the lowest expected ranging sidelobes for random data signals, for both periodic and aperiodic autocorrelations. The result simplifies 6G integrated sensing and communication design, and a zero-padding extension reveals a tradeoff between sidelobe level and range resolution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 (A-ACF global optimality) is not proved: Appendix D's Cauchy-Schwarz step compares a lower bound, not the actual even-column term, so uniform PA's optimality is unsupported.","rationale":"The reader's stated weakest assumption is Assumption 1, which excludes BPSK and 8-QAM; that is a real domain restriction, and the abstract's phrase 'all constellations' overstates it. However, it is explicitly disclosed in the paper, and the P-ACF formulas do not even require vanishing pseudo-variance. The more central issue is Theorem 3: the proof in Appendix D uses Cauchy-Schwarz in Eq. (87) to lower-bound the even-column term, then interprets the equality condition at uniform P as a minimization condition. That is logically reversed: equality makes the lower bound tight and, for a fixed sum of A_n, maximizes the lower bound; it does not show the actual term is minimized. The proof therefore never establishes that the even-column term at arbitrary P is at least as large as at uniform P. This affects the headline claim for A-ACF even under Assumption 1. Since the reader already gave a CONDITIONAL verdict and cited this A-ACF proof gap in their rationale, my stress-test does not move the verdict; it sharpens the specific step that must be fixed or replaced.","tokens_in":18965,"tokens_out":25341,"duration_ms":247638,"concrete_test":"Directly evaluate the normalized A-ACF EISL in Eq. (80) for N=4 and N=8, with mu4 in {1, 4/3, 2}, using a global optimization over the simplex P_i>=0, sum(P_i)=N (e.g., differential evolution plus a dense grid including extreme vectors such as p=(N,0,...,0)). If any feasible p gives a lower normalized EISL than uniform PA, Theorem 3 is false. If no such p is found, the theorem may still be true, but Appendix D must be repaired by proving a genuine inequality comparing the even-column sum at arbitrary P with its value at uniform P, e.g., via the Schur-convexity of sum(A_n) under the doubly stochastic matrix |H|^2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing defect is in Appendix D's proof of Theorem 3 (global optimality of uniform PA for the A-ACF), and it is independent of Assumption 1. In Eq. (80), the normalized EISL is split into odd- and even-column terms. After showing the odd part is minimized at uniform P, the proof handles the even part via the Cauchy-Schwarz inequality in Eq. (87), writing N*sum(A_n)/D >= (sum(sqrt(A_n)))^2/D. The text then states that this bound reaches its minimum only when all sqrt(A_n) are equal, which happens at uniform P. This is not a valid optimality argument: equality in Cauchy-Schwarz only makes the lower bound tight, and for fixed sum(A_n) it actually maximizes the lower bound rather than minimizing it. More importantly, the actual term N*sum(A_n)/D at a nonuniform P is never compared with its value at uniform P; the lower bound can lie below the uniform value while the true term lies above it. Thus Theorem 3 is unproven as written, and the claim that uniform PA is the only minimizer of the A-ACF EISL rests on a gap. This is a proof gap, not a demonstrated falsehood, but it affects one of the two central global-optimality claims even within the paper's stated constellation assumptions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies power allocation (PA) for OFDM-based monostatic ISAC ranging under random communication symbols. It derives closed-form expressions for the expected squared periodic and aperiodic autocorrelation functions (P-ACF and A-ACF), formulates the normalized expected integrated sidelobe level (EISL), and claims that uniform PA is globally optimal for both the P-ACF and A-ACF and for every P-ACF delay index. It then extends the analysis to frequency-domain zero-padding, where uniform PA is no longer optimal, and proposes projected gradient descent and successive convex approximation algorithms to optimize PA under a mainlobe-width constraint. Simulations validate the closed-form expressions and illustrate the sidelobe-versus-mainlobe tradeoff.","tokens_in":19262,"tokens_out":11664,"duration_ms":116029,"significance":"If the main claims are correct, the paper establishes a strong and useful negative result for basic OFDM ranging with random communication signals: no PA optimization is needed, since uniform PA minimizes the average sidelobe level of both P-ACF and A-ACF under the stated constellation assumptions. The closed-form characterizations in Propositions 1-3 are valuable and carefully derived. The zero-padding extension, with its explicit tradeoff between sidelobe level and mainlobe width, is also a worthwhile contribution. The paper is self-contained except for one decomposition from the authors' companion work [26], and the simulation results support the derived expressions. However, the proof of the A-ACF global-optimality theorem contains a substantial gap, and the theorem statements overclaim the constellation domain relative to Assumption 1.","major_comments":[{"comment":"The proof of Theorem 3 does not establish global optimality of uniform PA. The displayed Cauchy-Schwarz inequality is a lower bound on the even-column contribution, not an expression for it. Equality at uniform PA only makes the lower bound tight at that point; it does not show that the true even-column term at any nonuniform P is at least its value at uniform P. Moreover, for a fixed value of sum_n A_n, the Cauchy-Schwarz right-hand side is maximized, not minimized, when all A_n are equal, so the direction of the asserted minimization is incorrect. The theorem may be true, but it needs a direct proof, for example by writing the even-column term as a monotone function of S = sum P_n^2 once the column norms of W2 are taken into account.","section":"Appendix D, Eq. (87)"},{"comment":"The claim that uniform PA is the only power vector for which |sum P_n exp(j2pi k n / N)|^2 is minimized is false. For N=4 and k=2, p=(1.5, 0.5, 0.5, 1.5) satisfies sum P_n exp(j pi n)=0 while being nonuniform. The theorem is recoverable because the denominator grows with sum P_n^2, but the 'only when' conclusion requires an additional argument, and the current proof is incomplete as written.","section":"Section IV-A, Theorem 2 proof, Eq. (33)"},{"comment":"Assumption 1 restricts the analysis to zero-mean, zero-pseudo-variance constellations and explicitly excludes BPSK and 8-QAM, yet the theorem statements and the abstract claim optimality 'for all constellations.' The A-ACF derivations in Appendix C rely on the decomposition S = I + S1 + S2 in Eq. (62), which uses the vanishing pseudo-variance. The authors should either prove the extension to improper constellations or restate the theorems with the explicit restriction.","section":"Section II-A and Theorems 1-3"},{"comment":"The update rule in Algorithm 1 contains a leading minus sign before ProjB, which would drive every iterate negative and violate the constraint Pi >= 0. In addition, ProjB in Eq. (52) is a radial scaling that forces the sum to N but is not the standard Euclidean projection onto the simplex. The pseudocode should be corrected and the projection actually used in the simulations should be specified precisely.","section":"Section V, Algorithm 1, Eq. (52) and line 5"}],"minor_comments":[{"comment":"The same symbol for the normalized objective is reused for the P-ACF, the A-ACF, and the zero-padding cases; introducing separate notation for E(|r0|^2) in each case would improve readability.","section":"Section II-B, Eq. (16)"},{"comment":"The first-order Taylor surrogate in Eq. (54) is not necessarily a majorant of the nonconvex objective, and no convergence proof is given; the paper should state whether the reported convergence is empirical or provide a supporting argument.","section":"Section V, Algorithm 2, Eq. (54)"},{"comment":"The notation 'a_m.n' should be 'a_{m,n}', and the matrix norm in Eq. (39) should be written as ||FN ~F2N^H||_4^4 with parentheses to avoid ambiguity.","section":"Notations and Eq. (39)"}],"recommendation":"major_revision","confidential_remarks":"The paper is closely tied to the authors' companion work [26], and the new contribution is the power-allocation dimension. The proof gap in Theorem 3 and the overclaimed constellation scope should be fixed before acceptance. If the authors can supply a correct proof of Theorem 3 and correct the Algorithm 1 pseudocode, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The periodic-ACF result is the clean part of the paper: under Assumption 1 (zero pseudo-variance), the normalized EISL of the P-ACF is a monotone function of sum P_i^2, so uniform power is the unique global minimizer. Theorem 1 and Theorem 2 are correct as far as I can tell, and the closed-form expressions (Propositions 1 and 3, Corollary 1) are derived cleanly. The zero-padding analysis and the PGD/SCA algorithms are a genuinely useful extension, and the simulations back the theory. That is real progress for CP-OFDM ISAC: no PA optimization needed for the basic ranging setup.\n\nThe aperiodic case is where I part ways. Theorem 3 may be true—actually I think it is, because the even-column term reduces to N c S / ((mu4-1)S + N^2), which is increasing in S = sum P_i^2—but Appendix D does not prove it. The Cauchy-Schwarz step in (87) produces a lower bound L = (sum sqrt(A_n))^2 / D. For A_n = c P_n^2, L = c N^2 / D, which is maximal at uniform PA, not minimal. So the statement that the bound reaches its minimum when the A_n are equal is wrong in direction. Equality in Cauchy-Schwarz only makes the bound tight; it doesn't locate the minimum of the actual term. Because the actual even-column term is monotone in S, the theorem is salvageable with a one-line argument, but it is not proved as written. That is a load-bearing gap in one of the paper's two central claims.\n\nMinor caveat: Assumption 1 excludes BPSK and 8-QAM, so the abstract's 'arbitrary constellation mapping' overstates the scope. The body is explicit about the restriction, so this is a wording issue.\n\nWho this is for: ISAC people doing waveform design who want to know whether PA is worth optimizing under random signaling. For CP-OFDM, this paper gives a good answer. For the aperiodic case, they should wait for a repaired proof, or fix it themselves easily.\n\nRecommendation: send it to peer review, but require a corrected Appendix D (and a tighter argument for Theorem 2, which has a similar component-wise flavor). The P-ACF contribution stands on its own; the A-ACF claim needs the proof to match the truth.","headline":"The P-ACF half of the paper is solid and useful; the A-ACF global-optimality claim is true but not proved, because the Appendix D lower-bound argument minimizes the wrong quantity.","tokens_in":19741,"tokens_out":10632,"would_cite":true,"duration_ms":88356,"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":"Uniform power allocation is the unique optimum for random-OFDM ranging sidelobes, the paper proves.","keywords":["integrated sensing and communication","OFDM","power allocation","random signaling","ranging sidelobe","periodic autocorrelation function","aperiodic autocorrelation function","frequency zero-padding"],"falsifier":"For a small system, say $N=8$ subcarriers with 16-QAM symbols (satisfying Assumption 1), numerically minimize the normalized EISL in equation (16) over the simplex $\\sum_i P_i = N$, $P_i \\ge 0$ by exhaustive grid search or random sampling; the theorem says the global minimum occurs only at $P_1=\\cdots=P_8=1$, with value $N\\mu_4/((\\mu_4-1)+N)-1$. Finding any admissible power vector with a strictly smaller normalized EISL, or with a smaller normalized $\\mathbb{E}(|\\tilde{r}_k|^2)$ at some lag $k\\neq 0$ in the periodic case, would refute the uniqueness claim.","tokens_in":18774,"feed_emoji":"📡","tokens_out":11988,"duration_ms":95570,"temperature":0.7,"pith_summary":"This paper asks whether subcarrier power allocation can improve the ranging sidelobe behavior of OFDM signals whose subcarriers carry random communication data, a setup central to 6G integrated sensing and communication (ISAC). It establishes that, under the standard circular-symmetric constellation assumption (unit power, zero mean, zero pseudo-variance, which excludes BPSK and 8-QAM), uniform power allocation is the only scheme that minimizes the normalized expected integrated sidelobe level for both cyclic-prefix (periodic) and non-cyclic (aperiodic) autocorrelations of the OFDM signal. For the periodic case, uniform power also minimizes the average sidelobe at every delay lag. The practical consequence is that no power-allocation optimization is needed for the basic OFDM ranging problem, saving computation and preserving communication performance. The exception is frequency-domain zero-padding for finer range resolution, where uniform power is suboptimal and optimized allocations trade a wider mainlobe for lower sidelobes.","feed_headline":"Uniform power is already optimal for OFDM ranging sidelobes","feed_subtitle":"Proof: equal subcarrier powers minimize expected sidelobe energy for random ISAC signals, with a zero-padding caveat.","key_machinery":"The carrying object is the exact expression for the average squared periodic autocorrelation, $\\mathbb{E}(|\\tilde{r}_k|^2) = (\\mu_4 - 1)\\sum_{i=1}^N P_i^2 + |\\sum_{n=1}^N P_n e^{j2\\pi k n/N}|^2$, where $\\mu_4 = \\mathbb{E}(|s_n|^4)$ is the kurtosis of the unit-power, zero-pseudo-variance constellation. Summing this over nonzero lags and normalizing by the expected mainlobe makes the normalized EISL a strictly monotone function of $\\sum_i P_i^2$, and the Cauchy-Schwarz inequality forces that sum to be minimized only at $P_1 = \\cdots = P_N = 1$. The aperiodic case is reduced to the same structure by embedding the $N$-sample signal in a $2N$-periodic shift, after which a second application of Cauchy-Schwarz on the odd and even column blocks delivers the same uniqueness conclusion. The kurtosis parameter $\\mu_4$ is what carries constellation dependence: all PSK constellations have $\\mu_4 = 1$ and all QAM constellations have $1 \\le \\mu_4 \\le 2$, so the theorems cover the whole allowed family at once.","core_discovery":"The paper proves that uniform power allocation, $P_i = 1$ on every subcarrier, is the unique minimizer of the normalized expected integrated sidelobe level (EISL) for both the periodic and aperiodic autocorrelation functions of a random OFDM ISAC signal, whenever the constellation satisfies the unit-power, zero-mean, zero-pseudo-variance condition. It further proves that for the periodic case the same uniform scheme minimizes the average squared sidelobe at every delay lag, not just the integrated total. The proof works uniformly across all such constellations because the normalized EISL depends on the power vector only through $\\sum_i P_i^2$, which is minimized exactly when all subcarrier powers are equal. Frequency-domain zero-padding breaks this conclusion: the paper shows that with zero-padding, uniform power is no longer optimal, and it gives a projected-gradient-descent algorithm that lowers sidelobes at the cost of a wider mainlobe, plus a constrained variant that makes the tradeoff tunable.","pith_inferences":["Because the theorems depend on vanishing pseudo-variance, allowing improper constellations such as BPSK or 8-QAM is the natural stress test; extra conjugate-correlation terms may make non-uniform power strictly optimal even without zero-padding, which would extend rather than contradict the paper's domain.","The optimality applies to the expectation over random symbols; a sensing system concerned with per-realization peak sidelobes or worst-case ambiguity could still profit from non-uniform power, a different objective than EISL.","The zero-padding case effectively turns the power vector into a spectral window on the interpolated ranging response; the mainlobe-versus-sidelobe tradeoff found here has the same shape as classical window design, suggesting the PGD and SCA allocations could be compared against standard window families in future work."],"forward_implications":["For the basic zero-Doppler ranging setup with cyclic-prefix or non-cyclic OFDM, the power-allocation subproblem has a closed-form answer: set every subcarrier power to 1, so no iterative PA optimization is required.","Uniform power gives an average sidelobe floor of $(\\mu_4-1)N$ per lag for the periodic ACF; with PSK constellations, where $\\mu_4=1$, the expected periodic ACF is impulse-like with zero sidelobes.","Any non-uniform power vector strictly raises the normalized EISL for both P-ACF and A-ACF, so communication-oriented power loading carries a ranging sidelobe penalty in this metric.","Frequency zero-padding changes the optimal design: the PGD allocation reduces sidelobes relative to uniform power, and the SCA variant lets an operator choose how much mainlobe widening to accept for a given sidelobe reduction."],"supporting_citations":[{"why":"Prior companion result showing OFDM gives the lowest ranging sidelobe under random ISAC signaling; this paper extends that to subcarrier power allocation.","marker":"[26]"},{"why":"Establishes the ambiguity function and its zero-Doppler slice as the ranging performance metric.","marker":"[32]"},{"why":"Supplies the periodic shift matrix decomposition used to convert the P-ACF into an IDFT of subcarrier powers.","marker":"[33]"},{"why":"Defines the integrated sidelobe level and provides sequence-optimization background for the EISL metric.","marker":"[34]"},{"why":"Provides the convex-optimization tools behind the projected gradient descent and successive convex approximation algorithms for the zero-padding extension.","marker":"[36]"}],"fun_headline_variants":["Uniform power is proven optimal for OFDM ranging sidelobes","Equal subcarrier powers minimize OFDM sidelobe energy","Uniform power wins in OFDM ranging—except with zero-padding","Proof: uniform OFDM power beats all for sidelobes","Random ISAC: uniform power is the sidelobe minimizer"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proofs assume the random data symbols come from a constellation with zero mean, unit power, and no correlation between a symbol and its complex conjugate (zero pseudo-variance), which excludes BPSK and 8-QAM; if that assumption fails, the closed-form sidelobe expressions gain extra terms and uniform power may stop being optimal.","fun_headline_variants_meta":{"raw":{"variants":["Uniform power is proven optimal for OFDM ranging sidelobes","Equal subcarrier powers minimize OFDM sidelobe energy","Uniform power wins in OFDM ranging—except with zero-padding","Proof: uniform OFDM power beats all for sidelobes","Random ISAC: uniform power is the sidelobe minimizer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000416,"raw_usage":{"total_tokens":2204,"prompt_tokens":1057,"completion_tokens":1147,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":1061}},"tokens_in":673,"tokens_out":1147,"duration_ms":9927,"temperature":1.0,"reasoning_tokens":1061,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:27:59.051932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small system, say $N=8$ subcarriers with 16-QAM symbols (satisfying Assumption 1), numerically minimize the normalized EISL in equation (16) over the simplex $\\sum_i P_i = N$, $P_i \\ge 0$ by exhaustive grid search or random sampling; the theorem says the global minimum occurs only at $P_1=\\cdots=P_8=1$, with value $N\\mu_4/((\\mu_4-1)+N)-1$. Finding any admissible power vector with a strictly smaller normalized EISL, or with a smaller normalized $\\mathbb{E}(|\\tilde{r}_k|^2)$ at some lag $k\\neq 0$ in the periodic case, would refute the uniqueness claim.","supporting_citations":[{"cited_title":"Radar signals,","cited_arxiv_id":null,"evidence_quote":"Establishes the ambiguity function and its zero-Doppler slice as the ranging performance metric."},{"cited_title":"On designing sequences with impulse-like periodic correlation,","cited_arxiv_id":null,"evidence_quote":"Supplies the periodic shift matrix decomposition used to convert the P-ACF into an IDFT of subcarrier powers."},{"cited_title":"Optimization metho ds for design- ing sequences with low autocorrelation sidelobes,","cited_arxiv_id":null,"evidence_quote":"Defines the integrated sidelobe level and provides sequence-optimization background for the EISL metric."}],"review_version":1}