{"id":"398df580-5512-4aa9-94d5-60f51c67ad79","arxiv_id":"2501.11583","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Constellation kurtosis alone determines CFAR detection probability in OFDM-ISAC, and autoencoder-based joint geometric and probabilistic shaping provides a flexible communications-sensing trade-off.","lead":"This paper compares geometric, probabilistic, and joint constellation shaping for 6G OFDM radar-and-communication systems, and derives that target detection probability depends only on the constellation's kurtosis. It trains an autoencoder under a kurtosis constraint and finds that joint shaping gives the most flexible trade-off between data rate and sensing range.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The kurtosis-only PD claim rests on a Gaussian-tail approximation that is validated only in the bulk of the distribution; the P_FA=10^-3 operating point and finite CA-CFAR window are not covered.","rationale":"The reader's weakest assumption is the Gaussian approximation behind Eq. (9), and I agree that this is the most load-bearing premise. I sharpen it: the CLT argument in Sec. II-D concerns the bulk of the IFFT output distribution, whereas the CFAR operates at the 10^-3 tail. A histogram of Re{tilde w[k]} for 16-QAM and 64 sub-carriers does not establish tail Gaussianity, and shaped constellations with highly nonuniform probabilities can have slower tail convergence. If the tail is non-Gaussian, PD is not determined by kappa alone, which would invalidate the kurtosis-only formulation and the constraint (C1). The finite CA-CFAR reference-window effect is a smaller, separate correction also absent from Eq. (9). I also noticed an inconsistency between Eq. (2), which includes a 1/sqrt(N) factor in H_n, and Eqs. (11) and (13), which do not; since the simulations align with Eq. (13), this appears to be a typographical normalization error rather than the central weakness. The proposed test, comparing two equal-kurtosis constellations with different higher-order statistics, would settle whether the kurtosis-only claim holds. The reader's conditional verdict is appropriate: the concern is a request for validation, not a demonstrated failure.","tokens_in":8687,"tokens_out":21361,"duration_ms":236346,"concrete_test":"Generate two distinct 64-point constellations with identical unit-power kurtosis kappa = 1.5, e.g., a geometrically shaped multi-ring set and a probabilistically shaped 64-QAM with unequal probabilities, and run the exact CA-CFAR simulation of Fig. 5 with P_FA=10^-3 and N=100 reference cells. If the simulated PD values differ by more than the Monte Carlo uncertainty, the kurtosis-only claim fails. Repeat with N=1024 sub-carriers: convergence toward Eq. (9) would indicate a finite-N CLT/tail effect, while persistent differences would indicate a deeper modeling error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that PD depends only on the constellation kurtosis via Eqs. (9) and (13) requires the interference plus noise at the CFAR input to be approximately Gaussian with variance sigma_s^2 + (kappa-1)*sum_j |a_j|^2. The paper's evidence, Fig. 2, is a histogram of Re{tilde w[k]} for 16-QAM with 64 sub-carriers; it shows the center of the distribution, not the tail that sets a false-alarm probability of 10^-3. For probabilistically or jointly shaped constellations with kappa near 2, the per-subcarrier term H_n(|X_n|^2-1) can be heavy-tailed because a few low-probability, high-amplitude points dominate the fourth moment, and CLT convergence in the tail can be much slower than in the bulk. If the tail is non-Gaussian, PD is not a function of kappa alone, and the sensing loss (15) does not exactly enforce the intended detection-probability constraint. Separately, Eq. (9) is the known-noise-power CFAR expression; CA-CFAR with N=100 reference cells incurs a threshold-estimation loss that is not included in the analytical curves. Both effects may be small in the displayed operating range, but they are unquantified, and the markers in Fig. 5 have no error bars, so the claimed agreement cannot be independently assessed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies constellation shaping for a mono-static OFDM-ISAC system with a bit-interleaved coded-modulation communications receiver and a CA-CFAR sensing receiver. Its central analytical claim is that, after matched filtering and IFFT, the disturbance at the CFAR input is approximately Gaussian with variance sigma_s^2 + (kappa-1) sum_j |a_j|^2, so the SINR of a target of interest is gamma = N |a_TOI|^2 / (sum_j |a_j|^2 (kappa-1) + sigma_s^2) (Eq. 13) and the detection probability is P_D = P_FA^{1/(1+gamma)} (Eq. 9), where kappa is the fourth moment of the unit-power constellation; hence P_D depends on the constellation only through kappa. The paper then introduces an autoencoder that optimizes geometric, probabilistic, or joint constellation shaping under a kurtosis constraint, using a penalty-based sensing loss (15), and evaluates the resulting GMI-versus-kurtosis trade-off. Simulation results show a kurtosis-dependent detection range, agreement between derived and simulated P_D, and that joint shaping outperforms legacy 64-QAM/64-PSK.","tokens_in":8944,"tokens_out":19694,"duration_ms":209051,"significance":"The kurtosis-only characterization, if correct, is a genuinely useful reduction: it turns a detection-probability constraint into a scalar moment constraint and makes the ISAC trade-off analytically tractable. The disturbance-variance derivation in Eqs. (10)-(13) is transparent and parameter-free, and the kurtosis-to-detection-range mapping in Fig. 5 is a falsifiable prediction that the paper tests directly. The comparison of geometric, probabilistic, and joint shaping in Fig. 7 is also a useful design insight. However, two gaps affect the central claim: Eq. (9) is not consistent with the deterministic point-target model used in Sec. II-B, and the Gaussian-tail/finite-window approximations underlying the kurtosis-only statement are not quantitatively validated. The contribution is therefore significant but not yet fully established.","major_comments":[{"comment":"The formula P_D = P_FA^{1/(1+gamma)} is the CA-CFAR detection probability for a Swerling-1 (exponentially fluctuating) target in Gaussian interference, not for the deterministic point-target channel described in Sec. II-B, where each target has a fixed complex amplitude a_j. For a nonfluctuating target the probability of detection is the Marcum Q-function of the normalized threshold and SNR, which can differ substantially from (9) at the operating points of Fig. 5 (e.g., at P_FA=10^{-3} and gamma=10 dB, (9) gives about 0.53, whereas the nonfluctuating probability is considerably higher). The derivation of gamma in Eqs. (10)-(13) is built on a deterministic H_n, so it does not provide the fluctuation averaging that (9) requires. Please either re-derive (9) for the stated fixed-target model or explicitly adopt a Swerling-1 target model and show how the average target power enters (13). This is load-bearing because the kurtosis-only claim rests on (9).","section":"Sec. II-B, Eq. (9)"},{"comment":"The Gaussian approximation is motivated by a histogram of Re{tilde w[k]} in the bulk of the distribution for 64 sub-carriers, but the operating point of interest is the P_FA=10^{-3} tail, and the CA-CFAR uses a finite reference window of N=100 cells. For probabilistically or jointly shaped constellations with rare large-amplitude symbols, the terms H_n(|X_n|^2-1) can be heavy-tailed and the CLT convergence at the 10^{-3} tail can be much slower than in the bulk. The finite-window CA-CFAR threshold-estimation loss is also not included in the analytical curves. Please quantify the tail approximation error for the actual optimized constellations (e.g., with tail-region distribution tests), report Monte Carlo trial counts and error bars for the markers in Fig. 5, and bound or incorporate the finite-window correction. Without this, the statement that P_D is a function of kurtosis alone is an unverified asymptotic approximation.","section":"Sec. II-D and Fig. 2"},{"comment":"The validation of the central claim is difficult to assess as reported. The simulation parameters are only referenced to [14] and [15] rather than stated (number of sub-carriers, CP length, bandwidth, noise powers, and the mapping from RCS to |a_j|^2), and the Monte Carlo markers in Fig. 5 have no error bars or trial counts. Furthermore, the caption states that the TOI follows a Swerling-1 model while the analytical curves appear to use (9) and (13) with a fixed target power; the text does not explain how the fluctuation is averaged in the simulations or in the curves. The authors should state the complete parameter set, the number of Monte Carlo runs, and the precise fluctuation-averaging formula used for the analytical curves so that the claimed agreement can be independently checked.","section":"Sec. IV-B, Fig. 5"}],"minor_comments":[{"comment":"The notation N is used both for the number of OFDM sub-carriers and for the CA-CFAR sliding-window length; please rename one of the two to avoid ambiguity.","section":"Sec. II"},{"comment":"The phrase 'we increase the batch size increases' is a typo; it should read 'we increase the batch size'.","section":"Sec. IV"},{"comment":"Reference [12] lists the arXiv identifier as '407.06691', which appears malformed; it should likely be '2407.06691'.","section":"References"},{"comment":"The unbiasedness of the matched-filter channel estimate hat H_n requires E|X_n|^2=1, which is later enforced by constraint (C2); this condition should be stated explicitly before Eq. (4).","section":"Sec. II-B, Eq. (4)"},{"comment":"The sentence 'all shaping methods reduce the gap to capacity and outperform the conventional 64-QAM across an SNR range of 10 dB' is ambiguous; please specify the exact SNR range over which the claim holds.","section":"Fig. 6"},{"comment":"The paper does not provide a data or code availability statement; given the number of adjustable training hyperparameters (batch-size schedule, learning rate, penalty factor d), a reproducibility statement or code release would be helpful.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of eess.SP and the topic is timely. My main concern is the mismatch between the deterministic point-target model in Sec. II-B and the Swerling-1 detection formula in Eq. (9); I believe this is fixable in revision and do not see it as grounds for rejection. The related literature is cited, and there is no indication of missing prior work. The paper would benefit from a clearer separation between exact steps and the CLT/CFAR asymptotic approximations in the detection-probability derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Geiger et al. give ISAC constellation designers something they didn't have: a closed-form expression for CFAR detection probability that depends on the constellation only through its kurtosis. The derivation in Eqs. (10)–(13) is algebraically clean and correctly identifies the integration gain and the interference term. They then use that kurtosis as a differentiable penalty in an autoencoder shaping framework, which lets them optimize geometric, probabilistic, or joint shaping under a sensing constraint. That is a real extension of AE-based shaping from comms-only to ISAC, and the comparison across shaping methods is informative.\n\nThe paper does well on several fronts. The analytical curves in Fig. 5 track the simulated CA-CFAR markers across operating points, which gives independent support for the central formula. The discussion of when geometric versus probabilistic shaping wins under strict versus loose sensing constraints is useful, and the joint-shaping result—matching the better method at each operating point—is a sensible outcome. The related work is appropriately cited.\n\nThe soft spots are mostly presentation, but one is substantive. The Gaussian assumption behind Eq. (9) is validated with a histogram of the bulk of the distribution (Fig. 2), not the tail that sets P_FA=10^-3. For probabilistically shaped constellations with rare high-amplitude points, the fourth-moment contribution can be heavy-tailed, and CLT convergence in the tail can lag the bulk. The finite CA-CFAR reference window (N=100) also introduces a threshold offset that the analytical formula ignores. These effects are probably small at the displayed operating range, but the Monte Carlo markers have no error bars or trial counts, so we can't verify the agreement quantitatively. A short paragraph adding Berry-Esseen-style bounds or simulated tail comparison would close the gap.\n\nThe \"significantly outperforming legacy modulation\" claim rests on a 0.16 bit/symbol GMI gain with no statistical test. That's a minor overstatement in an otherwise measured paper. The authors also don't release code or full training hyperparameters, which would help reproducibility but is not a fatal omission for a conference-level submission.\n\nOverall, this is a competent, well-scoped engineering paper with a genuinely new and useful connection between constellation shaping and sensing performance. The central derivation holds up. With modest revisions (error bars, tail validation, toning down the significance claim), it would be a solid contribution. Yes, send it to peer review.","headline":"A kurtosis-only CFAR detection probability formula and its use as an AE sensing loss is the real contribution; the paper is worth citing and refereeing, with a few validation gaps.","tokens_in":9513,"tokens_out":4935,"would_cite":true,"duration_ms":44796,"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 derives that OFDM-ISAC target detection probability is fixed by the constellation's kurtosis alone, and shows joint geometric and probabilistic shaping beats legacy formats across the sensing-communications trade-off.","keywords":["integrated sensing and communications","OFDM","constellation shaping","geometric shaping","probabilistic shaping","joint shaping","kurtosis","detection probability"],"falsifier":"Run the same two-target OFDM-ISAC scenario with a much smaller number of subcarriers (e.g., 8 or 16) or with a high-kurtosis constellation and compare measured detection probability against Eqs. (9) and (13): a systematic offset would show where the Gaussian assumption breaks. Alternatively, compare two constellations with identical kurtosis but different higher-order moments; if their simulated $P_D$ differs, then kurtosis alone does not determine detection probability.","tokens_in":8462,"feed_emoji":"📡","tokens_out":5480,"duration_ms":53847,"temperature":0.7,"pith_summary":"Integrated sensing and communications (ISAC) wants one OFDM waveform to carry data and detect radar targets, but the best constellations for the two jobs pull in opposite directions: Gaussian-like signals maximize data rate, while constant-modulus signals maximize detection. This paper tries to settle how the choice of constellation changes radar sensitivity, and to shape constellations that manage the trade-off. It derives that, after matched filtering and the IFFT, the target detection probability depends on the transmit constellation only through its kurtosis, and it builds an autoencoder that optimizes geometric, probabilistic, or joint shaping under a kurtosis constraint while maximizing the generalized mutual information (GMI) of the communications link. The simulation results support the kurtosis-only formula and show that joint shaping gives the best of both methods, beating 64-QAM and 64-PSK in the considered regime.","feed_headline":"A single constellation statistic fixes ISAC detection probability","feed_subtitle":"Autoencoder-shaped constellations trade data rate against radar range, beating 64-QAM and 64-PSK in both directions.","key_machinery":"The load-bearing object is the kurtosis of the unit-power constellation, $\\kappa = \\mathbb{E}[|X|^4]$, which appears in the variance of the matched-filter output and therefore in the SINR fed to the cell-averaging constant false alarm rate (CA-CFAR) detector. The paper translates the detection-probability requirement $P_D \\geq \\alpha_D$ into the constraint $\\kappa \\leq \\tilde{\\kappa}$, which is easy to enforce during training. The optimization itself runs through a bitwise autoencoder whose trainable parameters are the constellation points, the symbol probabilities (via Gumbel-softmax), or both, with a Gaussian demapper that produces bit LLRs and a loss $L = (M - \\text{GMI})/M + L_{\\text{sens}}$, where $L_{\\text{sens}}$ penalizes violations of the kurtosis constraint.","core_discovery":"At the heart of the paper is the claim that for the considered monostatic OFDM-ISAC system, the detection probability of a target at the cell-averaging CFAR detector is fixed by the constellation's kurtosis alone. With $N$ subcarriers, a target of interest of complex amplitude $a_{\\text{TOI}}$, interferers of amplitudes $a_j$, noise variance $\\sigma_s^2$, and constellation kurtosis $\\kappa = \\mathbb{E}[|X|^4]$ for unit-power zero-mean symbols, the average SINR at the detector input is $\\gamma = N |a_{\\text{TOI}}|^2 / (\\sum_j |a_j|^2 (\\kappa - 1) + \\sigma_s^2)$, and the detection probability is $P_D = P_{\\text{FA}}^{1/(1+\\gamma)}$. Because $P_D$ depends on the constellation only through $\\kappa$, the sensing requirement can be written as a simple kurtosis constraint $\\kappa \\leq \\tilde{\\kappa}$ inside a differentiable autoencoder. The paper then optimizes constellation points (geometric shaping), symbol probabilities (probabilistic shaping), or both (joint shaping) to maximize GMI subject to that constraint. It reports that geometric shaping wins under strict sensing constraints, probabilistic shaping wins when sensing constraints are loose, and joint shaping tracks whichever method is better across the whole range, improving over 64-QAM and 64-PSK.","pith_inferences":["The Gaussian assumption behind the CFAR formula relies on the central limit theorem over $N$ subcarriers, so at the small-$N$ end of practical OFDM configurations the derived $P_D$ may need a finite-size correction; a good test would be to measure $P_D$ at $N=16$ and compare with the formula.","The kurtosis-only dependence suggests that constellations with identical kurtosis but different higher-order moments or peak-to-average ratios may still behave differently in a real CFAR with finite reference cells; comparing such pairs would reveal whether the fourth moment is truly sufficient.","The paper optimizes at a single communications SNR of 10 dB, so the reported GMI plateaus at high SNR; training over a range of SNRs could extend the joint-shaping advantage to rate-adaptive systems."],"forward_implications":["A system designer can predict radar detection range directly from the kurtosis of the chosen constellation, without Monte Carlo simulation of the CFAR.","The same constellation can be re-optimized for different operating points by changing one scalar constraint, giving a continuous S&C trade-off instead of the two discrete choices offered by 64-PSK and 64-QAM.","Geometric shaping should be preferred when sensing is prioritized and probabilistic shaping when data rate is prioritized, with joint shaping capturing the better of the two in both regimes.","Because $P_D$ depends only on $\\kappa$, any shaping method can be evaluated on the same sensing scale, making the comparison between methods a one-dimensional trade-off curve."],"supporting_citations":[{"why":"Supplies the CA-CFAR detection-probability formula $P_D = P_{\\text{FA}}^{1/(1+\\gamma)}$ and the Gaussian-disturbance assumption it relies on.","marker":"[10]"},{"why":"Establishes the random-deterministic trade-off and the opposing constellation extremes (Gaussian for communications, constant modulus for sensing) that motivate shaping.","marker":"[5]"},{"why":"Provides the autoencoder framework for jointly learning geometric and probabilistic constellation shaping that the paper adapts.","marker":"[6]"},{"why":"Shows end-to-end learning of joint geometric and probabilistic shaping, used as the optimization basis here.","marker":"[7]"},{"why":"Introduces probabilistic constellation shaping for OFDM-ISAC to trade mutual information against sidelobe level, the prior work this paper extends by targeting detection probability.","marker":"[8]"},{"why":"Used to state that a circular complex Gaussian distribution has kurtosis $\\kappa = 2$, the communications-optimal extreme.","marker":"[12]"},{"why":"Provides the multi-target simulation setup used for the detection-probability validation.","marker":"[14]"},{"why":"Supplies the FR2 scenario parameters used in the simulations.","marker":"[15]"}],"fun_headline_variants":["Kurtosis fixes radar detection, enabling targeted ISAC shaping","Kurtosis constraint unlocks optimal ISAC shaping","Autoencoder shapes constellations to trade data for sensing","Kurtosis-based shaping toggles ISAC priority","Kurtosis alone predicts ISAC detection probability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire detection-probability formula assumes that the disturbance entering the CA-CFAR detector is Gaussian after matched filtering and the IFFT, so the textbook relation $P_D = P_{\\text{FA}}^{1/(1+\\gamma)}$ holds; the paper supports this with the central limit theorem and a histogram at 64 subcarriers, but the finite reference-window size of the CFAR is not part of the formula.","fun_headline_variants_meta":{"raw":{"variants":["Kurtosis fixes radar detection, enabling targeted ISAC shaping","Kurtosis constraint unlocks optimal ISAC shaping","Autoencoder shapes constellations to trade data for sensing","Kurtosis-based shaping toggles ISAC priority","Kurtosis alone predicts ISAC detection probability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000785,"raw_usage":{"total_tokens":3483,"prompt_tokens":979,"completion_tokens":2504,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":2428}},"tokens_in":595,"tokens_out":2504,"duration_ms":28696,"temperature":1.0,"reasoning_tokens":2428,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:05:17.961884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same two-target OFDM-ISAC scenario with a much smaller number of subcarriers (e.g., 8 or 16) or with a high-kurtosis constellation and compare measured detection probability against Eqs. (9) and (13): a systematic offset would show where the Gaussian assumption breaks. Alternatively, compare two constellations with identical kurtosis but different higher-order moments; if their simulated $P_D$ differs, then kurtosis alone does not determine detection probability.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CA-CFAR detection-probability formula $P_D = P_{\\text{FA}}^{1/(1+\\gamma)}$ and the Gaussian-disturbance assumption it relies on."},{"cited_title":"From torch to projector: Fundamental tradeoff of integrated sensing and communications,","cited_arxiv_id":null,"evidence_quote":"Establishes the random-deterministic trade-off and the opposing constellation extremes (Gaussian for communications, constant modulus for sensing) that motivate shaping."},{"cited_title":"Joint learning of geometric and probabilistic constellation shaping,","cited_arxiv_id":null,"evidence_quote":"Provides the autoencoder framework for jointly learning geometric and probabilistic constellation shaping that the paper adapts."},{"cited_title":"End-to-end learning of joint geometric and probabilistic constellation shaping,","cited_arxiv_id":null,"evidence_quote":"Shows end-to-end learning of joint geometric and probabilistic shaping, used as the optimization basis here."},{"cited_title":"Reshaping the ISAC tradeoff under OFDM signaling: A probabilistic constellation shaping approach,","cited_arxiv_id":null,"evidence_quote":"Introduces probabilistic constellation shaping for OFDM-ISAC to trade mutual information against sidelobe level, the prior work this paper extends by targeting detection probability."},{"cited_title":"OFDM achieves the lowest ranging sidelobe under random ISAC sig- naling,","cited_arxiv_id":null,"evidence_quote":"Used to state that a circular complex Gaussian distribution has kurtosis $\\kappa = 2$, the communications-optimal extreme."},{"cited_title":"OFDM radar algorithms in mobile communication net- works,","cited_arxiv_id":null,"evidence_quote":"Provides the multi-target simulation setup used for the detection-probability validation."},{"cited_title":"Survey on integrated sensing and communication performance modeling and use cases feasibility,","cited_arxiv_id":null,"evidence_quote":"Supplies the FR2 scenario parameters used in the simulations."}],"review_version":1}