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REVIEW 3 major objections 6 minor 15 references

Joint Optimization of Geometric and Probabilistic Constellation Shaping for OFDM-ISAC Systems

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.11583 v1 pith:D7KB647J submitted 2025-01-20 eess.SP

Benedikt Geiger , Fan Liu , Shihang Lu , Andrej Rode , Laurent Schmalen This is my paper · ORCID
classification eess.SP
keywords integratedsensingandcommunicationsOFDMconstellationshapinggeometricprobabilisticjointkurtosisdetectionprobability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [Sec. II-B, Eq. (9)] 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).
  2. [Sec. II-D and Fig. 2] 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.
  3. [Sec. IV-B, Fig. 5] 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.
minor comments (6)
  1. [Sec. II] 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.
  2. [Sec. IV] The phrase 'we increase the batch size increases' is a typo; it should read 'we increase the batch size'.
  3. [References] Reference [12] lists the arXiv identifier as '407.06691', which appears malformed; it should likely be '2407.06691'.
  4. [Sec. II-B, Eq. (4)] 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).
  5. [Fig. 6] 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.
  6. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kurtosis-based detection probability is derived from a stated Gaussian/CLT model and independently validated by CA-CFAR simulation, not by construction.

full rationale

The paper's central derivation (Eqs. 9-13) computes PD from a textbook CFAR formula and the constellation kurtosis. The kurtosis enters through the variance of the matched-filter estimate, E|Hhat|^2 - |E Hhat|^2 = |Hn|^2(kappa-1)+sigma_s^2, which follows algebraically from E|X|^4. The CLT justification for Gaussianity is empirical (Fig. 2) but is not circular: it is a stated approximation and is independently tested by the simulated CFAR markers in Fig. 5. The optimization loss (15) uses kurtosis only as a constraint derived from that PD model; the simulated PD results do not feed back into the derivation or loss. Self-citations (e.g., [12] for the Gaussian kurtosis value, [6,7,13] for AE training) are background/implementation references and are not load-bearing. Any concern about CFAR window size or tail Gaussianity is a correctness/approximation risk, not a circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The analytical claim rests on standard signal-processing assumptions: static point targets, AWGN communications, i.i.d. symbols, and CLT-justified Gaussianity at the CFAR input. No physical entities are invented and no parameters are fitted to data; the hand-chosen loss penalty d=3 and training SNR of 10 dB are design choices that shape the reported trade-off curves.

free parameters (2)
  • Sensing loss penalty factor d = 3
    Hand-chosen in Eq. (15); large enough to dominate the communications loss when kappa exceeds the threshold, but the exact value affects the trained constellations and the reported trade-off curves.
  • Training communications SNR = 10 dB
    All constellations are optimized at SNR_c = 10 dB; Fig. 6 evaluates GMI at other SNRs, but the design point is fixed and results may not generalize to strongly different operating SNRs.
assumptions (5)
  • domain assumption All sub-carriers use the same constellation and transmit symbols are i.i.d. with distribution P(X).
    Used in Sec. II-A and II-D to make the kurtosis kappa = E|X|^4 the only constellation statistic entering the sensing SINR.
  • domain assumption The sensing channel consists of J static point targets with delays that are integer multiples of the sampling time within the CP; Doppler is neglected and a single OFDM symbol is used.
    Stated in Sec. II; required for Eq. (11), where the IFFT concentrates target power into one delay cell with gain sqrt(N).
  • domain assumption After matched filtering and IFFT, the residual disturbance at the CA-CFAR input is Gaussian with variance sigma_s^2 + (kappa-1)*sum_j |a_j|^2, so PD = P_FA^(1/(1+gamma)) applies.
    Justified by the central limit theorem and the histogram in Fig. 2; this is the statistical model behind Eqs. (9) and (13).
  • domain assumption The communications channel is AWGN after ideal equalization.
    Stated in Sec. II-C; it decouples sub-carriers and lets GMI per sub-carrier be the communications objective.
  • standard math PD = P_FA^(1/(1+gamma)) is the CA-CFAR detection probability.
    Taken from [10]; valid for Gaussian noise and interference with a sufficiently large reference window, but finite-N effects for the N=100 window are not included.

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Cite this review

Pith. "Pith review of Joint Optimization of Geometric and Probabilistic Constellation Shaping for OFDM-ISAC Systems." pith.science (2026). https://pith.science/paper/D7KB647J

@misc{pith2026250111583,
  author       = {Pith},
  title        = {Pith review of: Joint Optimization of Geometric and Probabilistic Constellation Shaping for OFDM-ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D7KB647J}},
  note         = {Machine review of arXiv:2501.11583}
}
read the original abstract

6G communications systems are expected to integrate radar-like sensing capabilities enabling novel use cases. However, integrated sensing and communications (ISAC) introduces a trade-off between communications and sensing performance because the optimal constellations for each task differ. In this paper, we compare geometric, probabilistic and joint constellation shaping for orthogonal frequency division multiplexing (OFDM)-ISAC systems using an autoencoder (AE) framework. We first derive the constellation-dependent detection probability and propose a novel loss function to include the sensing performance in the AE framework. Our simulation results demonstrate that constellation shaping enables a dynamic trade-off between communications and sensing. Depending on whether sensing or communications performance is prioritized, geometric or probabilistic constellation shaping is preferred. Joint constellation shaping combines the advantages of geometric and probabilistic shaping, significantly outperforming legacy modulation formats.

Figures

Figures reproduced from arXiv: 2501.11583 by the authors.

Figure 1
Figure 1. To reduce computational complexity and simplify [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 1
Figure 1. Block diagram of the considered monostatic bit-interleaved coded modulation OFDM-ISAC system [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Probability distribution of the noise at the input of the target detector, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: Proposed AE framework to shape constellations for ISAC. The trainable parameters are marked in orange and their normalization in gray. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 5
Figure 5. Figure 5: Derived and simulated detection probability of a TOI, e.g. a drone, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 4
Figure 4. Figure 4: Optimized constellations for geometric, probabilistic and joint [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: Simulated communications performance (AIR) of the optimized [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the S&C trade-off between the shaping methods. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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Reference graph

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