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arxiv: 2604.22533 · v1 · submitted 2026-04-24 · 💻 cs.IT · math.IT

Gamma-Distributed Geometric Constellation for ISAC: Design and Analysis

Amirhossein Keshavarzchafjiri , Janith K. Dassanayake , Gayan A. Aruma Baduge , Mojtaba Vaezi This is my paper

Pith reviewed 2026-05-08 09:43 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords ISACgeometric constellationGamma distributionparticle swarm optimizationmutual informationprobability of detectionsymbol error rateCramer-Rao bound
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The pith

A Gamma-distributed geometric constellation for ISAC jointly optimizes detection probability and mutual information via particle swarm optimization.

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

The paper proposes a framework that generates constellation points by drawing amplitudes from a Gamma distribution and phases from a uniform distribution. Parameters of this distribution are tuned by particle swarm optimization to improve the probability of detection for sensing and mutual information for communication. Analytical bounds are derived for the union bound on symbol error rate and the Cramer-Rao bound on sensing parameter estimation. The resulting design matches neural-network performance while using far fewer parameters and requiring no training data, and it fits more easily into conventional transmitter and receiver architectures.

Core claim

Constellation points modeled as samples from a parameterized Gamma-uniform two-dimensional distribution can be optimized through particle swarm search on the joint objective of sensing detection probability and communication mutual information. This yields a geometric constellation for which the authors derive a union bound on symbol error rate and a Cramer-Rao bound on sensing estimation error, producing competitive end-task performance with substantially reduced complexity relative to end-to-end neural designs.

What carries the argument

The Gamma-uniform distribution (Gamma amplitude, uniform phase) that generates the constellation points, with its two parameters adjusted by particle swarm optimization to maximize the combined sensing and communication objective.

If this is right

  • The design supplies a closed-form union bound on symbol error rate for communication analysis.
  • The design supplies a Cramer-Rao bound on sensing parameter estimation accuracy.
  • Performance remains competitive with neural-network constellations while using orders of magnitude fewer parameters and no training.
  • The geometric structure integrates directly into existing modulation and detection pipelines without retraining or probabilistic shaping blocks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Gamma-uniform generator could be re-optimized for other sensing metrics such as estimation variance or multi-target resolution.
  • Because the constellation remains deterministic once parameters are fixed, it may simplify certification and hardware calibration compared with learned probabilistic mappings.
  • Extending the framework to time-varying channels would require only re-running the particle swarm at channel coherence intervals rather than full neural retraining.

Load-bearing premise

Particle swarm optimization reliably locates the Gamma-uniform parameters that simultaneously improve detection probability and mutual information under the channel models used for optimization.

What would settle it

Hardware experiments that apply realistic impairments and show the proposed constellation failing to improve either detection probability or mutual information over standard or neural baselines would disprove the central claim.

Figures

Figures reproduced from arXiv: 2604.22533 by Amirhossein Keshavarzchafjiri, Gayan A. Aruma Baduge, Janith K. Dassanayake, Mojtaba Vaezi.

Figure 1
Figure 1. Figure 1: An ISAC system with one communication receiver and one target, where view at source ↗
Figure 2
Figure 2. Figure 2: Constellation points in the candidate set and their assigned likelihood that is Gamma-distributed in amplitude. view at source ↗
Figure 3
Figure 3. Figure 3: Sensing and communication performance for three different constellations: QAM, PSK, and randomly generated. view at source ↗
Figure 4
Figure 4. Figure 4: Constellation diagrams designed using the proposed method and NN–based method for different scenarios. Fig. 4: Constellation diagrams designed using the proposed method and NN–based method for different scenarios. view at source ↗
Figure 5
Figure 5. Figure 5: Trade-off between communication SER and sensing Detection Probability Pd view at source ↗
Figure 7
Figure 7. Figure 7: Verification of fitting the distribution of view at source ↗
Figure 8
Figure 8. Figure 8: KL divergence for different mixture components view at source ↗
Figure 10
Figure 10. Figure 10: The CRB for estimating the channel parameter view at source ↗
Figure 7
Figure 7. Figure 7: However, the KL divergence decreases below view at source ↗
read the original abstract

A novel Gamma-distributed geometric constellation design framework for integrated sensing and communication (ISAC) is proposed in this paper. In this framework, constellation points are modeled as samples drawn from a parameterized two-dimensional distribution, with a Gamma distribution for the amplitude and a uniform distribution for the phase. End-task performance metrics, namely, the probability of detection for sensing and mutual information for communication, are used as objective functions of the optimization problem, and the problem is solved via particle swarm optimization. We further derive analytical performance bounds for the proposed design, including the union bound on the symbol error rate for communication and the Cramer--Rao bound for sensing parameter estimation. The proposed method is compared with constellations obtained via end-to-end neural network design, demonstrating competitive performance while requiring significantly fewer parameters and no training data. Moreover, the proposed geometric constellation is more compatible with conventional system architectures than probabilistic or neural network-based designs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a Gamma-distributed geometric constellation for ISAC, with amplitudes drawn from a parameterized Gamma distribution and phases from a uniform distribution. Parameters are optimized via particle swarm optimization (PSO) using probability of detection and mutual information as joint objectives. Standard analytical bounds are derived (union bound on symbol error rate, Cramér-Rao bound for sensing), and the design is claimed to achieve competitive performance versus end-to-end neural-network constellations while using only two parameters, requiring no training data, and offering better compatibility with conventional transceivers.

Significance. If the PSO optimization reliably locates competitive operating points and the numerical comparisons hold under realistic impairments, the work would offer a low-parameter, analytically tractable alternative to data-driven constellation design in ISAC. The reduction to a two-parameter family and the explicit derivation of standard bounds are strengths that could aid interpretability and hardware implementation.

major comments (2)
  1. [§4] §4 (PSO Optimization Framework): The joint objective of detection probability and mutual information is non-convex; the manuscript reports a single optimized point without multiple random restarts, convergence diagnostics, or comparison to gradient-based or global optimizers. This directly affects the central claim of competitive performance versus neural-network designs, which explore a richer function class.
  2. [§5] §5 (Numerical Evaluation): The comparisons to end-to-end NN constellations lack sufficient detail on simulation parameters (SNR range, channel model, number of Monte Carlo trials, exact NN architecture and training procedure). Without these, it is impossible to verify whether the reported performance parity is robust or an artifact of the chosen operating point.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'significantly fewer parameters' should be quantified (e.g., exact parameter counts for both the Gamma-uniform design and the NN baseline).
  2. [§2] Notation: The mapping from Gamma shape/scale parameters to the two-dimensional constellation points should be stated explicitly in the system model, including any normalization or power constraint.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. The comments highlight important aspects of reproducibility and optimization validation that we will address in the revision. Below we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [§4] §4 (PSO Optimization Framework): The joint objective of detection probability and mutual information is non-convex; the manuscript reports a single optimized point without multiple random restarts, convergence diagnostics, or comparison to gradient-based or global optimizers. This directly affects the central claim of competitive performance versus neural-network designs, which explore a richer function class.

    Authors: We agree that the joint objective is non-convex and that additional validation of the PSO procedure would strengthen the central claims. The original manuscript reports results from a single PSO run that yielded competitive performance. In the revised manuscript we will include results from multiple independent random restarts (with different initializations), convergence diagnostics in the form of objective-value trajectories, and a comparison against an alternative global optimizer such as differential evolution. These additions will demonstrate that the reported operating point is reliably attained and not an artifact of a single run. revision: yes

  2. Referee: [§5] §5 (Numerical Evaluation): The comparisons to end-to-end NN constellations lack sufficient detail on simulation parameters (SNR range, channel model, number of Monte Carlo trials, exact NN architecture and training procedure). Without these, it is impossible to verify whether the reported performance parity is robust or an artifact of the chosen operating point.

    Authors: We acknowledge that the numerical evaluation section does not provide all the implementation details required for full reproducibility. In the revised manuscript we will expand §5 to explicitly state the SNR ranges evaluated, the channel models used for both the sensing and communication links, the number of Monte Carlo trials performed for each metric, the precise neural-network architecture and hyperparameters of the end-to-end baselines, and the training procedure (optimizer, learning rate, number of epochs, and dataset size). These clarifications will allow readers to assess the robustness of the reported performance parity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper explicitly optimizes the two parameters of the Gamma-uniform distribution via PSO to maximize the joint objective of detection probability and mutual information. The subsequent analytical results (union bound on SER, CRB on parameter estimation) are standard closed-form expressions derived from the constellation geometry and channel model, independent of the specific PSO solution. The performance comparison to end-to-end NN designs is an external benchmark rather than a fitted quantity renamed as a prediction. No self-definitional steps, load-bearing self-citations, or ansatz smuggling appear in the provided abstract or described chain; the optimization is the declared method, not a hidden reduction of outputs to inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on modeling constellation points as i.i.d. samples from a Gamma-uniform distribution whose parameters are tuned by PSO to ISAC metrics, plus standard analytical bounds whose validity depends on conventional assumptions about noise and channel models.

free parameters (1)
  • Gamma distribution shape and scale parameters
    Two parameters of the Gamma distribution for amplitude are chosen by particle swarm optimization to maximize the joint objective of detection probability and mutual information.
axioms (2)
  • standard math Union bound on symbol error rate
    Standard union bound applied to derive communication performance limit.
  • standard math Cramer-Rao bound for sensing parameter estimation
    Standard lower bound used for sensing performance analysis.

pith-pipeline@v0.9.0 · 5477 in / 1387 out tokens · 35758 ms · 2026-05-08T09:43:32.380171+00:00 · methodology

discussion (0)

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Reference graph

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