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Gamma-Distributed Geometric Constellation for ISAC: Design and Analysis

This paper proposes a novel Gamma-distributed geometric constellation design framework for Integrated Sensing and Communication (ISAC) that optimizes amplitude and phase distributions to balance sensing detection probability and communication mutual information, achieving competitive performance with fewer parameters and higher architectural compatibility than neural network-based methods.

Original authors: Amirhossein Keshavarzchafjiri, Janith K. Dassanayake, Gayan A. Aruma Baduge, Mojtaba Vaezi

Published 2026-04-27

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CC BY 4.0

Original authors: Amirhossein Keshavarzchafjiri, Janith K. Dassanayake, Gayan A. Aruma Baduge, Mojtaba Vaezi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a DJ at a massive outdoor festival. You have two very different jobs to do at the same time:

The Communication Job: You need to send clear, rhythmic messages to the crowd (like "The next song is a techno hit!") so everyone stays in sync.
The Sensing Job: You also need to act like a sonar system, sending out sound pulses to "feel" the crowd—detecting if they are moving, how many people are there, and where the gaps are.

The problem? If you play music that is too rhythmic and predictable (perfect for messages), it’s bad for "feeling" the crowd because it doesn't create enough "echo" variety. If you play chaotic, random noise (perfect for sensing), the crowd won't understand your messages.

This paper introduces a way to design the "music" (which engineers call a constellation) so that it perfectly balances these two jobs.

The Core Idea: The "Shape" of the Music

In digital communication, "music" is made of specific patterns called constellation points. Think of these as different "notes" you can play.

QAM (The Organized Orchestra): Notes are spread out in a neat grid. Great for messages, bad for sensing.
PSK (The Steady Drumbeat): Notes all have the same volume but different rhythms. Great for sensing, bad for complex messages.

The researchers decided not to pick a pre-made "orchestra" or "drumbeat." Instead, they used a mathematical tool called a Gamma Distribution to "sculpt" a brand-new set of notes.

The Analogy: The Clay Sculptor
Imagine you have a lump of clay.

If you want to be Communication-centric, you spread the clay out into a wide, flat pancake (like a QAM grid) so every note is distinct and easy to hear.
If you want to be Sensing-centric, you squeeze the clay into a thin, tight ring (like a PSK drumbeat) so the "echoes" are very sharp.
If you want a Trade-off, you shape it into several concentric rings—like a target or a ripple in a pond.

By simply adjusting two "knobs" (called shape and scale), the researchers can morph the constellation from a message-heavy shape to a sensing-heavy shape.

How They Do It: The "Smart Search"

Instead of using a massive, power-hungry Artificial Intelligence (like a giant supercomputer trying to learn everything from scratch), they used a clever shortcut called Particle Swarm Optimization (PSO).

The Analogy: The Golden Bird Search
Imagine a flock of birds flying around a dark forest, looking for the best spot to land. Each bird (a "particle") flies around, sharing information with its neighbors. If one bird finds a spot that is "good for both messages and sensing," the whole flock moves toward it. This is much faster and requires much less "brainpower" (data and computing) than training a massive AI.

Why This Matters (The "So What?")

It’s Efficient: It doesn't need a mountain of data to learn; it just needs a few mathematical rules.
It’s Practical: It works with the hardware we already have in our phones and towers.
It’s Smart: It provides a "mathematical guarantee" (the Union Bound and Cramér–Rao Bound) that tells engineers exactly how much error to expect, much like a weather forecast tells you exactly how likely it is to rain.

In short: The paper provides a "master dial" for 6G technology, allowing future devices to switch seamlessly between being a high-speed internet provider and a high-precision radar, all while using the same signal.

Technical Summary: Gamma-Distributed Geometric Constellation for ISAC

This paper proposes a novel framework for designing modulation constellations tailored for Integrated Sensing and Communication (ISAC) systems. The research addresses the fundamental tension between communication efficiency (which favors high-entropy, spread-out constellations like QAM) and sensing accuracy (which favors deterministic, constant-modulus structures like PSK).

1. The Problem: The ISAC Trade-off

In ISAC, a single waveform must perform two conflicting tasks:

Communication: Requires maximizing mutual information, typically achieved by maximizing the minimum distance between constellation points to combat noise.
Sensing (Radar): Requires high precision in parameter estimation (e.g., target presence or channel reflection coefficients), which is optimized by minimizing the variance of the estimator, often favoring specific amplitude distributions.

Existing solutions like Probabilistic Constellation Shaping (PCS) increase transceiver complexity, while Neural Network (NN)-based designs are "black boxes" that require massive training datasets and high computational overhead, making them difficult to implement in conventional hardware.

2. Methodology: Gamma-Distributed Geometric Shaping

The authors propose a Geometric Constellation Shaping (GCS) approach. Instead of optimizing individual points, they model the constellation as a set of samples drawn from a continuous two-dimensional (2D) probability density function (PDF):

Amplitude ( $\rho$ ): Modeled using a Gamma distribution, parameterized by shape ( $\alpha$ ) and scale ( $\beta$ ). This is highly expressive; adjusting these parameters can transition the constellation from a single-ring (PSK-like) to a multi-ring (QAM-like) structure.
Phase ( $\phi$ ): Modeled as a Uniform distribution in $[0, 2\pi)$ , as rotation does not affect performance.

Optimization Process:

Sampling: Constellation points are selected from an Amplitude Phase Shift Keying (APSK) candidate set by maximizing the Gamma-distributed likelihood, subject to a regularization term that ensures spatial diversity (preventing points from clustering too closely).
Objective Function: The design uses a weighted sum of two end-task metrics:
- Communication: Normalized Mutual Information (MI), estimated via Monte Carlo methods.
- Sensing: Average Probability of Detection ( $P_d$ ), evaluated using the closed-form Albersheim approximation.
Solver: Since the objective function is non-differentiable, the authors employ Particle Swarm Optimization (PSO) to find the optimal $\alpha$ and $\beta$ .

3. Key Contributions

Parameter-Efficient Framework: Reduces a high-dimensional constellation design problem to the optimization of just two parameters ( $\alpha, \beta$ ).
Analytical Bounds: The authors derive rigorous mathematical bounds to characterize performance limits:
- A Union Bound on the Symbol Error Rate (SER) for communication, using an $L$ -component mixture of Gamma distributions to accurately model the distance between points.
- A Cramér–Rao Bound (CRB) for sensing, proving that sensing performance is fundamentally governed by the shape parameter $\alpha$ .
Interpretability: Unlike NN-based designs, the resulting constellations are geometrically structured and easily integrated into existing hardware architectures.

4. Results and Discussion

Performance vs. NN: Numerical simulations demonstrate that the proposed method achieves performance comparable to end-to-end neural networks across all scenarios (radar-centric, communication-centric, and trade-off) but with significantly fewer parameters and no training data.
Scenario Adaptability:
- Radar-centric ( $\omega_d \approx 1$ ): Results in high $\alpha$ (PSK-like), minimizing the CRB.
- Comm-centric ( $\omega_d \approx 0$ ): Results in low $\alpha$ (QAM-like), maximizing mutual information.
- Trade-off: Produces multi-ring constellations that balance both metrics.
Validation of Bounds: The derived analytical SER union bound and CRB closely track the Monte Carlo simulation results, confirming the mathematical validity of the framework.

5. Significance

This work provides a principled, mathematically grounded alternative to data-driven constellation design. By bridging the gap between statistical distribution modeling and practical waveform design, it offers a scalable and hardware-friendly solution for the next generation of 6G wireless systems, where sensing and communication must coexist seamlessly.