⚡ electrical engineering

On Pareto-Optimal Estimation-Information Performance Limits of MIMO Integrated Sensing and Communications Systems

This paper investigates the fundamental joint estimation and information performance limits of MIMO integrated sensing and communications systems, specifically characterizing their Pareto-optimal trade-offs.

Original authors: Zi-Jie Wang, Xudong Wang, Giuseppe Caire

Published 2026-07-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Zi-Jie Wang, Xudong Wang, Giuseppe Caire

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 trying to send a secret message to a friend while simultaneously taking a high-resolution photo of a moving car. In the world of next-generation wireless networks (often called 6G), engineers are trying to build devices that do both at the exact same time using the same radio waves. This is called "Integrated Sensing and Communications" (ISAC). Think of it like a single flashlight that needs to be bright enough to read a book (communication) but also sharp enough to reveal the texture of a wall (sensing). For decades, scientists have tried to figure out the absolute best way to do this, asking: "What is the perfect signal shape that maximizes both tasks?"

To understand the challenge, you need to know two things. First, in regular phone calls, the best signal is usually a smooth, random "static" noise (mathematically called a Gaussian distribution) because it carries the most information. Second, for taking a clear picture (sensing), you usually want a very predictable, steady signal so you can measure exactly how it bounces off things. The big question has been: Can one signal be both perfectly random for talking and perfectly steady for sensing? This paper dives into the deep math of that question to find the theoretical "speed limit" of how well these dual-purpose systems can actually work.

The Great Signal Shape-Shift

In this paper, the authors, Zi-Jie Wang, Xudong Wang, and Giuseppe Caire, tackle the ultimate puzzle of the ISAC world: finding the perfect "shape" for the radio waves that carry both data and sensing information. They treat the signal not just as a wave, but as a character with a personality—a probability distribution that decides how likely the signal is to be strong, weak, or somewhere in between.

The researchers set up a virtual laboratory with a MIMO (Multiple-Input Multiple-Output) system. Imagine a transmitter with multiple antennas shouting out a signal that travels to two different places: a communication receiver (your phone) and a sensing receiver (a radar looking for targets). The goal is to find the "Pareto-optimal" frontier. Think of this frontier as a tightrope walk. If you lean too far toward making the signal perfect for talking, your radar picture gets blurry. If you lean too far toward making it perfect for radar, your phone connection slows down. The paper maps out every single point on that tightrope to see what the absolute best performance looks like.

The Shocking Discovery: The "Perfect" Signal Doesn't Exist (Yet)

Here is where the story gets twisty. The authors used advanced math (variational calculus) to ask: "What does the perfect signal distribution look like?" They found a surprising answer that breaks the rules of standard engineering.

For the extreme case where you only care about talking (communication), the perfect signal is indeed the smooth, random "static" noise everyone expects. However, for any other point on the tightrope—where you want some sensing and some talking—the math says the perfect signal cannot be a smooth, continuous wave.

The paper proves that the ideal signal distribution is "singular." To use an analogy: if a normal signal is like a smooth, flowing river, the perfect ISAC signal is more like a collection of distinct, frozen ice cubes floating in that river. It's not a smooth curve; it's a jagged, broken shape that lives on a very specific, tiny subset of possibilities. The authors show that if you try to force the signal to be smooth and continuous (like the standard Gaussian waves used in most current research), you will never reach the true theoretical limit. You will always be slightly off. The true limit is a "supremum"—a ceiling you can get infinitely close to, but never quite touch with standard, smooth signals.

The Two Great Trade-Offs

The paper identifies two main "tug-of-war" battles that dictate how the signal behaves:

The Water-Filling Trade-off: Imagine pouring water into a bumpy landscape. In communication, you pour more water (power) into the deep valleys (strong signal paths) to get the most data. In sensing, you also want to pour power into specific paths to get the best reflection. The paper shows you can't fill the valleys perfectly for both tasks at once. You have to choose which "valleys" to fill, balancing the power between the two goals.
The Waveform Uncertainty Trade-off: This is the personality shift. When you want to talk, the signal needs to be chaotic and random (high uncertainty) to carry secrets. When you want to sense, the signal needs to be predictable and steady (low uncertainty) to measure reflections accurately. The paper finds that as you move from "pure sensing" to "pure talking," the signal's personality morphs from a rigid, deterministic structure (like a perfect geometric shape) into a wild, Gaussian cloud. The most interesting part is that for the middle ground, the signal has to be a weird, hybrid creature that is neither fully random nor fully steady.

How They Solved the Unsolvable

Since the perfect signal is so weird (a jagged, singular shape) that you can't write down a simple formula for it, the authors couldn't just solve it with a pencil and paper. Instead, they built a digital "robot" called a Blahut-Arimoto-type algorithm.

Think of this algorithm as a sculptor who starts with a block of clay (a standard signal) and chisels away tiny pieces over and over again. With every chisel stroke, the signal gets closer to that perfect, jagged, theoretical shape. The paper proves that this robot converges to the true limit. They used this method to simulate the results on a computer, showing exactly how the signal shape changes as you adjust the balance between sensing and talking.

What the Numbers Say

The authors didn't just talk about theory; they ran simulations to show what happens in real-world scenarios.

In a simple SISO (Single-Input Single-Output) system, they showed that as you shift from sensing to talking, the signal distribution morphs from a "binary" shape (like a coin flip, either high or low) into a smooth Gaussian bell curve.
In a MIMO system (with multiple antennas), they found that at high signal-to-noise ratios (very strong signals), the gap between the best sensing performance and the best communication performance gets wider. This means the "Water-Filling" trade-off becomes even more critical; you have to make harder choices about where to put your power.

The Bottom Line

This paper doesn't just give you a new formula; it changes how we think about designing these signals. It tells us that the "standard" smooth signals we use today are likely not the best possible tools for ISAC. To reach the true limits of 6G, we might need to design "singular" or "non-standard" waveforms—signals that look like jagged ice cubes rather than smooth rivers. While the perfect signal might be mathematically impossible to achieve exactly with standard continuous waves, this work provides the map to get as close as humanly possible, guiding engineers on how to sculpt the next generation of wireless signals.

Problem Statement

The paper addresses the fundamental performance limits of Multiple-Input-Multiple-Output (MIMO) Integrated Sensing and Communications (ISAC) systems. Unlike previous works that often presume Gaussian ISAC waveforms or approximate limits by combining separate sensing and communication optimal strategies, this study investigates the true Pareto-optimal tradeoff between sensing and communication.

The core challenge is that the optimal distribution of an ISAC waveform is unknown. Existing approaches relying on Gaussian assumptions or second-order statistics (correlation matrices) yield loose bounds because they fail to account for the randomness of the waveform distribution itself. The paper seeks to characterize the Pareto boundary between the Average Minimum Mean Square Error (MMSE) for sensing channel estimation and the Ergodic Coherent Rate for communication, treating the waveform distribution as the optimization variable.

System Model and Metrics

The authors consider a generic MIMO ISAC setup where a dual-functional signal $X$ is transmitted to:

Communicate: Convey information over a channel $H$ to a communication receiver (Rx), yielding received signal $Y = HX + Z_c$ .
Sense: Estimate the sensing channel $G$ (target response) from the received sensing signal $S = GX + Z_s$ .

Key assumptions include:

The sensing channel $G$ follows a correlated MIMO model with known statistics.
The communication channel $H$ is known at the transmitter (CSIT).
The sensing receiver knows the transmitted waveform realization $X$ (acting as a reference signal).
Performance is measured by the Average MMSE (ensemble expectation of the estimation error) and the Ergodic Rate.

Both metrics are functionals of the input waveform distribution $p_X(X)$ . The goal is to solve a stochastic functional optimization problem to find the Pareto-optimal set of distributions.

Methodology

The paper employs variational calculus to derive the necessary conditions for optimal distributions.

Euler-Lagrange Conditions: The authors derive a high-dimensional convolutional equation (Theorem 1) that any optimal distribution must satisfy. This equation balances the mutual information term (communication) and the MMSE term (sensing).
Analysis of Achievability: By analyzing the derived condition, the authors prove that for any point on the Pareto boundary except the communication-optimal point, the optimal distribution cannot be a standard continuous, fully-supported probability density function (PDF). Instead, the optimal distributions are shown to be singular or supported on a proper subset of the signal space (e.g., manifolds).
Numerical Algorithm: Due to the intractability of analytically solving for these singular distributions, the authors propose a continuous-version Blahut-Arimoto (B-A) type algorithm. This iterative algorithm numerically approximates the optimal input distributions and the corresponding MMSE-Rate performance, with a proof of convergence to the limit.
Closed-Form Analysis: For the two extreme points of the Pareto boundary—pure sensing ( $\alpha=0$ ) and pure communication ( $\alpha=1$ )—the authors derive closed-form characterizations of the optimal waveforms and performance.

Key Contributions and Results

1. Structural Characterization of Optimal Distributions

Unachievability of Standard Distributions: The paper proves that except at the communication-optimal point (where Gaussian inputs are optimal), the Pareto-optimal ISAC limit is a theoretical supremum within the class of continuous, fully-supported distributions.
Singular Nature: To achieve the limit, the optimal waveform distribution must be discontinuous, singular, or supported on a restricted subset (e.g., requiring a deterministic sample correlation matrix for sensing). This implies that conventional waveforms (like Gaussian) are suboptimal for general ISAC tradeoffs.

2. Closed-Form Optimal Points

Communication-Optimal Point ( $P_c$ ): The optimal waveform is a Gaussian distribution with power allocation determined by the water-filling strategy over the communication channel eigenspace.
Sensing-Optimal Point ( $P_s$ ): The optimal waveform is an "isometry signal" where the sample correlation matrix is deterministic and unique. The power allocation follows a water-filling-like strategy based on the sensing channel statistics, but the randomness is confined to a semi-unitary matrix used for modulation.

3. Fundamental Tradeoffs
The analysis reveals two distinct tradeoffs governing ISAC performance:

Water-Filling Tradeoff (WFT): A conflict in power allocation between the sensing and communication channel eigenspaces. The signal power cannot be simultaneously optimally allocated for both channels.
Waveform Uncertainty Tradeoff (WUT): A tradeoff between the predictability required for sensing (low randomness) and the entropy required for communication (high randomness). As the system moves from sensing-dominant to communication-dominant, the optimal distribution shifts from singular/isometric structures toward Gaussian-like distributions.

4. Numerical Validation and Bounds

The proposed B-A algorithm successfully computes the MMSE-Rate limit curve for both Single-Input-Single-Output (SISO) and MIMO scenarios.
Numerical results demonstrate that standard "time-sharing" bounds and outer bounds based on second-order statistics (assuming Gaussian inputs) are loose and fail to capture the true limit, particularly in the intermediate tradeoff regions.
The results confirm that the performance gap between sensing and communication optima widens at high Signal-to-Noise Ratios (SNR) due to the divergence of optimal power allocation strategies.

Significance and Claims

The paper claims to be the first work to investigate the fundamental estimation-information performance limits of MIMO ISAC systems at the distribution level, considering general and random ISAC waveforms rather than restricting analysis to Gaussian inputs or second-order statistics.

The primary significance lies in:

Theoretical Rigor: Providing a necessary condition for optimality that reveals the singular nature of optimal ISAC waveforms, challenging the conventional assumption that Gaussian inputs are sufficient.
Design Implications: Highlighting that achieving optimal ISAC performance requires singular or non-standard waveform designs (e.g., signals with deterministic correlation structures) rather than standard continuous distributions.
Methodological Advance: Introducing a convergent numerical algorithm (B-A type) to approximate these limits where analytical solutions are intractable.
Insight into Tradeoffs: Explicitly characterizing the "Waveform Uncertainty Tradeoff," offering a structural understanding of how randomness must be engineered to balance sensing and communication tasks.

The authors conclude that while the exact analytical characterization of the Pareto boundary remains challenging, their framework provides the structural implications necessary for designing high-performance ISAC waveforms and motivates further research into singular signal designs.