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MJSAC: McCormick Relaxation-based Waveform Design for Joint Sensing and Communication

This paper proposes MJSAC, a novel waveform design method utilizing McCormick relaxation to generate beampattern-invariant covariance matrices with maximized inter-symbol distances without requiring channel state information, thereby outperforming conventional algorithms in joint sensing and communication performance.

Original authors: Bodhibrata Mukhopadhyay, Sajid Ahmed, Mohamed-Slim Alouini

Published 2026-06-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Bodhibrata Mukhopadhyay, Sajid Ahmed, Mohamed-Slim Alouini

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

The Big Picture: Doing Two Things at Once

Imagine you are a lighthouse keeper. Your job has two parts:

Sensing: You need to shine a specific pattern of light to spot ships (radar).
Communicating: You need to flash Morse code to talk to those ships (communication).

In the past, engineers tried to do these two things separately or by letting them interfere with each other. This new paper, MJSAC, proposes a smarter way to design the "light" (the radio wave) so it does both jobs perfectly at the same time, without needing to know exactly where the ships are before you start flashing.

The Problem: The "Fuzzy" Flash

The researchers explain that current methods have a flaw. They are great at creating the right shape of the light beam (so the radar works), but they are bad at making the messages clear.

Think of the different messages (symbols) as different colored balls.

Old Method: The balls are all the same color, or they are bunched so close together that if you throw them in the dark, the receiver can't tell which one is which. This leads to mistakes (errors).
The Goal: We want to throw balls that are very far apart from each other (easy to distinguish) but still land in the exact same spot on the ground (the same radar beam pattern).

The Solution: MJSAC (The "Smart Arranger")

The authors created a new algorithm called MJSAC (McCormick-based JSAC). Here is how it works, step-by-step:

The Challenge: Mathematically, trying to arrange these "balls" (covariance matrices) so they are as far apart as possible while keeping the beam shape perfect is a nightmare. It's like trying to solve a puzzle where the pieces keep changing shape, and the rules say the puzzle isn't "convex" (a fancy math word meaning it's not a smooth, easy bowl shape to roll a ball into). It's full of bumps and traps.
The Trick (McCormick Relaxation): To solve this impossible puzzle, the authors used a mathematical "trick" called McCormick relaxation.
- Analogy: Imagine you are trying to find the highest point on a jagged, rocky mountain. It's hard to climb. Instead, the authors built a smooth, curved ramp (a "relaxation") that wraps around the rocks. They solve the problem on the smooth ramp, which is easy, and that solution is so close to the real mountain peak that it works perfectly for their needs.
The Result: The algorithm generates a set of unique "symbols" (waveforms).
- They all look the same to the radar (same beam pattern).
- They are maximally different from each other (easy for the receiver to tell apart).
- Crucially: The transmitter doesn't need to know the weather or the location of the receiver (Channel State Information) to create these symbols. It can prepare them all in advance (offline).

Why It's Better (The Race Results)

The paper ran simulations to see how MJSAC compares to other methods (like "Fan" and "Sana" methods).

The Scoreboard (Symbol Error Rate): The graph in the paper shows that MJSAC makes far fewer mistakes than the others. It's like the runner who finishes the race with a clear lead, while the others are tripping over each other.
The "Distance" Factor: The paper shows that as the number of antennas increases, the "distance" between the symbols gets bigger.
- Analogy: With 8 antennas, the balls are a bit crowded. With 16 antennas, the balls are spread out across a huge field. The receiver can easily pick the right one. MJSAC makes the most of this space.
Efficiency: Some old methods require the transmitter to do heavy math while talking to the user (online), and they need to know exactly where the user is. MJSAC does all the heavy math before the conversation starts (offline) and doesn't need to know the user's location to work.

Summary

MJSAC is a new recipe for designing radio waves. It uses a clever math trick to arrange the "ingredients" (symbols) so they are:

Distinct: Easy to tell apart (low error rate).
Uniform: They all project the same radar beam (good for sensing).
Prepared: They can be made ahead of time without needing real-time information about the receiver.

The paper claims this method beats existing techniques, even those that try to use extra information about the channel, making it a strong candidate for future 5G and 6G networks.

Technical Summary: MJSAC – McCormick Relaxation-based Waveform Design for Joint Sensing and Communication

Problem Statement
In the context of 5G Advanced and 6G technologies, Joint Sensing and Communication (JSAC) aims to unify radar and communication capabilities to share hardware and spectrum resources efficiently. While existing waveform design approaches often focus on generating beampattern-invariant covariance matrices to transmit symbols without requiring Channel State Information (CSI) at the transmitter, they suffer from a critical limitation: poor communication performance. Specifically, current methods (such as those by Sana et al.) generate sets of covariance matrices that share the same beampattern but fail to maximize the geometric distance between them. This lack of separation results in a high Symbol Error Rate (SER) because the receiver cannot effectively distinguish between the transmitted symbols (covariance matrices). Furthermore, alternative approaches that do maximize performance (like Fan et al.) typically require full CSI at the transmitter and involve computationally intensive online waveform generation for every transmission, which is impractical for many scenarios.

Methodology
The paper proposes MJSAC (McCormick-based JSAC), a novel waveform design algorithm that generates a set of $L$ covariance matrices for an $L$ -alphabet modulation scheme. The core objective is to maximize the mutual distances (measured by the Frobenius norm) among these covariance matrices while strictly maintaining a consistent beampattern required for sensing.

The technical approach involves the following steps:

Formulation: The problem is defined as maximizing the sum of squared Frobenius norms between a reference covariance matrix $R$ and a set of $L-1$ new matrices $Z_i$ , as well as maximizing the pairwise distances among the $Z_i$ matrices themselves. This must be done subject to constraints ensuring the beampattern remains identical to $R$ (by matching the trace of diagonal and sub-diagonal elements) and that all matrices remain Positive Semidefinite (PSD).
Sequential Optimization: The original optimization problem is non-convex and intractable due to the maximization of a convex function over a spectrahedron. To solve this, the authors employ a sequential approach, estimating one matrix $Z_k$ at a time while keeping previously estimated matrices fixed.
McCormick Relaxation: To handle the non-convex objective function (which involves quadratic terms $Z_k(p,q)^2$ ), the authors apply McCormick relaxation. This technique linearizes the objective function by introducing bounded auxiliary variables to represent the non-linear terms. This transformation converts the non-convex problem into a standard Semidefinite Programming (SDP) problem with a linear objective and convex constraints.
Offline Generation: The resulting SDP is solved using standard solvers (e.g., CVX). Crucially, this process does not require channel information ( $h$ ) and is performed offline. The square roots of the resulting covariance matrices serve as the constellation points (symbols) for transmission.

Key Contributions

Novel Waveform Design: Introduction of MJSAC, which generates covariance matrices that are both beampattern-invariant (for sensing) and maximally separated (for communication).
Non-Convex Optimization Solution: The application of McCormick relaxation to transform a challenging non-convex waveform design problem into a tractable convex SDP.
Channel-Independent Operation: The method eliminates the need for Channel State Information (CSI) at the transmitter for symbol generation, allowing for offline computation and reducing online computational load.
Scalability: The algorithm scales with the number of antennas ( $N$ ) and modulation alphabet size ( $L$ ), generating symbols that are well-separated in the matrix space.

Numerical Results
Simulations were conducted comparing MJSAC against Binary Phase Shift Keying (BPSK), the Sana et al. method (Sana-Omni/Direc), and the Fan et al. method (Fan-Omni/Direc) across different antenna configurations ( $N=8, 12, 16$ ) and beam patterns (Omnidirectional and Directional).

Symbol Error Rate (SER): MJSAC consistently outperformed BPSK and the Sana et al. method across all scenarios. For $N=8$ , Fan-Omni performed slightly better, but for $N=12$ and $N=16$ , MJSAC surpassed Fan-Omni.
Distance Analysis: Heat maps of the Frobenius norm differences among symbols demonstrated that MJSAC generates significantly larger separations between symbols as the number of antennas increases, directly correlating to improved detection performance.
Complexity: While MJSAC has a higher computational complexity ( $O(N^7)$ ) compared to the online complexity of Fan-Omni/Direc, this computation is performed offline. In contrast, Fan-Omni/Direc requires online computation dependent on the number of transmissions and frame length, making MJSAC more scalable for repeated use.

Significance and Claims
The paper claims that MJSAC addresses the fundamental trade-off in JSAC waveform design between sensing performance (beampattern consistency) and communication reliability (symbol distinguishability). By maximizing the distance between covariance matrices without relying on CSI, MJSAC offers a robust solution that enhances communication performance while maintaining radar functionality. The authors assert that their method outperforms conventional algorithms, including those that utilize channel information at the transmitter, by providing a more efficient and scalable framework for joint sensing and communication systems. The work highlights that increasing the separation between symbols is critical for reducing SER, a factor often overlooked in previous beampattern-invariant designs.