# Modulus Waveform Design Based on Manifold ADMM Idea in Dual-Function Radar-Communication System
## 1 Introduction
To obtain an MIMO-based CM waveform with good beamforming traits, the authors in [19] tried to minimize the minimization square error of the designing beam and the desired one, and proposed an efficient alternating direction method of multipliers (ADMM) algorithm, while it cannot satisfy the strict CM constraint. But this idea gave us some enlightenment. In the context of DFRC systems, various approaches have been proposed to balance radar and communication performance. These approaches differ in the practical constraints they address and in the mathematical techniques they employ, as shown in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**References**} & **Practical Waveform and** & **Mathematical Optimization** & \multirow{2}{*}{**Key Findings**} \\  & **System Constraints** & **Approach** & \\ \hline
[8] & Low PAPR, low-range & & Optimized waveform with low PAPR and \\  & side lobes & & low-range side lobes. \\ \hline
[9] & PAPR constraint, & & \\  & minimize MUI & & \\ \hline
[11] & Space–time coding, & Constellation mapping & Proposed direct constellation mapping and \\  & constant-envelope constraint & methods & phase-rotation constellation \\ \hline
[12] & Multiple users, & & \\  & multiple targets & Fairness profile optimization & Characterized trade-offs between \\  & & & communication and radar sensing. \\ \hline
[19] & Clutter, SINR Metric & ADMM algorithm & Proposed efficient ADMM algorithm for \\  & & & CM waveform design. \\ \hline Proposed Work & Constant modulus, & & Achieved superior performance with \\  & multiuser interference & Manifold ADMM & DFRC-MA compared to \\  & & & prevalent methods. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Categorization of these studies. In this paper, we try to design a DFRC waveform based on a manifold-ADMM framework. The main contributions of this paper are summarized as follows:  
* **Novel manifold-based ADMM framework:** We introduce the manifold-ADMM framework, a novel approach that leverages manifold optimization techniques within the ADMM framework. This innovation addresses the complex problem of constant modulus waveform design in MIMO DFRC systems, ensuring optimal radar beamforming while maintaining stringent beampattern constraints. * **Enhanced radar-communication integration:** Our proposed method effectively balances the trade-off between radar and communication performance. By formulating and solving the DFRC optimization problem under the constant modulus constraint, DFRC-MA significantly mitigates MUI, resulting in superior integrated waveform performance compared to existing methods. * **Superior simulation results:** Through extensive simulations, we demonstrate the efficacy of the DFRC-MA approach. The integrated waveform designed by DFRC-MA not only exhibits excellent matching performance with the desired radar beampattern but also showcases enhanced radar communication integration, outperforming prevalent waveform design methods in both radar and communication functionalities. The organization of this paper is as follows. The system model and problem formulation are presented in Section 2. The DFRC waveform design by the manifold-ADMM idea is proposed in Section 3. Section 4 presents our numerical results. Finally, our conclusions are drawn in Section 5. _Notation:_ Lowercase letters \(\mathbf{x}\) and uppercase letters \(\mathbf{X}\) denote vectors and matrices, respectively. The symbols \((\cdot)^{\mathsf{T}}\), \((\cdot)^{\mathsf{H}}\), and \((\cdot)^{*}\) stand for the transpose, the conjugate transpose, and the conjugate operators, respectively. The \(n\)-th element of a vector \(\mathbf{x}\) is written as \(\mathbf{x}(n)\). \(vec(\mathbf{X})\) denotes the column vector obtained by stacking columns of matrix \(\mathbf{X}\) on top of one and another. The set of \(N\times N\) complex matrices and the set of \(n\)-dimensional complex vectors are denoted by \(\mathbb{C}^{N\times N}\) and \(\mathbb{C}^{n}\), respectively.