# Modulus Waveform Design Based on Manifold ADMM Idea in Dual-Function Radar-Communication System
## 1 Introduction  
Waveform design in dual-function radar-communication (DFRC) systems has drawn a lot of attention in the last 5 years [1; 2; 3]. Therein, the dual-function integrated waveform of multiple-input-multiple-output (MIMO)-based DFRC outperforming the traditional phased-array has been the focus [4; 5; 6]. Typically, in DFRC systems, most communication signals using multicarrier modulation would inevitably lead to high peak average power ratio (PAPR) and great distortion to affect the transmission efficiency. [PERSON] et al. aimed at optimizing the DFRC waveform with low PAPR and low-range side lobes [7]. The authors in [8] considered DFRC waveform designed with PAPR constraint to minimize the downlink multiuser multi-user interference (MUI), which can be regarded as a quadratically constrained quadratic programs (QCQP) problem and further derived into a convex problem. To solve this, the authors in [9] utilized the constant modulus (CM) waveform and proposed an efficient branch-and-bound (BnB) algorithm, while only minimizing MUI could satisfy the beampattern similarity constraint but with high computational complexity. In addition, the authors in [10] discussed the spatial beamforming problem to achieve the dual-functional base station (BS) detecting and communicating, by defining the achievable performance region and formulating a radar-communication-centric optimization model, respectively. Meanwhile, the authors in [11] developed a space-time coding scheme where their shape approximation and integrated power approximation criteria were adopted, respectively, under constant-envelope constraint. Based on this space-time coding scheme, the direct constellation mapping and phase-rotation constellation mapping methods are proposed. Moreover, [PERSON] and [PERSON] et al. characterized the multifold trade-offs betweencommunication and radar sensing by defining an achievable performance region, and also formulated a fairness profile optimization problem [12]. Different from using radar waveforms to achieve integration, the authors in [13; 14] used existing communication waveforms to achieve radar sensing tasks. From the communication perspective, DFRC signals need to have high-quality communication performance, such as a higher communication rate. The communication rate can be improved by minimizing multiuser interference. In [15], a DFRC waveform design model was proposed that reduces MUI under the constraint of similarity between the integrated waveform and the reference radar waveform. That is, under the given reference radar waveform covariance matrix and total power constraints, MUI is minimized to design a DFRC waveform. DFRC waveform design is more challenging when considering scenarios involving multiple communication users and multiple radar-sensing targets. For communication, the SNR of each user needs to be considered. For radar, the SNR of each target also needs to be considered. For the entire DFRC system, the performance of communication and sensing is coupled together, which means that any improvement in communication may deteriorate radar performance, and vice versa. Therefore, in reality, when the DFRC system works in a multiuser and multitarget scenario, it inevitably faces multiple performance trade-offs, including trade-offs between multiple users, multiple targets, and between communication and perception. In order to achieve a good performance trade-off between radar and communication, it is necessary to design a good DFRC waveform [16]. Usually, a DFRC waveform often develops a referenced radar waveform to satisfy radar beamforming patterns by formulating the covariance matrix and other constraints [17]. The authors in [18] investigated the CM waveform design for DFRC systems in the presence of clutter. To minimize the interference power and enhance the target acquisition performance, they tried to use the signal-to-interference-plus-noise-ratio (SINR) as the metric, and enforced a constraint on the synthesis error of every communication signal.