## 1 Introduction  
With the rapid development of information theory and electronic technology, the deployment and utilization of various electronic devices are showing exponential growth, and spectrum resources are becoming increasingly scarce. Radar and wireless communications, as important electronic systems, have been developing independently and each occupies a significant portion of spectrum resources. With the continuous improvement in information technology and the huge demand for high-quality services from infinite devices, the scope of their work is constantly penetrating into each other's traditional spectrum range, leading to an increasingly tense contradiction in spectrum resources [1]. Due to their many similarities in signal processing algorithms, device usage, and system architecture, they are likely to share spectrum, which has sparked a strong interest in the coexistence, collaboration, and collaborative design of these two systems [2]. The design concept of the Joint Radar and Communication System (JRCS) is to integrate these two functions into one system, and the initial concept can be traced back to the 1960s [3]. The design of JRCS can be divided into two categories [4]. The first type is the physical integration of two functions into a system, which uses two dedicated hardware components and/or two different waveforms, stacked and/or separated in the time domain, frequency domain, or spatial domain [5; 6; 7]. For example, JRCS based on the sampling technique is proposed in reference [8], which transmits two signals on different frequency bands to avoid mutual interference. Reference [9] embeds different communication information into radar pulses to achieve JRCS, which can provide different symbol error rate (SER) performance, but its communication rate is very low, only on the order of Kbps. Reference [10] designed JCRS that can perform radar and communication simultaneously by allocating subcarriers of orthogonal frequency division multiplexing (OFDM) waveforms. However, the uncertainty of radar and communication channels can affect the performance of this method and increase the computational complexity of the system. The integration of such JRCS methods is relatively loose, and the benefits obtained may be limited; furthermore, each system will cause additional mutual interference, requiring signal processing work to mitigate these interferences [11; 12]. The second type achieves more robust integration by sharing most hardware components and using the same waveform. Single spectrum shared transmission signals are used for both radar detection and data communication, known as dual function radar communication (DFRC). DFRC aims to optimize communication and radar performance, and most research has focused on designing such dual function waveforms [13]. Reference [14] provides a good review of the historical development and latest technologies of DFRC. Although DFRC can significantly improve spectral efficiency, its challenge lies in balancing the performance indicators of radar and communication systems, thereby improving radar resolution, communication, and bit error rate (BER) performance. For example, reference [15] proposed an algorithm for jointly designing transmission waveforms and reception filters, which maximizes the detection performance of DFRC systems and enables multi-user communication. However, as the number of communication users increases, the radar detection performance will decrease, and a basic balance is needed to ensure the joint performance of the system. In order to improve the detection/imaging performance of radar sensors and improve performance in multi-user communication, multiple input multiple output (MIMO) technology has been widely applied in DFRC systems [16; 17]. However, most of the dual function waveforms generated by this method are based on pre-coding methods [18; 19; 20] or OFDM format [21; 22; 23].