## 1 Introduction  
Over the past several years, integrated sensing and communications (ISAC) has attracted widespread attention. ISAC can reduce hardware costs and greatly improve frequency spectrum efficiency by reusing hardware and the same frequency spectrum [1, 2, 3]. As popularly seen in the literature, ISAC has broad applications, such as autonomous driving vehicles [4], vehicle-to-vehicle [5] systems, indoor locations [6], wireless radar sensor networks [7], and drone surveillance networks [8]. The main research work on ISAC is how to design the waveform to obtain good performance of both sensing and communication. Generally, the waveform design of ISAC can be mainly divided into three categories: sensing-centered [9, 10], communication-centered [11, 12], and joint design [13, 14]. The sensing-centered methods add communication functions by reusing the existed sensing system. It provides a signal processing perspective of mm-wave ISAC systems with an emphasis on waveform design in [15]. The communication-centered methods are contrasted to sensing-centered methods in that they add sensing functions by reusing the existed communication system. Sensing mutual information (MI) is applied as a performance metric to optimize the waveform by adding sensing paddings in the idle communication layers without degrading communication performance [16]. Based on the classical phase-coded waveform used in wireless communication, ref. [17] studied its requirements for high dynamic range radar measurement, and discussed a variety of possible radar processing algorithms. Based on the millimeter-wave wireless local area network standard, ref. [18] designed the joint waveform of automotive radar and potential millimeter-wave vehicle communication systems. The performance of the uplink integrated sensing and communication system in which communication users and radar targets share the same frequency band is analyzed in [19], and new expressions are derived to characterize the outage probability, ergodic communication rate, and sensing rate. In order to break the limitation of the cyclic prefix (CP) on the sensing range, the echo signal is evenly divided into sub-blocks in [20]. For each sub-block, the virtual cyclic prefix method is used to remove the communication data symbol, and two types of range-Doppler maps are generated for sensing. The sensing-centered methods face the problem of low communication rate, while the communication-centered methods bring problems such as peak-to-average-power ratio (PAPR) and random autocorrelation due to the randomness of communication symbols. Hence, it is necessary to jointly optimize the waveform design of ISAC to balance the performance of the two functions. A sensing integrated discrete Fourier transform spread orthogonal frequency division multiplexing (SI-DFT-s-OFDM) system is proposed for THz ISAC, which can provide a lower PAPR power ratio than OFDM and is adaptive to a flexible delay spread of the THz channel [21]. In [22], a novel waveform design method for Full-Duplex integration is proposed. The communication signal is transmitted by using the idle time of the traditional pulse radar, and can improve the detection probability of the sensing target and the communication rate of the system. It is usually required to have excellent self-ambiguity function and cross-ambiguity function to improve the performance of measurement accuracy and the suppression of clutter interference. Ref. [23] designed a synthesis signal with a strict low ambiguity function to improve sensing performance. With the rapid development of 5G technology, 5G communication systems are being widely commercially deployed. Especially, orthogonal frequency division multiplexing (OFDM) waveforms used in 5G communication systems provide high communication data rates and the ability to efficiently deal with frequency selective fading. Ref. [24] verifies the performance benefit and feasibility to enable ISAC in 5G for the sensing mode based on the base station. Ref.