## 1 Introduction  
Underwater acoustic (UWA) communication enables a wireless link in an underwater environment by the transmission of acoustic waves. Recently, UWA communication has become an essential necessity for underwater operations in marine science and ocean engineering. However, due to the disadvantages of severe transmission loss, time-varying multipath propagation, severe Doppler spread, limited and distance-dependent bandwidth, and high propagation delay, UWA channel is considered as one of the most challenging communication media in use [1]. In the past development process, underwater acoustic communication is mainly divided into two categories that include the single-carrier underwater acoustic (SC-UWA) communication technology and the multi-carrier underwater acoustic (MC-UWA) communication technology such as the orthogonal frequency division multiplexing (OFDM). Due to the high peak-to-average power ratio (PAPR), it is difficult to use MC-UWA communication technology for long-distance transmission. On the other hand, the SC-UWA signals are more suitable to keep a wide coverage for the lower PAPR. However, in high-speed underwater acoustic communication, multi-carrier (MC) modulation technology such as OFDM is superior to single-carrier (SC) modulation technology [2]. At present, considering the communication rate and distance coverage, the hybrid modulation of single- and multi-carrier technology has attracted the attention of researchers. In radio frequency wireless communication, a hybrid single- and multi-carrier system for next-generation mobile communication system was studied in [3], and both high data rate and wide coverage were obtained in the proposed hybrid system. The generalized frequency division multiplexing (GFDM) system which can achieve the unification of single- to multi-carrier systems is suitable for low-latency scenarios foreseen for 5G networks, especially for Tactile Internet [4]. In underwater acoustic communication, the single-carrier frequency-domain equalization (SC-FDE) is a typical hybrid modulation technology for high speed and wide coverage UWA communication, using the low PAPR of a single-carrier system and the relatively simple FDE technology of the OFDM system [5]. At the same time, adaptive multi-carrier modulation technology is also a research hotspot of high performance underwater acoustic communication [6]. Orthogonal signal division multiplex (OSDM), also known as vector OFDM, is an emerging signal modulation technology which can be seen as a bridge between the single-carrier modulation and the OFDM modulation in the frequency domain [7]. In 2001, a precoded OFDM technology, vector-OFDM (VOFDM), was proposed by [PERSON], which could make a traditional OFDM have good adaptability to spectral nulls [8]. Almost at the same time, in 2002, [PERSON] first used the Kronecker product with rows of discrete Fourier transform (DFT) matrix as a signal modulation method for data transmission [9; 10]. In 2007, the Kronecker product-based signal modulation method was called an orthogonal signal division multiplex (OSDM) [11]. In 2008, a novel constellation-rotated V-OFDM with improved diversity was proposed [12]. In 2009, in order to eliminate inter-carrier interference (ICI), a vector-OFDM system of non-contiguous carriers was proposed and the performance of the non-contiguous vector-OFDM (NC-vector-OFDM) was better than that of NC-OFDM [13]. In 2012, [PERSON] and [PERSON] analyzed vector-OFDM with linear receivers [14]. In 2014, the OSDM method was introduced into the field of UWA communication by Professor [PERSON] from Tsukuba University [15]. In 2016, an OSDM-based data transmission mode with an anti-Doppler shift of UWA channel was designed, which provided a reliable and flexible data transmission to overcome the time-frequency dual-spread channel [16]. [PERSON] et al. used space-time block code (STBC) and space-frequency block code (SFBC) to increase coding and diversity gain for the OSDM system [17; 18].