## 1 Introduction  
Underwater acoustic (UWA) communication plays a pivotal role in addressing the challenges of transmitting information through the complex underwater environment [1]. It leverages sound waves as a means of communication, making it a viable solution for various underwater applications, ranging from oceanographic research and environmental monitoring to defense and offshore industries [2]. UWA communication relies on the propagation of sound waves through the water medium. Sound travels faster in water than in air, with a speed of approximately 1500 m per second, allowing it to serve as an efficient carrier of information over long distances [3]. Acoustic waves are also preferred because light waves scatter underwater [4], whereas radio waves undergo absorption. Key factors affecting underwater acoustic communication include the acoustic channel's characteristics, which are affected by factors such as temperature, salinity, and depth, as well as the presence of acoustic noise [5]. Underwater communication is one of the most challenging tasks due to the very harsh nature of the underwater medium [6]. The UWA channel has a very large delay spread and suffers from frequency selectivity and multipath as well as time selectivity and Doppler spread. Usually, acoustic communication takes place at low frequencies and bandwidth is limited to the order of a few kilohertz, realizing a channel that is wideband in nature [7]. Single-carrier communication approaches, such as SC-FDE, are susceptible to inter-symbol interference (ISI), which requires complex equalization techniques. The increased complexity associated with such equalizers poses a significant obstacle to achieving high data rates in these systems [8; 9]. OFDM is the scheme of choice in traditional wireless communication and is reliably used for high-data-rate communication [10]. It has been employed in UWA channels for high-data-rate communication over longer distances [11]. A key benefit of OFDM in underwater acoustics is its ability to subdivide the available bandwidth into orthogonal subcarriers. The orthogonality in subcarriers eliminates ISI, allowing for the use of longer symbol durations, which enhances robustness against the multipath fading commonly encountered in underwater channels [12]. The equalizer complexity is significantly reduced; however, with the increase in high-data-rate communication requirements and limited bandwidth, such as in an UWA channel, the symbol duration must be reduced, which will, in turn, increase the equalizer complexity [13]. Another benefit of employing OFDM in an UWA channel is that the equalization can be performed in the frequency domain [14]. Due to Doppler shifts, the orthogonality of sub-carriers can be severely affected in a time-variant channel, and it must be countered at the receiver [15]. A time-frequency synchronization method for UWA OFDM is proposed in [16]. Authors compared their technique with the traditional methods and showed that their proposed method could obtain the signal timing position more accurately and estimate the carrier frequency offset in complex multipath channels. In [17], the authors propose an effective deep-learning-based channel estimation technique. Their BER results demonstrate the superiority of their algorithm over other methods. OFDM systems have traditionally relied on memoryless modulators such as QAM and PSK [18]. Memory-based modulation schemes such as continuous phase frequency shift keying (CPFSK) and continuous phase modulation (CPM) have been the subject of extensive research for several decades and possess a well-established foundation [19]. CPM is a digital modulation technique where the phase of the carrier signal is continuously varied according to the data to be transmitted, offering spectral efficiency and robustness. Scientists have employed it for its spectral efficiency and fewer out-of-band (OOB) leakages and for it having constant envelope together with phase continuity [20; 21; 22].