## 1 Introduction  
In recent years, underwater acoustic (UWA) communication has received much attention, with numerous applications emerging in environmental monitoring, ocean exploration, and military missions [1, 2, 3]. However, the UWA channel fluctuates and causes a time-varying multipath, which may result in intersymbol interference (ISI) and Doppler shifts and spreads [4, 5]. Recently, several commercial UWA modems have already been introduced. In the past, the Teledyne Benthos ATM-886 model had a baud rate of 360 bps, and it has 1/2 rate convolution coding, a multipath guard period, multiple frequency shift keying (MFSK), and phase shift keying (PSK) modulation schemes [6]. LinkQueset's SoundLink UWA modem uses broadband acoustic spread spectrum technology. The data rate achieved is up to 38,400 baud [7]. AquaSeNT provides a UWA communication modem using orthogonal frequency division multiplexing (OFDM) technology. These modems can operate in two modes: command mode and data mode [8]. DSPComm Aquacomm Gen2 modems can now operate below the noise floor at a signal-to-noise ratio (SNR) of \(-10\) dB to \(-15\) dB with \(+/-10\) knot Doppler compensation in highly reflective and noisy environments. It is available in 100 bps to 1000 bps by the direct sequence spread spectrum (DSSS)/OFDM. The possible distance was tested to 8 km [9]. The Micron modem working frequency band is 20-28 kHz for the chirp spread spectrum (CSS). The data rate is 40 bps,and the micron data modem provides multipath noise rejection and compact size, with low error rates and 500 m and 150 m horizontal and vertical ranges [10]. The EvoLogics underwater communication modem is based on sweep spread technology, which provides full duplex communication. The modem has data rates up to 13.9 kbps over a 3500 m range with a frequency of 18-34 kHz [11]. The spread spectrum technique is robust to underwater wireless channel distortions; for this reason, it has been widely used in UWA communications [12]. For example, there is the frequency hopping spread spectrum (FHSS), the DSSS, and the CSS method. The CSS compensates for the drawbacks of other spread spectrum techniques and uses a wide bandwidth against various frequency selective fading; thus, it is widely used. In general, the conventional CSS method consists of linear frequency modulation (LFM) waveforms. Low sidelobe levels can be achieved without SNR loss by employing a nonlinear frequency modulation (NLFM) waveform [13]. There are many types of NLFM, such as sinusoidal frequency modulation (SFM), in which the instantaneous frequency (IF) function has a sinusoidal form [14]. However, it contains many high sidelobes in the ACF due to the periodicity of the IF. For this reason, generalized sinusoidal frequency modulation (GSFM) was suggested to eliminate the drawbacks of the SFM [15; 16]. The GSFM waveform is a generalized form of the SFM waveform and uses a variable exponent parameter. Unlike the SFM waveform, the broadband ambiguity function (AF) of this variant GSFM waveform has a distinct mainlobe centered at the origin without a peak sidelobe, and the AF is very similar to a thumbtack. Designing GSFM waveforms with different parameter values (determined by the user) can produce a family of waveforms that occupy the same band of frequencies and are nearly orthogonal to each other [15]. For this reason, the GSFM waveform was recently researched in active sonar by [PERSON] and [PERSON] [16]. The integrated waveform based on the GSFM waveform was researched for continuous active sonar detection and communication, which was called GSFM-com [17]. That method was modulated by multiplying the baseband binary pulse amplitude modulation signal with the original GSFM signal. Therefore, it must consider phase synchronization, and the configuration of the receiver becomes complicated. In [18], a UWA communication using two types of GSFM signals was proposed.