## 2 UWA Communication by Generalized Sinusoidal Frequency Modulation
### Signal Design for UWA Communication  
The waveform signal \(s(t)\) and IF function \(f(t)\) are expressed as shown in Equations (1) and (2), respectively [16; 17].  
\[s(t)=Ke^{j\varphi(t)}e^{j2\pi f_{c}t}, \tag{1}\]  
\[f(t)=\frac{1}{2\pi}\frac{\partial\varphi(t)}{\partial t}+f_{c}. \tag{2}\]  
\(K\) in Equation (1) is the normalizing factor; \(f_{c}\) is the carrier frequency; and \(\varphi(t)\) is the instantaneous phase (IP) of the signal.  
\(\varphi_{G}(t)\) and \(f_{G}(t)\) are the IP and IF functions of the GSFM, respectively, and their expressions are as shown in Equations (3) and (4).  
\[\varphi_{G}(t)=\frac{\beta}{t^{\rho-1}}\sin\frac{(2\pi\alpha t^{\rho})}{\rho}, \tag{3}\]  
\[f_{G}(t)=\beta\bigg{[}\alpha\cos\bigg{(}\frac{2\pi\alpha t^{\rho}}{\rho}\bigg{)} -\frac{(\rho-1)}{2\pi t^{\rho}}\sin\bigg{(}\frac{2\pi\alpha t^{\rho}}{\rho} \bigg{)}\bigg{]}. \tag{4}\]  
\(\rho\) is a variable exponent parameter that can be used to adjust the asymmetric IF function of the waveform. \(\beta\) is a modulation index and \(\alpha\) is the frequency modulation term.  
In using the same index \(\rho=2\), the waveform of the GSFM can be used to generate various types of waveforms by reversing the time and frequency domains. In this paper, we use 4 types of waveforms, which are represented in the spectrograms of Figure 1. In Figure 1, (a) represents the forward type, (b) represents the flipped frequency of type (a) and is the same as the modulated cosine function, (c) is the reverse time variant of type (a), and (d) has reversed time and a flipped frequency compared with type (a).  
Figure 1: _Cont._  
In this case, we assume that \(f_{c}\) is 16 kHz, the bandwidth is 2 kHz and the waveform length T is 1 s. In Figure 1, \(\beta=50\), and \(\alpha=20\) is the same as \(\alpha=C/T^{\rho}\), which determines the number of cycles \(C=10\). The four types of GSFM waveforms are orthogonal to each other.  
We expect the UWA communication performance to be more effective as a result of using the orthogonal characteristic of the GSFM AFs than that of the conventional LFM. In this paper, we use Equation (5), which is expressed as  
\[s_{i}(t)=\left\{\begin{array}{ll}s_{0}(t),&if\ b_{2n-1}b_{2n}=00\\ s_{1}(t),&if\ b_{2n-1}b_{2n}=01\\ s_{2}(t),&if\ b_{2n-1}b_{2n}=10\\ s_{3}(t),&if\ b_{2n-1}b_{2n}=11\end{array}\right.,\ (n-1)T\leq t<nT. \tag{5}\]  
Here, \(s_{0}(t)\) is the forward-type waveform, \(s_{1}(t)\) is the forward-time/flipped-frequency-type waveform, \(s_{2}(t)\) is the reverse-time-type waveform and \(s_{3}(t)\) is the reverse-time/flipped-frequency-type waveform of the GSFM. \(b_{n}\) is the \(n\)-th bit sequence, T is the length of the symbol, and each symbol represents a 2-bit sequence.  
The block diagram of the proposed method is shown in Figure 2. The received signal that passes through a channel leaves only the band energy using a band pass filter and uses preamble for fine synchronization of the data packet. Finally, the outputs of each matched filter are compared to find the maximum value.  
Figure 1: Four types of GSFM waveform spectrograms: (**a**) forward type, (**b**) forward-time/flipped-frequency, (**c**) reverse time, and (**d**) reverse-time/flipped-frequency.  
Figure 2: Block diagram of UWA communication by using GSFM waveform.