# Generalized Chirp Spread Spectrum for Underwater Acoustic Communications
## 3 Generalized Chirp Spread Spectrum  
In this section, we describe the proposed G-CSS transmission technique. The G-CSS method configures frequency bins with different modulation orders at the start and end of a symbol duration. It then modulates the signal by assigning a chirp that starts and ends at specific frequencies based on the transmitted bits, as shown in Figures 3 and 4. In Figures 3 and 4, \(B\) represents the modulation bandwidth and \(T_{sym}\) represents the symbol duration. The modulation orders at the start and end of the symbol duration can be set differently. Let the modulation orders at the start and end of the symbol duration be \(M_{l}\) and \(M_{r}\), respectively. The modulation order of the proposed method is expressed as \((M_{l},M_{r})\). The number of bits per symbol (\(N_{bps}\)) is expressed as  
\[N_{bps}=\log_{2}(M_{l}\cdot M_{r}) \tag{5}\]  
Figures 3 and 4 have modulation orders of \((2,4)\) and \((4,8)\), respectively. In the case of Figure 3, 1 bit and 2 bits are allocated to the start and end points, respectively, enabling the  
Figure 4: Example of the proposed G-CSS with \((4,8)\). Figure 3: Example of the proposed G-CSS with \((2,4)\). Figure 2: Example of the conventional LoRa-CSS modulation. transmission of 3 bits per symbol. For Figure 4, 2 bits and 3 bits are allocated to the start and end points, allowing for the transmission of 5 bits per symbol. The center frequency of the G-CSS symbol is denoted as \(f_{c}\), and the start and end frequencies range between \(f_{c}-B/2\) and \(f_{c}+B/2\), with the frequency interval and the number of frequency bins determined by the modulation orders \(M_{l}\) and \(M_{r}\). The frequency bins for the start and end frequencies \(\boldsymbol{F}_{l}\) and \(\boldsymbol{F}_{r}\) are defined as follows:  
\[\boldsymbol{F}_{l}=\left[ \begin{array}{cccc}f_{l,0},&f_{l,1},&\ldots,&f_{l, n},&\ldots,&f_{l,M_{l}-1}\end{array} \right] \tag{6}\]  
\[\boldsymbol{F}_{r}=\left[ \begin{array}{cccc}f_{r,0},&f_{r,1},&\ldots,&f_{r,m},&\ldots,&f_{r,M_{r}-1}\end{array} \right] \tag{7}\]  
where  
\[f_{l,n}=f_{c}-\frac{B}{2}+\frac{nB}{M_{l}-1},\quad n=0,1,\ldots,M_{l}-1 \tag{8}\]  
\[f_{r,m}=f_{c}-\frac{B}{2}+\frac{mB}{M_{r}-1},\quad m=0,1,\ldots,M_{r}-1 \tag{9}\]  
The number of bits assigned to the start frequency in a G-CSS symbol is \(\log_{2}M_{l}\), and the number of bits assigned to the end frequency is \(\log_{2}M_{r}\). Therefore, the number of bits assigned to one symbol is divided into bits for modulation.Let the bits assigned to the start and end frequencies of the \(k\)-th symbol be \(\boldsymbol{B}_{k,l}\) and \(\boldsymbol{B}_{k,r}\), respectively. These are expressed as follows:  
\[\boldsymbol{B}_{k,l}=\left[b_{l,1}^{k},b_{l,2}^{k},\ldots,b_{l,\log_{2}M_{l}}^ {k}\right] \tag{10}\]  
\[\boldsymbol{B}_{k,r}=\left[b_{r,1}^{k},b_{r,2}^{k},\ldots,b_{r,\log_{2}M_{r}}^ {k}\right] \tag{11}\]  
To select the frequency bins corresponding to the bits to be transmitted, the binary bits are converted to decimal values as follows:  
\[d_{l,k}=\sum_{i=1}^{\log_{2}M_{l}}b_{l,i}^{k}\cdot 2^{\;i-1} \tag{12}\]  
\[d_{r,k}=\sum_{j=1}^{\log_{2}M_{r}}b_{r,\;j}^{k}\cdot 2^{\;j-1} \tag{13}\]  
Using the values calculated from Equations (12) and (13), the start and end frequencies are selected from the frequency bins defined in Equations (8) and (9), and the \(k\)-th modulated symbol, \(s_{k}(t)\), is expressed as follows. \[s_{k}(t)=\sqrt{2}\cos\left(2\pi\left(f_{l,d_{l,k}}t+\frac{\alpha_{k}}{2}t^{2} \right)\right) \tag{14}\]  
where \(\alpha_{k}\) is given by:  
\[\alpha_{k}=\frac{f_{r,d_{r,k}}-f_{l,d_{l,k}}}{T_{sym}} \tag{15}\]  
At the receiver, the energy of the received symbol is detected to perform demodulation. The \(k\)-th received symbol \(r_{k}(t)\) is expressed as follows:  
\[r_{k}(t)=h(t)\otimes s_{k}(t)+n(t) \tag{16}\]  
where \(h(t)\) is the underwater acoustic channel, \(n(t)\) is noise, and \(\otimes\) denotes the convolution operation.