# Generalized Chirp Spread Spectrum for Underwater Acoustic Communications
## 2 Related Works  
In this section, we briefly describe the conventional up/down-CSS and LoRa-CSS. The up/down-CSS modulates signals using up-chirp or down-chirp based on the transmitted bits. An example of the modulated signal is shown in Figure 1, and the modulated signal for the \(k\)-th symbol, \(s_{k,c}(t)\), can be expressed as:  
\[s_{k,c}(t)=\begin{cases}\sqrt{2}\cos\left(2\pi\Big{(}f_{c}-\frac{BW}{2}\Big{)} t+\frac{BW}{2I_{sym}}t^{2}\right)&\text{for $b_{k}=0$,}\\ \sqrt{2}\cos\left(2\pi\Big{(}f_{c}+\frac{BW}{2}\Big{)}t-\frac{BW}{2I_{sym}}t^ {2}\right)&\text{for $b_{k}=1$,}\end{cases} \tag{1}\]where \(f_{c}\), \(BW\), and \(b_{k}\) represent the center frequency, bandwidth, and the \(k\)-th transmitted bit, respectively.  
Up/down-CSS leverages the frequency characteristics of chirp signals to recover data using energy detection at the receiver. It is robust to multipath interference and provides a simple demodulation structure, making it advantageous in low-complexity environments. However, the modulation order of up/down-CSS is limited to 2, so achieving higher transmission rates requires reducing the symbol length. This weakens signal orthogonality and can degrade demodulation performance in multipath environments. Additionally, if the energy between symbols is low, BER increases significantly in noisy environments.  
LoRa-CSS is designed to support long transmission distances with low transmission power by performing circular shifts on chirp signals based on the modulation order and transmitted bits. In Figure 2, the example of LoRa-CSS modulation is depicted. The basis chirp of LoRa-CSS, \(s_{b}(t)\), is expressed as follows:  
\[s_{b}(t)=\sqrt{2}\cos\biggl{(}2\pi\biggl{(}f_{c}-\frac{BW}{2}\biggr{)}t+\frac{ BW}{2T_{sym}}t^{2}\biggr{)}. \tag{2}\]  
For a modulation order of \(M\), \(\log_{2}M\) bits can be mapped to a single symbol. If the bits to be transmitted for the \(k\)-th symbol are \(b_{k}=[b_{1}^{k},\ldots,b_{n}^{k},\ldots,b_{\log_{2}M}^{k}]\), the circular shift size of the basis chirp is calculated as follows:  
\[\Delta_{k}=\frac{T_{sym}}{M}\sum_{n=1}^{\log_{2}M}b_{n}^{k}2^{n-1}. \tag{3}\]  
By applying the circular shift \(\Delta_{k}\) calculated from Equation (3) to Equation (2), the \(k\)-th LoRa-CSS symbol is generated as follows:  
\[s_{k,L}(t)=s_{b}\bigl{(}(t-\Delta_{k})\mod T_{sym}\bigr{)}, \tag{4}\]  
where mod denotes the modulus operation.  
LoRa-CSS restores data based on the initial frequency and performs demodulation using FFT. This approach supports high modulation orders, allowing it to adapt to various data transmission rate requirements. However, as the modulation order increases, the spacing between the chirp signals narrows, reducing the demodulation accuracy due to the resolution limits of FFT in high-order modulation. Furthermore, LoRa-CSS is susceptible to distortions in the demodulated signal in environments with severe frequency-selective fading, leading to BER performance degradation due to multipath. These issues are particularly pronounced in complex channel environments such as underwater communication.  
Figure 1: Example of the conventional up/down-CSS modulation.