# Generalized Chirp Spread Spectrum for Underwater Acoustic Communications
## 4 Performance Evaluations
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**Values**} \\ \cline{2-4}  & **G-CSS (\(M_{I}\), \(M_{r}\))** & **Up/Down-CSS** & **LoRa-CSS** \\ \hline Sampling rate & 200 kHz & 200 kHz & 200 kHz \\ \hline Bandwidth & 4.5 kHz & 4.5 kHz & 4.5 kHz \\ \hline Center frequency & 20 kHz & 20 kHz & 20 kHz \\ \hline Code rate & 1/3 & 1/3 & 1/3 \\ \hline \multirow{4}{*}{Modulation order} & (2, 4) & 2 & 8 \\ \cline{2-4}  & (8, 2) & 2 & 16 \\ \cline{2-4}  & (2, 16) & 2 & 32 \\ \cline{2-4}  & (4, 8) & 2 & 32 \\ \hline \multirow{4}{*}{Symbol duration} & 8 msec & 2.6 msec & 8 msec \\ \cline{2-4}  & 8 msec & 2 msec & 8 msec \\ \cline{2-4}  & 8 msec & 1.6 msec & 8 msec \\ \cline{2-4}  & 8 msec & 1.6 msec & 8 msec \\ \hline Complexity & \(O(M_{I}\times M_{r}\times N)\) & \(O(N)\) & \(O(NlogN)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computer simulation parameters. Figure 5: East channel model: (**a**) SSP and (**b**) CIR. Figure 6: West channel model: (**a**) SSP and (**b**) CIR. a receiver depth of 20 m. In Figure 6, the main paths in the West Sea channel model are located at 0, 1, and 3.6 ms. At 1 ms, there are five paths with a 45% gain relative to the direct path, and at 3.6 ms, there is one path with a 35% gain relative to the direct path. These results indicate that the West Sea channel has more multipath components than the East Sea channel, leading to a more severe inter-symbol interference environment. The Rician channel model was used to generate the channel based on the channel impulse response shown in Figures 5 and 6, and the Doppler shift and spread were set to 5 Hz, and the K-factor to 10 dB. In Figure 7, the BER results of the proposed G-CSS for the East Sea channel presented in Figure 5 are shown for modulation orders of \((2,4)\), \((8,2)\), \((2,16)\), and \((4,8)\), respectively, under varying SNR conditions. The black solid line represents the conventional up/down-CSS, the blue solid line represents LoRa-CSS, and the red solid line represents the proposed G-CSS. As shown in Figure 7a, at low modulation orders like \((2,4)\), G-CSS may not provide significant performance advantages compared with LoRa-CSS. This is because at low modulation orders, symbol orthogonality and signal interference issues are relatively minor, limiting the range where G-CSS's design characteristics can be fully utilized. However, comparing Figure 7b-d, it can be observed that the conventional up/down-CSS exhibits a lower BER performance as the modulation order increases. This is because, due to the limited modulation order, achieving a higher transmission rate requires reducing the symbol length, which in turn reduces the symbol energy. The conventional LoRa-CSS performs demodulation using FFT, and when the circular shift intervals become smaller at higher modulation orders, the resolution capability deteriorates due to FFT resolution limitations, resulting in a degraded BER performance. The proposed G-CSS achieves a stable performance even as the modulation order increases, making it particularly suitable for high-speed data transmission environments requiring high modulation orders. Figure 7: BER in the East Sea with a modulation order of (**a**) (2, 4), (**b**) (8, 2) (**c**) (2, 16), and (**d**) (4, 8). The BER performance gap between the proposed method and the conventional methods becomes particularly significant for higher modulation orders in the SNR range above 0 dB. This superior performance can be attributed to the unique structure of G-CSS, which allows for the independent configuration of different modulation orders at the start and end points of the symbol. This feature enables the system to distribute frequency resources effectively, reducing inter-signal interference even in environments with severe multipath interference or Doppler effects.