# Generalized Chirp Spread Spectrum for Underwater Acoustic Communications
## 4 Performance Evaluations  
In this section, we compare and analyze the BER performance of the proposed G-CSS with the conventional up/down-CSS and LoRa-CSS through computer simulations and ocean experiments. First, we analyze the BER performance of the proposed method and the existing methods through computer simulations. The parameters used for the simulations are shown in Table 1. For a fair performance comparison between the proposed G-CSS and the existing methods, the symbol duration of up/down-CSS was adjusted to ensure the same transmission rate by adjusting the symbol length based on the modulation order of the proposed G-CSS. The modulation order of LoRa-CSS was also set accordingly. The modulation orders of the proposed G-CSS were set to \((2,4)\), \((8,2)\), \((2,16)\), and \((4,8)\). When the modulation orders were set to \((2,4)\), \(1\) bit could be allocated at the start and \(2\) bits at the end of the symbol, making it equivalent to a LoRa-CSS modulation order of \(8\). For the up/down-CSS, where only \(1\) bit could be assigned per symbol, the symbol length was shortened by three times to achieve the same transmission rate. When the modulation orders was set to \((8,2)\), \(3\) bits could be allocated at the start and \(1\) bit at the end of the symbol, making it equivalent to a LoRa-CSS modulation order of \(16\). For the up/down-CSS, where only \(1\) bit could be assigned per symbol, the symbol length was shortened by four times to achieve the same transmission rate. Similarly, when the modulation order of the proposed G-CSS was set to \((2,16)\), it allocated \(1\) bit at the start and \(4\) bits at the end of the symbol, equivalent to a LoRa-CSS modulation order of \(32\), and the symbol length for up/down-CSS was reduced by five times. And, when the modulation order of the proposed G-CSS was set to \((4,8)\), it allocated \(2\) bits at the start and \(3\) bits at the end of the symbol, equivalent to a LoRa-CSS modulation order of \(32\), and the symbol length for the up/down-CSS was reduced by five times. Additionally, all three modulation schemes (up/down-CSS, LoRa-CSS, and the proposed G-CSS) employed a \(1/3\) turbo code for forward error correction (FEC). The analysis of computational complexity for the three modulation techniques revealed that up/down-CSS exhibits a linear complexity of \(O(N)\) with respect to the symbol length \(N\), showing the simplest structure that only requires binary decision and energy detection. LoRa-CSS has a complexity of \(O(N\log N)\) due to FFT operations at the receiver, requiring additional computations based on the modulation order. The proposed G-CSS has a complexity of \(O(M_{l}\times M_{r}\times N)\), where \(M_{l}\) and \(M_{r}\) are the modulation orders at the start and end points, respectively, showing the highest computational load among the three techniques as it requires inner product operations for all possible frequency combinations. The underwater acoustic channels used in the computer simulations were generated using the Bellhop model based on the sound speed profiles (SSPs) measured with conductivity temperature depth (CTD) sensors at specific locations in the East and West Seas of South Korea. The SSPs and channel impulse responses (CIRs) of the East and West Seas are shown in Figures 5 and 6, respectively. at 0 and 50 msec, with two additional paths at 50 msec that have a 40% gain relative to the direct path. Figure 6 presents the SSP (Figure 6a) and CIR (Figure 6b) for the West Sea. The Bellhop model was configured with a horizontal distance of 1 km, a transmitter depth of 20 m, and