## 4 Simulation and Results  
We performed the simulation to analyze the performance of the proposed method. Figure 8 shows a packet structure of a transmission signal designed for the simulation. We used 1080 bits per frame, a center frequency of 16 kHz, a sampling frequency of 192 kHz, and a data rate of 100 bps. One chirp symbol band is 1.2 kHz and the total band is approximately 10 kHz. The LFM (linear frequency modulation) signals are used to distinguish the packets. The signal length is 0.1 s and the bandwidth is 4 kHz. A guard interval of 1 s is arranged to prevent an interference due to the signal. The pseudo-noise code uses 512 bits for 0.512 s, and no channel coding technique is applied. Figure 8: Packet design. Figure 7: Aggregate FrFT spectra for all frequency bins. There are several software packages for simulating channel transfer characteristics in underwater environments. Norwegian FFI provides the Watermark software that simulates transmission information in several underwater channels [31]. However, to verify the performance of the proposed method, a channel is simulated by applying the sound velocity distribution measured in the South Sea, South Korea, to the VirTEX (Virtual Time Series Experiment), which is a Bellhop-based underwater channel modeling program developed and released by the Scripps Institution of Oceanography (SIO) [32; 33]. The simulation environment is shown in Figure 9. In the simulation, it is assumed that the sea floor is inclined at a depth of 30 m from the transmitter to 40 m from the receiver. The position of the receiver is placed at a depth of 30 m and the transmitter is assumed to be moving horizontally with a fixed depth of 5 m. It is assumed that the distance between the transmitter and receiver moves from 300 m to 200 m linearly at approximately 12 knots towards the receiver and then moves in the opposite direction at the same speed from 200 m to 300 m.  
The channel response characteristics for the simulation are shown in Figure 10. The channel impulse response shows that not only the direct path, but also the sub-path, are reflected from the surface and the floor. Figure 10. Channel impulse response for simulation. Figure 9. Sound velocity profile for the simulation. Figure 11a shows the result of the matched filter for the termination by overlapping the received signal and the ideal signal when approaching between the transmitter and the receiver. As shown in Figure 11a, the signal is compressed since it is positioned backward, even though the packet synchronization is initially adjusted due to the Doppler frequency, which is caused by the mobility of the transmitter. The synchronization error due to the compression is approximately 0.056 s, and assuming a transmission rate of 100 bps, the error is approximately 5-6 symbols. On the other hand, Figure 11b shows an overlapping of the received signal and the ideal signal in the case of retreating, which expands the signal due to the Doppler frequency. The synchronization error due to the expansion is approximately 0.055 s and has an error of approximately 5 to 6 symbols, as in the previous case. Table 1 shows the bit error rate according to the signal-to-noise ratio (SNR) in the presence or absence of the symbol synchronization. The symbol length changes continuously due to the compression and expansion of the signal. Accordingly, if the synchronization is not performed, a large error occurs. On the other hand, when the symbol synchronization is performed using the proposed method, the uncoded bit error rate (BER) is considerably low. The simulation results show that the proposed method is robust to multipath delays and variability due to the Doppler shift. Figure 11: Matched filtering output for the received data and the ideal data. (**a**) The approach; (**b**) the retreat. The second simulation was performed using the underwater channel software released by FFI. Watermark is a benchmark for physical-layer schemes of underwater acoustic communications.