## 2 UWA OFDM Conventional Doppler Compensation Method  
Figure 1 shows an example of an Np multipath environment in underwater acoustic communication. At the autonomous underwater vehicle (AUV), Np overlapped, and Doppler-shifted and shrunk-or-expanded signals are received. The Path #p signal is characterized by complex attenuation \(r_{p}\), relative delay \(\tau_{p}\), normalized Doppler shift \(\alpha_{p}\), and sampling clock error \(\beta_{p}\) as explained in detail in [18]. Path #1 and Path #2 are affected by the backward Doppler shift when the receiving device moves. Therefore, as shown in Figure 2, the received signal is shrunk or expanded in underwater acoustic communication. In Figure 2, \(v\) represents the moving speed of the receiver and index k corresponds to the OFDM symbol number. Then, we previously introduced resampling and de-rotation to the OFDM receiver as shown in Figure 3[15]. The performance in the mobile reception is improved by detecting the desired path signal with high intensity and stretching the received signal in resample and de-rotation processing. However, if the Doppler shift of the second wave is different from the first wave, as in this case, the second wave is affected by the Doppler shift and generates ICI. In addition, channel estimation uses intermittently placed SPs to estimate the CTF for each carrier. In the conventional method, the channel estimation of the time and frequency directions are performed by applying a CTF interpolation filter in the time direction, as shown in the channel estimation block in Figure 3, followed by a CTF interpolation filter in the frequency direction. As shown in the channel estimation block in Figure 3, channel estimation was performed for each carrier within three symbols. In this method, carrier edge processing is required when performing CTF interpolation in the frequency direction, resulting in the degradation of the channel estimation accuracy at the carrier edges.  
Detailed system parameters are listed in Table 1. The subcarrier spacing of the OFDM is 50 Hz, and the effective OFDM symbol duration is 20 ms with 2048 points IFFT/FFT as modulation/demodulation. Since the total number of subcarriers is 161, the OFDM signal bandwidth is roughly 8 kHz. Figure 4 shows the time-frequency subcarriers' arrangement at the transmitter and receiver sides. To estimate CTF, two kinds of pilot signals are inserted. The blue circles are SPs which are inserted in every two rows and every two columns. The other yellow circles are continuous pilots (CPs) for tracking phase-change in the time domain every 12 rows. The \(d(k,n)\) modulated by 64 QAM is sending data symbols of the subcarrier index \(n\) at the \(k\)th OFDM symbol and \(\hat{d}(k,l)\) is receiving data symbols of subcarrier index \(l\) at the \(k\)th OFDM symbol.  
Figure 2: OFDM signal after the shrink and expansion processing.  
Figure 4: Time–frequency representation of OFDM signal of transmitter and receiver.  
Figure 3: Block diagram of previous receiver.  
Channel estimation uses intermittently placed SPs to estimate the CTF for each carrier. In the conventional method, channel estimation in the time and frequency directions is performed by applying a CTF interpolation filter in the time direction, as shown in the channel estimation block in Figure 3, followed by a CTF interpolation filter in the frequency direction. As shown in the channel estimation block in Figure 3, channel estimation was performed for each carrier within three symbols. In this method, carrier edge processing is required when performing CTF interpolation in the frequency direction, resulting in the degradation of channel estimation accuracy at the carrier edges.