## 4 Simulation and Experimental Results
The computational complexities of the main functional models for one packet are depicted in Table 2. The total complexity for the proposed scheme was approximately six million multiply-accumulate operations (MACs), which is 24% of that of a traditional scheme. It can be seen that the complexity of the DNN was negligible because of the usage of the expert knowledge of the timing estimation. The model with the most operations in the proposed scheme was the resampling model, which was \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Scaler of input vector & 1000 \\ \hline Scaler of output vector & 0.0001 \\ \hline Size of input layer & 15 \\ \hline Size of hidden layers & [128, 128] \\ \hline Size of output layer & 15 \\ \hline Activation function of hidden layers & ReLU \\ \hline Activation function of output layer & None \\ \hline Loss function & MSE \\ \hline Learning rate & 0.001 \\ \hline Batch size & 128 \\ \hline Optimizer & Adam \\ \hline Total number of parameters & 20,495 \\ \hline \end{tabular} Note: ReLU is rectified linear unit and Adam is adaptive moment estimation.  
\end{table} 
Table 1: Hyper-parameters of deep learning models. Figure 4: The timing offset estimation performances in the simulations. RMSE is root-mean-square error. also halved as a result of using the integer symbol-spaced sampling rate rather than the fractional one. The complexity of the equalizer was tremendously reduced because of the time-invariant structure. The proposed DNN-based scheme was verified in the experimental data packet, which was sampled in the 2011 sea trials of China's first deep manned submersible \"Jiaolong.\" The experimental condition was introduced by [PERSON] [1]. Unfortunately, the wind speed of the experiments was not recorded. However, from the timing estimation, it can be deduced that the surface wave height from the valley to the peak was 2.25 m and the period was approximately 8 s; these measurements can be used for accurate descriptions of the instant sea condition. The vertical and horizontal communication distances were 5030 m and 2390 m, respectively. The packet consisted of 15 LFMs and 14 data frames, and for each frame, the numbers of the training symbols and the information symbols were 200 and 1936, respectively. Only the internal 10 frames were used in the performance comparisons because the two boundary frames on each side had fewer LFM signals nearby for synchronization. During the experiments, the receiver consisted of four hydrophone channels, which were jointly processed using the model from [1]. The hydrophone in the first channel was omnidirectional, and the other three hydrophones were cone-directional; their signal noise ratios (SNRs) were 9.3, 20.1, 18.5, and 16.9 dB, respectively and the total SNR was 18.9 dB (as described in [1]), which was calculated using four-channel equal-gain combining. In this paper's comparisons of the traditional and proposed methods, we only used the second channel, which had the largest SNR, and the adaptive equalizer was replaced by the time-invariant one for the robustness of the impulsive noise. Figure 5 illustrates the timing offset before unbiasing and after interpolation, as well as the bias output of the DNN, which are marked in the previous section as \(\hat{\tau}_{\text{B}}(kT_{\text{Frm}})\), \(\hat{\tau}(nT_{\text{BB}})\), and \(\hat{\Delta}(kT_{\text{Frm}})\), respectively. We can see that, after spline interpolation, the timing offset was much smoother, which fits the true movement, and the bias estimated by the DNN was in proportion to the first-order derivative of the displacement.