# Double-Scale Adaptive Transmission in Time-Varying Channel for Underwater Acoustic Sensor Networks
## 7 Performance Evaluation
### Channel Prediction Performance  
Multiple sets of channel state series have been generated for simulation, and each series has 3000 channel measurements. And the results of two sets of series will be shown, which are called Data 1 and Data 2. The fluctuation of Data 1 is relatively regular, while the varying of Data 2 is relatively complicated.  
Figures 12 and 13 show the large-scale channel state prediction results of Data 1 and Data 2, respectively. The predicted large-scale channel states are very close to the real large-scale channel states. The root mean square error (RMSE) of large-scale channel state prediction are 1.25 dB for Data 1 and 1.32 dB for Data 2, respectively. The results reveal that the predicted large-scale channel states can reflect the channel trend in the future.  
Figures 14-17 show the large-scale channel prediction performance of Data 1 corresponding to the different lengths of storage series as training set and different lengths of the sliding window. The longer the stored historical series is, the smaller the prediction error is. When the stored historical series exceeds \(2T_{p}\), increasing the length of the reserved series has a smaller effect on improving the prediction accuracy. However, the longer the reserved historical series is, the more time the prediction consumes.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Transmission distance & 1 km \\ Acoustic speed & 1500 m/s \\ Carrier frequency & 10 kHz \\ bandwidth & 5 kHz \\ Time slot & 2 s \\ Block size & 1000 symbols \\ Buffer Capacity & 100 kb \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
As the length of the sliding window increases, the RMSE of prediction decreases gradually. When the length of the sliding window exceeds 6, the RMSE of prediction almost no longer decreases. The effect of the sliding window length on prediction performance can be explained as follows. Each channel state value is affected by previous channel states. In other words, the channel state might display certain time-delay effects on subsequent channel states. Obviously, the larger the vector length is, the more information it contains. Therefore, in order to cover the time-delay effects, the sliding window length \(n\) (number of selected values) should be large enough. However, overly large length of the sliding window may significantly increase computational complexity for prediction and lead to overfitting, which may cause poor performance [48; 49].  
The small-scale channel state is predicted by the decomposition-based prediction model and AR prediction separately. Figures 18 and 19 show the small-scale channel state prediction results of Data 1 and Data 2, respectively. The RMSEs of decomposition-based prediction is 15.7% and 9.2% lower than that of AR prediction for Data 1 and Data 2, respectively. In the same epoch, channel states of the decomposition-based prediction are relatively stable, while the predicted channel states of the auto-regression prediction fluctuate violently.  
Figure 12: Predicted large-scale channel state and real large-scale channel state of Data 1.  
Figure 13: Predicted large-scale channel state and real large-scale channel state of Data 2.  
Figure 14: Root mean square error (RMSE) of 1-step ahead prediction with different length of sliding window and stored series.  
Figure 15: RMSE of 5-step ahead prediction with different length of sliding window and stored series.  
Figure 16: RMSE of 15-step ahead prediction with different length of sliding window and stored series.  
Figure 17: RMSE of 25-step ahead prediction with different length of sliding window and stored series.  
In Figure 20, the performances of small-scale channel prediction by two methods are shown for Data 1. The predicted small-scale channel state is used to determine the actual transmission power. RMSE of decomposition-based prediction is smaller than RMSE of AR prediction. The decomposition-based prediction shows better performance.