## 4 Application of the \(EDNN\) in Denoising
### Denoising of Second-Order Damped Free Oscillation Signal via EDNN  
#### 4.2.1 Acquisition of Second-Order Damped Free Oscillation Signal  
The measured second-order damped free oscillation (_SODFO_) signal has errors, and the effect of the errors on the derivatives is difficult to evaluate. Therefore, a _SODFO_ signal is obtained via simulation. High-frequency noise and random noise were superposed on the simulated signal. Then, the performance of the _EDNN_ on denoising was studied. The simulation model of _SODFO_ signal is shown in expression (35). Additionally, the simulated signal is a cosine-shape curve with decreasing amplitude (shown in Figure 4a). The simulation model of the high-frequency noise is shown in expression (36). Moreover, the simulated high-frequency noise is a cosine signal with a frequency 10 times the frequency of the _SODFO_ signal (shown in Figure 4b). The simulation model of the random noise is shown in expression (37). Further, random noise is a uniformly distributed noise on [\(-\)0.01, 0.01] (shown in Figure 4c). The superposition of the higher-order noise and random noise on the _SODFO_ signal does not change the amplitude, apparently (shown in Figure 4d). \[y_{c}=10\cdot e^{(-0.3\cdot t)}\cdot cos(3\cdot t) \tag{35}\]  
\[h_{noise}=0.02\cdot cos(30\cdot t) \tag{36}\]  
\[r_{noise}\sim U(-0.01,0.01) \tag{37}\]  
\[y=y_{c}+h_{noise}+r_{noise} \tag{38}\]  
In expressions (35)-(38), the range of time \(t\) is [0.4, 6] seconds; \(y\) is the noisy second-order damped free oscillation (_NSODFO_) signal. The time \(t\), signal \(y\) and the signal \(y_{c}\) are vectors of length 491. 2.2 Comparative Research of \(EDNN\) with Shallow Neural Network on Denoising of Second-Order Damped Free Oscillation Signal  
Comparative research on denoising performance of the \(EDNN\) was conducted. The \(NSDFO\) signal processed using the shallow neural network and the \(EDNN\) were compared. The \(SODFO\) signal, the noise-removed signal from the shallow neural network (\(NRSNN\) signal) and the noise-removed signal from the \(EDNN\) (\(NREDNN\) signal) are shown in Figure 5.  
It can be found in Figure 5(a) that the \(SODFO\) signal, \(NRSNN\) signal and the \(NREDNN\) signal are well coincident. In Figure 5(b), the differences between the denoising signals and  
Figure 4: Second-order damped free oscillation signal and noise. (**a**) Simulated second-order damped free oscillation (\(SODFO\)) signal; (**b**) high-frequency noise; (**c**) random noise; (**d**) noisy second-order damped free oscillation (\(NSDFO\)) signal. Figure 5: \(SODFO\) signal, \(NRSNN\) signal, \(NREDNN\) signal and the errors. (**a**) \(SODFO\) signal, \(NRSNN\) signal, \(NREDNN\) signal; (**b**) errors between the \(NRSNN\) signal, \(NREDNN\) signal and \(SODFO\) signal. the _SODFO_ signal show that the errors of the _NRSNN_ signal and the _NREDNN_ signal are of the same magnitude. The maximum error of the _NRSNN_ signal is 0.09 percent of the maximum amplitude of the _SODFO_ signal. The maximum error of the _NREDNN_ signal is 0.08 percent of the maximum amplitude of the _SODFO_ signal. It indicates that the shallow neural networks and the _EDNN_ have similar performance in approximating the amplitude of the signal. The first- through fourth-order derivatives and the derivative errors are shown in Figure 6.  
Figure 6: The first- through fourth-order derivatives and the derivative errors. (**a**) First-order derivatives and errors; (**b**) second-order derivatives and errors; (**c**) third-order derivatives and errors; (**d**) magnification of the red circled section in (**c**); (**e**) fourth-order derivatives and errors; (**f**) magnification of the red circled section in (**e**). Figure 6a is the first-order derivative and the errors. The first-order derivatives of the _NRSNN_ signal and the _NREDNN_ signal coincide with the first-order derivative of the _SODFO_ signal. The maximum error of the first-order derivative of the _NREDNN_ signal is 0.16 percent of the maximum amplitude of the first-order derivative of the SODFO signal.