## 4 Application of the \(EDNN\) in Denoising
### Denoising of the Cylinder Pressure Signal with EDNN  
The internal combustion engine cylinder pressure trace is the variation of cylinder pressure with crank angle. Generally, the cylinder pressure trace is recorded using a piezoelectric high-temperature pressure sensor and smoothed via cyclic averaging [42]. The cyclically averaged cylinder pressure trace can satisfy the demands of calculation of indicated mean effective pressure. However, when the analysis involves derivatives \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **1 st Derivative** & **2 nd Derivative** & **3 rd Derivative** & **4 th Derivative** \\ \hline _SNN\_18_ neural network & 0.016 & 0.215 & 2.642 & 28.692 \\ _EDNN_ neural network & 0.010 & 0.074 & 0.751 & 8.414 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The standard deviation of the derivatives. beyond the second order, the noise often leads to deviation from the physical trend. The denoising performance of the _EDNN_ on cylinder pressure trace was studied. The cylinder signal is the in-cylinder pressure which varies with the crank angle. The pressure signal is between the \(-120\) degree CA and \(120\) degree CA, and the sample interval is \(0.1\) degrees CA. Additionally, the cylinder pressure signal is a vector of length \(2400\) whose maximum value is \(1.6946\times 10^{6}\) Pa. The single-input, single-hidden-layer, single-output _EDNN_ with eight nodes in the hidden layer was applied to reduce the noise of the cylinder pressure trace. Moreover, the noise reduction results were compared with that from a shallow neural network with \(8\) nodes (_SNN_8_), a shallow neural network with \(18\) nodes (_SNN_18_) and the smoothing splines that are traditionally adopted in smoothing cylinder pressure trace. The noise reduction results are shown in Figure 7. The noise-reduced signals of the four techniques are all consistent with the measured cylinder pressure (Figure 7a). Additionally, the errors between the noise-reduced signal and the measured signal in Figure 7b are all less than \(400\) Pa, and the relative error is less than \(1.2\times 10^{-6}\%\), which satisfies the accuracy requirements. Figure 8 shows the first- through fourth-order derivatives. It can be seen in Figure 8a that the first-order derivatives of the noise-reduced signal from the four techniques are all consistent with each other, and there are no fluctuations deviating from the physical trend. The first-order derivative of the measured signal shows a small fluctuation near the zero-degree crank angle, which deviates from the physical trend. It can be seen in Figure 8b that the second-order derivative of the noise-reduced signal from the _EDNN_ and the _SNN_18_ neural network is consistent with real trend and there is no fluctuations deviating from the physical trend. The second-order derivatives of the noise-reduced signal from the _SNN_8_ neural network and the smoothing splines showed small fluctuations. Further, the second-order derivative of the measured signal shows frequent fluctuations deviating from the physical trend in the whole crank angle range. It can be seen in Figure 8c that there are no fluctuations deviating from the physical trend occur on the third-order derivative of the noise-reduced signal, obtained from the _EDNN_. There are small fluctuations on the third-order derivative of the noise-reduced signal from the _SNN_18_ neural network. In addition, there are obvious fluctuations on the third-order derivative of the noise-reduced signal from the _SNN_8_ neural network. There are frequent fluctuations on the third-order derivative of the noise-reduced signal, obtained from the smoothing spline. Additionally, the third-order derivative of the measured signal is submerged in the noise. It can be seen in Figure 8d that there are no fluctuations deviating from the physical trend on the fourth-order derivative of the noise-reduced signal, obtained from the _EDNN_. Figure 7: Cylinder pressures and errors.