## 5 Simulation and Industrial Application
### Signal Denoising  
In this paper, the MSE \(\xi\), the PVE \(\eta\) and the CT \(t\) were selected as evaluation indexes of the denoising methods. MSE and the PVE are calculated as follows:  
\[\xi=\frac{1}{k}\sum\limits_{i=1}^{k}{[ds_{i}-s_{i}]^{2}} \tag{15}\]  
\[\eta=\frac{{|P_{o}-P_{d}|}}{{P_{o}}}\times 100\% \tag{16}\]  
where \(ds\) is denoised signal, \(s\) is original signal, \(k\) is the length of the signal, and \(P_{o}\) and \(P_{d}\) are, respectively, the peak value of the original signal and denoised signal. To test the performance of SST, TFB, WTD-GA, WTD-FOA and the proposed WTD-IFAO, Gauss white noise was added into the original sound. The signal to noise ratio (SNR) \(\zeta\) (dB) was introduced to measure the degree of noise, and SNR was defined as follows:  
\[\zeta=10\lg\frac{\sum\limits_{i=1}^{k}{s_{i}^{2}}}{\sum\limits_{i=1}^{k}{n_{i} ^{2}}} \tag{17}\]  
where \(s_{i}\) was the amplitude of the original signal and \(n_{i}\) was that of the added noise. The denoising process was conducted as follows:  
1. Add Gaussian white noise into the original sound and conduct wavelet decomposition. A noisy signal with \(\zeta\) = 5 dB was firstly analyzed and the synthesis was finished in Matlab 8.0 (MathWorks Inc., Natick, MA, USA, 2012). Then the synthetic signal was decomposed by wavelet decomposition with db2 wavelet at 5 levels [6]. The decomposition result is shown in Figure 7.
2. Denoise the noisy signal by SST. The value range of the wavelet threshold was firstly calculated according to Equation (3). The recommended threshold \(\lambda_{\text{rec}}\) was adopted according to Donoho, where \(q\) = 0.6745. The wavelet coefficients of each level were shrunk according to Equation (5). Then the signal was reconstructed by inverse WT. Figure 6: The extracted sound signal of the motor. 3. Denoise the noisy signal by TFB. An improved threshold function with continuous first and second order derivative was introduced in [10], and is presented as follows: \[\hat{d}_{i,j}=\left\{ \begin{array}{ll}d_{i,j}+\lambda-\frac{\lambda}{2k+1},&d_ {i,j}<-\lambda\\ \frac{1}{(2k+1)\lambda^{2k}}\cdot d_{i,j}^{2k+1},&|d_{i,j}|<\lambda\\ d_{i,j}-\lambda+\frac{\lambda}{2k+1},&d_{i,j}>\lambda\end{array} \right\}\] (18) where \(\lambda\) is determined according to Equation (3) and \(k=3\). 4. Denoise the noisy signal by WTD-GA. The maximum \(\lambda_{\max}\) appeared at \(q=1\) and \(\lambda_{\min}\) obtained when \(q=0.4\), so \(\lambda\in[\lambda_{min},\,\lambda_{max}]\). The population size was 100, each chromosome was a five-dimensional vector, the crossover probability was 0.7, the mutation probability was 0.01 and the most iteration generation was 100, as recommended by ref. [15]. 5. Denoise the noisy signal by WTD-FOA. The parameters were set as follows: \(\lambda\in[\lambda_{min},\,\lambda_{max}]\), \(\lambda_{\min}\) and \(\lambda_{\max}\) were calculated as previously. The fly distance obeys uniform distribution, \(FR\sim\)U\((-0.2,\,0.2)\). The population number was 5, each group contained 20 individuals and the iteration number was 100. The fitness of each fruit fly was calculated according to Equation (11). 6. Denoise the noisy signal by WTD-IFOA. The parameters were set as follows: \(\lambda\in[\lambda_{min},\,\lambda_{max}]\), the fly distance obeys normal distribution, \(FR\sim\)N\((0,\,0.2^{2})\). The population number was 5, each group contained 20 individuals and the iteration number was 100. The fitness of each fruit fly was calculated according to Equation (11). Subsequently, a comprehensive comparison was made for the five methods. The \(\xi\), \(\eta\) and \(t\) at \(\zeta=5\) dB of the average value of the five simulation results are presented in Table 2. And the denoised signals of the 5 solutions were shown in Figure 8. It can be seen in the table that adaptive denoising methods based on intelligent optimization had a better comprehensive performance than SST and TFB.