## 4 Simulation and Analysis
### Signal Decomposition and Denoising  
Five different denoising solutions with SNR = 5 dB were conducted as follows:  
1. Wavelet threshold denoising. The noisy signal was first decomposed by db2 wavelet at five levels [32], and the decomposition result was presented in Figure 5. Then, the soft threshold function was adopted to shrink the wavelet coefficients. Inverse wavelet transform was conducted consequently to reconstruct the signal. 2. EMD denoising. The noisy signal was first decomposed by EMD adaptively, which could be shown in Figure 6. The noisy signal could be decomposed as \(x=\mathit{imf}_{1}+\mathit{imf}_{2}+\mathit{imf}_{3}+\ldots+\mathit{imf}_{12}+ \mathit{res}\). Then, the shrinkage threshold of each IMF was determined according to Equations (5) and (6). Soft threshold function was applied and the denoised signal was reconstructed as Equation (9). Some parameters in Equations (5), (6) and (9) were set as follows [10]: \(C=0.7\), \(\alpha=0.719\), \(\beta=2.01\), \(M_{1}=2\) and \(M_{2}=4\). Figure 4: The original signal and white noise-added signal. SNR: signal-to-noise ratio. Figure 5: Wavelet decomposition result of the noisy signal. EMD-PSO denoising. The noisy signal was first decomposed by EMD adaptively. Then, the threshold of each IMF was determined according to PSO intelligently. Soft threshold function was applied and the denoised signal was reconstructed as Equation (9). Some parameters during the EMD-PSO were set as recommended in [23]: \(M_{1}=2\), \(M_{2}=4\), the dimensionality of each particle was \(M_{2}-M_{1}+1=3\), the learning factor \(c_{1}\) and \(c_{2}\) were set as 2, the iteration number was 100 and the particle number of the population was 10. 4. EMD-FOA denoising. The noisy signal was first decomposed by EMD adaptively. Then, the threshold of each IMF was determined according to FOA automatically. Soft threshold function was applied and the denoised signal was reconstructed as Equation (9). Some parameters during the EMD-FOA were set as follows: \(M_{1}=2\), \(M_{2}=4\), the number of fruit fly population was \(M_{2}-M_{1}+1=3\), the iteration number was 100, the fly number in each population was 10, the fruit fly location range and the random fly distance range was 1. 5. EMD-IFOA denoising. The noisy signal was first decomposed by EMD adaptively. Then, the threshold of each IMF was determined according to IFOA automatically. Soft threshold function was applied and the denoised signal was reconstructed as Equation (9). Some parameters during the EMD-IFOA were set as follows: \(M_{1}=2\), \(M_{2}=4\), the number of fruit fly population was \(M_{2}-M_{1}+1=3\), the iteration number was 100, the fly number in each population was 10, the fruit fly location range and the random fly distance range was 1, the variation coefficient \(C_{v}=0.2\) and disturbance coefficient \(C_{d}=5\). The denoised signal of the five noise elimination schemes are presented in Figure 7.  
Figure 6: Empirical mode decomposition (EMD) result of the noisy signal. IMF: intrinsic mode function. In order to evaluate the noise reduction performance of the five methods, three appraisal parameters such as the mean squared error (MSE), SNR improvement (SNR\({}_{\text{imp}}\)) and percent root mean square difference (PRD) were adopted and defined as follows:  
\[\text{MSE}=\frac{1}{N}\sum\limits_{i=1}^{N}{\left({y_{i}-\widetilde{x}_{i}} \right)^{2}} \tag{20}\]  
\[\text{SNR}_{\text{imp}}=10\text{g}\left(\frac{\sum\limits_{i=1}^{N}{\left({x_ {i}-y_{i}}\right)^{2}}}{\sum\limits_{i=1}^{N}{\left({\widetilde{x}_{i}-y_{i}} \right)^{2}}}\right)\left[\text{dB}\right] \tag{21}\]  
\[\text{PRD}=\sqrt{\frac{\sum\limits_{i=1}^{N}{\left({\widetilde{x}_{i}-y_{i}} \right)^{2}}}{\sum\limits_{i=1}^{N}{y_{i}}^{2}}} \tag{22}\]  
where \(y\) was the original signal, \(\widetilde{x}_{i}\) was the denoised signal and \(x\) was the noise-added signal.