# Fault Diagnosis of Vehicle Gerboxes Based on Adaptive Wavelet Threshold and LT-PCA-NGO-SVM
## 2 Principle of the Method
### Principle of Wavelet Threshold Denoising
#### 2.1.1 Optimization and Improvement of Threshold Function in Wavelet Denoising
 \tag{8}\]  
where \(a\) is the adjustment parameter, \(w\) is the wavelet coefficient of the original signal, \(T\) is the set threshold, and \(F_{T}\) is the wavelet coefficient of the noise reduction signal. Equation (8) approximates the hard threshold function as \(a\) tends to 0 and the soft threshold function as \(a\) tends to 1. Therefore, the adjustment parameter \(a\) can be specifically taken in the range of the (0, 1) interval. The properties of the threshold function are examined in terms of parity, continuity, and asymptote, as demonstrated below:  
(1) Proving the parity of the function  
When \(0<a<1\), \(F(-w)=-F(w)\), indicating that the function is odd. (2) Proving the continuity of the function  
When \(w\) approaches \(T^{+}\):  
\[\lim_{w\to T^{+}}w=\lim_{w\to T^{+}}sgn(w)\bigg{(}|w|-\frac{aT}{e^{(|w|-T)(1-a )}}\Big{)}=\lim_{w\to T^{+}}\bigg{(}|w|-\frac{aT}{e^{(|w|-T)(1-a)}}\bigg{)}=(1- a)T \tag{9}\]  
When \(w\) approaches \(T^{-}\):  
\[\lim_{w\to T^{-}}w=\lim_{w\to T^{-}}sgn(w)(1-a)|w|e^{(|w|-T)}=\lim_{w\to T^{-} }(1-a)|w|=(1-a)T \tag{10}\]  
From Equations (9) and (10), \(\lim_{w\to T^{+}}w=\lim_{w\to T^{-}}w=(1-a)T\), proving that the function is continuous at \(T\). Similarly, the function is also continuous at \(-T\). This threshold function addresses the issue where the hard threshold function causes oscillations in the reconstructed signal due to sudden signal changes at a fixed threshold. (3) Proving the asymptote of the function  
When \(w\) approaches \(+\infty\):  
\[\lim_{w\to+\infty}\frac{F}{w}=\frac{sgn(w)\Big{(}|w|-\frac{aT}{e^{(|w|-T)(1-a )}}\Big{)}}{w}=1-\frac{aT}{we^{(|w|-T)(1-a)}}=1 \tag{11}\]  
\[\lim_{w\to+\infty}(F-w)=\lim_{w\to\infty}\frac{aT}{e^{(|w|-T)(1-a)}}= 0 \tag{12}\]  
Similarly, when \(w\) approaches \(-\infty\):  
\[\lim_{w\to-\infty}\frac{F}{w}=1 \tag{13}\]  
\[\lim_{w\to-\infty}(F-w)=0 \tag{14}\]  
The asymptote of \(F\) is \(w\). As \(w\) tends to \(\infty\), \(F\) tends to \(w\), which effectively addresses the deficiency of deviation observed in the soft threshold function. In summary, the demonstrated proof establishes that the improved threshold function maintains continuity across the entire real number line (\(-\infty\),\(+\infty\)). This adherence to the general rule of wavelet threshold function optimization signifies excellent flexibility and usability. The improved threshold function enhances wavelet processing at critical thresholds compared to Equation (6) and introduces the adjustment parameter \(a\) for increased flexibility in threshold processing compared to Equation (7). Figure 1 displays the traditional soft and hard threshold functions and three improved threshold function images. Before optimizing the wavelet coefficient threshold, it is essential to adjust parameter \(a\) to achieve the optimal threshold function. Following the principles of wavelet noise reduction, a larger threshold than the fixed threshold results in obtaining more useful signals. In this case, parameter \(a\) needs to be adjusted to bring the function closer to the hard threshold function, thereby minimizing energy loss. Conversely, when the threshold is close to the fixed threshold, noise signals become more apparent, and the soft threshold function exhibits superior denoising and smoothness effects. Therefore, adjusting parameter \(a\) becomes necessary to align the function closer to the soft threshold function. To determine the optimal \(a\) for the proposed improved threshold function, the simulated annealing (SA) method can be employed.