## 2 Methods
### Mutation Interval Reconstruction Envelope
#### 2.2.2 Mutation Interval Reconstruction  
Based on the above analysis, we think the main reason for the demodulation mutation phenomenon is the intersection of upper and lower envelopes due to local envelope distortion. Therefore, an effective solution needs to be proposed to correct this issue for the stability of the demodulation process. The envelopes are created using extreme points in the interpolation-based LMD methods such as CSILMD and MPCHILMD. In some local interpolation intervals where the amplitude of the extreme points changes drastically, the information in the extreme point is not enough to ensure the accuracy of the envelope, which leads to the mutation of the envelope. Hence, a direct approach for this issue is to supplement more interpolation points to enrich the envelope construction information in these local interpolation intervals. [PERSON] [33] has proposed the concept of an extreme symmetry point when examining typical single-component signals with clear physical meaning (such as sine, FM, AM, and AM-FM signals) in the local characteristic-scale decomposition (LCD) method. As shown in Figure 6, a straight line is used to connect two adjacent maximum points (\(\tau_{k},X_{k}\)) and (\(\tau_{k+2},X_{k+2}\)). A point (\(\tau_{k+1},A_{k+1}\)) which has the same abscissa position as the minimum point (\(\tau_{k+1},X_{k+1}\)) between two adjacent maximum points is given on this line. The point (\(\tau_{k+1},A_{k+1}\)) and the minimum point (\(\tau_{k+1},X_{k+1}\)) are usually symmetrical about the horizontal axis, and we call this point (\(\tau_{k+1},A_{k+1}\)) the extreme symmetry point of the minimum point (\(\tau_{k+1},X_{k+1}\)). The specific calculation process of this extreme symmetry point (\(\tau_{k+1},A_{k+1}\)) is as follows:  
\[A_{k+1}=X_{k}+\frac{\tau_{k+1}-\tau_{k}}{\tau_{k+2}-\tau_{k}}(X_{K+2}-X_{K}) \tag{14}\]  
The PF component is one of the single-component signals with clear physical meaning, and its extreme symmetry points also contain inherent structural information. Therefore, we can employ the extreme symmetry point as interpolation points in these local interpolation intervals to enrich the envelope construction information. This paper uses extreme symmetry points to supplement the interpolation points of the mutation interval. As shown in Figure 7a, the upper and lower envelopes of a vibration signal from the reciprocating compressor are intersected in the mutation interval, and then the extreme symmetry points are supplemented into the original extreme points. The specific algorithm flow is as follows:  
Figure 6: AM-FM signal. 1. Calculate the local extreme point sequence \(n_{i}(i=1,\ldots,w)\) of the signal (\(n_{i}\) represents the local maximum point sequence \(nmax_{i}\) or local minimum point sequence \(nmin_{i}\)), using cubic spline interpolation (or cubic Hermite interpolation) to construct the upper and lower envelope of the signal, and record the first derivative value of the envelope. 2. Take the difference between the upper and lower envelope lines to find the local maximum point interval [ \(n_{k},n_{k+2}\)] and local minimum point interval [ \(n_{k+1},n_{k+3}\) ] (or [ \(n_{k-1},n_{k+1}\) ]) of the original signal where the intersection point is located. 3. Formula (12) is applied to calculate the extremum augmentation point \(L_{i}\) between the two maxima \(n_{k}\) and \(n_{k+2}\); calculate the extreme augmentation point \(L_{i}\) between the two minimum points \(n_{k+1}\) and \(n_{k+3}\) (or \(n_{k-1}\) and \(n_{k+1}\)). 4. According to \(n_{k},L_{i},n_{k+2}\) and the first derivative value here, cubic Hermite is used to reconstruct the local interval interpolation of the upper envelope; the lower envelope local interval interpolation is reconstructed according to \(n_{k+1},L_{j},n_{k+3}\) (or \(n_{k-1},L_{j},n_{k+1}\)). 5. Repeat step (3) and step (4) to reconstruct all the singular intervals in the construction process of the envelope.