(11) The second level decomposition aj−2,k and dj−2,k can be obtained after doing decomposition on the approximate sequence aj−1,k resulted from the first stage of decomposition results again, and the third level decomposition aj−3,k and dj−3,k can be obtained selleck chemicals after doing decomposition on the approximate sequence aj−2,k resulted from the second stage of decomposition results,

and so on, until the multiscale wavelet is decomposed into a specified stage. The decomposition process is called Mallat Pyramidal algorithm as shown in Figure 15. Mallat algorithm is inspired by the famous Pyramidal algorithm [34] for image decomposition and combined with multiresolution analysis, proposing signal tower multiresolution decomposition and synthesis algorithms. It is named after the data structure which is a tower structure in decomposition process. Decomposition and reconstruction process is shown in Figures ​Figures1414 and ​and1515. Figure 14 Process of wavelet

decomposition. Figure 15 Process of wavelet reconstruction. S is the original signal in figure, cd1 and ca1 are detail sequence and approximate sequence after level 1 decomposition, and cd2 and ca2 are detail sequence and approximate sequence after level 2 decomposition, and so on. Standard deviation reflects changes of the deviation from mean of track irregularity. When distribution of irregularity around the mean value is more discrete, the representation of average is poorer, and track irregularity will be in poorer state. Conversely, the smaller the standard deviation is, the smaller the variation between the track irregularity values is, the denser the irregularity distribution around the mean value is, and the better the representation of the mean and track state is. Changes of track irregularity standard deviation series data of the Beijing-Kowloon line K449+000–K450+000 sections within 44 times inspection are

selected as the research object, and the original signal is shown in Figure 16. Figure 16 Original waveform signal of track irregularity. Daubechies wavelet is chosen in signal decomposition of track irregularity standard deviation time series data, with the decomposition depth 3. Mallat tower algorithm is used for decomposition and reconstruction of track irregularity standard deviation time series. After wavelet decomposition, 1, 2, and 3 layers are the waveform signal (high frequency) of details, respectively, represented Carfilzomib by D1, D2, and D3; and approximate sequence waveform signal (LF) of layer 3 is represented by A3. The results of specific decomposition are shown in Figures ​Figures17,17, ​,18,18, ​,19,19, and ​and2020. Figure 17 The first layer detail waveform signal of track irregularity (HF). Figure 18 The second layer detail waveform signal of track irregularity (HF). Figure 19 The third layer detail waveform signal of track irregularity (HF).