# Train Speed Estimation from Track Structure Vibration Measurements
## 3 Procedures for Train Speed Estimation  
### Train Speed Estimation from Acceleration Dynamic Response Signals  
#### 3.1.1 Signal Pretreatment  
Figure 1a shows the typical time domain signals of vertical acceleration dynamic response of track structure. It can be seen from Figure 1 that the signal contains a lot of noise. Unfortunately, the true dynamic response signals and the disturbance signals are in the same frequency bands [18; 19].  
Wavelet transform, which inherits and develops the thought of localization of short-time Fourier transform, is a useful method of analyzing the non-stationary signals. It is the inner product of the wavelet basis function and the signals to be analyzed--that is, the signals are expressed as a linear combination of a series of wavelet functions with different scales and shifts. The coefficients of each term in the linear combination are called wavelet coefficients.  
This paper selected the Mallat algorithm to calculate the discrete wavelet transform. The Mallat algorithm utilizes the multi-resolution characteristic of the wavelet to decompose the signal into two parts of the high-frequency \(D_{1}\) and the low-frequency \(A_{1}\). Then, the part of the low-frequency \(A_{1}\) is continually decomposed layer by layer to obtain the signal characteristics. The formula for the n-layer decomposition of the original signal \(S\) is as shown in Equation (2).  
\[S=A_{n}+D_{n}+D_{n-1}+\ldots+D_{1} \tag{2}\]  
The wavelet transform coefficients reflect the similarity between signal locality and the wavelet basis function. Accordingly, the shape of the selected wavelet basis function should be similar to that of the target signal. The vertical acceleration signal diagram with bogie information is as shown in Figure 3, which is obtained from the second derivative of the filtered vertical displacement signals.  
Figure 2: The framework of the proposed train speed estimation algorithm.  
The Fejer-Korovkin wavelet with a length 22 filter (as shown in Figure 4) is selected as the base wavelet since the shapes and characteristics of the wavelet are similar to the vertical acceleration signals.  
The original vertical acceleration signals are decomposed using the Fejer-Korovkin wavelet with a length 22 filter, and then the high-frequency signal \(D_{1}\) is recombined. As shown in Figure 5, the low-frequency interferences in the reconstructed signals are suppressed compared to the original signals in Figure 1a, but the high-frequency interferences still exist in the signal graph. Therefore, the signals need to be further processed to obtain signal features consistently.  
After the wavelet transform, a mathematical morphological operation is performed on the binary image (Figure 5). The diamond structure element set \(S\) is selected, and the corrosion operation [20] is performed on the image. Then, the integrated grayscale vertical projection [21] is performed on the  
Figure 4: Fejer–Korovkin wavelet with a length 22 filter.  
Figure 5: High-frequency D\({}_{1}\) signal recombination waveform.  
Figure 3: Acceleration waveform diagram containing bogie information.  
image after corrosion operation, and the grayscale change diagram is shown in Figure 6. Noteworthily, to ensure that the different signals can be transformed linearly after processing, all signals are intercepted with the same length.