## 3 Acquired Data
### Disturbance Simulation  
The real signal contains also noise and distortions coming from the electric motors or railway infrastructure [19]. The basic approach to introducing this kind of disturbance is to add a noise and pulse d(n) to the wheel signature u(n). Such a model does not include the limitation of the bandwidth introduced by the hardware design of the wheel detector, but it is the worst-case scenario for testing. To include the impact of analog path of the detector, an additional low-pass filter for disturbances is used (Figure 15A). The solution presented in Figure 15B includes also the impact of this filter on the wheel signature signal for higher speed.  
The filter used shall reflect the frequency characteristic of the analog part of wheel detector.  
Figure 14: (**A**) Example of acceleration and (**B**) de-acceleration—step change of speed (cases B and D of Figure 12).  
Figure 15: Disturbance model with an equivalent filter. (**A**) Simplified model - filter for disturbances only; (**B**) Filter for the whole signal.  
In the case of the wheel detector concerned, it is a low-pass, Butterworth IIR digital filter, with the characteristic shown in Figure 16 and the following parameters:  
* cut-off frequency (F\({}_{\text{cutoff}}\)): 300 Hz,
* stop band edge (F\({}_{\text{stop}}\)): 600 Hz
* pass band attenuation (A\({}_{\text{pass}}\)): 3 dB
* stop band attenuation (A\({}_{\text{stop}}\)): 60 dB
* filter order: 10  
The pass band attenuation (A\({}_{\text{pass}}\)) of 3 dB for the cut-off frequency equals to 300 Hz (0.3 kHz) as well as stop band attenuation (A\({}_{\text{stop}}\)) of 60 dB for frequency of 600 Hz (0.6 kHz) are presented as a dashed line in Figure 16.  
Filter coefficients, 5x Second Order Section (SOS) are presented in Tables 3 and 4.  
For single SOS (y--output of section, x--input of section):  
\[y[n]=b(0)x[n]+b(1)x[n-1]+b(2)x[n-2]-a(1)y[n-1]-a(2)y[n-2] \tag{9}\]  
The filter is a digital equivalent of the hardware filter used in the design of the subjected wheel detector, 10 th order, low-pass filter (LTC1569-6). Impact of unilinear phase of the filter used during the signal preparation as it is applied for the randomly generated disturbances is considered negligible.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Filter** & **Section 1** & **Section 2** & **Section 3** \\
**Coeff.** & & & \\ \hline b(0) & 0.00879706726769769083 & 0.00834036423798943660 & 0.00798761629247967363 \\ b(1) & 0.01759413453539538166 & 0.01668072847597887320 & 0.01597523258495934725 \\ b(2) & 0.00879706726769769083 & 0.00834036423798943660 & 0.00798761629247967363 \\ a(1) & \(-1.90723559039974510654\) & \(-1.8082074306518110198\) & \(-1.73174372911906848671\) \\ a(2) & 0.94242385947053586293 & 0.84158220001713901492 & 0.76369419428898721591 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Filter coefficients for sections 1–3.  
Figure 16: Filter characteristic. Dashed line – filter design parameters (cut-off frequency and attenuation), Blue, solid line – filter characteristic.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter** & **Section 1** & **Section 2** & **Section 3** \\
**Coeff.** & & & \\ \hline b(0) & 0.00879706726769769083 & 0.00834036423798943660 & 0.00798761629247967363 \\ b(1) & 0.01759413453539538166 & 0.01668072847597887320 & 0.01597523258495934725 \\ b(2) & 0.00879706726769769083 & 0.00834036423798943660 & 0.00798761629247967363 \\ a(1) & \(-1.90723559039974510654\) & \(-1.8082074306518110198\) & \(-1.73174372911906848671\) \\ a(2) & 0.94242385947053586293 & 0.84158220001713901492 & 0.76369419428898721591 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Filter coefficients for section 4 and 5.