## 3 Modular Modeling for BTM System
### Modeling of Sensitive Equipment Module  
For getting the critical condition \(T\) for sensitive equipment, the sensitive equipment module is modeled in two parts, as shown in Figure 2 with the filter amplifier circuit and the part for decoding and determination.  
Ignoring cable loss, \(S_{r}\) is the input signal transmitting into the sensitive equipment module. It could be divided into two parts, useful signal \(S_{0}\) and disturbance signal \(S_{n}\). Since the modulation of BTM system signal is 2 FSK, the signal \(S_{r}\) received by BTM could express as [27],  
\[S_{r}(t)=S_{0}(t)+S_{n}(t), \tag{12}\]  
Figure 11: S-parameters of antenna system: (**a**) S12 with the location of on-board antenna. (**b**) S-parameters of antenna system when on-board antenna located right above the EMI radiation antenna.  
where,  
\[\begin{split} S_{0}(t)&=U_{\text{PP}}\sum_{k=0}^{N} \left[\text{Rect}\left(\frac{t-kT_{b}}{T_{b}}\right)\bullet\cos[2\pi[(f_{c}+ \Delta f_{c})+(2b_{k}-1)(f_{m}+\Delta f_{m})]t+\phi(kT_{b})]\right],\end{split} \tag{13}\]  
where \(f_{c}\) is the carrier frequency, \(f_{m}\) is frequency offset, while \(\Delta f_{c}\) and \(\Delta f_{m}\) are the instability of carrier frequency and frequency offset, respectively. \(k\) is the length of data. \(T_{b}\) is the cycle of the baseband data signal and \(b_{k}\) is the bit value. The useful signal \(S_{0}(f)\) in the frequency domain is shown in Figure 12.  
\(S_{u}\) is the output signal from filter amplifier circuit [28],  
\[S_{u}(f)=S_{0}(f)h+S_{n}(f)h, \tag{14}\]  
where \(h\) is the transfer function of the filter amplifier circuit. It could be expressed as the magnitude-frequency characteristic of the filter amplifier circuit. The disturbance signal \(S_{n}\) going into the sensitive equipment could be quantified by measurement.  
To get \(h\), black-box testing method is used [29]. The setup of the test is shown in Figure 13.  
As shown in Figure 14, the filter amplifier is a band pass filter working at the BTM band, and the amplifier factor is about 10 dB at 4.232 MHz. The BTM operating frequency range is defined as 3-6 MHz for this type of BTM system.  
Figure 12: The signal of BTM (\(\text{Upper}=1\) V).  
Figure 13: The setup of the test for getting \(h\).  
The information will be transferred to the Automatic Train Protection (ATP) system after decoding. The bit error rate could be obtained from formulations (12), (13), and (14) when the instability of carrier frequency and frequency offset is equal to zero [27].  
\[p_{e}=\frac{1}{2}erfc\bigg{(}\sqrt{\frac{E_{b}}{2E_{N}}}\bigg{)}, \tag{15}\]  
where the \(E_{b}\) is the energy of the useful signal \(S_{0}h\). \(E_{\text{N}}\) is the energy of the disturbance signal \(S_{n}h\)[24].  
\[\begin{array}{l}E_{b}=\int_{T}\frac{(S_{0}h)^{2}}{R_{\text{BM}}}\text{d}t\\ E_{N}=\int_{T}\frac{(S_{0}h)^{2}}{R_{\text{BM}}}\text{d}t\end{array}, \tag{16}\]  
where \(R_{\text{BTM}}\) is the input impedance of BTM.  
For getting the critical condition \(T\), the function of BTM system listed in Table 1 should be considered. Function F2 is simple comparing to the other two functions: F1 and F3, due to it does not involve signal analysis. Thus, the EMS for BTM system should be predicted in two ways for different functions.