## 4 Impact on Encoded RSTT under Pulsed Disturbances in HSR Environment  
In this section, the impact of pulsed disturbances on the encoded railway radio transmission is analyzed. Each \(RBIR(\sigma_{e})\) value of the received signal and its corresponding probability is analyzed, based on the statistical characteristics of disturbances, and the \(RBIR_{L}\) is obtained according to (14). Thus, the \(BLER\) of the on-board receiver is then achieved as a metric of radio transmission performance through the \(RBIR_{L}\). As for the pulsed disturbances affecting the RSTT, the width mainly depends on the time-domain property of the sources, while the amplitude is also influenced by the coupling path. As a result, the amplitude is usually arbitrarily distributed without specific rules, and the width is discretely distributed. Therefore, the bursts could be grouped according to the pulse width \(\{W_{i}\}[i=0,1,\ldots,N_{W})\), where \(W_{i}=0\) refers to the AWGN, and the amplitude of pulses in each group is arbitrarily distributed. Given the time allowance \(\theta\), the probability of each burst set whose pulse width \(W\in[W_{i}-\theta,W_{i}+\theta]\) is:  
\[p_{i}=\Pr(W_{i}-\theta<W<W_{i}+\theta)=\int_{0}^{\infty}\int_{W_{i}-\theta}^{ W_{i}+\theta}dPDD(r,\tau)dAPD(r) \tag{15}\]  
and the corresponding APD of each burst set is:  
\[APD_{W_{i}}(r)=\frac{\int_{r}^{\infty}\int_{W_{i}-\theta}^{W_{i}+\theta}dPDD( r,\tau)dAPD(r)}{\int_{0}^{\infty}\int_{W_{i}-\theta}^{W_{i}+\theta}dPDD(r,\tau)dAPD(r)} \tag{16}\]  
Since the sum of the interference power in one code block determines the actual transmission error, and the relative position of burst hardly affects the error-correcting capability of the FEC encoding [36], the pulse length and amplitude in every code block under \(n(t)\) should be analyzed to obtain the RBIR value. As for the RBIR of blocks containing pulsed interference, the number, the width and the amplitude of the bursts in one block are crucial factors. To distinguish the situation where affected blocks containing a different number of bursts, the relation between \(T_{p}\) and code block length \(Q\) should be taken into account. Thus, the calculation of RBIR\({}_{L}\) is discussed in the following three cases. Figure 6: Squared error of the \(BLER\) with \(t\) in the different distributions of the RBIR. Case \(T_{p}\approx Q\)  
As shown in Figure 7, each code block contains almost one burst, and the ratio of the block that contains a burst with the pulse width \(W_{i}\) is:  
\[\alpha_{i}=\left\{ \begin{array}{lr}\frac{p_{i}}{1-p_{0}}&,i=1,2,\cdots N_{W} \\ 0&,i=0\end{array} \right. \tag{17}\]  
the effective noise of blocks which contain a burst with a pulse width \(W_{i}\) and amplitude \(r\) is:  
\[\sigma_{e}(W_{i},r)=\sqrt{\frac{Q-W_{i}}{Q}\int_{0}^{\infty}r^{2}\cdot dAPD_{W _{0}}(r)+\frac{W_{i}}{Q}r^{2}} \tag{18}\]  
Case \(T_{p}>>Q\)  
As shown in Figure 8, there is a chance that some blocks are not influenced by the pulses, and their probability is \(\alpha_{0}\), with the noise power to be \(\sigma_{0}^{2}=\int_{0}^{\infty}r^{2}\cdot dAPD_{W_{0}}(r)\). The other code blocks contain only one burst, so the ratio of the blocks that contain a burst with width \(W_{i}\) is \(\alpha_{i}\), expressed as  
\[\alpha_{i}=\left\{ \begin{array}{lr}\frac{Qp_{i}}{T_{p}(1-p_{0})}&,i=1,2, \cdots N_{W}\\ 1-\frac{Q}{T_{p}}&,i=0\end{array} \right. \tag{19}\]  
and the corresponding effective value of noise is the same with (18). Case \(T_{p}<<Q\)  
As shown in Figure 9, the number of bursts in one code block is almost \(K=Q/T_{p}\), where \(x\) denotes the smallest integer greater than or equal to \(x\). It is assumed that the width and amplitude of bursts in the one block is the same. Therefore, the ratio of the blocks that contain bursts with the pulse  
Figure 8: Illustration of the case that the repeated cycle \(T_{p}\) is much larger than the length of code block \(Q\). Figure 7: Illustration of the case that the repeated cycle \(T_{p}\) is nearly equal to the length of the code block \(Q\).