## 3 Simulation Results
### Channel Capacity  
By using the Shannon-Hartley theorem, we calculated the theoretical capacities of the seven channels whose frequency responses are shown in Figure 7. The channel capacity is expressed as a function of the signal-to-noise power ratio _S_/_N_, where \(S\) and \(N\) represent, respectively, the power of the transmitted signal and the power of an additive white Gaussian noise at the receiver. If we consider both \(S\) and \(N\) independent from frequency, then the power of the received signal is given by \(\left|H(f)\right|^{2}\)_S_, where \(H(f)\) denotes the frequency response of the channel, and the capacity of the channel may be written as:  
\[C\bigg{(}\frac{S}{N}\bigg{)}=\int_{B_{\min}}^{B_{\max}}\log_{2}\bigg{(}1+|H(f) |^{2}\frac{S}{N}\bigg{)}df, \tag{8}\]  
where the capacity is expressed in bits per second.  
A numerical calculation of (6) was performed using a trapezoidal interpolation, and considering \(B_{\min}=2\) MHz and \(B_{\max}=30\) MHz, that is, the bandwidth limits of the HOMEPLUG AV protocol that uses OFDM. We note that the performance of the OFDM modulation method has been reported to be, in general, very close to the limit given by the theoretical channel capacity as calculated in (6).  
Figure 11: Crosstalk as a function of distance and frequency, characteristic impedance termination.  
Figure 12: Crosstalk as a function of distance and frequency, open circuit termination.  
We considered a transmission power of \(-55\) dBm/Hz over the entire frequency band, which is typical for PLC devices, and a background white Gaussian noise with power spectral density between \(-115\) dBm/Hz, which represents a typical low-noise scenario, and \(-80\) dBm/Hz, which represents a highly noisy channel. This corresponds to signal-to-noise power ratios (SNRs) at the transmitter between 25 db and 60 db, which at the receiver imply per-carrier SNRs between \(-35\) db and 50 db, considering the frequency responses previously obtained.  
Figure 13 shows the resulting channel capacities for increasing distance from the transmitter, considering signal-to-noise power ratio (SNR) values from 25 to 60 db.  
Considering a highly noisy scenario of SNR = 25 db, the capacity of the shortest-path channel (25 m) is about 65 Mbit/s, and the capacity of the longer-path channel (175 m) is 1.6 Mbit/s. In a low-noise scenario (SNR = 60 db), the capacities increase up to 220 Mbit/s for the short path and 57 Mbit/s for the long path. Moving from one coach to the successive one, the gain loss is on average 10 db. In addition, for shorter distances, the channel is adequate for a high-speed broadband transmission, for example for video broadcasting and infotainment applications. To better characterize the channel, many performance indicators were calculated, and a bit error rate analysis was carried out, as shown in the following sections.  
It is worth noticing that the transmission power of \(-55\) dBm/Hz used above is taken as reference value, but this value should be analyzed with respect to railway system EMC constraints; in addition, the real noise levels on UIC cables under regular operations should be measured in order to evaluate the actual operating point of Figure 13.