# Optical Channel Selection Avoiding DIPP in DSB-RFoF Fronthaul Interface
## 3 Calculation Results
### Relative Delay and CP-OFDM Modulation
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**5G** & **BB-** & **Subcarrier** & **OFDM** & **Modulation** & **Symbol-** & **1-Layer** & **FFT** & **CP** & **Duration** & **FFT + CP** \\
**(\(\mu\))** & **Bandwidth** & **(\(\Delta f_{\text{RF}}\))** & **Spacing** & **Subcarriers** & **(min-max)** & **Rate** & **(max)** & **Duration** & **(T\({}_{\text{CP}}\))** & **Duration** \\ \hline - & (MHz) & (kHz) & - & - & (Bd) & (Mbps) & (\(\mu\)s) & (\(\mu\)s) & (\(\mu\)s) \\ \hline \(0\) & \(5\) & \(15\) & \(300\) & 2–256 & 14,000 & 33.60 & 66.67 & 4.69 & \(\sim\)71.36 \\ \(0\) & \(10\) & \(15\) & 624 & 2–256 & 14,000 & 69.89 & 66.67 & 4.69 & \(\sim\)71.36 \\ \end{tabular}
\end{table} 
Table 8: Selected parameters of CP-OFDM modulation used in the 5G New Radio interface [49; 50]. Figure 16: Differential DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.655D optical fiber, and for selected radio channel widths: (**a**) 50 MHz; (**b**) 100 MHz; (**c**) 200 MHz; (**d**) 400 MHz. Figure 17: Differential DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.655E optical fiber, and for selected radio channel widths: (**a**) 50 MHz; (**b**) 100 MHz; (**c**) 200 MHz; (**d**) 400 MHz. Figure 15: Differential DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.652D or G.657A optical fiber, and for selected radio channel widths: (**a**) 50 MHz; (**b**) 100 MHz; (**c**) 200 MHz; (**d**) 400 MHz. In the next step, we look at the susceptibility of the CP-OFDM symbol to the multipath effect caused by the propagation of two optical sidebands in the DSB format. For this purpose, we determine the dependence on the total width of the optical channel (assuming a very narrow spectral line of the laser in relation to the radio frequency):  
\[\Delta f_{0}=2f_{RF}+\Delta f_{RF} \tag{18}\]  
where \(f_{RF}\) is the central radio frequency of the channel and \(\Delta f_{RF}\) is the radio frequency width of the signal containing OFDM subcarriers and side bands of the outer subcarriers. Then we move on to the wavelength domain, while taking into account the optical carrier wavelength that is used during transmission in a fiber of a certain standard. For this purpose, we use the following approximate relationship:  
\[\Delta\lambda_{0}=\lambda_{2}-\lambda_{1}=\frac{c\cdot\Delta f_{0}}{(f_{2}- \Delta f_{0})\cdot f_{2}}=\frac{c\cdot\Delta f_{0}}{(f_{1}+\Delta f_{0})\cdot f _{1}}\cong\frac{c\cdot\Delta f_{0}}{f_{0}^{2}}\cong\frac{\Delta f_{0}\cdot \lambda_{0}}{c} \tag{19}\]  
where \(f_{0}=f_{1}+\Delta f_{0}/2=f_{2}-\Delta f_{0}/2\), and \(f_{0}=c/\lambda_{0}\). After substituting the dependence (18) into (19), we obtain the following:  
\[\Delta\lambda_{0}=\frac{\lambda_{0}(2f_{RF}+\Delta f_{RF})}{c} \tag{20}\]  
The effective chromatic dispersion coefficient is used to calculate the relative delay as follows:  
\[D_{eff}(\lambda_{0},\Delta\lambda_{0})=D(\lambda_{0})+S(\lambda_{0})\cdot \left(\frac{\Delta\lambda_{0}}{2}\right) \tag{21}\]  
where \(S\) is the chromatic dispersion slope. Now, taking into account (21), the relative delay can be determined using the following equation:  
\[\tau_{DSB\max}(\lambda_{0})=D_{eff}(\lambda_{0})\Delta\lambda_{0}L=\left(|D( \lambda_{0})|\cdot\Delta\lambda_{0}+\frac{1}{2}S\cdot\Delta\lambda_{0}^{2} \right)\cdot L. \tag{22}\]  
After substituting (20) into (22) we obtain the following:  
\[\tau_{DSB\max}(\lambda_{0})=\left(|D(\lambda_{0})|(2f_{RF}+\Delta f_{RF})+ \frac{S\lambda_{0}(2f_{RF}+\Delta f_{RF})^{2}}{2c}\right)\frac{L\lambda_{0}}{ c}. \tag{23}\]  
Formula (23) can be used to calculate, with high accuracy, the relative delay in the optical channel, especially when the optical chromatic dispersion coefficient of the path is characterized by a small value below 1 ps/nm*km.