# Optical Channel Selection Avoiding DIPP in DSB-RFoF Fronthaul Interface
## 2 System, Methods, and Network Components
### Radio-Frequency-over-Fiber in DSB Format  
From the three A-RoF interface formats mentioned in the introduction above, the RFoF format is the most susceptible to the dispersion parameters of a single-mode optical fiber. This type of signal is present at the level of the proposed Option 10 interface (Figure 1). In the case of the classic and at the same time the simplest optical IM-DD link, the RFoF signal contains amplitude modulation (AM) products in the continuous version of the carrier. At the beginning of the analysis, we assume that the radio signal consists of a non-modulated single-tone carrier with a frequency \(f_{RF}\) that is a signal modulating an optical carrier with wavelength \(\lambda_{0}\). In the case of RF signals, we operate with frequency units and in the case of optical signals, with the wavelength units. At a later stage, we move on to the frequency domain also on the optical side. The signal modulated in the RFoF format, coming from the optical modulator, can be represented by the formula making the optical carrier signal dependent on the RF signal:  
\[E_{IM}(t)=\sqrt{1+m_{IM}\cos(2\pi f_{RF})}\exp\left(j2\pi\frac{c}{\lambda_{0} }t\right), \tag{4}\]  
Figure 5: Averaged characteristics of the chromatic dispersion coefficients for the exemplary optical fronthaul paths. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Variant**} & \multicolumn{2}{c}{**Link A**} & \multicolumn{2}{c}{**Link B**} & \multicolumn{2}{c}{**Link C**} \\ \cline{2-7}  & **Fiber** & **Length (km)** & **Fiber** & **Length (km)** & **Fiber** & **Length (km)** \\ \hline
1 & G.652D & 5 & G.655D & 12 & G.652D & 3 \\
2 & G.652D & 3 & G.655E & 10 & G.657A & 2 \\
3 & G.657A & 7 & G.655D & 8 & G.652D & 3 \\
4 & G.657A & 10 & G.655E & 3 & G.657A & 7 \\
5 & G.652D & 1 & G.655D & 17 & G.657A & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Examples of the optical paths with different standards of the single-mode optical fibers used in individual internodal links. where \(m_{IM}\) is the intensity modulation index. When \(m_{IM}\) is small, the optical field (4) can be written as follows:  
\[E_{IM}(t)=(1+m_{IM}\cos(2\pi f_{RF}t))\exp\left(j2\pi\frac{c}{\lambda_{0}}t \right). \tag{5}\]  
If we transform formula (5) trigonometrically, we obtain a relationship showing the frequency components that create the IM signal:  
\[E_{IM}(t)=\exp\left(j2\pi\frac{c}{\lambda_{0}}t\right)+\frac{m_{IM}}{2}\bigg{\{} \exp\left[j2\pi\left(\frac{c}{\lambda_{0}}+f_{RF}\right)t\right]+\exp\left[j2 \pi\left(\frac{c}{\lambda_{0}}-f_{RF}\right)t\right]\bigg{\}}. \tag{6}\]  
As we can see in formula (6), the signal takes the form of a DSB, where we can distinguish the optical carrier and two side fringes, which in the case of a modulated RF signal, are sidebands. Now the signal represented by (6) is introduced to the optical path with the dispersive optical fiber, while ignoring the absorption and scattering loss:  
\[ \begin{array}{c}E_{IM}(t,\cdot)=\exp\left(j2\pi\frac{c}{\lambda_{0}}t \right)+\frac{m_{IM}}{2}\bigg{\{}\exp\left[j2\pi\left(\frac{c}{\lambda_{0}}+f _{RF}\right)t-\Phi(\cdot)\right]+\\ \qquad\qquad+\exp\left[j2\pi\left(\frac{c}{\lambda_{0}}-f_{RF}\right)t-\Phi( \cdot)\right]\bigg{\}}.\end{array}  \tag{7}\]  
In the context of the phenomenon taking place in the photodetector working in the direct detection format, it can be assumed that the optical path is a band-pass filter with a flat amplitude response and a linear group delay factor. On this basis, we can assume that the fiber-optic transfer function can be represented by the following [45]:  
\[H(f_{RF},\lambda_{0})=\exp(-j\Phi(f_{RF},\lambda_{0})) \tag{8}\]  
where  
\[\Phi(f_{RF},\lambda_{0})=\pi D(\lambda_{0})L\frac{\lambda_{0}^{2}}{c}f_{RF}^{2} \tag{9}\]  
and \(D(\lambda_{0})\) is the chromatic dispersion coefficient changing with the optical carrier \(\lambda_{0}\) (Figure 3). \(L\) is the optical path length.