# Mitigation of Nonlinear Distortions for a 100 Gb/s Radio-Over-Fiber-Based WDM Network
## 3 Analytical Modeling
The gain of electrical amplifier which suppresses the high PAPR [36; 37; 38] is measured as  
\[\frac{P_{out}}{P_{in}}=10^{b/10} \tag{7}\]  
The outcomes of Equation (7) are measured in \(dB\), which gives the amplified electrical signals and is presented as  
\[E_{out}=E_{in}\surd b \tag{8}\]In the presence of PAPR induced distortions, the NLI and LD power spectral density (PSD) function is calculated as  
\[PSD=\surd P_{noise}.f_{s} \tag{9}\]  
where \(P_{noise}\) means the noise power and \(f_{s}\) denotes the frequency grid spacing which is defined as  
\[f_{s}=\frac{R_{s}}{\psi_{s}} \tag{10}\]  
In order to transmit the RF signals over OFTL, the laser source is modeled as a low-cost DML. The electric field intensity of DML is defined as  
\[E(t)=E_{0}e^{j2\pi f_{s}t} \tag{11}\]  
From Equation (1) to Equation (10), the cosine of RF signal \(S(t)\) is calculated as  
\[S(t)=V_{RF}cos(2\pi ft) \tag{12}\]  
where \(V_{RF}\) shows the driving voltage of RF. The optical field at output of intensity modulation (IM) is given as  
\[E(t)=\frac{E_{0}}{2}[e^{j\pi(\frac{V_{DC}+S(t)}{V_{\pi}})}+e^{-j\pi(\frac{V_{DC }+S(t)}{V_{\pi}})}] \tag{13}\]  
where \(V_{DC}\) is the DC bias voltage and \(V_{\pi}\) explains the IM half wave voltage. Assume \(\xi=\pi V_{RF}/V_{\pi}\) and \(\varpi=\pi V_{DC}/V_{\pi}\), which explain the modulation index and initial phase caused by \(V_{DC}\) of the IM, respectively. From Equation (13), the \(E_{1}\) can be extended as  
\[E_{1}(t)=E_{0}exp(j2\pi f_{c}t).\] \[ \begin{cases}cos\varpi[J_{0}(\xi)+2\Sigma_{i=1}^{\infty}(-1)^{i}J _{2i}(\xi)cos(4i\pi f_{s}t)]\\ +sin\varpi[2\Sigma_{i=1}^{\infty}(-1)^{i}J_{2i-1}(\xi)cos(2i-1)2\pi f_{s}t] \end{cases}  \tag{14}\]  
Here, \(J_{i}\) means the \(i\)th order of Bessel function. The transfer function of the transmitted signal is measured as  
\[H(f)= \begin{cases}\alpha&(|f|<f_{1})\\ \beta&(f_{1}\leq|f|<f_{2})\\ 0&f_{2}<|f|\end{cases}  \tag{15}\]  
where \(\beta\) is written as  
\[\beta=\surd 0.5\alpha^{2}.1+cos((|f|-f_{1}/r_{p}.\delta).\pi) \tag{16}\]  
Here, \(\alpha\) denotes insertion loss, \(r_{p}\) means roll of factor and \(f_{1},f_{2}\) are calculated as  
\[f_{1}=1-r_{p}f_{c}(0\leq r_{p}\leq 1) \tag{17}\]  
\[f_{2}=1+r_{p}f_{c}(0\leq r_{p}\leq 1) \tag{18}\]Measurements of the NLIs and LDs in the propagation of light wave in fiber are given below:  
\[i\partial A/\partial z-\beta_{2}(\partial^{2}A)/(\partial t^{2})-i\beta_{3} \partial^{3}A)/(\partial t^{3})+\gamma|A|^{2}A=-i\alpha/2A. \tag{19}\]  
where \(A\) represents complex terms of electromagnetic field, \(t\) represents time, \(z\) denotes distance of fiber, for group velocity dispersion (GVD), \(\beta_{2}\) is used which is also known as linear dispersion; \(\beta_{3}\) is the nonlinear dispersion parameter, attenuation is represented by \(\alpha\) and \(\gamma\) is used for nonlinear coefficient. The relation among nonlinear refractive index \(n_{2}\), carrier wavelength \(\lambda_{0}\) and \(\gamma\) is given by  
\[\gamma=(2\pi n_{2})/\lambda_{0}. \tag{20}\]  
The parameters which affect the amount of NLIs and LDs are cross-section area of SMF, amount of light intensity propagating through the fiber, number of channels in the WDM system and data rate. These factors generate an-harmonic motions of photons inside the SMF [31], where the behavior of induced polarization \(\Omega_{P}\) is no longer linear, which is given as  
\[\Omega_{P}=\epsilon_{0}\kappa^{\prime}E+\epsilon_{0}\kappa^{\prime\prime}E^{2 }+\epsilon_{0}\kappa^{\prime\prime\prime}E^{3}. \tag{21}\]  
Here, \(\epsilon_{0}\) describes free space permittivity, \(E\) means electrical field intensity and \(\kappa^{\prime}\), \(\kappa^{\prime\prime}\) and \(\kappa^{\prime\prime\prime}\) determine first, second and third order susceptibility, respectively.