# Investigation of OFDM-Based HS-PON Using Front-End LiFiSystem for 5G Networks
## 2 Proposed Model
### Mathematical Description  
In each TX, the generated optical signal from the laser diode is passed through the OFDM modulator having a 16-QAM modulation format. For a low-pass cosine roll off filter, the transfer function is given as follows [23]:  
\[X(f)=\left\{ \begin{array}{l}\sigma\\ \sqrt{0.5\sigma^{2}\Big{[}1+cos\Big{(}\pi.\frac{|f|-f_{1}}{f_{a}\Delta f_{FWHM} }\Big{)}\Big{]}}(f_{1}\leq|f|\leq f_{2})\\ 0\\ \end{array} \right. \tag{1}\]  
where \(\sigma\) is the insertion loss, and \(r_{a}\) is the roll off factor parameter. \(\Delta f_{FWHM}\) is the full width at a half-maximum frequency between \(f_{1}\) and \(f_{2}\). Here, for the cutoff frequency, \(f_{c}\), parameters \(f_{1}\) and \(f_{2}\) are given as follows [23]:  
\[f_{1}=1-r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{2}\]  
and  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]  
The received I and Q component signals are phase-shifted at 90\({}^{\circ}\) using a 90\({}^{\circ}\) phase shift at the coherent receiver, as shown in Figure 2. Mathematically, the optical hybrid outputs disregarding imbalance/loss can be given as follows [24]:  
\[E_{1}=0.707[E_{r}+E_{o}] \tag{4}\]  
\[E_{2}=0.707[E_{r}-E_{o}] \tag{5}\]  
\[E_{3}=0.707[E_{r}-jE_{o}] \tag{6}\]  
\[E_{4}=0.707[E_{r}+jE_{o}] \tag{7}\]  
where \(E_{r}\) is the received signal and \(E_{o}\) is the local oscillator input at RX. For two PDs, the generated photocurrents \(I_{1}\), \(I_{2}\), \(I_{3}\), and \(I_{4}\) are given as follows [24]:  
\[I_{1}=0.5\Big{\{}|E_{r}|^{2}+|E_{o}|^{2}+2 Re[E_{r}E_{o}^{*}]\Big{\}} \tag{8}\]  
\[I_{2}=0.5\Big{\{}|E_{r}|^{2}+|E_{o}|^{2}-2 Re[E_{r}E_{o}^{*}]\Big{\}} \tag{9}\]  
\[I_{3}=0.5\Big{\{}|E_{r}|^{2}+|E_{o}|^{2}+j[E_{r}E_{o}^{*}]\Big{\}} \tag{10}\]  
and  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]0  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]1 LiFi-LOS propagation scenario in the proposed model. Then, the detected signal comprising in-phase and quadrature-phase components is given as follows [24]:  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]2  
Again, the photocurrent noise contribution (shot-, thermal-, and phase intensity-induced noise) in the total photocurrent is expressed as follows [25]:  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]3  
where \(e\) is the electronic charge, \(B\) is the filter bandwidth, \(I\) is the PD current, \(K\) is the Boltzmann constant, \(T\) is the absolute temperature, \(R_{L}\) is the load resistance, and \(\tau_{c}\) is the optical source coherent time. Further, the BER for the proposed model using m-QAM OFDM modulation is evaluated as follows [22]:  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]4  
where \(erfc(. )\) represents the complementary error function, and \(m\) is the QAM modulation order. Here, \(SNR\) is given as follows [22]:  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]5  
where \(P_{T}\) is the average transmittance power, \(h\) is [PERSON]'s constant, \(f\) is the operating frequency, and \(\eta\) is the PD quantum efficiency. Furthermore, the LOS frequency response LiFi model is given as follows [26]:  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]6  
where \(X\), \(\theta\), and \(\mu\) indicate the detector area, TX radiance angle as well as RX incidence angle, respectively. Here, \(n\) indicates the Lambertian emission order. \(\vartheta\) means represents the blockage status. The symbol \(s\) presents the distance between the access point and user equipment. Also, \(rect\bigg{(}\frac{\mu}{\varphi}\bigg{)}=1\) for \(0\leq\mu\leq\varphi\) and \(0\) otherwise. \(\varphi\) means represent the field of view. Moreover, the LOS LiFi channel using an LED source is represented as follows [27]:  
\[f_{2}=1+r_{a}f_{c}(0\leq r_{a}\leq 1) \tag{3}\]7  
where \(p(t)\) means represents the received signal, \(q(t)\) is a transmitted signal, \(h(t)\) is an impulse response of the LiFi channel, \(n(t)\) is the noise signal, especially Gaussian signal-independent noise (ambient light, thermal noise, and shot noise), and * signifies the convolution operation.