## 3 Performance Analysis  
This section comprises a theoretical analysis of the hybrid optical CDMA/DWDM system performance under the influence of inter- and intra-channel FWM, receiver noise, and MAI. Considering the intensity modulation/direct detection (IM/DD) transmission based on CSRZ signaling, the hybrid system accommodates \(K\) channels; each channel carries \(M\) optical CDMA users; hence, the data corresponding to the \(m\)th user in \(k\)th channel are  
\[d_{k}^{m}(t)=\sqrt{P_{t}}b_{k}^{m}(t)\cos^{2}\Big{[}\frac{\pi}{2}(\sin(\pi Bt)+ 1)\Big{]}\text{where},0<t\leq T \tag{1}\]  
where \(b_{k}^{m}(t)\) is the data of the \(m\)th user in the \(k\)th DWDM channel. For optical CDMA user's signatures, several address codes can be used, including \(m\) sequence, double padded-modified prime code (DPMPC), modified quadratic congruence (MQC), extended grouped new modified prime code (EG-nMPC) and multi-diagonal (MD) code [29; 30]. Among all these codes, MD offers the best performance when MAI is considered due to the zero cross-correlation property of MD, where the length of the MD sequences is \(F=M\times w\), [31]. Hence, the signature code of the \(m\)th user in \(k\)th channel is  
\[C^{km}(t)=\sum_{f=1}^{M\times w}C_{f}^{km}P(t-fT_{c}) \tag{2}\]  
where \(C^{km}\in\{0,1\}\) and \(P(t)\) refers to the unit rectangular pulse with \(T_{c}\) duration. All \(M\) encoded users' data are combined, where the encoded optical CDMA signals corresponding to \(k\)th channel are mathematically expressed as:  
Figure 1: Hybrid Optical CDMA–DWDM system’s operational diagram. Single mode fiber (SMF) and dispersion compensating fiber (DCF). \[S_{k}(t)=\sum_{m=1}^{M}\sum_{f=1}^{M\times w}\sqrt{P_{b}}b_{k}^{m}(t)C_{f}^{km}P(t- fT_{c})\cos^{2}\Bigl{[}\frac{\pi}{2}\bigl{(}\sin(\pi Bt)+1\bigr{)}\Bigr{]} \tag{3}\]  
Finally, all the \(K\) channels are multiplexed, where each channel consisting of \(M\) optical CDMA users is  
\[S(t)=\sum_{k=1}^{K}S_{k}(t) \tag{4}\]  
Then, the output of the DWDM multiplexer is sent over to an EOPM to modulate simultaneously the signal's phase at frequency equal to a single channel's data rate, where such a process helps to suppress the effect of intra-channel FWM. Therefore, the phase of each channel is modulated as:  
\[\varphi(t)=\varphi_{\text{EOPM}}\sin(2\pi f_{\text{SCS}}t) \tag{5}\]  
where \(\varphi_{\text{EOPM}}\) and \(f_{\text{SCS}}\) are the phase deviation and the frequency of the sinusoidal clock signal, respectively. The output of the EOPM is then transmitted over the SMF as  
\[S(t)=\left[\sum_{k=1}^{K}S_{k}(t)\right]e^{i\varphi(t)} \tag{6}\]  
where, the \(q\)th channel signal of the \(m\)th user at the DWDM de-multiplexer's output is  
\[r_{i}(t)=\sum_{m=1}^{M}\sum_{f}^{M\times w}\sqrt{P_{s,\varphi m}}b_{q}^{m}(t) C_{f}^{qm}(t)\cos(\theta_{q})+\sqrt{P_{q}^{\text{FWM}}}(t)\cos(\theta_{q}- \theta_{\text{FWM}}) \tag{7}\]  
where \(P_{s}=\frac{P_{r}}{M\times w}\), \(P_{r}\), \(b_{q}^{m}\), and \(P_{q}^{\text{FWM}}=\sum\limits_{abc}P_{\text{OCDMA-DWDM}}^{abc}\) are the received power, the data bit of the \(m\)th user in the \(q\)th channel, and the total power of generated FWM products at \(f_{q}\), respectively. The generated FWM power at frequencies \(f_{a},f_{b}\), and \(f_{c}\) is [32]  
\[P_{\text{OCDMA-DWDM}}^{abc}=\frac{\eta}{9}(dy)^{2}(\frac{P_{a}}{N})(\frac{P_{ b}}{N})(\frac{P_{c}}{N})\exp[-\alpha L]\bigl{[}\frac{\left(1-e^{-\alpha L} \right)^{2}}{\alpha^{2}}\bigr{]} \tag{8}\]  
where \(\eta\) is the FWM efficiency and can be expressed as [33].