## 3 Analytical System Model  
The signal-to-noise ratio (SNR) expression as a function of simultaneous PON users can be expressed as follows:  
\[\text{SNR}=\frac{S}{(K-1)\sigma^{2}+N_{0}} \tag{1}\]  
where \(S\) is the received signal (i.e., corresponding to the transmitted symbol) and \(\sigma^{2}\) is the noise variance. In our model, the noise variance includes MAI and subcarriers overlapping, which can be defined as follows:  
\[\sigma^{2}=\sigma^{2}_{MAI}+\sigma^{2}_{overlap} \tag{2}\]  
where \(\sigma^{2}_{MAI}\) is related to 2D-HC codes and given by [23]:  
\[\sigma^{2}_{MAI}=\frac{H}{2P^{2}}\Big{(}1-\frac{H}{2P^{2}}\Big{)} \tag{3}\]  
In Equation (3), \(H\) and \(P\) are the average probability hit of chips related to 2D-HC and the prime number, respectively. In this study, we also consider the photo-detector noise (N\({}_{0}\)). In fact, 2D-HC is built as follows: the maximum-length (ML) sequence is employed for the wavelength hopping, and the prime number is used for temporal spreading (i.e., examples of seven 2D-HC codes are described in Table 1, where [1, 2, 3, 4, 5, 6, 7] symbolize the wavelengths {\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\), \(\lambda_{6}\), \(\lambda_{7}\)} and [23]). The 2D-HC code is defined with respect to wavelength hoping H\({}_{\text{j}}\) and temporal spreading S\({}_{\text{i}}\) as shown in Table 1 the different code sequences S\({}_{\text{i}}\)H\({}_{\text{j}}\) where i vary from 0 to 4 and j vary from 0 to 3). Figure 1: IM/DD Fast-OFDM-CDMA system architecture. Figure 2 represents the implementation of the OCDMA encoder and decoder based on the 2D-HC S\({}_{0}\)H\({}_{1}\) sequence. As shown in Figure 2, the 2D-OCDMA encoder and decoder are composed of the WDM multiplexer, time delay, Bessel filter and WDM de-multiplexer. One major drawback related to single-band (SB) and multi-band (MB) OFDM systems is their sensibility to fiber nonlinear effects due to the high PAPR of OFDM signals [24]. In fact, even in SB OFDM, four-wave mixing (FWM) happens among electronic OFDM subcarriers due to its multi-carrier nature producing inter-subcarrier interference (ICI) [25]. As long as the cyclic prefix is a time-shifted copy of a part of the OFDM signal in the \"observation\" period, the transmitted time domain waveform inside this period for one OFDM symbol can be expressed by Equation (4):  
\[s(t)=\sum_{k=1}^{N}X_{k}\ e^{(\frac{2\pi|(k-1)t}{T_{k}})} \tag{4}\]  
where \(X_{k}\) and \(N\) represent the input symbol and the subcarrier number of the classical OFDM symbol, respectively. Since the frequency gap between subcarriers is reduced to half (\(\frac{1}{2T_{s}}\)) for the Fast-OFDM symbol, the signal becomes [26]:  
\[s(t)=\sum_{k=1}^{N}X_{k}\ e^{(\frac{\pi|(k-1)t}{T_{s}})} \tag{5}\]  
The kth OCDMA encoder produces the user signature as follows:  
\[s_{k}(t)=\sum_{l=1}^{F}P_{T}c_{k}(t-IT_{c}) \tag{6}\]  
where \(F\) and \(T_{c}\) represent the code length and the time chip period, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**S\({}_{0}\)H\({}_{1}\)** & **30000000** & **20000000** & **70000000** & **00000000** & **00000000** & **10000000** & **0000000** \\ \hline S\({}_{2}\)H\({}_{1}\) & 50000000 & 40000000 & 20000000 & 00000000 & 00000000 & 300000000 & 00000000 \\ S\({}_{4}\)H\({}_{1}\) & 20000000 & 1000000 & 60000000 & 00000000 & 00000000 & 70000000 & 00000000 \\ S\({}_{1}\)H\({}_{2}\) & 10000000 & 70000000 & 50000000 & 00000000 & 00000000 & 60000000 & 00000000 \\ S\({}_{3}\)H\({}_{2}\) & 40000000 & 30000000 & 10000000 & 00000000 & 00000000 & 20000000 & 00000000 \\ S\({}_{0}\)H\({}_{3}\) & 60000000 & 5000000 & 30000000 & 00000000 & 00000000 & 40000000 & 00000000 \\ S\({}_{2}\)H\({}_{3}\) & 70000000 & 60000000 & 40000000 & 00000000 & 00000000 & 50000000 & 00000000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Seven 2D-HC sequences with prime number equal to 7. Figure 2: Implementation of 2D-HC for S\({}_{0}\)H\({}_{1}\) sequence.