# Development of New Spectral Amplitude Coding OCDMA Code by Using Polarization Encoding Technique
## 5 Performance Analysis
### Theoretical Analysis  
FDW code system analysis is assumed to have zero cross-correlation properties due to a new encoding technique (polarization technique). These properties use a direct detection technique for the decoding technique. In a theoretical analysis for the proposed system, a noise must be considered in the calculation, such as PINN (phase-induced intensity noise, \(I_{PIN}\)), shot noise (\(I_{SHOT}\)), and thermal noise (\(I_{TH}\)) [25; 26; 27]. Using direct detection technique, PINN noise is eliminated in the calculation due to zero cross-correlation properties. Let \(C_{K}(i)\) denote the ith element of the \(K_{th}\) FDW code sequence. The properties of FDW code can be expressed as:  
\[\sum_{1=K}^{N}C_{K}(i)C_{l}(i)= \begin{cases}W,&\text{for }K=l\\ 0,&\text{else}.\end{cases}  \tag{10}\]  
The power spectral density (PSD) of the received optical signal can be written as:  
\[r(v) = \frac{P_{sr}}{\Delta V}\sum_{k=1}^{K}d_{K}\sum_{i=1}^{L}C_{K}(i) \bigg{\{}u\bigg{[}V-V_{0}-\frac{\Delta V}{2N}(-N+2i-2)-u\bigg{[}v-v_{o}-\frac {\Delta V}{2N}(-N+2i)\bigg{]}\bigg{]}\bigg{]}\bigg{\}}\] \[\text{the unit step function }u(v)\text{ can be expressed as:}\] \[u(v)= \begin{cases}1,&v\geq 0\\ 0,&v<0.\end{cases}  \tag{12}\]  
From Equation (11), the power spectral density at the photodetector is:  
\[G(v) = \frac{P_{sr}}{\Delta V}\sum_{k=1}^{K}d_{K}\sum_{i=1}^{L}C_{K}(i)C _{l}(i)\bigg{\{}u\bigg{[}V-V_{0}-\frac{\Delta V}{2N}(-N+2i-2)-u\bigg{[}V-V_{o} -\frac{\Delta V}{2N}(-N+2i)\bigg{]}\bigg{]}\bigg{\}} \tag{13}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Units** \\ \hline Photodetector quantum efficiency (\(\eta\)) & 0.6 \\ Operating wavelength (\(\lambda_{O}\)) & 1550 nm \\ Line width broadband source (\(\triangle\,V\)) & 3.75 THz \\ Electrical bandwidth (B) & 311 MHz \\ Receiver load resistor (RL) & 1030 Ohm \\ Receiver noise temperature (\(T_{n}\)) & 300 K \\ Bit Rate & 622 Mbps \\ Fiber Length & 10 km \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Typical parameters used in the calculation. the equation can be simplified much further, let  
\[ \begin{split} M&=\bigg{\{}u\bigg{[}V-V_{0}-\frac{\Delta V }{2N}(-N+2i-2)-u\bigg{[}V-V_{o}-\frac{\Delta V}{2N}(-N+2i)\bigg{]}\bigg{]} \bigg{\}}\\ &=\bigg{\{}u\bigg{[}V-V_{0}-\frac{\Delta V}{2N}(-N+2i-2)-V+V_{o}+ \frac{\Delta V}{2N}(-N+2i)\bigg{]}\bigg{\}}\\ &=\bigg{\{}u\bigg{[}-\frac{\Delta V}{2N}(-N+2i-2)+\frac{\Delta V} {2N}(-N+2i)\bigg{]}\bigg{\}}\\ &=\bigg{\{}u\bigg{[}\frac{\Delta V}{2}-\frac{\Delta V}{N}i+\frac {\Delta V}{N}-\frac{\Delta V}{2}+\frac{\Delta V}{N}i\bigg{]}\bigg{\}}\\ M&=\bigg{\{}u\bigg{[}\frac{\Delta V}{N}\bigg{]}\bigg{\}} \end{split}  \tag{14}\]  
applying Equation (13) with Equation (14) to obtain the equation below:  
\[ \begin{split} G(v)&=\frac{P_{sr}}{\Delta V}\sum_{ k=1}^{K}d_{K}\sum_{i=1}^{L}C_{K}(i)C_{l}(i)[M]\\ &=\frac{P_{sr}}{\Delta V}\sum_{k=1}^{K}d_{K}\sum_{i=1}^{L}C_{K}(i )C_{l}(i)\bigg{\{}u\bigg{[}\frac{\Delta V}{N}\bigg{]}\bigg{\}}\end{split}  \tag{15}\]  
where \(P_{sr}\) is the effective power of a broadband source at the receiver, \(K\) is the number of active users, \(N\) is the code length of the FDW code and \(d_{k}\) is the data bit of the \(K_{th}\) user, which is \"1\" or \"0\". The photocurrent of the receiver, \(I_{p}\), is defined by [28]:  
\[I_{p}=\Re\int_{0}^{\infty}G(v)dV \tag{16}\]  
where \(\Re\) is the responsivity of the photodetector, and \(\int_{0}^{\infty}G(v)dVdV\) is the total power of the photodiode [26; 27].