# Visible-Light CDMA Communications Using Inverted Spread Sequences
## 5 Bit Error Rate Performance  
### System Model  
All the optical CDMA systems using the six MUI cancellation schemes introduced in Sections 2 and 3 have a common advantage, which is error-free performance under an ideal link where the noise can be considered negligible.  
This section investigates the BER performance of optical CDMA systems with the MUI cancellation schemes described above. We consider that the receiver uses a pin photodiode (pin-PD) to convert an optical signal into an electrical signal at every chip duration. The BERs of these schemes are evaluated by computer simulation. We assume that shot noise and thermal noise are dominant noise sources. It is known that the shot noise distribution can be assumed Gaussian [26; 27]. Since the thermal noise distribution is also Gaussian, the distribution of the pin-PD output current can be approximated by a Gaussian distribution with mean \(\mu\) and variance \(\sigma^{2}\).  
The mean of the output current is given by  
\[\mu=\rho I_{s}+I_{b}+I_{d}, \tag{16}\]  
where \(I_{s}\) is the photo-current when a single light source is on, \(I_{b}\) is the background light current, \(I_{d}\) is the dark current, and \(\rho\) is the average light intensity when the light intensity of a mark is assumed to be one. For each scheme, \(\rho\) is derived as the equations in Table 5. In each equation, the first term is a factor related to the \(N\) active users and the others are factors related to the \(N_{\rm L}-N\) non-active users. Each \(\rho\) corresponds to the average light intensity shown in Figure 5 when the modulation extinction ratio \(M_{e}\) is large enough. \(I_{s}\) and \(I_{b}\) are given by  
\[I_{s}=\frac{\eta P_{w}e}{hf}, \tag{17}\]  
\[I_{b}=\frac{\eta P_{b}e}{hf}, \tag{18}\]  
where \(\eta\) is the quantum efficiency, \(P_{w}\) is the received signal light power when a light source transmits marks, \(P_{b}\) is the received background light power, \(e\) is the elementary charge, \(h\) is [PERSON]'s constant, and \(f\) is the optical frequency.  
The variance of the pin-PD output current is given by  
\[\sigma^{2}=\sigma_{\rm sh}^{2}+\sigma_{\rm th}^{2}, \tag{19}\]  
where \(\sigma_{\rm sh}^{2}=2 eB(\rho I_{s}+I_{b}+I_{d})\) and \(\sigma_{\rm th}^{2}=4k_{B}BT_{r}/R_{L}\) are the variance in the shot noise and the variance in the thermal noise, respectively, \(B\) is the noise-equivalent bandwidth, \(k_{B}\) is [PERSON]'s constant, \(T_{r}\) is the receiver noise temperature, and \(R_{L}\) is the receiver load resistance. The nominal values of the parameters used in our computer simulation are given in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Code** & **Scheme** & \(\rho\) & **Code** & **Scheme** & \(\rho\) \\ \hline \multirow{4}{*}{MPSC} & EWO & \(\frac{N\left(q+\frac{2^{2}-t}{M_{e}}\right)}{q^{2}}+\frac{N_{\rm L}-N}{M_{e}}\) & & EWO & \(\frac{N\left(q^{2}-q+\frac{t}{M_{e}}\right)}{q^{2}}+N_{\rm L}-N\) \\ \cline{2-6}  & Shalaby & \(\frac{N\left(q+\frac{2^{2}-t}{M_{e}}\right)}{2q^{2}}+\frac{N_{\rm L}-N}{M_{e}}\) & & MPSC & Shalaby & \(\frac{N\left(2q^{2}-q+\frac{t}{M_{e}}\right)}{2q^{2}}+N_{\rm L}-N\) \\ \cline{2-6}  & Liu & \(\frac{N\left(q+1+\frac{2^{2}+q-t}{M_{e}}\right)}{2(q^{2}+q)}+\frac{N_{\rm L}-N}{M_ {e}}\) & & Modified Liu & \(\frac{N\left(2q^{2}+q+1+\frac{641}{M_{e}}\right)}{2(q^{2}+q+1)}+N_{\rm L}-N\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: The average light intensity \(\rho\) for the MUI cancellation schemes.