## 4 System Performance Analysis and Discussions
As some interference signals remain, the effective user number \(K_{\text{eff}}\) in the signal for the final decoding is expressed as:  
\[K_{\text{eff}}=1+\sum_{i=0}^{S-1}C_{i}^{S-1}i[BER(K)]^{i}[1-BER(K)]^{S-i-1}. \tag{11}\]  
According to the [PERSON] first theorem [33], the receiver bandwidth must be larger than one half of the bit-rate to avoid intersymbol interference (ISI). Therefore, under a given SNR, the maximum signal bit-rate \(R_{b}\) supported by the proposed receiver is:  
\[R_{b(\text{max})}=2B=\frac{l^{2}}{SNR(K_{\text{eff}})\big{[}S_{\text{TH}}+R^{ 2}P_{\text{L}}^{2}k(k+1)(N+1)/4 Nb\big{]}}. \tag{12}\]  
Figure 8 shows the relation between BER and the transmission rate of OCDMA users. The simulation parameters are listed as follow: \(R\) = 0.81 A/W, \(P_{\text{L}}\) = \(-\)10 dBm, \(N\) = 31, \(S_{\text{TH}}\) = 1.632 \(\times\) 10\({}^{-23}\) W/Hz, and \(v\) = 3.75 THz. The number of active users \(K\) is fixed to 20. The proposed RIC scheme with different stage numbers (\(S\) = 10, 15, and 20) is analyzed, where the RIC with a unit stage (\(S\) = 1) is equivalent to the conventional balanced detector. In previous publications related to SAC-OCDMA, balanced detector has been widely used for decoding as well as MAI eliminating [11; 12; 13]. It can be found that the transmission rates \(R_{b}\) supported by RIC are larger than the ones supported by the balanced detector. For PINN variance, it is both dependent and positively related to \(K\) and \(R_{b}\). Reducing the user number through RIC allows the network to send data bits with a higher speed under a given noise floor. In addition, as \(S\) increases, more interference users are removed, and RIC obtains a faster data speed for the same BER. Figure 9 shows the relationship between BER and the stage number \(S\) with \(K\) = 20. The results, which correspond to Figure 8, indicate that BER lowers as \(S\) increases for the same data bit-rate \(R_{b}\). Thus, RIC with a larger \(S\) benefits from greater BER performance at the expense of executing additional procedures of interference user elimination. In addition, RIC with a smaller \(R_{b}\) reveals a better performance when \(S\) is fixed, as less noise power is allocated in the receiver bandwidth. Figure 8: Bit-error rate (BER) versus the transmission data rate. Splitting loss is another issue that should be considered in RIC. At each RIC stage, the received signal is divided into two portions with equal power, one for MUI estimation and the other for MUI elimination. This results in a 3-dB power penalty. When the stage number \(S\) is small, the effect of splitting loss is not evident. However, when \(S\) is further increased, the splitting loss may dominate BER due to the low received power. Figure 10 shows the relationship of BER and the received power \(P_{\text{L}}\) for active user number \(K\) = 30 and bit-rate \(R_{b}\) = 2 Gb/s. When \(P_{\text{L}}\) is relatively small, both schemes suffer from thermal noise and have a large BER. For the large values of \(P_{\text{L}}\), where PIIN becomes dominant, the performance of RIC is better than that of the balanced detector. The performance improvement comes from the reduction of the interference users, resulting in less power contributing to PIIN. Based on this figure, although splitting loss degrades the BER performance of RIC, it can be compensated by launching a high-power light source or using an optical amplifier to recover the received power to its original level. As for the balanced detector, although it does not suffer from power loss, the BER is still large due to the high PIIN power contributed from the interference users. Figure 10: Bit-error rate (BER) versus the light source power. Figure 9: Bit-error rate (BER) versus the stage number of RIC. In this section, we mainly discussed the factors associated with the proposed receiver, such as transmission data-rate \(R_{b}\), receiver stage number \(S\), and BER. We fixed the number of users \(K\) to more effectively assess the extent of data-rate improvement.