# Tolerance-Aided Interference Degradation for Optical OFDM in Power-Constrained Systems
## 3 Principles of Tolerance-Aided Interference Degradation Algorithm
### Effective Noise Model and the Performance Analysis  
In (9), \(n_{\mathrm{eff}}\) represents the effective additive noise after reconstructing the received signal using TAID, which is modeled as a combination of the peak power clipping noise,  
Figure 4: The block diagram of the receiver for TAID algorithm.  
additive noise at the receiver, and the noise of using the decision tolerance. In this paper, we derive \(n_{\text{eff}}\) as  
\[n_{\text{eff}}=n_{\text{clip}}+n_{\text{add}}+n_{\text{tol}}, \tag{12}\]  
where \(n_{\text{clip}}\), \(n_{\text{add}}\), and \(n_{\text{tol}}\) represent the clipping noise, effective additive noise, and noise introduced by the decision tolerance, respectively.  
Despite CEO-OFDM being applied, it is noteworthy that the transmitted signals remain susceptible to peak clipping distortion. The severity of this peak power constraint-induced distortion escalates with the modulation index, denoted as \(c\), which governs the scale of the transmitted signals. To evaluate the system performances using TAID, we adopt an approach wherein we represent the clipping distortion as an additional source of noise. This distortion is characterized as a Gaussian-distributed random variable with a zero mean. Its variance can be calculated as in [15], which is a function of the modulation index, as  
\[\begin{split}\sigma_{\text{clip}}^{2}(c)&=\frac{2 cP _{\text{max}}\sigma_{d}}{N\sqrt{2\pi}}\exp\left(\frac{-2N^{2}P_{\text{max}}^{2} }{c^{2}\sigma_{d}^{2}}\right)\\ &+\frac{1}{2}\text{erfc}\left(\frac{2 NP_{\text{max}}}{c\sqrt{2 \sigma_{d}^{2}}}\right)\!\!\left(4P_{\text{max}}^{2}+\frac{c^{2}\sigma_{d}^{2 }}{N}\right),\end{split} \tag{13}\]  
where \(\sigma_{d}\) is the variance of \(X_{i},~{}\forall~{}i\).  
By implementing the proposed TAID algorithm, the reduction of additive noise across the three signal frames is remarkably achieved. Leveraging the central limit theorem (CLT), the effective noise subsequent to the application of the TAID algorithm can be suitably approximated as a Gaussian random variable characterized by a zero mean. The variance of this effective noise is intricately linked to the decision tolerance, which is a composite of two constituent components, represented as  
\[\sigma_{\text{tol}}^{2}(\Delta)=\sigma_{\text{r1}}^{2}(\Delta)+\sigma_{\text{ r2}}^{2}(\Delta). \tag{14}\]  
In Equation (14), \(\sigma_{\text{r1}}^{2}(\Delta)\) pertains to the scenario where the transmitted signals within frames 1 and 2 have not undergone peak clipping. However, in practice, the influence of noise can lead to instances where these signals, upon reception, exceed the defined decision threshold of \(P_{\text{max}}-\Delta\). In such situations, it becomes necessary to artificially consider these signals as having been clipped, requiring us to enforce a value of \(P_{\text{max}}\) for the received signals within frames 1 and 2. Consequently, the introduction of interference stemming from the decision tolerance must be taken into account, and \(\sigma_{\text{r1}}^{2}(\Delta)\) can be calculated as  
\[\begin{split}\sigma_{\text{r1}}^{2}(\Delta)&\!\! =\!\!\!\int_{0}^{P_{\text{max}}}\!\!\!\!\int_{P_{\text{max}}-\Delta-r}^{\infty }\  
are derived. For this problem, the exhaustive search method is a feasible way to find the optimal solution for the two parameters, \(c\) and \(\Delta\).