# Tolerance-Aided Interference Degradation for Optical OFDM in Power-Constrained Systems
## 3 Principles of Tolerance-Aided Interference Degradation Algorithm
### Received Signal Model and Effective Noise Models  
At the receiver, the incoming optical signals undergo a conversion process into electrical signals, followed by discrete-time sampling. The representation of the \(n\)th sample of the received discrete-time electrical signal can be expressed as  
\[y[n]=\rho\cdot h[n]*x_{t}[n]+n_{y}[n],\;n=0,1,\cdots,3N-1, \tag{4}\]  
where \(\rho\) represents the responsivity of the photodetector (PD), which denotes the optical-to-electrical power conversion ratio. \(n_{y}[n]\) is the additive noise, including thermal and shot noises. In this paper, we assume \(n_{y}[n]\) is a Gaussian white noise with zero mean. Its variance can be calculated as \(\sigma_{n}^{2}=\frac{3R_{s}}{2N}\mathcal{N}_{o}\), where \(R_{s}\) is the transmitted \(M\)-QAM symbol rate and \(\mathcal{N}_{o}\) is the single-sided noise power spectral density. \(h[n]\) is the \(n\)th sample of the discrete-time end-to-end system impulse response, which can be pre-estimated. The operation \"\(\ast\)\" represents the discrete-time convolution. Drawing upon the CEO-OFDM signal model, the received signal in one OFDM symbol contains three signal frames. Each frame is mixed with the additive noise, and the peak power clipping distortion happens in frame 3 only. Since the additive noise is mixed in all three signal frames, simply combining the three frames dramatically amplifies the effective noise. As shown in Figure 3, with the interference of the noise mixed in frames 1 and 2, the signals that should have been clipped at \(P_{\text{max}}\) are greater or less than the peak power. To address this challenge, we introduce a decision tolerance parameter denoted as \(\Delta\). By applying decision criteria to each received sample within frames 1 and 2, we can effectively reduce the impact of additive noise. Specifically, if the signals in frames 1 and 2 surpass the threshold of \(P_{\text{max}}-\Delta\), we enforce their value to be \(P_{\text{max}}\). Consequently, the corresponding signals in frame 3 are regarded as the clipped portion that necessitates combination during the demodulation process. Conversely, when the signals in frames 1 and 2 fall below the threshold of \(P_{\text{max}}-\Delta\), it is assumed that they have not been distorted by the peak power limit, and their associated clipped portion within frame 3 is deemed unnecessary. A comprehensive block diagram illustrating this decision process is provided in Figure 4 as a detailed representation, where \(\text{sign}(\cdot)\) denotes the unit sign function. It is imperative to recognize that the choice of the decision tolerance parameter \(\Delta\) can significantly impact the trade-off between the noise introduced in frame 3 and the interference stemming from clipping distortion within frames 1 and 2. Figure 3: An illustration of the received signals with noise. [PERSON] and green symbols represent the amplitude of the positive and negative signals, respectively. Blue symbols represent the additive noise. After applying the proposed TAID algorithm, the \(n\)th sample of the reconstructed signal is represented as  
\[r[n]=r_{\mathrm{s}}[n]+r_{\mathrm{c}}[n],\;\;n=0,1,\cdots,N-1, \tag{5}\]  
where  
\[r_{\mathrm{s}}[n]=\left(\phi_{\Delta}(y[n])-\phi_{\Delta}(y[n+N])\right)U_{N}[n], \tag{6}\]  
\[r_{\mathrm{c}}[n]=\left\{ \begin{array}{ll}0,\;\;\;y[n]<(P_{\mathrm{max}}- \Delta)||y[n+N]<(P_{\mathrm{max}}-\Delta)\\ -y[n+2N]U_{N}[n],\;\;\;y[n]<y[n+N]\\ y[n+2N]U_{N}[n],\;\;\;\;y[n]>y[n+N]\end{array} \right., \tag{7}\]  
and \(U_{N}[n]=u[n]-u[n-N]\). \(u[n]\) is the discrete-time unit step function. \(\phi_{\Delta}(\cdot)\) is a decision function controlled by the defined tolerance \(\Delta\), which is represented as  
\[\phi_{\Delta}(x)=\left\{ \begin{array}{ll}P_{\mathrm{max}},\;\;\;\;x\geq P_{ \mathrm{max}}-\Delta\\ x,\;\;\;\;\;\;0<x<P_{\mathrm{max}}-\Delta\end{array}