# RCLED Optimization and Nonlinearity Compensation in a Polymer Optical Fiber DMT System
## 3 Nonlinear Noise  
A DMT symbol with \(N\) sub-carriers is mapped onto \(X_{l}=A_{l}+jB_{l}\left(l=0,1,\cdots,N-1\right)\) based on the quadrature amplitude modulation (QAM) constellation. Then, the DMT symbol can be illustrated as  
\[x_{k}=\frac{1}{\sqrt{2N}}\sum_{n=0}^{2N-1}C_{n}\text{exp}\left(\frac{\left|2 \pi nk\right.}{2N}\right), \tag{3}\]  
\[C_{2N-l}=C_{l}^{*}=X_{l}^{*}, \tag{4}\]  
where \(x_{k}\) (\(k=0,1,\cdots,2N-1\)) denotes the real-value modulated sequence with the inverse fast Fourier transform (IFFT) length being \(2N\), and \((\cdot)^{*}\) denotes the complex conjugate. Additionally, \(C_{0}\) and \(C_{N}\) are set to be zero. After, the digital to analog conversion (DAC), \(x(t)\) is obtained from \(x_{k}\), then added by the bias current \(I_{\text{DC}}\). Assuming \(X_{l}\) to be statistically independent and identically distributed, both \(A_{l}\) and \(B_{l}\) follow the Gaussian distribution with the central limit theorem when \(N\) is large enough. Since \(x_{k}\), is a sum of \(N\) independent random variables \(X_{l}\), thus \(x(t)\) also approaches the Gaussian distribution with the mean \(\mathbb{E}\left\{\left.x(t)\right\}=I_{\text{DC}}\right.\) and the variance \(\mathbb{E}\left\{\left.x^{2}(t)\right\}=\sigma_{0}^{2}\right.\), where \(\mathbb{E}\{\cdot\}\) denotes the statistical expectation. Accordingly, the probability density function \(P_{r}\left[x\left(t\right)\right]\) of \(x(t)\) can be obtained as  
\[P_{r}\left[x\left(t\right)\right]=\frac{1}{\sqrt{2\pi\sigma_{0}^{2}}}\text{ exp}\left(\frac{\left(x\left(t\right)-I_{\text{DC}}\right)^{2}}{2\sigma_{0}^{2}} \right), \tag{5}\]  
According to the [PERSON] theorem [10], the nonlinear distortion of \(x(t)\) can be described by a gain factor \(k_{\text{op}}\) and an uncorrelated noise component \(n(t)\), i.e.,  
\[y(t)=D\left\{x(t)\right\}=k_{\text{op}}x(t)+n(t), \tag{6}\]  
where \(D\left\{\cdot\right\}\) denotes the nonlinear behaviors of RCLED. Figure 3: Measured results and polynomial fitting: (**a**) root mean squared error (RMSE) comparison; (**b**) 2-order fitting curves. With \(\mathbb{E}\left\{x(t)n(t)\right\}=0\), \(k_{\text{op}}\) can be calculated by  
\[k_{\text{op}}=\frac{\mathbb{E}\left\{x(t)y(t)\right\}}{\sigma_{0}^{2}}, \tag{7}\]  
Since the RCLED nonlinearity includes both the E/O nonlinear transition noise \(n_{\text{dis}}\) and the turn-ON voltage and SAV threshold clipping noise \(n_{\text{clip}}\), after the SI-POF transmission, the overall noise power can be written as  
\[P_{\text{noise}}=P_{\text{awgn}}+P_{\text{dis}}+P_{\text{clip}}, \tag{8}\]  
where \(P_{\text{dis}}\) and \(P_{\text{clip}}\) denote the power of \(n_{\text{dis}}\) and \(n_{\text{clip}}\), respectively, \(P_{\text{awgn}}\) denotes the background noise power in the receiver. Due to the deviation from the ideal linear E/O transition \(L\left(x\right)\), \(P_{\text{dis}}\) can be calculated as  
\[P_{\text{dis}}=\int\limits_{I_{-}}^{I_{+}}\left[g\left(x\right)-L\left(x \right)\right]^{2}P_{r}\left(x\right)dx, \tag{9}\]  
where \(g\left(x\right)\) denotes the practical E/O response of RCLED, \(I_{-}\) and \(I_{+}\) denote the threshold and maximal value, respectively. Similarly, \(P_{\text{clip}}\) can be obtained by  
\[ \begin{array}{ll}P_{\text{clip}}&=\int\limits_{I_{+}}^{+\infty}(x-I_{+})^{2 }\,P_{r}\left(x\right)dx+\int\limits_{-\infty}^{I_{-}}\left(x-I_{-}\right)^{2} \,P_{r}\left(x\right)dx\\ &=\int\limits_{k_{1}\sigma_{0}+I_{\text{DC}}}^{+\infty}\left(x-k_{1}\sigma_{0} -I_{\text{DC}}\right)^{2}\,P_{r}\left(x\right)dx+\int\limits_{-\infty}^{I_{ \text{DC}}-k_{2}\sigma_{0}}\left(x-I_{\text{DC}}+k_{2}\sigma_{0}\right)^{2} \,P_{r}\left(x\right)dx\\ &=\sqrt{\frac{2}{\pi}}\sigma_{0}^{2}\left[\frac{1}{k_{1}^{3}}\text{exp}\left( \frac{-k_{1}^{2}}{2}\right)+\frac{1}{k_{2}^{3}}\text{exp}\left(\frac{-k_{2}^ {2}}{2}\right)\right]\end{array}