## 4 PA DCO-OFDM Clipping Noise Analysis and Assessment  
The clipping of PA DCO-OFDM time-domain signal changes the signal mean and, as a result, its average optical power. Since \(x_{\mathrm{clip}}[n]\) is approximately Gaussian, the nonlinear distortion effects caused by the clipping operation can be modelled according to the Bussgang theorem, and expressed as follows [23]  
\[x_{\mathrm{clip}}[n]=\kappa\cdot x_{\mathrm{scaled}}[n]+w_{\mathrm{clip}}[n] \tag{15}\]  
where \(\kappa\) denotes the clipping attenuation factor and \(w_{\mathrm{clip}}\) is the nonlinear distortion, i.e., clipping noise imposed by the pre-clipping process and system front-end devices constraints. The nonlinear distortion \(w_{\mathrm{clip}}\) is uncorrelated with the transmitted signal \(x_{\mathrm{scaled}}[n]\), \(\mathrm{E}[x_{\mathrm{scaled}}[n]\cdot w_{\mathrm{clip}}[n]]=0\). The attenuation factor \(\kappa\) is a constant that is given as follows [1]:  
\[\kappa=\frac{\mathrm{E}[x_{\mathrm{scaled}}[n]\cdot x_{\mathrm{clip}}[n]]}{ \sigma_{x_{\mathrm{scaled}}}^{2}} \tag{16}\]  
where \(\sigma_{x_{\mathrm{scaled}}}^{2}\) is the variance of \(x_{\mathrm{scaled}}[n]\). The nonlinear distortion \(w_{\mathrm{clip}}\) is a non-Gaussian noise. However, due to the CLT, its representation in the frequency domain follows a  
Figure 3: CCDF plot for PAPR values for both conventional DCO-OFDM and PA DCO-OFDM, using \(N_{\mathrm{subs}}=1023\) data carrying subcarriers, with 4-QAM, up-sampling factor \(L=4\), data frame per block \(U=5\), and pilot sequence iteration \(R=10\) iterations. Gaussian distribution with a zero mean and a variance \(\sigma^{2}_{w_{\text{clip}}}\). The variance of the clipping noise is given as follows [1,8]:  
\[\text{E}[w_{\text{clip}}] =\text{E}[x_{\text{clip}}], \tag{17a}\] \[\sigma^{2}_{w_{\text{clip}}} =\text{E}[w^{2}_{\text{clip}}]-\text{E}[w_{\text{clip}}]^{2},\] (17b) \[\sigma^{2}_{w_{\text{clip}}} =\text{E}[x^{2}_{\text{clip}}]-\kappa^{2}\cdot\sigma^{2}_{x_{ \text{scaled}}}-\text{E}[x_{\text{clip}}]^{2}. \tag{17c}\]  
In the frequency domain, the clipping noise \(w_{\text{clip}}[n]\) is transformed into additive Gaussian noise according to CLT. Therefore, additive Gaussian noise component with zero-mean and variance \(\sigma^{2}_{w_{\text{clip}}}\) is present at each modulated subcarrier in addition to the additive white Gaussian noise \(w_{\text{rx}}[n]\). Therefore, the received signal in the frequency domain can be calculated as follows  
\[ \begin{split} Y[k]&=H[k]\Big{(}\kappa\cdot X[k]+W_{ \text{clip}}[k]\Big{)}+W_{\text{rx}}[k],\\ &=\kappa H[k]X[k]+H[k]W_{\text{clip}}[k]+W_{\text{rx}}[k].\end{split}  \tag{18}\]  
where \(Y[k]\) is the desired received signal, \(H[k]\) is the channel response, \(X[k]\) is the transmitted symbols, \(W_{\text{clip}}[k]\) is the clipping noise, and \(W_{\text{rx}}[k]\) is the additive white Gaussian noise. The received signal can be expressed as the summation of the desired signal, clipping noise, and the receiver noise with zero mean for both noise signals, as shown in (18). The SNR at the \(k\)th subcarrier is represented by \(\Gamma[k]\), and can be expressed as follows [8]  
\[\Gamma[k]=\frac{\kappa^{2}\frac{1}{C_{\text{B}}}|H[k]|^{2}}{\sigma^{2}_{w_{ \text{clip}}}|H[k]|^{2}+\sigma^{2}_{w_{\text{rx}}}} \tag{19}\]  
where \(G_{\text{B}}\) is the double-sided bandwidth utilisation factor of the optical OFDM frame and \(\sigma^{2}_{w_{\text{rx}}}\) is the receiver noise power. The attenuation factor can be simplified from (16) as follows [8]  
\[\kappa=Q(k_{\text{b}_{\text{norm}}})-Q(k_{\text{norm}}) \tag{20}\]  
where \(Q\) is the Q-function [8]. \(k_{\text{b}_{\text{norm}}}\) and \(k_{\text{norm}}\) are the normalised bottom and top clipping levels, respectively, and can be expressed as follows [8]  
\[k_{\text{b}_{\text{norm}}} =\frac{k_{\text{b}}}{\sigma_{x_{\text{scaled}}}} \tag{21}\] \[k_{\text{norm}} =\frac{k_{\text{t}}}{\sigma_{x_{\text{scaled}}}} \tag{22}\]  
The clipping noise variance is given as [8]  
\[