## 3 PAPR Reduction Procedure of PA DCO-OFDM  
DCO-OFDM is an optical OFDM variant that requires the addition of the DC bias to convert the negative peaks into a positive signal for intensity modulation. While DCO-OFDM is a spectral efficient modulation technique, it comprises the sum of independent subcarriers in the time-domain which results in the coherent addition of individual subcarriers to produce high PAPR values [20]. Consequently, the high PAPR electrical values require signal clipping at lower and/or upper levels to contain the signal swing within the linear dynamic region of the optical light source [21]. To benefit from the linear dynamic region of the light source in full, the high electrical PAPR peaks must be decreased which helps in minimising the clipping distortion of the transmitted signal [5]. To overcome the PAPR challenge, the PA technique is implemented to reduce the high electrical PAPR peaks of the time-domain signal of the system. In PA, the phase of individual subcarriers is rotated randomly to avoid their coherent addition as much as possible. The implementation process of PA in optical OFDM modulation technique is described in the following section. The electrical PAPR values of DCO-OFDM in the time-domain is evaluated as: [5]  
\[\text{PAPR}=\frac{\max\limits_{0\leq n\leq N-1}(|x[n]|^{2})}{\text{E}[|x[n]|^{ 2}]} \tag{11}\]  
where \(\text{E}[\cdot]\) denotes the statistical expectation. The complementary cumulative distribution function diagram (CCDF) is the most common metric that is used to evaluate the PAPR reduction [5]. The CCDF is defined as the probability that the PAPR of the optical OFDM frame exceeds a predefined reference value. The steps of reducing the electrical PAPR values of DCO-OFDM using PA algorithm is given as follows [5; 22]:  
* Group the QAM symbols into \(U\) blocks comprising active subcarriers, \(N_{\text{subs}}\), to obtain \(X^{u}[m]\), where \(u=1,2,\ldots,U.\) and \(m=0,1,\ldots,N_{\text{subs}}-1\). * Generate multiple iterations of pilot sequence candidates, \(R\), with \(N_{\text{subs}}\) length, \(X_{\text{p}}^{r}[m]\), where \(r=1,2,\ldots,R\). * Select the amplitude, \(A_{\text{p}}[m]\), of the pilot sequence, \(X_{\text{p}}^{r}[m]\), to be \(\pm 1\) only. * Set the phase value of the pilot sequence \(\theta_{\text{p}}[m]\) randomly to 0 or \(\pi\) only. * Rotate the phase of the current DCO-OFDM block, \(U\), by \(\theta_{\text{p}}[m]\) of every pilot sequence iteration \(r\). * Evaluate the PAPR\({}_{r}\) of each iteration of \(X_{\text{p}}^{r}[m]\) in the time domain using (11). * Select the pilot iteration with the minimum PAPR value, \(X_{\text{p}}[m]=X_{\text{p}}^{r}[m]\), to be used for PAPR reduction and to be transmitted with the corresponding frames blocks \(U\); where \[\tilde{r}=\underset{1\leq r\leq R}{\text{argmin}}(\text{PAPR}_{r})\] (12)
* Embedding the pilot sequence \(X_{\text{p}}^{p}[m]\) into the corresponding frames block, \(U\), and this will make the number of frames per block \(\tilde{U}=(U+1)\). In this manuscript, the high electrical PAPR values of the transmitted time-domain signal are reduced by PA, where the number of frames of DCO-OFDM per block, \(U\), is selected to be \(U=5\). This value of \(U\) is selected because when the number of \(U\) increases, the PAPR of the frames block increases [5]. The number of pilot sequence iterations, \(R\), is selected to be \(R=10\) iterations as this value shows a good trade-off between the PAPR reduction gain and computational complexity [15]. At the receiver side, the PA sequences are extracted from all the data blocks and then estimated using ML algorithm which is an optimum detection technique for the statistical parameter, \(\theta\), estimation [14]. The ML algorithm is used to estimate the pilot sequence in this work as follows [5]:  
The noise-corrupted phase \(\hat{\theta}_{\text{p}}[m]\) of the extracted pilot sequences is estimated using the ML algorithm to improve the recovery of the received data.