# Pilot-Aided Frame Synchronization in Optical OFDM Systems
## 2 Frame Synchronization in a Pilot-Assisted Optical OFDM System
The resulting time domain pilot symbol, \(\mathbf{p}\), after the IFFT operation is given as:  
\[\mathbf{p}=[p(0)\ \ \mathbf{p}_{N/2-1}\ \ p(N/2)\ \ \mathbf{p}_{N/2-1}^{ \text{mirror}}], \tag{1}\]  
where \(\mathbf{p}_{N/2-1}\) represents samples of length \(N/2-1\) and \(\mathbf{p}_{N/2-1}^{\text{mirror}}\) represents mirror symmetry of \(\mathbf{p}_{N/2-1}\), \(p(0)\) is the first pilot sample and \(p(N/2)\) is the \(N/2^{\text{th}}\) pilot sample. The mirror symmetry implies that \(p(n)=p(N-n)\) for \(n=1,2,\ldots N/2-1\). To generate a time domain pilot signal with these properties using an IFFT with Hermitian symmetry, \(\mathbf{X}_{p}(i)\), which is also used for data symbol phase rotation for the purpose of PAPR reduction, has its amplitude constrained to unity in order to  
Figure 1: An illustration of a pilot-assisted (PA) frame in (**a**) direct-current biased optical orthogonal frequency division multiplexing (DCO-OFDM) and (**b**) asymmetrically clipped O-OFDM (ACO-OFDM). preserve the electrical power of the data signals, while its phase can either be 0 or \(\pi\) so as to maintain the original constellation of the input data symbols [11]. The mirror symmetry implies that, \(\Delta_{\text{PA}}\), which is the summation of the differences between the first half samples and mirror image of the second half samples of the pilot symbol, is:  
\[ \begin{split}\Delta_{\text{PA}}&=\sum_{n=1}^{N/2-1}p (n)-p(N-n)\\ &=0.\end{split}  \tag{2}\]  
It should be noted that \(p(0)\) and \(p(N/2)\) are excluded from the evaluation of \(\Delta_{\text{PA}}\) for the definition in (2) to hold. After adding cyclic prefix to the pilot symbol, the resulting symbol is given as:  
\[ \begin{split}\mathbf{p}_{\text{cp}}=[\mathbf{p}_{cp}^{\text{ mirror}}\ \ p(0)\ \ \mathbf{p}_{N/2-1}\ \ p(N/2)\ \ \mathbf{p}_{N/2-1}^{\text{mirror}}],\end{split}  \tag{3}\]  
where \(cp\) is the length of the appended cyclic prefix and \(\mathbf{p}_{cp}^{\text{mirror}}\) represents the last \(cp\) samples of \(\mathbf{p}_{N/2-1}^{\text{mirror}}\). To obtain the unipolar signal required for IM, the time domain O-OFDM symbols are clipped at zero level in ACO-OFDM while adequate DC bias is added in DCO-OFDM. At the receiver, the proposed timing synchronization algorithm exploits the mirror symmetric property of the received pilot signal, according to (2), to determine the start of each pilot symbol within the PA O-OFDM frames. The frame synchronization scheme is correlation-based with a detection window of size \(N\) that slides through the received signal, \(y(m)\). The start of the pilot signal within a frame is found by searching \(y(m)\) for an O-OFDM symbol of \(N\) samples in which the first half is identical to the mirror image of the second half using the proposed synchronization timing metric given as:  
\[ \begin{split} P_{\text{PA}}(\tau)=\sum_{m=\tau+1}^{\frac{N}{2} +\tau}\mathbf{y}(m)-\mathbf{y}(N-m),\end{split}  \tag{4}\]  
where \(\tau\) is the PA timing metric sample index corresponding to the first sample in a window of \(N\) samples. The proposed PA timing metric in (4) has low computational complexity because it requires simple addition operation. The receiver uses the minimum point of \(P_{\text{PA}}(\tau)\) to indicate the start of a pilot symbol. In this work, the performance of the proposed PA method is compared with the Park's synchronization technique modified for O-OFDM system [4; 5; 9]. The modified Park's time domain training symbol sequence, after concatenation of cyclic prefix, is given as [5; 9]:  
\[ \begin{split}\mathbf{s}_{\text{cp}}=[\mathbf{C}_{\text{cp}}\ 0\ \mathbf{C}_{N/4-1}\ 0\ - \mathbf{C}_{N/4-1}^{\text{mirror}}\ 0\ -\mathbf{C}_{N/4-1}\ 0\ \mathbf{C}_{N/4-1}^{\text{mirror}}],\end{split}  \tag{5}\]  
where \(\mathbf{C}_{N/4-1}\) represents samples of length \(N/4-1\), \(\mathbf{C}_{N/4-1}^{\text{mirror}}\) represents mirror symmetry of \(\mathbf{C}_{N/4-1}\) and \(\mathbf{C}_{\text{cp}}\) is the last \(cp\) samples of \(\mathbf{C}_{N/4-1}^{\text{mirror}}\). The timing metric is given as [9]:  
\[