# Pilot-Aided Frame Synchronization in Optical OFDM Systems
## 3 Results and Discussions  
In this Section, we present the performance of the proposed PA frame synchronization timing metric by evaluating the average of the metric over 1000 PA O-OFDM frames. The performance evaluation is done in the presence of additive white Gaussian noise (AWGN) and indoor VLC channel [13; 14]. The channel consists of line-of-sight (LOS) and non-LOS (NLOS) components with the LOS components dominating and the channel impulse response is obtained from power delay profile in [13] and workplace with open office in [14]. The two O-OFDM frames considered are the DCO-OFDM and ACO-OFDM structure. In the simulations, the \(M\)-QAM average symbol energy is normalized to unity, such that \(\mathbb{E}[|\mathbf{X}_{\mathrm{s}}^{n}(i)|^{2}]=1\). Other parameters used in the simulation are indicated in the caption of each figure. Figure 2 shows the normalized average of the timing metric (4) for a PA ACO-OFDM. The plot shows that the timing metric varies as the detection window slides through the received signal in order to estimate the start of a pilot symbol. The start of a pilot symbol corresponds to the minimum point of the timing metric. The minimum average value of the metric is expected to be equal to zero, according to (2), when the detection window correlates with the received pilot symbol. Although this value of zero is not obtainable due to the effects of noise and indoor VLC channel on the received pilot signal, the proposed PA frame synchronization timing metric still results in a relatively large difference between the value at the start of a pilot symbol and other points within a frame. Figure 3 shows the normalized average of the modified Park's timing metric in ACO-OFDM with cyclic prefix length of \(N/8\). The correct location of the training symbol corresponds to the maximum point of the normalized average of the timing metric. To quantify the difference between the principal peak and the next peak, let \(\beta=|\beta_{\mathrm{P}}-\beta_{2}|\), where \(\beta_{\mathrm{P}}\) is the principal peak and \(\beta_{2}\) is the second corresponding peak. Based on the pattern of the training symbol according to (5) and the sum of products operation used in the timing metric (6), a cyclic prefix whose length is greater or close to \(N/8\) results into principal and second corresponding peaks that are almost similar. Comparing Figures 2 and 3 where \(cp=N/8\) is utilized for ACO-OFDM in both cases, it is shown that the pattern of the time domain pilot symbol in (3) and the PA frame timing metric (4) result in \(\beta=0.3\), whereas \(\beta=0.04\) is obtained using the modified [PERSON]'s method. This approach shows the robustness of the proposed PA synchronization algorithm to the effect of cyclic prefix length considered on the timing metric. The PA frame synchronization method also performs well for different ranges of \(N\) and \(U\). Figure 2: Plot of normalized average of the proposed timing metric (4) for a PA ACO-OFDM with \(N=256\), \(U=5\), \(cp=N/8\), signal-to-noise ratio (SNR) \(=10\) dB, and 4-quadrature amplitude modulation (QAM). In addition, the PA method has the lowest requirements in terms of computational complexity with respect to number of operations. For a pilot symbol with window size of \(N\) samples, the PA timing metric (4) requires \(N-3\) additions and no multiplication. The equivalent number of additions and multiplications in (6) is \(\frac{N}{2}-8\) and \(\frac{N}{4}+2\) respectively for preamble of a detection window of \(\frac{N}{2}-1\) samples. To investigate the rate of deviation of the PA frame synchronization timing metric (4) and modified Park's timing metric (6) from the exact symbol position, Figure 4 shows the mean of the timing offset in DCO-OFDM and ACO-OFDM for different SNR levels. In this work, a frame timing is considered to be estimated precisely when the timing metric gives its principal peak at the start of a pilot or training symbol without any tolerance margin.