# Phase Offset Tracking for Free Space Digital Coherent Optical Communication System
## 4 Number Results  
In this section, we present the simulation experience results of the VVPE-KF and the VVPE-MF is used as a reference. We consider the variance of the measurement noise \(R\) and the variance of prediction noise \(Q\) of the Kalman filter as the variance of \(n_{k}\) in Equation (5) and the variance of \(f_{i}\) described as Equation (7) approximately, respectively. The variance \(\sigma_{f}^{2}\) in Equation (8) is assumed as \(3.14\times 10^{-6}\). The parameters of the Kalman estimator \(P(k/k-1)\), \(P(k)\) and \(K_{f}\) are all initialized to be 0.1 and the estimator can get convergence within 30 iterations. The SNR in this paper is defined as the average ratio of the received symbol power to the noise variance. Figure 3 illustrates the effectiveness in tracking the phase offset by VVPE-MF (\(N=6\)), VVPE-MF (\(N=12\)) and VVPE-KF with various SNR (10 and 15 dB). The atmosphere turbulence is ignored, and the channel gain is assumed as a constant. The tracking curves of VVPE-KF can approximate the real phase offset closely and the VVPE-MF curves fluctuated severely. To evaluate the performance of the estimation method further, we demonstrate the estimation error of VVPE-KF and VVPE-MF against SNR in Figure 4. This error is defined as:  
\[Error=\frac{1}{L}\sum_{k=1}^{L}\left|\theta_{k}{}^{\prime}-\theta_{k}\right| \tag{21}\]  
which is the absolute value of the mean difference between the estimated phase offset \(\theta_{k}{}^{\prime}\) and the real phase offset \(\theta_{k}\). \(L\) is set to be \(10^{4}\). It is observed that the VVPE-KF has definite advantage over VVPE-MF especially with low SNR. For example, when the SNR is 10, the error produced by VVPE-KF is one third of VVPE-MF (\(N=12\)) error and one fourth of VVPE-MF (\(N=6\)) error. Figure 2: Schematic of the method combining Viterbi-Viterbi phase offset estimation algorithm with the Kalman filter (VVPE-KF). In coherent space optical signal receiver, the received modulated optical signal through the atmosphere is affected by the atmosphere turbulence. To evaluate the performance of VVPE-KF with the atmosphere turbulence, we employ a group of experience data collected by the ground station of the quantum communication satellite located in Xinjiang of China. Figure 5a shows a group of normalized channel gain (optical irradiance) values and its histogram. Processing this group of data using the method described in Section 2.1, we can obtain the data shown in Figure 5b,c. With the scintillation index increasing, more information symbols with low SNR appears and the more precise phase offset estimation method is demanded for. Figure 4: The error produced by VVPE-KF and VVPE-MF without atmosphere turbulence. Figure 3: The performance of tracking the phase offset by VVPE-KF and VVPE-MF (Viterbi-Viterbi phase offset estimation algorithm with the mean filter) without atmosphere turbulence. The SNR values of (**a**), (**b**), (**c**) and (**d**) are 10, 10, 15, 15 dB respectively. Figure 6 presents the curves of the tracking phase offset by VVPE-KF and VVPE-MF with atmosphere turbulence. The same as Figure 3, the real phase offset is simulated by Equation (7). The received signal containing channel gain and noise described as Equation (5) is used to simulate the performance of the phase offset estimation methods. The fluctuated optical irradiance in Figure 4(b) is used as channel gain \(A_{k}\). Figure 5: The normalized optical intensity (channel gain) curves and histogram. The scintillation indexes of (**a**), (**b**), and (**c**) are 0.015, 0.154, and 0.561, respectively. Figure 6: The performance of tracking the phase offset by VVPE-KF and VVPE-MF with atmosphere turbulence. The SNR vales of (**a**), (**b**), (**c**), and (**d**) are 10, 10, 15, and 15 dB, respectively. Compared with the performance without turbulence shown in Figure 3, the error of the VVPE-MF is greater within the local range as a result of the fluctuated optical intensity and the performance of VVPE-KF keeps better obviously.