## 3 Results  
### Simulation Results  
Firstly, the parameter optimization of the frame structure is discussed. The block length is \(2^{10}\), and the total length is \(2^{15}\). The power ratio between the QPSK and 16 QAM is set as 6.5 dB, which can maximize the power efficiency [16]. The key parameter to be discussed in the following is the format ratio (FR). The FR between QPSK and 16 QAM is set to {1:1 1:3 1:7} and the corresponding system symbol rate is 32.6 Gbaud, 28 Gbaud, and 26.13 Gbaud to obtain the same effective bit rate of 98 Gbit/s. Figure 3 depicts the bit error rate (BER) performance of AKF with pilot symbols within the optical signal to noise ratio (OSNR) range of 17-22 dB. The result shows that compared with pure 16 QAM case, the OSNR sensitivity gain is around 1 dB at FR of 1:1. For FR = 1:3 and 1:7, the OSNR sensitivity gain is 0.5 dB and 0.35 dB, respectively. While FR = 1:1 has better performance, it will significantly decrease spectrum efficiency. Therefore, a higher device bandwidth is required. In the following, the systems with FR = 1:3 and 1:7 is only considered for acceptable spectrum efficiency. Figure 2: The schematic diagram of the proposed adaptive Kalman filter scheme. Taking the tracking performance into consideration in Figure 4a, the horizontal axis is the polarization rotation frequency drift rate, and the vertical axis is log10 (BER). The system target BER is 1 \(\times\) 10\({}^{-2}\). With the increasing polarization rotation frequency drift rate, the performance at FR = 1:3 is the best, and the maximum tracking polarization rotation frequency drift rate is 220 rad/us\({}^{2}\). The pure-16 QAM demonstrates the worst performance, and the maximum tracking polarization rotation frequency drift rate is 74 rad/us\({}^{2}\). With FR = 1:7, moderate performance is obtained and the allowable drift rate is 128 rad/us\({}^{2}\). When it comes to the phase noise, the same conclusion is obtained from Figure 4b. The tolerable laser linewidth when FR = 1:3 is 10.7 MHz. The pure-16 QAM system has the worst tolerance of 3.8 MHz. When FR = 1:7, the laser linewidth of 8.2 MHz can be tolerable. Now, we can conclude that with an increase in FR, the tracking performance is improved. In a practical situation, we need to get a good trade-off between the tracking ability and spectrum efficiency. In the case of desiring improvements on convergence and tracking with sacrificing spectrum resource at an acceptable level, the FR of 1:7 is preferred and adopted in the following discussion. The convergence speed is evaluated by mean square error (MSE) of the recovered signals, which is presented in Figure 5. The polarization crosstalk is set to be pi/4, and laser linewidth is 100 kHz. With FR = 1:7, AKF with pilot symbols can converge within 50 symbols, and AKF without pilot symbols need around 2500-symbols to be stable. Figure 4: **(a)** BER performance of FR = 1:3 and 1:7 compared with pure-16 QAM of dynamic polarization rotation and **(b)** laser linewidth. Figure 3: OSNR-BER curves of different FR. The dynamic polarization rotation tolerance is investigated in Figure 6. With the polarization rotation frequency drift rate increasing, pilot AKF and EKF have different performance dependence on Qpol. The EKF is strongly dependent on Qpol. With smaller Qpol values, higher estimation accuracy is achieved with the penalty of reduced estimation range. The proposed pilot AKF with different Qpol have the same estimation accuracy and range, which indicates that there is no need to adjust the initial Qpol value manually under different application scenarios. In this part, the laser linewidth tolerance under a different Q value is carried out. Figure 7 presents the BER performance as a function of laser linewidth for AKF and EKF with different initial Qphase values. The result is similar to the polarization tracking case. EKF with a smaller Qphase value shows higher estimation accuracy and degraded estimation ranges.