## 3 Simulation Setup and Results  
Extensive simulations were carried out in VPItransmissionMakerTM (VPIphotonics GmbH, Berlin, Germany) [16] to evaluate the performance of the proposed CPR scheme. The simulation setup corresponds to the transmission of a pseudorandom bit sequence (PRBS) with a sequence length of 2\({}^{15}\) - 1 bits mapped onto 2\({}^{17}\) symbols in a 28-Gbaud back-to-back transmission system. The outgoing signal after the transmitter is loaded with AWGN emulating erbium-doped fiber amplifier noise. Then, the signal is directly fed in the receiver and passed to a DSP-based demodulator where different CPR algorithms are applied for their relative performance evaluation. In order to mitigate the effect of cycle slips, the symbols are then differentially decoded [14], and the number of bit errors is counted for the bit error rate (BER) evaluation. Figure 2: Block diagram of the proposed two-stage n-PSK partitioning carrier phase recovery (CPR) scheme. (**a**) Input symbols. (**b**) Estimated constellation after first stage. (**c**) Detection of symbols with optimal decision boundaries. (**d**) Detection of symbols using sub-optimal decision boundaries (an alternative option to (**c**) for complexity reduction). (**e**) Final corrected symbols. The performance of the proposed algorithm is compared to the single-stage n-PSK partitioning algorithm, BPS algorithm in Sq-mQAM constellations (BPS\({}_{\text{Sq-mQAM}}\)), and BPS in C-mQAM constellations (BPS\({}_{\text{C-mQAM}}\)). The bit mapping employed for Sq-QAM constellations can be seen in [2], while the bit mapping for C-mQAM constellations is illustrated in Figure 1. The performance of all the algorithms is evaluated at BER target limits of \(1\times 10^{-2}\) and \(3.8\times 10^{-3}\) assuming the use of forward error correction (FEC). The number of test phases \(\beta\) in BPS\({}_{\text{Sq-16 QAM}}\) is set to 32 while \(\beta=64\) for BPS\({}_{\text{Sq-6 QAM}}\). \(\beta\) is set to 32 for both BPS\({}_{\text{C-16 QAM}}\) and BPS\({}_{\text{C-6 QAM}}\) due to a \(\pi/4\) rotational symmetry of C-64 QAM constellations. The block length of all the algorithms has been optimized to show the best performance for each of the points in the figures. Figure 3 shows the OSNR sensitivity penalties _versus_ the \(\Delta vT_{s}\) for C-16 QAM and Sq-16 QAM employing different CPR schemes at BER targets of \(1\times 10^{-2}\) (Figure 3a) and \(3.8\times 10^{-3}\) (Figure 3b). The proposed two-stage n-PSK partitioning improves the performance of the single-stage n-PSK partitioning and achieves a performance superior to the BPS\({}_{\text{C-16 QAM}}\) algorithm. The probability of wrongly classifying symbols during the ring selection process in the single-stage n-PSK algorithm increases for low OSNR values. This process is avoided in the proposed two-stage n-PSK partitioning, as symbol classification is performed using optimal decision boundaries resulting in an improved performance of the algorithm. The use of sub-optimal boundaries in Figure 2 (inset \(d\)) results in a similar performance and is proposed here to reduce the computational complexity of the algorithm as explained in Section 5. Figure 4: OSNR sensitivity penalty _versus_\(\Delta vT_{s}\) for C-64 QAM and Sq-64 QAM utilizing different CPR schemes at a bit error rate (BER) of \(1\times 10^{-2}\) (**a**) and \(3.8\times 10^{-3}\) (**b**). Figure 3: Optical signal-to-noise ratio (OSNR) sensitivity penalty _versus_\(\Delta vT_{s}\) for C-16 QAM and Sq-16 QAM utilizing different CPR schemes at a BER of \(1\times 10^{-2}\) (**a**) and \(3.8\times 10^{-3}\) (**b**). Figure 4 shows the OSNR sensitivity penalties _versus_ the \(\Delta vT_{s}\) for C-64 QAM and Sq-64 QAM employing different CPR schemes at BER targets of \(1\times 10^{-2}\) (Figure 4a) and \(3.8\times 10^{-3}\) (Figure 4b). As in the previous case, the proposed two-stage n-PSK partitioning CPR scheme outperforms the n-PSK partitioning algorithm and provides higher performance than the BPSC-64 QAM.