Using coherent optical detection and digital signal processing, laser phase noise and equalization enhanced phase noise can be effectively mitigated using the feed-forward and feed-back carrier phase recovery approaches. In this paper, theoretical analyses of feed-back and feed-forward carrier phase recovery methods have been carried out in the long-haul high-speed n-level phase shift keying (n-PSK) optical fiber communication systems, involving a one-tap normalized least-mean-square (LMS)

more »

... ithm, a block-wise average algorithm, and a Viterbi-Viterbi algorithm. The analytical expressions for evaluating the estimated carrier phase and for predicting the bit-error-rate (BER) performance (such as the BER floors) have been presented and discussed in the n-PSK coherent optical transmission systems by considering both the laser phase noise and the equalization enhanced phase noise. The results indicate that the Viterbi-Viterbi carrier phase recovery algorithm outperforms the one-tap normalized LMS and the block-wise average algorithms for small phase noise variance (or effective phase noise variance), while the one-tap normalized LMS algorithm shows a better performance than the other two algorithms for large phase noise variance (or effective phase noise variance). In addition, the one-tap normalized LMS algorithm is more sensitive to the level of modulation formats. Photonics 2016, 3, 51 2 of 18 channels and laser sources, such as chromatic dispersion (CD), polarization mode dispersion (PMD), laser phase noise (PN) and fiber nonlinearities (FNLs) [4] [5] [6] [7] [8] . Using coherent optical detection and digital signal processing, the powerful equalization and effective mitigation of the communication system impairments can be implemented in the electrical domain [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] , which has become one of the most promising techniques for the next-generation optical fiber communication networks to achieve a performance very close to the Shannon capacity limit [19, 20] , with an entire capture of the amplitude and phase of the optical signals. Using high-level modulation formats such as the n-level phase shift keying (n-PSK) and the n-level quadrature amplitude modulation (n-QAM), the performance of optical fiber transmission systems will be degraded seriously by the phase noise from the transmitter (Tx) lasers and the local oscillator (LO) lasers [21, 22] . To compensate the phase noise from the laser sources, some feed-forward and feed-back carrier phase recovery (CPR) approaches have been proposed to estimate and remove the phase of optical carriers [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] . Among these carrier phase estimation (CPE) methods, the one-tap normalized least-mean-square (LMS) algorithm, the block-wise average (BWA) algorithm, and the Viterbi-Viterbi (VV) algorithm have been validated for mitigating the laser phase noise effectively, and are also regarded as the most promising DSP algorithms in the real-time implementation of the high-speed coherent optical fiber transmission systems [27] [28] [29] [30] [31] [32] . Thus it will be of importance and interest to investigate the performance of these three carrier phase recovery algorithms in long-haul high-speed optical communication systems. In electronic dispersion compensation (EDC) based coherent optical fiber communication systems, an effect of equalization enhanced phase noise (EEPN) will be generated due to the interactions between the electronic dispersion equalization module and the laser phase noise (in the post-EDC case the EEPN comes from the LO laser) [33] [34] [35] [36] [37] [38] . The performance of long-haul optical fiber communication systems will be degraded seriously due to the equalization enhanced phase noise, with the increment of fiber dispersion, laser linewidths, modulation levels, symbol rates and system bandwidths [33] [34] [35] [36] . The impacts of EEPN have been investigated in the single-channel, the WDM multi-channel, the orthogonal frequency division multiplexing (OFDM), the dispersion pre-distorted, and the multi-mode optical fiber transmission systems [39] [40] [41] [42] [43] [44] [45] [46] . In addition, some investigations have been carried out to study the influence of EEPN in the carrier phase recovery in long-haul high-speed optical communication systems [47] [48] [49] [50] . Considering the equalization enhanced phase noise, the traditional analyses of the carrier phase recovery approaches are not suitable any longer for the design and the optimization of long-haul high-speed optical fiber networks. Therefore, it will also be interesting and useful to investigate the performance of the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi carrier phase recovery algorithms, when the influence of equalization enhanced phase noise is taken into account. In previous reports, the analytical derivations and numerical studies for the one-tap normalized LMS, block-wise average, and Viterbi-Viterbi carrier phase recovery methods have been carried out based on the quadrature phase shift keying (QPSK) coherent optical transmission system [26, 37, 51] . However, with the development of the optical fiber networks and the increment of transmission data capacity, the QPSK modulation format can no longer satisfy the demand for high-speed optical fiber communication systems. Therefore, the analyses on the carrier phase recovery approaches should also be updated accordingly for the optical fiber transmission systems using higher-level modulation formats, such as the n-PSK communication systems. In this paper, built on the previous work in [26, 37, 51] , the theoretical assessments of the carrier phase recovery using the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi algorithms are extended and analyzed in detail for the long-haul high speed n-PSK coherent optical fiber communication systems, considering both the intrinsic laser phase noise and the equalization enhanced phase noise. The analytical expressions for the estimated carrier phase in the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi algorithms has been derived, and the bit-error-rate (BER) performance such as the BER floors in these three carrier phase recovery approaches has been predicted for the n-PSK coherent optical transmission systems. Our results indicate that Photonics 2016, 3, 51 3 of 18 the Viterbi-Viterbi carrier phase recovery algorithm outperforms the one-tap normalized LMS and the block-wise average algorithms for small phase noise variance (or effective phase noise variance), while the one-tap normalized LMS carrier phase recovery algorithm shows a better performance than the other two algorithms for large phase noise variance (or effective phase noise variance). It is also found that the one-tap normalized LMS algorithm is more sensitive to the level of the modulation formats than the other two algorithms. Laser Phase Noise and Equalization Enhanced Phase Noise As shown in Figure 1 , the origin of equalization enhanced phase noise in the coherent optical communication systems using electronic dispersion compensation and carrier phase recovery is schematically illustrated. In such systems, the transmitter laser phase noise goes through both the transmission fiber and the EDC module, and therefore the net dispersion experienced by the transmitter laser phase noise is close to zero. However, the LO laser phase noise only goes through the EDC module, where the transfer function is heavily dispersed in the transmission system without using any optical dispersion compensation (ODC) techniques. As a result, the LO laser phase noise will interplay with the dispersion equalization module, and will significantly degrade the performance of the long-haul high-speed coherent optical fiber communication systems, with the increment of fiber dispersion, laser linewidths, modulation formats, and symbol rates [33, 34, 36] .