## 1 Introduction  
Since the first generation of optical fiber communication systems was deployed over 30 years ago, the achievable data rates carried by a single optical fiber have been raised over 10,000 times, and the data network traffic has also been increased by over a factor of 100 [1; 2]. To date, more than 90% of digital data is transmitted over optical fibers, to constitute the greater part of national and international telecommunication infrastructures. The effective information capacity of these networks has been widely increased over the past three decades with the introduction and development of wavelength division multiplexing (WDM), higher-level modulation formats, digital signal processing (DSP), advanced optical fibers and amplification technologies [3; 4]. These developments promoted the revolution of communication systems and the growth of the Internet, towards the direction of higher-speed and longer-distance transmissions [2; 3]. The performance of long-haul high-speed optical fiber communication systems can be significantly degraded by the impairments in the transmission 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].