Access the full text.

Sign up today, get DeepDyve free for 14 days.

Photonics
, Volume 3 (4) – Sep 24, 2016

/lp/multidisciplinary-digital-publishing-institute/analytical-investigations-on-carrier-phase-recovery-in-dispersion-8pdrqDkE0Y

- Publisher
- Multidisciplinary Digital Publishing Institute
- Copyright
- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
- ISSN
- 2304-6732
- DOI
- 10.3390/photonics3040051
- Publisher site
- See Article on Publisher Site

hv photonics Article Analytical Investigations on Carrier Phase Recovery in Dispersion-Unmanaged n-PSK Coherent Optical Communication Systems 1 , 2 , 3 , 4 4 3 2 2 Tianhua Xu *, Gunnar Jacobsen , Sergei Popov , Jie Li , Tiegen Liu , Yimo Zhang and Polina Bayvel Optical Networks Group, Department of Electronic & Electrical Engineering, University College London, London WC1E 7JE, UK; p.bayvel@ucl.ac.uk Department of Optical Engineering, Tianjin University, Tianjin 300072, China; tgliu@tju.edu.cn (T.L.); ymzhang@tju.edu.cn (Y.Z.) Acreo Swedish ICT AB, Stockholm SE-16425, Sweden; gunnar.jacobsen@acreo.se (G.J.); Jie.Li@acreo.se (J.L.) Royal Institute of Technology, Stockholm SE-16440, Sweden; sergeip@kth.se * Correspondence: tianhua.xu@ucl.ac.uk or xutianhua@tju.edu.cn; Tel.: +44-770-966-2611 Received: 22 August 2016; Accepted: 21 September 2016; Published: 24 September 2016 Abstract: 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) algorithm, 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. Keywords: coherent optical detection; optical ﬁber communication; carrier phase recovery; feed-back and feed-forward; laser phase noise; equalization enhanced phase noise; n-level phase shift keying 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 Photonics 2016, 3, 51; doi:10.3390/photonics3040051 www.mdpi.com/journal/photonics 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) [4–8]. 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 [9–18], 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 [23–32]. 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 [27–32]. 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) [33–38]. 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 [33–36]. 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 [39–46]. 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 [47–50]. 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 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), indicate that the Viterbi-Viterbi carrier phase recovery algorithm outperforms the one-tap while the one-tap normalized LMS carrier phase recovery algorithm shows a better performance than normalized LMS and the block-wise average algorithms for small phase noise variance (or effective the other two algorithms for large phase noise variance (or effective phase noise variance). It is also phase noise variance), while the one-tap normalized LMS carrier phase recovery algorithm shows a found that the one-tap normalized LMS algorithm is more sensitive to the level of the modulation better performance than the other two algorithms for large phase noise variance (or effective phase formats than the other two algorithms. 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. 2. Laser Phase Noise and Equalization Enhanced Phase Noise 2. Laser Phase Noise and Equalization Enhanced Phase Noise As shown in Figure 1, the origin of equalization enhanced phase noise in the coherent optical 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 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 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 transmission fiber and the EDC module, and therefore the net dispersion experienced by the laser phase noise is close to zero. However, the LO laser phase noise only goes through the EDC transmitter laser phase noise is close to zero. However, the LO laser phase noise only goes through module, where the transfer function is heavily dispersed in the transmission system without using the EDC module, where the transfer function is heavily dispersed in the transmission system any optical dispersion compensation (ODC) techniques. As a result, the LO laser phase noise will without using any optical dispersion compensation (ODC) techniques. As a result, the LO laser interplay with the dispersion equalization module, and will signiﬁcantly degrade the performance of phase noise will interplay with the dispersion equalization module, and will significantly degrade the long-haul high-speed coherent optical ﬁber communication systems, with the increment of ﬁber the performance of the long-haul high-speed coherent optical fiber communication systems, with the dispersion, incremen laser t of fi linewidths, ber dispersimodulation on, laser linewidth formats, s, mo and dulation symbol forma rates ts, a[ n 33 d s ,34 ymb ,36 ol rates ]. [33,34,36]. Figure 1. Principle of equalization enhanced phase noise in electronic dispersion compensation Figure 1. Principle of equalization enhanced phase noise in electronic dispersion compensation based based n-PSK coherent optical transmission system. PRBS: pseudo random bit sequence, N(t): n-PSK coherent optical transmission system. PRBS: pseudo random bit sequence, N(t): additive white additive white Gaussian noise (AWGN), e.g., amplified spontaneous emission (ASE) noise from Gaussian noise (AWGN), e.g., ampliﬁed spontaneous emission (ASE) noise from optical ampliﬁers, optical amplifiers, ADC: analog-to-digital convertor. ADC: analog-to-digital convertor. In coherent optical communication systems, the variance of the phase noise coming from the transmitter laser and the LO laser follows a Lorentzian distribution and can be described using the In coherent optical communication systems, the variance of the phase noise coming from the following equation [4,5]: transmitter laser and the LO laser follows a Lorentzian distribution and can be described using the following equation [4,5]: 2 2 f f T , (1) Laser Tx LO S = 2 (D f + D f ) T , (1) T x LO S Laser where ΔfTx and ΔfLO are the 3-dB linewidths (assuming the Lorentzian distribution) of the transmitter laser and the LO laser, respectively, and TS is the symbol period of the coherent transmission system. where Df and Df are the 3-dB linewidths (assuming the Lorentzian distribution) of the transmitter Tx LO It can be found that the variance of the laser phase noise decreases with the increment of the signal laser and the LO laser, respectively, and T is the symbol period of the coherent transmission system. symbol rate RS = 1/TS. It can be found that the variance of the laser phase noise decreases with the increment of the signal Considering the interplay between the electronic dispersion compensation module and the LO symbol rate R = 1/T . S S laser phase noise, the noise variance of the equalization enhanced phase noise in the long-haul Considering the interplay between the electronic dispersion compensation module and the LO high-speed optical fiber communication systems can be expressed as follows [33,37,40]: laser phase noise, the noise variance of the equalization enhanced phase noise in the long-haul high-speed optical ﬁber communication systems can be expressed as follows [33,37,40]: Photonics 2016, 3, 51 4 of 18 2 2 = D L D f /2cT , (2) LO S EEPN where f is the central frequency of the LO laser, which is equal to the central frequency of the LO transmitter laser f in the homodyne optical communication systems, D is the CD coefﬁcient of Tx the transmission ﬁber, L is the length of the transmission ﬁber, R is the signal symbol rate of the communication system, and l = c/f = c/f is the central wavelength of the optical carrier wave. Tx LO When the equalization enhanced phase noise is taken into account in the carrier phase recovery, the total noise variance (or effective phase noise variance) in the long-haul high-speed n-PSK optical ﬁber transmission systems can be calculated and described using the following expression [37,40]: 2 2 2 2 + = 2T (D f + D f ) + D L D f /2cT (3) S T x LO LO S T Laser EEPN The equalization enhanced phase noise differs from the laser phase noise, and the noise variance in Equation (2) has two-thirds contribution in the phase noise and one-third contribution in the amplitude noise [33,37,40]. Therefore, Equation (3) is only valid for n-PSK communication systems, and the performance of n-QAM transmission systems has to be assessed based on the evaluation of error vector magnitudes [52]. Corresponding to the deﬁnition of the laser phase noise from the transmitter and the LO lasers, an effective linewidth Df can be employed to describe the total phase noise variance in the EDC-based Eff n-PSK coherent optical communication systems and it can be expressed as follows: 2 2 D f + /2T . (4) E f f Laser EEPN When the impact from laser phase noise and the inﬂuence from EEPN give an equal contribution in 2 2 the n-PSK optical ﬁber communication systems, namely = , we will have the transmission Laser EEPN 2 2 distance of L = 8cT / D. Take the 32-Gbaud n-PSK coherent optical ﬁber transmission system as an example, and we assume that the ﬁber CD coefﬁcient is 17 ps/nm/km and the central wavelengths of the transmitter and the LO lasers are both 1550 nm. In this case, we have L = 60.69 km. It means that at this transmission distance, the laser phase noise and the EEPN will have the same impact on the degradation of the performance of the 32-Gbaud n-PSK optical transmission systems. 3. Analysis of Carrier Phase Recovery Approaches 3.1. One-Tap Normalized Least-Mean-Square (LMS) Carrier Phase Recovery As a feed-back carrier phase recovery approach [27,28], which is schematically shown in Figure 2, the transfer function of the one-tap normalized LMS algorithm in the n-PSK coherent optical communication systems can be expressed using the following equations: y (k) = w (k) x (k), (5) N L MS w (k + 1) = w (k) + e (k) x (k)/jx (k)j , (6) N L MS N L MS e (k) = d (k) y (k), (7) where x(k) is the complex input symbol, k is the index of the symbol, y(k) is the complex output symbol, w (k) is the tap weight of the one-tap normalized LMS equalizer, d(k) is the desired output symbol NLMS after the carrier phase recovery, e(k) is the estimation error between the output symbol and the desired output symbol, and m is the step size of the one-tap normalized LMS algorithm. It has been veriﬁed that the optimized one-tap normalized LMS carrier phase recovery in the QPSK optical transmission systems behaves similarly to the ideal differential carrier phase recovery [24,37], Photonics 2016, 3, 51 5 of 18 It has been verified that the optimized one-tap normalized LMS carrier phase recovery in the QPSK optical transmission systems behaves similarly to the ideal differential carrier phase recovery Photonics 2016, 3, 51 5 of 18 [24,37], and the BER floor of the one-tap normalized LMS carrier phase recovery in the QPSK transmission systems can be approximately described as follows [37]: and the BER ﬂoor of the one-tap normalized LMS carrier phase recovery in the QPSK transmission 1 NLMS _ QPSK systems can be approximately described as follows [37]: BER erfc . (8) floor 42 T N L MS_QPSK BER er f c p . (8) f loor Therefore, the BER floor of the one-tap norm2 alized LMS carrier phase recovery for the n-PSK 4 2 optical fiber communication systems can be derived accordingly, and can be expressed using the Therefore, the BER ﬂoor of the one-tap normalized LMS carrier phase recovery for the n-PSK following equation: optical ﬁber communication systems can be derived accordingly, and can be expressed using the following equation: 1 NLMS BER erfc , (9) floor log n N L MS n 2 T BER er f c p , (9) f loor log n n 2 2 T where is the total phase noise variance (or effective phase noise variance) in the long-haul where is the total phase noise variance (or effective phase noise variance) in the long-haul high-speed high-speed n-PSK optical transmission systems. n-PSK optical transmission systems. Figure 2. Schematic of one-tap normalized LMS carrier phase recovery algorithm. Figure 2. Schematic of one-tap normalized LMS carrier phase recovery algorithm. 3.2. Block-Wise Average Carrier Phase Recovery 3.2. Block-Wise Average Carrier Phase Recovery As an n-th power feed-forward carrier phase recovery approach, which is schematically shown As an n-th power feed-forward carrier phase recovery approach, which is schematically shown in in Figure 3, the block-wise average algorithm calculates the n-th power of the received symbols to Figure 3, the block-wise average algorithm calculates the n-th power of the received symbols to remove remove the information of the modulated phase in the n-PSK coherent transmission systems, and the the information of the modulated phase in the n-PSK coherent transmission systems, and the computed computed phase (n-th power) data are summed and averaged over a certain block. The averaged phase (n-th power) data are summed and averaged over a certain block. The averaged phase value is phase value is then divided over n, and the final result is regarded as the estimated phase for the then divided over n, and the ﬁnal result is regarded as the estimated phase for the received symbols received symbols within the entire block [29,30]. For the n-PSK coherent optical communication within the entire block [29,30]. For the n-PSK coherent optical communication systems, the estimated systems, the estimated carrier phase for each process block using the block-wise average algorithm carrier phase for each process block using the block-wise average algorithm can be expressed as: can be expressed as: 8 9 qN < = BWA qN BWA ' (k) = ar 1g x ( p) , (10) BWA å n : ; k arg x p , (10) BWA p=1+(q1)N BWA p11 q N BWA q = dk/N e, (11) BWA (11) q k N , BWA where k is the index of the received symbol, N is the block length in the block-wise average BWA algorithm, and dxe means the closest integer lager than x. where k is the index of the received symbol, NBWA is the block length in the block-wise average The BER ﬂoor of the block-wise average carrier phase recovery in the n-PSK coherent optical algorithm, and means the closest integer lager than x. communication systems can be derived using the Taylor series expansion, and can be approximately The BER floor of the block-wise average carrier phase recovery in the n-PSK coherent optical described using the following equation: communication systems can be derived using the Taylor series expansion, and can be approximately described using the following equation: BWA BWA BER er f c , (12) f loor N log n BWA n 2 ( p) 2 p=1 BWA h i 3 2 3 2 2 ( p 1) + 3 ( p 1) + 2 (N p) + 3 (N p) + N 1 BWA BWA BWA ( p) = , (13) BWA 6N BWA Photonics 2016, 3, 51 6 of 18 Photonics 2016, 3, 51 6 of 18 N BWA BWA 1 BWA 1 BWA BER erfc , (12) BER erfc , (12) floor floor Nn log p1 Nn log np 2 p1 np 2 BWA 2 BWA 2 BWA BWA 3 2 3 2 3 2 3 2 2 2 p 1 3 p 1 2 N p 3 N p N 1 2 p 1 3 p 1 2 N p 3 N p N 1 T BWA BWA BWA T BWA BWA BWA (13) 2 (13) 2 p , , p BWA 2 BWA Photonics 2016, 3, 51 6N 6 of 18 6N BWA BWA where is the total phase noise variance (or effective phase noise variance) in the long-haul where is the total phase noise variance (or effective phase noise variance) in the long-haul 2 T where is the total phase noise variance (or effective phase noise variance) in the long-haul high-speed high-speed n-PSK optical transmission systems. high-speed n-PSK optical transmission systems. n-PSK optical transmission systems. Figure 3. Schematic of block-wise average carrier phase recovery algorithm. Figure 3. Schematic of block-wise average carrier phase recovery algorithm. Figure 3. Schematic of block-wise average carrier phase recovery algorithm. 3.3. Viterbi-Viterbi Carrier Phase Recovery 3.3. Viterbi-Viterbi Carrier Phase Recovery 3.3. Viterbi-Viterbi Carrier Phase Recovery As another n-th power feed-forward carrier phase recovery approach, which is schematically As another n-th power feed-forward carrier phase recovery approach, which is schematically As another n-th power feed-forward carrier phase recovery approach, which is schematically shown in Figure 4, the Viterbi-Viterbi algorithm also calculates the n-th power of the received shown in Figure 4, the Viterbi-Viterbi algorithm also calculates the n-th power of the received shown in Figure 4, the Viterbi-Viterbi algorithm also calculates the n-th power of the received symbols symbols to remove the information of the modulated phase. The computed phase data are also symbols to remove the information of the modulated phase. The computed phase data are also to remove the information of the modulated phase. The computed phase data are also summed summed and averaged over the processing block (with a certain block length). Compared to the summed and averaged over the processing block (with a certain block length). Compared to the and averaged over the processing block (with a certain block length). Compared to the block-wise block-wise average algorithm, the Viterbi-Viterbi algorithm just treats the extracted phase as the block-wise average algorithm, the Viterbi-Viterbi algorithm just treats the extracted phase as the average algorithm, the Viterbi-Viterbi algorithm just treats the extracted phase as the estimated estimated phase for the central symbol in each processing block [31,32]. The estimated carrier phase estimated phase for the central symbol in each processing block [31,32]. The estimated carrier phase phase for the central symbol in each processing block [31,32]. The estimated carrier phase in the in the Viterbi-Viterbi algorithm in the n-PSK coherent optical transmission systems can be described in the Viterbi-Viterbi algorithm in the n-PSK coherent optical transmission systems can be described Viterbi-Viterbi algorithm in the n-PSK coherent optical transmission systems can be described using using the following equation: using the following equation: the following equation: 8 9 N 12 N VV12 VV 1 (N 1)/2 1 < n = VV k arg x k q , (14) k arg x nk q , (14) VV ' VV (k) = arg x (k + q) , (14) VV å n qN 12 n : qN VV12 ; VV q=(N 1)/2 VV where NVV is the block length of the Viterbi-Viterbi algorithm, and should be an odd value of e.g., 1, where NVV is the block length of the Viterbi-Viterbi algorithm, and should be an odd value of e.g., 1, where N is the block length of the Viterbi-Viterbi algorithm, and should be an odd value of e.g., 1, 3, VV 3, 5, 7… 3, 5, 7… 5, 7. . . Figure 4. Schematic of Viterbi-Viterbi carrier phase recovery algorithm. Figure 4. Schematic of Viterbi-Viterbi carrier phase recovery algorithm. Figure 4. Schematic of Viterbi-Viterbi carrier phase recovery algorithm. Using the Taylor expansion, the BER floor of the Viterbi-Viterbi carrier phase recovery in the n-PSK Using the Taylor expansion, the BER floor of the Viterbi-Viterbi carrier phase recovery in the n-PSK Using the Taylor expansion, the BER ﬂoor of the Viterbi-Viterbi carrier phase recovery in the coherent optical communication systems can be assessed analytically, and can be expressed coherent optical communication systems can be assessed analytically, and can be expressed n-PSK coherent optical communication systems can be assessed analytically, and can be expressed approximately using the following equation: approximately using the following equation: approximately using the following equation: 0 1 B C VV B C BER er f c r , (15) f loor @ A log n 2 2 N 1 VV 6N VV where is the variance of the total phase noise (or effective phase noise) in the long-haul high-speed n-PSK optical ﬁber transmission systems. Photonics 2016, 3, 51 7 of 18 1 VV BER erfc , (15) floor log n N 1 VV n 6N VV where is the variance of the total phase noise (or effective phase noise) in the long-haul Photonics 2016, 3, 51 7 of 18 high-speed n-PSK optical fiber transmission systems. 4. Results and Discussion 4. Results and Discussion 4.1. Results 4.1. Results I In n this this section, section, the the performance performance o of f ca carrier rrier ph phase ase re recovery covery in in the the long-haul long-haul high-speed high-speed n n- -PSK PSK coherent optical transmission systems using the one-tap normalized LMS, the block-wise average, coherent optical transmission systems using the one-tap normalized LMS, the block-wise average, and and th the e V Viterbi-V iterbi-Viterbi iterbi algorithms algorithms ar are e investigated investigated based based on on the the ab above ove th theor eoret etical ical st studies, udies, w which hich c can an be regarded as extensions of the QPSK results in previously reported work [26,37,51]. The be regarded as extensions of the QPSK results in previously reported work [26,37,51]. The comparisons com of the pari thr son ee carrier s of the phase three recovery carrier ph appr ase oaches recovery have appr also oaches been carried have also out been in detail. carried In all out these in de analyses tail. In all these analyses and discussions, the standard single mode fiber (SSMF) has been employed in the and discussions, the standard single mode ﬁber (SSMF) has been employed in the n-PSK coherent n optical -PSK coh transmission erent optical systems, transmission wheresthe ystem ﬁber s, where dispersion the fiis ber 17 di ps/nm/km, spersion is 17 the ps/ central nm/km wavelengths , the central wavelengths of both the transmitter laser and the LO laser are 1550 nm, and the signal symbol rate is of both the transmitter laser and the LO laser are 1550 nm, and the signal symbol rate is 32-Gbaud. The 32-Gﬁber baud. attenuation, The fiber atten the u PMD, ation, and the PM the nonlinear D, and the no effects nline arar e all effects are neglected. all neglected. Based on Equation (9), the performance of the one-tap normalized LMS carrier phase recovery Based on Equation (9), the performance of the one-tap normalized LMS carrier phase recovery in the in the cohere coherent nt opt optical ical fi ﬁber ber com communication munication s systems ystems usi using ng di dif ff fer ere ent nt modu modulation lation form formats ats is is s shown hown i in n Figure 5. The optimization of the one-tap normalized LMS carrier phase recovery has been Figure 5. The optimization of the one-tap normalized LMS carrier phase recovery has been investigated investig and discussed ated and in detail discuin sse Refer d in ence detai[l27 in, 37 Ref ], erence where the [27,37], step size whevarying re the step from size 0.01 varying to 1 has from been0.01 applied. to 1 has been applied. A smaller step size will degrade the BER floor due to the fast phase changing A smaller step size will degrade the BER ﬂoor due to the fast phase changing occurring in the long occurrin effectiveg symbol in the average-span, long effective while symbo a llar aver gerag step e-sp size an, will whidegrade le a larger thestep signal-to-noise size will degr ratio ad (SNR) e the signal-to-noise ratio (SNR) sensitivity in the one-tap normalized LMS carrier phase recovery. The sensitivity in the one-tap normalized LMS carrier phase recovery. The optimal step sizes for different op effective timal step linewidths sizes for have dibeen fferent studied, effectiv with e linewi a resolution dths have of 0.005 been us sted udi in ed, the woptimization ith a resolution process of 0.0 [37 05 ]. used in the optimization process [37]. In this section we assume that all the operations of the one-tap In this section we assume that all the operations of the one-tap normalized LMS algorithm have norm been optimized. alized LMS Italgorith can be found m have in been Figur e op 5timize that the d. one-tap It can be normalized found in LMS Figure carrier 5 thphase at the recovery one-tap normalized LMS carrier phase recovery algorithm is very sensitive to the phase noise variance and algorithm is very sensitive to the phase noise variance and the modulation formats, especially when phase the modul noise ati variance on formats, is less especi than ally 0.1. when phase noise variance is less than 0.1. Figure Figure 5. 5. Bit-err Bit-err or or -rate -rate (BER) (BERﬂoors ) floors versus versus phase phase noise noi variance se variance in the in one-tap the one normalized -tap normalized LMS carrier LMS phase recovery in the coherent optical transmission systems using different modulation formats. carrier phase recovery in the coherent optical transmission systems using different modulation formats. In the case of back-to-back (BtB) or without considering EEPN, the performance of BER ﬂoors versus laser linewidths in the one-tap normalized LMS carrier phase estimation has been studied in Figure 6 based on the analysis in Equations (1) and (9). The indicated linewidth value in Figure 6 is the 3-dB linewidth for both the transmitter laser and the LO laser. It can be seen that the BER ﬂoors of the one-tap normalized LMS carrier phase recovery are deteriorated with the increment of the modulation format levels and the laser linewidths. The degradations due to the laser phase noise (laser linewidths) are more drastic for higher-level modulation formats. Considering the impact of EEPN, the effective phase noise variance will increase with the increment of the transmission distance and the laser linewidth. As shown in Figure 7, the BER ﬂoors Photonics 2016, 3, 51 8 of 18 Photonics 2016, 3, 51 8 of 18 In the case of back-to-back (BtB) or without considering EEPN, the performance of BER floors In the case of back-to-back (BtB) or without considering EEPN, the performance of BER floors versus laser linewidths in the one-tap normalized LMS carrier phase estimation has been studied in versus laser linewidths in the one-tap normalized LMS carrier phase estimation has been studied in Figure 6 based on the analysis in Equations (1) and (9). The indicated linewidth value in Figure 6 is Figure 6 based on the analysis in Equations (1) and (9). The indicated linewidth value in Figure 6 is the 3-dB linewidth for both the transmitter laser and the LO laser. It can be seen that the BER floors the 3-dB linewidth for both the transmitter laser and the LO laser. It can be seen that the BER floors of the one-tap normalized LMS carrier phase recovery are deteriorated with the increment of the of the one-tap normalized LMS carrier phase recovery are deteriorated with the increment of the modulation format levels and the laser linewidths. The degradations due to the laser phase noise modulation format levels and the laser linewidths. The degradations due to the laser phase noise (laser linewidths) are more drastic for higher-level modulation formats. (laser linewidths) are more drastic for higher-level modulation formats. Photonics 2016, 3, 51 8 of 18 Considering the impact of EEPN, the effective phase noise variance will increase with the Considering the impact of EEPN, the effective phase noise variance will increase with the increment of the transmission distance and the laser linewidth. As shown in Figure 7, the BER floors increment of the transmission distance and the laser linewidth. As shown in Figure 7, the BER floors of of the the one-tap one-tap normalized normalized LMSLM carrier S carrier phase rph ecovery ase recov has been ery h investigated as been inve forstiga differ ted ent fo transmission r different of the one-tap normalized LMS carrier phase recovery has been investigated for different transmission distances, when the linewidths of both the Tx and the LO lasers are set to 1 MHz. It can distances, when the linewidths of both the Tx and the LO lasers are set to 1 MHz. It can be found that transmission distances, when the linewidths of both the Tx and the LO lasers are set to 1 MHz. It can the be fou performance nd that the ofperfo the one-tap rmance normalized of the one-LMS tap norm carrier alize phase d LM recovery S carrier is ph degraded ase recov signiﬁcantly ery is degra with ded be found that the performance of the one-tap normalized LMS carrier phase recovery is degraded significantly with the increment of transmission distances, and this effect is more severe for the increment of transmission distances, and this effect is more severe for higher-level modulation significantly with the increment of transmission distances, and this effect is more severe for formats higher-leve duel m to odul less tolerance ation form to ats laser due phase to less noise tolera and nceEEPN. to laser phase noise and EEPN. higher-level modulation formats due to less tolerance to laser phase noise and EEPN. Figure 6. BER floors versus laser linewidths in the one-tap normalized LMS carrier phase recovery in Figure 6. BER floors versus laser linewidths in the one-tap normalized LMS carrier phase recovery in Figure 6. BER ﬂoors versus laser linewidths in the one-tap normalized LMS carrier phase recovery in the optical fiber transmission systems using different modulation formats. The indicated linewidth the optical fiber transmission systems using different modulation formats. The indicated linewidth the optical ﬁber transmission systems using different modulation formats. The indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. value is the 3-dB linewidth for both the Tx and the LO lasers. value is the 3-dB linewidth for both the Tx and the LO lasers. Figure 7. BER floors versus transmission distances in the one-tap normalized LMS carrier phase Figure 7. BER floors versus transmission distances in the one-tap normalized LMS carrier phase Figure 7. BER ﬂoors versus transmission distances in the one-tap normalized LMS carrier phase recovery in the coherent optical transmission systems using different modulation formats, recovery in the coherent optical transmission systems using different modulation formats, recovery in the coherent optical transmission systems using different modulation formats, considering considering the equalization enhanced phase noise. Both the Tx and LO lasers linewidths are 1 MHz. considering the equalization enhanced phase noise. Both the Tx and LO lasers linewidths are 1 MHz. the equalization enhanced phase noise. Both the Tx and LO lasers linewidths are 1 MHz. The performance of the block-wise average carrier phase recovery approach has also been The performance of the block-wise average carrier phase recovery approach has also been The performance of the block-wise average carrier phase recovery approach has also been investigated, as shown in Figures 8–10. The BER floors versus different phase noise variances (or investigated, as shown in Figures 8–10. The BER floors versus different phase noise variances (or investigated, as shown in Figures 8–10. The BER ﬂoors versus different phase noise variances effective phase noise variance) in the block-wise average carrier phase recovery is described in Figure 8. effective phase noise variance) in the block-wise average carrier phase recovery is described in Figure 8. (or effective phase noise variance) in the block-wise average carrier phase recovery is described in Figure 8. In Figure 8a, the performance of the block-wise average algorithm is studied in terms of different block lengths in the 8-PSK optical transmission system. It can be found that the phase noise (or effective phase noise) induced BER ﬂoor in the block-wise average carrier phase recovery algorithm is increased with the increment of block length. Generally, a smaller block length will lead to a lower phase noise induced BER ﬂoor due to a more accurate estimation of carrier phase, while a larger block length is more effective for mitigating the amplitude noise (such as the ampliﬁed spontaneous emission—ASE—noise) to improve the SNR sensitivity. In practical transmission systems, the optimal Photonics 2016, 3, 51 9 of 18 In Figure 8a, the performance of the block-wise average algorithm is studied in terms of different block lengths in the 8-PSK optical transmission system. It can be found that the phase noise (or effective phase noise) induced BER floor in the block-wise average carrier phase recovery algorithm is increased with the increment of block length. Generally, a smaller block length will lead to a lower phase noise induced BER floor due to a more accurate estimation of carrier phase, while a Photonics 2016, 3, 51 9 of 18 larger block length is more effective for mitigating the amplitude noise (such as the amplified spontaneous emission—ASE—noise) to improve the SNR sensitivity. In practical transmission systems, the optimal block length is determined by considering the trade-off between the phase block length is determined by considering the trade-off between the phase noise and the amplitude noise and the amplitude noise. As an example, the block length of NBWA = 11 is employed in all noise. As an example, the block length of N = 11 is employed in all subsequent analyses, if the BWA subsequent analyses, if the value is not specified. Based on Equations (12) and (13), the performance value is not speciﬁed. Based on Equations (12) and (13), the performance of the block-wise average of the block-wise average carrier phase recovery in the coherent optical communication systems carrier phase recovery in the coherent optical communication systems using different modulation using different modulation formats is shown in Figure 8b, where the block length is 11. It can be formats is shown in Figure 8b, where the block length is 11. It can be found in Figure 8b that the found in Figure 8b that the block-wise average carrier phase recovery algorithm is also very sensitive block-wise average carrier phase recovery algorithm is also very sensitive to the phase noise variance to the phase noise variance and the modulation formats, when phase noise variance is less than 0.1. and the modulation formats, when phase noise variance is less than 0.1. Based on the analyses in Equations (1), (12) and (13), the BER floors versus laser linewidths in Based on the analyses in Equations (1), (12) and (13), the BER ﬂoors versus laser linewidths in the block-wise average carrier phase recovery for the back-to-back case or without considering the block-wise average carrier phase recovery for the back-to-back case or without considering EEPN EEPN has been studied in Figure 9, where the indicated linewidth value is the 3-dB linewidth for has been studied in Figure 9, where the indicated linewidth value is the 3-dB linewidth for both the both the transmitter laser and the LO laser. It can be found that the BER floors in the block-wise transmitter laser and the LO laser. It can be found that the BER ﬂoors in the block-wise average carrier average carrier phase recovery are also degraded with the increment of modulation format levels phase recovery are also degraded with the increment of modulation format levels and laser linewidths. and laser linewidths. The degradations due to the laser phase noise (laser linewidths) are also more The degradations due to the laser phase noise (laser linewidths) are also more severe for higher-level severe for higher-level modulation formats. modulation formats. (a) (b) Figure 8. BER floors versus phase noise variances in the block-wise average carrier phase recovery in Figure 8. BER ﬂoors versus phase noise variances in the block-wise average carrier phase recovery in the coherent optical transmission systems. (a) Different block lengths in the 8-PSK transmission the coherent optical transmission systems. (a) Different block lengths in the 8-PSK transmission system; system; (b) different modulation formats with the block length of 11. (b) different modulation formats with the block length of 11. Photonics 2016, 3, 51 10 of 18 Photonics 2016, 3, 51 10 of 18 Photonics 2016, 3, 51 10 of 18 Figure 9. BER floors versus laser linewidths in the block-wise average carrier phase recovery in the Figure 9. BER floors versus laser linewidths in the block-wise average carrier phase recovery in the Figure 9. BER ﬂoors versus laser linewidths in the block-wise average carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, coher coheren ent t optical optical transmission transmission syste systems ms using using dif different ferent modulation modulation formats. formats. The The b block lock llength ength is is 11 11, , and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. Figure 10. BER floors versus transmission distances in the block-wise average carrier phase recovery Figure 10. BER floors versus transmission distances in the block-wise average carrier phase recovery Figure 10. BER ﬂoors versus transmission distances in the block-wise average carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is in the coherent optical transmission systems using different modulation formats. The block length is in the coherent optical transmission systems using different modulation formats. The block length is 11, and the linewidth of both the Tx and the LO lasers are 1 MHz. 11, and the linewidth of both the Tx and the LO lasers are 1 MHz. 11, and the linewidth of both the Tx and the LO lasers are 1 MHz. As shown in Figure 10, the BER floors of the block-wise average carrier phase recovery have As shown in Figure 10, the BER floors of the block-wise average carrier phase recovery have As shown in Figure 10, the BER ﬂoors of the block-wise average carrier phase recovery have been investigated for different transmission distances by considering the impact of EEPN, where the been investigated for different transmission distances by considering the impact of EEPN, where the been investigated for different transmission distances by considering the impact of EEPN, where the linewidths of both the Tx and the LO lasers are set to 1 MHz. It can be found that the performance of linewidths of both the Tx and the LO lasers are set to 1 MHz. It can be found that the performance of linewidths of both the Tx and the LO lasers are set to 1 MHz. It can be found that the performance of the the block-wise average carrier phase recovery is degraded significantly with the increment of the block-wise average carrier phase recovery is degraded significantly with the increment of block-wise average carrier phase recovery is degraded signiﬁcantly with the increment of transmission transmission distances, and this effect is more severe for higher-level modulation formats, since the transmission distances, and this effect is more severe for higher-level modulation formats, since the distances, and this effect is more severe for higher-level modulation formats, since the EEPN inﬂuence EEPN influence will scale with the increment of transmission distances and modulation formats. EEPN influence will scale with the increment of transmission distances and modulation formats. will scale with the increment of transmission distances and modulation formats. From Figures 11–13, the performance of the Viterbi-Viterbi carrier phase recovery approach has From Figures 11–13, the performance of the Viterbi-Viterbi carrier phase recovery approach has From Figures 11–13, the performance of the Viterbi-Viterbi carrier phase recovery approach has been investigated in terms of the phase noise variances, the laser linewidths and the transmission been investigated in terms of the phase noise variances, the laser linewidths and the transmission been investigated in terms of the phase noise variances, the laser linewidths and the transmission distances. The BER floors versus the phase noise variances in the Viterbi-Viterbi carrier phase distances. The BER floors versus the phase noise variances in the Viterbi-Viterbi carrier phase distances. The BER ﬂoors versus the phase noise variances in the Viterbi-Viterbi carrier phase recovery recovery algorithm is studied in Figure 11. In Figure 11a, the BER floors are also studied in terms of recovery algorithm is studied in Figure 11. In Figure 11a, the BER floors are also studied in terms of algorithm is studied in Figure 11. In Figure 11a, the BER ﬂoors are also studied in terms of different different block lengths in the Viterbi-Viterbi CPR algorithm for the 8-PSK optical transmission different block lengths in the Viterbi-Viterbi CPR algorithm for the 8-PSK optical transmission block lengths in the Viterbi-Viterbi CPR algorithm for the 8-PSK optical transmission system. Similar system. Similar to the block-wise average algorithm, a smaller block length in the Viterbi-Viterbi system. Similar to the block-wise average algorithm, a smaller block length in the Viterbi-Viterbi to the block-wise average algorithm, a smaller block length in the Viterbi-Viterbi carrier phase recovery carrier phase recovery will generate a lower phase noise induced BER floor; in contrast, a larger carrier phase recovery will generate a lower phase noise induced BER floor; in contrast, a larger will generate a lower phase noise induced BER ﬂoor; in contrast, a larger block length is more tolerant block length is more tolerant to the amplitude noise and will lead to better SNR sensitivity. The block length is more tolerant to the amplitude noise and will lead to better SNR sensitivity. The to the amplitude noise and will lead to better SNR sensitivity. The optimal block length is again optimal block length is again determined by considering the trade-off between phase noise and optimal block length is again determined by considering the trade-off between phase noise and determined by considering the trade-off between phase noise and amplitude noise. It can be found amplitude noise. It can be found that the phase noise induced BER floors in the Viterbi-Viterbi amplitude noise. It can be found that the phase noise induced BER floors in the Viterbi-Viterbi carrier phase recovery algorithm are also deteriorated with the increment of block lengths. Similar to carrier phase recovery algorithm are also deteriorated with the increment of block lengths. Similar to Photonics 2016, 3, 51 11 of 18 that Photonics the 2016 phase , 3, 5noise 1 induced BER ﬂoors in the Viterbi-Viterbi carrier phase recovery algorithm 11 of ar 18 e also deteriorated with the increment of block lengths. Similar to the block-wise average algorithm, the block-wise average algorithm, the block length of NVV = 11 is also selected as an example in the the block length of N = 11 is also selected as an example in the Viterbi-Viterbi carrier phase recovery VV Viterbi-Viterbi carrier phase recovery to consider the mitigation of both the phase noise and the to consider the mitigation of both the phase noise and the amplitude noise in practical applications. amplitude noise in practical applications. Based on Equation (15), the performance of the Based on Equation (15), the performance of the Viterbi-Viterbi carrier phase recovery in the coherent Viterbi-Viterbi carrier phase recovery in the coherent optical communication systems using different optical communication systems using different modulation formats is shown in Figure 11b, where the modulation formats is shown in Figure 11b, where the block length is 11. It is found in Figure 11b block length is 11. It is found in Figure 11b that the Viterbi-Viterbi carrier phase recovery algorithm is that the Viterbi-Viterbi carrier phase recovery algorithm is also very sensitive to the phase noise also very sensitive to the phase noise variance and the modulation formats, when phase noise variance variance and the modulation formats, when phase noise variance is less than 0.15. is less than 0.15. (a) (b) Figure 11. BER floors versus phase noise variances in the Viterbi-Viterbi carrier phase recovery in the Figure 11. BER ﬂoors versus phase noise variances in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems. (a) Different block lengths in the 8-PSK transmission system; coherent optical transmission systems. (a) Different block lengths in the 8-PSK transmission system; (b) different modulation formats with the block length of 11. (b) different modulation formats with the block length of 11. Without considering EEPN (or for the back-to-back case), the BER floors versus laser linewidths Without considering EEPN (or for the back-to-back case), the BER ﬂoors versus laser linewidths in the Viterbi-Viterbi carrier phase recovery have been studied in Figure 12 based on the analyses in in the Viterbi-Viterbi carrier phase recovery have been studied in Figure 12 based on the analyses in Equations (1) and (15), where the indicated linewidth value is again the 3-dB linewidth for both the Equations (1) and (15), where the indicated linewidth value is again the 3-dB linewidth for both the transmitter laser and the LO laser. It can be found that the BER floors are also degraded significantly transmitter laser and the LO laser. It can be found that the BER ﬂoors are also degraded signiﬁcantly with the increment of laser linewidths, and this degradation is also more severe with the increment with the increment of laser linewidths, and this degradation is also more severe with the increment of of modulation format levels. modulation format levels. As shown in Figure 13, the BER floors of the Viterbi-Viterbi carrier phase recovery have also As shown in Figure 13, the BER ﬂoors of the Viterbi-Viterbi carrier phase recovery have also been investigated for different transmission distances considering the impact of EEPN, where the been investigated for different transmission distances considering the impact of EEPN, where the linewidths of the Tx and the LO lasers are both set to 1 MHz. It can be found that in the linewidths of the Tx and the LO lasers are both set to 1 MHz. It can be found that in the Viterbi-Viterbi Viterbi-Viterbi carrier phase recovery, the EEPN influence will also increase with the increment of carrier phase recovery, the EEPN inﬂuence will also increase with the increment of transmission transmission distances and modulation formats. The performance of the Viterbi-Viterbi algorithm is degraded significantly with the increment of transmission distances, and this effect is more serious for higher-level modulation formats. Photonics 2016, 3, 51 12 of 18 distances and modulation formats. The performance of the Viterbi-Viterbi algorithm is degraded signiﬁcantly with the increment of transmission distances, and this effect is more serious for higher-level modulation formats. Photonics 2016, 3, 51 12 of 18 Photonics 2016, 3, 51 12 of 18 Figure 12. BER floors versus laser linewidths in the Viterbi-Viterbi carrier phase recovery in the Figure 12. BER ﬂoors versus laser linewidths in the Viterbi-Viterbi carrier phase recovery in the Figure 12. BER floors versus laser linewidths in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, coherent optical transmission systems using different modulation formats. The block length is 11, coherent optical transmission systems using different modulation formats. The block length is 11, and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. Figure 13. BER floors versus transmission distances in the Viterbi-Viterbi carrier phase recovery in Figure 13. BER floors versus transmission distances in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, Figure 13. BER ﬂoors versus transmission distances in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the linewidth of both the Tx and the LO lasers are 1 MHz. the coherent optical transmission systems using different modulation formats. The block length is 11, and the linewidth of both the Tx and the LO lasers are 1 MHz. and the linewidth of both the Tx and the LO lasers are 1 MHz. The comparisons of the one-tap normalized LMS, the block-wise average, and the The comparisons of the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi carrier phase recovery algorithms have also been investigated in detail. The BER The comparisons of the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi Viterbi-Viterbi carrier phase recovery algorithms have also been investigated in detail. The BER floors versus different phase noise variances in the above three carrier phase recovery algorithms in carrier phase recovery algorithms have also been investigated in detail. The BER ﬂoors versus different floors versus different phase noise variances in the above three carrier phase recovery algorithms in the 8-PSK optical fiber communication system have been studied and are shown in Figure 14, where phase noise variances in the above three carrier phase recovery algorithms in the 8-PSK optical the 8-PSK optical fiber communication system have been studied and are shown in Figure 14, where the block length in the block-wise average and the Viterbi-Viterbi algorithms varies from 5 to 17, in ﬁber communication system have been studied and are shown in Figure 14, where the block length the block length in the block-wise average and the Viterbi-Viterbi algorithms varies from 5 to 17, in Figure 14a–c, respectively. It can be seen that the phase noise induced BER floors in the block-wise in the block-wise average and the Viterbi-Viterbi algorithms varies from 5 to 17, in Figure 14a–c, Figure 14a–c, respectively. It can be seen that the phase noise induced BER floors in the block-wise average and Viterbi-Viterbi algorithms are degraded with increment of the block length, and the respectively. It can be seen that the phase noise induced BER ﬂoors in the block-wise average and average and Viterbi-Viterbi algorithms are degraded with increment of the block length, and the one-tap normalized LMS algorithm keeps the same performance due to its optimized operation. It is Viterbi-V one-tap iterbi norm algorithms alized LMS a are lgorithm degraded keep with s th incr e sa ement me performanc of the block e du length, e to its and optimi theze one-tap d operation normalized . It is also found in Figure 14 that for the small phase noise variance (or effective phase noise variance), the LMS alsalgorithm o found in keeps Figure the 14 th same at for performance the small ph due ase nois to its e optimized variance (or operation. effective ph It ase is also noise found varia in nc Figur e), the e 14 Viterbi-Viterbi carrier phase recovery algorithm outperforms the one-tap normalized LMS and the Viterbi-Viterbi carrier phase recovery algorithm outperforms the one-tap normalized LMS and the that for the small phase noise variance (or effective phase noise variance), the Viterbi-Viterbi carrier block-wise average algorithms, while for the large phase noise variance (or effective phase noise block-wise average algorithms, while for the large phase noise variance (or effective phase noise phase recovery algorithm outperforms the one-tap normalized LMS and the block-wise average variance), the one-tap normalized LMS algorithm shows a better performance than the other two variance), the one-tap normalized LMS algorithm shows a better performance than the other two algorithms, while for the large phase noise variance (or effective phase noise variance), the one-tap algorithms in the carrier phase recovery. algorithms in the carrier phase recovery. normalized LMS algorithm shows a better performance than the other two algorithms in the carrier phase recovery. Photonics 2016, 3, 51 13 of 18 Photonics 2016, 3, 51 13 of 18 (a) (b) (c) Figure 14. BER floors versus different phase noise variances in the three carrier phase recovery Figure 14. BER ﬂoors versus different phase noise variances in the three carrier phase recovery algorithms in the 8-PSK optical fiber communication systems. (a) Block length of the BWA and VV algorithms in the 8-PSK optical ﬁber communication systems. (a) Block length of the BWA and VV algorithms is 5; (b) block length of the BWA and VV algorithms is 11; (c) block length of the BWA and algorithms is 5; (b) block length of the BWA and VV algorithms is 11; (c) block length of the BWA and VV algorithms is 17. VV algorithms is 17. As shown in Figures 15 and 16, the comparison of the one-tap normalized LMS, the block-wise As shown in Figures 15 and 16, the comparison of the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi carrier phase recovery algorithms has also been investigated in terms of different modulation formats. Here a block length of 11 is used in the block-wise average average, and the Viterbi-Viterbi carrier phase recovery algorithms has also been investigated in terms and the Viterbi-Viterbi algorithms. It can be found in Figure 15 that the BER floors in all the three of different modulation formats. Here a block length of 11 is used in the block-wise average and the algorithms are increased with the increment of modulation formats, but the variation in the one-tap Viterbi-Viterbi algorithms. It can be found in Figure 15 that the BER ﬂoors in all the three algorithms are increased with the increment of modulation formats, but the variation in the one-tap normalized LMS CPR algorithm is larger than in the other two algorithms. Therefore, the one-tap normalized LMS algorithm is more sensitive to the level of the modulation formats. Photonics 2016, 3, 51 14 of 18 Photonics 2016, 3, 51 14 of 18 normalized LMS CPR algorithm is larger than in the other two algorithms. Therefore, the one-tap Photonics 2016, 3, 51 14 of 18 norm normalized alized L LMS MS al CPR gorithm algorithm is more s is la ensi rgetive t r than o th ine l thev e el other of th tw e mod o algulation orithm s. forma Ther ts. efore, the one-tap normalized LMS algorithm is more sensitive to the level of the modulation formats. (a) (b) (a) (b) (c) (d) (c) (d) Figure 15. BER floors versus different phase noise variances in the three carrier phase recovery Figure 15. BER ﬂoors versus different phase noise variances in the three carrier phase recovery Figure algorithm 15.s BER in the flooptic ors vers al fib us e r dco iffm erent munph icati ase on noi syst seems varia usi nces ng di in fferent the thr m ee od carrier ulation ph for ase m ats rec . ove Blory ck algorithms in the optical ﬁber communication systems using different modulation formats. Block algorithm lengths of s the in tBWA he optic and al VV fib e algorithm r commun s ic are atiboth on syst 11.ems (a) QP usiSK ng syst different em, (b m ) od 8-PS ulation K systfor em, m(ats c) .16 B-lo PS ck K lengths of the BWA and VV algorithms are both 11. (a) QPSK system; (b) 8-PSK system; (c) 16-PSK system, (d) 32-PSK system. lengths of the BWA and VV algorithms are both 11. (a) QPSK system, (b) 8-PSK system, (c) 16-PSK system; (d) 32-PSK system. system, (d) 32-PSK system. (a) (b) (a) (b) (c) (d) (c) (d) Figure 16. BER floors versus different transmission distances in the three carrier phase recovery Figure algorithm 16.s BER in the floors optic vers al us f ibe different r comm tra un nsmiss ication ion syst dist em ances s us ing in the di ff thr erent ee carrier modulatio phase n rec form ove ats ry. Figure 16. BER ﬂoors versus different transmission distances in the three carrier phase recovery algorithm Linewidths s of in the the tran optic smitt al er fibe and r c LO omm lase un rs icat are ioboth n syst 1 em MHz, s us and ing blo dick fferent lengths mod of the BWA ulation form and ats VV . algorithms in the optical ﬁber communication systems using different modulation formats. Linewidths Li CPR newidt algorithm hs of the s are tran both smitt 1 er 1. and (a) QP LOS la K se srs ystem are ;both (b) 8 1 -MHz, PSK sys and tem b;lo (ck c) le 16 ng -PS ths K of syst the BWA em; (d) a32 nd -P V SK V of the transmitter and LO lasers are both 1 MHz, and block lengths of the BWA and VV CPR algorithms CPR system. algorithm s are both 11. (a) QPSK system; (b) 8-PSK system; (c) 16-PSK system; (d) 32-PSK are both 11. (a) QPSK system; (b) 8-PSK system; (c) 16-PSK system; (d) 32-PSK system. system. Figure 16 shows the performance of BER floors versus transmission distances in the three Figure 16 shows the performance of BER ﬂoors versus transmission distances in the three carrier carrier Figph ure as 1 e6 recov shows ery th met e per hods form unance der do ifferent f BER floo mod rs ulation versus forma transm ts, is where sion d th is et an linew ces idths in the ofth bo ree th carrier phase recovery methods under different modulation formats, where the linewidths of both phase recovery methods under different modulation formats, where the linewidths of both the transmitter and the LO lasers are set as 1 MHz and the transmission distance are set from 0 km to 5000 km. It can also be found in Figure 16 that, similar to Figure 15, the one-tap normalized LMS algorithm is more sensitive to the level of the modulation formats than the block-wise average and the Viterbi-Viterbi CPR algorithms. In addition, the Viterbi-Viterbi algorithm outperforms the other two carrier phase recovery approaches when the transmission distance varies from 0 km to 5000 km, Photonics 2016, 3, 51 15 of 18 while the differences between the three carrier phase recovery algorithms become smaller with the increment of modulation formats. 4.2. Ideal Spectral Efﬁciency in Carrier Phase Recovery For the binary symmetric channel (binary input and output alphabets, symmetric transition probability), the coding rate R in the n-PSK optical ﬁber communication systems (assuming an ideal hard-decision forward error correction coding) can be expressed as follows [53], R = 1 + BER log (BER) + (1 BER) log (1 BER). (16) 2 2 Correspondingly, the ideal spectral efﬁciency (assuming an ideal hard-decision forward error correction coding) in the carrier phase recovery in the n-PSK coherent optical ﬁber communication systems can be calculated as: SE = R N log (n) , (17) C p where SE is the ideal spectral efﬁciency, n is the modulation format level of the communication system, N is the number of polarization states. The BER limits in Equation (16) can be obtained from the BER ﬂoors in the three carrier phase recovery approaches according to Equations (9), (12) and (15), respectively. 4.3. Complexity of Carrier Phase Recovery Approaches The computational complexity is always a signiﬁcant consideration and criterion for the DSP algorithms. Here the complexity of the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi carrier phase recovery algorithms has been investigated in terms of the number of the complex multiplications per recovered symbol, which are shown in Table 1 (n is the level of modulation formats). It is found that the computational complexity of the one-tap normalized LMS algorithm is independent from the modulation formats, while the complexity of the block-wise average and the Viterbi-Viterbi CPR algorithms scales linearly with the level of modulation formats. Note that the computations in the pre-convergence of the one-tap normalized LMS algorithm also has to be considered in the practical applications. Table 1. Complexity of carrier phase recovery approaches (complex multiplications per symbol). One-Tap Normalized LMS Block-Wise Average Viterbi-Viterbi 5 n n All the above analyses are based on the carrier phase recovery in the n-PSK coherent optical communication systems; however, all these discussions can be directly extended into the circular n-QAM transmission systems. Meanwhile, although the n-PSK signals have a lower tolerance to the ASE noise than the multi-amplitude signals (such as n-QAM signals), the n-PSK signals will have a better tolerance to ﬁber nonlinearities due to the constant amplitudes [54,55]. 5. Conclusions Theoretical analyses of the carrier phase recovery in long-haul high-speed n-PSK (n-level phase shift keying) coherent optical ﬁber communication systems, using the one-tap normalized least-mean-square (LMS), block-wise average, and Viterbi-Viterbi algorithms, have been investigated and described in detail, considering both the laser phase noise and the equalization enhanced phase noise. The expressions for the estimated carrier phase in these three algorithms have been presented, and the bit-error-rate (BER) performance such as the BER ﬂoors, has been predicted analytically. Comparative studies of the one-tap normalized LMS, block-wise average, and Viterbi-Viterbi algorithms have also been carried out. It has been found that the Viterbi-Viterbi carrier phase recovery Photonics 2016, 3, 51 16 of 18 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 in the carrier phase recovery for large phase noise variance (or effective phase noise variance). In addition, the one-tap normalized LMS carrier phase recovery algorithm is more sensitive to the level of modulation formats than the other two algorithms. The BER ﬂoors in this paper were discussed and analyzed based on the inﬂuence from laser phase noise and equalization enhanced phase noise in the long-haul n-PSK transmission systems, and this represents the system limits from laser phase noise and equalization enhanced phase noise. In addition, signal degradation from ﬁber nonlinearities is also a signiﬁcant effect in such communication systems. Therefore, the actual BER ﬂoors will be determined by involving the impacts from ampliﬁed spontaneous emission (ASE) noise, laser phase noise, equalization enhanced phase noise and ﬁber nonlinearities, which will be investigated in our future work. Acknowledgments: This work is supported in part by UK Engineering and Physical Sciences Research Council (project UNLOC EP/J017582/1), in part by European Commission Research Council FP7-PEOPLE-2012-IAPP (project GRIFFON, No. 324391), in part by European Commission Research Council FP7-PEOPLE-2013-ITN (project ICONE, No. 608099), and in part by Swedish Research Council Vetenskapsradet (No. 0379801). Author Contributions: T.X. presented the basic idea and carried out the analytical calculations and discussions. G.J., S.P., J.L., T.L., Y.Z. and P.B. contributed to developing the research ideas and were involved in the discussion of results. T.X. wrote the main manuscript and prepared the ﬁgures. All authors reviewed the manuscript and gave the ﬁnal approval for publication. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Essiambre, R.J.; Foschini, G.J.; Kramer, G.; Winzer, P.J. Capacity limits of information transport in ﬁber-optic networks. Phys. Rev. Lett. 2008, 101, 163901. [CrossRef] [PubMed] 2. Bayvel, P.; Maher, R.; Xu, T.; Liga, G.; Shevchenko, N.A.; Lavery, D.; Alvarado, A.; Killey, R.I. Maximising the optical network capacity. Philos. Trans. R. Soc. A 2016, 374, 20140440. [CrossRef] [PubMed] 3. Essiambre, R.J.; Tkach, R.W. Capacity trends and limits of optical communication networks. Proc. IEEE 2012, 100, 1035–1055. [CrossRef] 4. Kaminow, I.; Li, T.; Willner, A.E. Optical Fiber Telecommunications VB: System and Networks; Academic Press: Oxford, UK, 2010. 5. Agrawal, G.P. Fiber-Optic Communication Systems, 4th ed.; John Wiley & Sons, Inc.: New York, NY, USA, 2010. 6. Li, Y.; Xu, T.; Jia, D.; Jing, W.; Hu, H.; Yu, J.; Zhang, Y. Dynamic dispersion compensation in a 40 Gb/s single-channeled optical ﬁber communication system. Acta Opt. Sin. 2007, 27, 1161–1165. 7. Xu, T.; Li, J.; Jacobsen, G.; Popov, S.; Djupsjöbacka, A.; Schatz, R.; Zhang, Y.; Bayvel, P. Field trial over 820 km installed SSMF and its potential Terabit/s superchannel application with up to 57.5-Gbaud DP-QPSK transmission. Opt. Commun. 2015, 353, 133–138. [CrossRef] 8. Galili, M.; Hu, H.; Mulvad, H.C.H.; Medhin, A.K.; Clausen, A.; Oxenløwe, L.K. Optical systems for ultra-high-speed TDM networking. Photonics 2014, 1, 83–94. [CrossRef] 9. Ip, E.; Lau, A.P.T.; Barros, D.J.F.; Kahn, J.M. Coherent detection in optical ﬁber systems. Opt. Express 2008, 16, 753–791. [CrossRef] [PubMed] 10. Xu, T.; Jacobsen, G.; Popov, S.; Li, J.; Vanin, E.; Wang, K.; Friberg, A.T.; Zhang, Y. Chromatic dispersion compensation in coherent transmission system using digital ﬁlters. Opt. Express 2010, 18, 16243–16257. [CrossRef] [PubMed] 11. Taylor, M.G. Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments. IEEE Photonics Technol. Lett. 2004, 16, 674–676. [CrossRef] 12. Savory, S.J. Digital ﬁlters for coherent optical receivers. Opt. Express 2008, 16, 804–817. [CrossRef] [PubMed] 13. Xu, T.; Jacobsen, G.; Popov, S.; Li, J.; Wang, K.; Friberg, A.T. Normalized LMS digital ﬁlter for chromatic dispersion equalization in 112-Gbit/s PDM-QPSK coherent optical transmission system. Opt. Commun. 2010, 283, 963–967. [CrossRef] Photonics 2016, 3, 51 17 of 18 14. Kudo, R.; Kobayashi, T.; Ishihara, K.; Takatori, Y.; Sano, A.; Miyamoto, Y. Coherent optical single carrier transmission using overlap frequency domain equalization for long-haul optical systems. J. Lightwave Technol. 2009, 27, 3721–3728. [CrossRef] 15. Ip, E.; Kahn, J.M. Digital equalization of chromatic dispersion and polarization mode dispersion. J. Lightwave Technol. 2007, 25, 2033–2043. [CrossRef] 16. Xu, T.; Jacobsen, G.; Popov, S.; Li, J.; Friberg, A.T.; Zhang, Y. Carrier phase estimation methods in coherent transmission systems inﬂuenced by equalization enhanced phase noise. Opt. Commun. 2013, 293, 54–60. [CrossRef] 17. Liga, G.; Xu, T.; Alvarado, A.; Killey, R.; Bayvel, P. On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission. Opt. Express 2014, 22, 30053–30062. [CrossRef] [PubMed] 18. Maher, R.; Xu, T.; Galdino, L.; Sato, M.; Alvarado, A.; Shi, K.; Savory, S.J.; Thomsen, B.C.; Killey, R.I.; Bayvel, P. Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back propagation. Sci. Rep. 2015, 5, 08214. [CrossRef] [PubMed] 19. Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 379–423. [CrossRef] 20. Essiambre, R.J.; Kramer, G.; Winzer, P.J.; Foschini, G.J.; Goebel, B. Capacity limits of optical ﬁber networks. J. Lightwave Technol. 2010, 28, 662–701. [CrossRef] 21. Kazovsky, L.G. Impact of laser phase noise on optical heterodyne communication systems. J. Opt. Commun. 1986, 7, 66–78. [CrossRef] 22. Zhang, S.; Kam, P.Y.; Yu, C.; Chen, J. Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems. IEEE Photonics Technol. Lett. 2009, 21, 1075–1077. [CrossRef] 23. Taylor, M.G. Phase estimation methods for optical coherent detection using digital signal processing. J. Lightwave Technol. 2009, 17, 901–914. [CrossRef] 24. Fatadin, I.; Ives, D.; Savory, S.J. Differential carrier phase recovery for QPSK optical coherent systems with integrated tunable lasers. Opt. Express 2013, 21, 10166–10171. [CrossRef] [PubMed] 25. Ip, E.; Kahn, J.M. Feedforward carrier recovery for coherent optical communications. J. Lightwave Technol. 2007, 25, 2675–2692. [CrossRef] 26. Goldfarb, G.; Li, G. BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing. Opt. Express 2006, 14, 8043–8053. [CrossRef] [PubMed] 27. Mori, Y.; Zhang, C.; Igarashi, K.; Katoh, K.; Kikuchi, K. Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver. Opt. Express 2009, 17, 1435–1441. [CrossRef] [PubMed] 28. Fatadin, I.; Ives, D.; Savory, S.J. Blind equalization and carrier phase recovery in a 16-QAM optical coherent system. J. Lightwave Technol. 2009, 27, 3042–3049. [CrossRef] 29. Kikuchi, K. Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation. IEEE J. Sel. Top. Quant. Electron. 2006, 12, 563–570. [CrossRef] 30. Ly-Gagnon, D.S.; Tsukamoto, S.; Katoh, K.; Kikuchi, K. Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation. J. Lightwave Technol. 2006, 24, 12–21. [CrossRef] 31. Viterbi, A.J.; Viterbi, A.M. Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission. IEEE Trans. Inf. Theory 1983, 29, 543–551. [CrossRef] 32. Jacobsen, G.; Xu, T.; Popov, S.; Sergeyev, S. Study of EEPN mitigation using modiﬁed RF pilot and Viterbi-Viterbi based phase noise compensation. Opt. Express 2013, 21, 12351–12362. [CrossRef] [PubMed] 33. Shieh, W.; Ho, K.P. Equalization-enhanced phase noise for coherent detection systems using electronic digital signal processing. Opt. Express 2008, 16, 15718–15727. [CrossRef] [PubMed] 34. Xie, C. Local oscillator phase noise induced penalties in optical coherent detection systems using electronic chromatic dispersion compensation. In Proceedings of the Conference on Optical Fiber Communication (OFC), San Diego, CA, USA, 22–26 March 2009. 35. Lau, A.P.T.; Shen, T.S.R.; Shieh, W.; Ho, K.P. Equalization-enhanced phase noise for 100 Gb/s transmission and beyond with coherent detection. Opt. Express 2010, 18, 17239–17251. [CrossRef] [PubMed] 36. Fatadin, I.; Savory, S.J. Impact of phase to amplitude noise conversion in coherent optical systems with digital dispersion compensation. Opt. Express 2010, 18, 16273–16278. [CrossRef] [PubMed] Photonics 2016, 3, 51 18 of 18 37. Xu, T.; Jacobsen, G.; Popov, S.; Li, J.; Friberg, A.T.; Zhang, Y. Analytical estimation of phase noise inﬂuence in coherent transmission system with digital dispersion equalization. Opt. Express 2011, 19, 7756–7768. [CrossRef] [PubMed] 38. Ho, K.P.; Lau, A.P.T.; Shieh, W. Equalization-enhanced phase noise induced time jitter. Opt. Lett. 2011, 36, 585–587. [CrossRef] [PubMed] 39. Jacobsen, G.; Xu, T.; Popov, S.; Li, J.; Friberg, A.T.; Zhang, Y. EEPN and CD study for coherent optical nPSK and nQAM systems with RF pilot based phase noise compensation. Opt. Express 2012, 20, 8862–8870. [CrossRef] [PubMed] 40. Xu, T.; Jacobsen, G.; Popov, S.; Li, J.; Sergeyev, S.; Friberg, A.T.; Zhang, Y. Analytical BER performance in differential n-PSK coherent transmission system inﬂuenced by equalization enhanced phase noise. Opt. Commun. 2015, 334, 222–227. [CrossRef] 41. Xu, T.; Liga, G.; Lavery, D.; Thomson, B.C.; Savory, S.J.; Killey, R.I.; Bayvel, P. Equalization enhanced phase noise in Nyquist-spaced superchannel transmission systems using multi-channel digital back-propagation. Sci. Rep. 2015, 5, 13990. [CrossRef] [PubMed] 42. Zhuge, Q.; Morsy-Osman, M.H.; Plant, D.V. Low overhead intra-symbol carrier phase recovery for reduced-guard-interval CO-OFDM. J. Lightwave Technol. 2013, 31, 1158–1169. [CrossRef] 43. Jacobsen, G.; Lidón, M.; Xu, T.; Popov, S.; Friberg, A.T.; Zhang, Y. Inﬂuence of pre- and post-compensation of chromatic dispersion on equalization enhanced phase noise in coherent multilevel systems. J. Opt. Commun. 2011, 32, 257–261. [CrossRef] 44. Kakkar, A.; Navarro, J.R.; Schatz, R.; Pang, X.; Ozolins, O.; Louchet, H.; Jacobsen, G.; Popov, S. Equalization enhanced phase noise in coherent optical systems with digital pre- and post-processing. Photonics 2016, 3, 12. [CrossRef] 45. Ho, K.P.; Shieh, W. Equalization-enhanced phase noise in mode-division multiplexed systems. J. Lightwave Technol. 2013, 31, 2237–2243. 46. Shieh, W. Interaction of laser phase noise with differential-mode-delay in few-mode ﬁber based MIMO systems. In Proceedings of the Conference on Optical Fiber Communication (OFC), Los Angeles, CA, USA, 4–8 March 2012. 47. Colavolpe, G.; Foggi, T.; Forestieri, E.; Secondini, M. Impact of phase noise and compensation techniques in coherent optical systems. J. Lightwave Technol. 2011, 29, 2790–2800. [CrossRef] 48. Farhoudi, R.; Ghazisaeidi, A.; Rusch, L.A. Performance of carrier phase recovery for electronically dispersion compensated coherent system. Opt. Express 2012, 20, 26568–26582. [CrossRef] [PubMed] 49. Jacobsen, G.; Xu, T.; Popov, S.; Li, J.; Friberg, A.T.; Zhang, Y. Receiver implemented RF pilot tone phase noise mitigation in coherent optical nPSK and nQAM systems. Opt. Express 2011, 19, 14487–14494. [CrossRef] [PubMed] 50. Yoshida, T.; Sugihara, T.; Uto, K. DSP-based optical modulation technique for long-haul transmission. In Proceedings of the Next-Generation Optical Communication: Components, Sub-Systems, and Systems IV, San Francisco, CA, USA, 7 February 2015. 51. Vanin, E.; Jacobsen, G. Analytical estimation of laser phase noise induced BER ﬂoor in coherent receiver with digital signal processing. Opt. Express 2010, 18, 4246–4259. [CrossRef] [PubMed] 52. Kakkar, A.; Navarro, J.R.; Schatz, R.; Louchet, H.; Pang, X.; Ozolins, O.; Jacobsen, G.; Popov, S. Comprehensive study of equalization-enhanced phase noise in coherent optical systems. J. Lightwave Technol. 2015, 33, 4834–4841. [CrossRef] 53. Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2006. 54. Kojima, K.; Koike-Akino, T.; Millar, D.S.; Parsons, K. BICM capacity analysis of 8QAM-alternative modulation formats in nonlinear ﬁber transmission. In Proceedings of the IEEE Tyrrhenian International Workshop on Digital Communications, Florence, Italy, 22 September 2015; pp. 57–59. 55. Kojima, K.; Koike-Akino, T.; Millar, D.S.; Pajovic, M.; Parsons, K.; Yoshida, T. Investigation of low code rate DP-8PSK as an alternative to DP-QPSK. In Proceedings of the Conference on Optical Fiber Communication (OFC), Anaheim, CA, USA, 20–22 March 2016. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Sep 24, 2016

Loading...

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

Access the full text.

Sign up today, get DeepDyve free for 14 days.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.