We propose a simple non-data-aided (or unsupervised) and universal cycle slip detection and correction (CS-DC) technique based on locating the minimum of the sliding average of twice estimated phase noise. The CS-DC can be appended to any carrier phase estimation(CPE) technique and is modulation format independent. We analytically derive the probability density function of the CS detection metric and study how the sliding window length and detection threshold affects CS detection performance. Simulation results reveal significant cycle slips reduction for various modulation formats with a residual CS probability of 2 × 10−7 for single carrier system even in unrealistic highly nonlinear system setups. In addition, we show that a second stage of CS-DC with a different sliding window length can further reduce the cycle slip probability by at least an order of magnitude. We also show that CS-DC is tolerant to inter-channel nonlinearities and residue frequency offset effects. Overall, the proposed CS-DC technique can be used in conjunction to other CS reduction techniques to maximize the ability of CS mitigation in next generation optical transceivers.
© 2014 Optical Society of America
Carrier phase estimation (CPE) is an integral part of any coherent communication systems. Over the past few years, numerous digital signal processing (DSP) techniques have been developed to estimate and recover the laser phase for QPSK, 16-QAM systems and beyond. However, most CPE techniques suffer from the problem of cycle slips (CS) in which the received signal are phase rotated by integer multiples of π/2 and can potentially lead to catastrophic detection errors.
To cope with this problem, one can use differential encoding where information is encoded in the difference between neighboring bits/symbols. On the other hand, as soft decision-forward error correction (SD-FEC)-based transmission system become the standard choice for systems beyond 100Gb/s, differential encoding is not preferred as it induces an extra OSNR penalty for signals entering into the SD-FEC decoder [1–3]. To this end, turbo decoding was proposed to reduce the extra differential encoding penalty . However, it was also shown that the correction capability of FEC depends on the CS occurrence probability. For example, turbo decoding exhibits an error floor when CS probability exceeds 10−4 and might cause total decoder failure.
Alternatively, numerous other differential-encoding-free techniques were proposed which basically involves either inserting pilot tones or pilot symbols at regular intervals [5, 6]. Pilot symbols are slightly easier to be incorporated while inserting pilot tones may require additional transmitter complexity and inevitably sacrifices signal power. However, trade-off has to be made between CS robustness and spectral efficiency. In most recent high spectral efficiency system demonstrations, the amount of pilot symbols are kept sufficiently low to minimize the impact to spectral efficiency, rendering the system susceptible to burst errors and hence require more robust FECs with increased code interleaving depth . Several pilot-aided CPEs have already been modified in an attempt to mitigate the CS-induced burst-error impact [8–10]. In addition, a fully blind or non-data-aided CS compensation technique has recently been proposed by using block polarization coding which, however, is only applicable to BPSK systems . This is becoming a serious drawback as optical communications are moving towards flexible transmissions and elastic optical networks(EON) which simultaneously support signal transmissions with multiple modulation formats. Modulation-format independence or universality has become an important attribute to any digital signal processing(DSP) algorithm in next generation transceivers . Thus, it would be highly desirable to have a non-data-aided and universal CS detection and correction technique with minimal additional transceiver complexities.
In , we have proposed a blind CS detection and correction (CS-DC) technique that can be appended to any CPE algorithm. The CS-DC examines the sliding average of twice estimated phase noise and it is shown numerically that the magnitude of such average undergo an abrupt drop when CS occurs. Therefore, locating the abrupt drops and studying the corresponding phase evolution at that time allow one to detect and correct CS. In this paper, we extend our investigation and analytically derive the probability density function of the sliding average, which allow us to optimize the sliding window length and threshold for CS detection. Simulation results show that even for unrealistic long-haul systems with high nonlinearity, the proposed CS-DC can substantially reduce the CS probability in both single carrier and Nyquist wavelength division multiplexing (WDM) systems. Furthermore, we show that cascading a second stage of CS-DC can successfully detect and correct multiple cycle slips that are close to each other and further reduce the CS probability by at least one order of magnitude. In the end, the tolerance of CS-DC to residue frequency offset (FO) is also studied. The proposed CS-DC is simple, non-data-aided and universal and can be used in conjunction with turbo differential decoding  or other pilot symbol/tone based techniques to mitigate CS to the largest possible extent.
2. Principle of non-data-aided and universal cycle slip detection and correction(CS-DC)
Consider a digital coherent system using QPSK, 16-QAM or other common single-carrier modulation formats. For simplicity purpose, we neglect fiber nonlinearity and WDM effects and assume that linear transmission impairments such as chromatic dispersion and polarization-mode dispersion has been compensated by appropriate signal processing algorithms preceding the CPE. With symbol-rate sampling, the ith signal in one polarization going into the CPE unit of the overall DSP platform is given by
The block diagram of cycle-slip detection and correction is shown in Fig. 1. For practical 100Gb/s, 400Gb/s and super-channel transmission systems, is small enough such that the laser phase can be considered as a slowly varying process. When a cycle slip occurs, however, the estimated phase noise will considerably deviate from its true value . In particular, let di be the detected symbols from the CPE output and considerFig. 2. Such minimum can be used to detect cycle slips. To correct such cycle slips, one can study the evolution of the estimate phase around ics and determine whether the CPE has incorrectly rotated the signals by –π/2 or π/2.
3. Theoretical analysis of CS-DC
We will begin our analysis with the summation of yi over a window with length K + 1. Since K + 1 is typically larger than 50, the effect of ASE noise will be considerably averaged out and suppressed and hence can be neglected. In this analysis, nonlinear effects are not considered for simplicity purposes and simulation results on nonlinear transmission systems (described in the next section) show that nonlinear impairments are second-order effects of the analytical insights developed herein. In addition, we assume no detection errors in the window of K + 1 symbols except when a cycle slip occurred at the center of the window, which will be followed by catastrophic detection errors till the end. When there are no cycle slips, the sum of yi over a window with length K + 1 is given by
When a cycle slip occurs at the center of the summing window followed by catastrophic detection errors, one can setwithout loss of generality and
In this case, the two summation terms in (9) are i.i.d. with pdfand the pdf of can be obtained by similar derivations outlined above. Figures 3(a) and (b) depicts the pdf of x61 with and without CS and compared with Monte Carlo simulation results obtained from 109 independent realizations and the pdf of and are shown in Fig. 4. It can be seen that the analytical derivations agree well with simulation results. Also, the analytical derivations are particularly useful for our CS analysis as they enable us to study scenarios with low CS occurrences, which are otherwise prohibited by simulations that require excessively long times.
The proposed CS-DC technique detects cycle slips by examining whether is smaller than a threshold. The threshold should be set to minimize the cycle slip probability after CS-DC, given by
Figure 5 shows the CS probability Ppost after CS-DC for different thresholdsand K. It can be seen that the optimum threshold becomes smaller with increasing averaged window length. With Ppre of 10−5, for example, the optimum threshold decreases from 0.6 to 0.5 when the average window length doubled from 41 to 81. Also, Ppost increases with K as the laser phase noise correlation weakens with time. The results further suggest that the optimum threshold does not change drastically with respect to window length and thus we can fix our threshold to be 0.5 for the rest of this paper.
4. Simulation results
Simulations are conducted to investigate the performance of the proposed CS-DC technique for polarization multiplexed(PM)-QPSK and PM-16-QAM systems. The CS occurrence probability before and after CS-DC is shown in Fig. 6 for 112 Gb/s PM-QPSK transmissions over 2400km and 7200 km with various optical signal to noise ratio (OSNR, in dB/0.1nm) and CPE lengths. In particular, 340M symbol sequences for each data point with gray coding were transmitted through a link consisting of multiple spans of 80-km SMF with inline optical amplifiers. The signal launched power is chosen to be 4 dBm, significantly more than the typical launch power of 0 dBm or below so as to increase the amount of CS occurring instances. The laser linewidths of both transmitter and local oscillator are 100kHz which is typical for external cavity lasers(ECL). The receiver DSP unit includes chromatic dispersion compensation , polarization de-multiplexing  and CPE using Viterbi & Viterbi phase estimation (VVPE) method . The window length for CS-DC is K + 1 = 401. The missed cycle slips are identified by examining the bit errors patterns in a 200-symbol sliding window after carrier phase recovery or CS-DC. If the BER in the window is more than expected, i.e. 0.1, a cycle slip is assumed to have ocurred in this window. Then, the correction algorithm de-rotates the recovered data at the middle of the window with three other possible ambiguous phases and assumes the one with lowest BER to be the actual phase estimate. The whole process is carried out until the whole set of data is correctly decoded. The cycle slip probability (CSP) is calculated by dividing the amount of missed CSs with the length of the whole data set. It is obvious from Fig. 6 that CS-DC can appropriately detect and correct cycle slips and substantially reduce the CS probability by nearly two orders of magnitudes. In addition, when the CPE half-length exceeds 50, no CS has been identified except for the case of 16dB OSNR at 7200km in which the BER is as high as 5 × 10−2, an impractical system setup. In more realistic system configurations with moderate launch power and adequate OSNR for a given transmission distance, all cycle slips are perfectly detected and corrected through CS-DC.
The performance of the proposed CS-DC technique for 224bit/s PM-16QAM transmissions over 1200 km and 2400 km are shown in Fig. 7. 340M 16-QAM symbols are generated and the receiver DSP include CD compensation , constant modulus algorithm for pre-convergence followed by the cascaded multi-modulus algorithm  and the CPE used is QPSK partitioning + maximum likelihood phase estimation method . Other 16-QAM CPE techniques such as blind phase search  also gives similar CS-DC occurrences. The window length for CS-DC is K + 1 = 301. The proposed CS-DC technique can reduce CS probability by at least an order of magnitude and essentially eliminate all cycle slips as long as the CPE half-filter length exceeds 30 taps.
To study the impact of inter-channel nonlinearities, simulations have been conducted for a 5 × 112Gbit/s PM-QPSK Nyquist-WDM system over a 2400km SMF link and a 5 × 224Gbit/s Nyquist-WDM PM-16QAM system over a 1200km SMF link. The channel spacing and bandwidth are set to be 50GHz and 40GHz respectively. The launch power for each channel is 4dBm and the laser linewidths are 100 kHz. Figure 8(a) shows the CSP versus CPE half-filter length for the middle channel. Although more CSs have been observed comparing with single channel cases, the CS-DC algorithm still achieves notable CS reduction of at least one- and two-orders of magnitudes respectively for the one-stage and two-stage strategies. Without CS-DC, the amount of CS for each data point ranges from 10s to more than 2400. With 2-stage CS-DC, the CSP is driven down to 0 most of the time and at most 8 × 10−6 under highly unrealistic system conditions. Figure 8(b) shows the CSP versus CPE half-filter length. Comparing with single-carrier systems. Without CS-DC, the amount of CS for each data point ranges from 10s to more than 1700. With the proposed 2-stage CS-DC, the CSP is driven down to 0 most of the time and at most 7 × 10−6 under highly unrealistic conditions. This proves that the algorithm works effectively for the PM-QPSK signal in presence of inter-channel nonlinearities.
To further improve the CS mitigation performance, we can cascade an additional stage of CS-DC with a different window length K2 + 1 after a first stage of CS-DC with window length K1 + 1 as shown in Fig. 9. This helps to detect and correct multiple cycle-slips that occurred close to each other and a first stage of CS-DC fail to correctly identify and correct all of them. With the two-stage CS-DC, Figs. 6-8 show the final CS probability for QPSK and 16-QAM transmissions can be further reduced by ten times to below 10−6 and 10−5 for single channel and WDM systems respectively. The window length for the second-stage CS-DC is 600 and 450 for QPSK and 16-QAM transmissions respectively. Furthermore, the required CPE average length to achieve zero CS probability is shortened. This offers more flexibility for CPE length design and is beneficial to the systems with large laser linewidths. We emphasize that we intended to exemplify CS occurrences by studying highly non-linear and unrealistic system setups and the proposed CS-DC can virtually detect and correct all CS for more realistic and practical scenarios.
It is unquestionably clear from the simulation results above that a longer CPE filter length is desirable in suppressing cycle slip occurrences. However, it is also known that increasing CPE filter length affects phase tracking capability especially in the highly nonlinear transmission scenarios . To this end, Bisplinghoff et al. proposed to use a shorter CPE filter with half-filter length N1 in addition to a longer one in parallel to form a slip-reduced CPE (SR-CPE) . We can show that one can append CS-DC to another other algorithms to further improve CS mitigation performance. We reproduced the channel model in , simulate a single channel 112 Gbit/s PM-QPSK system with 210M symbols on each polarization and compare the performance between SR-CPE and SR-CPE + CS-DC. For various N1, the required OSNR for a differentially decoded BER of 0.04 (target for turbo decoding)  and the CS probability is studied and shown in Fig. 10. Combined laser linewidth is set to be 200 kHz and the correlated nonlinear phase noise standard deviation is set from 0.2 rad to 0.8 rad  to investigate worst case scenarios. Compared with SR-CPE, the CS probability of SR-CPE + CS-DC is reduced by more than 10 times to well below 2x10−7 which is much lower than the upper tolerance of turbo differential decoding of 10−4 . In addition, the best OSNR can be achieved by optimizing N1 to be 4.
In the practical systems, frequency offset cannot be fully compensated before carrier phase recovery. Thus, it is of great importance to study CS-DC’s tolerance to residue FO. For residue-FO tolerance investigation, tens of normally distributed frequency offsets are added to the input signal of carrier phase recovery. Here, the mean square error (MSE) of residue FO is defined as 
Figure 11(a) shows the CSP versus the MSE of residue frequency offset for a 112Gbit/s PM-QPSK system over 7200km SMF link. The MSE of the residue FO is swept from 10−11 to 7 × 10−9. The launch power is 4dBm. The OSNR is set to be 16 dB which is the worst case scenario. For the 1-stage CS-DC, no penalty has been noticed until the MSE becomes larger than 6 × 10−9. For the 2-stage CS-DC, the missed CS rate does not change much until MSE is larger than 4 × 10−9. This is equivalent to a FO estimation error of 1.77 MHz for a 28GBaud system. Since the achievable MSE is around 10−12, it is safe to conclude that CS-DC is robust against residue FO . We have also studied the impact of residue FO on the optimum average length. Extensive simulations with average lengths of 100, 200, 300 and 400 symbols have been conducted. Due to the residue FO, the best average length is 100 for the 1-stage CS-DC while the optimum length for the 2-stage CS-DC is found to be 200. This is because the phase noise evolution dynamics of most of the missed CS after the first stage CS-DC happens gradually, and therefore, average window with a length of 100 is not sufficient anymore. It also explains why the 2-stage CS-DC is more susceptible to residue FO.
Then, the same investigation has been conducted for a 224 Gbit/s PM-16QAM transmission system over 2400km SMF link as shown in Fig. 11(b). The launch power is 4dBm and the OSNR is set to be 18 dB. The tolerable residue FO is compromised as the modulation order becomes higher. Nonetheless, the 1-stage CS-DC is still quite robust when the MSE is smaller than 10−8 while the 2-stage CS-DC can tolerate residue FO with MSE up to 10−9. This is still 1000 times larger than the tolerable MSE considered in , and is equivalent to a FO estimation error of 885 kHz for a 28GBaud system. The best filter length is 100 for the 1-stage CS-DC while the optimum filter length for 2-stage CS-DC is 200.
We proposed a non-data-aided (or unsupervised) and universal cycle-slip detection and correction (CS-DC) technique based on locating the minimum of the sliding average of twice estimated phase noise. The technique is independent of modulation format and choice of preceding CPE algorithms, which potentially enable the use of soft decision-forward error correction (SD-FEC) without regularly inserted pilot symbols. We analytically derived the probability density function of the CS identification metric and characterize the amount of CS reduction with the proposed CS-DC technique. Simulation results show that the CS-DC can reduce CS occurrence probability by orders of magnitudes even in systems with excessive noise and nonlinear impairments. In addition, the proposed CS-DC technique is tolerant to inter-channel nonlinearities and residue frequency offset effects and can be appended to other CS mitigation techniques reported in the literature. As CS mitigation is the primary objective of regularly inserting pilot symbols in practice, the blindness and universality of the proposed CS-DC technique may hold key to finally break such conventional wisdom and pave the way to practical implementation of non-data-aided and universal transceivers for coherent communications.
The authors would like to acknowledge the support of the Hong Kong Government General Research Fund (GRF) under project number PolyU 152079/14E and Hong Kong Polytechnic University project 4-ZZ7U and H-ZDA9.
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