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Two slipless schemes are compared with application to single carrier 30 Gbaud quadrature phase shift keying (QPSK) system. An equivalent linewidth model considering the phase noise induced by both the laser linewidth and fiber nonlinearity is applied in the performance analysis. The simulation results show that it is possible to mitigate cycle slip (CS) using only 0.39% pilot overhead for the proposed blind carrier phase recovery (CPR) + pilot-symbols-aided phase unwrapping (PAPU) scheme within 1dB signal-to-noise ratio (SNR) penalty limit at the bit error ratio (BER) of 10 −3 with 4 MHz equivalent linewidth.
© 2014 Optical Society of America
1. Introduction
Carrier phase estimation (CPE) or carrier phase recovery (CPR) is one of the key procedure in digital signal processing (DSP) in 100 Gb/s or beyond coherent transmission system [1–3]. Several feedforward blind CPE algorithms, such as 4th power and blind phase search (BPS), have been proposed and it has been demonstrated that with feedforward CPE, MHz level cost-efficient laser could be used in coherent system [4]. However cycle slip (CS), an inherent property of blind CPE, will lead to error propagation [5–9]. In [5] it is stated that in hybrid transmission system nonlinearity induced phase noise is a major source of CS besides laser linewidth. The typical way to deal with CS is to use differential encoding/decoding, but this technique can lead to error duplication (approximately) [10]. Therefore, specially designed slipless CPR schemes are usually used in current coherent systems [11].
Most of the recent solutions to CS are based on in-band pilot symbols [5–9]. In [7] a two stage CPR scheme was proposed with pilot symbols-aided CPR (PA-CPR) and successive blind CPR, such as 4th power or blind phase search (BPS) [4, 11] (denoted as PA-CPR + Blind CPR). For this scheme large carrier phase error can be compensated with unambiguous carrier phase estimated from the 1st PA-CPR, and then the phase error can be refined by the 2nd stage blind CPR without phase unwrapper. One crucial issue for PA-CPR + Blind CPR scheme is the pilot rate or pilot overhead [11]. In [7] a pilot rate of no less than 3% is used for homogeneous transmission in the standard single mode fiber (SSMF) link. In presence of highly nonlinear transmission, such as hybrid transmission on nonzero dispersion shifted fiber (NZDSF) link, a pilot rate of as large as 10% must be used. In this case this scheme performs similarly with differential coding.
In our previous work [8, 9] a slipless CPR scheme, blind CPR together with pilot-symbols-aided phase unwrapping (PAPU) (denoted as Blind CPR + PAPU), is proposed and demonstrated in a 28 Gbaud single polarization QPSK system. In this paper, an equivalent linewidth model considering both the laser linewidth and fiber nonlinearity induced phase noise is applied and the performance of Blind CPR + PAPU scheme at lower pilot rate is further investigated by the comparisons with PA-CPR + Blind CPR scheme. The simulation results for 30 Gbaud single carrier quadrature phase shift keying (QPSK) system show that at the bit error ratio (BER) of 10 −3 within 1dB signal-to-noise ratio (SNR) penalty limit, it is possible to prevent CS with only 0.39% pilot overhead and 4 MHz equivalent linewidth for the Blind CPR + PAPU scheme.
2. Principles
2.1 Equivalent linewidth model
The input symbols to the CPE have the following expression, assuming all linear optical impairments have been compensated (as well as ideal timing recovery and synchronization):
where is the complex-valued transmitted symbol at the k-th time instant, is the additive channel noise, andis the cumulated noise phase, which is composed of phase noise caused by laser linewidth and fiber nonlinearity. The phase noise caused by laser linewidth is usually modeled as a Wiener process [4]. In [12] an empirical phase noise model of using Wiener process for reproducing fiber nonlinearities has been experimentally verified. This suggests that we can regard laser phase noise and nonlinear phase noise equally and characterized them using a uniform Wiener process:where is white noise with zero mean and variance . Here is the equivalent linewidth parameter and is the symbol period. In the deployed coherent system 100 KHz level laser is widely adopted, however if the phase noise induced by fiber nonlinearity is also taken into consideration, the equivalent linewidth may be broadened to MHz level [12].2.2 Slipless CPR scheme
For the PA-CPR used in the slipless CPR schemes, we can only get the phase error estimation at the pilot position by correlating the received symbols with the known pilot sequences [5–9]. It is necessary to use a realizable interpolation filter to get the continuous carrier phase and minimize the noise effect. Wiener’s method can be used to determine the optimal filter for continuous interpolation, however it is too complex to implement for practical applications and a suboptimum filter by cascade of two moving averages is used to approximate that of the optimal filter without much performance loss [13, 14]. The PA-CPR can track the laser PN from 0 to , which is unambiguous [6] and can be used to compensate laser PN [7] or as an absolute carrier phase reference to compensate CS to the CPR [8, 9].
In the proposed Blind CPR + PAPU scheme in [8, 9] it seeks to modify the usual phase unwrap procedure, which is the source of CS, to make it immune to CS. After interpolating the usual phase unwrapped carrier phase by in-band pilot symbols, the obtained phase (reference carrier phase) can be thought an approximation to the true carrier phase without ambiguity. In addition the blind estimated phase is another approximation to the true carrier phase, however, it may experience CS in any position. The idea of PAPU is to do the usual phase wrapping first and then compare each unwrapped phase with the reference carrier phase, and compensate the slipped phase at the same time. The key point of the proposed slipless CPR scheme, as shown in Fig. 1(a), is the PAPU which carry out CS detection and correction in a completely feedforward way and combine itself with the usual phase unwrapping.
For the PA-CPR + Blind CPR scheme in [7], as shown in Fig. 1(b), after large carrier phase error is removed by the interpolated phase of the pilot symbols then the output of the 1st stage CPR assumed to be subject only to a small phase offset limited to the range [−π/4, π/4), then a relatively less-complexity blind CPE algorithm without phase unwrapping is required to compensate the residual phase error and no CS introduced.
The crucial issue for the two slipless schemes is the choice of the pilot rate and the interpolation filter design, which depends on the linewidth and the signal-to-noise ratio (SNR). Generally the pilot rate should not be too low for the two stage PA-CPR + Blind CPR scheme. If pilot rate in the 1st stage is not sufficient for the amount of phase noise present in the system the inaccurate CPE may enhance the phase noise effect for the 2nd stage CPE [6]. But for the Blind CPR + PAPU scheme, PA-CPE is not directly exponentially multiplied to symbols to compensate the phase error, the above disadvantage can be avoided and lower pilot rate may be used. Without loss of generality we use 4th power algorithm (VVPE) as the blind CPE for QPSK. According to our observations CS usually occurs consecutively and the steady slipped phase can only be an integer multiple of π/2. In this case a relatively lower pilot-rate may be used to capture this phase slip.
3. Simulation results
As in [5, 7], considering the QPSK modulation format, the symbol rate of 30 Gbaud and a total number of 218 symbol is used in a single simulation. In a long-haul 100 Gb/s PDM-QPSK system [12] an expansion of a factor 10 compared to the nominal back to back linewidth for SSMF homogeneous transmission and larger broadening for NZDSF hybrid transmission has been observed. In a more realistic system configuration MHz equivalent linewidth, e.g. 2 MHz, is chosen for the simulations. In order to investigate the pilot rate’s influence to the performance of the CPR schemes, interpolation filter design and CPE filter length must be optimized simultaneously. For simplicity, the duration of the pilot period N of the interpolation filter will be determined firstly before the CPE filter length being analyzed.
In simulations we use a similar symbol definition as in [13, 14] and SNR (or energy per symbol to noise density, Es/N0) is used to describe the noise level [15]. The interpolation filter is obtained by cascading two moving averages each one of length MN [13, 14] with pilot spacing M (1/M equal to pilot rate or pilot overhead) and duration of the pilot period N. Generally a larger N will lead to better tolerance to the additive noise but more smooth effect to the phase noise. The influence of the duration of pilot period N on the accuracy of PA-CPE is shown in Fig. 2. Block averaging for VVPE blind CPE is used here with a fixed filter length of 48 and a pilot rate of 1.56% (equivalent to M = 64, similar to [5, 6]). When N = 1, where the interpolated filter has the least additive noise tolerance, the interpolated carrier phase deviates the most from the true carrier phase. While N becomes larger, e.g. N = 8, the additive noise is averaged however the carrier phase is also smoothed out, which also results in a deviation from the true carrier phase. Roughly when N = 3 the two effects can be balanced.
Fig. 2 The influence of the duration of the pilot period N on the accuracy of PA-CPE. The linewidth is 2MHz, SNR is 9.78dB, and fixed filter length is 48.
As shown in Fig. 3 the duration of the pilot period’s influence to BER is investigated with various linewidth, SNR and pilot rate. The equivalent linewidth parameters are chosen to be 200 KHz and 2 MHz, respectively. The pilot rates are 1.56% and 0.39% respectively. The SNRs are 7.33dB and 9.78dB respectively, which is theoretical limit for 10 −2 and 10 −3. The performance of the slipless CPE scheme is investigated at the BER of 10 −2 assuming the use of hard decision forward error correction (FEC) with 20% coding overhead [15]. At the BER of 10 −3 the linewidth induced SNR penalty is also studied. In Figs. 3(a) and 3(b) the equivalent linewidth is too small to show the difference between different CPR scheme and pilot rate. In Fig. 3(c) though the equivalent linewidth has a tenfold increase to 2MHz, adequate pilot rate (1.56%) provides accurate CPE without much degradation for the two CPR scheme. Howeverin Fig. 3(d) the pilot rate is reduced to as low as 0.39% the difference between the two CPR schemes become significant. As analyzed in last section, the low pilot rate (0.39%) cannot provide the accurate PA-CPE for the PA-CPR + Blind CPR scheme in presence of 2 MHz equivalent linewidth. The pilot period N also has large impact on the BER performance of the PA-CPR + Blind CPR scheme. The BER of the Blind CPR + PAPU scheme seems less affected by the pilot rate reduction, as shown in Figs. 3(c) and 3(d). And the influence of pilot period on BER is much less than the PA-CPR + Blind CPR scheme. It can be seen that N = 3 almost leads to the minimum BER for all the cases in Fig. 3. For simplicity we choose the optimum N = 3 for all the following simulations.
Fig. 3 The duration of pilot period’s influence to BER for the PA-CPR + VVPE and VPE + PAPU scheme with various linewidth (200KHz and 2MHz), pilot rate (1.56% and 0.39%) and SNR (7.33dB and 9.78dB).
The optimum filter length at different SNR and pilot rate is studied as shown in Table 1 and Table 2 respectively. The equivalent linewidth is 2 MHz. With 30 times extensive simulations, filter length of the blind CPR for two schemes is optimized between 4 and 128 with step 4 under different pilot rate. 30 optimized filter length is counted and shown with minimum, maximum and mean value. It is found that the optimum filter length alters randomly in each simulation, but the range is quite regular, which is roughly 40 to 100 with 7.33dB SNR and about 24 to 72 with 9.78dB SNR respectively. From Fig. 4 it can also be seen that the BER performance is quite stable in a large range of filter length. Therefore a fixed optimum CPE filter length of 48 could be adopted for the interpolation filter optimization.
Table 1. Pilot-rate versus optimum CPE filter length for the two slipless CPR scheme @ 2MHz equivalent linewidth and 7.33dB SNR
Table 2. Pilot-rate versus optimum CPE filter length for the two slipless CPR scheme @ 2MHz equivalent linewidth and 9.78dB SNR
Fig. 4 Contour plots of CPE filter length’s influence to log(BER) performance using block averaging, and 0.39% pilot rate. Four different situations are considered. (a) The linewidth is 3MHz, with PA-CPR + VVPE scheme. (b) The linewidth is 3MHz, with VVPE + PAPU scheme. (c) The linewidth is 4MHz, with PA-CPR + VVPE scheme. (d) The linewidth is 4MHz, with VVPE + PAPU scheme.
The impact of CPE filter length to the BER between a large ranges of SNR is studied in Fig. 4. It can be seen that with a pilot rate as low as 0.39% and an equivalent linewidth of 3MHz or 4MHz, VVPE + PAPU scheme almost always outperforms PA-CPR + VVPE scheme under various SNR values. Significant BER performance improvement can be observed especially in the high SNR region.
Figure 5 shows the SNR penalty over block averaging for PA-CPR + VVPE and VVPE + PAPU scheme at the target BER of 10 −2 and 10 −3 in presence of various equivalent linewidth symbol duration product and optimized CPE filter length. The SNR loss due to pilot overhead is not taken into account to the SNR penalty. In case of pilot rate larger than1.56%, no obvious performance difference between the two schemes is observed through Figs. 5(a) and 5(b). While pilot rate is equal or lower than 0.78%, as shown in Fig. 5(c), with the increase of the SNR penalty difference between the two schemes becomes more significant regardless of which BER level. From Fig. 5(d) when the pilot rate lower to 0.39% VVPE + PAPU outperforms PA-CPR + VVPE scheme by more than 0.5dB with >10−4 (equivalent linewidth > 2.8 MHz). And the speed of SNR penalty increase is much less than the PA-CPR + VVPE scheme. If the linewidth induced penalty is limited to 1dB at the BER of 10−3, it is possible to prevent CS with only 0.39% pilot overhead for the VVPE + PAPU scheme under the condition of = 1.43 × 10−4 (4 MHz equivalent linewidth).
Fig. 5 Penalty of block averaging for the PA-CPR + VVPE and VVPE + PAPU scheme at the target BER of 10−2 and 10−3 with various phase noise represented by and pilot rate.
4. Conclusions
In this paper we study a low overhead slipless carrier phase estimation schemes applied to single carrier 30 Gbaud QPSK system. An equivalent linewidth model is applied in the performance analysis for the slipless CPR scheme. By simulations we confirm that low pilot rate (0.39%) degrades the performance of the PA-CPR + Blind CPR scheme in presence of MHz level equivalent linewidth due to the inaccurate PA-CPE. However, the Blind CPR + PAPU scheme is less affected by such pilot rate with the different usage of the PA-CPE. And it is possible to prevent CS with only 0.39% pilot overhead for the Blind CPR + PAPU scheme within 1dB SNR penalty limit at the BER of 10−3 with 4 MHz equivalent linewidth. Considering the application of long-haul coherent system, MHz equivalent linewidth should be considered for the CPR if the system is working in the nonlinearity region, the Blind CPR + PAPU scheme realizes slipless operation with lower pilot rate and more overhead could be reserved for FEC coding.
Acknowledgments
This work was partly supported by 863 program 2013AA014202, 2012AA011303, 973 program 2014CB340100, 2011CB301700, NSFC program 61331008, 61205031, 61307055, fund of State Key Laboratory of Information Photonics and Optical Communications (BUPT), and the fundamental research funds for the central universities.
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