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Abstract
Although fruitful investigations of carrier phase estimation (CPE) have been conducted for a traditional coherent fiber optical transmission, there are few studies on the CPE for a nonlinear Fourier transform (NFT) based transmission. A laser linewidth induced phase noise leads to a phase rotation of the nonlinear spectra and the scattering data, which is similar to its effect on the linear spectra. Here, we first identify that both feed forward the M-th power, and the blind phase search (BPS)-based CPE can function well in the nonlinear frequency division multiplexing (NFDM) transmission with discrete spectrum modulation. Then, a performance comparison between two CPE schemes is presented for various modulation formats under the scenario of a single eigenvalue NFDM transmission. Our simulation results indicate that the laser linewidth tolerances of 2 GBaud quadrature phase shift keying (QPSK), 8-phase shift keying (8-PSK), and 16-amplitude phase shift keying (16-APSK) are 2.3 MHz, 1.05 MHz, and 250 KHz, respectively, given a 1-dB optical signal to noise ratio (OSNR) penalty at BER = 10−3. Finally, the BPS algorithm is experimentally verified under the same scenario of a 2 GBaud back-to-back transmission, due to the use of a semiconductor laser with a 100 KHz linewidth.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the past few decades, the great progress of fiber optical transmission has brought the system capacity gradually close to the Shannon channel capacity [1–3]. When the launched power is further increased, the optical signal is badly damaged by the nonlinear transmission impairments, thus limiting the possible enhancement of transmission capacity [4,5]. Fiber nonlinearity becomes the main obstacle for high-speed long-haul fiber optical transmission. Several mitigation and compensation techniques have been proposed over past few years in both the digital domain [6] and the optical domain [7]. However, no clear winner stands out. Recently, an elegant method, nonlinear frequency division multiplexing (NFDM), has attracted worldwide research interests, which can effectively address the nonlinear transmission impairments arising in standard single mode fiber (SSMF) transmission [8,9]. In such an approach, information is encoded onto the nonlinear spectrum of signal, which is generated by nonlinear Fourier transform (NFT) [10,11]. At the nonlinear frequency domain, the nonlinear transmission impairments arising in the SSMF are converted to a phase shift, which can be completely compensated in the digital domain. Generally, the nonlinear spectrum can be divided into two parts, including the continuous spectrum and the discrete spectrum where eigenvalue lies on the upper-half complex plane. Recent experimental results have demonstrated 4 GBaud 16-amplitude phase shift keying (16-APSK) with single eigenvalue discrete spectrum modulation by using semiconductor laser with a linewidth of 100 Hz [12] or 500 MBaud quadrature phase shift keying (QPSK) with independent discrete spectrum modulation of 7 discrete eigenvalues by using a semiconductor laser with a linewidth of 1 KHz [13]. Meanwhile, coding on the continuous spectrum has achieved a transmission capacity of 220 Gb/s with the QPSK format [14]. The polarization division multiplexing (PDM) technique is also introduced into the NFDM system to further increase its transmission capacity, using either discrete, continuous spectrum, or both parts [15–17]. Since the continuous spectrum better resembles regular orthogonal frequency division multiplexing (OFDM) subcarriers in the linear system, large transmission capacity can be expected. Alternatively, the discrete spectrum modulation has more dimensions to be explored. The digital signal processing (DSP) of NFDM system is quite mature, whereas the phase noise mitigation of NFDM with discrete spectrum modulation is still unclear, especially the laser linewidth tolerance.
Most existing NFDM transmission results rely on the use of excellent semiconductor laser with a linewidth of less than few KHz, without considering the effect of phase noise. Generally, phase noise due to the occurrence of laser linewidth is the main linear transmission impairments to be properly mitigated. Although traditional carrier phase estimation (CPE) algorithm is claimed to be effective for the NFDM transmission [18], the effect of phase noise on the performance of NFDM transmission is still unclear. Until now, only a few investigations about the phase noise on the performance of NFDM transmission have been reported. The blind phase search (BPS) algorithm has been applied in two-eigenvalue dual-polarization NFDM transmission with QPSK format at 250 MBaud. The laser linewidth tolerance has been identified to be 750 KHz under the condition of back-to-back transmission [19]. Though the phase noise can be well solved for traditional coherent fiber optical transmission [20–22], the tolerance of laser linewidth for the NFDM transmission still needs to be identified. As for the NFDM system with discrete spectrum modulation, the more eigenvalues are multiplexed, the more sensitive the NFDM signal is to the noise [23–25]. Thus, the NDFM transmission with single eigenvalue has theoretically maximum laser linewidth tolerance. Meanwhile, as for the NFDM system with multiple eigenvalues multiplexing, after calculating eigenvalue and nonlinear frequency spectrum by nonlinear Fourier transform (NFT), the spectrum of each eigenvalue is processed by DSP [15] and the phase noise can be mitigated by applying the blind phase search (BPS) algorithm to each eigenvalue individually. Therefore, the laser linewidth tolerance of NDFM transmission with a single eigenvalue is helpful to provide a guidance for the research of the NFDM system with multiple eigenvalues multiplexing.
In this current submission, since the commonly used modulation formats in the NFDM transmission, including QPSK, 8-phase shift keying (8-PSK), and 16-APSK, are all cyclic phase modulations, we carry out performance comparisons between the feed forward M-th power and BPS-based CPE algorithms to deal with the phase noise arising in the NFDM transmission. Then, we identify the laser linewidth tolerances and the correspondingly optimal block sizes for 2 GBaud QPSK, 8-PSK, and 16-APSK, respectively. Finally, the BPS algorithm is experimentally verified under the scenario of 2 GBaud back-to-back single eigenvalue NFDM transmissions, due to the use of a semiconductor laser with 100 KHz linewidth.
2. Operation principle
Generally, NFT is a method to solve the nonlinear Schrödinger equation (NLSE), which is often referred as the inverse scattering transform (IST) in mathematics and physics. Under the assumption of noiseless and lossless, NLSE can be normalized [10]
where $q(t,z)$ is the evolution of a slow-varying optical field envelope, z plays the role of distance along the SSMF while t is the time variable. Equation (1) can be reduced to an eigenvalue equation by a linear partial differential operator L, so that $q(t,z)$ varies with the transmission distance while the eigenvalue of L remains constant [11]. The simplified eigenvalue equation corresponding to Eq. (1) with initial conditions isAfter the successful compensation of nonlinear transmission impairments and frequency offset, the signal samples can be simplified to
where A represents its amplitude with the additive noise and ${\theta _m}$ represents its phase modulation, while ${\varphi _k}$ and ${\varphi _n}$ are the phase noise induced by laser linewidth and additive noise, respectively. The phase noise of semiconductor laser is generally modeled as a wiener process, leading to a random phase rotation of signal nonlinear spectrum. Figure 1 illustrates the phase noise effect on the scattering dada $b(\lambda )$ and the points of various colors and shapes indicates different modulation phases. Under the back-to-back configuration, when the optical signal to noise ratio (OSNR) is 30 dB, in comparison with the 16-APSK signal without considering the laser linewidth, the 16-APSK with 100 KHz laser linewidth has severe random phase shift.
Fig. 1. Phase noise on the scattering dada $b(\lambda )$ under the back-to-back configuration, (a) without considering the laser linewidth and (b) with a linewidth of 100 KHz.
3. CPE implementation strategy
In order to effectively mitigate the phase noise, we can employ either feed forward M-th power or BPS scheme to estimate the phase noise in traditional coherent optical communication system. Multiplying an estimated phase value with opposite sign can compensate the phase noise. Since the phase noise also causes a random phase rotation in the nonlinear frequency domain, we propose to use the feed forward M-th power or BPS scheme to deal with the phase noise arising in the NFDM transmission. Since the commonly-used modulation formats in the NFDM transmission, including QPSK, 8-PSK, and 16-APSK, are all cyclic phase modulation, according to Eq. (10), it is nature to choose the feed forward M-th power scheme first.
3.1. M-th power algorithm
The algorithm of feed forward M-th power is illustrated in Fig. 2. According to Eq. (10) and Eq. (11), after the preliminary processing, the signal Xk only contains the amplitude A, M-ary phase modulation $\theta $, and phase noise ${\varphi _k}$. For the QPSK modulation, M is 4 while M is 8 for 8-PSK or 16-APSK modulation. Since the phase modulation is equal to $2\pi /M$, the complex samples are first raised to the M-th power, taking advantage of the geometric properties of a constellation in order to wipe-off the M-ary phase modulation. To improve the CPE accuracy out of the shot-noise, the raised samples within a block length Nblock are added. The phase shift of signal Xk is estimated by averaging a total block length Nblock=2N + 1 considering the preceding and following N samples. After raising the samples to M-th power, we can perform summation, argument calculation and phase unwrapping. Then, a common CPE for such specific block is obtained by calculating the argument of the complex sum vector.
As for 16-APSK format, the samples have to be partitioned into two subgroups, as shown in Figs. 2(b). Please note that we perform the amplitude normalization prior to the M-th power based CPE, which aims to reduce the ambiguity during the phase estimation, due to additive noise and different amplitude levels of 16-APSK signal.3.2. BPS algorithm
The main idea of feed forward BPS algorithm is to discretize the phase space and make a minimum error, as shown in Fig. 3. The samples are firstly rotated by B uniformly distributed test phase ${\phi _b} = (\frac{{2b}}{B} - 1)\frac{\pi }{M}(b = 1,2, \cdots ,B).$ Because of the rotational symmetry of the constellation, the searching angle ranges from $\textrm{ - }{\pi \mathord{\left/ {\vphantom {\pi M}} \right.} M}$ to ${\pi \mathord{\left/ {\vphantom {\pi M}} \right.} M}$. ${X_{k,b}}$ is fed into a decision circuit and the squared distance to the closest constellation point is calculated in the complex plane, as shown in Eq. (12)
In order to mitigate the additive noise, N consecutive input samples with the same test phases are summed. The test angle, corresponding to the minimum of ${e_{k,b}}$, is the estimated phase noise for the k-th samples.
4. Results and discussion
The numerical simulation setup and corresponding DSP flow at the transmitter and receiver sides are shown in Fig. 4. At the transmitter-side, all symbols are modulated in a b-coefficient manner on the single eigenvalue of λ=0.3j in its nonlinear frequency domain by utilizing the Darboux transformation. Each pulse is de-normalized to 0.5 ns time window, corresponding to a pulse repetition rate of 2 GHz. Those pulses are then generated in the arbitrary waveform generator (AWG) with a sampling rate of 64 GSa/s and consequently drive the IQ modulator. The transmitter laser has a central wavelength of 1550 nm and a frequency offset of less than few MHz. Then, amplified spontaneous emission (ASE) noise loading is used to adjust the OSNR. Finally, the received signal is coherently detected and sampled at 80 GSa/s sampling rate. Each pulse is transformed by NFT to obtain its eigenvalue and corresponding scattering data $b(\lambda )$. After the compensation of nonlinear phase shift and frequency offset, phase noise is compensated by either M-th power or BPS scheme. Finally, after the symbol decisions, the bit error ratio (BER) is counted. As the generated pulse is limited to a finite time slot for practical applications, the time window must cover 99.99% pulse energy for the correct NFT operation. The parameters satisfying such pulse energy requirement for different situations are summarized in Table 1. The sampling number D indicates the number of points utilized to describe a pulse in each time window.
Table 1. Parameters of different modulation schemes.
The phase noise of semiconductor laser is generally modeled as a Wiener process. Large linewidth leads to fast-varied phase noise over a short time. Thus, longer block size will degrade the CPE performance. For traditional fiber optical coherent transmission system, linewidth and symbol duration product is an important metric to evaluate the capability of phase noise compensation [20–22]. When the symbol rate increases, the tolerance of maximum laser linewidth decreases accordingly. Thus, we choose linewidth and symbol duration product to characterize the laser linewidth tolerance of NDFM transmission with single eigenvalue. For the ease of performance comparison, the OSNR penalty at BER=$\textrm{1}{\textrm{0}^{\textrm{ - 3}}}$ is referred to a phase noise free transmission system without using the CPE.
For feed forward BPS algorithm, the number of test angle B is a significant parameter. The larger number of test angle can achieve better compensation effect, while bring greater computational complexity. Thus, a trade-off occurs between the system performance and the implementation complexity. Figure 5 shows the OSNR penalty at BER = 10−3 with respect to the number of test angle, given the laser linewidth $\Delta v = 2MHz$ for 2 GBaud QPSK, $\Delta v = 500KHz$ for 2 GBaud 8-PSK, and $\Delta v = 150KHz$ for 2 GBaud 16-PASK. With the growing number of test angle, the OSNR penalty starts to converge, indicating of less than 0.1 dB OSNR penalty fluctuations with steady-state value. Taking both the implementation complexity and CPE performance into account, the number of test angle $B = 8$ is the optimal choice for QPSK, 8-PSK and 16-APSK, and fixed in our next investigation.
Then, the optimization of block length for both two CPE algorithms is investigated. For small laser linewidth, longer block length is preferred to mitigate the additive noise, especially when the phase noise varies slowly. With the growing linewidth, the optimal block length becomes smaller, because the rapidly varied phase noise becomes less correlated over the block. Thus, there occurs a trade-off between the mitigation of additive noise and the CPE capability. The contour plots of the OSNR penalty are shown in Fig. 6 for each modulation format with respect to both the laser linewidth and the block length. Figures 6(a) and 6(b) correspond to the feed forward M-th power and feed forward BPS schemes, respectively. The numbers on different colored curves correspond to various OSNR penalties. From the view of modulation format, the constellation of QPSK shows a larger Euclidean distance. In addition, as for the NFDM transmission, the amplitude of the discrete nonlinear spectrum corresponds to the center of pulses in the time domain [12]. Pulse modulated by QPSK or 8-PSK has a single time-center, while pulses modulated by 16-APSK has two time-centers, which is more sensitive to the noise. The CPE algorithm can only compensate the phase noise and can’t compensate the amplitude noise. From another perspective of CPE algorithm, an optimal block length is necessary to maintain the BER performance. Large block length is preferred for narrow laser linewidth while small block length is preferred under the condition of wide laser linewidth. Compared to feed forward M-th power based CPE, the feed forward BPS is more sensitive to the length of block. When the block length is larger than the optimal one, the tolerance of laser linewidth drops rapidly under the same ONSR penalty. Although the phase noise estimation within a specific range brings some computational complexity, the BPS has better CPE performance, regardless of the modulation format. In order to keep the OSNR penalty less than 1dB, the optimal block lengths are identified in Table 2.
Fig. 6. OSNR penalty at BER = 10−3 with respect to the laser linewidth and block length for each modulation format under the condition of individual CPE algorithm, (a) feed forward M-th power and (b) feed forward BPS.
Table 2. Optimal block length for different modulation formats using tow CPE algorithms.
Table 3. Laser linewidth tolerance under the condition of 1 dB OSNR penalty.
The OSNR penalty at BER = 10−3 with respect to the product of the laser linewidth and the symbol duration is presented in Fig. 7, under the condition of optimal block length for each modulation format. The required laser linewidth decreases with the growing number of phase states. Since both 16-APSK and 8-PSK format have eight phase states, their curves have similar trends in Fig. 7. With the help of feed forward M-th power CPE, the limitations of product of linewidth per laser and symbol duration leading to 1 dB OSNR penalty are identified to be $\textrm{1}\textrm{.12} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, $\textrm{3}\textrm{.75} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, and $\textrm{1} \times \textrm{1}{\textrm{0}^{\textrm{ - 3}}}$ for 16-APSK, 8-PSK, and QPSK. As a result, the linewidth tolerance of 225 KHz, 750 KHz, and 2 MHz for each modulation format can be secured at 2 GBaud. As for the feed forward BPS, the limitations of the product of linewidth per laser and symbol duration leading to 1 dB OSNR penalty are found to be $\textrm{1}\textrm{.25} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, $\textrm{5}\textrm{.25} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, and $\textrm{1}\textrm{.15} \times \textrm{1}{\textrm{0}^{\textrm{ - 3}}}$ for 16-APSK, 8-PSK, and QPSK. Consequently, the linewidth tolerance of 250 KHz, 1.05 MHz, and 2.3 MHz for each modulation format is identified at 2 GBaud. The results of the laser linewidth tolerance are summarized in Table 3.
Fig. 7. OSNR penalty at BER = 10−3 vs the laser linewidth and symbol duration product under the condition of optimal block length (dashed lines denote BPS and solid lines denotes M-th power, respectively).
Figure 8 shows the BER performance under the scenario of 2GBaud back-to-back single eigenvalue NFDM transmission, when the experimental setup is consistent with the simulation setup. The linewidth of used laser is around 100 KHz and signal is modulated by 2 GBaud QPSK, 8-PSK and 16-APSK on the discrete spectrum with single eigenvalue. At the receiver, we choose BPS algorithm to compensate phase noise. In comparison with simulation results, the experimental results of three modulation formats have obvious OSNR penalty in order to reach the threshold of BER = 10−3. We infer that the dominant impairments are due to the amplitude noise and limited bandwidth of transmitter and receiver, which have been ignored in the numerical simulation. However, it is clearly observed that, the phase noise caused by a laser with 100 KHz linewidth can be fully compensated with the BPS algorithm, indicating of the possible use of ordinary semiconductor laser in NFDM transmission experiment.
Fig. 8. BER performance for three modulation formats under the B2B transmission after the use of BPS (dashed lines are experiment results and solid lines represent the simulation results).
5. Conclusion
The same as traditional coherent fiber optical communication, phase noise play a vital role for the NFDM transmission system. We successfully apply both the M-th power and BPS based CPE algorithms to the NFDM transmission system. Regardless of computational complexity, BPS has a better performance of phase noise estimation. The tolerance of laser linewidth for NFDM with discrete spectrum modulation is investigated and our results indicate that, under the condition of 2 GBaud symbol rate, the laser linewidth tolerances for QPSK, 8-PSK and 16-APSK formats are within the range of 2.3 MHz, 1.05 MHz, and 250 KHz, respectively. Finally, the BPS algorithm is experimentally verified under the same scenario of 2 GBaud back-to-back transmission, due to the use of a semiconductor laser with 100 KHz linewidth.
Funding
National Key Research and Development Program of China (2018YFB1801301); National Natural Science Foundation of China (61875061); Key project of R&D Program of Hubei Province (2018AAA041).
Acknowledgments
The authors would like to thank people who are fighting on the frontlines during the epidemic, especially in Wuhan. Without their brave contributions, the revision is impossible. Go, Wuhan! Stay strong, China!
Disclosures
The authors declare no conflicts of interest.
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