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Abstract
We propose a fast and blind chromatic dispersion (CD) estimation method by one sample per symbol after coherent detection. The CD estimation process is non-data aided, without the iterative scanning to obtain the CD values. Moreover, we identify that the proposed CD estimation method is transparent to the used modulation format and robust to the transmission impairments, including amplified spontaneous emission (ASE) noise and fiber nonlinearity. When the 35-GBaud DP-16QAM signal with a roll-off factor of 0.1 is transmitted over standard single mode fiber (SSMF) with a range from 320-km to 560-km, the error of CD estimation is less than 150-ps/nm under the condition of 8192 symbols used.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
To achieve both adaptive optical path routing and flexible optical wavelength switching, dynamically reconfigurable digital signal processing (DSP) is desired in the near future, which allows high capacity transmission and agile optical connections. Due to the variable fiber reaches in optical networks, chromatic dispersion (CD) is the primary impairment to be considered and the use of dispersion compensating fiber (DCF) is not a flexible solution in next-generation optical networks. Moreover, as individual channels among the wavelength-division multiplexing (WDM) transmission can experience incomplete CD compensation, due to the occurrence of dispersion slope for standard single-mode fiber (SSMF) [1], precise CD estimation for each wavelength channel is an urgent need. Accumulated CD arising in the SSMF can be estimated via different strategies, which can be generally divided into the data-aided CD estimation [2–7] and the blind CD estimation [8–16]. Overhead due to the use of training sequences reduces the spectral efficiency and transmission rate, while blind CD estimation is preferred for initialization of the digital equalization especially for the long-haul dynamic transmission. One of blind CD estimation methods relies on the iteration of CD compensation in term of the error criterion and has to scan over a wide CD range with different step sizes [8,9]. Such solutions suffer from consecutively loading equalizer coefficients and relatively long acquisition time with a CD estimation error of around 400-ps/nm [8–11]. Other blind CD estimation methods avoid scanning over a wide range of CD values. Some take advantages of fractional Fourier transform (FrFT) to obtain the chirp parameter with a relationship to the designated CD values [12]. However, the specific order of FrFT needs to be identified. Some utilize both Godard clock tone algorithm and fast Fourier transform (FFT) to obtain the best-match peak [13], meanwhile requiring a high sampling rate of analog to digital converter (ADC) to improve the accuracy of CD estimation. Calculating either the auto-correlation or the variance of signal power waveform has been reported for the estimation of CD values [14,15]. The computation of temporal auto-correlation after digital spectrum superposition has also been experimentally verified [16]. All those solutions without the scanning need to perform FFT, both the range and the accuracy of CD estimation are limited by the length of data block. Therefore, those solutions are constrained by the sampling rate of analog-to-digital convertor (ADC). Recently, the sampling rate and operation bandwidth of ADC become the bottleneck of high baud-rate fiber optical transmission, because generally the sampling rate of coherent transceiver is assumed to be twice of the symbol-rate. Under the condition of under-sampling, several CD estimation methods have been demonstrated. By using the anti-aliasing filter, CD estimation can be implemented with one sample per symbol [17]. Training sequence-based CD estimation can splice points with the periodic sequence, leading to the successful implementation of under-sampling [5]. Based on delay-tap sampling (DTS) technique in combination with periodic training sequences, equivalent sampling can realize dynamical CD evaluation with an ultra-low sampling rate [18]. However, to the best of our knowledge, no experimental demonstration of blind CD estimation with one sample per symbol has been reported.
In this paper, we propose a blind CD estimation method with only one sample per symbol. Our method takes advantage of strong periodicity of the overlapped spectrum for a block of received symbols. In addition, the proposed method is analytically identified to be insensitive to the order of quadrature amplitude modulation (QAM) format and has a robust tolerance of the amplified spontaneous emission (ASE) noise and polarization mode dispersion (PMD). Simulation results show that, there only need 4096 symbols for the 56-Gbaud non-return-to-zero (NRZ) format while 8192 symbols are necessary for the same NRZ signal when the roll-off factor of root raised cosine filter is 0.1. Since we only choose a periodical segment of frequency spectrum to obtain the unique peak, the computational complexity is substantially reduced. When the 35-Gbaud DP-16QAM signal with a roll-off factor of 0.1 is transmitted over SSMF with a range from 280-km to 560-km, the error of CD estimation is less than 150-ps/nm by the use of 8192 symbols, even in presence of fiber nonlinearity. Furthermore, our proposed CD estimation method has a low implementation complexity.
2. Operation principle
Generally, the transmitter sends complex-valued signals $S(f )= {[{{S_X}(f ),\; \; {S_Y}\; (f )} ]^T}$ on two orthogonal states of polarization (SOPs). For the ease of discussion, we first take single polarization transmission into consideration and assume the channel transfer function ${\boldsymbol H}(f )$ as the CD transfer function, regardless of the effect of PMD, fiber nonlinearity, frequency offset, and phase noise. Hence the optical signal at the receiver side in time domain can be written as,
Under the condition of under-sampling, the overlapped frequency spectrum can be calculated from Eq. (2),
Fig. 1. Frequency spectrum of baseband signal and the spectrum aliasing under the condition of ${F_s}$ $= \; $ ${R_s}$. Inset shows the accumulated CD induced periodicity.
As shown in Eq. (4), there exists periodic property for the first-order term. Then, a segment of frequency spectrum from $f ={-} \frac{{{R_s}}}{2}\; $ to $f ={-} \frac{{{R_s}}}{4}\; $ is chosen to estimate CD values, as shown in Fig. 1. Thus, we can treat the overlapped spectrum intensity terms $G(f )$ and $G({f + {R_s}} )$ as constant G, respectively. From Eq. (3) and Eq. (4), we can derive
The waveforms of $P(f )$ are schematically shown in Fig. 2, under the condition of accumulated variable CD, which show strong periodicity. In particular, the period of term $\frac{{2\pi DL{\lambda ^2}}}{c}{R_s}$ arising in $P(f )$ results in a CD relevant peak after the inverse Fourier transform (IFT) operation.
Fig. 2. Frequency spectrum of $P(f )$ under the condition of fiber length (a) 1000-km, (b) 2000-km, and (c) 3000-km. The gray line is the periodic envelope and CD induced period value $\textrm{T}{\mathrm{\omega}_1} = 3\textrm{T}{\omega _3}$, $\textrm{T}{\omega _2} = 2\textrm{T}{\omega _3}$.
Since the sampling rate of ADC may not be consistent with the symbol rate of transmitted signal, we need to consider the case of fractional sampling rate. When the sampling rates of ADC are 28-GSa/s, 32-GSa/s, and 42-GSa/s, respectively, we numerically examine the CD estimation results for 28-Gbaud DP-16QAM signals with an optical signal-to-noise ratio (OSNR) of 20-dB. As shown in Fig. 4, our CD estimation method can function well when it comes to the fractional sampling rate. As for Nyquist-WDM transmission systems, the electrical signals to be transmitted are usually pulse shaped by a root raised cosine (RRC) filter with a roll-off factor varying from 0.01 to 0.2, for the realization of high spectral efficiency and mitigating inter-channel-interference (ICI). The proposed method can also estimate the CD values from the Nyquist-WDM signal. In such case, the frequency spectrum segment will be selected depending on various roll-off factors, instead of from $f = \; - \frac{{{R_s}}}{2}\; $ to $f = \; - \frac{{{R_s}}}{4}$. Typically, when the roll-off factor is small, the selected frequency spectrum segment for the CD estimation becomes smaller, in comparison with the condition of large roll-off factors. Please note that all the analytical derivations above are completed in the analog domain, hence the resolution in the digital domain after the FFT needs to be taken into consideration to determine the resolution of the proposed CD estimation. Now in the digital domain we consider the tradeoff between the CD estimation resolution and the length of sampling points. The CD estimation resolution mainly influences the CD estimation error, while the length of sampling points determines the computation complexity. Firstly, Eq. (2) can be rewritten after the digital sampling as
where N is the length of sampling points. Then, the resolution of frequency spectrum is where ${F_s}$ is the same as ${R_s}$. Since we only choose partial frequency spectrum to obtain the CD peak after IFFT, the sampling points in time domain are assumed to be M, instead of N, leading to the change of resolution in time domain. Then the CD resolution in time domain can be derived asAs shown in Eq. (9), the resolution of CD estimation is proportional to the length of sampling points N and inversely proportional to the length of spectrum segment M and sampling rate ${F_s}^2$. Therefore, the resolution of CD estimation becomes better, as the sampling rate (symbol rate) increases, indicating that the proposed method performs better for high baud-rate transmission. Meanwhile, we need to consider the values of N and M to balance the computation complexity and the resolution of CD estimation.
Fig. 3. CD estimation results for the 28-Gbaud DP-16QAM signal with different SSMF lengths, under conditions of $D = 16ps/nm/km$, OSNR = 20 dB and ${R_s}$ = 28Gsa/s.
Fig. 4. CD estimation results for 28-Gbaud DP-16QAM signals under the condition of (a) ${R_s}$ = 28Gsa/s, (b) ${R_s}$ = 32Gsa/s and (c) ${R_s}$ = 42Gsa/s.
3. Simulation results and discussion
In order to investigate the performance of the proposed CD estimation method, we carry out numerical simulations for single carrier 56-GBaud DP-QPSK, DP-16QAM and DP-64QAM signals. The simulation parameters are presented in Table 1. We intend to investigate the effects of OSNR, the number of used symbols, and the roll-off factor on the performance of the proposed CD estimation method. For each modulation format, we carry out 50 times of Monte-Carlo simulation and take an average of those results, for the ease of presentation. After the SSMF transmission, 4096 symbols are selected for the blind CD estimation and the corresponding flow of CD estimation is shown in Fig. 5. We perform FFT on the 4096 symbols and then 1024 points out of 4096 are chosen to do the IFFT, for the ease of finding the CD peak. To establish a criterion of performance comparison, we first define the error of CD estimation and peak to average ratio (PAR). Here PAR is a metric to evaluate whether the amplitude of CD peak is robust to the OSNR variation.
where $C{D_{est}}$ is the estimated CD value, and ${A_{peak}}$ and ${A_{average}}$ represent the peak amplitude and the average of amplitude, respectively. During the simulation, we identify that the peak can be correctly secured, when PAR is more than 6-dB.
Table 1. Simulation parameter for 56-Gbaud DP-mQAM transmission.
We first investigate the accuracy of the proposed CD estimation method when the CD is varied from 8000 to 40000-ps/nm. The OSNR of 56-GBaud DP-QPSK is set to be 15-dB. The estimated CD values with respect to the SSMF length is shown in Fig. 6(a). The error of CD estimation is less than 200-ps/nm. Then, we investigate the robustness of the proposed method under the condition of low OSNR and the effect of sample points N. As shown in Fig. 6(b), enlarging N can effectively improve PAR. Meanwhile, we can observe that 4096 symbols are enough to achieve a PAR of more than 10-dB, when OSNR is varied from 8-dB to 20-dB, leading to the correct identification of the CD peak. Increasing both sampling points N and the OSNR can improve the performance of the proposed CD estimation method. Hence, an accurate CD estimation can be realized by the optimization of sampling points N and the OSNR to ensure PAR more than 6-dB.
Fig. 6. (a) 56-GBaud DP-QPSK transmission with different accumulated CD values varying from 8000 to 40000-ps/nm, under conditions of OSNR = 15 dB and N = 4096. (b) Contour of PAR variation with respect to the sampling points and the OSNR, when the SSMF length is 500-km.
We next investigate the proposed CD estimation method for 56-Gbaud DP-QPSK, DP-16QAM and DP-64QAM, in order to verify the modulation format transparent operation. When the OSNR of all modulation formats are 10-dB, the mean error of CD estimation results for three modulation formats is less than 250-ps/nm, as shown in Fig. 7(a). CD estimation error becomes worse for high-order modulation formats, because high-order modulation formats normally require higher OSNR for successful transmission. Figure 7(b) presents the variation of PAR with respect to each modulation format. Definitely, the reduction of PAR under the condition of fixed CD values with respect to modulation formats leads to a CD estimation error. PAR rapidly decreases as the order of modulation format increases, however during all simulations PARs are more than 6-dB. Moreover, both sampling points N and the OSNR range can be jointly optimized in order to achieve precise CD estimation, in order to realize the modulation format transparent operation.
Fig. 7. CD estimation results for various modulation formats, (a) mean CD estimation errors and (b) PARs under conditions of OSNR = 10 dB and N = 4096
We further examine the performance of the proposed CD estimation method for Nyquist-DP-16QAM transmission, when the signal to be transmitted is shaped with a RRC filter. Generally, the roll-off factor $\mathrm{\alpha}$ varies from 0.01 to 0.2. As shown in Fig. 8(a), we can observe that the required N increases when the roll-off factor becomes small. Typically, when the roll-off factor is less than 0.05, the sampling points N rapidly increases. Because we choose a smaller segment of frequency spectrum when the method is applied to Nyquist-WDM signals, the sampling points need to be increased to 8194. The CD estimation error with respect to the roll-off factor under the condition of N = 8194 is shown in Fig. 8(b). Although it is challenging to achieve the same PAR for various roll-off factors, the CD estimation error is less than 280-ps/nm when the CD value ranges from 4000 to 32000-ps/nm.
Fig. 8. (a) Required sampling points N to achieve PAR = 8-dB, when CD is 16000-ps/nm and OSNR is 15-dB. (b) CD estimation errors with respect to the roll-off factor under the condition of 15-dB OSNR.
Finally, we compare the computation complexity of our proposed CD estimation method with two similar CD estimation methods based on time-frequency domain analysis, as shown in Table 2. It is obvious that the computation complexity of our proposed CD estimation method is greatly reduced.
Table 2. Performance comparison among three similar CD estimation methods for 112-Gbps DP-QPSK signal, under the condition of 10-dB OSNR.
4. Experimental results and discussions
The experimental setup is shown in Fig. 9. At the transmitter side, electrical signals are offline generated and filtered by an RRC with a roll-off factor of 0.1, and then loaded into an AWG operated at 35-GSa/s sampling rate with 6-bit resolution. Thus, the baud rate of DP-16QAM signal is 35-GBaud. The output of continuous wave (CW) laser with operation wavelength of 1550.2-nm and linewidth of less than 100-kHz is modulated by a dual-polarization I/Q modulator driven by the AWG. An erbium-doped fiber amplifier (EDFA) is used to boost the optical power before the optical signal is introduced into the SSMF recirculation loop. The fiber loop includes two sections of 40-km SSMF and an EDFA to compensate the transmission attenuation. To eliminate out-of-band ASE noise, an optical bandpass filter (OBPF) is used. At the receiver side, a coherent receiver is used to realize the polarization and phase diversity detection and four electrical outputs are recorded by a 40-GSa/s real-time oscilloscope, for the offline verification of our proposed CD estimation scheme. We experimentally investigate the performance of the proposed method for various SSMF lengths. The results are summarized in Fig. 10. In Fig. 10(a), after 320-km, 400-km, 480-km and 560-km SSMF transmission, CD peaks with almost the same PAR of 10-dB are, respectively, obtained under the condition of N = 8192. Since the theoretical CD resolution is 173.4-ps/nm, which is derived from Eq. (9), the CD estimation error is smaller than the theoretical resolution. Generally, we can further improve the resolution of CD estimation by either enlarging the data block length N or selecting a larger segment of frequency spectrum. However, selecting a larger segment of frequency spectrum leads to a sharp reduction of PAR due to the deterioration of periodicity. According to our theoretical and experimental investigations, 8192 symbols are sufficient for practical one sample per symbol implementation.
Fig. 9. Experimental setup of 35-GBaud DP-16QAM signal transmission. AWG: arbitrary waveform generator, VOA: variable optical attenuator, SW: optical switch, ECL: external cavity laser, IQM: I/Q modulator.
Fig. 10. (a) Experimental CD estimation peaks for 35-Gbaud Nyquist shaped DP-16QAM signal after 320-km, 400-km, 480-km and 560-km fiber link. Launched power of 1dBm, N = 8192 (b) Histogram of the CD estimation error with 200 independent trials for each launch power after 320-km, 400-km, 480-km and 560-km fiber link.
Moreover, the CD estimation error with respect to the launched power is also investigated, as shown in Fig. 10(b). Here we take -2-dBm, 1-dBm and 4-dBm into consideration and 300 independent trials are performed for each launched power. The mean error of CD estimation among three typically launched power is trivial, and the maximum error is only 150-ps/nm. Therefore, we can conclude that the proposed CD estimation method is robust to the nonlinear transmission impairments arising in Nyquist-WDM transmission.
5. Conclusion
We propose a fast and blind CD estimation method with only one sample per symbol. Owing to the selection of a segment of overlapped frequency spectrum, the proposed method has low implementation complexity and high PAR. Our proposed CD estimation method can be implemented without tentative scanning procedures, leading to a fast response. We numerically verify that our proposed CD estimation method is insensitive to the ASE noise, fiber nonlinearity, and the use of modulation format. In the experiment, when a 35-GBaud Nyqiust shaped DP-16QAM signal is transmitted over the SSMF, we can successfully estimate the CD values with an error of less than 150-ps/nm, proving its feasibility for practical Nyquist-WDM systems. Therefore, the proposed CD estimation scheme with one sample per symbol is promising for the future integrated coherent receiver.
Funding
National Key Research and Development Program of China (2018YFB1801301); National Natural Science Foundation of China (62025502).
Disclosures
The authors declare no conflicts of interest.
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