Abstract
1 Introduction
Optical field reconstruction, including both intensity and phase information, has been a long-standing issue in many fields of physics and their applications. Generally, coherent detection has been commonly used to recover the optical field in optical communications with the help of a local oscillator (LO) laser. Thus, both the use of complex modulation formats and the mitigation of various transmission impairments arising in the standard single-mode fiber (SSMF), such as chromatic dispersion (CD) and polarization mode dispersion (PMD), become realistic,1
To circumvent the requirement of a strong CW optical carrier, a carrier-free phase-retrieval (CF-PR) receiver has been proposed to reconstruct the full optical field of the DSB signal via the intensity-only measurements of different projections. Then, a PR algorithm, such as the modified Gerchberg–Saxton (GS) algorithm, runs iteratively, until the numerically reconstructed intensities at all projection planes match the measurements. In terms of recovering quadrature amplitude modulation (QAM) signals, the CF-PR receiver has the simplest hardware configuration, including an optical splitter, a dispersive element, two single-ended PDs, and two-channel analog-to-digital converters (ADCs), in comparison with the CADD and ASCD receivers. However, the CF-PR receiver based on the modified GS algorithm usually suffers from slow convergency and the stagnant issue. Normally, around 1000 iterations are essential to approach a steady PR performance for the quadrature phase-shift keying (QPSK) signal.10 The use of more intensity measurements is helpful to improve the PR performance with fewer iterations and steady bit error ratio (BER) at the cost of additional utilization of PDs and dispersion elements. The required number of iterations can be reduced to around 500 by utilizing three intensity measurements for the QPSK signal.11 Alternatively, a PR receiver variant by adding a CW optical carrier, for the ease of PR initialization, has been investigated to realize the effective PR acceleration.12
In this work, we propose a digital subcarrier multiplexing (DSM)-enabled CF-PR receiver, with hardware-efficient and modulation format-transparent merits. We identify that the DSM-enabled CF-FR receiver outperforms the conventional single-carrier counterpart when various modulation formats are employed by significantly reducing the implementation complexity of each PR iteration. Moreover, the DSM-enabled CF-PR receiver is helpful to enhance the robust operation towards various transmission impairments, including Tx laser linewidth and wavelength drift, Rx time skew, and the imbalance between two intensity tributaries. This paper is organized as follows. In Sec. 2, we introduce the generation of DSM signals at Tx and the corresponding operation principle of DSM-enabled CF-PR receiver. The simulation setup and comprehensive comparison between traditional single-carrier- and DSM-enabled CF-PR receivers are presented in Sec. 3. In Sec. 4, we carry out experimental verification for the DSM-enabled CF-PR receiver. Finally, we summarize our conclusion in Sec. 5.
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2 Operation Principle
2.1 Generation of DSM Signals
The DSM technique is based on dividing a high baud-rate single-carrier signal into multiple lower baud-rate subcarrier signals, which has been extensively explored in coherent fiber optical transmission systems.17
Figure 1.Generation of DSM signals at Tx.
2.2 DSM-Enabled CF-PR Receiver
The objective of the CF-PR receiver is to reconstruct the optical field of QAM signals from two intensity measurements, and , where and are the undispersed signal and the dispersed signal, respectively. We set and for the ease of discussion. The solution for the reconstruction of the optical field can be represented as a problem of identifying a phase signal to satisfy the following equation:
Function |
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2. |
3. |
4. |
5. |
6. |
7. |
8. |
9. |
10. |
11. |
12. |
13. |
14. |
15. |
16. |
Table 1. PR_AIT algorithm for DSM signals.
3 Simulation Results and Discussions
3.1 Simulation Setup
To examine the performance of the proposed DSM-enabled CF-PR receiver, we carry out numerical simulations for single-polarization QPSK/16QAM/32QAM signals. The simulation setup and the corresponding DSP stack are shown in Fig. 2. The RRC-shaped DSM signals are first generated offline with an aggregate baud rate of 56 GBaud, as shown in the following equation:
Figure 2.Simulation setup and corresponding DSP stack.
The transmission link includes a span of 80-km SSMF with a CD parameter of . We ignore the nonlinearity induced by the SSMF for simplicity. After the SSMF transmission, a noise-loading module is used to adjust the optical signal-to-noise ratio (OSNR) of received signals. After passing through an optical bandpass filter (OBPF) with a 3-dB bandwidth of 100 GHz, the signal is fed into the CF-PR receiver, including an optical splitter, a dispersive element, two single-ended PDs, whose response is emulated as a fourth-order Bessel filter with a 3-dB bandwidth of 60 GHz, and two ADCs with a resolution of 8 bits and a sampling rate of . Once the optical field reconstruction is realized by the PR_AIT algorithm, the following Rx DSP flow for the DSM signals first begins with the subcarrier demultiplexing, CDC, and the matched RRC filtering. Afterwards, the downsampling to one Sps and the symbol decision are implemented under the subcarrier basis. Finally, we calculate the BER for each subcarrier and record the averaged BER for the DSM signals.
3.2 PR Performance Comparison
First, we investigate the effect of the dispersive element on the performance of the DSM-enabled CF-PR receiver, under the condition of 19, 27, and 33 dB OSNR for QPSK/16QAM/32QAM signals, respectively. The CF-PR receiver with traditional single-carrier modulation is presented, for ease of fair comparison.16 The achieved BER with respect to the CD value of the dispersive element is summarized in Fig. 3. The total number of iterations is set to 500 to ensure full convergence, and 20% periodically inserted pilot symbols are used. We find that the PR performance improves with the growing CD value due to the involvement of more symbols. We credit such a phenomenon to the fact that the introduction of a large CD is helpful to decorrelate two intensity measurements and prevent the PR_AIT algorithm from being trapped into a suboptimal solution. Moreover, the BER starts to saturate, after the use of a sufficiently large amount of CD. The minimum CD value required to ensure optimal PR performance is found to be . For ease of discussion, we fix the CD value of the dispersive element at for next investigations. In particular, the BER performance is almost the same for single-carrier- and DSM-enabled CF-PR receivers under various CD values.
Figure 3.Simulated BER with respect to the dispersion of dispersion element used when the modulation format is (a) QPSK, (b) 16QAM, and (c) 32QAM.
Next, we investigate the BER performance with respect to various OSNRs for the single carrier- and the DSM-enabled CF-PR receivers; the theoretical performance is also included for the purpose of comparison. As shown in Fig. 4, the BER performance becomes better along with the growing OSNR. Again, the BER performance is almost the same for single-carrier- and DSM-enabled CF-PR receivers, after the convergency. However, both the single-carrier- and DSM-enabled CF-PR receivers experience a performance penalty in comparison with the theory curve. To reach the threshold of 7% hard-decision FEC (HD-FEC) at , the required OSNRs for 56 GBaud QPSK/16QAM/32QAM signals are 18.5, 26.2, and 32.4 dB, respectively.
Figure 4.Simulated BER with respect to the OSNR of received signals when the modulation format is (a) QPSK; (b) 16QAM; (c) 32QAM.
To investigate the performance of the convergence speed, we investigate the BER performance with respect to the number of iterations, as shown in Fig. 5, under the condition of 19, 27, and 33 dB OSNR for QPSK/16QAM/32QAM signals, respectively. The BER performance becomes better with the growing number of iterations and is finally saturated to a steady value. Note that an optimal number of iterations occur, which are determined by the AIT parameters and OSNR as a result of the interaction between the noise and intensity signal when the PR_AIT algorithm is applied.16 Similarly, the single-carrier- and DSM-enabled CF-PR receivers have the same BER performance under various numbers of iterations, revealing that both CF-PR receivers have the same convergency speed.
Figure 5.Simulated BER as a function of iteration when the modulation format is (a) QPSK; (b) 16QAM; (c) 32QAM.
3.3 Tolerance of Transmission Impairments
Intuitively, the phase noise induced by the laser linewidth is inessential during the intensity measurements, without impact on the PR performance. For the first time, we investigate the impact of phase noise on the PR performance by tuning the Tx laser linewidth. When the required OSNR without phase noise is considered as the benchmark, the required OSNR penalty with respect to the laser linewidth is summarized in Fig. 6. The PR performance is quite susceptible to the phase noise, because the interaction between phase noise and CD generates the phase-to-amplitude conversion noise.23 In addition, the required OSNR penalty becomes larger along with the use of higher-order QAM, because the higher-order QAM is sensitive to the noise. However, we find that, the DSM-enabled CF-PR receiver has a better tolerance toward phase noise because the DSM signals have less CD-induced impairment.24,25 The larger number of subcarriers we set, the better the PR performance can be always guaranteed under sufficiently large linewidth.
Figure 6.Simulated tolerance toward the laser linewidth when the modulation format is (a) QPSK; (b) 16QAM; (c) 32QAM.
Since the uncooled lasers are preferred for the reduction of power consumption, the operation wavelength of the laser may vary with respect to the time and environment. Such a wavelength drift further worsens the CD-induced phase distortions, which are proportional to the square of laser wavelength,26 as shown in the following equation:
Figure 7.Simulated tolerance toward the wavelength drift when the modulation format is (a) QPSK; (b) 16QAM; (c) 32QAM.
Finally, we investigate the tolerance toward the imperfection of the CF-PR receiver, including the time skew and the amplitude imbalance between two tributary signals, and . Note that we load additional noise to the signal to emulate the amplitude imbalance after the amplitude normalization, which indeed results in an OSNR difference. Figures 8 and 9 show the required OSNR penalty for the signal to reach the 7% HD-FEC threshold. Evidently, the PR performance degrades along with receiver imperfections for both the single-carrier- and DSM-enabled receivers. Considering 1 dB required OSNR penalty, the skew tolerance of 7.10, 3.37, and 1.43 ps is obtained for the single-carrier-enabled CF-PR receiver employing QPSK/16QAM/32QAM modulation, respectively, as shown in Fig. 8. Meanwhile, the skew tolerance can be enhanced to 7.43, 4.21, and 1.63 ps for the eight-subcarrier DSM-enabled CF-PR receiver utilizing QPSK/16QAM/32QAM modulation, respectively. The reason the DSM-enabled CF-PR receiver guarantees an improved skew tolerance is that the symbol duration of each subcarrier of DSM signals is much longer than that of the single-carrier signal. Similarly, the amplitude imbalance tolerance is found to be 2.40, 2.02, and 1.53 dB for the single-carrier-enabled CF-PR receiver with QPSK/16QAM/32QAM modulation, respectively. Alternatively, it can be improved to 2.42, 2.33, and 1.71 dB for the eight-subcarrier DSM-enabled CF-PR receiver with QPSK/16QAM/32QAM modulation, respectively. Therefore, the DSM-enabled CF-PR receiver outperforms its traditional single-carrier counterpart in terms of the tolerance towards receiver-side imperfections. Although the PR_AIT algorithm is taken into account during our investigation, the same conclusion can be drawn for other CF-PR algorithms.
Figure 8.Simulated tolerance toward the receiver skew when the modulation format is (a) QPSK; (b) 16QAM; (c) 32QAM.
Figure 9.Simulated tolerance toward the amplitude imbalance when the modulation format is (a) QPSK; (b) 16QAM; (c) 32QAM.
3.4 Comparison of Calculation Complexity
Since the calculation complexity of the CF-PR receiver is critical for the hardware implementation, it is essential to compare the single-carrier- and DSM-enabled CF-PR receivers. The PR_AIT algorithm is implemented in a block-wise manner with a block size of and for the single-carrier- and DSM-enabled receivers, respectively.23
Therefore, the AIT_PR algorithm for each iteration requires real multiplications, real adders, and a LUT for the DSM-enabled CF-PR receiver with blocks ( samples in total), respectively. Since the block size is determined by the accumulated CD from the fiber and dispersion element, we choose it to be 1024 by taking 50% overlapping frequency domain equalization into account. Therefore, we can obtain the calculation complexity of the PR_AIT algorithm, as shown in Fig. 10. The calculation complexity can be greatly reduced by the utilization of the DSM-enabled CF-PR receiver. As for the eight-subcarrier DSM-enabled CF-PR receiver, we can reduce the calculation complexity by 39.13% and 40.90%, respectively, in terms of the real multiplications and real adders, in comparison with the single-carrier-enabled CF-PR receiver.
Figure 10.Implementation complexity comparison, in terms of (a) number of adders and (b) number of multiplications.
4 Experimental Results and Discussions
We carry out a proof-of-concept experimental verification, as shown in Fig. 11. A C-band external cavity laser with the central wavelength of 1546.4771 nm and a linewidth of less than 100 kHz is chosen as the laser source. At the Tx, the single-carrier and DSM 16QAM electrical signals are generated offline and filtered by an RRC filter with a roll-off factor of 0.01. Afterwards, a pre-emphasis module is applied to compensate for the spectral attenuation from the Tx. After the resampling, the signal is loaded into the arbitrary waveform generator (AWG) operated at sampling rate with 8-bit resolution. The aggregate baud rate of the DSM signal is 25 GBaud, which is the same as that of single-carrier signals. Afterwards, the electrical signal is first amplified and then fed to the I/Q modulator for the electrical-to-optical conversion. The I/Q modulator output is boosted to 5.7 dBm by an EDFA, and then introduced into a span of 40 km SSMF.
Figure 11.Experimental setup of 25 GBaud 16QAM fiber optical transmission.
At the Rx, the optical signal is first amplified to 24 dBm, and the noise loading is applied to manage the OSNR. Afterwards, the optical signal is filtered with an OBPF and then split into two parts with a 30:70 optical coupler. The 70% power tributary is dispersed by two cascaded dispersion compensation modules (DCMs) with a tunable CD value within the range from 0 to . The total insertion loss of two cascaded DCMs is 8.5 dB. The received optical power of both the undispersed signal and the dispersed signal is . Then, two optical signals are detected by two PDs and digitized by an real-time oscilloscope. As for the Rx DSP, the signals are first resampled to two Sps, followed by the PR_AIT algorithm. Then, the reconstructed signal is fed to the CDC module, along with subcarrier demultiplexing and matched RRC filtering. Thereafter, a feedforward equalizer based on the least mean square algorithm is applied for each subcarrier before the symbol decision. Finally, we calculate the BER for ease of performance comparison.
We first investigate the convergency for the single-carrier- and DSM-enabled CF-PR receivers, as shown in Fig. 12(a). Obviously, the single-carrier- and DSM-enabled receivers have the same BER performance, under various number of iterations, indicating the same convergency speed. Additionally, the obtained BER for both receivers first decreases, then increases, and finally gets saturated along with increasing the number of iterations, which is consistent with that in simulations. To ensure the optimal performance, we fix the number of iterations at 60, to experimentally guarantee the optimal PR performance. We then investigate the BER performance with respect to the OSNR, as shown in Fig. 12(b). The obtained BER becomes gradually better along with the growing OSNR. Both the single-carrier- and DSM-enabled receivers show the same BER performance under various OSNRs. Therefore, the single-carrier- and DSM-enabled CF-PR receivers have the same PR performance in terms of convergency speed and steady BER performance.
Figure 12.Achieved BER as a function of (a) number of iterations and (b) OSNR.
Thereafter, the impact of laser linewidth on the CF-PR imperfections on the single-carrier- and DSM-enabled receivers is investigated. Note that we manually add phase noise in the digital domain after the pre-emphasis to emulate the tunable linewidth. Figure 13(a) summarizes the relationship between the laser linewidth and the BER under the condition of OSNR of 27 dB. The use of the DSM-enabled CF-PR receiver can enhance the tolerance toward the laser linewidth. Finally, we examine the impact of skew on the CF-PR receiver, as shown in Fig. 13(b). Similarly, better BER performance can be guaranteed for the DSM-enabled CF-PR receiver under the condition of variable skew.
Figure 13.Achieved BER as a function of (a) laser linewidth, and (b) receiver skew.
5 Conclusion
We propose a DSM-enabled CF-PR receiver with both the reduction of implementation complexity and the transparent operation of modulation format. When the optical field of 56 GBaud QPSK/16QAM/32QAM signals is numerically reconstructed after 80-km SSMF transmission, we identify that the DSM-enabled CF-PR receiver outperforms its traditional single-carrier counterpart in terms of implementation complexity and the tolerance towards various transmission impairments, including the transmitter laser linewidth, the transmitter wavelength drift, the receiver skew, and the amplitude imbalance between two intensity measurements. Finally, we carry out the experimental demonstration of recovering 25 GBaud 16QAM signals after a 40-km SSMF transmission, verifying the superiority of the DSM-enabled CF-PR receiver.
Biographies of the authors are not available.
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