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Abstract
We numerically and experimentally assessed novel optical nonlinear compensation systems and modified field-intensity averaging digital back propagation (mFIA-DBP). mFIA-DBP achieves a higher performance than DBP with lower computational complexity. For the 50 Gbaud-DP-16QAM system in the simulation, a higher performance was obtained by mFIA-DBP with one steps/span (StPS) over DBP with 2StPS. A comparison of the computational complexity in terms of the number of real multiplications shows that mFIA-DBP can compensate for the signal nonlinearity after a 4000 km transmission with 66.5% of the DBP. We confirmed the superior performance of mFIA-DBP in the high-power range and showed that this trend was consistent in experiments with the simulations.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The increasing global demand for data communication has led to rapid development of optical networks. Spectral efficiency is severely limited by the impact of nonlinear distortion owing to the optical Kerr effect in optical communication systems [1]. Digital signal processing (DSP)-based nonlinearity compensation methods have received considerable attention as an approach for mitigating fiber nonlinearities [2]. The DBP method mitigates the distortion caused by fiber dispersion and nonlinearity by solving the nonlinear Schrödinger equation (NLSE) for single polarization and the Manakov equation for polarization multiplexing, so that the received signal propagates in the opposite direction in the virtual link. To find the inverse solution of the NLSE based on the split-step Fourier method (SSFM), we continuously analyzed the propagating signal in the frequency domain for dispersion compensation and in the time domain for nonlinear compensation [3–7]. Contrary to the high compensation performance, however, DBP has an implementation challenge of high computational cost. Because the signal needs to be transformed repeatedly into the frequency and time domains, the fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) need to be executed repeatedly. The number of FFTs/IFFTs must increase as the transmission distance increases. Several low-complexity and advanced methods have been proposed to improve nonlinear compensation performance and reduce computational complexity. Volterra series nonlinear equalization (VSNE) uses the Volterra series expansion [8–10], which expands the linear and nonlinear terms in parallel, whereas SSFM expands them serially. The computational complexity of VLSE is lower than that of SSFM-based DBP. Advanced DBP algorithms have been proposed, and their complexity has been reduced [11–15].
Mateo et al. found from coupled nonlinear Schrödinger equation (C-NLSE) that cross phase modulation (XPM) has a much larger impact on the signal than four wave mixing (FWM), when comparing the computational complexity to be compensated [4], and also proposed an enhanced C-NLSE that compensates for part of the FWM [6]. The efficiency of XPM compensation by SSFM was improved by considering the effect of inter-channel walk-off [12]. The computational load was reduced to one-fourth that of the conventional DBP. However, the perturbation method can effectively compensate for the nonlinear interaction in the channel and is effective for compensating intra-channel XPM (IXPM) and intra-channel FWM (IFWM). It is also effective for polarization-multiplexed signals and systems [13,14]. In [14], a 32 Gbaud-dual polarized-16 quadrature amplitude modulation (DP-16QAM) system was used for a transmission distance of 2000 km, and the number of complex multiplications was reduced by 82% compared to the conventional DBP. The weighted DBP considers the correlation between adjacent symbols by including weighting in the nonlinear step. In contrast to SSFM, the nonlinear phase shift in one symbol is related to the power of various consecutive symbols, thus accounting for pulse spread due to chromatic dispersion (CD) [15]. By developing further improvements in DBP technologies, further increases in the capacity and speed of optical transmission systems can be realized. Therefore, the computational complexity of DSP is expected to continue to increase.
In this paper, we propose a novel nonlinear compensation method for SSFM. We aim to reduce the computational complexity of DBP by exploring new methods that can improve upon the existing effective methods, while keeping in mind the improved performance of the nonlinear compensation method. The remainder of this paper is organized as follows. In Section 2, the principle of the proposed modified field-intensity averaging (mFIA)-DBP is explained and compared with our previously proposed field intensity averaging-DBP (FIA-DBP) method [16]. The weighted average of mFIA-DBP is also explained. Section 3 presents the simulation results. 50 Gbaud-DP-16QAM signals were transmitted over 100 km-per-span standard single-mode fiber (SSMF), and nonlinear compensation was performed at the receiver. mFIA-DBP was compared to DBP in terms of performance and the real number of multiplications. In Section 4, the measurement results are presented. 10 Gbaud-DP-QPSK signals were transmitted through the SSMFs, and then an offline DSP was performed. Section 5 provides a summary of this study.
2. Operating principle
In the SSFM-based analysis, DBP mitigates nonlinear distortion by transmitting in the direction opposite to the virtual transmission in the DSP. A polarization-multiplexed optical signal distorted by transmission in a fiber link can be compensated for by DBP using the inverse Manakov equation in Eq. (1).
The conventional DBP method is accomplished by splitting Eq. (1) into linear and nonlinear steps. The linear step compensates for the dispersion in the frequency domain and the nonlinear step compensates for the fiber nonlinearity in the time domain. The proposed DBP method is also split into linear and nonlinear steps, as shown in Fig. 1. In FIA-DBP, the compensation performance was improved by averaging the field intensity obtained for each span, as shown in the block diagram in Fig. 1(a). The received signal, ${E_{x/y}}$ is input to the FIA-DBP compensator. In each step, the input signal is nonlinearly compensated by the DBP over all required steps. In each span, the DBP adapted signal is stored in the parameter ${E_{x/y,DBP,\; n}}$, and $E{^{\prime}_{x/y,DBP,\; n}}$ is stored in the memory after the residual linear distortion between the current calculated span and launched point are compensated. Here, the residual linear distortion is the distortion due to the wavelength dispersion generated during transmission minus the dispersion compensated by the DBP up to the current span. Thus, the residual linear distortion is the dispersion that occurs between the current DBP span and launched point. After processing the number of compensated steps and spans necessary to obtain sufficient performance, the field intensity estimated by FIA-DBP was derived by applying the averaging in Eq. (2) to $E{^{\prime}_{x/y,DBP,\; n}}$ stored in the memory.
Fig. 1. Block diagrams of the proposed DBP method. (a) field intensity averaging; ${N_{span}}$: number of DBP span, ${E_{x/y}}$: input field intensity to compensator, ${E_{x/y,DBP,{n_{span}}}}$: field intensity calculated by DBP at ${n_{span}}$ span, $E{^{\prime}_{x/y,DBP,{n_{span}}}}$: field intensity with $\textrm{L}{\textrm{C}_{\textrm{res}}}$ applied, ${E_{x/y,FIA - DBP}}$: field intensity with FIA-DBP applied, $\textrm{L}{\textrm{C}_{\textrm{res}}}$: residual linear distortion compensation. (b) modified field intensity averaging; ${E_{x/y,{n_{span}}}}$: field intensity at each span, ${E_{x/y,\textrm{mFIA} - \textrm{DBP},{n_{span}}}}$: field intensity with averaging applied at each span, ${E_{x/y,\textrm{mFIA} - \textrm{DBP}}}$: field intensity obtained with mFIA-DBP.
In contrast, mFIA-DBP uses a nonlinear operator $\Delta {\Phi _{\textrm{mFIA} - \textrm{DBP},{n_{span}}}}$ which averages the field intensity for each span, defined by
The simulation setup is shown in Fig. 2. In the transmitter, a pseudo-random code with a word length of ${2^{16}} - 1$ was generated, and a single wavelength 50 Gbaud-DP-16QAM signal with a carrier wavelength of 1550 nm was modulated. The signal is transmitted over a 100 km/span SSMF, and the loss is compensated using EDFAs. The parameters of the fiber for a propagation loss of 0.16 dB/km, dispersion parameter of 16 ps/nm/km, nonlinear coefficient of 1.33 W−1 km−1, and EDFA noise figure of 5 dB were used. The following process was performed in the receiver after the transmission. Phase noise generated by the LO used in the coherent receiver is added, linear (LC) or nonlinear compensation (NLC) and phase estimation are performed, and the compensation performance is compared using the ${Q^2}$ value. The error vector magnitude (EVM) was calculated from the constellation, and the bit error rate (BER) and ${Q^2}$ values were calculated using Eqs. (6) and (7) [20,21],
Figure 3 shows the optical field intensity transition of a certain 16QAM symbol. The field intensity transitions shown in Fig. 3 are the transmission (ideal), DBP, and field intensity $E{^{\prime}_{x/y,DBP,\; {n_{span}}}}$ using Eq. (2). The field intensity of the launched symbol was $E = \sqrt 5 $. The obtained field intensity compensated for the residual linear distortion under all conditions. At span number 20, where the calculation at the DBP ends, the field intensity exceeds the estimation limit of the adjacent field intensity; thus, the field intensity of DBP (▪) for each span is judged as an error. The field intensity of the FIA (▴) is the average field intensity of the DBP (▪) for each span. The field intensity transition using FIA (▴) is judged as the correct signal because its fluctuation is suppressed compared to that of DBP (▪), and it is within the estimation limit. The field intensity of the FIA (▴) is closer to the intensity $E = \sqrt 5 $ of the transmitted field than that of DBP (▪). We just show an example of electric field intensity transition in Fig. 3, but similar results were obtained in other symbols.
Fig. 3. Optical field intensity transition: the minimum field intensity of a 16QAM symbol is normalized to $|E |= 1.0$. Estimation limit is defined as the correctness boundary between the target field intensity and the adjacent field intensity defined on the constellation map.
Figure 4 shows the values when the weight w is carried. For ${Q^2} = 8.22$ dB, the value $w = 1$ is equivalent to the unweighted case. The peak weight $w = {w_{peak}}$ was found to be 8, where ${Q^2}$ reaches a maximum value of 8.60 dB. Changing the weight w changes the effect of the optical field ${E_{x/y,{n_{span}}}}$ on the nonlinear operator $\Delta {\Phi _{\textrm{mFIA} - \textrm{DBP},{n_{span}}}}$. When $w = 0$, only the optical field before ${n_{span}}$ is considered and ${E_{x/y,{n_{span}}}}$ is not considered, the ${Q^2}$ value is the smallest. When $w = 1$, the effect of ${E_{x/y,{n_{span}}}}$ is equivalent to the optical field before ${n_{span}}$ and is an additive average of the optical field intensity for each span. When $w > 1$, it is a weighted average, and when $w = {w_{peak}}$, the effect of ${E_{x/y,{n_{span}}}}$ is the best and the ${Q^2}$ value is the largest among the values of w.
3. Simulation results
3.1 Compensation performance
Figures 5(a)–(d) show the constellations of the compensated signals by linear compensation only, by nonlinear compensation using DBP (1 step per span (StPS)), FIA-DBP (1 StPS), and mFIA-DBP (1 StPS, $w = {w_{peak}}$), respectively. As can be seen in these figures, the broadening of the symbols with FIA-DBP and mFIA-DBP is suppressed compared with the other schemes.
Fig. 5. Constellation of received signals: (a) linear compensation only (${Q^2} = 7.95\; dB$); (b) DBP (1 StPS, ${Q^2} = 7.86\; dB$); (c) FIA-DBP (1 StPS, ${Q^2} = 8.17\; dB$); (d) mFIA-DBP (1 StPS, $w = {w_{peak}},\; {Q^2} = 8.60\; dB$)
The compensation performance (${Q^2}$ value) as a function of the optical power launched at a transmission distance of 2000 km is shown in Fig. 6. 1StPS and 2StPS were used for both DBP and FIA-DBP, respectively. 1StPS with $w$ = 1, and 1StPS and 2StPS with $w$ = ${w_{peak}}$ were used for mFIA-DBP. The peak power of the LC was 2 dBm, where ${Q^2}$ = 7.95 dB. The peak power of DBP (1StPS) was 1 dBm, which was less than that of the LC, and ${Q^2}$ = 7.86 dB. The degradation of the ${Q^2}$ value of DBP (1StPS) from LC was due to the fact that the range of symbols affected by CD increased due to higher baud rates. As a result, the signals had a large amount of nonlinearity and ASE coupling noise, which limited the performance of the DBP with a small number of StPSs. In contrast, the ${Q^2}$ value of the DBP (2StPS) was better than that of the LC, with a peak power of 2 dBm and ${Q^2}$ = 8.29 dB, which is 0.43 dB better than the ${Q^2}$ value over DBP (1StPS). The ${Q^2}$ value of the FIA-DBP improved over the LC for both the 1StPS and 2StPS cases. The peak power of FIA-DBP (1StPS and 2StPS) was 2 dBm, where ${Q^2}$ = 8.17 dB for 1StPS and 8.44 dB for 2StPS. Compared to DBP, the ${Q^2}$ value is 0.31 dB higher at 1StPS and 0.15 dB higher at 2StPS. In contrast, in the case of mFIA-DBP, the peak power was 2 dBm for both $w$ = 1 and $w$ = ${w_{peak}}$. When $w$ = 1, ${Q^2}$ = 8.22 dB, which is an improvement by 0.36 and 0.05 dB over DBP (1StPS) and FIA-DBP (1StPS), respectively. Furthermore, when $w$ = ${w_{peak}}$, ${Q^2}$ = 8.60 dB, which is a gain of 0.65 dB, 0.31 and 0.16 dB compared to LC, DBP (2StPS) and FIA-DBP (2StPS), respectively. Despite 1StPS, mFIA-DBP achieved a better compensation performance than DBP with 2StPS. Furthermore, mFIA-DBP can further improve ${Q^2}$ value by increasing StPS. When using mFIA-DBP at 2StPS, the maximum ${Q^2}$ value was 8.89 dB at an output power of 3 dBm. In these simulations, we did not consider the intrachannel nonlinearity coming from the orthogonal polarization tributaries, and we expect the performance further by introducing such terms [13].
Figure 7 shows the compensation performance (${Q^2}$ value) versus transmission distance. The launched optical power was set to 2 dBm, and the transmission distance on the horizontal axis was plotted at intervals of 500 km from 1000 to 5000 km. The horizontal dotted line indicates ${Q^2}$ = 5.90 dB (BER = $2.43 \times {10^{ - 2}}$), and the reachable transmission distance is examined based on this value. The reachable transmission distances were 3350 km for LC, 3700 km for DBP (2StPS), 3800 km for FIA-DBP (2StPS), 3600 km for mFIA-DBP (1StPS, $w$ = 1), 4000 km for mFIA-DBP (1StPS, $w = {w_{peak}}$), and 4250 km for mFIA-DBP (2StPS, $w = {w_{peak}}$). Therefore, mFIA-DBP (1StPS, $w = {w_{peak}}$) achieved increases of 650 km over LC, 300 km over DBP (2StPS), and 200 km over FIA-DBP (2StPS). The ${Q^2}$ values for a transmission distance of 4000 km are 5.23 dB for LC, 5.60 dB for DBP (2StPS), and 5.70 dB for FIA-DBP (2StPS). To achieve ${Q^2}$ = 5.90 dB for DBP and FIA-DBP, calculations of 3 StPS for DBP and 3 StPS and 27 spans for FIA-DBP, respectively are needed, resulting in higher computational complexity than mFIA-DBP as explained in the next subsection.
3.2 Computational complexity
Next, the computational complexity of each compensation method was compared in terms of the number of real multiplications (RM) [22]. The numbers of RM for each method are listed in Table 1. ${N_{FFT}}$ is the number of FFT blocks, ${N_s}$ is the number of DBP spans, s is the number of steps per span, ${N_{\textrm{span}}}$ is the number of FIA-DBP spans, ${N_{sym}}$ is the number of transmitted symbols. The first term of LC ($8{N_{FFT}}{\log _2}({{N_{FFT}}} )$) is the sum of the FFT and IFFT for x and y polarizations, assuming the Cooley-Tukey radix-2 FFT algorithm [23,24]. The second term ($8{N_{FFT}}$) is the number of times the rotation matrix is multiplied by the CD of the x and y polarizations. The second term of DBP is the sum of the nonlinear phase rotation ($13{N_s}s{N_{FFT}}$) and phase rotation due to CD ($8{N_s}s{N_{FFT}}$) [25]. The number of RM in FIA-DBP is the first term plus the number of calculations of the residual linear compensation ($L{C_{res}}$) for each span. The third terms of FIA-DBP (${N_{sym}}$) and mFIA-DBP (2 ${N_s}{N_{FFT}}$) are calculated by averaging the field intensity.
Table 1. Number of RM for each compensation method; ${{N}_{{FFT}}}$: the number of FFT blocks, ${{N}_{s}}$: the number of DBP spans, ${s}$: the number of steps per span, ${{N}_{{\rm span}}}$: the number of FIA-DBP spans, ${{N}_{{sym}}}$: the number of transmitted symbols.
Figure 8 shows the variation in the number of RM for each compensation method when the number of transmission spans was varied. In these simulations, ${N_{FFT}} = {2^{17}}$ and ${N_{sym}} = {2^{13}}$ were used. The horizontal axis represents the number of transmission spans, and the vertical axis represents the number of RM calculated from Table 1. LC has the lowest compensation performance among the plotted compensation methods, and thus, it is superior in terms of computational complexity, requiring $1.89 \times {10^7}$ RM. FIA-DBP (2StPS, full span) uses a slightly higher RM than DBP for the same number of transmission spans, whereas FIA-DBP (2StPS, half span) uses fewer RM. When the number of transmitted spans is 20 spans, the number of RM with FIA-DBP (2StPS, full span), DBP(2StPS), FIA-DBP (2StPS, half span), and mFIA-DBP are $1.18 \times {10^9}$, $8.23 \times {10^8}$, $5.90 \times {10^8}$, and $4.12 \times {10^8}$, respectively.
The difference in the number of RM between the mFIA-DBP and the other compensation methods was compared using the following equation:
3.3 Baud rate dependence
In studies of optical nonlinear compensation, various baud rates, such as 10 Gbaud [5,26], 28 Gbaud [13], 30 Gbaud [14,15], 40 Gbaud [27], and 50 Gbaud [28] have been investigated with the development of communication technology. Figure 9 shows the dependence of ${Q^2}$ values on the launched optical power at baud rates of 10, 25, and 50 Gbaud at a transmission distance of 2000 km. At 10, 25, and 50Gbaud, 1 StPS was used, and at 100Gbaud, it was 8 StPS. At 10 Gbaud, the peak power of DBP was 5 dBm, whereas it was −3 dBm for LC, showing a significant improvement. In addition, the ${Q^2}$ value of mFIA-DBP was almost identical to that of DBP and provides an advantage of 0.01 dB in the high-power region of 8 dBm. The differences in the nonlinear compensation characteristics of DBP, FIA-DBP, and mFIA-DBP at different baud rates were investigated by comparing the results. At 25 Gbaud, the peak power of LC was −1 dBm, and that for DBP, FIA-DBP, and mFIA-DBP was 2 dBm, each. The maximum ${Q^2}$ values, were 10.43 dB for DBP, 10.36 dB for FIA-DBP, and 10.56 dB for mFIA-DBP, confirming the improvement in the ${Q^2}$ values of mFIA-DBP over DBP. At 50 Gbaud, the peak power of the LC increased to 2 dBm, as also in the case of 25 Gbaud, whereas the peak power of DBP degraded to 1 dBm. In contrast, the peak powers of FIA-DBP, mFIA-DBP ($w$ = 1), and mFIA-DBP ($w = {w_{peak}}$) were 2 dBm each. There was an improvement in the ${Q^2}$ values of mFIA-DBP ($w$ = 1) and mFIA-DBP ($w = {w_{peak}}$) over LC and DBP at 1StPS; hence, the compensation performance of mFIA-DBP was larger than that at 25 Gbaud. To summarize the results of Fig. 9, in case of using nonlinear compensation, the launched power at lower baud rate is drastically improved, and on the other hand, improvement at higher baud rate becomes smaller because of residual distortion that is not completely compensated. In addition, the performances of FIA-DBP and mFIA-DBP are advantageous at higher baud rates. From the above, the larger the baud rate, the larger the impact of distortion caused by the interaction of the waveform broadening between the adjacent symbols owing to chromatic dispersion and nonlinear phase rotation.
4. Experimental results
To evaluate the effectiveness of the proposed method, we used a recirculating loop system, which is commonly used for long-haul transmission experiments on optical signals. Figure 10 illustrates the experimental setup. On the transmitter side, a pseudo-random code with a word length of ${2^{15}} - 1$ was generated using a signal generator. An external cavity laser (ECL) with a linewidth of 100 kHz was utilized as the light source, and DP-QPSK signals were generated using a dual-polarization IQ modulator (DP-IQM) at a baud rate of 10 Gbaud (with a delay between IQ components to eliminate the effect of correlation) [29]. The generated DP-QPSK signals were sent to the recirculating loop controlled by two acousto-optic switches (AOSs) while controlling the input optical power with an erbium-doped fiber amplifier (EDFA) and a variable optical attenuator (VOA). The recirculating loop comprised four cascaded $25\; \textrm{km}$ fiber spools which are characterized by a loss of $\alpha = 0.25$ dB/km on an average, a dispersion parameter of 16.54 ps/nm/km, and a nonlinear coefficient of 2 W−1 km−1. The loss was compensated for using an EDFA installed per span. An optical bandpass filter (OBPF) was inserted to restrict the bandwidth of the amplified spontaneous emission (ASE) generated by the EDFA. At the receiver side, an EDFA and VOA were utilized to control the received optical power, and an OBPF was used to remove excessive ASE before a coherent receiver. Finally, a real-time oscilloscope was used to digitize the received signals, and an offline DSP was performed. The offline DSP consists of frequency offset compensation, nonlinear compensation, polarization demultiplexing [30], phase estimation, and a final ${Q^2}$ calculation.
Figures 11(a) and (b) show the $x$- and $y$-polarization constellations under back-to-back conditions, and it can be seen that the QPSK signals can be generated clearly. For the instances shown, they are 11.1% and 12.0%, respectively. Figures 11(c) and (d) show the constellations of the $x$- and $y$-polarizations after the 10-loop transmission. It can be seen that the symbols of both polarizations are distorted owing to chromatic dispersion and nonlinearity, and it is necessary to demodulate offline DSP.
Fig. 11. Constellations for DP-QPSK signal; (a) and (b) are $x$- and $y$- polarization under back to back condition, (c) and (d) are $x$- and $y$- polarization after $10\; \textrm{loops} \times 100\; \textrm{km}$ transmission.
Figure 12(a) shows the ${Q^2}$ value obtained for each compensation method with respect to the number of loops. In Fig. 12, 1 StSP was used. When the number of loops is two, the effect of noise and nonlinear distortion is small; thus, the signal quality is good and ${Q^2} > 16.0\; \textrm{dB}$ for all four compensation methods. As the number of loops increases, the overall ${Q^2}$ value decreases due to the greater effect of noise, and the difference between the ${Q^2}$ values of the LC and nonlinear compensation methods becomes larger, with the mFIA-DBP improving the ${Q^2}$ value by 0.61 dB when the number of loops is 20. The ${Q^2}$ value difference between the nonlinear compensation methods is very small for loops up to 20 in number. The performance of the experiments is worse than that in simulation because of the optical loss in one recirculating loop in the optical coupler and the AOS.
Fig. 12. Performance of nonlinear equalization as a function of the number of loops (a) ${Q^2}$ vs. number of loops (b) The ${Q^2}$ difference of mFIA-DBP compared with LC, DBP, and FIA-DBP.
Figure 12(b) shows the difference in the ${Q^2}$ values between DBP and FIA-DBP, and $\Delta {Q^2}$ is defined as follows:
The largest effect is a ${Q^2}$ gain of 0.07 dB for DBP and 0.06 dB for FIA-DBP when the number of loops is 20. This result is consistent with the trend shown in Fig. 9, and therefore, we could expect that mFIA-DBP would be more effective under highly nonlinear conditions, such as higher baud rate and larger launched power.
5. Summary
In summary, we numerically and experimentally demonstrated that the mFIA-DBP has lower computational complexity and higher nonlinearity compensation performance. The proposed method improves the performance of DBP by providing a nonlinear operator that averages the field intensity for each span and introduces an optimized weight factor. In simulations for 50 Gbaud DP-16QAM at a transmission distance of 2000 km, a ${Q^2}$ value of 8.60 dB at $w = {w_{peak}}\; $ with mFIA-DBP is achieved, which shows a gain of 0.65, 0.31 and 0.16 dB compared to LC, DBP (2StPS) and FIA-DBP (2StPS), respectively. Owing to the superior ${Q^2}$, mFIA-DBP ($w = {w_{\textrm{peak}}})$ for 4000 km transmission achieved an increase of 650 km over LC, 300 km over DBP (2StPS), and 200 km over FIA-DBP (2StPS). Comparing the number of RM in a simulated 50 Gbaud-DP-16QAM system to achieve ${Q^2}$ = 5.90 dB at a distance of 4000 km and launching an optical power of 2 dBm, mFIA-DBP achieved a complexity reduction of 66.5% against DBP. The measurement of 10 Gbaud-DP-QPSK shows that the advantage of the mFIA-DBP performance is obtained mainly under highly nonlinear conditions, which is consistent with the simulation trend. It is also expected that the proposed method will show significant improvement in performance in higher baud rate systems.
Acknowledgments
The authors would like to thank Prof. Emeritus K. Iga, Prof. Emeritus K. Kobayashi, Prof. F. Koyama, and Assoc. Prof. T. Miyamoto of Tokyo Institute of Technology for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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