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Abstract
We investigate the rate adaptability of quadrature amplitude modulation (QAM)-based probabilistic constellation shaping (PCS) using a fixed forward error correction (FEC) scheme over a wide range of information rates (IRs). Blind adaptive equalization that does not sacrifice any of the IRs was adopted. We show that the conventional decision directed least mean square (DDLMS) algorithm can cause a problem of mis-convergence when it is applied to the PCS of a low IR. To avoid the mis-convergence of DDLMS, we propose a DDLMS-based algorithm that simultaneously minimizes the error between the average symbol power of filter outputs and that of a transmitted PCS signal. Using this technique, we conducted a wavelength-division multiplexed transmission experiment with 32-Gbaud 16/64QAM-based PCS and a fixed FEC of a low-density parity-check code for DVBS-2, where the IR of PCS was optimized at each transmission distance. We confirmed that the data rate of PCS with a fixed FEC and DSP configuration could be improved up to 1.9 times compared with that of QAM-only rate adaptation and that 64QAM-based PCS could provide a wider range of transmission distance and IR while almost covering that of the 16QAM-based one.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
There is in general a trade-off between spectral efficiency and the transmission reach of modulation formats [1,2]. Therefore, rate adaptation is required if we want to maximize the data rate for a given channel. It is however difficult for conventional optical fiber communication systems that adopt high-speed analog devices and digital circuits to deal with flexible rate adaptation. Dedicated systems for specific transmission distance would not be promising from an economic point of view since optical communication systems cover from short reach to long-haul submarine systems. Switching the order of QAM provides a very limited data rate. Set-partitioning or trellis coded modulation enables more flexible tuning, though it is still coarse [3,4].
Probabilistic constellation shaping (PCS) is an attractive technology for providing fine rate adaptability together with high sensitivity approaching the Shannon limit [5–7]. PCS uses a certain base modulation format, for which square quadrature amplitude modulation (QAM) formats such as 16QAM and 64QAM are often used, and it changes the probability of the occurrence of constellation points. By adjusting the variance of the probability distribution, PCS enables fine rate adaptation and generation of a signal close to the optimum Gaussian distribution for a linear channel. To maximize the performance for a certain channel condition, the probability distribution of PCS and forward error correction (FEC) should be optimized ideally, whereas it has been numerically shown that the optimal FEC coding rate for each QAM-based PCS is nearly constant in a practically-interesting signal-to-noise ratio (SNR) range and that rate adaptation yielding near the channel capacity can be performed only by PCS with a fixed FEC code rate [8]. Several remarkable transmission results have been achieved by using PCS, including a C- and L-band wavelength-division multiplexed (WDM) transmission of 70.4 Tb/s together with geometric constellation shaping over 7,600 km [9], a field trial of a high spectral efficiency transmission of 5.68 b/s/Hz over 11,064 km of submarine cables [10], and a 50-km transmission of 4096QAM-based PCS with a spectral efficiency of 17.3 b/s/Hz [11], while a relatively narrow range of a high information rate (IR) has been of interest, and a few values of IR have been evaluated in these experiments.
For demodulation of a PCS signal, it has been considered that digital signal processing (DSP) for QAM can basically be used [7,12], which is one of the merits of PCS over other capacity-approaching modulation schemes such as geometrical shaping. One important DSP step required in optical fiber communications is adaptive polarization demultiplexing and equalization, where filter coefficients must be controlled adaptively to track the variation of the polarization state through a fiber. In previous experiments using PCS, this was accomplished mainly with pilot signals [10,11,13–15]. The pilot-based approach might be reliable, especially for higher order QAM-based PCS; however, it requires a pilot signal to be inserted, which sacrifices a portion of the data rate more or less. Alternatively, the blind approach without pilot signals does not degrade the data rate, whereas it must work with PCS regardless of the probability distribution. In this category, polarization demultiplexing in Stokes space and radius-directed equalization that uses prior probability of PCS to decide the symbol amplitude has been proposed [16].
In this study, we investigate the rate adaptability of QAM-based PCS with a fixed FEC and blind adaptive equalization without pilot signals over a wide range of IR. The decision directed least mean square (DDLMS) algorithm is widely used for blind adaptive filter control for QAM signals [17]; however, we show that it can cause a problem when it is applied to higher order QAM-based PCS with a low variance of the probability distribution in order to set a low IR and high SNR tolerance. In some cases, most of the symbols can be mapped on the constellation points of QAM, even with a particular phase offset and amplitude ratio, since a few constellation points are mainly used by PCS from those of QAM. This leads to the presence of local minima for the filter coefficients controlled with DDLMS, which causes the PCS signals to mis-converge. To avoid mis-convergence of DDLMS, we propose a DDLMS-based algorithm that simultaneously minimizes the error between the average symbol power of the filter outputs and that of the transmitted QAM-based PCS signal to avoid the mis-convergence of DDLMS applied to PCS. Using this technique, we conducted a WDM experiment with 16 channels of 32 Gbaud 16/64QAM-based PCS on a 50-GHz grid to assess the wide range rate adaptability of PCS with a fixed FEC of low-density parity-check code (LDPC) for DVBS-2. Optimizing the IR of PCS at each transmission distance, we confirmed that PCS with a fixed FEC, symbol rate, and DSP configuration could provide fine rate adaptability and that its data rate could be improved up to 1.9 times compared with that of rate adaptation by changing the modulation level of only QAM. We also experimentally demonstrate that 64QAM-based PCS could provide a wider range of the transmission distance and IR while almost covering that of 16QAM-based one.
The paper is organized as follows. First, we show numerical simulation results of the performance of rate adaptation with PCS with a fixed FEC over an additive white Gaussian noise (AWGN) channel. Then, we assess the performance of blind adaptive equalization of the conventional DDLMS for QAM-based PCS and explain the proposed blind DSP for a wide range of IR to avoid mis-convergence. Finally, we show the results of the WDM transmission experiment.
2. Rate adaptation with PCS
We first evaluated the performance of rate adaptation with PCS with a fixed FEC in numerical simulations. An AWGN channel is assumed here, and modulation or demodulation are performed ideally without considering DSP for demodulation.
2.1 Data frame structure of PCS
PCS is generated by the probabilistic amplitude shaping (PAS) architecture [5,18]. The data frame structure of $M^{2}$-QAM-based PCS with the PAS architecture is shown in Fig. 1, where $L_{\mathrm {FEC}}$ and $R$ are the FEC frame length and FEC coding rate. In the PAS architecture, PCS and FEC encoding are carried out jointly in each in-phase (I) and quadrature (Q) dimension. The frame length of a generated PCS signal becomes $L_{\mathrm {FEC}}/\log _{2} M$. There are two parts to the payload for uniform binary data bits. One part is converted by a distribution matcher (DM) for constructing unilateral pulse-amplitude modulation (PAM) signals with a desired probability density. FEC parity is added to the total of the converted non-uniform binary bits and the rest of the data bits. FEC parity bits and the rest of the data bits act as sign bits to construct bilateral PAM signals. Combining PAM signals for I and Q, a probabilistically shaped (PS)-QAM signal is obtained. Through this procedure, the probability distribution of PCS becomes a Boltzmann distribution, whose variance is controlled by the rate parameter $\lambda$.
Fig. 1. Data frame structure of PCS based on PAS architecture. DM: distribution matcher, FEC: forward error correction.
The IR of PCS is adjustable even at a fixed FEC. The entropy $\mathbb {H}$ of the PCS signal is related to the IR as [18]
In the case of 64QAM-based PCS, the entropy ranges from 2 bit/symbol to 6 bit/symbol, and thus the data rate ranges from 51.2 Gb/s to 307.2 Gb/s assuming a polarization-division multiplexed (PDM) signal and a symbol rate of 32 Gbaud at an FEC coding rate of 4/5. In the case of 16QAM-based PCS, that ranges from 76.8 Gb/s to 204.8Gb/s. Higher order QAM-based PCS provides both a lower and higher data rate at a fixed FEC.2.2 Simulation on AWGN channel
In this study, constant composition distribution matching [19] was used for DM and LDPC for DVBS-2 with a frame length of 64,800 and a code rate of 4/5 for FEC. 64QAM and 16QAM were used as a base constellation of PCS.
The performance of PCS with a fixed FEC was evaluated over an AWGN channel without effects of fiber propagation, opto-electronic devices, and DSP performance. Random data bits were loaded in the frame shown in Fig. 1, though PDM was not considered in this simulation. Fifty frames were generated for each I and Q (100 frames in total). After white Gaussian noise was added, the signal was received with a soft-decision bit-metric receiver calculating log-likelihood ratios. With FEC decoding and inverse DM, the post-decoding bit error rate (BER) was calculated. The pre-decoding BER was calculated by comparing the received pattern to that of the transmitted one with the bit-symbol mapping of QAM and information of the prior probability of bits [7]. The number of data bits for evaluating the post-decoding BER depended on the IR of PCS, which corresponds to about 0.8 to 5 Mbits. The generalized mutual information (GMI) and the normalized GMI (NGMI) [20,21] were also calculated by using the AWGN channel model and SNR. For comparison, the performances of conventional 64QAM, 16QAM, and QPSK were also evaluated with the same FEC.
Figure 2 shows the pre/post-decoding BERs of QPSK, whose IR was 1.6 bit/symbol with an FEC coding rate of 4/5, 16QAM (3.2 bit/symbol), 64QAM (4.8 bit/symbol), PS16QAM with IR of 1.6 bit/symbol, PS64QAM (1.6 bit/symbol), and PS64QAM (3.2 bit/symbol), against the SNR. 16QAM and PS64QAM (3.2 bit/symbol) had the same IR and performed similarly, though the latter was post-decoding error-free at a slightly lower SNR, which is the gain by PCS. Three formats yielding 1.6 bit/symbol performed almost the same. Figure 3 shows the GMI and the NGMI. When SNR increased, the GMI and NGMI approached asymptotically to the entropy of the transmitted signal and unity, respectively. Figure 4 shows the relation of the NGMI and the post-decoding BER. These curves are close around the BER cliff, which confirms that the NGMI is a good indicator for the post-decoding BER [21]. PCS provides higher post-decoding BERs compared with QAM at a low NGMI region, because bit errors remained after FEC decoding disperse through inverse distribution matching. In this simulation, no error post-decoding was achieved above an NGMI of 0.85.
Fig. 2. Received (a) pre-decoding BER and (b) post-decoding BER of various modulation formats over AWGN channel.
The performance evaluated while changing the IR of PS64QAM at a constant SNR of 5 dB, which is near the lowest value of SNR where QPSK achieves post-decoding error-free, is shown in Fig. 5. Decreasing the IR from 4.8 bit/symbol, post-decoding error free was achieved below an IR of 1.7 bit/symbol, whereas the maximum GMI was obtained at an IR of 2.3 bit/symbol. These two did not match probably because of the discrepancy between the performance of the fixed FEC of DVBS-2 and that of an ideal FEC with infinite frame length. The IR which resulted in an NGMI above 0.85 corresponded to 1.7 bit/symbol. Hereinafter, we focus on the upper limit of the IR achieving no error post-decoding.
Fig. 5. Optimization of information rate of PS64QAM at fixed SNR of 5 dB: (a) pre/post-decoding BER and (b) GMI/NGMI as function of information rate.
The results of the optimized IR achieving no error post-decoding of PS64QAM and PS16QAM for each SNR are shown in Fig. 6. Those of QAM are also plotted. The larger the SNR, the higher an IR achieving no error post-decoding could be obtained. QAM-based PCS with a fixed FEC enabled fine rate tuning in contrast that changing the modulation level of QAM took only few values. Around the SNR range from 3 dB to 10 dB, 64QAM-based and 16QAM-based PCS provided almost the same IR. PS64QAM could deal with a lower and higher SNR, because a fixed FEC frame considered here enables a higher order QAM-based PCS to set both a higher and lower IR. These simulation results confirm that higher order QAM-based PCS with a fixed FEC has flexible rate adaptability and performs well for a wide range of SNR. To realize this rate adaptation practically, demodulation DSP, which works with PCS regardless of the variance of the probability distribution, or a high to low IR, is crucial.
3. Blind adaptive equalization for QAM-based PCS with wide range of information rate
We assessed the performance of the DDLMS algorithm, which is widely used for blind adaptive equalization and polarization demultiplexing for QAM signals, when applied to higher order QAM-based PCS with a wide range of IR. The offline DSP used in the experimental evaluation is shown in Fig. 7(a). The received signals were first normalized and resampled to two-fold oversampling in which the sampling phase was optimized manually in this experiment. Then, chromatic dispersion (CD) compensation and a low-pass filter with matched root raised cosine were performed. After the compensation of the frequency offset, which was estimated with the fourth power of the adaptive filter outputs after pre-convergence [22], polarization demultiplexing and equalization were performed with adaptive 2 $\times$ 2 MIMO filters, where the frequency domain filters shown in Fig. 7(b) were used to calculate the block average symbol power, which is explained later. Coefficients of frequency domain filters were adaptively controlled. The received signals were converted to blocks with overlap to perform the fast Fourier transform (FFT). The filter outputs were transformed into the time domain by the inverse FFT (IFFT). They were downsampled to a symbol rate with the overlap removed, and carrier phase estimation (CPE) and compensation were performed with a decision directed phase locked loop. After adaptive 2 $\times$ 2 MIMO filters, the Gram-Schmidt orthogonalization procedure [23] was performed to compensate for the distortion from the deviation of modulator biases in the transmitter.
Fig. 7. Offline DSP for demodulation: (a) schematic diagram, (b) adaptive 2 $\times$ 2 MIMO with frequency domain filters. CD: chromatic dispersion, LPF: low-pass filter, FFT: fast Fourier transform, CPE: carrier phase estimation.
Figure 8 shows an example of mis-convergence when the conventional DDLMS algorithm for QAM was used for filter control. Plotted are the constellations of X and Y polarizations after demodulation of PDM PS64QAM with IR of 1.6 bit/symbol per polarization in a back-to-back configuration. Such a signal of low IR with high order QAM-based PCS uses only a few constellation points near the origin with high probability. The received symbols shown in Fig. 8 suffer a particular phase offset and amplitude ratio, while they mostly remain on any of the constellation points of 64QAM. Therefore, the symbol decision for 64QAM returns a small error for these received symbols; nevertheless, they are not the desired ones. In other words, this situation is one of the local solutions of DDLMS, and filter coefficients are not left by DDLMS itself. This is the cause of the mis-convergence of DDLMS for PCS. One approach to resolving this problem might be to use the maximum a posteriori decision rule with a prior probability distribution [16], whereas the prior probability is not as influential in terms of decision as the distance to the constellation point candidates.
Fig. 8. Example of mis-convergence of conventional DDLMS applied to PDM-PS64QAM with IR of 1.6 bit/symbol.
In this study, we used a modified DDLMS while simultaneously minimizing the error between the average symbol power of the filter outputs and that of the PCS signal. The DDLMS algorithm operates with the average symbol power forced to the desired value determined by the probability distribution of PCS so that mis-convergence can be avoided while polarization demultiplexing and equalization are accomplished. The outputs $Y_{i}$ of frequency domain 2 $\times$ 2 MIMO filters are described as
where $X_{j}$ are the inputs of the filters that are converted from the received signal $x_{j}$ of two-fold oversampling by FFT as $X_{j}[n] = \mathcal {F}[x_{j}[k]] = \sum _{k} x_{i}[k] \exp (i 2 \pi n k/N)$, and $H_{ij}$ are the filter coefficients in the frequency domain. The filter outputs in the time domain are $y_{i}[k] = \mathcal {F}^{-1}[Y_{i}[n]]$. The downsampled $y_{i}[2 l]$ are the samples at the symbol timing. The DDLMS algorithm updates the filter coefficients to minimize the magnitude of the error between the filter outputs and the decided symbols with the instantaneous error values through the stochastic gradient algorithm. For frequency domain filters, the coefficients are updated with an appropriate constraint [24], or overlap-save and zero-padding, by where $E_{i}[n] = \mathcal {F}[e_{i}[k]]$, andThe modified DDLMS updates the filter coefficients while simultaneously minimizing the magnitude of the error between the average symbol power of the filter outputs and that of the transmitted PCS signal $P$,
The brackets represent averaging in the block of interest. Here, to understand how the modified DDLMS avoids mis-convergence from an intuitive point of view, we consider a simple example of $y=h x$, where $x$ is PS64QAM with IR of 1.6 bit/symbol. The mean squared error profile of $e_{i}$ and $\tilde {e}_{i}$ calculated for this condition are plotted in Fig. 9. In Fig. 9(a), there are multiple local minima other than those originating from the four-fold symmetry of QAM. The profile of Fig. 9(b) has the minimum at $|h|=1$ and a slight peak at the zero. Figure 10 shows the number of local minima of $|e_{i}|^{2} + \alpha |\tilde {e}_{i}|^{2}$ as a function of the IR of PS64QAM under the condition that $y=h x$ and $|h| \geq 1$. $e_{i}$ has four minima under a high IR in this condition, and they merely come from the four-fold symmetry of QAM. At a lower IR, the number of local minima exceeds four, and thus mis-convergence of DDLMS for 64QAM can occur. In contrast, the number of local minima shrinks to four over any IR when combining these errors. In this case, the coefficient update becomesFig. 9. Mean squared error profile of (a) $e_{i}$ and (b) $\tilde {e}_{i}$ calculated for condition that $y=h x$ and $x$ is PS64QAM with IR of 1.6 bit/symbol.
Fig. 10. Number of local minima of $|e_{i}|^{2} + \alpha |\tilde {e}_{i}|^{2}$ as function of IR of PS64QAM calculated for condition that $y=h x$ and $|h| \geq 1$.
In the case of blind adaptive filter control with DDLMS, the filter coefficients are usually pre-converged with a more robust method such as the constant modulus algorithm (CMA). In the evaluation, pre-convergence was done with CMA while the amplitude criterion was set to the value determined by the average symbol power of the transmitted PCS signal. After pre-convergence with CMA, the modified DDLMS was applied.
Figure 11 shows the constellations after demodulation using the modified DDLMS for the same received samples for Fig. 8. The transmitted signal of PS64QAM with IR of 1.6 bit/symbol was properly obtained, with mis-convergence avoided, in contrast to Fig. 8.
4. Experiment on WDM transmission and rate adaptation of QAM-based PCS signals
Next, we experimentally evaluated the performance of the WDM transmission of 32-Gbaud PDM QAM-based PCS signals and its rate adaptability with the same fixed FEC of LDPC with a frame length of 64,800 and a code rate of 4/5 as in the previous simulation. Blind adaptive equalization and polarization demultiplexing were performed with the same DSP configuration by using the modified DDLMS while simultaneously minimizing the error of the average symbol power of the filter outputs.
4.1 Experimental setup
A schematic diagram of the experimental setup is shown in Fig. 12. On the transmitter side, WDM signals of 16 channels with frequencies ranging from 192.90 to 193.65 THz on a 50-GHz grid were generated. The signal under evaluation was at a frequency of 193.3 THz, and it was generated by modulating a laser source having a linewidth of about 100 kHz with waveforms generated by a four-channel digital-to-analog converter (DAC) at a sampling rate of 64 GS/s with a vertical resolution of eight bits (Keysight: M8195A). The rest of the 15 channels were generated with a modulator and DAC. Because of the limitation of the DAC prepared at this time, the symbol rate of these 15 signals was 33 Gbaud. After equalization of the optical channel power, they were multiplexed with the evaluated signal to generate 16-channel WDM signals. In this evaluation, PDM 16/64QAM-based PCS of various IR, QPSK, 16QAM, and 64QAM were used as the modulation format, while the same modulation format was used for all of the WDM signals.
Fig. 12. Experimental setup for WDM transmission and rate adaptation of 32 Gbaud PDM QAM-based PCS signals. LD: laser diode, PM-AWG: polarization-maintaining arrayed waveguide grating, DAC: digital-to-analog converter, MOD: modulator, EQ: equalizer, PS: polarization scrambler, AOM: acousto-optic modulator, EDFA: erbium-doped fiber amplifier, OBPF: optical bandpass filter, ADC: analog-to-digital converter.
The data frame structure was the same in the simulation. Four successive frames with random data bits in their payload were generated for each I and Q. For the orthogonal polarization, the same data was used with sufficient decorrelation for simplicity. These data were upsampled to two-fold oversampling for DAC. Then, the root raised cosine filter with a roll-off factor of 0.1 and the compensation for the frequency characteristics of the transmitter devices were performed. In the case of PS64QAM, the generated waveform had a length of 86,400 symbol time slots, and there were 129,600 symbol time slots in the case of PS16QAM.
The rate adaptability was evaluated on a loop transmission setup. The loop consisted of four standard single mode fiber (SMF) spans with a length of 75 km and erbium doped fiber amplifiers (EDFAs) compensating for the span loss. An optical channel equalizer was used to compensate for the differences in gain and attenuation in a loop. The span input optical power was set to −1 dBm per channel, which was estimated as the optimum on the basis of the GN-model [25]. WDM signals were loaded to the loop transmission line with slow polarization scrambling at a rate of 10 $\times$ 2$\pi$ rad/s. Figure 13 shows the optical spectrum with a resolution bandwidth of 0.1 nm before transmission and after 10,200 km where the signals were PDM PS64QAM with IR of 1.6 bit/symbol. The optical signal-to-noise ratio (OSNR) was 16.5 dB/0.1 nm after 10,200-km transmission. The optical power difference of the WDM signals was within $\pm$ 2.6 dB.
Fig. 13. Optical spectrum of 50-GHz-grid WDM of 32-Gbaud PDM-PS64QAM with IR of 1.6 bit/symbol (a) before transmission and (b) after 10,200 km with resolution bandwidth of 0.1 nm.
After transmission, the signal under evaluation at a frequency of 193.3 THz was demultiplexed with an optical bandpass filter (OBPF) and then received with a polarization diversity coherent receiver with a laser source having a linewidth of about 100 kHz as a local oscillator. The outputs of the coherent receiver were sampled with a digital oscilloscope (Keysight: DSOX92504Q) at a sampling rate of 80 GS/s with a vertical resolution of eight bits. After offline DSP described in the previous section, pre- and post-decoding BER were calculated for the transmitted four frames. The frame size and the overlap size of the frequency domain MIMO filters were set to 128 and 64, respectively. The step sizes of $\mu$ and $\tilde {\mu }$ were normalized by the average power of the filter inputs [24], considering that it differs depending on the probability distribution of PCS. It should be noted that the PS64QAM with IR around 3.2 bit/symbol required more symbols for pre-convergence in this evaluation.
4.2 Back-to-back OSNR sensitivity
The evaluation results of back-to-back OSNR sensitivity of 32-Gbaud PDM QPSK (1.6 bit/symbol), 16QAM (3.2 bit/symbol), 64QAM (4.8 bit/symbol), PS16QAM with IR of 1.6 bit/symbol, PS64QAM (1.6 bit/symbol), and PS64QAM (3.2 bit/symbol) are shown in Fig. 14. The error-free results are plotted at 10$^{-5}$ for visibility. A pre-decoding BER floor was observed for 64QAM and 64QAM-based PCS at a high OSNR. The OSNR achieving the post-decoding error free was 23 dB/0.1 nm for 64QAM, 17 dB/0.1 nm for 16QAM, 16 dB/0.1 nm for PS64QAM (3.2 bit/symbol), and 10 dB/0.1 nm for QPSK, PS16QAM (1.6 bit/symbol), and PS64QAM (1.6 bit/symbol).
Fig. 14. Evaluation results of back-to-back OSNR sensitivity of (a) pre-decoding BER and (b) post-decoding BER.
4.3 WDM transmission and rate adaptation
We finally conducted the WDM transmission and evaluated the rate adaptation performance. PDM QPSK, 16QAM, 64QAM, 16QAM-based PCS, and 64QAM-based PCS were evaluated. In the case of PCS, the IR was optimized for each transmission distance in steps of 0.2 bit/symbol to achieve post-decoding error free. The constellations for 64QAM-based PCS of several IRs received at the input of the loop are show in Fig. 15. The received pre/post-decoding BER of PS64QAM (4.0 bit/symbol) after transmission are shown in Fig. 16, for example. Although the adaptive filters sometimes failed to converge when the received OSNR was considerably low, mis-convergence like in Fig. 8 never occurred in the evaluation.
Fig. 15. Constellations received before transmission of PDM PS64QAM. (a) 4.8 bit/symbol, (b) 4.0 bit/symbol, (c) 3.2 bit/symbol, (d) 2.4 bit/symbol, and (e) 1.6 bit/symbol.
Fig. 16. Received pre/post-decoding BER of PDM PS64QAM with IR of 4.0 bit/symbol after WDM transmission.
Figure 17 shows the evaluation results of the data rate achieving no error post-decoding for QPSK, 16QAM, 64QAM, and PS16QAM/64QAM with its IR optimized for each transmission distance. PS64QAM provided a larger and finer-grained data rate than any of the QAMs over a wide range of transmission distance. PS16QAM demonstrated fine rate adaptability from a transmission distance of 3,600 km, and there it provided a similar or slightly larger data rate compared with PS64QAM. At a transmission distance of 4,200 km, QPSK (102.4 Gb/s) could be transmitted with post-decoding error free, while 16QAM could not. PS64QAM and PS16QAM enabled 192 Gb/s to be provided under this condition, which is 1.9 times compared with that of QPSK. According to these results, the rate adaptability of PCS with a fixed FEC of LDPC for DVBS-2, a symbol rate of 32 Gbaud, and a DSP configured in a blind manner was confirmed.
Fig. 17. Experimental results for data rate of QAM-based PCS achieving post-decoding error free in WDM transmission.
In this experiment, the rate optimization of PCS was done after obtaining the transmission results. Figure 18 is the data rate achieving post-decoding error free for PS64QAM as a function of received OSNR with -1 dBm/ch. The curve shows behavior similar to the simulation results of Fig. 6, though it suffers from degradation including a fiber nonlinear penalty and back-to-back penalty. It indicates that monitoring OSNR, or more precisely the generalized one including the nonlinear penalty [25], would enable the optimal IR or $\lambda$ to be chosen in advance.
Fig. 18. Data rate achieving post-decoding error free for PDM PS64QAM as function of received OSNR with −1 dBm/ch.
5. Conclusion
Rate adaptability of QAM-based PCS with a fixed FEC of LDPC for DVBS-2 and blind equalization was investigated in a numerical simulation and WDM transmission experiment. To avoid the mis-convergence of the conventional DDLMS applied to PCS for adaptive polarization demultiplexing and equalization, DDLMS modified by using the average symbol power of PCS was proposed. WDM transmission of 16 channels with 32 Gbaud PDM 16/64QAM-based PCS signals on a 50-GHz grid was conducted with the IR of PCS optimized at each transmission distance. We confirmed that PCS with a fixed FEC and a DSP configured in a blind manner could provide fine rate adaptability, and that the data rate of PCS with a fixed FEC and a DSP configuration could be improved up to 1.9 times compared with that of QAM-only rate adaptation. We also experimentally demonstrated that 64QAM-based PCS could provide a wider range of transmission distance and IR while almost covering that of 16QAM-based PCS.
Appendix: Calculation of filter coefficient update
For MIMO processing with time-domain $M$-tap $T/2$ finite impulse response filters, the outputs $y_{i}$ are described as $y_{i}[k] = \sum _{j} \boldsymbol {h}_{ij}^{T} \boldsymbol {x}_{j}[k]$, where $\boldsymbol {h}_{ij} = (h_{ij}[0], \ldots , h_{ij}[M-1])^{T}$ are the filter coefficients, and $\boldsymbol {x}_{j}[k] = (x_{j}[k], \ldots , x_{j}[k-M+1])^{T}$ are the inputs. The DDLMS algorithm minimizes the magnitude of error $e_{i}[k] = d(y_{i}[k])-y_{i}[k]$ (the phase offset of the output is neglected for simplicity) in a stochastic gradient decent manner [17] as
For minimizing the error of the average symbol power of the filter outputs to that of the transmitted PCS signal $P$, $\varepsilon _{i} = \tilde {e}_{i}^2$ should be minimized, where
Differentiation with respect to the filter coefficients becomesFunding
Ministry of Internal Affairs and Communications (Research and development of innovative optical network technology as a new social infrastructure).
Acknowledgments
We thank Hidemi Noguchi, Norifumi Kamiya, and Emmanuel Le Taillandier de Gabory for the interesting discussions.
Disclosures
The authors declare no conflicts of interest.
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