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Abstract
Since the timing error detectors sensitivity (TEDS) of the timing recovery algorithm is close to zero under the singularity condition of azimuth θ about ±π/4 and differential group delay (DGD) about n × 1/2T (n is an odd number, T is the symbol period), it makes the squared Gardner phase detector (SGPD) timing recovery algorithm fail to achieve timing synchronization. What's worse, in the faster-than-Nyquist wavelength division multiplexing (FTN-WDM) systems, the tight filtering introduces inter-symbol-interference (ISI) so severe that the convergence cost of the SGPD timing recovery algorithm is extremely large even under the non-singularity condition. This paper proposes a joint timing recovery and adaptive equalization scheme for FTN coherent systems based on training sequences that could calculate channel matrix and indicate polarization characteristics, thereby avoiding the influence of azimuth on adaptive equalization and polarization demultiplexing (AEPD) embedded in the timing recovery feedback loop. Since embedded AEPD could compensate for most of DGD, the TEDS could be restored and timing synchronization could be achieved under the above adverse conditions. Thanks to the innovative scheme, which equalizes ISI and DGD during the feedback process of the loop, the convergence cost of timing recovery could be reduced with similar computational complexity compared with the conventional one. The simulation results of 128 GBaud polarization multiplexing (PM) 16-quadrature amplitude modulation (QAM) FTN-WDM transmission systems demonstrate that the proposed scheme could stably achieve timing synchronization under the singularity condition. And compared with the conventional scheme, the convergence cost is reduced by at least 42% @ 0.9 acceleration factor. In addition, 40 GBaud PM-16QAM FTN experiment results show that the proposed scheme could not only achieve timing synchronization stably but also exhibit an optical signal-to-noise ratio tolerance gain of 0.8 dB compared with the conventional scheme under 800 km transmission.
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1. Introduction
With the development of optical fiber communication technologies, various new applications and technologies such as virtual reality, cloud computing, self-driving car and high-definition television are emerging, and the communication network traffic keeps growing rapidly [1–3]. However, as the available spectrum in C band is almost fully utilized, it is difficult to increase the system capacity by increasing the number of multiplexed wavelengths or reducing the channel spacing in wavelength division multiplexing (WDM) systems. In addition, because the high-order modulation signal with high spectrum efficiency (SE) is sensitive to various transmission damages e.g., polarization crosstalk-related damages, effective number of bits of analog-to-digital/digital-to-analog converters (ADCs/DACs), IQ imbalance, frequency offset, linewidth and noise, etc. And it is difficult to compensate for those damages, enhancing system capacity by increasing modulation order has many technical difficulties and limitations. Nyquist WDM systems is a potential solution to enabling large-capacity optical transmission. However, the orthogonal transmission rule limits the system to further improve SE, so Nyquist WDM systems is still difficult to meet the development needs for future high SE and large capacity optical transmission systems. Faster-than-Nyquist (FTN) technology compresses the signal spectrum through digital domain or electrical domain tight filtering, making the symbol rate greater than the channel spacing and introducing controlled inter-symbol-interference (ISI), which theoretically provides higher spectral efficiency compared to Nyquist [4,5]. Although FTN tight filtering introduces serious ISI, the powerful digital signal process (DSP) algorithms can compensate for various transmission damages [6–9]. The probabilistic shaping based on higher-order modulation need to be completely reset due to the corresponding key algorithm parameters after changing the parameters of the scheme, which can lead to interruption of the service signal. However, FTN technology can dynamically adjust the spectral efficiency without interrupting the service signal by adjusting the taps of the digital filtering at the transmitter and the impairment equalization algorithm at the receiver. On the other hand, FTN technology has an advantage compared to higher-order modulation formats in passing through multi-reconfigurable optical add-drop multiplexer (ROADM) [10]. Therefore, FTN technology could be a potential development direction in the field of ultra-high SE and large capacity coherent optical transmissions. The FTN acceleration factor is defined as the ratio of the symbol period TF of the FTN system to the symbol period TN of the Nyquist system, expressed in the paper as $\tau $ ($\tau \le 1$), e.g., ${{T}_{F}}{ = }\tau {{T}_{N}}$ means that the spectral efficiency of the FTN signal is ${1 / \tau }$ times the spectral efficiency of the Nyquist signal.
In coherent optical systems, digital timing recovery is essential to compensate for the timing error between a transmitter and a receiver. Particularly, in a high symbol rate FTN system with low acceleration factor. FTN tight filtering introduces serious ISI, making it difficult to accurately and reliably enable the timing recovery loop to achieve synchronization by using Gardner timing recovery algorithm [11,12]. To solve the above problems, some solutions have been proposed in the relevant literature. In [13], it proposed to add a pilot sequence to recover the timing error in FTN systems. Compared with Gardner timing recovery algorithm, this scheme has a higher timing error detector sensitivity (TEDS) with the cost of higher computational complexity and lower SE. In [14], the authors demonstrated the effectiveness of the timing recovery based on the squared Gardner phase detector (SGPD) scheme in Nyquist systems. The scheme could also be used in FTN systems in theory, but the influence of ISI introduced by tight filtering will lead to unbearable convergence cost. Feedforward timing error detector (TED) such as absolute value nonlinearity timing error detector [15], fourth law nonlinearity timing error detector [16] and logarithmic nonlinearity timing error detector [17] were proposed for FTN systems. However, these schemes all need four samples per symbol, thereby increasing computational complexity and system costs. In [18], it proposed a novel digital phase detector that is based on Gardner timing recovery algorithm cloud recovery timing error in the FTN systems. However, this algorithm does not work under the singularity condition. In [19], an adaptive timing error detector was proposed. Although it could solve the problem that the timing recovery algorithm does not work under the singularity condition, there is still a problem of bigger convergence cost. In [20], a timing recovery scheme based on polarization transformation was proposed to solve the singularity problem. These two schemes could track and compensate for timing error under the singularity condition, but they were not suitable for FTN systems.
In this paper, we propose a joint scheme of timing recovery and adaptive equalization based on training sequences (TSs) to solve the above problems. The proposed scheme uses TSs to calculate channel matrix to avoid the influence of azimuth $\theta $ about ${\pm} {\pi / 4}$ on adaptive equalization and polarization demultiplexing (AEPD) embedded in timing recovery feedback loop, e.g., the constant modulus algorithm (CMA) and radius directed equalization (RDE) algorithm. Due to that differential group delay (DGD) could be compensated by AEPD, the TEDS could be restored and timing synchronization could be achieved. In addition, CMA & RDE algorithm embedded in the timing recovery feedback loop could equalize ISI and DGD during the feedback process of the loop, thereby reducing the convergence cost remarkably. This paper is organized as follows. Section 2 introduces the principle of the proposed joint scheme of timing recovery and adaptive equalization. In addition, the CC of the proposed algorithm is analyzed. Section 3 presents the simulation results under 128GBaud polarization multiplexing (PM) 16-quadrature amplitude modulation (QAM) FTN-WDM long-distance transmission. Section 4 shows the experimental results under three-carrier 40 GBaud PM-16QAM FTN-WDM. Lastly, Section 5 summarizes the paper.
2. Principle of the proposed scheme
2.1 Joint timing recovery and adaptive equalization based on training sequences
As shown in Fig. 1, although the SGPD timing recovery algorithm in most cases is applicable to FTN systems with different acceleration factors, the TEDS is close to zero in conditions of $\theta $ about ${\pm} {\pi / 4}$ and ${DGD}$ about ${n} \times \textrm{0}{.5T}$ (n is an odd number, T is the symbol period), making SGPD timing recovery algorithm very difficult to achieve timing synchronization [21]. Moreover, the large convergence cost of the timing recovery feedback loop is mainly caused by serious ISI, DGD, rotation of state of polarization (RSOP) and timing error. In addition, the SGPD timing recovery algorithm and the adaptive equalization algorithm influences each other. The serious ISI, dynamic DGD and RSOP will damage timing error detector's ability to track timing error, so the SGPD timing recovery algorithm needs the input signal to be processed by AEPD to achieve good synchronization. However, large timing error will damage AEPD performance, and it needs timing recovery to be implemented first. Meanwhile, the AEPD has singularity problem which the two polarization tributaries tend to converge to the same source under the condition that $\theta $ is about ${\pm} {\pi / 4}$ [22]. Therefore, it is necessary to joint timing recovery and adaptive equalization to alleviate the mutual restriction.
The principle of joint timing recovery and adaptive equalization based on training sequences (TSs) is shown in Fig. 2. Firstly, schmidl algorithm is used to determine the frame header position within one symbol period, under the presence of impairments such as timing error, ISI, and DGD [23]. The timing measurement function of schmidl algorithm is shown in Eq. (1)
Fig. 2. (a) Frame structures (b) The joint scheme for timing recovery and adaptive equalization based on TSs
Secondly, channel matrix calculated based on the training sequences is used to roughly complete polarization demultiplexing and avoid the influence of $\theta $ about ${\pm} {\pi / 4}$ on AEPD. However, because of timing errors and the Gaussian white noise, polarization demultiplexing is not perfectly accomplished through the channel matrix. In addition, channel matrix sets initial values for the tap coefficients of AEPD to speed up the convergence, as shown in Fig. 2(b) red line. Thirdly, AEPD is embedded into the timing recovery feedback loop, where AEPD is used to compensate ISI and DGD during the feedback process. To adjust the timing interpolation, the compensated signal is fed into the timing error detector, loop filter and numerically controlled oscillator. After the convergence of AEPD embedded in the timing recovery feedback loop, most of ISI and DGD are compensated, so that the TEDS could be restored and timing synchronization could be achieved by the proposed scheme.
The proposed scheme includes the following two main points.
i) Frame structure design and channel matrix calculation
Figure 2(a) shows the frame structure, which is composed of frame header sequence, TSs and payload. The frame header is used to roughly frame synchronization to indicate the start of the burst after synchronization. The TSs is used to calculate channel matrix. Since the proposed scheme use channel matrix to roughly realize polarization demultiplexing of payload. If the frame length is too long, the polarization crosstalk and other characteristics will change within one frame period, resulting in that channel matrix is not effective any more. The frame length is mainly dependent on symbol rate and RSOP speed. In our 128 GBaud PM-16QAM FTN-WDM simulation systems the frame length is designed as16840 symbols. The performance of frame synchronization is affected by the polarization crosstalk caused by azimuth, DGD, RSOP and so on. The combining scheme can improve the tolerance to polarization crosstalk is shown in Eq. (2), so the frame synchronization could be accomplished [24].
Meanwhile, in order to avoid the effect of timing error on schmidl algorithm, e.g., large timing error results in a rapid accumulation of symbol timing position offsets over time, to determine the frame header position within one symbol period, the length of the frame header cannot be too long. And short frame headers can affect the performance of the schmidl algorithm. Therefore, the length of the frame header for X and Y polarization is designed to be 200 symbols. The same length of two parts training sequences in a frame is used to roughly calculate channel matrix for polarization demultiplexing. TS1x and TS2x are two parts of the X-polarization TSs in a frame structure, and similarly TS1y and TS2y are two parts of the Y-polarization TSs. TS2x and TS1y are designed as the same length all-0 sequences. Due to that time domain averaging could reduce the effect of random noise on the accuracy of the calculated channel matrix, the long enough training sequences are required to calculate channel matrix accurately. However, when the training sequences are excessively long, it is more heavily influenced by timing error, making it difficult to achieve perfect polarization demultiplexing. Meanwhile in order to maintain the SE, it is necessary to comprehensively consider the proportion of cost. Therefore, the length of the TSs is designed to 256 symbols. The training sequences cost accounts for 1.5% of one frame.
The channel matrix could be calculated by Eq. (4) as:
If the signal was affected by timing error, which will cause channel matrix calculated by Eq. (3) to be less accurate. However, under the adverse condition of timing error and azimuth (even if $\theta $ is about ${\pm} {\pi / 4}$), the channel matrix could still complete roughly polarization demultiplexing and avoid the effect of azimuth on CMA & RDE algorithm. Thus, CMA & RDE algorithm embedded in the timing recovery feedback loop could work stably. Due to DGD could be compensated by CMA & RDE algorithm during the feedback process of the loop, the TEDS could be restored and timing synchronization could be achieved.
ii) Embedding AEPD in SGPD timing recovery feedback loop
As shown in Fig. 2(b), CMA & RDE algorithm is embedded in the timing recovery feedback loop, and most of ISI and DGD impairments could be compensated by CMA & RDE algorithm during the feedback process of the loop. Since the DGD is compensated, the TEDS could be restored even under singular DGD value (odd multiple of half symbol period) and the SGPD timing recovery algorithm could work correctly. In addition, the SGPD algorithm in the proposed scheme needs 2sps, so the CMA and RDE algorithms need to output 2sps accordingly. In the two-fold sampling condition, the first sample of a symbol is used to update the tap coefficients, the second sample uses the previous tap coefficients to complete the equalization. Meanwhile, the computational complexity of CMA and RDE algorithms can be effectively reduced.
Where the timing error detector base on the SGPD is shown in Eq. (5) [25]:
The CMA’s tap coefficients are initialized by channel matrix to speed up convergence by reducing the impact of ISI and DGD on the timing error detector. In addition, after CMA algorithm reaches the target mean-square error threshold, it switches to the RDE algorithm to better compensate for ISI and DGD impairments in the timing recovery feedback loop. This results in lower convergence cost required to achieve timing synchronization.
2.2 Complexity analysis and comparison
The computational complexity of the proposed scheme is compared with that of the conventional scheme. Generally, the computational complexity of a multiplier is approximately ten times that of an adder in hardware circuit implementation [26,27], and the multiplier is the dominant source of complexity for the proposed scheme. Therefore, the computational complexity here is evaluated in terms of the required number of real multiplications per symbol.
The joint timing recovery and adaptive equalization based on training sequences scheme includes the following parts:
①Finding the frame header position based on schmidl algorithm needs 4 real multiplications per symbol.
②Calculating channel matrix based on the TSs, 9 real multiplications per symbol are required.
③The timing recovery feedback loop of the SGPD timing recovery algorithm is consisted of timing error detector, loop filter, control unit and interpolator. The timing error detector needs 7 real multiplications per symbol. The loop filter consists of a proportional unit and an integral unit. The output of the proportional unit and integral unit require 2 and 2 real multiplications per symbol respectively, so the loop filter requires 4 real multiplications per symbol. The control unit uses the control word W(n) output by the loop filter to determine the two control variables required for interpolator, the basic pointer mk and fractional interval μk of the k-th interpolation point. In the control unit, the output of numerical control oscillator (NCO) is ${N}({n}){ = }[{{N}({n - }1) - {W}({n - }1)} ]\bmod 1$, and the fractional interval is ${\mu _{k}}{ = }{{{N}({k})} / {{W}({k})}}$, requiring 4 real number multiplications per symbol. The interpolator adopts Lagrange square interpolation method, receives asynchronous sampling sequence, obtains a continuous curve according to Lagrange interpolation formula, and then re-samples the sequence after synchronization to complete the process of sampling rate conversion. For a certain interpolation point X(kTi), four basic samples need to be selected: x(mk−2), x(mk−1), x(mk), and x(mk + 1). The waveform curve of the signal during this period:
④Equalizing signal damage and polarization demultiplexing based on CMA and RDE algorithm. A finite impulse response filter with an impulse response taps length of N in the CMA algorithm needs N real multiplications per symbol at the output. In the time domain CMA algorithm, the polarization demultiplexing is done through a butterfly structure of four finite impulse response filters. So, the butterfly structure needs 4N real multiplications per symbol. In the CMA and RDE algorithms the tap update module needs 4N real multiplications per symbol. In addition, in the error calculation module needs 2 real multiplications per symbol. The CMA algorithm needs a total of 8N + 2 real multiplications per symbol. After CMA algorithm reaches the target mean-square error threshold, it switches to the RDE algorithm to better compensate for ISI and DGD impairments. By the same principle, the RDE algorithm output also needs 8N + 2 real multiplications per symbol.
The conventional scheme in this paper only includes the SGPD timing recovery algorithm and CMA & RDE algorithm, whose computational complexity is also shown in Table 1.
Table 1. Computational complexity of different schemes (e.g., N = 25)
Through the above analysis, the proposed scheme has 4.15% increase in computational complexity compared with the conventional scheme due to the use of frame header and TSs, when the N is 25. And the proposed scheme has 1.5% training sequences cost. However, the proposed scheme could solve the problem that the timing recovery algorithm does not work under the singularity condition, and it can effectively reduce the convergence cost of timing synchronization.
3. Simulation results and discussions
Figure 3 shows a three-wavelength 128 GBaud PM-16QAM FTN-WDM coherent optical transmission system. At a transmitter, after the pseudo-random bit sequence (PRBS) is mapped to 16QAM symbols, FTN shaping is realized by root raised cosine (RRC) in which the cut-off bandwidth is the product of acceleration factor and symbol rate, and the roll-off factor is 0.01. Pre-equalization is performed in the electrical domain for compensating insufficient frequency response caused by some devices, such as the drivers, DACs and so on. Three lasers are used to generate optical carriers with a channel spacing of 137.5 GHz for three optical IQ modulators. The modulated three optical carriers enter the optical fiber transmission through a multiplexer (MUX), and the optical transmission link consists of multiple fiber spans with span length of 80 km, ROADM and erbium-doped optical fiber amplifier (EDFA). At a receiver, a de-multiplexer (DMUX) is used to filter out the desired wavelength signal. After the signal is converted into a digital signal by the optical coherent receiver and ADCs, it enters the DSP for processing and restores the original signal. Receiver DSP mainly includes IQ imbalance compensation, chromatic dispersion compensation, frame synchronization, the joint scheme of timing recovery and adaptive equalization based on TSs, frequency offset estimation, matching filter, cycle-slip-free phase estimation [28], the post-filter and Maximum-likelihood-sequence detection (PF & MLSD) and finally bit error rate (BER) counting.
Figure 4(a) shows the s-curve of the timing error detector based on SGPD with different acceleration factors under back-to-back (BTB). Through the results, it could be seen that the s-curve maintains the sinusoidal shape and the zero-crossing point position with different acceleration factors, which means timing error detector could effectively track the timing error. From Fig. 4(b), it could be seen that the TEDS of the conventional scheme is close to zero under the singularity condition, so the SGPD timing recovery algorithm fails to achieve timing synchronization. And the proposed scheme uses channel matrix to avoid the effect of azimuth on CMA & RDE algorithm, which is embedded in the timing recovery loop. Therefore, most of DGD could be compensated during the feedback process. And the TEDS is restored simultaneously to a correct value, thus the proposed scheme could achieve timing synchronization stably.
Figure 5(a-c) shows the fraction interval of 128 GBaud PM-16QAM FTN-WDM systems with three kinds of timing recovery schemes under the conditions of clock offset (CO) = 50 ppm, DGD = 0 T and $\tau $=0.9. After several times of timing recovery feedback loop the fractional interval will form a stable period determined by the clock offset and the RMS of the detected timing error converges towards zero, which means the timing is synchronized, and the interpolator starts to output synchronous samples. The RMS of the timing error is measured as:
Fig. 5. (a), (b) and (c) show the convergence cost in FTN systems (d) convergence cost versus different DGD
Figure 6 shows the impact of RSOP at different speeds on the proposed scheme based on the TSs. It could be seen that as the RSOP speed increases, the convergence speed of CMA & RDE algorithm is affected, thus the convergence cost required to achieve timing synchronization increases. In addition, as RSOP speed increases to 3 Mrad/s which exceeds the limits of traditional CMA & RDE algorithms for tracking RSOP, and the performance of CMA & RDE algorithm decreases significantly at this condition [30,31], thereby leading to the proposed scheme fails to achieve timing synchronization.
Figure 7 depicts the BER versus OSNR curve for 128 GBaud PM-16QAM FTN-WDM under BTB and 800 km transmission, respectively. The proposed scheme and the conventional scheme could effectively track and compensate clock offset under the above conditions. And the simulation results show that under the BTB and 800 km transmission, the proposed scheme compared with the conventional scheme has 0.15 dB and 0.45 dB OSNR tolerance gain, respectively. The simulation results have shown that OSNR required for SD-FEC limit are improved compared with the conventional scheme, specially under long-distance transmission. Since the proposed scheme equalizes ISI and DGD during the feedback process of the loop, which relieves the influence of ISI and DGD on the SGPD timing recovery algorithm, making the timing recovery algorithm provide more accurate estimates of clock offset.
4. Experimental results and discussions
To further investigate the performance of the proposed scheme, 40 GBaud PM-16QAM experiments have been carried out. Figure 8 shows the experimental setup for 40 GBaud PM-16QAM FTN coherent transmission systems for BTB and different span number transmission. In the offline transmitter DSP, firstly 219−1 PRBS are generated and then mapped into Gray mapped 16QAM symbols. Note that the FEC encoding is not employed in the experiment, and FTN shaping is realized by RRC in which the cut-off bandwidth is the product of acceleration factor and symbol rate, and the roll-off factor is 0.01. Then the discrete samples after FTN shaping are pre-equalized to alleviate high-frequency damage and sent into 4 synchronized channels of arbitrary waveform generator (AWG, Agilent 8194A) operating at 120 Gsample/s with ∼45 GHz −3 dB bandwidth. Three oversampling are performed to generate 40 GBaud electrical signals. Three-channel carriers, spaced at 50 GHz centered at 193.414 THz, are emitted from three tunable external cavity lasers (ECLs) at ∼50 kHz measured linewidth. These three channel carriers are divided into two groups the odd and even channels, which are respectively sent to the dual polarization IQ modulators 1 and 2 driven by I and Q electrical signals generated by AWG. Then the modulated carrier is launched into a re-circulating transmission fiber loop, which consists of optical couplers, optical switch (OS), ROADM, a fiber span of 80 km standard single-mode fiber (SSMF G.652D) with an attenuation of 0.17 dB/km and PMD of 0.2 ps/km1/2, and EDFAs. The influence of azimuth angle and DGD on the proposed scheme are not demonstrated due to the limitation of offline experimental conditions. However, the performance of the proposed scheme under DGD and random azimuth conditions is verified by long-distance fiber transmission. After fiber loop transmission, PM-16QAM FTN signals are sent to a programmable optical band pass filter (OBPF) with 0.5 nm −3 dB bandwidth and 1 GHz resolution. At receiver, the local oscillating (LO) laser is the same as ECL with ∼50 kHz linewidth at 193.414 THz to realize coherent detection. The FTN PM-16QAM signal is sampled by an 80 Gsample/s real-time sampling oscilloscope (Lecroy 10Zi-A) after photoelectric conversion for offline processing. The offline receiver DSP is also shown in Fig. 8. Although the AWG and the oscilloscope operate independently under different clock sources, which are not adjustable. Therefore, the clock offset cannot be adjusted in the experimental system to evaluate the performance of the proposed scheme under different clock offset, here specific clock offset was introduced by resampling in the offline receiver DSP. Then, after chromatic dispersion compensation and frame synchronization algorithm, the proposed scheme is employed for polarization demultiplexing and timing synchronization. The frequency offset estimation, matching filter and cycle-slip-free phase estimation [28] are subsequently carried out before performing the PF & MLSD scheme. Finally, more than 106 symbols are used for BER counting.
As shown in Fig. 9, there is an inherent clock offset of 4.2 ppm in the experiment when additional clock offset is not introduced by receiver DSP. Figure 9(a-b) shows the fractional interval curves with using the conventional scheme and the proposed scheme in the experiment with inherent clock offset under BTB. The experimental results shows that the proposed scheme and the conventional scheme could track and compensate the inherent clock offset. However, compare with the conventional scheme, the proposed scheme reduces the convergence cost by 45% under 0.9 acceleration factor in BTB. Figure 9(c) shows the comparison of convergence cost required to achieve timing synchronization between the proposed scheme and the conventional scheme under different transmission distances with 0.9 acceleration factor. Compared with the conventional scheme, the proposed scheme could reduce the convergence cost by 43% under transmission 640 km and acceleration factor 0.9.
Fig. 9. (a) and (b) are show convergence cost in FTN systems (c) convergence cost versus different span number
Figure 10 shows the influence on system performance by using the proposed scheme and the conventional scheme to track and compensate the clock offset. The experimental data is processed by the proposed scheme and the conventional scheme to perform timing recovery after equivalently adding 100 ppm clock offset which at the very beginning offline receiver DSP, in order to more fully verify the performance of the algorithm under larger clock offset. It could be seen from the experiment results that the performance of the proposed scheme is similar to that of the conventional scheme in BTB transmission. However, when the transmission distance is 800 km, the proposed scheme has 0.8 dB OSNR tolerance gain compared with the conventional scheme. It is difficult for the conventional scheme to fully compensate for ISI, DGD, timing error and other damages, so the performance of the conventional scheme is relatively poor, thus the OSNR cost of experiment results is more than the simulation results.
5. Conclusion
In this paper, a joint timing recovery and adaptive equalization based on TSs for FTN coherent optical systems has been proposed and demonstrated. The proposed scheme joints the SGPD timing recovery with AEPD based on TSs to achieve timing synchronization of high symbol rate and high SE PM-16QAM FTN systems. The proposed scheme not only achieves timing synchronization stably under the singularity condition, but also could reduce the convergence cost remarkably. The simulation results of 128 GBaud PM-16QAM FTN systems have demonstrated that the proposed scheme has strong robustness under the singularity condition, and the convergence cost of timing recovery could be reduced by at least 42% compared with the conventional SGPD scheme under 0.9 acceleration factor. And the proposed scheme exhibits an OSNR tolerance gain of 0.45 dB compared to the conventional scheme under the condition of 800 km SSMF transmission. Meanwhile, the experimental results of three-carrier 40 GBaud PM-16QAM show that the proposed scheme not only could achieve timing synchronization, but also could reduce the convergence cost by at least 38% under 0.9 acceleration factor. The proposed scheme exhibits an OSNR tolerance gain of 0.8 dB compared with the conventional scheme under 800 km SSMF transmission. In a conclusion, the proposed scheme is of great potential to realize efficient timing recovery in the practical application of FTN systems.
Funding
National Natural Science Foundation of China (No. 62001045); State Key Laboratory of Information Photonics and Optical Communications (No. IPOC2021ZT17); Fundamental Research Funds for the Central Universities (No. 2022RC09); ZTE’s Industry, University and Research Cooperation Foundation (No. HC-CN-20201208015).
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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