1. Introduction

To meet the rapidly growing bandwidth requirements of optical communication systems, probabilistically shaped (PS), as a breakthrough technology, makes the reachable information rate of signals be close to the Shannon limit by changing the probability of uniformly distributed constellation points. Theoretical analyses have shown that the PS technique can achieve a gain of 1.53 dB in additive white Gaussian noise (AWGN) channel [1]. Under the condition of fixed average optical transmission power, PS can bring shaping gain to the noise contributions caused by fiber amplifier and modulation-independent nonlinear interference [2]. In addition, it can achieve an arbitrarily fine balance between bit rate and transmission distance [3], and eliminate the shaping gap to approach the ultimate performance [4]. Hence, PS has been regarded as one of the promising technologies to improve the performance of optical communication system in the future [5–7].

Since PS technology changes the occurrence probability of different amplitude symbols, some classical digital signal processing (DSP) solutions for uniformly distributed M quadrature amplitude modulation (QAM) systems are facing with the dilemma of failure, so it is essential to perform DSP processing research within the coherent receiver of PS-MQAM system. In this regard, many related works have been carried out for PS system, including clock recovery [8,9], polarization demultiplexing [10–13] and carrier phase recovery (CPR) [14–18], etc. It is well known that due to the inevitable frequency offset (FO) between transmitter laser and local oscillator (LO) laser, a fast time-varying rotation of constellation points of received symbols will be induced. Accordingly, frequency offset estimation (FOE) before CPR algorithm is also one of the key DSP processes for PS-MQAM system.

Until now, many works have proven that under the moderate or strong shaping conditions, the traditional FOE schemes, such as time-domain 4^th power feedforward scheme [19], peak finding scheme in the 4^th power periodogram [20], and the circular harmonic expansion (CHE) scheme [21], could not work properly for high-order PS-MQAM system [22,23]. To address this issue, the reported FOE schemes for PS-MQAM system could be roughly divided into two categories (1) pilot-aided FOE scheme: for example, a peak search scheme of spectrum assisted by 4QAM pilot symbol has been proposed [23]. Its realization principle is simple, while the spectral efficiency maybe reduced to a certain extent. The FOE range of this scheme is limited within [-R_s /2P, R_s/2P], where P denotes the number of pilot symbols, generally equals to 32. (2) blind FOE schemes: Ref. [24] has proposed a spectrum-based FOE scheme for coherent passive optical network (CPON). It firstly transforms 4096 data samples into frequency domain using fast Fourier transform (FFT), then performs channel boundary search and calibrates the frequency window. This scheme has achieved good FOE performance in 80 km uplink transmission of CPON. Nevertheless, when it is applied to PS long-distance transmission system, the inevitable chromatic dispersion (CD) and fiber nonlinearity effects would cause boundary search of the spectrum to fail. These problems need to be further studied. Besides that, the basic innovation ideas of another type of blind FOE schemes are to highlight or restore the intensity peak in the periodogram after the 4^th power of received symbols. For instance, Yan et al. have proposed a FOE scheme using generalized circular harmonic (GCHE) algorithm [22], which performs maximum likelihood (ML) estimation with a theoretically derived nonlinear radius transfer function. This scheme obtains a better periodogram to facilitate the peak searching. Moreover, Yao et al. have investigated Fourier transform based blind FOE scheme in fiber-mm Wave integrated system [25]. The scheme selects and enlarges the two rings formed by quadrature phase shift keying (QPSK)-shaped constellation points in order to restore the intensity peak in the 4^th power discrete-frequency spectrum of received symbols. It is considerably clear that these blind FOE schemes have gained good FOE results, whereas high calculation complexity is inevitable since these schemes are highly dependent on the size of FFT and the search process of the maximum value in the 4^th power periodogram. Furthermore, we propose a QPSK-partition assisted (QPSK-AR) FOE scheme in Ref. [26]. It utilizes the Euclidean distance for ring judgment, which would inevitably lead to the misjudgment under low optical signal-to-noise ratio (OSNR), eventually resulting in the reduction of FOE accuracy. Based on these unsolved problems, it is still worthy of further proposing a low complexity blind FOE scheme for PS-MQAM system.

In this paper, we have proposed a new optimal decision threshold assisted QPSK-partition (ODT-QP) blind FOE scheme for PS-16/64QAM system. Firstly, an optimal decision threshold method, which is closely related to the occurrence probability of PS-16/64QAM ring radius, is utilized to reduce the error probability of ring judgment. Secondly, the received symbols on multiple QPSK-shaped rings are uniformly augmented to the same value to eliminate the amplitude noise. Thirdly, an improved time-domain 4^th power feedforward method, which eliminates the time intervals caused by symbol selection, is implemented to estimate the FO of PS-16/64QAM signals. The effectiveness of this ODT-QP FOE scheme is firstly verified by 28 GBaud polarization division multiplexing (PDM) PS-16/64QAM back-to-back (BTB) and fiber transmission simulation scenarios, and the generalized mutual information (GMI) and normalized mean square error (NMSE) results are also given under different shaping strengths, OSNR, FO and other conditions. Furthermore, 28 GBaud PS-16QAM and 8 GBaud PS-64QAM experiments are carried out to further investigate the effectiveness of ODT-QP FOE scheme. The results manifest that high-precision FOE accuracy can be achieved, the FOE range of this scheme is [-R_s/8, R_s/8], and the overall complexity of the scheme is on the order of O(N), which is even lower than 26.5% of the FFT FOE scheme.

The rest of the paper is arranged as follows: firstly, the principle of ODT-QP FOE scheme for PS-MQAM system is described in detail in Section 2. Then the effectiveness of ODT-QP FOE scheme is fully investigated by 28 GBaud PDM PS-16/64QAM simulation system and 28/8 GBaud PS-16/64QAM experiments in Sections 3 and 4, respectively. Subsequently, the computational complexity of this proposed scheme is analyzed in Section 5. Finally, the conclusions are given in Section 6.

2. Principles of the proposed ODT-QP scheme

2.1 Basic principle of probabilistic shaping

As we know, PS optimizes the channel capacity by changing the priori probability with the Maxwell-Boltzmann (MB) distribution in AWGN channel [1]. The probability mass function (PMF) of MB distribution is given by:

Besides that, the notation PS-MQAM-H(X) represents the PS-MQAM constellations with an entropy of H(X) throughout this paper. Normally, PS-MQAM system utilizes the metric of achievable information rate (AIR) to measure the maximum transmission capacity of fiber channel [28]. The specific representations of AIR include GMI based on bit metric and normalized generalized mutual information (NGMI). The GMI metric can be written as [7]:

NGMI can be obtained from the GMI using [4,29]:

2.2 Principle of the proposed ODT-QP FOE scheme

According to the principle of PS technology, we know that under the condition of moderate or strong shaping, the probability of the innermost QPSK-shape constellation points is much higher than that of the outer ones for PS-MQAM signals. Accordingly, the DSP algorithms originally designed for uniform distributed QPSK modulation format can obtain better performance for PS-MQAM signals after being improved [30]. Based on this idea, we propose the ODT-QP FOE scheme based on the time-domain 4^th power feedforward method. The overall flow of the off-line DSP at the receiving end is shown in Fig. 1(a). It depicts that after normalization and standardization, CD compensation, clock recovery and polarization demultiplexing, the proposed ODT-QP scheme is carried out for FOE. After that, CPR is executed and GMI/NGMI are calculated to evaluate the scheme performance.

 

Fig. 1. (a) The overall Rx-DSP process of PS-16/64QAM system, (b) the flow chart of the ODT-QP FOE scheme, (c) under conditions of 30 dB OSNR and 300 MHz FO, the input and output constellation diagrams of PS-16QAM-3.6 in each stage of ODT-QP FOE scheme.

Download Full Size | PDF

Specifically, the detailed schematic diagram of ODT-QP FOE scheme is shown in Fig. 1(b), which could be further divided into three stages. On the first stage, all the received PS-16/64QAM symbols are judged using an optimal decision threshold assisted method, then the symbols on multiple QPSK-shape rings are filtrated. For the second stage, the symbol amplitudes on multiple QPSK-shape rings are normalized and augmented to the same value. Finally, an improved time-domain 4^th power feedforward method that eliminates the time interval is used for FOE on the third stage. In order to depict the ODT-QP FOE scheme, taking PS-16QAM as an example, Fig. 1(c) illustrates four insets of constellation diagrams which correspond to the input and output on the above three stages respectively.

To accurately describe the proposed scheme, it is assumed that the k-th received symbol ${y_k}$ entering the FOE process is only affected by FO and phase noise of lasers, which can be expressed as:

Stage 1. Symbol selection assisted by optimal decision threshold

The purpose of the first stage is to judge all received symbols through the optimal decision threshold assist method, which is shown in inset ① of Fig. 1(c), and then filtrate QPSK-shaped rings for PS-MQAM signals (as shown in Fig. 2(a)). Finally, the constellation distribution could be obtained, and inset ② of Fig. 1(c) illustrates an example of PS-16QAM-3.6.

 

Fig. 2. (a) Partition of QPSK-shape constellation points selected in PS-64QAM, (b) schematic diagram of the comparison between the optimal decision threshold method (solid line) and the Euclidean distance-to-threshold method (dotted line) in PS-64QAM-5 with SNR of 20 dB. (c) Enlargement of the decision boundary of the amplitude radii $\sqrt {10} $ and $\sqrt {18} $ in Fig. 2(b).

Download Full Size | PDF

The optimal decision threshold is obtained by the lowest intersection between the adjacent amplitude radius curves. In the presence of an AWGN with variance of ${\sigma ^2}$, the probability density function (PDF) $P(R|{{A_n},\sigma } )$ can be computed for a received symbol with radius of R, where R belongs to the set ${A_n}({n \in 1,2,\ldots U} )$ of amplitude radius. $P(R|{{A_n},\sigma } )$ follows the Rician distribution [10], which is denoted as:

In order to more intuitively present the effect of the optimal decision threshold method based on Rician distribution, when signal-to-noise ratio (SNR) is set to 20 dB, Fig. 2(b) illustrates a comparison example using Euclidean distance and optimal decision methods for PS-64QAM-5 system, and Fig. 2(c) is an enlargement of part area of Fig. 2(b). Taking Fig. 2(c) as an example, it is clear that the difference is 0.2 between the optimal threshold (solid line) and the symbol decision threshold obtained by Euclidean distance (dashed line). In the case of low SNR, using the Euclidean distance method will bring many unnecessary errors to the symbol decision, which is illustrated as shaded area in Fig. 2(c).

Stage 2. Amplitude normalization and augmentation

The purpose of the second stage is to remove the effects of amplitude noise. The selected symbols (shown in inset ② of Fig. 1(c)) are divided by their corresponding amplitudes, and then the amplitude radius is amplified to $\sqrt {18} $ (see 3.1 Parameter optimization for details), finally a QPSK-shaped ring C_QPSK is formed (shown in inset ③ of Fig. 1(c)).

After symbol selection, when using the time-domain 4^th power feedforward scheme for FOE, the ring with amplitude normalization has better FOE accuracy than the ring without. Here we use the geometric vector algorithm to explain this conclusion. The traditional time-domain 4^th power feedforward scheme calculates the estimated FO value through Eq. (8) and Eq. (9).

Figures 3(a) and (b) present the constellation diagrams calculated by Eq. (8), respectively. The difference between the two figures is whether the influence of amplitude noise is eliminated. We use two symbols D(k) and D(k-1) that are adjacent in time as an example to enter Eq. (9). In (a) and (b) of Fig. 3, the symbols used are the same. The two phase angles obtained by ${\alpha _1}$ in Fig. 3(a) and ${\alpha _2}$ of Fig. 3(b) exist with or without amplitude noise, respectively. There is a certain angular difference between these two angles. And through the calculation of Eq. (9), we can get that ${\alpha _2}$ is closer to the ideal phase angle. According to the above analysis, it can be concluded that the method of eliminating the amplitude noise can improve the FOE performance of the time-domain 4^th power feedforward scheme.

 

Fig. 3. After the first stage of processing, the constellation diagrams of present or absent amplitude noise are compared after Eq. (8) is calculated for the symbol ${y_{QPSK}}$ on the ring with the QPSK-shape. (a) Amplitude noise is present, (b) amplitude noise is absent.

Download Full Size | PDF

Stage 3. Frequency offset estimation

As shown in ③ in Fig. 1(c), after a ring C_QPSK is formed, on the third stage, the time-domain 4^th power feedforward scheme is improved to perform FOE. For the traditional time-domain 4^th power feedforward scheme, in order to obtain $\Delta f$, it needs to first multiply the current symbol with the complex conjugate of the symbol of adjacent previous moment, next calculate the average value of the above products at multiple moments, and finally divide by $8\pi {T_\textrm{s}}$.

In contrast, as shown in Fig. 4(a), because the optimal decision threshold assisted method is utilized for QPSK-shaped rings filtration at the stage 1 of this proposed scheme, the two symbols entering the third stage in sequence are not necessarily closely adjacent in time. Assuming that two adjacent symbols are ${y_{QPSK}}({k - 1} )$ and ${y_{QPSK}}(k )$, the time interval between them is L_k (${L_k} \ge 1$) as represented in this figure. Figure 4(b) illustrates an example of PS-64QAM-5.4. It can be observed that time interval of 1 occupies a larger proportion of all-time intervals. As the time interval increases, the total number of symbols corresponding to larger time intervals also decrease. Furthermore, the distribution of time intervals is closely related to the number of symbols entering the third stage, both of them are strongly influenced by information source entropy. We would discuss it for details in the complexity analysis of Section 5.

 

Fig. 4. (a) The time interval between two symbols after symbol selection, (b) the distribution of the number of symbols for different time intervals in PS-64QAM-5 with OSNR of 28 dB.

Download Full Size | PDF

According to the above analysis, the estimated value $\Delta \hat{f}$ of FO can be calculated by:

3. Simulation setup and results analysis

To verify the effectiveness of the proposed FOE scheme, we conduct 28 GBaud PDM PS-16/64QAM simulation system using VPI TransmissionMaker 11.1 and MATLAB, as shown in Fig. 5. At first, the transmitter generates 28 GBaud PDM PS-16/64QAM signals. The bit sequence enters the constant composition distribution matching (CCDM) module to generate the required probability distribution, and then generates symbols through steps such as symbol mapping. Different shaping strengths can be achieved by changing the size of the entropy. The entropies of PDM PS-16QAM and PDM PS-64QAM vary within the range from 2 bit/symbol to 3.8 bit/symbol and from 4 bit/symbol to 5.8 bit/symbol, respectively, with a step interval of 0.2 bit/symbol. The transmitter uses a square root raised cosine (SRRC) filter with a roll-off factor of 0.2 for pulse shaping. The central wavelength of laser at the transmitter is fixed at 1550 nm, and the line width is set to 100 kHz. Secondly, two scenarios of BTB and optical fiber transmission are used to verify the performance of the proposed scheme. In BTB scenario, the performance of ODT-QP FOE scheme are investigated in presence of amplifier spontaneous emission (ASE) noise and phase noise. For PDM PS-16QAM, the OSNR range changes from 12 dB to 26 dB, and for PDM PS-64QAM, the OSNR variation range is 17 dB to 30 dB, both the step size of OSNR is 1 dB. In optical fiber transmission scenario, a fiber span length of 80 km is used in the transmission loop. The dispersion, PMD and nonlinear coefficients of the fiber are set as 16 ${{ps} / {({nm \cdot km} )}}$, 0.1 ${{ps} / {\sqrt {km} }}$ and 1.3 ${{{W^{ - 1}}} / {km}}$, respectively. An erbium doped fiber amplifier (EDFA) with noise figure of 4 dB is used in the fiber link. The PDM PS-16/64QAM signals are transmitted through optical fibers with total lengths of 800 km and 480 km respectively. Besides that, an optical bandpass filter (OBPF) with bandwidth of 39.2 GHz is used before the coherent receiver. For the receiver, the FOs of LO laser vary in the range of [-3.6 GHz, 3.6 GHz]. After coherent reception, 60 groups of symbols are totally acquired and each group owns 65536 symbols for the following simulations. For the received PS symbols of each group, they would undergo off-line DSPs, which include CD compensation, clock recovery, polarization demultiplexing, FOE schemes and CPR algorithms. It should be noted that we have used four kinds of FOE schemes to compare the performance, which are the proposed ODT-QP FOE scheme, QPSK-AR FOE scheme, FFT FOE scheme and the time-domain 4^th power feedforward scheme (this scheme is hereinafter referred to as feedforward FOE scheme), respectively.

 

Fig. 5. Schematic diagram of 28 Gbaud PDM PS-16/64QAM simulation system.

Download Full Size | PDF

3.1 Parameter optimization

This part focuses on the influence of important parameters of ODT-QP scheme on the final FOE performance, so as to obtain the optimal parameters for subsequent simulation and experiments. To measure and compare the FOE accuracy, the NMSE metric is define as follows:

According to the principle description of Section 2.2, the ring radius of C_QPSK is one of the most important parameters of the ODT-QP FOE scheme, which has a significant impact on FOE performance. For PDM PS-16/64QAM system, Fig. 6 exhibits the achievable NMSE performance with different ring radii of C_QPSK. Figure 6(a) shows the NMSEs of PDM PS-16QAM-3/3.6 system with three C_QPSK ring radii. For PDM PS-16QAM-3/3.6 systems under conditions of OSNR 26 dB, the NMSEs with C_QPSK radius of $\sqrt 2 $ can achieve 5e-9 (3.7e-8). While the amplitude radius is $\sqrt {18} $, the NMSE can be reduced to 3.8e-9 (1.5e-8). As illustrated in Fig. 6(b), for PDM PS-64QAM-4.4/5 system with nine different C_QPSK ring radii, the same trend can be obtained for PDM PS-16QAM. When OSNR is set 30 dB, the NMSE of radius $\sqrt {18} $ is reduced to 3.7e-8(4e-8) compared with the case of radius $\sqrt 2 $ for PDM PS-64QAM-4.4/5. However, if the amplitude radius is increased from $\sqrt {10} $ to $\sqrt {18} $ for PS-16QAM or larger than $\sqrt {18} $ for PS-64QAM, the NMSE improvement is very limited. Accordingly, on the stage 2 of the ODT-QP FOE scheme, the radius of the ring for PDM PS-16/64QAM C_QPSK is uniformly determined as $\sqrt {18} $ in the following simulations and experiments.

 

Fig. 6. The effects of amplitude radius of C_QPSK on the NMSE performance using the ODT-QP FOE scheme. (a) PDM PS-16QAM-3/3.6, (b) PDM PS-64QAM-4.4/5.

Download Full Size | PDF

3.2 BTB scenarioc

Firstly, to verify the effect of ASE noise on FOE performance, we have compared the GMI performance curves under different OSNR conditions using four kinds of FOE schemes, which are the ODT-QP FOE scheme, QPSK-AR FOE scheme, FFT FOE scheme and feedforward FOE scheme. The results are shown in Fig. 7. Meanwhile, the ideal curve of this system is given as a reference when it is not affected by FO. The dotted line in the legend is the GMI threshold for measuring performance, which is converted to NGMI equal to 0.9 [31].

 

Fig. 7. GMI performance of FFT FOE scheme, feedforward FOE scheme, QPSK-AR FOE scheme, and ODT-QP FOE scheme under different OSNR conditions. (a) PDM PS-16QAM-3, (b) PDM PS-64QAM-4.4, (c) PDM PS-64QAM-5.

Download Full Size | PDF

As shown in Fig. 7(a), for the PDM PS-16QAM-3 system, when OSNR equals to 24 dB, both the feedforward FOE scheme and FFT FOE scheme still cannot meet the GMI threshold requirement. In contrast, the GMIs of the ODT-QP FOE scheme and QPSK-AR FOE scheme could exceed the GMI threshold of 5.2 bits/symbol when OSNRs are higher than 15 dB, and they finally converge to the ideal GMI value. Compared with the ideal curve, the OSNR penalty of ODT-QP FOE scheme is 0.75 dB lower than that of the QPSK-AR FOE scheme at the GMI threshold of 5.2 bits/symbol. Besides that, Figs. 7(b) and (c) demonstrate the GMI curves of PDM PS-64QAM-4.4 and PDM PS-64QAM-5, respectively. It is obvious that both the feedforward FOE scheme and FFT FOE scheme cannot work properly for PS-64QAM-4.4 and PS-64QAM-5 systems. Nevertheless, the GMIs obtained by the ODT-QP FOE scheme and QPSK-AR FOE scheme could exceed the corresponding GMI thresholds. Especially, the OSNR penalty of ODT-QP FOE scheme could be reduced by 1.5 dB (1 dB) compared to QPSK-AR FOE scheme. The above results show that the ODT-QP FOE scheme is better than the other three schemes under different conditions of modulation formats, entropies and OSNRs.

Secondly, the range of FOE is investigated for 28 GBaud PDM PS-16/64 systems. The detailed results are illustrated in Fig. 8. Due to the 4^th power operation in the proposed scheme, QPSK-AR FOE scheme, FFT FOE scheme and feedforward FOE scheme, their estimation range is limited to [-R_s/8, R_s/8], namely [-3.5 GHz, 3.5 GHz]. Since the final reaching of the NGMI threshold needs to go through all DSP processes rather than only the FOE scheme, and the CPR algorithm used after the FOE algorithm has a certain tolerance effect on the FO, and the same NGMI threshold in different systems does not require the same NMSE. Therefore, within the test range, the NMSE of the proposed scheme can meet the performance requirements of NGMI. The results show that the QPSK-AR FOE scheme and the ODT-QP FOE scheme can perform stably in the range of [-3.5 GHz, 3.5 GHz], but the FFT FOE scheme and the feedforward FOE scheme fail. Specifically, Fig. 8(a) gives the NMSE results for PDM PS-16QAM-3 with OSNR of 19 dB. The average NMSE of the QPSK-AR FOE scheme and the proposed scheme can reach 9.1e-8 and 2.2e-8, respectively. Under the condition of OSNR 24 dB for PDM PS-64QAM-4.4 and PS-64QAM-5 systems, Figs. 8(b) and (c) illustrate the corresponding NMSEs. It can be found that the average NMSE of the two schemes achieve 1.9e-7 and 5.6e-8 (1.6e-7 and 7.7e-8) for PDM PS-64QAM-4.4 (PS-64QAM-5), respectively, and the proposed ODT-QP FOE scheme outperforms the QPSK-AR FOE scheme.

 

Fig. 8. The NMSE versus the actual FO using four kinds of FOE schemes in BTB scenario. (a) PDM PS-16QAM-3 with OSNR of 19 dB, (b) PDM PS-64QAM-4.4 with OSNR of 24 dB, (c) PDM PS-64QAM-5 with OSNR of 24 dB.

Download Full Size | PDF

3.3 Optical fiber transmission scenario

Thirdly, when there are impairments such as CD, PMD, and nonlinearity in fiber transmission scenario, Fig. 9 shows the NMSE results using four kinds of schemes. Figure 9(a) presents the FO ranges for PDM PS-16QAM-3 system, we can observe that the FFT FOE scheme and the feedforward FOE scheme are still in an invalid state. In contrast, the average NMSEs of the ODT-QP FOE scheme are still better than that of QPSK-AR FOE scheme, and they are can be stabilized at 1.2e-8 and 3.3e-9 within the theoretical ranges. Besides that, as shown in Figs. 9(b) and (c), for PDM PS-64QAM-4.4 (PDM PS-64QAM-5), the NMSEs obtained by QPSK-AR FOE scheme and ODT-QP FOE scheme can stabilize at 1e-7 and 4.3e-8 (2.7e-7 and 1e-7) in the range of [-3.5 GHz,3.5 GHz], respectively.

 

Fig. 9. The NMSE versus the actual FO using four kinds of FOE schemes in fiber transmission scenario. (a) PDM PS-16QAM-3, (b) PDM PS-64QAM-4.4, (c) PDM PS-64QAM-5.

Download Full Size | PDF

Finally, to demonstrate the applicability and flexibility of ODT-QP FOE scheme for PS system, Fig. 10 illustrates the GMI curves with different shaping strengths using these four kinds of schemes. As shown in Fig. 10(a), it is clearly observed that the FFT FOE scheme and feedforward FOE scheme could perform frequency offset estimation only under tiny minority entropies for PDM PS-16QAM system. Especially, the FFT FOE scheme can only be used for very strong shaping strengths (H < 2.4 bits/symbol) and very weak shaping strengths(H > 3.4 bits/symbol). The reason is that in both conditions the peaks of the 4^th power periodogram are not drowned out. By contrast, both the QPSK-AR FOE scheme and ODT-QP FOE scheme exceed the corresponding GMI thresholds within the entropy range of 2 bit/symbol to 3.8 bit/symbol. Moreover, Fig. 10(b) shows the GMI curve with different entropies for PDM PS-64QAM system. It indicates that the FFT FOE scheme cannot work properly when the entropy is less than 5.3 bits/symbol and greater than 2.5 bits/symbol, and the feedforward FOE scheme only achieves the target GMI under an entropy of 2 bits/symbol. In comparison, the QPSK-AR FOE scheme exceeds the corresponding target GMIs within entropies range from 4 bit/symbol to 5.45 bit/symbol, and our proposed ODT-QP FOE scheme could further increase this range from 4 bit/symbol to 5.7 bit/symbol for PDM PS-64QAM systems. Therefore, the advantages of the proposed scheme are more manifested under moderate or strong shaping conditions for PS system.

 

Fig. 10. The GMI curves with different shaping strengths using these four kinds of schemes. (a) PDM PS-16QAM, (b) PDM PS-64QAM.

Download Full Size | PDF

4. Experimental results

In order to further prove the FOE performance of the ODT-QP FOE scheme for PS system, we have built an experimental system under 28 GBaud PS-16QAM and 8 GBaud PS-64QAM BTB scenarios. The system block diagram is shown in Fig. 11. Firstly, the transmitter generates symbols with probability distribution after CCDM, symbol mapping, etc., and then enters the 65 GSa/s arbitrary waveform generator (AWG) for pulse shaping, predistortion and other preprocessing to generate 28 GBaud PS-16QAM and 8 GBaud PS-64QAM electrical signals. The entropy of PS-16QAM (PS-64QAM) varies from 2 bit/symbol to 3.8 bit/symbol (from 4 bit/symbol to 5.8 bit/symbol) with interval of 0.2 bit/symbol. The pulse shaping stage uses a SRRC with roll-off factor of 0.2. It uses a transmitter laser with a fixed center wavelength at 1550 nm, and the FO and laser linewidth are varied in the range of 100 MHz and 100 kHz, respectively. Secondly, OSNR is adjusted by varying the output power of ASE noise source during transmission, meanwhile the OSNRs are changed within the range of 17 dB to 23 dB and 19 dB to 25 dB with 1 dB step for PS-16QAM and PS-64QAM, respectively. The actual OSNR is measured by using an OSA with 0.1 nm resolution. Subsequently, an OBPF with bandwidth of 39.2 GHz is utilized to filter the ASE noise. Finally, a narrow linewidth laser with a maximum linewidth of 100 Hz is adopted as the LO of coherent receiver, the central wavelengths of this LO is varied within the range of [1549.972 nm, 1550.028 nm] and [1549.992 nm, 1550.008 nm] for PS-16QAM and PS-64QAM, respectively. The FO ranges of [-3.65 GHz, 3.65 GHz] and [-1.25 GHz, 1.25 GHz] are loaded through adjusting the deviation between the transmitter laser and LO laser for PS-16QAM and PS-64QAM, respectively. The receiving end uses a polarization diversity coherent receiver to obtain data for off-line DSP processing through a real-time oscilloscope with sampling rate of 80 GS/s and electrical bandwidth of 36 GHz. To get relatively stable and correct results, 20 random data samples are performed on each dataset, each dataset contains 32768 symbols.

 

Fig. 11. Block diagram of 28 GBaud PS-16QAM and 8 GBaud PS-64QAM experimental systems.

Download Full Size | PDF

To further verify the tolerance of the proposed scheme to ASE noise in the experiment, Fig. 12 presents the GMI performance between the experimental and simulated systems for 28/8 GBaud PS-16/64QAM systems. In order to make the simulation results can be compared with the experimental results, the parameters of the simulation system are the same as those of the experimental system. As exhibited in Fig. 12(a), for PS-16QAM-3 system, the GMI of 2.6 bits/symbol is seen as the performance measurement standard. It is obvious that the experimental results of QPSK-AR FOE scheme and ODT-QP FOE scheme bring 1.1 dB OSNR penalty than the simulation results. Figure 12(b) shows that when GMI in PS-64QAM-4.4 system is equal to 3.8 bits/symbol, the experimental result of QPSK-AR FOE scheme brings 5.7 dB OSNR penalty than the corresponding simulation result, while the ODT-QP FOE scheme only brings 5.2 dB. Hence, the proposed scheme always outperforms the QPSK-AR FOE scheme at low SNR, and this advantage is more pronounced in PS-64QAM.

 

Fig. 12. GMI changes under different OSNR conditions in simulation and experiment scenarios. (a) PS-16QAM-3, (b) PS-64QAM-4.4.

Download Full Size | PDF

In addition, the NGMI performance with different entropies is tested in Fig. 13. As exhibited in Fig. 13(a), for PS-16QAM with OSNR of 19 dB, the NGMI results of the QPSK-AR FOE scheme and the ODT-QP FOE scheme can meet the threshold regardless of the entropy conditions. But through the given constellation diagram under the same conditions, it can be concluded that the simulation results are better than the experimental results. As shown in Fig. 13(b), for the PS-64QAM system with OSNR of 24 dB, the NGMIs of the QPSK-AR FOE scheme and the ODT-QP FOE scheme can both reach around 1 under the entropy tested by the simulation system. The experimental results show that the threshold requirement of NGMI of 0.9 can be reached before the entropy is 4.8 bit/symbol. After the entropy of the FOE scheme reaches 4.8 bit/symbol, the proposed scheme has better anti-noise ability than the QPSK-AR FOE scheme.

 

Fig. 13. NGMI curves with different shaping strengths in the experimental system. (a) PS-16QAM with OSNR of 19 dB, (b) PS-64QAM with OSNR of 24 dB.

Download Full Size | PDF

Finally, the FOE range is verified using the proposed scheme, and the obtained results are depicted in Fig. 14. It should be noted that the examples of constellation diagrams correspond to the experimental result before calculating the GMI. When OSNR is set to 17 dB and the FO test range is set [-3.625 GHz, 3.625 GHz] for 28 GBaud PS-16QAM-3 system, as illustrated in Fig. 14(a), our proposed scheme can meet the GMI threshold within the range of [-3.5 GHz, 3.5 GHz], and it can estimate the FO value stably at 3 bits/symbol. As depicted in Fig. 14(b), for 8 GBaud PS-64QAM-4.4/5 with OSNR of 27 dB, when the range of tested FO is [-1.125 GHz, 1.125 GHz], the proposed scheme can stably reach the target GMI of 4.4/3.8 bits/symbol within the FO range of [-1 GHz, 1 GHz].

 

Fig. 14. The GMI versus the actual FO with the proposed scheme in experimental transmission system. (a) PS-16QAM-3 with OSNR of 17 dB, (b) PS-64QAM-4.4/5 with OSNR of 27 dB.

Download Full Size | PDF

5. Complexity analysis

Computational complexity is one of the important indicators for evaluating FOE schemes. Based on the idea of optimal operation, the calculation in ODT-QP FOE scheme can be divided into real multiplication, real addition, comparison, and look-up table (LUT) operations. Table 1 summarizes the complexity analysis of each stage of this scheme. We take the analysis of the complexity of PS-64QAM as an example. The proposed FOE scheme requires a total of 2N+7B+3 real multiplications, 2N+5B real additions, 5N comparisons, and 9N+2B+1 LUTs. As a consequence, the overall computational complexity of the proposed FOE scheme can be denoted as O(N).

Table 1. Computational complexity analysis of each stage using the proposed ODT-QP scheme

View Table | View all tables in this article

Besides that, the ratio of the number of symbols B used in C_QPSK to the total number of symbols N varies with the entropy. According to Eq. (1), the theoretical curves of the change are shown in Fig. 15. With the increase of entropies of PS system, the ratio of symbol number in C_QPSK to the total number of symbols decreases. As shown in Fig. 15(a), when the entropy is equal to 2 bit/symbol for PS-16QAM, the constellations completely degenerate into a shape of QPSK, all the symbols would participate in all these three stages for the proposed FOE scheme, and the B/N reaches 100%. When the source entropy of PS system finally reaches 3.8 bit/symbol, B is only as 56.77% of N. Similar relationship and trend of entropy versus B/N can also be observed in Fig. 15(b) for PS-64QAM system. It indicates that when the entropy is increased to 5.8 bit/symbol, the ratio of B to N is only 23.78%.

 

Fig. 15. Under ideal conditions of different entropies, the ratio of symbols in C_QPSK to the total number of symbols. (a) PS-16QAM, (b) PS-64QAM.

Download Full Size | PDF

Furthermore, according to the proportional relationship between B and N, we find that the number of symbols entering the third stage of the proposed scheme also has a significant impact on the FOE performance. To verify this finding, the number of symbols is tested for PDM PS-16/64QAM. As shown in Fig. 16(a), for PDM PS-16QAM-3 system, when the number of symbols entering the third stage decreases from 52701 to 3294, the GMI threshold for the same target would bring an OSNR penalty of 2.1 dB. When OSNR is set to 14 dB, 52701 and 26351 symbols could already reach the target GMIs. But at this time, the symbol number of the third stage is 3294, and there is a distance of 3.7 bits/symbol (dual polarization) from the target GMI. The PDM PS-64QAM-4.4/5 system depicted in Figs. 16(b) and (c) also demonstrate the same trend as Fig. 16(a). As the number of symbols in C_QPSK increases, the OSNR value required to achieve the same GMI condition also decreases. After the above analysis, for different requirements, a trade-off between complexity, existence conditions and performance can be made.

 

Fig. 16. The effect of different number of symbols on the GMI performance using ODT-QP FOE scheme. The total number of symbols is set to 4096, 8192, 16384, 32768, and 65536, respectively. The number of symbols shown in the legend is obtained in the third stage according to the proportional relationship of B/N. (a) PDM PS-16QAM-3, (b) PDM PS-64QAM-4.4, (c) PDM PS-64QAM-5.

Download Full Size | PDF

Table 2 lists the complexity comparison between ODT-QP FOE scheme and FFT FOE scheme. We have tested the complexity of the total number of fixed symbols of 4096, 8192, 16384, 32768 and 65536 for PS-64QAM-4.4. The results show that the real multiplications of the ODT-QP FOE scheme are 16.25%, 15.35%, 14.54%, 13.81%, and 13.15% as those of the FFT FOE scheme, respectively. They also do not exceed 16.25% of the FFT FOE scheme in the worst case. Even under the most extreme with B equal to N for PS-16QAM-2 system, the test results demonstrate that although 4096, 8192, 16384, 32768 and 65536 symbols are used, the real multiplication of ODT-QP FOE scheme still does not exceed 26.5% of FFT FOE scheme.

Table 2. Complexity comparison of FFT FOE and ODT-QP FOE

View Table | View all tables in this article

6. Conclusion

In this paper, we propose a blind optimal decision threshold assisted QPSK-partition FOE for PS-MQAM systems. The effectiveness of this scheme has been verified by 28 GBaud PDM PS-16/64QAM simulation system and 28/8 GBaud PS-16/64QAM experimental system. The results show that this scheme can stably achieve high NMSE accuracy in the range of [-R_s/8, R_s/8]. Specifically, for the optical fiber link transmission system of PDM PS-64QAM-4.4, under the condition of 28 GBaud, the NMSE can stabilize around 4.3e-8 in the range of [-3.5 GHz, 3.5 GHz]. In addition, the proposed scheme is especially suitable for moderately or strongly shaped PS-MQAM systems. For the PS system with weak shaping strength or higher-order modulation format, the FOE accuracy can be improved by increasing the number of symbols entering the third stage of the scheme.

Our propose ODT-QP scheme is more tolerant to ASE noise than the scheme of symbol decision by Euclidean distance. And it improves the traditional time-domain 4^th power feedforward scheme to make the FOE more accurate. The overall computational complexity of the proposed scheme can be as low as O(N), which is reduced at least 73.5% compared to FFT FOE scheme. The scheme can reduce the complexity while meeting the performance requirements. Therefore, we believe that the proposed blind and low-complexity FOE scheme for PS-MQAM systems will have a good development prospect in the future optical communication field.