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We show for the first time through comprehensive simulations under both uncompensated transmission (UT) and dispersion managed transmission (DMT) systems that the statistical distribution of the nonlinear interference (NLI) within the polarization multiplexed 16-state quadrature amplitude modulation (PM-16QAM) Coherent Optical OFDM (CO-OFDM) system deviates from Gaussian distribution in the absence of amplified spontaneous emission (ASE) noise. We also observe that the dependences of the variance of the NLI noise on both the launch power and the transmission distance (logrithm) seem to be in a simple linear way.
© 2013 optical society of america
1. Introduction
Constrained by the limitation of transmission bandwidth, the continuing growth of network traffic ignite a strong motivation to employ high spectral efficient (SE) transmission systems which aims at reducing the cost per bit as well as up-scaling the total transmission capacity. The coherent detection systems marked a rapid progress during the last few years. While polarization-division-multiplexed quadrature phase shift keying (PDM QPSK) has shown good transmission performance with its high spectral efficiency (SE) [1] higher level modulation formats such as PDM quadrature amplitude-modulation with 16 symbols (PDM 16QAM) has been attracting great interests too [2–5], as well as some other higher cardinality formats. As the principle and compensation of amplified spontaneous emission (ASE) noise has been well understood, the nonlinear interference may become the final obstacle to the development of the long-haul broadband optical transmission [6, 7]. Traditionally, researchers employs numerical simulations to analyze the effects of fiber nonlinearity by solving non-linear Schrodinger equation (NLSE) which is very time-consuming. Closed-form expression for nonlinearity in Dense Spaced-OFDM (DS-OFDM) systems with QPSK modulation format has been proposed based on the assumption that all the nonlinear effects such as XPM, FWM and SPM can be considered as FWM between all the subcarriers within a big “single band” [8]. They claimed that the approximated “FWM” can be modeled as an additive Gaussian noise and that the theory suits for all modulation formats by merely presenting the simulation results in the case of QPSK. Recent research shows that it may be not that intuitive to extend such property directly to 16QAM systems [9]. In 16QAM systems deviation from Gauss distribution for the nonlinear interference in the absence of ASE noise has been observed and the results are presented in this paper.
This paper is devoted to make a comprehensive study of the statistical characterization of the nonlinear noise within PDM 16QAM CO-OFDM systems under the circumstances of both UT and DMT. We carried out a broad range of simulations by analyzing in both linear and nonlinear regime under both UT and DMT fiber links in order to gain a thorough insight. We focused merely on the NLI itself by removing the ASE noise sources during the simulation process.
2. Simulation set-up
The parameters for the analyzed dual-polarization transmission systems are as follows illustrated by Fig. 1: 11 wavelength channels, each covering 32-GHz bandwidth, no frequency guard band between wavelength channels giving total bandwidth B of 352GHz; OFDM sub-carrier number of 4096 for the whole bandwidth; 16QAM modulation for each subcarrier. As you can see from Fig. 1, given no guard band in between, the spectrum structure on the left is completely identical to that on the right only that they are viewed from different aspects.
The simulation system is shown in Fig. 2. Two uncorrelated Pseudo Bandom Binary Sequences (PRBSs) with the order of 32 were employed: each for one polarization and from which 524288 bits were chosen and mapped into 16QAM constellation before IFFT. We did zero padding (ZP) with the ratio of 1/4 before the IFFT process to avoid the influence of unideal filters [10]. After IFFT cyclic prefix (CP) with the ratio of 1/16 was added to remove the channel-dispersion induced inter-symbol interference (ISI) and inter-carrier interference (ICI) [11]. The baseband OFDM signal was then converted to the analog domain by the digital-to-analog converter (DAC) and directly to the optical domain by an optical IQ modulator. After that the two polarized signals were mixed by polarization beam combiner (PBC).
We employed a transmission link composed of multiple spans, each including a 100-km length transmission fiber followed by a fixed length DCF fiber and an Erbium Doped Fiber Amplifier (EDFA) without ASE noise. The Scheme of one fiber span is pictorially described by Fig. 3. We considered two types of fibers: Standard Single Mode Fiber (SSMF) and Non-Zero Dispersion Shifted Fiber (NZDSF) with identical fiber loss but different chromatic dispersion coeficients of 16 ps/nm/km and 4 ps/nm/km respectively. The nonlinear coefficients for both kinds of fibers are 1.22 w−1km−1.
Fig. 3 Schematic of a single fiber span including a transmission fiber and a dispersion compensation fiber (DCF) followed by an Erbium-doped fiber amplifier (EDFA) with no ASE noise.
In order to gain a comprehensive simulation result we investigated into three combinations of fiber dispersion and compensation ratios: I) SMF with no dispersion compensation, abbreviated as “SMF 0%”; II) NSDSF with no dispersion compensation, abbreviated as “NZDSF 0%” and III) SMF with 95% dispersion compensated each span, abbreviated as “SMF 95%”. The details of these three cases are shown by Table 1.
Table 1. Three types of fiber links in detail
At the receiver part, the signal was then detected and received by two coherent optical receivers and then converted to electrical signal for further digital processing. Before DSP processing, chromatic dispersion were fully compensated. After removing CP, applying FFT and deleting the zero padding, the signals were equalized by (Minimum Mean Square Error) MMSE algorithm and then demodulated and processed to calculate the Q factor based on the Bit Error Rate (BER) according to Eq. (1).
The absence of ASE noise throughout the simulation enables a simple way of extracting the NLI by subtracting originally transmitted data from the recovered data at the receiver side as illustrated by Eq. (2) where RX, TX and NLI are all complex signals.
3. Simulation results
3.1. Comparison between closed-form theory and simulation results
The theoretical Q-factor QTheo. is estimated through Eq. (3) with the reference to [8]. Here, a little modification is done to the originally proposed equation as we omit the ASE noise and Noise Figure.
where I is defined as and I0 is defined as Eq. (4). where he is defined as Eq. (5). where B is the system bandwidth and fw is the walkoff bandwidth.As we can see from Fig. 4, the simulation result goes well with the theoretical performance estimated based on the proposed closed-form expression [8]. This comparison results indicate the validity of the system setup and the accuracy of the closed-form expression when applied to the 16 QAM system under the assumption of densely spaced subcarriers.
3.2. NLI distribution variance versus Transmission Length
Figure 5 shows the averaged variance of the NLI at the X polarization with the fixed launch power of 12dBm as a function of Transmission Length (km).
- SMF 0%: line with solid “Circle” marker;
- NZDSF 0%: line with solid “Triangle” marker;
- SMF 95%: line with solid “Diamond” marker;
As we can see from Fig. 5, the average variances increase at different speeds with the increase of transmission length (logrithm) in a smiple linear way. And if we analyze at the fixed transmission length such as 1200km, we can always conclude that the distribution variance within the three cases follow the relationship as SMF95% > NZDSF0% > SMF0%.
3.3. NLI distribution variance versus the Launch power
Figure 6 shows the averaged variance of the Nonlinear Interference (NLI) at the X polarization after a propagation of 1200 km as a function of launch power. The transmission is conducted in the absence of ASE noise thus enable us to focus merely on the nonlinearity induced noise within the system. As we can see in Fig. 6, the slopes of the three red lines are all very close to 2 (circle: 1.9, triangle: 2.05, diamond: 2.08 respectively), which indicate that 1-dB increase in launch power will introduce about 2-dB increase in the average variance. Thus we can conclude that under all cases, the variance of NLI noise will increase proportionally to the square of the launch power.
The group of blue lines within Fig. 6 shows the Gauss Fitting Error of the NLI calculated by Eq. (8) as a function of the launch power under the same system configuration. From the trend of these blue lines, the Gauss Fitting Error increases along with the increase of the launch power which indicates a trend of deviating from the gauss distribution especially for the “SMF 0%” case.
As shown in Fig. 7, GTheo.(n) and GSimu.(n) represent the number of the theoretical and simulation gaussian distribution values located at the nth hist bin respectively. We use 1000 bins for about 90000 samples during the simulation under the case of “SMF 0%” with the launch power of 12dBm and the histogram shows obvious diviation from the red Gaussian fitting line.3.4. Recovered constellation after propagation of 1200km
Figure 8(a) is the recovered constellation after propagation of 1200 km SMF fiber with no dispersion compensation. As we can see, the distributions of the constellation points at the outer ring are not isotropic while the inner four points display isotropic scattering. We colored the constellation into four groups with different colors in order to illustrate their difference clearly. Figure 8(b) is obtained from Fig. 8(a) by merging the four constellation points with the same color into one point located in the top right corner through rotating. As we can see from Fig. 8(b), the BLUE star shows the most severe deviation from the isotropic distribution. This conclusion goes well with the results obtained within the single carrier system by researchers from Fujitsu R&D Center (FRDC) [9]. The BLACK and GREEN stars show similar distribution with a deviation from the isotropic scattering in a milder way. A proper reason accounting for this deviation may lies in the non-constant envelop of 16QAM symbols. Figure 8(c) is used to depict the distribution of the density within a single compounded star. We have a red central disc and two interconnected ring forms with blue and green color respectively. Each of these three colors accounts for 30 percent of the 16QAM symbols. Thus the density of the 16QAM symbols decreases continuously along with the radius of the star points.
Fig. 8 Recovered constellation after transmission of 1200km at the launch power of 9 dBm in the case of SMF 0%.
Figure 9(a) is the recovered constellation after propagation of 1200 km NZDSF fiber with no dispersion compensation. As we can see, the distributions of the constellation points at the outer ring are not isotropic while the inner four points display isotropic scattering. This is similar to that of the case of “SMF 0%” but in a much lighter degree. We differenciate the constellation into four groups with different colors to magnify their difference. Figure 9(b) is generated from Fig. 9(a) by merging the four constellation points with the same color into one point located in the top right corner through rotating. As we can see from Fig. 9(b), the blue shows the most severe deviation from the isotropic distribution compared with the BLACK and GREEN stars which is similar to the conclusion in the case of “SMF 0%”. Figure 9(c) gives a similar conclusion that the density of the 16QAM symbols decreases continuously along with the radius of the star points.
Fig. 9 Recovered constellation after transmission of 1200km at the launch power of 9 dBm in the case of NZDSF 0%.
Figure 10(a) is the recovered constellation after propagation of 1200 km SMF fiber with 95% dispersion compensated. As we can see, all constellation points display isotropic scattering. This is very different to those of the case of “SMF 0%” and “NZDSF 0%”. Figure 10(a) is also classified into four groups with different colors. Then Fig. 10(b) can be obtained from Fig. 10(a) by merging the four constellation points with the same color into one point located in the top right quadrant through rotating. As we can see from Fig. 10(b), the four colored stars show almost the same isotropic distribution. Figure 10(c) gives a similar conclusion that the density of the 16QAM symbols decreases continuously along with the radius of the star points.
Fig. 10 Recovered constellation after transmission of 1200km at the launch power of 9 dBm in the case of SMF 95%.
Given the results of Figs. 8 and 10 and note that the reduction of the value of Chromatic Dispersion residue, we may draw the conclusion that for the 16QAM constellation the distributions of the constellation points at the outer ring will deviate from isotropic while the inner four points maintain with the increase of system dispersion residues.
4. Conclusion
We have for the first time shown through comprehensive simulation under both UT and DMT systems that the statistical distribution of the nonlinear noise within the PM-16QAM CO-OFDM system derivates from Gaussian distribution in the absence of ASE noise even in the UT circumstance. Our simulation results also showed that the dependences of the variance of the NLI noise on both the launch power and transmission distance(logrithm) seems to be in a simple linear way.
Acknowledgments
Project supported by National Natural Science Foundation of China (Grant No 61271192 and 60932004), National High-Tech Research and Development Program of China (Grant No 2013AA013401). We would like to thank Dr. Meng Yan and Dr. Zhenning Tao from Fujitsu Research & Development Center (FRDC) for useful discussions.
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