WRELM based CO-OFDM system with improved performance, spectral efficiency and computational complexity

Coherent Optical Orthogonal Frequency Division Multiplexing (CO-OFDM) system with Pilot-free phase noise compensator was introduced in order to accomplish the need of high spectral efficiency in the optical communication. In CO-OFDM system Discrete wavelet packet transforms (DWPTs) in place of Fast Fourier Transforms (FFTs) had attracted more attention since it has removed the need of cyclic prefix used to compensate fiber dispersion. In this paper, a DWPT based CO-OFDM system with Wilcoxon Robust Extreme Learning Machine based pilot-free phase noise compensator using multi-level QPSK partitioning of 16-QAM has been proposed. From the results of this work it has been seen that the percentage improvement in performance (in terms of Q-Factor) and spectral efficiency over traditional pilot-aided techniques is approximately 6 and 21 respectively. Moreover, this proposed work will comparatively reduce the overall system complexity.


Introduction
High-speed processing of signals is increasingly becoming a vital factor for several applications such as 5G Internet and cloud computing. The growing requirement of various services of the Internet are producing the need for the development of high spectral efficiency optical communication networks. Coherent Optical Orthogonal Frequency Division Multiplexing (CO-OFDM) system was introduced in order to achieve high spectral efficiency both in flexible passive optical networks and long-haul optical communication systems (Armstrong 2009;Shieh and Djordjevic 2010). Laser phase noise caused due to longer symbol duration has become a critical issue in CO-OFDM systems since it results in a degradation in overall system performance (Wu and Bar-Ness 2004). Various compensation techniques such as Pilot-aided (PA) and RF-pilot (RFP) was introduced for the mitigation of phase noise in a CO-OFDM system and verified for the compensation capacity (Yi et al. 2007, Zhu et al. 2015Randel et al. 2010). A key problem with pilot-based compensation techniques is the reduction in spectral efficiency because used pilot subcarriers require extra bandwidth. Therefore, various non-pilot aided CPE techniques (Mousa- Pasandi and Plant 2010;Ha and Chung 2013;Nguyen et al. 2017;Venkatasubramani et al. 2019) were introduced in order to overcome the problem of spectral efficiency decline. The main downside of these methods is computational expensiveness since the phase noise estimation is performed using the information of all the subcarriers. Various other techniques such as adaptive EKF Based estimator (Balogun et al. 2018) and Gaussian Wavelet Basis Expansion based phase noise compensator were proposed (Chen et al. 2020).
In this paper, CP-and pilot-free Wilcoxon Robust Extreme Learning Machine (WRELM) based Phase noise compensator has been proposed. In this technique, prior knowledge of channel statistics in the form of pilot subcarriers is not needed, since the received training symbols will be used in the compensator network. It has been shown in results that the proposed technique in comparison to the PA technique improves approximately 4 spans of the transmission reach. Further, it has been noticed from the results that at optimum launch power of 4dBm the improvement in performance is approximately 0.6 dB (in terms of Q-Factor). The most noteworthy observation received from this work is that by using the WRELM phase noise compensation technique the results will be more robust and are 21% high spectral efficient as compared to the PA technique.
This paper is divided into six sections. The first section gives an introduction. The second section explains the simulation setup with mathematical modeling of the CO-OFDM system. In the third section, mathematical modeling of proposed work is presented. The fourth section compares the spectral efficiency and overall system complexity of the proposed work with previous techniques. The fifth section discusses various simulation results. The main conclusions are drawn in the final section.

Simulation setup
The transceiver design of a basic CO-OFDM system used for simulation work has been shown in Fig. 1. Before applying the input data at the serial-to-parallel converter of the transmitter side, 16-QAM modulation using multi-level QPSK partitioning with sliding window (Fatadin et al. 2014) has been performed mapping of data. After that for generating the subcarriers, inverse discrete wavelet packet transform (IDWPT) has been used (Li et al. 2010). In our simulation model, a total number of subcarriers is equal to data subcarriers i.e.,128. Since with the use DWPT in place of FFT, chromatic dispersion and polarization mode dispersion has been mitigated (Li et al. 2010;Kattoush et al. 2013;Bulakci et al. 2009). Therefore the OFDM frame consists of a preamble (training symbols for channel equalization) and data symbols only (means there is no need for cyclic prefix symbols). These OFDM frames has been applied to a digital to analog converter of clipping ratio 4 dB. And low pass filter has been used to avoid aliasing.
For the electrical to optical conversion, a pair of Mach-Zehnder modulators (MZMs) with a phase shift of 90 degrees has been used, to provide smooth frequency response over a range of multiple GHZ. The optical carrier line-width used in the IQ modulator is 1550 nm. Then this modulated signal has been transmitted over multiple spans of single-mode fiber (one span is of 80 km). Erbium-doped fiber amplifiers has been used after each span for the compensation of link losses. The total amplifier gain G for an EDFA of length L is obtained using equation; sT 1 a d hv p P p z , and a d = Γ s a s = Γ p a p N t is the total ion density, a d is the cross-sectional area of the doped portion of the fiber core, P s and P p are input and pump signal powers, T 1 is the spontaneous lifetime of the excited state (10 ms for EDFAs), v s and v p are input and pump signal frequency, e s and a s are the emission and absorption cross section at the angular frequency of s , and Γ s is the input confinement factor.
At the receiver side, coherent detection in place of direct detection is used since it has more spectral efficiency. Analog to digital converter with a sampling rate of 6.3 GSamples/s has been used. In this work, our main focus is on phase noise compensator block. Various simulation parameters used for this work is presented in Table 1.

Mathematical modeling
One of the main aims of this work is to improve the spectral efficiency of a CO-OFDM system. This is possible by avoiding the Cyclic prefix (CP). It has been seen in the previous studies that the need of CP can be eliminated by replacing FFT/IFFT based CO-OFDM systems with wavelet packet transform (WPT) based CO-OFDM systems (Li et al. 2010;Kattoush et al. 2013;Bulakci et al. 2009). The length of sinusoids functions of FFT in time domain is infinite while the length of wavelets both in time and frequency domain is finite. Moreover, the speed of computation with WPT is faster than FFT since we can use effective computational algorithm. In this paper, we have used IDWPT/ DWPT are at the transmitter and receiver end, respectively. The transmitted CO-OFDM signal after using Jones vector of IDWPT is expressed as follows: (2) The time domain polarization components of s(n) has been represented by s x and s y . Jones vector of transmitted i th OFDM information symbol for k th subcarrier has been denoted by − c (k, i) with the polarization components c(k, i) x and c(k, i) y . Here, f k is the k th subcarrier frequency, T s denotes a period of OFDM symbol, t s is the observation period, Δ G is the length of Guard interval, and N sc is the number of subcarriers i.e., 128.
For handling the fiber dispersion i.e., CD and PMD, the length of a guard interval has been selected by using the following marginal timing condition; Here c is the speed of light, f is the optical carrier frequency, D t is the total accumulated chromatic dispersion, Δf is the spacing between subcarrier channels, and DGD max is the maximum value of differential group delay. For generating a sufficient margin, the chosen value of DGD max should be equal to 3.5 times the mean value of PMD.
For the efficient computation of DWPT, FIR filter bank i.e., quadrature mirror filter (QMFs) has been used. The received CO-OFDM signal in the form of wavelet packet coefficients, can be expressed as follows: t represents the transpose of Jones vector of received ith OFDM information symbol for kth subcarrier. The inline optical amplifiers based ASE noise with polarization components has been represented by The optical fiber link Jones matrix of the channel has been denoted by T(k) which can be expressed as follows: where N is a number of spans or cascading elements, l is the birefringence vector, and l is the Polarization dependent loss (PDL) vector, is the Pauli matrix.
The chromatic dispersion based phase change D f k can be easily estimated as follows: where f LD is the laser frequency.
(k, i) represents the phase noise in i th OFDM symbol and k th subcarrier due to phase noises generated at both transmitter and receiver lasers and local oscillators. The estimated phase drift of the i th OFDM symbol can be written as; where arg(.) is the phase angle of the information symbol, ŝ ik is the known transmitted training symbol, and r ik is received symbol, N SC the number of OFDM subcarriers. Moreover, laser phase noises are more dominating in (k, i) . The focus of this work is to estimate and compensate the effects of laser phase noise. The I(i) is the Inter-carrier Interference arg r ik − arg ŝ ik noise in ith OFDM symbol generated from phase noise and can be calculated as (Zhou et al. 2012); ICI coefficient is represented by m with m difference between two subcarriers; the value of normalized RFO ≪ 1.
The frequency domain representation of subcarriers has been illustrated in Fig. 2a, b without any phase noise and with phase noise respectively. In Fig. 2b, it has been shown that due to the effect of inter-carrier interference generated from laser phase noise the orthogonality between the subcarriers of the OFDM system is lost. The main purpose of this work is to propose an improved phase noise mitigation technique and compare it with existing techniques.
The variance of the interference I(i) can be calculated as (Zhou et al. 2012); The effective signal to noise ratio (SNR ' ) has been used as performance metric in this work and can be calculated as (Wan and Banta 2006); is the original channel SNR without the effect of inter-carrier interference . 2 w is the variances of the ASE noise. (11)

Fig. 2
Frequency domain picture of subcarriers a without phase noise corruption b with phase noise corruption

Proposed phase noise estimation technique
Machine learning is attracting much attention over the last several decades (Wan and Banta 2006;Hogg et al. 2005;Chuang et al. 2004). It has been noticed that the convergence rate of Wilcoxon robust extreme learning machine (WRELM) algorithm is very fast as compared to other available robust machine learning algorithms (Xie et al. 2016).
A real-complex extreme learning machine (RC-ELM) based phase noise compensation technique was proposed which has used the pilot subcarriers as a training set for learning (Blanco et al. 2020). But, this technique has many drawbacks such as difficult to implement, less robustness, and less spectral efficiency. For removing these drawbacks, WRELM based algorithm for phase noise mitigation has been used in work. Since we are using channel equalizer training symbols instead of pilot subcarriers, therefore, the proposed work has high spectral efficiency than previously proposed RC-ELM technique. In this section, the detailed mathematical modeling for proposed technique has been presented.

Basic concepts of Wilcoxon norm
Before discussing the proposed work some basic concepts of Wilcoxon norm have been presented in this section. In statistics, due to its robustness against outliers, Wilcoxon learning over linear regression methods is attracting widespread interest (Hsieh et al. 2008). In Wilcoxon learning, as the main function Wilcoxon norm is calculated for a vector (Hogg et al. 2005). The Wilcoxon norm of a score function φ: (Armstrong 2009) → R (which is a non-decreasing function) can be given by The value of this score function ( ) has been calculated as (Hsieh et al. 2008); where l denotes fixed value positive integer. To get the Wilcoxon norm of any function, a pseudo-norm (semi-norm) is utilized (Wan and Banta 2006;Hogg et al. 2005;Chuang et al. 2004;Xie et al. 2016;Blanco et al. 2020;Hsieh et al. 2008) which is calculated as follows;

Mathematical modeling of the proposed work
In this work, the phase noise for i th OFDM symbol has been modeled as wiener process; where Δ (k, i) is a Gaussian random variable. It has been represented in matrix H and Penrose-Moore pseudo inverse H + for k number of subcarriers and i number of OFDM symbols. The matrix H has been calculated using following equation (Huang et al. 2004); where a j and v j are the mean and variance. In phase noise estimation, the value of a j and v j is zero and 2 Δ T s respectively. In this work, Radial Basis Function (RBF) network based Gaussian activation function i.e., G(a j , v j , ) has been calculated using following equation (Liang et al. 2006); The equation used for the calculation of gaussian activation function is; where i is the OFDM symbol number and it varies from 1 to o . In this work, phase noise has been predicted by using training symbols (preamble) utilized for channel equalization. The value of common phase noise error has been estimated by using the Kalman filtering (Weiner process) based recursive formulae presented below: where the initial value of apriori and the aposteriori phase change i.e., (k, i|i − 1) and (k, i) has been estimated by using training symbol and zero respectively.K(k, i) denotes the Kalman gain. The formula used to calculate the value of (k, i) mean rotation angle is expressed as follows: In this technique, for preventing blind estimation based phase slip all symbols has been first rotated by apriori phase value. Then, these symbols has been raised to the power of four for estimating the in m and out m mean rotation angles. e.g., weight value w = 0.2, Rotation angle of outer constellation points i.e., (k, i) out is 40 0 , Rotation angle of outer constellation points i.e., (k, i) in is 10 0 and apriori phase change (k, i|i − 1) of training OFDM symbol is 5 0 . So the mean rotation angle is 21 o . For a large number of subcarriers, the estimation of the rotation angle is more accurate. Since a number of points required for the inner and outer layer of constellation diagram has been accurately calculate for QPSK partitioning of QAM modulation format.
The weight w has been initially selected randomly between 0 to 1 and updated using following formula (Xie et al. 2016); the value of scale factor (i) was calculated; The Residual values ( (i)) has been calculated using following equation; where (k, i) = ( (k, i) d − (k, i) ) , here (k, i) d is the desired rotation angle, (k, i) is the actual rotation angle, and R( (k, i)) denotes the rank of the residual (k, i) . Moreover, c is a fixed constant value and I(f k ) is the Fisher information, which has been calculated from the applied density function f k as (Xie et al. 2016); This procedure of angle rotation and phase estimation has been repeated for the initially selected number of epochs. At the end, the response of WRELM algorithm for phase noise compensation in the CO-OFDM system has been checked. The number of epochs can be increased according to the required set SD-FEC limit of BER. The proposed compensator is working on many equations. For better clarification of the proposed work one flow chart is added in paper as Fig. 3.

Spectral efficiency calculation
The spectral efficiency (amount of information transmitted per unit bandwidth) of a traditional CO-OFDM system is; The spectral efficiency of a proposed CO-OFDM system is; The data subcarriers by using 16-QAM and QPSK samples with PRBS-15 bit stream has been loaded in N filled subcarriers. For the generation of 128 number of subcarriers the value of N SC has been selected to 128. N CP i.e., length of cyclic prefix has been set at 0 since the use of DWPT has removed the need of CP to compensate dispersion (linear impairments). One frame of OFDM has been comprised of the preamble (training symbols) used for channel estimation followed by data symbols. In this study, 32GBd CO-OFDM system has been considered. The symbol b denotes the number of bits per symbol.
To calculate the total occupied bandwidth (BW) following formula has been used: where t s is the observation period and T s is the OFDM symbol period. And Total symbol rate R is; t s is the observation period and T s is the OFDM symbol period. For the generation of 128 number of subcarriers the value of N SC has been selected to 128. The % improvement in SE with the proposed technique is approximately 21 as compared to conventional pilot aided technique which had reported equal to18 (Venkatasubramani et al. 2019) in Pilot-free K4P technique by L. N. Venkatasubramani and co-authors.

Overall complexity comparison
The step by step comparison of system complexity of the proposed work and other techniques has been presented in Table 2. The complexity with proposed work to compensate linear impairments such as CD and PMD is N 2 sc which was equal to 4N 2 sc with conventional PA and Pilot free K4P technique.
The complexity for splitting the band into a number of subcarriers using a transform with proposed work is equal to N sc − 1 ⋅ L , which was N sc ⋅ (log 2 N sc ) in conventional PA and pilot-free K4P technique. Here, L is a length of the QMF used for DWPT.
In the conventional Pilot-Aided phase noise compensation technique, the complexity was dependent on the number of pilots ( N P ) used. The complexity in Conventional Pilot-Aided technique and Pilot-free K4P algorithm for the compensation of phase noise was equal to 9 × N P and (12 ∧ +1)N sc , which with the proposed technique is equal to ((12 ∧ +1)N sc L)-3L. Here, ∧ is the ratio of the outermost circle constellation points to the total number of constellation points of QPSK. The value of ∧ is equal to 0.25 for 16-QAM. From this Table 2 it has been observed that the overall system complexity will reduce with the proposed work.

Simulation results and discussions
The simulation results obtained using various phase noise compensation techniques to improve the performance of the CO-OFDM system have been summarized in Figs. 4-11. In this section various parameters used for performance measurement such as optical signal to noise ratio (OSNR), error vector magnitude (EVM), Bit error rate (BER) and Q-Factor has been presented.
The relationship between the optical SNR (OSNR) and effective SNR presented has been given by following equation; where B f denotes the central bandwidth while R s denotes the symbol rate.
In this work, the multi-level QPSK partitioning of 16-QAM modulation format has been used. Here in 16-QAM modulation format, outermost and innermost constellations points has been calculated at a phase angle of (2n + 1) ∕4 using QPSK modulation technique, where n has been varied from 0 to 3. For five-channel CO-OFDM system, at a combined laser line-width of 200 kHz, the received constellation diagram with the proposed work has been presented in Fig. 4.
To calculate the numerical value of CO-OFDM system performance from the spreading of points in the received constellation diagram, error vector magnitude (EVM) has been used. The formula to calculate the EVM has been presented below (Schmogrow et al. 2012): where x k and y k represents the ideal and received constellation points, respectively. And S max denotes the maximum number of points in the constellation diagram over which the EVM has been calculated.
The formula used for the calculation of BER from the EVM in M-ary square QAM format has been presented below (Fatadin 2016;Shafik et al. 2006). Now let us discuss the simulation results of the proposed work. First of all, the effect of variation in normalized line-width on B2B performance of the CO-OFDM system has been studied. For the measurement of performance at each line-width, the value of BER has been calculated from OSNR. The normalized line width versus OSNR penalty, at BER of value 1 × 10 −3 using proposed technique as compared to previously reported techniques in back to back transmission has been presented in Fig. 5. It has been observed in this plot, that by increasing in a number of pilots is reducing the OSNR penalty. But it has reduced the spectral efficiency. Moreover, it has been seen in the results of Fig. 5, that the proposed technique is significantly reduce the OSNR penalty at smaller values of Δ T s .
The improvement in OSNR penalty (dB) and spectral efficiency (%) at high value of normalized line-width i.e., 2 × 10 −3 has been presented in Table 3. From this Table 3, two main observations have been concluded. First observation is that the proposed work is better in terms of spectral efficiency as compared to previously reported techniques for phase noise compensation with respect to the pilot-aided technique. Second observation from this Table 3 is that proposed technique is better in terms of OSNR penalty as compared to earlier techniques. For the simulation of the multi-channel CO-OFDM system, the split-step Fourier method of nonlinear Schrödinger equation has been used to solve the propagation of the optical signal through the optical fiber channel. This signal has been passed through multiple spans of fiber with optical amplifiers for the compensation of loss from each span. For choosing the desired band, the received signal has been passed from the filter and applied to a coherent receiver. In the end, the received signal has been applied to the phase compensation module and it has been assumed that the received signal is perfectly synchronized in time and frequency using Schmidl-Cox based synchronization algorithm.
The received modulated spectrum in terms of power spectral density for x-polarization with 5 channel CO-OFDM system is shown in Fig. 6. The center carrier frequency has been set at 1550 nm. One channel has been comprising bitrate of 200Gbps, and the net data rate for the multichannel CO-OFDM system has been set at 1 Tbps.
OSNR (dB) versus BER using proposed technique as compared to theoretical results and previously reported techniques in 5 channel CO-OFDM system has been presented in Fig. 7. It has been observed in this plot, that by using proposed technique the obtained results are close to the theoretical results as compared to all the previously used methods.
The comparison of launched power versus BER for 5 channel CO-OFDM system without phase noise compensator (PNC) and with different techniques has been presented in Fig. 8. From this Figure, it has been observed that the performance is improving by increasing in launch power up to the optimum value of launch power (~ 4dBm). The BER performance, beyond this value of optimum launch power has been degrading which is due  to the nonlinear effects of fiber. It has been seen in this plot that in the linear regime the performance with proposed work has been better as compared to other pilot-aided or nonpilot techniques. Till now, in most of the studies, the Q-Factor was the main factor under consideration for the measurement of CO-OFDM system performance. Therefore in this work, Q-Factor has also been calculated for comparison of proposed work with the previous techniques of phase noise compensation. The value of the Q-factor can be calculated by the formula given as (2013); For this work, the margin value of Q-factor has been chosen to 6.2 dB which has been obtained with 25% FEC soft-decision (SD) on the standard value of BER (i.e., approximately 10 −3 ). The plot of launch power versus Q-Factor for five-channel CO-OFDM, without PNC and with different techniques such as PA-12 pilots, K4P, RC-ELM, and proposed work, has been presented in Fig. 9. At optimal launch power of 4dBm, the percent improvement in Q -Factor as compared to PA-12 pilots with K4P, RC-ELM, and proposed work, has been equal to is 1.25, 4.61, and 6.013 respectively. At 200 kHz combined laser line-width, the performance in BER versus fiber length of 800 km to 1600 km (10 to 20 number of spans) at a bitrate of 1Tbps for multi-channel CO-OFDM system has been presented in Fig. 10. At the pre-FEC limit of BER value of 1 × 10 3 , it has been observed that the improvement in transmission reach using the proposed technique is approximately 300 km more in comparison to PA-12 pilots, while K4P and ELM technique was equal to 100 km and 240 km respectively.
The performance in BER versus fiber length of 800 km to 1600 km at a bitrate of 1Tbps for different values of linewidths i.e., 200 kHz, 500 kHz, and 1000 kHz using proposed technique in multi-channel CO-OFDM system has been presented in Fig. 11. From this

Conclusion
A DWPT based CO-OFDM system with Wilcoxon Robust Extreme Learning Machine based pilot-free phase noise compensator using multi-level QPSK partitioning of 16-QAM has been proposed in this paper. From the results of this work it has been seen that there is an improvement in performance (in terms of Q-Factor), OSNR penalty and spectral efficiency over traditional phase noise compensation techniques. Moreover, this proposed work comparatively reduced the overall system complexity since DWPT has been used in place of FFT that removed the need of cyclic prefix. Wilcoxon Extreme learning machine has been used for weights updation in phase noise compensator that offers an extremely fast speed of training and improved generalization performance with less number of user interventions with more robust results.