Gaussian wavelet basis expansion based phase noise suppression method for coherent optical OFDM systems

In this paper, we developed a phase noise suppression method based on Gaussian wavelet basis expansion (GWBE) for coherent optical orthogonal frequency-division-multiplexing (CO-OFDM) systems. Compared with traditional phase noise suppression methods, the GWBE method significantly enhances the system’s robustness to laser phase noise, providing improved phase noise suppression, especially in scenarios with high linewidth. Additionally, the method demonstrates excellent performance in 32-QAM and 64-QAM. By employing the proposed GWBE method, the impact of laser phase noise on CO-OFDM can be effectively mitigated.


Introduction
CO-OFDM is a modulation and multiplexing technique for optical communication systems.Compared with OFDM, CO-OFDM has the advantages of high spectral efficiency, strong robustness to dispersion and polarization mode dispersion [1][2][3][4].Due to the characteristics of fiber optic transmission medium, the phase noise in CO-OFDM system may be more serious, and the impact on system performance is more significant.Common phase error (CPE) and inter-carrier interference (ICI) caused by laser phase noise are important interferences in CO-OFDM transmission system [3,5].It causes the receiver to be unable to accurately demodulate and decode the signal, thus increasing the bit error rate (BER) and reducing the capacity and reliability of the system [6].
In recent years, there have been many reports on phase noise estimation schemes for OFDM systems.In [7], T. Nguyen et al. proposed a method that utilizes a simplified extended Kalman filter (S-EKF) to be applicable in both pilot-aided and blind modes for phase noise compensation.This method is suitable for transmission systems using a small-to-moderate number of subcarriers.In [8], B. Sokal et al. demonstrated that the received signal of a pilot subcarrier can be modeled as a third-order PARAFAC tensor.And two algorithms for channel and phase noise estimation of pilot subcarriers are proposed.The two algorithms achieve similar performance.In [9], Yi et al. proposed the pilot-aided and dataaided methods.They identified the CPE and approximated the ICI as additive Gaussian noise for effective CPE compensation.In [10], Fang proposed a phase noise suppression method based on orthogonal basis expansion (OBE), which has been shown to be effective in suppressing ICI caused by laser phase noise in CO-OFDM system.In [11], the presence of Carrier Frequency Offset (CFO) and Linear Phase Noise (LPN) introduces Inter-Carrier Interference (ICI) and leads to constellation rotation, which degrades the system performance.Liu, Shuai et al. proposed to utilise a Gaussian particle filter (GPF) and an extended Kalman filter (EKF) to dynamically and jointly track and compensate for carrier frequency offsets and linear phase noise."Pilot-Based Phase Noise Tracking for Uplink DFT-s-OFDM in 5G" by Jean-Christophe Sibel: This paper compares various phase noise tracking algorithms for 5G, focusing on low-complexity methods like average phase error estimation and linear interpolation between pilot observations and high-complexity methods such as projection on the discrete cosine transform basis and Kalman filtering.In this paper, we investigated models for phase noise suppression methods in 16-QAM, 32-QAM, and 64-QAM CO-OFDM systems.We leveraged the Gaussian characteristics of inter-carrier interference (ICI) noise.A novel phase noise estimation method for CO-OFDM systems based on Gaussian wavelets was proposed for estimating and compensating phase noise.Compared to alternative approaches, it demonstrated superior phase noise fitting capabilities for additive white Gaussian noise.Particularly, in comparison to the OBE [10] scheme, GWBE maintained similar computational complexity while exhibiting better phase noise suppression, especially in scenarios with high linewidth.According to simulation results, this method significantly enhanced phase noise suppression in CO-OFDM systems.

Principle analysis
The digital signal processing (DSP) diagram of the CO-OFDM system is shown in Figure 1.
The transmitted signal chant of the OFDM system affected by phase noise is represented as: In (1), φ(t)is the laser phase noise should obey the Wiener distribution.N is the number of subcarriers within an OFDM symbol.α m is the first OFDM symbol in any m subcarriers in any OFDM symbol.T s is the time for an OFDM symbol period.ω(t) is the additive ASE noise obeying a Gaussian distribution in the channel.

Theory dedution
Considering the presence of linear transmission impairments such as CD and PMD in the fiber channel, laser phase noise, and amplified spontaneous emission (ASE) noise caused by periodically placed optical amplifiers in the fiber link.
( 2 ) where   0  1 ⋯   1 denotes the i-th OFDM block received after discrete Fourier transform (DFT) operation.  0  1 ⋯   1 indicates ASE noise in the fiber link.H is the N×N diagonal matrix representing the linear transmission impairments associated with CD, PMD.We consider the fiber optic channel as a retarded channel, then we ignore the subscript i in the channel transmission matrix.F and F H denote the matrices realizing DFT and IDFT operations, respectively, with superscripts H denotes the conjugate operation.we define φi as the diagonal element of   0  1 ⋯   1 to represent the phase noise.
Here the G matrix is a matrix of dimension N×L.It can be derived from the Gaussian wavelet function.The time domain expression for the Gaussian wavelet function can be expressed as: ( 5 ) we have selected L to be an odd number, so t = (L-1)/2.The Gaussian wavelet basis expansion set is the n-th order derivative of the frequency domain expression.For example, if five basis are used, the GWBE is expressed as   ,   , 1,  ,  .Afterwards the elements within the matrix are normalised. is the column vector of L×1.denoted as the coefficients of the GWBE.According to the above equation, we can express the phase noise as: The signal obtained after channel estimation and channel equalization through the training sequence can be represented as: where * denotes the conjugate operation, and i  denotes the residual error due to ASE noise and other noise after channel equalization.For ease of computation we denote the signal as: where  can be expressed as: Since a is a diagonal matrix, then by the law of exchange we can express the signal as: finally, we are left with only two positional variables, i  and  .We define: * =F ( ) We use pilots for phase noise estimation, where the pilots are inserted uniformly in each OFDM signal.We denote the position of the pilot frequency by  .the pilot is extracted from the OFDM symbols after channel equalization as: Using the principle of least mean squared error, we can compute an estimate of the i From this we obtain the coefficients of the Gaussian wavelet basis expansion in the presence of phase noise, and we can represent the phase noise compensated signal as:

Complexity analysis
The main cause of the computation complexity in the GWBE is primarily attributed to the matrix multiplications and matrix inversions present in equations ( 11), (13), and (14).In practical scenarios.

Simulation results and discussion
The GWBE method was validated through simulations of a CO-OFDM system using the commercial software VPI TransmissionMaker 9.9.
The OFDM frame in the simulation was designed as follows: The FFT/IFFT size was set to 256.The 16-QAM signal was mapped and converted to the time-domain using a 256-point IFFT.40 guard subcarriers were added and allocated at both ends of the band to provide approximately 20% oversampling for spectral separation of aliasing products with a low-pass filter (LPF).A cyclic prefix (CP) and cyclic suffix (CS) of length 16 each were inserted into every data block before transmission.The sampling rate Rs was chosen as 10 GS/s.The system configuration for 32-QAM and 64-QAM is similar to that of 16-QAM, with the exception that 40 pilot symbols have been uniformly set due to the increased impact of phase noise.Figure 2 represents the BER as a function of optical signal-to-noise ratio (OSNR) for different number of bases, when the laser linewidth is 800 kHz and the pilot number is 40.With the increase of OSNR, the accuracy of GWBE (L=5) improved evidently.The performance of GWBE (L=5) outperforms GWBE (L=3) due to the improved accuracy brought by 5 basis.For example, when the BER is 3.8×10 -3 for the GWBE, the required OSNR is 18.8dB for L=3 and 17.2dB for L=5.It is seen that the increase in the number of basics can significantly improve the system performance.Figures 3 and 4 demonstrate the transmission performances of three methods: GWBE, CPE [9] and OBE [10].The evaluations were conducted in back-to-back (BTB) scenarios, where the system was affected by the disturbance of laser phase noise.Figure 3 shows the BER of the three methods as a function of the OSNR for laser linewidth of 500 kHz.It can be seen that the GWBE method is significantly superior to both OBE and CPE when pilots is taken as 8, 16 and 24 respectively.For example, at an OSNR of 23 dB, the BER of CPE [9] and OBE [10] with 16 pilots are 4.6 × 10 -3 and 1.5 × 10 -3 , respectively, while the BER of GWBE with 16 pilots is 7.2 × 10 -4 .It can be seen that the GWBE method brought a significant improvement on the phase noise robustness compared with the CPE and OBE methods.
In Figure 4, we compare the required OSNR as a function of linewidth for the three methods CPE [9], OBE [10] and GWBE with a BER target of 3.8 × 10 -3 .We find that GWBE has a lower required OSNR at higher linewidth.At the linewidth of 500kHz, when the required OSNR of GWBE method is 16dB, the required OSNR of CPE method is 21.7dB and the required OSNR of OBE method is 17.3dB.In summary, The CPE [9] is more effective at low to moderate linewidth, and the GWBE can have significant advantages in combating laser phase noise at moderate to high linewidth.In Figure 5, we used a system with an 800 kHz laser linewidth and 20 pilot symbols.The green color represents the GWBE method, and the red color represents the OBE method.From the images, it is evident that under the same system conditions, the constellation diagram of the GWBE method converges more effectively than that of the OBE [10] method.This indicates that the GWBE method provides better robustness to the system, highlighting its stronger suppression effect on phase noise.
Compared to the conventional 16-QAM modulation, higher-order modulation schemes like 32-QAM and 64-QAM face increased challenges in terms of phase noise mitigation.These challenges encompass greater complexity, higher data transmission rates, stricter signal-to-noise ratio requirements, and a heightened susceptibility to error propagation.These factors necessitate more robust mitigation measures in high-order modulation schemes to ensure communication reliability and performance in the presence of phase noise.Therefore, the deployment of high-order modulation schemes requires careful consideration and the incorporation of effective phase noise suppression techniques to address these challenges.In Figure 6 and 7, we conducted performance tests for GWBE in 32-QAM and 64-QAM systems.Despite using 40 pilot symbols, it is still evident that GWBE exhibits good phase noise suppression.

Figure 6.
Comparing the BER performance of the GWBE method in the presence of interference caused by laser phase noise when using the 32-QAM modulation format in BTB transmission scenarios.
In Figure 6, we employed a 32-QAM system with 40 pilots and linewidth of 300 kHz and 400 kHz.We compared the impact of different linewidth on this system with a BER target of 3.8 × 10 -3 .It can be observed that for the 32-QAM system, with a laser linewidth of 300 kHz, the required OSNR of is 18.6 dB, and with a laser linewidth of 400 kHz, the required OSNR of is 21.1 dB.This indicates that GWBE effectively mitigates the effects of phase noise in the 32-QAM system.In Figure 7, we utilized a 64-QAM system with 40 pilots and linewidth of 100 kHz and 200 kHz.We compared the influence of different linewidth on this system to meet the BER target of 3.8 × 10 -3 .It can be seen that for the 64-QAM system, with a laser linewidth of 100 kHz, the required OSNR of is 19.7 dB, and with a laser linewidth of 200 kHz, the required OSNR of is 21.3 dB.Even in the 64-QAM system, GWBE continues to exhibit effective phase noise suppression.
These results strongly indicate that, both in the case of 32-QAM and 64-QAM systems, GWBE method effectively mitigates the adverse impact of phase noise on communication performance in highorder modulation schemes, offering a robust solution to enhance data transmission reliability and quality.
In this study, we simulated the CO-OFDM system and validated the Gaussian wavelet basis expansion (GWBE) method.The simulation involved detailed OFDM frame design, considering different numbers of basis functions, laser linewidths, and pilot symbols.Our results indicate that, compared to other methods such as CPE [9] and OBE [10], GWBE exhibits superior performance, especially under higher laser linewidths, demonstrating its effectiveness in suppressing phase noise.Furthermore, the method was successfully extended to 32-QAM and 64-QAM systems, showcasing consistent and effective phase noise suppression across different modulation schemes.Overall, our research highlights GWBE as a robust and promising solution for mitigating phase noise in high-order modulation schemes, contributing to improved reliability and quality of data transmission.

Conclusion
In this paper, we propose a phase noise suppression method for CO-OFDM systems using Gaussian wavelet basis expansion (GWBE).As demonstrated by numerous simulation results, our proposed method exhibits significantly improved robustness against laser phase noise compared to OBE and CPE methods.When constraining the BER to 3.8 × 10 -3 , it achieves a lower required OSNR compared to other methods, yielding superior outcomes.It is noteworthy that GWBE demonstrates exceptional proficiency in eliminating phase noise, especially in scenarios characterized by high linewidth.
Using the Fourier transform and Euler's formula, we can obtain the expression:

Figure 2 .
Figure 2. Comparison of BTB transmission with different numbers of basis for laser linewidth = 800 kHz and the number of pilots of 40.

Figure 3 .
Figure 3.Comparison of GWBE, OBE and CPE performances in BTB transmission with different number of pilots for laser linewidth = 500 kHz.

Figure 5 .
Figure 5.In a system with an 800 kHz laser linewidth and 20 pilot symbols, we conducted a constellation diagram comparison between the OBE method and the GWBE method.

Figure 7 .
Figure 7. Comparing the BER performance of the GWBE method in the presence of interference caused by laser phase noise when using the 64-QAM modulation format in BTB transmission scenarios.