Adaptive CKF phase noise compensation method in coherent light OFDM system

Abstract

The invention relates to a self-adaptive CKF phase noise compensation method in a coherent light OFDM system. The method comprises the following steps of: estimating a phase noise value of each OFDM symbol at a signal receiving end through known pilot frequency information, performing first phase noise fitting by using a second-order Hermite interpolation function, performing pre-judgment and secondary symbol partitioning on a compensated signal, performing second phase noise fitting by using an updated second-order Hermite interpolation function, and performing phase noise fitting on the compensated signal. And finally, the residual phase noise is suppressed by using the characteristics that the adaptive CKF can adaptively update the process noise and reduce the estimation error. Simulation results show that the method can effectively compensate CPE and ICI phase noise, meanwhile, the tolerance of the CO-OFDM system to the laser line width is also improved, and the performance of the system is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of phase noise compensation in a coherent optical orthogonal frequency division multiplexing (CO-OFDM) system, and is a method for compensating phase noise by combining Block Sub-symbol Second-order Hermite Interpolation (BSSHI) and Adaptive Cubature Kalman Filter (ACKF). Background Art

[0002] The Coherent Optical Orthogonal Frequency-Division Multiplexing (CO-OFDM) system was proposed in 2006. It is based on the Orthogonal Frequency Division Multiplexing (OFDM) system and combines the optical communication system and coherent detection technology. Therefore, the CO-OFDM system has the advantages of "coherent detection" and "OFDM" modulation, high spectrum utilization, the photoelectric conversion at the receiving end is approximately linear transformation, and there is no beat noise at the receiving end. When the system transmission rate is high, the system performance will not deteriorate sharply. At the same time, the CO-OFDM system has high receiving sensitivity, good tolerance to chromatic dispersion (CD), and strong ability to resist polarization mode dispersion (PMD).

[0003] A major drawback of the CO-OFDM system is that it is very sensitive to the phase noise introduced by the local oscillator. The noise is mainly divided into two categories: one is the common phase error (CPE) caused by the zero-order spectral component of the phase noise, which will cause the signal constellation point to rotate; the other is the inter-carrier interference (ICI) caused by the non-zero-order spectral component, which will cause the cloud-like divergence of the signal constellation point. Both will lead to the degradation of CO-OFDM performance. In order to solve the problem caused by phase noise, it is necessary to use narrow linewidth lasers with a linewidth of hundreds of kHz or less at the CO-OFDM transmitter and receiver, but as the laser linewidth decreases, the system cost will increase significantly. In addition to using narrow linewidth lasers, the phase noise compensation method of digital signal processing has become an economical and efficient solution. Therefore, using a good phase noise compensation method in the CO-OFDM system can greatly reduce the requirements for the laser linewidth. Summary of the invention

[0004] In view of this, an object of the present invention is to provide a phase noise compensation algorithm of an adaptive Kalman filter in a coherent optical OFDM system.

[0005] To achieve the above object, the present invention provides the following technical methods:

[0006] In the first stage, the pilot is inserted uniformly at the signal transmitting end, and at the signal receiving end, after channel equalization, the phase noise estimate φ of the i-th OFDM symbol is estimated using the known pilot. i Then, the second-order Hermite Interpolation (SHI) function is used to obtain the OFDM symbol phase noise estimate of the kth sampling point of the i-th OFDM symbol in the time domain. This compensates for the phase noise.

[0007] In the second stage, the signal processed by the first SHI algorithm is converted into a frequency domain signal Z through Fast Fourier Transform (FFT) i,k , and perform tentative decision processing on the frequency domain signal, and then use the decision result Converted into time domain signal through Inverse Fast Fourier Transform (IFFT) The OFDM symbol is then divided into several sub-symbols, and the average phase deviation σ of each sub-symbol is estimated using the least squares (LS) criterion. n That is, phase noise, and σn As the phase noise estimation value of the intermediate sampling point of each sub-symbol, SHI is then used to perform phase noise fitting, and then the phase noise compensation of the signal is obtained.

[0008] In the third stage, the compensated signal is filtered using the adaptive Kalman filter (ACKF). and the signal after the tentative decision Further compensation is performed, which mainly includes three parts: time update, measurement update and process noise adaptive update. (1) Time update: The phase noise value and covariance estimate of the previous sampling point are subjected to Cholesky decomposition to obtain volume points and state volume points, and then the state volume points are used to update the predicted state and predicted variance. (2) Measurement update: Cholesky decomposition is performed based on the predicted variance and pre-judgment to obtain the corresponding volume points and measurement volume points. The measurement prediction value, new information variance and covariance are calculated through the volume points, and the CKF gain is calculated to update the phase noise value and covariance value. (3) Process noise adaptive update: The covariance of the noise is updated using the forgetting factor and the weighted combination of the current and past estimated data, and the updated process noise covariance is used for the estimation at the next moment until the algorithm ends.

[0009] The beneficial effects of the present invention are:

[0010] The present invention relates to a phase noise compensation method combining second-order Hermite interpolation with adaptive cubature Kalman filtering. The invented method eliminates the influence of CPE and ICI in a CO-OFDM system, improves the problem of low tolerance of the system to phase noise, and improves spectrum utilization. The simulation results show that the bit error rate performance of this method is improved to a certain extent compared with the pilot-assisted algorithm in the literature [1] "HONG XZ, HONG XJ, HE S. Linearly interpolated sub-symbol optical phase noise suppression in CO-OFDM system[J]. Optics Express, 2015, 23(4): 4691-4702.", the extended Kalman filter algorithm in the literature [2] "Bao WB, Xiao MH, Fang SL, et al. Lagrange interpolation based extended Kalman filter for phase noise suppression in CO-OFDM system[J]. Optics Communications, 2019, 435(3): 221-226.", and the cubature Kalman filter algorithm in the literature [3] "Yuan Jianguo, Li Yangxu, Zuo Zhihang. CKF phase noise compensation algorithm based on block Lagrangian interpolation in CO-OFDM system[J]. Acta Optica Sinica, 2022, 42(16): 1606003.". In summary, the algorithm proposed in the present invention has a relatively obvious effect on phase noise compensation, still has good bit error rate performance under large line width conditions and effectively reduces the error floor. Therefore, in practical applications, the method of the present invention has high utilization value and practical significance for compensating the phase noise of the CO-OFDM system. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] In order to make the purpose, technical method and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0012] Figure 1 It is a technical roadmap of the method of the present invention;

[0013] Figure 2 This is the constellation diagram under different phase noise compensation methods proposed in this paper;

[0014] Figure 3 (a) and (b) are the SNR-BER comparison diagrams of various methods under 16QAM / 32QAM when the laser line width is 800kHz;

[0015] Figure 4(a) and (b) are the bit error rate curves of each method changing with the laser line width under 16QAM / 32QAM; DETAILED DESCRIPTION

[0016] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0017] Step 1: After performing quadrature amplitude modulation (QAM) mapping and serial-to-parallel conversion at the transmitter in the CO-OFDM system, the known pilot signal S is uniformly inserted. p Then, the phase noise of the i-th orthogonal frequency-division multiplexing (OFDM) symbol is estimated in the frequency domain of the receiving end using the least squares (LS) criterion. The estimated value of the phase noise CPE is φ i As shown in formula (1):

[0018]

[0019] In formula (1), angle(·) is the argument operation of a complex number, (·) * To take the conjugate operation, I i,0 The Fourier transform of the phase noise on each subcarrier, the receiving signal, the transmitting signal, and the channel response of the kth sampling point in the i-th OFDM symbol are Y i,k , X i,k , H i,k ; By combining the least squares method with the second-order Hermite Interpolation (SHI) algorithm, the phase noise estimation of the adjacent OFDM symbols is used to perform the first SHI processing to obtain the phase noise values ​​of all sampling points of the current OFDM symbol, thereby obtaining the phase noise estimation value of the kth sampling point in the OFDM symbol. The SHI function used is as follows:

[0020]

[0021] In formula (2), φ i ′ represents the derivative of the phase noise estimation value at the sampling point, and λ is the second-order Hermite basis as shown in formula (3):

[0022]

[0023] In formula (3), N is the number of subcarriers, N cpis the length of the cyclic prefix (CP); the phase noise estimate obtained after SHI processing is used For the receiving end signal y i,k Compensate and get the compensated signal z i,k As shown in formula (4):

[0024]

[0025] Step 2: In the rough phase noise compensation, the CPE and part of the ICI are compensated by Hermite interpolation. In order to improve the compensation accuracy of the phase noise, the compensated OFDM symbols are processed in blocks and Hermite interpolation is performed for the second time.

[0026] The roughly compensated signal z i,k Convert to frequency domain and make tentative decisions to get the signal Then the judgment result Converted to the time domain, it is expressed as formula (5):

[0027]

[0028] In formula (5), F H is the inverse fast Fourier transform, Δξ is the decision error of the tentative decision; the decision result The i-th OFDM signal is divided into N blocks b There are T=tN sub-symbols in t OFDM symbols. b The average phase noise of the nth (1≤n≤T) symbol is obtained according to the LS criterion. It is expressed as formula (6):

[0029]

[0030] In formula (6) is the signal z i,k The transpose of the nth signal after blocking; The time domain decision result The nth signal after block division;

[0031] Will get As the phase noise estimation value of the middle sampling point of each sub-symbol, the second SHI processing is performed between the middle sampling points of all sub-symbols to obtain the lth (0≤l≤N / N) of the nth sub-symbol. b ) The phase noise estimate of the sampling point The updated SHI function is adopted as formula (7):

[0032]

[0033] In formula (7), represents the derivative of the phase noise estimate at the sampling point, and γ is the updated second-order Hermite basis as shown in equation (8):

[0034]

[0035] use Obtaining a Phase Noise Estimate The compensated signal is calculated using formula (9):

[0036]

[0037] Step 3: The main idea of ​​the traditional Kalman filter algorithm is to use the estimated value of the previous moment and the measured value of the current moment to calculate the state value of the current moment; the system equation of the OFDM symbol and the measurement equation It is expressed as formula (10):

[0038]

[0039] In the formula, u i,k-1 is the difference in phase noise between the current moment and the previous moment, satisfying u i,k-1 ~N(0,Q k-1 ),in v i,k is the observation noise of the system, satisfying v i,k-1 ~N(0,R k ),in The signal after compensation The pre-decision value of

[0040] However, the traditional Kalman filter is only applicable to the estimation of linear systems, and it is difficult to achieve a good estimation effect for the phase noise in the CO-OFDM system. Therefore, the CKF based on the third-order spherical radial volume rule and the calculation of the state estimation value through the volume point is adopted based on the Bayesian theory.

[0041] In the CKF algorithm, process noise reflects the uncertainty caused by modeling errors, sensor errors, etc. in the change of system state at each time point. It is usually set as a tuning parameter in the CKF algorithm, and its initial value is set by empirical value. The proposed adaptive CKF algorithm can dynamically adjust the process noise to enable the CKF algorithm to run stably under different process noise initial values. The method mainly includes three parts, namely time update, measurement update and process noise adaptive update.

[0042] The phase noise value and CKF variance after initialization are and p1(1), as shown in formula (11):

[0043]

[0044] 1) Time update:

[0045] By calculating the variance p of the previous sampling point i,k-1 S obtained by Cholesky decomposition i,k-1 and the phase noise value of the previous sampling point Calculate volume point ξ i,(k-1,m) :

[0046]

[0047] In formula (12), m is the number of volume points, m = 1, 2..., n, n is the total number of volume points, satisfying m = 2n; is the basic sampling point;

[0048] Through the volume point ξ i,(k-1,m) , obtain the volume point ξ after state transfer i,(k|k-1,m) :

[0049]

[0050] Volume point ξ after state transfer i,(k|k-1,m) Calculate the predicted value of state quantity and error covariance predicted value P i,k|k-1 :

[0051]

[0052] In formula (14), Q k-1 is the process noise;

[0053] 2) Measurement update:

[0054] The error covariance prediction value P in the time update i,k|k-1 Cholesky decomposition; and update the state quantity prediction value in time Substitute (15) to obtain the volume point ξ * i,(k|k-1,m) :

[0055]

[0056] The compensated signal Make a pre-judgment to get the signal The volume point after state transfer is obtained by the volume point in formula (15)

[0057]

[0058] Calculate the measurement prediction value by the volume point after the state transition

[0059]

[0060] According to formula (15), formula (16) and formula (17), the new information covariance is calculated respectively and covariance estimates

[0061]

[0062] Where R k represents the measurement noise at the kth moment;

[0063] Calculate the Kalman gain K i,k , and the phase noise estimate and the covariance matrix P i,k To update:

[0064]

[0065] 3) Process noise adaptive update:

[0066] The traditional CKF algorithm requires complete dynamic prior information, measurement model and noise parameters, which has certain limitations;

[0067] Covariance matching adjusts the noise covariance matrix according to online statistics and updates the noise covariance using a forgetting factor and a weighted combination of current and past estimation data.

[0068] The residual sequence is defined as the difference d between the estimated measurement state and the actual received measurement value k , as shown in formula (20):

[0069]

[0070] The current residual value is updated with the adaptive process noise covariance as shown in (21):

[0071]

[0072] Where α represents the forgetting factor; and the updated process noise covariance is used for the estimation at the next moment until the algorithm ends;

[0073] The final output signal of the receiving end after the phase noise compensation algorithm is as follows:

[0074]

[0075] Finally, the signal after phase noise compensation by this method is expressed as

[0076] 2. Combine with attachment Figure 2 , Figure 3 and Figure 4 Description, in order to further illustrate that the method of the present invention has a relatively significant improvement in the compensation performance of phase noise to a certain extent.

[0077] Table 1 Simulation parameter settings

[0078] Parameters Set Size Number of subcarriers N 256 Cyclic prefix length 64 Baseband sampling frequency 20GHz Modulation 16QAM / 32QAM IFFT / FFT 256 Number of OFDM symbols 5000 Number of pilots 16 Number of sub-symbols 4 Channel AWGN

[0079] The parameters used in the experimental simulation are shown in Table 1.

[0080] Figure 2 Under 16QAM modulation, when the laser linewidth is 800kHz and the signal-to-noise ratio (SNR) is 15dB, the signal constellation diagram after phase noise compensation is based on the Block Sub-symbol Second-order Hermite Interpolation (BSSHI) algorithm, the Cubature Kalman Filter (CKF) algorithm of BSSHI and the Adaptive Cubature Kalman Filter (ACKF) algorithm of BSSHI.

[0081] Figure 2 (a) shows the constellation diagram of the original CPE and ICI signals without any phase noise compensation algorithm. Figure 2 (b) is the constellation diagram after compensation using the BSSHI algorithm in this paper. The divergence of the constellation diagram is significantly reduced, but there is still residual phase noise. Figure 2 (c) is the constellation diagram after further compensation using the BSSHI-CKF algorithm. Its constellation points are significantly more concentrated than those of BSSHI, and the CPE and ICI noises are better compensated. Figure 2 (d) is the constellation diagram after compensation using the BSSHI-ACKF algorithm. The concentration of its constellation points is more obvious than that of BSSHI-CKF. This is because the ACKF algorithm continuously and adaptively updates the process noise variance in the CKF filter, so that CPE and ICI are better compensated. It can be seen that the algorithm proposed by the method of the present invention has a good compensation effect for the phase noise in the CO-OFDM system.

[0082] Figure 3(a) is a SNR-BER comparison diagram of different phase noise compensation methods under 16QAM modulation and line width of 800kHz, where NPHN is the BER curve when there is no phase noise. It can be seen from the figure that the BSSHI-ACKF algorithm proposed by the method of the present invention has a bit error rate of 10 -5 Compared with the LI-SCPEC algorithm proposed in the pilot-assisted algorithm in the literature [1] "HONG XZ, HONG XJ, HE S. Linearly interpolated sub-symbol optical phase noise suppression in CO-OFDM system [J]. Optics Express, 2015, 23 (4): 4691-4702.", the performance is improved by about 1.8 dB; compared with the literature [2] "Bao WB, Xiao MH, Fang SL, et al. Lagrange interpolationbased extended Kalman filter for phase noise suppression in CO-OFDM system [J]. Optics Communications, 2019, 435(3): 221-226." The performance of the LRI-EKF algorithm proposed in the extended Kalman filter based on Lagrangian interpolation is improved by about 0.8 dB; compared with the BSSLI-CKF algorithm proposed in the literature [3] "Yuan Jianguo, Li Yangxu, Zuo Zhihang. CKF phase noise compensation algorithm based on block Lagrangian interpolation in CO-OFDM system [J]. Acta Optica Sinica, 2022, 42(16): 1606003." The performance is improved by about 0.4 dB. Figure 3 (b) Under 32QAM, BER is 10 -2.5 When , the BSSHI-ACKF algorithm is improved by 1.5dB, 0.8dB, and 0.6dB compared with the LI-SCPEC, LRI-CKF, and BSSLI-CKF algorithms, respectively.

[0083] Figure 4 (a) shows the BER versus laser line width curve when the SNR is 15dB and 16QAM modulation is used. Figure 4(a) It can be seen that the performance of the BSSHI-ACKF algorithm is better than the LI-SCPEC proposed in the pilot-assisted algorithm in the literature [1] "HONG XZ, HONG XJ, HE S. Linearly interpolated sub-symbol optical phase noise suppression in CO-OFDM system [J]. Optics Express, 2015, 23 (4): 4691-4702." and the literature [2] "Bao WB, Xiao MH, Fang SL, et al. Lagrange interpolation based extended Kalman filter for phase noise suppression in CO-OFDM system [J]. Optics Express, 2015, 23 (4): 4691-4702." Communications, 2019, 435(3): 221-226. The LRI-EKF proposed in the extended Kalman filter based on Lagrangian interpolation and the BSSLI-CKF algorithm proposed in the literature [3] "Yuan Jianguo, Li Yangxu, Zuo Zhihang. Block Lagrangian interpolation CKF phase noise compensation algorithm in CO-OFDM system [J]. Acta Optica Sinica, 2022, 42(16): 1606003.". At a BER of 10 -5 When compared with LI-SCPEC, LRI-EKF, and BSSLI-CKF algorithms, the laser linewidth tolerance of BSSHI-ACKF is increased by 500kHz, 380kHz, and 320kHz, respectively. When the laser linewidth exceeds 1500kHz, the FEC upper limit can be reached. Figure 4 (b) shows that under 32QAM, the line width tolerance of LI-SCPEC, LRI-EKF, and BSSLI-CKF algorithms is 760kHz, 880kHz, and 950kHz, and the line width tolerance of BSSHI-ACKF exceeds 1100kHz. In general, the BSSHI-ACKF algorithm adaptively updates the process noise, making the updated process noise more accurate than the process noise obtained based on experience, reducing the error caused by the process noise, enhancing the phase noise compensation effect, and reducing the cost of the system.

[0084] Based on the above simulation verification, it can be seen that the present invention proposes a phase noise compensation method of second-order Hermite interpolation and adaptive cubature Kalman filtering in coherent optical OFDM system for the CO-OFDM system which is sensitive to phase noise. The simulation results show that under the same conditions, the BSSHI-ACKF proposed in the present invention greatly improves the system performance and reduces the error floor compared with the LI-SCPEC, LRI-EKF and BSSLI-CKF algorithms.

[0085] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made in form and details without departing from the scope defined by the claims of the present invention.

Claims

1. A method for compensating phase noise of an adaptive cubature Kalman filter (ACKF) in a coherent optical orthogonal frequency-division multiplexing (CO-OFDM) system; the method compensates phase noise through three stages. In the first stage, the signal is processed at the receiving end, and the common phase error (CPE) and inter-carrier interference (ICI) phase noise are preliminarily compensated using the second-order Hermite interpolation algorithm; in the second stage, the signal after the preliminarily compensated phase noise is judged and sub-symbol divided, and the phase noise is further compensated using the Hermite interpolation algorithm; in the third stage, the signal after pre-judgment and sub-symbol block processing is input into the ACKF to process the residual phase noise; the specific steps are as follows: Step 1: After performing quadrature amplitude modulation (QAM) mapping and serial-to-parallel conversion at the transmitter in the CO-OFDM system, the known pilot signal S is uniformly inserted. p Then, the phase noise of the i-th orthogonal frequency-division multiplexing (OFDM) symbol is estimated in the frequency domain of the receiving end using the least squares (LS) criterion. The estimated value of the phase noise CPE is φ i As shown in formula (1): In formula (1), angle(·) is the argument operation of a complex number, (·) * To take the conjugate operation, I i,0 The Fourier transform of the phase noise on each subcarrier, the receiving signal, the transmitting signal, and the channel response of the kth sampling point in the i-th OFDM symbol are Y i,k , X i,k , H i,k ; By combining the least squares method with the second-order Hermite Interpolation (SHI) algorithm, the phase noise estimation of the adjacent OFDM symbols is used to perform the first SHI processing to obtain the phase noise values ​​of all sampling points of the current OFDM symbol, thereby obtaining the phase noise estimation value of the kth sampling point in the OFDM symbol. The SHI function used is as follows: In formula (2), φ i ′ represents the derivative of the phase noise estimation value at the sampling point, and λ is the second-order Hermite basis as shown in formula (3): In formula (3), N is the number of subcarriers, N cp is the length of the cyclic prefix (CP); the phase noise estimate obtained after SHI processing is used For the receiving end signal y i,k Compensate and get the compensated signal z i,k As shown in formula (4): Step 2: In the rough phase noise compensation, the CPE and part of the ICI are compensated by Hermite interpolation. In order to improve the compensation accuracy of the phase noise, the compensated OFDM symbols are processed in blocks and Hermite interpolation is performed for the second time. The roughly compensated signal z i,k Convert to frequency domain and make tentative decisions to get the signal The judgment result Converted to the time domain, it is expressed as formula (5): In formula (5), F H is the inverse fast Fourier transform, Δξ is the decision error of the tentative decision; the decision result The i-th OFDM signal is divided into N blocks b There are T=tN sub-symbols in t OFDM symbols. b The average phase noise of the nth (1≤n≤T) symbol is obtained according to the LS criterion. It is expressed as formula (6): In formula (6) is the signal z i,k The transpose of the nth signal after blocking; The time domain decision result The nth signal after block division; Will get As the phase noise estimation value of the middle sampling point of each sub-symbol, the second SHI processing is performed between the middle sampling points of all sub-symbols to obtain the lth (0≤l≤N / N) of the nth sub-symbol. b ) The phase noise estimate of the sampling point The updated SHI function is adopted as formula (7): In formula (7), represents the derivative of the phase noise estimate at the sampling point, and γ is the updated second-order Hermite basis as shown in equation (8): use Obtaining a Phase Noise Estimate The compensated signal is calculated using formula (9): Step 3: The main idea of ​​the traditional Kalman filter algorithm is to use the estimated value of the previous moment and the measured value of the current moment to calculate the state value of the current moment; the system equation of the OFDM symbol and the measurement equation It is expressed as formula (10): In the formula, u i,k-1 is the difference in phase noise between the current moment and the previous moment, satisfying u i,k-1 ~N(0,Q k-1 ),in v i,k is the observation noise of the system, satisfying v i,k-1 ~N(0,R k ),in The signal after compensation The pre-decision value of However, the traditional Kalman filter is only applicable to the estimation of linear systems, and it is difficult to achieve a good estimation effect for the phase noise in the CO-OFDM system. Therefore, the CKF based on the third-order spherical radial volume rule and the calculation of the state estimation value through the volume point is adopted based on the Bayesian theory. In the CKF algorithm, process noise reflects the uncertainty caused by modeling errors, sensor errors, etc. in the change of system state at each time point. It is usually set as a tuning parameter in the CKF algorithm, and its initial value is set by empirical value. The proposed adaptive CKF algorithm can dynamically adjust the process noise to enable the CKF algorithm to run stably under different process noise initial values. The method mainly includes three parts, namely time update, measurement update and process noise adaptive update. The phase noise value and CKF variance after initialization are and p1(1), as shown in formula (11): 1) Time update: By calculating the variance p of the previous sampling point i,k-1 S obtained by Cholesky decomposition i,k-1 and the phase noise value of the previous sampling point Calculate volume point ξ i,(k-1,m) : In formula (12), m is the number of volume points, m = 1, 2..., n, n is the total number of volume points, satisfying m = 2n; is the basic sampling point; Through the volume point ξ i,(k-1,m) , obtain the volume point ξ after state transfer i,(k|k-1,m) : x i,(k|k-1,m) =ξ i,(k-1,m) (13) Volume point ξ after state transfer i,(k|k-1,m) Calculate the predicted value of state quantity and error covariance predicted value P i,k|k-1 : In formula (14), Q k-1 is the process noise; 2) Measurement update: The error covariance prediction value P in the time update i,k|k-1 Cholesky decomposition; and update the state quantity prediction value in time Substitute (15) to obtain the volume point The compensated signal Make a pre-judgment to get the signal The volume point after state transfer is obtained by the volume point in formula (15) Calculate the measurement prediction value by the volume point after the state transition According to formula (15), formula (16) and formula (17), the new information covariance is calculated respectively and covariance estimates Where R k represents the measurement noise at the kth moment; Calculate the Kalman gain K i,k , and the phase noise estimate and the covariance matrix P i,k To update: 3) Process noise adaptive update: The traditional CKF algorithm requires complete dynamic prior information, measurement model and noise parameters, which has certain limitations; Covariance matching adjusts the noise covariance matrix according to online statistics and updates the noise covariance using a forgetting factor and a weighted combination of current and past estimation data; The residual sequence is defined as the difference d between the estimated measurement state and the actual received measurement value k , as shown in formula (20): The current residual value is updated with the adaptive process noise covariance as shown in (21): Where α represents the forgetting factor; and the updated process noise covariance is used for the estimation at the next moment until the algorithm ends; The final output signal of the receiving end after the phase noise compensation algorithm is as follows: Finally, the signal after phase noise compensation by this method is expressed as

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