Method for estimating RSOP, frequency offset and phase in PDM system based on H-infinity filtering

Abstract

The invention relates to a method for estimating RSOP, frequency offset and phase in a PDM system based on H-infinity filtering. Comprising the following steps: signal preprocessing: preprocessing a received signal in the PDM system; establishing a state space model: establishing a state space model of polarization state rotation RSOP, frequency offset and phase in the PDM system, wherein the state space model comprises a state equation and an observation equation; designing an H-infinity filter: designing the H-infinity filter, and ensuring the robustness of the filter in a nonlinear system and a non-Gaussian noise environment by selecting proper filter parameters; and state estimation: performing real-time estimation on the RSOP, the frequency offset and the phase by using an H-infinity filter, and outputting an estimation result. According to the method, stable estimation performance can be provided in a nonlinear system and a non-Gaussian noise environment, meanwhile, the calculation complexity is low, and the method is suitable for real-time application. By establishing the state space model and designing the H-infinity filter, the parameters can be estimated in real time, so that the overall performance of the optical communication system is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of optical communication signal processing, and more specifically, to a method for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering. Background Art

[0002] In modern optical communication systems, Polarization Division Multiplexing (PDM) technology is widely used to improve the capacity and spectral efficiency of fiber optic communication systems. The PDM technology transmits data on two orthogonal polarization states, effectively utilizing the polarization characteristics of optical fibers. However, the state of polarization changes during fiber transmission, and this change is called Rotation of State of Polarization (RSOP). In addition, optical communication systems also face problems such as frequency offset (FO) and phase noise (PN), which can seriously affect the signal quality and system performance.

[0003] Existing digital signal processing (DSP) algorithms for compensating the rotation of the state of polarization (RSOP) can be roughly divided into three categories: the constant modulus algorithm (CMA), the Stokes space (SS)-based method, and the Bayesian filtering-based method. The constant modulus algorithm (CMA) aims to minimize the difference between the received signal and the estimated signal by iteratively adjusting the weights of a finite impulse response (FIR) filter while satisfying the constant modulus constraint condition. However, the performance of CMA is limited by its slow convergence speed and singularity problem. For modulation formats with multiple amplitudes, a multi-mode algorithm (MMA) has been proposed in the existing literature to achieve reliable initial convergence and greater flexibility by modifying the cost function. CMA and MMA can provide computational efficiency and robustness, making them popular choices in practical transmission. However, they both suffer from the limitations of slow convergence speed and the inability to effectively track ultrafast RSOP. The Stokes space (SS)-based method uses the representation of the signal in the Stokes space for polarization demultiplexing. For example, polarization demultiplexing is achieved by performing a least-squares fit of the symmetry plane, and its convergence speed is faster than that of CMA. However, the performance of SS-based polarization demultiplexing may be sensitive to changes in system parameters such as polarization mode dispersion (PMD) and residual chromatic dispersion (CD). The SS method improves the efficiency and robustness of polarization demultiplexing, making it more suitable for practical transmission. However, in scenarios where the signal conditions change, the implementation of the SS signal representation process may become more complex and less robust in a rapidly changing environment. The Bayesian filtering-based method can track ultrafast RSOP with high precision, but its convergence speed is slower than that of the SS-based method. The recursive update of state estimation in the Bayesian filtering process increases the computational complexity. In addition, the use of an approximate function in the extended Kalman filter (EKF) results in a reduction in estimation accuracy. Summary of the Invention

[0004] An object of the present invention is to overcome the deficiencies of low estimation accuracy, high computational complexity, and low robustness of RSOP, FO, and PN in the existing PDM system, and provide a method for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering, which can effectively improve the estimation accuracy, improve the robustness, and effectively reduce the computational complexity.

[0005] To solve the above technical problems, the technical solution adopted by the present invention is:

[0006] Provide a method for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering, including the following steps:

[0007] S1. Signal preprocessing: Preprocess the received signal in the PDM system;

[0008] S2. Establish the state - space model: Establish the state - space model of the polarization - state rotation (RSOP), frequency offset, and phase in the PDM system. The state - space model includes the state equation and the observation equation;

[0009] S3. Design the H∞ filter: Design the H∞ filter. By selecting appropriate filter parameters, ensure the robustness of the filter in a non - linear system and non - Gaussian noise environment;

[0010] S4. State estimation: Use the H∞ filter to perform real - time estimation of RSOP, frequency offset, and phase, and output the estimation results.

[0011] A method for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering according to the present invention can provide stable estimation performance in a non - linear system and non - Gaussian noise environment, and at the same time has a low computational complexity, making it suitable for real - time applications. By establishing the state - space model of RSOP, frequency offset, and phase in the PDM system and designing the H∞ filter, the present invention can estimate these parameters in real time, thereby improving the overall performance of the optical communication system.

[0012] Further, the step S2 includes:

[0013] In the PDM system, the transmitted signal at time n is represented by the two - dimensional vector s n =[s x,n ; s y,n , where x and y represent two polarizations respectively; the polarization - state rotation (RSOP) at time n is represented by the 2×2 Jones matrix J n in the Cayley - Klein form where the sum of the squares of the polarization - state rotation parameters is 1, that is ∈ is the frequency offset normalized by the sampling rate; the time - varying phase noise φ n is modeled as a Wiener process: φ n =φ n-1 +ν n , ν n is a Gaussian random variable with a mean of 0 and a variance of , where Δν is the laser combined linewidth (CLW) and T is the sampling time interval;

[0014] Considering the influence of RSOP, frequency offset, and phase noise, the received signal at time n is expressed as where w n =[w x,n ; w y,n represents the complex additive Gaussian white noise vector with a mean of 0 and a covariance matrix of The state at time n is represented as a five - dimensional vector x n =[a n ,b n ,cn , ∈ n , φ n T ;

[0015] The state - space model is represented as:

[0016] State equation: x n = [a n , b n , c n , ∈ n , φ n T = f n-1 (x n-1 , v n ) = x n-1 + v n ;

[0017] Observation equation:

[0018] where v n represents the state - noise vector.

[0019] Furthermore, the step S3 includes:

[0020] Introduce a threshold to ensure the robustness of the H∞ filter; apply the performance boundary to find an estimation strategy that meets the threshold, and define the optimization problem as:

[0021]

[0022] where P0 is the initial estimation error covariance matrix, Q n is the covariance matrix of the state - noise vector v n , R n is the covariance matrix of w n , that is N represents the pilot - symbol length; in addition, P0 and R n are symmetric positive - definite matrices; the threshold parameter λ specifies the maximum allowable gain from the disturbance to the estimation error;

[0023] The estimation process is regarded as a min - max optimization problem, whose goal is to find an estimation to obtain the minimum value of the objective function, and at the same time, the following constraints must be satisfied to ensure that the H∞ gain can be calculated:

[0024] where H n represents the partial derivative of the observation equation h n-1 with respect to x in the prior.

[0025] Furthermore, the step S4 includes: ​​

[0026] In the initialization step, the parameters to be estimated are considered to be independent and all uniformly distributed, i.e., a0, b0, c0 ∼ U(0, 1), ∈0 ∼ U(-0.5, 0.5), φ0 ∼ U(-π, π); thus, the initial values are determined first. and the initial covariance matrix Then, recursive updates are performed.

[0027] For n = 1, …, N - 1, the following steps are performed to iteratively update the estimation vector:

[0028] Based on the five - variable normal density function p(x n |x n-1 ) the prior state estimate is updated

[0029] The H∞ gain is calculated

[0030] The posterior estimation error covariance matrix is updated

[0031] The posterior state estimate is updated

[0032] In the algorithm, the F n-1 term represents the partial derivative matrix of f n-1 with respect to x in the posterior while H n represents the partial derivative of the observation equation h n-1 with respect to x in the prior (·) H represents the conjugate transpose operation on the matrix; by using the H∞ gain K n , it is possible to decide whether to trust more the observation results or the prediction results of the model; the final state estimate is the vector which is the final result obtained by the joint estimation based on the H∞ filter.

[0033] Furthermore, it also includes step S5 for performance optimization, in which according to the estimation result of step S4, the parameters of the filter are adjusted to further optimize the estimation performance.

[0034] Furthermore, the step S5 includes:

[0035] Adjust the initial state estimation vector and the initial covariance matrix P0, and propose a new hypothesis for the distribution of the vector ;

[0036] Adjust the H∞ gain K n : According to the simulation results, if the estimation error is large, increase the filtering gain to make the filter more dependent on the measurement data;

[0037] Adjust performance metrics: On the premise of ensuring system stability, adjust the threshold parameter λ through an optimization method to minimize the estimation error in the worst case;

[0038] After each parameter adjustment, re - conduct the simulation and evaluate the new performance of the filter; stop the adjustment when the estimation error and robustness reach the expected goals.

[0039] The present invention also provides a system for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering, including:

[0040] Signal reception module: Used to pre - process the received signal in the PDM system;

[0041] State - space model module: Used to establish the state - space model of the polarization state rotation (RSOP), frequency offset, and phase in the PDM system. The state - space model includes a state equation and an observation equation;

[0042] H∞ filter design module: Used to design an H∞ filter. By selecting appropriate filter parameters, ensure the robustness of the filter in a non - linear system and non - Gaussian noise environment;

[0043] State estimation module: Used to use the H∞ filter to estimate RSOP, frequency offset, and phase in real - time and output the estimation results.

[0044] Furthermore, the state - space model establishment module establishes the state - space model according to the following method:

[0045] In the PDM system, the transmitted signal at time n is represented by the two - dimensional vector s n =[s x,n ; s y,n , where x and y represent two polarizations respectively; the polarization state rotation (RSOP) at time n is represented by the 2×2 Jones matrix J n , and the Cayley - Klein form is where the sum of the squares of the polarization state rotation parameters is 1, that is ∈ is the frequency offset normalized by the sampling rate; the time - varying phase noise φ n is modeled as a Wiener process: φ n =φ n-1 +v n , v n is a Gaussian random variable with a mean of 0 and a variance of , where Δν is the laser combined linewidth (CLW) and T is the sampling time interval;

[0046] Considering the influence of RSOP, frequency offset, and phase noise, the received signal at time n is expressed as where w n =[wx,n ; w y,n represents an additive white Gaussian noise vector with a mean of 0 and a covariance matrix of The state at time n is represented as a five-dimensional vector x n = [a n , b n , c n , ∈ n , φ n T ;

[0047] The state space model is represented as:

[0048] State equation: x n = [a n , b n , c n , ∈ n , φ n T = f n-1 (x n-1 , v n ) = x n-1 + v n ;

[0049] Observation equation:

[0050] where v n represents the state noise vector.

[0051] Furthermore, the H∞ filter design module designs the H∞ filter in the following manner:

[0052] Introduce a threshold to ensure the robustness of the H∞ filter; apply performance bounds to find an estimation strategy that meets the threshold, and define the optimization problem as:

[0053]

[0054] where P0 is the initial estimation error covariance matrix, Q n is the covariance matrix of the state noise vector v n , R n is the covariance matrix of w n , that is N represents the pilot symbol length; in addition, P0 and R n are symmetric positive definite matrices; the threshold parameter λ specifies the maximum allowable gain from the disturbance to the estimation error;

[0055] The estimation process is regarded as a min-max optimization problem, the goal of which is to find an estimation to obtain the minimum value of the objective function, and at the same time, the following constraints must be satisfied to ensure that the H∞ gain can be calculated:​​

[0056] Among them, H n represents the observation equation h n-1 in the prior partial derivative with respect to x.

[0057] Furthermore, the state estimation module performs estimation according to the following method:

[0058] In the initialization step, it is considered that the parameters to be estimated are independent of each other and all uniformly distributed, that is, a0, b0, c0 ∼ U(0, 1), ∈0 ∼ U(-0.5, 0.5), φ0 ∼ U(-π, π); therefore, the initial values and the initial covariance matrix are first determined. Then, recursive updates are performed;

[0059] For n = 1, …, N - 1, the following steps are performed to iteratively update the estimation vector:

[0060] Based on the five - variable normal density function p(x n |x n-1 ) to update the prior state estimation

[0061] Calculate the H∞ gain

[0062] Update the posterior estimation error covariance matrix

[0063] Update the posterior state estimation

[0064] In the algorithm, the F n-1 term represents the partial derivative matrix of f n-1 with respect to x in the posterior , while H n represents the partial derivative of the observation equation h n-1 with respect to x in the prior ; (·) H represents the conjugate transpose operation on the matrix; by using the H∞ gain K n , it is possible to decide whether to trust more the observation results or the prediction results of the model; the final state estimation is the vector which is the final result obtained by the joint estimation based on the H∞ filter.

[0065] Compared with the prior art, the beneficial effects of the present invention are:

[0066] A method for estimating RSOP, frequency offset and phase in a PDM system based on H∞ filtering according to the present invention can provide stable estimation performance in a non-linear system and non-Gaussian noise environment, and at the same time has a low computational complexity, making it suitable for real-time applications. By establishing a state space model of RSOP, frequency offset and phase in the PDM system and designing an H∞ filter, the present invention can estimate these parameters in real time, thereby improving the overall performance of the optical communication system. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a schematic flow chart of the method according to the first embodiment of the present invention;

[0068] Figure 2 is a schematic structural diagram of the system according to the second embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0069] The present invention will be further described below in conjunction with the specific embodiments. Among them, the drawings are only for illustrative purposes, showing only schematic diagrams, not physical diagrams, and should not be construed as a limitation to the present invention; in order to better illustrate the embodiments of the present invention, some components in the drawings will be omitted, enlarged or reduced, and do not represent the size of the actual product; for those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0070] In the drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components; in the description of the present invention, it should be understood that if there are terms such as "upper", "lower", "left", "right", etc. indicating the orientation or positional relationship, it is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, so the terms describing the positional relationship in the drawings are only for illustrative purposes and should not be construed as a limitation to the present invention. For those of ordinary skill in the art, the specific meanings of the above terms can be understood according to specific circumstances.

[0071] Embodiment 1

[0072] This embodiment is the first embodiment of a method for estimating RSOP, frequency offset and phase in a PDM system based on H∞ filtering, and includes the following steps:

[0073] Step S1. Signal preprocessing: Preprocess the received signal in the PDM system; the received signal at time n can be expressed as a two-dimensional vector r n = [r x,n ; r y,n , where the subscripts x and y represent two polarizations respectively.

[0074] Step S2. Establish a state space model: Establish a state space model for the polarization state rotation RSOP, frequency offset, and phase in the PDM system. The state space model includes a state equation and an observation equation.

[0075] In the PDM system, the transmission signal at time n is represented by the two-dimensional vector s n =[s x,n ; s y,n , where x and y represent two polarizations respectively; the polarization state rotation RSOP at time n is represented by the 2×2 Jones matrix J n , and the Cayley-Klein form is where the sum of the squares of the polarization state rotation parameters is 1, that is ∈ is the frequency offset normalized by the sampling rate; the time-varying phase noise φ n is modeled as a Wiener process: φ n =φ n-1 +v n , where v n is a Gaussian random variable with a mean of 0 and a variance of , where Δv is the laser combined linewidth CLW and T is the sampling time interval;

[0076] It should be noted that: RSOP can be written as the matrix and satisfies In the process of jointly estimating RSOP, frequency offset, and phase based on H∞ filtering, real parameters are usually considered to form the state matrix. Since the elements J xx , J xy , J yx , J yy of the matrix J are complex numbers, it is considered to write them in the form of the real part plus the imaginary part, that is: j represents the imaginary unit. Taking J xx as an example, J xx =a + jb, the real part of the complex number J xx is a, and the imaginary part is jb. Therefore, the RSOP at time n can be expressed as The phase noise φ n at time n follows a Wiener process. Therefore, the relationship expression between φ n and the phase noise φ n-1 at the previous moment is: φ n =φ n-1 +ν n . Here, v n is a random variable with a mean of 0 and a variance of

[0077] Considering the influence of RSOP, frequency offset, and phase noise, the received signal at time n is expressed as where wn = [w x,n ; w y,n represents the complex additive white Gaussian noise vector with a mean of 0 and a covariance matrix of The state at time n is represented as a five-dimensional vector x n = [a n , b n , c n , ∈ n , φ n T .

[0078] The state space model is expressed as:

[0079] State equation: x n = [a n , b n , c n , ∈ n , φ n T = f n-1 (x n-1 , v n ) = x n-1 + v n ;

[0080] Observation equation:

[0081] where v n represents the state noise vector.

[0082] It should be noted that: Since there are two polarizations x and y in the PDM system, the transmitted signal, received signal, and additive white Gaussian noise at time n are all two-dimensional vectors. The complex additive white Gaussian noise vector w n =

[0083] [w x,n ; w y,n , w x,n is the complex additive white Gaussian noise on the x polarization, while w y,n represents the complex additive white Gaussian noise on the y polarization. Due to the inclusion of dual polarization, the covariance matrix of w n is I represents the identity matrix, that is, the elements on the diagonal are 1 and the elements off the diagonal are 0. Here, I is a 2*2 identity matrix. That is, the covariance matrix of w n is The state at time n can be expressed as an equation of the state at time n-1: x n = [a n , b n , c n , ∈ n , φ n ​​T = f n-1 (x n-1 , v n ) = x n-1 + v n ; f n-1 (·) represents the state equation at time n, where v n represents the state noise vector.

[0084] Step S3. Design an H∞ filter: Design an H∞ filter. By selecting appropriate filter parameters, ensure the robustness of the filter in a non - linear system and non - Gaussian noise environment.

[0085] In the estimation algorithm based on the H∞ filter, this embodiment introduces a threshold to ensure the robustness of the estimator. Since the direct minimization of the estimation error is difficult to handle, this embodiment applies a performance bound to find an estimation strategy that meets the threshold. Define the optimization problem as:

[0086]

[0087] where P0 is the initial estimation error covariance matrix, Q n is the covariance matrix of the state noise vector v n , R n is the covariance matrix of w n , that is N represents the length of the pilot symbol; in addition, P0 and R n are symmetric positive - definite matrices; is the state estimation at time n; the threshold parameter λ specifies the maximum allowable gain from the disturbance to the estimation error;

[0088] The filter should be robust to uncertainties and disturbances. This is particularly important in practical applications where system parameters may vary or be subject to external disturbances. Therefore, the estimation process can be regarded as a min - max optimization problem, whose goal is to find an estimation to obtain the minimum value of the objective function, while satisfying the following constraints to ensure that the H∞ gain can be calculated: where H n represents the partial derivative of the observation equation h n-1 with respect to x a priori .

[0089] Step 4. State estimation: Use the H∞ filter to perform real - time estimation on the RSOP, frequency offset, and phase, and output the estimation results.

[0090] ​In the initialization step, the parameters to be estimated are considered to be independent and all uniformly distributed, i.e., a0, b0, c0 ∼ U(0, 1), ∈0 ∼ U(-0.5, 0.5), φ0 ∼ U(-π, π); thus, the initial values are determined first. and the initial covariance matrix Then, recursive updates are performed.

[0091] For n = 1, …, N-1, the following steps are performed to iteratively update the estimation vector:

[0092] Update the prior state estimate based on the five-variable normal density function p(x n |x n-1 )

[0093] Calculate the H∞ gain

[0094] Update the posterior estimation error covariance matrix

[0095] Update the posterior state estimate

[0096] In the algorithm, the F n-1 term represents the partial derivative matrix of f n-1 with respect to x in the posterior , and F n represents the partial derivative matrix of f n-1 with respect to x in the posterior , while H n represents the partial derivative of the observation equation h n-1 with respect to x in the prior ; represents the conjugate transpose operation on H n ; (·) H represents the conjugate transpose operation on a matrix; R n is the covariance matrix of w n , i.e., refers to the inverse matrix of R n . All (·) -1 represent the operation of finding the inverse matrix.

[0097] By using the H∞ gain K n , it is possible to decide whether to trust the observation results or the prediction results of the model more; the final state estimate is the vector which is the final result obtained by the joint estimation based on the H∞ filter.

[0098] Step S5. Performance optimization: According to the estimation results in step S4, adjust the parameters of the filter to further optimize the estimation performance.

[0099] In actual optical transmission, the distribution and covariance of noise are usually unknown, and this uncertainty will affect the estimation accuracy and stability of the algorithm. The H∞ filter makes no assumptions about the noise distribution, and its goal is to minimize the worst-case estimation error. By solving the min-max optimization problem proposed above, the minimum mean square error (MMSE) estimation under extreme noise conditions can be found based on the H∞ filter, thereby greatly improving the robustness of the algorithm. After obtaining the parameter estimation, this embodiment will analyze the estimation error and the robustness of the filter to uncertainty. If the estimation error is large or the robustness is insufficient, the parameters need to be adjusted to optimize its estimation performance. Specifically, it includes the following steps:

[0100] Adjust the initial state estimation vector and the initial covariance matrix P0, and propose a new hypothesis about the distribution of the vector ;

[0101] Adjust the H∞ gain K n : According to the simulation results, if the estimation error is large, increase the filtering gain to make the filter more dependent on the measurement data;

[0102] Adjust the performance index: On the premise of ensuring system stability, adjust the threshold parameter λ through an optimization method to minimize the estimation error in the worst case;

[0103] After each parameter adjustment, re-run the simulation and evaluate the new performance of the filter; when the estimation error and robustness reach the expected goals, stop the adjustment.

[0104] A method for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering according to the present invention can provide stable estimation performance in a non-linear system and non-Gaussian noise environment, and at the same time has a low computational complexity and is suitable for real-time applications. By establishing a state-space model of RSOP, frequency offset, and phase in a PDM system and designing an H∞ filter, the present invention can estimate these parameters in real time, thereby improving the overall performance of the optical communication system.

[0105] The method proposed in this embodiment effectively improves the estimation accuracy: The H∞ filtering algorithm can provide more accurate estimation results in a non-linear system and non-Gaussian noise environment. Compared with the Kalman filter and the adaptive filter, this embodiment has a significant improvement in estimation accuracy.

[0106] The method proposed in this embodiment effectively enhances the robustness: The H∞ filtering algorithm ensures stability in a complex environment by minimizing the worst-case performance of the system. Compared with the particle filter, this embodiment has an obvious advantage in robustness.

[0107] The method proposed in this embodiment effectively reduces the computational complexity: The H∞ filtering algorithm has a relatively low computational complexity and is suitable for real-time applications. Compared with the particle filter, this embodiment shows better performance in terms of real-time performance.

[0108] The method proposed in this embodiment is adaptable to multiple modulation formats: The H∞ filtering algorithm is not only applicable to traditional constant modulus modulation formats but also capable of adapting to modulation formats with multiple amplitudes. By dynamically adjusting the filter parameters, the H∞ filtering algorithm can better adapt to the signal characteristics of different modulation formats, improving the flexibility and applicability of the system.

[0109] The method proposed in this embodiment improves the overall performance of the system: Through the comprehensive application of the above technical means, this embodiment significantly enhances the overall performance of the optical communication system. In an actual optical communication system, accurate RSOP, frequency offset, and phase estimation are crucial for high-quality signal transmission. In this embodiment, through the H∞ filtering algorithm, not only the estimation accuracy and robustness are improved, but also the computational complexity is reduced, enabling the system to operate more efficiently and stably in practical applications.

[0110] Embodiment 2

[0111] This embodiment is the second embodiment of a system for estimating RSOP, frequency offset, and phase in a PDM system based on H∞ filtering, including:

[0112] Signal reception module: Used to preprocess the received signal in the PDM system; The received signal at time n can be expressed as a two-dimensional vector r n =[r x,n ; r y,n , where the subscripts x and y represent two polarizations respectively;

[0113] State space model module: Used to establish the state space model of the polarization state rotation RSOP, frequency offset, and phase in the PDM system. The state space model includes a state equation and an observation equation;

[0114] H∞ filter design module: Used to design the H∞ filter. By selecting appropriate filter parameters, the robustness of the filter in a non-linear system and non-Gaussian noise environment is ensured;

[0115] State estimation module: Used to use the H∞ filter to perform real-time estimation of RSOP, frequency offset, and phase and output the estimation results;

[0116] Performance optimization module: Used to adjust the parameters of the filter according to the estimation results of the state estimation module to further optimize the estimation performance.

[0117] In this embodiment: The state space model establishment module establishes the state space model according to the following method:

[0118] In the PDM system, the transmission signal at time n is represented by a two-dimensional vector s n =[s x,n ; s y,n , where x and y represent two polarizations respectively; the polarization state rotation RSOP at time n is represented by a 2×2 Jones matrix J n , and the Cayley-Klein form is where the sum of the squares of the polarization state rotation parameters is 1, that is ∈ is the frequency offset normalized by the sampling rate; the time-varying phase noise φ n is modeled as a Wiener process: φ n =φ n-1 +v n , where v n is a Gaussian random variable with a mean of 0 and a variance of , where Δv is the laser combined linewidth CLW and T is the sampling time interval;

[0119] Considering the effects of RSOP, frequency offset, and phase noise, the received signal at time n is expressed as where w n =[w x,n ; w y,n represents a complex additive Gaussian white noise vector with a mean of 0 and a covariance matrix of The state at time n is represented by a five-dimensional vector x n =[a n , b n , c n , ∈ n , φ n T ;

[0120] The state space model is expressed as:

[0121] State equation: x n =[a n , b n , c n , ∈ n , φ n T =f n-1 (x n-1 , v n ) = x n-1 +v n ;

[0122] Observation equation:

[0123] where v n represents the state noise vector.

[0124] ​​In this embodiment, the H∞ filter design module designs the H∞ filter in the following manner:

[0125] Introduce a threshold to ensure the robustness of the H∞ filter; apply the performance boundary to find the estimation strategy that meets the threshold, and define the optimization problem as:

[0126]

[0127] where P0 is the initial estimation error covariance matrix, Q n is the covariance matrix of the state noise vector v n , R n is the covariance matrix of w n , that is N represents the pilot symbol length; in addition, P0 and R n are symmetric positive definite matrices; the threshold parameter λ specifies the maximum allowable gain from the interference to the estimation error;

[0128] The estimation process is regarded as a min-max optimization problem, and its goal is to find an estimation to obtain the minimum value of the objective function, and at the same time, the following constraints must be satisfied to ensure that the H∞ gain can be calculated:

[0129] where H n represents the partial derivative of the observation equation h n-1 with respect to x in the prior.

[0130] In this embodiment, the state estimation module estimates according to the following manner:

[0131] In the initialization step, it is considered that the parameters to be estimated are independent of each other and are all uniformly distributed, that is, a0, b0, c0 ∼ U(0, 1), ∈0 ∼ U(-0.5, 0.5), φ0 ∼ U(-π, π); therefore, first determine the initial value and the initial covariance matrix Then, perform recursive updates;

[0132] For n = 1, …, N - 1, perform the following steps to iteratively update the estimation vector:

[0133] Update the prior state estimation based on the five-variable normal density function p(x n |x n-1 )

[0134] Calculate the H∞ gain

[0135] Update the posterior estimation error covariance matrix

[0136] Update the posterior state estimate

[0137] In the algorithm, the F n-1 term represents the partial derivative matrix of f n-1 with respect to x in the posterior , while the H n represents the partial derivative of the observation equation h n-1 with respect to x in the prior ; (·) H represents the conjugate transpose operation on the matrix; by using the H∞ gain K n , it is possible to decide whether to trust the observation results or the prediction results of the model more; the final state estimate is the vector which is the final result obtained by the joint estimation based on the H∞ filter.

[0138] In this embodiment, the performance optimization module adjusts the parameters according to the following steps to achieve system optimization:

[0139] Adjust the initial state estimate vector and the initial covariance matrix P0, and propose a new hypothesis for the distribution of the vector ;

[0140] Adjust the H∞ gain K n : According to the simulation results, if the estimation error is large, increase the filtering gain to make the filter more dependent on the measurement data;

[0141] Adjust the performance index: On the premise of ensuring the system stability, adjust the threshold parameter λ through the optimization method to minimize the estimation error in the worst case;

[0142] After each parameter adjustment, re - conduct the simulation and evaluate the new performance of the filter; when the estimation error and robustness reach the expected goals, stop the adjustment.

[0143] Embodiment Three

[0144] This embodiment is an embodiment of a computer device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the method described in Embodiment One are implemented.

[0145] Embodiment Four

[0146] This embodiment is an embodiment of a computer - readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the method described in Embodiment One are implemented.

[0147] In the specific content of the above specific embodiments, the technical features can be combined arbitrarily without contradiction. For the sake of concise description, not all possible combinations of the above technical features are described. However, as long as the combinations of these technical features do not exist in contradiction, they should all be considered as the scope recorded in this specification.

[0148] Obviously, the above embodiments of the present invention are merely examples given to clearly illustrate the present invention, rather than limitations on the embodiments of the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all the embodiments here. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included in the protection scope of the claims of the present invention.

Claims

1. A method for estimating RSOP, frequency deviation and phase in a PDM system based on H∞ filtering, characterized in that: The following steps are involved: S1. Signal preprocessing: preprocessing the received signal in the PDM system; S2. Establishing a state space model: Establishing a state space model of polarization rotation RSOP, frequency offset and phase in the PDM system. The state space model includes state equations and observation equations. S3. Design H∞ filter: Design H∞ filter and ensure the robustness of the filter in nonlinear system and non-Gaussian noise environment by selecting appropriate filter parameters; S4. State estimation: Use H∞ filter to estimate RSOP, frequency offset and phase in real time and output the estimation results.

2. The method for estimating RSOP, frequency deviation and phase in a PDM system based on H∞ filtering according to claim 1, characterized in that: The step S2 comprises: In the PDM system, the transmission signal at time n is represented by a two-dimensional vector s n =[s x,n ;s y,n ] indicates that x and y represent two polarizations respectively; the polarization state at time n is rotated by RSOP with a 2×2 Jones matrix J n Indicates that the Cayley-Klein form is The sum of the squares of the polarization state rotation parameters is 1, that is, ∈ is the frequency deviation normalized by the sampling rate; the time-varying phase noise φ n Modeled as a Wiener process: φ n =φ n-1 +ν n , ν n The mean is 0 and the variance is Gaussian random variable, where Δν is the laser combined line width CLW, and T is the sampling time interval; Considering the influence of RSOP, frequency offset and phase noise, the received signal at time n is expressed as where w n =[w x,n ;w y,n ] represents a complex Gaussian white noise vector with a mean of 0 and a covariance matrix of w x,n is the complex white Gaussian noise on the x polarization, w y,n is the complex white Gaussian noise on the y polarization, I represents the unit matrix, and the state at time n is represented by a five-dimensional vector x n =[a n ,b n ,c n ,∈ n ,φ n ] T ; The state space model is expressed as: Equation of state: x n =[a n ,b n ,c n ,∈ n ,φ n ] T =f n-1 (x n-1 ,v n )=x n-1 +v n ; Observation equation: where v n represents the state noise vector, f n-1 (·) represents the state equation at time n.

3. The method for estimating RSOP, frequency deviation and phase in a PDM system based on H∞ filtering according to claim 2, characterized in that: The step S3 comprises: Introducing a threshold To ensure the robustness of the H∞ filter; apply the performance boundary to find an estimation strategy that meets the threshold, and define the optimization problem as: Where P0 is the initial estimation error covariance matrix, Q n is the state noise vector v n The covariance matrix, R n w n The covariance matrix of N represents the pilot symbol length; in addition, P0 and R n is a symmetric positive definite matrix; the threshold parameter λ specifies the maximum allowable gain from interference to estimation error; is the state estimate at time n; The estimation process is considered as a min-max optimization problem, where the goal is to find an estimate that minimizes the objective function while satisfying the following constraints to ensure that the H∞ gain can be calculated: Among them, H n Represents the observation equation h n-1 In the prior The partial derivative with respect to x.

4. The method for estimating RSOP, frequency deviation and phase in a PDM system based on H∞ filtering according to claim 3, characterized in that: The step S4 comprises: In the initialization step, the parameters to be estimated are considered to be independent of each other and are uniformly distributed, that is, a0, b0, c0 ~ U (0, 1), ∈0 ~ U (-0.5, 0.5), φ0 ~ U (-π, π); therefore, the initial value is first determined And the initial covariance matrix Then, perform a recursive update; For n=1,…,N-1, perform the following steps to iteratively update the estimated vector: Based on the five-variable normal density function p(x n |x n-1 ) Update the prior state estimate Calculate H∞ gain Update the posterior estimation error covariance matrix Update the posterior state estimate In the algorithm, F n-1 The term represents f n-1 In the posterior The partial derivative matrix with respect to x, F n Then it means f n-1 In the posterior The partial derivative matrix of x, and H n It represents the observation equation h n-1 In the prior The partial derivative with respect to x; (·) H Represents the conjugate transpose operation on the matrix; Refers to R n The inverse matrix of n , which can decide whether to trust the observation or prediction of the model more; the final state estimate is the vector This is the final result obtained based on the joint estimation of the H∞ filter.

5. The method for estimating RSOP, frequency deviation and phase in a PDM system based on H∞ filtering according to claim 4, characterized in that: The method further includes step S5 of performance optimization, in which the parameters of the filter are adjusted according to the estimation result of step S4 to further optimize the estimation performance.

6. The method for estimating RSOP, frequency deviation and phase in a PDM system based on H∞ filtering according to claim 5, characterized in that: The step S5 comprises: Adjust the initial state estimate vector And the initial covariance matrix P0, the vector Propose new assumptions about the distribution of Adjust H∞ gain K n : According to the simulation results, if the estimation error is large, increase the filter gain to make the filter more dependent on the measured data; Adjust performance indicators: Under the premise of ensuring system stability, adjust the threshold parameter λ through optimization method to minimize the estimation error in the worst case; After each parameter adjustment, the simulation is re-performed and the new performance of the filter is evaluated; when the estimation error and robustness reach the expected goals, the adjustment is stopped.

7. A system for estimating RSOP, frequency offset and phase in a PDM system based on H∞ filtering, characterized in that: include: Signal receiving module: used to pre-process the received signal in the PDM system; State space model module: used to establish the state space model of polarization rotation RSOP, frequency offset and phase in the PDM system. The state space model includes state equation and observation equation. H∞ filter design module: used to design H∞ filters and ensure the robustness of the filters in nonlinear systems and non-Gaussian noise environments by selecting appropriate filter parameters; State estimation module: used to use H∞ filter to estimate RSOP, frequency offset and phase in real time and output the estimation results.

8. The system for estimating RSOP, frequency offset and phase in a PDM system based on H∞ filtering according to claim 7, characterized in that: The state space model building module builds the state space model according to the following method: In the PDM system, the transmission signal at time n is represented by a two-dimensional vector s n =[s x,n ;s y,n ] indicates that x and y represent two polarizations respectively; the polarization state at time n is rotated by RSOP with a 2×2 Jones matrix J n Indicates that the Cayley-Klein form is The sum of the squares of the polarization state rotation parameters is 1, that is, ∈ is the frequency deviation normalized by the sampling rate; the time-varying phase noise φ n Modeled as a Wiener process: φ n =φ n-1 +ν n , v n The mean is 0 and the variance is Gaussian random variable, where Δv is the laser combined line width CLW, T is the sampling time interval; Considering the influence of RSOP, frequency offset and phase noise, the received signal at time n is expressed as where w n =[w x,n ;w y,n ] represents a complex Gaussian white noise vector with a mean of 0 and a covariance matrix of w x,n is the complex white Gaussian noise on the x polarization, w y,n is the complex white Gaussian noise on the y polarization, I represents the unit matrix, and the state at time n is represented by a five-dimensional vector x n =[a n ,b n ,c n ,∈ n ,φ n ] T ; The state space model is expressed as: Equation of state: x n =[a n ,b n ,c n ,∈ n ,φ n ] T =f n-1 (x n-1 ,v n )=x n-1 +v n ; Observation equation: where v n represents the state noise vector, f n-1 (·) represents the state equation at time n.

9. The system for estimating RSOP, frequency offset and phase in a PDM system based on H∞ filtering according to claim 8, characterized in that: The H∞ filter design module designs the H∞ filter in the following way: Introducing a threshold To ensure the robustness of the H∞ filter; Applying performance bounds to find estimation strategies that satisfy the threshold, the optimization problem is defined as: Where P0 is the initial estimation error covariance matrix, Q n is the state noise vector v n The covariance matrix, R n w n The covariance matrix of N represents the pilot symbol length; in addition, P0 and R n is a symmetric positive definite matrix; the threshold parameter λ specifies the maximum allowable gain from interference to estimation error; The estimation process is considered as a min-max optimization problem, where the goal is to find an estimate that minimizes the objective function while satisfying the following constraints to ensure that the H∞ gain can be calculated: Among them, H n Represents the observation equation h n-1 In the prior The partial derivative with respect to x.

10. The system for estimating RSOP, frequency offset and phase in a PDM system based on H∞ filtering according to claim 9, characterized in that: The state estimation module performs estimation according to the following method: In the initialization step, the parameters to be estimated are considered to be independent of each other and are uniformly distributed, that is, a0, b0, c0 ~ U (0, 1), ∈0 ~ U (-0.5, 0.5), φ0 ~ U (-π, π); therefore, the initial value is first determined And the initial covariance matrix Then, perform a recursive update; For n=1,…,N-1, perform the following steps to iteratively update the estimated vector: Based on the five-variable normal density function p(x n |x n-1 ) Update the prior state estimate Calculate H∞ gain Update the posterior estimation error covariance matrix Update the posterior state estimate In the algorithm, F n-1 The term represents f n-1 In the posterior The partial derivative matrix with respect to x, F n Then it means f n-1 In the posterior The partial derivative matrix of x, and H n It represents the observation equation h n-1 In the prior The partial derivative with respect to x; (·) H Represents the conjugate transpose operation on the matrix; Refers to R n The inverse matrix of n , which can decide whether to trust the observation or prediction of the model more; the final state estimate is the vector This is the final result obtained based on the joint estimation of the H∞ filter.

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