A fiber nonlinear equalization method based on perturbation theory and complex-valued convolutional neural networks.

Abstract

This invention discloses a fiber optic nonlinear equalization method based on perturbation theory using a complex-valued convolutional neural network, comprising the following steps: S1, constructing a complex-valued channel feature map from symbol samples using perturbation theory to form an original training set, and inputting it into the neural network; S2, training the complex-valued convolutional neural network using the original training dataset, fitting the training data, and finally obtaining the most suitable nonlinear equalizer model; S3, constructing a complex-valued channel input feature map for the test signal, and inputting it as the test set into the fitted nonlinear equalizer model for equalization, obtaining the output result. The fiber optic nonlinear equalization method based on perturbation theory proposed in this invention utilizes perturbation theory combined with a complex-valued convolutional neural network to extract more useful feature information, solving the problems of high bit error rate and poor compensation effect during nonlinear equalization, improving the nonlinear compensation effect, and greatly enhancing system performance.

Description

Technical Field

[0001] This invention relates to the field of optical fiber communication technology, and in particular to an optical fiber nonlinear equalization method based on perturbation theory using complex-valued convolutional neural networks (CvCNN). Background Technology

[0002] Currently, global data is growing explosively, and people's demand for communication capacity is increasing dramatically. Linear and nonlinear impairments in optical fiber communication are the main factors restricting the development of long-distance, large-capacity, and high-speed coherent optical communication.

[0003] Currently, digital signal processing techniques can compensate for deterministic linear and nonlinear impairments in optical fibers. However, nonlinear phase noise caused by amplified spontaneous emission (ASE) noise and the fiber Kerr nonlinearity effect in the system cannot be compensated. Therefore, addressing nonlinear transmission impairments, including self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM), has become a key research area for further improving the performance of modern optical fiber communication systems.

[0004] Traditional digital signal processing algorithms commonly used to handle nonlinear transmission impairments include digital backpropagation (DBP), Volterra series transfer function (VSTF), and perturbation theory-based equalization (PB). Among these, digital backpropagation (DBP) is an effective nonlinear compensation (NLC) method, and perturbation theory has also been widely applied in NLC. However, both of these methods require significant computational resources and are not practically convenient for real-world applications.

[0005] Machine learning (ML) algorithms can improve the performance of fiber optic communication systems and are considered an effective method for solving nonlinear equilibrium problems. These algorithms are mainly divided into two categories: classification-based clustering and neural network-based compensation methods. Representative classification-based clustering compensation algorithms include Support Vector Machines (SVM), K-Nearest Neighbors (KNN), and K-means. Neural networks (NNs) are mathematical or computational models that mimic the structure and function of biological neural networks and can be used to estimate or approximate functions. Neural networks include Convolutional Neural Networks (CNNs), Fully Connected Neural Networks (FNNs), and Recurrent Neural Networks (RNNs). In particular, Convolutional Neural Networks (CNNs) are widely used in image recognition and object detection due to their excellent parameter reduction, automatic parameter updating capabilities, and good curve fitting ability. Furthermore, because CNNs have good adaptability to linear and nonlinear relationships between input and output and can effectively extract high-dimensional features, they have excellent prospects in the field of fiber optic communication systems.

[0006] In optical signal processing of fiber optic communication systems, digital signals are usually represented in complex form. However, most existing neural networks use real number form to represent digital signals, ignoring the correlation between the real and imaginary parts of complex signals. This leads to high bit error rate and poor compensation effect when using existing neural networks to solve nonlinear equalization methods, which in turn affects system performance. Summary of the Invention

[0007] This invention addresses the problems of high bit error rate and poor compensation effect when using existing neural networks to solve nonlinear equalization methods. It proposes a fiber nonlinear equalization method based on perturbation theory using complex-valued convolutional neural networks, which extracts more useful feature information and greatly improves system performance.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] This invention provides a fiber nonlinear equalization method based on perturbation theory using complex-valued convolutional neural networks, comprising the following steps:

[0010] S1. Using perturbation theory, construct complex-valued channel feature maps for symbol samples to form the original training set, and input them into the neural network;

[0011] S2. Train a complex-valued convolutional neural network using the original training dataset, fit the training data, and finally obtain the most suitable nonlinear equalizer model.

[0012] S3. Construct a complex-valued channel input feature map for the signal to be tested, and use it as the test set input to the fitted nonlinear equalizer model for equalization to obtain the output result.

[0013] Furthermore, the specific process of step S1 is as follows:

[0014] When compensating the kth symbol, the I and Q component data of the symbol and the m, n, and m+n adjacent symbols before and after the symbol are obtained respectively. When processing information, the I component is regarded as the real part of the signal and the Q component is regarded as the imaginary part of the signal.

[0015] Assuming the transmitted optical signal in the optical fiber is a Gaussian pulse sequence, based on first-order perturbation theory and the assumption of large dispersion in optical fiber links, we obtain the nonlinear perturbation triple product term of the current symbol k, which is subject to cross-phase modulation and four-wave mixing within the channel, i.e., the characteristic unit.

[0016]

[0017] Among them, A * Denotes the complex conjugate of A, B * To represent the complex conjugate of B, m and n take values ​​according to the following rules:

[0018]

[0019] Where T is a restriction on the meaningful combinations of values ​​for m and n;

[0020] The characteristic unit F of the received complex symbol is retained s The correspondence between the real and imaginary parts of F, and the different F s By analyzing the positional relationships between the real and virtual channels, a complex-valued channel feature map is obtained. After performing the above processing on all symbol sample sets in sequence, the original training dataset is obtained.

[0021] Furthermore, the fitting process in step S2 includes two stages: forward propagation and backward propagation, specifically as follows:

[0022] Forward propagation stage: The complex-valued feature map of the current input signal is passed to the complex-valued convolutional neural network, and the data is classified or regressed at the output layer through the input layer, convolutional layer, and fully connected layer respectively;

[0023] Backpropagation phase: The results obtained in the forward propagation phase are compared with the correct results in the original training dataset. The error between the estimated value and the actual value is calculated, and this error is propagated backward from the output layer to the previous layer until it reaches the input layer. During the backpropagation process, the values ​​of the weight parameters and bias parameters are adjusted according to the error. The above process is iterated continuously. Finally, when the loss value no longer changes, the complex-valued convolutional neural network equalizer tends to converge, the training ends, and the most suitable nonlinear equalizer model is obtained.

[0024] Furthermore, during the forward propagation phase, the input X = Xn to the complex-valued convolutional neural network... R +iX I Output Y = Y R +iY I Bias b = b R +ib I And weight ω=ω R +iω I All are in the complex domain, and the operations satisfy complex operations, X R Let i be the real part of the input X, and i be the imaginary unit. I Let X be the imaginary part of the input, and Y be the imaginary part of the input. R To output the real part of Y, Y I To output the imaginary part of Y, b R Let b be the real part of the bias b. I Let ω be the imaginary part of the bias b. R Let ω be the real part of the weight w. I Let w be the imaginary part of the weight w.

[0025] Furthermore, during the forward propagation phase, the output expression obtained from the convolution operation of the complex-valued convolutional neural network is:

[0026] Y = f(ωX + b)

[0027] =f[(X R ω R -X I ω I +b R )]+if[(X R ω I +X I ω R +b I )]

[0028] =f(Y) R )+if(Y I )

[0029] Where ω is the complex-valued weight parameter, b is the complex-valued bias parameter, and f represents the complex-valued nonlinear activation function, chosen as the complex ReLU function:

[0030] CRelu = max(0, Y) R )+imax(0,Y I )

[0031] Where C is the complex value.

[0032] Furthermore, the complex-valued convolutional neural network structure in step S2 includes an input layer, two convolutional layers, and a fully connected layer. Each convolutional layer contains multiple filters. The specific process of the forward propagation stage is as follows: the obtained complex-valued feature map is fed into the input layer, and then through two convolutional layers, the convolutional kernels are shared across all spatial locations of the input. The input feature map and the convolutional kernels are summed by an inner product, and a bias is added to obtain a new output feature map. In this process, the number of convolutional kernels represents the number of channels in the new feature map. Then, after the new feature map is activated by the complex-valued nonlinear activation function f, the high-dimensional feature vector of the output is input into the fully connected layer for feature computation. Finally, by selecting the type of loss function in the output layer, the complex-valued convolutional neural network can act as either a classifier or a regressor.

[0033] Furthermore, in the backpropagation stage, complex cross-entropy loss is chosen as the loss function for the classifier, complex mean square error is chosen as the loss function for the regressor, and the Adam optimizer is used to optimize the complex-valued convolutional neural network.

[0034] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0035] The fiber nonlinear equalization method based on perturbation theory proposed in this invention utilizes perturbation theory combined with complex-valued convolutional neural networks to extract more useful feature information, solving the problems of high bit error rate and poor compensation effect in nonlinear equalization. Experimental results show that the bit error rate of the complex-valued convolutional neural network structure is lower than that of the real-valued structure, and the bit error rate of the classifier is lower than that of the regressor, thus improving the nonlinear compensation effect and greatly enhancing the system performance. Attached Figure Description

[0036] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0037] Figure 1 The replica channel feature map provided in the embodiments of the present invention.

[0038] Figure 2 This is a schematic diagram of the convolution expression process provided in an embodiment of the present invention.

[0039] Figure 3 This is a block diagram of the CvCNN structure provided in an embodiment of the present invention.

[0040] Figure 4 This is a schematic diagram comparing the performance of nonlinear equalization algorithms provided in an embodiment of the present invention. Detailed Implementation

[0041] To better understand this technical solution, the technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described examples are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art based on this application are within the scope of protection of the present invention.

[0042] This embodiment employs a 120Gb / s dual-polarization 64QAM coherent optical communication system. At the transmitter, a random bit sequence is generated using MATLAB, mapped to 64QAM, and loaded into an arbitrary waveform generator (AWG). The I and Q channel signals are amplified by electrical amplifiers (EA) and then fed into the I / Q modulator by an external cavity laser (ECL). At the coherent receiver, the digital signal processing (DSP) section includes I / Q imbalance compensation, dispersion (CD) compensation, clock recovery, polarization demultiplexing, polarization mode dispersion (PMD) compensation, frequency offset estimation (FOE), carrier phase recovery, and CvCNN equalization.

[0043] The fiber nonlinear equalization method based on perturbation theory and complex-valued convolutional neural network proposed in this embodiment includes the following steps:

[0044] S1. Using perturbation theory, construct complex-valued channel feature maps for symbol samples to form the original training set, and input them into the neural network.

[0045] Specifically, when compensating the kth symbol, the I and Q component data of the symbol and the m, n, and m+n adjacent symbols before and after the symbol are obtained respectively. When processing information, the I component can be regarded as the real part of the signal and the Q component can be regarded as the imaginary part of the signal.

[0046] The propagation function of an optical pulse in a typical single-mode optical fiber is expressed by the nonlinear Schrödinger equation as follows:

[0047]

[0048] in Let α represent the optical field, β2 represent the fiber loss, β2 represent the second-order group velocity dispersion coefficient, and γ represent the nonlinear Kerr coefficient.

[0049] Treating the nonlinear term on the right-hand side of the equation as a perturbation term, and assuming that the transmitted optical signal in the optical fiber is a Gaussian pulse sequence, based on first-order perturbation theory and the assumption of large dispersion in optical fiber links, we can obtain the nonlinear perturbation triple product term of the current symbol k being subjected to cross-phase modulation and four-wave mixing within the channel:

[0050]

[0051] Among them, A * Denotes the complex conjugate of A, B * Denotes the complex conjugate of B, the perturbation term. It is a feature unit and perturbation coefficient C mn The vector dot product, the nonlinear perturbation coefficient, can be calculated by the following formula:

[0052]

[0053] Where τ represents the pulse width, and E1(x) represents the exponential integral function. m and n are selected according to the following rules:

[0054]

[0055] Here, T is a restriction on the meaningful combinations of values ​​for m and n.

[0056] The characteristic unit is obtained from equation (2). The characteristic unit F of the received complex symbol is retained s The correspondence between the real and imaginary parts of F, and the different F sThe positional relationship between them combines the real channel and the virtual channel to construct a structure such as Figure 1 The complex-valued channel feature map is shown. After repeating the process on all symbol sample sets, the original training dataset is obtained.

[0057] S2. Train a complex-valued convolutional neural network using the original training dataset, fit the training data, and finally obtain the most suitable nonlinear equalizer model.

[0058] Specifically, in the forward propagation phase: the complex-valued feature map of the current input signal is passed to CvCNN, where it is used for data classification or regression through the input layer, convolutional layer, fully connected layer, and output layer. The input X = X... R +iX I Output Y = Y R +iY I Bias b = b R +ib I And weight ω=ω R +iω I Both are in the complex domain, and the operations satisfy complex operations.

[0059] Figure 2 This is a schematic diagram of the convolution operation process. By activating the real and imaginary parts respectively, the output expression is obtained as follows:

[0060] Y = f(ωX + b)

[0061] =f[(X R ω R -X I ω I +b R )]+if[(X R ω I +X I ω R +b I )]

[0062] =f(Y) R )+if(Y I (5) where f represents the complex-valued nonlinear activation function, and the complex ReLU function is chosen:

[0063] CRelu = max(0, Y) R )+imax(0,Y I (6)

[0064] Figure 3The diagram illustrates the basic structure of the CvCNN designed in this invention. The obtained feature map is fed into the input layer, then passed through two convolutional layers. Each convolutional layer contains multiple filters, and the convolutional kernels are shared across all spatial locations of the input. The input feature map and the convolutional kernels are summed by an inner product, and a bias is added to obtain a new output feature map. In this process, the number of convolutional kernels represents the number of channels in the new feature map. The new feature map is then activated by a complex-valued nonlinear activation function f, and the high-dimensional feature vector of the output is input into a fully connected layer for feature computation. Finally, by selecting the type of loss function in the output layer, CvCNN can function as either a classifier or a regressor.

[0065] Backpropagation Phase: The results obtained in the forward propagation phase are compared with the correct results on the original training dataset. The error between the estimated and actual values ​​is calculated, and this error is propagated backward from the output layer to the previous layer, until it reaches the input layer. During backpropagation, the values ​​of the weight and bias parameters are adjusted based on the error. This process is iterated continuously until the loss value no longer changes, at which point the CvCNN equalizer converges, and the training process is terminated. In this model, complex cross-entropy loss is chosen as the loss function for the classifier, complex mean squared error is chosen as the loss function for the regressor, and the Adam optimizer is used to optimize the CvCNN network.

[0066] S3. Following step S1, construct complex-valued channel input feature maps for all signals to be tested, and use them as test sets to input into a fitted nonlinear equalizer model for equalization. Obtain the output results through the complete propagation process. Compare the output results with the original data to calculate key indicators such as the bit error rate (BER) of the optical fiber communication system.

[0067] Figure 4 This paper presents the compensation effects of different equalization algorithms on the received signal under different transmit optical power (LOP). When the LOP is 1 dBm, the bit error rate (BER) of the complex-valued convolutional neural network classifier (CvCNNC) is 31% lower than that of the real-valued convolutional neural network classifier (RvCNNC), and the BER of the complex-valued convolutional neural network regressor (CvCNNR) is 18% lower than that of the real-valued convolutional neural network regressor (RvCNNR). This indicates that the complex-valued neural network has a better compensation effect than the real-valued neural network. When the LOP is 1 dBm, the BER of CvCNNC is 16% lower than that of CvCNNR. When the LOP is 0 dBm, the BER of RvCNNC is 10% lower than that of RvCNNR. This indicates that the classifier has a better compensation effect than the regressor.

[0068] In summary, the results of this invention show that the bit error rate of complex-valued structures is lower than that of real-valued structures, and the bit error rate of classifiers is lower than that of regressors. Therefore, this invention uses perturbation theory combined with complex-valued convolutional neural networks to reduce the bit error rate, improve the nonlinear compensation effect, and enhance system performance.

[0069] The above description is merely a detailed explanation of preferred embodiments and principles of the present invention and is not intended to limit the scope of protection of the present invention. For those skilled in the art, any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention, based on the ideas provided by the present invention, should be considered within the scope of protection of the present invention.

Claims

1. A fiber nonlinear equalization method based on perturbation theory using complex-valued convolutional neural networks, characterized in that, Includes the following steps: S1. Construct complex-valued channel feature maps from symbol samples using perturbation theory to form the original training set, and input it into the neural network; the specific process of step S1 is as follows: When compensating the kth symbol, the I and Q component data of the symbol and the m, n, and m+n adjacent symbols before and after the symbol are obtained respectively. When processing information, the I component is regarded as the real part of the signal and the Q component is regarded as the imaginary part of the signal. Assuming the transmitted optical signal in the optical fiber is a Gaussian pulse sequence, based on first-order perturbation theory and the assumption of large dispersion in optical fiber links, we obtain the nonlinear perturbation triple product term of the current symbol k, which is subject to cross-phase modulation and four-wave mixing within the channel, i.e., the characteristic unit. Among them, A * Denotes the complex conjugate of A, B * Indicates the complex conjugate of B; The characteristic unit F of the received complex symbol is retained s The correspondence between the real and imaginary parts of F, and the different F s The positional relationship between the real and virtual channels is used to combine them to obtain a complex-valued channel feature map; after performing the above processing on all symbol sample sets in sequence, the original training dataset is obtained. S2. Train a complex-valued convolutional neural network using the original training dataset, fit the training data, and finally obtain the most suitable nonlinear equalizer model; the fitting process in step S2 includes two stages: forward propagation and backward propagation, specifically as follows: Forward propagation stage: The complex-valued feature map of the current input signal is passed to the complex-valued convolutional neural network, and the data is classified or regressed at the output layer through the input layer, convolutional layer, and fully connected layer respectively; Backpropagation phase: The results obtained in the forward propagation phase are compared with the correct results in the original training dataset. The error between the estimated value and the actual value is calculated, and this error is propagated backward from the output layer to the previous layer until it reaches the input layer. During the backpropagation process, the values ​​of the weight parameters and bias parameters are adjusted according to the error. The above process is iterated continuously. Finally, when the loss value no longer changes, the complex valued convolutional neural network equalizer tends to converge, the training ends, and the most suitable nonlinear equalizer model is obtained. During the forward propagation phase, the input X = Xn to the complex-valued convolutional neural network R +iX I Output Y = Y R +iY I Bias b = b R +ib I And weight ω=ω R +iω I All are in the complex domain, and the operations satisfy complex operations, X R Let i be the real part of the input X, and i be the imaginary unit. I Let X be the imaginary part of the input, and Y be the imaginary part of the input. R To output the real part of Y, Y I To output the imaginary part of Y, b R Let b be the real part of the bias b. I Let ω be the imaginary part of the bias b. R Let ω be the real part of the weight w. I Let w be the imaginary part of the weight w; During the forward propagation phase, the convolution operation of the complex-valued convolutional neural network yields the following output expression: Y = f(ωX + b) =f[(X R oh R -X I oh I +b R )]+if[(X R oh I +X I oh R +b I )] =f(Y R )+if(Y I ) Where ω is the complex-valued weight parameter, b is the complex-valued bias parameter, and f represents the complex-valued nonlinear activation function, chosen as the complex ReLU function: CRElu=max(0,Y R )+imax(0,Y I ) Where C is a complex value; S3, construct a complex value channel input feature map for the signal to be tested, and use it as a test set input to the fitted nonlinear equalizer model for equalization to obtain the output result.

2. The fiber nonlinear equalization method based on perturbation theory using complex-valued convolutional neural networks according to claim 1, characterized in that, The complex-valued convolutional neural network structure in step S2 includes an input layer, two convolutional layers, and a fully connected layer. Each convolutional layer contains multiple filters. The specific process of the forward propagation stage is as follows: the obtained complex-valued feature map is fed into the input layer, and then through two convolutional layers, the convolutional kernels are shared across all spatial locations of the input. The input feature map and the convolutional kernels are summed by an inner product, and a bias is added to obtain a new output feature map. In this process, the number of convolutional kernels represents the number of channels in the new feature map. Then, the new feature map is activated by a complex-valued nonlinear activation function f, and the high-dimensional feature vector of the output is input into the fully connected layer for feature computation. Finally, by selecting the type of loss function in the output layer, the complex-valued convolutional neural network can act as either a classifier or a regressor.

3. The fiber nonlinear equalization method based on perturbation theory using complex-valued convolutional neural networks according to claim 1, characterized in that, In the backpropagation stage, complex cross-entropy loss is chosen as the loss function for the classifier, complex mean squared error is chosen as the loss function for the regressor, and the Adam optimizer is used to optimize the complex-valued convolutional neural network.

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