FI-DAC system phase frequency error correction method based on BO-Adabound

Abstract

The invention provides an FI-DAC system phase frequency error correction method based on BO-Adabound, belongs to the technical field of phase frequency error calibration, and aims at solving the problems that a traditional nonlinear phase frequency error correction method is poor in performance of phase frequency characteristics of sub-band target frequency and low in correction precision, and the method comprises the steps that an initial target function is calculated according to an input signal; calculating an initial objective function output based on an AdaBound algorithm; performing Bayesian loop optimization on the initial data set to obtain a new evaluation point of an integral adjustment factor; obtaining a new integral adjustment factor based on the new evaluation point of the integral adjustment factor, and obtaining an Adabound initial parameter through a table look-up method; carrying out Adabound training, updating solving parameters, iteratively updating target functions corresponding to the parameters, and obtaining current optimal filter parameters and the target functions; and updating the data set based on the optimal objective function and carrying out Bayesian loop optimization again until a global convergence condition is met or the maximum number of iterations is reached, and outputting an optimal all-pass filter parameter.

Description

TECHNICAL FIELD

[0001] The application relates to a BO-Adabound-based phase frequency error correction method for an FI-DAC system and belongs to the technical field of phase frequency error correction. BACKGROUND

[0002] The non-ideal characteristics of various analog modules in the FI-DAC, especially the group delay characteristics of the analog filter, can cause inconsistency of signals of different frequencies in the output time. The inconsistency is respectively reflected in the nonlinear group delay characteristics of the sub-path signal, the inconsistency of the arrival time of the low-frequency sub-path signal and the high-frequency sub-path signal at the input port of the analog combiner, and the inconsistency of the arrival time of the high-frequency sub-path baseband signal and the local oscillator signal at the input port of the analog mixer. The phase frequency error mainly affects the accuracy of signal output and is obvious in the overlapping band frequency interval. The phase frequency error directly causes the deterioration of the flatness of the overlapping band and the amplitude and phase distortion of the combined signal. At the same time, for communication signals, especially in high-speed and high-precision communication systems, the effective correction of the phase frequency error is of great significance. Accurate phase frequency error compensation can improve the quality of signals, reduce the bit error rate, and ensure the reliability and performance of the system in actual application. In the application, the nonlinear phase frequency error performance and its correction method are particularly concerned.

[0003] The traditional nonlinear phase frequency error correction method focuses on the phase frequency characteristics of the full-band signal. Although these methods improve the phase frequency consistency of the system signal to a certain extent, the correction accuracy, especially the phase frequency characteristics of the sub-band target frequency, still has room for improvement. SUMMARY

[0004] To solve the problem of poor phase frequency characteristics of the sub-band target frequency and low correction accuracy of the traditional nonlinear phase frequency error correction method, the application provides a BO-Adabound-based phase frequency error correction method for an FI-DAC system.

[0005] The technical solution adopted by the application to solve the above problem is as follows: Step 1: calculating an initial target function according to an input signal; Step 2: calculating an initial integral adjustment factor based on the AdaBound algorithm The initial target function output when the value is 1 generates an initial data set; Step 3: selecting Gaussian process regression as a proxy model to perform Bayesian cycle optimization on the initial data set, obtaining a posterior distribution according to the prior distribution of the initial data set, and selecting a new evaluation point of the next integral adjustment factor in the global range through the acquisition function based on the mean and variance of the posterior distribution optimization search strategy; Step 4: Obtain the corresponding new integral adjustment factor based on the new evaluation point of the integral adjustment factor, calculate and update the target group delay performance based on the new integral adjustment factor, obtain the initial parameters of Adabound by looking up the table and perform Adabound training. Step 5: In Adabound training, calculate the objective function and obtain its gradient performance, update the first and second moment estimates, update the solution parameters by pruning the learning rate, and iteratively update the objective function corresponding to the parameters. When the convergence condition or the maximum number of iterations is reached, return the current optimal filter parameters and their corresponding objective function. Update the dataset based on the optimal objective function and perform Bayesian loop optimization again until the global convergence condition is met or the maximum number of iterations is reached, and output the optimal all-pass filter parameters.

[0006] Furthermore, step 2 specifically includes: Set the initial integral adjustment factor The value is 1, combined with the current all-pass filter parameters. The output of the initial objective function is obtained. , ,in, Radius of the poles of the all-pass filter Phase angle with poles The combination of these factors, based on the output of the initial objective function. Get the initial dataset .

[0007] Furthermore, step 3 specifically includes: Step 3.1: Set the joint distribution of the initial dataset to conform to the condition of a multivariate normal distribution, and combine this with the initial dataset that conforms to a multivariate normal distribution to set the ideal training sample dataset. The ideal training sample dataset It follows a Gaussian prior distribution with a mean of 0 and a covariance matrix of K; Step 3.2: Based on the ideal training sample dataset Calculate new input posterior distribution Receive new input Normally distributed data at any data point, with probability enhancement applied, combined with new input. The optimal function value in the current dataset from the normal distribution number at any data point. Predicted mean and prediction variance Calculate the acquisition function, and select a new evaluation point for the next integral adjustment factor in the global scope based on the acquisition function; The expression for the joint distribution satisfying the condition of a multivariate normal distribution is: (1); In formula (1), For Gaussian processes, It is a mean function. Let covariance function be used. The magnitude factor of the kernel function is used to control the overall output scale of the covariance function of the prior distribution of the Gaussian process. The standard deviation of the random variable corresponding to the integral adjustment factor; The expression for the Gaussian prior distribution that the ideal training sample dataset follows is: (2); In formula (2), Input to the training dataset, For training dataset The corresponding set of objective function values, For kernel function hyperparameters, The training dataset is obtained from the kernel function hyperparameters. The covariance matrix of its objective function value set.

[0008] Furthermore, step 3.2 specifically includes: Based on the ideal training sample dataset Using a Gaussian process, construct a joint multivariate normal distribution. Based on the conditions of the multivariate normal distribution, derive the new input from the constructed joint multivariate normal distribution. The posterior probability distribution is used to obtain the new input. Normally distributed data at any data point; Improved application of probability, combined with new input The optimal function value in the current dataset from the normal distribution number at any data point. Predicted mean and prediction variance Calculate the acquisition function To predict the mean and prediction variance Construct a search strategy based on the collection function. Maximize the collection function values ​​of all candidate points to obtain new evaluation points for the next integral adjustment factor; The expression for the joint multivariate normal distribution is: (3); In formula (3), For new input data Corresponding to the set of function values ​​to be predicted, For new input data With training dataset The covariance matrix between them For new input data With new input data The covariance matrix between them for The transpose of the matrix; The formula for calculating normally distributed data at any data point is: (4); In formula (4), The set of function values ​​to be predicted The predicted mean, The set of function values ​​to be predicted The prediction variance is used to measure the uncertainty of the predicted value. Acquisition function The calculation formula is: (5); In formula (5), The cumulative distribution function of the standard normal distribution. These are balancing parameters used to weigh exploration against exploitation. The optimal function value in the current dataset; The formula for calculating the new evaluation points of the next integral adjustment factor is as follows: (6); In formula (6), For the entire input space, For the current dataset, For the current dataset The following acquisition function, This will serve as a new assessment point for the next step of the integral adjustment factor.

[0009] Furthermore, in step 4, a new integral adjustment factor is obtained based on the new evaluation point, and the target group delay performance is calculated and updated based on the new integral adjustment factor. The initial parameters of Adabound are obtained using a lookup table method, including: New assessment points for integral adjustment factors Calculate the current linear delay performance to be compensated Based on the linear delay performance to be compensated Update target group latency Combined with the updated target group latency The initial parameters of Adabound are obtained by looking up a table. ; Current linear delay performance to be compensated The calculation formula is: (7); In formula (7), and These represent the start and end points of the frequency sampling interval, indicating the lowest and highest frequencies, respectively. For the target group delay at frequency point The actual value at that location; Target group latency The update formula is: (8).

[0010] Furthermore, step 5 specifically includes: Step 5.1: Obtain the current parameters by looking up the table. Calculate the objective function gradient Introducing the momentum factor and root mean square momentum factor The gradient of the objective function computed by the set Update the first-order matrix estimate and the second-order matrix estimate; Step 5.2: Dynamically change the lower boundary function of the learning rate and the upper boundary function of the dynamically changing learning rate Learning rate after clipping Perform a correction update based on the corrected learning rate. The updated first-order matrix estimate and second-order matrix estimate are iteratively updated to solve for the objective parameters. Based on target parameters New assessment points for integral adjustment factors Update parameters corresponding to the objective function Continue until the global convergence condition is met or the maximum number of iterations is reached, then output the current optimal all-pass filter parameters. And obtain the optimal objective function. ; Step 5.3: Based on the optimal objective function New assessment points for integral adjustment factors Update the dataset and perform Bayesian loop optimization again until the global convergence condition is met or the maximum number of iterations is reached, and output the globally optimal all-pass filter parameters. The update formulas for first-order and second-order matrix estimates are as follows: (9); In formula (9), For the first n The gradient of the objective function with respect to the parameters at the next iteration. The exponentially weighted moving average of the gradient represents the first moment estimate of the gradient. The exponentially weighted moving average of the squared gradient represents the second moment estimate of the gradient. Learning rate after cropping The corrected update expression is: (10); In formula (10), For the first n The actual learning rate after pruning at the next iteration; The formula for updating the target parameter is: (11); In formula (11), F For the feasible region or constraint set of parameters, Let represent the diagonal preconditioning matrix composed of the inverse estimates of the learning rate; The formula for calculating the optimal objective function is: (12); The expression for updating the dataset is: (13).

[0011] The beneficial effects of this invention are: This invention uses a Bayesian optimization algorithm combined with an Adabound algorithm to determine the optimal performance of the integral compensation factor and pre-equalizer parameters. By optimizing the integral compensation factor using the Bayesian optimization algorithm and combining it with the Adabound algorithm to obtain the target nonlinear group delay performance of the current integral compensation factor, the optimal equalizer parameters and objective function are calculated. The Bayesian optimization dataset is supplemented, and the optimization is iteratively performed to finally obtain the optimal phase frequency error pre-equalizer parameters, thereby further improving the correction accuracy of the phase frequency response of the FI-DAC system. Attached Figure Description

[0012] Figure 1 This is a flowchart illustrating the phase frequency error correction method for a FI-DAC system based on BO-Adabound provided by the present invention. Figure 2 This is a schematic diagram of the sub-path nonlinear phase frequency error test results of the FI-DAC system after processing by the method proposed in this invention; Figure 3 A flowchart illustrating the phase frequency error correction of FI-DAC signals using the method proposed in this invention; Figure 4 This is a schematic diagram of the group delay error test results of the FI-DAC system after processing by the method proposed in this invention; Figure 5 This is a schematic diagram showing the low-frequency sub-path characteristics and target correction pre-equalization performance of the FI-DAC system after processing by the method proposed in this invention; Figure 6 This is a schematic diagram showing the high-frequency sub-path characteristics and target correction pre-equalization performance of the FI-DAC system after processing by the method proposed in this invention; Figure 7 When the pre-equalizer order N=8, the group delay characteristics of the low-frequency sub-path nonlinear phase frequency error pre-equalizer after processing by the method proposed in this invention are shown. Figure 8 When the pre-equalizer order N=16, the group delay characteristics of the low-frequency sub-path nonlinear phase frequency error pre-equalizer after processing by the method proposed in this invention are shown. Figure 9 A schematic diagram showing the group delay characteristics of the high-frequency sub-path nonlinear phase frequency error pre-equalizer after processing by the method proposed in this invention when the pre-equalizer order N=8. Figure 10 A schematic diagram showing the group delay characteristics of the high-frequency sub-path nonlinear phase frequency error pre-equalizer after processing by the method proposed in this invention when the pre-equalizer order N=16. Detailed Implementation Combination Figures 1-3 This implementation method is described as follows: Figure 3 As shown, the nonlinear phase frequency error correction of the FI-DAC system in this embodiment includes the following components: The integrated local oscillator clock in the FI-DAC hardware module is input to the external trigger port and external reference clock port of the AWG backplane via a power divider. The AWG selects the external clock as the DAC sampling clock. When the external trigger edge signal is detected, the signals from the low-frequency and high-frequency sub-paths are output to the inputs of the FI-DAC hardware module. Subsequently, the sub-path signals undergo frequency conversion and combining in the analog domain, and finally, the output signal is detected by an oscilloscope, and its phase-frequency characteristics are obtained through an FFT operation.

[0013] The digital domain includes generating the target waveform table and performing frequency division and downsampling on the input signal. A nonlinear phase-frequency error pre-equalizer is designed to compensate for this error before the signal enters the DAC, reducing the nonlinear phase-frequency error introduced by subsequent analog processing. The analog domain consists of a DAC module, mixer, filters, and combiner. These modules work together to convert the discrete signal output from the digital domain into an analog signal, performing up-conversion, filtering, and amplification before signal combining.

[0014] This implementation scheme designs a nonlinear phase-frequency error pre-equalizer for a target frequency band with K discrete angular frequency points within a known distribution. The latency of the target group and its corresponding target group is as follows To ensure accurate correction of group delay in broadband systems and overcome limitations of data testing systems, interpolation is required to improve signal frequency resolution. When designing an N-stage cascaded second-order all-pass filter, the integral characteristic of the target group delay is affected by the integral of each second-order section over the full band. Due to limitations, it is necessary to calculate the group delay integral over the entire frequency band and introduce linear group delay compensation. To ensure that the total score reaches times .

[0015] Considering that the core issue in time delay compensation design is how to provide appropriate linear time delay compensation for the nonlinear response of the target frequency band without introducing new nonlinear errors, so as to obtain the best fitting effect, this implementation proposes the following design method: First, determine the minimum number of second-order all-pass filters required corresponding to the total order of the all-pass filters, by integrating the target group time delay performance by... The result is then rounded up. To meet design standards and achieve sufficient correction accuracy, the order of the all-pass filter is set to an integer multiple of 2.

[0016] Second, linear time delay compensation is set for the target frequency band, and the integral values ​​of each sub-segment are reasonably allocated. Implementing a simplified average integral frequency division strategy combined with a global optimization algorithm is an efficient solution. Specifically, after determining the specific order of the all-pass filter, an integral adjustment factor ranging from 0 to 1 is introduced. This is to optimize the target frequency band fitting performance obtained through a global algorithm. The total integral after target frequency band compensation is defined as... The target subband compensation characteristics of each second-order all-pass filter are obtained through an average allocation strategy. The subband integral equals... At this point, the target frequency band delay value to be compensated satisfies: (1); At this point, the key to improving fitting accuracy shifts to obtaining the integral adjustment factor. The optimal solution. The target group delay performance after integral correction includes the compensation delay and the integral adjustment factor. Relevant, satisfying the following definition: (2); To design a cascaded second-order nodal-pass filter such that its group delay in the target frequency band is as follows: To approximate the preset target group delay performance in order to achieve nonlinear phase correction, the sos matrix of the pre-equalizer to be solved satisfies the condition.

[0017] (3); The design of an all-pass filter aims to minimize the error between the group delay and the target group delay across the entire target frequency band, i.e., to minimize the sum of the squared errors of the group delay at each frequency point. At this point, the design problem is transformed into a constrained nonlinear minimization problem, which aims to minimize the sum of squared errors between the calculated group delay and the target group delay by optimizing the pole radius and pole angle parameters, as shown in formula (4): (4); In formula (4), Let be the loss function, representing the sum of squared errors at the target sampling frequency points. Frequency point Error at that point Frequency point Compensation for delay Post-target group delay, Frequency point The fitting group delay, Accumulate the second-order sections, For the first The second-order section at frequency The group delay performance at the target frequency, where K is the total number of sampling points at the target frequency, and N is the order of the all-pass filter, including... There are two second-order sections, where w is the radius of the poles of the all-pass filter. Phase angle with poles Combine and simplify the writing format.

[0018] For the aforementioned nonlinear optimization problems with simple boundary constraints, the objective function typically exhibits continuity and differentiability under the influence of decision variables, and its derivative can be explicitly calculated. However, the inclusion of trigonometric terms in the objective function may lead to non-convexity, resulting in multiple local optima. In such cases, the optimization algorithm may converge to a local optimum rather than a global optimum. Therefore, this implementation proposes a phase-frequency error correction method for FI-DAC systems based on BO-Adabound. This implementation method addresses the design of a nonlinear phase error predistortion all-pass filter under normal temperature testing conditions, proposing an optimization of the integral adjustment factor. The core design approach aims to achieve optimal group delay fitting. This method comprises two nested global optimization processes. Algorithm 1, used for all-pass filter parameter optimization, is included within each single-parameter optimization iteration loop. In this case, a global optimization algorithm is used to find the optimal all-pass filter parameters and minimize the group delay fitting error. This process requires the optimization algorithm to have fast convergence to efficiently approximate the optimal solution and avoid local optima. Therefore, the algorithm must possess strong global search capabilities and sufficient exploratory power to handle complex non-convex optimization problems. Algorithm 2 is used as an integral adjustment factor. Single-parameter optimization, where the parameter directly affects the size of the compensation region. Nested algorithms are used to find the corresponding all-pass filter parameters, and the optimal parameter is selected based on the fitting error performance. The cost function depends on the filter parameters obtained from the nested global optimization. Although the optimization objective is clear, it cannot be directly derived. The explicit relationship between the filter and the error makes the specific behavior of each iteration opaque, ultimately yielding the optimal all-pass filter and its fitting performance, and enabling adjustments to the all-pass filter parameters and integral factors. Joint optimization.

[0019] The flowchart of the BO-Adabound algorithm proposed in this embodiment is shown in Table 1: Table 1

[0020] like Figure 1 As shown, the steps of the BO-Adabound-based FI-DAC system phase frequency error correction method described in this embodiment include: S1: Calculate the initial objective function based on the input signal; S2: Calculate the initial integral adjustment factor based on the AdaBound algorithm. The output of the initial objective function when the value is 1 is used to generate the initial dataset; This implementation method sets an initial integral adjustment factor. The value is 1, combined with the current all-pass filter parameters. The output of the initial objective function is obtained. , ,in, Radius of the poles of the all-pass filter Phase angle with poles The combination of these factors, based on the output of the initial objective function. Get the initial dataset .

[0021] S3: Perform Bayesian loop optimization on the initial dataset and conduct a global search to obtain new evaluation points for the next integral adjustment factor; S301: Integral adjustment factor in this implementation method The initial value is set to 1, and the range of values ​​satisfies The aim is to quickly find the optimal value through Bayesian optimization, minimizing the fitting error. Bayesian optimization utilizes prior knowledge to construct a probabilistic surrogate model, dynamically updates the posterior distribution of the objective function based on observed data, and then selects the next optimal evaluation point globally through a search strategy based on the posterior mean and variance of the acquisition function, thereby gradually approaching the optimal solution of the objective function.

[0022] Probabilistic surrogate models are used to approximate an objective function by constructing a probability distribution of the objective function given limited observation data. A Gaussian process, a nonparametric method, is typically employed. A Gaussian process is a type of stochastic process where the function value at any data point follows a Gaussian distribution, and the joint distribution of multiple data points conforms to a multivariate normal distribution. It can provide mean prediction and uncertainty quantification, and is derived from the mean function. Sum of covariance functions The unique condition is that it satisfies the following formula: (5); In formula (5), For Gaussian processes, It is a mean function. Let covariance function be used. The magnitude factor of the kernel function is used to control the overall output scale of the covariance function of the prior distribution of the Gaussian process. Let be a random variable with an integral adjustment factor. The standard deviation of the random variable corresponding to the integral adjustment factor; S302: In the modeling process, the first step is to make a prior assumption, assuming that the ideal training sample dataset is known. It follows a Gaussian prior distribution with mean 0 and covariance matrix K, satisfying: (6); In formula (6), Input to the training dataset, For training dataset The corresponding set of objective function values, For kernel function hyperparameters, The training dataset is obtained from the kernel function hyperparameters. The covariance matrix of its objective function value set.

[0023] S303: Based on the ideal training sample dataset Gaussian processes are used to construct a joint multivariate normal distribution: (7); In formula (7), For new input data Corresponding to the set of function values ​​to be predicted, For new input data With training dataset The covariance matrix between them For new input data With new input data The covariance matrix between them for To avoid ambiguity, this implementation uses an apostrophe (') to mark the optimal input point and the optimal observation value for the transpose matrix: This represents the optimal evaluation point selected through the acquisition function. This represents the largest known function value in the current observation dataset.

[0024] S304: Based on the conditions of the multivariate normal distribution, derive the new input from the constructed joint multivariate normal distribution. The posterior probability distribution is used to obtain the new input. Normally distributed data at any data point: (8); In formula (8), The set of function values ​​to be predicted The predicted mean, The set of function values ​​to be predicted The prediction variance is used to measure the uncertainty of the predicted value. S305: Applying probability improvements, combined with new inputs The optimal function value in the current dataset from the normal distribution number at any data point. Predicted mean and prediction variance Calculate the acquisition function ; During optimization, the sampling function balances exploration and utilization, avoids repeatedly querying already sampled points, prioritizes regions with higher variance to enhance exploration, and performs queries in regions with higher mean to improve optimization efficiency. Different sampling functions reflect different optimization strategies. Taking Probability of Improvement (PI) as an example, its goal is to select the point with the highest probability of improvement, and its sampling function is defined as... (9); In formula (9), The cumulative distribution function of the standard normal distribution. These are balancing parameters used to weigh exploration against exploitation. The optimal function value in the current dataset; S306: The acquisition function is based on the posterior distribution of the surrogate model, using the mean and variance to construct a search strategy, accelerating the optimization of the objective function. The next optimal evaluation point is obtained by maximizing the acquisition function values ​​of all candidate points. (10); In formula (10), For the entire input space, For the current dataset, For the current dataset The following acquisition function, This will serve as a new assessment point for the next step of the integral adjustment factor.

[0025] S4: Obtain the corresponding new integral adjustment factor based on the new evaluation point of the integral adjustment factor, calculate and update the target group delay performance based on the new integral adjustment factor, and obtain the Adabound initial parameters by table lookup method; S401: Adjusting new assessment points for factors through integration. Calculate the current linear delay performance to be compensated : (11); In formula (11), and These represent the start and end points of the frequency sampling interval, indicating the lowest and highest frequencies, respectively. For the target group delay at frequency point The actual value at that location; S402: Based on linear delay performance to be compensated Update target group latency : (12).

[0026] S403: Combined with updated target group latency The initial parameters of Adabound are obtained by looking up a table. ; S5: Perform Adabound training, update the solution parameters and iteratively update the objective function corresponding to the parameters. When the convergence condition or the maximum number of iterations is reached, return the current optimal filter parameters and their corresponding objective function. This implementation method selects a gradient-based mathematical optimization algorithm. The Adam optimizer is an adaptive learning rate method. In each calculation process, the algorithm calculates the first and second moments of the current gradient by using the exponentially weighted moving average and its square, respectively. It uses this information to dynamically control the learning rate, thereby improving the convergence speed and stability.

[0027] S501: Step 5.1: Obtain the current parameter by looking up the table. Calculate the objective function gradient Introducing the momentum factor and root mean square momentum factor The gradient of the objective function computed by the set Update the first-order matrix estimate and the second-order matrix estimate: (13); In formula (13), For the first n The gradient of the objective function with respect to the parameters at the next iteration. The exponentially weighted moving average of the gradient represents the first moment estimate of the gradient. The exponentially weighted moving average of the squared gradient represents the second moment estimate of the gradient. S502: Using a dynamically changing learning rate and lower boundary function and the upper boundary function of the dynamically changing learning rate Learning rate after clipping Make corrections and updates: (14); In formula (14), For the first n The actual learning rate after pruning at the next iteration; S503: Based on the corrected and updated learning rate The updated first-order matrix estimate and second-order matrix estimate are iteratively updated to solve for the objective parameters. : (15); In formula (15), F For the feasible region or constraint set of parameters, Let represent the diagonal preconditioning matrix composed of the inverse estimates of the learning rate; S504: Based on target parameters New assessment points for integral adjustment factors Update parameters corresponding to the objective function Continue until the global convergence condition is met or the maximum number of iterations is reached, then output the current optimal all-pass filter parameters. And obtain the optimal objective function. : (16).

[0028] S6: Update the dataset based on the optimal objective function and perform Bayesian loop optimization again until the global convergence condition is met or the maximum number of iterations is reached, and output the optimal all-pass filter parameters.

[0029] The expression for updating the dataset is: (17).

[0030] In summary, this implementation adds the new evaluation point and its objective function value calculated by the embedded algorithm to the dataset. The sampling function iterative process is repeated until a predetermined number of iterations is reached or the optimal solution is found, thus implementing the Bayesian optimization process. The BO-Adabound algorithm combines Bayesian optimization with Adabound adaptive optimization to search for the optimal all-pass filter parameters. First, the initial integral adjustment factor is calculated using the Adabound algorithm. The objective function performance is evaluated to generate an initial dataset. A Bayesian optimization loop is then initiated. In each iteration, Gaussian process regression is selected as the surrogate model. The posterior distribution is calculated based on the prior distribution of the existing dataset. A probability improvement is chosen as the acquisition function, and a new evaluation point with an integral adjustment factor is selected. The target group's time delay performance is calculated and updated based on the new integral adjustment factor. Initial Adabound hyperparameters are obtained using a lookup table, and Adabound training begins.

[0031] In Adabound training, the objective function is calculated and its gradient is obtained. The first and second moment estimates are updated. By pruning the learning rate, the solution parameters are updated iteratively, and the objective function value is calculated. When the convergence condition or the maximum number of iterations is reached, the current optimal filter parameters and their corresponding objective function are returned. This objective function is used to update the dataset, and the Bayesian optimization loop continues until the global convergence condition is met or the maximum number of iterations is reached. Finally, the optimal all-pass filter parameters obtained by Adabound are output.

[0032] Through the above steps, this invention uses a Bayesian optimization algorithm combined with an Adabound algorithm to determine the optimal performance of the integral compensation factor and pre-equalizer parameters. By optimizing the integral compensation factor using the Bayesian optimization algorithm and combining it with the Adabound algorithm to obtain the target nonlinear group delay performance of the current integral compensation factor, the optimal equalizer parameters and objective function are calculated. The Bayesian optimization dataset is supplemented, and the optimization is iteratively performed to finally obtain the optimal phase frequency error pre-equalizer parameters, thereby further improving the correction accuracy of the phase frequency response of the FI-DAC system.

[0033] Example Combination Figures 4-10 This embodiment describes the process of verifying the nonlinear phase-frequency error characteristics of the subchannel after phase-frequency correction according to the present invention. Phase-frequency tests were performed on both the low-frequency and high-frequency subchannels, followed by linear time delay removal of the data. The processed subchannel nonlinear phase-frequency error test results are as follows: Figure 4 As shown, the group delay characteristic corresponding to the nonlinear phase frequency error is as follows: Figure 5 As shown.

[0034] Figure 5 and Figure 6This paper compares the time delay characteristics of ideal signals, test signals, and nonlinear phase-frequency error pre-equalizers for both the low-frequency and high-frequency sub-channels. The time delay of the ideal signal should be constant; to accurately extract the time delay characteristics of the pre-equalizer, the ideal time delay is set to the maximum value of the test time delay characteristics. In contrast, the test signal is affected by nonlinear phase-frequency errors, exhibiting fluctuating time delay characteristics with frequency. By calculating the difference between the ideal and test time delays, the response curve of the nonlinear phase-frequency error pre-equalizer is constructed. This curve visually reflects the compensation effect of the pre-equalizer at different frequencies, providing a basis for further optimization of the correction strategy.

[0035] To verify the fitting performance of the proposed BO-Adabound algorithm on the group delay characteristics of the nonlinear phase-frequency error pre-equalizer at different orders, this embodiment selects the fitting results of 8th and 16th orders for comparative analysis. The fitting effects of the low-frequency sub-path and the high-frequency sub-path are shown respectively. The comparison methods include Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), Simulated Annealing (SA), and the AdaBound adaptive optimization method. The comparison mainly focuses on commonly used heuristic intelligent optimization algorithms and the mathematical methods used before optimization, and error analysis is performed with the target black dotted line to evaluate the fitting performance of different optimization methods at different orders. The pre-equalizer fitting effect for the low-frequency sub-path is shown below. Figure 7 and Figure 8 As shown, the pre-equalizer fitting effect for the high-frequency sub-path is as follows: Figure 9 and Figure 10 As shown.

[0036] Root Figures 7-10 The fitting results show that, under both 8th and 16th order filter structures, the BO-AdaBound algorithm can achieve highly accurate group delay responses within the target frequency band of both low-frequency and high-frequency sub-paths. Especially at the frequency endpoints and boundary transition regions, the error between the group delay curve fitted by BO-AdaBound and the target characteristic curve (black dotted line) is significantly smaller than other optimization methods, demonstrating stronger local fine-grained fitting capabilities. Furthermore, compared to the control algorithm which exhibits local oscillations or abrupt changes at certain frequency points, BO-AdaBound demonstrates superior performance in terms of overall smoothness and continuity of the group delay curve. In summary, the BO-AdaBound algorithm proposed in this invention shows significant advantages in fitting accuracy, full-band robustness, local detail fidelity, and adaptability to high-order structures, providing a more practical solution for high-precision pre-equalization correction of nonlinear phase-frequency errors.

[0037] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent substitutions, and improvements made to the above embodiments without departing from the scope of the present invention, based on the technical essence of the present invention and within the spirit and principles of the present invention, shall still fall within the protection scope of the present invention.

Claims

1. A phase frequency error correction method for a FI-DAC system based on BO-Adabound, characterized in that, include: Step 1: Calculate the initial objective function based on the input signal; Step 2: Calculate the initial integral adjustment factor based on the AdaBound algorithm The output of the initial objective function when the value is 1 is used to generate the initial dataset; Step 3: Select Gaussian process regression as the surrogate model, perform Bayesian iterative optimization on the initial dataset, obtain the posterior distribution based on the prior distribution of the initial dataset, and select the new evaluation point of the next integral adjustment factor in the global range by optimizing the search strategy based on the mean and variance of the posterior distribution through the acquisition function. Step 4: Obtain the corresponding new integral adjustment factor based on the new evaluation point of the integral adjustment factor, calculate and update the target group delay performance based on the new integral adjustment factor, obtain the initial parameters of Adabound by looking up the table and perform Adabound training. Step 5: In Adabound training, calculate the objective function and obtain its gradient performance, update the first and second moment estimates, update the solution parameters by pruning the learning rate, and iteratively update the objective function corresponding to the parameters. When the convergence condition or the maximum number of iterations is reached, return the current optimal filter parameters and their corresponding objective function. Update the dataset based on the optimal objective function and perform Bayesian loop optimization again until the global convergence condition is met or the maximum number of iterations is reached, and output the optimal all-pass filter parameters.

2. The phase frequency error correction method for a FI-DAC system based on BO-Adabound according to claim 1, characterized in that, Step 2 specifically includes: Set the initial integral adjustment factor The value is 1, combined with the current all-pass filter parameters. The output of the initial objective function is obtained. , ,in, Radius of the poles of the all-pass filter Phase angle with poles The combination of these factors, based on the output of the initial objective function. Get the initial dataset .

3. The phase frequency error correction method for a FI-DAC system based on BO-Adabound according to claim 1, characterized in that, Step 3 specifically includes: Step 3.1: Set the joint distribution of the initial dataset to conform to the condition of a multivariate normal distribution, and combine this with the initial dataset that conforms to a multivariate normal distribution to set the ideal training sample dataset. The ideal training sample dataset It follows a Gaussian prior distribution with a mean of 0 and a covariance matrix of K; Step 3.2: Based on the ideal training sample dataset Calculate new input posterior distribution Receive new input Normally distributed data at any data point, with probability enhancement applied, combined with new input. The optimal function value in the current dataset from the normal distribution number at any data point. Predicted mean and prediction variance Calculate the acquisition function, and select a new evaluation point for the next integral adjustment factor in the global scope based on the acquisition function; The expression for the joint distribution satisfying the condition of a multivariate normal distribution is: （1）； In formula (1), For Gaussian processes, It is a mean function. Let covariance function be used. The magnitude factor of the kernel function is used to control the overall output scale of the covariance function of the prior distribution of the Gaussian process. Let be a random variable that is an integral adjustment factor. The standard deviation of the random variable corresponding to the integral adjustment factor; The expression for the Gaussian prior distribution that the ideal training sample dataset follows is: （2）； In formula (2), Input to the training dataset, For training dataset The corresponding set of objective function values, For kernel function hyperparameters, The training dataset is obtained from the kernel function hyperparameters. The covariance matrix of its objective function value set.

4. The phase frequency error correction method for a FI-DAC system based on BO-Adabound according to claim 3, characterized in that, Step 3.2 specifically includes: Based on the ideal training sample dataset A Gaussian process is used to construct a joint multivariate normal distribution. Based on the conditions of the multivariate normal distribution, a new input is derived from the constructed joint multivariate normal distribution. The posterior probability distribution is used to obtain the new input. Normally distributed data at any data point; Improved application of probability, combined with new input The optimal function value in the current dataset from the normal distribution number at any data point. Predicted mean and prediction variance Calculate the acquisition function To predict the mean and prediction variance Construct a search strategy based on the collection function. Maximize the collection function values ​​of all candidate points to obtain new evaluation points for the next integral adjustment factor; The expression for the joint multivariate normal distribution is: （3）； In formula (3), For new input data Corresponding to the set of function values ​​to be predicted, For new input data With training dataset The covariance matrix between them For new input data With new input data The covariance matrix between them for The transpose of the matrix; The formula for calculating normally distributed data at any data point is: （4）； In formula (4), The set of function values ​​to be predicted The predicted mean, The set of function values ​​to be predicted The prediction variance is used to measure the uncertainty of the predicted value. Acquisition function The calculation formula is: （5）； In formula (5), The cumulative distribution function of the standard normal distribution. These are balancing parameters used to weigh exploration against exploitation. The optimal function value in the current dataset; The formula for calculating the new evaluation points of the next integral adjustment factor is as follows: （6）； In formula (6), For the entire input space, For the current dataset, For the current dataset The following acquisition function, This will serve as a new assessment point for the next step of the integral adjustment factor.

5. The phase frequency error correction method for a FI-DAC system based on BO-Adabound according to claim 1, characterized in that, In step 4, a new integral adjustment factor is obtained based on the new evaluation point. The target group delay performance is calculated and updated based on this new integral adjustment factor. The initial Adabound parameters are obtained using a lookup table method, including: New assessment points for integral adjustment factors Calculate the current linear delay performance to be compensated Based on the linear delay performance to be compensated Update target group latency Combined with the updated target group latency The initial parameters of Adabound are obtained by looking up a table. ; Current linear delay performance to be compensated The calculation formula is: （7）； In formula (7), and These represent the start and end points of the frequency sampling interval, indicating the lowest and highest frequencies, respectively. For the target group delay at frequency point The actual value at that location; Target group latency The update formula is: （8）。 6. The phase frequency error correction method for a FI-DAC system based on BO-Adabound according to claim 1, characterized in that, Step 5 specifically includes: Step 5.1: Obtain the current parameters by looking up the table. Calculate the objective function gradient Introducing the momentum factor and root mean square momentum factor The gradient of the objective function computed by the set. Update the first-order matrix estimate and the second-order matrix estimate; Step 5.2: Dynamically change the lower boundary function of the learning rate and the upper boundary function of the dynamically changing learning rate Learning rate after clipping Perform a correction update based on the corrected learning rate. The updated first-order matrix estimate and second-order matrix estimate are iteratively updated to solve for the objective parameters. Based on target parameters New assessment points for integral adjustment factors Update parameters corresponding to the objective function Continue until the global convergence condition is met or the maximum number of iterations is reached, then output the current optimal all-pass filter parameters. And obtain the optimal objective function. ; Step 5.3: Based on the optimal objective function New assessment points for integral adjustment factors Update the dataset and perform Bayesian loop optimization again until the global convergence condition is met or the maximum number of iterations is reached, and output the globally optimal all-pass filter parameters. The update formulas for first-order and second-order matrix estimates are as follows: （9）； In formula (9), For the first n The gradient of the objective function with respect to the parameters at the next iteration. The exponentially weighted moving average of the gradient represents the first moment estimate of the gradient. The exponentially weighted moving average of the squared gradient represents the second moment estimate of the gradient. Learning rate after cropping The corrected update expression is: （10）； In formula (10), For the first n The actual learning rate after pruning at the next iteration; The formula for updating the target parameter is: （11）； In formula (11), F For the feasible region or constraint set of parameters, Let represent the diagonal preconditioning matrix composed of the inverse estimates of the learning rate; The formula for calculating the optimal objective function is: （12）； The expression for updating the dataset is: （13）。

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