FI-DAC system nonlinear phase error calibration method based on GA-Dogleg-BFGS

Abstract

The invention provides an FI-DAC system nonlinear phase error calibration method based on GA-Dogleg-BFGS, belongs to the technical field of FI-DAC system phase error calibration, and aims to solve the problems that the existing multichannel FI-DAC system phase error calibration is difficult to adapt to multi-band synchronous correction of broadband signals and lacks flexible expansion capability, and the method comprises the following steps: S1, adding a multi-tone test signal in an FPGA, phase unwrapping processing is carried out on the collected phase data; s2, performing nonlinear phase correction on the second-order section cascade type all-pass filter, and calculating a phase correction error; s3, constructing a weighted nonlinear minimum mean square problem; s4, the weighted nonlinear minimum mean square problem is solved in a staged mode based on a GA-dog-BFGS algorithm; and S5, according to the solving result, the coefficient of the digital all-pass filter is re-issued to each sub-DAC channel for verification.

Description

TECHNICAL FIELD

[0001] The application relates to a FI-DAC system nonlinear phase error calibration method based on GA-Dogleg-BFGS, and belongs to the technical field of FI-DAC system phase error calibration. BACKGROUND

[0002] In the prior art, the phase mismatch between channels of a multi-channel FI-DAC system needs to be measured and adjusted manually frequently, consumes a large amount of manpower and time, and is difficult to adapt to multi-frequency band synchronous correction of a wideband signal, and needs to rely on a special hardware scheme which is expensive and not universal, thereby increasing the system cost and lacking flexible expansion capability, and failing to realize automatic, real-time and high-precision online phase calibration in a dynamic environment such as manufacturing tolerance and temperature drift, so that a low-cost, wideband and fast-deployable full-automatic multi-channel phase calibration technology is urgently needed. SUMMARY

[0003] The application is to solve the problems that the prior art multi-channel FI-DAC system phase error calibration is difficult to adapt to multi-frequency band synchronous correction of a wideband signal and lacks flexible expansion capability, and further provides a FI-DAC system nonlinear phase error calibration method based on GA-Dogleg-BFGS.

[0004] The technical scheme adopted by the application to solve the above problems is as follows: Step 1: adding a multi-tone test signal in an FPGA, connecting analog outputs of four sub-DACs to a high-bandwidth oscilloscope respectively and synchronously collecting phase values of each channel at each test frequency point, and performing phase unwrapping processing on the collected phase data; Step 2: performing nonlinear phase correction on a second-order notch all-pass filter, and calculating a phase correction error; Step 3: constructing an error vector based on the phase correction error, limiting a pole modulus of a digital all-pass filter to be less than 1, and constructing a weighted nonlinear least square problem in combination with the error vector; Step 4: adopting a GA-dogleg-BFGS algorithm to solve the weighted nonlinear least square problem in stages, in the GA algorithm stage, encoding parameters of the digital all-pass filter as a chromosome, constructing a solution vector to form an initial population matrix, and performing exploration in a global solution space through selection, crossover and mutation operations to obtain a high-quality initial solution, and in the dogleg-BFGS algorithm stage, performing fine search near the high-quality initial solution to obtain a phase offset value ; Step 5: inputting the phase offset value The coefficients are converted into digital all-pass filter coefficients and sent to each sub-DAC channel in the FPGA to generate the same multi-tone test signal as in step 1. The output phase of the four sub-DACs is measured repeatedly. If the maximum deviation between each sub-DAC is within the allowable range of the FI-DAC system, the nonlinear phase error calibration of the FI-DAC system is completed. If it is not within the allowable range of the FI-DAC system, steps 2-3 are repeated until the maximum deviation between each sub-DAC is within the allowable range of the FI-DAC system.

[0005] Furthermore, step 2 specifically includes: Step 2.1: Set the order P and poles of the second-order cascaded all-pass filter. The order P and poles of a second-order cascaded all-pass filter Obtain the transfer function of a second-order cascaded all-pass filter. ; Step 2.2: Set the p-th pole of the second-order cascaded all-pass filter. ,in, For radius, For phase, Let be the imaginary unit, and be the independent variable of the z-transform response. Substitute into the transfer function Obtain the amplitude-frequency response of a second-order cascaded all-pass filter. ; Step 2.3: Amplitude-frequency response based on a second-order cascaded all-pass filter and acquisition of phase data P Phase response function of an all-pass filter ; Step 2.4: Perform nonlinear phase correction on the second-order cascaded all-pass filter, combined with the phase response function. Obtain the phase frequency response of the compensation structure And construct the phase frequency response The conditions for satisfying the nonlinear part are based on the phase frequency response. Obtaining the phase correction error by satisfying the conditions of the nonlinear part ; Transfer function of a second-order cascaded all-pass filter The calculation formula is: (1); In formula (1), Let the filter order be . For the first An extreme point, Let z be the independent variable of the z-transform response; Amplitude-frequency response of a second-order cascaded all-pass filter The expression is: (2); In formula (2), is the passband frequency; P Phase response function of the all-pass filter The calculation formula of is: (3); Phase frequency response of the compensation structure The calculation formula of is: (4); In formula (4), is the linear time delay of the system, is the nonlinear phase error of the system; Phase frequency response The condition that the nonlinear part in satisfies is: (5); (6); In formulas (5) and (6), is the nonlinear phase response of the all-pass filter; Phase correction error The calculation formula of is: (7).

[0006] Further, step 3 specifically comprises: Step 3.1: dividing the passband frequency range of the first sub-band into parts, that is, based on the passband frequency of the first sub-band after the division and the phase correction error, an error vector is obtained. m Step 3.2: limiting the poles of the digital all-pass filter in the unit circle in the Z domain, the pole modulus m , and constructing a weighted nonlinear least square problem in combination with the error vector, wherein the solution of the weighted nonlinear least square problem is the phase offset value. The expression of the error vector is: Step 3.2: limiting the poles of the digital all-pass filter in the unit circle in the Z domain, the pole modulus , and constructing a weighted nonlinear least square problem in combination with the error vector, wherein the solution of the weighted nonlinear least square problem is the phase offset value. The expression of the error vector is: (8); In formula (8), is the radius of the first pole, is the radius of the second pole, is the phase of the first pole,​​​​​​​ the phase of the second pole, The expression of the weighted nonlinear least mean square problem is: (9); In formula (9), .

[0007] Further, the obtaining of the high-quality initial solution in step 4 specifically comprises: Step 4.1.1: taking the pole parameters of the all-pass filter as a chromosome to construct a solution vector , and forming an initial population matrix ; Step 4.1.2: constructing a target function based on the phase error corresponding to each chromosome in the population, taking the negative of the target function as the fitness of the chromosome in the population , and obtaining the fitness corresponding to each chromosome ; Step 4.1.3: judging whether the current optimal individual of the fitness meets any one of the three global search termination conditions, if yes, proceeding to the Dogleg algorithm stage, and taking the selected optimal individual as the starting point of the Dogleg-BFGS algorithm stage , if not, proceeding to step 4.1.4; Step 4.1.4: using the inverse sorting roulette method, combining the fitness corresponding to each chromosome and the numerical insurance to obtain the parent selection probability and performing inverse sorting roulette selection; Step 4.1.5: according to the probability , sampling the t-th generation individual several times with replacement, and writing the obtained index order into a one-dimensional sequence , taking the adjacent two items as a pair as the individual number of the parent population required by the crossover operator, and obtaining the corresponding parent vector ; Step 4.1.6: using the simulated binary crossover algorithm to perform crossover operation on the first dimension of the selected parent individual k , and applying Gaussian mutation to the offspring after crossover, evolving the parent population into the offspring population, repeating step 4.1.2 to obtain the optimal individual in the evolved offspring population , directly copying the optimal individual in the evolved offspring population to the next generation, and filling the remaining positions with offspring generated by crossover-variation, and judging whether it meets any one of the three global search termination conditions, if yes, proceeding to the Dogleg-BFGS algorithm stage, and taking the selected optimal individual as the starting point of the Dogleg-BFGS algorithm stage If not, repeat steps 4.1.4-4.1.6 until the termination condition of the global search is satisfied. solution vector The expression of the solution vector is: (10). initial population matrix The expression of the initial population matrix is: (11). In formula (11), N is the number of chromosomes. chromosome fitness The calculation formula of the chromosome fitness is: (12). In formula (12), f is the nonlinear phase error of the frequency point f, is the nonlinear phase response of the all-pass filter at the frequency point f, parent selection probability The calculation formula of the parent selection probability is: (13). In formula (13), f is the fitness of the i-th chromosome, is the fitness of the j-th chromosome, , indicates numerical insurance; parent vector The expression of the parent vector is: (14). parent individual The expression of the i-th dimension of the offspring of the parent individual is: k (15). (16). In formulas (15) and (16), , is used to control the compactness of the crossover, when the offspring basically surrounds the midpoint of the parent; when a long jump outside the parent interval can be generated; The expression of applying Gaussian mutation to the offspring after the crossover is: (17). (18). In formula (18), is the centroid of the offspring population obtained by the crossover,​​​​​ is a constant, positively correlated with chromosome fitness, for controlling the covariance update speed; The expression of the termination condition of the global search is: (19); In formula (19), is the first global search termination condition, is the second global search termination condition, is the third global search termination condition, , Can be set to 5% -10% of the engineering tolerance, is the maximum running time of the program, t is the program running time; Starting point of Dogleg algorithm The expression of is: (20).

[0008] Further, the fine search around the high-quality initial solution in step 4 includes: Step 4.2.1: Initial value of the Hessian matrix of the starting point of the Dogleg-BFGS algorithm is calculated, and the trust region radius is initialized; Step 4.2.2: Repeat step 4.1.2 to obtain the fitness of the current iteration point , set the gradient matrix of the current iteration point , according to the fitness of the current iteration point , the Hessian matrix value , the gradient matrix , and the trust region radius of the iteration point , obtain the quadratic approximation ; Step 4.2.3: Determine whether the current quadratic approximation satisfies the convergence condition of the Dogleg-BFGS algorithm, if it does, output the final solution of the weighted nonlinear least squares problem , if not, proceed to step 4.2.4, wherein the convergence condition of the Dogleg-BFGS algorithm is that the gradient and the trust region of the current round iteration point are both converged to a preset threshold; Step 4.2.4: According to the gradient matrix of the current iteration point , obtain the Cauchy step along the negative gradient Step 4.2.4: According to the gradient matrix of the current iteration point , obtain the Cauchy step along the negative gradient ​​, the gradient matrix at the current iteration point and the Hessian matrix value , the Newton step is calculated Step 4.2.5: If and is positive definite, it is determined whether is greater than the trust region radius , if it is greater, the Cauchy step is scaled to obtain the Cauchy step , and step 4.2.7 is performed, if is less than the trust region radius , then is introduced, and the final Cauchy step is calculated, and step 4.2.7 is performed Step 4.2.6: If and is not positive definite, then is introduced, and step 4.2.7 is performed Step 4.2.7: According to the Cauchy step , the gradient matrix , the actual reduction ratio is calculated, the threshold value , the threshold value , the scaling coefficient and the scaling coefficient are introduced, if , then the Cauchy step is rejected, the trust region radius is updated as , if , then the Cauchy step is accepted, the trust region radius is updated as , if , then the Cauchy step is accepted, the trust region radius is updated as ; Step 4.2.8: After accepting the Cauchy step , the next iteration point is selected according to the Cauchy step and the current iteration point , and and are calculated according to the current iteration point , the corresponding gradient matrix , the next iteration point and the corresponding gradient matrix ; Step 4.2.9: If the curvature condition is met, then the next iteration point Hessian matrix Update the settings; if the conditions are not met, introduce a damping coefficient. calculate and will Replace the calculation in step 4.2.8 ,make sure Maintain positive stillness and repeat steps 4.2.2-4.2.3; Initial value of Hessian matrix The calculation formula is: (twenty one); In formula (21), Starting point fitness T It is the transpose symbol; Initialize trust region radius The expression is: (twenty two); fitness of the current iteration point The calculation formula is: (twenty three); Second approximation The calculation formula is: (twenty four); In formula (24), , indicating about The gradient matrix, , indicating about The Hessian matrix; Cauchy step size along the negative gradient The calculation formula is: (25); In formula (25), Cauchy step size scaling factor; Newton's stride The calculation formula is: (26); Cauchy step length The calculation formula is: (27); The final formula for calculating the Cauchy step size is: (28); (29); (30); In formula (30), The final Cauchy step size; actual decline ratio The calculation formula is: (31); Next iteration point The calculation formula is: (32); and The calculation formula is: (33); In formula (33), To solve for the vector difference, The difference between the gradient vectors; Next iteration point Hessian matrix The calculation formula is: (34); Damping coefficient as well as The calculation formula is: (35); In formula (35), To introduce the gradient difference of the damping coefficient, and use replace Enter equation (34) to ensure that the updated Hessian matrix remains positive definite.

[0009] The beneficial effects of this invention are: 1. This invention achieves fully automated online phase calibration without the need for additional dedicated calibration hardware or manual intervention, reducing system costs, enhancing the flexibility and scalability of the calibration scheme, and significantly simplifying the calibration process and maintenance workload.

[0010] 2. This invention uses an intelligent optimization algorithm to quickly identify and compensate for static and dynamic phase deviations between channels, enabling synchronous calibration of multi-frequency components of broadband signals and achieving online real-time updates. This effectively resists the influence of measurement noise, device temperature drift, and manufacturing tolerances. Simultaneously, it balances global optimal search and local fine adjustment, significantly shortening convergence time and reducing computational resource consumption while ensuring extremely high calibration accuracy. This greatly improves the output consistency and system reliability of multi-channel DAC systems in fields such as communication, radar, and testing. Attached Figure Description

[0011] Figure 1 A schematic flowchart illustrating the nonlinear phase error calibration method for a FI-DAC system based on GA-Dogleg-BFGS provided by this invention; Figure 2 This is a flowchart illustrating the GA-Dogleg-BFGS algorithm provided by the present invention. Detailed Implementation

[0012] Combination Figure 1 and Figure 2 This implementation method is described as follows: Figure 1 As shown, the steps of the nonlinear phase error calibration method for the FI-DAC system based on GA-Dogleg-BFGS described in this embodiment include: S1: Add a multi-tone test signal to the FPGA and perform phase unwrapping processing on the acquired phase data; A wideband multi-tone test signal containing several sinusoidal components is generated in the FPGA, covering the entire operating bandwidth of the FI-DAC. The analog outputs of the four sub-DACs are connected to a high-bandwidth oscilloscope (or high-speed ADC), and the phase values ​​of each channel at each test frequency are acquired synchronously. The acquired phase data is then unwrapped to eliminate interference from cross-cycle transitions on subsequent fitting.

[0013] S2: Perform nonlinear phase correction on a second-order cascaded all-pass filter and calculate the phase correction error; The nonlinear phase compensation module consists of a digital all-pass filter (APF). APFs possess constant system gain and phase delay characteristics, making them widely used in various nonlinear phase correction applications. This characteristic allows the all-pass filter to only alter the phase frequency characteristics of the sub-band without affecting its amplitude frequency characteristics, making it suitable for sub-band nonlinear phase correction in FI-DAC systems. Since direct-type all-pass filters are sensitive to effective word length effects and relatively unstable, a second-order cascaded all-pass filter is selected for nonlinear phase error correction of the sub-band channels in this embodiment. The transfer function of the second-order cascaded all-pass filter is: (1); In formula (1), Let the filter order be . For the first An extreme point, Let z be the independent variable of the z-transform response; make , For radius, For phase. as well as Substituting into the equation, we obtain the amplitude-frequency response of the all-pass filter as follows: (2); In formula (2), is the passband frequency; Amplitude-frequency response based on second-order cascade all-pass filter and the acquired phase data P Phase response function of the all-pass filter : (3); After adding the all-pass filter for phase correction, the phase-frequency response of the compensation structure can be calculated as: (4); In formula (4), is the linear time delay of the system, is the nonlinear phase error of the system; The purpose of nonlinear phase correction is to make the subband have a linear phase-frequency response, so the nonlinear part in formula (4) should satisfy: (5); At this time: (6); In formulas (5) and (6), is the nonlinear phase response of the all-pass filter; But there will be errors in the actual fitting process, so the phase correction error can be represented as: (7).

[0014] S3: Construct a weighted nonlinear least square problem; Divide the passband frequency range of the mth subband into equal parts, denoted as Therefore, the error vector can be represented as: (8); In formula (8), is the radius of the first pole, is the radius of the second pole, is the phase of the first pole, is the phase of the second pole; In order to ensure the stability of the all-pass filter, each pole should be within the unit circle in the Z domain, so the modulus of all poles must be less than 1: (9); Then a weighted nonlinear least square (NLS) problem is defined: (10); In formula (10), .

[0015] S4: solving the weighted nonlinear least square problem in stages based on the GA-dogleg-BFGS algorithm; Gradient-based Newton iterative optimization algorithms are effective means for solving the NLS problem in formula (10), such as Gradient Descend Method (GD), Gauss Newton Method (GN), and Levenberg Marquardt Method (LM). Based on the above research, the present embodiment proposes a sub-band phase frequency response compensation technology for optimizing the design of APF based on the GA-dogleg algorithm, which uses the GA-dogleg algorithm for fine search based on the second-order node cascade structure of APF to realize the design of high-order APF.

[0016] The GA-Dogleg algorithm adopts a two-stage optimization framework, combining the complementary advantages of the two algorithms: Genetic Algorithm (GA) stage: simulating the natural evolution process, exploring the global solution space through selection, crossover, and mutation operations. The advantage is strong global search capability and not easy to fall into local optimum, but the convergence speed is slow and the accuracy is limited.

[0017] Dogleg stage: a local optimization algorithm based on the trust region method, which performs fine search near the high-quality initial solution provided by GA. The advantage is fast local convergence speed and high accuracy, but it is sensitive to the initial point. The specific steps are as shown in Figure 2 , including: S401: Genetic Algorithm Stage: Each chromosome is represented as a solution vector: (11); At this time, the population matrix is: (12); In formula (12), N is the number of chromosomes; Parameter setting: for example, the population size is about 60 individuals, the number of parents is about 30, the number of iterations is about 100 generations, the crossover probability is higher than 0.8, the mutation probability is about 1, and the heuristic selection uses SBX crossover and Gaussian mutation strategy.

[0018] In each generation, an adaptive function is needed to calculate the fitness of each chromosome in the population, so as to evaluate the quality of the chromosome. In the algorithm, the quality of the chromosome is related to the approximation accuracy of the phase response of the target system. The smaller the phase approximation error is, the better the quality of the chromosome is. Therefore, the target function can be mapped to the fitness by taking the opposite number, as shown in formulas (13) and (14): (13); (14); In formulas (13) and (14), is the nonlinear phase error of the frequency point , and is the nonlinear phase response of the all-pass filter at the frequency point ; The higher the adaptive function value is, the higher the quality of the chromosome is. At the same time, in order to suppress the “premature” phenomenon, the algorithm uses the inverse sorting roulette method for parent selection, and the probability distribution definition is shown in formula (15): (15); In formula (15), is the fitness of the i-th chromosome, is the fitness of the j-th chromosome, , and the expected parent vector satisfies the expected relationship and can be maintained by the expected relationship .

[0019] After completing the inverse sorting roulette selection, the algorithm performs resampling on the t-th generation individuals several times according to the probability , and writes the obtained index order into a one-dimensional sequence in turn. , as the parent number required by the crossover operator. The corresponding parent vector is denoted as: (16); Since the sampling is with replacement, the same number can appear repeatedly in different positions, which ensures that individuals with lower fitness still have a non-zero probability of entering the mating pool, thereby maintaining genetic diversity.

[0020] After evaluating the chromosomes of the initial population, the population needs to be evolved and upgraded according to the evaluation of the chromosomes. The evolution strategy of the population is the core of GA, and the evolution strategy defines how the population evolves from the parent population to the child population, and also determines the convergence speed and solution quality of GA. In the evolution stage, the simulated binary crossover (SBX) algorithm is selected in the embodiment. Assuming that the parent individual whose k-th dimension offspring is derived from equation (17) and equation (18): (17); (18); In equation (17) and (18), , for controlling the compactness of the crossover, when the offspring basically revolves around the midpoint of the parent; when a long jump outside the parent's interval can be generated, which helps to cross the saddle point or shallow local minimum.

[0021] After crossover, the offspring is subjected to Gaussian mutation again: (19); (20); In equation (20), is the centroid of the offspring population obtained by crossover, is a constant, which is positively correlated with the fitness of the chromosome, for controlling the speed of covariance update; In order to ensure the monotonic improvement of the optimal solution, this algorithm will copy the current optimal individual directly to the next generation, and the remaining positions are filled by the offspring generated by crossover-mutation. The stopping conditions of the global stage are comprehensive consideration of space contraction, target flatness and iteration upper limit: (21); In equation (21), is the first global search termination condition, is the second global search termination condition, is the third global search termination condition, , can be set to 5%–10% of the engineering tolerance, is the maximum running time of the program, t is the program running time; When any condition of equation (21) is met, output the global stage optimal individual and the corresponding Hessian approximation matrix and use it as the starting point of local refinement: (22).

[0022] S402: Dogleg-BFGS algorithm stage; The initial value of the Hessian matrix of is calculated, and the trust region radius is initialized: (23); (24); In formulas (23) and (24), is the starting point of the adaptive step size, T is the transpose symbol; In the local precise search phase, the dogleg algorithm constructs a quadratic approximation around the current iteration point of each round: (25); (26); In formula (26), denotes the gradient matrix with respect to , denotes the Hessian matrix with respect to ; It is judged whether the current quadratic approximation satisfies the convergence condition of the Dogleg-BFGS algorithm: the gradient of the current round iteration point and the trust region are both converged to the preset threshold, if satisfied, the final solution of the weighted nonlinear least squares problem is output , if not satisfied, the following steps are performed.

[0023] Compared with the traditional line search, the trust region method directly constrains the step size in the Euclidean sphere, which can explicitly control the model extrapolation error, especially when the Hessian matrix is singular or non-positive definite. In order to avoid additional one-dimensional search on (26), Dogleg first calculates two typical directions: one is the Cauchy step along the negative gradient: (27); In formula (27), is the Cauchy step length scaling factor; The second is the unconstrained Newton step: (28); If and is positive definite, then is the optimal solution of the subproblem, at this time: .

[0024] If or is not positive definite, it needs to be modified through a dogleg path. At this time, Dogleg connects the above two steps end to end to form a segmented dogleg path: (29); When has exceeded the trust region, it can be scaled to obtain : (30); Otherwise, we need to solve the intersection of the sphere and the second segment. Let be the intersection point, then we can expand and arrange it as a quadratic equation, i.e. (31); Its explicit root is (32); At this time , the final step size is (33); Geometrically, is the intersection point of the line connecting the Cauchy point and the Newton point and the sphere of the trust region.

[0025] In order to determine the reliability of the quadratic model for the original function, the algorithm introduces the actual reduction ratio to determine whether to adopt the step size: (34); Where, measures the prediction accuracy of the quadratic model for the true reduction. The threshold value , and the scaling coefficient , , the step size update rule is shown in Table 1: Table 1

[0026] This "three-color light" strategy realizes adaptive balance between convergence rate and robustness. Accept after this time: (35); In order to continuously improve the quality of the Hessian matrix, the algorithm uses a BFGS update strategy with damping: let (36); In formula (36), is the difference value of the solution vector, is the difference value of the gradient vector; If the curvature condition is satisfied, the Hessian is updated: (37); Otherwise, introduce the damping coefficient , at this time (38); In formula (38), is the gradient difference value with the introduction of the damping coefficient, and use substitute Enter formula (34) to ensure that the updated Hessian matrix remains positive definite and the quadratic approximation of the next iteration point And re-determine whether the convergence condition is met.

[0027] In summary, the present application quickly identifies and compensates for static and dynamic phase deviations between channels through intelligent optimization algorithms, can synchronize and calibrate the multi-frequency components of wideband signals, realizes online real-time updating, effectively resists the error influence caused by measurement noise, device temperature drift and manufacturing tolerance; At the same time, global optimal search and local fine adjustment are taken into account, which can significantly shorten the convergence time and reduce the calculation resource consumption while ensuring high accuracy, thereby greatly improving the output consistency and system reliability of multi-channel DAC system in the fields of communication, radar, testing, etc.

[0028] S5: The solution result is used to reissue the coefficient of the digital all-pass filter to each sub-DAC channel for verification; The final phase offset value is converted into the coefficient of the digital all-pass filter, which is issued to each sub-DAC channel in FPGA or DSP. The same multi-tone test signal is generated again, and the relative phase deviation between channels is verified by repeating the measurement of the four-way output phase. If the maximum deviation is within the system allowable range, the calibration is completed.

[0029] The above is only a preferred embodiment of the present application, and does not limit the present application in any form. Although the present application has been disclosed as above, it is not intended to limit the present application. Any skilled person in the art can make some changes or modifications to the above disclosed technical content without departing from the scope of the technical solution of the present application, and any simple modification, equivalent replacement and improvement of the above embodiments within the scope of the technical solution of the present application, according to the technical essence of the present application, within the spirit and principles of the present application, are all within the protection scope of the present application.

Claims

1. A nonlinear phase error calibration method for a FI-DAC system based on GA-Dogleg-BFGS, characterized in that, include: Step 1: Add a multi-tone test signal to the FPGA, connect the analog outputs of the four sub-DACs to a high-bandwidth oscilloscope and synchronously acquire the phase value of each channel at each test frequency point, and perform phase unwrapping processing on the acquired phase data. Step 2: Perform nonlinear phase correction on the second-order cascaded all-pass filter and calculate the phase correction error; Step 3: Construct an error vector based on the phase correction error, restrict the pole magnitude of the digital all-pass filter to be less than 1, and construct a weighted nonlinear minimum mean square problem in combination with the error vector; Step 4: The GA-dogleg-BFGS algorithm is used to solve the weighted nonlinear least mean square problem in stages. In the GA algorithm stage, the parameter encoding of the digital all-pass filter is used as chromosomes to construct solution vectors and form an initial population matrix. Selection, crossover, and mutation operations are used to explore the global solution space to obtain high-quality initial solutions. In the dogleg-BFGS algorithm stage, a fine-grained search is performed near the high-quality initial solutions to obtain the phase shift values. ; Step 5: Set the phase offset value The coefficients are converted into digital all-pass filter coefficients and sent to each sub-DAC channel in the FPGA to generate the same multi-tone test signal as in step 1. The output phase of the four sub-DACs is measured repeatedly. If the maximum deviation between each sub-DAC is within the allowable range of the FI-DAC system, the nonlinear phase error calibration of the FI-DAC system is completed. If it is not within the allowable range of the FI-DAC system, steps 2-4 are repeated until the maximum deviation between each sub-DAC is within the allowable range of the FI-DAC system.

2. The nonlinear phase error calibration method for a FI-DAC system based on GA-Dogleg-BFGS according to claim 1, characterized in that, Step 2 specifically includes: Step 2.1: Set the order P and poles of the second-order cascaded all-pass filter. The order P and poles of a second-order cascaded all-pass filter Obtain the transfer function of a second-order cascaded all-pass filter. ; Step 2.2: Set the p-th pole of the second-order cascaded all-pass filter. ,in, For radius, For phase, Let be the imaginary unit, and be the independent variable corresponding to the z-transform response. Substitute into the transfer function Obtain the amplitude-frequency response of a second-order cascaded all-pass filter. ; Step 2.3: Amplitude-frequency response based on a second-order cascaded all-pass filter and acquisition of phase data P Phase response function of an all-pass filter ; Step 2.4: Perform nonlinear phase correction on the second-order cascaded all-pass filter, combined with the phase response function. Obtain the phase frequency response of the compensation structure And construct the phase frequency response The conditions for satisfying the nonlinear part are based on the phase frequency response. Obtaining the phase correction error by satisfying the conditions of the nonlinear part ; Transfer function of a second-order cascaded all-pass filter The calculation formula is: （1）； In formula (1), Let the filter order be . For the first An extreme point, Let z be the independent variable of the z-transform response; The amplitude-frequency response of a second-order cascaded all-pass filter The expression is: （2）； In formula (2), The passband frequency; P Phase response function of an all-pass filter The calculation formula is: （3）； Phase frequency response of the compensation structure The calculation formula is: （4）； In formula (4), For the system's linear time delay, This refers to the nonlinear phase error of the system. Phase frequency response The condition for satisfying the nonlinear part is: （5）； （6）； In formulas (5) and (6), This represents the nonlinear phase response of the all-pass filter. Phase correction error The calculation formula is: （7）。 3. The nonlinear phase error calibration method for a FI-DAC system based on GA-Dogleg-BFGS according to claim 1, characterized in that, Step 3 specifically includes: Step 3.1: Place the first m passband frequency of the path scope All divided into A portion, that is Based on the partitioned first m passband frequency of the path and phase correction error Obtain the error vector ; Step 3.2: Restrict the poles of the digital all-pass filter to be inside the unit circle in the Z-domain, and limit the pole magnitudes. And combine the error vector to construct a weighted nonlinear minimum mean square problem, where the solution of the weighted nonlinear minimum mean square problem is the phase offset value; Error vector The expression is: （8）； In formula (8), The radius of the first pole is... The radius of the second pole. The phase of the first pole, The phase of the second pole, The expression for the weighted nonlinear least mean square problem is: （9）； Formula (9) .

4. The nonlinear phase error calibration method for a FI-DAC system based on GA-Dogleg-BFGS according to claim 1, characterized in that, Step 4, obtaining a high-quality initial solution, specifically includes: Step 4.1.1: Construct the solution vector using the pole parameters of the all-pass filter as a chromosome. To form the initial population matrix ; Step 4.1.2: Construct an objective function based on the phase error corresponding to each chromosome in the population, and map the inverse of the objective function to the fitness of the chromosomes in the population. And obtain the fitness of each chromosome. ; Step 4.1.3: Determine whether the currently fitness-optimal individual satisfies any of the three global search termination conditions. If so, proceed to the Doggleg algorithm phase, and use the selected optimal individual as the starting point of the Doggleg–BFGS algorithm phase. If the conditions are not met, proceed to step 4.1.4; Step 4.1.4: Use the reverse sorting roulette wheel method, combined with the fitness of each chromosome and numerical safety. Obtain the parent selection probability And perform a reverse sorting roulette selection; Step 4.1.5: According to probability For the t-th generation individuals, sample with replacement several times, and write the obtained indexes in order as a one-dimensional sequence. The individuals in the parent population required for the crossover operator are selected by pairing adjacent pairs, and the corresponding parent vectors are obtained. ; Step 4.1.6: Use the simulated binary crossover algorithm to crossover the selected parent individuals. The k Perform a crossover operation on the offspring, and apply Gaussian mutation to the offspring after crossover to evolve the parent population into the offspring population. Repeat step 4.1.2 to obtain the best individual in the evolved offspring population. The best individual in the offspring population after evolution Copy directly to the next generation, the rest Each position is filled with offspring generated by crossover-mutation, and it is determined whether any of the three global search termination conditions are met. If so, the Doglg-BFGS algorithm phase is initiated, and the selected optimal individual is taken as the starting point of the Doglg-BFGS algorithm. If not, repeat steps 4.1.4-4.1.6 until the termination condition of the global search is met; Solution vector The expression is: （10）； Initial population matrix The expression is: （11）； In formula (11), N is the number of chromosomes; Chromosome fitness The calculation formula is: （12）； In formula (12), For frequency point Nonlinear phase error, For the all-pass filter at frequency point The nonlinear phase response; Parent selection probability The calculation formula is: （13）； In formula (13), Let be the fitness of the i-th chromosome. Let be the fitness of the j-th chromosome. , indicating numerical insurance; Parent vector The expression is: （14）； Parental individuals The k The expression for the child is: （15）； （16）； In formulas (15) and (16), , Used to control the compactness of the crossover. The offspring generation basically revolves around the midpoint of the parent generation; It can generate long jumps that cross the parent's interval; The expression for applying Gaussian mutation to the offspring after crossover is: （17）； （18）； In formula (18), The centroid of the offspring population obtained by crossover. It is a constant and positively correlated with chromosome fitness. Used to control the rate of covariance update; The expression for the termination condition of the global search is: （19）； In formula (19), This is the first global search termination condition. This serves as the second global search termination condition. This is the third global search termination condition. , It can be set to 5%–10% of the engineering tolerance. The maximum program execution time. t This refers to the program's runtime. The starting point of the Dogleg algorithm The expression is: （20）。 5. The nonlinear phase error calibration method for a FI-DAC system based on GA-Dogleg-BFGS according to claim 4, characterized in that, Step 4 involves a refined search around a high-quality initial solution, specifically including: Step 4.2.1: Starting point of the Doglg–BFGS algorithm Initialize the Hessian matrix Calculate and initialize the trust region radius. ; Step 4.2.2: Repeat step 4.1.2 to obtain the fitness of the current iteration point. Set the current iteration point gradient matrix Based on the current iteration point fitness Hessian matrix values gradient matrix and iteration points Trust region radius Obtaining a second approximation ; Step 4.2.3: Determine the current quadratic approximation Does the Dogleg–BFGS algorithm meet its convergence criteria? If so, output the final solution to the weighted nonlinear least mean square problem. If not satisfied, proceed to step 4.2.4, where the convergence condition of the Doglg–BFGS algorithm is that the gradient and trust region of the current iteration point converge to a preset threshold. Step 4.2.4: Based on the gradient matrix of the current iteration point Obtain the Cauchy step size along the negative gradient Based on the gradient matrix of the current iteration point and Hessian matrix values Calculate Newton step size ; Step 4.2.5: If and If positive definite, then determine Is it greater than the trust region radius? If it is greater than, then adjust the Cauchy step size. The Cauchy step size is obtained by scaling proportionally. And proceed to step 4.2.7, if Smaller than the trust region radius Then introduce , and calculate the final Cauchy step size, and proceed to step 4.2.7; Step 4.2.6: If and If the positive definiteness does not hold, then let Then proceed to step 4.2.7; Step 4.2.7: Based on the Cauchy step size gradient matrix Calculate the actual decrease ratio Introducing a threshold Threshold Scaling factor and scaling factor ,like Then reject Cauchy step size Trust region radius updated to ,like Then accept Cauchy step size Trust region radius updated to ,like Then accept Cauchy step size Trust region radius updated to ; Step 4.2.8: Accept Cauchy step size Then, based on the Cauchy step size and the current iteration point Select the next iteration point Based on the current iteration point and the corresponding gradient matrix Next iteration point and the corresponding gradient matrix calculate and ; Step 4.2.9: If the curvature condition is satisfied Then for the next iteration point Hessian matrix Update the settings; if the conditions are not met, introduce a damping coefficient. calculate and will Replace the calculation in step 4.2.8 ,make sure Maintain positive stillness and repeat steps 4.2.2-4.2.3; Initial value of Hessian matrix The calculation formula is: （21）； In formula (21), Starting point fitness T It is the transpose symbol; Initialize trust region radius The expression is: （22）； fitness of the current iteration point The calculation formula is: （23）； Second approximation The calculation formula is: （24）； In formula (24), , indicating about The gradient matrix, , indicating about The Hessian matrix; Cauchy step size along the negative gradient The calculation formula is: （25）； In formula (25), Cauchy step size scaling factor; Newton's stride The plan T For transpose: （26）； Cauchy step length The calculation formula is: （27）； The final formula for calculating the Cauchy step size is: （28）； （29）； （30）； In formula (30), The final Cauchy step size; actual decline ratio The calculation formula is: （31）； Next iteration point The calculation formula is: （32）； and The calculation formula is: （33）； In formula (33), To solve for the vector difference, The difference between the gradient vectors; Next iteration point Hessian matrix The calculation formula is: （34）； Otherwise, a damping coefficient is introduced. as well as The calculation formula is: （35）； In formula (35), To introduce the gradient difference of the damping coefficient, and use replace Enter equation (34) to ensure that the updated Hessian matrix remains positive definite.

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