ZOH characteristic frequency response compensation optimization method for TI-DAC system based on CMA-ES

Abstract

The invention provides a CMA-ES-based TI-DAC system ZOH characteristic frequency response compensation optimization method, belongs to the technical field of filter optimization, and aims to solve the problems that a traditional pre-compensation filter is designed for correction in a digital domain, the global search capability is weak, local extremum is likely to be trapped in, and the correction efficiency is low. The method comprises the following steps: S1, constructing a pre-compensation filter; S2, constructing a pre-compensation filter; s2, a filter is pre-compensated based on a CMA-ES method; s3, convolution is carried out on the input signal and a compensation filter, and compensation is carried out on the convolved digital signal in a digital domain through a pre-coding filter, so that sinc attenuation caused by ZOH is effectively compensated, and the problem of frequency response distortion in a TI-DAC system is solved; according to the method, the CMA-ES method is introduced, the coefficient of the FIR filter is optimized within the constraint frequency band range, and then convolution is carried out on the input signal and the compensation filter, so that sinc attenuation caused by ZOH is effectively compensated, and the problem of frequency response distortion in a TI-DAC system is solved.

Description

Technical Field

[0001] The present invention relates to a method for frequency response compensation optimization of ZOH characteristics of a TI-DAC system based on CMA-ES, and belongs to the technical field of filter optimization. Background Art

[0002] As applications such as broadband communications, radar imaging, and electronic testing continue to demand higher-frequency, high-speed, and high-precision signal sources, the performance requirements for arbitrary waveform generators (AWGs), core signal synthesis devices, are also rising. As a key component of an AWG system, the sampling rate and signal fidelity of the digital-to-analog converter (DAC) directly determine the system's spectrum quality and modulation capabilities.

[0003] Due to current limitations in semiconductor processes and circuit design, increasing the sampling rate of a single-channel DAC has reached a bottleneck. However, time-interleaved DAC (TI-DAC) technology, through parallel interleaved sampling of multiple DAC channels, can effectively overcome the single-channel sampling rate limitation and double the system sampling rate. However, the TI-DAC architecture inherently relies on the DAC's zero-order hold (ZOH) output signal, which exhibits a sinc-shaped frequency response in the frequency domain. This limits the analog output bandwidth, resulting in a non-flat amplitude-frequency response of the TI-DAC output signal and, in particular, severe attenuation at high frequencies, significantly reducing the output signal fidelity. Currently, the most common approach to addressing this amplitude-frequency attenuation in TI-DAC systems is to design a pre-compensation filter in the digital domain. However, traditional designs often struggle to achieve accurate high-frequency compensation while maintaining low hardware resource consumption. Furthermore, existing research lacks a quantitative analysis of how the ZOH frequency response affects the frequency response flatness of TI-DAC systems. Summary of the Invention

[0004] In order to solve the problems of traditional pre-compensation filters designed in the digital domain for correction, such as weak global search capability, easy falling into local extreme values, and difficulty in achieving accurate compensation of high-frequency response while maintaining low hardware resource consumption, the present invention proposes a TI-DAC system ZOH characteristic frequency response compensation optimization method based on CMA-ES.

[0005] The technical solution adopted by the present invention to solve the above problems is: the present invention comprises the following steps:

[0006] Step 1: Build a pre-compensation filter;

[0007] Step 2: Set the optimization goal to minimize the mean square error between the actual frequency response and the ideal frequency response, and calculate the cosine weighting coefficient coe of the pre-compensation filter based on the CMA-ES method. k Optimize and use the optimized cosine weighting coefficient coe kGet the optimized coefficient f(n) of the pre-compensation filter;

[0008] Step 3: Convolve the input digital signal with the optimized pre-compensation filter. Compensate the convolved digital signal in the digital domain using the precoding filter. Extract the compensated digital signal and send it to multiple sub-DACs to achieve parallel digital-to-analog conversion. Interleave the analog outputs of all sub-DACs in a time sequence to obtain the output signal of the TI-DAC system after ZOH frequency response compensation.

[0009] Furthermore, the transfer function of the pre-compensation filter constructed in step 1 is F(z), the order N of the compensation filter is an even number and satisfies the symmetry condition f(n)=f(Nn), and the frequency response of the pre-compensation filter is F(e jω ), the amplitude-frequency response is A(w), and the cosine weighting coefficient is coe k , the mean square error of the frequency response of the ideal compensation filter is J MSE ;

[0010] The expression of the transfer function F(z) is:

[0011]

[0012] In formula (1), f(n) is the pre-compensation filter coefficient;

[0013] The frequency response of the pre-compensation filter F(e jω ) is:

[0014]

[0015] In formula (2), N is an even number, indicating the order of the precompensation filter;

[0016] Let the weighting coefficient be coe k ,but The expression of the amplitude-frequency response A(w) is:

[0017]

[0018] Mean square error J MSE The expression is:

[0019]

[0020] In formula (4), L is the number of discrete frequency points, and I(w) is the frequency response of the ideal compensation filter.

[0021] Furthermore, step 2 specifically includes:

[0022] Step 2.1: Initialize the multivariate normal distribution N(μ,σ 2C) and determine the input parameter system sampling rate f s , filter order N, ideal frequency response I(w) and minimum mean square error ε, where μ is the current mean vector, representing the search center position; σ>0 is the scale range of the global search, that is, the step size, and C is the covariance matrix, used to control the search direction;

[0023] Step 2.2: Set the maximum number of iterations MaxIter and the population size PopSize in the iterative optimization process, and generate the candidate solution population coe through each iterative optimization k ~N(μ,σ 2 ,C), the weighted coefficient coe in the candidate solution population k Mapped to a symmetrical filter coefficient f(n), the actual frequency response A(w) is calculated based on the filter coefficient f(n) and formula (3), and the mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) is minimized. MSE And as the fitness value;

[0024] Step 2.3: Sort by fitness, update the weighted mean μ, and adjust the covariance matrix C to capture the weighted coefficient coe k The nonlinear relationship between them is considered, and the step size σ is adaptively adjusted to balance the global search and local refinement, and the optimization coefficient f(n) is output, so that the constructed compensation filter frequency response A(w) approaches the ideal compensation filter frequency response I(w).

[0025] Furthermore, in step 2.2, the mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) is minimized. MSE The steps include:

[0026] The weighted coefficient coe in the candidate solution population k Map the filter coefficient f(n) to a symmetrical value, and determine whether f(n) is equal to f(Nn). If so, calculate the actual frequency response A(w) based on the filter coefficient f(n) and formula (3), and calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w). MSE If it is not equal, let f(n)←f(n)+flip·f(n) / 2, and calculate the actual frequency response A(w) according to the filter coefficient f(n) and formula (3), and calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) MSE .

[0027] Furthermore, the step of outputting the optimization coefficient f(n) in step 2.3 includes:

[0028] Based on the fitness J calculated in the current iteration MSE Update the multivariate normal distribution N(μ,σ 2distribution parameters μ, σ and C in C), and determining whether an iteration termination condition is met, the iteration termination condition being that a maximum iteration number MaxIter is reached or |J MSE |≤ε, if the iteration termination condition is met, obtaining the optimized coefficient f(n) according to the distribution parameters μ, σ and C, and if the iteration termination condition is not met, repeating steps 2.2-2.3 for iteration update until the iteration termination condition is met.

[0029] Further, step 3 specifically comprises:

[0030] convolving the digital signal x(n) with the optimized pre-compensation filter, wherein the digital signal x(n) is a sampling sequence of the analog signal x(t) with a sampling frequency of Mfs, and performing compensation on the convolved digital signal x(n) in the digital domain through a pre-coding filter, performing M times decimation on the compensated digital signal, and then inputting the digital signal into M sub-DACs for parallel digital-to-analog conversion, and superimposing analog outputs of all sub-DACs in time sequence to obtain a signal Y(f) output by the TI-DAC system after ZOH characteristic frequency response compensation, wherein an equivalent sampling period of the output signal is shortened to Ts / M, and the overall sampling rate of the TI-DAC system is increased from fs to Mfs;

[0031] The expression of the frequency response H(f) of the ZOH of a single sub-DAC is:

[0032] H(f)=T s e^{-jπfT s )sinc(fT s ) (5);

[0033] In formula (5),

[0034] The expression of the frequency response of the transfer function of the pre-coding filter is:

[0035]

[0036] Let z=e jw , The expression of the equivalent analog frequency response I(f) of the transfer function of the pre-coding filter is:

[0037]

[0038] In combination with formula (5) and formula (6), the frequency response of the TI-DAC system after pre-coding filtering is:

[0039]

[0040] The present application has the following beneficial effects:

[0041] 1. This paper introduces the CMA-ES method, namely the covariance matrix adaptive evolutionary strategy optimization method, to optimize the FIR filter coefficients within the constrained frequency band. It then convolves the input signal with the compensation filter, effectively compensating for the sinc attenuation caused by ZOH, solving the frequency response distortion problem in the TI-DAC system. This makes the system output signal spectrum flatter and achieves higher-precision signal processing. By minimizing the frequency response error or in-band fluctuation under the objective function, it achieves a higher-precision and higher-degree-of-freedom compensation design for the ZOH frequency response characteristics.

[0042] 2. This invention does not require gradient information when solving filter coefficients, has excellent global search capabilities, and can effectively avoid falling into local extreme values. The adaptive mechanism can dynamically adjust the search step size and direction, significantly improving the filter design accuracy and passband frequency response fluctuation, thereby effectively suppressing the amplitude-frequency distortion of the TI-DAC system output signal and improving the accuracy of the system output signal. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 A flow chart of the method for optimizing the ZOH frequency response compensation of the TI-DAC system based on CMA-ES provided by the present invention;

[0044] Figure 2 A schematic diagram of the process of optimizing the pre-compensation filter using the CMA-ES method provided by the present invention;

[0045] Figure 3 The structure and timing diagram of the M=4 channel TI-DAC system provided by the present invention;

[0046] Figure 4 Frequency response diagram of the two-channel TI-DAC system provided by the present invention;

[0047] Figure 5 A comparison chart of the frequency responses of the pre-compensation filters designed using the CMA-ES proposed in this invention and other methods;

[0048] Figure 6 Schematic diagram comparing the amplitude-frequency flatness of the output signal of the TI-DAC system using the CMA-ES proposed in the present invention and pre-compensation filters designed by other methods. DETAILED DESCRIPTION

[0049] Specific implementation method 1: Combination Figure 1-4 This embodiment is described as follows. Figure 1 As shown, the steps of the TI-DAC system ZOH characteristic frequency response compensation optimization method based on CMA-ES described in this embodiment include:

[0050] S1: build pre-compensation filter;

[0051] The pre-compensation filter constructed in the embodiment is shown in the Pre-filter module in Figure 3 The embodiment adopts a FIR filter to approximate the ideal compensation frequency response to realize a high-precision TI-DAC system, avoiding the use of an IIR filter to realize frequency response compensation, which may introduce a pole outside the unit circle (such as a pole at the π frequency point in a two-way TI-DAC system), resulting in a significant deterioration of system stability.

[0052] The transfer function thereof can be expressed as:

[0053]

[0054] If the filter order N is even and satisfies the symmetry condition f(n) = f(N-n), the frequency response expression of the FIR filter is obtained as:

[0055]

[0056] Let the weighting coefficient be coe k , then wherein k = 1, 2, …, N / 2. Therefore, the amplitude-frequency response of the filter can be expressed as:

[0057]

[0058] In formula (3), w ∈ [0, π]. Therefore, the expression of the mean square error is:

[0059]

[0060] In formula (4), L is the number of discrete frequency points, and I(w) is the ideal compensation filter frequency response.

[0061] S2: Optimizing the pre-compensation filter based on the CMA-ES method;

[0062] The CMA-ES method is a global optimization method without gradient, based on multivariate Gaussian distribution and only relying on the value of the objective function. Since the method has inherent invariance to translation, scaling and orthogonal transformation of variables, it can adapt to different optimization problems and reduce the dependence on problem characteristics. At the same time, it can automatically adjust the step size and covariance matrix without manual parameter tuning, and can adaptively balance global search and local search during optimization to improve optimization efficiency and accuracy. The core idea is to efficiently explore the solution space by dynamically adjusting the mean and covariance matrix of the search distribution. The key of the embodiment is to optimize the cosine weighting coefficient coe k based on formula (4) by using the CMA-ES method, and to set the optimization goal as minimizing the mean square error of the actual frequency response and the ideal frequency response. The process of optimizing the CMA-ES method is as follows Figure 2 As shown, the key mechanisms are as follows:

[0063] S201: Initialize multivariate normal distribution parameters and determine input parameters;

[0064] Initialize the multivariate normal distribution N(μ,σ 2 C) and determine the input parameter system sampling rate f s , filter order N, ideal frequency response I(w) and minimum mean square error ε, where μ is the current mean vector, representing the search center position; σ>0 is the scale range of the global search, that is, the step size, and C is the covariance matrix, used to control the search direction;

[0065] S202: Generate candidate population;

[0066] Set the maximum number of iterations MaxIter and the population size PopSize in the iterative optimization process, and generate the candidate solution population coe through each iterative optimization k ~N(μ,σ 2 ,C).

[0067] S203: Evaluate fitness;

[0068] The weighted coefficient coe in the candidate solution population k Map the filter coefficient f(n) to a symmetrical value, and determine whether f(n) is equal to f(Nn). If so, calculate the actual frequency response A(w) based on the filter coefficient f(n) and formula (3), and calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w). MSE If it is not equal, let f(n)←f(n)+flip·f(n) / 2, and calculate the actual frequency response A(w) according to the filter coefficient f(n) and formula (3), and calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) MSE And used as the fitness value.

[0069] S204: Update parameters. Based on fitness ranking, weighted update mean μ (approximate optimal coe k distribution center), adjust the covariance matrix C to capture the coe k The nonlinear relationship between them is calculated and the step size σ is adjusted adaptively to balance the global search and local refinement. It is determined whether the iteration termination condition is met. The iteration termination condition is to reach the maximum number of iterations MaxIter or |J MSE |≤ε, if the iteration termination condition is met, the optimization coefficient f(n) is obtained according to the distribution parameters μ, σ and C. If the iteration termination condition is not met, repeat steps 2.2-2.3 for iterative update until the iteration termination condition is met. This process is achieved by constraining coe kThe symmetry of the filter ensures that the filter satisfies the linear phase structure, and ultimately makes the designed compensation filter frequency response A(w) close to the ideal compensation filter frequency response I(w), achieving high-precision compensation of complex frequency responses.

[0070] The present invention introduces the CMA-ES method, namely the covariance matrix adaptive evolution strategy optimization method, to optimize the FIR filter coefficients within the constrained frequency band, and then convolve the input signal with the compensation filter, thereby effectively compensating for the sinc attenuation caused by the ZOH, solving the frequency response distortion problem in the TI-DAC system, making the signal spectrum output by the system flatter, and achieving higher-precision signal processing. By minimizing the frequency response error or in-band fluctuation under the objective function, a higher-precision and higher-degree-of-freedom compensation design for the ZOH frequency response characteristics is achieved.

[0071] S3: Performs convolution operations on the input digital signal in the digital domain through a precoding filter to effectively compensate for the sinc attenuation caused by ZOH and solve the frequency response distortion problem in the TI-DAC system.

[0072] like Figure 2 As shown, in this embodiment, the number of sub-DACs is set to 4, x(t) is set to be an ideal analog signal, and x(n) is set to be its ideal sampling sequence, and the sampling frequency is Mfs. After convolving the digital signal with the compensation filter, the impulse response of the sub-DAC zero-order hold is defined as h(t), which is a width of one sampling period T. s , a rectangular function with a height of 1, then the expression of its frequency response H(f) is:

[0073] H(f)=T s e^{-jπfT s}sinc(fT s ) (5);

[0074] In formula (5), Assume that the delay difference between two adjacent channels of the TI-DAC system is ΔT = Ts / M, then the time domain signal y output by the mth sub-DAC is m (t) is the convolution of the sampling sequence and the rectangular pulse, and its expression is:

[0075]

[0076] In formula (6), m=0,1,...,M-1. According to the Fourier property of convolution, the frequency domain signal Y output by the mth sub-DAC is m The expression of (f) is:

[0077]

[0078] According to the Poisson summation formula, we can Consider the paired sampling interval as T s Pulse sampling.

[0079]

[0080] In formula (8), X(f) is the Fourier transform of the input signal x(t). Therefore, (3) can be further simplified as:

[0081]

[0082] Therefore, the frequency response of the signal output by the M-channel TI-DAC system is the sum of the frequency responses of the DAC output signals of all sub-channels. The frequency response expression is:

[0083]

[0084] In formula (10), k = ±M, ±2M, .... As can be seen from the above, if the signals output by M phase-interleaved low-speed DACs are added together, the mirror components with k≠±M, ±2M, ... will cancel each other out, leaving only the frequency component at 2ifs±fin, which is equivalent to the system sampling at a high sampling rate (Mfs), where i = 0, 1, 2, ..., fin is the frequency of the input signal. Moreover, compared to a low-speed single DAC, after passing through the TI-DAC system, the nearest mirror image also moves from the frequency point fs-fin to the farther frequency point Mfs-fin, greatly reducing the difficulty of designing the subsequent analog low-pass filter. The frequency response diagram of the two-way TI-DAC system is shown below. Figure 3 shown.

[0085] This method eliminates the need for gradient information when solving filter coefficients, exhibits excellent global search capabilities, and effectively avoids falling into local extrema. Furthermore, its adaptive mechanism dynamically adjusts the search step size and direction, significantly improving both filter design accuracy and passband frequency response fluctuations. This effectively suppresses amplitude-frequency distortion in the TI-DAC system's output signal, improving the accuracy of the system's output signal.

[0086] Although the TI-DAC system can eliminate some image components, the attenuation characteristics of the sinc envelope cause the amplitude and frequency of the output signal within the system passband to be attenuated, and the flatness within the band is reduced. Therefore, to ensure the fidelity of the system output signal, this embodiment compensates the convolved signal in the digital domain. The frequency response expression of the transfer function of the precoding filter is:

[0087]

[0088] Let z = e jw , Then the equivalent analog frequency response I(f) expression of the precoding filter transfer function is:

[0089]

[0090] Therefore, the frequency response of the TI-DAC system output after precoding filtering is:

[0091]

[0092] Substituting (5) and (11) into (13) yields:

[0093] Specific implementation method 2

[0095] Combine Figure 4-6 This embodiment is explained. To verify the frequency response compensation optimization effect of the present invention, this embodiment compares and tests the frequency response compensation optimization method proposed by the present invention with the commonly used FIR filter design method. The comparison process includes: commonly used FIR filter design methods include frequency sampling method (Frequency Sampling), minimum maximum value method (Minimax), least squares method (LS), error back propagation method (BP) and differential evolution-error back propagation hybrid optimization (DE-BP) method. The performance of the pre-compensation filter designed by the CMA-ES method adopted in the present invention is significantly better than other methods, as shown in the following: Figure 4 shown.

[0096] Under the condition of the same filter order and a passband frequency range of 0.1 to 0.4 GHz, the CMA-ES method achieves the best fitting effect. The specific performance indicators are shown in Table 1.

[0097] Table 1

[0098]

[0099] Although the Minimax method achieved minimal ripple within the passband, at only 1.7dB, it also exhibited the largest mean square error (MSE) from the ideal frequency response (MSE = 0.0441), and the maximum stopband attenuation was -2.16dB, making it difficult to meet high rejection requirements. The CMA-ES optimization method, on the other hand, achieved the smallest mean square error (MSE = 0.0088) within the passband, reducing the MSE by 11.11% compared to the BP method, which achieved the best stopband attenuation. This method strikes a balance between passband flatness and stopband rejection, meeting the requirements for high-precision filtering. Furthermore, the precompensation filters designed by each method exhibited constant group delay, indicating that the designed precompensation filters possess linear phase characteristics.

[0100] The test includes: This implementation is tested on a two-channel TI-DAC system. The amplitude-frequency flatness of the system output signal within the passband is significantly better than other methods. The specific performance is as follows: Figure 5 shown.

[0101] Within the passband, the BP, DE-BP, and CMA-ES methods achieve the best fit to the input signal's frequency response, while the frequency sampling method falls significantly short. A detailed comparison of performance metrics is shown in Table 2.

[0102] Table 2

[0103]

[0104]

[0105] Under the same filter order, the Frequency Sampling method achieves the highest mean square error (MSE) of the output signal within the passband (MSE = 0.0820). The CMA-ES optimization method not only reduces the mean square error (MSE) to a global minimum of 0.0055 (a reduction of approximately 93.3% compared to the Frequency Sampling method), but also improves passband flatness by approximately 1.8 dB compared to the pre-compensated method, converging the amplitude-frequency fluctuation range to [–2.68 dB, 0 dB]. Compared to the BP method with the same amplitude-frequency fluctuation within the passband, its mean square error (MSE) is further reduced by 12.7%, demonstrating superior fitting accuracy. While DE-BP achieves slightly better fluctuation flatness (Amp-FreRange = 2.59 dB), its amplitude-frequency mean square error (MSE) is 0.0061, and it requires repeated fitness evaluation over thousands of generations and hundreds of dimensions. Therefore, its computational overhead is extremely high, making it unsuitable for real-time calibration scenarios.

[0106] The filter order is a key factor affecting the flatness of the passband amplitude-frequency response, so Figure 6 The amplitude-frequency response fluctuations of the compensated TI-DAC system under different filter orders are compared. When the filter order is between 20 and 40, the amplitude fluctuation is relatively smooth, with a maximum of about 3dB; when the order increases to 100, the fluctuation range can converge to about 2.5dB. As the filter order increases, although higher orders can further reduce the amplitude-frequency fluctuations, they will also significantly increase the computational complexity and implementation cost of the filter. Therefore, the optimal order range should be selected based on the balance between performance and efficiency according to the application scenario.

[0107] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical content disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement of the above embodiments made according to the technical essence of the present invention, within the spirit and principles of the present invention, without departing from the content of the technical solution of the present invention, shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. The TI-DAC system ZOH characteristic frequency response compensation optimization method based on CMA-ES is characterized by: include: Step 1: Build a pre-compensation filter; Step 2: Set the optimization goal to minimize the mean square error between the actual frequency response and the ideal frequency response, and calculate the cosine weighting coefficient coe of the pre-compensation filter based on the CMA-ES method. k Optimize and use the optimized cosine weighting coefficient coe k Get the optimized coefficient f(n) of the pre-compensation filter; Step 3: Convolve the input digital signal with the optimized pre-compensation filter. Compensate the convolved digital signal in the digital domain using the precoding filter. Extract the compensated digital signal and send it to multiple sub-DACs to achieve parallel digital-to-analog conversion. Interleave the analog outputs of all sub-DACs in a time sequence to obtain the output signal of the TI-DAC system after ZOH frequency response compensation.

2. The method for optimizing the ZOH frequency response compensation of the TI-DAC system based on CMA-ES according to claim 1, characterized in that: The transfer function of the pre-compensation filter constructed in step 1 is F(z), the order N of the compensation filter is an even number, and satisfies the symmetry condition f(n) = f(Nn), and the frequency response of the pre-compensation filter is F(e jω ), the amplitude-frequency response is A(w), and the cosine weighting coefficient is coe k , the mean square error of the frequency response of the ideal compensation filter is J MSE ; The expression of the transfer function F(z) is: In formula (1), f(n) is the pre-compensation filter coefficient; The frequency response of the pre-compensation filter F(e jω ) is: In formula (2), N is an even number, indicating the order of the precompensation filter; Let the weighting coefficient be coe k ,but The expression of the amplitude-frequency response A(w) is: Mean square error J MSE The expression is: In formula (4), L is the number of discrete frequency points, and I(w) is the frequency response of the ideal compensation filter.

3. The method for optimizing the ZOH frequency response compensation of the TI-DAC system based on CMA-ES according to claim 2, characterized in that: Step 2 specifically includes: Step 2.1: Initialize the multivariate normal distribution N(μ,σ 2 C) and determine the input parameter system sampling rate f s , filter order N, ideal frequency response I(w) and minimum mean square error ε, where μ is the current mean vector, representing the search center position, σ>0 is the scale range of the global search, that is, the step size, and C is the covariance matrix, which is used to control the search direction; Step 2.2: Set the maximum number of iterations MaxIter and the population size PopSize in the iterative optimization process, and generate the candidate solution population coe through each iterative optimization k ~N(μ,σ 2 ,C), the weighted coefficient coe in the candidate solution population k Mapped to a symmetrical filter coefficient f(n), the actual frequency response A(w) is calculated based on the filter coefficient f(n) and formula (3), and the mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) is minimized. MSE And as the fitness value; Step 2.3: Sort by fitness, update the weighted mean μ, and adjust the covariance matrix C to capture the weighted coefficient coe k The nonlinear relationship between them is considered, and the step size σ is adaptively adjusted to balance the global search and local refinement, and the optimization coefficient f(n) is output, so that the constructed compensation filter frequency response A(w) approaches the ideal compensation filter frequency response I(w).

4. The method for optimizing the ZOH frequency response compensation of the TI-DAC system based on CMA-ES according to claim 3, characterized in that: In step 2.2, calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) MSE The steps include: The weighted coefficient coe in the candidate solution population k Map the filter coefficient f(n) to a symmetrical value, and determine whether f(n) is equal to f(Nn). If so, calculate the actual frequency response A(w) based on the filter coefficient f(n) and formula (3), and calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w). MSE If it is not equal, let f(n)←f(n)+flip·f(n) / 2, and calculate the actual frequency response A(w) according to the filter coefficient f(n) and formula (3), and calculate the minimum mean square error J between the actual frequency response A(w) and the ideal frequency response I(w) MSE .

5. The method for optimizing the ZOH frequency response compensation of the TI-DAC system based on CMA-ES according to claim 3, characterized in that: The steps of outputting the optimization coefficient f(n) in step 2.3 include: Based on the fitness J calculated in the current iteration MSE Update the multivariate normal distribution N(μ,σ 2 C) in the distribution parameters μ, σ and C, to determine whether the iteration termination condition is met. The iteration termination condition is to reach the maximum number of iterations MaxIter or |J MSE |≤ε, if the iteration termination condition is met, the optimization coefficient f(n) is obtained according to the distribution parameters μ, σ and C. If the iteration termination condition is not met, steps 2.2-2.3 are repeated for iterative update until the iteration termination condition is met.

6. The method for optimizing the ZOH frequency response compensation of the TI-DAC system based on CMA-ES according to claim 1, characterized in that: Step 3 specifically includes: The digital signal x(n) is convolved with the optimized pre-compensation filter, where the digital signal x(n) is a sampling sequence of the analog signal x(t) with a sampling frequency of Mfs. The convolved digital signal x(n) is compensated in the digital domain by the precoding filter. The compensated digital signal is decimated M times and sent to M sub-DACs for parallel digital-to-analog conversion. The analog outputs of all sub-DACs are interleaved and superimposed in time sequence to obtain the output signal Y(f) of the TI-DAC system after ZOH frequency response compensation. The equivalent sampling period of the output signal is shortened to Ts / M, increasing the overall sampling rate of the TI-DAC system from fs to Mfs. The frequency response H(f) of ZOH of a single sub-DAC is expressed as: H(f)=T s e^{-jπfT s }sinc(fT s ) (5); In formula (5), The frequency response expression of the transfer function of the precoding filter is: Let z = e jw , The expression of the equivalent analog frequency response I(f) of the precoding filter transfer function is: Combining formula (5) and formula (6), the frequency response of the TI-DAC system output after precoding filtering is:

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