A modeling method for suppressing aliasing error of FI-DAC system

Abstract

The application discloses a modeling method for aliasing error suppression of an FI-DAC system and relates to a modeling method. In order to solve the problem that the prior art excessively depends on digital pre-calibration to compensate for system errors, which is not only high in calculation complexity, but also occupies a large amount of calculation resources, and is difficult to meet the real-time application requirements, the application comprises the following steps: step 1, a digital domain and analog domain joint modeling method is designed through systematic analysis of aliasing error of the FI-DAC system; step 2, a frequency matching iterative optimization algorithm combining ADMM and SQP algorithms is introduced to optimize key frequency parameters of the system, reduce resource consumption of digital domain error calibration, and improve the real-time performance of signal processing; step 3, under a distributed verification framework, transition from a module level to complete hardware design is completed, and channel energy distribution and impedance matching problems are optimized; and step 4, system complexity is reduced through integrated hardware design. The application belongs to the technical field of frequency band interlaced digital-to-analog conversion.

Description

Technical Field

[0001] This invention relates to a modeling method, belonging to the field of frequency band interleaving digital-to-analog conversion technology. Background Technology

[0002] Currently, frequency-interleaved digital-to-analog converter (FI-DAC) technology has been theoretically proven to overcome the physical limitations of DAC sampling rates, thereby significantly improving the output bandwidth of arbitrary waveform generators (AWGs). However, this technology still faces several key challenges in practical applications. First, due to the sampling characteristics of DACs and the non-ideal nature of analog mixers, severe aliasing errors are inevitably introduced into the output signal of FI-DAC systems. These errors include in-band signal overlap errors and out-of-band image signal leakage, directly affecting the quality and reliability of the output signal. Second, the modeling methods of existing FI-DAC systems largely rely on empirical design, failing to fully integrate the signal characteristics of the digital and analog domains, resulting in high complexity and poor real-time performance in error calibration. Furthermore, the hardware implementation of FI-DAC technology is still immature, with most research remaining at the simulation verification stage. When the hardware is implemented, problems such as impedance mismatch and uneven channel energy distribution cause system signal distortion, making it difficult to meet the high-precision output requirements of AWGs. Finally, existing technologies rely excessively on digital pre-calibration to compensate for system errors. This approach is not only computationally complex but also consumes a large amount of computing resources, making it difficult to meet the requirements of real-time applications. Summary of the Invention

[0003] To address the problem that existing technologies rely excessively on digital pre-calibration to compensate for system errors, which is not only computationally complex but also consumes a large amount of computing resources and is difficult to meet the requirements of real-time applications, this invention proposes a modeling method for aliasing error suppression in FI-DAC systems.

[0004] The technical solution adopted by the present invention to solve the above problems is as follows: The steps of the present invention include:

[0005] Step 1: Through systematic analysis of aliasing errors in the FI-DAC system, design a joint modeling method for the digital and analog domains;

[0006] Step 2: Introduce a frequency matching iterative optimization algorithm that combines ADMM and SQP algorithms to optimize the key frequency parameters of the system, reduce the resource consumption of digital domain error calibration, and thus improve the real-time performance of signal processing.

[0007] Step 3: Under the distributed verification framework, complete the transition from module level to complete hardware design, optimize channel energy distribution and impedance matching issues, and ensure high quality and high accuracy of system signal output;

[0008] Step 4: Reduce system complexity through integrated hardware design.

[0009] Furthermore, step 1 specifically includes:

[0010] An ideal digital mixer output includes upper and lower sideband signals, and the mixer output signal is:

[0011]

[0012] Select the signal with the input signal as the target down-conversion signal; in formula (1), A RF ω is the amplitude of the radio frequency signal. RF Indicates the frequency of radio frequency, A LO ω represents the amplitude of the local oscillator signal. LO Indicates the frequency of the local oscillator;

[0013] Both the digital signal and the local oscillator signal have a sampling frequency of f. s Due to signal sampling, the output spectrum is periodically based on the digital domain sampling frequency f. s To extend;

[0014] When the extended signal is greater than the down-converted signal;

[0015]

[0016] In formula (2), f represents the transition bandwidth of the digital low-pass filter H0. LO ω represents the local oscillator frequency, ω2 represents the high-frequency sub-channel termination frequency, i.e., the total output bandwidth of the FI-DAC system. H1 f represents the starting frequency of the FI-DAC high-frequency sub-channel. s This represents the digital domain sampling frequency, which is equal to M times the DAC sampling frequency f. sm And satisfy

[0017]

[0018] When the extended signal completely overlaps with the target down-conversion signal;

[0019] The extended signal caused by sampling completely overlaps with the obtained target down-conversion signal, at which point the following condition is met.

[0020] f s -f LO -ω2=f LO -ω2(4),

[0021] f s -f LO -ω H1 =f LO -ω H1 (5),

[0022] That is, the digital domain sampling frequency f s Equal to twice the local oscillator frequency f LO ;

[0023] Mf sm =2f LO (6),

[0024] In formula (6), M represents the downsampling factor, f sm DAC chip sampling frequency.

[0025] Furthermore, in step 2, the aliasing error is corrected by changing the modeling system and setting a guard band, which is used to place the filter transition band.

[0026] When the sampling characteristics of the DAC chip cause aliasing;

[0027] The stopband frequencies of low-pass filters LPF_L1 and LPF_H1 are lower than the first extension frequencies caused by DAC sampling of the low-frequency and high-frequency baseband signals output in the digital domain, respectively; the passband should encompass all frequencies of the baseband signal, i.e., the stopband and passband cutoff frequencies satisfy:

[0028]

[0029]

[0030] In formulas (7) and (8), Ω L1 This indicates the passband cutoff frequency of the low-pass filter LPF_L1 in the low-frequency sub-path. ω represents the transition band bandwidth of the low-pass filter LPF_L1 in the low-frequency sub-path. L1 This indicates the termination frequency of the low-frequency sub-path. Ω H1 This indicates the passband cutoff frequency of the low-pass filter LPF_H1 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter LPF_H1 in the high-frequency sub-path;

[0031] The restrictions on filter LPF_H1 in the high-frequency sub-path are not as stringent as those on filter LPF_L1 in the high-frequency sub-path, so LPF_L1 can be used instead.

[0032] When aliasing occurs due to the hardware characteristics of the mixer;

[0033] The high-frequency sub-channel signal is up-converted and restored using an actual analog mixer. This analog mixer is a nonlinear device, and compared to an ideal multiplier, the output signal includes both upper and lower sideband signals. For any dual-frequency excitation input nonlinear device, assuming A... IF ω represents the amplitude of the intermediate frequency signal. IF For the intermediate frequency, ALO ω represents the amplitude of the local oscillator signal. LO Let A be the frequency of the local oscillator. For ease of explanation, let's assume A is... IF =A LO =A;

[0034] v in1 =Acos(ω) IF t)

[0035] v in2 =Acos(ω) LO t)

[0036] v i =v in1 +v in2 =A[cos(ω IF t)+cos(ω LO t)](9),

[0037] Its output response is expressed as:

[0038]

[0039] In formulas (9) and (10), v in1 This represents the input signal at port 1 of the three-port network; in this paper, it is the input intermediate frequency signal, v. in2 This represents input port 2 of the three-port network, which in this paper is the input local oscillator signal, v. i This represents the combined input signal of port 1 and port 2, where t represents time and v represents the input signal. out The output signal of the port 3 nonlinear transform of the three-port network represents the signal. The output of a nonlinear system includes the fundamental frequency component, harmonic components, and intermodulation components introduced by the intermodulation between the input signals. k1 is a first-order coefficient representing the system's linear gain, the direct contribution of the fundamental frequency component of the input signal to the output. k2 is a second-order coefficient representing the system's second-order nonlinear response characteristics, including the second harmonic and intermodulation components. k3 is a third-order coefficient representing the system's third-order nonlinear response characteristics, including the third harmonic and higher-order intermodulation components.

[0040] The low-pass filter LPF_H2 and the high-pass filter HPF_H3 constitute a bandpass filter. Their cutoff frequencies should be less than the local oscillator leakage frequency caused by mixer non-ideality and the upper sideband image frequency obtained from up-conversion, respectively, and greater than the high-frequency baseband signal frequency. The passband should encompass all frequencies of the baseband signal; that is, the stopband and passband cutoff frequencies should satisfy the following:

[0041]

[0042] In formulas (11) and (12), Ω H2 This indicates the passband cutoff frequency of the low-pass filter LPF_H2 in the high-frequency sub-path. ω2 represents the transition bandwidth of the low-pass filter LPF_H2 in the high-frequency sub-path, and Ω represents the termination frequency of the high-frequency sub-path. H3 This indicates the passband cutoff frequency of the high-pass filter HPF_H3 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter HPF_H3 in the high-frequency sub-path.

[0043] Furthermore, the specific methods for using ADMM-SQP to solve for the system's key frequency points include:

[0044] The modeling process is transformed into an optimization problem, namely, modeling a FI-DAC system that maximizes output bandwidth while satisfying certain constraints. The mathematical programming problem with nonlinear constraints satisfies the following:

[0045]

[0046] stc1(x)=-ω H1 +ω L1 -Ω TRAM =0

[0047]

[0048] In formula (13), Ω TRAM c1(x) represents the transition bandwidth of the frequency divider filter, that is, the overlap bandwidth between the low-frequency sub-path and the high-frequency sub-path. c1(x) represents constraint 1, c2(x) represents constraint 2, c3(x) represents constraint 3, c4(x) represents constraint 4, c5(x) represents constraint 5, and c6(x) represents constraint 6.

[0049] Because the transition band bandwidth of the filter and the passband cutoff frequency domain of the filter satisfy... Therefore, we introduce parameters. Among them (R) mn ) N Let be the Jacobian elliptic function of the corresponding filter.

[0050] s L1 (N) represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the low-frequency sub-path LPF_L1 filter, s H2 (N) represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the high-frequency sub-path LPF_H2 filter, s H3 (N) represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the high-frequency sub-channel HPF_H3 filter;

[0051] Solve QP subproblems of the following form:

[0052]

[0053] In formula (14), d represents the current iteration point x. k Based on the search direction vector, H k Let c represent the Hessian matrix with respect to Lagrange. i (x k Let ) represent the i-th constraint. At the beginning of each iteration, the SQP algorithm constructs an approximate Lagrangian function using a quadratic approximation of the objective function and the constraint function, and calculates the Hessian matrix corresponding to the function during each iteration.

[0054]

[0055] In formula (15), λ k Let represent the Lagrange multiplier, and I represent the active constraint index set, which plays a filtering role in constrained optimization problems to determine the constraint terms to be considered in the Hessian matrix.

[0056] The objective variable x is decomposed into subproblems for optimization with respect to variables xx, yy, and zz, respectively. The objective variable x refers to the total output bandwidth ω2, and xx, yy, and zz correspond to the transition bandwidth of the target filter, respectively. Consider the following linearly constrained multi-block optimization:

[0057] min f(x)=f(xx,yy,zz)=θ1(xx)+θ2(yy)+θ3(zz)

[0058] stc(x)=c(xx,yy,zz)=A1xx+A2yy+A3zz-b=0,xx∈X,yy∈Y,zz∈Z (16)

[0059] when The condition holds true for all conditions, meaning that after optimizing the inequality constraints, the original c_4 constraint always holds true. This can be represented as a matrix:

[0060]

[0061] when The condition holds true consistently, meaning that after optimizing the inequality constraints, the constraints are simplified and modified, and can be represented using matrix notation as follows:

[0062]

[0063] The augmented Lagrangian function for formula (16) is:

[0064]

[0065] Where λ is the Lagrange multiplier and β is the quadratic penalty parameter, the ADMM algorithm solves the problem step by step by alternately optimizing xx, yy, zz and updating λ;

[0066]

[0067] The objective function is rewritten as:

[0068]

[0069] st A1xx k +A2yy k +A3zz k -b=0, xx∈X,yy∈Y,zz∈Z (21)

[0070] Among them, H k It is a symmetric approximation of the complete Lagrangian function with respect to the Hessian matrix of the original variables, and has a block diagonal feature, which is used to describe the local curvature of the objective function at that point;

[0071]

[0072] The augmented Lagrangian function for the QP subproblem is now:

[0073]

[0074] Therefore, the ADMM-SQP mathematical model is transformed into three QP subproblems.

[0075]

[0076] The solution to each QP subproblem satisfies the KKT conditions by introducing multipliers α and Lagrange multipliers λ, ensuring that the optimal solution to the problem is found at point k+1;

[0077]

[0078] The descent property of the augmented Lagrangian function at k in the search direction satisfies:

[0079]

[0080] The descent property of the search direction of the multiplier λ at k satisfies:

[0081]

[0082] The beneficial effects of this invention are as follows: Firstly, through a systematic analysis of aliasing errors in FI-DAC systems, this invention designs a method for joint modeling of the digital and analog domains to optimize the system structure and significantly reduce the impact of aliasing errors. Secondly, by introducing a frequency matching iterative optimization algorithm combining ADMM and SQP algorithms, this invention efficiently optimizes key frequency parameters of the system, reducing resource consumption for digital domain error calibration and thus improving the real-time performance of signal processing. Thirdly, within a distributed verification framework, this invention gradually completes the transition from module-level to complete hardware design, optimizing channel energy distribution and impedance matching to ensure high-quality and high-precision system signal output. Fourthly, through integrated hardware design, this invention reduces system complexity and improves signal stability, thereby providing solid technical support for the commercialization of FI-DAC technology.

[0083] This invention targets high-frequency sub-paths. Due to the limitation that frequency domain downconversion would consume a large amount of digital resources and could not meet the real-time requirements of AWG signals, the downconversion operation is achieved through digital mixing.

[0084] In the simulation test, the present invention performed well in suppressing aliasing error, verifying the effectiveness of the filter design and highlighting the key role of system modeling in ensuring the quality of output signal. By precisely controlling the filter parameters and system configuration, the interference that may be caused by non-target frequency signals is minimized, further improving the performance and reliability of the entire system. Attached Figure Description

[0085] Figure 1 This is a schematic diagram of the structure and error of a two-channel FI-DAC signal generator;

[0086] Figure 2 This is a schematic diagram of digital domain transform mixing.

[0087] Figure 2 (a) is a schematic diagram of the ideal downconversion principle in the digital domain;

[0088] Figure 2 (b) is a schematic diagram of the actual downconversion aliasing behavior in the digital domain;

[0089] Figure 2 (c) is a schematic diagram showing the general relationship between the digital domain sampling frequency and the mixing frequency;

[0090] Figure 3 This is a schematic diagram illustrating the aliasing behavior of analog domain transform;

[0091] Figure 3 (a) is a schematic diagram of the analog filter protection band setting to avoid signal aliasing caused by low-frequency sub-channel DAC conversion;

[0092] Figure 3(b) is a schematic diagram of the analog filter protection band setting to avoid signal aliasing caused by high-frequency sub-channel DAC conversion;

[0093] Figure 3 (c) is a schematic diagram of the analog filter guard band setting to avoid signal aliasing caused by mixer characteristics;

[0094] Figure 4 This is a comparison of the amplitude-frequency response of the low-frequency subband signal of the true model before and after passing through LPF_1;

[0095] Figure 4 (a) is a schematic diagram of direct output of low-frequency subband signals;

[0096] Figure 4 (b) is a schematic diagram of the output of the low-frequency subband signal after LPF_1 correction;

[0097] Figure 5 This is a comparison diagram of the amplitude-frequency response of the high-frequency subband signal in the simulation model before and after passing through LPF_1;

[0098] Figure 5 (a) is a schematic diagram of direct output of high-frequency subband signals;

[0099] Figure 5 (b) is a schematic diagram of the output of the high-frequency subband signal after LPF_1 correction;

[0100] Figure 6 This is a comparison diagram of the amplitude-frequency response of the high-frequency subband signal in the simulation model before and after passing through HPF_3 and LPF_3;

[0101] Figure 6 (a) is a schematic diagram of the direct output of the up-converted signal;

[0102] Figure 6 (b) is a schematic diagram of the output of the up-converted signal after correction by HPF_3 and LPF_3;

[0103] Figure 7 This is a schematic diagram of the amplitude-frequency response of the combined signal output from the simulation model;

[0104] Figure 8 This is a schematic diagram illustrating the specific implementation method of ADMM-SQP. Detailed Implementation

[0105] Specific implementation method one: as follows Figures 1 to 8 As shown, a modeling method for aliasing error suppression in a FI-DAC system includes the following steps:

[0106] Step 1: Through systematic analysis of aliasing errors in the FI-DAC system, design a joint modeling method for the digital and analog domains; the ideal digital mixer output includes upper and lower sideband signals, and the mixer output signal is:

[0107]

[0108] Select the signal with the input signal as the target down-conversion signal; in formula (1), A RF ω is the amplitude of the radio frequency signal. RF Indicates the frequency of radio frequency, A LO ω represents the amplitude of the local oscillator signal. LO Indicates the frequency of the local oscillator;

[0109] Both the digital signal and the local oscillator signal have a sampling frequency of f. s Due to signal sampling, the output spectrum is periodically based on the digital domain sampling frequency f. s To extend;

[0110] When the extended signal is greater than the down-converted signal;

[0111]

[0112] In formula (2), f represents the transition bandwidth of the digital low-pass filter H0. LO ω represents the local oscillator frequency, ω2 represents the high-frequency sub-channel termination frequency, i.e., the total output bandwidth of the FI-DAC system. H1 f represents the starting frequency of the FI-DAC high-frequency sub-channel. s This represents the digital domain sampling frequency, which is equal to M times the DAC sampling frequency f. sm And satisfy

[0113]

[0114] When the extended signal completely overlaps with the target down-conversion signal;

[0115] The extended signal caused by sampling completely overlaps with the obtained target down-conversion signal, at which point the following condition is met.

[0116] f s -f LO -ω2=f LO -ω2(4),

[0117] f s -f LO -ω H1 =f LO -ω H1 (5),

[0118] That is, the digital domain sampling frequency f s Equal to twice the local oscillator frequency f LO ;

[0119] Mf sm =2fLO (6),

[0120] In formula (6), M represents the downsampling factor, f sm DAC chip sampling frequency.

[0121] Step 2: Introduce a frequency matching iterative optimization algorithm that combines ADMM and SQP algorithms to optimize key frequency parameters of the system, reduce the resource consumption of digital domain error calibration, and thus improve the real-time performance of signal processing; By changing the modeling system, aliasing error is corrected by setting a guard band, which is used to place the filter transition band.

[0122] When the sampling characteristics of the DAC chip cause aliasing;

[0123] The stopband frequencies of low-pass filters LPF_L1 and LPF_H1 are lower than the first extension frequencies caused by DAC sampling of the low-frequency and high-frequency baseband signals output in the digital domain, respectively; the passband should encompass all frequencies of the baseband signal, i.e., the stopband and passband cutoff frequencies satisfy:

[0124]

[0125] In formulas (7) and (8), Ω L1 This indicates the passband cutoff frequency of the low-pass filter LPF_L1 in the low-frequency sub-path. ω represents the transition band bandwidth of the low-pass filter LPF_L1 in the low-frequency sub-path. L1 This indicates the termination frequency of the low-frequency sub-path. Ω H1 This indicates the passband cutoff frequency of the low-pass filter LPF_H1 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter LPF_H1 in the high-frequency sub-path;

[0126] The restrictions on filter LPF_H1 in the high-frequency sub-path are not as stringent as those on filter LPF_L1 in the high-frequency sub-path, so LPF_L1 can be used instead.

[0127] When aliasing occurs due to the hardware characteristics of the mixer;

[0128] The high-frequency sub-channel signal is up-converted and restored using an actual analog mixer. This analog mixer is a nonlinear device, and compared to an ideal multiplier, the output signal includes both upper and lower sideband signals. For any dual-frequency excitation input nonlinear device, assuming A... IF ω represents the amplitude of the intermediate frequency signal. IF For the intermediate frequency, A LO ω represents the amplitude of the local oscillator signal. LO Let A be the frequency of the local oscillator. For ease of explanation, let's assume A is... IF =A LO =A;

[0129] v in1 =Acos(ω) IF t)

[0130] v in2 =Acos(ω) LO t)

[0131] v i =v in1 +v in2 =A[cos(ω IF t)+cos(ω LO t)](9),

[0132] Its output response is expressed as:

[0133]

[0134] In formulas (9) and (10), v in1 This represents the input signal at port 1 of the three-port network; in this paper, it is the input intermediate frequency signal, v. in2 This represents input port 2 of the three-port network, which in this paper is the input local oscillator signal, v. i This represents the combined input signal of port 1 and port 2, where t represents time and v represents the input signal. out The output signal of the port 3 nonlinear transform of the three-port network represents the signal. The output of a nonlinear system includes the fundamental frequency component, harmonic components, and intermodulation components introduced by the intermodulation between the input signals. k1 is a first-order coefficient representing the system's linear gain, the direct contribution of the fundamental frequency component of the input signal to the output. k2 is a second-order coefficient representing the system's second-order nonlinear response characteristics, including the second harmonic and intermodulation components. k3 is a third-order coefficient representing the system's third-order nonlinear response characteristics, including the third harmonic and higher-order intermodulation components.

[0135] The low-pass filter LPF_H2 and the high-pass filter HPF_H3 constitute a bandpass filter. Their cutoff frequencies should be less than the local oscillator leakage frequency caused by mixer non-ideality and the upper sideband image frequency obtained from up-conversion, respectively, and greater than the high-frequency baseband signal frequency. The passband should encompass all frequencies of the baseband signal; that is, the stopband and passband cutoff frequencies should satisfy the following:

[0136]

[0137] In formulas (11) and (12), Ω H2 This indicates the passband cutoff frequency of the low-pass filter LPF_H2 in the high-frequency sub-path. ω2 represents the transition bandwidth of the low-pass filter LPF_H2 in the high-frequency sub-path, and Ω represents the termination frequency of the high-frequency sub-path. H3This indicates the passband cutoff frequency of the high-pass filter HPF_H3 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter HPF_H3 in the high-frequency sub-path.

[0138] Step 3: Within the distributed verification framework, complete the transition from module-level to full hardware design, optimize channel energy distribution and impedance matching to ensure high quality and accuracy of system signal output; the ADMM-SQP method is used to solve for key system frequency points, specifically including:

[0139] The modeling process is transformed into an optimization problem, namely, modeling a FI-DAC system that maximizes output bandwidth while satisfying certain constraints. The mathematical programming problem with nonlinear constraints satisfies the following:

[0140]

[0141] stc1(x)=-ω H1 +ω L1 -Ω TRAM =0

[0142]

[0143] In formula (13), Ω TRAM c1(x) represents the transition bandwidth of the frequency divider filter, that is, the overlap bandwidth between the low-frequency sub-path and the high-frequency sub-path. c1(x) represents constraint 1, c2(x) represents constraint 2, c3(x) represents constraint 3, c4(x) represents constraint 4, c5(x) represents constraint 5, and c6(x) represents constraint 6.

[0144] Because the transition band bandwidth of the filter and the passband cutoff frequency domain of the filter satisfy... Therefore, we introduce parameters. Among them (R) mn ) N Let be the Jacobian elliptic function of the corresponding filter.

[0145] s L1 (N) represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the low-frequency sub-path LPF_L1 filter, s H2 (N) represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the high-frequency sub-path LPF_H2 filter, s H3 (N) represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the high-frequency sub-channel HPF_H3 filter;

[0146] Solve QP subproblems of the following form:

[0147]

[0148] In formula (14), d represents the current iteration point x. k Based on the search direction vector, H k Let c represent the Hessian matrix with respect to Lagrange. i (x k Let ) represent the i-th constraint. At the beginning of each iteration, the SQP algorithm constructs an approximate Lagrangian function using a quadratic approximation of the objective function and the constraint function, and calculates the Hessian matrix corresponding to the function during each iteration.

[0149]

[0150] In formula (15), λ k Let represent the Lagrange multiplier, and I represent the active constraint index set, which plays a filtering role in constrained optimization problems to determine the constraint terms to be considered in the Hessian matrix.

[0151] The objective variable x is decomposed into subproblems for optimization with respect to variables xx, yy, and zz, respectively. The objective variable x refers to the total output bandwidth ω2, and xx, yy, and zz correspond to the transition bandwidth of the target filter, respectively. Consider the following linearly constrained multi-block optimization:

[0152] min f(x)=f(xx,yy,zz)=θ1(xx)+θ2(yy)+θ3(zz)

[0153] stc(x)=c(xx,yy,zz)=A1xx+A2yy+A3zz-b=0,xx∈X,yy∈Y,zz∈Z (16)

[0154] when The condition holds true for all conditions, meaning that after optimizing the inequality constraints, the original c_4 constraint always holds true. This can be represented as a matrix:

[0155]

[0156] when The condition holds true consistently, meaning that after optimizing the inequality constraints, the constraints are simplified and modified, and can be represented using matrix notation as follows:

[0157]

[0158] The augmented Lagrangian function for formula (16) is:

[0159]

[0160] Where λ is the Lagrange multiplier and β is the quadratic penalty parameter, the ADMM algorithm solves the problem step by step by alternately optimizing xx, yy, zz and updating λ;

[0161]

[0162] The objective function is rewritten as:

[0163]

[0164] st A1xx k +A2yy k +A3zz k -b=0, xx∈X,yy∈Y,zz∈Z (21)

[0165] Among them, H k It is a symmetric approximation of the complete Lagrangian function with respect to the Hessian matrix of the original variables, and has a block diagonal feature, which is used to describe the local curvature of the objective function at that point;

[0166]

[0167] The augmented Lagrangian function for the QP subproblem is now:

[0168]

[0169] Therefore, the ADMM-SQP mathematical model is transformed into three QP subproblems.

[0170]

[0171] The solution to each QP subproblem satisfies the KKT conditions by introducing multipliers α and Lagrange multipliers λ, ensuring that the optimal solution to the problem is found at point k+1;

[0172]

[0173] The descent property of the augmented Lagrangian function at k in the search direction satisfies:

[0174]

[0175] The descent property of the search direction of the multiplier λ at k satisfies:

[0176]

[0177] Step 4: Reduce system complexity through integrated hardware design.

[0178] Proof of effectiveness

[0179] like Figure 4 and Figure 5 As shown, the modeling method's ability to suppress aliasing errors was tested under simulation conditions. The system modeling verification under simulation conditions showed that the low-pass filter (LPF_1) can achieve a suppression effect of more than -45dB against aliasing errors caused by DAC sampling extension.

[0180] like Figure 6 As shown, the high-pass filter (HPF_2) and low-pass filter (LPF_3) can achieve a suppression effect of more than -45dB against aliasing errors caused by the non-ideal characteristics of the mixer. Compared with the processing of out-of-band mirror signals, the filter bank shows better suppression performance in the leakage processing of in-band input signals.

[0181] like Figure 7 As shown, during the combining process of the two signals, the combined output signal and each sub-signal exhibit good flatness and smoothness within the target frequency band. The aliasing error caused by signal leakage can be ignored, thus further confirming the effectiveness of system modeling and filter design in reducing aliasing errors caused by non-target frequency signals.

[0182] 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 modeling method for aliasing error suppression in a FI-DAC system, characterized in that, The specific steps include: Step 1: Through systematic analysis of aliasing errors in the FI-DAC system, design a joint modeling method for the digital and analog domains; Step 2: Introduce a frequency matching iterative optimization algorithm combining ADMM and SQP algorithms to optimize the key frequency parameters of the system, reduce the resource consumption of digital domain error calibration, and thus improve the real-time performance of signal processing; use ADMM-SQP to solve for the key frequency points of the system; the SQP algorithm uses a quadratic approximation of the objective function and constraint function to construct an approximate Lagrangian function, and calculates the Hessian matrix corresponding to the function in each iteration; the ADMM algorithm solves the problem step by step by alternately optimizing xx, yy, zz and updating λ; xx, yy, zz correspond to the transition bandwidth of the target filter, and λ represents the Lagrange multiplier; Step 3: Under the distributed verification framework, complete the transition from module level to complete hardware design, optimize channel energy distribution and impedance matching issues, and ensure high quality and high accuracy of system signal output; Step 4: Reduce system complexity through integrated hardware design.

2. The modeling method for aliasing error suppression in a FI-DAC system according to claim 1, characterized in that, Step 1 specifically includes: An ideal digital mixer output includes both upper and lower sideband signals. The amplitude of the radio frequency signal. For radio frequency, The amplitude of the local oscillator signal. Given the local oscillator frequency, the mixer output signal is: （1）， Select the signal with the input signal as the target down-conversion signal; in formula (1), Indicates the frequency of radio frequency. Indicates the frequency of the local oscillator; Both the digital signal and the local oscillator signal have a sampling frequency of 100 MHz. f s Due to signal sampling, the output spectrum is periodically based on the digital domain sampling frequency. f s To extend; When the extended signal is greater than the down-converted signal; （2）， In formula (2), This represents the transition bandwidth of the digital low-pass filter H0. Indicates the frequency of the local oscillator. This indicates the high-frequency sub-channel termination frequency, which is the total output bandwidth of the FI-DAC system. Indicates the starting frequency of the FI-DAC high-frequency sub-channel. This represents the digital domain sampling frequency, which is equal to M times the DAC sampling frequency. fsm And satisfy （3）； Indicates the sampling frequency. Indicates the termination frequency of the high-frequency sub-path. This indicates the starting frequency of the FI-DAC high-frequency sub-channel, when the extended signal and the target down-converted signal completely overlap. The extended signal caused by sampling completely overlaps with the obtained target down-conversion signal, at which point the following condition is met. （4）， （5）， That is, digital domain sampling frequency f s Equal to twice the local oscillator frequency ; （6）， In formula (6), Indicates the downsampling factor. This indicates the sampling frequency of the DAC chip.

3. The modeling method for aliasing error suppression in a FI-DAC system according to claim 1, characterized in that, In step 2, the aliasing error is corrected by changing the modeling system and setting a guard band. The guard band is used to place the filter transition band. When the sampling characteristics of the DAC chip cause aliasing; The stopband frequencies of low-pass filters LPF_L1 and LPF_H1 are lower than the first extension frequencies caused by DAC sampling of the low-frequency and high-frequency baseband signals output in the digital domain, respectively; the passband should encompass all frequencies of the baseband signal, i.e., the stopband and passband cutoff frequencies satisfy: （7）， （8）， In formulas (7) and (8), This indicates the passband cutoff frequency of the low-pass filter LPF_L1 in the low-frequency sub-path. This represents the transition band bandwidth of the low-pass filter LPF_L1 in the low-frequency sub-path. Indicates the termination frequency of the low-frequency sub-path; This indicates the passband cutoff frequency of the low-pass filter LPF_H1 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter LPF_H1 in the high-frequency sub-path; The restrictions on filter LPF_H1 in the high-frequency sub-path are not as stringent as those on filter LPF_L1 in the high-frequency sub-path, so LPF_L1 can be used instead. When aliasing occurs due to the hardware characteristics of the mixer; The high-frequency sub-channel signal is up-converted and restored using an actual analog mixer. This analog mixer is a nonlinear device, and compared to an ideal multiplier, the output signal includes both upper and lower sideband signals. For any dual-frequency excitation input nonlinear device, it is assumed that… The amplitude of the intermediate frequency signal. The intermediate frequency, The amplitude of the local oscillator signal. Let be the frequency of the local oscillator. For ease of explanation, assume... ; （9）， Its output response is expressed as: （10）， In formulas (9) and (10), This represents the input signal at port 1 of the three-port network; in this paper, it is the input intermediate frequency signal. This represents input port 2 of the three-port network, which in this article is the input local oscillator signal. This represents the combined input signal of port 1 and port 2. Indicates time, This represents the output signal of the nonlinear transformation at port 3 of a three-port network. The output of a nonlinear system includes the fundamental frequency component, harmonic components, and intermodulation components introduced by the intermodulation between the input signals. The first-order coefficient represents the linear gain of the system, and is the direct contribution of the fundamental frequency component of the input signal to the output. The second-order coefficients represent the second-order nonlinear response characteristics of the system, including the second harmonic and intermodulation components. The third-order coefficients represent the third-order nonlinear response characteristics of the system, including the third harmonic and higher-order intermodulation components. The low-pass filter LPF_H2 and the high-pass filter HPF_H3 constitute a bandpass filter. Their cutoff frequencies should be less than the local oscillator leakage frequency caused by mixer non-ideality and the upper sideband image frequency obtained from up-conversion, respectively, and greater than the high-frequency baseband signal frequency. The passband should encompass all frequencies of the baseband signal; that is, the stopband and passband cutoff frequencies should satisfy the following: （11）， （12）， In formulas (11) and (12), This indicates the passband cutoff frequency of the low-pass filter LPF_H2 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter LPF_H2 in the high-frequency sub-path. Indicates the termination frequency of the high-frequency sub-path. This indicates the passband cutoff frequency of the high-pass filter HPF_H3 in the high-frequency sub-path. This represents the transition band bandwidth of the low-pass filter HPF_H3 in the high-frequency sub-path.

4. The modeling method for aliasing error suppression in a FI-DAC system according to claim 1, characterized in that, The ADMM-SQP method is used to solve for the key frequency points of the system, specifically including: The modeling process is transformed into an optimization problem, namely, modeling a FI-DAC system that maximizes output bandwidth while satisfying certain constraints. The mathematical programming problem with nonlinear constraints satisfies the following: （13）， In formula (13), This indicates the transition band bandwidth of the frequency divider filter. This represents the bandwidth of the overlap band between the low-frequency sub-path and the high-frequency sub-path. This indicates constraint 1. This indicates constraint condition 2. This indicates constraint 3. This indicates constraint 4. This indicates constraint 5. This indicates constraint 6; Indicates the termination frequency of the high-frequency sub-path. Indicates the sampling frequency of the DAC chip. Indicates the local oscillator frequency. Indicates the starting frequency of the FI-DAC high-frequency sub-channel. Indicates the termination frequency of the low-frequency sub-path; Because the transition band bandwidth of the filter and the passband cutoff frequency domain of the filter satisfy... Therefore, parameters are introduced. ,in This is the Jacobian elliptic function of the corresponding filter; This represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the low-frequency sub-channel LPF_L1 filter. This represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the high-frequency sub-channel LPF_H2 filter. This represents the conversion factor between the passband cutoff frequency and the transition band bandwidth of the HPF_H3 filter in the high-frequency sub-channel. This represents the median reflectance value in decibels. This represents the median reflectance in decibels from the standard reference sample. This represents the normalized median response reduction factor. Indicates the transition band bandwidth of the filter. Indicates the passband cutoff frequency of the filter; Solve QP subproblems of the following form: （14）， In formula (14), Indicates the current iteration point Based on the search direction vector, Let Hesse be the matrix in Lagrange. This represents the i-th constraint condition, located at the starting point of each iteration; Indicates the number of iteration points. This indicates the total number of constraints. （15）， In formula (15), Represents the Lagrange multipliers. This represents the active constraint index set, which plays a filtering role in constrained optimization problems, determining the constraint terms to be considered in the Hessian matrix. express The augmented Lagrange function, Representing the iteration point The objective function, Denote the Lagrange multiplier for iteration point i; The objective variable x is decomposed into subproblems for optimization of variables xx, yy, and zz, respectively, where the objective variable x refers to the total output bandwidth. , xx, yy, zz correspond to the transition bandwidth of the target filter, respectively. , , Consider the following linearly constrained multi-block optimization: （16） A1 represents signal 1, A2 represents signal 2, and A3 represents signal 3. when The condition holds true for all conditions, meaning that after optimizing the inequality constraints, the original c_4 constraint always holds true. This can be represented as a matrix: （17） when The condition holds true consistently, meaning that after optimizing the inequality constraints, the constraints are simplified and modified, and can be represented using matrix notation as follows: （18） The augmented Lagrangian function for formula (16) is: （19） Where λ is a Lagrange multiplier. For the secondary penalty parameter, , , The tables correspond to the transition bandwidth of the target filter, and the ADMM algorithm solves the problem step by step by alternately optimizing xx, yy, zz and updating λ. （20） The objective function is rewritten as: （21） Where b represents a constant vector, Describe the objective function exist The gradient at the iteration point, Describe the objective function exist The gradient at the iteration point, Describe the objective function exist The gradient at the iteration point, It is a symmetric approximation of the complete Lagrangian function with respect to the Hessian matrix of the original variables, possessing a block diagonal characteristic, and is used to describe the objective function in... Iteration point Iteration point The local curvature corresponding to the iteration point; （22） The augmented Lagrangian function for the QP subproblem is now: （23） Therefore, the ADMM-SQP mathematical model is transformed into three QP subproblems. （24） The solution to each QP subproblem satisfies the KKT conditions by introducing multipliers α and Lagrange multipliers λ, ensuring that the optimal solution to the problem is found at point k+1; （25） The descent property of the augmented Lagrangian function at k in the search direction satisfies: （26） This represents the diagonal Hessian matrix component of the x component. This represents the diagonal Hessian matrix component of the yy component. This represents the diagonal Hessian matrix component of the z-component. Denotes the iteration point x. Represents the y-iteration point. Represents the iteration point zz; express The augmented Lagrange function, This represents the search direction vector at iteration point xx. This represents the search direction vector for the y-y iteration point. This represents the search direction vector at the zz iteration point. Denotes the complementary relaxation multiplier at iteration point xx. The complementary relaxation multiplier represents the iteration point of yy. Represents the complementary relaxation multiplier at the zz iteration point; The descent property of the search direction of the multiplier λ at k satisfies: （27） This represents the step size scaling hyperparameter for multiplier updates. This indicates the direction of the iterative search for the multiplier λ.

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