Modeling method for FI-DAC system aliasing error suppression

Abstract

The invention discloses a modeling method for FI-DAC system aliasing error suppression, and relates to a modeling method. The invention aims to solve the problems that in the prior art, system errors are compensated by excessively depending on digital pre-calibration, the calculation complexity is high, a large number of calculation resources are occupied, and the real-time application requirement is difficult to meet. The method comprises the following steps: step 1, designing a digital domain and analog domain joint modeling method through systematic analysis of FI-DAC system aliasing errors; step 2, introducing a frequency matching iterative optimization algorithm combining ADMM and SQP algorithms, optimizing key frequency parameters of the system, and reducing resource consumption of digital domain error calibration, thereby improving real-time performance of signal processing; 3, under a distributed verification framework, completing transition from a module level to a complete hardware design, and optimizing channel energy distribution and impedance matching problems; and step 4, reducing system complexity through integrated hardware design. The invention belongs to the technical field of frequency band interleaving digital-to-analog conversion.

Description

Technical Field

[0001] The invention relates to a modeling method, and belongs to the technical field of frequency band interleaving digital-to-analog conversion. Background Art

[0002] At present, the frequency-interleaved digital-to-analog conversion (FI-DAC) technology has been theoretically proven to be able to break through the physical limitations of the DAC sampling rate, thereby significantly improving the output bandwidth of the arbitrary waveform generator (AWG). However, this technology still faces many key challenges in practical applications. First, due to the sampling characteristics of the DAC and the non-ideality of the analog mixer, serious aliasing errors are inevitably introduced into the output signal of the FI-DAC system. These errors include the overlap error of the in-band signal and the leakage of the out-of-band image signal, which directly affects the quality and reliability of the output signal. Secondly, the modeling method of the existing FI-DAC system relies heavily on empirical design and fails to fully integrate the signal characteristics of the digital domain and the analog domain, resulting in high error calibration complexity and poor real-time performance. In addition, the hardware implementation of FI-DAC technology is not yet mature, and the main research remains in the simulation verification stage. When the hardware is implemented, due to problems such as impedance mismatch and uneven channel energy distribution, the system signal is distorted, which is difficult to meet the AWG's demand for high-precision output. Finally, the existing technology relies too much on digital pre-calibration to compensate for system errors. This method not only has high computational complexity, but also occupies a large amount of computing resources, making it difficult to meet the requirements of real-time applications. Summary of the invention

[0003] The present invention aims to solve the problem that the prior art over-relies on digital pre-calibration to compensate for system errors, which not only has high computational complexity but also occupies a large amount of computing resources and is difficult to meet the requirements of real-time applications. A modeling method for suppressing aliasing errors in a FI-DAC system is proposed.

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

[0005] Step 1: Design a joint modeling method between digital domain and analog domain by systematically analyzing the aliasing error of the FI-DAC system;

[0006] Step 2: Introduce a frequency matching iterative optimization algorithm that combines ADMM and SQP algorithms to optimize key frequency parameters of the system, reduce 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] The ideal digital mixing output includes upper and lower sideband signals, and the mixing output signal is:

[0011]

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

[0013] The sampling frequency of the digital signal and the local oscillator signal is f s , affected by signal sampling, the output spectrum is periodically sampled at 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), represents the transition bandwidth of the digital low-pass filter H0, f LO represents the frequency of the local oscillator, ω 2 Indicates the high-frequency sub-path termination frequency, that is, the total output bandwidth of the FI-DAC system, ω H1 Indicates the starting frequency of the FI-DAC high-frequency sub-circuit, f s Represents the digital domain sampling frequency, which is equal to M times the DAC sampling frequency f sm , and satisfies

[0017]

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

[0019] The extended signal caused by sampling completely overlaps with the target down-converted signal obtained, and at this time,

[0020] f s -f LO -ω 2 =f LO -ω 2 (4)

[0021] f s -f LO -ω H1 =fLO -ω H1 (5)

[0022] That is, the digital domain sampling frequency f s Equal to 2 times the local oscillator frequency f LO ;

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

[0024] In formula (6), M represents the downsampling multiple, 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, and the guard band is used to place the filter transition band;

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

[0027] The stopband frequencies of the low-pass filters LPF_L1 and LPF_H1 are respectively less than the first extension frequency caused by the DAC sampling of the low-frequency signal and the high-frequency signal baseband signal output in the digital domain; the passband should include all frequencies of the baseband signal, that is, the stopband and passband cutoff frequencies meet the following requirements:

[0028]

[0029]

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

[0031] The restriction on filter LPF_H1 in the high-frequency sub-circuit is not as strict as that on filter LPF_L1 in the high-frequency sub-circuit, so LPF_L1 is used instead;

[0032] When mixer hardware characteristics introduce aliasing;

[0033] The high-frequency sub-path signal is restored to an up-converted signal through an actual analog mixer, where the analog mixer is a nonlinear device. Compared with an ideal multiplier, the output signal of the mixer includes upper and lower sideband signals. For any dual-frequency excitation input nonlinear device, assuming AIF is the amplitude of the intermediate frequency signal, ω IF is the frequency of the intermediate frequency, A LO is the amplitude of the local oscillator signal, ω LO is the frequency of the local oscillator. For the convenience of description, it is assumed that A 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 represents the input signal of port 1 of the three-port network, which is the input intermediate frequency signal in this paper, v in2 represents the input port 2 of the three-port network, which is the input local oscillator signal in this paper, v i represents the combined input signal of port 1 and port 2, t represents time, v out k represents the nonlinear transformation output signal of port 3 of the three-port network. The output of the nonlinear system will contain the fundamental frequency component and harmonic component of the input signal, as well as the intermodulation component introduced by the intermodulation between the input signals. 1 The first-order coefficient represents the linear gain of the system, the direct contribution of the fundamental frequency component of the input signal to the output. 2 The second-order coefficient represents the second-order nonlinear response characteristics of the system, including the second harmonic and intermodulation components. 3 The third-order coefficient represents the third-order nonlinear response characteristics of the system, including third-order harmonics and high-order intermodulation components.

[0040] The low-pass filter LPF_H2 and the high-pass filter HPF_H3 form a bandpass filter, whose cutoff frequency should be smaller than the local oscillator leakage frequency caused by the mixer's non-ideal behavior and the upper sideband image frequency obtained by up-conversion, and larger than the high-frequency baseband signal frequency; the passband includes all frequencies of the baseband signal, that is, the stopband and passband cutoff frequencies meet the following requirements:

[0041]

[0042] In formulas (11) and (12), Ω H2 Indicates the passband cutoff frequency of the low-pass filter LPF_H2 in the high-frequency sub-path, Indicates the transition band bandwidth of the low-pass filter LPF_H2 in the high-frequency sub-path, ω 2 Indicates the high-frequency sub-path termination frequency, Ω H3 Indicates the passband cutoff frequency of the high-pass filter HPF_H3 in the high-frequency sub-circuit, Represents the transition band bandwidth of the low-pass filter HPF_H3 in the high-frequency sub-path.

[0043] Furthermore, ADMM-SQP is used to solve the key frequency points of the system, including:

[0044] The modeling process is converted into an optimization problem, that is, to achieve the FI-DAC system modeling with the maximum output bandwidth while satisfying certain constraints. The mathematical programming problem of nonlinear constraints satisfies:

[0045]

[0046] In formula (13), Ω TRAM represents the transition bandwidth of the crossover filter, that is, the overlapping bandwidth of the low-frequency sub-path and the high-frequency sub-path, c 1 (x) represents constraint 1, c 2 (x) represents constraint 2, c 3 (x) represents constraint 3, c 4 (x) represents constraint 4, c 5 (x) represents constraint 5, c 6 (x) represents constraint 6.

[0047] Since the transition band bandwidth of the filter is consistent with the filter passband cutoff frequency domain, So we introduce the parameter Among them (R mn ) N is the Jacobi elliptic function of the corresponding filter.

[0048] s L1 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the low-frequency sub-path LPF_L1 filter, s H2 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the high-frequency sub-path LPF_H2 filter, s H3 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the high-frequency sub-path HPF_H3 filter;

[0049] Solve a QP subproblem of the form:

[0050]

[0051] In formula (14), d represents the value at the current iteration point x k Based on the search direction vector, H k represents the Hesse matrix with respect to Lagrange, c i (x k ) represents the i-th constraint condition. At the starting point of each iteration, the SQP algorithm constructs an approximate Lagrangian function using the quadratic approximation of the objective function and the constraint function, and calculates the Hessian matrix corresponding to the function in each iteration process;

[0052]

[0053] In formula (15), λ k represents the Lagrange multiplier, I represents the active constraint index set, which plays a screening role in the constrained optimization problem and is used to determine the constraints that need to be considered in the Hessian matrix.

[0054] The target variable x is decomposed into sub-problems about variables xx, yy and zz, respectively. The target variable x refers to the total output bandwidth ω 2 , xx, yy, zz correspond to the transition bandwidth of the target filter respectively, Consider the following linearly constrained multi-block optimization:

[0055] min f(x)=f(xx,yy,zz)=θ 1 (xx)+θ 2 (yy)+θ 3 (zz)

[0056] stc(x)=c(xx,yy,zz)=A 1 xx+A 2 yy+A 3 zz-b=0,xx∈X,yy∈Y,zz∈Z(16)

[0057] when Always holds true, that is, after optimizing the inequality constraints, the original c_4 constraints always hold true, which can be converted to the following using the matrix expression method:

[0058]

[0059] when Always holds true, that is, after optimizing the inequality constraints, the constraints are simplified and modified and converted into the matrix expression method:

[0060]

[0061] The augmented Lagrangian function for formula (16) is

[0062]

[0063] Where λ is the Lagrange multiplier, β is the quadratic penalty parameter, and the ADMM algorithm gradually solves the problem by alternately optimizing xx, yy, zz and updating λ;

[0064]

[0065] The objective function is rewritten as:

[0066]

[0067] Among them, H k It is a symmetric approximation of the Hessian matrix of the complete Lagrangian function with respect to the original variable, with block diagonal characteristics, and is used to describe the local curvature of the objective function at that point;

[0068]

[0069] At this time, the augmented Lagrangian function for the QP subproblem is:

[0070]

[0071] Therefore, the following ADMM-SQP mathematical model is converted into three QP sub-problems:

[0072]

[0073] The solution of each QP subproblem satisfies the KKT condition by introducing multipliers α and Lagrange multipliers λ to ensure that the optimal solution of the problem is found at point k+1;

[0074]

[0075] The descent property of the search direction of the augmented Lagrangian function at k satisfies:

[0076]

[0077] The descent property of the search direction of the multiplier λ at k satisfies:

[0078]

[0079] The beneficial effects of the present invention are as follows: the present invention designs a method for joint modeling of digital domain and analog domain through systematic analysis of aliasing errors of FI-DAC system, so as to optimize the system structure and significantly reduce the influence of aliasing errors; the present invention introduces a frequency matching iterative optimization algorithm combining ADMM and SQP algorithms to efficiently 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; the present invention gradually completes the transition from module level to complete hardware design under the distributed verification framework, optimizes channel energy distribution and impedance matching problems, and ensures high quality and high precision of system signal output; the present invention reduces system complexity and improves signal stability through integrated hardware design, thereby providing solid technical support for the commercialization of FI-DAC technology;

[0080] The present invention is aimed at high-frequency sub-paths. Due to the limitation that frequency domain down-conversion will occupy a large amount of digital resources and cannot meet the real-time requirements of AWG for signals, down-conversion operation is achieved through digital mixing;

[0081] In the simulation test phase, the present invention performed well in suppressing aliasing errors, which verified the effectiveness of the filter design and emphasized the key role of system modeling in ensuring the quality of the output signal. By precisely controlling the filter parameters and system configuration, the interference that may be caused by non-target frequency signals was minimized, further improving the performance and reliability of the entire system. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] Figure 1 It is based on the structure and error diagram of two-way FI-DAC signal generator;

[0083] Figure 2 It is a schematic diagram of the mixing performance of digital domain conversion;

[0084] Figure 2 (a) is a schematic diagram of the ideal down-conversion principle in the digital domain;

[0085] Figure 2 (b) is a schematic diagram of the actual down-conversion aliasing performance in the digital domain;

[0086] Figure 2 (c) is a schematic diagram showing the general relationship between the sampling frequency and the mixing frequency in the digital domain;

[0087] Figure 3 It is a schematic diagram of the aliasing performance of analog domain transformation;

[0088] Figure 3 (a) is a schematic diagram of the analog filter guard band setting to avoid signal aliasing caused by low-frequency sub-channel DAC conversion;

[0089] Figure 3(b) is a schematic diagram of the analog filter guard band setting to avoid signal aliasing caused by high-frequency sub-channel DAC conversion;

[0090] Figure 3 (c) is a schematic diagram of setting the analog filter guard band to avoid signal aliasing caused by mixer characteristics;

[0091] Figure 4 This is the comparison diagram of the amplitude-frequency response of the low-frequency subband signal of the true model before and after passing through LPF_1;

[0092] Figure 4 (a) is a schematic diagram of direct output of low-frequency sub-band signals;

[0093] Figure 4 (b) is a schematic diagram of the output of the low-frequency subband signal after LPF_1 correction;

[0094] Figure 5 It is the comparison diagram of the amplitude-frequency response of the high-frequency sub-band signal of the simulation model before and after passing through LPF_1;

[0095] Figure 5 (a) is a schematic diagram of direct output of high-frequency sub-band signals;

[0096] Figure 5 (b) is a schematic diagram of the output of the high-frequency sub-band signal after LPF_1 correction;

[0097] Figure 6 It is the comparison diagram of the amplitude-frequency response of the high-frequency sub-band signal of the simulation model before and after being up-converted by HPF_3 and LPF_3;

[0098] Figure 6 (a) is a schematic diagram of direct output of up-converted signals;

[0099] Figure 6 (b) is a schematic diagram of the output of the up-converted signal after correction by HPF_3 and LPF_3;

[0100] Figure 7 It is a schematic diagram of the amplitude-frequency response of the combined signal output by the simulation model;

[0101] Figure 8 It is a schematic diagram of the specific implementation method for ADMM-SQP. DETAILED DESCRIPTION

[0102] Specific implementation method 1: Figures 1 to 8 As shown, a modeling method for aliasing error suppression of a FI-DAC system includes the following specific steps:

[0103] Step 1: Through the systematic analysis of the aliasing error of the FI-DAC system, a joint modeling method of the digital domain and the analog domain is designed; the ideal digital mixing output includes upper and lower sideband signals, and the mixing output signal is:

[0104]

[0105] Select the band signal as the target down-conversion signal; in formula (1), A RF is the amplitude of the RF signal, ω RF Indicates the frequency of the radio frequency, A LO is the amplitude of the local oscillator signal, ω LO Indicates the frequency of the local oscillator;

[0106] The sampling frequency of the digital signal and the local oscillator signal is f s , affected by signal sampling, the output spectrum is periodically sampled at the digital domain sampling frequency f s To extend;

[0107] When the extended signal is greater than the down-converted signal;

[0108]

[0109] In formula (2), represents the transition bandwidth of the digital low-pass filter H0, f LO represents the frequency of the local oscillator, ω 2 Indicates the high-frequency sub-path termination frequency, that is, the total output bandwidth of the FI-DAC system, ω H1 Indicates the starting frequency of the FI-DAC high-frequency sub-circuit, f s Represents the digital domain sampling frequency, which is equal to M times the DAC sampling frequency f sm , and satisfies

[0110]

[0111] When the extended signal completely overlaps with the target down-converted signal;

[0112] The extended signal caused by sampling completely overlaps with the target down-converted signal obtained, and at this time,

[0113] f s -f LO -ω 2 =f LO -ω 2 (4)

[0114] f s -f LO -ω H1 =f LO -ω H1 (5)

[0115] That is, the digital domain sampling frequency f s Equal to 2 times the local oscillator frequency f LO ;

[0116] Mf sm =2f LO (6)

[0117] In formula (6), M represents the downsampling multiple, f sm DAC chip sampling frequency.

[0118] 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; by changing the modeling system, set a guard band to correct the aliasing error, and the guard band is used to place the filter transition band;

[0119] When the sampling characteristics of the DAC chip bring about aliasing;

[0120] The stopband frequencies of the low-pass filters LPF_L1 and LPF_H1 are respectively less than the first extension frequency caused by the DAC sampling of the low-frequency signal and the high-frequency signal baseband signal output in the digital domain; the passband should include all frequencies of the baseband signal, that is, the stopband and passband cutoff frequencies meet the following requirements:

[0121]

[0122] In formulas (7) and (8), Ω L1 Indicates the passband cutoff frequency of the low-pass filter LPF_L1 in the low-frequency sub-path, In the low-frequency sub-path, the transition band bandwidth of the low-pass filter LPF_L1, ω L1 Indicates the end frequency of the low-frequency sub-path. Ω H1 Indicates the passband cutoff frequency of the low-pass filter LPF_H1 in the high-frequency sub-circuit, Indicates the transition band bandwidth of the low-pass filter LPF_H1 in the high-frequency sub-path;

[0123] The restriction on filter LPF_H1 in the high-frequency sub-circuit is not as strict as that on filter LPF_L1 in the high-frequency sub-circuit, so LPF_L1 is used instead;

[0124] When mixer hardware characteristics introduce aliasing;

[0125] The high-frequency sub-path signal is restored to an up-converted signal through an actual analog mixer, where the analog mixer is a nonlinear device. Compared with an ideal multiplier, the output signal of the mixer includes upper and lower sideband signals. For any dual-frequency excitation input nonlinear device, assuming A IF is the amplitude of the intermediate frequency signal, ω IF is the frequency of the intermediate frequency, A LO is the amplitude of the local oscillator signal, ω LO is the frequency of the local oscillator. For the convenience of description, it is assumed that AIF =A LO =A;

[0126] v in1 =Acos(ω IF t)

[0127] v in2 =Acos(ω LO t)

[0128] v i =v in1 +v in2 =A[cos(ω IF t)+cos(ω LO t)](9),

[0129] Its output response is expressed as:

[0130]

[0131] In formulas (9) and (10), v in1 represents the input signal of port 1 of the three-port network, which is the input intermediate frequency signal in this paper, v in2 represents the input port 2 of the three-port network, which is the input local oscillator signal in this paper, v i represents the combined input signal of port 1 and port 2, t represents time, v out k represents the nonlinear transformation output signal of port 3 of the three-port network. The output of the nonlinear system will contain the fundamental frequency component and harmonic component of the input signal, as well as the intermodulation component introduced by the intermodulation between the input signals. 1 The first-order coefficient represents the linear gain of the system, the direct contribution of the fundamental frequency component of the input signal to the output. 2 The second-order coefficient represents the second-order nonlinear response characteristics of the system, including the second harmonic and intermodulation components. 3 The third-order coefficient represents the third-order nonlinear response characteristics of the system, including third-order harmonics and high-order intermodulation components.

[0132] The low-pass filter LPF_H2 and the high-pass filter HPF_H3 form a bandpass filter, whose cutoff frequency should be smaller than the local oscillator leakage frequency caused by the mixer's non-ideal behavior and the upper sideband image frequency obtained by up-conversion, and larger than the high-frequency baseband signal frequency; the passband includes all frequencies of the baseband signal, that is, the stopband and passband cutoff frequencies meet the following requirements:

[0133]

[0134] In formulas (11) and (12), Ω H2 Indicates the passband cutoff frequency of the low-pass filter LPF_H2 in the high-frequency sub-path, Indicates the transition band bandwidth of the low-pass filter LPF_H2 in the high-frequency sub-path, ω 2 Indicates the high-frequency sub-path termination frequency, Ω H3 Indicates the passband cutoff frequency of the high-pass filter HPF_H3 in the high-frequency sub-circuit, Represents the transition band bandwidth of the low-pass filter HPF_H3 in the high-frequency sub-path.

[0135] Step 3: In 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; use ADMM-SQP to solve the key frequency points of the system, including:

[0136] The modeling process is converted into an optimization problem, that is, to achieve the FI-DAC system modeling with the maximum output bandwidth while satisfying certain constraints. The mathematical programming problem of nonlinear constraints satisfies:

[0137]

[0138] In formula (13), Ω TRAM represents the transition bandwidth of the crossover filter, that is, the overlapping bandwidth of the low-frequency sub-path and the high-frequency sub-path, c 1 (x) represents constraint 1, c 2 (x) represents constraint 2, c 3 (x) represents constraint 3, c 4 (x) represents constraint 4, c 5 (x) represents constraint 5, c 6 (x) represents constraint 6.

[0139] Since the transition band bandwidth of the filter is consistent with the filter passband cutoff frequency domain, So we introduce the parameter Among them (R mn ) N is the Jacobi elliptic function of the corresponding filter.

[0140] s L1 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the low-frequency sub-path LPF_L1 filter, s H2 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the high-frequency sub-path LPF_H2 filter, s H3 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the high-frequency sub-path HPF_H3 filter;

[0141] Solve a QP subproblem of the form:

[0142]

[0143] In formula (14), d represents the value at the current iteration point x k Based on the search direction vector, H k represents the Hesse matrix with respect to Lagrange, c i (x k ) represents the i-th constraint condition. At the starting point of each iteration, the SQP algorithm constructs an approximate Lagrangian function using the quadratic approximation of the objective function and the constraint function, and calculates the Hessian matrix corresponding to the function in each iteration process;

[0144]

[0145] In formula (15), λ k represents the Lagrange multiplier, I represents the active constraint index set, which plays a screening role in the constrained optimization problem and is used to determine the constraints that need to be considered in the Hessian matrix.

[0146] The target variable x is decomposed into sub-problems about variables xx, yy and zz, respectively. The target variable x refers to the total output bandwidth ω 2 , xx, yy, zz correspond to the transition bandwidth of the target filter respectively, Consider the following linearly constrained multi-block optimization:

[0147] min f(x)=f(xx,yy,zz)=θ 1 (xx)+θ 2 (yy)+θ 3 (zz)

[0148] stc(x)=c(xx,yy,zz)=A 1 xx+A 2 yy+A 3 zz-b=0,xx∈X,yy∈Y,zz∈Z(16)

[0149] when Always holds true, that is, after optimizing the inequality constraints, the original c_4 constraints always hold true, which can be converted to the following using the matrix expression method:

[0150]

[0151] when Always holds true, that is, after optimizing the inequality constraints, the constraints are simplified and modified and converted into the matrix expression method:

[0152]

[0153] The augmented Lagrangian function for formula (16) is

[0154]

[0155] Where λ is the Lagrange multiplier, β is the quadratic penalty parameter, and the ADMM algorithm gradually solves the problem by alternately optimizing xx, yy, zz and updating λ;

[0156]

[0157] The objective function is rewritten as:

[0158]

[0159] Among them, H k It is a symmetric approximation of the Hessian matrix of the complete Lagrangian function with respect to the original variable, with block diagonal characteristics, and is used to describe the local curvature of the objective function at that point;

[0160]

[0161] At this time, the augmented Lagrangian function for the QP subproblem is:

[0162]

[0163] Therefore, the following ADMM-SQP mathematical model is converted into three QP sub-problems:

[0164]

[0165] The solution of each QP subproblem satisfies the KKT condition by introducing multipliers α and Lagrange multipliers λ to ensure that the optimal solution of the problem is found at point k+1;

[0166]

[0167] The descent property of the search direction of the augmented Lagrangian function at k satisfies:

[0168]

[0169] The descent property of the search direction of the multiplier λ at k satisfies:

[0170]

[0171] Step 4: Reduce system complexity through integrated hardware design.

[0172] Proven results

[0173] like Figure 4 and Figure 5As shown, the modeling method is tested under simulation conditions for its aliasing error suppression effect. The system modeling verification under simulation conditions shows that the low-pass filter (LPF_1) can achieve a suppression effect of more than -45dB for the aliasing error caused by DAC sampling extension.

[0174] like Figure 6 As shown in the figure, the high-pass filter (HPF_2) and the low-pass filter (LPF_3) can achieve a suppression effect of more than -45dB for the aliasing error caused by the non-ideal characteristics of the mixer. Compared with the processing of out-of-band image signals, the filter group shows better suppression performance in the leakage processing of in-band input signals.

[0175] like Figure 7 As shown in the figure, during the combining process of the two signals, the combined output signal and each sub-signal show good flatness and smoothness within the target frequency band, and the aliasing error caused by signal leakage can be ignored, which once again confirms the effectiveness of system modeling and filter design in reducing the aliasing error caused by non-target frequency signals.

[0176] The above is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement made to the above embodiments without departing from the content of the technical solution of the present invention, based on the technical essence of the present invention, within the spirit and principles of the present invention, still fall within the protection scope of the technical solution 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: Design a joint modeling method between digital domain and analog domain by systematically analyzing the aliasing error of the FI-DAC system; Step 2: Introduce a frequency matching iterative optimization algorithm that combines ADMM and SQP algorithms to optimize key frequency parameters of the system, reduce resource consumption of digital domain error calibration, and thus improve the real-time performance of signal processing; 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 precision of system signal output; Step 4: Reduce system complexity through integrated hardware design.

2. A modeling method for aliasing error suppression of a FI-DAC system according to claim 1, characterized in that: Step 1 specifically includes: The ideal digital mixer output includes upper and lower sideband signals, A RF is the amplitude of the RF signal, f RF is the frequency of the radio frequency, A LO is the amplitude of the local oscillator signal, f LO is the frequency of the local oscillator, and the mixing output signal is: Select the band signal as the target down-conversion signal; in formula (1), A RF is the amplitude of the RF signal, ω RF Indicates the frequency of the radio frequency, A LO is the amplitude of the local oscillator signal, ω LO Indicates the frequency of the local oscillator; The sampling frequency of the digital signal and the local oscillator signal is f s , affected by signal sampling, the output spectrum is periodically sampled at the digital domain sampling frequency f s To extend; When the extended signal is greater than the down-converted signal; In formula (2), represents the transition bandwidth of the digital low-pass filter H0, f LO represents the frequency of the local oscillator, ω2 represents the termination frequency of the high-frequency sub-path, that is, the total output bandwidth of the FI-DAC system, ω H1 Indicates the starting frequency of the FI-DAC high-frequency sub-circuit, f s Represents the digital domain sampling frequency, which is equal to M times the DAC sampling frequency f sm , and satisfies When the extended signal completely overlaps with the target down-converted signal; The extended signal caused by sampling completely overlaps with the target down-converted signal obtained, and at this time, f s -f LO -ω2=f LO -ω2(4), f s -f LO -oh H1 =f LO -oh H1 (5), That is, the digital domain sampling frequency f s Equal to 2 times the local oscillator frequency f LO ; Mf sm =2f LO (6), In formula (6), M represents the downsampling multiple, f sm DAC chip sampling frequency.

3. A modeling method for aliasing error suppression of 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, which is used to place the filter transition band; When the sampling characteristics of the DAC chip bring about aliasing; The stopband frequencies of the low-pass filters LPF_L1 and LPF_H1 are respectively less than the first extension frequency caused by the DAC sampling of the low-frequency signal and the high-frequency signal baseband signal output in the digital domain; the passband should include all frequencies of the baseband signal, that is, the stopband and passband cutoff frequencies meet the following requirements: In formulas (7) and (8), Ω L1 Indicates the passband cutoff frequency of the low-pass filter LPF_L1 in the low-frequency sub-path, In the low-frequency sub-path, the transition band bandwidth of the low-pass filter LPF_L1, ω L1 Indicates the end frequency of the low-frequency sub-path. Ω H1 Indicates the passband cutoff frequency of the low-pass filter LPF_H1 in the high-frequency sub-circuit, Indicates the transition band bandwidth of the low-pass filter LPF_H1 in the high-frequency sub-path; The restriction on filter LPF_H1 in the high-frequency sub-circuit is not as strict as that on filter LPF_L1 in the high-frequency sub-circuit, so LPF_L1 is used instead; When mixer hardware characteristics introduce aliasing; The high-frequency sub-path signal is restored to an up-converted signal through an actual analog mixer, where the analog mixer is a nonlinear device. Compared with an ideal multiplier, the output signal of the mixer includes upper and lower sideband signals. For any dual-frequency excitation input nonlinear device, assuming A IF is the amplitude of the intermediate frequency signal, ω IF is the frequency of the intermediate frequency, A LO is the amplitude of the local oscillator signal, ω LO is the frequency of the local oscillator. For the convenience of description, it is assumed that A IF =A LO =A; v in1 =Acos(ω IF t) v in2 =Acos(ω LO t) v i =v in1 +v in2 =A[cos(ω IF t)+cos(ω LO t)](9), Its output response is expressed as: In formulas (9) and (10), v in1 represents the input signal of port 1 of the three-port network, which is the input intermediate frequency signal in this paper, v in2 represents the input port 2 of the three-port network, which is the input local oscillator signal in this paper, v i represents the combined input signal of port 1 and port 2, t represents time, v out Represents the nonlinear transformation output signal of port 3 of the three-port network. The output of the nonlinear system will contain the fundamental frequency component and harmonic component of the input signal, as well as the intermodulation component introduced by the intermodulation between the input signals. k1 is the first-order coefficient, which represents the linear gain of the system and the direct contribution of the fundamental frequency component of the input signal to the output. k2 is the second-order coefficient, which represents the second-order nonlinear response characteristics of the system, including the second harmonic and intermodulation components. k3 is the third-order coefficient, which represents 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 form a bandpass filter, whose cutoff frequency should be smaller than the local oscillator leakage frequency caused by the mixer's non-ideal behavior and the upper sideband image frequency obtained by up-conversion, and larger than the high-frequency baseband signal frequency; the passband includes all frequencies of the baseband signal, that is, the stopband and passband cutoff frequencies meet the following requirements: In formulas (11) and (12), Ω H2 Indicates the passband cutoff frequency of the low-pass filter LPF_H2 in the high-frequency sub-path, In the high-frequency sub-path, the transition band bandwidth of the low-pass filter LPF_H2 is represented, ω2 represents the termination frequency of the high-frequency sub-path, Ω H3 Indicates the passband cutoff frequency of the high-pass filter HPF_H3 in the high-frequency sub-circuit, Represents the transition band bandwidth of the low-pass filter HPF_H3 in the high-frequency sub-path.

4. A modeling method for aliasing error suppression of a FI-DAC system according to claim 1, characterized in that: Using ADMM-SQP to solve the key frequency points of the system specifically includes: The modeling process is converted into an optimization problem, that is, to achieve the FI-DAC system modeling with the maximum output bandwidth while satisfying certain constraints. The mathematical programming problem of nonlinear constraints satisfies: In formula (13), Ω TRAM represents the transition band bandwidth of the crossover filter, that is, the overlapping band bandwidth of 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. Since the transition band bandwidth of the filter is consistent with the filter passband cutoff frequency domain, Therefore, the parameter Among them (R mn ) N is the Jacobi elliptic function of the corresponding filter. s L1 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the low-frequency sub-path LPF_L1 filter, s H2 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the high-frequency sub-path LPF_H2 filter, s H3 (N) represents the conversion coefficient between the passband cutoff frequency and transition band bandwidth of the high-frequency sub-path HPF_H3 filter; Solve a QP subproblem of the form: In formula (14), d represents the value at the current iteration point x k Based on the search direction vector, H k represents the Hesse matrix about Lagrange, c i (x k ) represents the i-th constraint condition. At the starting point of each iteration, the SQP algorithm constructs an approximate Lagrangian function using the quadratic approximation of the objective function and the constraint function, and calculates the Hessian matrix corresponding to the function in each iteration process; In formula (15), λ k represents the Lagrange multiplier, I represents the active constraint index set, which plays a screening role in the constrained optimization problem and is used to determine the constraints that need to be considered in the Hessian matrix; The target variable x is decomposed into sub-problems about variables xx, yy and zz. The target 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: min f(x)=f(xx,yy,zz)=θ1(xx)+θ2(yy)+θ3(zz) st c(x)=c(xx,yy,zz)=A1xx+A2yy+A3zz-b=0, xx∈X,yy∈Y,zz∈Z (16) when Always holds true, that is, after optimizing the inequality constraints, the original c_4 constraints always hold true, which can be converted to the following using the matrix expression method: when Always holds true, that is, after optimizing the inequality constraints, the constraints are simplified and modified and converted into the matrix expression method: The augmented Lagrangian function for formula (16) is Where λ is the Lagrange multiplier, β is the quadratic penalty parameter, and the ADMM algorithm gradually solves the problem by alternately optimizing xx, yy, zz and updating λ; The objective function is rewritten as: Among them, H k It is a symmetric approximation of the Hessian matrix of the complete Lagrangian function with respect to the original variable, with block diagonal characteristics, and is used to describe the local curvature of the objective function at that point; At this time, the augmented Lagrangian function for the QP subproblem is: Therefore, the following ADMM-SQP mathematical model is converted into three QP sub-problems: The solution of each QP subproblem satisfies the KKT condition by introducing multipliers α and Lagrange multipliers λ to ensure that the optimal solution of the problem is found at point k+1; The descent property of the search direction of the augmented Lagrangian function at k satisfies: The descent property of the search direction of the multiplier λ at k satisfies:

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