Multi-channel FIADC design method based on second-order cone programming

Abstract

The invention provides a multi-channel FIADC design method based on second-order cone programming, belongs to the technical field of filter optimization, and aims to solve the problem that distortion-free recovery of signals cannot be realized due to errors in a reconstruction process of a digital reconstruction filter in an FIADC model. Comprising the following steps: S1, constructing initial perfect reconstruction conditions of a sampling filter bank and a digital reconstruction filter; s2, a sampling filter module is designed by adopting a standard filter design method, output signal frequency response of the sampling filter module is subjected to instantiation, and preliminary perfect reconstruction conditions are rewritten; s3, designing a digital reconstruction filter module in combination with an SOCP algorithm and the rewritten perfect reconstruction condition; and S4, completing the design of a multi-channel FIADC model, and verifying the error between the frequency spectrum of the reconstructed signal and the frequency spectrum obtained by directly sampling the broadband signal. Compared with an existing algorithm, the reconstruction error of the whole FIADC system is greatly reduced and is only 0.03, and distortion-free recovery of signals is better facilitated.

Description

Technical Field

[0001] The invention relates to a multi-channel FIADC design method based on second-order cone programming, and belongs to the technical field of filter optimization. Background Art

[0002] In recent years, continuous breakthroughs in global semiconductor manufacturing processes have significantly improved ADC performance. However, due to the fundamental constraints of the Nyquist sampling theorem, the effective bandwidth of a single ADC device is typically limited to only 50% of its sampling frequency. This inherent limitation makes it difficult for traditional single-channel ADCs to meet the dual requirements of high sampling rate and high resolution for modern wideband signal processing. To improve this situation, the sampling rate needs to be increased. A popular approach in the current technological landscape is multi-channel interleaved sampling, which utilizes multiple low-speed ADCs for parallel sampling to increase the overall sampling rate. This involves two steps: sampling and reconstruction. Two typical types of this technique are time-interleaved and frequency-interleaved.

[0003] In frequency-interleaving technology, each sub-ADC is controlled by the same operating clock during operation. Compared with time-interleaving technology, the sensitivity to ADC operating clock phase matching is greatly reduced, and the system processing bandwidth is equal to the sum of the device bandwidths of all channel ADCs. It is particularly suitable for high-speed sampling scenarios of broadband signals.

[0004] Acquisition systems built using frequency-interleaved technology are often called frequency-interleaved analog-to-digital conversion systems (FIADC). With the rapid development of modern communication systems, FIADC has become widely used. While this approach effectively distributes the overall high sampling rate burden to individual sub-ADCs, it introduces a new challenge: the output signals of the individual low-speed sub-ADCs must be integrated under certain conditions, ensuring that the output digital signal is a distortion-free reproduction of the original signal and minimizing reconstruction errors.

[0005] The analog sampling filter in the FIADC model is a continuous-time filter with continuously varying amplitude and time dimensions. However, this analog characteristic also makes it almost impossible to achieve perfect reconstruction. Therefore, the greatest expected effect of this model is to design and optimize the analog sampling filter and the digital reconstruction filter in order to minimize the reconstruction error. Therefore, when the analog filter is determined, the error caused by the synthesis of the digital filter should be minimized. Summary of the Invention

[0006] In order to solve the problem that the error exists in the reconstruction process of the digital reconstruction filter in the FIADC model, resulting in the inability to achieve distortion-free signal recovery, the present invention proposes a multi-channel FIADC design method based on second-order cone programming.

[0007] The technical solution adopted by the present invention to solve the above problems is: the present invention comprises the following steps:

[0008] Step 1: The broadband analog input signal x(t) is input to the FIADC model and passes through the analog filter module, sampling filter module, and digital reconstruction filter module in sequence. The output signal of the digital reconstruction filter module is merged to construct the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter.

[0009] Step 2: Use the standard filter design method to design a sampling filter module, instantiate the output signal frequency response of the designed sampling filter module, and rewrite the preliminary perfect reconstruction condition;

[0010] Step 3: Design a digital reconstruction filter module by combining the SOCP algorithm and the rewritten perfect reconstruction condition;

[0011] Step 4: Pass the broadband analog input signal x(t) through the analog filter module, the designed sampling filter module, and the designed digital reconstruction filter module in sequence, synthesize the output results of the digital reconstruction filter module and perform inverse Fourier transform to reconstruct the original broadband signal spectrum characteristics that are completely consistent with the results of direct sampling of the original broadband signal.

[0012] Furthermore, the FIADC model in step 1 includes an analog filter module, a sampling filter module, and a digital reconstruction filter module;

[0013] The analog filter module consists of M, M∈[1,2,...,m] analog filters, each analog filter is a channel, and the sub-band bandwidth corresponding to each analog filter is 1 / M of the total bandwidth;

[0014] The sampling filter module is composed of M parallel sampling filters, which is used to convert the analog sub-band signal output by the analog filter into a digital signal from the frequency dimension;

[0015] The digital reconstruction filter module is composed of M digital synthesis filters designed using the SOCP algorithm and is used to suppress noise on digital signals.

[0016] Furthermore, the steps of constructing the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter in step 1 include:

[0017] Step 1.1: Use an M-channel analog filter bank to perform frequency division and filtering on the broadband analog input signal x(t) to obtain the analog subband signal x m (t);

[0018] Step 1.2: Simulate subband signal x m (t) After the sampling period is MTS The sampling filter group performs low-speed ADC quantization conversion. In each channel, the analog sub-band signal is converted into a digital signal from the frequency dimension through Fourier transform.

[0019] Step 1.3: Perform M-fold interpolation upsampling on the digital signal of each channel to a high-speed sampling rate, and pass it through a digital reconstruction filter bank to obtain the noise-suppressed signal Y m (e jω );

[0020] Step 1.4: Limit the bandwidth of the input signal x(t) to |Ω| ≤ π / T S , r value∈[0,M-1], where Ω is the bandwidth of the signal, T S is the sampling period of the sampling filter, and repeat steps 1.1 to 1.3 to reconstruct the output signal Y of all digital reconstruction filters. m (e jω ) are combined to obtain signal Y m ′(e jω ), based on the combined signal Y m ′(e jω ) in the system transfer function T r (e jω ), and obtain the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter;

[0021] The Fourier transform expression of the mth channel is:

[0022]

[0023] In formula (1), φ m (jΩ) is the frequency response of the sampling filter of the mth channel, X(jΩ) is the frequency response of the input signal, T S is the sampling period of the sampling filter, MT S is the sampling period of the sampling filter bank, is the bandwidth of the input signal, X m (e jω ) is the digital signal converted by the mth channel;

[0024] The signal Y after noise suppression m (e jω ) is calculated as:

[0025]

[0026] In formula (2), Ψ m (e jω ) is the frequency response of the digital reconstruction filter bank;

[0027] The combined signal Y m ′(e jω ) is calculated as:

[0028]

[0029] In formula (3), T r (e jω ) is the system transfer function, and its calculation formula is:

[0030]

[0031] The expression of the preliminary perfect reconstruction condition is:

[0032]

[0033] In formula (5), τ is the delay factor. When r = 0, the system transfer function is T0(e jω ), T0(e jω ) is the system distortion function, corresponding to the gain and phase of the digital reconstruction filter. When r∈[1,M-1], the system transfer function is T r (e jω ), T r (e jω ) is the system aliasing function, which corresponds to the aliasing error of the digital reconstruction filter.

[0034] Furthermore, the rewriting of the preliminary perfect reconstruction conditions in step 2 specifically includes:

[0035] Set the digital reconstruction filter Ψ m The number of taps of (z) is L. The frequency response of the digital reconstruction filter is adjusted based on the tap operation. The frequency response of the output signal of the designed sampling filter module is instantiated and combined with the frequency response of the adjusted digital reconstruction filter to rewrite the preliminary perfect reconstruction condition.

[0036] The expression for adjusting the frequency response of the digital reconstruction filter is:

[0037]

[0038] In formula (6), is the digital reconstruction filter coefficient of the mth channel to be solved;

[0039] The expression of the perfect refactoring condition after rewriting is:

[0040]

[0041] In formula (7), A is the coefficient matrix of the analog filter, T r (e jω) is the system transfer function.

[0042] Furthermore, step 3 specifically includes:

[0043] Step 3.1: Construct a convex optimization problem based on the rewritten perfect reconstruction condition Where A is the coefficient matrix of the known analog filter, is the FIR structure coefficient matrix of the digital reconstruction filter, T is the system transfer function in formula (7);

[0044] Step 3.2: Convert the convex optimization problem into a second-order cone programming problem;

[0045] Step 3.3: Based on the SOCP algorithm, integrate the objective function and all constraints in the second-order cone programming problem into the standard form of the SOCP problem;

[0046] Step 3.4: Use the SOCP solver to solve the SOCP problem and obtain the synthesis coefficients of M groups of digital reconstruction filters. Based on the synthesis coefficients of the digital reconstruction filters, synthesize the frequency responses of the M signals and reconstruct the spectral characteristics of the original broadband signal through inverse Fourier transform. Compare the spectral characteristics of the original broadband signal with the result of direct sampling of the original broadband signal. If the error is not greater than the preset value, adjust the constraints and the optimization objectives in the objective function until the error is less than the preset value, and complete the design of the digital reconstruction filter module.

[0047] Furthermore, the expression of the second-order cone programming problem in step 3.2 is:

[0048]

[0049] In formula (8), c T x is the objective function, x is the decision variable vector of the second-order cone programming problem, c is the coefficient vector of the objective function, is a linear constraint, representing a second-order cone constraint, A i is the coefficient matrix of linear constraints, b i is the constant vector of linear constraints, d i is the coefficient constant of the linear term associated with the second-order cone constraint, e i is a constant term associated with the second-order cone constraint.

[0050] Furthermore, the integration of SOCP problems in step 3.3 specifically includes:

[0051] Step 3.3.1: Discretize the frequency response of the continuous analog filter into a set of discrete frequency points. For each discrete point, calculate the frequency response of the digital reconstruction filter.

[0052] Step 3.3.2: Use the minimax criterion to obtain the maximum error at all discrete frequency points and construct the minimum value problem of the maximum error;

[0053] Step 3.3.3: Integrate the objective function and all constraints, and combine them with the large error minimization problem to obtain the standard form of the SOCP problem, where the standard form of the SOCP problem is used to minimize the maximum sum of the errors of each channel;

[0054] The objective function expression of the minimax criterion is:

[0055] min h max ω∈Ω |H(e jω )-D(e jω )| (9);

[0056] In formula (9), H(e jω ) is the frequency response of the digital reconstruction filter, D(e jω ) is the expected frequency response of the digital reconstruction filter;

[0057] The expression of the maximum error minimization problem is:

[0058]

[0059] In formula (10), E k (e jω ) is the frequency response error value of the digital reconstruction filter, and W(ω) is a positive weighting parameter used to express the relative importance between distortion and aliasing errors. Its expression is:

[0060]

[0061] In formula (11), f is the FIR filter coefficient vector, t k is the ideal value vector, T idea,k is the ideal frequency response value;

[0062] The standard form of the SOCP problem is:

[0063]

[0064] The beneficial effects of the present invention are:

[0065] 1. The present invention dynamically balances the optimization priorities of different frequency bands by adjusting the weight vector and applying the SOCP algorithm, which can be quickly solved with the help of the interior point method, and is particularly suitable for high-order filters.

[0066] 2. The FIADC model designed in the present invention can handle multiple types of constraints simultaneously, allowing designers to optimize the design of the FIR filter while meeting multiple performance indicators. The reconstruction error of the entire system is reduced, which is more conducive to distortion-free signal restoration. The coefficients of the synthesized digital filter are closer to the ideal values, and the spectral ripple is significantly reduced. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 A schematic flow chart of a multi-channel FIADC design method based on second-order cone programming provided by the present invention;

[0068] Figure 2 The spectrum diagram of the low-pass filter obtained by directly sampling the broadband signal provided by the present invention;

[0069] Figure 3 The spectrum diagram of the high-pass filter obtained by directly sampling the broadband signal provided by the present invention;

[0070] Figure 4 This is a spectrum diagram of the low-pass filter obtained by the minimax method provided by the present invention.

[0071] Figure 5 This is a spectrum diagram of the high-pass filter obtained by the minimax method provided by the present invention. DETAILED DESCRIPTION

[0072] Combine Figure 1-5 This embodiment is described as follows. Figure 1 As shown, the steps of a multi-channel FIADC design method based on second-order cone programming described in this embodiment include:

[0073] S1: Constructing the initial perfect reconstruction conditions of the sampling filter bank and digital reconstruction filter;

[0074] S101: A broadband analog input signal x(t) is input to the FIADC model. The FIADC model consists of an analog filter module, a sampling filter module, and a digital reconstruction filter module. The analog filter module consists of M, M∈[1,2,...,m] analog filters, each analog filter being a channel, and the subband bandwidth corresponding to each analog filter being 1 / M of the total bandwidth. The sampling filter module consists of M parallel sampling filters, and is used to convert the broadband analog signal into a digital signal from the frequency dimension for the analog subband signal output by the analog filter. The digital reconstruction filter module consists of M digital synthesis filters designed using the SOCP algorithm, and is used to suppress noise on the digital signal.

[0075] S102: Perform frequency division and filtering on the broadband analog input signal x(t) through an M-channel analog filter bank to obtain an analog sub-band signal x m(t), analog subband signal x m (t) After the sampling period is MT S The sampling filter group performs low-speed ADC quantization conversion. In each channel, the analog sub-band signal is converted into a digital signal from the frequency dimension through Fourier transform.

[0076] The Fourier transform expression of the mth channel is:

[0077]

[0078] In formula (1), φ m (jΩ) is the frequency response of the sampling filter of the mth channel, X(jΩ) is the frequency response of the input signal, T S is the sampling period of the sampling filter, MT S is the sampling period of the sampling filter bank, is the bandwidth of the input signal, X m (e jω ) is the digital signal converted by the mth channel;

[0079] S103: Perform M-fold interpolation upsampling on the digital signal of each channel to a high-speed sampling rate, and pass it through a digital reconstruction filter bank to obtain a noise-suppressed signal Y m (e jω ), which is calculated as follows:

[0080]

[0081] In formula (2), Ψ m (e jω ) is the frequency response of the digital reconstruction filter bank;

[0082] S104: Limit the bandwidth of the input signal x(t) to set |Ω|≤π / T S , r value∈[0,M-1], where Ω is the bandwidth of the signal, T S is the sampling period of the sampling filter, and repeats S102-S103 to reconstruct the output signal Y of all digital reconstruction filters. m (e jω ) are combined to obtain signal Y m ′(e jω ), which is calculated as follows:

[0083]

[0084] Based on the combined signal Y m ′(e jω ) in the system transfer function T r (e jω), as shown in formula (4), the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter are obtained, as shown in formula (5);

[0085]

[0086] In formula (5), τ is the delay factor. When r = 0, the system transfer function is T0(e jω ), T0(e jω ) is the system distortion function, corresponding to the gain and phase of the digital reconstruction filter. When r∈[1,M-1], the system transfer function is T r (e jω ), T r (e jω ) is the system aliasing function, which corresponds to the aliasing error of the digital reconstruction filter.

[0087] S2: Design a sampling filter module using the standard filter design method, instantiate its output signal frequency response, and rewrite the preliminary perfect reconstruction condition;

[0088] For the FIADC model, a key design task is to design an analog sampling filter and a digital reconstruction filter that meet the perfect reconstruction condition. These two sets of filters can be designed sequentially or jointly. Because analog circuits are more complex to implement than digital circuits, a sequential design approach is typically used: the analog sampling filter is designed first, followed by the digital reconstruction filter, constrained by the perfect reconstruction condition.

[0089] Sequential design means that the analog filter needs to be designed independently first. The independent design of the analog sampling filter can be directly implemented based on the filter's decisive parameter indicators without being affected by other components in the system. This implementation adopts the standard filter design method. Once the analog filter design is completed, the optimal solution for the digital filter can be found based on the analog filter's frequency response and perfect reconstruction conditions. The details are as follows:

[0090] FIR structure is a common form of digital filter. Assuming that the digital reconstruction filter Ψ m (z) has L taps, and the function of the taps is to enhance or weaken the frequency response of the digital reconstruction filter, as shown in formula (6):

[0091]

[0092] In formula (6), is the digital reconstruction filter coefficient of the mth channel to be solved;

[0093] The input signal frequency is instantiated as P test points, and the preliminary perfect reconstruction condition can be rewritten as formula (7):

[0094]

[0095] In formula (7), T r (e jω ) is the system transfer function.

[0096] Let A be the coefficient matrix of the analog filter, is the FIR structure coefficient matrix of the digital reconstruction filter, then the perfect reconstruction condition can be written as Therefore, the solution of the digital synthesis filter coefficients is transformed into solving the matrix when A and T are known. Optimal solution. This is a convex optimization problem. Compared with directly using QR and matrix transformation methods to operate on the matrix, the SOCP algorithm can obtain better results.

[0097] S3: Design a digital reconstruction filter module by combining the SOCP algorithm and the rewritten perfect reconstruction condition;

[0098] Convex optimization is an important branch of mathematical optimization that studies optimization algorithms for convex problems. Convex optimization problems have convex objective functions and convex constraint sets, which means that there is a global optimal solution in the solution space of the problem. This solution can be efficiently found through a variety of numerical methods, as follows:

[0099] S301: Define the objective function and establish constraints: Second-order cone programming is a convex optimization problem. Its objective function should be to minimize the error. The maximum error is expressed by the ideal frequency response and the obtained frequency response of the analog filter. It has a specific standard form and can be expressed as:

[0100]

[0101] In formula (8), c T x is the objective function, x is the decision variable vector of the second-order cone programming problem, c is the coefficient vector of the objective function, is a linear constraint, representing a second-order cone constraint, A i is the coefficient matrix of linear constraints, b i is the constant vector of linear constraints, d i is the coefficient constant of the linear term associated with the second-order cone constraint, e i is a constant term associated with the second-order cone constraint.

[0102] S302: Using the SOCP algorithm, the objective function and all constraints in the second-order cone programming problem are integrated into the standard form of the SOCP problem;

[0103] The SOCP algorithm has two criteria to define the objective function, namely the minimax criterion and the least squares criterion, each of which has different optimization goals and application scenarios. When designing an FIR filter, only one of them needs to be used. The least squares criterion aims to minimize the sum of squares of the errors between the filter response and the ideal response over the entire frequency range. It is suitable for applications that are more sensitive to overall errors rather than single maximum errors. For example, in signal processing, you may pay more attention to the overall average performance rather than the performance in extreme cases. The minimax criterion aims to minimize the maximum deviation between the filter response and the ideal response within the frequency range. It is particularly suitable for applications that have strict requirements on worst-case performance. For example, in communication systems, it may be necessary to ensure that the performance of the filter is as consistent as possible throughout the passband. Taking into account the design background, this embodiment adopts the minimax criterion.

[0104] First, the continuous frequency response is discretized into a set of discrete frequency points, which allows the problem to be transformed into a form suitable for numerical solution. The following are the detailed steps of discretization and solution when designing FIR filters using the SOCP method:

[0105] Select a set of discrete frequency points that cover the frequency range of interest in the filter design, such as the passband and stopband. For each discrete frequency point, calculate the filter's frequency response. This typically involves evaluating the product of the filter coefficients and the complex exponential of the frequency point. For the minimax criterion, find the maximum error across all discrete frequency points and minimize this maximum error. Convert the continuous constraints to discrete form.

[0106] The objective function of the minimax criterion is shown in formula (9):

[0107] min h max ω∈Ω |H(e jω )-D(e jω )| (9);

[0108] In formula (9), H(e jω ) is the frequency response of the digital reconstruction filter, D(e jω ) is the expected frequency response of the digital reconstruction filter;

[0109] When designing the digital synthesis filter in the FIADC model using the minimax criterion, the frequency response of the required filter and the frequency response of the ideal filter should be expressed. Formula (9) can be simplified to the minimum value problem shown in formula (10):

[0110]

[0111] In formula (10), E k (ejω ) is the frequency response error value of the digital reconstruction filter, and W(ω) is a positive weighting parameter used to express the relative importance between distortion and aliasing errors. Its expression is:

[0112]

[0113] In formula (11), f is the FIR filter coefficient vector, t k is the ideal value vector, T idea,k is the ideal frequency response value;

[0114] Therefore, the minimax problem can be restated as minimizing the maximum value of the sum of the errors of each channel. The standard form of the SOCP problem is shown in formula (12):

[0115]

[0116] S304: Solve using the SOCP solver. This embodiment uses CVXPY in Python to solve the optimization problem shown in formula (11). This method can handle linear and convex quadratic constraints and obtain M groups of digital synthesis filter coefficients.

[0117] S4: Complete the design of the multi-channel FIADC model and verify the error between its reconstructed signal spectrum and the spectrum obtained by directly sampling the broadband signal;

[0118] After completing the design of the above-mentioned sampling filter module and digital reconstruction filter module, re-input the analog input signal into the FIADC model, combine the reconstructed signal through the M groups of digital synthesis filter coefficients obtained by solving, calculate the actual error, and verify whether these coefficients meet all the design constraints. If the results do not meet the requirements, it may be necessary to adjust the constraints or optimization objectives and re-solve.

[0119] In order to verify the accuracy of SOCP algorithm in solving filter coefficients, this embodiment adopts QR method and matrix transformation to directly solve filter coefficients. The spectrum of the obtained low-pass filter is as follows: Figure 2 As shown, the spectrum of the high-pass filter is Figure 3 As shown in the figure, the calculated error from the ideal value is greater than 1, which is relatively large and has more ripples.

[0120] The spectrum of the low-pass filter obtained by the minimax method is as follows Figure 4 As shown, the spectrum of the high-pass filter is Figure 5 As shown, the error from the ideal value is only 0.03, and it can be clearly seen that the ripples have been reduced.

[0121] From the above analysis, it can be seen that this embodiment uses the SOCP algorithm to solve the filter coefficients and complete the design of the sampling filter module and the digital reconstruction filter module. The reconstruction error of the designed entire FIADC system is greatly reduced compared with the existing algorithm, only 0.03, which is more conducive to the distortion-free restoration of the signal. At the same time, the coefficients of the synthesized digital filter are closer to the ideal value, and the spectral ripple is significantly reduced.

[0122] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical content disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement of the above embodiments made according to the technical essence of the present invention, within the spirit and principles of the present invention, without departing from the content of the technical solution of the present invention, shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A multi-channel FIADC design method based on second-order cone programming, characterized in that: include: Step 1: The broadband analog input signal x(t) is input to the FIADC model and passes through the analog filter module, sampling filter module, and digital reconstruction filter module in sequence. The output signal of the digital reconstruction filter module is merged to construct the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter. Step 2: Use the standard filter design method to design a sampling filter module, instantiate the output signal frequency response of the designed sampling filter module, and rewrite the preliminary perfect reconstruction condition; Step 3: Design a digital reconstruction filter module by combining the SOCP algorithm and the rewritten perfect reconstruction condition; Step 4: Pass the broadband analog input signal x(t) through the analog filter module, the designed sampling filter module, and the designed digital reconstruction filter module in sequence, synthesize the output results of the digital reconstruction filter module and perform inverse Fourier transform to reconstruct the original broadband signal spectrum characteristics that are completely consistent with the results of direct sampling of the original broadband signal.

2. The multi-channel FIADC design method based on second-order cone programming according to claim 1, characterized in that: In step 1, the FIADC model includes an analog filter module, a sampling filter module, and a digital reconstruction filter module; The analog filter module is composed of M, M∈[1,2,...,m] analog filters, each analog filter is a channel, and the sub-band bandwidth corresponding to each analog filter is 1 / M of the total bandwidth; The sampling filter module is composed of M parallel sampling filters, which are used to convert the analog sub-band signal output by the analog filter into a digital signal from the frequency dimension; The digital reconstruction filter module is composed of M digital synthesis filters designed using the SOCP algorithm and is used for noise suppression of digital signals.

3. The multi-channel FIADC design method based on second-order cone programming according to claim 1, characterized in that: The steps for constructing the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter in step 1 include: Step 1.1: Use an M-channel analog filter bank to perform frequency division and filtering on the broadband analog input signal x(t) to obtain the analog subband signal x m (t); Step 1.2: Simulate subband signal x m (t) After the sampling period is MT S The sampling filter group performs low-speed ADC quantization conversion. In each channel, the analog sub-band signal is converted into a digital signal from the frequency dimension through Fourier transform. Step 1.3: Perform M-fold interpolation upsampling on the digital signal of each channel to a high-speed sampling rate, and pass it through a digital reconstruction filter bank to obtain the noise-suppressed signal Y m (e jω ); Step 1.4: Limit the bandwidth of the input signal x(t) to |Ω| ≤ π / T S , r value∈[0,M-1], where Ω is the bandwidth of the signal, T S is the sampling period of the sampling filter, and repeat steps 1.1 to 1.3 to reconstruct the output signal Y of all digital reconstruction filters. m (e jω ) are combined to obtain signal Y′ m (e jω ), based on the combined signal Y′ m (e jω ) in the system transfer function T r (e jω ), and obtain the preliminary perfect reconstruction conditions of the sampling filter bank and the digital reconstruction filter; The Fourier transform expression of the mth channel is: In formula (1), φ m (jΩ) is the frequency response of the sampling filter of the mth channel, X(jΩ) is the frequency response of the input signal, T S is the sampling period of the sampling filter, MT S is the sampling period of the sampling filter bank, is the bandwidth of the input signal, X m (e jω ) is the digital signal converted by the mth channel; The signal Y after noise suppression m (e jω ) is calculated as: In formula (2), Ψ m (e jω ) is the frequency response of the digital reconstruction filter bank; The combined signal Y′ m (e jω ) is calculated as: In formula (3), T r (e jω ) is the system transfer function, and its calculation formula is: The expression of the preliminary perfect reconstruction condition is: In formula (5), τ is the delay factor. When r = 0, the system transfer function is T0(e jω ), T0(e jω ) is the system distortion function, corresponding to the gain and phase of the digital reconstruction filter. When r∈[1,M-1], the system transfer function is T r (e jω ), T r (e jω ) is the system aliasing function, which corresponds to the aliasing error of the digital reconstruction filter.

4. The multi-channel FIADC design method based on second-order cone programming according to claim 1, characterized in that: The rewriting of the preliminary perfect reconstruction conditions in step 2 specifically includes: Set the digital reconstruction filter Ψ m The number of taps of (z) is L. The frequency response of the digital reconstruction filter is adjusted based on the tap operation. The frequency response of the output signal of the designed sampling filter module is instantiated and combined with the frequency response of the adjusted digital reconstruction filter to rewrite the preliminary perfect reconstruction condition. The expression for adjusting the frequency response of the digital reconstruction filter is: In formula (6), is the digital reconstruction filter coefficient of the mth channel to be solved; The expression of the perfect refactoring condition after rewriting is: In formula (7), A is the coefficient matrix of the analog filter, T r (e jω ) is the system transfer function.

5. The multi-channel FIADC design method based on second-order cone programming according to claim 1, characterized in that: Step 3 specifically includes: Step 3.1: Construct a convex optimization problem based on the rewritten perfect reconstruction condition Where A is the coefficient matrix of the known analog filter, is the FIR structure coefficient matrix of the digital reconstruction filter, T is the system transfer function in formula (7); Step 3.2: Convert the convex optimization problem into a second-order cone programming problem; Step 3.3: Based on the SOCP algorithm, integrate the objective function and all constraints in the second-order cone programming problem into the standard form of the SOCP problem; Step 3.4: Use the SOCP solver to solve the SOCP problem and obtain the synthesis coefficients of M groups of digital reconstruction filters. Based on the synthesis coefficients of the digital reconstruction filters, synthesize the frequency responses of the M signals and reconstruct the spectral characteristics of the original broadband signal through inverse Fourier transform. Compare the spectral characteristics of the original broadband signal with the result of direct sampling of the original broadband signal. If the error is not greater than the preset value, adjust the constraints and the optimization objectives in the objective function until the error is less than the preset value, and complete the design of the digital reconstruction filter module.

6. The multi-channel FIADC design method based on second-order cone programming according to claim 5, characterized in that: The expression of the second-order cone programming problem in step 3.2 is: In formula (8), c T x is the objective function, x is the decision variable vector of the second-order cone programming problem, c is the coefficient vector of the objective function, ||A i x+b i ||2≤d i T x+e i is a linear constraint, representing a second-order cone constraint, A i is the coefficient matrix of linear constraints, b i is the constant vector of linear constraints, d i is the coefficient constant of the linear term associated with the second-order cone constraint, e i is a constant term associated with the second-order cone constraint.

7. The multi-channel FIADC design method based on second-order cone programming according to claim 5, characterized in that: The integration of SOCP problems in step 3.3 specifically includes: Step 3.3.1: Discretize the frequency response of the continuous analog filter into a set of discrete frequency points. For each discrete point, calculate the frequency response of the digital reconstruction filter. Step 3.3.2: Use the minimax criterion to obtain the maximum error at all discrete frequency points and construct the minimum value problem of the maximum error; Step 3.3.3: Integrate the objective function and all constraints, and combine them with the large error minimization problem to obtain the standard form of the SOCP problem, where the standard form of the SOCP problem is used to minimize the maximum sum of the errors of each channel; The objective function expression of the minimax criterion is: bad h maximum ω∈Ω |H(e jω )-D(e jω )| (9); In formula (9), H(e jω ) is the frequency response of the digital reconstruction filter, D(e jω ) is the expected frequency response of the digital reconstruction filter; The expression of the maximum error minimization problem is: In formula (10), E k (e jω ) is the frequency response error value of the digital reconstruction filter, and W(ω) is a positive weighting parameter used to express the relative importance between distortion and aliasing errors. Its expression is: In formula (11), f is the FIR filter coefficient vector, t k is the ideal value vector, T idea,k is the ideal frequency response value; The standard form of the SOCP problem is:

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