A critical sampling circuit bandwidth extension method and system

Abstract

A method and system for bandwidth extension of a critical sampling circuit, belonging to the field of signal sampling technology. The implementation method is as follows: S1, estimating the model coefficients of the squared amplitude response of the simulated channel of the critical sampling circuit through ultra-relaxed primordial duality, constructing the simulated channel response matrix, and obtaining the system response matrix by row decimation; S2, establishing the high-dimensional response matrix of the critical sampling circuit and completing parameter initialization; S3, performing iterative calculation of the extension features through elastic network regression, checking the convergence of the extension residual or the number of iterations, if satisfied, proceeding to S4, otherwise continuing iteration; S4, reconstructing the high-dimensional signal using the extension features calculated in S3, and obtaining the reconstructed signal after bandwidth extension of the critical sampling circuit through sampling rate conversion. This invention, by iteratively calculating the spectral characteristics within the target extended bandwidth of the critical sampling circuit, can recover out-of-band lost spectral information without increasing the sampling clock frequency and analog input bandwidth, achieving low-cost high-frequency signal sampling.

Description

Technical Field

[0001] This invention belongs to the field of signal sampling technology and relates to a method and system for extending the bandwidth of a critical sampling circuit. Background Technology

[0002] The sampling rate of the critical sampling circuit is twice the system's analog input bandwidth. If the input signal bandwidth exceeds the system's analog bandwidth, the "insufficient bandwidth" of the critical sampling circuit may cause signal energy outside the 3dB cutoff band to be suppressed by the system's frequency response transition band and stopband, resulting in the loss of high-frequency information. Even if the out-of-band signal is not effectively suppressed and residual energy remains, undersampling will still occur due to the "insufficient sampling rate" of the critical sampling circuit, leading to aliasing. The bandwidth extension problem of the critical sampling circuit refers to the distortionless recovery of high-frequency information outside the 3dB cutoff band of the critical sampling circuit through back-end digital signal processing without changing the sampling rate of the ADC of the critical sampling circuit. Unlike the undersampling problem of simply "insufficient sampling rate" or the bandwidth extension problem of "insufficient bandwidth," the characteristic of "insufficient bandwidth and insufficient sampling rate" makes it severely ill-posed and therefore more challenging. Currently, there is no effective solution for this type of problem. Summary of the Invention

[0003] The purpose of this invention is to provide a method and system for extending the bandwidth of a critical sampling circuit, which extends the sampling bandwidth of the circuit without increasing the sampling rate, thereby achieving low-cost high-frequency signal sampling.

[0004] The objective of this invention is achieved through the following technical solution.

[0005] This invention discloses a method for extending the bandwidth of a critical sampling circuit, comprising the following steps:

[0006] S1. Measure the amplitude-squared response of the simulated channel of the critical sampling circuit to be extended using a multi-tone or linear frequency modulated signal; using the l1 norm of the difference between the amplitude-squared response to be estimated and the measured amplitude-squared response as the objective function, by introducing an over-relaxation parameter, transform the original non-convex, non-concave fractional polynomial nonlinear programming problem with respect to the objective function into a constrained convex optimization dual problem, and use the interior-point method to obtain the coefficient vector of the amplitude-squared response to be estimated; construct a linear equation using the coefficient vector of the amplitude-squared response to be estimated, and solve it to obtain the frequency response of the simulated channel of the critical sampling circuit; construct the response matrix of the simulated channel of the critical sampling circuit, and perform row decimation on the response matrix of the simulated channel of the critical sampling circuit to obtain the system response matrix of the critical sampling circuit.

[0007] S1.1 Generate a multi-tone or linear frequency modulated (LFM) signal as the test signal. The generated test signal has I frequency points uniformly distributed within the frequency range (0, MB), where M is an integer greater than 1, and B = F.s / 2 is the analog input bandwidth of the critical sampling circuit, F s Let be the sampling rate of the ADC in the critical sampling circuit. The test signal is input into the analog channel of the critical sampling circuit, and simultaneously, the output of the analog channel is oversampled by a factor of M using an additional reference ADC, i.e., the reference ADC sampling rate is MF. s The measured amplitude-squared response of the analog channel was obtained based on the test signal and the oversampled output signal.

[0008] S1.2. The following optimization problem is constructed using the l1 norm of the difference between the squared amplitude response to be estimated and the measured squared amplitude response as the objective function.

[0009]

[0010] The squared magnitude response to be estimated is modeled using an L-order autoregressive moving average model, with the sign... For example Corresponding to the I frequency points in S1.1,

[0011]

[0012] Among them 0 L×1 Represents an L-row vector with all zeros. Let be the coefficient vector of the Lth-order squared magnitude response to be estimated;

[0013] S1.3 For i = 0, 1, ..., I-1, use the interior point method or gradient descent method to solve the following 2I convex optimization problems.

[0014]

[0015] S1.4, Introducing the over-relaxation parameter λ i The l obtained by solving using S1.3 i and υ i As λ i The upper and lower boundary constraints transform the original non-convex, non-concave fractional polynomial nonlinear programming problem constructed in S1.2 into the following constrained hyperrelaxed convex optimization dual problem.

[0016]

[0017] The coefficient vector of the squared magnitude response to be estimated can be obtained using the interior-point method or the augmented Lagrange method.

[0018] S1.5 Constructing a system of linear equations

[0019]

[0020] in

[0021]

[0022] The parameters of the frequency response to be estimated are obtained by solving (5) using the least squares method. and The frequency response of the analog channel of the critical sampling circuit is obtained by the following equation (7).

[0023]

[0024] Performing an inverse discrete Fourier transform on equation (7) yields the unit impulse response h[n] of the analog channel;

[0025] S1.6. Construct the response matrix of the simulated channel using the unit impulse response h[n] of the simulated channel:

[0026]

[0027] Where K is the order of the square matrix H. The response matrix H of the analog channel of the critical sampling circuit. K×K By performing M-fold row decimation, the system response matrix of the critical sampling circuit is obtained.

[0028] S2. Construct the feature transformation space, and combine it with the critical sampling circuit system response matrix constructed in S1 to establish the response matrix model of the critical sampling circuit to the high-dimensional input space; initialize the number of iterations, high-dimensional input space, number of extended features, index space of unlocated extended features, index space of located extended features, extended feature response matrix, and bandwidth extension residual.

[0029] S2.1 Constructing the Feature Transformation Space

[0030]

[0031] in The response matrix of the critical sampling circuit system constructed in conjunction with S1.3 Establish the response matrix model of the critical sampling circuit to the high-dimensional input space.

[0032]

[0033] Where a k Let Ξ be the kth atom of the high-dimensional input space response matrix Ξ, hereinafter referred to as Ξ as the high-dimensional response matrix;

[0034] S2.2 Let the iteration number j = 0, and the high-dimensional input space ξ = 0. K×1The number of extended features is S; the initial index space of unlocalized extended features is K = {k | k = 0, 1, ..., K / 2(1-1 / M), K / 2(1+1 / M), ..., K-2, K-1}, which is composed of the indices of unlocalized features within the target extended bandwidth in the high-dimensional input space; the index space of localized extended features... The extended feature response matrix is ​​composed of the indices of the located extended features within the target extended bandwidth in the high-dimensional input space. Bandwidth extension residual Where y is the output signal of the critical sampling circuit.

[0035] S3. Iteratively solve the extended feature vector: In each iteration, update the index space of the unlocated extended features by removing the indices of the located extended features in the high-dimensional input space within the target extended bandwidth; in the updated unlocated extended feature index space, relocate the feature atoms with the highest correlation to the bandwidth extension residual and update the feature index space; based on the updated feature index space, update the extended feature response matrix, construct the elastic network regression objective function, and use the interior point method to solve the extended feature vector; update the bandwidth extension residual based on the extended feature vector, perform a convergence check, and if it is satisfied, terminate the iteration and transfer the finally solved extended feature vector to S4; otherwise, continue the iteration.

[0036] S3.1 Let the iteration number j = j + 1, remove the index of the located extended features in the high-dimensional input space from the index space of the unlocated extended features within the target extended bandwidth, that is, update K = K\w;

[0037] S3.2, Construct matrix Λ={a} using unlocated extended features corresponding to atoms. k |k∈K}, through c=Λ T r calculates the correlation between the atoms corresponding to the unlocated extension features and the bandwidth extension residuals, and finds the atom a with the highest correlation to the bandwidth extension residuals. k’ Update the located extended feature index space w[i] = k';

[0038] S3.3, Atom a found according to S3.2 k’ Update the extended feature response matrix Ξ n =[Ξ n ,a k' Construct the following objective function based on elastic network regression.

[0039]

[0040] Where η is the regularization parameter and μ∈[0,1] is the mixing parameter. The extended feature vector is obtained by solving equation (11) using the interior point method.

[0041] S3.4, Update bandwidth extension residuals

[0042] S3.5 Convergence check: If ||r||2 is less than the set threshold or j≥S, then execute S4; otherwise, return to S3.1.

[0043] S4. Update the high-dimensional input space using the extended feature vector calculated in S3, and obtain the high-dimensional reconstructed signal by combining the feature transformation space. Perform sampling rate conversion on the high-dimensional reconstructed signal to obtain the reconstructed signal after bandwidth extension of the critical sampling circuit.

[0044] S4.1. Update the high-dimensional input space using the extended feature vectors obtained in S3:

[0045]

[0046] S4.2. Obtain the high-dimensional reconstructed signal using the high-dimensional input space and feature transformation space, i.e., x' = E H ξ; Perform sampling rate conversion on the high-dimensional reconstructed signal x', with a conversion factor of MF. s / 2B u The sampling rate is 2B. u The low-dimensional reconstructed signal x, at this time the sampling bandwidth of the critical sampling circuit is extended from the original B to B. u .

[0047] This invention also discloses a critical sampling circuit bandwidth extension system for implementing the critical sampling circuit bandwidth extension method described above. The critical sampling circuit bandwidth extension system includes a system response estimation module, an extension initialization module, an extension feature iterative calculation module, and a broadband signal reconstruction module.

[0048] The system response estimation module measures the amplitude-squared response of the simulated channel of the critical sampling circuit to be extended. It solves for the model coefficients of the amplitude-squared response to be estimated by minimizing the l1 norm of the difference between the amplitude-squared response to be estimated and the measured amplitude-squared response, thereby obtaining the frequency response and unit impulse response of the simulated channel of the critical sampling circuit. Then, it constructs the response matrix of the simulated channel of the critical sampling circuit and extracts the system response matrix of the critical sampling circuit by row extraction of the response matrix of the simulated channel of the critical sampling circuit, which is used for parameter initialization of the extension initialization module.

[0049] The extended initialization module is used to initialize the parameters of the extended feature iterative solution module using the system response matrix output by the system response estimation module, and input the initialized parameters into the extended feature iterative solution module.

[0050] The extended feature iterative solution module is used to locate the extended feature most relevant to the current extended residual within the target extended bandwidth in each iteration. It uses the located extended features to reduce the dimensionality of the high-dimensional response matrix to obtain a low-dimensional extended feature response matrix. It solves the extended feature vector through elastic network regression and updates the extended residual for the extended feature location and residual update in the next iteration. It also performs a convergence check on the extended residual or the number of iterations. If the convergence is satisfied, the solved extended feature vector is input into the broadband signal reconstruction module; otherwise, the iteration continues.

[0051] The broadband signal reconstruction module is used to update the high-dimensional input space using the extended feature vector output by the extended feature iterative solution module, and obtain the high-dimensional reconstructed signal by combining the feature transformation space. The sampling rate of the high-dimensional reconstructed signal is then converted to obtain the reconstructed signal after bandwidth extension of the critical sampling circuit.

[0052] Beneficial effects:

[0053] 1. The present invention discloses a method and system for extending the bandwidth of a critical sampling circuit. By using super-relaxed primal duality, the non-convex and non-concave fractional polynomial nonlinear programming problem for estimating the frequency response parameters of the analog channel of the critical sampling circuit is transformed into a constrained convex optimization problem, thereby reducing the computational complexity of estimating the frequency response parameters of the analog channel of the critical sampling circuit and improving the efficiency of extending the bandwidth of the critical sampling circuit.

[0054] 2. The present invention discloses a critical sampling circuit bandwidth extension method and system, which, by constraining the feature search range within the target extension bandwidth during the extension initialization stage, suppresses the aliasing effect caused by insufficient sampling rate when the critical sampling circuit samples the high-frequency components of the signal, thereby improving the reconstruction accuracy of the bandwidth extension signal.

[0055] 3. The present invention discloses a method and system for extending the bandwidth of a critical sampling circuit. By performing dimensionality reduction processing on the high-dimensional response matrix of the critical sampling circuit, it can solve the problem of severely ill-posed extension feature calculation caused by insufficient bandwidth and loss of high-frequency signal information in the critical sampling circuit, thereby improving the accuracy and efficiency of bandwidth extension of the critical sampling circuit.

[0056] 4. The present invention discloses a critical sampling circuit bandwidth extension method and system. By using the critical sampling circuit bandwidth extension method for signal sampling, the out-of-band lost spectrum information can be recovered without increasing the sampling clock frequency and analog input bandwidth, thus realizing low-cost high-frequency signal sampling. Attached Figure Description

[0057] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0058] Figure 1 Flowchart of the method for extending the bandwidth of the critical sampling circuit;

[0059] Figure 2 The example shows the time-frequency domain waveforms of a 64QAM multicarrier signal before and after bandwidth extension, where the multicarrier signal has 2 carriers and the center frequencies of the subcarriers are located at 100MHz and 230MHz, respectively.

[0060] Figure 3 The residual convergence performance is shown for different η and μ values ​​in the examples. Detailed Implementation

[0061] The present invention will be further described and illustrated below with reference to the accompanying drawings and embodiments.

[0062] like Figure 1 As shown in the figure, the specific implementation steps of the critical sampling circuit bandwidth extension method disclosed in this embodiment are as follows:

[0063] S1. System response estimation, as detailed below:

[0064] S1.1. Generate a multi-tone signal with I = 600 as the test signal, with its frequency uniformly distributed within (0, 1 GHz). Input this test signal into the analog channel of the critical sampling circuit, which has a sampling rate of 400 MSPS and an analog input bandwidth of 200 MHz. Oversample the output of the analog channel by a factor of 5 using a reference ADC with a sampling rate of 2 GSPS. Obtain the measured amplitude-squared response of the analog channel based on the test signal and the oversampled output signal.

[0065] S1.2. The following optimization problem is constructed using the l1 norm of the difference between the squared amplitude response to be estimated and the measured squared amplitude response as the objective function.

[0066]

[0067] The squared response of the magnitude to be estimated is modeled using an L=15 order autoregressive moving average model.

[0068] S1.3 Solve the following convex optimization problem using gradient descent.

[0069]

[0070] S1.4, Introducing the over-relaxation parameter λ i The l obtained by solving using S1.3 i and υ i As λ i The upper and lower boundary constraints transform the non-convex, non-concave fractional multinomial nonlinear programming problem constructed in S1.2 into the following constrained convex optimization problem.

[0071]

[0072] λ i ∈(l i ,υ i )

[0073] The coefficient vector of the squared magnitude response to be estimated is obtained by using the augmented Lagrange method.

[0074] S1.5 Constructing a system of linear equations

[0075]

[0076] in

[0077]

[0078] The parameters of the frequency response to be estimated are obtained using the least squares method. and The frequency response of the analog channel of the critical sampling circuit is obtained by the following formula.

[0079]

[0080] The unit impulse response h[n] of the analog channel is obtained by performing an inverse discrete Fourier transform;

[0081] S1.6. Construct the response matrix of the simulated channel using the unit impulse response h[n] of the simulated channel:

[0082]

[0083] The order H is K, which is 4000. The response matrix H of the simulated channel for the critical sampling circuit. 4000×4000 By performing a 5x row decimation, the system response matrix of the critical sampling circuit is obtained.

[0084] S2, Extended initialization, as follows:

[0085] S2.1 Constructing the Feature Transformation Space

[0086]

[0087] The response matrix of the critical sampling circuit system constructed in conjunction with S1.3 Establish the high-dimensional response matrix of the critical sampling circuit

[0088]

[0089] S2.2, Let the iteration number j = 0, ξ = 0 4000×1 S = 200; initialize K = {k | k = 0, 1, ..., 1600, 2400, ..., 3998, 3999};

[0090] S3. Iterative solution of extended features, as detailed below:

[0091] S3.1 Let the iteration number j = j + 1, remove the index of the located extended features in the high-dimensional input space from the index space of the unlocated extended features within the target extended bandwidth, that is, update K = K\w;

[0092] S3.2, Construct matrix Λ={a} using unlocated extended features corresponding to atoms. k |k∈K}, through c=Λ T r calculates the correlation between the atoms corresponding to the unlocated extension features and the bandwidth extension residuals, and finds the atom a with the highest correlation to the bandwidth extension residuals. k’ Update the located extended feature index space w[i] = k';

[0093] S3.3, Update the extended characteristic response matrix Ξ n =[Ξ n ,a k' Construct the following objective function based on elastic network regression.

[0094]

[0095] Where η is the regularization parameter and μ∈[0,1] is the mixing parameter. The extended feature vector is obtained by solving equation (11) using the interior point method.

[0096] S3.4, Update bandwidth extension residuals

[0097] S3.5 Convergence check: If ||r||2 is less than the set threshold or j≥S, then execute S4; otherwise, return to S3.1.

[0098] S4. Broadband signal reconstruction, as detailed below:

[0099] S4.1. Update the high-dimensional input space using the extended feature vectors obtained in S3:

[0100]

[0101] S4.2. Obtain the high-dimensional reconstructed signal using the high-dimensional input space and feature transformation space, i.e., x' = E H ξ; The high-dimensional reconstructed signal x' is downsampled by 4 times to obtain a low-dimensional reconstructed signal x with a sampling rate of 500MSPS. At this time, the sampling bandwidth of the critical sampling circuit is extended from the original 200MHz to 250MHz.

[0102] In this embodiment, the critical sampling circuit has a sampling rate of 400 MSPS and an analog input bandwidth of 200 MHz. x(t) is a 64QAM multi-carrier signal with two carriers, and the center frequencies of the subcarriers are located at 100 MHz and 230 MHz respectively. The bandwidth extension target is 250 MHz, resulting in a signal sampling rate of 500 MSPS after extension. A comparison of the time-domain waveforms before and after bandwidth extension is provided. Figure 2 As shown, at the same time Figure 3 The residual convergence performance for different η and μ values ​​in the examples is shown.

[0103] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for extending the bandwidth of a critical sampling circuit, characterized in that: Includes the following steps, S1. Measure the amplitude square response of the simulated channel of the critical sampling circuit to be extended using a multi-tone or linear frequency modulated signal; Using the l1 norm of the difference between the squared amplitude response to be estimated and the measured squared amplitude response as the objective function, the original non-convex and non-concave fractional polynomial nonlinear programming problem with respect to the objective function is transformed into a constrained convex optimization dual problem by introducing an over-relaxation parameter. The coefficient vector of the squared amplitude response to be estimated is obtained by solving the interior-point method. A linear equation is constructed using the coefficient vector of the squared amplitude response to be estimated, and the frequency response of the simulated channel of the critical sampling circuit is obtained by solving it. The response matrix of the simulated channel of the critical sampling circuit is constructed, and the system response matrix of the critical sampling circuit is obtained by row decimation of the response matrix of the simulated channel of the critical sampling circuit. S2. Construct the feature transformation space, and combine it with the critical sampling circuit system response matrix constructed in S1 to establish the response matrix model of the critical sampling circuit to the high-dimensional input space; initialize the number of iterations, high-dimensional input space, number of extended features, index space of unlocated extended features, index space of located extended features, extended feature response matrix, and bandwidth extension residual; S3. Iteratively solve the extended feature vector: In each iteration, update the index space of the unlocated extended features by removing the indexes of the located extended features in the high-dimensional input space within the target extended bandwidth; In the updated unlocated extended feature index space, the feature atom with the highest correlation to the bandwidth extended residual is relocated, and the feature index space is updated. Based on the updated feature index space, update the extended feature response matrix, construct the elastic network regression objective function, and use the interior point method to solve for the extended feature vector; update the bandwidth extended residual based on the extended feature vector, perform a convergence check, and if it is satisfied, terminate the iteration and transfer the finally solved extended feature vector to S4; otherwise, continue the iteration. S4. Update the high-dimensional input space using the extended feature vector calculated in S3, and obtain the high-dimensional reconstructed signal by combining the feature transformation space. Perform sampling rate conversion on the high-dimensional reconstructed signal to obtain the reconstructed signal after bandwidth extension of the critical sampling circuit.

2. The method for extending the bandwidth of a critical sampling circuit as described in claim 1, characterized in that: The method for implementing step S1 is as follows: S1.1 Generate a multi-tone or linear frequency modulated (LFM) signal as the test signal. The generated test signal has I frequency points uniformly distributed within the frequency range (0, MB), where M is an integer greater than 1, and B = F. s / 2 is the analog input bandwidth of the critical sampling circuit, F s The sampling rate of the ADC in the critical sampling circuit is given. The test signal is input into the analog channel of the critical sampling circuit, and the output of the analog channel is oversampled by a factor of M using an additional reference ADC, i.e., the reference ADC sampling rate is MF. s The measured amplitude squared response of the analog channel is obtained based on the test signal and the oversampled output signal. i = 0, 1, ..., I-1; S1.

2. The following optimization problem is constructed using the l1 norm of the difference between the squared amplitude response to be estimated and the measured squared amplitude response as the objective function. The squared magnitude response to be estimated is modeled using an L-order autoregressive moving average model, with the sign... For example Corresponding to the I frequency points in S1.1, Among them 0 L×1 Represents an L-row vector with all zeros. Let be the coefficient vector of the Lth-order squared magnitude response to be estimated; S1.3 For i = 0, 1, ..., I-1, use the interior point method or gradient descent method to solve the following 2I convex optimization problems. S1.4, Introducing the over-relaxation parameter λ i The solution obtained using S1.3 and υ i As λ i The upper and lower boundary constraints transform the original non-convex, non-concave fractional polynomial nonlinear programming problem constructed in S1.2 into the following constrained hyperrelaxed convex optimization dual problem. The coefficient vector of the squared magnitude response to be estimated can be obtained using the interior-point method or the augmented Lagrange method. S1.5 Constructing a system of linear equations in The parameters of the frequency response to be estimated are obtained by solving (5) using the least squares method. and The frequency response of the analog channel of the critical sampling circuit is obtained by the following equation (7). Performing an inverse discrete Fourier transform on equation (7) yields the unit impulse response h[n] of the analog channel; S1.

6. Construct the response matrix of the simulated channel using the unit impulse response h[n] of the simulated channel: Where K is the order of the square matrix H; the response matrix H of the analog channel of the critical sampling circuit. K×K By performing M-fold row decimation, the system response matrix of the critical sampling circuit is obtained.

3. The critical sampling circuit bandwidth extension method as described in claim 2, characterized in that: The implementation method for step S2 is as follows: S2.1 Constructing the Feature Transformation Space in The response matrix of the critical sampling circuit system constructed in conjunction with S1.3 Establish the response matrix model of the critical sampling circuit to the high-dimensional input space. Where a k Let Ξ be the kth atom of the high-dimensional input space response matrix Ξ, hereinafter referred to as Ξ as the high-dimensional response matrix; S2.2 Let the iteration number j = 0, and the high-dimensional input space ξ = 0. K×1 The number of extended features is S; the initial index space of unlocalized extended features is K = {k | k = 0, 1, ..., K / 2(1-1 / M), K / 2(1+1 / M), ..., K-2, K-1}, which is composed of the indices of unlocalized features within the target extended bandwidth in the high-dimensional input space; the index space of localized extended features... It consists of the indices of the located extended features within the target extended bandwidth in the high-dimensional input space; Extended characteristic response matrix Bandwidth extension residual Where y is the output signal of the critical sampling circuit.

4. The critical sampling circuit bandwidth extension method as described in claim 3, characterized in that: The implementation method for step S3 is as follows: S3.1 Let the iteration number j = j + 1, remove the index of the located extended features in the high-dimensional input space from the index space of the unlocated extended features within the target extended bandwidth, that is, update K = K\w; S3.2, Construct matrix Λ={a} using unlocated extended features corresponding to atoms. k |k∈K}, through c=Λ T r calculates the correlation between the atoms corresponding to the unlocated extension features and the bandwidth extension residuals, and finds the atom a with the highest correlation to the bandwidth extension residuals. k’ Update the located extended feature index space w[i] = k'; S3.3, Atom a found according to S3.2 k’ Update the extended feature response matrix Ξ n =[Ξ n ,a k' Construct the following objective function based on elastic network regression. Where η is the regularization parameter, and μ∈[0,1] is the mixing parameter; the extended feature vector is obtained by solving equation (11) using the interior point method. S3.4, Update bandwidth extension residuals S3.5 Convergence check: If ||r||2 is less than the set threshold or j≥S, then execute S4; otherwise, return to S3.

1.

5. The critical sampling circuit bandwidth extension method as described in claim 4, characterized in that: The implementation method for step S4 is as follows: S4.

1. Update the high-dimensional input space using the extended feature vectors obtained in S3: S4.

2. Obtain the high-dimensional reconstructed signal using the high-dimensional input space and feature transformation space, i.e., x' = E H ξ; Perform sampling rate conversion on the high-dimensional reconstructed signal x', with a conversion factor of MF. s / 2B u The sampling rate is 2B. u The low-dimensional reconstructed signal x, at this time the sampling bandwidth of the critical sampling circuit is extended from the original B to B. u .

6. A critical sampling circuit bandwidth extension system, used to implement the critical sampling circuit bandwidth extension method as described in claim 1, 2, 3, 4 or 5, characterized in that: It includes a system response estimation module, an extended initialization module, an extended feature iterative solution module, and a broadband signal reconstruction module; The system response estimation module is used to measure the amplitude square response of the analog channel of the critical sampling circuit to be extended. It solves the model coefficients of the amplitude square response to be estimated by minimizing the l1 norm of the difference between the amplitude square response to be estimated and the measured amplitude square response, and then obtains the frequency response and unit impulse response of the analog channel of the critical sampling circuit. Then, the response matrix of the simulated channel of the critical sampling circuit is constructed, and the system response matrix of the critical sampling circuit is obtained by row extraction of the response matrix of the simulated channel of the critical sampling circuit, which is used to extend the parameter initialization of the initialization module. The extended initialization module is used to initialize the parameters of the extended feature iterative solution module using the system response matrix output by the system response estimation module, and input the initialized parameters into the extended feature iterative solution module. The extended feature iterative solution module is used to locate the extended feature most relevant to the current extended residual within the target extended bandwidth in each iteration. It uses the located extended features to reduce the dimensionality of the high-dimensional response matrix to obtain a low-dimensional extended feature response matrix. It then solves the extended feature vector through elastic network regression and updates the extended residual for use in the next iteration's extended feature location and residual update. It also performs a convergence check on the extended residual or the number of iterations. If convergence is satisfied, the solved extended feature vector is input into the broadband signal reconstruction module; otherwise, the iteration continues. The broadband signal reconstruction module is used to update the high-dimensional input space using the extended feature vector output by the extended feature iterative solution module, and obtain the high-dimensional reconstructed signal by combining the feature transformation space. The sampling rate of the high-dimensional reconstructed signal is then converted to obtain the reconstructed signal after bandwidth extension of the critical sampling circuit.

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