Range super-resolution method for pulse compression radar based on sparse recovery

Abstract

The present invention proposes a pulse compression radar range super-resolution method based on sparse recovery, which is mainly used to solve the problems that the existing method is not applicable to narrow-band pulse compression radar and the amount of calculation is too large. The implementation steps are: pulse compression processing without windowing radar echo data; constructing a subdivided time matrix using radar parameter information; obtaining a simulated pulse compression data matrix by subdividing the time matrix, intercepting the data on both sides of the pulse compression peak to obtain a peak data matrix, constructing a peak matching sparse dictionary matrix, using an orthogonal matching tracking algorithm to process the echo data after pulse compression to obtain a recovery matrix, extracting the target information of the recovery matrix, and obtaining a one-dimensional super-resolution range image of the radar. The present invention can improve the resolution by more than 1 times based on the theoretical resolution of traditional pulse compression processing.

Description

Technical Field

[0001] The present invention belongs to the field of radar technology, and further relates to a pulse compression radar distance super-resolution method based on sparse recovery in the field of radar signal processing technology. The present invention can be used to perform super-resolution processing on pulse compression data of a pulse compression radar, thereby obtaining a distance super-resolution capability that breaks through the Rayleigh limit when radar bandwidth resources are limited. Background Art

[0002] In modern radar systems, range resolution is the basis for target imaging, recognition, and parameter estimation. Higher range resolution is also an effective means for more efficient target detection and analysis. Radar transmits a signal with a certain bandwidth, and uses pulse compression to receive the received signal to obtain a one-dimensional range image. Its range resolution is inversely proportional to the radar signal bandwidth. To further improve the resolution, the signal bandwidth needs to be increased, which is costly in actual engineering applications.

[0003] Nanjing Guorui Defense System Co., Ltd. proposed a radar distance super-resolution method based on frequency domain de-skewing and compressed sensing in its patent document "A radar distance super-resolution method" (patent application number: 202111431117.1, application publication number: CN 114167415.A). The specific steps of this method are as follows: first, the radar receiving signal is de-skewed in the frequency domain, a sparse mathematical model of the distance dimension of the multi-target echo signal is established, and a dictionary matrix is ​​constructed using the frequency domain de-skewing steering vector. The objective function about the target distance is constructed and solved to obtain a super-resolution range image, thereby separating targets that are close in distance dimension and difficult to separate, and realizing radar distance super-resolution. However, the disadvantage of this method is that the method uses the frequency domain de-skewing steering vector to construct a dictionary matrix, which makes the method not applicable to narrow-band pulse compression radars without de-skewing, and can only achieve distance super-resolution in broadband de-skewing radars.

[0004] The Air Force Engineering University of the Chinese People's Liberation Army disclosed a radar target distance super-resolution method based on adaptive pulse compression in its applied patent document "Radar target distance super-resolution method based on adaptive pulse compression" (patent application number: 202110609279.3, application publication number: CN113325407.A). The specific steps of this method are as follows: first, the transmitted linear frequency modulation waveform is oversampled, extracted and shifted, and then the radar echo data is iteratively adaptively pulse compressed using the obtained extraction and shift sequence, and then the result after adaptive pulse compression is used to reorganize according to the oversampling interval and sampling timing, thereby achieving super-resolution of multiple targets within a distance resolution unit. This method can effectively distinguish multiple targets within a distance resolution unit, but the method still has the disadvantage that it is difficult to apply to engineering implementation due to the large amount of computation required for adaptive pulse compression processing. Summary of the invention

[0005] The purpose of the present invention is to address the deficiencies of the above-mentioned prior art and propose a pulse compression radar range super-resolution method based on sparse recovery, which is used to solve the problem that the existing broadband de-slant range super-resolution method is not suitable for narrowband pulse compression radar and the existing adaptive pulse compression processing method has too much computational complexity.

[0006] The technical solution for achieving the purpose of the present invention is that the present invention constructs a subdivided time matrix that is not aligned with the front edge by using the parameters of the radar transmission signal, and obtains simulated pulse pressure data by using the obtained subdivided time matrix and the pulse pressure data formula. The data near the pulse pressure peak is intercepted to obtain a peak data matrix, and a peak matching dictionary matrix is ​​constructed according to the division of the range gate. The super-resolution range image is obtained by using a sparse recovery algorithm through the constructed peak matching dictionary matrix. The existing de-slant idea distance super-resolution method uses a dictionary matrix constructed by de-slant steering vectors, so it cannot be applied to narrow-band pulse compression radar. The adaptive pulse compression processing algorithm designs a corresponding adaptive filter for each distance unit. The processing of each distance unit needs to use all the data in the pulse compression echo. The processing result is obtained by repeated iteration. The total amount of calculation of the adaptive pulse compression processing algorithm is relatively high. The present invention uses the peak matching dictionary matrix as the dictionary matrix in the orthogonal matching tracking (Orthogonal Matching Pursuit, OMP) algorithm, and uses the OMP algorithm to sparsely restore the echo data after the radar pulse compression to obtain a one-dimensional super-resolution range image, thereby avoiding the problem of excessive calculation of the existing adaptive pulse compression processing distance super-resolution method, and realizing the improvement of radar range resolution.

[0007] The implementation steps of the present invention are as follows:

[0008] Step 1: Pulse compression of radar echo data:

[0009] The radar received echo data is pulse compressed by using the pulse compression coefficient without adding a window to obtain the radar echo data after pulse compression;

[0010] Step 2, construct the peak data matrix:

[0011] Step 2.1, calculate the value of each element in each column of the segmented time matrix, and form all columns into a segmented time matrix;

[0012] Step 2.2, calculating the simulated pulse pressure data value of each element in each column of the subdivided time matrix, and using the simulated pulse pressure data value of the element to update the corresponding element value in the subdivided time matrix to obtain a simulated pulse pressure data matrix;

[0013] Step 2.3, find the peak point position z of the first column of the simulated pulse pressure data matrix, intercept the elements from the zqth row to the z+qth row in the first column, and use all the elements in the intercepted length as the first column of the peak data matrix. Similarly, in the kth column of the simulated data matrix, intercept the elements from the zqth row to the z+qth row in the kth column, and use all the elements in the intercepted length as the kth column of the peak data matrix; the dimension of the peak data matrix is ​​(2q+1)×K, and the value of q is any integer from 2 to 4 times the number of sampling points in the main lobe of the radar pulse pressure;

[0014] Step 3, construct the dictionary matrix for peak matching:

[0015] Step 3.1, construct a dictionary matrix of dimension U×N, where each element of the matrix is ​​initially 0; U=M+2q, where the value of M is equal to the length of the radar echo data after pulse compression, and N=MK;

[0016] Step 3.2, in the first to Kth columns of the dictionary matrix, the first to (2q+1)th rows of the peak data matrix are sequentially placed in the first to (2q+1)th rows of the dictionary matrix, in the K+1th to 2Kth columns of the dictionary matrix, the first to (2q+1)th rows of the peak data matrix are sequentially placed in the second to (2q+2)th rows of the dictionary matrix, and so on, in the (m-1)K+1th to mKth columns of the dictionary matrix, the first to (2q+1)th rows of the peak data matrix are sequentially placed in the mth to m+2qth rows of the dictionary matrix, m=1,2,...,M;

[0017] Step 4, adding 2q zeros to the echo data after pulse compression to obtain zero-filled echo data after pulse compression;

[0018] Step 5: Use the peak matching dictionary matrix to perform sparse recovery on the zero-filled post-pulse pressure echo data:

[0019] The peak matching dictionary matrix is ​​used as the dictionary matrix in the orthogonal matching tracking algorithm, and the orthogonal matching tracking algorithm is used to perform sparse recovery on the zero-filled post-pulse pressure echo data to obtain the recovery vector;

[0020] Step 6: Extract the target information of the recovery vector:

[0021] The position of each element in the restored vector is converted into the distance position of the target, and the mapping of the radar one-dimensional super-resolution range image is realized to obtain the radar one-dimensional super-resolution range image.

[0022] Compared with the prior art, the present invention has the following advantages:

[0023] First, since the present invention uses the radar's own parameter information to construct simulated pulse pressure data and uses the pulse pressure peak data to construct a dictionary matrix, it overcomes the disadvantage that the existing de-slant distance super-resolution method cannot be applied to narrow-band pulse compression radars, so that the present invention can be applied to narrow-band pulse compression radars and achieves distance super-resolution of pulse compression radars. It should be noted that in current radar systems, most radars use pulse compression systems.

[0024] Second, since the present invention adopts the OMP algorithm to perform sparse recovery on the radar echo data after pulse compression to obtain a super-resolution range image, it overcomes the defect of the prior art that the adaptive pulse algorithm has a large amount of computation, making the present invention suitable for engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 is a flow chart of the present invention;

[0026] Figure 2 This is the result of pulse compression using the existing radar receiving echo data without windowing pulse compression coefficient;

[0027] Figure 3 This is a result diagram of the method of the present invention processing the radar echo data after pulse compression. DETAILED DESCRIPTION

[0028] The present invention is further described in detail below in conjunction with the accompanying drawings and embodiments.

[0029] Reference Figure 1 The specific implementation steps of the present invention are further described in detail with reference to the accompanying drawings and embodiments.

[0030] Step 1: Perform pulse compression processing on the radar echo signal.

[0031] By using the pulse compression coefficient without windowing to perform pulse compression processing on the radar received echo data, the echo data after radar pulse compression is obtained, and the signal-to-noise ratio gain of pulse compression can be obtained. The detailed steps are as follows:

[0032] Step 1.1, construct the unwindowed pulse compression coefficient of the radar transmit signal according to the radar transmit signal:

[0033]

[0034] Where h represents the unwindowed pulse compression coefficient of the radar transmission signal, represents the jth element of s, j = 1, 2, ..., J, J represents the total length of the discrete sequence emitted by the radar in one pulse time, Indicates the rounding operation, T p represents the pulse width of the radar, f s represents the sampling rate of the radar, s=[s1,s2,...,s J ] represents the discrete sequence of radar pulse time transmission, S(·) represents the radar transmission waveform function, Δt=1 / f s represents the radar sampling time interval, f s represents the sampling rate of the radar, T p represents the pulse width of the radar, (·) * represents the conjugation operation.

[0035] Step 1.2, by convolving the radar echo data with the pulse compression coefficient, the echo data after radar pulse compression is obtained:

[0036]

[0037] Among them, y represents the radar echo data after pulse compression, represents the convolution operation, and r represents the echo data received by the radar.

[0038] Step 2: Construct the peak data matrix.

[0039] Step 2.1, calculate the value of each element in each column of the segmented time matrix according to the following formula, and form all columns into a segmented time matrix.

[0040]

[0041] Among them, D k Represents the value of each element in the kth column of the subdivision time matrix, k = 1, 2, ..., K, K represents the number of subdivisions of the radar sampling interval, and the total number of rows of the subdivision time matrix is ​​equal to the total length of the discrete sequence emitted by the radar in one pulse time.

[0042] In the embodiment of the present invention, the radar pulse width T p =40μs, the sampling rate of radar receiving echo data is f s =1MHz, K=5, Δt=1 / f s=1μs, the unit of each element of the subdivision time matrix is ​​microseconds, and the subdivision time matrix is ​​as follows:

[0043]

[0044] Step 2.2, calculate the simulated pulse pressure data value of each element in each column of the subdivided time matrix according to the following formula, and use the simulated pulse pressure data value of the element to update the corresponding element value in the subdivided time matrix to obtain the simulated pulse pressure data matrix.

[0045]

[0046] Among them, O k represents the kth column of the simulated pulse pressure data matrix, k=1, 2, ..., K, and T represents the transposition operation.

[0047] Step 2.3, find the peak point position z of the first column of the simulated pulse pressure data matrix, intercept the elements from the zqth row to the z+qth row in the first column, and use all the elements in the intercepted length as the first column of the peak data matrix. Similarly, in the kth column of the simulated data matrix, intercept the elements from the zqth row to the z+qth row in the kth column, and use all the elements in the intercepted length as the kth column of the peak data matrix. The dimension of the peak data matrix is ​​(2q+1)×K, and the value of q is any integer from 2 to 4 times the number of sampling points in the main lobe of the radar pulse pressure.

[0048] Step 3: Construct a dictionary matrix for peak matching.

[0049] Step 3.1, construct a dictionary matrix of dimension U×N, where each element of the matrix is ​​initially 0. U=M+2q, where the value of M is equal to the length of the radar echo data after pulse compression, and N=MK.

[0050] Step 3.2, in the first to Kth columns of the dictionary matrix, place the first to (2q+1)th rows of the peak data matrix into the first to (2q+1)th rows of the dictionary matrix in sequence; in the K+1th to 2Kth columns of the dictionary matrix, place the first to (2q+1)th rows of the peak data matrix into the second to (2q+2)th rows of the dictionary matrix in sequence; and so on; in the (m-1)K+1th to mKth columns of the dictionary matrix, place the first to (2q+1)th rows of the peak data matrix into the mth to m+2qth rows of the dictionary matrix in sequence, where m=1,2,...,M.

[0051] Step 4, adding 2q zeros to the echo data after pulse compression to obtain zero-filled echo data after pulse compression;

[0052] Step 5: Use the peak matching dictionary matrix to perform sparse recovery on the zero-filled post-pulse compression echo data.

[0053] The peak matching dictionary matrix is ​​used as the dictionary matrix in the orthogonal matching tracking algorithm. The orthogonal matching tracking algorithm is used to perform sparse recovery on the zero-filled post-pulse compression echo data to obtain the recovery vector. The specific steps are as follows:

[0054] Step 5.1, calculate the inner product vector of the dictionary matrix and the residual vector according to the following formula:

[0055] Φ i =A H r i-1

[0056] Among them, Φ i represents the inner product vector of the dictionary matrix and the residual vector in the i-th iteration, i represents the number of iterations, the initial value of i is 1, A represents the dictionary matrix, (·) H represents the conjugate transpose operation, r i-1 Represents the residual vector after the i-1th iteration, r0 = Y H , Y represents the pulse pressure echo data after zero padding.

[0057] Step 5.2, find the position l corresponding to the maximum element value of the inner product vector in the i-th iteration i , l i Represents the index of the column in the dictionary matrix corresponding to the maximum correlation with the residual vector in the i-th iteration.

[0058] Step 5.3, i Add to the index set in the current iteration, and i The column vector of the corresponding dictionary matrix is ​​added to the column vector set of the dictionary matrix corresponding to the index set, so as to obtain the column vector set of the dictionary matrix corresponding to the index set of the current iteration.

[0059] Step 5.4, calculate the estimated vector of the pulse pressure echo data according to the following formula:

[0060]

[0061] Among them, x i represents the estimated vector of the echo data after pulse pressure in the i-th iteration, (·) -1 represents the inverse operation, A set of column vectors representing the dictionary matrix corresponding to the index set of the i-th iteration.

[0062] Step 5.5, update the residual vector according to the following formula:

[0063]

[0064] Among them, r i represents the residual vector after the i-th iteration.

[0065] Step 5.6, determine whether the current iteration number is less than the sparsity of the radar echo data. If so, set i=i+1 and execute the first step and then step 4.1. Otherwise, use the index set and estimated vector of this iteration as the final index set Γ and estimated vector x, and execute step 4.7. In this embodiment of the present invention, the sparsity

[0066] Step 5.7, construct a recovery vector of dimension 1×N, each element of the recovery vector is initially 0, find the position corresponding to each element of the index set Γ in the recovery vector, and use the estimated vector x to update each element of the corresponding position to obtain the recovery vector.

[0067] In the embodiment of the present invention, the index set Γ = {100, 96}, x = [1, 0.4] T , the data of the 100th element of the restored vector is updated to 1, and the data of the 96th element of the restored vector is updated to 0.4.

[0068] Step 6: extract the target information of the restored vector and obtain the one-dimensional super-resolution range image of the radar.

[0069] The position of each element in the recovery vector is converted into the distance position of the target to realize the mapping of the radar one-dimensional super-resolution range image and obtain the radar one-dimensional super-resolution range image. The position of an element in the recovery vector is divided by K to obtain the position of the corresponding distance unit of the radar one-dimensional super-resolution range image. In the embodiment of the present invention, K=5, and the recovery vector is [0,...,0,0.4,0,0,0,1,...,0,0], where the position of element 0.4 is 96 and the position of element 1 is 100. Element 0.4 corresponds to the 19.2th distance unit on the radar range image, and element 1 corresponds to the 20th distance unit on the radar range image.

[0070] The effect of the present invention is further described below in conjunction with simulation experiments:

[0071] 1. Conditions of simulation experiment:

[0072] The platforms for the simulation experiment of the present invention are: Windows 10 operating system and Matlab R2019b.

[0073] The simulation experiment data of the present invention is: the time width T of the LFM signal emitted by the radar p =40×10 -6 s, bandwidth B = 1 × 10 6 Hz, sampling frequency f s =2×10 6Hz, conventional pulse compression distance resolution Δr=c / 2B=150m, radar echo signal contains target signal and Gaussian white noise signal, detection signal-to-noise ratios of two targets are SNR1=46dB, SNR2=46dB, target 1 is located in distance unit 19, target 2 is located in distance unit 20, the distance between the two targets is d=75m, the distance interval between the two targets in the experiment is 0.5 times of the theoretical resolution of traditional pulse compression processing, and the targets are both stationary.

[0074] 2. Simulation content and results analysis:

[0075] There are two simulation experiments of the present invention.

[0076] Simulation experiment 1, under the above simulation conditions, the radar received echo signal is processed by the pulse compression coefficient without windowing in the prior art to obtain a one-dimensional range image representing the target distribution in the range dimension, such as Figure 2 . Figure 2 The horizontal axis represents the distance unit, the vertical axis represents the amplitude value, and the curve represents the one-dimensional range image of the target distribution in the distance dimension. Figure 2 It can be seen from the curve shown that after the radar received echo signal is processed by the existing pulse compression coefficient without adding a window, the processing result only forms a peak, and the two targets in the echo signal cannot be detected separately.

[0077] In simulation experiment 1, the prior art used refers to a pulse compression processing method for radar.

[0078] Simulation experiment 2, under the conditions of the above simulation experiment, the pulse compression radar range super-resolution method based on sparse recovery in the present invention is used to process the echo data after radar pulse compression to obtain a super-resolution range image representing the target distribution in the range dimension, such as Figure 3 . Figure 3 The horizontal axis represents the distance unit, the vertical axis represents the amplitude value, and the peak represents the spectral peak of the target distribution in the distance dimension.

[0079] from Figure 3 It can be seen that obvious peaks appear near the distance units 19 and 20 where the two targets are located in the super-resolution range image, and the resolution of two targets with a distance interval of 0.5 times the theoretical resolution of traditional pulse compression processing is achieved. Therefore, the range super-resolution method based on sparse recovery in the present invention is more than 1 times better than the traditional pulse compression processing in terms of resolution performance.

[0080] The above simulation experiments show that: in order to solve the problem that the target distance resolution capability of conventional pulse compression processing is limited by the signal bandwidth and the target resolution and identification is difficult, the present invention proposes a pulse compression radar distance super-resolution method based on sparse recovery. Compared with the traditional pulse compression processing, the resolution effect of the pulse compression radar distance super-resolution method based on sparse recovery can be improved by more than 1 times.

Claims

1. A pulse compression radar range super-resolution method based on sparse recovery, characterized in that: Construct a peak data matrix and a peak matching dictionary matrix; the specific steps of this method are as follows: Step 1: Pulse compression of radar echo data: The radar received echo data is pulse compressed by using the pulse compression coefficient without adding a window to obtain the radar echo data after pulse compression; Step 2, construct the peak data matrix: Step 2.1, calculate the value of each element in each column of the segmented time matrix, and form all columns into a segmented time matrix; Step 2.2, calculating the simulated pulse pressure data value of each element in each column of the subdivided time matrix, and using the simulated pulse pressure data value of the element to update the corresponding element value in the subdivided time matrix to obtain a simulated pulse pressure data matrix; Step 2.3, find the peak point position z of the first column of the simulated pulse pressure data matrix, intercept the elements from the zqth row to the z+qth row in the first column, and use all the elements in the intercepted length as the first column of the peak data matrix. Similarly, in the kth column of the simulated data matrix, intercept the elements from the zqth row to the z+qth row in the kth column, and use all the elements in the intercepted length as the kth column of the peak data matrix; the dimension of the peak data matrix is ​​(2q+1)×K, and the value of q is any integer from 2 to 4 times the number of sampling points in the main lobe of the radar pulse pressure; Step 3, construct the dictionary matrix for peak matching: Step 3.1, construct a dictionary matrix of dimension U×N, where each element of the matrix is ​​initially 0; U=M+2q, where the value of M is equal to the length of the radar echo data after pulse compression, and N=MK; Step 3.2, in the first to Kth columns of the dictionary matrix, the first to (2q+1)th rows of the peak data matrix are sequentially placed in the first to (2q+1)th rows of the dictionary matrix, in the K+1th to 2Kth columns of the dictionary matrix, the first to (2q+1)th rows of the peak data matrix are sequentially placed in the second to (2q+2)th rows of the dictionary matrix, and so on, in the (m-1)K+1th to mKth columns of the dictionary matrix, the first to (2q+1)th rows of the peak data matrix are sequentially placed in the mth to m+2qth rows of the dictionary matrix, m=1,2,...,M; Step 4, adding 2q zeros to the echo data after pulse compression to obtain zero-filled echo data after pulse compression; Step 5: Use the peak matching dictionary matrix to perform sparse recovery on the zero-filled post-pulse pressure echo data: The peak matching dictionary matrix is ​​used as the dictionary matrix in the orthogonal matching tracking algorithm, and the orthogonal matching tracking algorithm is used to perform sparse recovery on the zero-filled post-pulse pressure echo data to obtain the recovery vector; Step 6: Extract the target information of the recovery vector: The position of each element in the restored vector is converted into the distance position of the target, and the mapping of the radar one-dimensional super-resolution range image is realized to obtain the radar one-dimensional super-resolution range image.

2. The pulse compression radar range super-resolution method based on sparse recovery according to claim 1 is characterized by: The steps of the pulse compression process described in step 1 are as follows: The first step is to construct the unwindowed pulse compression coefficient of the radar transmit signal according to the radar transmit signal as follows: Where h represents the unwindowed pulse compression coefficient of the radar transmission signal, represents the jth element of s, j = 1, 2, ..., J, J represents the total length of the discrete sequence emitted by the radar in one pulse time, Indicates the rounding operation, T p represents the pulse width of the radar, f s represents the sampling rate of the radar, s=[s1,s2,...,s J ] represents the discrete sequence of radar pulse time transmission, S(·) represents the radar transmission waveform function, Δt=1 / f s represents the sampling time interval of the radar, (·) * represents the conjugation operation; In the second step, the radar echo data is convolved with the pulse compression coefficient to obtain the radar pulse compressed echo data.

3. The pulse compression radar range super-resolution method based on sparse recovery according to claim 2 is characterized in that: The simulated pulse pressure data value of each element in each column of the subdivided time matrix described in step 2.2 is obtained by the following formula: Among them, O k represents the kth column of the simulated pulse pressure data matrix, k = 1, 2, ..., K, the value of K is equal to the total number of subdivided time matrix columns, D k represents the kth column of the segmented time matrix, and T represents the transpose operation.

4. The pulse compression radar range super-resolution method based on sparse recovery according to claim 1 is characterized by: The specific steps of the orthogonal matching pursuit algorithm described in step 5 are as follows: The first step is to calculate the inner product vector of the dictionary matrix and the residual vector in the current iteration according to the following formula: F i =A H r i-1 Among them, Φ i represents the inner product vector of the dictionary matrix and the residual vector in the i-th iteration, i represents the number of iterations, the initial value of i is 1, A represents the dictionary matrix, (·) H represents the conjugate transpose operation, r i-1 Represents the residual vector after the i-1th iteration, r0 = Y H , Y represents the pulse pressure echo data after zero padding; The second step is to find the position l corresponding to the maximum element value of the inner product vector in the i-th iteration. i , l i Represents the serial number of the column in the dictionary matrix corresponding to the maximum correlation with the residual vector in the i-th iteration; The third step is to i Add to the index set in the current iteration, add the column vector of the dictionary matrix corresponding to the column number to the column vector set of the dictionary matrix corresponding to the index set, and obtain the column vector set of the dictionary matrix corresponding to the index set of the current iteration; The fourth step is to calculate the estimated vector of the pulse pressure echo data according to the following formula: Among them, x i represents the estimated vector of the echo data after pulse pressure in the i-th iteration, (·) -1 represents the inverse operation, Represents the column vector set of the dictionary matrix corresponding to the index set of the i-th iteration; Step 5: Update the residual vector according to the following formula: Among them, r i represents the residual vector after the i-th iteration; Step 6: Determine whether the current number of iterations is less than the sparsity of the radar echo data. If so, set i=i+1 and then execute the first step. Otherwise, use the index set and estimated vector of this iteration as the final index set Γ and estimated vector x, and execute the seventh step. The seventh step is to construct a recovery vector of dimension 1×N. Each element of the recovery vector is initially 0. Find the position corresponding to each element of the index set Γ in the recovery vector, and use the estimated vector x to update each element of the corresponding position to obtain the recovery vector.

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