Off-lattice compressed sensing DOA (direction of arrival) estimation method based on rear nested array

Abstract

The invention relates to an off-grid compressed sensing DOA (direction of arrival) estimation method based on a rear nested array, and belongs to the technical field of array signal processing. Two uniform linear arrays are utilized to construct a rear nested array, a signal covariance matrix is constructed, an initial estimation angle is obtained, off-grid correction is performed through a first-order Taylor expansion method, and final angle estimation is realized after iteration is completed. The method has the advantages that the array aperture is expanded by using the improved rear nested array structure, the degree of freedom of the array is improved, and the direction of arrival of more incident signals can be estimated; through Taylor out-of-lattice correction, the problem that the accuracy of a compressed sensing DOA estimation method is reduced in a grid mismatch scene is successfully solved, and compared with a traditional method, the method can still keep the estimation performance of high accuracy and high success rate on the incident angle of the information source under the low signal-to-noise ratio.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and specifically relates to an off-grid compressive sensing direction of arrival (DOA) estimation method based on a post-nested array, which is applicable to high-precision positioning of multiple targets in scenarios such as radar, sonar, and wireless communication. Background Art

[0002] As an important branch in the field of signal processing, array signal processing has received extensive attention and developed rapidly in recent years. Its core idea is to use multiple sensors arranged in a specific pattern in three-dimensional space to receive target signals and collect discrete observation data. Due to the different spatial positions of the sensors, the signals received by them have differences in phase and amplitude, which provide favorable conditions for designing various signal parameter estimation algorithms, enabling accurate extraction of signal feature information from complex environments. Compared with a single sensor, array signal processing exhibits obvious advantages in many aspects. First, due to the collaborative work of multiple sensors, it has high performance in spatial resolution, interference suppression, and beam control. Array signal processing can more effectively separate target signals from interference signals, significantly improving the accuracy and reliability of signal detection and positioning. At the same time, through reasonable algorithm design and processing, the influence of noise can be reduced to a certain extent, thereby improving the overall signal-to-noise ratio. However, traditional array signal processing methods also have some limitations in practical applications. One of the main problems is that the estimation of the number of signal sources is limited by the number of array elements. That is to say, in the traditional model, the number of distinguishable signal sources often cannot exceed the number of sensors in the array. If more signal sources need to be detected, usually the number of sensors needs to be increased, but this will lead to a significant increase in hardware costs. To solve this problem, various sparse array designs have been proposed in recent years. Among them, the nested array, as a typical representative, can achieve a larger array aperture under the condition of the same number of array elements, providing a higher degree of freedom, thus breaking through the limitation of traditional arrays in the estimation of the number of signal sources and having better performance in terms of resolution and estimation.

[0003] Traditional direction of arrival (DOA) estimation methods, such as MUSIC (Multiple Signal Classification) and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques), are widely used in the field of signal processing. These methods usually rely on subspace decomposition of the received signals to extract the direction information of the signals. However, the performance of these methods is poor in low signal-to-noise ratio (SNR) or multi-target scenarios. Especially in complex environments, due to the mutual interference between signals and the influence of noise, the estimation accuracy of traditional methods often fails to meet the actual requirements. To overcome the limitations of traditional methods, in recent years, the theory of compressed sensing (CS) has been introduced into DOA estimation. The compressed sensing theory improves the resolution of DOA estimation by introducing a sparse signal model and leveraging the sparsity characteristics of signals. In this framework, through compressed sensing technology, the direction information of signals can be effectively recovered from a small amount of observed data, thereby improving the accuracy and resolution of DOA estimation. Existing compressed sensing DOA estimation algorithms based on compressed sensing still have a series of prominent problems in off-grid scenarios. First, the grid mismatch problem is very obvious. Since the actual signal direction usually deviates from the preset discrete grid points, this deviation will directly lead to a decrease in the reconstruction accuracy, especially in complex scenarios, where the impact is more serious. Second, traditional off-grid correction methods mostly rely on repeatedly refining the grid to gradually reduce the grid deviation. Although the estimation accuracy can be improved to a certain extent, its computational complexity is relatively high and cannot meet the requirements of real-time processing when the number of targets is large or the scenario is complex. Summary of the Invention

[0004] The present invention provides an off-grid compressed sensing direction of arrival estimation method based on a post-nested array to solve the problems of low degrees of freedom, insufficient accuracy and success rate in estimating the parameters of incident signals in traditional signal processing methods and compressed sensing DOA processing methods under low signal-to-noise ratio and off-grid conditions.

[0005] The solution adopted by the present invention includes the following steps:

[0006] Step 1: Construct a post-nested array using two uniform linear arrays;

[0007] Step 2: Construct a signal covariance matrix according to the signals received by the post-nested array in Step 1;

[0008] Step 3: Obtain the sparse representation of the signal by vectorizing and de-duplicating and sorting the covariance matrix, set the number of iterations and the convergence threshold, and input the sparse representation of the signal into the OMP algorithm to obtain the initial estimated angle;

[0009] Step 4: For the initial estimated angle, off-grid correction is performed by the first-order Taylor expansion method, and the final angle estimation is achieved after iteration.

[0010] In the post-nested array with M array elements in Step 1 of the present invention, the first array element of the first-level array in the second-level nested array is moved to the end of the second-level array. Among them, the element spacing of sub-array 1 is d1, and the number of array elements is M1; the element spacing of sub-array 2 is d2, and the number of array elements is M2; the element spacing d1 of sub-array 1 is used as the unit spacing d, the total number of array elements M of the post-nested array is M = M1 + M2, the spacing between the two sub-arrays is the element spacing d1 of the previous-level sub-array, d1 = λ / 2, d2 = (M1 + 1)d1, and λ is the wavelength of the incident signal.

[0011] In Step 2 of the present invention, if there are K far-field narrowband signal sources incident on the post-nested array, the mathematical model expression of the array received signal is:

[0012] Y(t) = A(θ)X(t) + N(t)

[0013] In the formula, Y(t) = [y1(t), …, y m (t), … y M (t)] T is the M×L-dimensional signal data matrix received by the created antenna array, where M is the total number of array elements, L is the number of snapshots, y m (t) represents the signal received by the m-th array element, [·] T represents the transpose operation, X(t) = [x1(t), …, x k (t), … x K (t)] T is the K×L-dimensional data matrix of the spatial target incident signal source, where K is the total number of signal sources, x k (t) represents the signal emitted by the k-th signal source, N(t) = [n1(t), …, n m (t), … n M (t)] T is the mutually independent noise data matrix of the array received signal, n m (t) represents the noise received by the m-th array element, the noise mean value is 0, A(θ) = [a(θ1), …, a(θ k ), … a(θ K )] is the array manifold matrix, where a(θ k ) is the steering vector of the k-th incident signal source, as shown in the following formula:

[0014]

[0015] In the formula, j is a complex number, d mis the spatial distance of the m-th array element relative to the reference array element, d m ∈P;

[0016] P is the set of physical array element positions of the array. The post-nested array has:

[0017]

[0018] And the following assumptions are made about the signal sources: The signal sources are cyclostationary signals, the signal sources are independent of each other, the noise is Gaussian noise and is independent of the signal sources;

[0019] The covariance matrix R of the received signal is obtained from the signal data received by the post-nested array, as shown in the following equation:

[0020]

[0021] where (·) H represents the conjugate transpose operation.

[0022] In the third step of the present invention, the covariance matrix is vectorized to obtain a single-block snapshot signal model Z, as shown in the following equation:

[0023] Z = vec(R)

[0024] where vec(·) represents vectorization;

[0025] Then, a difference set Ω is generated based on the positions P of the physical array elements of the post-nested array, as shown in the following equation:

[0026]

[0027] where both i and o are sequence numbers with values between 1 and M;

[0028] The difference set Ω is then de-duplicated to obtain a unique difference set Ω un i que ;

[0029] For the unique difference set Ω unique is sorted in ascending order to obtain an ordered difference set Ω sort , and the index value Ι of the sorting of the ordered difference set is recorded. According to the index value Ι, the single-block snapshot signal model Z that has completed synchronization update is sorted to obtain the final signal sparse model The obtained ordered difference set Ω sort is the set of positions of the virtual array elements included in the virtual array;

[0030] The sparse representation of the signal is input into the OMP algorithm for initial incident angle estimation.

[0031] The present invention obtains the unique difference set Ωun i que , the specific method is as follows:

[0032] Traverse each element in Ω. If it already exists in Ω un i que , then skip it; otherwise, add it to Ω un i que ;

[0033] And synchronously update the single-block snapshot signal model Z, and delete the data items in Z that are the same sequences as the elements of the repeated difference set.

[0034] The sparse representation of the signal described in the present invention is input into the OMP algorithm for estimating the initial incident angle. The specific steps are as follows:

[0035] First, set the maximum number of iterations u and the convergence threshold β, and use the sparse representation of the signal as the initial residual r;

[0036] Then, evenly divide the angle range [-90°, 90°] into N parts, where the number of divided grids N is greater than the number of signal sources K. Each grid corresponds to a unique number, and a grid set and the array manifold matrix of the corresponding virtual array are obtained. Use as an overcomplete dictionary, and use the candidate angles in the grid set to provide the initial estimated angle range for the OMP algorithm;

[0037] To obtain the matching degree between each candidate angle and the residual, screen out the angle direction most relevant to the current residual, and calculate the inner product index of the grid array manifold matrix and the residual r as shown in the following formula:

[0038]

[0039] index is the angle direction most relevant to the current residual. If index is used as the angle index value in the grid set, the initial estimated angle of this iteration can be obtained as shown in the following formula:

[0040]

[0041] For the initial estimated angle of the grid in step four of the present invention, off-grid correction is performed through the first-order Taylor expansion method to achieve the final angle estimation;

[0042] First, construct a steering vector a according to the initial estimated angle, as shown in the following formula:

[0043]

[0044] The first-order Taylor expansion of the steering vector a is performed to obtain its first derivative da with respect to the angle, as shown in the following equation:

[0045]

[0046] Thus, through correcting the phase change, local linear approximation is achieved;

[0047] Then, the signal x is recovered by the least squares method, as shown in the following equation:

[0048]

[0049] Furthermore, the residual r is updated, as shown in the following equation:

[0050]

[0051] That is, the estimated signal component is subtracted from the sparse model of the signal to reflect the remaining signal energy;

[0052] To determine the correction direction of the initial estimated angle, the inner product of the derivative da and the residual r is calculated to obtain the gradient g that can determine the angle correction direction, as shown in the following equation:

[0053] g = -2·real[(da) T r]

[0054] To accelerate convergence and improve the accuracy of angle correction, the second derivative of the steering vector a with respect to the angle is approximated by the first order to obtain the approximate Hessian matrix H, as shown in the following equation:

[0055] H = 2real(da T ·da)

[0056] To prevent numerical instability of the matrix and improve the iterative convergence, the approximate Hessian matrix H is regularized to obtain the regularized matrix as shown in the following equation:

[0057]

[0058] From the gradient and the regularized Hessian matrix obtained above the angle correction delta is calculated, as shown in the following equation:

[0059] delta = g / H

[0060] Then, the initial estimated angle is updated, as shown in the following equation:

[0061]

[0062] Determine whether the angle correction amount delta is less than the threshold β. If it is not less, continue with the following steps; if it is less than the threshold, terminate the iteration;

[0063] Finally, update the residual orthogonal projection to obtain the initial residual r for the next iteration next As shown in the following formula:

[0064]

[0065] where A(:,index) represents the index-th column vector of the two-dimensional array A, and norm(·) represents the l2 norm;

[0066] After the update is completed, the obtained r next is used as the initial residual for the next iteration. After all iterations and angle updates are completed, the set of θ est obtained in each iteration is the finally estimated angle.

[0067] The present invention has the following advantages: By using the improved post-nested array structure, the array aperture is expanded, the degrees of freedom of the nested array are increased, and more incident signal sources can be estimated; through Taylor off-grid correction, the problem of accuracy degradation of the traditional compressive sensing DOA estimation method in the grid mismatch scenario is solved. Compared with the traditional method, the present invention can still maintain high accuracy and high success rate in estimating the incident angle of the signal source under low signal-to-noise ratio. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 is a schematic diagram of the post-array structure used in the embodiment of the present invention;

[0069] Figure 2 is the flowchart of the present invention;

[0070] Figure 3 is the DOA estimation result diagram of the present invention provided by Experimental Example 1 when the number of incident signal sources is more than the number of array elements;

[0071] Figure 4 is the root mean square error comparison diagram of the present invention, the orthogonal matching pursuit (OMP) method, and the NA-OMP method provided by Experimental Example 2 under multiple signal sources and different signal-to-noise ratios;

[0072] Figure 5 is the success probability comparison diagram of parameter estimation of the present invention, the orthogonal matching pursuit (OMP) method, and the NA-OMP method provided by Experimental Example 3 under multiple signal sources and different signal-to-noise ratios. DETAILED DESCRIPTION OF THE INVENTION

[0073] As Figure 2 shown, it includes the following steps:

[0074] Step 1: Construct a post nested array using two uniform linear arrays;

[0075] The post nested array with M array elements is obtained by moving the first array element of the first-level array in the second-level nested array to the end of the second-level array. Among them, the element spacing of sub-array 1 is d1, and the number of array elements is M1; the element spacing of sub-array 2 is d2, and the number of array elements is M2; taking the element spacing d1 of sub-array 1 as the unit spacing d, the total number of array elements M of the post nested array is M = M1 + M2, and the spacing between the two sub-arrays is the element spacing d1 of the previous-level sub-array, d1 = λ / 2, d2 = (M1 + 1)d1, where λ is the wavelength of the incident signal. Taking the case of M1 = M2 = 3, a physical model of the post nested array as shown in Figure 1 is given;

[0076] Step 2: Construct a signal covariance matrix according to the signals received by the post nested array in Step 1;

[0077] If there are K far-field narrowband signal sources incident on the post nested array, the mathematical model expression of the array received signal is:

[0078] Y(t) = A(θ)X(t) + N(t)Y(t) = A(θ)X(t) + N(t)

[0079] In the formula, Y(t) = [y1(t), …, y m (t), … y M (t)] T is the M×L dimensional signal data matrix received by the created antenna array, where M is the total number of array elements and L is the number of snapshots. y m (t) represents the signal received by the m-th array element, and [·] T represents the transpose operation. X(t) = [x1(t), …, x k (t), … x K (t)] T is the K×L dimensional data matrix of the spatial target incident signal source, where K is the total number of signal sources, x k (t) represents the signal emitted by the k-th signal source, N(t) = [n1(t), …, n m (t), … n M (t)] T is the mutually independent noise data matrix of the array received signal, n m (t) represents the noise received by the m-th array element, the mean value of the noise is 0, and A(θ) = [a(θ1), …, a(θ k ), … a(θ K )] is the array manifold matrix of the array, where a(θ kis the steering vector of the k-th incident source, as shown in the following equation

[0080]

[0081] where j is a complex number, and d m is the spatial distance of the m-th array element relative to the reference array element, and d m ∈P, where P is the set of physical array element positions of the array. For Figure 1 the post nested array shown below:

[0082]

[0083] And the following assumptions are made for the sources: the sources are cyclostationary signals, the sources are independent of each other, the noise is Gaussian noise and independent of the sources;

[0084] The covariance matrix R of the received signal is obtained from the signal data received by the post nested array, as shown in the following equation:

[0085]

[0086] where (·) H represents the conjugate transpose operation;

[0087] Step 3: Obtain the sparse representation of the signal by vectorizing and de-duplicating and sorting the covariance matrix, set the number of iterations and the convergence threshold, and input the sparse representation of the signal into the OMP algorithm to obtain the initial estimated angle;

[0088] Vectorize the covariance matrix to obtain the single-block snapshot signal model Z, as shown in the following equation:

[0089] Z = vec(R)

[0090] where vec(·) represents vectorization.

[0091] Then generate the difference set Ω based on the positions P of the physical array elements of the post nested array, as shown in the following equation:

[0092]

[0093] where both i and o are sequence numbers with values between 1 and M;

[0094] Then de-duplicate the difference set Ω to obtain the unique difference set Ω un i que , and the specific method is as follows:

[0095] Traverse each element in Ω. If it already exists in Ω un i que , then skip it; otherwise add it to Ω un i que ;

[0096] And synchronously update the single-block beat signal model Z, and delete the data items in Z whose sequences are the same as the elements in the duplicate difference set;

[0097] Finally, for the unique difference set Ω unique Perform ascending sorting to obtain the ordered difference set Ω sort , and record the index value Ι of the ordered difference set sorting. Sort the single-block beat signal model Z that has completed synchronous update according to the index value Ι to obtain the final signal sparse model The obtained ordered difference set Ω sort is the position set of the virtual array elements included in the virtual array;

[0098] Input the sparse representation of the signal into the OMP algorithm for initial incident angle estimation. The specific steps are as follows:

[0099] First, set the maximum number of iterations u and the convergence threshold β, and use the sparse representation of the signal as the initial residual r;

[0100] Then, evenly divide the angle range [-90°, 90°] into N parts, where the number of divided grids N is greater than the number of signal sources K. Each grid corresponds to a unique number, and the grid set and its corresponding array manifold matrix of the virtual array will be used as an overcomplete dictionary, and use the candidate angles in the grid set to provide the initial estimated angle range for the OMP algorithm;

[0101] In order to obtain the matching degree between each candidate angle and the residual, screen out the angle direction most relevant to the current residual, and calculate the inner product index of the grid array manifold matrix and the residual r, as shown in the following formula:

[0102]

[0103] index is the angle direction most relevant to the current residual. If index is used as the angle index value in the grid set, the initial estimated angle of this iteration can be obtained as shown in the following formula:

[0104]

[0105] Step 4: For the initial estimated angle, perform off-grid correction through the first-order Taylor expansion method, and finally achieve the estimation of the signal direction of arrival angle after iteration;

[0106] For the initial estimated angle of the grid, off-grid correction is performed through the first-order Taylor expansion method to achieve the final angle estimation;

[0107] First, construct the steering vector a according to the initial estimated angle, as shown in the following formula:

[0108]

[0109] Perform the first-order Taylor expansion on the steering vector a to obtain its first derivative da with respect to the angle, as shown in the following formula:

[0110]

[0111] Thus, through correcting the phase change, local linear approximation is achieved;

[0112] Then, the signal x is recovered by the least squares method, as shown in the following formula:

[0113]

[0114] Further update the residual r, as shown in the following formula

[0115]

[0116] That is, subtract the estimated signal component from the sparse model of the signal to reflect the remaining signal energy;

[0117] To determine the correction direction of the initial estimated angle, calculate the inner product of the derivative da and the residual r to obtain the gradient g that can determine the angle correction direction, as shown in the following formula:

[0118] g = -2·real[(da) T r]

[0119] To accelerate convergence and improve the accuracy of angle correction, perform a first-order approximation on the second derivative of the steering vector a with respect to the angle to obtain the approximate Hessian matrix H, as shown in the following formula:

[0120] H = 2real(da T ·da)

[0121] To prevent matrix numerical instability and improve iterative convergence, regularize the approximate Hessian matrix H to obtain the regularized matrix as shown in the following formula:

[0122]

[0123] From the gradient and the regularized Hessian matrix obtained above Calculate the angle correction delta as shown below:

[0124] delta=g / H

[0125] Then update the initial estimated angle as shown below:

[0126]

[0127] Determine whether the angle correction delta is less than the threshold β. If not, continue with the following steps. If less than the threshold, terminate the iteration.

[0128] Finally, the residual orthogonal projection is updated to obtain the initial residual r for the next iteration. next As shown below:

[0129]

[0130] in A(:,index) represents the index-th column vector of the two-dimensional array A, and norm(·) represents the l2 norm;

[0131] After the update is completed, the obtained r next As the initial residual for the next iteration, after completing all iterations and angle updates, the θ obtained in each iteration is est The collection of is the final estimated angle.

[0132] The effects of the present invention are further illustrated by experimental examples below.

[0133] Experimental Example 1:

[0134] To verify that the post-nested array can increase the degree of freedom of the array, the arrival directions of more sources can be estimated. According to the above steps, a simulation experiment is designed. Eleven signals to be tested are incident in the far field with the angles between -60° and 60°. The number of snapshots is 512, the total number of array elements is 6, the noise is Gaussian white noise, and the signal-to-noise ratio SNR = 5dB. The simulation results are as follows: Figure 3 shown.

[0135] Depend on Figure 3 The simulation results show that even when the number of incident signal sources to be measured is much larger than the number of array elements and the signal-to-noise ratio is low, the DOA parameter estimation effect is still ideal, which verifies the feasibility of this method.

[0136] Experimental Example 2:

[0137] Based on the core steps in Embodiment 1, the number of incident signals set in Embodiment 1 is 6, and the measured signals are incident within the angular range of -60° to 60°. In this embodiment, the Root Mean Square Error (RMSE) is used as the performance index to evaluate the DOA parameter estimation method. Under the same Signal-to-Noise Ratio (SNR), the smaller the RMSE, the higher the algorithm accuracy and the better the performance. The RMSE is obtained in Q = 300 Monte Carlo experiments, and the RMSE calculation formula is as follows:

[0138]

[0139] where K represents the number of angles, and Q represents the number of experiments. represents the true value of the k-th parameter in the q-th trial, the estimated value of the k-th parameter in the q-th trial. Monte Carlo experiments are carried out with a step size of 1 dB in the SNR range of [-10dB, 10dB], and compared with other algorithms. The comparison curves of the mean square error of the three algorithms at different signal-to-noise ratios are as Figure 4 shown.

[0140] From Figure 4 the simulation results show that, compared with other algorithms, the method proposed in this patent still has high estimation accuracy and robustness in the case of low signal-to-noise ratio.

[0141] Embodiment 3: Based on the core steps in Embodiment 1, a comparative experiment on the success probability of DOA parameter estimation by three algorithms is carried out. If the average deviation between the estimated angle and the true angle is less than 0.5°, the estimation is successful; otherwise, it is a failure. The angles set in Embodiment 1 are set as random angles, seven incident signal sources are taken within the angular range of [-60, 60°], the SNR range is [-10dB, 10dB], and Monte Carlo experiments are carried out with a step size of 1 dB. In this embodiment, the Monte Carlo success probability is used as the performance index to evaluate the DOA method algorithm. The comparison diagram of the DOA parameter estimation success probability of the three algorithms is as Figure 5 shown.

[0142] From Figure 5 the simulation results show that, compared with other algorithms, the present invention still has a high success probability of parameter estimation in the case of low signal-to-noise ratio, indicating the superiority of the present invention in algorithm accuracy and robustness.

[0143] In summary of the above embodiments, the method proposed in the present invention improves the nested array, increases the degrees of freedom of the array, and adds an off-grid correction part to the Orthogonal Matching Pursuit (OMP) algorithm, thereby effectively improving the number of estimable signal sources of the algorithm and the accuracy of parameter estimation.

Claims

1. A method for off-grid compressive sensing direction of arrival estimation based on a post-nested array, characterized in that, It includes the following steps: Step 1: Construct a post-nested array using two uniform linear arrays; Step 2: Construct a signal covariance matrix based on the signals received by the post-nested array in Step 1; Step 3: Obtain the sparse representation of the signal by vectorizing and de-duplicating and sorting the covariance matrix, set the number of iterations and the convergence threshold, and input the sparse representation of the signal into the OMP algorithm to obtain the initial estimated angle; Step 4: For the initial estimated angle, perform off-grid correction through the first-order Taylor expansion method, and achieve the final angle estimation after iteration.

2. A method for off-grid compressive sensing direction-of-arrival estimation based on a post nested array according to claim 1, wherein: The post-nested array with M array elements in Step 1 is obtained by moving the first array element of the first-level array in the second-level nested array to the end of the second-level array. Among them, the element spacing of sub-array 1 is d1, and the number of array elements is M1; the element spacing of sub-array 2 is d2, and the number of array elements is M2; take the element spacing d1 of sub-array 1 as the unit spacing d, the total number of array elements M of the post-nested array = M1 + M2, the spacing between the two sub-arrays is the element spacing d1 of the previous-level sub-array, d1 = λ / 2, d2 = (M1 + 1)d1, and λ is the wavelength of the incident signal.

3. A method for off-grid compressive sensing direction-of-arrival estimation based on a post-nested array according to claim 1, characterized in that: In Step 2, if there are K far-field narrowband signal sources incident on the post-nested array, the mathematical model expression of the array received signal is: Y(t) = A(θ)X(t) + N(t) where \(Y(t)=[y_1(t),\ldots,y m (t),\ldots,y M (t)] T is the \(M\times L\) - dimensional signal data matrix received by the created antenna array, where \(M\) is the total number of array elements, \(L\) is the number of snapshots, \(y m (t)\) represents the signal received by the \(m\) - th array element, \([\cdot] T represents the transpose operation, \(X(t)=[x_1(t),\ldots,x k (t),\ldots,x K (t)] T is the \(K\times L\) - dimensional data matrix of the spatial target incident source, where \(K\) is the total number of sources, \(x k (t)\) represents the signal emitted by the \(k\) - th source, \(N(t)=[n_1(t),\ldots,n m (t),\ldots,n M (t)] T is the mutually independent noise data matrix of the array - received signal, \(n m (t)\) represents the noise received by the \(m\) - th array element, the mean value of the noise is \(0\), \(A(\theta)=[a(\theta_1),\ldots,a(\theta k ),\ldots,a(\theta K )]\) is the array manifold matrix, where \(a(\theta k )\) is the steering vector of the \(k\) - th incident source, as shown in the following formula: where j is a complex number, and d m is the spatial distance of the m-th array element relative to the reference array element, d m ∈P; P is the set of physical array element positions of the array, and the post-nested array has: And the following assumptions are made for the signal sources: the signal sources are cyclostationary signals, the signal sources are independent of each other, the noise is Gaussian noise and independent of the signal sources; The covariance matrix R of the received signal is obtained from the signal data received by the post-nested array, as shown in the following formula: where (·) H represents the conjugate transpose operation.

4. A method for off-grid compressive sensing direction-of-arrival estimation based on a post-nested array according to claim 1, characterized in that: In Step 3, the covariance matrix is vectorized to obtain a single-block snapshot signal model Z, as shown in the following formula: Z = vec(R) Among them, vec(·) represents vectorization; Then a difference set Ω is generated based on the positions P of the physical array elements of the post-nested array, as shown in the following formula: Among them, both i and o are serial numbers with values between 1 and M; Then, duplicate elements in the difference set Ω are removed to obtain the unique difference set Ω unique ; For the unique difference set Ω unique perform ascending sorting to obtain the ordered difference set Ω sort , and record the index value Ι of the sorting of the ordered difference set. Sort the single-block snapshot signal model Z that has completed synchronous update according to the index value Ι to obtain the final signal sparse model The obtained ordered difference set Ω sort is the position set of the virtual array elements included in the virtual array; Sparse representation of signals is input into the OMP algorithm for estimating the initial incident angle.

5. A method for off-grid compressive sensing direction of arrival estimation based on a post-nested array according to claim 4, characterized in that: The obtained unique difference set Ω unique , and the specific method is as follows: Traverse each element in Ω. If it already exists in Ω unique , then skip it; otherwise, add it to Ω unique ; And the single-block snapshot signal model Z is updated synchronously, and the data items with the same sequence as the elements of the repeated difference set in Z are deleted.

6. A method for off-grid compressive sensing direction-of-arrival estimation based on a post-nested array according to claim 4, characterized in that: The sparse representation of the signal is input into the OMP algorithm for estimating the initial incident angle, and the specific steps are as follows: First, set the maximum number of iterations \(u\) and the convergence threshold \(\beta\), and use the sparse representation of the signal as the initial residual \(r\). Then it is evenly divided into N parts according to the angular range [-90°, 90°], where the number of divided grids N is greater than the number of signal sources K, and each grid corresponds to a unique number, obtaining a grid set and the array manifold matrix of its corresponding virtual array Take as an overcomplete dictionary, and use the candidate angles in the grid set to provide the initial estimated angular range for the OMP algorithm; To obtain the matching degree between each candidate angle and the residual, and screen out the angular direction most relevant to the current residual, calculate the manifold matrix of the grid array The inner product index with the residual r is shown in the following formula: The index is the angular direction most relevant to the current residual. If the index is used as the angular index value in the grid set, the initial estimated angle of this iteration can be obtained. As shown in the following formula:

7. A method for off-grid compressive sensing direction of arrival estimation based on a post-nested array according to claim 1, characterized in that: In Step 4, for the initial estimated angle of the grid, off-grid correction is performed through the first-order Taylor expansion method to achieve the final angle estimation; First, a steering vector a is constructed according to the initial estimated angle, as shown in the following formula: The first-order derivative da of the steering vector a with respect to the angle is obtained by performing the first-order Taylor expansion on a, as shown in the following formula: Thereby, by correcting the phase change, local linear approximation is achieved; Then the signal x is recovered by the least squares method, as shown in the following formula: The residual r is further updated, as shown in the following formula: That is, subtracting the estimated signal components from the sparse model of the signal to reflect the remaining signal energy; To determine the correction direction of the initial estimated angle, the inner product of the derivative da and the residual r is calculated to obtain a gradient g that can determine the angle correction direction, as shown in the following formula: g = -2·real[(da) T r] To accelerate convergence and improve the accuracy of angle correction, the second-order derivative of the steering vector a with respect to the angle is approximately first-order approximated to obtain an approximate Hessian matrix H, as shown in the following formula: H = 2 real(da T ·da) To prevent numerical instability of the matrix and improve iterative convergence, the approximate Hessian matrix H is regularized to obtain a regularized matrix as shown in the following equation: The gradient obtained above and the regularized Hessian matrix Calculate the angular correction delta as shown in the following equation: delta = g / H Then the initial estimated angle is updated, as shown in the following formula: Judge whether the angle correction amount delta is less than the threshold β. If it is not less than, continue the following steps; if it is less than the threshold, terminate the iteration; Finally, the update of the residual orthogonal projection is performed to obtain the initial residual r for the next iteration next As shown in the following formula: Among them A(:, index) represents the index-th column vector of the two-dimensional array A, and norm(·) represents the l2 norm; After the update is complete, the obtained r next is used as the initial residual for the next iteration. After all iterations and angle updates are completed, the set of θ est obtained in each iteration is the finally estimated angle.

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