A grid-based underdetermined DOA estimation algorithm for wideband signals in sparse arrays

Abstract

The present invention belongs to the technical field of sparse array signal processing, and specifically relates to an underdetermined DOA estimation algorithm for sparse array broadband signals based on a grid. First, a corresponding sparse array is designed according to a given number of array elements W; second, a discrete Fourier transform is used to divide the incident broadband signal into multiple narrowband signals and a narrowband signal receiving model of different frequency bands is established; finally, the DOA estimation value of the incident signal is solved using the idea of ​​sparse reconstruction. Compared with traditional broadband signal DOA estimation algorithms, the present invention has higher estimation accuracy and can realize DOA estimation under underdetermined conditions. The present invention is simple in design, reasonable, and highly feasible, and can effectively solve the problem of underdetermined DOA estimation of broadband signals.

Description

Technical Field

[0001] The invention belongs to the field of sparse array signal processing and relates to an underdetermined DOA estimation algorithm for sparse array broadband signals based on a grid. Background Art

[0002] The algorithm and theory of DOA estimation for broadband signals are developed based on DOA estimation for narrowband signals. However, the array manifolds corresponding to different frequencies of broadband signals are different, which leads to different signal subspaces corresponding to different frequencies. Therefore, the original narrowband DOA estimation algorithm cannot be directly used for DOA estimation of broadband signals. The classic DOA estimation algorithms for broadband signals are divided into two categories: one is the incoherent signal subspace method (ISSM); the other is the coherent signal subspace method (CSSM). The principle of the ISSM algorithm is to decompose the broadband signal into multiple frequency subbands and perform DOA estimation separately, and then perform weighted calculation on the results to obtain the final result of DOA estimation of the broadband signal. Obviously, the ISSM algorithm only uses the information of each frequency subband, but not all the information of the broadband signal. Therefore, the DOA estimation performance of this type of algorithm is poor and cannot resolve the coherent source. The CSSM algorithm introduces the idea of ​​"focusing", aligning different frequency components on a subspace of a certain frequency component, and then averaging the covariance matrices of each subband to obtain the focused covariance matrix for DOA estimation. Since the DOA estimation performance of the CSSM method is better than that of the ISSM method and has the ability to process coherent signals, CSSM algorithms under different constraint criteria are proposed based on CSSM, mainly including the two-side correlation transformation algorithm (TCT), the rotating signal subspace algorithm (RSS) and the signal subspace transformation algorithm (SST). However, the focusing matrix of this type of algorithm mostly requires pre-estimation of the angle of the incident signal, and the focusing transformation process is relatively complicated. This type of algorithm can only have a good estimation result under the condition of a large number of snapshots, and its computational complexity is very large. The above broadband DOA estimation algorithms are all proposed for uniform linear arrays. In order to achieve underdetermined DOA estimation of broadband signals, the present invention combines sparse array technology and sparse signal reconstruction algorithm and extends them to underdetermined DOA estimation of broadband signals. Summary of the invention

[0003] In order to overcome the shortcomings of the existing algorithms, the purpose of the present invention is to propose an underdetermined DOA estimation algorithm for broadband signals in sparse arrays based on grids, so as to realize the underdetermined DOA estimation of broadband signals. Firstly, the corresponding sparse array is designed according to the given total number of array elements, and then the broadband signal is decomposed into narrowband signals by discrete Fourier transform. Secondly, a narrowband signal receiving model based on sparse array is established, and finally the DOA estimation value of each incident signal is solved by using the idea of ​​joint sparsity.

[0004] The present invention and technical solutions are described as follows:

[0005] The present invention provides an underdetermined DOA estimation algorithm for sparse array broadband signals based on off-grid, comprising:

[0006] S1: Design the corresponding sparse array according to the given total number of array elements;

[0007] S2: Decompose the broadband signal of the sparse array into the narrowband signal of the sparse array by discrete Fourier transform, and establish a receiving model based on the narrowband signal of the sparse array;

[0008] S3: Use the idea of ​​joint sparsity to solve the DOA estimation value of each incident signal.

[0009] Optionally, the designing of a corresponding sparse array according to a given total number of array elements in S1 specifically includes:

[0010] The total number of array elements of the sparse array S is W, including sub-array S1 and sub-array S2, sub-array S1 has N physical array elements, sub-array S2 has M physical array elements, N≥M≥2 and W=M+N; the values ​​of M, N and W are natural numbers;

[0011] The array element spacing of subarray S1 is R1=(M+1)d; the array element spacing of subarray S2 is R2=R1+d, d=λ2, λ represents the wavelength corresponding to the highest frequency of the incident signal;

[0012] S1d and S2d represent the array element positions of sub-array S1 and sub-array S2 respectively, then:

[0013]

[0014] Subarray S1 and subarray S2 have 2P array elements intersecting each other, where The position of the cross-element is expressed as Sd=S1d∪S2d.

[0015] Optionally, the step of decomposing the sparse array broadband signal into a sparse array narrowband signal by discrete Fourier transform in S2 specifically includes:

[0016] Assume that there are K far-field broadband signals with angles θ = [θ1, θ2, …, θ K ]T Incident on the sparse array, T represents the transpose of the matrix; with f s The incident signal is sampled at the sampling frequency, and then the signal received by each array element is divided into Q sub-segments in time, each segment is L in length, and each segment is subjected to L-point discrete Fourier transform to obtain the frequency domain observation signal vector of the lth frequency band of the qth segment signal, q = {1, 2, 3...Q}, l = {1, 2, 3...L}:

[0017]

[0018] In formula (3), X[l,q], S[l,q] and denote the received signal vector, incident source signal vector and noise signal vector of the lth frequency subband of the qth segment signal, respectively. A(l,θ)=[a(l,θ1),a(l,θ2),…,a(l,θ K )] represents the steering vector matrix corresponding to the lth frequency subband, and the column vector a(l,θ k ) represents the steering vector corresponding to the lth frequency subband and the kth incident signal; k = {1, 2, 3 ... K};

[0019] Calculate the signal covariance matrix R of the lth frequency subband corresponding to the sparse array xx [l], its expression is:

[0020]

[0021] Among them, E{·} and {·} H denote expectation and Hermitian transpose respectively, and represents the power and noise power of the kth incident signal in the corresponding lth frequency subband, I W represents the W×W dimensional identity matrix;

[0022] in,

[0023]

[0024] e represents a natural number; j represents an imaginary unit; s = {1, 2, 3...S}; s W represents the corresponding array element; λ l represents the signal wavelength corresponding to the lth frequency subband, and T represents the transpose of the matrix.

[0025] Optionally, establishing a receiving model based on a sparse array narrowband signal in S2 includes:

[0026]

[0027] Wherein B[l] represents the equivalent steering vector matrix of the virtual array corresponding to the optimal interleaved array, and its kth column vector represents the steering vector of the kth incident signal on the lth frequency subband, where represents the Kronecker product, and the superscript * indicates taking the conjugate; represents the equivalent incident signal vector, T represents the transpose of the matrix; Represents the vectorized identity matrix I W The obtained W 2 ×1-dimensional column vector;

[0028] Assume that the incident broadband signal contains V ≤ L frequency sub-bands, and v (v=0,1,…,V-1) represents the index of the included frequency subband; the virtual array observation vectors corresponding to all V frequency subbands are merged into a column vector T represents the transpose of the matrix and constructs a block diagonal matrix Its expression is:

[0029]

[0030] Then the broadband signal virtual array model of the incident signal can be expressed as

[0031]

[0032] in, represents a virtual signal vector; W 2 ×V-dimensional block diagonal matrix, Represents the noise signal power vector within the corresponding frequency band;

[0033] Diagonal Matrix The equivalent steering matrix Performing a first-order Taylor expansion, the obtained off-grid signal source broadband virtual array receiving model is expressed as:

[0034]

[0035] in, represents a block diagonal matrix, where The expression is:

[0036] H g represents the number of predefined discrete grids; θ g Predefined signal incident angle; (1) The first row of the matrix represented by; It means to seek the partial derivative of the opposite meaning; represents a diagonal matrix, where

[0037] Constructing the Matrix in Indicates H g ×V-dimensional complex matrix; h={1, 2, 3…H}; the hth g The row can be represented as right beg The norm can be obtained as formula (9);

[0038]

[0039] Among them, ||·||2 means norm;

[0040] Constructing column vectors And define the matrix Y g , whose expression is:

[0041] Y g =[u g [l0],u g [l1],…,u g [l V-1 ],β[l0],β[l1],…,β[l V-1 ]](10);

[0042] Constructing column vectors Its expression is:

[0043]

[0044] in, Represents Y g h g row, h = {1, 2, 3...H}.

[0045] Furthermore, the S3 uses the idea of ​​joint sparsity to solve the DOA estimation value of each incident signal, which specifically includes:

[0046] The DOA estimation algorithm can be expressed as:

[0047]

[0048] Among them, ||·||1 represents the l1 norm, ε is the upper limit parameter of the error fitting, ≤ means that the corresponding elements in the matrix or vector should satisfy the relationship ≤, and the column vector

[0049] In formula (9) Indicated by H g The initial DOA estimation result is composed of discrete angles, and the corresponding off-grid deviation vector can be expressed as:

[0050]

[0051] in, Represents the division operation between corresponding elements of two matrices or vectors; only Corresponding to The non-zero terms of realize the calibration of DOA estimation results, m = {1, 2, 3…M}.

[0052] Furthermore, mandatory constraints And adopt α g [l v ] instead of β g [l v ] to reduce the computational complexity of the algorithm, where Represents the off-grid deviation vector to be estimated, construct Its expression is:

[0053]

[0054] in, Represents H consisting of all 1s g ×1-dimensional column vector, represents the Kronecker product;

[0055] Two convex optimization problems are used to calculate and The expressions are:

[0056]

[0057] Among them, the column vector ⊙ represents the multiplication operation of corresponding elements of two matrices or vectors; st represents the constraint, and r represents the step length of the grid.

[0058] Compared with the existing broadband signal DOA estimation algorithm, the present invention has the following advantages:

[0059] The present invention provides a sparse array setting method, which is simple and effective. The present invention establishes a DOA estimation model based on joint sparseness based on a signal reception model under off-grid conditions. The present invention utilizes the high degree of freedom characteristics of a virtual array corresponding to a sparse array to achieve DOA estimation of more sources. The present invention is simple in design, reasonable, and highly feasible, and can effectively solve the problem of underdetermined DOA estimation of broadband signals. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] The accompanying drawings are used to provide a further understanding of the present disclosure and constitute a part of the specification. Together with the following specific embodiments, they are used to explain the present disclosure but do not constitute a limitation of the present disclosure. In the accompanying drawings:

[0061] Figure 1 The flowchart of the underdetermined DOA estimation algorithm for sparse array broadband signals based on grid separation is shown in Figure 1.

[0062] Figure 2 is a pair of sparse uniform subarrays;

[0063] Figure 3 is a non-uniform sparse array composed of;

[0064] Figure 4 Simulation results for DOA estimation. DETAILED DESCRIPTION

[0065] In order to make the technical solutions and advantages of the present invention more clearly understood, the present invention is further described below in conjunction with the accompanying drawings:

[0066] The underdetermined DOA estimation algorithm of the sparse array broadband signal based on the grid of the present invention specifically includes:

[0067] Step 1: Design the corresponding sparse array according to the given total number of array elements;

[0068] The total number of array elements of the sparse array is W, including sub-array S1 and sub-array S2. Sub-array S1 has N physical array elements, sub-array S2 has M physical array elements, N≥M≥2 and W=M+N; the values ​​of M, N and W are natural numbers;

[0069] The array element spacing of subarray S1 is R1=(M+1)d; the array element spacing of subarray S2 is R2=R1+d, d=λ / 2, λ represents the wavelength corresponding to the highest frequency of the incident signal;

[0070] S1d and S2d represent the array element positions of sub-array S1 and sub-array S2 respectively, then:

[0071]

[0072] Subarray S1 and subarray S2 have 2P array elements intersecting each other, where The position of the cross array element is expressed as Sd=S1d∪S2d, where S represents a sparse array.

[0073] Step 2: Decompose the broadband signal of the sparse array into a narrowband signal of the sparse array by discrete Fourier transform, and establish a receiving model based on the narrowband signal of the sparse array;

[0074] 2.1: Establish a data reception model and calculate the signal covariance matrix R of the lth frequency subband xx [l];

[0075] Assume that there are K far-field broadband signals with angles θ = [θ1, θ2, …, θ K ] Tis incident on the sparse array. s The incident signal is sampled at the sampling frequency, and then the signal received by each array element (the superposition of multiple far-field broadband signals) is divided into Q sub-segment broadband signals in units of time, each segment is L (L points), and a discrete Fourier transform (DFT) of L points is performed on each segment signal to obtain the frequency domain observation signal vector of the lth frequency band of the qth segment signal: q = {1, 2, 3...Q}, l = {1, 2, 3...L};

[0076]

[0077] Among them, X[l,q], S[l,q] and denote the received signal vector, incident source signal vector and noise signal vector of the lth frequency subband of the qth segment signal, respectively. A(l,θ)=[a(l,θ1),a(l,θ2),…,a(l,θ K )] represents the steering vector matrix corresponding to the lth frequency subband, and the column vector a(l,θ k ) represents the steering vector corresponding to the lth frequency subband and the kth incident signal. k = {1, 2, 3, ... K}.

[0078] Calculate the signal covariance matrix R of the lth frequency subband corresponding to the optimal interleaved array xx [l], whose expression is

[0079]

[0080] Among them, E{·} and {·} H denote expectation and Hermitian transpose respectively. and represents the power and noise power of the kth incident signal in the corresponding lth frequency subband, I W represents the W×W dimensional identity matrix.

[0081]

[0082] e represents a natural number; j represents an imaginary unit; s = {1, 2, 3...S}; s W represents the corresponding array element; λ l represents the signal wavelength corresponding to the lth frequency subband, and T represents the transpose of the matrix.

[0083] 2.2: R xx [l] is vectorized to obtain z[l] and a wideband virtual array receiving model of off-grid signal source is constructed;

[0084] Vectorized signal covariance matrix R xx [l], we can get

[0085]

[0086] Wherein B[l] represents the equivalent steering vector matrix of the virtual array corresponding to the optimal interleaved array, and its kth column vector represents the steering vector of the kth incident signal on the lth frequency subband, where represents the Kronecker product, Represents the equivalent incident signal vector, the superscript T represents the transpose of the matrix, Represents the vectorized identity matrix I W The obtained W 2 ×1-dimensional column vector.

[0087] Assume that the incident broadband signal contains V ≤ L frequency sub-bands, and v (v = 0, 1, ..., V-1) represents the index of the included frequency subband. The virtual array observation vectors corresponding to all V frequency subbands are merged into a column vector T represents the transpose of the matrix and constructs a block diagonal matrix Its expression is

[0088]

[0089] Then the broadband signal virtual array model of the incident signal can be expressed as:

[0090]

[0091] in, represents a virtual signal vector; W 2 ×V-dimensional block diagonal matrix, Represents the noise signal power vector within the corresponding frequency band.

[0092] However, the incident signal angle often cannot fall exactly on the discrete grid. In this case, the equivalent steering matrix Performing a first-order Taylor expansion, the off-grid broadband signal model can be expressed as

[0093]

[0094] in, represents a block diagonal matrix, where The expression is:

[0095] H g represents the number of predefined discrete grids; θ g Predefined signal incident angle; (1) The first row of the matrix represented by; It means to seek the partial derivative of the opposite meaning; represents a diagonal matrix, where The subscript g generally indicates predefined.

[0096] Constructing the Matrix in Indicates H g ×V-dimensional complex matrix; h={1, 2, 3…H}; the hth g The row can be represented as right By calculating the l2 norm, we can obtain formula (9);

[0097]

[0098] Among them, ||·||2 means Norm.

[0099] Column vector and matrix Joint reconstruction is a non-convex optimization problem. To convert it into a convex optimization problem, we construct the column vector And define the matrix Y g , whose expression is

[0100] Y g =[u g [l0],u g [l1],…,u g [l V-1 ],β[l0],β[l1],…,β[l V-1 ]](10);

[0101] Constructing column vectors Its expression is

[0102]

[0103] in, Represents Y g h g OK.

[0104] Step 3: Use the idea of ​​joint sparsity to solve the DOA estimate of each incident signal.

[0105] The DOA estimation algorithm can be expressed as:

[0106]

[0107] Among them, ||·||1 means norm, ε is the upper limit parameter of error fitting, ≤ means that the corresponding elements in the matrix or vector should satisfy the relationship ≤, and the column vector

[0108] In formula (9), Indicated by H g The initial DOA estimation result is composed of discrete angles, and the corresponding off-grid deviation vector can be expressed as:

[0109]

[0110] in, Represents the division operation between corresponding elements of two matrices or vectors; only Corresponding to The non-zero terms of realize the calibration of DOA estimation results, m = {1, 2, 3…M}.

[0111] Since the column vector and The solution of is very complicated. In order to reduce the computational complexity of equation (12), this section will simplify the column vector and solution process to reduce the computational complexity.

[0112] Enforce constraints And adopt α g [l v ] instead of β g [l v ] to reduce the computational complexity of the algorithm, where Represents the off-grid deviation vector to be estimated, construct Its expression is:

[0113]

[0114] in, Represents H consisting of all 1s g ×1-dimensional column vector, represents the Kronecker product;

[0115] Two convex optimization problems are used to calculate and The expressions are:

[0116]

[0117] Among them, the column vector ⊙ represents the multiplication operation of the corresponding elements of two matrices or vectors; st represents the constraint, and r represents the step length of the grid. Then the DOA estimation value of the incident signal can be solved by equations (15) and (16). Equation (15) represents the broadband DOA estimation algorithm based on joint sparseness, which aims to achieve Reconstruction; Formula (16) is to solve the minimization problem under bounded constraints, and realize the vector Estimates.

[0118] Embodiment 1:

[0119] Reference Figure 1 , i.e. the algorithm flow chart of the present invention, the specific implementation steps are as follows:

[0120] S1: Design sparse array;

[0121] Reference Figure 2 and 3 Design a sparse array. Assuming that the total number of given array elements is 12, the number of array elements in subarray 1 is N = 9, and the number of array elements in subarray 2 is M = 3. That is, the array element spacing of subarray 1 is R1 = 4d, and the array element spacing of subarray 2 is R2 = 5d. The number of cross array elements between subarray 1 and subarray 2 is The element positions of the optimal interleaved array are:

[0122] S2: Decompose the broadband signal of the sparse array into the narrowband signal of the sparse array by discrete Fourier transform, and establish a receiving model based on the narrowband signal of the sparse array;

[0123] 2.1: Establish a data reception model and calculate the signal covariance matrix R of the lth frequency subband xx [l];

[0124] The signal received by each array element is divided into Q sub-segments, where the length of each segment is L, and each segment signal is subjected to L-point discrete Fourier transform (DFT), and the frequency domain observation signal vector of the lth frequency band of the qth segment signal can be obtained as According to formula (4), the signal covariance matrix R of the lth frequency subband is calculated xx [l].

[0125] 2.2: R xx [l] is vectorized to obtain z[l] and a wideband virtual array receiving model of off-grid signal source is constructed;

[0126] According to formula (4), the signal covariance matrix R of the vectorized frequency subband xx [l], according to equations (6) and (7), the block diagonal matrices are constructed respectively and the broadband signal virtual array model, and construct the off-grid broadband signal model according to equation (8);

[0127] S3: Use the idea of ​​joint sparsity to solve the DOA estimation value of each incident signal;

[0128] Two convex optimization problems are used to calculate and Then the DOA estimation value of the incident signal can be solved by equations (15) and (16).

[0129] Reference Figure 4 , assuming that there are K = 16 far-field broadband signals with incident angles uniformly distributed between -59.25° and 58.75° incident on a certain optimal interleaved array, and L = 128 points are used to perform discrete Fourier transform on the incident signal. It is assumed that the frequency of the incident signal contains V = 31 frequency subbands from the 33rd to the 63rd, and its normalized frequency range is 0.5π to π. The simulation results are shown in the figure. The dotted line in the figure represents the true angle of the incident signal, and the solid line represents the estimated angle of the algorithm. It can be observed that the broadband algorithm proposed in the present invention can correctly estimate the angles of K = 16 incident signal sources.

[0130] The above description is only a preferred specific implementation manner of the present invention, and the protection scope of the present invention is not limited thereto. Within the technical concept of the present disclosure, a variety of simple modifications can be made to the technical solution of the present disclosure, and these simple modifications all belong to the protection scope of the present disclosure.

Claims

1. An underdetermined DOA estimation algorithm for sparse array broadband signals based on grid separation, characterized in that: include: S1: Design the corresponding sparse array according to the given total number of array elements; S2: Decompose the broadband signal of the sparse array into the narrowband signal of the sparse array by discrete Fourier transform, and establish a receiving model based on the narrowband signal of the sparse array; S3: Use the idea of ​​joint sparsity to solve the DOA estimation value of each incident signal; The establishment of a receiving model based on a sparse array narrowband signal in S2 includes: Where B[l] represents the equivalent steering vector matrix of the virtual array corresponding to the optimal interleaved array, and its kth column vector represents the steering vector of the kth incident signal on the lth frequency subband, where represents the Kronecker product, and the superscript * indicates taking the conjugate; represents the equivalent incident signal vector, T represents the transpose of the matrix; Represents the vectorized identity matrix I W The obtained W 2 ×1-dimensional column vector; Assume that the incident broadband signal contains V ≤ L frequency sub-bands, and v (v=0,1,…,V-1) represents the index of the included frequency subband; the virtual array observation vectors corresponding to all V frequency subbands are merged into a column vector T represents the transpose of the matrix and constructs a block diagonal matrix Its expression is: Then the broadband signal virtual array model of the incident signal can be expressed as in, represents a virtual signal vector; W 2 ×V-dimensional block diagonal matrix, Represents the noise signal power vector within the corresponding frequency band; Diagonal Matrix The equivalent steering matrix Performing a first-order Taylor expansion, the obtained off-grid signal source broadband virtual array receiving model is expressed as: in, represents a block diagonal matrix, where The expression is: H g represents the number of predefined discrete grids; θ g Predefined signal incident angle; (1) The first row of the matrix represented by; It means to seek the partial derivative of the opposite meaning; represents a diagonal matrix, where Constructing the Matrix in Indicates H g ×V-dimensional complex matrix; h={1, 2, 3…H}; the hth g The row can be represented as right beg The norm can be obtained as formula (9); Among them, ||·||2 means norm; Constructing column vectors And define the matrix Y g , whose expression is: Y g =[in g [l0],u g [l1],…,u g [l V-1 ],β[l0],β[l1],…,β[l V-1 ]](10); Constructing column vectors Its expression is: in, Represents Y g h g row, h = {1, 2, 3...H}.

2. The underdetermined DOA estimation algorithm for sparse array broadband signals based on off-grid according to claim 1, characterized in that: The design of the corresponding sparse array according to the given total number of array elements in S1 specifically includes: The total number of array elements of the sparse array S is W, including sub-array S1 and sub-array S2, sub-array S1 has N physical array elements, sub-array S2 has M physical array elements, N≥M≥2 and W=M+N; the values ​​of M, N and W are natural numbers; The array element spacing of subarray S1 is R1=(M+1)d; the array element spacing of subarray S2 is R2=R1+d, d=λ2, λ represents the wavelength corresponding to the highest frequency of the incident signal; S1d and S2 d Respectively represent the array element positions of sub-array S1 and sub-array S2, then: Subarray S1 and subarray S2 have 2P array elements intersecting each other, where The position of the cross-element is expressed as Sd=S1d∪S2d.

3. The underdetermined DOA estimation algorithm for sparse array broadband signals based on off-grid according to claim 1 or 2, characterized in that: Decomposing the sparse array broadband signal into the sparse array narrowband signal by discrete Fourier transform in S2 specifically includes: Assume that there are K far-field broadband signals with angles θ = [θ1, θ2, …, θ K ] T Incident on the sparse array, T represents the transpose of the matrix; with f s The incident signal is sampled at the sampling frequency, and then the signal received by each array element is divided into Q sub-segments in time, each segment is L in length, and each segment is subjected to L-point discrete Fourier transform to obtain the frequency domain observation signal vector of the lth frequency band of the qth segment signal, q = {1, 2, 3...Q}, l = {1, 2, 3...L}: In formula (3), X[l,q], S[l,q] and denote the received signal vector, incident source signal vector and noise signal vector of the lth frequency subband of the qth segment signal, respectively. A(l,θ)=[a(l,θ1),a(l,θ2),…,a(l,θ K )] represents the steering vector matrix corresponding to the lth frequency subband, and the column vector a(l,θ k ) represents the steering vector corresponding to the lth frequency subband and the kth incident signal; k = {1, 2, 3 ... K}; Calculate the signal covariance matrix R of the lth frequency subband corresponding to the sparse array xx [l], its expression is: Among them, E{·} and {·} H denote expectation and Hermitian transpose respectively, and represents the power and noise power of the kth incident signal in the corresponding lth frequency subband, I W represents the W×W dimensional identity matrix; in, e represents a natural number; j represents an imaginary unit; s = {1, 2, 3...S}; s W represents the corresponding array element; λ l represents the signal wavelength corresponding to the lth frequency subband, and T represents the transpose of the matrix.

4. The underdetermined DOA estimation algorithm for sparse array broadband signals based on off-grid according to claim 1 or 2, characterized in that: The S3 uses the idea of ​​joint sparseness to solve the DOA estimation value of each incident signal, which specifically includes: The DOA estimation algorithm can be expressed as: Among them, ||·||1 means norm, ε is the upper limit parameter of error fitting, Indicates that the corresponding elements in the matrix or vector should satisfy the relationship ≤, and the column vector In formula (9) Indicated by H g The initial DOA estimation result is composed of discrete angles, and the corresponding off-grid deviation vector can be expressed as: in, Represents the division operation between corresponding elements of two matrices or vectors; only Corresponding to The non-zero term of realizes the calibration of DOA estimation results.

5. The underdetermined DOA estimation algorithm for sparse array broadband signals based on off-grid according to claim 4, characterized in that: Enforce constraints And adopt α g [l v ] instead of β g [l v ] to reduce the computational complexity of the algorithm, where Represents the off-grid deviation vector to be estimated, construct Its expression is: in, Represents H consisting of all 1s g ×1-dimensional column vector, represents the Kronecker product; Two convex optimization problems are used to calculate and The expressions are: Among them, the column vector ⊙ represents the multiplication operation of corresponding elements of two matrices or vectors; st represents the constraint, and r represents the step length of the grid.

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