A three-layer nested array design method based on non-Gaussian signal sources

Abstract

The present invention discloses a three-layer nested array design method based on a non-Gaussian signal source. The array structure proposed by the method utilizes the continuous characteristics of the nested array difference array, and designs an array structure so that there is a nested relationship in its discontinuous sum array, so that a virtual array with a continuous sum and difference array can be finally obtained. The array can provide a large degree of freedom while continuously deriving its closed-form solution. The array of the present invention utilizes the continuous characteristics of the nested array difference array, and the proposed array can derive a provably completely continuous fourth-order virtual array element position, that is, the array has a closed-form solution, and the array has strong ductility and usability.

Description

Technical Field

[0001] The invention belongs to the field of array signal processing, and in particular to a three-layer nested array design method based on a non-Gaussian signal source. Background Art

[0002] Direction of Arrival (DOA) estimation is one of the fundamental problems in array signal processing and plays an important role in various fields such as radar systems, acoustics, navigation, and wireless communications. Compared with the traditional uniform linear array (ULA), sparse arrays can obtain less mutual coupling and higher degrees of freedom (DOF), and can perform DOA estimation for signals from more sources. Nested arrays (NA) are sparse arrays composed of two ULAs with different spacings, which have strong scalability in DOA estimation algorithms based on fourth order cumulant (FOC). Coprime arrays (CPA) are composed of two ULAs with coprime element spacings, and have less coprime coupling compared to NA. These arrays can generate a set of uniformly distributed virtual second order difference co-arrays (2-DC), especially the 2-DC of NA has no holes. However, these sparse arrays have limitations due to their array geometry. The 2-DC of CPA generates a large number of voids, which significantly reduces the continuous degree of freedom, and the dense part of NA leads to more serious mutual coupling than that of CPA.

[0003] Traditional DOA parameter estimation algorithms, such as the Multiple Signal Classification (MUSIC) algorithm and the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm, mostly use the second-order characteristics of the signal to estimate signal parameters. When the signal obeys a Gaussian distribution, it can be described by first-order or second-order statistics. However, the situation in actual applications is more complicated. Most of the signal sources to be processed are non-Gaussian signals, and their statistical characteristics can be described by fourth-order cumulants (FOC), which makes the array aperture larger and the performance better. Compared with the second-order cumulant (SOC), the DOA estimation performance is better. Considering non-Gaussian signals, the FOC-based DOA estimation method is mainly used, including the MUSIC-LIKE and virtual-ESPRIT algorithms. The virtual array used by the vectorized FOC method can be obtained from the fourth order difference array (Fourth Order Difference Co-array, 4-DC) or its second order difference array (Second Order Difference Co-array of Sum Co-array, 2-DCSC) of the physical sensor. FOC can not only suppress Gaussian white noise or color noise, but also effectively expand the length of the array aperture. Therefore, the DOA estimation performance based on FOC is greatly improved.

[0004] Combining the design of sparse arrays and the development of 4-DC or 2-DCSC, scholars have proposed many array structures, such as the four-level nested array (FL-NA) and the fourth-order differential array based on mutually prime arrays (SAFE-CPA). In order to deal with the above sparse array structures, relevant scholars have proposed a series of FOC estimation algorithms based on spatial smoothing subspace methods, such as the multiple signal classification algorithm based on spatial smoothing (SS-MUSIC) and the rotational invariant subspace algorithm based on spatial smoothing (SS-ESPRIT). Summary of the invention

[0005] Purpose of the invention: In view of the above-mentioned prior art, a three-layer nested array design method based on non-Gaussian signal sources is proposed.

[0006] Technical solution: A three-layer nested array design method based on non-Gaussian signal sources includes the following steps:

[0007] Step 1: Establish a DOA estimation model under non-Gaussian sources;

[0008] Step 2, constructing a three-layer nested array model with a three-layer nested relationship;

[0009] Step 3: Find the fourth-order cumulant matrix R of the array 4 , vectorize it to get vector z, sort it and remove redundancy to get z 1 ;

[0010] Step 4: Use the spatial smoothing method to obtain the spatial smoothing covariance matrix R;

[0011] Step 5: Use the classic MUSIC algorithm to obtain the final angle estimate and compare the root mean square error between arrays.

[0012] Furthermore, the process of establishing the DOA estimation model under the non-Gaussian source in step 1 is as follows:

[0013] Assume that there are K narrowband far-field incoherent signal sources in space, and their one-dimensional wave arrival direction is θ k (k=1,2,…,K), the signal source is incident on a hypothetical array containing P array elements. The received signal model of the hypothetical array is expressed as:

[0014] X=A(θ)S+N

[0015] in, represents the direction matrix, and represents a non-Gaussian signal source with a mean of 0, and L represents the number of snapshots and k = 1, 2, ..., K; It is represented by a mean of 0 and a variance of σ 2 Additive Gaussian white noise; λ represents half wavelength, d represents the spacing between array elements; for a zero-mean random process X, given a random variable x k1 ,x k2 , Their fourth-order cumulants are expressed as:

[0016]

[0017] Applying this definition to the entire hypothetical array, the fourth-order signal accumulant of the received signal for the entire hypothetical array is expressed as:

[0018]

[0019] X is the overall matrix representation of x, which is a calculation of the fourth-order cumulant of the matrix as a whole.

[0020] Furthermore, the positions of the P elements in the assumed array in step 1 are represented by the set {v p ,p=1,2,...,P} means that after calculating the fourth-order cumulant of the received signal, the corresponding element position of the fourth-order differential virtual array is expressed as: Where 1≤v p1 ,v p2 ,v p3 ,v p4 ≤P.

[0021] Furthermore, the three-layer nested array with three-layer nested relationship described in step 2 is the array structure (THL-NA) proposed in this paper, and the array element position corresponding to the array is expressed as:

[0022] S 1 ={n 1 d,1≤n 1 ≤N 1}

[0023] S 2 ={n 2 (N 1 +1)d,1≤n 2 ≤N 2}

[0024] S 3 ={n 3 (N 2 +1)(N 1 +1)d,1≤n 3 ≤N 3}

[0025] P THL-NA =S 1 ∪S 2 ∪S 3

[0026] The schematic diagram of the array element position distribution is as follows Figure 1 As shown, the array P has a total of N = N 1 +N 2 +N 3 The 2-DCSC of the array is in the range -2N. 3 (N 2 +1)(N 1 +1)+2 to 2N 3 (N 2 +1)(N 1 +1)-2 must be continuous, and the continuous degree of freedom is cDOF = 2[2N 3 (N 2 +1)(N 1 +1)-2]+1=4N 3 (N 2 +1)(N 1 +1)-3.

[0027] Furthermore, step 3 includes the following steps:

[0028] Step 31: First, solve the fourth-order cumulant matrix R of the received signal 4 ;

[0029] Step 32: To further improve the DOA estimation accuracy, a virtualization method is used to calculate the fourth-order cumulant matrix R 4 Processing is performed to obtain vector z;

[0030] Step 33: Sort z according to the positions of the secondary virtual array elements in L and remove redundant array elements to construct z 1 .

[0031] Furthermore, the step 31 includes the following steps:

[0032]

[0033] in

[0034]

[0035]

[0036] B(θ) is the array manifold after expansion using the fourth-order cumulant array b(θ k ) is a(θ k ) is the expanded array manifold, where X represents the matrix as a whole and x is a vector in the matrix.

[0037] Step 32: To further improve the DOA estimation accuracy, we use a virtualization method to calculate the fourth-order cumulant matrix R 4 Processing is performed to obtain vector z;

[0038] z=vec(R 4 )=vec[B(θ)C s B H (θ)]=Λ(θ)p

[0039] in

[0040]

[0041] and represents the fourth-order cumulant of the k-th source vector, where Represents the fourth-order cumulant of the kth source.

[0042] Vectorized R 4 The corresponding direction vector matrix Λ(θ) can be considered to be expanded by the two-level differential array A(θ) B(θ) corresponds to the first-level continuous virtual array element position L = {-M b d,-(M b -1)d,...,0,...,(Mb -1)d,M b d},M b =2N 3 (N 2 +1)(N 1 +1)-2 Set the corresponding parameters to N 1 =3,N 2 =3,N 3 =4; the corresponding parameters of the secondary virtual array element position corresponding to Λ(θ) are set to N 1 =3,N 2 =3,N 3 =4, the closed-form solution derived in step 2 can be expressed as

[0043] The positions of the secondary virtual array elements within the closed-form solution are completely continuous; b It represents the maximum number of positions of virtual array elements.

[0044] Further, the step 33 includes the following steps:

[0045] z 1 =Λ 1 p

[0046] in, is the matrix of Λ(θ) after removing redundancy and sorting; Based on the above, we can know that the direction matrix of the differential array Λ 1 for

[0047]

[0048] Beneficial effects:

[0049] 1. The array of the present invention utilizes the continuous characteristics of nested array difference arrays. The proposed array can derive a provably completely continuous fourth-order virtual array element position, that is, the array has a closed-form solution, and the array has strong scalability and availability;

[0050] 2. The array THL-NA designed by the present invention has a large degree of freedom, and compared with the newly proposed sparse array SAFE-CPA, the traditional two-level nested array TL-NA, the four-level nested array FL-NA, and the traditional augmented coprime array ACA, it has a larger continuous degree of freedom;

[0051] 3. When the array THL-NA designed by the present invention is compared with the above-mentioned four array structures SAFE-CPA, TL-NA, FL-NA and ACA with the same number of array elements and other conditions, the angle estimation performance of the array proposed in this paper is better than that of other array structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1It is a schematic diagram of the array element distribution of the three-layer nested array designed by the present invention;

[0053] Figure 2 It is a comparison diagram of several different array element positions under the array structure proposed by the present invention;

[0054] Figure 3 It is a schematic diagram of the positions and weights of the first-level virtual array elements of the array proposed by the present invention;

[0055] Figure 4 It is a schematic diagram of the positions and weights of the secondary virtual array elements of the array proposed by the present invention;

[0056] Figure 5 is a schematic diagram of virtual array element distribution after adopting the spatial smoothing algorithm;

[0057] Figure 6 This is a comparison diagram of the array proposed by the present invention and several other array structures in terms of degrees of freedom;

[0058] Figure 7 It is a performance comparison diagram of the root mean square error of the array proposed by the present invention and other array structures using SS-MUSIC (spatial smoothing MUSIC algorithm) as the signal-to-noise ratio changes;

[0059] Figure 8 This is a performance comparison chart of the root mean square error of the array proposed in the present invention and other array structures using the SS-MUSIC algorithm as the number of snapshots changes. DETAILED DESCRIPTION

[0060] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0061] Symbols: In the present invention, (·) T ,(·) H (·) -1 and(·) * They represent transpose, conjugate transpose, inversion, and conjugate operations, respectively. Bold capital letters represent matrices, bold lowercase letters represent vectors, the symbol cum(·) represents fourth-order cumulant calculations, and E(·) represents expectation calculations. represents the Kronecker product, ⊙ represents the Khatri-Rao product, vec(·) represents matrix vectorization, and angle(·) represents the phase angle of a complex number.

[0062] The specific implementation of the present invention is as follows:

[0063] The method for designing nested arrays with coprime relationships based on non-Gaussian signal sources comprises the following steps:

[0064] Step 1: Establish a DOA estimation model under non-Gaussian sources:

[0065] Assume that there are K narrowband far-field incoherent signal sources in space, and their one-dimensional wave arrival direction is θ k (k=1,2,…,K), the signal source is incident on the assumed array containing P array elements. The received signal model of the assumed array can be expressed as:

[0066] X=A(θ)S+N

[0067] in, represents the direction matrix, and represents a non-Gaussian signal source with a mean of 0, and L represents the number of snapshots and k=1,2,…,K. It is represented by a mean of 0 and a variance of σ 2 Additive Gaussian white noise. λ represents half wavelength, d represents the spacing between array elements; for a zero-mean random process X, given a random variable x k1 ,x k2 , Their fourth-order cumulants are expressed as:

[0068]

[0069] Applying this definition to the entire hypothetical array, the fourth-order signal accumulant of the received signal for the entire hypothetical array is expressed as:

[0070]

[0071] In step 1, the positions of the P elements in the assumed array are represented by the set {v p ,p=1,2,...,P} means that after calculating the fourth-order cumulant of the received signal, the corresponding element position of the fourth-order differential virtual array can be expressed as:

[0072]

[0073] Where 1≤v p1 ,v p2 ,v p3 ,v p4 ≤P. The following discussion is based on the above definition. The following steps will specifically analyze the arrangement of the fourth-order differential virtual array elements corresponding to the proposed nested array with a coprime relationship.

[0074] Step 2: Construct a three-layer nested array model with three-layer nested relationships:

[0075] The nested array is composed of two uniform linear arrays, and the array element distribution position can be expressed as:

[0076] P NA ={n1 d,0≤n 1 ≤N 1 -1}∪{[n 2 (N 1 +1)-1]d,1≤n 2 ≤N 2}

[0077] The spacing between array elements is d = λ / 2, which is equal to half the wavelength of the incident signal. 2 (N 1 +1)-1]d to [N 2 (N 1 +1)-1]d are all continuous.

[0078] The array element position corresponding to the three-layer nested array (THL-NA) proposed in this paper is expressed as:

[0079] S 1 ={n 1 d,1≤n 1 ≤N 1}

[0080] S 2 ={n 2 (N 1 +1)d,1≤n 2 ≤N 2}

[0081] S 3 ={n 3 (N 2 +1)(N 1 +1)d,1≤n 3 ≤N 3}

[0082] P THL-NA =S 1 ∪S 2 ∪S 3

[0083] The schematic diagram of the array element position distribution is as follows Figure 1 As shown, the array P has a total of N = N 1 +N 2 +N 3 Next, this paper will prove that the 2-DCSC of the array must be continuous.

[0084] First we can get the 2-SC of the array, which ranges from 2d to 2N 3 (N 2 +1)(N 1+1)d, and the sum matrix is ​​not continuous. We can observe that although the 2-SC of the array is not continuous, there is a nested relationship. In order to observe more clearly, we divide the sum matrix into several parts and express the position of the array elements of each part in the form of a set.

[0085] S 1 ={n 1 d,2≤n 1 ≤(N 1 +1)(N 2 +1)}

[0086] S 2 =(N 1 +1)(N 2 +1)d+{n 1 d,1≤n 1 ≤N 1}∪{n 2 (N 1 +1)d,1≤n 2 ≤N 2 +1}

[0087] S 3 =2(N 1 +1)(N 2 +1)d+{n 1 d,1≤n 1 ≤N 1}∪{n 2 (N 1 +1)d,1≤n 2 ≤N 2 +1}

[0088]

[0089]

[0090]

[0091] The 2-SC of the proposed array is divided into N 3 +1 part, where set S 1 Continuous, Collection The sub-matrices are (N 1 ,N 2 ) of nested arrays, sets The sub-matrices are (N 1 ,N 2 ,N 3 ). Based on the continuity of the difference matrix of the nested array, we get the range of array element positions of 2-DCSC, that is, the 2-DCSC of the three-layer nested array is continuous, and the position set of the virtual array is

[0092] S 2-DCSC ={-2N 3 (N 2 +1)(N 1 +1)+2,…,-1,0,1,…,2N 3 (N 2 +1)(N 1 +1)-2}

[0093] From this we can deduce that the closed-form solution of the continuous degrees of freedom of the physical array structure is:

[0094] cDOF = 2[2N 3 (N 2 +1)(N 1 +1)-2]+1=4N 3 (N 2 +1)(N 1 +1)-3

[0095] Below we cite a specific array for further explanation. Figure 2 The physical array element positions, 2-SC virtual array element positions, and 2-DCSC virtual array element positions of the three-layer nested array with array element distribution (3,3,4) are shown. We can observe that the 2-SC of the array is discontinuous and there is a nested relationship. The position distribution of the sensors is {0.5,1,1.5,2,4,6,8,16,24,32}, and the range of 2-SC is 1 to 64, which divides the array into five parts, namely

[0096] S 1 ={1,1.5,...,8}

[0097] S 2 =8+{0.5,1,1.5,2,4,6,8}

[0098] S 3 =16+{0.5,1,1.5,2,4,6,8}

[0099] S 4 =24+{0.5,1,1.5,2,4,6,8}

[0100] S 5 =32+{0.5,1,1.5,2,4,6,8,16,24,32}

[0101] The set S 1 Continuous, set S 2 ,S 3 ,S 4 The nested array of array configuration (3,3), set S5 The array configuration is (3,3,4) with three levels of nested arrays. Figure 2 There is a nested relationship in the set enclosed by the box on the 2-SC sequence. According to the continuous property of the difference matrix of the nested matrix, we can obtain a continuous three-layer nested matrix 2-DCSC, whose continuous degree of freedom is

[0102] cDOF=4N 3 (N 2 +1)(N 1 +1)-3=253

[0103] Comparison Figure 2 The values ​​shown verify that our derivation is completely correct. The 2-DCSC of the three-layer nested array proposed in this paper is continuous.

[0104] Step 3: Find the fourth-order cumulant matrix R 4 , vectorize it to get vector z, sort it and remove redundancy to get z 1 :

[0105] Step 31: First, solve the fourth-order cumulant matrix R of the received signal 4 ;

[0106]

[0107] in

[0108]

[0109]

[0110] B(θ) is the array manifold after expansion using the fourth-order cumulant array b(θ k ) is a(θ k )The expanded array manifold.

[0111] Step 32: To further improve the DOA estimation accuracy, we use a virtualization method to calculate the fourth-order cumulant matrix R 4 Processing is performed to obtain vector z;

[0112] z=vec(R 4 )=vec[B(θ)C s B H (θ)]=Λ(θ)p

[0113] in

[0114]

[0115] and represents the fourth-order cumulant of the k-th source vector, where Represents the fourth-order cumulant of the kth source.

[0116] Vectorized R 4 The corresponding direction vector matrix Λ(θ) can be considered to be expanded from the two-level differential array A(θ) The position of the first-order continuous virtual array element corresponding to B(θ) is as follows: Figure 3 As shown, the corresponding parameters are set to N 1 =3,N 2 =3,N 3 =4; the position of the secondary virtual array element corresponding to Λ(θ) is as follows Figure 4 As shown, the corresponding parameters are set to N 1 =3,N 2 =3,N 3 =4, the closed-form solution derived in step 2 can be expressed as

[0117] L={-M b d,-(M b -1)d,...,0,...,(M b -1)d,M b d},M b =2N 3 (N 2 +1)(N 1 +1)-2

[0118] The positions of the secondary virtual array elements within the closed-form solution are completely continuous. b It indicates the maximum number of positions of virtual array elements. The position comparison of several different array elements under this parameter configuration is shown in the figure below. Figure 2 As shown.

[0119] Step 33: Sort z according to the positions of the secondary virtual array elements in L and remove redundant array elements to construct z 1 ,

[0120] z 1 =Λ 1 p

[0121] in, It is the matrix of Λ(θ) after removing redundancy and sorting. Based on the above, we can know that the direction matrix Λ of the differential array 1 for

[0122]

[0123] Step 4: Use the spatial smoothing method to obtain the spatial smoothing covariance matrix R:

[0124] The basic idea of ​​the spatial smoothing algorithm is to divide the equidistant linear array into several overlapping sub-arrays. If the sub-arrays have the same structure, their covariance matrices can be added and averaged to replace the original received signal covariance matrix.

[0125] The range of the fully continuous secondary virtual array corresponding to the z-direction matrix Λ(θ) is [-M b d,M b d] A long uniform linear array with an element spacing of d and an element number of T = 2M b +1=4N 3 (N 2 +1)(N 1 +1)-3. Using spatial smoothing algorithms, such as Figure 5 As shown, the secondary virtual array is divided into M b +1 overlapping subarrays, each containing M b +1 array element. Among them, the array element position of the i-th sub-array is

[0126] {(-i+1+n)d,n=0,1,…,M b}

[0127] The received signal matrix is ​​z 1 The Mth b +2-i row to 2Mth row b +2-i row, recorded as z 1i , and construct the covariance matrix:

[0128]

[0129] All M b The covariance matrix of the +1 sub-matrix is ​​summed and the mean is calculated to obtain the spatial smoothing covariance matrix:

[0130]

[0131] Thus we obtain the spatial smoothing covariance matrix R, which has the same form as the signal covariance matrix based on the classical subspace algorithm. Next, we will use the classical MUSIC algorithm on the matrix R to obtain the DOA estimate of the spatial source.

[0132] Step 5: Use the classic MUSIC algorithm to obtain the final angle estimate and compare the root mean square error between arrays.

[0133] Perform eigenvalue decomposition on the matrix R to obtain the signal subspace U N and the noise subspace U S .

[0134]

[0135] Where: Σ S is the diagonal matrix consisting of the eigenvalues ​​corresponding to the signal subspace vector, Σ N is a diagonal matrix consisting of the eigenvalues ​​corresponding to the noise subspace vectors.

[0136] The spatial spectrum function of the MUSIC algorithm can be constructed as

[0137]

[0138] Where a(θ) is the direction vector of the array when the signal arrival angle is assumed to be θ, and Change the value of θ and search in the spatial domain. When the denominator of the spectrum function approaches 0, that is, when the spatial spectrum function reaches a peak, the noise vector is orthogonal to the signal vector. At this time, θ is the signal arrival angle.

[0139] Figure 6 The figure is a comparison of the proposed array and several array structures in terms of degrees of freedom. It can be seen from the figure that the degrees of freedom of THL-NA are always greater than those of the other four arrays. When the number of array elements exceeds 12, the degrees of freedom of CP-NA begin to be lower than that of SAFE-CPA.

[0140] Figure 7 This is a performance comparison chart of the root mean square error of the proposed array and the above-mentioned array structures using the spatial smoothing MUSIC algorithm as the signal-to-noise ratio changes. The corresponding parameters are set to the number of snapshots L = 500, the angle parameter of the incident signal (θ 1 ,θ 2 )=(10°,20°), the system signal-to-noise ratio changes from -15dB to 5dB. The number of array elements is selected as 10, and the fourth-order difference virtual array degrees of freedom of THL-NA, SAFE-CPA, FL-NA, TL-NA, and ACA can reach 253, 241, 215, 117, and 85 respectively. The performance of these five arrays in DOA angle estimation through spatial smoothing MUSIC algorithm is compared. It can be seen from the figure that the array still has better performance than other arrays even in the case of low signal-to-noise ratio.

[0141] Figure 8 This is a performance comparison chart of the proposed array and the above-mentioned array structures using the spatial smoothing MUSIC algorithm as the root mean square error changes with the number of snapshots. The number of array elements and other parameter settings are the same as above, the system signal-to-noise ratio SNR = 0dB, and the number of snapshots changes from 200 to 1000. It can be seen from the figure that the array still has better performance than other arrays even when the number of snapshots is small.

[0142] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A three-layer nested array design method based on non-Gaussian sources, characterized in that: The following steps are involved: 1) Assuming an array, establish a model of the array receiving signals; 2) The array is constructed as a three-layer nested array array P having a three-layer nested relationship THL-NA ; 3) Calculate the fourth-order cumulant matrix R4 of the three-layer nested array, perform vectorization processing to obtain a vector z, and sort and remove redundancy to obtain z1; 4) Using the spatial smoothing method to obtain the spatial smoothing covariance matrix R; 5) Use the MUSIC algorithm to obtain the final angle estimation value, and compare the root mean square error of the angle estimation values ​​of different arrays to verify the three-layer nested array array P THL-NA performance; In step 2), the three-layer nested array array P THL-NA The corresponding array element position is expressed as: S1={n1d,1≤n1≤N1} S2={n2(N1+1)d, 1≤n2≤N2}, S3={n3(N2+1)(N1+1)d,1≤n3≤N3} P THL-NA =S1∪S2∪S3; Among them: S1, S2, S3 represent three-layer nested array array P THL-NA There are three sub-arrays, d represents the spacing of the array elements in sub-array S1, and n1, n2, and n3 are variables; The three-layer nested array array P THL-NA There are N=N1+N2+N3 array elements in total, and the second-order sum difference matrix 2-DCSC of the array is continuous in the range -2N3(N2+1)(N1+1)+2 to 2N3(N2+1)(N1+1)-2, and the continuous degree of freedom is cDOF=2[2N3(N2+1)(N1+1)-2]+1=4N3(N2+1)(N1+1)-3; Step 3) is implemented as follows: 3.1) Solve the three-layer nested array array P THL-NA The fourth-order cumulant matrix R4 of the received signal: X = A(θ)S + N; B(θ) is the array manifold after expansion using the fourth-order cumulant array R4; b(θ k ) is a(θ k ) The expanded array manifold; C s is the fourth-order cumulant of the non-Gaussian signal source s; in: represents the Kronecker product, (·) H represents conjugate transpose, ⊙ represents Khatri-Rao product; 3.2) The fourth-order cumulant matrix R4 in step 3.1) is processed by virtualization to obtain the vector z: z=vec(R4)=vec[B(θ)C s B H (θ)]=Λ(θ)p; Head represents the fourth-order cumulant of the k-th source vector, where represents the fourth-order cumulant of the k-th source, vec(·) represents matrix vectorization; The corresponding parameters of the first-order continuous virtual array element positions corresponding to the array manifold B(θ) are set as: N1=3, N2=3, N3=4; The corresponding parameters of the secondary virtual array element position corresponding to the direction vector matrix Λ(θ) are set to: N1=3, N2=3, N3=4; its closed-form solution is: L={-M b d,-(M b -1)d,…,0,…,(M b -1)d,M b d},M b =2N3(N2+1)(N1+1)-2 The positions of the secondary virtual array elements within the closed-form solution are completely continuous; b It represents the maximum number of positions of the secondary virtual array element; Step 3.3) z is sorted and redundant array elements are removed to construct z1: z1=Λ1p in, The matrix of Λ(θ) after removing redundancy and sorting is: M b is the maximum number of positions of virtual array elements; The implementation process of step 4) is: The range of the completely continuous secondary virtual array corresponding to the direction vector matrix Λ(θ) of vector z is [-M b d,M b d] A long uniform linear array with an element spacing of d and an element number of T = 2M b +1=4N3(N2+1)(N1+1)-3, using the spatial smoothing algorithm, the secondary virtual array is divided into M b +1 overlapping subarrays, each containing M b +1 array element, where M b is the maximum number of positions of the virtual array elements, and the position of the elements of the i-th sub-array is: {(-i+1+n)d,n=0,1,…,M b }; Its received signal matrix is ​​the Mth b +2-i row to 2Mth row b +2-i row, recorded as z 1i , and construct the covariance matrix: All M b The covariance matrix of the +1 sub-matrix is ​​summed and the mean is calculated to obtain the spatial smoothing covariance matrix: The implementation process of step 5) is: Perform eigenvalue decomposition on the matrix R to obtain the signal subspace U N and the noise subspace U S , Where: Σ S is the diagonal matrix consisting of the eigenvalues ​​corresponding to the signal subspace vector, Σ N is the diagonal matrix consisting of the eigenvalues ​​corresponding to the noise subspace vectors, The spatial spectrum function of the MUSIC algorithm is constructed as: Where a(θ) is the direction vector of the array when the signal arrival angle is assumed to be θ, and Change the value of θ and search in the spatial domain. When the denominator of the spectrum function approaches 0, that is, when the spatial spectrum function reaches a peak, the noise vector is orthogonal to the signal vector. At this time, θ is the angle estimate. The angle estimation values ​​obtained by spatial smoothing and MUSIC algorithm for different arrays are calculated, and the root mean square error of the angle estimation values ​​of different arrays under the same number of snapshots is compared to verify the three-layer nested array P. THL-NA performance.

2. The three-layer nested array design method based on non-Gaussian source according to claim 1, characterized in that: The implementation process of step 1) is as follows: Assume that there are K narrowband far-field incoherent signal sources in space, and their one-dimensional wave arrival direction is: θ k (k=1, 2, ..., K), K signal sources are incident on the array, the array includes P array elements, and the received signal model of the array is expressed as: X = A(θ)S + N; represents the direction matrix; Indicates a non-Gaussian signal source with a mean of 0; Where: d represents the spacing between the array elements, λ represents the wavelength, L represents the number of snapshots, It is represented by a mean of 0 and a variance of σ 2 Additive Gaussian white noise, s k (1) represents the source of the first snapshot number, 1 = (1, 2, 3, ..., L); () T Represents the transpose operation; Given a random variable The random variable The fourth-order cumulant is expressed as: The fourth-order signal accumulation of the entire array received signal is expressed as: Where Cum(·) represents the fourth-order cumulant calculation, (·) * Represents the conjugate operation.

3. The three-layer nested array design method based on non-Gaussian source according to claim 2 is characterized in that: In step 1), the positions of the P elements in the array are grouped as follows: p , p = 1, 2, ..., P}, the array element position of the fourth-order virtual array corresponding to the array is expressed as: Where 1≤v p1 , v p2 , v p3 , v p4 ≤P,v p1 、v p2 、v p3 、v p4 It represents the position number of the first, second, third, and fourth array element in the array.

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