A three-order sparse array design method based on sum-difference analysis

Abstract

The application discloses a third-order sparse array design method based on sum-difference analysis, selects two linear arrays, a first linear array is a dense uniform linear array, calculates a second-order sum and a second-order covariance of the first linear array, calculates a second-order sum, a second-order covariance and a second-order difference covariance of a second linear array, obtains a longest section in the covariance, and constructs a third-order sparse array based on the longest section; and according to the third-order sparse array, a third-order virtual array, namely a third-order exhaustive covariance, is constructed by merging four third-order cumulants; the method has the advantages that the second-order sparse array is mapped to the third order to adapt to the third-order cumulant model through expansion and shifting of the second linear array, the continuous section of the sum-difference covariance of the second linear array is fully utilized, the third-order sparse array has a longer continuous part in the virtual array (namely the third-order exhaustive covariance), and the number of degrees of freedom of the array is significantly improved.

Description

TECHNICAL FIELD

[0001] The present application relates to an array signal processing technology, in particular to a third-order sparse array design method based on sum-difference analysis. BACKGROUND

[0002] Array signal processing is one of the key research directions in the field of wireless communication. It arranges multiple antenna sensors (array elements) in a specific geometric layout and jointly processes the signals collected by these array elements to achieve signal detection, parameter estimation, and target direction finding. Traditional array direction finding methods usually use one-dimensional uniform linear arrays (ULAs), which have constant element spacing and simple array layout, but have limited degrees of freedom. For a one-dimensional uniform linear array with N elements, the traditional subspace-based array signal processing method can analyze at most N-1 signal sources. Although increasing the number of elements can improve the resolution and estimation accuracy, it will significantly increase the system hardware cost, making it difficult to implement large-scale and large-aperture uniform linear arrays.

[0003] Sparse arrays are increasingly valued in array signal processing due to their high degrees of freedom and low mutual coupling characteristics. Compared with uniform linear arrays, the element spacing of sparse arrays is non-uniform, and the array structure is more diversified. For a sparse array with N elements, the second-order statistics (i.e., covariance matrix) of the received data can be treated as a set of single-shot data received by a virtual array after vectorization processing. From this, an equivalent virtual difference co-array is constructed, which can achieve O(N 2 ) degrees of freedom, thereby analyzing a larger number of signal sources and improving direction finding accuracy. Following this approach, a virtual array constructed based on higher-order cumulants can provide more virtual array elements, further improving the number of array degrees of freedom, achieving better direction finding accuracy with fewer physical array elements, and having practical application value. However, existing sparse array design mainly focuses on second-order sparse arrays, and there is less research on higher-order sparse array design methods.

[0004] Third-order sparse arrays use third-order cumulants to construct third-order exhaustive co-arrays, obtain more virtual array elements, achieve a higher upper limit of uniform degrees of freedom, and thus improve the direction finding performance of the array. Existing third-order sparse arrays (i.e., third-order nested arrays) adjust the element spacing based on the prototype two-level nested array, thereby adapting to the third-order cumulant model, constructing a third-order exhaustive co-array with a relatively long continuous segment, and achieving a considerable uniform degree of freedom with fewer physical array elements. However, as the number of elements increases, its degree of freedom advantage gradually weakens. In addition, it also has the shortcomings of the prototype two-level nested array, i.e., poor mutual coupling resistance, and serious DOA estimation performance loss in a strong mutual coupling environment. SUMMARY

[0005] The technical problem to be solved by the present application is to provide a third-order sparse array design method based on sum-difference analysis, which can design a third-order sparse array with higher degrees of freedom and better mutual coupling resistance through analysis of the array sum-difference performance, and can achieve higher direction finding precision and more stable DOA estimation performance.

[0006] The technical solution adopted by the present application to solve the above technical problem is a third-order sparse array design method based on sum-difference analysis, characterized by comprising the following steps:

[0007] Step 1: selecting a first linear array, the first linear array being a dense uniform linear array, the number of physical array elements of the first linear array being N1, the physical array element interval being d, and the physical array element position set Z1 being represented as Z1=P1·d={0,1,...,N1-1}d, wherein P1 represents an index set of the physical array elements in the dense uniform linear array, P1={0,1,...,N1-1}; and selecting a second linear array, the number of array elements of the second linear array being N2, and the physical array element position set Z2 being represented as wherein P2 represents an index set of the physical array elements in the second linear array,

[0008] Step 2: calculating the second-order sum co-array of the dense uniform linear array wherein g and h are any two indices in P1; and calculating the second-order sum co-array of the second linear array and the second-order difference co-array wherein a and b are any two indices in P2; then the subset composed of the longest continuous numerical segment in is recorded as the subset composed of the longest continuous numerical segment in is recorded as the subset composed of the longest continuous index segment in P2 is recorded as

[0009] Step 3: constructing a third-order sparse array according to the dense uniform linear array and the second linear array, the number of physical array elements of the third-order sparse array being N, N=N1+N2, the physical array element interval being non-uniform, and the physical array element position set Z being represented as Z=P·d, wherein P represents an index set of the physical array elements in the third-order sparse array, P=P1∪P EXT , P EXT represents an extended index set, P EXT =(λP2+s), λ represents an expansion coefficient, s represents a shift factor for maximizing the third-order uniform degrees of freedom of the third-order sparse array, s is an optimization variable, and the optimal solution of s is obtained through P1, calculation, and "∪" is a set union operator; ​​

[0010] Step 4: Based on the third-order sparse array, a third-order virtual array, i.e., a third-order exhaustive co-array, is constructed by merging four third-order cumulants.

[0011] Preferably, in step 2, The way to obtain is: Sort the values ​​in from small to large, and then find the longest continuous value segment with an interval of 1 from the sorted values, which is the longest continuous value segment, and then construct the longest continuous value segment. The way to obtain is: Sort the values ​​in from small to large, and then find the longest continuous value segment with an interval of 1 from the sorted values, which is the longest continuous value segment, and then construct the longest continuous value segment. The method of obtaining is: find the longest continuous index segment with an interval of 1 from P2, which is the longest continuous index segment, and then construct the longest continuous index segment

[0012] Preferably, in step 3, λ=2N1-1.

[0013] As an example, in step 3, the process of obtaining the optimal solution of s is as follows: define Then according to U D 、U P and U S , construct s optimal solution to solve the problem, described as: Among them, uniform(·) indicates finding the subset consisting of the longest continuous numerical segment in the set, and “|·|” indicates finding the number of elements in the set; then the optimal solution problem of s is solved to obtain the optimal solution of s.

[0014] Preferably, in step 4, the construction process of the third-order exhaustive co-matrix is:

[0015] Step 4.1: Assume that G incoherent far-field narrowband signal sources are incident on the third-order sparse array. Then, the vector x(k) consisting of the observation values ​​of all elements in the third-order sparse array at time k is expressed as x(k) = As(k) + n(k). Then, the l+1th third-order cumulant of the observation values ​​of all elements in the third-order sparse array is Expressed as

[0016] Where A represents the array manifold matrix, which is composed of the steering vectors of G far-field narrowband signal sources, that is, A=[a(θ1),a(θ2),…,a(θ G )],a(θ irepresents the steering vector of the i-th far-field narrowband signal source, θ i represents the signal angle of arrival of the i-th far-field narrowband signal source, e represents the natural base, j is a virtual number, [·] T represents the transpose operation, q1, q2, q N represents the index of the 1st, 2nd, Nth element in the third-order sparse array, q1d, q2d, q N d represents the position of the 1st, 2nd, Nth element in the third-order sparse array, m represents the wavelength of the incident signal of the far-field narrowband signal source, s(k) represents the value of the signal source vector at time k, s(k) = [s1(k), s2(k), …, s i (k), …, s G (k)] T , n(k) represents the value of the noise vector at time k, l = 0, 1, 2, 3, E[·] represents the mathematical expectation, the superscript “*” is the conjugate operation, the “○” is the tensor product operator symbol, the “×” is the scalar multiplication operator symbol, when l = 0, x(k) °(0) = 1, when l = 1, x(k) when l = 2, 3, x(k) represents the tensor product is used l-1 times, when l = 0, s i (k) ×(0) = 1, when l = 1, s i (k) ×(1) = s i (k), when l = 2, 3, s represents the scalar multiplication × is used l-1 times;

[0017] Step 4.2: Vectorize each third-order cumulant, the vectorization result of the l+1th third-order cumulant is denoted as wherein, the dimension of is N 3 ×1, vec(·) represents the vectorization operation, the dimension of is N 3 ×1, is the Kronecker product operator symbol, when l = 0, when l = 1, when l = 2, 3, represents the Kronecker product is used l-1 times, the dimension of is G × 1,

[0018] Step 4.3: Combine the vectorization results of the four third-order cumulants to obtain the combined cumulant c x , wherein the dimension of c x is 4N 3 ×1, the dimension of R is 4N 3 ×4G, and the dimension of f is 4G×1.

[0019] Step 4.4: regarding R as an equivalent array manifold matrix and regarding f as an equivalent source vector, c x is regarded as single-shot data received by a virtual array, which is a third-order exhaustive common array, and the position set Z vir of the virtual array elements of the third-order exhaustive common array is represented as Z vir =P vir ·d, wherein P vir represents the index set of the virtual array elements in the third-order exhaustive common array, x1, x2, x3 are any three indices in P.

[0020] Compared with the prior art, the present application has the following advantages:

[0021] By extending and shifting the second linear array, the low-order array is mapped to the third order to adapt to the third-order cumulant model, and the present application makes full use of the continuous segments of the sum-difference common array of the second linear array, and the virtual array (i.e. the third-order exhaustive common array) of the third-order sparse array designed by the present application has a longer continuous part, and the number of degrees of freedom of the array is significantly improved; based on the third-order cumulant model, the third-order exhaustive common array can provide O(N 3 ) degrees of freedom, which is much higher than the method based on the second-order statistics. When the number of array elements increases, the difference between the degrees of freedom of the third-order sparse array designed by the present application and the second-order sparse array will further expand, and at the same time, the third-order sparse array designed by the present application is superior to the existing third-order nested array, so the third-order sparse array designed by the present application can achieve better DOA estimation performance at a lower hardware cost. In addition, the array element positions of the third-order sparse array designed by the present application have a closed-form expression and can be generated from many commonly used second-order sparse arrays; thanks to the extension of the subarray spacing, the mutual coupling effect of the third-order sparse array designed by the present application is also significantly improved, making the DOA estimation performance in a strong mutual coupling environment more stable. BRIEF DESCRIPTION OF DRAWINGS

[0022] Figure 1 is a schematic diagram of the overall implementation process of the method of the present application;

[0023] Figure 2 is a schematic diagram of the structure of the third-order sparse array designed by the method of the present application when the second linear array is selected as the transposed nested array;

[0024] Figure 3 The root mean square error (RMSE) of the three-order sparse array designed by the method of the application and the existing three-order sparse array and two commonly used four-order sparse arrays in DOA estimation changes with the signal-to-noise ratio (SNR) and the comparison schematic diagram is shown in the figure. DETAILED DESCRIPTION

[0025] The application will be further described with reference to the drawings and in conjunction with the specific embodiments, and the protection scope of the application is not limited to the specific embodiments.

[0026] The application provides a three-order sparse array design method based on sum-difference analysis, and the overall implementation process is as shown in the figure. Figure 1 The method comprises the following steps:

[0027] Step 1: selecting a first linear array, the first linear array is a dense uniform linear array, the number of physical array elements is N1, the physical array element interval is d, and the physical array element position set Z1 is represented as Z1=P1·d={0,1,...,N1-1}d, wherein N1 is greater than or equal to 2, and d is generally set to half a wavelength, that is, d=m / 2. m represents the wavelength of the incident signal of a far-field narrowband signal source, P1 represents the index set of the physical array elements in the dense uniform linear array, P1={0,1,...,N1-1}, that is, the indexes of the N1 physical array elements are 0, 1,..., N1-1; and a second linear array is selected, the number of array elements is N2, and the physical array element position set Z2 is represented as Wherein N2 is greater than or equal to 2, P2 represents the index set of the physical array elements in the second linear array, p1,p2,p N2 Corresponding to the indexes of the first physical array element, the second physical array element and the N2th physical array element in the second linear array, the positions of each physical array element in the dense uniform linear array and the second linear array are all integer multiples of d.

[0028] Step 2: calculating the second-order sum and covariance of the dense uniform linear array Wherein g and h are any two indexes in P1 (which can be repeatedly selected and all possible indexes are exhausted); and the second-order sum and covariance of the second linear array is calculated and the second-order difference covariance Wherein a and b are any two indexes in P2 (which can be repeatedly selected and all possible indexes are exhausted); then the subset composed of the longest continuous numerical segment in S P2 is recorded as The subset composed of the longest continuous numerical segment in D P2 is recorded as The subset composed of the longest continuous index segment in P2 is recorded as

[0029] As preferred, in step 2, is obtained by: sorting the values in from small to large, then finding the longest segment of adjacent continuous values with interval 1 from the sorted values, which is the longest continuous value segment, and then constructing is obtained by: sorting the values in from small to large, then finding the longest segment of adjacent continuous values with interval 1 from the sorted values, which is the longest continuous value segment, and then constructing is obtained by: finding the longest segment of adjacent continuous indexes with interval 1 from P2, which is the longest continuous index segment, and then constructing Here, the continuous index segment is composed of continuous indexes 4, 5, 6, 7, and 8.

[0030] Step 3: Constructing a third-order sparse array based on the dense uniform linear array and the second linear array, with the number of physical array elements N, N=N1+N2, the physical array element interval being non-uniform, and the physical array element position set Z being represented as Z=P·d, where P represents the index set of the physical array elements in the third-order sparse array, P=P1∪P EXT , P EXT represents the extended index set, P EXT =(λP2+s), λ represents the expansion coefficient, λ=2N1-1, s represents the shift factor for maximizing the third-order uniform degree of freedom of the third-order sparse array, s is an optimization variable, and the optimal solution of s is obtained by P1, calculation, and “∪” is the set union operator.

[0031] As preferred, in step 3, the optimal solution of s is obtained by: defining and then constructing the s optimal solution solving problem based on U D , U P , and U S , described as: where uniform(·) represents the subset composed of the longest continuous value segment in the set, the process of uniform(·) is the same as the acquisition process of , and “|·|” represents the number of elements in the set; and then solving the s optimal solution solving problem to obtain the optimal solution of s.

[0032] Step 4: Based on the third-order sparse array, a third-order virtual array, i.e., a third-order exhaustive co-array, is constructed by combining four third-order cumulants. Continuous virtual array elements in the third-order exhaustive co-array can be used for DOA estimation. The index range of the continuous virtual array elements is [-L, L], and the third-order uniform degree of freedom is 2L+1, where max(·) is the maximum value function, The third-order cumulant model has a much higher upper limit of degrees of freedom than the physical array and the second-order virtual co-array, and can analyze more signal sources. Compared with the fourth-order cumulant model, the third-order exhaustive co-array can achieve a higher number of degrees of freedom when the number of array elements is small, and has a very large advantage in computational complexity.

[0033] Preferably, in step 4, the construction process of the third-order exhaustive co-matrix is:

[0034] Step 4.1: Assume that G incoherent far-field narrowband signal sources are incident on the third-order sparse array. Then, the vector x(k) consisting of the observation values ​​of all elements in the third-order sparse array at time k is expressed as x(k) = As(k) + n(k). Then, the l+1th third-order cumulant of the observation values ​​of all elements in the third-order sparse array is Expressed as Where G ≥ 1, A represents the array manifold matrix, which is composed of the steering vectors of G far-field narrowband signal sources, that is, A = [a(θ1), a(θ2), …, a(θ G )],a(θ i ) represents the steering vector of the i-th far-field narrowband signal source, a(θ i ) is determined by the signal arrival angle θ i And the array element position determines, 1≤i≤G, θ i represents the signal arrival angle of the i-th far-field narrowband signal source, e represents the natural base, e = 2.71…, j represents the imaginary number, [·] T Indicates transpose operation, q1,q2,q N Indicates the index of the first array element, the second array element, and the Nth array element in the third-order sparse array, q1d, q2d, q N d represents the positions of the first, second, and Nth elements in the third-order sparse array, m represents the wavelength of the incident signal from the far-field narrowband signal source, s(k) represents the value of the source vector at time k, s(k) = [s1(k), s2(k), …, s i (k),…,s G (k)] T , n(k) represents the value of the noise vector at time k, l=0,1,2,3, To define the symbols, E[·] means finding the mathematical expectation, and the superscript “*” is the conjugate operation. is the tensor product operator symbol, "×" is the scalar multiplication operator symbol, when l = 0 When l=1 When l=2,3 Represents tensor product It is used l-1 times in total. When l=0, s i (k) ×(0) =1, when l=1, s i (k) ×(1) =s i (k), when l = 2, 3 Indicates that the scalar multiplication × is used l-1 times.

[0035] Step 4.2: Vectorize each third-order cumulant and convert the l+1th third-order cumulant The vectorized result is recorded as in, The dimension is N 3 ×1, vec(·) represents vectorized operation, r (l) (θ i ) has a dimension of N 3 ×1, r1 (l) (θ i ), Corresponding representation r (l) (θ i ) in the first, second, and Nth elements 3 elements, is the Kronecker product operator symbol, when l = 0 When l=1 When l=2,3 represents the Kronecker product Used 1-1 times in total, The dimension is G×1,

[0036] Step 4.3: Combine the vectorized results of the four third-order cumulants to obtain the combined cumulant c x , Among them, c x The dimension is 4N 3 ×1, The dimension of R is 4N 3 ×4G, The dimension of f is 4G×1.

[0037] Step 4.4: Let R be regarded as an equivalent array manifold matrix and f be regarded as an equivalent source vector, then c xas a single snapshot of data received by a virtual array, which is a third-order exhaustive co-array, whose virtual element position set Z vir is expressed as Z vir = P vir · d, where P vir represents the index set of the virtual elements in the third-order exhaustive co-array, x1, x2, x3 are any three indices in P (which can be repeatedly selected to exhaust all possibilities).

[0038] The superiority of the method of the present application is illustrated below through simulation experiments.

[0039] In the simulation experiment, three different second linear arrays are selected, which are a dense uniform linear array, a nested array with a shift subarray (NADiS), and a transposed nested array (TNA), the latter two of which are based on a second-order sum-difference co-array design and have high second-order sum-difference degrees of freedom. Among them, the nested array with a shift subarray (NADiS) is derived from P. Gupta and M. Agrawal, Design And Analysis of the Sparse Array for DoA Estimation of Noncircular Signals, IEEE Transactions on Signal Processing, vol. 67, no. 2, pp. 460-473, 15 Jan. 15, 2019. (Design and analysis of sparse array for DoA estimation of noncircular signals, IEEE Signal Processing Journal), and the transposed nested array (TNA) is derived from Y. Wang, W. Wu, X. Zhang and W. Zheng, Transformed nested array designed for DOA estimation of non-circular signals: Reduced sum-difference co-array redundancy perspective, IEEE Communications Letters, vol. 24, no. 6, pp. 1262-1265, June 2020. (Transformed nested array designed for DOA estimation of non-circular signals: Reduced sum-difference co-array redundancy perspective, IEEE Communications Letters).

[0040] Under the three different second linear arrays, three third-order sparse arrays are designed using the method of the present application, which are represented as TO-SDA (ULA), TO-SDA (NADiS), and TO-SDA (TNA), Figure 2A structure diagram of the third-order sparse array designed by the method of the present application when the second linear array selects the transposed nested array is given, Figure 2 Part (a) shows a dense uniform linear array (first linear array) composed of four physical array elements, part (b) shows a transposed nested array (second linear array) composed of four physical array elements, and part (c) shows the third-order sparse array TO-SDA (TNA) obtained by construction, wherein the expansion coefficient λ = 2N1-1 = 7, and the shift factor s = 25. It is compared with the existing third-order sparse array, i.e., the third-order nested array (TONA). In addition, a four-level nested array (FL-NA) and a simplified and improved four-level nested array (SE-FL-NA) are introduced as a comparison. Among them, the third-order nested array (TONA) is derived from U. Sharma and M. Agrawal, Third-order nested array: an optimal geometry for third-order cumulants based array processing, IEEE Transactions on Signal Processing, 2023, 71: 2849-2862, the four-level nested array (FL-NA) is derived from P. Pal and P. P. Vaidyanathan, Multiple level nested array: An efficient geometry for 2qth order cumulant based array processing, IEEE Transactions on Signal Processing, 2012, 60(3): 1253-1269, and the simplified and improved four-level nested array (SE-FL-NA) is derived from Q. Shen, W. Liu, W. Cui, S. Wu and P. Pal, Simplified and enhanced multiple level nested arrays exploiting high-order difference co-arrays, IEEE Transactions on Signal Processing, 2019, 67(13): 3502-3515.

[0041] The physical element quantity of the third-order sparse array is set as N=9, the non-coherent far-field narrowband signal source quantity is G=16, the incidence angles of the far-field narrowband signal sources are uniformly distributed between-60 degrees and 60 degrees, the number of snapshots is set as 7500, the signal-to-noise ratio is taken as a step of 5dB, 500 Monte Carlo simulations are performed under each signal-to-noise ratio, the root mean square error (RMSE) of DOA estimation of each third-order sparse array under-5dB to 20dB is tested, the DOA estimation algorithm is uniformly taken as the root MUSIC algorithm of smoothing space, and the obtained results are as shown in Figure 3 From Figure 3 it can be seen that the third-order sparse array (TO-SDA) designed by the method has the lowest root mean square error and the best performance. Specifically, the transposed nested array (TNA) has the highest second-order sum difference uniform degrees of freedom, and also has a continuous part in the physical array, which can be taken as a second linear array, so that the highest third-order uniform degrees of freedom can be obtained, thereby the lowest estimation error is shown; the nested array with displacement subarray (NADiS) has higher second-order sum difference uniform degrees of freedom, but cannot utilize the continuous part in the physical array, the physical array of the uniform linear array (ULA) is completely continuous, but the second-order sum difference uniform degrees of freedom is lower, the TO-SDA (NADiS) and the TO-SDA (ULA) obtained by taking them as the second linear array respectively have similar third-order uniform degrees of freedom, and the estimation performance is slightly lower than that of the TO-SDA (TNA), but is significantly better than that of the existing array. In summary, the third-order sparse array designed by the method has good universality, and many existing arrays can be directly taken as the second linear array for construction and excellent direction finding precision can be achieved.

Claims

1. A third-order sparse array design method based on sum-difference analysis, characterized in that The following steps are involved: Step 1: Select the first linear array, which is a dense uniform linear array with N1 as the number of physical elements and d as the spacing between physical elements. The physical element position set Z1 is expressed as Z1=P1·d={0,1,…,N1-1}d, where P1 represents the index set of the physical elements in the dense uniform linear array, P1={0,1,…,N1-1}; and select the second linear array with N2 as the number of elements. The physical element position set Z2 is expressed as Wherein, P2 represents the index set of physical array elements in the second linear array, Step 2: Calculate the second order and co-array of the dense uniform linear array Where g,h are any two indices in P1; and calculate the second order and co-matrix of the second linear array and second-order difference matrix Among them, a and b are any two indexes in P2; then The subset consisting of the longest continuous numerical segment in is recorded as Will The subset consisting of the longest continuous numerical segment in is recorded as The subset consisting of the longest continuous index segments in P2 is recorded as Step 3: Based on the dense uniform linear array and the second linear array, a third-order sparse array is constructed. The number of physical array elements is N, N = N1 + N2, the physical array elements are spaced non-uniformly, and the physical array element position set Z is expressed as Z = P·d, where P represents the index set of the physical array elements in the third-order sparse array, P = P1∪P EXT , P EXT Represents the extended index set, P EXT =(λP2+s), λ represents the expansion coefficient, s represents the shift factor used to maximize the third-order uniform degree of freedom of the third-order sparse array, s is the optimization variable, and the optimal solution of s is obtained by P1, It is calculated that "∪" is the set union operator symbol; Step 4: Based on the third-order sparse array, a third-order virtual array, i.e., a third-order exhaustive co-array, is constructed by merging four third-order cumulants.

2. A third-order sparse array design method based on sum-difference analysis according to claim 1, characterized in that In the step 2, The way to obtain is: Sort the values ​​in from small to large, and then find the longest continuous value segment with an interval of 1 from the sorted values, which is the longest continuous value segment, and then construct the longest continuous value segment. The way to obtain is: Sort the values ​​in from small to large, and then find the longest continuous value segment with an interval of 1 from the sorted values, which is the longest continuous value segment, and then construct the longest continuous value segment. The method of obtaining is: find the longest continuous index segment with an interval of 1 from P2, which is the longest continuous index segment, and then construct P2 with the longest continuous index segment. U .

3. The method for designing a third-order sparse array based on sum-difference analysis according to claim 1, characterized in that In step 3, λ=2N1-1.

4. The method for designing a third-order sparse array based on sum-difference analysis according to claim 3, characterized in that In step 3, the process of obtaining the optimal solution of s is as follows: define Then according to U D 、U P and U S , construct s optimal solution to solve the problem, described as: Where uniform(·) indicates finding the subset consisting of the longest continuous numerical segment in a set, and "|·|" indicates finding the number of elements in a set. The optimal solution problem for s is then solved to obtain the optimal solution for s.

5. The method for designing a third-order sparse array based on sum-difference analysis according to claim 4, characterized in that In step 4, the construction process of the third-order exhaustive co-matrix is: Step 4.1: Assume that G incoherent far-field narrowband signal sources are incident on the third-order sparse array. Then, the vector x(k) consisting of the observation values ​​of all elements in the third-order sparse array at time k is expressed as x(k) = As(k) + n(k). Then, the l+1th third-order cumulant of the observation values ​​of all elements in the third-order sparse array is Expressed as Where A represents the array manifold matrix, which is composed of the steering vectors of G far-field narrowband signal sources, that is, A=[a(θ1),a(θ2),…,a(θ G )],a(θ i ) represents the steering vector of the i-th far-field narrowband signal source, θ i represents the signal arrival angle of the i-th far-field narrowband signal source, e represents the natural base, j represents the imaginary number, [·] T Indicates transpose operation, q1,q2,q N Indicates the index of the first array element, the second array element, and the Nth array element in the third-order sparse array, q1d, q2d, q N d represents the positions of the first, second, and Nth elements in the third-order sparse array, m represents the wavelength of the incident signal from the far-field narrowband signal source, s(k) represents the value of the source vector at time k, s(k) = [s1(k), s2(k), …, s i (k),…,s G (k)] T , n(k) represents the value of the noise vector at time k, l=0,1,2,3, E[·] represents the mathematical expectation, and the superscript "*" is the conjugate operation. is the tensor product operator symbol, "×" is the scalar multiplication operator symbol, when l = 0 When l=1 When l=2,3 Represents tensor product It is used l-1 times in total. When l=0, s i (k) ×(0) =1, when l=1, s i (k) ×(1) =s i (k), when l = 2, 3 Indicates that scalar multiplication × is used l-1 times in total; Step 4.2: Vectorize each third-order cumulant and convert the l+1th third-order cumulant The vectorized result is recorded as in, The dimension is N 3 ×1, vec(·) represents vectorized operation, r (e) (θ i ) has a dimension of N 3 ×1, is the Kronecker product operator symbol, when l = 0 When l=1 When l=2,3 represents the Kronecker product Used 1-1 times in total, The dimension is G×1, Step 4.3: Combine the vectorized results of the four third-order cumulants to obtain the combined cumulant c x , Among them, c x The dimension is 4N 3 ×1, The dimension of R is 4N 3 ×4G, The dimension of f is 4G×1; Step 4.4: Let R be regarded as an equivalent array manifold matrix and f be regarded as an equivalent source vector, then c x It is considered as a single snapshot data received by a virtual array. This virtual array is a third-order exhaustive co-array, and its virtual array element position set Z vir Represented as Z vir =P vir ·d, where P vir represents the index set of virtual array elements in the third-order exhaustive common matrix, x1, x2, x3 are any three indices in P.

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