Three-order sparse array design method based on sum and difference analysis

Abstract

The invention discloses a sum and difference analysis-based three-order sparse array design method, which comprises the following steps of: selecting two linear arrays, calculating a second-order sum common array of the first linear array, calculating a second-order sum common array and a second-order difference common array of the second linear array, acquiring a longest section in the common arrays, and constructing a three-order sparse array on the basis of the longest section; according to the third-order sparse array, constructing a third-order virtual array, namely a third-order exhaustion common array, by combining the four third-order cumulants; the method has the advantages that the second-order sparse array is mapped to the third order through expansion and displacement of the second linear array so as to adapt to the third-order cumulant model, the continuous section of the sum-difference common array of the second linear array is fully utilized, and the designed virtual array (namely the third-order exhaustion common array) of the third-order sparse array has a long continuous part, so that the method is suitable for the three-order cumulant model. And the number of degrees of freedom of the array is obviously improved.

Description

Technical Field

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

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

[0003] Sparse arrays are gaining more and more attention in array signal processing due to their high degrees of freedom and low mutual coupling characteristics. Compared with uniform linear arrays, sparse arrays have non-uniform element spacing and more diverse array structures. For a sparse array with N elements, after vectorizing the second-order statistics of the received data (i.e., the covariance matrix), it can be regarded as a set of single-snapshot data received by a virtual array, thereby constructing an equivalent virtual differential co-array, which can achieve O(N 2 ) degrees of freedom, thereby resolving more signal sources and improving direction finding accuracy. Following this idea, a virtual array constructed based on high-order cumulants can provide more virtual array elements, thereby further increasing the number of array degrees of freedom and achieving better direction finding accuracy with fewer physical array elements, which has practical application value. However, existing sparse array designs are mainly focused on second-order sparse arrays, and there is little research on higher-order sparse array design methods.

[0004] The third-order sparse array constructs a third-order exhaustive common array by using the third-order cumulants to obtain more virtual array elements and achieve a higher upper limit of uniform degrees of freedom, thereby improving the direction-finding performance of the array. The existing third-order sparse array (i.e., third-order nested array) readjusts the array element spacing on the basis of the prototype two-level nested array, thereby adapting the third-order cumulant model and constructing a third-order exhaustive common array with a long continuous segment, achieving considerable uniform degrees of freedom when the number of physical array elements is small. However, as the number of array elements increases, its degree of freedom advantage gradually weakens. In addition, it also has the disadvantages of the prototype two-level nested array, namely, poor mutual coupling resistance, and there is a serious loss of DOA estimation performance in a strong mutual coupling environment. Summary of the invention

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

[0006] The technical solution adopted by the present invention 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: Select the first linear array, which 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 expressed as Z1=P1·d={0,1,...,N1-1}d, where P1 represents the index set of the physical array elements in the dense uniform linear array, P1={0,1,...,N1-1}; and select the second linear array, the number of array elements is N2, and the physical array element position set Z2 is expressed as Wherein, P2 represents the index set of the physical array elements in the second linear array,

[0008] Step 2: Calculate the second order and co-array of the dense uniform linear array Among them, g, h are any two indices in P1; and the second-order and co-array of the second linear array are calculated. and second-order difference matrix Among them, a and b are any two indexes in P2; then The subset consisting of the longest continuous value segment in is denoted as Will The subset consisting of the longest continuous value segment in is denoted as The subset consisting of the longest continuous index segments in P2 is recorded as

[0009] Step 3: Based on the dense uniform linear array and the second linear array, a third-order sparse array is constructed, wherein the number of physical array elements is N, N = N1 + N2, the physical array element spacing is non-uniform, 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 degrees 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;

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

[0011] Preferably, in step 2, The method to obtain is: Sort the values ​​in from small to large, 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 to obtain is: Sort the values ​​in from small to large, 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

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

[0013] Preferably, 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(·) means finding the subset consisting of the longest continuous numerical segment in the set, and “|·|” means finding the number of elements in the set; then the optimal solution to s is solved to obtain the optimal solution to s.

[0014] Preferably, in step 4, the construction process of the third-order exhaustive common 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 array 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 array 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 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 represents the transpose operation, q1,q2,q N Represents 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 of 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, the superscript “*” is the conjugate operation, “○” is the tensor product operation symbol, and “×” is the scalar multiplication operation symbol. When l = 0, x(k) °(0) =1, 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;

[0017] 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(·) indicates vectorized operation, The dimension is N 3 ×1, is the Kronecker product operator. When l = 0 When l = 1 When l=2,3 represents the Kronecker product Used l-1 times in total, The dimension is G×1,

[0018] 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;

[0019] Step 4.4: Assume that R is regarded as an equivalent array manifold matrix and f is regarded as an equivalent source vector, then c x It is regarded 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.

[0020] Compared with the prior art, the advantages of the present invention are:

[0021] By expanding and shifting the second linear array, the low-order array is mapped to the third order to adapt to the third-order cumulant model. The present invention makes full use of the continuous segment of the sum and difference co-array of the second linear array. The virtual array (i.e., the third-order exhaustive co-array) of the designed third-order sparse array has a longer continuous section, and the number of degrees of freedom of the array is significantly improved. Based on the third-order cumulant model, the third-order exhaustive co-array can provide O(N 3 ) is much higher than the method based on second-order statistics. When the number of array elements increases, the gap in degrees of freedom between the third-order sparse array designed by the present invention and the second-order sparse array will be further expanded. At the same time, it is also better than the existing third-order nested array. Therefore, the third-order sparse array designed by the present invention 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 invention have closed expressions and can be generated by many commonly used second-order sparse arrays; thanks to the expansion of the sub-array spacing, the mutual coupling effect of the third-order sparse array designed by the present invention is also significantly improved, making the DOA estimation performance in a strong mutual coupling environment more stable. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0024] Figure 3 The present invention is a schematic diagram showing the comparison of the root mean square error (RMSE) versus signal-to-noise ratio (SNR) curves when performing DOA estimation on a third-order sparse array designed using the method of the present invention, an existing third-order sparse array, and two commonly used fourth-order sparse arrays. DETAILED DESCRIPTION

[0025] The invention will be further described below with reference to the accompanying drawings and in combination with specific implementations, so that those skilled in the art can implement the invention with reference to the description. The protection scope of the invention is not limited to the specific implementations.

[0026] The present invention proposes a third-order sparse array design method based on sum-difference analysis, and its overall implementation process is as follows: Figure 1 As shown, it includes the following steps:

[0027] Step 1: Select the 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 expressed as Z1=P1·d={0,1,...,N1-1}d, where N1≥2 and d is generally set to half the wavelength. m represents the wavelength of the incident signal from the 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 respectively; and the second linear array is selected, the number of its array elements is N2, and the physical array element position set Z2 is expressed as Wherein, N2≥2, P2 represents the index set of the physical array elements in the second linear array, p1,p2,p N2 Corresponding to the indexes representing the 1st physical array element, the 2nd physical array element, and the N2th physical array element in the second linear array, the positions of the physical array elements in the dense uniform linear array and the second linear array are both integer multiples of d.

[0028] Step 2: Calculate the second order and co-array of the dense uniform linear array Where g and h are any two indices in P1 (can be selected repeatedly to exhaust all possibilities); and calculate the second-order and co-matrix of the second linear array and second-order difference matrix Where a and b are any two indexes in P2 (they can be selected repeatedly to exhaust all possibilities); then S P2 The subset consisting of the longest continuous value segment in is denoted as D P2 The subset consisting of the longest continuous value segment in is denoted as The subset consisting of the longest continuous index segments in P2 is recorded as

[0029] Preferably, in step 2, The method to obtain is: Sort the values ​​in from small to large, 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 to obtain is: Sort the values ​​in from small to large, 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 Here, the continuous index segment is composed of continuous indexes 4, 5, 6, 7, and 8, for example.

[0030] Step 3: Based on the dense uniform linear array and the second linear array, a third-order sparse array is constructed, wherein the number of physical array elements is N, N = N1 + N2, the physical array element spacing is non-uniform, 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, λ=2N1-1, s represents the shift factor used to maximize the third-order uniform degrees 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.

[0031] Preferably, 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(·) means to find the subset consisting of the longest continuous value segment in the set. The process of uniform(·) is the same as The process of obtaining s is the same as that of obtaining s, where “|·|” indicates the number of elements in the set; then the optimal solution problem of s is solved 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 merging four third-order cumulants. The continuous virtual array elements in the third-order exhaustive co-array can be used to estimate DOA. 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 common array, and can analyze more signal sources. Compared with the fourth-order cumulant model, the third-order exhaustive common array can achieve a higher number of degrees of freedom when the number of array elements is small, and has a great advantage in computational complexity.

[0033] Preferably, in step 4, the construction process of the third-order exhaustive common 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 array 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 array 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 position of the array element is determined, 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 represents the transpose operation, q1,q2,q N Represents 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 of 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 to find the mathematical expectation, and the superscript “*” is the conjugate operation. is the tensor product symbol, and “×” is the scalar multiplication 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 in total.

[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(·) indicates vectorized operation, r (l) (θ i ) has a dimension of N 3 ×1, r1 (l) (θ i ), Corresponding to r (l) (θ i ) in the first, second, and Nth elements. 3 elements, is the Kronecker product operator. When l = 0 When l = 1 When l=2,3 represents the Kronecker product Used l-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: Assume that R is regarded as an equivalent array manifold matrix and f is regarded as an equivalent source vector, then c xIt is regarded 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 (which can be selected repeatedly to exhaust all possibilities).

[0038] The superiority of the method of the present invention is illustrated by simulation experiment analysis below.

[0039] In the simulation experiment, three different second linear arrays are selected, namely dense uniform linear array, nested array with shifted subarrays (NADiS) and transposed nested array (TNA). The latter two are based on second-order sum and difference common array design and have higher second-order sum and difference degrees of freedom. Among them, the nested array with shifted subarrays (NADiS) comes from P.Gupta and M.Agrawal, Design And Analysis of the Sparse Array for DoA Estimation of NoncircularSignals, IEEE Transactions on Signal Processing, vol.67, no.2, pp.460-473, 15Jan.15, 2019. (Design and analysis of sparse array for DoA estimation of non-circular signals, IEEE Transactions on Signal Processing), and the transposed nested array (TNA) comes 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. (Transposed nested array designed for DOA estimation of non-circular signals: Reduced sum-difference co-array redundancy perspective, IEEE Communications Letters).

[0040] Under three different second linear arrays, three third-order sparse arrays are designed using the method of the present invention, which are represented as TO-SDA (ULA), TO-SDA (NADiS) and TO-SDA (TNA). Figure 2A schematic diagram of the structure of the third-order sparse array designed by the method of the present invention when the second linear array uses 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 constructed third-order sparse array TO-SDA (TNA), where the expansion coefficient λ = 2N1-1 = 7 and the shift factor s = 25. It is compared with the existing third-order sparse array, namely the third-order nested array (TONA). In addition, a four-layer nested array (FL-NA) and a simplified and improved four-layer nested array (SE-FL-NA) are introduced for comparison. Among them, the third-order nested array (TONA) comes 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. (Third-order nested array: the optimal geometry for array processing based on third-order cumulants, IEEE Transactions on Signal Processing), the four-level nested array (FL-NA) comes from P.Pal andP.P.Vaidyanathan, Multiple level nested array: An efficient geometry for 2qthorder cumulant based array processing, IEEE Transactions on Signal Processing, 2012, 60(3): 1253-1269. (Multiple level nested array: an efficient array geometry based on 2qthorder cumulants, IEEE Transactions on Signal Processing), and the simplified improved four-level nested array (SE-FL-NA) comes from Q.Shen, W.Liu, W.Cui, S.Wu andP.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 number of physical array elements of the third-order sparse array is set to N = 9, the number of incoherent far-field narrowband signal sources is set to G = 16, the incident angle of the far-field narrowband signal source is uniformly distributed between -60 degrees and 60 degrees, the number of snapshots is set to 7500, the signal-to-noise ratio is set to 5 dB, and 500 Monte Carlo simulations are performed at each signal-to-noise ratio to test the root mean square error (RMSE) of DOA estimation of each third-order sparse array from -5 dB to 20 dB. The DOA estimation algorithm uniformly adopts the root-finding MUSIC algorithm in smooth space, and the results are shown in the figure. Figure 3 As shown. Figure 3 It can be seen that the third-order sparse array (TO-SDA) designed by the method of the present invention has the lowest RMS error and the best performance. Specifically, the transposed nested array (TNA) has the highest second-order and difference uniform degrees of freedom, and the physical array also has a continuous part. Using it as the second linear array can obtain the highest third-order uniform degrees of freedom, thereby showing the lowest estimation error; the nested array with shifted subarrays (NADiS) has a higher second-order and difference uniform degrees of freedom, but cannot use the continuous part in the physical array. The physical array of the uniform linear array (ULA) is completely continuous, but the second-order and difference uniform degrees of freedom are low. The TO-SDA (NADiS) and TO-SDA (ULA) constructed by using them as the second linear array have similar third-order uniform degrees of freedom, and their estimation performance is slightly lower than that of TO-SDA (TNA), but it is also significantly better than the existing arrays. In summary, the third-order sparse array designed by the present invention has good versatility, and many existing arrays can be directly constructed as the second linear array to achieve excellent direction finding accuracy.

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, the number of physical array elements is N1, the physical array element interval is d, and the physical array element position set Z1 is expressed as Z1=P1·d={0,1,…,N1-1}d, where P1 represents the index set of the physical array elements in the dense uniform linear array, P1={0,1,…,N1-1}; and select the second linear array, the number of array elements is N2, and the physical array element position set Z2 is expressed as Wherein, P2 represents the index set of the 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 and h are any two indices in P1; and the second-order and co-array of the second linear array are calculated. and second-order difference matrix Among them, a and b are any two indexes in P2; then The subset consisting of the longest continuous value segment in is recorded as Will The subset consisting of the longest continuous value 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, wherein the number of physical array elements is N, N = N1 + N2, the physical array element spacing is non-uniform, 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 degrees 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; Step 4: According to the third-order sparse array, a third-order virtual array, i.e., a third-order exhaustive common 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 method to obtain is: Sort the values ​​in from small to large, 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 to obtain is: Sort the values ​​in from small to large, 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. A third-order sparse array design method based on sum-difference analysis according to claim 1, characterized in that In the step 3, λ=2N1-1.

4. A third-order sparse array design method 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: Among them, uniform(·) means finding the subset consisting of the longest continuous numerical segment in the set, and "|·|" means finding the number of elements in the set; then the optimal solution of s is solved to obtain the optimal solution of s.

5. A third-order sparse array design method based on sum-difference analysis according to claim 4, characterized in that In step 4, the construction process of the third-order exhaustive common 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 array 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 array 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 represents the transpose operation, q1,q2,q N Represents 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 of 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 symbol, "×" is the scalar multiplication 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(·) indicates vectorized operation, r (e) (θ i ) has a dimension of N 3 ×1, is the Kronecker product operator. When l = 0 When l = 1 When l=2,3 represents the Kronecker product Used l-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: Assume that R is regarded as an equivalent array manifold matrix and f is regarded as an equivalent source vector, then c x It is regarded 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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