Sparse array design method based on coprime array redundancy analysis

Abstract

The application discloses a kind of sparse array design method based on coprime array redundancy analysis, it is related to DOA estimation field.The steps include: step 1: the signal model of coprime array is established;Step 2: coprime array is analyzed to redundancy virtual element;Step 3: based on the analysis result of step 2, two shift coprime arrays are formed by moving subarray, which are CATrS-I array structure and CATrS-II array structure respectively;Step 4: using CATrS-I array structure or CATrS-II array structure carries out DOA estimation.The application gives two kinds of subarray shift coprime arrays CATrS-I and CATrS-II, can effectively increase degree of freedom and continuous degree of freedom under the condition of keeping the total number of physical elements unchanged, and suppresses element mutual coupling.

Description

Technical Field

[0001] This invention relates to the field of DOA estimation, and more particularly to a sparse array design method based on coprime array redundancy analysis. Background Technology

[0002] Uniform linear arrays (ULAs) are widely used in direction-of-arrival (DOA) estimation due to their regular and uniform array configuration. However, for a ULA structure with N elements, subspace algorithms such as Multiple Signal Classification (MUSIC) and Estimating Signal Parameters via Rotational Invariance Techniques (ESPRIT) can only estimate a maximum of N-1 signal sources. Furthermore, the dense element spacing in ULAs leads to severe mutual coupling effects. In summary, limited by factors such as array configuration and the number of physical elements, ULA structures are no longer sufficient to meet the current demands for resolution and accuracy in array direction finding. In contrast, sparse arrays offer significant advantages in terms of degrees of freedom, element mutual coupling, and redundancy, making them an effective solution for achieving high-resolution and high-precision DOA estimation.

[0003] Coprime arrays (CAs) are a representative sparse array design, consisting of two sparse subarrays whose element spacing and number satisfy coprime. Compared to nested arrays (NAs), CAs can effectively suppress element coupling, but their limited continuous degrees of freedom and holes in differential coarrays significantly reduce the estimation performance of coarray MUSIC algorithms. To address this issue, researchers have conducted in-depth studies on coprime array structure optimization and DOA estimation algorithm design. However, considering real-time requirements, optimizing the coprime array structure to improve angle estimation performance is the simplest and most direct method. For example, the Augmented Coprime Array (ACA) structure has been proposed in the prior art, which increases the continuous degrees of freedom by increasing the number of elements in one subarray of the CA from M to 2M. Other existing technologies have further extended the ACA structure from a generalized perspective. Furthermore, the design of Thinned Coprime Array (TCA) shows that redundant physical elements exist in both the ACA and its generalized structures, thus providing the same virtual aperture and array degrees of freedom while reducing the number of physical elements. Existing research on array robustness indicates that redundant physical elements also exist in the CA. Meanwhile, increasing the subarray spacing eliminates redundant virtual elements in the difference set, but reduces the continuous degrees of freedom. In summary, since only the continuous portion can be utilized, the co-array MUSIC algorithm's estimation performance deteriorates when the virtual array has holes. Therefore, there is an urgent need to design an array structure that can both improve the continuous degrees of freedom and suppress mutual coupling. Summary of the Invention

[0004] The purpose of this invention is to provide a sparse array design method based on coprime array redundancy analysis, which can reduce the number of redundant virtual array elements in the subarray mutual difference set, thereby improving the degrees of freedom and continuous degrees of freedom, and effectively suppressing the mutual coupling of array elements.

[0005] To achieve the above objectives, this invention provides a sparse array design method based on coprime array redundancy analysis, characterized by comprising the following steps:

[0006] Step 1: Based on the signal model of coprime arrays, perform redundant virtual array element analysis on traditional coprime arrays;

[0007] Step 2: Based on the analysis results of Step 1, two shifted coprime arrays are formed by moving subarrays, namely CATrS-Ⅰ array structure and CATrS-Ⅱ array structure;

[0008] Step 3: Perform DOA estimation using either the CATrS-Ⅰ array structure or the CATrS-Ⅱ array structure.

[0009] Therefore, the sparse array design method based on coprime array redundancy analysis described above in this invention has the following beneficial effects:

[0010] This invention proposes two CATrS structures by moving the subarrays of a coprime array by an appropriate distance. These array structures can reduce the number of redundant virtual array elements in the subarray cross-difference set, thereby significantly improving the degrees of freedom and continuous degrees of freedom, and effectively suppressing the mutual coupling of array elements. Using the two CATrS structures proposed in this invention for DOA estimation can improve the direction finding accuracy and resolution.

[0011] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of a coprime array.

[0013] Figure 2 This is a schematic diagram of the CATrS-Ⅰ structure.

[0014] Figure 3 This is a schematic diagram of the CATrS-Ⅱ structure.

[0015] Figure 4 (a)-(b) show the comparison results of the continuous degrees of freedom and the changes in degrees of freedom with the number of array elements for different coprime arrays; among them, Figure 4 (a) shows the variation of continuous degrees of freedom with the number of array elements. Figure 4 (b) shows how the degrees of freedom change with the number of array elements.

[0016] Figure 5 The results show the comparison of coupling leakage of different coprime arrays with the number of array elements.

[0017] Figure 6 (a)-(f) are the difference comatrices of different coprime arrays; where, Figure 6 (a) is the difference comatrix of CA. Figure 6 (b) is the difference comatrix of ACA. Figure 6 (c) is the difference comatrix of RSRCA-Ⅰ. Figure 6 (d) is the difference comatrix of RSRCA-II. Figure 6 (e) is the difference comatrix of CATRS-I. Figure 6 (f) is the differential comatrix of CATRS-II.

[0018] Figure 7 (a)-(f) are the mapping diagrams of mutually coupled matrix elements for different coprime arrays; where, Figure 7 (a) is the mapping diagram of the mutual coupling matrix elements of CA. Figure 7 (b) is the mapping diagram of the mutual coupling matrix elements of ACA. Figure 7 (c) is the mapping diagram of the mutual coupling matrix elements of RSRCA-I. Figure 7 (d) is the mapping diagram of the mutual coupling matrix elements of RSRCA-II. Figure 7 (e) is the mapping diagram of the mutual coupling matrix elements of CATRS-I. Figure 7 (f) is the mapping diagram of the mutual coupling matrix elements of CATRS-II.

[0019] Figure 8 (a)-(f) represent the spatial spectra of the nine targets estimated by different coprime arrays; where... Figure 8 (a) Estimate the spatial spectrum of the nine targets for CA. Figure 8 (b) Estimate the spatial spectrum of the nine targets for ACA. Figure 8 (c) Estimating the spatial spectrum of the nine targets for RSRCA-Ⅰ. Figure 8 (d) The spatial spectrum of the nine targets estimated by RSRCA-II. Figure 8 (e) Estimates the spatial spectrum of the nine targets for CATRS-I. Figure 8 (f) Estimates the spatial spectrum of the nine targets for CATRS-II.

[0020] Figure 9 (a)-(f) represent the spatial spectra of the 11 targets estimated by different coprime arrays; where... Figure 9 (a) Estimate the spatial spectrum of 11 targets for CA. Figure 9 (b) Estimate the spatial spectrum of 11 targets for ACA. Figure 9 (c) Estimating the spatial spectrum of 11 targets for RSRCA-Ⅰ. Figure 9 (d) Spatial spectra of 11 targets estimated by RSRCA-II. Figure 9 (e) Estimate the spatial spectrum of 11 targets for CATRS-I. Figure 9 (f) Estimates the spatial spectrum of 11 targets for CATRS-II. Figure 10 (a)-(b) show the comparison of RMSE results for DOA estimation of different coprime arrays; where... Figure 10 (a) shows the change of RMSE with signal-to-noise ratio. Figure 10 (b) shows how RMSE changes with the number of snapshots. Detailed Implementation

[0021] In the description of this invention, it should also be noted that, unless otherwise expressly specified and limited, these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various alterations or modifications to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0022] This invention, based on CA (Carry-Age Array) and redundant virtual array element analysis, proposes two types of coprime arrays with translated subarrays (CATrS) by moving subarrays. The two CArS array structures proposed in this invention can improve continuous degrees of freedom and reduce element mutual coupling while increasing the number of degrees of freedom, thereby improving DOA (Domain of Ability) estimation performance.

[0023] 1. Signal model of existing coprime arrays

[0024] For ease of description, this invention normalizes the positions of all array elements with respect to the unit array element spacing d = λ / 2, where λ represents the source wavelength.

[0025] Suppose K far-field, narrowband, uncorrelated signal sources originate from direction {θ} k ∈[-π / 2,π / 2)|k=1,2,…,K} is incident on a sparse array with a total of T elements, and the set of element positions is P={p i |i=1,2,…,T}. Under the influence of array mutual coupling, the array output at the l-th sampling snapshot time is expressed as:

[0026]

[0027] In the formula, A=[a(θ1),a(θ2),…,a(θ K ] represents the physical array manifold matrix, and The steering vector of the k-th signal is represented by s(l) = [s1(l), s2(l), ..., s K (l)] T and n(l)=[n1(l),n2(l),…,n T (l)] T Let B represent the signal vector and the Gaussian white noise vector, respectively, and the two are uncorrelated; C represents the mutual coupling matrix, whose elements can be approximated by the B-band symmetric Toeplitz matrix:

[0028]

[0029] In the formula, p i ,p j ∈P; c b Denotes the coupling coefficient, and 1 = c0 > |c1| > |c2| > ... > |c B |>|c B+1 |=0, usually set B = 100, 2≤l≤B;

[0030] The strength of array mutual coupling effect can be specifically quantified by coupling leakage:

[0031]

[0032] Clearly, 0 ≤ L ≤ 1, and the smaller L is, the weaker the mutual coupling.

[0033] According to equation (1), the theoretical covariance matrix of the array output can be obtained as follows:

[0034]

[0035] In the formula, Let represent the source covariance matrix, and σ is the power of the k-th source; 2 Indicates noise power.

[0036] In practical applications, since the theoretical covariance matrix cannot be obtained, it can be approximately estimated from the sample covariance matrix.

[0037]

[0038] In the formula, L represents the total number of sample snapshots.

[0039] By vectorizing the covariance matrix, the following virtual domain signal measurement model is obtained:

[0040]

[0041] In the formula, This represents the equivalent virtual signal vector composed of the source power; in This represents a column vector where the i-th element is 1 and all other elements are 0. Represents the virtual array manifold matrix, and Clearly, the position of the virtual array element corresponding to B can be given by the following difference comatrix:

[0042]

[0043] In the formula, Indicates the position of the virtual array element.

[0044] The difference comatrix of a sparse array is defined as follows:

[0045] Definition 1 (Degrees of Freedom): The number of distinct elements in a difference comatrix D is defined as the degrees of freedom of the sparse array P, denoted as |D|.

[0046] Definition 2 (Continuous Degrees of Freedom) Suppose that U represents the largest central continuous segment in the difference comatrix D, then the number of different elements in U is defined as the continuous degrees of freedom of the sparse array P, denoted as |U|.

[0047] Definition 3 (hole) Assume the shortest ULA containing D is Then for any integer when and hour, The hole is referred to as D.

[0048] Definition 4 (weights) defines the positions of virtual matrix elements that can be generated in the difference matrix. The number of physical array element pairs is defined as weight

[0049]

[0050] Combining equation (2), it can be seen that the weights reflect the strength of the mutual coupling effect. When comparing array mutual coupling, the first three weights, namely ω(1), ω(2), and ω(3), are usually considered. Additionally, when... When, it indicates There are redundant virtual array elements.

[0051] 2. Redundancy analysis of coprime arrays

[0052] like Figure 1 As shown, CA is composed of alternating sparse subarrays with a number of elements and an element spacing that satisfy a coprime relationship. One subarray has N elements and an element spacing of Md, while the other subarray has M elements and an element spacing of Nd, where gcd(M,N) = 1 and M < N. gcd(M,N) = 1 indicates that M and N are coprime. Due to the coprime arrangement, all elements except the reference element do not overlap. Therefore, CA contains a total of T = M + N - 1 elements, and its position set is as follows:

[0053]

[0054] Combining equation (7), we can see that the difference comatrix of CA is:

[0055]

[0056] In the formula, and Denotes the difference set of two subarrays. and The mutual difference set of two subarrays can be represented as:

[0057]

[0058] Due to the uniformity of each subarray, and Redundant virtual array elements inevitably exist in the array, among which... There exists N 2 -2N+1 redundant virtual array elements, M exists in 2 -2M+1 redundant virtual array elements. Since the two subarrays satisfy the coprime property... and There are no redundant virtual array elements, but but There are redundant virtual array elements, and their positions are:

[0059]

[0060] In the formula, m1∈<0,M-1>, m2∈<0,M-1>, n1∈<0,N-1>, n2∈<0,N-1>;

[0061] The equivalent form of equation (12) is:

[0062] M(n1+n2)=N(m1+m2) (13)

[0063] Equation (13) holds true when one of the following conditions is met:

[0064] (a) m1=m2=n1=n2=0. At this time, There is a redundant virtual array element at this location;

[0065] (b) n1+n2=τN and m1+m2=τM, where τ is a positive integer; when (n1,n2) takes

[0066] This condition holds true when (m1,m2) takes any one of (1,N-1), (2,N-2), ..., (N-1,1), and (m1,m2) takes any one of (1,M-1), (2,M-2), ..., (M-1,1). In this case, there are a total of (M-1)(N-1) redundant virtual array elements.

[0067] As can be seen from the above analysis, there are a large number of redundant virtual array elements in the differential coarray of CA, which limits the improvement of the degree of freedom. Therefore, this invention further improves it and proposes the following subarray shift coprime array.

[0068] 3. Subarray displacement coprime array

[0069] Generally speaking, given a certain number of physical array elements, the fewer redundant virtual array elements in a sparse array, the higher its degrees of freedom. Therefore, reducing redundant virtual array elements in a sparse array (CA) can effectively improve the degrees of freedom. Based on this idea, this invention appropriately shifts the subarrays of a CA, resulting in two types of shifted coprime arrays. By optimizing the array deployment, the number of redundant virtual array elements is reduced, thereby increasing the array's degrees of freedom and suppressing element coupling.

[0070] (1)CATrS-Ⅰ

[0071] like Figure 2 As shown, after shifting the last N-1 elements of the CA subarray with a spacing of Md to the right by a distance αNd, a CATrS-Ⅰ structure is formed. At this point, the set of physical element positions can be represented as:

[0072]

[0073] In the formula, α is an integer and 1≤α≤(M-1) / 2.

[0074] Combining equation (7), the difference comatrix of CATrS-Ⅰ is:

[0075]

[0076] In the formula,

[0077]

[0078] First, let's analyze the range of values ​​for α. Since the subarray is shifted to the right, α ≥ 1. Second, The positions of redundant virtual array elements are:

[0079]

[0080] In the formula,

[0081] The equivalent form of equation (17) is:

[0082]

[0083] Obviously, when Choose any one of (1, N-1), (2, N-2), ..., (N-1, 1), and Equation (18) holds true when any one of (2α+1,M-1), (2α+2,M-2), ..., (M-1,2α+1). Compared with (m1,m2) in equation (13), the (m1,m2) in equation (18) The number of possible values ​​decreases by 2α. For equation (18) to hold, 2α ≤ M-1, i.e., α ≤ (M-1) / 2. Therefore, 1 ≤ α ≤ (M-1) / 2.

[0084] Property 1: Sets and There are M(N-1) different virtual array elements in each array.

[0085] Proof: Because Therefore, taking the former as an example, we will use the method of inversion to prove it.

[0086] definition and It is a set Any two elements in, where

[0087] If p = q, then we have Right now:

[0088]

[0089] because Since gcd(M,N)=1, equation (19) does not hold, i.e., p≠q. Therefore, There are M(N-1) different virtual array elements.

[0090] Proposition 1: Difference comatrix D CATrS-Ⅰ The number of different virtual array elements in the array is MN+M+(2α+1)N-2(α+1), and D CATrS-Ⅰ There exists M2+N 2 +MN-3M-(2α+3)N+2(α+1)+1 redundant virtual array elements.

[0091] Proof due to Equation (15) can be simplified to:

[0092]

[0093] in, There are a total of 2M-1 different virtual array elements, which are set up There are a total of 2M(N-1)-(M-2α-1)(N-1)=(M+2α+1)(N-1) different virtual array elements.

[0094] Furthermore, set D CATrS-Ⅰ It exists in:

[0095] (M+N-1) 2 -(MN+M+(2α+1)N-2(α+1))

[0096] =M 2 +N 2 +MN-3M-(2α+3)N+2(α+1)+1 (21)

[0097] A redundant virtual array element.

[0098] Proposition 2: Difference comatrix D CATrS-Ⅰ The hole in H is located CATrS-Ⅰ ={±(αN+xM+yN)|x≥1,y≥1}.

[0099] Proof: Set It can be restated as:

[0100]

[0101] because but:

[0102]

[0103] Similarly,

[0104]

[0105] therefore,

[0106]

[0107] According to existing technology, the set The positive hole in the middle is located in {aM+bN|a≥0,b≥1}, therefore The positive void in the equation is located in {αN+aM+bN|a≥0,b≥1}. Because... Just enough to fill the space located The hole at that location, therefore D CATrS-Ⅰ The first positive hole in the middle is located Therefore, D CATrS-Ⅰ The positive hole in the middle is located at H CATrS-Ⅰ = αN + xM + yN, where x ≥ 1, y ≥ 1. Due to symmetry, the difference comatrix D... CATrS-Ⅰ The hole in H is located CATrS-Ⅰ ={±(αN+xM+yN)|x≥1,y≥1}.

[0108] Proposition 3: Difference comatrix D CATrS-Ⅰ The range of continuous virtual array elements is <-αN-M-N+1, αN+M+N-1>.

[0109] Proof: By Proposition 2, D CATrS-Ⅰ The first positive and negative holes in D are located at ±(αN+M+N), therefore D CATrS-Ⅰ The range of continuous virtual array elements is <-αN-M-N+1, αN+M+N-1>.

[0110] The above-mentioned property 1 and propositions 1-3 demonstrate that the CATrS-Ⅰ structure proposed in this invention has more continuous degrees of freedom.

[0111] Proposition 4: When α = (M-1) / 2, the difference comatrix D CATrS-Ⅰ The weight expression is:

[0112]

[0113] In the formula, i∈<1,N-1>, j∈<1,M-1>, x≥1, y≥1.

[0114] Proof: Since the total number of physical array elements is M+N-1, ω(0)=M+N-1. After subarray shifting, there are two ULA structures in CATrS-Ⅰ, namely {αN+Mn|n∈<0,N-1>} and Therefore, ω(|iM|)=Ni, ω(|jN|)=Mj. From Proposition 2, it is easy to see that ω(|αN+xM+yN|)=0.

[0115] In addition to the aforementioned virtual array element positions, D CATrS-Ⅰ The following remains:

[0116] (MN+M+(2α+1)N-2(α+1))-2(M-1+N-1)-1=(M-1)(N-1)+2α(N-1) (27) virtual array element positions are unassigned weights, and the number of remaining virtual array elements is:

[0117] (M+N-1) 2 -M(M-1)-N(N-1)-(M+N-1)=2(M-1)(N-1) (28)

[0118] From equations (27) and (28), it can be seen that when α = (M-1) / 2, the weights of the remaining virtual array elements are all 1.

[0119] As can be seen from the proof of Proposition 4, when 1≤α≤(M-1) / 2, the weight of some remaining virtual array elements will also decrease from 2 to 1, which shows that CATrS-Ⅰ can reduce the mutual coupling of array elements.

[0120] (2)CATrS-II

[0121] like Figure 3 As shown, this invention shifts the last M-1 array elements of the CA subarray with a spacing of Nd to the right by a distance of βMd, forming a CATrS-Ⅱ structure. At this point, the set of physical array element positions can be represented as:

[0122]

[0123] In the formula, β is an integer and 1≤β≤(N-1) / 2.

[0124] Combining equation (7), we can see that the difference comatrix of CATrS-Ⅱ is:

[0125]

[0126] In the formula,

[0127]

[0128] Difference comatrix D CATrS-Ⅱ The following properties and propositions hold true.

[0129] Property 2: Sets and There are N(M-1) different virtual array elements in each array.

[0130] Proposition 5 Difference Comatrix D CATrS-Ⅱ The number of different virtual array elements in the array is MN+N+(2β+1)M-2(β+1), and D CATrS-Ⅱ M exists in 2 +N 2 +MN-3N-(2β+3)M+2(β+1)+1 redundant virtual array elements.

[0131] Proposition 6 Difference Comatrix D CATrS-Ⅱ The hole in H is located CATrS-Ⅱ ={±(βM+xM+yN)|x≥1,y≥1}.

[0132] Proposition 7 Difference Comatrix D CATrS-Ⅱ The range of continuous virtual array elements is <-βM-M-N+1,βM+M+N-1>.

[0133] Proposition 8 When β = (N-1) / 2, the difference comatrix D CATrS-Ⅱ The weight expression is:

[0134]

[0135] In the formula, i∈<1,M-1>, j∈<1,N-1>, x≥1, y≥1.

[0136] 4. Performance Analysis

[0137] When α is an integer and α = (M-1) / 2, the continuous degrees of freedom and the number of degrees of freedom of CATrS-Ⅰ are MN+2M+N-1 and 2MN-1, respectively. When the total number of physical array elements is given as T = M+N-1, the optimal array configuration parameters can be solved by maximizing the continuous degrees of freedom:

[0138]

[0139] Using the arithmetic-geometric mean inequality, the optimal values ​​for M and N can be obtained as follows:

[0140]

[0141] Finally, the solution to equation (33) is:

[0142]

[0143] Similarly, the maximum degree of freedom of CATrS-Ⅰ can be obtained as:

[0144]

[0145] When β is an integer and β = (N-1) / 2, the continuous degrees of freedom and the number of degrees of freedom of CATrS-Ⅱ are MN+M+2N-1 and 2MN-1, respectively. Using the same optimization method, the maximum continuous degrees of freedom of CATrS-Ⅱ can be obtained as:

[0146]

[0147] Similarly, the maximum degree of freedom of CATrS-II is:

[0148]

[0149] Below, CA, ACA, RSRCA-Ⅰ, and RSRCA-Ⅱ are selected as comparison objects to illustrate the performance advantages of CATrS-Ⅰ and CATrS-Ⅱ in terms of continuous degrees of freedom, degrees of freedom, and element coupling. Table 1 summarizes the maximum continuous degrees of freedom and closed-form analytical expressions for the maximum degrees of freedom of different coprime arrays under optimal array configuration parameters. Since ACA is difficult to solve for the optimal array configuration parameters, Figure 4 The continuous degrees of freedom and the relationship between the degrees of freedom and the number of physical array elements for the above six coprime arrays were further plotted. (See Table 1 and...) Figure 4 It can be seen that, with the same number of physical array elements, the two coprime arrays designed in this invention can obtain more continuous degrees of freedom and degrees of freedom compared to CA, ACA, RSRCA-Ⅰ, and RSRCA-Ⅱ. Table 2 summarizes the expressions for the first three weights of different coprime arrays to measure their ability to suppress mutual coupling effects, where CATrS-Ⅰ and CATrS-Ⅱ are the cases when α and β take their maximum values, respectively. It can be seen that the reduction in the number of redundant virtual array elements by ACA constructed by expanding the number of subarrays and RSRCA-Ⅰ and RSRCA-Ⅱ constructed by repositioning the reference array element positions is limited (especially difficult to reduce the weights corresponding to small hysteresis), and therefore has considerable mutual coupling with CA; while the two coprime arrays designed in this invention, by moving a subarray as a whole to an appropriate position, destroy the conditions for the generation of redundant virtual array elements in the subarray mutual difference set, effectively reducing the number of redundant virtual array elements (significantly reducing the weights corresponding to small hysteresis), and can reduce array element mutual coupling to a large extent. In addition, Figure 5The curves depicting the coupling leakage as a function of the number of physical array elements clearly show that the two coprime arrays designed in this invention have lower element mutual coupling than CA, ACA, RSRCA-Ⅰ, and RSRCA-Ⅱ. This is because the reduction in the number of redundant virtual array elements by ACA (constructed by expanding the number of subarrays) and RSRCA-Ⅰ and RSRCA-Ⅱ (constructed by moving the reference array element positions) is limited (especially difficult to reduce the weight of small hysteresis), thus having comparable mutual coupling to CA. However, the coprime array designed in this invention, by moving a subarray as a whole to an appropriate position, destroys the conditions for the generation of redundant virtual array elements in the subarray mutual difference set, thereby significantly reducing the number of redundant virtual array elements and reducing mutual coupling to a large extent.

[0150] Table 1. Optimal array arrangement, maximum continuous degrees of freedom, and maximum degrees of freedom for relevant coprime matrices.

[0151]

[0152]

[0153] Table 2. Expressions for the first three weights of different coprime matrices.

[0154]

[0155] Example

[0156] To verify the superior performance of the two coprime arrays proposed in this invention in terms of array properties and DOA estimation, the following simulation experiments were conducted.

[0157] With the total number of physical array elements set to 10, the configuration parameters for CA, RSRCA-Ⅰ, RSRCA-Ⅱ, CATrS-Ⅰ, and CATrS-Ⅱ are M=5 and N=6, respectively, and the configuration parameter for ACA is... and

[0158] 1. Array properties

[0159] After determining the array element positions based on the configuration parameters, the continuous degrees of freedom, degrees of freedom, first three weights, and coupling leakage of each coprime array were calculated. The results are shown in Table 3. Meanwhile, Figure 6 Differential comatrices for different coprime arrays are presented. It can be seen that, compared to the contrasting coprime arrays, the CATrS-Ⅰ and CATrS-Ⅱ designed in this invention not only improve the continuous degrees of freedom and the number of degrees of freedom, but also reduce the virtual element weights and significantly suppress element mutual coupling.

[0160] Figure 7The mapping diagrams of the mutual coupling matrix elements for different coprime arrays are further illustrated. The color intensity of the off-diagonal components reflects that CATrS-Ⅰ and CATrS-Ⅱ have fewer small-pitch elements than the other coprime arrays, thus indicating lower mutual coupling. Table 3 shows the element positions, continuous degrees of freedom, degrees of freedom, first three weights, and coupling leakage of different coprime arrays.

[0161]

[0162] 2. Spatial spectrum

[0163] The spatial spectrum estimation performance of different coprime arrays was compared using the co-array MUSIC algorithm. It was assumed that the nine targets were uniformly distributed between -30° and 30°, and the signal-to-noise ratio (SNR) and sampling snapshot number were set to 0 dB and 500, respectively. The estimation results are shown below. Figure 8 As can be seen, except for CA, the other five coprime arrays can accurately estimate the above nine targets. This is because CA has only 21 continuous degrees of freedom, and theoretically can estimate a maximum of 10 targets. Due to the limited number of sampling snapshots and the low continuous degrees of freedom, its estimation results deviate significantly from the true angles of the targets.

[0164] Then, the number of targets was increased to 11, while the remaining simulation parameters remained unchanged. The estimation results are shown below. Figure 9 Clearly, CA, due to its low continuous degrees of freedom, cannot estimate all 11 targets. The spectral peaks of ACA, RSRCA-Ⅰ, and RSRCA-Ⅱ show significant attenuation and large estimation biases, while CATrS-Ⅰ and CATrS-Ⅱ can still accurately estimate the 11 targets. From the perspective of continuous degrees of freedom, ACA, RSRCA-Ⅰ, and RSRCA-Ⅱ are fully capable of estimating all 11 targets, but strong element coupling prevents them from obtaining accurate estimation results. The CATrS-Ⅰ and CATrS-Ⅱ designed in this invention not only increase the number of continuous degrees of freedom but also significantly suppress element coupling, thus achieving the best spatial spectrum estimation results.

[0165] 3. Root mean square error

[0166] To further evaluate the superior performance of CATrS-Ⅰ and CATrS-Ⅱ, this group of experiments compared the root mean square error (RMSE) of DOA estimation for six coprime arrays under different conditions. It was assumed that the four targets were located at -20°, -8°, 5°, and 10°, and the number of Monte Carlo experiments was set to 500. Figure 10 (a) The curve of RMSE as a function of SNR was plotted, where the number of sampling snapshots was 500; Figure 10 (b) A curve showing the change in RMSE as a function of the number of sampling snapshots was plotted, where SNR = -5dB. From Figure 10 As can be seen, the estimation error of each coprime array decreases with the increase of SNR and sampling snapshots. However, due to its lower continuous degrees of freedom, the estimation accuracy of CA is much lower than that of the other five types of coprime arrays. Since the continuous degrees of freedom and mutual coupling of ACA, RSRCA-Ⅰ, and RSRCA-Ⅱ are basically at the same level, their estimation accuracy is basically the same. CATrS-Ⅰ and CATrS-Ⅱ have the most continuous degrees of freedom and the lowest element mutual coupling, and their estimation errors are always smaller than those of the coprime arrays under the same conditions. Therefore, the two coprime arrays designed in this invention have better DOA estimation performance. In addition, under the assumption that the element parameters are M=5 and N=6, CATrS-Ⅱ cannot obtain the optimal array configuration, so its estimation performance is weaker than that of the CATrS-Ⅰ structure. Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A sparse array design method based on coprime array redundancy analysis, characterized in that, Includes the following steps: Step 1: Based on the signal model of coprime arrays, perform redundant virtual array element analysis on traditional coprime arrays; Step 2: Based on the analysis results of Step 1, two shifted coprime arrays are formed by moving subarrays, namely CATrS-Ⅰ array structure and CATrS-Ⅱ array structure; Step 3: Perform DOA estimation using either the CATrS-Ⅰ array structure or the CATrS-Ⅱ array structure; The CATrS-Ⅰ array structure uses a spacing of [missing information] between coprime arrays. The rear of the sub-array The entire array element shifts to the right. After obtaining a distance, where, Indicates the number of array elements. The spacing between the elements of the subarray, Indicates the number of array elements. The spacing between the elements of the subarray, and Coprime, Integer and ; The set of physical element locations for the CATrS-Ⅰ array structure is: ； in, This represents the first subarray in the CATrS-Ⅰ array structure. This represents the second subarray in the CATrS-Ⅰ array structure; Furthermore, the differential co-array of the CATrS-Ⅰ array structure for: ； In the formula, ； in, Indicates the operator and ; , .

2. The sparse array design method based on coprime array redundancy analysis according to claim 1, characterized in that, The CATrS-Ⅱ array structure uses a spacing of [missing information] between coprime arrays. The rear of the sub-array The entire array element shifts to the right. After a distance, Integer and .

3. The sparse array design method based on coprime array redundancy analysis according to claim 2, characterized in that, The differential co-array structure of CATrS-Ⅰ has the following properties: 1) and There are respectively in A different virtual array element; 2) The number of different virtual array elements in the array is ,and There exists One redundant virtual array element; 3) The hole in the middle is located ; 4) The range of continuous virtual array elements in the middle is ; 5) When At that time, difference coarray The weight expression is: ； in, , , , .

4. The sparse array design method based on coprime array redundancy analysis according to claim 3, characterized in that, The set of physical element locations for the CATrS-Ⅱ array structure is: ； in, This represents the first subarray in the CATrS-Ⅱ array structure. This represents the second subarray in the CATrS-Ⅱ array structure; Furthermore, the differential co-array of the CATrS-II array structure for: ； In the formula, 。 5. The sparse array design method based on coprime array redundancy analysis according to claim 4, characterized in that, The differential co-array structure of CATrS-II has the following properties: 1) and There are respectively in A different virtual array element; 2) The number of different virtual array elements in the array is ,and There exists One redundant virtual array element; 3) The hole in the middle is located ; 4) The range of continuous virtual array elements in the middle is ; 5) When At that time, difference coarray The weight expression is: ； in, , , , .

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