Sparse array design method based on co-prime array redundancy analysis

Abstract

The invention discloses a sparse array design method based on co-prime array redundancy analysis, and relates to the field of DOA estimation. Comprising the steps of 1, establishing a signal model of a co-prime array; 2, redundant virtual array element analysis is carried out on the co-prime array; step 3, based on an analysis result of the step 2, forming two shift co-prime arrays, namely a CATrS-I array structure and a CATrS-II array structure, by moving the sub-arrays; and step 4, carrying out DOA (Direction of Arrival) estimation by adopting a CATrS-I array structure or a CATrS-II array structure. According to the invention, two sub-array shift co-prime arrays CATrS-I and CATrS-II are provided, the degree of freedom and the continuous degree of freedom can be effectively increased under the condition that the total number of physical array elements is kept unchanged, and mutual coupling of the array elements is inhibited.

Description

Technical Field

[0001] The invention relates to the field of DOA estimation, and in particular to a sparse array design method based on coprime array redundancy analysis. Background Art

[0002] Uniform Linear Array (ULA) is widely used in Direction of Arrival (DOA) estimation due to its regular and uniform array. However, for a ULA structure with N array elements, subspace algorithms such as Multiple Signal Classification (MUSIC) and Estimating Signal Parameters via Rotational Invariance Techniques (ESPRIT) can only estimate N-1 signal sources at most. In addition, the dense array element spacing causes serious mutual coupling effects in ULA. In short, limited by factors such as the array layout and the number of physical array elements, the ULA structure can no longer meet the actual requirements of resolution and accuracy in the current array direction finding field. In contrast, sparse arrays have significant advantages in terms of degrees of freedom, element mutual coupling and redundancy, and have become an effective solution for achieving high-resolution and high-precision DOA estimation.

[0003] Coprime Array (CA) is one of the representative sparse array designs. The structure consists of two sparse sub-arrays with mutually prime spacing and number of array elements. Compared with Nested Array (NA), CA can effectively suppress the mutual coupling of array elements, but its limited continuous degrees of freedom and holes in the differential co-array greatly reduce the estimation performance of the co-array MUSIC algorithm. To solve this problem, scholars have conducted in-depth research from two aspects: coprime array structure optimization and coprime array DOA estimation algorithm design. However, considering the real-time requirements, improving the angle estimation performance by optimizing the coprime array structure 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 array elements in a sub-array of CA from M to 2M. Other prior arts further expand the ACA structure from a generalized perspective. In addition, the design of Thin Coprime Array (TCA) shows that there are redundant physical array elements in ACA and its generalized structure, which can provide the same virtual aperture and array degrees of freedom while reducing the number of physical array elements. Research on array robustness in the prior art shows that redundant physical array elements also exist in CA. At the same time, the redundant virtual array elements in the mutual difference set are eliminated by increasing the sub-array spacing, but the continuous degrees of freedom are reduced. In summary, since only the continuous part can be used, the estimation performance of the co-array MUSIC algorithm will be reduced when there are holes in the virtual array. Therefore, there is an urgent need to design an array structure that can both improve the continuous degrees of freedom and suppress mutual coupling at the same time. Summary of the invention

[0004] The purpose of the present 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 sub-array mutual difference set, thereby improving the degree of freedom and continuous degree of freedom, and effectively suppressing the mutual coupling of array elements.

[0005] To achieve the above object, the present invention provides a sparse array design method based on coprime array redundancy analysis, characterized in that it comprises the following steps:

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

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

[0008] Step 3: Use the CATrS-Ⅰ array structure or the CATrS-Ⅱ array structure to estimate DOA.

[0009] Therefore, the present invention adopts the above-mentioned sparse array design method based on coprime array redundancy analysis, which has the following beneficial effects:

[0010] The present invention proposes two CATrS structures by moving the subarrays of the coprime arrays by an appropriate distance. The array structure can reduce the number of redundant virtual array elements in the subarray mutual difference set, thereby greatly improving the degree of freedom and continuous degree of freedom, and effectively suppressing the mutual coupling of the array elements. The two CATrS structures proposed by the present invention are used for DOA estimation, which can improve the direction finding accuracy and resolution.

[0011] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 Schematic diagram of a coprime array.

[0013] Figure 2 Schematic diagram of the structure of CATrS-Ⅰ.

[0014] Figure 3 Schematic diagram of the structure of CATrS-Ⅱ.

[0015] Figure 4 (a)-(b) are the comparison results of the continuous degrees of freedom and the degrees of freedom changing with the number of array elements of different coprime arrays; Figure 4 (a) shows the change of continuous degrees of freedom with the number of array elements. Figure 4 (b) The degree of freedom varies with the number of array elements.

[0016] Figure 5 Comparison results of coupling leakage of different coprime arrays as the number of array elements changes.

[0017] Figure 6 (a)-(f) are differential co-arrays of different coprime arrays; among them, Figure 6 (a) is the differential matrix of CA, Figure 6 (b) is the differential matrix of ACA, Figure 6 (c) is the differential common array of RSRCA-Ⅰ, Figure 6 (d) is the differential common array of RSRCA-Ⅱ. Figure 6 (e) is the differential common array of CATRS-Ⅰ. Figure 6 (f) is the differential common array of CATRS-Ⅱ.

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

[0019] Figure 8 (a)-(f) are the spatial spectra of 9 targets estimated by different coprime arrays; Figure 8 (a) The spatial spectrum of 9 targets estimated by CA, Figure 8 (b) ACA estimates the spatial spectrum of 9 targets. Figure 8 (c) The spatial spectrum of 9 targets estimated by RSRCA-Ⅰ, Figure 8 (d) The spatial spectrum of 9 targets estimated by RSRCA-Ⅱ, Figure 8 (e) Estimation of the spatial spectrum of 9 targets for CATRS-I, Figure 8 (f) Spatial spectrum of 9 targets estimated by CATRS-Ⅱ.

[0020] Fig. 9 (a)-(f) are the spatial spectra of 11 targets estimated by different coprime arrays; Fig. 9 (a) Estimation of the spatial spectrum of 11 targets for CA, Fig. 9 (b) ACA estimates the spatial spectra of 11 targets. Fig. 9 (c) The spatial spectrum of 11 targets estimated by RSRCA-I, Fig. 9 (d) The spatial spectrum of 11 targets estimated by RSRCA-Ⅱ, Fig. 9 (e) Estimation of the spatial spectrum of 11 targets for CATRS-I, Fig. 9 (f) Spatial spectra of 11 targets estimated by CATRS-Ⅱ. Fig.10 (a)-(b) are the RMSE comparison results of DOA estimation of different coprime arrays; Fig.10 (a) shows the change of RMSE with signal-to-noise ratio. Fig.10 (b) shows the change of RMSE with the number of snapshots. DETAILED DESCRIPTION

[0021] In the description of the present invention, it is also necessary to explain that, unless otherwise clearly specified and limited, these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art make various changes or modifications to the present invention, and these equivalent forms also fall within the scope limited by the claims attached to the application.

[0022] Based on CA and redundant virtual array element analysis, the present invention proposes two coprime arrays with translated subarrays (CATrS) by moving subarrays. The two CAtrS array structures proposed in the present invention can reduce the mutual coupling of array elements while improving the continuous degree of freedom and degree of freedom, thereby improving the DOA estimation performance.

[0023] 1. Signal model of existing coprime arrays

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

[0025] Assume that K far-field, narrow-band, uncorrelated signal sources are emitted from the direction {θ k ∈[-π / 2,π / 2)|k=1,2,…,K} is incident on a sparse array with a total number of array elements of T, and the array element position set is P={p i |i=1,2,…,T}. Under the influence of array mutual coupling, the array output at the lth sampling snapshot is expressed as:

[0026]

[0027] In the formula, A=[a(θ1),a(θ2),…,a(θ K )] represents the physical array flow matrix, and represents the steering vector of the kth signal; s(l) = [s1(l), s2(l), …, s K (l)] T and n(l)=[n1(l),n2(l),…,n T (l)] T They represent the signal vector and Gaussian white noise vector respectively, and the two are unrelated; 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 represents 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 the array mutual coupling effect can be quantified by the coupling leakage as follows:

[0031]

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

[0033] According to formula (1), the theoretical covariance matrix of the array output is:

[0034]

[0035] In the formula, represents the source covariance matrix, and is the power of the kth source; σ 2 Represents the noise power.

[0036] In actual processing, the theoretical covariance matrix cannot be obtained, but it can be approximately estimated by the sample covariance matrix:

[0037]

[0038] Where L represents the total number of sampled snapshots.

[0039] The vectorized covariance matrix gives the following virtual domain signal measurement model:

[0040]

[0041] In the formula, represents the equivalent virtual signal vector composed of the source power; in represents a column vector whose i-th element is 1 and the rest of the elements are 0; represents the virtual array flow matrix, and Obviously, the virtual array element position corresponding to Β can be given by the following differential matrix:

[0042]

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

[0044] The differential matrix of a sparse array is defined as follows:

[0045] Definition 1 (Degree of Freedom) The number of different elements in the differential matrix D is defined as the degree of freedom of the sparse array P, denoted by |D|.

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

[0047] Definition 3 (hole) Assume that the shortest ULA containing D is Then for any integer when and hour, are called holes in D.

[0048] Definition 4 (weight) is the position of virtual array elements that can be generated in the differential common array The number of physical array element pairs is defined as Weight

[0049]

[0050] Combining with formula (2), we can see that the weight can reflect the strength of the mutual coupling effect. When comparing array mutual coupling, the first three weights are usually considered, namely ω(1), ω(2) and ω(3). When There are redundant virtual array elements.

[0051] 2. Redundancy analysis of mutually prime arrays

[0052] like Figure 1 As shown in FIG, CA is composed of a pair of sparse sub-arrays whose number of array elements and array element spacing satisfy the coprime relationship. The number of array elements and array element spacing of one sub-array are N and Md respectively, and the number of array elements and array element spacing of the other sub-array are M and Nd, where gcd(M,N)=1 and M<N, and gcd(M,N)=1 indicates that M and N are coprime. Since the array arrangement satisfies the coprime property, except for the reference array element, the remaining array elements do not overlap. Therefore, CA contains a total of T=M+N-1 array elements, and its position set is:

[0053]

[0054] Combining formula (7), we can see that the differential matrix of CA is:

[0055]

[0056] In the formula, and represents the difference set of two sub-matrices, and represents the mutual difference set of two sub-matrices, which can be specifically expressed as:

[0057]

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

[0059]

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

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

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

[0063] When one of the following conditions is met, formula (13) is established:

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

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

[0066] This situation is true when (m1,m2) is 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] From the above analysis, it can be seen that there are a large number of redundant virtual array elements in the differential co-array of CA, which limits the improvement of the degree of freedom. Therefore, the present invention further improves it and proposes the following sub-array shifted coprime array.

[0068] 3. Subarray displacement coprime array

[0069] Generally speaking, given the number of physical array elements, the fewer redundant virtual array elements in a sparse array, the higher its degree of freedom. Therefore, reducing redundant virtual array elements in CA can effectively improve the degree of freedom. Based on this idea, the present invention appropriately shifts the subarrays of CA to obtain two subarray shifted coprime arrays. By optimizing the array deployment, the number of redundant virtual array elements is reduced, thereby improving the array degree of freedom and suppressing the mutual coupling of array elements.

[0070] (1) CATrS-Ⅰ

[0071] like Figure 2 As shown in the figure, the last N-1 array elements of the sub-array with a spacing of Md in CA are shifted to the right by a distance of αNd to form a CATrS-I structure. At this time, the set of physical array element positions can be expressed as:

[0072]

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

[0074] Combining formula (7), we can see that the differential matrix of CATrS-Ⅰ is:

[0075]

[0076] In the formula,

[0077]

[0078] First, analyze the range of α. Since the sub-matrix is ​​shifted to the right, α≥1. The redundant virtual array element positions in are:

[0079]

[0080] In the formula,

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

[0082]

[0083] Obviously, when Take any one of (1,N-1),(2,N-2),…,(N-1,1), and When any one of (2α+1,M-1), (2α+2,M-2), ..., (M-1,2α+1) is selected, equation (18) holds. Compared with (m1,m2) in equation (13), The number of possible values ​​is reduced by 2α, and in order for equation (18) to hold, 2α≤M-1, that is, α≤(M-1) / 2. In summary, 1≤α≤(M-1) / 2.

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

[0085] Proof: Due to Therefore, taking the former as an example, we use the reverse method to prove it.

[0086] definition and Is a collection Any two elements in

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

[0088]

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

[0090] Proposition 1: Differential Common Array D CATrS-Ⅰ The number of different virtual array elements in 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 Formula (15) can be simplified as:

[0092]

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

[0094] Furthermore, the set D CATrS-Ⅰ There are:

[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] Redundant virtual array elements.

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

[0099] Proof: Set can be restated as:

[0100]

[0101] because but:

[0102]

[0103] Similarly

[0104]

[0105] therefore,

[0106]

[0107] According to the prior art, the collection The positive holes in are located at {aM+bN|a≥0,b≥1}, so The positive holes in are located at {αN+aM+bN|a≥0,b≥1}. Just enough to fill the The hole at CATrS-Ⅰ The first positive hole in Therefore, we can know that D CATrS-Ⅰ The positive hole in H CATrS-Ⅰ =αN+xM+yN, where x≥1, y≥1. According to the symmetry, the differential matrix D CATrS-Ⅰ The hole in H CATrS-Ⅰ ={±(αN+xM+yN)|x≥1,y≥1}.

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

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

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

[0111] Proposition 4: When α = (M-1) / 2, the differential matrix 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 the subarray shift, 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 above virtual array element positions, D CATrS-Ⅰ There are still remaining:

[0116] (MN+M+(2α+1)N-2(α+1))-2(M-1+N-1)-1=(M-1)(N-1)+2α(N-1) (27) The virtual array element positions are not assigned 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] It can be seen from equations (27) and (28) that when α = (M-1) / 2, the weights of the remaining virtual array elements are all 1.

[0119] From the proof process of Proposition 4, we can see that when 1≤α≤(M-1) / 2, the weights 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, the present invention shifts the last M-1 array elements of the sub-array with a spacing of Nd in CA to the right by a distance of βMd to form a CATrS-Ⅱ structure. At this time, the physical array element position set can be expressed as:

[0122]

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

[0124] Combining formula (7), we can see that the differential matrix of CATrS-Ⅱ is:

[0125]

[0126] In the formula,

[0127]

[0128] Differential Common Array D CATrS-Ⅱ The following properties and propositions hold.

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

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

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

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

[0133] Proposition 8 When β = (N-1) / 2, the differential matrix 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 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 ​​of M and N are:

[0140]

[0141] Finally, the solution of formula (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 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] Next, 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 array element mutual coupling. Table 1 summarizes the closed analytical expressions of the maximum continuous degrees of freedom and maximum degrees of freedom of different coprime arrays under the optimal array configuration parameters. Since ACA is difficult to solve 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 of the above six coprime arrays are further plotted. Figure 4 It can be seen that under the same number of physical array elements, the two coprime arrays designed by the present invention can obtain more continuous degrees of freedom and degrees of freedom than CA, ACA, RSRCA-Ⅰ and RSRCA-Ⅱ. Table 2 summarizes the expressions of 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 the maximum value respectively. It can be seen that the ACA constructed by expanding the number of subarrays and the RSRCA-Ⅰ and RSRCA-Ⅱ constructed by repositioning the reference array element positions have limited reduction in the number of redundant virtual array elements (especially it is difficult to reduce the weights corresponding to small lags), and therefore have comparable mutual coupling with CA; and the two coprime arrays designed by the present invention destroy the conditions for the generation of redundant virtual array elements in the subarray mutual difference concentration by moving a certain subarray as a whole to an appropriate position, effectively reducing the number of redundant virtual array elements (significantly reducing the weights corresponding to small lags), and can reduce the mutual coupling of array elements to a large extent. In addition, Figure 5The curve of coupling leakage versus the number of physical array elements depicted in the figure clearly shows that the two coprime arrays designed in the present invention have lower element mutual coupling than CA, ACA, RSRCA-Ⅰ and RSRCA-Ⅱ. The reason is that the ACA constructed by expanding the number of subarrays and the RSRCA-Ⅰ and RSRCA-Ⅱ constructed by moving the reference array element position have limited reduction in the number of redundant virtual array elements (especially it is difficult to reduce the weight of small lags), and therefore have comparable mutual coupling with CA; while the coprime array designed in the present invention destroys the conditions for generating redundant virtual array elements in the subarray mutual difference concentration by moving a certain subarray as a whole to an appropriate position, thereby significantly reducing the number of redundant virtual array elements and being able to reduce mutual coupling to a large extent.

[0150] Table 1 The best array arrangement, maximum continuous degree of freedom and maximum degree of freedom of related coprime arrays

[0151]

[0152]

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

[0154]

[0155] Example

[0156] In order to verify the superior performance of the two coprime arrays proposed in the present invention in terms of array properties and DOA estimation, the following simulation experiments are carried out.

[0157] Assume that the total number of physical array elements is 10, then the configuration parameters of CA, RSRCA-Ⅰ, RSRCA-Ⅱ, CATrS-Ⅰ and CATrS-Ⅱ are M=5 and N=6, and the configuration parameters of ACA are and

[0158] 1. Array properties

[0159] After determining the array element positions according to the configuration parameters, the continuous degrees of freedom, degrees of freedom, the first three weights and coupling leakage of each coprime array are calculated, and the results are shown in Table 3. Figure 6 The differential co-arrays of different coprime arrays are given. It can be seen that compared with the comparative coprime arrays, the CATrS-Ⅰ and CATrS-Ⅱ designed by the present invention can not only improve the continuous degree of freedom and degree of freedom, but also reduce the weight of virtual array elements and significantly suppress the mutual coupling of array elements.

[0160] Figure 7The mutual coupling matrix element mapping diagrams of different coprime arrays are further shown. The color depth of the off-diagonal components reflects that CATrS-Ⅰ and CATrS-Ⅱ have fewer small-pitch array elements than the other coprime arrays, so the mutual coupling is lower. Table 3 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 co-array MUSIC algorithm is used to compare the spatial spectrum estimation performance of different coprime arrays. Assuming that 9 targets are evenly distributed between -30° and 30°, the signal-to-noise ratio (SNR) and the number of sampling snapshots are set to 0dB and 500 respectively, the estimation results are shown in Figure 8 It can be seen that, except for CA, the other five coprime arrays can accurately estimate the above nine targets. This is because the continuous degrees of freedom of CA are only 21, and the maximum number of targets that can be estimated in theory is 10. Due to the limited number of sampling snapshots and the low continuous degrees of freedom, the estimation results are seriously deviated from the true angle of the target.

[0164] Then, the number of targets is increased to 11, and the other simulation parameters remain unchanged. The estimation results are shown in Fig. 9 . Obviously, CA cannot estimate 11 targets due to its low continuous degrees of freedom. The spectral peaks of ACA, RSRCA-Ⅰ and RSRCA-Ⅱ are obviously attenuated, and the estimation deviation is large, while CATrS-Ⅰ and CATrS-Ⅱ can still accurately estimate the above 11 targets. From the perspective of continuous degrees of freedom, ACA, RSRCA-Ⅰ and RSRCA-Ⅱ are fully capable of estimating 11 targets, but the strong mutual coupling of array elements makes it impossible to obtain accurate estimation results. The CATrS-Ⅰ and CATrS-Ⅱ designed in the present invention not only increase the number of continuous degrees of freedom, but also can significantly suppress the mutual coupling of array elements, thereby obtaining 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 of six coprime arrays under different conditions. Assuming that the four targets are located at -20°, -8°, 5° and 10°, the number of Monte Carlo experiments is set to 500. Fig.10 (a) The RMSE versus SNR curve is plotted, where the number of sampling snapshots is 500; Fig.10 (b) shows the variation of RMSE with the number of sampled snapshots, where SNR = -5dB. Fig.10 It can be seen that with the increase of SNR and the number of sampling snapshots, the estimation errors of each coprime array show a downward trend, but due to the low 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 maintained 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 array element mutual coupling, and their estimation errors are always smaller than the compared coprime arrays under the same conditions. Therefore, the two coprime arrays designed by the present invention have better DOA estimation performance. In addition, under the assumption that the array element parameters are M=5 and N=6, CATrS-Ⅱ cannot obtain the optimal array configuration, so the estimation performance is weaker than the CATrS-Ⅰ structure. Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A sparse array design method based on coprime array redundancy analysis, characterized in that: The following steps are involved: Step 1: Based on the signal model of coprime array, redundant virtual array element analysis is performed on the traditional coprime array; Step 2: Based on the analysis results of step 1, two shifted coprime arrays are formed by moving the subarrays, namely the CATrS-Ⅰ array structure and the CATrS-Ⅱ array structure; Step 3: Use the CATrS-Ⅰ array structure or the CATrS-Ⅱ array structure to estimate DOA.

2. The sparse array design method based on coprime array redundancy analysis according to claim 1, characterized in that: The CATrS-I array structure is obtained by translating the last N-1 array elements in a sub-array with a spacing of Md in a coprime array to the right as a whole by a distance of αNd, wherein Md represents the array element spacing of a sub-array with N array elements, Nd represents the array element spacing of a sub-array with M array elements, M and N are coprime, α is an integer and 1≤α≤(M-1) / 2.

3. The sparse array design method based on coprime array redundancy analysis according to claim 2, characterized in that: The CATrS-Ⅱ array structure is obtained by translating the last M-1 array elements in the subarray with a spacing of Nd in the coprime array to the right by a distance of βMd, where β is an integer and 1≤β≤(N-1) / 2.

4. The sparse array design method based on coprime array redundancy analysis according to claim 2, characterized in that: The physical array element position set of the CATrS-Ⅰ array structure is: in, represents the first subarray in the CATrS-Ⅰ array structure, represents the second subarray in the CATrS-Ⅰ array structure; Moreover, the differential common array D of the CATrS-Ⅰ array structure CATrS-Ⅰ for: In the formula, Among them, ! represents an operation symbol and 5. The sparse array design method based on coprime array redundancy analysis according to claim 4, characterized in that: The differential co-array of the CATrS-Ⅰ array structure has the following properties: 1) and There are M(N-1) different virtual array elements in each; 2) D CATrS-Ⅰ The number of different virtual array elements in is MN+M+(2α+1)N-2(α+1), and D CATrS-Ⅰ There is M 2 +N 2 +MN-3M-(2α+3)N+2(α+1)+1 redundant virtual array elements; 3) D CATrS-Ⅰ The hole in H CATrS-Ⅰ ={±(αN+xM+yN)x≥1,y≥1}; 4) D CATrS-Ⅰ The range of continuous virtual array elements in is <-αN-M-N+1,αN+M+N-1>; 5) When α = (M-1) / 2, the differential matrix D CATrS-Ⅰ The weight expression is: Among them, i∈<1,N-1>, j∈<1,M-1>, x≥1, y≥1.

6. The sparse array design method based on coprime array redundancy analysis according to claim 3, characterized in that: The physical array element position set of the CATrS-Ⅱ array structure is: in, represents the first subarray in the CATrS-Ⅱ array structure, Represents the second subarray in the CATrS-Ⅱ array structure; Moreover, the differential common array D of the CATrS-Ⅱ array structure CATrS-Ⅱ for: In the formula, 7. The sparse array design method based on coprime array redundancy analysis according to claim 3, characterized in that: The differential common array of CATrS-Ⅱ array structure has the following properties: 1) and There are N(M-1) different virtual array elements in each; 2) D CATrS-Ⅱ The number of different virtual array elements in is MN+N+(2β+1)M-2(β+1), and D CATrS-Ⅱ There exists M2+N 2 +MN-3N-(2β+3)M+2(β+1)+1 redundant virtual array elements; 3) D CATrS-Ⅱ The hole in H CATrS-Ⅱ ={±(βM+xM+yN)x≥1,y≥1}; 4) D CATrS-Ⅱ The range of continuous virtual array elements in is <-βM-M-N+1,βM+M+N-1>; 5) When β = (N-1) / 2, the differential matrix D CATrS-Ⅱ The weight expression is: Among them, i∈<1,M-1>, j∈<1,N-1>, x≥1, y≥1.

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