Sparse array arrangement method for direction of arrival estimation

Abstract

The invention belongs to the technical field of array signal processing, and particularly relates to a sparse array arrangement method for direction of arrival estimation, which comprises the following steps: S1, determining a co-prime number pair (M, N), selecting two co-prime positive integers M and N, and constructing an extended co-prime array structure; s2, determining the optimal displacement, maximizing the center degree of freedom (CDoF) of the shifted array, avoiding the sub-arrays from sharing array elements, traversing the L value for different co-prime pairs (M, N), calculating the continuous part of the second-order virtual differential common array center, selecting the optimal displacement capable of maximizing the center degree of freedom (CDoF), and giving the corresponding optimal displacement expression; and S3, determining a displacement distance according to the derived optimal displacement expression, and constructing a new array structure. Under the condition that the number of physical sensors is the same, the center freedom degree of the array can be effectively improved, the space utilization rate is increased, the coupling effect of the array is weakened, and therefore the DOA estimation performance is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of array signal processing, and particularly to a sparse array arrangement method for direction of arrival (DOA) estimation. Background Art

[0002] Array signal processing, as an important research branch in the field of modern signal processing, arranges several sensors according to a certain topological structure to receive and process incoming signals, achieving objectives such as signal detection, parameter estimation, and tracking and positioning. The DOA estimation algorithm based on a uniform array requires that the element spacing does not exceed half of the wavelength of the incident signal. For a general uniform linear array, only by increasing the number of elements can a larger array be obtained, and it cannot perform direction finding on signals exceeding the number of elements. The non-uniformity of the sparse array overcomes this difficulty. With the same number of elements, due to its larger spacing, it can generate a larger central degree of freedom of the array, improve the continuous degree of freedom of the array, weaken the mutual coupling effect, and achieve DOA estimation of the incident signal, with improved estimation performance. However, when performing DOA estimation on incident signals based on a coprime array, due to the relatively small continuous degree of freedom of the array and the strong mutual coupling effect between elements, the estimation accuracy of DOA is not high. Therefore, aiming at the problems of low continuous degree of freedom of the sparse coprime array structure, low space utilization rate of the array structure, and strong coupling effect, a sparse arrangement method of a coprime array for direction of arrival estimation is proposed. Summary of the Invention

[0003] (1) Technical Problems to be Solved

[0004] In view of the deficiencies of the prior art, the present invention provides a sparse array arrangement method for direction of arrival estimation, which solves the problems raised in the above background art; without increasing the number of physical elements, it increases the virtual array aperture and the continuous degree of freedom of the array, enhances the space utilization rate of the array structure, reduces the coupling effect, and improves the DOA estimation performance.

[0005] (2) Technical Solutions

[0006] The present invention specifically adopts the following technical solutions to achieve the above objectives:

[0007] A sparse array arrangement method for direction of arrival estimation, comprising the following steps:

[0008] S1: Determine the pair of relatively prime numbers (M, N), select two relatively prime positive integers M and N, and construct an extended coprime array structure;

[0009] S2: Determine the optimal displacement to maximize the central degrees of freedom (CDoF) of the shifted array while avoiding sub - arrays sharing elements. For different pairs of relatively prime numbers (M, N), traverse the value of L, calculate the continuous part of the center of the second - order virtual difference co - array, select the optimal displacement that can maximize the central degrees of freedom (CDoF), and give its corresponding optimal displacement expression;

[0010] S3: Determine the shift distance according to the derived optimal displacement expression and construct a new array structure;

[0011] S4: Analyze the central degrees of freedom (CDoF) of the shifted array structure and the specific distribution of holes in the difference array;

[0012] S5: Under the coupled model, use the spatial spectrum estimation algorithm for direction - of - arrival estimation;

[0013] S6: Compare the direction - of - arrival estimation performance of the shifted co - prime array with other co - prime array structures under both non - coupled and coupled models.

[0014] Furthermore, the specific content of S1 is as follows: Select two relatively prime integers M and N, where N > M, and construct an extended co - prime array structure;

[0015] The element positions of sub - array 1 of the extended co - prime array: S1 = {Mn|n ∈ [0, N - 1]};

[0016] The element positions of sub - array 2 of the extended co - prime array: S2 = {Nm|m ∈ [0, 2M - 1]};

[0017] The positions of all elements of the extended co - prime array can be expressed as:

[0018] S = S1 ∪ S2 = {Mn|n ∈ [0, N - 1]} ∪ {Nm|m ∈ [0, 2M - 1]}.

[0019] Furthermore, the specific content of S2 is as follows: After selecting the pair of relatively prime numbers (M, N), on the basis of the extended co - prime array structure, move the whole of sub - array 2 to the right; Considering different situations, after determining the values of M and N, traverse the value range of L through the algorithm, calculate the length of the continuous part of the center of the second - order virtual difference co - array, and use the displacement corresponding to its maximum value as the optimal shift distance; To apply to all possible pairs of relatively prime numbers (M, N), its corresponding mathematical expression of the optimal displacement is proposed to efficiently determine the best shift scheme;

[0020] The specific steps are as follows: After determining the values of M and N, traverse the value of L in the range of (0, N(M - 1)), extend it to the second - order virtual domain, construct the second - order virtual difference co - array structure, and obtain the set D of element positions of the virtual difference co - array diff, denoted as D diff = D s ∪D c ;

[0021] D s = {M(n i - n j ), 0 ≤ n i , n j ≤ N - 1}

[0022] ∪{N(m i - m j ), 0 ≤ m i , m j ≤ 2M - 1}

[0023] = {±Mn, 0 ≤ n ≤ N - 1} ∪ {±Nm, 0 ≤ m ≤ 2M - 1}

[0024] D c = {Nm i - Mn j + L, 0 ≤ m i ≤ 2M - 1, 0 ≤ n j ≤ N - 1}

[0025] ∪{Mn j - Nm i - L, 0 ≤ m i ≤ 2M - 1, 0 ≤ n j ≤ N - 1}

[0026] = {±(Nm - Mn + L, 0 ≤ m ≤ 2M - 1, 0 ≤ n ≤ N - 1}

[0027] D s is the self - lag of two sub - arrays, and the corresponding mirror position is D s - ;

[0028] D c is the cross - lag of two sub - arrays, and the corresponding mirror position is D c - , where 0 ≤ n ≤ N - 1, 0 ≤ m ≤ 2M - 1, and the positive part of the difference array is

[0029] D diff + = (D s ∪D c ) +

[0030] Expand from the center of the array to both sides until a hole is encountered or a preset condition is reached, and record the maximum continuous expansion lengths on the left and right sides respectively. For each possible shift distance, repeat the above process to calculate the length of the central continuous part in each shift configuration; select the shift distance that maximizes the length of the central continuous part as the optimal displacement to maximize the central degree of freedom of the array.

[0031] According to different values of the relatively prime pair (M, N), the relatively prime pair (M, N) is divided into the following cases and the corresponding mathematical expressions for the optimal displacement are given: when N is odd, M is odd or even, when N is even and M is odd, and N > 2M, when N is even and M is odd, and M < N < 2M,

[0032] Furthermore, the specific content of S3 is: on the basis of expanding the relatively prime array, move the entire sub-array 2 to the right by the corresponding optimal displacement amount L s , and construct a shifted relatively prime array structure. At this time, there are no shared array elements among the two sub-arrays of the shifted relatively prime array;

[0033] The element positions of sub-array 1 of the shifted relatively prime array: S1 = {Mn | n ∈ [0, N - 1]};

[0034] The element positions of sub-array 2 of the shifted relatively prime array: S2 = {Nm + L s | m ∈ [0, 2M - 1]};

[0035] The positions of all elements of the shifted relatively prime array can be expressed as:

[0036] S shift = S1 ∪ S2 = {Mn | n ∈ [0, N - 1]} ∪ {Nm + L s | m ∈ [0, 2M - 1]};

[0037] In the above formula, the specific form of L s is selected according to the selected relatively prime number combination to obtain the corresponding optimal displacement amount.

[0038] Furthermore, the specific content of S4 is: analyze the central degree of freedom (CDoF) of the shifted relatively prime array and the specific distribution of holes in the difference array under different conditions;

[0039] Specific analysis for different cases of relatively prime pairs (M, N) is mainly divided into the following three cases:

[0040] Ⅰ: N is odd, M is odd or even;

[0041] Property 1: In the cross-lag set D cThere are 2MN unique integers in it;

[0042] Property two:

[0043] Property three: In D diff + the hole positions are: M, 2M, ···, ((N - 1) / 2)M;

[0044] Property four: The difference co - array D diff contains integers in the range of -(MN + M + L s - 1) ≤ l c ≤ MN + M + L s - 1;

[0045] Property five: The hole positions of the entire difference array D diff are {±(aM + bN + L s ), a ≥ 0, b > 0};

[0046] Ⅱ: When N is even, M is odd, and N > 2M;

[0047] Property one: In the cross - lag set D c there are 2MN unique integers;

[0048] Property two:

[0049] Property three: In D diff + the hole positions are: M, 2M, ···, ((N - 1) / 2)M;

[0050] Property four: The difference co - array D diff contains integers in the range of -(MN + M + L s - 1) ≤ l c ≤ MN + M + L s - 1;

[0051] Property five: The hole positions of the entire difference array D diff are {±(aM + bN + L s ), a ≥ 0, b > 0}

[0052] Ⅲ: When M is even, N is odd, and M < N < 2M;

[0053] Property one: In the cross - lag set D c there are 2MN unique integers;

[0054] Property two:

[0055] Property three: In D diff+ Inside, the hole positions are: M, 2M, ···, ((N - 1) / 2)M;

[0056] Property Four: Difference Coarray D diff Contains -(MN + M + L s -1) ≤ l c ≤ MN + M + L s Integers in the range of -1;

[0057] Property Five: The hole positions of the entire difference array D diff Are {±(aM + bN + L s ), a ≥ 0, b > 0}.

[0058] Furthermore, the specific content of S5 is: In the coupled state, that is, considering the coupling effect between array elements, use the spatial spectrum estimation algorithm for direction-of-arrival estimation;

[0059] The mutual coupling signal model of the shifted co-prime array is as follows:

[0060] y(t) = A(θ)Cs(t) + n(t)

[0061] Where A(θ) = [a(θ1), a(θ2), … a(θ k )] is the manifold matrix of the co-prime array, C is the coupling matrix, s(t) is the transmitted signal matrix, expressed as s(t) = [s(t1), s(t2), … s(t k )] Τ , and n(t) is Gaussian white noise;

[0062] The covariance matrix of the received signal y(t) of the shifted co-prime array is expressed as follows:

[0063]

[0064] In the formula, P k Is the power of the kth incident signal, a(θ k ) is the steering vector of the kth incident signal, E[·] is the expectation, (·) H Is the conjugate transpose operation, Represents the noise power; The sample covariance is used to replace the ideal covariance matrix, expressed as follows:

[0065]

[0066] y(t) H Is the conjugate transpose of y(t), Is the sample covariance matrix of the received signal, and T is the number of snapshots;

[0067] Perform eigenvalue decomposition on the covariance matrix to obtain the noise subspace and the signal subspace, which are expressed as follows:

[0068]

[0069] where \(a(\theta)\) is the array manifold vector, \(a H *(\theta)\) represents the conjugate transpose of \(a(\theta)\), and \(U n is the noise subspace; since the signal subspace and the noise subspace are orthogonal, spectral peak search is used for direction of arrival (DOA) estimation.

[0070] Furthermore, the specific content of S6 is as follows: in the non-coupled model, it is assumed that there is no mutual influence between array elements; while in the coupled model, the interaction between array elements is considered; under these two models, the DOA estimation performance of the shifted co-prime array is compared with that of other co-prime array structures.

[0071] (III) Advantageous Effects

[0072] Compared with the prior art, the present invention provides a sparse array arrangement method for direction of arrival estimation, which has the following advantageous effects:

[0073] 1. The present invention aims to improve the array performance by optimizing the layout between sub-arrays. According to different co-prime number pairs \((M, N)\), the mathematical expressions of the corresponding optimal displacement amounts are derived; and the central degrees of freedom (CDoF) of the shifted co-prime array and the specific distribution of holes in the difference array are analyzed in detail. Without increasing the number of physical array elements, the continuous degrees of freedom of the virtual array are increased, the space utilization efficiency of the array structure is improved, and the coupling effect between array elements is reduced, thereby improving the direction of arrival estimation (DOA) performance.

[0074] 2. The present invention first determines the optimal displacement amount according to different co-prime number pairs \((M, N)\), and appropriately shifts the sub-array 2 of the extended co-prime array to form an improved shifted co-prime array; then, it is extended to an augmented virtual array through the difference combination operation, during which the central degrees of freedom and the array space utilization rate are improved; under this model, the SS-MUSIC algorithm is used for angle estimation, and the estimation performance is evaluated; the present invention not only increases the continuous degrees of freedom of the array, but also effectively reduces the coupling effect between array elements, thereby improving the accuracy of DOA estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 is a schematic diagram of the shifted co-prime array constructed in the present invention;

[0076] Figure 2 is a schematic diagram of the non-negative part of the co-prime array constructed in the present invention;

[0077] Figure 3 Power spectrum estimation comparison images for DOA estimation of several different array structures of the present invention;

[0078] Figure 4 Simulation diagram of the relationship curve between the root mean square error of DOA estimation and the number of snapshots under the uncoupled model of the present invention;

[0079] Figure 5 Simulation diagram of the relationship curve between the root mean square error of DOA estimation and the signal-to-noise ratio under the uncoupled model of the present invention;

[0080] Figure 6 Simulation diagram of the relationship curve between the root mean square error of DOA estimation and the number of snapshots under the coupled model of the present invention;

[0081] Figure 7 Simulation diagram of the relationship curve between the root mean square error of DOA estimation and the signal-to-noise ratio under the coupled model of the present invention;

[0082] Figure 8 Flowchart of the method of the present invention. Detailed implementation manners

[0083] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0084] Embodiment

[0085] As Figure 1-8 shown, a sparse array arrangement method for direction of arrival estimation proposed in an embodiment of the present invention includes the following steps:

[0086] S1: Determine a pair of relatively prime numbers (M, N), select two relatively prime integers M and N, and N > M, and construct an extended co-prime array structure;

[0087] Element positions of sub-array one of the extended co-prime array: S1 = {Mn | n ∈ [0, N - 1]};

[0088] Element positions of sub-array two of the extended co-prime array: S2 = {Nm | m ∈ [0, 2M - 1]};

[0089] All element positions of the extended co-prime array can be expressed as:

[0090] S = S1 ∪ S2 = {Mn | n ∈ [0, N - 1]} ∪ {Nm | m ∈ [0, 2M - 1]}.

[0091] S2: After determining the pair of relatively prime numbers (M, N), based on the extended co-prime array structure, move the whole sub-array two to the right, and determine an optimal displacement to maximize the central degrees of freedom (CDoF) of the shifted array while avoiding element sharing between sub-arrays.

[0092] Considering different situations, after determining the values of M and N, traverse the value range of L through an algorithm, calculate the length of the continuous part of the center of the second-order virtual difference co-array, and use the displacement corresponding to its maximum value as the optimal shift distance; in order to be applicable to all possible pairs of relatively prime numbers (M, N), a set of optimal displacement expressions are proposed to efficiently determine the best shift scheme.

[0093] The specific steps are as follows: After determining the values of M and N, traverse the value of L within the range of (0, N(M - 1)), extend it to the second-order virtual domain, construct the second-order virtual difference co-array structure, perform difference combination, and obtain the set D of the element positions of the virtual difference co-array diff , denoted as D diff = D s ∪ D c

[0094] D s = {M(n i - n j ), 0 ≤ n i , n j ≤ N - 1}

[0095] ∪ {N(m i - m j ), 0 ≤ m i , m j ≤ 2M - 1}

[0096] = {±Mn, 0 ≤ n ≤ N - 1} ∪ {±Nm, 0 ≤ m ≤ 2M - 1}

[0097] D c = {Nm i - Mn j + L, 0 ≤ m i ≤ 2M - 1, 0 ≤ n j ≤ N - 1}

[0098] ∪ {Mn j - Nm i - L, 0 ≤ m i ≤ 2M - 1, 0 ≤ n j ≤ N - 1}

[0099] = {±(Nm - Mn + L, 0 ≤ m ≤ 2M - 1, 0 ≤ n ≤ N - 1}

[0100] D sis the self-lag of two sub-arrays, and the corresponding mirror position is D s - ;

[0101] D c is the cross-lag of two sub-arrays, and the corresponding mirror position is D c - , where 0 ≤ n ≤ N - 1, 0 ≤ m ≤ 2M - 1. The positive part of the difference array is

[0102] D diff + = (D s ∪ D c ) +

[0103] Expand from the center of the array to both sides until a hole is encountered or a preset condition is reached, and record the maximum continuous expansion lengths on both the left and right sides respectively. For each possible shift distance, repeat the above process to calculate the length of the central continuous part under each shift configuration; select the shift distance that maximizes the length of the central continuous part as the optimal displacement amount to maximize the central degree of freedom of the array.

[0104] According to different cases of the relatively prime pair (M, N), the relatively prime pair (M, N) is divided into the following cases and the corresponding mathematical expressions of the optimal displacement are given: When N is odd, M is odd or even, When N is even and M is odd, and N > 2M, When N is even and M is odd, and M < N < 2M,

[0105] S3: On the basis of expanding the relatively prime array, move the whole of sub-array 2 to the right by the optimal displacement amount L s , to construct the structure of the shifted relatively prime array. At this time, there are no shared array elements among the two sub-arrays of the shifted relatively prime array;

[0106] The element positions of sub-array 1 of the shifted relatively prime array: S1 = {Mn | n ∈ [0, N - 1]};

[0107] The element positions of sub-array 2 of the shifted relatively prime array: S2 = {Nm + L s | m ∈ [0, 2M - 1]};

[0108] All the element positions of the shifted relatively prime array can be expressed as:

[0109] S shift = S1 ∪ S2 = {Mn | n ∈ [0, N - 1]} ∪ {Nm + L s | m ∈ [0, 2M - 1]}

[0110] In the above formula, Ls For the specific form, select the corresponding optimal displacement according to the selected pair of relatively prime numbers.

[0111] S4: Analyze the central degrees of freedom (CDoF) of the shifted co-prime array and the specific distribution of holes in the difference array under different conditions;

[0112] The specific analysis for different pairs of relatively prime numbers (M, N) is mainly divided into the following three cases:

[0113] Ⅰ: N is odd, and M is odd or even;

[0114] Property 1: There are 2MN unique integers in the cross-lag set D c ;

[0115] Property 2: (The self-difference set is not a subset of the cross-difference set);

[0116] Property 3: In D diff + the hole positions are: M, 2M, ···, ((N - 1) / 2)M;

[0117] Property 4: The difference co-array D diff contains integers in the range of -(MN + M + L s - 1) ≤ l c ≤ MN + M + L s - 1;

[0118] Property 5: The hole positions of the entire difference array D diff are {±(aM + bN + L s ), a ≥ 0, b > 0}.

[0119] Ⅱ: When N is even and M is odd, and N > 2M;

[0120] Property 1: There are 2MN unique integers in the cross-lag set D c ;

[0121] Property 2: (The self-difference set is not a subset of the cross-difference set);

[0122] Property 3: In D diff + the hole positions are: M, 2M, ···, ((N / 2) - 1)M;

[0123] Property 4: The difference co-array D diff contains integers in the range of -(MN + M + L s - 1) ≤ l c ≤ MN + M + L s - 1;

[0124] Property 5: The hole positions of the entire difference matrix D diff are {±(aM + bN + L s ), a ≥ 0, b > 0}.

[0125] Ⅲ: When M is even and N is odd, and M < N < 2M;

[0126] Property 1: There are 2MN unique integers in the cross - lag set D c ;

[0127] Property 2: (The autocorrelation set is not a subset of the cross - correlation set);

[0128] Property 3: Inside D diff + the hole positions are: N, 2N, ···, (M - 1)N / 2;

[0129] Property 4: The difference co - matrix D diff contains integers in the range of -(MN + M + L s - 1) ≤ l c ≤ MN + M + L s - 1;

[0130] Property 5: The hole positions of the entire difference matrix D diff are {±(aM + bN + L s ), a ≥ 0, b > 0};

[0131] S5: In the coupled state, that is, considering the coupling effect between array elements, use the spatial spectrum estimation algorithm for direction - of - arrival estimation;

[0132] The mutual - coupling signal model of the shifted co - prime array is as follows:

[0133] y(t) = A(θ)Cs(t)+n(t)

[0134] where A(θ) = [a(θ1), a(θ2),…a(θ k )] is the manifold matrix of the co - prime array, C is the coupling matrix, s(t) is the transmitted signal matrix, expressed as s(t) = [s(t1), s(t2),…s(t k )] Τ , and n(t) is Gaussian white noise.

[0135] The covariance matrix of the received signal y(t) of the shifted co - prime array is expressed as follows:

[0136]

[0137] In the formula, P kis the power of the k-th incident signal, a(θ k ) is the steering vector of the k-th incident signal, E[·] is the expectation, (·) H is the conjugate transpose operation, represents the noise power; The sampling covariance is adopted to replace the ideal covariance matrix, which is expressed as follows:

[0138]

[0139] y(t) H is the conjugate transpose of y(t), is the sampling covariance matrix of the received signal, and T is the number of snapshots;

[0140] The covariance matrix is eigen-decomposed to obtain the noise subspace and the signal subspace, which are expressed as follows:

[0141]

[0142] where a(θ) is the array manifold vector, a H (θ) represents the conjugate transpose of a(θ), and U n is the noise subspace; Since the signal subspace and the noise subspace are orthogonal, spectral peak search is used for direction of arrival (DOA) estimation.

[0143] S6: In the non-coupled model, it is assumed that there is no mutual influence between array elements; while in the coupled model, the interaction between array elements is considered. The DOA estimation performance of the shifted co-prime array and other co-prime array structures is compared under these two models.

[0144] As Figure 1 shown, the number of physical array elements of sub-array 1 is N = 5, the number of physical array elements of sub-array 2 is 2M = 6, the distance unit is d = λ / 2, and sub-array 2 is shifted to the right by L s unit distances, where the optimal shift distance L s = 11, to construct an improved shifted co-prime array.

[0145] As Figure 2 shown, there are K = 22 mutually independent far-field narrowband signal sources incident on the shifted co-prime array in the spatial domain, the incident angles of the signal sources are [-45, 45], the number of snapshots is T = 500, and the signal-to-noise ratio is SNR = 10 dB.

[0146] As Figure 3As shown, in the simulation result graph, the abscissa represents the angle value and the ordinate represents the power spectrum amplitude value. By observing the image, it can be seen that the present invention can correctly estimate all angle values. Compared with array structures such as the augmented co-prime array (ACA), the sliding extended co-prime array (SECA), and the thinned co-prime array (TCA), the proposed shifted co-prime array structure has sharper spectral peaks and higher accuracy.

[0147] As Figure 4 and Figure 5 shown, for the simulation experiments of the relationship curves between the root mean square error of DOA estimation, the number of snapshots, and the signal-to-noise ratio, 200 Monte Carlo experiments are all adopted. The root mean square error (RMSE) is defined as follows:

[0148]

[0149] where K is the number of signal sources, Q is the number of Monte Carlo experiments, the estimated angle of the k-th signal in the q-th Monte Carlo experiment, θ k is the k-th angle value; the experimental average value with Q = 200 is taken as the final result.

[0150] The simulation experiment results are as Figure 4 shown. Under the uncoupled model, to explore the root mean square error performance at different numbers of snapshots, 10 angles uniformly distributed in the angle range [-45, 45] are selected, SNR = 10, and the number of snapshots range is set to [200:200:2000], and 200 Monte Carlo experiments are carried out. The RMSE of various array structures monotonically decreases with the increase of the number of snapshots (Snapshots). When the number of snapshots is greater than 1600, the RMSE of each algorithm tends to be stable. The RMSE of the shifted co-prime array (Proposed) is always lower than that of other structures, that is, the RMSE performance of the shifted co-prime array is higher than that of other array structures.

[0151] The simulation experiment results are as Figure 5 shown. Under the uncoupled model, to explore the root mean square error performance at different signal-to-noise ratios, 10 angles uniformly distributed in the angle range [-45, 45] are selected, the number of snapshots is fixed at 500, and the signal-to-noise ratio range is [0:5:30], and 200 Monte Carlo experiments are carried out. This figure compares the root mean square error (RMSE) performance of various array structures at different signal-to-noise ratios (SNR). As the SNR increases from 0 dB to 30 dB, the RMSE of various array structures monotonically decreases and tends to be stable at high SNR. Throughout the SNR range, the RMSE of the shifted co-prime array structure is always the lowest, especially in the low SNR scenario, showing a more significant advantage compared with other structures.

[0152] The simulation experiment results are as follows Figure 6 shown. Under the coupled model, the root mean square error (RMSE) performance under different numbers of snapshots is studied. Ten target angles uniformly distributed within the [-45, 45] angular range are selected, the signal-to-noise ratio (SNR) is set to 10 dB, the number of snapshots ranges from [200:200:2200], and 200 Monte Carlo experiments are carried out. As the number of snapshots increases from 200 to 2200, the RMSE of several array structures continuously decreases and finally stabilizes; within the entire range, the RMSE of the shifted co-prime array (Proposed) is always lower than that of other array structures, and even when the number of snapshots is small, it is still significantly better than several other arrays.

[0153] The simulation experiment results are as follows Figure 7 shown. Under the coupled model, the root mean square error (RMSE) performance under different signal-to-noise ratios is studied; ten target angles uniformly distributed within the [-45, 45] angular range are selected, the number of snapshots is fixed at 500, the signal-to-noise ratio range is set to [0:5:30], and 200 Monte Carlo experiments are carried out. As the signal-to-noise ratio increases from 0 dB to 30 dB, the RMSE values of the six curves all show a stepwise decrease and finally stabilize. Compared with the augmented co-prime array (ACA), smoothed co-prime array (SECA), and thinned co-prime array (TCA), the shifted co-prime array has a larger continuous degree of freedom and a smaller coupling effect, resulting in a smaller RMSE error and better DOA estimation performance.

[0154] Finally, it should be noted that the above are only the preferred embodiments of the present invention and are not used to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, for those skilled in the art, they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A sparse array arrangement method for direction of arrival estimation, characterized in that It includes the following steps: S1: Determine the pair of relatively prime numbers (M, N), select two relatively prime positive integers M and N, and construct an extended co-prime array structure; S2: Determine the optimal displacement amount to maximize the central degrees of freedom (CDoF) of the shifted array while avoiding sub-arrays sharing array elements. When L < N(M - 1), for different pairs of relatively prime numbers (M, N), traverse L, calculate the continuous part of the center of the second-order virtual difference co-array, select the optimal displacement amount that can maximize the central degrees of freedom (CDoF), and give its corresponding optimal displacement expression; S3: According to the derived optimal displacement expression, determine the displacement distance and construct a shifted co-prime array structure; S4: Analyze the central degrees of freedom (CDoF) of the shifted array structure and the specific distribution of holes in the difference array; S5: Under the coupling model, use the spatial spectrum estimation algorithm for direction-of-arrival estimation; S6: Under both non-coupling and coupling models, compare the direction-of-arrival estimation performance of the shifted co-prime array with other co-prime array structures.

2. The sparse array arrangement method for direction-of-arrival estimation according to claim 1, characterized in that: The specific content of S1 is: Select two relatively prime integers M and N, where N > M, and construct an extended co-prime array structure; The element positions of sub-array one of the extended co-prime array: S1 = {Mn|n ∈ [0, N - 1]}; The element positions of sub-array two of the extended co-prime array: S2 = {Nm|m ∈ [0, 2M - 1]}; All the element positions of the extended co-prime array can be expressed as: S = S1 ∪ S2 = {Mn|n ∈ [0, N - 1]} ∪ {Nm|m ∈ [0, 2M - 1]}.

3. A sparse array arrangement method for direction-of-arrival estimation according to claim 2, characterized in that: The specific content of S2 is: After selecting the pair of relatively prime numbers (M, N), on the basis of the extended co-prime array structure, move sub-array two as a whole to the right; Considering different situations, after determining the values of M and N, traverse the value range of L through the algorithm, calculate the length of the continuous part of the center of the second-order virtual difference co-array, and use the displacement amount corresponding to its maximum value as the optimal displacement distance; In order to be applicable to all possible pairs of relatively prime numbers (M, N), its corresponding mathematical expression of the optimal displacement is proposed to efficiently determine the best displacement scheme; The specific steps are as follows: After determining the values of M and N, within the range of (0, N(M - 1)), traverse the values of L, extend it to the second-order virtual domain, construct the second-order virtual difference co-array structure, and obtain the set D of the array element positions of the virtual difference co-array diff , denoted as D diff = D s ∪ D c ; D s = {M(n i - n j ), 0 ≤ n i , n j ≤ N - 1} ∪{N(m i -m j ),0≤m i ,m j ≤2M - 1} = {±Mn, 0 ≤ n ≤ N - 1} ∪ {±Nm, 0 ≤ m ≤ 2M - 1} D c = {Nm i - Mn j + L, 0 ≤ m i ≤ 2M - 1, 0 ≤ n j ≤ N - 1} ∪{Mn j -Nm i -L, 0 ≤ m i ≤ 2M - 1, 0 ≤ n j ≤ N - 1} = {±(Nm - Mn + L, 0 ≤ m ≤ 2M - 1, 0 ≤ n ≤ N - 1} D s is the self-lag of the two sub-arrays, and the corresponding mirror position is D s - ; D c is the cross lag of two sub-arrays, and the corresponding mirror position is D c - , where 0 ≤ n ≤ N - 1, 0 ≤ m ≤ 2M - 1, and the positive part of the difference array is D diff + = (D s ∪ D c ) + Expand from the center of the array to both sides until holes are encountered or a preset condition is reached, and record the maximum continuous expansion lengths on both the left and right sides respectively. For each possible displacement distance, repeat the above process and calculate the length of the central continuous part for each displacement configuration; Select the displacement distance that makes the length of the central continuous part the largest as the optimal displacement amount to maximize the central degrees of freedom of the array; According to different values of the relatively prime pair (M, N), the relatively prime pair (M, N) is divided into the following cases and the mathematical expressions of their corresponding optimal displacements are given: when N is odd, M is odd or even, when N is even and M is odd, and N > 2M, when N is even and M is odd, and M < N < 2M, 4. A sparse array arrangement method for direction-of-arrival estimation according to claim 3, characterized in that: The specific content of S3 is as follows: Based on the extended co-prime array, the entire sub-array 2 is shifted to the right by the corresponding optimal displacement amount L s , and a shifted co-prime array structure is constructed. At this time, there are no shared array elements among the two sub-arrays of the shifted co-prime array; The element positions of sub-array one of the shifted co-prime array: S1 = {Mn|n ∈ [0, N - 1]}; Array element positions of array two of the shifted co-prime array: S2 = {Nm + L s | m ∈ [0, 2M - 1]}; All the element positions of the shifted co-prime array can be expressed as: S shift = S1 ∪ S2 = {Mn | n ∈ [0, N - 1]} ∪ {Nm + L s | m ∈ [0, 2M - 1]}; In the above formula, L s In its specific form, according to the selected combination of relatively prime numbers, select the corresponding optimal displacement amount.

5. A sparse array arrangement method for direction of arrival estimation according to claim 4, characterized in that: The specific content of S4 is: Analyze the central degrees of freedom (CDoF) of the shifted co-prime array and the specific distribution of holes in the difference array under different situations; The specific analysis for different pairs of relatively prime numbers (M, N) mainly falls into the following three situations: Ⅰ: N is odd, and M is odd or even; Property 1: In the cross-lagged set D c there are 2MN unique integers; Property 2: Property 3: Inside D diff + The positions of the holes are: M, 2M, ···, ((N - 1) / 2)M; Property 4: Difference coarray D diff contains -(MN + M + L s - 1) ≤ l c ≤ MN + M + L s - 1 range of integers; Property 5: The hole positions of the entire difference array D diff are {±(aM + bN + L s ), a ≥ 0, b > 0}; Ⅱ: When N is even and M is odd, and N > 2M; Property 1: In the cross-lagged set D c There are 2MN unique integers; Property 2: Property Three: Inside D diff + the positions of the holes are: M, 2M, ···, ((N - 1) / 2)M; Property 4: Difference coarray D diff contains -(MN + M + L s -1) ≤ l c ≤ MN + M + L s -1 range of integers; Property 5: The hole positions of the entire difference array D diff are {±(aM + bN + L s ), a ≥ 0, b > 0} Ⅲ: When M is even and N is odd, and M < N < 2M; Property 1: In the cross-lagged set D c there exist 2MN unique integers; Property 2: Property Three: In D diff + the positions of the holes are: M, 2M, ···, ((N - 1) / 2)M; Property 4: Difference coarray D diff Contains -(MN + M + L s -1) ≤ l c ≤ MN + M + L s -1; Property 5: The hole positions of the entire difference array D diff are {±(aM + bN + L s ), a ≥ 0, b > 0}.

6. A sparse array arrangement method for direction-of-arrival estimation according to claim 5, characterized in that: The specific content of S5 is as follows: In the coupled state, that is, considering the coupling effect between array elements, the spatial spectrum estimation algorithm is used for direction-of-arrival estimation; The mutual coupling signal model of the shifted co-prime array is as follows: y(t) = A(θ)Cs(t) + n(t) where \(A(\theta)=[a(\theta_1),a(\theta_2),\cdots,a(\theta k )]\) is the manifold matrix of the coprime array, \(C\) is the coupling matrix, \(s(t)\) is the transmitted signal matrix, expressed as \(s(t)=[s(t_1),s(t_2),\cdots,s(t k )] Τ , and \(n(t)\) is white Gaussian noise; The covariance matrix of the received signal y(t) of the shifted co-prime array is expressed as follows: where P k is the power of the k-th incident signal, a(θ k ) is the steering vector of the k-th incident signal, E[·] is the expectation, (·) H is the conjugate transpose operation, represents the noise power; The sampling covariance is adopted to replace the ideal covariance matrix, which is expressed as follows: y(t) H is the conjugate transpose of y(t), is the sampling covariance matrix of the received signal, and T is the number of snapshots; Performing eigenvalue decomposition on the covariance matrix to obtain the noise subspace and the noise subspace, which are expressed as follows: where \(a(\theta)\) is the array manifold vector, \(a\) H \((\theta)\) represents the conjugate transpose of \(a(\theta)\), and \(U\) n is the noise subspace; since the signal subspace and the noise subspace are orthogonal, spectral peak search is used for direction of arrival (DOA) estimation.

7. A sparse array arrangement method for direction of arrival estimation according to claim 6, characterized in that: The specific content of S6 is as follows: In the non-coupled model, it is assumed that there is no mutual influence between array elements; while in the coupled model, the mutual interaction between array elements is considered; Under these two models, the direction-of-arrival estimation performance of the shifted co-prime array and other co-prime array structures is compared.

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