Method for two-dimensional direction of arrival estimation of space-time far-field source based on three parallel relatively prime arrays

Abstract

This invention discloses a two-dimensional direction-of-arrival (DOA) estimation method for a space-time far-field source based on a three-parallel coprime array. The method constructs virtual received data by cross-correlating the far-field signal data received by any two elements of the three subarrays in the time domain. By changing the physical positions of the two elements, the time correlation function of the far-field signal data received by the two elements is calculated. Then, pseudo-snapshot sampling is performed at uniform time intervals to obtain the first and second virtual received data. Cross-correlation of the first and second virtual received data yields the covariance matrix. After vectorization of the covariance matrix, a virtual output vector is obtained. A second-order Toeplitz matrix is ​​obtained from the virtual output vector, and the azimuth and elevation angles are obtained using Vandermonde decomposition. The advantages are that the azimuth and elevation angles of the far-field signal source are estimated by constructing virtual received data based on second-order correlation operations and combining the space-time information of multi-element received data, and the estimation accuracy is high.

Description

Technical Field

[0001] This invention relates to a two-dimensional direction-of-arrival (DOA) estimation technique for far-field signal sources, and more particularly to a two-dimensional DOA estimation method for space-time far-field sources based on a three-parallel coprime array. Background Technology

[0002] Two-dimensional far-field signal source direction-of-arrival (DOA) estimation is an important aspect of array signal processing, widely applied in radar, wireless communication, and other fields. Traditional ODA methods often utilize uniform linear arrays and employ conventional subspace-based algorithms to estimate the two-dimensional angles. However, using traditional uniform linear arrays and subspace-based algorithms requires the number of far-field signal sources to be less than the number of physical array elements. In contrast, non-uniform linear arrays can estimate more far-field signal sources with the same number of physical array elements. In recent years, many ODA algorithms based on non-uniform linear arrays have been proposed, such as those based on L-shaped nested arrays or double parallel coprime arrays. However, these algorithms only utilize the spatial information of the received signal, resulting in insufficient estimation accuracy. Combining the timing information of the received signal can improve the estimation performance of the algorithm. However, most two-dimensional far-field signal source direction-of-arrival estimation algorithms based on non-uniform linear arrays and the spatiotemporal characteristics of the received signal only utilize the spatiotemporal characteristics of the received signal of some array elements. Currently, no algorithm has been proposed that utilizes the spatiotemporal characteristics of the received signal of multiple array elements in the scenario of two-dimensional far-field signal source direction-of-arrival estimation based on non-uniform linear arrays. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a two-dimensional direction of arrival estimation method for a space-time far-field source based on a three-parallel coprime array. It uses second-order correlation operation and combines the space-time information of multi-element received data to construct virtual received data to estimate the azimuth and elevation angles of the far-field signal source, and the estimation accuracy is high.

[0004] The technical solution adopted by this invention to solve the above-mentioned technical problems is as follows: a two-dimensional direction-of-arrival estimation method for space-time far-field sources based on a three-parallel coprime array, characterized by the following steps:

[0005] Step 1: Establish a three-dimensional far-field spatial propagation model based on three parallel coprime arrays. This model consists of three mutually parallel subarrays parallel to the y-axis of the three-dimensional space. Each subarray is a coprime array with parameters (M, N) in a non-uniform linear array. The first subarray is located between the second and third subarrays with a spacing of d. The first subarray has a total of 2M elements with an element spacing of Nd. The second and third subarrays each have a total of N elements with an element spacing of Md. Under this model, the set of positions of all elements in the three-dimensional space of the three subarrays is represented as: There are K positions, respectively. Uncorrelated narrowband far-field signal sources arrive at the model; where M and N are coprime and 1 <M<N, λ represents the wavelength of the narrowband far-field signal source, 1≤m≤2M, 1≤n≤N, K≥1, 1≤k≤K, θ k This represents the elevation angle of the k-th narrowband far-field signal source from the reference point. θ represents the azimuth angle from the k-th narrowband far-field signal source to the reference point, where the reference point is the origin of the three-dimensional coordinate system. k and This constitutes a pair of two-dimensional direction of arrival angles;

[0006] Step 2: After K narrowband far-field signal sources arrive at the three-dimensional far-field spatial propagation model based on a three-parallel coprime array, the electrical angle between the k-th narrowband far-field signal source and the y-axis of the three-dimensional space, i.e., the angle of arrival, is denoted as α. k Let β be the electrical angle between the k-th narrowband far-field signal source and the x-axis in three-dimensional space, i.e., the angle of arrival. k The electric angle, azimuth angle, and elevation angle have the following equivalent relationship: Then, the far-field signal data received by the first, second, and third subarrays are further denoted as z1(t), z2(t), and z3(t), respectively, where z1(t) = A1s(t) + n1(t), z2(t) = A2Φs(t) + n2(t), and z3(t) = A2Φ * s(t) + n3(t); where t is the time variable, s(t) represents the far-field signal source vector, and s(t) = [s1(t), s2(t), ..., s k (t),…,s K (t)] T s k (t) represents the k-th narrowband far-field signal source. σ sk Let ω represent the average power of the k-th narrowband far-field signal source, e be the natural constant, j be the imaginary part, and ω be the value of ω. k Let ω represent the angular frequency of the k-th narrowband far-field signal source. k =2πf k f kLet nk represent the frequency of the k-th narrowband far-field signal source, n1(t) represent the additive Gaussian noise vector of the far-field signal data z1(t) received by the first subarray, n2(t) represent the additive Gaussian noise vector of the far-field signal data z2(t) received by the second subarray, n3(t) represent the additive Gaussian noise vector of the far-field signal data z3(t) received by the third subarray, and A1 represent the manifold matrix of the first subarray, A1 = [a1(α1), a1(α2), ..., a1(αk), ..., a1(αk)]. K A2 represents the popular matrix of the second and third subarrays, A2 = [a2(α1), a2(α2), ..., a2(αk), ..., a2(αk)]. K )],a1(α k ) represents the first subarray with respect to α k The guide vector, a2(α k ) represents the relationship between α in the second and third subarrays. k The guide vector, Φ represents the rotation matrix. (·) T Represents the transpose of a vector or matrix, (·) * The conjugate of a vector or matrix is ​​represented by diag(·), which constructs a diagonal matrix.

[0007] Step 3: Construct virtual received data by cross-correling the far-field signal data received by any two elements from all elements of the three subarrays in the time domain. Assume these two elements are the v-th and c-th elements among all elements. Then, denote the constructed virtual received data as... Where τ represents the time delay, E{·} represents the statistical expectation function, and x v (t) represents the far-field signal data received by the v-th array element, x c (t+τ) represents the far-field signal data received by the c-th array element after a delay of τ. Let v represent the noise correlation function between the v-th array element and the c-th array element. n v (t) represents x v Additive Gaussian noise of (t), n c (t+τ) represents x c Additive Gaussian noise of (t+τ), additive Gaussian noise n v (t) and n c The average power of each (t+τ) is σ. n δ(·) represents the impulse function, δ(τ) represents the impulse function at time τ, and δ(vc) represents the impulse function at time (vc). This represents the time autocorrelation function of the k-th narrowband far-field signal source. s k (t+τ) represents the k-th narrowband far-field signal source after a delay of τ, and exp represents an exponential function with the natural constant e as the base. v ,q v (p) represents the position of the v-th element in three-dimensional space. c ,q c () represents the position of the c-th array element in three-dimensional space.

[0008] Step 4: Virtual data reception constructed in step 3 Based on this, let (p c ,q c If )=(0,0), then we have: Then let r1(τ) represent the first subarray and the position at (p c ,q c Let r2(τ) represent the time correlation function of the array element at position (p) = (0,0). c ,q c Let r3(τ) represent the time-dependent function of the array element at position (p) = (0,0). c ,q c The time dependence function of the array element ) = (0,0), let r s (τ) represents the time autocorrelation function of a narrowband far-field signal source. Then we have: r1(τ)=A1r s (τ), r2(τ)=A2Φr s (τ), r3(τ)=A2Φ*rs(τ); then according to the characteristics rs(τ)=(rs(-τ))*, there is: (r1(-τ)) * =A1 * r s (τ), (r3(-τ)) * =A2 * Φr s (τ); then use A1 - Indicates A1 * The last 2M-1 lines, use r1 - (τ) represents (r1(-τ)) * The last 2M-1 lines, use A2 - A2 * The last N-1 lines, use r3 - (τ) represents (r3(-τ)) * The following N-1 rows then have: r1 - (τ)=A1 -r s (τ), r3 - (τ)=A2 - Φr s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain.

[0009] The virtual received data constructed in step 3 Based on this, let (p c ,q c If )=(d,0), then we have: Then let r1′(τ) represent the first subarray and the position at (p c ,q c Let r2′(τ) represent the time-dependent function of the array element at position (p) = (d,0). c ,q c The time dependence function of the array element ) = (d,0) Then we have: r1′(τ)=A1Φ * r s (τ), r2′(τ)=A2r s (τ); then, based on the characteristic r s (τ)=(r s (-τ)) * , we have: (r1′(-τ)) * =A1 * Φr s (τ), (r2′(-τ)) * =A2 * r s (τ); then use r1′ - (τ) represents (r1′(-τ)) * The last 2M-1 lines, use r2′ - (τ) represents (r2′(-τ)) * For the last N-1 rows, we have: r1′ - (τ)=A1 - Φr s (τ), r2′ - (τ)=A2 - r s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q cThe array element ) = (d,0) is the virtual received data constructed by cross-correlation of the far-field signal data received by the first array element of the second subarray in the time domain;

[0010] The virtual received data constructed in step 3 Based on this, let (p c ,q c If ) = (-d, 0), then we have: Then let r1″(τ) represent the first subarray and the position at (p c ,q c The time dependence function of the array element ) = (-d, 0) Then we have: r1″(τ)=A1Φr s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The array element with the value (-d,0) is the virtual received data constructed by cross-correlation of the far-field signal data received by the first array element of the third subarray in the time domain.

[0011] According to r1 - (τ)=A1 - r s (τ), r1(τ)=A1r s (τ), r2′ - (τ)=A2 - r s (τ), r2′(τ)=A2r s (τ), let According to r2(τ)=A2Φr s (τ), r3 - (τ)=A2-Φr s (τ), r1″(τ)=A1Φr s (τ), r1′-(τ)=A1-Φr s (τ), let

[0012] Step 5: For r (1) (τ) Perform Ns pseudo-snap samplings at uniform time intervals, and denote the first virtual received data obtained after sampling as R1. For r (2) (τ) is performed at uniform time intervals N s The second virtual received data obtained after the "pseudo-snapshot" sampling is denoted as R2. Where, N s T represents the number of pseudo-snapshots. s Indicates the pseudo-sampling period.

[0013] Step 6: Perform cross-correlation on R1 and R2 to obtain the covariance matrix, denoted as Rcovariance. 12 , in, The covariance matrix of the virtual far-field signal source is represented. (·) H Represents the conjugate transpose of a vector or matrix;

[0014] Step 7: For R 12 Perform vectorization to obtain a virtual output vector, denoted as r. Where vec(·) denotes the vectorization operation, and A denotes the manifold matrix of the virtual spatiotemporal array. “⊙” is the Khatri-Rao product operator;

[0015] Step 8: Let R = E{rr} H} = AP Σ A H Then construct a solution for P. Σ The optimization problem is described as follows: Then, using SDP techniques to solve the optimization problem, we obtain P. Σ The optimal solution is denoted as in, Let be a second-order Toeplitz matrix, min be the minimum function, w be the intermediate variable for optimization, "||·||2" represent the l2 norm, and tr(·) represent finding the trace of the matrix;

[0016] Step 9: [Regarding...] Using the Vandermonde decomposition technique, solutions for α and β are obtained, denoted as follows: and Then, based on the equivalent relationship between electric angle, azimuth angle, and elevation angle... and Get θ and The respective solutions are denoted as follows: and Where α=[α1,α2,…,α k ,…,α K ], β=[β1,β2,…,β k ,…,β K ], θ=[θ1,θ2,…,θ k ,…,θ K ], α k The solution, Indicates β k The solution, Represents θ k The solution, express The solution.

[0017] Compared with the prior art, the advantages of the present invention are as follows:

[0018] Compared to traditional uniform linear arrays, the method of this invention utilizes a three-parallel coprime array, enabling the estimation of more far-field signal source direction-of-arrival angles with the same number of physical array elements. Simultaneously, this method leverages the spatiotemporal information of multi-element received data and second-order correlation operations to estimate the azimuth and elevation angles of the far-field signal source, achieving high estimation accuracy. Furthermore, in the two-dimensional angle estimation process, this method directly utilizes existing techniques for angle estimation, eliminating the need for angle matching. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of the structure of the three-dimensional far-field spatial propagation model based on a three-parallel coprime matrix established in the method of this invention;

[0020] Figure 2 This is a schematic diagram showing the elevation and azimuth angles from the narrowband far-field signal source to the reference point, estimated using the method of this invention. Detailed Implementation

[0021] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0022] This invention proposes a two-dimensional direction-of-arrival estimation method for space-time far-field sources based on a three-parallel coprime array, which includes the following steps:

[0023] Step 1: Establish a three-dimensional far-field spatial propagation model based on a three-parallel coprime matrix, such as... Figure 1 As shown, the model consists of three parallel subarrays parallel to the y-axis in three-dimensional space. Each subarray is a coprime matrix with parameters (M, N) in a non-uniform linear array. The first subarray is located between the second and third subarrays with a spacing of d. The first subarray has a total of 2M elements with an element spacing of Nd, while the second and third subarrays each have N elements with an element spacing of Md. In this model, the set of positions of all elements in the three-dimensional space of the three subarrays is represented as... There are K positions respectively Uncorrelated narrowband far-field signal sources arrive at the model; where M and N are coprime and 1 <M<N, λ represents the wavelength of the narrowband far-field signal source, the symbol "∪" is the union operator of sets, 1≤m≤2M, 1≤n≤N, K≥1, 1≤k≤K, θ k This represents the elevation angle of the k-th narrowband far-field signal source from the reference point. Let θk represent the azimuth angle from the k-th narrowband far-field signal source to the reference point, where the reference point is the origin of the three-dimensional coordinate system. Let θ1 represent the elevation angle from the 1-th narrowband far-field signal source to the reference point. θ represents the azimuth angle from the first narrowband far-field signal source to the reference point. K This represents the elevation angle of the Kth narrowband far-field signal source from the reference point. θ represents the azimuth angle from the Kth narrowband far-field signal source to the reference point, where the reference point is the origin of the three-dimensional coordinate system. k and This constitutes a pair of two-dimensional direction of arrival angles.

[0024] Step 2: After K narrowband far-field signal sources arrive at the three-dimensional far-field spatial propagation model based on a three-parallel coprime array, the electrical angle between the k-th narrowband far-field signal source and the y-axis of the three-dimensional space, i.e., the angle of arrival, is denoted as α. k Let β be the electrical angle between the k-th narrowband far-field signal source and the x-axis in three-dimensional space, i.e., the angle of arrival. k According to spatial geometry, there is an equivalent relationship between electric angle, azimuth angle, and elevation angle as follows: Then, the far-field signal data received by the first, second, and third subarrays are further denoted as z1(t), z2(t), and z3(t), respectively, where z1(t) = A1s(t) + n1(t), z2(t) = A2Φs(t) + n2(t), and z3(t) = A2Φ * s(t) + n3(t); where t is the time variable, s(t) represents the far-field signal source vector, and s(t) = [s1(t), s2(t), ..., s k (t),…,s K (t)] T s1(t) represents the first narrowband far-field signal source, s2(t) represents the second narrowband far-field signal source, s K (t) represents the Kth narrowband far-field signal source, s k (t) represents the k-th narrowband far-field signal source. σ sk Let represent the average power of the k-th narrowband far-field signal source, e be the natural constant, e = 2.71…, j be the imaginary part, and ω be the value of ω. k Let ω represent the angular frequency of the k-th narrowband far-field signal source. k =2πf k f kLet nk represent the frequency of the k-th narrowband far-field signal source, n1(t) represent the additive Gaussian noise vector of the far-field signal data z1(t) received by the first subarray, n2(t) represent the additive Gaussian noise vector of the far-field signal data z2(t) received by the second subarray, n3(t) represent the additive Gaussian noise vector of the far-field signal data z3(t) received by the third subarray, and A1 represent the manifold matrix of the first subarray, A1 = [a1(α1), a1(α2), ..., a1(αk), ..., a1(αk)]. K A2 represents the popular matrix of the second and third subarrays, A2 = [a2(α1), a2(α2), ..., a2(α)]. k ),…,a2(α K )],a1(α k ) represents the first subarray with respect to α k The guide vector, a2(α k ) represents the relationship between α in the second and third subarrays. k The guide vector, a1(α1) represents the steering vector of the electrical angle α1 between the first narrowband far-field signal source and the y-axis of three-dimensional space in the first subarray, i.e., the angle of arrival α1. a1(α2) represents the steering vector of the electrical angle α2 between the second narrowband far-field signal source and the y-axis of three-dimensional space in the first subarray. K α represents the electrical angle α between the Kth narrowband far-field signal source in the first subarray and the y-axis in three-dimensional space. K The steering vectors, a2(α1) and a2(α2), represent the steering vectors of the electric angle α1 between the first narrowband far-field signal source and the y-axis of three-dimensional space in the second and third subarrays, respectively. K The angle α represents the electrical angle, or angle of arrival, between the Kth narrowband far-field signal source and the y-axis in three-dimensional space in the second and third subarrays. K The guiding vector, Φ represents the rotation matrix. β1 represents the electrical angle (angle of arrival) between the first narrowband far-field signal source and the x-axis of three-dimensional space, and β2 represents the electrical angle (angle of arrival) between the second narrowband far-field signal source and the x-axis of three-dimensional space. K Let represent the electrical angle (·) between the Kth narrowband far-field signal source and the x-axis in three-dimensional space, i.e., the angle of arrival. T Represents the transpose of a vector or matrix, (·) * The conjugate of a vector or matrix is ​​represented by diag(·), which means constructing a diagonal matrix.

[0025] Step 3: Construct virtual received data by cross-correling the far-field signal data received by any two elements from all elements of the three subarrays in the time domain. Assume these two elements are the v-th and c-th elements among all elements. Then, denote the constructed virtual received data as... Here, the far-field signal data received by a certain array element is a portion of the far-field signal data received by the subarray to which that element belongs, τ represents the time delay, E{·} represents the statistical expectation function, and x v (t) represents the far-field signal data received by the v-th array element, x c (t+τ) represents the far-field signal data received by the c-th array element after a delay of τ. Let v represent the noise correlation function between the v-th array element and the c-th array element. n v (t) represents x v Additive Gaussian noise of (t), n c (t+τ) represents x c Additive Gaussian noise of (t+τ), additive Gaussian noise n v (t) and n c The average power of each (t+τ) is σ. n δ(·) represents the impulse function, δ(τ) represents the impulse function at time τ, and δ(vc) represents the impulse function at time (vc). This represents the time autocorrelation function of the k-th narrowband far-field signal source. s k (t+τ) represents the k-th narrowband far-field signal source after a delay of τ, and exp represents an exponential function with the natural constant e as the base. v ,q v (p) represents the position of the v-th element in three-dimensional space. c ,q c () represents the position of the c-th array element in three-dimensional space.

[0026] Step 4: By changing (p) v ,q v ) and (p c ,q c Let ) represent the different physical positions of the two array elements, and then calculate the time correlation function of the far-field signal data received by the two array elements.

[0027] The virtual received data constructed in step 3 Based on this, let (p c ,q c If )=(0,0), then we have: Then let r1(τ) represent the first subarray and the position at (pc ,q c Let r2(τ) represent the time correlation function of the array element at position (p) = (0,0). c ,q c Let r3(τ) represent the time-dependent function of the array element at position (p) = (0,0). c ,q c The time dependence function of the array element ) = (0,0), let r s (τ) represents the time autocorrelation function of a narrowband far-field signal source. Then we have: r1(τ)=A1r s (τ), r2(τ)=A2Φr s (τ), r3(τ)=A2Φ*rs(τ); then according to the characteristics rs(τ)=(rs(-τ))*, there is: (r1(-τ)) * =A1 * r s (τ), (r3(-τ)) * =A2 * Φr s (τ); then use A1 - Indicates A1 * The last 2M-1 lines, use r1 - (τ) represents (r1(-τ)) * The last 2M-1 lines, use A2 - A2 * The last N-1 lines, use r3 - (τ) represents (r3(-τ)) * The following N-1 rows then have: r1 - (τ)=A1 - r s (τ), r3 - (τ)=A2 - Φr s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. This represents the first element of the first subarray, i.e., the element at position (0,0) and the element at position (p). c ,q c The virtual received data is constructed by autocorrelation of the far-field signal data received by the array element (0,0) in the time domain. The second element of the first subarray is represented by its position (p). c ,qc The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The position of the 2Mth element of the first subarray is (p c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The first element and position of the second subarray are represented by (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The second element of the second subarray is represented by its position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The position of the Nth element of the second subarray is (p) c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The first element and position of the third subarray are represented by (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The second element of the third subarray is represented by the position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. The position of the Nth element of the 3rd subarray is (p) c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain. This represents the time autocorrelation function of the first narrowband far-field signal source. This represents the time autocorrelation function of the second narrowband far-field signal source. Let represent the time autocorrelation function of the Kth narrowband far-field signal source.

[0028] The virtual received data constructed in step 3 Based on this, let (p c ,q c If )=(d,0), then we have: Then let r1′(τ) represent the first subarray and the position at (p c ,qc Let r2′(τ) represent the time-dependent function of the array element at position (p) = (d,0). c ,q c The time dependence function of the array element ) = (d,0) Then we have: r1′(τ)=A1Φ * r s (τ), r2′(τ)=A2r s (τ); then, based on the characteristic r s (τ)=(r s (-τ)) * , we have: (r1′(-τ)) * =A1 * Φr s (τ), (r2′(-τ)) * =A2 * r s (τ); then use r1′ - (τ) represents (r1′(-τ)) * The last 2M-1 lines, use r2′ - (τ) represents (r2′(-τ)) * For the last N-1 rows, we have: r1′ - (τ)=A1 - Φr s (τ), r2′ - (τ)=A2 - r s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The array element (d, 0) is the virtual received data constructed by cross-correlation of the far-field signal data received by the first element of the second subarray in the time domain. This represents the first element of the first subarray, i.e., the element at position (0,0) and the element at position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (d,0) in the time domain. The second element of the first subarray is represented by its position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (d,0) in the time domain. The position of the 2Mth element of the first subarray is (p c ,q cThe virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (d,0) in the time domain. The first element and position of the second subarray are represented by (p). c ,q c The virtual received data is constructed by autocorrelation of the far-field signal data received by the array element (d,0) in the time domain. The second element of the second subarray is represented by its position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (d,0) in the time domain. The position of the Nth element of the second subarray is (p) c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (d,0) in the time domain.

[0029] The virtual received data constructed in step 3 Based on this, let (p c ,q c If ) = (-d, 0), then we have: Then let r1″(τ) represent the first subarray and the position at (p c ,q c The time dependence function of the array element ) = (-d, 0) Then we have: r1″(τ)=A1Φr s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The array element with the value (-d, 0) is the virtual received data constructed by cross-correlation of the far-field signal data received by the first element of the third subarray in the time domain. This represents the first element of the first subarray, i.e., the element at position (0,0) and the element at position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (-d, 0) in the time domain. The second element of the first subarray is represented by its position (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (-d, 0) in the time domain. The position of the 2Mth element of the first subarray is (p c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (-d, 0) in the time domain.

[0030] According to r1 - (τ)=A1 - r s (τ), r1(τ)=A1r s (τ), r2′ - (τ)=A2-r s (τ), r2′(τ)=A2r s (τ), let According to r2(τ)=A2Φr s (τ), r3 - (τ)=A2 - Φr s (τ), r1″(τ)=A1Φr s (τ), r1′ - (τ)=A1 - Φr s (τ), let

[0031] Step 5: For r (1) (τ) Perform Ns pseudo-snap samplings at uniform time intervals, and denote the first virtual received data obtained after sampling as R1. For r (2) (τ) is performed at uniform time intervals N s The second virtual received data obtained after the "pseudo-snapshot" sampling is denoted as R2. Where, N s T represents the number of pseudo-snapshots. s Indicates the pseudo-sampling period. r (1) (T s ),r (1) (2T s ),...,r (1) (N s T s ) corresponds to r (1) (τ) Ns data points obtained by Ns pseudo-fast samplings, r (2) (T s ),r (2) (2T s ),...,r (2) (N s T s ) corresponds to r (2) (τ) Ns data points obtained by Ns pseudo-fast sampling, r s (T s ),r s (2T s ),...,r s (Ns T s ) corresponds to r s (τ) Ns data points are obtained by performing Ns pseudo-fast samplings.

[0032] Step 6: Perform cross-correlation on R1 and R2 to obtain the covariance matrix, denoted as Rcovariance. 12 , in, The covariance matrix of the virtual far-field signal source is represented. σ s1 σ represents the average power of the first narrowband far-field signal source. sK This represents the average power of the Kth narrowband far-field signal source. It is also a diagonal matrix, (·) H Represents the conjugate transpose of a vector or matrix.

[0033] Step 7: For R 12 Perform vectorization to obtain a virtual output vector, denoted as r. Where vec(·) denotes the vectorization operation, and A denotes the manifold matrix of the virtual spatiotemporal array. “⊙” is the Khatri-Rao product operator.

[0034] Step 8: Let R = E{rr} H} = AP Σ A H R contains the matrix P used to estimate the two-dimensional direction of arrival angle. Σ All elements in P; then construct a solution P Σ The optimization problem is described as follows: Then, using the SDP (semi-definite programming) technique to solve the optimization problem, we obtain P. Σ The optimal solution is denoted as in, Let be a second-order Toeplitz matrix, min is the minimum function, w is the intermediate variable for optimization, "||·||2" represents the l2 norm, tr(·) represents finding the trace of the matrix, and "st" means "constrained by...".

[0035] Step 9: [Regarding...] Using the Vandermonde decomposition technique, solutions for α and β are obtained, denoted as follows: and Then, based on the equivalent relationship between electric angle, azimuth angle, and elevation angle... and Get θ and The respective solutions are denoted as follows: and Where α=[α1,α2,…,αk ,…,α K ], β=[β1,β2,…,β k ,…,β K ], θ=[θ1,θ2,…,θ k ,…,θ K ], α k The solution, Indicates β k The solution, Represents θ k The solution, express The solution. Since the two-dimensional direction-of-arrival angles obtained using the Vandermonde decomposition technique are automatically paired, therefore... and It also pairs automatically.

[0036] In this embodiment, the Vandermonde decomposition technique is specifically the matrix pencil and pairing method.

[0037] To verify the effectiveness and accuracy of the method of the present invention, simulation experiments were conducted on the method of the present invention, as follows:

[0038] Assume seven independent narrowband far-field signal sources are uniformly distributed at azimuths {17°, 17°} and {81°, 81°} and incident on a three-dimensional far-field spatial propagation model based on a three-parallel coprime array. The first subarray of the three-parallel coprime array has four elements, and the second and third subarrays have three elements each, for a total of ten physical elements. The signal-to-noise ratio is set to 35 dB, and the number of snapshots is set to 5000. Experimental results are as follows: Figure 2 As shown, when seven narrowband far-field signal sources are incident simultaneously, the azimuth and elevation angles estimated by the method of this invention are basically consistent with the true values. This indicates that the method of this invention can correctly estimate all azimuth and elevation angles, and the estimation parameters of the seven narrowband far-field signal sources can be correctly matched, which fully demonstrates that the method of this invention has good estimation accuracy under multi-signal source conditions.

Claims

1. A two-dimensional direction-of-arrival estimation method for a space-time far-field source based on a three-parallel coprime array, characterized in that... The steps include the following: Step 1: Establish a three-dimensional far-field spatial propagation model based on three parallel coprime arrays. This model consists of three mutually parallel subarrays parallel to the y-axis of the three-dimensional space. Each subarray is a coprime array with parameters (M, N) in a non-uniform linear array. The first subarray is located between the second and third subarrays with a spacing of d. The first subarray has a total of 2M elements with an element spacing of Nd. The second and third subarrays each have a total of N elements with an element spacing of Md. Under this model, the set of positions of all elements in the three-dimensional space of the three subarrays is represented as: There are K positions respectively Uncorrelated narrowband far-field signal sources arrive at the model; where M and N are coprime and 1 <M<N, λ represents the wavelength of the narrowband far-field signal source, 1≤m≤2M, 1≤n≤N, K≥1, 1≤k≤K, θ k This represents the elevation angle of the k-th narrowband far-field signal source from the reference point. θ represents the azimuth angle from the k-th narrowband far-field signal source to the reference point, where the reference point is the origin of the three-dimensional coordinate system. k and This constitutes a pair of two-dimensional direction of arrival angles; Step 2: After K narrowband far-field signal sources arrive at the three-dimensional far-field spatial propagation model based on a three-parallel coprime array, the electrical angle between the k-th narrowband far-field signal source and the y-axis of the three-dimensional space, i.e., the angle of arrival, is denoted as α. k Let β be the electrical angle between the k-th narrowband far-field signal source and the x-axis in three-dimensional space, i.e., the angle of arrival. k The electric angle, azimuth angle, and elevation angle have the following equivalent relationship: Then, the far-field signal data received by the first, second, and third subarrays are further denoted as z1(t), z2(t), and z3(t), respectively, where z1(t) = A1s(t) + n1(t), z2(t) = A2Φs(t) + n2(t), and z3(t) = A2Φ * s(t) + n3(t); where t is the time variable, s(t) represents the far-field signal source vector, and s(t) = [s1(t), s2(t), ..., s k (t),…,s K (t)] T s k (t) represents the k-th narrowband far-field signal source. σ sk Let ω represent the average power of the k-th narrowband far-field signal source, e be the natural constant, j be the imaginary part, and ω be the value of ω. k Let ω represent the angular frequency of the k-th narrowband far-field signal source. k =2πf k f k Let nk represent the frequency of the k-th narrowband far-field signal source, n1(t) represent the additive Gaussian noise vector of the far-field signal data z1(t) received by the first subarray, n2(t) represent the additive Gaussian noise vector of the far-field signal data z2(t) received by the second subarray, n3(t) represent the additive Gaussian noise vector of the far-field signal data z3(t) received by the third subarray, and A1 represent the manifold matrix of the first subarray, A1 = [a1(α1), a1(α2), ..., a1(αk), ..., a1(αk)]. K A2 represents the popular matrix of the second and third subarrays, A2 = [a2(α1), a2(α2), ..., a2(αk), ..., a2(αk)]. K )],a1(α k ) represents the first subarray with respect to α k The guide vector, a2(α k ) represents the relationship between α in the second and third subarrays. k The guide vector, Φ represents the rotation matrix. (·) T Represents the transpose of a vector or matrix, (·) * The conjugate of a vector or matrix is ​​represented by diag(·), which means constructing a diagonal matrix. Step 3: Construct virtual received data by cross-correling the far-field signal data received by any two elements from all elements of the three subarrays in the time domain. Assume these two elements are the v-th and c-th elements among all elements. Then, denote the constructed virtual received data as... Where τ represents the time delay, E{·} represents the statistical expectation function, and x v (t) represents the far-field signal data received by the v-th array element, x c (t+τ) represents the far-field signal data received by the c-th array element after a delay of τ. Let v represent the noise correlation function between the v-th array element and the c-th array element. n v (t) represents x v Additive Gaussian noise of (t), n c (t+τ) represents x c Additive Gaussian noise of (t+τ), additive Gaussian noise n v (t) and n c The average power of each (t+τ) is σ. n δ(·) represents the impulse function, δ(τ) represents the impulse function at time τ, and δ(vc) represents the impulse function at time (vc). This represents the time autocorrelation function of the k-th narrowband far-field signal source. s k (t+τ) represents the k-th narrowband far-field signal source after a delay of τ, and exp represents an exponential function with the natural constant e as the base. v ,q v (p) represents the position of the v-th element in three-dimensional space. c ,q c () represents the position of the c-th array element in three-dimensional space. Step 4: Virtual data reception constructed in step 3 Based on this, let (p c ,q c If )=(0,0), then we have: Then let r1(τ) represent the first subarray and the position at (p c ,q c Let r2(τ) represent the time correlation function of the array element at position (p) = (0,0). c ,q c Let r3(τ) represent the time-dependent function of the array element at position (p) = (0,0). c ,q c The time dependence function of the array element ) = (0,0), let r s (τ) represents the time autocorrelation function of a narrowband far-field signal source. r1(τ)=A1r s (τ), r2(τ)=A2Φr s (τ), r3(τ)=A2Φ * r s (τ); then, based on the characteristic r s (τ)=(r s (-τ)) * , we have: (r1(-τ)) * =A1 * r s (τ), (r3(-τ)) * =A2 * Φr s (τ); then use A1 - Indicates A1 * The last 2M-1 lines, use r1 - (τ) represents (r1(-τ)) * The last 2M-1 lines, use A2 - A2 * The last N-1 lines, use r3 - (τ) represents (r3(-τ)) * The following N-1 rows then have: r1 - (τ)=A1 - r s (τ), r3 - (τ)=A2 - Φr s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The virtual received data is constructed by cross-correlation of the far-field signal data received by the array element (0,0) in the time domain; The virtual received data constructed in step 3 Based on this, let (p c ,q c If )=(d,0), then we have: Then let r1′(τ) represent the first subarray and the position at (p c ,q c Let r2′(τ) represent the time-dependent function of the array element at position (p) = (d,0). c ,q c The time dependence function of the array element ) = (d,0) Then we have: r1′(τ)=A1Φ * r s (τ), r2′(τ)=A2r s (τ); then, based on the characteristic r s (τ)=(r s (-τ)) * , we have: (r1′(-τ)) * =A1 * Φr s (τ), (r2′(-τ)) * =A2 * r s (τ); then use r1′ - (τ) represents (r1′(-τ)) * The last 2M-1 lines, use r2′ - (τ) represents (r2′(-τ)) * For the last N-1 rows, we have: r1′ - (τ)=A1 - Φr s (τ), r2′ - (τ)=A2 - r s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The array element ) = (d,0) is the virtual received data constructed by cross-correlation of the far-field signal data received by the first array element of the second subarray in the time domain; The virtual received data constructed in step 3 Based on this, let (p c ,q c If ) = (-d, 0), then we have: Then let r1″(τ) represent the first subarray and the position at (p c ,q c The time dependence function of the array element ) = (-d, 0) Then we have: r1″(τ)=A1Φr s (τ); where, Let the position of the v-th element in the three subarrays be (p). c ,q c The array element with the value (-d,0) is the virtual received data constructed by cross-correlation of the far-field signal data received by the first array element of the third subarray in the time domain. According to r1 - (τ)=A1 - r s (τ), r1(τ)=A1r s (τ), r2′ - (τ)=A2 - r s (τ), r2′(τ)=A2r s (τ), let According to r2(τ)=A2Φr s (τ), r3 - (τ)=A2 - Φr s (τ), r1″(τ)=A1Φr s (τ), r1′ - (τ)=A1 - Φr s (τ), let Step 5: For r (1) (τ) Perform Ns pseudo-snap samplings at uniform time intervals, and denote the first virtual received data obtained after sampling as R1. For r (2) (τ) is performed at uniform time intervals N s The second "pseudo-snapshot" sampling is used to obtain the second virtual received data, which is denoted as R2. Where, N s T represents the number of pseudo-snapshots. s Indicates the pseudo-sampling period. Step 6: Perform cross-correlation on R1 and R2 to obtain the covariance matrix, denoted as Rcovariance. 12 , in, The covariance matrix of the virtual far-field signal source is represented. (·) H Represents the conjugate transpose of a vector or matrix; Step 7: For R 12 Perform vectorization to obtain a virtual output vector, denoted as r. Where vec(·) denotes the vectorization operation, and A denotes the manifold matrix of the virtual spatiotemporal array. "⊙" is the Khatri-Rao product operator; Step 8: Let R = E{rr} H } = AP Σ A H Then construct a solution for P. Σ The optimization problem is described as follows: Then, using SDP techniques to solve the optimization problem, we obtain P. Σ The optimal solution is denoted as in, Let be a second-order Toeplitz matrix, min be the minimum function, w be the intermediate variable for optimization, "||·||2" represent the l2 norm, and tr(·) represent finding the trace of the matrix; Step 9: [Regarding...] Using the Vandermonde decomposition technique, solutions for α and β are obtained, denoted as follows: and Then, based on the equivalent relationship between electric angle, azimuth angle, and elevation angle... and Get θ and The respective solutions are denoted as follows: and Where α=[α1,α2,…,α k ,…,α K ], β=[β1,β2,…,β k ,…,β K ], θ=[θ1,θ2,…,θ k ,…,θ K ], α k The solution, Indicates β k The solution, Represents θ k The solution, express The solution.

Images