V-shaped sparse array based on four-stage uniform linear array and DOA estimation method

Abstract

The invention relates to the field of arrays, in particular to a V-shaped sparse array based on four-level uniform linear arrays and a DOA estimation method, the V-shaped sparse array comprises two four-level uniform linear arrays, the included angle of the two four-level uniform linear arrays is omega, and the intersection point of the two four-level uniform linear arrays is an original point; and each four-stage uniform linear array comprises four sub-arrays. The DOA estimation method is aimed at the V-shaped sparse array. For the V-shaped sparse array, the invention also provides a DOA estimation method. According to the method, redundancy in the virtual array element generation process is reduced, the array aperture is expanded, virtual array element full coverage from 0 to the maximum array element position can be achieved, and the incoming wave direction can be estimated more accurately.

Description

Technical Field

[0001] The present invention relates to the field of arrays, and in particular to a V-shaped sparse array based on a four-level uniform linear array and a DOA estimation method. Background Art

[0002] Direction of Arrival (DOA) estimation plays a crucial role in array signal processing, primarily responsible for signal source localization. One-dimensional (1-D) DOA can only estimate the azimuth of the signal source but cannot estimate the elevation angle. Two-dimensional (2-D) DOA technology overcomes this limitation and can provide estimated information for both azimuth and elevation angles. Currently, 2D DOA technology has been widely used in a variety of fields, including wireless communications, fifth-generation mobile communications (5G), and navigation and positioning. In addition to improvements in arrays, researchers have developed a variety of algorithms to further improve the accuracy of DOA estimation. Among them, the MUSIC (Multiple Signal Classification) algorithm has become one of the most powerful methods in this context due to its simplicity and asymptotic performance relative to the corresponding performance bounds. The effectiveness of the algorithm is attributed to the orthogonality between the signal space and the noise space.

[0003] Most traditional arrays adapted for far-field signal sources are based on uniform linear arrays, using half the wavelength of the received signal as the standard element spacing unit, and the receiving array arranges the elements at equal intervals. However, traditional design methods will cause the receiving array to face a series of problems, such as severe mutual coupling between elements; limited physical aperture of the receiving array; low spatial resolution of the receiving array. Compared with traditional uniform linear arrays, sparse arrays have many advantages, such as high array degrees of freedom, strong underdetermined estimation capabilities, large array physical aperture, high spatial resolution, and low hardware cost. However, the existing V-shaped sparse array has limited degrees of freedom and spatial resolution. How to further improve the element spacing in the V-shaped sparse array to increase the degrees of freedom and spatial resolution is of great significance for the promotion and application of sparse matrices. Summary of the Invention

[0004] In response to the above problems, the present invention provides a V-shaped sparse array based on a four-level uniform linear array, wherein the V-shaped sparse array includes two four-level uniform linear arrays, the angle between the two four-level uniform linear arrays is Ω, and the intersection of the two four-level uniform linear arrays is the origin;

[0005] Each of the four-level uniform linear arrays includes a sub-array K1, a sub-array K2, a sub-array K3, and a sub-array K4, and the four sub-arrays have a total of M elements; the four-level uniform linear array K VFLA And sub-matrix K1, sub-matrix K2, sub-matrix K3, sub-matrix K4 satisfy: K VFLA =K1∪K2∪K3∪K4;

[0006] The distance between the elements in the submatrix and the origin satisfies:

[0007]

[0008] Among them, M k and They represent the number of elements and the element spacing of the kth sub-array, respectively, where k∈[1,2,3,4].

[0009]

[0010] Preferably, the angle Ω is: L is the number of consecutive virtual sensors in the V-shaped array. To simplify the following, we define p=cos(Ω / 2) and q=sin(Ω / 2).

[0011] For the V-type sparse array based on the four-level uniform linear array provided by the present invention, the present invention also provides a V-type sparse array DOA estimation method, wherein the V-type sparse array includes a four-level uniform linear array on the U axis and a four-level uniform linear array on the V axis, and the signal output of the U axis and the V axis is:

[0012]

[0013] Where i=1,...,T, T is the number of snapshots, are the temporal and spatial white noise vectors,

[0014] {s k (t i )} k∈[1,K],i＝∈[1,T] is a collection of uncorrelated signal sources, represents the steering vector corresponding to the kth source, and The i-th element is given by:

[0015]

[0016] The covariance matrix corresponding to two four-level uniform linear arrays is:

[0017]

[0018] Among them, A u ,A v is a steering matrix of size M×K, whose kth column elements are and is the signal correlation matrix of size K×K, I is the identity matrix, is the noise variance. Then R u ,R v After vectorization, spatial smoothing technology is introduced to process the covariance matrix to avoid interference of false peaks on DOA estimation:

[0019]

[0020] Then, by averaging all sub-matrices we obtain

[0021]

[0022] R temp =R temp (m:m+l+1,m:m+l+1)

[0023] R temp It is used to store the sub-matrix of the covariance matrix. m is an index variable, which means extracting the sub-matrix starting from the m-th row and m-th column of the covariance matrix. That is, the sub-matrix is continuously taken out from the original covariance matrix and averaged to obtain the spatially smoothed covariance matrix for two-dimensional DOA estimation.

[0024] Where L is the aperture of the array, 1≤m≤L; similarly, the spatial smoothing covariance matrix of the V axis can be obtained. Once the spatially smoothed covariance matrix of the U-axis and V-axis outputs is calculated, it is applied to the MUSIC algorithm. By searching the spatial spectrum function, the spectral peak corresponding to the target source can be obtained, and the angles α and β used for conversion can be obtained as shown below:

[0025]

[0026] Where α, β are the relevant DOA angles used to compare with the azimuth and elevation angles θ, To convert, and R u-ss and R v-ss The noise subspace eigenvector matrix of u (α) and P v (β), then from P u (α) and P v The highest peak of (β) finds the associated DOA angle {α k ,β k} k＝[1,K] ; Then, the estimated DOA azimuth angle is obtained as

[0027]

[0028] The cross-covariance matrix is calculated as:

[0029]

[0030] Among them, the array steering matrix

[0031] Au Each column element of A v One-to-one correspondence, guide matrix A v By its corresponding matrix A u Equivalently, the steering matrix A is estimated by solving the following least squares problem: v :

[0032]

[0033] To estimate R s , the covariance matrix R u The characteristic decomposition of is:

[0034]

[0035] in is the signal subspace eigenvector matrix, is the noise subspace eigenvector matrix, Λ is the matrix of R u The diagonal matrix composed of the eigenvalues of s ∈K×K,Λ n ∈(MK)×(MK) are R u Regarding the diagonal matrix composed of the eigenvalues of the signal and noise subspaces, R s It can be estimated by the following formula:

[0036]

[0037] in represents the Moore-Penrose pseudo-inverse operation; by using the following closed-form expression, i.e.,

[0038]

[0039] Since the estimated steering matrix The size is M×K, you can use The columns of are used to estimate the elevation angle one by one, so that each elevation angle is paired with the corresponding azimuth angle. Therefore, the elevation angle can be obtained by the MUSIC algorithm using the covariance matrix To estimate, so Estimated by formula (15). Once obtained You can find the elevation angle

[0040]

[0041] The V-shaped sparse array of the present invention reduces redundancy in the virtual element generation process and expands the array aperture through two four-level uniform linear arrays and the spacing setting of the array elements in each four-level uniform linear array. It can achieve full coverage of virtual array elements from 0 to the maximum array element position, and can more accurately estimate the direction of the incoming wave. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 Schematic diagram of VFLA array element distribution;

[0043] Figure 2 Schematic diagram of the distribution of actual array elements and virtual array elements of VFLA;

[0044] Figure 3 Define schematic diagrams for azimuth and elevation angles;

[0045] Figure 4 Schematic diagram of simulation results of simulation 1;

[0046] Figure 5 Schematic diagram of simulation results of simulation 2;

[0047] Figure 6 Schematic diagram of simulation results of simulation 3;

[0048] Figure 7 Schematic diagram of simulation results of simulation 4. DETAILED DESCRIPTION

[0049] In the embodiments of the present invention, words such as "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of this application should not be construed as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner to facilitate understanding.

[0050] It will be understood that the “embodiment” mentioned throughout the specification means that the specific features, structures or characteristics related to the embodiment are included in at least one embodiment of the present application. Therefore, the various embodiments throughout the specification do not necessarily refer to the same embodiment. In addition, these specific features, structures or characteristics can be combined in one or more embodiments in any suitable manner. It will be understood that in the various embodiments of the present application, the size of the sequence number of each process does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiment of the present application.

[0051] In the present invention, unless otherwise specified, the same or similar parts between the various embodiments can refer to each other. In the various embodiments of the present invention, and the various implementation methods / implementation methods / implementation methods in each embodiment, if there is no special explanation and logical conflict, the terms and / or descriptions between different embodiments and the various implementation methods / implementation methods / implementation methods in each embodiment are consistent and can be referenced to each other. The technical features in different embodiments and the various implementation methods / implementation methods / implementation methods in each embodiment can be combined to form new embodiments, implementation methods, implementation methods, or implementation methods according to their inherent logical relationships. The implementation methods of the present application described below do not constitute a limitation on the scope of protection of the present application.

[0052] Embodiment 1: In this embodiment, a V-shaped sparse array based on a four-level uniform linear array is provided. The V-shaped sparse array includes two four-level uniform linear arrays. The angle between the two four-level uniform linear arrays is Ω, and the intersection of the two four-level uniform linear arrays is the origin.

[0053] Each of the four-level uniform linear arrays includes a sub-array K1, a sub-array K2, a sub-array K3, and a sub-array K4, and the four sub-arrays have a total of M elements; the four-level uniform linear array K VFLA And sub-matrix K1, sub-matrix K2, sub-matrix K3, sub-matrix K4 satisfy: K VFLA =K1∪K2∪K3∪K4.

[0054] The distance between the elements in the submatrix and the origin satisfies:

[0055]

[0056] Among them, M k and Respectively represent the number of array elements and the array element spacing of the kth sub-array:

[0057]

[0058] Among them, k∈[1,2,3,4].

[0059] Specifically, the VFLA array consists of two parts, each of which is a four-level uniform linear array (FLA), with the angle between the two parts being Ω. The center of the intersection of two identical FLAs is defined as the 0 element. Assume that the spacing between elements is represented by d, where d is equal to half a wavelength. The specific array element arrangement structure of the VFLA is as follows: Figure 1 As shown, it consists of four sub-arrays with a total of M elements. Sub-arrays 1, 2, 3, and 4 are uniform linear arrays (ULAs) with a total of M elements. The following is the expression for the element arrangement of the VFLA array, where K VFLA, K1, K2, K3, K4 represent array, subarray 1, subarray 2, subarray 3 and subarray 4 respectively:

[0060] K VFLA =K1∪K2∪K3∪K4 (1)

[0061] in The definition is as follows

[0062]

[0063] Where M represents the number of array elements, L represents the distance between array elements, and M k and L k Respectively represent the number of array elements and the array element spacing of the kth, k∈[1,2,3,4]th sub-array, and their values are

[0064]

[0065] Assume that the positions of subarrays K1, K2, K3, and K4 are determined by the set and The self-difference set and mutual difference set of the array can be expressed by the following two formulas:

[0066] Definition 1 (self-difference set): self-difference coarray of an array It is defined based on the position set of array elements and is expressed as:

[0067]

[0068] Definition 2 (mutual difference set): the mutual difference co-array of arrays It is defined based on the position set of array elements and is expressed as:

[0069]

[0070] gather Can be considered as a set and The union of . Set The elements in represent the hysteresis values of the array, where all possible hysteresis values correspond to the degrees of freedom (DOF) of the array, and the continuous hysteresis values correspond to the unambiguity of the array (uDOF). The aperture of the array is determined by the difference between the maximum and minimum values in V, which is denoted in this paper.

[0071] Therefore, the following lemma can be derived:

[0072] Lemma 1. The continuous hysteresis range of VFLA is

[0073]

[0074] Specifically, if Figure 2 As shown, Figure 2 The green, yellow, purple, and blue colors in the middle represent the first to fourth sub-arrays, respectively. The right side shows a schematic diagram of the array's primary difference virtual array element. The V-shaped VFLA array consists of two FLA arrays, with an angle of Ω between them, arranged along the yoz plane. For simplicity, the axes where the sensors are placed are called the U axis and the V axis. The intersection of the U axis and the V axis is at the coordinate origin O, where the u and v axis coordinates are defined as follows:

[0075]

[0076] where u i ,v i is an integer representing the distribution of array elements on the U and V axes. This characteristic makes the resulting virtual array closer to the Vandermonde model, which helps to increase the array aperture of the sparse array. The overall array distribution consists of two parts, each of which has four sub-arrays. The number of overall array elements in each part is M, and the array element distribution is shown in Equations (2) and (3).

[0077] like Figure 3 As shown, suppose there are K signal sources from K different directions The angle between the incident path of the signal and the Y plane of the array is defined as the azimuth angle θ. k , the angle between the incident path of the signal and the Z plane of the array is defined as the pitch angle This allows for unique transformations between them without loss of generality. In this paper, the number of signal sources is considered known a priori and has the following characteristics: the signals are narrowband and uncorrelated; and the sources of these signals are located in the far field of the array.

[0078] Preferably, the angle Ω is: L is the number of consecutive virtual sensors in the V-shaped array. Define p=cos(Ω / 2), q=sin(Ω / 2).

[0079] In the second embodiment, a V-type sparse array based on a four-level uniform linear array provided in the first embodiment is provided. This embodiment provides a DOA estimation method for a V-type sparse array, wherein the V-type sparse array includes a four-level uniform linear array on the U axis and a four-level uniform linear array on the V axis. The signal outputs of the U axis and the V axis are:

[0080]

[0081] Where i=1,...,T, T is the number of snapshots, are the temporal and spatial white noise vectors,

[0082] {s k (t i )} k∈[1,K],i＝∈[1,T] is a collection of uncorrelated signal sources, represents the steering vector corresponding to the kth source, and The i-th element is given by:

[0083]

[0084] The covariance matrix corresponding to two four-level uniform linear arrays is:

[0085]

[0086] Among them, A u ,A v is a steering matrix of size M×K, whose kth column elements are and is the signal correlation matrix of size K×K, I is the identity matrix, is the noise variance. Then R u ,R v After vectorization, spatial smoothing technology is introduced to process the covariance matrix to avoid interference of false peaks on DOA estimation:

[0087]

[0088] Then, by averaging all sub-matrices we obtain

[0089]

[0090] R temp =R temp (m:m+l+1,m:m+l+1)

[0091] R temp The submatrix used to store the covariance matrix is as follows. m is an index variable, which means that the submatrix is extracted starting from the mth row and mth column of the covariance matrix. That is, the submatrix is continuously taken out from the original covariance matrix and averaged to obtain the spatially smoothed covariance matrix for two-dimensional DOA estimation.

[0092] Where L is the aperture of the array, 1≤m≤L; similarly, the spatial smoothing covariance matrix of the V axis can be obtained. Once the spatially smoothed covariance matrix of the U-axis and V-axis outputs is calculated, it is applied to the MUSIC algorithm. By searching the spatial spectrum function, the spectral peak corresponding to the target source can be obtained, and the angles α and β used for conversion can be obtained as shown below:

[0093]

[0094] Where α, β are the relevant DOA angles used to compare with the azimuth and elevation angles θ, To convert, and R u-ss and R v-ss The noise subspace eigenvector matrix of u (α) and P v (β), then from P u (α) and P v The highest peak of (β) finds the associated DOA angle {α k ,β k} k＝[1,K] ; Then, the estimated DOA azimuth angle is obtained as

[0095]

[0096] The cross-covariance matrix is calculated as:

[0097]

[0098] Among them, the array steering matrix

[0099] A u Each column element of A v One-to-one correspondence, guide matrix A v By its corresponding matrix A u Equivalently, the steering matrix A is estimated by solving the following least squares problem: v :

[0100]

[0101] To estimate R s , the covariance matrix R u The characteristic decomposition of is:

[0102]

[0103] in is the signal subspace eigenvector matrix, is the noise subspace eigenvector matrix, Λ is the matrix of R u The diagonal matrix composed of the eigenvalues of s ∈K×K,Λ n ∈(MK)×(MK) are R u Regarding the diagonal matrix composed of the eigenvalues of the signal and noise subspaces, R s It can be estimated by the following formula:

[0104]

[0105] in represents the Moore-Penrose pseudo-inverse operation;

[0106] By using the following closed form expression, i.e.,

[0107]

[0108] Since the estimated steering matrix The size is M×K, you can use The columns of are used to estimate the elevation angle one by one, so that each elevation angle is paired with the corresponding azimuth angle. Therefore, the elevation angle can be obtained by the MUSIC algorithm using the covariance matrix To estimate, so Estimated by formula (15). Once obtained You can find the elevation angle

[0109]

[0110] Simulation experiments were conducted to evaluate the two-dimensional DOA estimation performance of various array configurations. The arrays involved included the proposed novel V-type array, a V-type CA array, a V-type TCA array, a V-type CACIS array, a V-type ANA11 array, a V-type NA array, and a V-type uniform array. To ensure a fair comparison by controlling variables, the number of elements in each array was set to 17, and each signal had the same transmission power. The array estimation performance is represented by the RMSE curve, which is calculated using the following formula:

[0111]

[0112] in and represents the Monte Carlo time, azimuth, and elevation estimates for the jth trial of the kth signal.

[0113] Table 1 shows the distribution of several sparse array elements involved in the simulation. In the subsequent simulation process, the distribution of different arrays is shown in Table 1.

[0114] Table 1: Actual array element distribution when the total number of array elements M = 17

[0115] Array Name Array element distribution (U axis or V axis) V-ULA {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16} VCA {0,7,11,14,21,22,28,33,35,42,44,49,55,56,63,66,70} V-TCA {0,9,18,27,36,45,54,63,72,73,74,75,76,77,78,79,80} V-CACIS {0,7,9,14,18,21,27,28,35,42,49,56,72,81,90,99,108,117} V-NA {0,1,2,3,4,5,6,7,8,17,26,36,44,53,62,71,80} V-ANA11 {1,2,3,4,10,20,30,40,50,60,70,80,81,82,83,84,85} VFLA {0,1,2,3,8,13,18,23,32,41,50,59,68,72,76,80,84}

[0116] Simulation 1, 2D DOA estimation by matching azimuth and elevation angles. The main purpose of this experiment is to verify the effectiveness of the proposed V-shaped VFLA for 2D DOA estimation and to verify its estimated azimuth and elevation angles for K sources. The number of snapshots is set to 100, and the signal-to-noise ratio (SNR) is set to 5dB. Five pairs of test signals are incident on the array from the directions [-0.3799, -0.4248], [0, 0], and [0.3799, 0.4248]. The test signal inputs are arranged in sequence. Figure 4 As shown in Figure 1, the spatial spectrum of the azimuth and elevation angles of the test signal is successfully plotted. The peak of the spectrum represents the estimated value, and the curve is clear with a distinct peak. The spatial spectrum of the elevation angle is plotted in order according to the corresponding azimuth angle, as shown in Figure 1. Figure 4 As shown in Figure 2, the estimated azimuth and elevation angles are thus successfully paired.

[0117] Simulation 2, comparison of the number of estimated sources. The purpose of this experiment is to explore the multi-signal or underdetermined estimation capabilities of each V-shaped array. Since the U-axis and V-axis sensor positions are the same, it is sufficient to evaluate the DOA estimation of only the test azimuth. For multi-signal estimation, a higher number of snapshots is required. Therefore, the snapshot count is set to 1000 and the SNR is reduced to 0dB. To ensure a fair comparison, all arrays are configured with 11 elements. Since uniform linear arrays cannot perform underdetermined estimation, 10 test signals are selected in this experiment, and the directions of the target sources are evenly spaced from -1 to 1, as shown in Figure 2. Figure 5 shown. Figure 5 The azimuth spatial spectra of the four arrays in a multi-source estimation scenario are shown. Compared with the V-shaped uniform array and the sparse arrays (VCA and VNA), the V-shaped VFLA can estimate more signal sources. It also provides better spectral clarity and estimation accuracy than other V-shaped sparse arrays, achieving superior overall estimation performance.

[0118] In simulation 3, different signal sources are used to observe the RMSE curves of different arrays as the signal-to-noise ratio changes. All array parameter estimation algorithms uniformly use the above-mentioned DOA estimation algorithm. The simulation sets the snapshot number T = 1000, the SNR ranges from -10 to 10dB, the Monte Carlo number γ = 500, and three signal sources are selected. The signal directions used for estimation are [-0.38, 0.45], [0, 0], and [0.38, 0.45]. All arrays, including V-shaped uniform antenna array (VA), VCA, VNA, CACIS, TCA, ANA11, and VFLA, are configured with a total of 17 array elements to maintain consistent hardware costs. The simulation results are shown in Figure 2. Figure 6 As shown in the figure, it can be seen that with the increase of signal-to-noise ratio and the number of snapshots, RMSE tends to decrease. Experiments show that the VFLA array has the best performance.

[0119] In simulation 4, multiple signal sources are used to observe the RMSE curves of different arrays as the number of snapshots changes. All array parameter estimation algorithms use the spatial smoothing MUSIC algorithm. The simulation settings are snapshot number T = 1000, SNR from -10 to 10dB, Monte Carlo order γ = 500, and three signal sources are selected. The signal directions used for estimation are [-0.38, 0.45], [0, 0],

[0120] [0.38,0.45]. The simulation results are as follows Figure 7 As shown in the figure, it can be seen that as the number of snapshots increases, the RMSE decreases. Experiments show that the VFLA array has the best performance.

[0121] The VFLA achieves the best distance estimation accuracy because the array has more virtual elements, resulting in a larger continuous virtual aperture and higher degrees of freedom. This results in higher estimation accuracy and a lower RMSE curve. The VFLA has the largest array aperture of the six arrays, enabling it to achieve higher estimation accuracy.

[0122] Although the present application has been described with reference to specific features and embodiments thereof, it is apparent that various modifications and combinations may be made thereto without departing from the spirit and scope of the present application. Accordingly, this specification and the drawings are merely illustrative of the present application as defined by the appended claims and are deemed to cover any and all modifications, variations, combinations or equivalents within the scope of the present application. Obviously, those skilled in the art may make various modifications and variations to the present application without departing from the scope of the present application. Thus, the present application is intended to include such modifications and variations if they fall within the scope of the claims of the present application and their equivalents.

Claims

1. A V-shaped sparse array based on a four-level uniform linear array, characterized in that: The V-shaped sparse array includes two four-level uniform linear arrays, the angle between the two four-level uniform linear arrays is Ω, and the intersection of the two four-level uniform linear arrays is the origin; Each of the four-level uniform linear arrays includes a sub-array K1, a sub-array K2, a sub-array K3, and a sub-array K4, and the four sub-arrays have a total of M elements; the four-level uniform linear array K VFLA And sub-matrix K1, sub-matrix K2, sub-matrix K3, sub-matrix K4 satisfy: K VFLA =K1∪K2∪K3∪K4; The distance between the elements in the submatrix and the origin satisfies: Among them, M k and Respectively represent the number of array elements and the array element spacing of the kth sub-array: Among them, k∈[1,2,3,4].

2. The method according to claim 1, wherein The angle Ω is: L is the number of consecutive virtual sensors in the V-shaped array.

3. A method for DOA estimation of a V-shaped sparse array, wherein the V-shaped sparse array comprises a four-level uniform linear array on the U axis and a four-level uniform linear array on the V axis, characterized in that: The signal output of the U axis and the V axis is: Where i=1,...,T, T is the number of snapshots, are the temporal and spatial white noise vectors, {s k (t i )} k∈[1,K],i＝∈[1,T] is a collection of uncorrelated signal sources, represents the steering vector corresponding to the kth source, and The i-th element is given by: The covariance matrix corresponding to two four-level uniform linear arrays is: Among them, A u ,A v is a steering matrix of size M×K, whose kth column elements are and is the signal correlation matrix of size K×K, I is the identity matrix, is the noise variance; then R u ,R v After vectorization, spatial smoothing technology is introduced to process the covariance matrix to avoid interference of false peaks on DOA estimation: Then, by averaging all sub-matrices we obtain R temp =R temp (m:m+l+1,m:m+l+1) R temp It is used to store the sub-matrix of the covariance matrix. m is an index variable, which means extracting the sub-matrix starting from the m-th row and m-th column of the covariance matrix. That is, the sub-matrix is continuously taken out from the original covariance matrix and averaged to obtain the spatially smoothed covariance matrix for two-dimensional DOA estimation. Where L is the aperture of the array, 1≤m≤L; similarly, the spatial smoothing covariance matrix of the V axis can be obtained. Once the spatially smoothed covariance matrix of the U-axis and V-axis outputs is calculated, it is applied to the MUSIC algorithm. By searching the spatial spectrum function, the spectral peak corresponding to the target source can be obtained, and the angles α and β used for conversion can be obtained as shown below: Where α, β are the relevant DOA angles used to compare with the azimuth and elevation angles θ, To convert, and R u-ss and R v-ss The noise subspace eigenvector matrix of u (α) and P v (β), then from P u (α) and P v The highest peak of (β) finds the associated DOA angle {α k ,β k } k＝[1,K] ; Then, the estimated DOA azimuth angle is obtained as The cross-covariance matrix is calculated as: Among them, the array steering matrix A u Each column element of A v One-to-one correspondence, guide matrix A v By its corresponding matrix A u Equivalently, the steering matrix A is estimated by solving the following least squares problem: v : To estimate R s , the covariance matrix R u The characteristic decomposition of is: in is the signal subspace eigenvector matrix, is the noise subspace eigenvector matrix, Λ is the matrix of R u The diagonal matrix composed of the eigenvalues of s ∈K×K,Λ n ∈(MK)×(MK) are R u Regarding the diagonal matrix composed of the eigenvalues of the signal and noise subspaces, R s It can be estimated by the following formula: in represents the Moore-Penrose pseudo-inverse operation; By using the following closed form expression, i.e., Since the estimated steering matrix The size is M×K, you can use The columns of are used to estimate the elevation angle one by one, so that each elevation angle is paired with the corresponding azimuth angle. Therefore, the elevation angle can be obtained by the MUSIC algorithm using the covariance matrix To estimate, so Estimated by formula (15). Once obtained You can find the elevation angle

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