A far-field target azimuth estimation method applicable to small samples

Abstract

The present invention proposes a far-field target direction angle estimation method applicable to small samples, and the implementation steps are as follows: obtaining a data matrix; constructing a kernel matrix; constructing an eigenvalue matrix and an eigenvector matrix; estimating the number of far-field targets by using an improved minimum description length criterion; estimating the noise subspace; and obtaining the estimation result of the far-field target direction angle. The present invention constructs a sufficiently estimated kernel matrix through the data matrix, estimates the number of signal sources by using the eigenvalue decomposition result of the kernel matrix and an improved minimum description length criterion to estimate the noise subspace, obtains a noise subspace that is closer to the true value due to a smaller deviation value, and finally uses the noise subspace to obtain the estimation result of the far-field target direction angle, improving the estimation accuracy of the far-field target direction angle and reducing the estimation complexity, and can be used in fields such as radar detection, sonar navigation, and multi-channel communication.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and relates to a method for estimating the azimuth angle of a far-field target, in particular to a method for estimating the azimuth angle of a far-field target suitable for small samples, which can be used in fields such as radar detection, sonar navigation, and multi-channel communication. Background Art

[0002] The target azimuth angle refers to the angle between the incident direction of the target signal on the antenna array and the normal direction of the array. It needs to be estimated in passive positioning, sonar direction finding, electronic or communication interference reconnaissance: There is a wave path difference when the signal emitted by the target signal source reaches different array elements of the antenna array. This wave path difference causes a spatial phase difference between different array elements, and the azimuth angle of the target signal source can be obtained by using this phase difference. According to the distance r between the target signal source and the antenna array, the target signal source can be divided into a near-field signal source and a far-field signal source. When r satisfies the target signal source is a far-field target signal source. At this time, the incident signal of the target signal source received by the antenna array is considered to be a plane wave, that is, the incident signals reaching each array element in the array are parallel.

[0003] The estimation of the far-field target azimuth angle can be divided into an estimation method based on the beamforming method and an estimation method based on the subspace class. For a single far-field target signal source, the estimation method based on the beamforming method can well estimate the far-field target azimuth angle. However, when the difference in the incident angles of two or more far-field target signal sources is less than the minimum beam width of the array, this method cannot distinguish these signal sources, resulting in an increase in the error of the azimuth angle estimation. In the estimation methods based on the subspace class, typical ones include the Multiple Signal Classification (MUSIC) method and the Estimation of Signal Parameters via Rotational Invariance (ESPRIT) method. The MUSIC method obtains the noise subspace by performing eigenvalue decomposition on the sample covariance matrix of the received signal, constructs a spatial spectrum function by using the orthogonality of the subspace, and the azimuth angle corresponding to the larger spatial spectrum peak is the azimuth angle of the far-field target.

[0004] When the number of samples of the antenna array is greater than the number of array elements, that is, in the case of large samples, the sample covariance matrix is a fully estimated matrix. The MUSIC method can obtain relatively accurate signal subspace and noise subspace through the sample covariance matrix, and the estimation accuracy is relatively high. However, in the case of small samples, the sample covariance matrix becomes a singular matrix, and the signal subspace and noise subspace obtained through the sample covariance matrix will deviate seriously from their true values. Noise components will be mixed into the signal subspace, and the estimation accuracy will drop severely. In recent years, in order to obtain higher resolution, stronger interference suppression ability and longer detection range, modern antenna arrays have adopted an increasing number of array elements, but the required number of samples has not been improved synchronously. Therefore, many scientific researchers have studied and explored the MUSIC method in the case of small samples, and proposed some improved MUSIC methods applicable to small samples to improve the estimation accuracy. For example, Wang Juan published a "Far-field Target Azimuth Estimation Method Based on Improved MUSIC and Applicable to Small Samples" in the Journal of Electronics & Information Technology, Vol. 42, No. 2, February 2020. This method first roughly estimates the target angle using the noise subspace obtained by the eigen-decomposition of the sample covariance matrix, then constructs a new noise subspace through the steering vectors corresponding to the target angle and its neighboring angles and optimizes and corrects it, and finally constructs a spatial spectrum function to estimate the far-field target azimuth angle. This method improves the azimuth angle estimation accuracy, but the sample covariance matrix it uses is a singular matrix in the case of small samples, resulting in the obtained noise subspace deviating from the true value, which affects the further improvement of the azimuth angle estimation accuracy. At the same time, the sample covariance is a high-dimensional matrix, and the eigenvalue decomposition of it leads to too high computational complexity of the estimation. Summary of the Invention

[0005] The purpose of the present invention is to propose a far-field target azimuth angle estimation method applicable to small samples in view of the deficiencies of the above-mentioned existing technologies, so as to solve the technical problems of low azimuth angle estimation accuracy and high computational complexity in the existing technologies in the case of small samples.

[0006] To achieve the above purpose, the technical solution adopted by the present invention includes the following steps:

[0007] (1) Obtain the data matrix X:

[0008] Construct a uniform linear array including M array elements with an interval of d between adjacent array elements, and incident the target signals generated by P far-field target signal sources onto each array element of the array, then sample the output signals of each array element L times, and finally form all the sampling results into a data matrix where M≥800, λ is the wavelength of the target signal, L<M, represents the complex number field;

[0009] (2) Construct the kernel matrix R:

[0010] Perform conjugate transpose on the data matrix X and multiply it on the right by the data matrix X to obtain the kernel matrix R:

[0011]

[0012] where X H represents the conjugate transpose result of X;

[0013] (3) Construct the eigenvalue matrix and eigenvector matrix:

[0014] Perform eigenvalue decomposition on the kernel matrix R, and construct an eigenvalue matrix Λ with the L eigenvalues s1, …, s l , …, s L as the main diagonal elements and the remaining elements being 0. At the same time, form an eigenvector matrix V from the L eigenvectors obtained by eigenvalue decomposition:

[0015]

[0016]

[0017] where diag(s1, …, s l , …, s L ) represents a matrix with s1, …, s l , …, s L as the main diagonal elements and the remaining elements all being 0, s l represents the l-th eigenvalue, and v l represents the eigenvector corresponding to s l ;

[0018] (4) Estimate the number of far-field targets P using the improved MDL criterion:

[0019] Take the kernel matrix R as the covariance matrix of the minimum description length (MDL) criterion, take the number of array elements M as the number of sampling points of the MDL criterion, and take the number of sampling points L as the number of array elements of the MDL criterion to improve the MDL criterion. Then estimate the number of far-field targets P through the improved MDL criterion:

[0020] P = arg min{f(P)}

[0021]

[0022] where f(P) represents the objective function, and arg min{·} represents the operation of taking the minimum value;

[0023] (5) Estimate the noise subspace U n :

[0024] Arrange the L eigenvalues contained in the eigenvalue matrix Λ in descending order, and right-multiply the eigenvectors corresponding to the first P eigenvalues by the data matrix X to obtain the signal subspace U s , and then normalize the column vectors of U s . Finally, estimate the noise subspace U s through the normalized result of U : n :

[0025] U s = X[v1,…,v P

[0026]

[0027] where I M represents the M×M dimensional identity matrix;

[0028] (6) Obtain the estimated result of the azimuth angle of the far-field target:

[0029] Construct the spatial spectrum function S(θ), and then select the P largest spectral peaks. The corresponding abscissa values are the azimuth angles of the P far-field target signal sources, where:

[0030]

[0031]

[0032] where a(θ) represents the steering vector of the array, represents the phase change caused by the time delay of the received signal at the m-th array element. exp represents the exponential function with the natural constant e as the base, j represents the imaginary unit, and [·] T represents the transpose operation.

[0033] Compared with the prior art, the present invention has the following advantages:

[0034] (1) The present invention constructs a kernel matrix through a data matrix, then performs eigenvalue decomposition on the kernel matrix, and then estimates the projection matrix of the noise subspace through some eigenvalues contained in the eigenvalue matrix obtained by eigenvalue decomposition. Since the kernel matrix is obtained by conjugate transposing the data matrix and right-multiplying the data matrix, it is a sufficient estimation matrix. The obtained noise subspace is closer to the true value because of the smaller deviation value, avoiding the defect that the sample covariance matrix adopted in the prior art belongs to a singular matrix, effectively improving the estimation accuracy. At the same time, since the dimension of the kernel matrix is much smaller than that of the sample covariance matrix, performing eigenvalue decomposition on it does not require a large amount of calculation and can reduce the complexity of estimation.

[0035] ​(2) In the present invention, the improved Minimum Description Length (MDL) criterion is adopted to estimate the number P of far-field targets, which makes up for the singularity of the sample covariance matrix caused by insufficient sampling number, and improves the estimation accuracy of the number of signal sources in the case of small samples. Then, by estimating the relatively accurate number of signal sources and the kernel matrix to estimate the noise subspace, the estimation accuracy can be further improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is the implementation flowchart of the present invention;

[0037] Figure 2 is the relationship diagram between the root mean square error of the present invention and the prior art and the input signal-to-noise ratio;

[0038] Figure 3 is the relationship diagram between the calculation time of the present invention and the prior art and the sampling number. DETAILED DESCRIPTION OF THE INVENTION

[0039] The present invention will be further described in detail below with reference to the drawings and specific embodiments:

[0040] Refer to Figure 1 , the present invention includes the following steps:

[0041] Step 1) Obtain the data matrix X:

[0042] Construct a uniform linear array including M array elements with an interval of d between adjacent array elements, and let the target signals generated by P far-field target signal sources be incident on each array element of the array. Taking the first array element as a reference, the target signal received by the m-th array element at time t for the P signal sources can be obtained as x m (t). Arrange the target signals x m (t) received by each array element at time t into an M×1-dimensional column vector by rows, and perform L samplings on it to obtain the sampling results x(1), …, x(l), …, x(L). Finally, form an M×L-dimensional data matrix X = [x(1), …, x(l), …, x(L)] by arranging the L sampling results by columns, where M≥800, λ is the wavelength of the target signal, 0<P≤100 and P is unknown at this time, L<M, represents the complex number field.

[0043] In this embodiment, M = 800, L = 240, and the number of samplings is less than the number of array elements, which is a small sample case.

[0044] Step 2) Construct the kernel matrix R:

[0045] Perform conjugate transpose on the data matrix X and multiply it on the right by the data matrix X to obtain the kernel matrix R:

[0046]

[0047] Among them, X H represents the conjugate transpose result of X;

[0048] In the prior art, the sample covariance matrix has the following expression:

[0049]

[0050] It can be seen from the definitions of the kernel matrix and the sample covariance matrix that the dimension of the kernel matrix is L×L, and the dimension of the sample covariance matrix is M×M. In the case of small samples, i.e., L < M, the sample covariance matrix is a singular matrix, and the noise covariance matrix separated from the sample covariance matrix may not be an effective diagonal matrix. Even if it is a diagonal matrix, the main diagonal elements are not exactly the same, and the noise subspace obtained from the sample covariance matrix deviates seriously from its true value. The kernel matrix is a fully estimated matrix, and the noise components of the kernel matrix are approximately uniform, that is, the noise correlation part is a diagonal matrix composed of the same main diagonal elements, which ensures good separability between the signal subspace and the noise subspace, and the noise subspace obtained from the kernel matrix is closer to its true value.

[0051] Step 3) Construct the eigenvalue matrix and the eigenvector matrix:

[0052] Step 3a) Construct the eigenmatrix R-s of the kernel matrix R l I L , and let det[R-s l I L = 0 to obtain L eigenvalues s1,…,s l ,…,s L , where det[·] represents the value of taking the determinant.

[0053] Step 3b) Substitute each eigenvalue s l of the kernel matrix R into the linear equation system (R-s l I L )v l = 0 to obtain L eigenvectors v1,…,v l ,…,v L .

[0054] Step 3c) Construct an eigenvalue matrix Λ with the L eigenvalues s1,…,s l ,…,s L obtained from the eigenvalue decomposition as the main diagonal elements and the remaining elements being 0. At the same time, form an eigenvector matrix V with the L eigenvectors v1,…,v l ,…,v L :

[0055]

[0056]

[0057] Among them, diag(·) represents a matrix with all elements being 0 except the main diagonal elements, and s l represents the l-th eigenvalue, and v l represents the eigenvector corresponding to s l .

[0058] (4) Estimate the number P of far-field targets by using the improved MDL criterion:

[0059] Take the kernel matrix R as the covariance matrix of the minimum description length (MDL) criterion, take the number M of array elements as the number of sampling points of the MDL criterion, and take the number L of sampling points as the number of array elements of the MDL criterion to improve the MDL criterion, and then estimate the number P of far-field targets through the improved MDL criterion:

[0060] P = arg min{f(P)}

[0061]

[0062] Among them, f(P) represents the objective function, and arg min{·} represents the operation of taking the minimum value;

[0063] The MDL criterion is a method for estimating the number of information sources based on information theory. In the case of large samples, the sample covariance matrix is a sufficient estimation matrix. The MDL criterion has advantages such as estimation consistency, no need for a decision threshold, and small computational complexity in target estimation, and is widely used in signal processing. However, in the case of small samples, the sample covariance matrix is a singular matrix, and its eigenvalues contain zero values, resulting in a serious reduction in the estimation accuracy of the MDL criterion. Its formula is:

[0064]

[0065] Among them, μ m represents the m-th eigenvalue of the sample covariance matrix with dimensions M×M.

[0066] The kernel matrix constructed in the present invention is a sufficient estimation, that is, the cross-term is approximately a zero matrix, and the noise kernel matrix is approximately a scalar matrix. Therefore, in the present invention, by using the MDL algorithm of the sufficient estimation kernel matrix, the number of samples of the MDL algorithm based on the kernel matrix is much larger than the number of array elements, so the number P of far-field targets can be estimated sufficiently and accurately.

[0067] Step 5) Estimate the noise subspace U n :

[0068] Arrange the L eigenvalues contained in the eigenvalue matrix Λ in descending order, and right-multiply the eigenvectors corresponding to the first P eigenvalues by the data matrix X to obtain the signal subspace U s , and then normalize the column vectors of U s . Finally, estimate the noise subspace U s through the normalized result of U : n :

[0069] U s = X[v1,…,v P

[0070]

[0071] where I M represents the M×M dimensional identity matrix;

[0072] Normalization is a method to simplify calculations in data processing. It can map data into the range (0,1) for fast processing, or convert dimensional expressions into dimensionless expressions for easy comparison and calculation. For the normalization of vectors, commonly used are L1 norm normalization and L2 norm normalization. In this embodiment, the L2 norm is used to normalize the column vectors of U s , that is, each element of each column vector is divided by the L2 norm of the vector, which can not only prevent overfitting but also make the solution more stable and fast.

[0073] In the case of small samples, the signal subspace can be expressed as a linear combination of the received data matrix, and there is a one-to-one correspondence between the first P largest eigenvalues of the kernel matrix and the first P largest eigenvalues of the covariance matrix. Therefore, the noise subspace can be indirectly obtained through the eigenvalue decomposition of the fully estimated kernel matrix and the MDL algorithm based on the kernel matrix. Compared with the noise subspace obtained through the singular sample covariance matrix, the noise subspace obtained from the kernel matrix has a smaller computational complexity and higher accuracy.

[0074] Step 6) Obtain the estimated result of the far-field target azimuth angle:

[0075] Construct the spatial spectrum function S(θ), and then select the P largest spectral peaks. The corresponding abscissa values are the azimuth angles of the P far-field target signal sources, where:

[0076]

[0077]

[0078] where a(θ) represents the steering vector of the array, ​It represents the phase change brought about by the time delay of the received signal at the m-th array element. exp represents the exponential function with the natural constant e as the base, and j represents the imaginary unit, [·] T It represents the transpose operation.

[0079] The technical effects of the present invention are described below in conjunction with simulation experiments:

[0080] 1. Simulation conditions and content:

[0081] The simulation uses a CPU of Intel Core i7-7700, 8GB of RAM, a 64-bit operating system and Microsoft Windows 10 Professional Edition, and Matlab R2015b simulation software.

[0082] A uniform linear array is constructed, with the total number of array elements M = 800, and the interval between adjacent array elements There are 24 far-field target signal sources incident on the array from directions of -30°, -27°, -25°, -21°, -16°, -15°, -12.5°, -10.6°, -8°, -6.6°, -3°, 0°, 2.1°, 5°, 6°, 8°, 10°, 12.3°, 14°, 15.9°, 20°, 23°, 26.6° and 30° respectively. The number of samples L = 240, and a total of 100 independent experiments are carried out.

[0083] The root mean square error RMSE and computational complexity of the present invention and the existing far-field target azimuth estimation method based on improved MUSIC and applicable to small samples are simulated, and the results are as Figure 2 、 Figure 3 shown:

[0084] 2. Analysis of simulation results:

[0085] Referring to Figure 2 , the abscissa in the figure represents the signal-to-noise ratio SNR, with the unit of dB, and the ordinate represents the root mean square error RMSE, with the unit of dB. The curves marked with cross signs and hollow circles respectively represent the root mean square error curves of estimating the far-field target azimuth angle using the present invention and the existing technology under a uniform linear array. Among them, the root mean square error RMSE represents the deviation degree between the estimated azimuth angle and the true azimuth angle, and is defined as:

[0086]

[0087] Among them, represents the estimated azimuth angle of the p-th far-field target signal source in the n-th independent experiment, represents the true azimuth angle of the p-th far-field target signal source.

[0088] From Figure 2It can be seen that when the signal-to-noise ratio is less than 0 dB, the root mean square error of the present invention is much smaller than that of the prior art. When the signal-to-noise ratio is greater than or equal to 0 dB, the root mean square error curves of the two are relatively close, which proves that the estimation accuracy of the azimuth angle of the far-field target in the case of small samples of the present invention is better than that of the prior art.

[0089] Referring to Figure 3 , the abscissa represents the number of samples, and the ordinate represents the calculation time, with the unit of dBs. The curves marked with cross symbols and hollow circles respectively represent the calculation complexity curves of estimating the azimuth angle of the far-field target using the present invention and the prior art under a uniform linear array.

[0090] From Figure 3 it can be seen that the calculation time of the present invention is much less than that required for the prior art calculation. It proves that compared with the prior art, the present invention greatly reduces the required calculation complexity.

[0091] Based on the comprehensive Figures 2 to 3 results, it is proved that the estimation accuracy of the azimuth angle of the far-field target in the case of small samples of the present invention is better than that of the prior art, and at the same time, the calculation complexity is much lower than that of the prior art.

Claims

1. A far-field target azimuth estimation method applicable to small samples, characterized in that, Including the following steps: (1) Obtain the data matrix X: Construct a uniform linear array consisting of M array elements with an interval of d between adjacent array elements, and incident the target signals generated by each of the P far-field target signal sources onto each array element of the array. Then, sample the output signals of each array element L times, and finally form a data matrix from all the sampling results where M ≥ 800, λ is the wavelength of the target signal, L < M, represents the complex domain; (2) Construct the kernel matrix R: Perform conjugate transpose on the data matrix X and right-multiply it by the data matrix X to obtain the kernel matrix R: where X H denotes the conjugate transpose result of X; (3) Construct the eigenvalue matrix and eigenvector matrix: Perform eigen - decomposition on the kernel matrix \(R\), and construct an eigenvalue matrix \(\Lambda\) with the \(L\) eigenvalues \(s_1,\ldots,s\) obtained from the eigenvalue decomposition as the main diagonal elements and the remaining elements being \(0\). At the same time, form an eigenvector matrix \(V\) from the \(L\) eigenvectors obtained from the eigenvalue decomposition: l ,\ldots,s L The above - mentioned eigenvalue matrix \(\Lambda\) has the \(L\) eigenvalues \(s_1,\ldots,s\) obtained from the eigenvalue decomposition as the main diagonal elements and the remaining elements being \(0\). At the same time, the \(L\) eigenvectors obtained from the eigenvalue decomposition are used to form the eigenvector matrix \(V\): where, diag(s1,…,s l ,…,s L ) represents a matrix with main diagonal elements s1,…,s l ,…,s L and all other elements being 0, s l represents the l-th eigenvalue, and v l represents the eigenvector corresponding to s l ; (4) Estimate the number P of far-field targets using the improved MDL criterion: Take the kernel matrix R as the covariance matrix of the minimum description length (MDL) criterion, take the number M of array elements as the number of sampling points of the MDL criterion, and take the number L of sampling points as the number of array elements of the MDL criterion to improve the MDL criterion, and then estimate the number P of far-field targets through the improved MDL criterion: P = arg min{f(P)} where f(P) represents the objective function, and arg min{·} represents the operation of taking the minimum value; (5) Estimate the noise subspace U n : Arrange the L eigenvalues contained in the eigenvalue matrix Λ in descending order, and right-multiply the eigenvectors corresponding to the first P eigenvalues by the data matrix X to obtain the signal subspace U s , then normalize the column vectors of U s , and finally estimate the noise subspace U s through the normalized result of U : n ​ U s = X[v1,…,v P ​ Among them, I M represents an M×M dimensional identity matrix; (6) Obtain the estimated result of the azimuth angle of the far-field target: Construct the spatial spectrum function S(θ), and then select the P largest spectral peaks, and the corresponding abscissa values are the azimuth angles of the P far-field target signal sources, where: where \(a(\theta)\) represents the steering vector of the array, represents the phase change caused by the time delay of the received signal at the \(m\) -th array element, \(\exp\) represents the exponential function with the natural constant \(e\) as the base, \(j\) represents the imaginary unit, and \([\cdot]\) T represents the transpose operation.

2. The far-field target angle estimation method applicable to small samples according to claim 1, wherein, The implementation steps for performing eigenvalue decomposition on the kernel matrix R described in step (3) are: (3a) Construct the kernel matrix R of the eigenmatrix R-s l I L , and let det[R-s l I L = 0, obtaining L eigenvalues s1,…,s l ,…,s L , where det[·] represents the value of taking the determinant; (3b) Substitute each eigenvalue s l into the linear equation system (R - s l I L )v l = 0, and obtain the eigenvectors v1, …, v l , …, v L .

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